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 <li><a href="rust/">Rust</a><ul>
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+    
+    
+      
+    
+    <li class="md-nav__item md-nav__item--section md-nav__item--nested">
+      
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5_3" >
+        
+          
+          <label class="md-nav__link" for="__nav_5_3" id="__nav_5_3_label" tabindex="">
+            
+  
+  <span class="md-ellipsis">
+    算法
+  </span>
+  
+
+            <span class="md-nav__icon md-icon"></span>
+          </label>
+        
+        <nav class="md-nav" data-md-level="2" aria-labelledby="__nav_5_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5_3">
+            <span class="md-nav__icon md-icon"></span>
+            算法
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+    
+    
+      
+      
+        
+      
+    
+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../dsa/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    数据结构
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
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+    
+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../leetcode/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    Leetcode题解
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+          
+          
+        
+      
+    
+    
+      
+    
+    <li class="md-nav__item md-nav__item--section md-nav__item--nested">
+      
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5_4" >
+        
+          
+          <label class="md-nav__link" for="__nav_5_4" id="__nav_5_4_label" tabindex="">
+            
+  
+  <span class="md-ellipsis">
+    机器学习
+  </span>
+  
+
+            <span class="md-nav__icon md-icon"></span>
+          </label>
+        
+        <nav class="md-nav" data-md-level="2" aria-labelledby="__nav_5_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5_4">
+            <span class="md-nav__icon md-icon"></span>
+            机器学习
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+      
+    
+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../machine-learning/linear-models/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    机器学习
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+      
+    
+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../dl-from-scratch/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    深度学习
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+    
+    
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+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../reinforcement-learning/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    强化学习
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+          
+          
+        
+      
+    
+    
+      
+    
+    <li class="md-nav__item md-nav__item--section md-nav__item--nested">
+      
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5_5" >
+        
+          
+          <label class="md-nav__link" for="__nav_5_5" id="__nav_5_5_label" tabindex="">
+            
+  
+  <span class="md-ellipsis">
+    其他
+  </span>
+  
+
+            <span class="md-nav__icon md-icon"></span>
+          </label>
+        
+        <nav class="md-nav" data-md-level="2" aria-labelledby="__nav_5_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5_5">
+            <span class="md-nav__icon md-icon"></span>
+            其他
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../svg/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    SVG动画
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+      
+    
+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../configuration/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    系统配置
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+    
+    
+      
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+    
+      
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+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../../japanese/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    日语
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../../about/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    关于我
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+    
+  </ul>
+</nav>
+        </div>
+    </div>
+    </div>
+
+
+    
+    <div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
+    <div class="md-sidebar__scrollwrap">
+        <div class="md-sidebar__inner">
+            
+
+<nav class="md-nav md-nav--secondary" aria-label="目录">
+  
+  
+  
+    
+  
+  
+</nav>
+            <!-- <iframe frameborder="no" border="0" marginwidth="0" marginheight="0" width=230 height=360 src="//music.163.com/outchain/player?type=0&id=5371395831&auto=1&height=430"></iframe> -->
+        </div>
+    </div>
+    </div>
+
+
+          
+            <div class="md-content" data-md-component="content">
+              <article class="md-content__inner md-typeset">
+                
+
+<a href="rust-jupyter.ipynb" title="Download Notebook" class="md-content__button md-icon">
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M5 20h14v-2H5m14-9h-4V3H9v6H5l7 7z"/></svg>
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+  
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+
+
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+    function copyNode(node) {
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+        if (button instanceof HTMLElement) {
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+      })
+</script>
+<style type="text/css">
+    pre { line-height: 125%; }
+td.linenos .normal { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+span.linenos { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+td.linenos .special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+span.linenos.special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+.highlight-ipynb .hll { background-color: var(--jp-cell-editor-active-background) }
+.highlight-ipynb { background: var(--jp-cell-editor-background); color: var(--jp-mirror-editor-variable-color) }
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+.highlight-ipynb .il { color: var(--jp-mirror-editor-number-color) } /* Literal.Number.Integer.Long */
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#4e342e;--md-brown-900: #3e2723;--md-grey-50: #fafafa;--md-grey-100: #f5f5f5;--md-grey-200: #eeeeee;--md-grey-300: #e0e0e0;--md-grey-400: #bdbdbd;--md-grey-500: #9e9e9e;--md-grey-600: #757575;--md-grey-700: #616161;--md-grey-800: #424242;--md-grey-900: #212121;--md-blue-grey-50: #eceff1;--md-blue-grey-100: #cfd8dc;--md-blue-grey-200: #b0bec5;--md-blue-grey-300: #90a4ae;--md-blue-grey-400: #78909c;--md-blue-grey-500: #607d8b;--md-blue-grey-600: #546e7a;--md-blue-grey-700: #455a64;--md-blue-grey-800: #37474f;--md-blue-grey-900: #263238}.jupyter-wrapper{--jp-shadow-base-lightness: 0;--jp-shadow-umbra-color: rgba( var(--jp-shadow-base-lightness), var(--jp-shadow-base-lightness), var(--jp-shadow-base-lightness), .2 );--jp-shadow-penumbra-color: rgba( var(--jp-shadow-base-lightness), var(--jp-shadow-base-lightness), var(--jp-shadow-base-lightness), .14 );--jp-shadow-ambient-color: rgba( var(--jp-shadow-base-lightness), var(--jp-shadow-base-lightness), var(--jp-shadow-base-lightness), .12 );--jp-elevation-z0: none;--jp-elevation-z1: 0px 2px 1px -1px var(--jp-shadow-umbra-color), 0px 1px 1px 0px var(--jp-shadow-penumbra-color), 0px 1px 3px 0px var(--jp-shadow-ambient-color);--jp-elevation-z2: 0px 3px 1px -2px var(--jp-shadow-umbra-color), 0px 2px 2px 0px var(--jp-shadow-penumbra-color), 0px 1px 5px 0px var(--jp-shadow-ambient-color);--jp-elevation-z4: 0px 2px 4px -1px var(--jp-shadow-umbra-color), 0px 4px 5px 0px var(--jp-shadow-penumbra-color), 0px 1px 10px 0px var(--jp-shadow-ambient-color);--jp-elevation-z6: 0px 3px 5px -1px var(--jp-shadow-umbra-color), 0px 6px 10px 0px var(--jp-shadow-penumbra-color), 0px 1px 18px 0px var(--jp-shadow-ambient-color);--jp-elevation-z8: 0px 5px 5px -3px var(--jp-shadow-umbra-color), 0px 8px 10px 1px var(--jp-shadow-penumbra-color), 0px 3px 14px 2px var(--jp-shadow-ambient-color);--jp-elevation-z12: 0px 7px 8px -4px var(--jp-shadow-umbra-color), 0px 12px 17px 2px var(--jp-shadow-penumbra-color), 0px 5px 22px 4px var(--jp-shadow-ambient-color);--jp-elevation-z16: 0px 8px 10px -5px var(--jp-shadow-umbra-color), 0px 16px 24px 2px var(--jp-shadow-penumbra-color), 0px 6px 30px 5px var(--jp-shadow-ambient-color);--jp-elevation-z20: 0px 10px 13px -6px var(--jp-shadow-umbra-color), 0px 20px 31px 3px var(--jp-shadow-penumbra-color), 0px 8px 38px 7px var(--jp-shadow-ambient-color);--jp-elevation-z24: 0px 11px 15px -7px var(--jp-shadow-umbra-color), 0px 24px 38px 3px var(--jp-shadow-penumbra-color), 0px 9px 46px 8px var(--jp-shadow-ambient-color);--jp-border-width: 1px;--jp-border-color0: var(--md-grey-400);--jp-border-color1: var(--md-grey-400);--jp-border-color2: var(--md-grey-300);--jp-border-color3: var(--md-grey-200);--jp-inverse-border-color: var(--md-grey-600);--jp-border-radius: 2px;--jp-ui-font-scale-factor: 1.2;--jp-ui-font-size0: .83333em;--jp-ui-font-size1: 13px;--jp-ui-font-size2: 1.2em;--jp-ui-font-size3: 1.44em;--jp-ui-font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, 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var(--md-grey-700);--jp-inverse-layout-color4: var(--md-grey-600);--jp-brand-color0: var(--md-blue-900);--jp-brand-color1: var(--md-blue-700);--jp-brand-color2: var(--md-blue-300);--jp-brand-color3: var(--md-blue-100);--jp-brand-color4: var(--md-blue-50);--jp-accent-color0: var(--md-green-900);--jp-accent-color1: var(--md-green-700);--jp-accent-color2: var(--md-green-300);--jp-accent-color3: var(--md-green-100);--jp-warn-color0: var(--md-orange-900);--jp-warn-color1: var(--md-orange-700);--jp-warn-color2: var(--md-orange-300);--jp-warn-color3: var(--md-orange-100);--jp-error-color0: var(--md-red-900);--jp-error-color1: var(--md-red-700);--jp-error-color2: var(--md-red-300);--jp-error-color3: var(--md-red-100);--jp-success-color0: var(--md-green-900);--jp-success-color1: var(--md-green-700);--jp-success-color2: var(--md-green-300);--jp-success-color3: var(--md-green-100);--jp-info-color0: var(--md-cyan-900);--jp-info-color1: var(--md-cyan-700);--jp-info-color2: var(--md-cyan-300);--jp-info-color3: var(--md-cyan-100);--jp-cell-padding: 5px;--jp-cell-collapser-width: 8px;--jp-cell-collapser-min-height: 20px;--jp-cell-collapser-not-active-hover-opacity: .6;--jp-cell-editor-background: var(--md-grey-100);--jp-cell-editor-border-color: var(--md-grey-300);--jp-cell-editor-box-shadow: inset 0 0 2px var(--md-blue-300);--jp-cell-editor-active-background: var(--jp-layout-color0);--jp-cell-editor-active-border-color: var(--jp-brand-color1);--jp-cell-prompt-width: 64px;--jp-cell-prompt-font-family: var(--jp-code-font-family-default);--jp-cell-prompt-letter-spacing: 0px;--jp-cell-prompt-opacity: 1;--jp-cell-prompt-not-active-opacity: .5;--jp-cell-prompt-not-active-font-color: var(--md-grey-700);--jp-cell-inprompt-font-color: #307fc1;--jp-cell-outprompt-font-color: #bf5b3d;--jp-notebook-padding: 10px;--jp-notebook-select-background: var(--jp-layout-color1);--jp-notebook-multiselected-color: var(--md-blue-50);--jp-notebook-scroll-padding: calc( 100% - var(--jp-code-font-size) * var(--jp-code-line-height) - var(--jp-code-padding) - var(--jp-cell-padding) - 1px );--jp-rendermime-error-background: #fdd;--jp-rendermime-table-row-background: var(--md-grey-100);--jp-rendermime-table-row-hover-background: var(--md-light-blue-50);--jp-dialog-background: rgba(0, 0, 0, .25);--jp-console-padding: 10px;--jp-toolbar-border-color: var(--jp-border-color1);--jp-toolbar-micro-height: 8px;--jp-toolbar-background: var(--jp-layout-color1);--jp-toolbar-box-shadow: 0px 0px 2px 0px rgba(0, 0, 0, .24);--jp-toolbar-header-margin: 4px 4px 0px 4px;--jp-toolbar-active-background: var(--md-grey-300);--jp-statusbar-height: 24px;--jp-input-box-shadow: inset 0 0 2px var(--md-blue-300);--jp-input-active-background: var(--jp-layout-color1);--jp-input-hover-background: var(--jp-layout-color1);--jp-input-background: var(--md-grey-100);--jp-input-border-color: var(--jp-inverse-border-color);--jp-input-active-border-color: var(--jp-brand-color1);--jp-input-active-box-shadow-color: rgba(19, 124, 189, .3);--jp-editor-selected-background: #d9d9d9;--jp-editor-selected-focused-background: #d7d4f0;--jp-editor-cursor-color: var(--jp-ui-font-color0);--jp-mirror-editor-keyword-color: #008000;--jp-mirror-editor-atom-color: #88f;--jp-mirror-editor-number-color: #080;--jp-mirror-editor-def-color: #00f;--jp-mirror-editor-variable-color: var(--md-grey-900);--jp-mirror-editor-variable-2-color: #05a;--jp-mirror-editor-variable-3-color: #085;--jp-mirror-editor-punctuation-color: #05a;--jp-mirror-editor-property-color: #05a;--jp-mirror-editor-operator-color: #aa22ff;--jp-mirror-editor-comment-color: #408080;--jp-mirror-editor-string-color: #ba2121;--jp-mirror-editor-string-2-color: #708;--jp-mirror-editor-meta-color: #aa22ff;--jp-mirror-editor-qualifier-color: #555;--jp-mirror-editor-builtin-color: #008000;--jp-mirror-editor-bracket-color: #997;--jp-mirror-editor-tag-color: #170;--jp-mirror-editor-attribute-color: #00c;--jp-mirror-editor-header-color: blue;--jp-mirror-editor-quote-color: #090;--jp-mirror-editor-link-color: #00c;--jp-mirror-editor-error-color: #f00;--jp-mirror-editor-hr-color: #999;--jp-collaborator-color1: #ffad8e;--jp-collaborator-color2: #dac83d;--jp-collaborator-color3: #72dd76;--jp-collaborator-color4: #00e4d0;--jp-collaborator-color5: #45d4ff;--jp-collaborator-color6: #e2b1ff;--jp-collaborator-color7: #ff9de6;--jp-vega-background: white;--jp-sidebar-min-width: 250px;--jp-search-toggle-off-opacity: .5;--jp-search-toggle-hover-opacity: .8;--jp-search-toggle-on-opacity: 1;--jp-search-selected-match-background-color: rgb(245, 200, 0);--jp-search-selected-match-color: black;--jp-search-unselected-match-background-color: var( --jp-inverse-layout-color0 );--jp-search-unselected-match-color: var(--jp-ui-inverse-font-color0);--jp-icon-contrast-color0: var(--md-purple-600);--jp-icon-contrast-color1: var(--md-green-600);--jp-icon-contrast-color2: var(--md-pink-600);--jp-icon-contrast-color3: var(--md-blue-600);--jp-jupyter-icon-color: #f37626;--jp-notebook-icon-color: #f37626;--jp-json-icon-color: var(--md-orange-700);--jp-console-icon-background-color: var(--md-blue-700);--jp-console-icon-color: white;--jp-terminal-icon-background-color: var(--md-grey-800);--jp-terminal-icon-color: var(--md-grey-200);--jp-text-editor-icon-color: var(--md-grey-700);--jp-inspector-icon-color: var(--md-grey-700);--jp-switch-color: var(--md-grey-400);--jp-switch-true-position-color: var(--md-orange-900)}[data-md-color-scheme=slate] .jupyter-wrapper{--jp-shadow-base-lightness: 32;--jp-shadow-umbra-color: rgba( var(--jp-shadow-base-lightness), var(--jp-shadow-base-lightness), var(--jp-shadow-base-lightness), .2 );--jp-shadow-penumbra-color: rgba( var(--jp-shadow-base-lightness), var(--jp-shadow-base-lightness), var(--jp-shadow-base-lightness), .14 );--jp-shadow-ambient-color: rgba( var(--jp-shadow-base-lightness), var(--jp-shadow-base-lightness), var(--jp-shadow-base-lightness), .12 );--jp-elevation-z0: none;--jp-elevation-z1: 0px 2px 1px -1px var(--jp-shadow-umbra-color), 0px 1px 1px 0px var(--jp-shadow-penumbra-color), 0px 1px 3px 0px var(--jp-shadow-ambient-color);--jp-elevation-z2: 0px 3px 1px -2px var(--jp-shadow-umbra-color), 0px 2px 2px 0px var(--jp-shadow-penumbra-color), 0px 1px 5px 0px var(--jp-shadow-ambient-color);--jp-elevation-z4: 0px 2px 4px -1px var(--jp-shadow-umbra-color), 0px 4px 5px 0px var(--jp-shadow-penumbra-color), 0px 1px 10px 0px var(--jp-shadow-ambient-color);--jp-elevation-z6: 0px 3px 5px -1px var(--jp-shadow-umbra-color), 0px 6px 10px 0px var(--jp-shadow-penumbra-color), 0px 1px 18px 0px var(--jp-shadow-ambient-color);--jp-elevation-z8: 0px 5px 5px -3px var(--jp-shadow-umbra-color), 0px 8px 10px 1px var(--jp-shadow-penumbra-color), 0px 3px 14px 2px var(--jp-shadow-ambient-color);--jp-elevation-z12: 0px 7px 8px -4px var(--jp-shadow-umbra-color), 0px 12px 17px 2px var(--jp-shadow-penumbra-color), 0px 5px 22px 4px var(--jp-shadow-ambient-color);--jp-elevation-z16: 0px 8px 10px -5px var(--jp-shadow-umbra-color), 0px 16px 24px 2px var(--jp-shadow-penumbra-color), 0px 6px 30px 5px var(--jp-shadow-ambient-color);--jp-elevation-z20: 0px 10px 13px -6px var(--jp-shadow-umbra-color), 0px 20px 31px 3px var(--jp-shadow-penumbra-color), 0px 8px 38px 7px var(--jp-shadow-ambient-color);--jp-elevation-z24: 0px 11px 15px -7px var(--jp-shadow-umbra-color), 0px 24px 38px 3px var(--jp-shadow-penumbra-color), 0px 9px 46px 8px var(--jp-shadow-ambient-color);--jp-border-width: 1px;--jp-border-color0: var(--md-grey-700);--jp-border-color1: var(--md-grey-700);--jp-border-color2: var(--md-grey-800);--jp-border-color3: var(--md-grey-900);--jp-inverse-border-color: var(--md-grey-600);--jp-border-radius: 2px;--jp-ui-font-scale-factor: 1.2;--jp-ui-font-size0: .83333em;--jp-ui-font-size1: 13px;--jp-ui-font-size2: 1.2em;--jp-ui-font-size3: 1.44em;--jp-ui-font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol";--jp-ui-font-color0: rgba(255, 255, 255, 1);--jp-ui-font-color1: rgba(255, 255, 255, .87);--jp-ui-font-color2: rgba(255, 255, 255, .54);--jp-ui-font-color3: rgba(255, 255, 255, .38);--jp-ui-inverse-font-color0: rgba(0, 0, 0, 1);--jp-ui-inverse-font-color1: rgba(0, 0, 0, .8);--jp-ui-inverse-font-color2: rgba(0, 0, 0, .5);--jp-ui-inverse-font-color3: rgba(0, 0, 0, .3);--jp-content-line-height: 1.6;--jp-content-font-scale-factor: 1.2;--jp-content-font-size0: .83333em;--jp-content-font-size1: 14px;--jp-content-font-size2: 1.2em;--jp-content-font-size3: 1.44em;--jp-content-font-size4: 1.728em;--jp-content-font-size5: 2.0736em;--jp-content-presentation-font-size1: 17px;--jp-content-heading-line-height: 1;--jp-content-heading-margin-top: 1.2em;--jp-content-heading-margin-bottom: .8em;--jp-content-heading-font-weight: 500;--jp-content-font-color0: rgba(255, 255, 255, 1);--jp-content-font-color1: rgba(255, 255, 255, 1);--jp-content-font-color2: rgba(255, 255, 255, .7);--jp-content-font-color3: rgba(255, 255, 255, .5);--jp-content-link-color: var(--md-blue-300);--jp-content-font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol";--jp-code-font-size: 13px;--jp-code-line-height: 1.3077;--jp-code-padding: 5px;--jp-code-font-family-default: Menlo, Consolas, "DejaVu Sans Mono", monospace;--jp-code-font-family: var(--jp-code-font-family-default);--jp-code-presentation-font-size: 16px;--jp-code-cursor-width0: 1.4px;--jp-code-cursor-width1: 2px;--jp-code-cursor-width2: 4px;--jp-layout-color0: #111111;--jp-layout-color1: var(--md-grey-900);--jp-layout-color2: var(--md-grey-800);--jp-layout-color3: var(--md-grey-700);--jp-layout-color4: var(--md-grey-600);--jp-inverse-layout-color0: white;--jp-inverse-layout-color1: white;--jp-inverse-layout-color2: var(--md-grey-200);--jp-inverse-layout-color3: var(--md-grey-400);--jp-inverse-layout-color4: var(--md-grey-600);--jp-brand-color0: var(--md-blue-700);--jp-brand-color1: var(--md-blue-500);--jp-brand-color2: var(--md-blue-300);--jp-brand-color3: var(--md-blue-100);--jp-brand-color4: var(--md-blue-50);--jp-accent-color0: var(--md-green-700);--jp-accent-color1: var(--md-green-500);--jp-accent-color2: var(--md-green-300);--jp-accent-color3: var(--md-green-100);--jp-warn-color0: var(--md-orange-700);--jp-warn-color1: var(--md-orange-500);--jp-warn-color2: var(--md-orange-300);--jp-warn-color3: var(--md-orange-100);--jp-error-color0: var(--md-red-700);--jp-error-color1: var(--md-red-500);--jp-error-color2: var(--md-red-300);--jp-error-color3: var(--md-red-100);--jp-success-color0: var(--md-green-700);--jp-success-color1: var(--md-green-500);--jp-success-color2: var(--md-green-300);--jp-success-color3: var(--md-green-100);--jp-info-color0: var(--md-cyan-700);--jp-info-color1: var(--md-cyan-500);--jp-info-color2: var(--md-cyan-300);--jp-info-color3: var(--md-cyan-100);--jp-cell-padding: 5px;--jp-cell-collapser-width: 8px;--jp-cell-collapser-min-height: 20px;--jp-cell-collapser-not-active-hover-opacity: .6;--jp-cell-editor-background: var(--jp-layout-color1);--jp-cell-editor-border-color: var(--md-grey-700);--jp-cell-editor-box-shadow: inset 0 0 2px var(--md-blue-300);--jp-cell-editor-active-background: var(--jp-layout-color0);--jp-cell-editor-active-border-color: var(--jp-brand-color1);--jp-cell-prompt-width: 64px;--jp-cell-prompt-font-family: var(--jp-code-font-family-default);--jp-cell-prompt-letter-spacing: 0px;--jp-cell-prompt-opacity: 1;--jp-cell-prompt-not-active-opacity: 1;--jp-cell-prompt-not-active-font-color: var(--md-grey-300);--jp-cell-inprompt-font-color: #307fc1;--jp-cell-outprompt-font-color: #bf5b3d;--jp-notebook-padding: 10px;--jp-notebook-select-background: var(--jp-layout-color1);--jp-notebook-multiselected-color: rgba(33, 150, 243, .24);--jp-notebook-scroll-padding: calc( 100% - var(--jp-code-font-size) * var(--jp-code-line-height) - var(--jp-code-padding) - var(--jp-cell-padding) - 1px );--jp-rendermime-error-background: rgba(244, 67, 54, .28);--jp-rendermime-table-row-background: var(--md-grey-900);--jp-rendermime-table-row-hover-background: rgba(3, 169, 244, .2);--jp-dialog-background: rgba(0, 0, 0, .6);--jp-console-padding: 10px;--jp-toolbar-border-color: var(--jp-border-color2);--jp-toolbar-micro-height: 8px;--jp-toolbar-background: var(--jp-layout-color1);--jp-toolbar-box-shadow: 0px 0px 2px 0px rgba(0, 0, 0, .8);--jp-toolbar-header-margin: 4px 4px 0px 4px;--jp-toolbar-active-background: var(--jp-layout-color0);--jp-statusbar-height: 24px;--jp-input-box-shadow: inset 0 0 2px var(--md-blue-300);--jp-input-active-background: var(--jp-layout-color0);--jp-input-hover-background: var(--jp-layout-color2);--jp-input-background: var(--md-grey-800);--jp-input-border-color: var(--jp-inverse-border-color);--jp-input-active-border-color: var(--jp-brand-color1);--jp-input-active-box-shadow-color: rgba(19, 124, 189, .3);--jp-editor-selected-background: var(--jp-layout-color2);--jp-editor-selected-focused-background: rgba(33, 150, 243, .24);--jp-editor-cursor-color: var(--jp-ui-font-color0);--jp-mirror-editor-keyword-color: var(--md-green-500);--jp-mirror-editor-atom-color: var(--md-blue-300);--jp-mirror-editor-number-color: var(--md-green-400);--jp-mirror-editor-def-color: var(--md-blue-600);--jp-mirror-editor-variable-color: var(--md-grey-300);--jp-mirror-editor-variable-2-color: var(--md-blue-400);--jp-mirror-editor-variable-3-color: var(--md-green-600);--jp-mirror-editor-punctuation-color: var(--md-blue-400);--jp-mirror-editor-property-color: var(--md-blue-400);--jp-mirror-editor-operator-color: #aa22ff;--jp-mirror-editor-comment-color: #408080;--jp-mirror-editor-string-color: #ff7070;--jp-mirror-editor-string-2-color: var(--md-purple-300);--jp-mirror-editor-meta-color: #aa22ff;--jp-mirror-editor-qualifier-color: #555;--jp-mirror-editor-builtin-color: var(--md-green-600);--jp-mirror-editor-bracket-color: #997;--jp-mirror-editor-tag-color: var(--md-green-700);--jp-mirror-editor-attribute-color: var(--md-blue-700);--jp-mirror-editor-header-color: var(--md-blue-500);--jp-mirror-editor-quote-color: var(--md-green-300);--jp-mirror-editor-link-color: var(--md-blue-700);--jp-mirror-editor-error-color: #f00;--jp-mirror-editor-hr-color: #999;--jp-collaborator-color1: #ad4a00;--jp-collaborator-color2: #7b6a00;--jp-collaborator-color3: #007e00;--jp-collaborator-color4: #008772;--jp-collaborator-color5: #0079b9;--jp-collaborator-color6: #8b45c6;--jp-collaborator-color7: #be208b;--jp-vega-background: var(--md-grey-400);--jp-sidebar-min-width: 250px;--jp-search-toggle-off-opacity: .6;--jp-search-toggle-hover-opacity: .8;--jp-search-toggle-on-opacity: 1;--jp-search-selected-match-background-color: rgb(255, 225, 0);--jp-search-selected-match-color: black;--jp-search-unselected-match-background-color: var( --jp-inverse-layout-color0 );--jp-search-unselected-match-color: var(--jp-ui-inverse-font-color0);--jp-scrollbar-background-color: #3f4244;--jp-scrollbar-thumb-color: 88, 96, 97;--jp-scrollbar-endpad: 3px;--jp-scrollbar-thumb-margin: 3.5px;--jp-scrollbar-thumb-radius: 9px;--jp-icon-contrast-color0: var(--md-purple-600);--jp-icon-contrast-color1: var(--md-green-600);--jp-icon-contrast-color2: var(--md-pink-600);--jp-icon-contrast-color3: var(--md-blue-600);--jp-jupyter-icon-color: #f37626;--jp-notebook-icon-color: #f37626;--jp-json-icon-color: var(--md-orange-500);--jp-console-icon-background-color: var(--md-blue-500);--jp-console-icon-color: white;--jp-terminal-icon-background-color: var(--md-grey-200);--jp-terminal-icon-color: var(--md-grey-800);--jp-text-editor-icon-color: var(--md-grey-200);--jp-inspector-icon-color: var(--md-grey-200);--jp-switch-color: var(--md-grey-400);--jp-switch-true-position-color: var(--md-orange-700)}.jupyter-wrapper 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+<!-- End of mermaid configuration --><div class="jupyter-wrapper">
+<div class="jp-Notebook" data-jp-theme-light="true" data-jp-theme-name="JupyterLab Light">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<h1 id="jupyter-for-rust">Jupyter for Rust<a class="anchor-link" href="#jupyter-for-rust">¶</a></h1><p>Rust的<code>evcxr_jupyter</code>包提供了对jupyter notebook的支持，可以在notebook中运行Rust代码。使用如下命令安装：</p>
+<div class="highlight"><pre><span></span>cargo<span class="w"> </span>install<span class="w"> </span>--locked<span class="w"> </span>evcxr_jupyter
+evcxr_jupyter<span class="w"> </span>--install
+</pre></div>
+<p>另外，还需要安装Rust标准库的源代码：</p>
+<div class="highlight"><pre><span></span>rustup<span class="w"> </span>component<span class="w"> </span>add<span class="w"> </span>rust-src
+</pre></div>
+<p>之后，就可以在notebook中运行Rust代码了。</p>
+<div class="highlight"><pre><span></span>jupyter<span class="w"> </span>notebook
+</pre></div>
+</div>
+</div>
+</div>
+</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell">
+<div class="jp-Cell jp-CodeCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea">
+<div class="jp-InputPrompt jp-InputArea-prompt">In [2]:</div><div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
+<div class="CodeMirror cm-s-jupyter">
+<div class="zeroclipboard-container">
+<clipboard-copy ,="" aria-label="Copy to Clipboard" for="cell-1">
+<div>
+<span class="notice" hidden="">Copied!</span>
+<svg aria-hidden="true" class="clipboard-copy-icon" data-view-component="true" height="20" version="1.1" viewbox="0 0 16 16" width="20">
+<path d="M0 6.75C0 5.784.784 5 1.75 5h1.5a.75.75 0 010 1.5h-1.5a.25.25 0 00-.25.25v7.5c0 .138.112.25.25.25h7.5a.25.25 0 00.25-.25v-1.5a.75.75 0 011.5 0v1.5A1.75 1.75 0 019.25 16h-7.5A1.75 1.75 0 010 14.25v-7.5z" fill="currentColor" fill-rule="evenodd"></path>
+<path d="M5 1.75C5 .784 5.784 0 6.75 0h7.5C15.216 0 16 .784 16 1.75v7.5A1.75 1.75 0 0114.25 11h-7.5A1.75 1.75 0 015 9.25v-7.5zm1.75-.25a.25.25 0 00-.25.25v7.5c0 .138.112.25.25.25h7.5a.25.25 0 00.25-.25v-7.5a.25.25 0 00-.25-.25h-7.5z" fill="currentColor" fill-rule="evenodd"></path>
+</svg>
+</div>
+</clipboard-copy>
+</div>
+<div class="highlight-ipynb hl-rust"><pre><span></span><span class="k">fn</span> <span class="nf">add</span><span class="p">(</span><span class="n">a</span>: <span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">b</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span>-&gt; <span class="kt">i32</span> <span class="p">{</span>
+<span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">b</span>
+<span class="p">}</span>
+
+<span class="n">add</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="c1">// The last expression is evaluated and printed</span>
+</pre></div>
+<div class="clipboard-copy-txt" id="cell-1">fn add(a: i32, b: i32) -&gt; i32 {
+    a + b
+}
+
+add(1, 2) // The last expression is evaluated and printed</div>
+</div>
+</div>
+</div>
+</div>
+<div class="jp-Cell-outputWrapper">
+<div class="jp-Collapser jp-OutputCollapser jp-Cell-outputCollapser">
+</div>
+<div class="jp-OutputArea jp-Cell-outputArea">
+<div class="jp-OutputArea-child jp-OutputArea-executeResult">
+<div class="jp-OutputPrompt jp-OutputArea-prompt">Out[2]:</div>
+<div class="jp-RenderedText jp-OutputArea-output jp-OutputArea-executeResult" data-mime-type="text/plain" tabindex="0">
+<pre>3</pre>
+</div>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<p>使用<code>:dep &lt;package&gt;</code>来导入包。<code>evcxr_input</code>包提供了<code>get_string</code>和<code>get_password</code>函数，可以在notebook中输入字符串和密码。</p>
+</div>
+</div>
+</div>
+</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell">
+<div class="jp-Cell jp-CodeCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea">
+<div class="jp-InputPrompt jp-InputArea-prompt">In [3]:</div><div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
+<div class="CodeMirror cm-s-jupyter">
+<div class="zeroclipboard-container">
+<clipboard-copy ,="" aria-label="Copy to Clipboard" for="cell-2">
+<div>
+<span class="notice" hidden="">Copied!</span>
+<svg aria-hidden="true" class="clipboard-copy-icon" data-view-component="true" height="20" version="1.1" viewbox="0 0 16 16" width="20">
+<path d="M0 6.75C0 5.784.784 5 1.75 5h1.5a.75.75 0 010 1.5h-1.5a.25.25 0 00-.25.25v7.5c0 .138.112.25.25.25h7.5a.25.25 0 00.25-.25v-1.5a.75.75 0 011.5 0v1.5A1.75 1.75 0 019.25 16h-7.5A1.75 1.75 0 010 14.25v-7.5z" fill="currentColor" fill-rule="evenodd"></path>
+<path d="M5 1.75C5 .784 5.784 0 6.75 0h7.5C15.216 0 16 .784 16 1.75v7.5A1.75 1.75 0 0114.25 11h-7.5A1.75 1.75 0 015 9.25v-7.5zm1.75-.25a.25.25 0 00-.25.25v7.5c0 .138.112.25.25.25h7.5a.25.25 0 00.25-.25v-7.5a.25.25 0 00-.25-.25h-7.5z" fill="currentColor" fill-rule="evenodd"></path>
+</svg>
+</div>
+</clipboard-copy>
+</div>
+<div class="highlight-ipynb hl-rust"><pre><span></span>:<span class="nc">dep</span><span class="w"> </span><span class="n">evcxr_input</span>
+<span class="kd">let</span><span class="w"> </span><span class="n">name</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">evcxr_input</span>::<span class="n">get_string</span><span class="p">(</span><span class="s">"Name?"</span><span class="p">);</span>
+<span class="kd">let</span><span class="w"> </span><span class="n">password</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">evcxr_input</span>::<span class="n">get_password</span><span class="p">(</span><span class="s">"Password?"</span><span class="p">);</span>
+
+<span class="fm">println!</span><span class="p">(</span><span class="s">"Hello, {name}! Your password is {password}"</span><span class="p">,</span><span class="w"> </span><span class="n">name</span><span class="o">=</span><span class="n">name</span><span class="p">,</span><span class="w"> </span><span class="n">password</span><span class="o">=</span><span class="n">password</span><span class="p">);</span>
+</pre></div>
+<div class="clipboard-copy-txt" id="cell-2">:dep evcxr_input
+let name = evcxr_input::get_string("Name?");
+let password = evcxr_input::get_password("Password?");
+
+println!("Hello, {name}! Your password is {password}", name=name, password=password);</div>
+</div>
+</div>
+</div>
+</div>
+<div class="jp-Cell-outputWrapper">
+<div class="jp-Collapser jp-OutputCollapser jp-Cell-outputCollapser">
+</div>
+<div class="jp-OutputArea jp-Cell-outputArea">
+<div class="jp-OutputArea-child">
+<div class="jp-OutputPrompt jp-OutputArea-prompt"></div>
+<div class="jp-RenderedText jp-OutputArea-output" data-mime-type="text/plain" tabindex="0">
+<pre>Hello, Alice! Your password is alice123
+</pre>
+</div>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<p>使用如下命令卸载Rust</p>
+<div class="highlight"><pre><span></span>evcxr_jupyter<span class="w"> </span>--uninstall
+cargo<span class="w"> </span>uninstall<span class="w"> </span>evcxr_jupyter
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+</div> <!-- jp-Notebook -->
+</div> <!-- jupyter-wrapper -->
+
+        <style>
+        ['pre { line-height: 125%; }\ntd.linenos .normal { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }\nspan.linenos { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }\ntd.linenos .special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }\nspan.linenos.special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }\n.highlight-ipynb .hll { background-color: var(--jp-cell-editor-active-background) }\n.highlight-ipynb { background: var(--jp-cell-editor-background); color: var(--jp-mirror-editor-variable-color) }\n.highlight-ipynb .c { color: var(--jp-mirror-editor-comment-color); font-style: italic } /* Comment */\n.highlight-ipynb .err { color: var(--jp-mirror-editor-error-color) } /* Error */\n.highlight-ipynb .k { color: var(--jp-mirror-editor-keyword-color); font-weight: bold } /* Keyword */\n.highlight-ipynb .o { color: var(--jp-mirror-editor-operator-color); font-weight: bold } /* Operator */\n.highlight-ipynb .p { color: var(--jp-mirror-editor-punctuation-color) } /* Punctuation */\n.highlight-ipynb .ch { color: var(--jp-mirror-editor-comment-color); font-style: italic } /* Comment.Hashbang */\n.highlight-ipynb .cm { color: var(--jp-mirror-editor-comment-color); font-style: italic } /* Comment.Multiline */\n.highlight-ipynb .cp { color: var(--jp-mirror-editor-comment-color); font-style: italic } /* Comment.Preproc */\n.highlight-ipynb .cpf { color: var(--jp-mirror-editor-comment-color); font-style: italic } /* Comment.PreprocFile */\n.highlight-ipynb .c1 { color: var(--jp-mirror-editor-comment-color); font-style: italic } /* Comment.Single */\n.highlight-ipynb .cs { color: var(--jp-mirror-editor-comment-color); font-style: italic } /* Comment.Special */\n.highlight-ipynb .kc { color: var(--jp-mirror-editor-keyword-color); font-weight: bold } /* Keyword.Constant */\n.highlight-ipynb .kd { color: var(--jp-mirror-editor-keyword-color); font-weight: bold } /* Keyword.Declaration */\n.highlight-ipynb .kn { color: var(--jp-mirror-editor-keyword-color); font-weight: bold } /* Keyword.Namespace */\n.highlight-ipynb .kp { color: var(--jp-mirror-editor-keyword-color); font-weight: bold } /* Keyword.Pseudo */\n.highlight-ipynb .kr { color: var(--jp-mirror-editor-keyword-color); font-weight: bold } /* Keyword.Reserved */\n.highlight-ipynb .kt { color: var(--jp-mirror-editor-keyword-color); font-weight: bold } /* Keyword.Type */\n.highlight-ipynb .m { color: var(--jp-mirror-editor-number-color) } /* Literal.Number */\n.highlight-ipynb .s { color: var(--jp-mirror-editor-string-color) } /* Literal.String */\n.highlight-ipynb .ow { color: var(--jp-mirror-editor-operator-color); font-weight: bold } /* Operator.Word */\n.highlight-ipynb .pm { color: var(--jp-mirror-editor-punctuation-color) } /* Punctuation.Marker */\n.highlight-ipynb .w { color: var(--jp-mirror-editor-variable-color) } /* Text.Whitespace */\n.highlight-ipynb .mb { color: var(--jp-mirror-editor-number-color) } /* Literal.Number.Bin */\n.highlight-ipynb .mf { color: var(--jp-mirror-editor-number-color) } /* Literal.Number.Float */\n.highlight-ipynb .mh { color: var(--jp-mirror-editor-number-color) } /* Literal.Number.Hex */\n.highlight-ipynb .mi { color: var(--jp-mirror-editor-number-color) } /* Literal.Number.Integer */\n.highlight-ipynb .mo { color: var(--jp-mirror-editor-number-color) } /* Literal.Number.Oct */\n.highlight-ipynb .sa { color: var(--jp-mirror-editor-string-color) } /* Literal.String.Affix */\n.highlight-ipynb .sb { color: var(--jp-mirror-editor-string-color) } /* Literal.String.Backtick */\n.highlight-ipynb .sc { color: var(--jp-mirror-editor-string-color) } /* Literal.String.Char */\n.highlight-ipynb .dl { color: var(--jp-mirror-editor-string-color) } /* Literal.String.Delimiter */\n.highlight-ipynb .sd { color: var(--jp-mirror-editor-string-color) } /* Literal.String.Doc */\n.highlight-ipynb .s2 { color: var(--jp-mirror-editor-string-color) } /* Literal.String.Double */\n.highlight-ipynb .se { color: var(--jp-mirror-editor-string-color) } /* Literal.String.Escape */\n.highlight-ipynb .sh { color: var(--jp-mirror-editor-string-color) } /* Literal.String.Heredoc */\n.highlight-ipynb .si { color: var(--jp-mirror-editor-string-color) } /* Literal.String.Interpol */\n.highlight-ipynb .sx { color: var(--jp-mirror-editor-string-color) } /* Literal.String.Other */\n.highlight-ipynb .sr { color: var(--jp-mirror-editor-string-color) } /* Literal.String.Regex */\n.highlight-ipynb .s1 { color: var(--jp-mirror-editor-string-color) } /* Literal.String.Single */\n.highlight-ipynb .ss { color: var(--jp-mirror-editor-string-color) } /* Literal.String.Symbol */\n.highlight-ipynb .il { color: var(--jp-mirror-editor-number-color) } /* Literal.Number.Integer.Long */']
+        </style>
+        
+
+
+
+
+
+
+
+  
+    
+  
+  
+    
+  
+
+
+  <aside class="md-source-file">
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+    </span>
+    <span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-timeago"><span class="timeago" datetime="2024-12-16T12:47:23+00:00" locale="zh"></span></span><span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-iso_date">2024-12-16</span>
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+    </span>
+    <span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-timeago"><span class="timeago" datetime="2024-12-16T12:47:23+00:00" locale="zh"></span></span><span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-iso_date">2024-12-16</span>
+  </span>
+
+    
+    
+    
+  </aside>
+
+
+
+
+<h2 id="__comments">评论</h2>
+<script src="https://giscus.app/client.js"
+        data-repo="HuangFuSL/HuangFuSL.github.io"
+        data-repo-id="MDEwOlJlcG9zaXRvcnkzMTQyMTQ4MTE="
+        data-category="Comments"
+        data-category-id="DIC_kwDOErqJm84CShoc"
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+<!-- Synchronize Giscus theme with palette -->
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+                frame.contentWindow.postMessage(
+                    { giscus: { setConfig: { theme } } },
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+                );
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+    </a>
+  
+    
+    
+    
+    
+      
+      
+    
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+    </a>
+  
+    
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+    
+    <a href="https://space.bilibili.com/387905326" target="_blank" rel="noopener" title="space.bilibili.com" class="md-social__link">
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diff --git a/coding/rust/0/index.html b/coding/rust/0/index.html
index 8254539ed..15c0e5a04 100644
--- a/coding/rust/0/index.html
+++ b/coding/rust/0/index.html
@@ -813,6 +813,8 @@
         
       
         
+      
+        
       
     
     
@@ -1207,6 +1209,27 @@
 
               
             
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../../rust-jupyter/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Jupyter for Rust
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
           </ul>
         </nav>
       
diff --git a/coding/rust/1/index.html b/coding/rust/1/index.html
index d24bddb64..f5c2994af 100644
--- a/coding/rust/1/index.html
+++ b/coding/rust/1/index.html
@@ -811,6 +811,8 @@
         
       
         
+      
+        
       
     
     
@@ -1256,6 +1258,27 @@
 
               
             
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../../rust-jupyter/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Jupyter for Rust
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
           </ul>
         </nav>
       
diff --git a/coding/rust/2/index.html b/coding/rust/2/index.html
index 3e7953191..163a871c0 100644
--- a/coding/rust/2/index.html
+++ b/coding/rust/2/index.html
@@ -813,6 +813,8 @@
         
       
         
+      
+        
       
     
     
@@ -1240,6 +1242,27 @@
 
               
             
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../../rust-jupyter/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Jupyter for Rust
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
           </ul>
         </nav>
       
diff --git a/coding/rust/3/index.html b/coding/rust/3/index.html
index 28115b43d..01f618bd9 100644
--- a/coding/rust/3/index.html
+++ b/coding/rust/3/index.html
@@ -813,6 +813,8 @@
         
       
         
+      
+        
       
     
     
@@ -1246,6 +1248,27 @@
 
               
             
+              
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+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../../rust-jupyter/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Jupyter for Rust
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
           </ul>
         </nav>
       
diff --git a/coding/rust/4/index.html b/coding/rust/4/index.html
index 71a0e1fc4..5cc6681e3 100644
--- a/coding/rust/4/index.html
+++ b/coding/rust/4/index.html
@@ -813,6 +813,8 @@
         
       
         
+      
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@@ -1198,6 +1200,27 @@
 
               
             
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+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../../rust-jupyter/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Jupyter for Rust
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
           </ul>
         </nav>
       
diff --git a/coding/rust/5/index.html b/coding/rust/5/index.html
index fe38d0527..5a6b934bd 100644
--- a/coding/rust/5/index.html
+++ b/coding/rust/5/index.html
@@ -813,6 +813,8 @@
         
       
         
+      
+        
       
     
     
@@ -1261,6 +1263,27 @@
 
               
             
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../../rust-jupyter/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Jupyter for Rust
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
           </ul>
         </nav>
       
diff --git a/coding/rust/6/index.html b/coding/rust/6/index.html
index 1ef47bf29..373ab215b 100644
--- a/coding/rust/6/index.html
+++ b/coding/rust/6/index.html
@@ -811,6 +811,8 @@
         
       
         
+      
+        
       
     
     
@@ -1187,6 +1189,27 @@
 
               
             
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../../rust-jupyter/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Jupyter for Rust
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
           </ul>
         </nav>
       
diff --git a/coding/rust/index.html b/coding/rust/index.html
index 33eb1b48f..40bd98910 100644
--- a/coding/rust/index.html
+++ b/coding/rust/index.html
@@ -813,6 +813,8 @@
         
       
         
+      
+        
       
     
     
@@ -1131,6 +1133,27 @@
 
               
             
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../rust-jupyter/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Jupyter for Rust
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
           </ul>
         </nav>
       
diff --git a/coding/svg/index.html b/coding/svg/index.html
index df437000e..d498fa9af 100644
--- a/coding/svg/index.html
+++ b/coding/svg/index.html
@@ -810,6 +810,8 @@
       
         
       
+        
+      
     
     
       
diff --git a/css/jupyter.css b/css/jupyter.css
index ab32158f3..ac19bd761 100644
--- a/css/jupyter.css
+++ b/css/jupyter.css
@@ -7,6 +7,7 @@ body {
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     --jp-cell-editor-background: var(--override-code-bg-color);
     --jp-cell-prompt-not-active-opacity: 1;
+    --jp-code-font-size: .85em;
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     --jp-content-font-family: var(--md-text-font-family);
     --jp-ui-font-family: var(--md-text-font-family);
@@ -26,10 +27,15 @@ span.MathJax_CHTML {
     font-family: inherit;
 }
 
+.jupyter-wrapper .highlight pre {
+    font-size: .85em !important;
+    background-color: var(--md-code-bg-color) !important;
+    border-radius: 0.4rem !important;
+}
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-    font-size: .8rem;
-    line-height: 1.6;
-    font-family: inherit;
+    font-size: .85em;
+    background-color: var(--md-code-bg-color);
 }
 
 small {
diff --git a/index.html b/index.html
index 4c23bab2c..d2c954a5f 100644
--- a/index.html
+++ b/index.html
@@ -1064,6 +1064,11 @@ <h3 id="blog-timeline"><span class="twemoji"><svg xmlns="http://www.w3.org/2000/
 <div class="nt-timeline-before"></div>
 <div class="nt-timeline-items">
 <div class="nt-timeline-item">
+<p class="nt-timeline-title"><a href="coding/rust-jupyter/">Jupyter</a></p>
+<span class="nt-timeline-sub-title">2024-12-16</span><p class="nt-timeline-content">Rust的evcxr_jupyter包提供了对jupyter notebook的支持，可以在notebook中运行Rust代码。使用如下命令安装：</p>
+<div class="nt-timeline-dot "></div>
+</div>
+<div class="nt-timeline-item">
 <p class="nt-timeline-title"><a href="coding/machine-learning/gaussian-process/">高斯过程</a></p>
 <span class="nt-timeline-sub-title">2024-12-16</span><p class="nt-timeline-content">高斯过程（Gaussian Process，GP）是将多变量高斯分布推广到无限维度<span class="arithmatex">\(\mathcal X\subseteq \mathbb R^n\)</span>的概率分布。具体地，对于<span class="arithmatex">\(X = \{x_1, \ldots, x_n\}\subseteq \mathcal X\)</span>，随机变量<span class="arithmatex">\(f(x_1), \ldots, f(x_n)\)</span>服从多元高斯分布<span class="arithmatex">\(\mathcal N(\mu(X), \Sigma(X))\)</span>。其中，<span class="arithmatex">\(\mu(X)\)</span>为均值函数，<span class="arithmatex">\(\Sigma(X)\)</span>为协方差函数。因此，高斯分布研究的是函数的概率分布。</p>
 <div class="nt-timeline-dot "></div>
@@ -1108,17 +1113,12 @@ <h3 id="blog-timeline"><span class="twemoji"><svg xmlns="http://www.w3.org/2000/
 <span class="nt-timeline-sub-title">2024-09-07</span><p class="nt-timeline-content">Attention机制虽然能捕捉序列中不同位置的依赖关系，但是无法区分不同位置的元素。为了解决这个问题，Transformer模型引入了位置编码（Positional Encoding）。</p>
 <div class="nt-timeline-dot "></div>
 </div>
-<div class="nt-timeline-item">
-<p class="nt-timeline-title"><a href="coding/dl-from-scratch/transformer-variants/">transformer变种</a></p>
-<span class="nt-timeline-sub-title">2024-09-07</span><p class="nt-timeline-content">Transformer变种主要包含如下几种：</p>
-<div class="nt-timeline-dot "></div>
-</div>
 </div>
 <div class="nt-timeline-after"></div>
 </div>
 <h3 id="recent-updates"><span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20.7 7c-.3.4-.7.7-.7 1s.3.6.6 1c.5.5 1 .9.9 1.4 0 .5-.5 1-1 1.5L16.4 16 15 14.7l4.2-4.2-1-1-1.4 1.4L13 7.1l4-3.8c.4-.4 1-.4 1.4 0l2.3 2.3c.4.4.4 1.1 0 1.4M3 17.2l9.6-9.6 3.7 3.8L6.8 21H3zM7 2v3h3v2H7v3H5V7H2V5h3V2z"/></svg></span> Recent updates<a class="headerlink" href="#recent-updates" title="Permanent link">&para;</a></h3>
 <ul>
-<li>2024-12-16：<a href="coding/machine-learning/">机器学习</a>，<a href="coding/machine-learning/gaussian-process/">高斯过程</a></li>
+<li>2024-12-16：<a href="coding/rust-jupyter/">Jupyter</a>，<a href="coding/machine-learning/">机器学习</a>，<a href="coding/machine-learning/gaussian-process/">高斯过程</a></li>
 <li>2024-12-09：<a href="coding/machine-learning/bayesian-optimization/">算法</a></li>
 <li>2024-12-04：<a href="coding/dl-from-scratch/">深度学习</a>，<a href="coding/dsa/">数据结构</a>，<a href="coding/python/">Python</a>，<a href="coding/reinforcement-learning/">强化学习</a>，<a href="coding/reinforcement-learning/multi-armed-bandit/">多臂老虎机</a>，<a href="coding/reinforcement-learning/upper-confidence-bound/">置信上界算法</a>，<a href="coding/rust/">Rust</a></li>
 <li>2024-11-25：<a href="coding/dl-from-scratch/llama-2/">实现Llama-2</a>，<a href="coding/dl-from-scratch/llm-inference/">模型推理</a></li>
@@ -1257,14 +1257,14 @@ <h2 id="acknowledgements"><span class="twemoji"><svg xmlns="http://www.w3.org/20
 </ul>
 <p>Unless noted, content in this blog are shared under <a href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC-BY-NC-SA 4.0</a> license.</p>
 <h2 id="version-information"><span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M21 10.12h-6.78l2.74-2.82c-2.73-2.7-7.15-2.8-9.88-.1a6.887 6.887 0 0 0 0 9.8c2.73 2.7 7.15 2.7 9.88 0 1.36-1.35 2.04-2.92 2.04-4.9h2c0 1.98-.88 4.55-2.64 6.29-3.51 3.48-9.21 3.48-12.72 0-3.5-3.47-3.53-9.11-.02-12.58a8.987 8.987 0 0 1 12.65 0L21 3zM12.5 8v4.25l3.5 2.08-.72 1.21L11 13V8z"/></svg></span> Version information<a class="headerlink" href="#version-information" title="Permanent link">&para;</a></h2>
-<div class="highlight"><pre><span></span><code>commit 4b7d0e344ecccd970fead21fc513bcd9dee83585
-Merge: 57013cd0 949c8cdc
+<div class="highlight"><pre><span></span><code>commit 3634d38c7e95f633ed80e816b6dbfa4e3edce5ac
+Merge: 4b7d0e34 98aca170
 Author: github-actions[bot] &lt;41898282+github-actions[bot]@users.noreply.github.com&gt;
-Date:   Mon Dec 16 17:33:30 2024 +0800
+Date:   Mon Dec 16 20:57:31 2024 +0800
 
-    Merge pull request #421 from HuangFuSL/gaussian-process
+    Merge pull request #422 from HuangFuSL/rust-jupyter
 
-    Update: Gaussian Process
+    Update: Rust Jupyter
 </code></pre></div>
 
 
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-                    <span>HuangFuSL</span>
-                </h1>
-                <div class="row">
-                    <section>
-                        <div class="field ">
-                            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
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-                            </svg>
-                            Joined GitHub 5 years ago
-                        </div>
-                        <div class="field">
-                            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                <path fill-rule="evenodd" d="M5.5 3.5a2 2 0 100 4 2 2 0 000-4zM2 5.5a3.5 3.5 0 115.898 2.549 5.507 5.507 0 013.034 4.084.75.75 0 11-1.482.235 4.001 4.001 0 00-7.9 0 .75.75 0 01-1.482-.236A5.507 5.507 0 013.102 8.05 3.49 3.49 0 012 5.5zM11 4a.75.75 0 100 1.5 1.5 1.5 0 01.666 2.844.75.75 0 00-.416.672v.352a.75.75 0 00.574.73c1.2.289 2.162 1.2 2.522 2.372a.75.75 0 101.434-.44 5.01 5.01 0 00-2.56-3.012A3 3 0 0011 4z"/>
-                            </svg>
-                            Followed by 12 users
-                        </div>
-                    </section>
-                    <section>
-                        <div class="field calendar">
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-                            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
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-                            </svg>
-                            Contributed to 18 repositories
-                        </div>
-                    </section>
-                </div>
-            </section>
-            <section>
-                <h2 class="field">
-                    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
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-                    </svg>
-                    About me
-                </h2>
-                <div class="row fill-width">
-                    <section>
-                        <div class="introduction">PhD student in School of Economics and Management, Tsinghua University.</div>
-                    </section>
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-                            Activity
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-                            1644 Commits
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-                            0 Pull requests reviewed
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-                            189 Pull requests opened
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-                            79 Issues opened
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-                            131 issue comments
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-                    </section>
-                    <section>
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-                            Community stats
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-                            Member of 2 organizations
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-                            Following 18 users
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-                            Sponsoring 0 repositories
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-                            Starred 185 repositories
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-                            Watching 26 repositories
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-                            31 Repositories
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-                                    Prefers MIT license
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-                                        <path fill-rule="evenodd" d="M2.5 7.775V2.75a.25.25 0 01.25-.25h5.025a.25.25 0 01.177.073l6.25 6.25a.25.25 0 010 .354l-5.025 5.025a.25.25 0 01-.354 0l-6.25-6.25a.25.25 0 01-.073-.177zm-1.5 0V2.75C1 1.784 1.784 1 2.75 1h5.025c.464 0 .91.184 1.238.513l6.25 6.25a1.75 1.75 0 010 2.474l-5.026 5.026a1.75 1.75 0 01-2.474 0l-6.25-6.25A1.75 1.75 0 011 7.775zM6 5a1 1 0 100 2 1 1 0 000-2z"/>
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-                                    19 Releases
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-                                    1 Package
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-                                    571 MB used
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-                                    0 added, 0 removed
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-                                    0 Sponsors
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-                                <div class="field">
-                                    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                        <path fill-rule="evenodd" d="M8 .25a.75.75 0 01.673.418l1.882 3.815 4.21.612a.75.75 0 01.416 1.279l-3.046 2.97.719 4.192a.75.75 0 01-1.088.791L8 12.347l-3.766 1.98a.75.75 0 01-1.088-.79l.72-4.194L.818 6.374a.75.75 0 01.416-1.28l4.21-.611L7.327.668A.75.75 0 018 .25zm0 2.445L6.615 5.5a.75.75 0 01-.564.41l-3.097.45 2.24 2.184a.75.75 0 01.216.664l-.528 3.084 2.769-1.456a.75.75 0 01.698 0l2.77 1.456-.53-3.084a.75.75 0 01.216-.664l2.24-2.183-3.096-.45a.75.75 0 01-.564-.41L8 2.694v.001z"/>
-                                    </svg>
-                                    11 Stargazers
-                                </div>
-                                <div class="field">
-                                    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                        <path fill-rule="evenodd" d="M5 3.25a.75.75 0 11-1.5 0 .75.75 0 011.5 0zm0 2.122a2.25 2.25 0 10-1.5 0v.878A2.25 2.25 0 005.75 8.5h1.5v2.128a2.251 2.251 0 101.5 0V8.5h1.5a2.25 2.25 0 002.25-2.25v-.878a2.25 2.25 0 10-1.5 0v.878a.75.75 0 01-.75.75h-4.5A.75.75 0 015 6.25v-.878zm3.75 7.378a.75.75 0 11-1.5 0 .75.75 0 011.5 0zm3-8.75a.75.75 0 100-1.5.75.75 0 000 1.5z"/>
-                                    </svg>
-                                    7 Forkers
-                                </div>
-                                <div class="field">
-                                    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                        <path fill-rule="evenodd" d="M1.679 7.932c.412-.621 1.242-1.75 2.366-2.717C5.175 4.242 6.527 3.5 8 3.5c1.473 0 2.824.742 3.955 1.715 1.124.967 1.954 2.096 2.366 2.717a.119.119 0 010 .136c-.412.621-1.242 1.75-2.366 2.717C10.825 11.758 9.473 12.5 8 12.5c-1.473 0-2.824-.742-3.955-1.715C2.92 9.818 2.09 8.69 1.679 8.068a.119.119 0 010-.136zM8 2c-1.981 0-3.67.992-4.933 2.078C1.797 5.169.88 6.423.43 7.1a1.619 1.619 0 000 1.798c.45.678 1.367 1.932 2.637 3.024C4.329 13.008 6.019 14 8 14c1.981 0 3.67-.992 4.933-2.078 1.27-1.091 2.187-2.345 2.637-3.023a1.619 1.619 0 000-1.798c-.45-.678-1.367-1.932-2.637-3.023C11.671 2.992 9.981 2 8 2zm0 8a2 2 0 100-4 2 2 0 000 4z"/>
-                                    </svg>
-                                    32 Watchers
-                                </div>
-                                <div class="field ">
-                                    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                        <path fill-rule="evenodd" d="M0 1.75A.75.75 0 01.75 1h4.253c1.227 0 2.317.59 3 1.501A3.744 3.744 0 0111.006 1h4.245a.75.75 0 01.75.75v10.5a.75.75 0 01-.75.75h-4.507a2.25 2.25 0 00-1.591.659l-.622.621a.75.75 0 01-1.06 0l-.622-.621A2.25 2.25 0 005.258 13H.75a.75.75 0 01-.75-.75V1.75zm8.755 3a2.25 2.25 0 012.25-2.25H14.5v9h-3.757c-.71 0-1.4.201-1.992.572l.004-7.322zm-1.504 7.324l.004-5.073-.002-2.253A2.25 2.25 0 005.003 2.5H1.5v9h3.757a3.75 3.75 0 011.994.574z"/>
-                                    </svg>
-                                    127 views in last two weeks
-                                </div>
-                            </section>
-                        </div>
-                    </section>
-                </div>
-            </section>
-            <section>
-                <h2 class="field">
-                    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                        <path fill-rule="evenodd" d="M1.5 2.75a.25.25 0 01.25-.25h12.5a.25.25 0 01.25.25v8.5a.25.25 0 01-.25.25h-6.5a.75.75 0 00-.53.22L4.5 14.44v-2.19a.75.75 0 00-.75-.75h-2a.25.25 0 01-.25-.25v-8.5zM1.75 1A1.75 1.75 0 000 2.75v8.5C0 12.216.784 13 1.75 13H3v1.543a1.457 1.457 0 002.487 1.03L8.061 13h6.189A1.75 1.75 0 0016 11.25v-8.5A1.75 1.75 0 0014.25 1H1.75zm5.03 3.47a.75.75 0 010 1.06L5.31 7l1.47 1.47a.75.75 0 01-1.06 1.06l-2-2a.75.75 0 010-1.06l2-2a.75.75 0 011.06 0zm2.44 0a.75.75 0 000 1.06L10.69 7 9.22 8.47a.75.75 0 001.06 1.06l2-2a.75.75 0 000-1.06l-2-2a.75.75 0 00-1.06 0z"/>
-                    </svg>
-                    13 Languages
-                </h2>
-            </section>
-            <section class="column">
-                <h3 class="field">Most used languages</h3>
-                <svg class="bar" xmlns="http://www.w3.org/2000/svg" width="460" height="8">
-                    <mask id="languages-bar">
-                        <rect x="0" y="0" width="460" height="8" fill="white" rx="5"/>
-                    </mask>
-                    <rect mask="url(#languages-bar)" x="0" y="0" width="0" height="8" fill="#d1d5da"/>
-                    <rect mask="url(#languages-bar)" x="0" y="0" width="230.99213084005729" height="8" fill="#3D6117"/>
-                    <rect mask="url(#languages-bar)" x="230.99213084005729" y="0" width="157.10479397591212" height="8" fill="#3572A5"/>
-                    <rect mask="url(#languages-bar)" x="388.0969248159694" y="0" width="41.525597319841275" height="8" fill="#dea584"/>
-                    <rect mask="url(#languages-bar)" x="429.62252213581064" y="0" width="30.377477864189338" height="8" fill="#3178c6"/>
-                </svg>
-                <div class="row fill-width">
-                    <section>
-                        <div class="field language details">
-                            <div class="field">
-                                <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                    <path fill="#3D6117" fill-rule="evenodd" d="M8 4a4 4 0 100 8 4 4 0 000-8z"/>
-                                </svg>
-                                TeX
-                            </div>
-                            <small>
-                                <div>784 kB</div>
-                                <div>50.22%</div>
-                            </small>
-                        </div>
-                        <div class="field language details">
-                            <div class="field">
-                                <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                    <path fill="#dea584" fill-rule="evenodd" d="M8 4a4 4 0 100 8 4 4 0 000-8z"/>
-                                </svg>
-                                Rust
-                            </div>
-                            <small>
-                                <div>141 kB</div>
-                                <div>9.03%</div>
-                            </small>
-                        </div>
-                    </section>
-                    <section>
-                        <div class="field language details">
-                            <div class="field">
-                                <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                    <path fill="#3572A5" fill-rule="evenodd" d="M8 4a4 4 0 100 8 4 4 0 000-8z"/>
-                                </svg>
-                                Python
-                            </div>
-                            <small>
-                                <div>533 kB</div>
-                                <div>34.15%</div>
-                            </small>
-                        </div>
-                        <div class="field language details">
-                            <div class="field">
-                                <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                    <path fill="#3178c6" fill-rule="evenodd" d="M8 4a4 4 0 100 8 4 4 0 000-8z"/>
-                                </svg>
-                                TypeScript
-                            </div>
-                            <small>
-                                <div>103 kB</div>
-                                <div>6.6%</div>
-                            </small>
-                        </div>
-                    </section>
-                </div>
-            </section>
-            <div class="row">
-                <section>
-                    <h2 class="field">
-                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M14.064 0a8.75 8.75 0 00-6.187 2.563l-.459.458c-.314.314-.616.641-.904.979H3.31a1.75 1.75 0 00-1.49.833L.11 7.607a.75.75 0 00.418 1.11l3.102.954c.037.051.079.1.124.145l2.429 2.428c.046.046.094.088.145.125l.954 3.102a.75.75 0 001.11.418l2.774-1.707a1.75 1.75 0 00.833-1.49V9.485c.338-.288.665-.59.979-.904l.458-.459A8.75 8.75 0 0016 1.936V1.75A1.75 1.75 0 0014.25 0h-.186zM10.5 10.625c-.088.06-.177.118-.266.175l-2.35 1.521.548 1.783 1.949-1.2a.25.25 0 00.119-.213v-2.066zM3.678 8.116L5.2 5.766c.058-.09.117-.178.176-.266H3.309a.25.25 0 00-.213.119l-1.2 1.95 1.782.547zm5.26-4.493A7.25 7.25 0 0114.063 1.5h.186a.25.25 0 01.25.25v.186a7.25 7.25 0 01-2.123 5.127l-.459.458a15.21 15.21 0 01-2.499 2.02l-2.317 1.5-2.143-2.143 1.5-2.317a15.25 15.25 0 012.02-2.5l.458-.458h.002zM12 5a1 1 0 11-2 0 1 1 0 012 0zm-8.44 9.56a1.5 1.5 0 10-2.12-2.12c-.734.73-1.047 2.332-1.15 3.003a.23.23 0 00.265.265c.671-.103 2.273-.416 3.005-1.148z"/>
-                        </svg>
-                        PageSpeed Insights
-                    </h2>
-                    <div class="field">
-                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M7.775 3.275a.75.75 0 001.06 1.06l1.25-1.25a2 2 0 112.83 2.83l-2.5 2.5a2 2 0 01-2.83 0 .75.75 0 00-1.06 1.06 3.5 3.5 0 004.95 0l2.5-2.5a3.5 3.5 0 00-4.95-4.95l-1.25 1.25zm-4.69 9.64a2 2 0 010-2.83l2.5-2.5a2 2 0 012.83 0 .75.75 0 001.06-1.06 3.5 3.5 0 00-4.95 0l-2.5 2.5a3.5 3.5 0 004.95 4.95l1.25-1.25a.75.75 0 00-1.06-1.06l-1.25 1.25a2 2 0 01-2.83 0z"/>
-                        </svg>
-                        https://blog.huangfusl.net
-                    </div>
-                </section>
-            </div>
-            <section>
-                <div class="row fill-width">
-                    <section class="categories">
-                        <div class="category column">
-                            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 120 120" width="50" height="50" class="gauge high">
-                                <circle class="gauge-base" r="53" cx="60" cy="60"/>
-                                <circle class="gauge-arc" transform="rotate(-90 60 60)" r="53" cx="60" cy="60" stroke-dasharray="325.71 329"/>
-                                <text x="60" y="60" dominant-baseline="central">99</text>
-                            </svg>
-                            <span class="title">Performance</span>
-                        </div>
-                        <div class="category column">
-                            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 120 120" width="50" height="50" class="gauge high">
-                                <circle class="gauge-base" r="53" cx="60" cy="60"/>
-                                <circle class="gauge-arc" transform="rotate(-90 60 60)" r="53" cx="60" cy="60" stroke-dasharray="305.97 329"/>
-                                <text x="60" y="60" dominant-baseline="central">93</text>
-                            </svg>
-                            <span class="title">Accessibility</span>
-                        </div>
-                        <div class="category column">
-                            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 120 120" width="50" height="50" class="gauge high">
-                                <circle class="gauge-base" r="53" cx="60" cy="60"/>
-                                <circle class="gauge-arc" transform="rotate(-90 60 60)" r="53" cx="60" cy="60" stroke-dasharray="315.84 329"/>
-                                <text x="60" y="60" dominant-baseline="central">96</text>
-                            </svg>
-                            <span class="title">Best Practices</span>
-                        </div>
-                        <div class="category column">
-                            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 120 120" width="50" height="50" class="gauge high">
-                                <circle class="gauge-base" r="53" cx="60" cy="60"/>
-                                <circle class="gauge-arc" transform="rotate(-90 60 60)" r="53" cx="60" cy="60" stroke-dasharray="296.1 329"/>
-                                <text x="60" y="60" dominant-baseline="central">90</text>
-                            </svg>
-                            <span class="title">SEO</span>
-                        </div>
-                    </section>
-                </div>
-            </section>
-            <div class="audits row">
-                <section>
-                    <div class="field">
-                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M5.75.75A.75.75 0 016.5 0h3a.75.75 0 010 1.5h-.75v1l-.001.041a6.718 6.718 0 013.464 1.435l.007-.006.75-.75a.75.75 0 111.06 1.06l-.75.75-.006.007a6.75 6.75 0 11-10.548 0L2.72 5.03l-.75-.75a.75.75 0 011.06-1.06l.75.75.007.006A6.718 6.718 0 017.25 2.541a.756.756 0 010-.041v-1H6.5a.75.75 0 01-.75-.75zM8 14.5A5.25 5.25 0 108 4a5.25 5.25 0 000 10.5zm.389-6.7l1.33-1.33a.75.75 0 111.061 1.06L9.45 8.861A1.502 1.502 0 018 10.75a1.5 1.5 0 11.389-2.95z"/>
-                        </svg>
-                        Time to interactive
-                    </div>
-                    <div class="field">
-                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M8 1.5a6.5 6.5 0 106.016 4.035.75.75 0 011.388-.57 8 8 0 11-4.37-4.37.75.75 0 01-.569 1.389A6.479 6.479 0 008 1.5zm6.28.22a.75.75 0 010 1.06l-4.063 4.064a2.5 2.5 0 11-1.06-1.06L13.22 1.72a.75.75 0 011.06 0zM7 8a1 1 0 112 0 1 1 0 01-2 0z"/>
-                        </svg>
-                        Speed Index
-                    </div>
-                    <div class="field">
-                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M1.643 3.143L.427 1.927A.25.25 0 000 2.104V5.75c0 .138.112.25.25.25h3.646a.25.25 0 00.177-.427L2.715 4.215a6.5 6.5 0 11-1.18 4.458.75.75 0 10-1.493.154 8.001 8.001 0 101.6-5.684zM7.75 4a.75.75 0 01.75.75v2.992l2.028.812a.75.75 0 01-.557 1.392l-2.5-1A.75.75 0 017 8.25v-3.5A.75.75 0 017.75 4z"/>
-                        </svg>
-                        Total Blocking Time
-                    </div>
-                </section>
-                <section>
-                    <div class="field">
-                        <div class="audit text">0.69 s</div>
-                        <svg xmlns="http://www.w3.org/2000/svg" class="audit high" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M1.5 8a6.5 6.5 0 1113 0 6.5 6.5 0 01-13 0zM0 8a8 8 0 1116 0A8 8 0 010 8zm11.78-1.72a.75.75 0 00-1.06-1.06L6.75 9.19 5.28 7.72a.75.75 0 00-1.06 1.06l2 2a.75.75 0 001.06 0l4.5-4.5z"/>
-                        </svg>
-                    </div>
-                    <div class="field">
-                        <div class="audit text">0.69 s</div>
-                        <svg xmlns="http://www.w3.org/2000/svg" class="audit high" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M1.5 8a6.5 6.5 0 1113 0 6.5 6.5 0 01-13 0zM0 8a8 8 0 1116 0A8 8 0 010 8zm11.78-1.72a.75.75 0 00-1.06-1.06L6.75 9.19 5.28 7.72a.75.75 0 00-1.06 1.06l2 2a.75.75 0 001.06 0l4.5-4.5z"/>
-                        </svg>
-                    </div>
-                    <div class="field">
-                        <div class="audit text">0</div>
-                        <svg xmlns="http://www.w3.org/2000/svg" class="audit high" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M1.5 8a6.5 6.5 0 1113 0 6.5 6.5 0 01-13 0zM0 8a8 8 0 1116 0A8 8 0 010 8zm11.78-1.72a.75.75 0 00-1.06-1.06L6.75 9.19 5.28 7.72a.75.75 0 00-1.06 1.06l2 2a.75.75 0 001.06 0l4.5-4.5z"/>
-                        </svg>
-                    </div>
-                </section>
-            </div>
-            <div class="audits row">
-                <section>
-                    <div class="field">
-                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M11.134 1.535C9.722 2.562 8.16 4.057 6.889 5.312 5.8 6.387 5.041 7.401 4.575 8.294a3.745 3.745 0 00-3.227 1.054c-.43.431-.69 1.066-.86 1.657a11.982 11.982 0 00-.358 1.914A21.263 21.263 0 000 15.203v.054l.75-.007-.007.75h.054a14.404 14.404 0 00.654-.012 21.243 21.243 0 001.63-.118c.62-.07 1.3-.18 1.914-.357.592-.17 1.226-.43 1.657-.861a3.745 3.745 0 001.055-3.217c.908-.461 1.942-1.216 3.04-2.3 1.279-1.262 2.764-2.825 3.775-4.249.501-.706.923-1.428 1.125-2.096.2-.659.235-1.469-.368-2.07-.606-.607-1.42-.55-2.069-.34-.66.213-1.376.646-2.076 1.155zm-3.95 8.48a3.76 3.76 0 00-1.19-1.192 9.758 9.758 0 011.161-1.607l1.658 1.658a9.853 9.853 0 01-1.63 1.142zM.742 16l.007-.75-.75.008A.75.75 0 00.743 16zM12.016 2.749c-1.224.89-2.605 2.189-3.822 3.384l1.718 1.718c1.21-1.205 2.51-2.597 3.387-3.833.47-.662.78-1.227.912-1.662.134-.444.032-.551.009-.575h-.001V1.78c-.014-.014-.112-.113-.548.027-.432.14-.995.462-1.655.942zM1.62 13.089a19.56 19.56 0 00-.104 1.395 19.55 19.55 0 001.396-.104 10.528 10.528 0 001.668-.309c.526-.151.856-.325 1.011-.48a2.25 2.25 0 00-3.182-3.182c-.155.155-.329.485-.48 1.01a10.515 10.515 0 00-.309 1.67z"/>
-                        </svg>
-                        First Contentful Paint
-                    </div>
-                    <div class="field">
-                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M1.75 2.5a.25.25 0 00-.25.25v10.5c0 .138.112.25.25.25h.94a.76.76 0 01.03-.03l6.077-6.078a1.75 1.75 0 012.412-.06L14.5 10.31V2.75a.25.25 0 00-.25-.25H1.75zm12.5 11H4.81l5.048-5.047a.25.25 0 01.344-.009l4.298 3.889v.917a.25.25 0 01-.25.25zm1.75-.25V2.75A1.75 1.75 0 0014.25 1H1.75A1.75 1.75 0 000 2.75v10.5C0 14.216.784 15 1.75 15h12.5A1.75 1.75 0 0016 13.25zM5.5 6a.5.5 0 11-1 0 .5.5 0 011 0zM7 6a2 2 0 11-4 0 2 2 0 014 0z"/>
-                        </svg>
-                        Largest Contentful Paint
-                    </div>
-                    <div class="field">
-                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M7.75 14A1.75 1.75 0 016 12.25v-8.5C6 2.784 6.784 2 7.75 2h6.5c.966 0 1.75.784 1.75 1.75v8.5A1.75 1.75 0 0114.25 14h-6.5zm-.25-1.75c0 .138.112.25.25.25h6.5a.25.25 0 00.25-.25v-8.5a.25.25 0 00-.25-.25h-6.5a.25.25 0 00-.25.25v8.5zM4.9 3.508a.75.75 0 01-.274 1.025.25.25 0 00-.126.217v6.5a.25.25 0 00.126.217.75.75 0 01-.752 1.298A1.75 1.75 0 013 11.25v-6.5c0-.649.353-1.214.874-1.516a.75.75 0 011.025.274zM1.625 5.533a.75.75 0 10-.752-1.299A1.75 1.75 0 000 5.75v4.5c0 .649.353 1.214.874 1.515a.75.75 0 10.752-1.298.25.25 0 01-.126-.217v-4.5a.25.25 0 01.126-.217z"/>
-                        </svg>
-                        Cumulative Layout Shift
-                    </div>
-                </section>
-                <section>
-                    <div class="field">
-                        <div class="audit text">0.69 s</div>
-                        <svg xmlns="http://www.w3.org/2000/svg" class="audit high" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M1.5 8a6.5 6.5 0 1113 0 6.5 6.5 0 01-13 0zM0 8a8 8 0 1116 0A8 8 0 010 8zm11.78-1.72a.75.75 0 00-1.06-1.06L6.75 9.19 5.28 7.72a.75.75 0 00-1.06 1.06l2 2a.75.75 0 001.06 0l4.5-4.5z"/>
-                        </svg>
-                    </div>
-                    <div class="field">
-                        <div class="audit text">0.69 s</div>
-                        <svg xmlns="http://www.w3.org/2000/svg" class="audit high" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M1.5 8a6.5 6.5 0 1113 0 6.5 6.5 0 01-13 0zM0 8a8 8 0 1116 0A8 8 0 010 8zm11.78-1.72a.75.75 0 00-1.06-1.06L6.75 9.19 5.28 7.72a.75.75 0 00-1.06 1.06l2 2a.75.75 0 001.06 0l4.5-4.5z"/>
-                        </svg>
-                    </div>
-                    <div class="field">
-                        <div class="audit text">0</div>
-                        <svg xmlns="http://www.w3.org/2000/svg" class="audit high" viewBox="0 0 16 16" width="16" height="16">
-                            <path fill-rule="evenodd" d="M1.5 8a6.5 6.5 0 1113 0 6.5 6.5 0 01-13 0zM0 8a8 8 0 1116 0A8 8 0 010 8zm11.78-1.72a.75.75 0 00-1.06-1.06L6.75 9.19 5.28 7.72a.75.75 0 00-1.06 1.06l2 2a.75.75 0 001.06 0l4.5-4.5z"/>
-                        </svg>
-                    </div>
-                </section>
-            </div>
-            <section class="habits">
-                <h2 class="field wrap">
-                    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                        <path fill-rule="evenodd" d="M8 1.5c-2.363 0-4 1.69-4 3.75 0 .984.424 1.625.984 2.304l.214.253c.223.264.47.556.673.848.284.411.537.896.621 1.49a.75.75 0 01-1.484.211c-.04-.282-.163-.547-.37-.847a8.695 8.695 0 00-.542-.68c-.084-.1-.173-.205-.268-.32C3.201 7.75 2.5 6.766 2.5 5.25 2.5 2.31 4.863 0 8 0s5.5 2.31 5.5 5.25c0 1.516-.701 2.5-1.328 3.259-.095.115-.184.22-.268.319-.207.245-.383.453-.541.681-.208.3-.33.565-.37.847a.75.75 0 01-1.485-.212c.084-.593.337-1.078.621-1.489.203-.292.45-.584.673-.848.075-.088.147-.173.213-.253.561-.679.985-1.32.985-2.304 0-2.06-1.637-3.75-4-3.75zM6 15.25a.75.75 0 01.75-.75h2.5a.75.75 0 010 1.5h-2.5a.75.75 0 01-.75-.75zM5.75 12a.75.75 0 000 1.5h4.5a.75.75 0 000-1.5h-4.5z"/>
-                    </svg>
-                    Recent coding habits
-                    <small class="h-details">(computed from last 7 commits)</small>
-                </h2>
-            </section>
-            <section class="habits">
-                <div class="column chart largeable">
-                    <h3>Commit activity per hour of day</h3>
-                    <div class="chart-bars">
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            00
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            01
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            02
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            03
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            04
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            05
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            06
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            07
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            08
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            09
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            10
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            11
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            12
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            13
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            14
-                        </div>
-                        <div class="entry">
-                            <span class="value">1</span>
-                            <div class="bar" style="height: 10px; background-color: var(--color-calendar-graph-day-L1-bg)"></div>
-                            15
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            16
-                        </div>
-                        <div class="entry">
-                            <span class="value">5</span>
-                            <div class="bar" style="height: 50px; background-color: var(--color-calendar-graph-day-L4-bg)"></div>
-                            17
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            18
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            19
-                        </div>
-                        <div class="entry">
-                            <span class="value">1</span>
-                            <div class="bar" style="height: 10px; background-color: var(--color-calendar-graph-day-L1-bg)"></div>
-                            20
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            21
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            22
-                        </div>
-                        <div class="entry">
-                            <span class="value"></span>
-                            <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                            23
-                        </div>
-                    </div>
-                </div>
-                <div class="row largeable">
-                    <section class="column chart">
-                        <h3>Commit activity per day of week</h3>
-                        <div class="chart-bars">
-                            <div class="entry">
-                                <span class="value">1</span>
-                                <div class="bar" style="height: 8.333333333333332px; background-color: var(--color-calendar-graph-day-L1-bg)"></div>
-                                Sun
-                            </div>
-                            <div class="entry">
-                                <span class="value"></span>
-                                <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                                Mon
-                            </div>
-                            <div class="entry">
-                                <span class="value"></span>
-                                <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                                Tue
-                            </div>
-                            <div class="entry">
-                                <span class="value">6</span>
-                                <div class="bar" style="height: 50px; background-color: var(--color-calendar-graph-day-L4-bg)"></div>
-                                Wed
-                            </div>
-                            <div class="entry">
-                                <span class="value"></span>
-                                <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                                Thu
-                            </div>
-                            <div class="entry">
-                                <span class="value"></span>
-                                <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                                Fri
-                            </div>
-                            <div class="entry">
-                                <span class="value"></span>
-                                <div class="bar" style="height: 0px; background-color: var(--color-calendar-graph-day-L0-bg)"></div>
-                                Sat
-                            </div>
-                        </div>
-                    </section>
-                </div>
-            </section>
-            <section>
-                <h2 class="field">
-                    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                        <path fill-rule="evenodd" d="M14.184 1.143a1.75 1.75 0 00-2.502-.57L.912 7.916a1.75 1.75 0 00-.53 2.32l.447.775a1.75 1.75 0 002.275.702l11.745-5.656a1.75 1.75 0 00.757-2.451l-1.422-2.464zm-1.657.669a.25.25 0 01.358.081l1.422 2.464a.25.25 0 01-.108.35l-2.016.97-1.505-2.605 1.85-1.26zM9.436 3.92l1.391 2.41-5.42 2.61-.942-1.63 4.97-3.39zM3.222 8.157l-1.466 1a.25.25 0 00-.075.33l.447.775a.25.25 0 00.325.1l1.598-.769-.83-1.436zm6.253 2.306a.75.75 0 00-.944-.252l-1.809.87a.75.75 0 00-.293.253L4.38 14.326a.75.75 0 101.238.848l1.881-2.75v2.826a.75.75 0 001.5 0v-2.826l1.881 2.75a.75.75 0 001.238-.848l-2.644-3.863z"/>
-                    </svg>
-                    Starred topics
-                </h2>
-                <div class="row">
-                    <section>
-                        <div class="topics fill-width">
-                            <img src="data:image/png;base64,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" width="24" height="24" alt="Python" title="Python" />
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" width="24" height="24" alt="C++" title="C++" />
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" width="24" height="24" alt="Linux" title="Linux" />
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" width="24" height="24" alt="C" title="C" />
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" width="24" height="24" alt="Rust" title="Rust" />
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" width="24" height="24" alt="PostgreSQL" title="PostgreSQL" />
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" width="24" height="24" alt="Electron" title="Electron" />
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" 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-            <section>
-                <h2 class="field">
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-                    <section>
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-                            Commits streaks
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-                                <path d="M8.5.75a.75.75 0 00-1.5 0v5.19L4.391 3.33a.75.75 0 10-1.06 1.061L5.939 7H.75a.75.75 0 000 1.5h5.19l-2.61 2.609a.75.75 0 101.061 1.06L7 9.561v5.189a.75.75 0 001.5 0V9.56l2.609 2.61a.75.75 0 101.06-1.061L9.561 8.5h5.189a.75.75 0 000-1.5H9.56l2.61-2.609a.75.75 0 00-1.061-1.06L8.5 5.939V.75z"/>
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-                            Best streak 5 days
-                        </div>
-                        <h3 class="field">
-                            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
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-                            Commits per day
-                        </h3>
-                        <div class="field">
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-                                <path d="M7.823 1.677L4.927 4.573A.25.25 0 005.104 5H7.25v3.236a.75.75 0 101.5 0V5h2.146a.25.25 0 00.177-.427L8.177 1.677a.25.25 0 00-.354 0zM13.75 11a.75.75 0 000 1.5h.5a.75.75 0 000-1.5h-.5zm-3.75.75a.75.75 0 01.75-.75h.5a.75.75 0 010 1.5h-.5a.75.75 0 01-.75-.75zM7.75 11a.75.75 0 000 1.5h.5a.75.75 0 000-1.5h-.5zM4 11.75a.75.75 0 01.75-.75h.5a.75.75 0 010 1.5h-.5a.75.75 0 01-.75-.75zM1.75 11a.75.75 0 000 1.5h.5a.75.75 0 000-1.5h-.5z"/>
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-                            Highest in a day at 20
-                        </div>
-                        <div class="field">
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-                                <path d="M10.896 2H8.75V.75a.75.75 0 00-1.5 0V2H5.104a.25.25 0 00-.177.427l2.896 2.896a.25.25 0 00.354 0l2.896-2.896A.25.25 0 0010.896 2zM8.75 15.25a.75.75 0 01-1.5 0V14H5.104a.25.25 0 01-.177-.427l2.896-2.896a.25.25 0 01.354 0l2.896 2.896a.25.25 0 01-.177.427H8.75v1.25zm-6.5-6.5a.75.75 0 000-1.5h-.5a.75.75 0 000 1.5h.5zM6 8a.75.75 0 01-.75.75h-.5a.75.75 0 010-1.5h.5A.75.75 0 016 8zm2.25.75a.75.75 0 000-1.5h-.5a.75.75 0 000 1.5h.5zM12 8a.75.75 0 01-.75.75h-.5a.75.75 0 010-1.5h.5A.75.75 0 0112 8zm2.25.75a.75.75 0 000-1.5h-.5a.75.75 0 000 1.5h.5z"/>
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-                            Average per day at ~0.59
-                        </div>
-                    </section>
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-                                        0
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-                                    <div class="name">
-                                        <span>NLPJCL/RAG-Retrieval</span>
-                                        <span>starred 17 days ago</span>
-                                    </div>
-                                </div>
-                                <div class="field description">Unify Efficient Fine-tuning of  RAG Retrieval, including Embedding, ColBERT, ReRanker.</div>
-                                <div class="field infos">
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-                                        </svg>
-                                        Python
-                                    </div>
-                                    <div>
-                                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                            <path fill-rule="evenodd" d="M8.75.75a.75.75 0 00-1.5 0V2h-.984c-.305 0-.604.08-.869.23l-1.288.737A.25.25 0 013.984 3H1.75a.75.75 0 000 1.5h.428L.066 9.192a.75.75 0 00.154.838l.53-.53-.53.53v.001l.002.002.002.002.006.006.016.015.045.04a3.514 3.514 0 00.686.45A4.492 4.492 0 003 11c.88 0 1.556-.22 2.023-.454a3.515 3.515 0 00.686-.45l.045-.04.016-.015.006-.006.002-.002.001-.002L5.25 9.5l.53.53a.75.75 0 00.154-.838L3.822 4.5h.162c.305 0 .604-.08.869-.23l1.289-.737a.25.25 0 01.124-.033h.984V13h-2.5a.75.75 0 000 1.5h6.5a.75.75 0 000-1.5h-2.5V3.5h.984a.25.25 0 01.124.033l1.29.736c.264.152.563.231.868.231h.162l-2.112 4.692a.75.75 0 00.154.838l.53-.53-.53.53v.001l.002.002.002.002.006.006.016.015.045.04a3.517 3.517 0 00.686.45A4.492 4.492 0 0013 11c.88 0 1.556-.22 2.023-.454a3.512 3.512 0 00.686-.45l.045-.04.01-.01.006-.005.006-.006.002-.002.001-.002-.529-.531.53.53a.75.75 0 00.154-.838L13.823 4.5h.427a.75.75 0 000-1.5h-2.234a.25.25 0 01-.124-.033l-1.29-.736A1.75 1.75 0 009.735 2H8.75V.75zM1.695 9.227c.285.135.718.273 1.305.273s1.02-.138 1.305-.273L3 6.327l-1.305 2.9zm10 0c.285.135.718.273 1.305.273s1.02-.138 1.305-.273L13 6.327l-1.305 2.9z"/>
-                                        </svg>
-                                        MIT
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-                                        553
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-                                    <div>
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-                                        49
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-                                        43
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-                                        11
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-                            </section>
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-                                        <path fill-rule="evenodd" d="M7.177 3.073L9.573.677A.25.25 0 0110 .854v4.792a.25.25 0 01-.427.177L7.177 3.427a.25.25 0 010-.354zM3.75 2.5a.75.75 0 100 1.5.75.75 0 000-1.5zm-2.25.75a2.25 2.25 0 113 2.122v5.256a2.251 2.251 0 11-1.5 0V5.372A2.25 2.25 0 011.5 3.25zM11 2.5h-1V4h1a1 1 0 011 1v5.628a2.251 2.251 0 101.5 0V5A2.5 2.5 0 0011 2.5zm1 10.25a.75.75 0 111.5 0 .75.75 0 01-1.5 0zM3.75 12a.75.75 0 100 1.5.75.75 0 000-1.5z"/>
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-                                    <div class="content">
-                                        Opened
-                                        <span class="issue">#419 Update: Bayesian optimization</span>
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-                                <div class="details">
-                                    <div> in <span class="repo">HuangFuSL/HuangFuSL.github.io</span></div>
-                                    <div>7 files changed <span class="code">++7888 --6</span></div>
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-                                    <div class="content">
-                                        Created new
-                      branch
-                                        <span class="code">bayesian-optimization</span>
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-                                        <span class="repo">HuangFuSL/HuangFuSL.github.io</span>
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-                            </section>
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-                                        <path fill-rule="evenodd" d="M7.177 3.073L9.573.677A.25.25 0 0110 .854v4.792a.25.25 0 01-.427.177L7.177 3.427a.25.25 0 010-.354zM3.75 2.5a.75.75 0 100 1.5.75.75 0 000-1.5zm-2.25.75a2.25 2.25 0 113 2.122v5.256a2.251 2.251 0 11-1.5 0V5.372A2.25 2.25 0 011.5 3.25zM11 2.5h-1V4h1a1 1 0 011 1v5.628a2.251 2.251 0 101.5 0V5A2.5 2.5 0 0011 2.5zm1 10.25a.75.75 0 111.5 0 .75.75 0 01-1.5 0zM3.75 12a.75.75 0 100 1.5.75.75 0 000-1.5z"/>
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-                                    <div class="content">
-                                        Opened
-                                        <span class="issue">#418 Update: re-organize navigation and clean up</span>
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-                                <div class="details">
-                                    <div> in <span class="repo">HuangFuSL/HuangFuSL.github.io</span></div>
-                                    <div>26 files changed <span class="code">++46 --3733</span></div>
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-                        <div class="row fill-width">
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-                                        <path fill-rule="evenodd" d="M11.75 2.5a.75.75 0 100 1.5.75.75 0 000-1.5zm-2.25.75a2.25 2.25 0 113 2.122V6A2.5 2.5 0 0110 8.5H6a1 1 0 00-1 1v1.128a2.251 2.251 0 11-1.5 0V5.372a2.25 2.25 0 111.5 0v1.836A2.492 2.492 0 016 7h4a1 1 0 001-1v-.628A2.25 2.25 0 019.5 3.25zM4.25 12a.75.75 0 100 1.5.75.75 0 000-1.5zM3.5 3.25a.75.75 0 111.5 0 .75.75 0 01-1.5 0z"/>
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-                                    <div class="content">
-                                        Created new
-                      branch
-                                        <span class="code">re-organize</span>
-                                        in
-                                        <span class="repo">HuangFuSL/HuangFuSL.github.io</span>
-                                    </div>
-                                </div>
-                            </section>
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-                        <div class="row fill-width">
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-                                        <path fill-rule="evenodd" d="M7.177 3.073L9.573.677A.25.25 0 0110 .854v4.792a.25.25 0 01-.427.177L7.177 3.427a.25.25 0 010-.354zM3.75 2.5a.75.75 0 100 1.5.75.75 0 000-1.5zm-2.25.75a2.25 2.25 0 113 2.122v5.256a2.251 2.251 0 11-1.5 0V5.372A2.25 2.25 0 011.5 3.25zM11 2.5h-1V4h1a1 1 0 011 1v5.628a2.251 2.251 0 101.5 0V5A2.5 2.5 0 0011 2.5zm1 10.25a.75.75 0 111.5 0 .75.75 0 01-1.5 0zM3.75 12a.75.75 0 100 1.5.75.75 0 000-1.5z"/>
-                                    </svg>
-                                    <div class="content">
-                                        Opened
-                                        <span class="issue">#417 Update: Multi armed bandit and UCB</span>
-                                    </div>
-                                </div>
-                                <div class="details">
-                                    <div> in <span class="repo">HuangFuSL/HuangFuSL.github.io</span></div>
-                                    <div>6 files changed <span class="code">++436 --0</span></div>
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-            <section class="steam">
-                <h2 class="field">
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-                    Steam
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-                            huangfusl
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-                                <path d="m10.41.24 4.711 2.774c.544.316.878.897.879 1.526v5.01a1.77 1.77 0 0 1-.88 1.53l-7.753 4.521-.002.001a1.769 1.769 0 0 1-1.774 0H5.59L.873 12.85A1.761 1.761 0 0 1 0 11.327V6.292c0-.304.078-.598.22-.855l.004-.005.01-.019c.15-.262.369-.486.64-.643L8.641.239a1.752 1.752 0 0 1 1.765 0l.002.001ZM9.397 1.534l-7.17 4.182 4.116 2.388a.27.27 0 0 0 .269 0l7.152-4.148-4.115-2.422a.252.252 0 0 0-.252 0Zm-7.768 10.02 4.1 2.393V9.474a1.807 1.807 0 0 1-.138-.072L1.5 7.029v4.298c0 .095.05.181.129.227Zm8.6.642 1.521-.887v-4.45l-1.521.882ZM7.365 9.402h.001c-.044.026-.09.049-.136.071v4.472l1.5-.875V8.61Zm5.885 1.032 1.115-.65h.002a.267.267 0 0 0 .133-.232V5.264l-1.25.725Z"/>
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-                            32 games
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-                            Steam level 6
-                        </div>
-                        <div class="field">
-                            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                <path d="m.427 1.927 1.215 1.215a8.002 8.002 0 1 1-1.6 5.685.75.75 0 1 1 1.493-.154 6.5 6.5 0 1 0 1.18-4.458l1.358 1.358A.25.25 0 0 1 3.896 6H.25A.25.25 0 0 1 0 5.75V2.104a.25.25 0 0 1 .427-.177ZM7.75 4a.75.75 0 0 1 .75.75v2.992l2.028.812a.75.75 0 0 1-.557 1.392l-2.5-1A.751.751 0 0 1 7 8.25v-3.5A.75.75 0 0 1 7.75 4Z"/>
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-                            5.65k hours played
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-                        Most played
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-                    <div class="medias largeable-flex-wrap">
-                        <div class="media largeable-width-half">
-                            <img src="data:image/jpg;base64,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" alt="" />
-                            <div class="about">
-                                <div class="name">Age of Empires II: Definitive Edition</div>
-                                <div class="infos">
-                                    <div>
-                                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                            <path fill-rule="evenodd" d="M2.5 7.775V2.75a.25.25 0 01.25-.25h5.025a.25.25 0 01.177.073l6.25 6.25a.25.25 0 010 .354l-5.025 5.025a.25.25 0 01-.354 0l-6.25-6.25a.25.25 0 01-.073-.177zm-1.5 0V2.75C1 1.784 1.784 1 2.75 1h5.025c.464 0 .91.184 1.238.513l6.25 6.25a1.75 1.75 0 010 2.474l-5.026 5.026a1.75 1.75 0 01-2.474 0l-6.25-6.25A1.75 1.75 0 011 7.775zM6 5a1 1 0 100 2 1 1 0 000-2z"/>
-                                        </svg>
-                                        Strategy
-                                    </div>
-                                </div>
-                                <div class="infos">
-                                    <div>
-                                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                            <path d="M8 0a8 8 0 1 1 0 16A8 8 0 0 1 8 0ZM1.5 8a6.5 6.5 0 1 0 13 0 6.5 6.5 0 0 0-13 0Zm7-3.25v2.992l2.028.812a.75.75 0 0 1-.557 1.392l-2.5-1A.751.751 0 0 1 7 8.25v-3.5a.75.75 0 0 1 1.5 0Z"/>
-                                        </svg>
-                                        1.83k hours played
-                                    </div>
-                                </div>
-                                <div class="infos">
-                                    <div>
-                                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                            <path d="M4.75 0a.75.75 0 0 1 .75.75V2h5V.75a.75.75 0 0 1 1.5 0V2h1.25c.966 0 1.75.784 1.75 1.75v10.5A1.75 1.75 0 0 1 13.25 16H2.75A1.75 1.75 0 0 1 1 14.25V3.75C1 2.784 1.784 2 2.75 2H4V.75A.75.75 0 0 1 4.75 0ZM2.5 7.5v6.75c0 .138.112.25.25.25h10.5a.25.25 0 0 0 .25-.25V7.5Zm10.75-4H2.75a.25.25 0 0 0-.25.25V6h11V3.75a.25.25 0 0 0-.25-.25Z"/>
-                                        </svg>
-                                        Last played on 10 Dec 2024
-                                    </div>
-                                </div>
-                                <div class="achievements">
-                                    <div class="infos">
-                                        <div>
-                                            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                                <path d="M3.217 6.962A3.75 3.75 0 0 1 0 3.25v-.5C0 1.784.784 1 1.75 1h1.356c.228-.585.796-1 1.462-1h6.864c.647 0 1.227.397 1.462 1h1.356c.966 0 1.75.784 1.75 1.75v.5a3.75 3.75 0 0 1-3.217 3.712 5.014 5.014 0 0 1-2.771 3.117l.144 1.446c.005.05.03.12.114.204.086.087.217.17.373.227.283.103.618.274.89.568.285.31.467.723.467 1.226v.75h1.25a.75.75 0 0 1 0 1.5H2.75a.75.75 0 0 1 0-1.5H4v-.75c0-.503.182-.916.468-1.226.27-.294.606-.465.889-.568.139-.048.266-.126.373-.227.084-.085.109-.153.114-.204l.144-1.446a5.015 5.015 0 0 1-2.77-3.117ZM4.5 1.568V5.5a3.5 3.5 0 1 0 7 0V1.568a.068.068 0 0 0-.068-.068H4.568a.068.068 0 0 0-.068.068Zm2.957 8.902-.12 1.204c-.093.925-.858 1.47-1.467 1.691a.766.766 0 0 0-.3.176c-.037.04-.07.093-.07.21v.75h5v-.75c0-.117-.033-.17-.07-.21a.766.766 0 0 0-.3-.176c-.609-.221-1.374-.766-1.466-1.69l-.12-1.204a5.064 5.064 0 0 1-1.087 0ZM13 2.5v2.872a2.25 2.25 0 0 0 1.5-2.122v-.5a.25.25 0 0 0-.25-.25H13Zm-10 0H1.75a.25.25 0 0 0-.25.25v.5c0 .98.626 1.813 1.5 2.122Z"/>
-                                            </svg>
-                                            178 / 301 achievements unlocked
-                                        </div>
-                                    </div>
-                                    <div class="achievement">
-                                        <img src="data:image/jpg;base64,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" alt="" />
-                                        <div>
-                                            <div class="name">
-                                                <div>Malleus Scotorum</div>
-                                                <div class="unlocked">27 Nov 2024</div>
-                                            </div>
-                                            <div class="description">Kill William Wallace before Robert the Bruce betrays you in "Hammer of the Scots".</div>
-                                        </div>
-                                    </div>
-                                    <div class="achievement">
-                                        <img src="data:image/jpg;base64,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" alt="" />
-                                        <div>
-                                            <div class="name">
-                                                <div>Throne Thief</div>
-                                                <div class="unlocked">16 Sept 2024</div>
-                                            </div>
-                                            <div class="description">Steal the Stone of Scone in the fourth Edward Longshanks mission "Toom Tabard".</div>
-                                        </div>
-                                    </div>
-                                    <div class="achievement">
-                                        <img src="data:image/jpg;base64,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" alt="" />
-                                        <div>
-                                            <div class="name">
-                                                <div>He Won't Be Back</div>
-                                                <div class="unlocked">13 Sept 2024</div>
-                                            </div>
-                                            <div class="description">Assist King Bela and defeat the Austrian Duke Frederick in the fifth Kotyan mission "A New Home" before 45 minutes have elapsed.</div>
-                                        </div>
-                                    </div>
-                                    <div class="achievement">
-                                        <img src="data:image/jpg;base64,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" alt="" />
-                                        <div>
-                                            <div class="name">
-                                                <div>Cuman Flush</div>
-                                                <div class="unlocked">12 Sept 2024</div>
-                                            </div>
-                                            <div class="description">Win the second Kotyan mission "The Battle at the Kalka River" without advancing to the Castle Age.</div>
-                                        </div>
-                                    </div>
-                                    <div class="achievement">
-                                        <img src="data:image/jpg;base64,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" alt="" />
-                                        <div>
-                                            <div class="name">
-                                                <div>No Royal Hair Touched</div>
-                                                <div class="unlocked">31 Mar 2024</div>
-                                            </div>
-                                            <div class="description">Reach the King before 10 minutes have elapsed in the second Gajah Mada mission "Unconditional Loyalty".</div>
-                                        </div>
-                                    </div>
-                                    <div class="achievement">
-                                        <div>
-                                            <div class="description ellipsis">+173 others...</div>
-                                        </div>
-                                    </div>
-                                </div>
-                            </div>
-                        </div>
-                        <div class="media largeable-width-half">
-                            <img src="data:image/jpg;base64,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" alt="" />
-                            <div class="about">
-                                <div class="name">Cities: Skylines</div>
-                                <div class="infos">
-                                    <div>
-                                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                            <path fill-rule="evenodd" d="M2.5 7.775V2.75a.25.25 0 01.25-.25h5.025a.25.25 0 01.177.073l6.25 6.25a.25.25 0 010 .354l-5.025 5.025a.25.25 0 01-.354 0l-6.25-6.25a.25.25 0 01-.073-.177zm-1.5 0V2.75C1 1.784 1.784 1 2.75 1h5.025c.464 0 .91.184 1.238.513l6.25 6.25a1.75 1.75 0 010 2.474l-5.026 5.026a1.75 1.75 0 01-2.474 0l-6.25-6.25A1.75 1.75 0 011 7.775zM6 5a1 1 0 100 2 1 1 0 000-2z"/>
-                                        </svg>
-                                        Simulation, Strategy
-                                    </div>
-                                </div>
-                                <div class="infos">
-                                    <div>
-                                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                            <path d="M8 0a8 8 0 1 1 0 16A8 8 0 0 1 8 0ZM1.5 8a6.5 6.5 0 1 0 13 0 6.5 6.5 0 0 0-13 0Zm7-3.25v2.992l2.028.812a.75.75 0 0 1-.557 1.392l-2.5-1A.751.751 0 0 1 7 8.25v-3.5a.75.75 0 0 1 1.5 0Z"/>
-                                        </svg>
-                                        1.06k hours played
-                                    </div>
-                                </div>
-                                <div class="infos">
-                                    <div>
-                                        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                            <path d="M4.75 0a.75.75 0 0 1 .75.75V2h5V.75a.75.75 0 0 1 1.5 0V2h1.25c.966 0 1.75.784 1.75 1.75v10.5A1.75 1.75 0 0 1 13.25 16H2.75A1.75 1.75 0 0 1 1 14.25V3.75C1 2.784 1.784 2 2.75 2H4V.75A.75.75 0 0 1 4.75 0ZM2.5 7.5v6.75c0 .138.112.25.25.25h10.5a.25.25 0 0 0 .25-.25V7.5Zm10.75-4H2.75a.25.25 0 0 0-.25.25V6h11V3.75a.25.25 0 0 0-.25-.25Z"/>
-                                        </svg>
-                                        Last played on 1 May 2023
-                                    </div>
-                                </div>
-                                <div class="achievements">
-                                    <div class="infos">
-                                        <div>
-                                            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="16" height="16">
-                                                <path d="M3.217 6.962A3.75 3.75 0 0 1 0 3.25v-.5C0 1.784.784 1 1.75 1h1.356c.228-.585.796-1 1.462-1h6.864c.647 0 1.227.397 1.462 1h1.356c.966 0 1.75.784 1.75 1.75v.5a3.75 3.75 0 0 1-3.217 3.712 5.014 5.014 0 0 1-2.771 3.117l.144 1.446c.005.05.03.12.114.204.086.087.217.17.373.227.283.103.618.274.89.568.285.31.467.723.467 1.226v.75h1.25a.75.75 0 0 1 0 1.5H2.75a.75.75 0 0 1 0-1.5H4v-.75c0-.503.182-.916.468-1.226.27-.294.606-.465.889-.568.139-.048.266-.126.373-.227.084-.085.109-.153.114-.204l.144-1.446a5.015 5.015 0 0 1-2.77-3.117ZM4.5 1.568V5.5a3.5 3.5 0 1 0 7 0V1.568a.068.068 0 0 0-.068-.068H4.568a.068.068 0 0 0-.068.068Zm2.957 8.902-.12 1.204c-.093.925-.858 1.47-1.467 1.691a.766.766 0 0 0-.3.176c-.037.04-.07.093-.07.21v.75h5v-.75c0-.117-.033-.17-.07-.21a.766.766 0 0 0-.3-.176c-.609-.221-1.374-.766-1.466-1.69l-.12-1.204a5.064 5.064 0 0 1-1.087 0ZM13 2.5v2.872a2.25 2.25 0 0 0 1.5-2.122v-.5a.25.25 0 0 0-.25-.25H13Zm-10 0H1.75a.25.25 0 0 0-.25.25v.5c0 .98.626 1.813 1.5 2.122Z"/>
-                                            </svg>
-                                            36 / 127 achievements unlocked
-                                        </div>
-                                    </div>
-                                    <div class="achievement">
-                                        <img src="data:image/jpg;base64,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" alt="" />
-                                        <div>
-                                            <div class="name">
-                                                <div>Beam Me Up</div>
-                                                <div class="unlocked">27 Jan 2022</div>
-                                            </div>
-                                            <div class="description">Build the Space Elevator Monument</div>
-                                        </div>
-                                    </div>
-                                    <div class="achievement">
-                                        <img src="data:image/jpg;base64,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" alt="" />
-                                        <div>
-                                            <div class="name">
-                                                <div>Airline Tycoon </div>
-                                                <div class="unlocked">27 Jan 2022</div>
-                                            </div>
-                                            <div class="description">Have an airport area reach level 3.</div>
-                                        </div>
-                                    </div>
-                                    <div class="achievement">
-                                        <img src="data:image/jpg;base64,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" alt="" />
-                                        <div>
-                                            <div class="name">
-                                                <div>Airlifter</div>
-                                                <div class="unlocked">26 Jan 2022</div>
-                                            </div>
-                                            <div class="description">Have 10 000 total passengers travel through an airport area. </div>
-                                        </div>
-                                    </div>
-                                    <div class="achievement">
-                                        <img src="data:image/jpg;base64,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" alt="" />
-                                        <div>
-                                            <div class="name">
-                                                <div>High-cost-carrier </div>
-                                                <div class="unlocked">25 Jan 2022</div>
-                                            </div>
-                                            <div class="description">Set the airline ticket price slider to the maximum price. </div>
-                                        </div>
-                                    </div>
-                                    <div class="achievement">
-                                        <img src="data:image/jpg;base64,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" alt="" />
-                                        <div>
-                                            <div class="name">
-                                                <div>Airfield Expertise</div>
-                                                <div class="unlocked">25 Jan 2022</div>
-                                            </div>
-                                            <div class="description">Build an airport area with an area of 25000 cells.</div>
-                                        </div>
-                                    </div>
-                                    <div class="achievement">
-                                        <div>
-                                            <div class="description ellipsis">+31 others...</div>
-                                        </div>
-                                    </div>
-                                </div>
-                            </div>
-                        </div>
-                    </div>
-                </div>
-            </section>
-            <footer>
-                <span>These metrics include private contributions</span>
-                <span>Last updated 16 Dec 2024, 17:43:04 (timezone Asia/Shanghai) with lowlighter/metrics@3.34.0</span>
-            </footer>
-        </div>
-        <div xmlns="http://www.w3.org/1999/xhtml" id="metrics-end"></div>
-    </foreignObject>
-</svg>
\ No newline at end of file
diff --git a/search/search_index.json b/search/search_index.json
index 520138f8b..e312a65c7 100644
--- a/search/search_index.json
+++ b/search/search_index.json
@@ -1 +1 @@
-{"config":{"lang":["en"],"separator":"[\\s\\u200b\\-]","pipeline":["stemmer"]},"docs":[{"location":"","title":"Welcome to HuangFuSL's blog","text":""},{"location":"#table-of-contents","title":"Table of contents","text":"<ul> <li>\u200b\u63a8\u9001\u200b</li> <li>LaTeX</li> <li>\u200b\u6570\u5b66\u200b</li> <li>Coding</li> <li>\u200b\u65e5\u8bed\u200b</li> <li>\u200b\u5173\u4e8e\u200b\u6211\u200b</li> </ul>"},{"location":"#blog-timeline","title":"Blog timeline","text":"<p>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b</p> 2024-12-16<p>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\uff08Gaussian Process\uff0cGP\uff09\u200b\u662f\u200b\u5c06\u200b\u591a\u200b\u53d8\u91cf\u200b\u9ad8\u65af\u5206\u5e03\u200b\u63a8\u5e7f\u200b\u5230\u200b\u65e0\u9650\u200b\u7ef4\u5ea6\u200b\\(\\mathcal X\\subseteq \\mathbb R^n\\)\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b\u3002\u200b\u5177\u4f53\u200b\u5730\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\\(X = \\{x_1, \\ldots, x_n\\}\\subseteq \\mathcal X\\)\uff0c\u200b\u968f\u673a\u53d8\u91cf\u200b\\(f(x_1), \\ldots, f(x_n)\\)\u200b\u670d\u4ece\u200b\u591a\u5143\u200b\u9ad8\u65af\u5206\u5e03\u200b\\(\\mathcal N(\\mu(X), \\Sigma(X))\\)\u3002\u200b\u5176\u4e2d\u200b\uff0c\\(\\mu(X)\\)\u200b\u4e3a\u200b\u5747\u503c\u200b\u51fd\u6570\u200b\uff0c\\(\\Sigma(X)\\)\u200b\u4e3a\u200b\u534f\u65b9\u5dee\u200b\u51fd\u6570\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u9ad8\u65af\u5206\u5e03\u200b\u7814\u7a76\u200b\u7684\u200b\u662f\u200b\u51fd\u6570\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b\u3002</p> <p>\u200b\u7b97\u6cd5\u200b</p> 2024-12-09<p>\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u662f\u200b\u4e00\u7c7b\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u4f18\u5316\u200b\u9ed1\u76d2\u200b\u51fd\u6570\u200b\u7684\u200b\u4f18\u5316\u200b\u65b9\u6cd5\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u9ed1\u76d2\u200b\u51fd\u6570\u200b\u6765\u8bf4\u200b\uff0c\u200b\u6211\u4eec\u200b\u65e0\u6cd5\u200b\u5f97\u77e5\u200b\u5176\u200b\u663e\u5f0f\u200b\u7684\u200b\u8868\u8fbe\u5f0f\u200b\uff0c\u200b\u53ea\u80fd\u200b\u901a\u8fc7\u200b\u8f93\u5165\u200b\u4e00\u4e2a\u200b\u6570\u636e\u200b\uff0c\u200b\u89c2\u6d4b\u200b\u5176\u200b\u8f93\u51fa\u200b\u7684\u200b\u65b9\u6cd5\u200b\u5bfb\u627e\u200b\u5b83\u200b\u7684\u200b\u6700\u5927\u503c\u200b\u3002\u200b\u501f\u52a9\u4e8e\u200b\u8d1d\u53f6\u65af\u200b\u7edf\u8ba1\u200b\u7684\u200b\u601d\u60f3\u200b\uff0c\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u6700\u521d\u200b\u8ba4\u4e3a\u200b\u51fd\u6570\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5b8c\u5168\u200b\u968f\u673a\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4e0d\u65ad\u200b\u5730\u200b\u89c2\u6d4b\u200b\u6765\u200b\u66f4\u65b0\u200b\u5bf9\u200b\u51fd\u6570\u200b\u7684\u200b\u4f30\u8ba1\u200b\uff08\u200b\u540e\u9a8c\u200b\uff09\u3002\u200b\u5728\u200b\u6bcf\u200b\u4e00\u6b65\u200b\u8fed\u4ee3\u200b\u4e2d\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u4f1a\u200b\u6839\u636e\u200b\u5f53\u524d\u200b\u7684\u200b\u540e\u9a8c\u200b\uff0c\u200b\u901a\u8fc7\u200b\u91c7\u96c6\u200b\u51fd\u6570\u200b\u6765\u200b\u9009\u62e9\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u89c2\u6d4b\u200b\u7684\u200b\u70b9\u200b\u3002\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u7684\u200b\u4f18\u70b9\u200b\u662f\u200b\u53ef\u4ee5\u200b\u5728\u200b\u8f83\u200b\u5c11\u200b\u7684\u200b\u8fed\u4ee3\u200b\u6b21\u6570\u200b\u5185\u200b\u627e\u5230\u200b\u5168\u5c40\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5728\u200b\u5b9e\u9645\u200b\u5e94\u7528\u200b\u4e2d\u200b\u88ab\u200b\u5e7f\u6cdb\u200b\u4f7f\u7528\u200b\u3002</p> <p>\u200b\u5f3a\u5316\u200b\u5b66\u4e60\u200b</p> 2024-12-04<p>\u200b\u8bb0\u5f55\u200b\u5f3a\u5316\u200b\u5b66\u4e60\u200b\u7684\u200b\u5b66\u4e60\u200b\u8fc7\u7a0b\u200b\u3002</p> <p>\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b</p> 2024-12-04<p>\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\uff08Multi-armed bandit\uff09\u200b\u95ee\u9898\u200b\u6765\u81ea\u200b\u4e8e\u200b\u73b0\u5b9e\u200b\u4e2d\u200b\u7684\u200b\u8001\u864e\u673a\u200b\u95ee\u9898\u200b\u3002\u200b\u6bcf\u53f0\u200b\u8001\u864e\u673a\u200b\u90fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u62c9\u6746\u200b\uff0c\u200b\u62c9\u52a8\u200b\u62c9\u6746\u200b\u540e\u200b\uff0c\u200b\u8001\u864e\u200b\u673a\u4f1a\u200b\u968f\u673a\u200b\u7ed9\u51fa\u200b\u4e00\u4e2a\u200b\u5956\u52b1\u200b\u3002\u200b\u73a9\u5bb6\u200b\u7684\u200b\u4efb\u52a1\u200b\u662f\u200b\u5728\u200b\u6709\u9650\u200b\u7684\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u5185\u200b\u5c3d\u53ef\u80fd\u200b\u83b7\u5f97\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u5956\u52b1\u200b\u3002</p> <p>\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u7b97\u6cd5\u200b</p> 2024-12-04<p>\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\uff08Upper Confidence Bound, UCB\uff09\u200b\u7b97\u6cd5\u200b\u7684\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u5c06\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u7eb3\u5165\u200b\u8003\u8651\u200b\uff0c\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u8f83\u5c11\u200b\u7684\u200b\u52a8\u4f5c\u200b\u6709\u200b\u66f4\u200b\u5927\u200b\u7684\u200b\u4e0d\u786e\u5b9a\u6027\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u4f1a\u200b\u5c3d\u53ef\u80fd\u200b\u4e50\u89c2\u200b\u5730\u200b\u8003\u8651\u200b\u8fd9\u4e9b\u200b\u6ca1\u6709\u200b\u5c1d\u8bd5\u200b\u8fc7\u200b\u7684\u200b\u52a8\u4f5c\u200b\uff0c\u200b\u5e76\u200b\u4f18\u5148\u200b\u63a2\u7d22\u200b\u8fd9\u4e9b\u200b\u52a8\u4f5c\u200b\u3002\u200b\u5728\u200b\u6bcf\u4e2a\u200b\u52a8\u4f5c\u200b\u63a2\u7d22\u200b\u6bd4\u8f83\u200b\u6df1\u5165\u200b\u540e\u200b\uff0c\u200b\u518d\u200b\u8f6c\u5411\u200b\u5229\u7528\u200b\u5956\u52b1\u200b\u8f83\u200b\u9ad8\u200b\u7684\u200b\u52a8\u4f5c\u200b\u3002</p> <p>\u200b\u6a21\u578b\u200b\u63a8\u7406\u200b</p> 2024-11-25<p>\u200b\u5728\u200b\u63a8\u7406\u200b\u9636\u6bb5\u200b\uff0c\u200b\u5927\u200b\u6a21\u578b\u200b\u6839\u636e\u200b\u8f93\u5165\u200b\u7684\u200b\u6587\u672c\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4e0d\u65ad\u200b\u9884\u6d4b\u200b\u8f93\u5165\u200b\u6587\u672c\u200b\u7684\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u8bcd\u200b\uff0c\u200b\u76f4\u5230\u200b\u9884\u6d4b\u51fa\u200b\u7ed3\u675f\u200b\u7b26\u53f7\u200b\u4e3a\u6b62\u200b\u3002\u200b\u4e0d\u8fc7\u200b\uff0c\u200b\u5728\u200b\u9884\u200b\u8bad\u7ec3\u200b\u9636\u6bb5\u200b\u4e2d\u200b\uff0c\u200b\u4e0d\u200b\u5305\u542b\u200b\u5bf9\u200bbos\uff08begin of sentence\uff09\u200b\u548c\u200beos\uff08end of sentence\uff09\u200b\u7684\u200b\u9884\u6d4b\u200b\u3002\u200b\u6240\u4ee5\u200b\uff0c\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b\u57fa\u5ea7\u200b\u6a21\u578b\u200b\uff0c\u200b\u6a21\u578b\u200b\u4f1a\u200b\u4e0d\u65ad\u200b\u8f93\u51fa\u200btoken\u3002</p> <p>\u200b\u5b9e\u73b0\u200bLlama-2</p> 2024-09-07<p>\u200b\u5728\u200b\u672c\u8282\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u5b9e\u73b0\u200b\u4e00\u4e2a\u200bLlama-2\u200b\u6a21\u578b\u200b\u3002</p> <p>\u200b\u5f52\u4e00\u5316\u200b</p> 2024-09-07<p>\u200b\u5f52\u4e00\u5316\u200b\u662f\u200b\u6307\u200b\u5c06\u200b\u6570\u636e\u200b\u6309\u7167\u200b\u6bd4\u4f8b\u200b\u7f29\u653e\u200b\uff0c\u200b\u6539\u53d8\u200b\u5176\u200b\u5206\u5e03\u200b\u3002\u200b\u6309\u7167\u200b\u5f52\u4e00\u5316\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u4e0d\u540c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u4ee5\u4e0b\u200b\u51e0\u79cd\u200b\uff1a</p> <p>\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b</p> 2024-09-07<p>Attention\u200b\u673a\u5236\u200b\u867d\u7136\u200b\u80fd\u200b\u6355\u6349\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u4e0d\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u4f9d\u8d56\u200b\u5173\u7cfb\u200b\uff0c\u200b\u4f46\u662f\u200b\u65e0\u6cd5\u200b\u533a\u5206\u200b\u4e0d\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\u200b\u4e3a\u4e86\u200b\u89e3\u51b3\u200b\u8fd9\u4e2a\u200b\u95ee\u9898\u200b\uff0cTransformer\u200b\u6a21\u578b\u200b\u5f15\u5165\u200b\u4e86\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff08Positional Encoding\uff09\u3002</p> <p>transformer\u200b\u53d8\u79cd\u200b</p> 2024-09-07<p>Transformer\u200b\u53d8\u79cd\u200b\u4e3b\u8981\u200b\u5305\u542b\u200b\u5982\u4e0b\u200b\u51e0\u79cd\u200b\uff1a</p>"},{"location":"#recent-updates","title":"Recent updates","text":"<ul> <li>2024-12-16\uff1a\u200b\u673a\u5668\u200b\u5b66\u4e60\u200b\uff0c\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b</li> <li>2024-12-09\uff1a\u200b\u7b97\u6cd5\u200b</li> <li>2024-12-04\uff1a\u200b\u6df1\u5ea6\u200b\u5b66\u4e60\u200b\uff0c\u200b\u6570\u636e\u7ed3\u6784\u200b\uff0cPython\uff0c\u200b\u5f3a\u5316\u200b\u5b66\u4e60\u200b\uff0c\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\uff0c\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u7b97\u6cd5\u200b\uff0cRust</li> <li>2024-11-25\uff1a\u200b\u5b9e\u73b0\u200bLlama-2\uff0c\u200b\u6a21\u578b\u200b\u63a8\u7406\u200b</li> <li>2024-09-07\uff1aCustom transformer\uff0c\u200b\u5f52\u4e00\u5316\u200b\uff0c\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff0ctransformer\u200b\u53d8\u79cd\u200b\uff0c\u200b\u5b9e\u73b0\u200btransformer</li> </ul>"},{"location":"#currently-working-on","title":"Currently working on","text":"<ul> <li>Vim</li> </ul>"},{"location":"#customization","title":"Customization","text":"<p>Click on the buttons to change the primary color.</p> Red Pink Purple Deep Purple Indigo Blue Light Blue Cyan Teal Green Light Green Lime Yellow Amber Orange Deep Orange Brown Grey Blue Grey White <p>Click on the buttons to change the accent color.</p> Red Pink Purple Deep Purple Indigo Blue Light Blue Cyan Teal Green Light Green Lime Yellow Amber Orange Deep Orange <p>However, if you try to switch from dark mode to light mode or reversed, changes to the primary color and accent color will lose.</p>"},{"location":"#building-documentation","title":"Building documentation","text":"<p>Run <code>git clone https://github.com/HuangFuSL/HuangFuSL.github.io.git</code> to get the source code.</p>"},{"location":"#bootstrap-icon-installation","title":"Bootstrap icon installation","text":"<p>The site uses bootstrap icons, which are added as submodules in <code>third_party/icons</code>. You have to manually initialize the submodule.</p> <pre><code>git submodule update --recursive --remote\n</code></pre>"},{"location":"#latex-support","title":"LaTeX support","text":"<p>The site uses <code>xelatex</code> and <code>dvisvgm</code> to render tex document to SVG images embedded in the markdown files. However, as the SVG images are ignored by <code>.gitignore</code>, you have to manually perform the conversion.</p> <p>For GitHub repository clones:</p> <ul> <li>Run <code>git submodule update --recursive --remote</code> to receive the template.</li> <li>Make sure you have installed and correctly configured <code>xelatex</code> and <code>dvisvgm</code>.</li> <li>Add <code>./template</code> directory to <code>$TEXINPUTS</code> environmental variable.</li> <li>Execute <code>ci/convert.py</code> in the root directory of the repository.</li> <li>Run <code>mkdocs serve</code> to view the images.</li> </ul> <p>The template is located at HuangFuSL/latex-template</p>"},{"location":"#github-workflow","title":"GitHub workflow","text":"<p>You need to install the dependencies stored in <code>requirements.txt</code> before you can  start building the site:</p> <pre><code>pip install -r requirements.txt\n</code></pre> <p>There are cross-links in the site which require metadata defined in the page, so the project should be built before <code>mkdocs serve</code> is executed. The exported metadata is saved in <code>meta.json</code> after a build is successfully executed. To build the site, execute the following command:</p> <pre><code>mkdocs build -d build\n</code></pre> <p>Execute <code>mkdocs serve</code>, the built site will appear at http://127.0.0.1:8000</p>"},{"location":"#acknowledgements","title":"Acknowledgements","text":"<p>The blog relies on the following open-source projects:</p> <ul> <li>mkdocs</li> <li>Python Markdown</li> <li>Python Markdown Extension</li> <li>jupyter</li> </ul> <p>The blog uses the following mkdocs plugins to function correctly.</p> <ul> <li>Neoteroi/mkdocs-plugins</li> <li>lukasgeiter/mkdocs-awesome-pages-plugin</li> <li>timvink/mkdocs-git-revision-date-localized-plugin</li> <li>zhaoterryy/mkdocs-git-revision-date-plugin</li> <li>squidfunk/mkdocs-material</li> <li>facelessuser/mkdocs-material-extensions</li> <li>fralau/mkdocs_macros_plugin</li> <li>danielfrg/mkdocs-jupyter</li> <li>prcr/mkdocs-meta-descriptions-plugin</li> </ul> <p>Unless noted, content in this blog are shared under CC-BY-NC-SA 4.0 license.</p>"},{"location":"#version-information","title":"Version information","text":"<pre><code>commit 4b7d0e344ecccd970fead21fc513bcd9dee83585\nMerge: 57013cd0 949c8cdc\nAuthor: github-actions[bot] &lt;41898282+github-actions[bot]@users.noreply.github.com&gt;\nDate:   Mon Dec 16 17:33:30 2024 +0800\n\n    Merge pull request #421 from HuangFuSL/gaussian-process\n\n    Update: Gaussian Process\n</code></pre>"},{"location":"about/","title":"\u5173\u4e8e\u200b\u6211","text":"<p>Guess who is here?</p> <p></p>"},{"location":"coding/","title":"Coding","text":"<p>\u200b\u7f16\u7a0b\u200b\u76f8\u5173\u200b</p>"},{"location":"coding/#_1","title":"\u7f16\u7a0b\u8bed\u8a00","text":"<ul> <li>Python</li> <li>Rust<ul> <li>GUI</li> </ul> </li> </ul>"},{"location":"coding/#_2","title":"\u7b97\u6cd5","text":"<ul> <li>Leetcode</li> <li>\u200b\u6570\u636e\u7ed3\u6784\u200b</li> </ul>"},{"location":"coding/#_3","title":"\u673a\u5668\u200b\u5b66\u4e60","text":"<ul> <li>\u200b\u673a\u5668\u200b\u5b66\u4e60\u200b</li> <li>\u200b\u6df1\u5ea6\u200b\u5b66\u4e60\u200b</li> <li>\u200b\u5f3a\u5316\u200b\u5b66\u4e60\u200b</li> </ul>"},{"location":"coding/#_4","title":"\u5176\u4ed6","text":"<ul> <li>\u200b\u7cfb\u7edf\u914d\u7f6e\u200b</li> <li>SVG</li> </ul>"},{"location":"coding/svg/","title":"SVG\u200b\u52a8\u753b","text":"<p>SVG\u200b\u7684\u200b\u52a8\u753b\u200b\u4f7f\u7528\u200b<code>&lt;animate&gt;</code>\u200b\u6807\u7b7e\u200b\uff0c\u200b\u5728\u200b\u4f7f\u7528\u200b\u65f6\u200b\uff0c<code>&lt;animate&gt;</code>\u200b\u6807\u7b7e\u200b\u9700\u8981\u200b\u653e\u5728\u200b\u52a8\u753b\u200b\u64cd\u4f5c\u200b\u5bf9\u8c61\u200b\u7684\u200b\u5185\u90e8\u200b\u3002</p> <p>\u200b\u4f5c\u4e3a\u200b\u5bf9\u6bd4\u200b\uff0c\u200b\u6211\u4eec\u200b\u5148\u200b\u6784\u9020\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u77e9\u5f62\u200b\uff0c\u200b\u5728\u200b\u77e9\u5f62\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\u6dfb\u52a0\u200b\u52a8\u753b\u200b\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#_1","title":"\u7b80\u5355\u200b\u52a8\u753b","text":"<p>\u200b\u5728\u200b<code>&lt;rect&gt;</code>\u200b\u6807\u7b7e\u200b\u4e2d\u200b\u6dfb\u52a0\u200b<code>&lt;animate&gt;</code>\u200b\u6807\u7b7e\u200b\uff0c\u200b\u6307\u5b9a\u200b\u52a8\u753b\u200b\u7684\u200b\u7ed1\u5b9a\u200b\u5c5e\u6027\u200b\u3001\u200b\u5f00\u59cb\u200b\u7ed3\u675f\u200b\u7684\u200b\u53d6\u503c\u200b\u548c\u200b\u65f6\u957f\u200b\u5373\u53ef\u200b\u3002</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" from=\"20\" to=\"150\" begin=\"click\" dur=\"1.5s\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre> <p>\u200b\u901a\u8fc7\u200b<code>attributeName</code>\u200b\u5c5e\u6027\u200b\u8bbe\u7f6e\u200b\u4ea4\u4e92\u200b\u7ed1\u5b9a\u200b\u7684\u200b\u5c5e\u6027\u200b\uff0c\u200b\u5141\u8bb8\u200b\u7684\u200b\u5c5e\u6027\u200b\u5217\u4e8e\u200b\u4e0b\u8868\u200b\uff1a</p> \u200b\u6807\u7b7e\u200b \u200b\u5c5e\u6027\u200b\u540d\u200b \u200b\u5907\u6ce8\u200b <code>&lt;animate&gt;</code> <code>x</code> <code>&lt;animate&gt;</code> <code>y</code> <code>&lt;animate&gt;</code> <code>width</code> <code>&lt;animate&gt;</code> <code>height</code> <code>&lt;animate&gt;</code> <code>r</code> <code>cx</code>, <code>cy</code> <code>&lt;animate&gt;</code> <code>opacity</code> <code>&lt;animate&gt;</code> <code>d</code> <code>&lt;animate&gt;</code> <code>points</code> <code>&lt;animate&gt;</code> <code>stroke</code> <code>stroke-width</code>, <code>stroke-linecap</code>, <code>stroke-dashoffset</code> <code>&lt;animate&gt;</code> <code>fill</code> <code>&lt;set&gt;</code> <code>visibility</code> <code>&lt;animateTransform&gt;</code> <code>translate</code> <code>&lt;animateTransform&gt;</code> <code>transform</code> <code>&lt;animateTransform&gt;</code> <code>scale</code> <code>&lt;animateTransform&gt;</code> <code>rotate</code> <code>&lt;animateTransform&gt;</code> <code>skewX</code> <code>&lt;animateTransform&gt;</code> <code>skewY</code> <code>&lt;animateMotion&gt;</code> <code>path</code> <code>&lt;animateMotion&gt;</code> <code>rotate</code> <code>&lt;animateMotion&gt;</code> <code>keypoints</code> <code>&lt;animateMotion&gt;</code> <code>mpath</code>"},{"location":"coding/svg/#_2","title":"\u81ea\u5b9a\u4e49\u200b\u52a8\u753b","text":"<p><code>&lt;animate&gt;</code>\u200b\u6807\u7b7e\u200b\u652f\u6301\u200b\u591a\u79cd\u200b\u81ea\u5b9a\u4e49\u200b\u5c5e\u6027\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u5c5e\u6027\u200b\u5141\u8bb8\u200b\u5728\u200b\u52a8\u753b\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u5173\u952e\u5e27\u200b\u3001\u200b\u8ba9\u200b\u52a8\u753b\u200b\u54cd\u5e94\u200b\u7528\u6237\u200b\u4ea4\u4e92\u200b\u7b49\u200b\u9ad8\u7ea7\u200b\u529f\u80fd\u200b\u7684\u200b\u5b9e\u73b0\u200b</p>"},{"location":"coding/svg/#fill","title":"<code>fill</code>\u200b\u5c5e\u6027","text":"<p><code>fill</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u52a8\u753b\u200b\u7ed3\u675f\u200b\u65f6\u200b\u52a8\u753b\u200b\u5bf9\u8c61\u200b\u7684\u200b\u72b6\u6001\u200b\uff0c\u200b\u53d6\u503c\u200b\u53ef\u4ee5\u200b\u4e3a\u200b<code>remove</code>\u200b\u6216\u200b<code>freeze</code>\uff0c\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c<code>fill</code>\u200b\u7684\u200b\u53d6\u503c\u200b\u4e3a\u200b<code>remove</code>\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b<code>fill</code>\u200b\u53d6\u503c\u200b\u4e3a\u200b<code>freeze</code>\u200b\u7684\u200bSVG\u200b\u52a8\u753b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u770b\u51fa\u200b\uff0c\u200b\u52a8\u753b\u200b\u7ed3\u675f\u200b\u540e\u200b\u77e9\u5f62\u200b\u6ca1\u6709\u200b\u56de\u5230\u200b\u5f00\u59cb\u200b\u7684\u200b\u72b6\u6001\u200b\u3002</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" from=\"20\" to=\"150\" begin=\"click\" dur=\"1.5s\" fill=\"freeze\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#repeatcount","title":"<code>repeatCount</code>\u200b\u5c5e\u6027","text":"<p><code>repeatcount</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200b\u52a8\u753b\u200b\u5728\u200b\u5f00\u59cb\u200b\u4ee5\u540e\u200b\u91cd\u590d\u200b\u7684\u200b\u6b21\u6570\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4e3a\u200b\u6574\u6570\u200b\u503c\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4e3a\u200b<code>indefinite</code>\u200b\u8868\u793a\u200b\u65e0\u9650\u200b\u6b21\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4f7f\u200b\u77e9\u5f62\u200b\u79fb\u52a8\u200b\u7684\u200b\u52a8\u4f5c\u200b\u91cd\u590d\u200b\u4e09\u6b21\u200b\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" from=\"20\" to=\"150\" begin=\"click\" dur=\"1.5s\" repeatcount=\"3\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#values","title":"<code>values</code>\u200b\u5c5e\u6027","text":"<p><code>values</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200b\u52a8\u753b\u200b\u7684\u200b\u5c5e\u6027\u200b\u5173\u952e\u5e27\u200b\uff0c\u200b\u5c5e\u6027\u200b\u503c\u4e3a\u200b\u5404\u200b\u5173\u952e\u5e27\u200b\u7684\u200b\u5c5e\u6027\u200b\u53d6\u503c\u200b\uff0c\u200b\u4ee5\u5206\u200b\u53f7\u200b\u5206\u9694\u200b\u3002<code>values</code>\u200b\u5c5e\u6027\u200b\u53ef\u4ee5\u200b\u540c\u65f6\u200b\u66ff\u6362\u200b<code>from</code>\u200b\u5c5e\u6027\u200b\u4e0e\u200b<code>to</code>\u200b\u5c5e\u6027\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4f7f\u7528\u200b<code>values</code>\u200b\u5c5e\u6027\u200b\uff0c\u200b\u4ee4\u200b\u77e9\u5f62\u200b\u5728\u200b\u79fb\u52a8\u200b\u540e\u200b\u6298\u8fd4\u200b\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" values=\"20;150;20\" begin=\"click\" dur=\"3s\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#keytimes","title":"<code>keyTimes</code>\u200b\u5c5e\u6027","text":"<p><code>keyTimes</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200b\u5173\u952e\u5e27\u200b\u7684\u200b\u65f6\u95f4\u200b\u5206\u914d\u200b\uff0c\u200b\u5f53\u200b<code>calcMode</code>\u200b\u5c5e\u6027\u200b\u4e3a\u200b<code>linear</code>\u200b\u6216\u200b<code>spline</code>\u200b\u65f6\u200b\uff0c\u200b\u8be5\u200b\u5c5e\u6027\u200b\u6709\u6548\u200b\uff0c\u200b\u5c5e\u6027\u200b\u7684\u200b\u503c\u200b\u4e3a\u200b\u5404\u4e2a\u200b\u5173\u952e\u5e27\u200b\u5728\u200b\u6574\u4e2a\u200b\u52a8\u753b\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u7684\u200b\u8fdb\u5ea6\u200b\uff08\u200b\u53d6\u503c\u200b\u5728\u200b0-1\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5c0f\u6570\u200b\uff09\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4f7f\u7528\u200b<code>keyTimes</code>\u200b\u5c5e\u6027\u200b\uff0c\u200b\u4ee4\u200b\u77e9\u5f62\u200b\u5728\u200b\u53f3\u79fb\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u7684\u200b\u901f\u5ea6\u200b\u662f\u200b\u5de6\u79fb\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u901f\u5ea6\u200b\u7684\u200b\u4e09\u500d\u200b\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" values=\"20;150;20\" keyTimes=\"0;0.25;1\" begin=\"click\" dur=\"3s\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#calcmodekeysplines","title":"<code>calcMode</code>\u200b\u5c5e\u6027\u200b\u4e0e\u200b<code>keySplines</code>\u200b\u5c5e\u6027","text":"<p><code>calcMode</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200b\u5173\u952e\u5e27\u200b\u63d2\u503c\u200b\u7684\u200b\u8ba1\u7b97\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u8fdb\u4e00\u6b65\u200b\u5730\u200b\uff0c<code>keySplines</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200b\u5173\u952e\u5e27\u200b\u63d2\u503c\u200b\u6240\u7528\u200b\u7684\u200b\u51fd\u6570\u200b\uff08\u200b\u8d1d\u585e\u5c14\u200b\u66f2\u7ebf\u200b\uff09\u3002<code>calcMode</code>\u200b\u5c5e\u6027\u200b\u5141\u8bb8\u200b\u7684\u200b\u53d6\u503c\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li><code>paced</code>\uff1a\u200b\u5ffd\u7565\u200b<code>keyTimes</code>\u200b\u5c5e\u6027\u200b\uff0c\u200b\u5c5e\u6027\u200b\u503c\u200b\u6309\u7167\u200b<code>values</code>\u200b\u4e2d\u200b\u5b9a\u4e49\u200b\u7684\u200b\u5173\u952e\u5e27\u200b\u7ebf\u6027\u200b\u53d8\u5316\u200b</li> <li><code>linear</code>\uff1a\u200b\u9ed8\u8ba4\u200b\u53d6\u503c\u200b\uff0c\u200b\u5c5e\u6027\u200b\u503c\u200b\u6309\u7167\u200b<code>values</code>\u200b\u4e2d\u200b\u5b9a\u4e49\u200b\u7684\u200b\u5173\u952e\u5e27\u200b\u5728\u200b<code>keyTimes</code>\u200b\u65f6\u95f4\u200b\u5185\u200b\u7ebf\u6027\u200b\u53d8\u5316\u200b</li> <li><code>discrete</code>\uff1a\u200b\u5c5e\u6027\u200b\u503c\u200b\u9636\u8dc3\u200b\u53d8\u5316\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8ba4\u4e3a\u200b\u6ca1\u6709\u200b\u63d2\u503c\u200b</li> <li><code>spline</code>\uff1a\u200b\u5c5e\u6027\u200b\u503c\u200b\u6309\u7167\u200b<code>keySplines</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\u5b9a\u4e49\u200b\u7684\u200b\u8d1d\u585e\u5c14\u200b\u66f2\u7ebf\u200b\u53d8\u5316\u200b</li> </ul> <p>\u200b\u4ee5\u4e0b\u200b\u5206\u522b\u200b\u6f14\u793a\u200b<code>calcMode</code>\u200b\u5c5e\u6027\u200b\u56db\u79cd\u200b\u53d6\u503c\u200b\u7684\u200b\u6548\u679c\u200b\uff0c<code>keySplines</code>\u200b\u5c5e\u6027\u200b\u503c\u200b\u8bbe\u5b9a\u200b\u4e3a\u200b<code>1 0 1 1; 0 0 0 1</code>\uff1a</p> SVG\u200b\u56fe\u50cf\u200b pacedlineardiscretespline <p> </p> <p> </p> <p> </p> <p> </p>"},{"location":"coding/svg/#beginend","title":"<code>begin</code>\u200b\u5c5e\u6027\u200b\u4e0e\u200b<code>end</code>\u200b\u5c5e\u6027","text":"<p><code>begin</code>\u200b\u5c5e\u6027\u200b\u4e0e\u200b<code>end</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200bSVG\u200b\u52a8\u753b\u200b\u5728\u200b\u4f55\u65f6\u200b\u5f00\u59cb\u200b\uff0c\u200b\u6b64\u65f6\u200b\u7684\u200b\u65f6\u95f4\u200b\u96f6\u70b9\u200b\u4e3a\u200b\u9875\u9762\u200b\u6e32\u67d3\u200b\u5b8c\u6210\u200b\u7684\u200b\u65f6\u95f4\u200b\uff0c\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5982\u679c\u200b\u4e0d\u200b\u5728\u200b<code>begin</code>\u200b\u4e2d\u200b\u6307\u5b9a\u200b\u7528\u6237\u200b\u4ea4\u4e92\u200b\u4e8b\u4ef6\u200b\uff0c\u200b\u5219\u200b\u52a8\u753b\u200b\u5728\u200b\u6e32\u67d3\u200b\u5b8c\u6210\u200b\u540e\u200b\u5373\u200b\u5f00\u59cb\u200b\u8ba1\u65f6\u200b\u3002<code>end</code>\u200b\u5c5e\u6027\u200b\u53ef\u4ee5\u200b\u63a7\u5236\u200b\u52a8\u753b\u200b\u7684\u200b\u5f3a\u5236\u200b\u7ed3\u675f\u200b\u65f6\u95f4\u200b\uff0c\u200b\u5982\u679c\u200b\u52a8\u753b\u200b\u5728\u200b\u8be5\u200b\u65f6\u95f4\u200b\u5185\u200b\u6ca1\u6709\u200b\u7ed3\u675f\u200b\uff0c\u200b\u5219\u200b\u7acb\u5373\u200b\u7ed3\u675f\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4f7f\u7528\u200b<code>begin</code>\u200b\u5c5e\u6027\u200b\u63a7\u5236\u200b\u52a8\u753b\u200b\u5ef6\u8fdf\u200b\u5f00\u59cb\u200b2s\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" from=\"20\" to=\"150\" begin=\"click+2s\" dur=\"1.5s\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#additive","title":"<code>additive</code>\u200b\u5c5e\u6027","text":"<p><code>additive</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b<code>values</code>\u200b\u53ca\u200b<code>from</code>\u200b\u4e0e\u200b<code>to</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\u7684\u200b\u5c5e\u6027\u200b\u503c\u200b\u662f\u200b\u7edd\u5bf9\u503c\u200b\u8fd8\u662f\u200b\u76f8\u5bf9\u503c\u200b\u3002\u200b\u5141\u8bb8\u200b\u7684\u200b\u53d6\u503c\u200b\u4e3a\u200b<code>sum</code>\u200b\u4e0e\u200b<code>replace</code>\uff0c\u200b\u5206\u522b\u200b\u8868\u793a\u200b\u76f8\u5bf9\u503c\u200b\u4e0e\u200b\u7edd\u5bf9\u503c\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c<code>additive</code>\u200b\u5c5e\u6027\u200b\u53d6\u200b<code>replace</code>\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4f7f\u7528\u200b\u76f8\u5bf9\u503c\u200b\u63a7\u5236\u200b\u77e9\u5f62\u200b\u7684\u200b\u79fb\u52a8\u200b\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" from=\"0\" to=\"130\" additive=\"sum\" begin=\"click\" dur=\"1.5s\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#accumulate","title":"<code>accumulate</code>\u200b\u5c5e\u6027","text":"<p><code>accumulate</code>\u200b\u5c5e\u6027\u200b\u5f53\u200b\u52a8\u753b\u200b\u91cd\u590d\u200b\u65f6\u200b\uff0c\u200b\u5c5e\u6027\u200b\u503c\u200b\u662f\u5426\u200b\u7d2f\u52a0\u200b\u3002\u200b\u5141\u8bb8\u200b\u7684\u200b\u53d6\u503c\u200b\u4e3a\u200b<code>sum</code>\u200b\u4e0e\u200b<code>none</code>\u3002\u200b\u5206\u522b\u200b\u8868\u793a\u200b\u4e0a\u200b\u4e00\u6b21\u200b\u52a8\u753b\u200b\u7ed3\u675f\u200b\u7684\u200b\u503c\u200b\u4e0e\u200b\u52a8\u753b\u200b\u539f\u672c\u200b\u7684\u200b\u5c5e\u6027\u200b\u503c\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c<code>accumulate</code>\u200b\u5c5e\u6027\u200b\u53d6\u200b<code>none</code>\u3002\u200b\u5f53\u200b\u5c5e\u6027\u200b\u503c\u200b\u4e0d\u200b\u652f\u6301\u200b\u76f8\u52a0\u200b\u6216\u200b\u52a8\u753b\u200b\u4e0d\u200b\u91cd\u590d\u200b\u65f6\u200b\uff0c\u200b\u8be5\u200b\u5c5e\u6027\u200b\u88ab\u200b\u5ffd\u7565\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u5b9e\u73b0\u200b\u6bcf\u6b21\u200b\u70b9\u51fb\u200b\u540e\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u77e9\u5f62\u200b\u4e24\u6b21\u200b\uff0c\u200b\u6bcf\u6b21\u200b65\u200b\u5355\u4f4d\u200b\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" values=\"0;65\" begin=\"click\" keyTimes=\"0;1\" keySplines=\"1 0 1 1\" calcMode=\"spline\" additive=\"sum\" accumulate=\"sum\" dur=\"1.5s\" fill=\"freeze\" repeatCount=\"2\"/&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#_3","title":"\u4ea4\u4e92\u200b\u52a8\u753b","text":"<p><code>begin</code>\u200b\u5c5e\u6027\u200b\u7684\u200b\u53d6\u503c\u200b\u53ef\u4ee5\u200b\u4e3a\u200b<code>click</code>\u200b\u6216\u200b<code>touch</code>\u200b\u4ee5\u200b\u54cd\u5e94\u200b\u7528\u6237\u200b\u8f93\u5165\u200b\u3002<code>click</code>\u200b\u65e2\u200b\u53ef\u4ee5\u200b\u54cd\u5e94\u200b\u9f20\u6807\u200b\u70b9\u51fb\u200b\u4e8b\u4ef6\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u54cd\u5e94\u200b\u7528\u6237\u200b\u89e6\u6478\u200b\u3002<code>touch</code>\u200b\u53ea\u80fd\u200b\u54cd\u5e94\u200b\u7528\u6237\u200b\u89e6\u6478\u200b\u3002</p>"},{"location":"coding/svg/#svg_1","title":"\u5e38\u89c1\u200b\u7684\u200bSVG\u200b\u52a8\u753b\u200b\u6548\u679c","text":"<ol> <li>\u200b\u4f4d\u79fb\u200b</li> <li>\u200b\u53d8\u6362\u200b</li> <li>\u200b\u6de1\u5165\u6de1\u51fa\u200b</li> <li>\u200b\u63cf\u8fb9\u200b</li> <li>\u200b\u8def\u5f84\u200b\u53d8\u5316\u200b</li> <li>\u200b\u989c\u8272\u200b\u53d8\u5316\u200b</li> </ol>"},{"location":"coding/svg/#_4","title":"\u4f4d\u79fb","text":"<p>\u200b\u4f4d\u79fb\u200b\u64cd\u4f5c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4f7f\u7528\u200b<code>&lt;animate&gt;</code>\u200b\u63a7\u5236\u200b\u5bf9\u8c61\u200b\u7684\u200b<code>x</code>\u3001<code>y</code>\u200b\u5c5e\u6027\u200b\u4e0e\u200b<code>width</code>\u3001<code>height</code>\u200b\u5c5e\u6027\u200b\u5b9e\u73b0\u200b\u3002\u200b\u53ef\u80fd\u200b\u7684\u200b\u5e94\u7528\u200b\u573a\u666f\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u8fdb\u5ea6\u6761\u200b\u3001\u200b\u76f4\u65b9\u56fe\u200b\u7b49\u200b\u7528\u4e8e\u200b\u4ea4\u4e92\u200b\u7684\u200b\u6570\u636e\u200b\u56fe\u8868\u200b</li> <li>\u200b\u4ea4\u901a\u5de5\u5177\u200b\u6216\u200b\u5176\u4ed6\u200b\u5bf9\u8c61\u200b\u7684\u200b\u79fb\u52a8\u200b</li> </ul> <p>\u200b\u8fdb\u5ea6\u6761\u200b\u793a\u4f8b\u200b\uff1a</p> \u200b\u8fdb\u5ea6\u6761\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p></p> <pre><code>&lt;svg id=\"\u200b\u56fe\u5c42\u200b_1\" data-name=\"\u200b\u56fe\u5c42\u200b 1\" xmlns=\"http://www.w3.org/2000/svg\" width=\"700\" height=\"195\" viewBox=\"0 0 700 195\"&gt;\n    &lt;rect x=\"0.5\" y=\"0.5\" width=\"699\" height=\"58\" fill=\"#fff\"/&gt;&lt;path d=\"M699,1V58H1V1H699m1-1H0V59H700V0Z\"/&gt;\n    &lt;rect x=\"1\" y=\"1\" width=\"446\" height=\"57\" fill=\"red\"&gt;\n        &lt;animate attributeName=\"width\" values=\"0;698\" keytimes=\"0;1\" calcmode=\"spline\" keysplines=\"0.44 0.27 0.21 0.99\" begin=\"click\" dur=\"1s\" fill=\"freeze\" repeatcount=\"3\"/&gt;\n    &lt;/rect&gt;\n    &lt;rect x=\"0.5\" y=\"68.5\" width=\"699\" height=\"58\" fill=\"#fff\"/&gt;\n    &lt;path d=\"M699,69v57H1V69H699m1-1H0v59H700V68Z\"/&gt;\n    &lt;rect x=\"1\" y=\"69\" width=\"446\" height=\"57\" fill=\"#dd2\"&gt;\n        &lt;animate attributeName=\"width\" values=\"0;698\" keytimes=\"0;1\" calcmode=\"spline\" keysplines=\"0 0 0.5 1\" begin=\"click\" dur=\"1s\" fill=\"freeze\" repeatcount=\"3\"/&gt;\n    &lt;/rect&gt;\n    &lt;rect x=\"0.5\" y=\"136.5\" width=\"699\" height=\"58\" fill=\"#fff\"/&gt;\n    &lt;path d=\"M699,137v57H1V137H699m1-1H0v59H700V136Z\"/&gt;\n    &lt;rect x=\"1\" y=\"137\" width=\"446\" height=\"57\" fill=\"lime\"&gt;\n        &lt;animate attributeName=\"width\" values=\"0;698\" keytimes=\"0;1\" calcmode=\"spline\" keysplines=\"0.5 0 1 1\" begin=\"click\" dur=\"1s\" fill=\"freeze\" repeatcount=\"3\"/&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre> <p>\u200b\u4e00\u4e2a\u200b\u79fb\u52a8\u200b\u7684\u200b\u5706\u70b9\u200b\uff1a</p> \u200b\u5706\u70b9\u200b\u52a8\u753b\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p></p> <pre><code>&lt;svg id=\"\u200b\u56fe\u5c42\u200b_1\" data-name=\"\u200b\u56fe\u5c42\u200b 1\" xmlns=\"http://www.w3.org/2000/svg\" width=\"150\" viewBox=\"0 0 700 195\"&gt;\n    &lt;rect width=\"700\" height=\"195\" fill=\"none\"/&gt;\n    &lt;circle cx=\"279.5\" cy=\"97.5\" r=\"25.5\"&gt;\n        &lt;animate attributeName=\"cx\" values=\"0;90;0\" keytimes=\"0;0.5;1\" calcmode=\"spline\" keysplines=\"0.5 0 0.5 1; 0.5 0 0.5 1\" begin=\"0s\" dur=\"2s\" fill=\"freeze\" repeatcount=\"indefinite\" additive=\"sum\"/&gt;\n    &lt;/circle&gt;\n&lt;/svg&gt;\n</code></pre> <p>\u200b\u6b64\u5916\u200b\uff0c\u200b\u90e8\u5206\u200b\u4f4d\u79fb\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>&lt;animateMotion&gt;</code>\u200b\u6807\u7b7e\u200b\u8fdb\u884c\u200b\u63a7\u5236\u200b\uff0c<code>&lt;animateMotion&gt;</code>\u200b\u6807\u7b7e\u200b\u652f\u6301\u200b\u5bf9\u8c61\u200b\u6309\u7167\u200b\u8def\u5f84\u200b\u8fdb\u884c\u200b\u4f4d\u79fb\u200b\u3002</p> <p>\u200b\u590d\u6742\u200b\u7684\u200b\u4f4d\u79fb\u200b\u9700\u8981\u200b\u63a7\u5236\u200b\u5bf9\u8c61\u200b\u7684\u200b<code>translate</code>\u200b\u5c5e\u6027\u200b\u3002</p>"},{"location":"coding/svg/#_5","title":"\u53d8\u6362","text":"<p>\u200b\u4ee5\u4e0b\u200b\u5c5e\u6027\u200b\u5f71\u54cd\u200b\u56fe\u5f62\u200b\u7684\u200b\u53d8\u6362\u200b\uff1a</p> <ul> <li><code>r</code>\uff1a\u200b\u534a\u5f84\u200b\uff08\u200b\u5bf9\u4e8e\u200b\u5706\u200b\uff09</li> <li><code>rotate</code>\uff1a\u200b\u65cb\u8f6c\u200b\uff08\u200b\u4ee5\u200b\u4e2d\u5fc3\u70b9\u200b\uff09</li> <li><code>skewX</code>\u3001<code>skewY</code>\uff1a\u200b\u503e\u659c\u200b</li> <li><code>transform</code>\uff1a\u200b\u590d\u6742\u200b\u53d8\u6362\u200b</li> </ul> <p>\u200b\u4ee5\u4e0b\u200b\u7ed9\u51fa\u200b<code>r</code>\u200b\u53d8\u6362\u200b\u7684\u200b\u793a\u4f8b\u200b\uff1a</p> r\u200b\u53d8\u6362\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p>     &gt;&gt;&gt; \u200b\u70b9\u51fb\u200b\u5706\u70b9\u200b \u200b\u67e5\u770b\u200b\u9690\u85cf\u200b\u5185\u5bb9\u200b &lt;&lt;&lt;      Hello              World!      <pre><code>&lt;svg id=\"\u200b\u56fe\u5c42\u200b_1\" data-name=\"\u200b\u56fe\u5c42\u200b 1\" xmlns=\"http://www.w3.org/2000/svg\" width=\"350\" viewBox=\"0 0 749 1334\"&gt;\n    &lt;rect width=\"749\" height=\"1334\" fill=\"#fff\"/&gt;\n    &lt;text transform=\"translate(221.4 1070.91)\" font-size=\"21\" font-family=\"PingFang-SC-Regular, PingFang SC\"&gt;\n        &amp;gt;&amp;gt;&amp;gt; \u200b\u70b9\u51fb\u200b\u5706\u70b9\u200b \u200b\u67e5\u770b\u200b\u9690\u85cf\u200b\u5185\u5bb9\u200b &amp;lt;&amp;lt;&amp;lt;\n    &lt;/text&gt;\n    &lt;circle cx=\"375\" cy=\"992\" r=\"29\" fill=\"#ed1e79\"&gt;\n        &lt;animate attributeName=\"r\" values=\"0;1100\" keytimes=\"0;1\" calcmode=\"spline\" keysplines=\"0 0 0.5 1\" begin=\"click\" dur=\"0.75s\" fill=\"freeze\" additive=\"sum\" restart=\"never\"&gt;\n    &lt;/circle&gt;\n    &lt;text transform=\"translate(0 295.12)\" font-size=\"192\" fill=\"#fff\" font-family=\"PingFang-SC-Heavy, PingFang SC\" font-weight=\"800\"&gt;\n        Hello\n        &lt;tspan x=\"0\" y=\"230.4\"&gt;\n            World!\n        &lt;/tspan&gt;\n    &lt;/text&gt;\n&lt;/svg&gt;\n</code></pre> <p><code>transform</code>\u200b\u5c5e\u6027\u200b\u5728\u200b<code>animateTransform</code>\u200b\u6807\u7b7e\u200b\u4e2d\u200b\u66f4\u6539\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u8f83\u4e3a\u200b\u590d\u6742\u200b\u7684\u200b\u53d8\u6362\u200b\uff0c\u200b\u5982\u7ed5\u200b\u67d0\u200b\u4e00\u4e2a\u200b\u7279\u5b9a\u200b\u70b9\u200b\u8fdb\u884c\u200b\u65cb\u8f6c\u200b\uff1a</p> transform\u200b\u53d8\u6362\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p></p> <pre><code>&lt;svg id=\"\u200b\u56fe\u5c42\u200b_1\" data-name=\"\u200b\u56fe\u5c42\u200b 1\" xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 749 1340\" width=\"350\"&gt;\n    &lt;rect y=\"6\" width=\"749\" height=\"1334\" fill=\"#fff\"/&gt;\n    &lt;g&gt;\n        &lt;animateTransform attributeName=\"transform\" type=\"rotate\" values=\"0 375 0; -10 375 0; 10 375 0; 0 375 0\" keytimes=\"0;0.25;0.75;1\" calcmode=\"spline\" keysplines=\"0 0 0.5 1; 0.5 0 0.5 1; 0.5 0 1 1\" start=\"0s\" dur=\"2s\" repeatcount=\"indefinite\" /&gt;\n        &lt;line x1=\"375\" y1=\"6\" x2=\"375\" y2=\"969.5\" fill=\"none\" stroke=\"#000\" stroke-linecap=\"round\" stroke-miterlimit=\"10\" stroke-width=\"12\"/&gt;\n        &lt;circle cx=\"375\" cy=\"969\" r=\"45\" stroke=\"#000\" stroke-linecap=\"round\" stroke-miterlimit=\"10\" stroke-width=\"12\"/&gt;\n    &lt;/g&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/configuration/","title":"\u7cfb\u7edf\u914d\u7f6e","text":""},{"location":"coding/configuration/#_2","title":"\u547d\u4ee4\u884c","text":"<ul> <li>apt</li> <li>Docker</li> <li>ssh</li> <li>Vim</li> <li>update-alternatives</li> </ul>"},{"location":"coding/configuration/#_3","title":"\u8f6f\u4ef6\u200b\u7f16\u8bd1\u200b\u6d41\u7a0b","text":"<ul> <li>\u200b\u6e90\u7801\u200b\u7f16\u8bd1\u200bnginx\u200b\u5e76\u200b\u6dfb\u52a0\u200bRTMP\u200b\u6a21\u5757\u200b</li> </ul>"},{"location":"coding/configuration/apt-get/","title":"apt\uff1aUbuntu \u200b\u5305\u200b\u7ba1\u7406\u5de5\u5177","text":"<p>Ubuntu\u200b\u4e0b\u200b\u7684\u200b\u5305\u200b\u7ba1\u7406\u5de5\u5177\u200b\u662f\u200b<code>dpkg</code>\uff0c\u200b\u6709\u200b\u4e24\u4e2a\u200b\u524d\u7aef\u200b<code>apt</code>\u200b\u548c\u200b<code>apt-get</code>\u3002</p>"},{"location":"coding/configuration/apt-get/#_1","title":"\u5b89\u88c5\u200b\u4e2d\u65ad\u200b\u540e\u200b\u7684\u200b\u6062\u590d","text":"<p>\u200b\u5728\u200b\u67d0\u4e2a\u200b\u8f6f\u4ef6\u200b\u5b89\u88c5\u200b\u4e2d\u65ad\u200b\uff08\u200b\u5982\u200bCtrl+C\u200b\u6216\u200bkill\uff09\u200b\u540e\u200b\uff0c\u200b\u8f6f\u4ef6\u200b\u7684\u200b\u5b89\u88c5\u200b\u5e76\u200b\u6ca1\u6709\u200b\u5b9e\u9645\u200b\u5b8c\u6210\u200b\uff0c\u200b\u9700\u8981\u200b\u7ee7\u7eed\u6267\u884c\u200b\u5b89\u88c5\u64cd\u4f5c\u200b\u6216\u200b\u56de\u6eda\u200b\u5230\u200b\u5b89\u88c5\u200b\u4e4b\u524d\u200b\u7684\u200b\u72b6\u6001\u200b\u3002</p> <p>\u200b\u82e5\u200b\u8981\u200b\u7ee7\u7eed\u200b\u5b89\u88c5\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6267\u884c\u200b\u5982\u4e0b\u200b\u547d\u4ee4\u200b\uff0c<code>dpkg</code>\u200b\u4f1a\u200b\u5c1d\u8bd5\u200b\u5bf9\u200b\u5df2\u7ecf\u200b\u89e3\u538b\u200b\u672a\u200b\u5b8c\u6210\u200b\u5b89\u88c5\u200b\u7684\u200b\u8f6f\u4ef6\u5305\u200b\u8fdb\u884c\u200b\u6062\u590d\u200b\u3002</p> <pre><code>sudo dpkg --configure -a\n</code></pre> <p>\u200b\u82e5\u8981\u200b\u56de\u6eda\u200b\u5b89\u88c5\u200b\uff0c\u200b\u9700\u8981\u200b\u5148\u200b\u5728\u200b<code>dpkg</code>\u200b\u7684\u200b\u5b89\u88c5\u200b\u8bb0\u5f55\u200b\u4e0b\u200b\u627e\u5230\u200b\u672a\u200b\u5b8c\u5168\u5b89\u88c5\u200b\u7684\u200b\u5305\u200b\uff0c\u200b\u5c06\u200b\u5176\u200b\u5220\u9664\u200b\u540e\u200b\u91cd\u65b0\u914d\u7f6e\u200b<code>apt</code>\u3002\u200b\u6b64\u5904\u200b\u4ee5\u200b<code>nvidia-cudnn</code>\u200b\u4e3a\u4f8b\u200b\uff1a</p> <pre><code>sudo ls -l /var/lib/dpkg/info | grep -i nvidia-cudnn\nsudo mv /var/lib/dpkg/info/nvidia-cudnn.* /tmp\nsudo dpkg --configure -a\nsudo apt-get autoremove\n</code></pre>"},{"location":"coding/configuration/docker/","title":"Docker","text":"<p><code>docker</code>\u200b\u662f\u200b\u7528\u4e8e\u200b\u63d0\u4f9b\u200b\u5bb9\u5668\u200b\u955c\u50cf\u200b\u7684\u200b\u8f6f\u4ef6\u200b\u3002\u200b\u955c\u50cf\u200b\u662f\u200b\u4e00\u5957\u200b\u914d\u7f6e\u200b\u597d\u200b\u7684\u200b\u8fd0\u884c\u200b\u73af\u5883\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5feb\u901f\u200b\u90e8\u7f72\u200b\u4e3a\u200b\u5e94\u7528\u7a0b\u5e8f\u200b\u3002</p>"},{"location":"coding/configuration/docker/#docker-registry","title":"Docker registry","text":"<p>Docker registry\u200b\u662f\u200bdocker\u200b\u7684\u200b\u955c\u50cf\u200b\u6e90\u200b\u3002</p>"},{"location":"coding/configuration/docker/#_1","title":"\u767b\u5f55","text":"<p>\u200b\u767b\u5f55\u200b\u7ec8\u7aef\u200b\u540e\u200b\uff0c\u200b\u624d\u80fd\u200b\u4ece\u200b\u7f51\u7edc\u200b\u4e0a\u200bpush\u200b\u548c\u200bpull\u200b\u955c\u50cf\u200b\u3002\u200b\u767b\u5f55\u200b\u7684\u200b\u547d\u4ee4\u200b\u4e3a\u200b</p> <pre><code>docker login &lt;registry-url&gt;\n</code></pre> <p>\u200b\u5982\u679c\u200b\u6ca1\u6709\u200b\u6307\u5b9a\u200b<code>&lt;registry-url&gt;</code>\uff0c\u200b\u5219\u200b\u9ed8\u8ba4\u200b\u4f7f\u7528\u200bDocker registry <code>docker.io</code>\u3002</p> <p>\u200b\u767b\u5f55\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u4f1a\u200b\u8bf7\u6c42\u200b\u7528\u6237\u200b\u8f93\u5165\u200b\u7528\u6237\u540d\u200b\u548c\u200b\u5bc6\u7801\u200b\uff08\u200b\u6216\u200bAccess Token\uff09\u3002\u200b\u5982\u679c\u200b\u8d26\u6237\u200b\u914d\u7f6e\u200b\u4e86\u200b2FA\uff0c\u200b\u5219\u200b\u53ea\u80fd\u200b\u901a\u8fc7\u200bAccess Token\u200b\u7684\u200b\u65b9\u5f0f\u200b\u767b\u5f55\u200b\u3002\u200b\u767b\u5f55\u200b\u4fe1\u606f\u200b\u4fdd\u5b58\u200b\u5728\u200b<code>~/.docker/config.json</code>\u200b\u6216\u200b\u7cfb\u7edf\u200b\u7684\u200b\u94a5\u5319\u200b\u4e32\u4e2d\u200b\u3002\u200b\u5982\u679c\u200b\u52a0\u4e0a\u200b<code>--password-stdin</code>\u200b\u53c2\u6570\u200b\uff0cdocker\u200b\u4f1a\u200b\u4ece\u200b\u547d\u4ee4\u884c\u200b\u4e2d\u200b\u8bfb\u53d6\u200b\u5bc6\u7801\u200b\u3002</p> <p>\u200b\u914d\u7f6e\u200bdocker\u200b\u4f7f\u7528\u200bmacOS\u200b\u94a5\u5319\u200b\u4e32\u200b</p> <ol> <li>\u200b\u6267\u884c\u200b<code>docker logout</code>\uff0c\u200b\u5220\u9664\u200b\u539f\u5148\u200b\u7684\u200b\u767b\u5f55\u200b\u51ed\u8bc1\u200b\uff1b</li> <li>\u200b\u4ece\u200bHomebrew\u200b\u5b89\u88c5\u200b<code>docker-credential-helper</code>\uff1b</li> <li>\u200b\u5728\u200b<code>~/.docker/config.json</code>\u200b\u4e2d\u200b\u52a0\u5165\u200b\u5982\u4e0b\u200b\u4e00\u884c\u200b\uff1a</li> </ol> <pre><code>{\n    \"credsStore\": \"osxkeychain\"\n}\n</code></pre> <ol> <li>\u200b\u91cd\u65b0\u200b\u6267\u884c\u200b<code>docker login</code>\u200b\u767b\u5f55\u200b\uff1b</li> <li>\u200b\u786e\u8ba4\u200b<code>~/.docker/config.json</code>\u200b\u91cc\u200b\u6ca1\u6709\u200b\u767b\u5f55\u200b\u51ed\u8bc1\u200b\uff1a</li> </ol> <pre><code>{\n    \"auths\": {\n            \"https://index.docker.io/v1/\": {}\n    }\n}\n</code></pre>"},{"location":"coding/configuration/docker/#_2","title":"\u7b2c\u4e09\u65b9\u200b\u955c\u50cf\u200b\u6e90\u200b\u5217\u8868","text":"<p>ghcr.io</p> <p><code>ghcr.io</code>\u200b\u662f\u200bGitHub\u200b\u63d0\u4f9b\u200b\u7684\u200b\u955c\u50cf\u200b\u6e90\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u65e0\u7f1d\u200b\u96c6\u6210\u200b\u81f3\u200bGitHub Actions\u3002</p> <p>\u200b\u9a8c\u8bc1\u200b\u65b9\u5f0f\u200b\uff1a</p> <ol> <li>\u200b\u4f7f\u7528\u200bGitHub Access Token\u200b\u9a8c\u8bc1\u200b\uff0cAccess Token\u200b\u5fc5\u987b\u200b\u81f3\u5c11\u200b\u5177\u5907\u200b<code>read:packages</code>\u200b\u6743\u9650\u200b\uff0c\u200b\u5982\u679c\u200b\u8981\u200b\u63a8\u9001\u200b\u955c\u50cf\u200b\uff0c\u200b\u5219\u200b\u8fd8\u200b\u9700\u8981\u200b<code>write:packages</code>\u200b\u6743\u9650\u200b</li> <li>\u200b\u5728\u200bGitHub Actions\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>GITHUB_TOKEN</code>\u200b\u9a8c\u8bc1\u200b\u3002</li> </ol>"},{"location":"coding/configuration/docker/#dockerfile","title":"Dockerfile","text":"<p><code>dockerfile</code>\u200b\u6587\u4ef6\u200b\u7528\u6765\u200b\u5c06\u200b\u4e00\u5957\u200b\u73af\u5883\u200b\u6253\u5305\u200b\u6210\u200b\u955c\u50cf\u200b\u3002\u200b\u6bcf\u200b\u4e00\u6761\u200b\u6307\u4ee4\u200b\u90fd\u200b\u5bf9\u5e94\u200b\u4e86\u200b\u5bb9\u5668\u200b\u955c\u50cf\u200b\u7684\u200b\u4e00\u5c42\u200b\u3002</p>"},{"location":"coding/configuration/docker/#docker_1","title":"Docker\u200b\u955c\u50cf\u200b\u57fa\u7840","text":"<p>Docker\u200b\u955c\u50cf\u200b\u7684\u200b\u57fa\u672c\u200b\u683c\u5f0f\u200b\u4e3a\u200b<code>[&lt;registry-name&gt;]/&lt;namespace&gt;/&lt;image-name&gt;:[&lt;tag&gt;]</code></p> <ul> <li><code>&lt;registry-name&gt;</code>\uff1a\u200b\u53ef\u200b\u9009\u200b\uff0c\u200b\u51b3\u5b9a\u200bDocker\u200b\u955c\u50cf\u200b\u53d1\u5e03\u200b\u5728\u200b\u54ea\u4e2a\u200bRegistry\u200b\u4e0a\u200b\uff1b</li> <li><code>&lt;namespace&gt;</code>\uff1a\u200b\u4e00\u822c\u200b\u662f\u200b\u7528\u6237\u540d\u200b\uff1b</li> <li><code>&lt;image-name&gt;</code>\uff1a\u200b\u955c\u50cf\u200b\u540d\u79f0\u200b\uff1b</li> <li><code>&lt;tag&gt;</code>\uff1a\u200b\u53ef\u200b\u9009\u200b\uff0c\u200b\u955c\u50cf\u200b\u7248\u672c\u200b\uff0c\u200b\u9ed8\u8ba4\u200b\u4e3a\u200b<code>latest</code></li> </ul> <p>\u200b\u4f7f\u7528\u200b<code>docker pull</code>\u200b\u547d\u4ee4\u200b\u4ece\u200bregistry\u200b\u4e0b\u8f7d\u200b\u955c\u50cf\u200b\u3002</p> <p>\u200b\u6267\u884c\u200b<code>docker build &lt;path&gt;</code>\u200b\u6765\u200b\u4ece\u200bDockerfile\u200b\u6784\u5efa\u200b\u955c\u50cf\u200b\u3002\u200b\u7528\u200b<code>-t</code>\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u6307\u5b9a\u200b\u955c\u50cf\u200b\u7684\u200b\u540d\u79f0\u200b\u548c\u200b\u7248\u672c\u200b\u3002</p> <p>\u200b\u5728\u200b\u955c\u50cf\u200b\u6784\u5efa\u200b\u5b8c\u6210\u200b\u540e\u200b\uff0c\u200b\u6267\u884c\u200b<code>docker push &lt;image-name&gt;</code>\u200b\u5c06\u200b\u955c\u50cf\u200b\u63a8\u9001\u200b\u5230\u200b\u5bf9\u5e94\u200b\u7684\u200bregistry\u200b\u4e0a\u200b\u3002</p>"},{"location":"coding/configuration/docker/#add","title":"<code>ADD</code>","text":"<p><code>ADD</code>\uff08\u200b\u6216\u200b<code>COPY</code>\uff09\u200b\u547d\u4ee4\u200b\u5c06\u200b\u672c\u5730\u200b\u6216\u200b\u4e92\u8054\u7f51\u200b\u4e0a\u200b\u7684\u200b\u6587\u4ef6\u200b\u590d\u5236\u5230\u200b\u5bb9\u5668\u200b\u955c\u50cf\u200b\u4e2d\u200b\u3002</p> <pre><code>ADD &lt;local-path&gt; &lt;image-path&gt;\n</code></pre> <ul> <li>\u200b\u5982\u679c\u200b\u8def\u5f84\u200b\u4e2d\u200b\u5305\u542b\u200b\u7a7a\u683c\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b\u53cc\u5f15\u53f7\u200b\u628a\u200b\u8def\u5f84\u200b\u62ec\u200b\u8d77\u6765\u200b\uff1b</li> <li><code>&lt;local-path&gt;</code>\u200b\u652f\u6301\u200b\u901a\u914d\u7b26\u200b<code>*?</code>\uff1b</li> </ul>"},{"location":"coding/configuration/docker/#from","title":"<code>FROM</code>","text":"<p><code>FROM</code>\u200b\u547d\u4ee4\u200b\u6307\u660e\u200b\u5f53\u524d\u200b\u7684\u200b<code>dockerfile</code>\u200b\u662f\u200b\u5728\u200b\u54ea\u200b\u4e00\u4e2a\u200b\u955c\u50cf\u200b\u4e0a\u200b\u6267\u884c\u200b\u7684\u200b\u3002\u200b\u6bcf\u4e2a\u200bDockerfile\u200b\u5fc5\u987b\u200b\u6709\u200b\u4e00\u6761\u200b<code>FROM</code>\u200b\u8bed\u53e5\u200b\u3002</p> <p>\u200b\u7a7a\u767d\u200bDocker\u200b\u955c\u50cf\u200b</p> <p><code>scratch</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u9884\u5b9a\u200b\u4e49\u200b\u7684\u200b\u7a7a\u767d\u200bDocker\u200b\u955c\u50cf\u200b\uff0c\u200b\u4e5f\u200b\u662f\u200b\u6240\u6709\u200bDocker\u200b\u955c\u50cf\u200b\u7684\u200b\u8d77\u59cb\u200b\u3002</p> <pre><code>FROM scratch\n</code></pre>"},{"location":"coding/configuration/docker/#run","title":"<code>RUN</code>","text":"<p><code>RUN</code>\u200b\u6267\u884c\u200b\u4e00\u4e2a\u200b\u547d\u4ee4\u200b\uff0c\u200b\u5982\u679c\u200b\u8981\u200b\u6267\u884c\u200b\u591a\u4e2a\u200b\u547d\u4ee4\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>&amp;&amp;</code>\u200b\u8fde\u63a5\u200b</p> <pre><code>RUN command-1 &amp;&amp; \\\n    command-2 &amp;&amp; \\\n    ...\n</code></pre>"},{"location":"coding/configuration/docker/#tls","title":"\u914d\u7f6e\u200bTLS\u200b\u8fdc\u7a0b\u200b\u8bbf\u95ee","text":"<p>Docker\u200b\u662f\u200bc/s\u200b\u67b6\u6784\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u53ef\u4ee5\u200b\u4ece\u200b\u8fdc\u7a0b\u200b\u8bbf\u95ee\u200b\u4e3b\u673a\u200b\u7684\u200bDocker daemon\uff0c\u200b\u4ece\u200b\u8fdc\u7a0b\u200b\u76f4\u63a5\u200b\u64cd\u4f5c\u200bDocker\u200b\u73af\u5883\u200b\uff0c\u200b\u5bb9\u5668\u200b\u548c\u200b\u955c\u50cf\u200b\u5747\u200b\u4fdd\u5b58\u200b\u5728\u200b\u8fdc\u7a0b\u200b\u4e3b\u673a\u200b\u4e0a\u200b\u3002\u200b\u4f46\u200b\u5728\u200b\u7f51\u7edc\u200b\u4e0a\u200b\u9700\u8981\u200bTLS\u200b\u52a0\u5bc6\u4f20\u8f93\u200b\u4ee5\u200b\u4fdd\u8bc1\u200b\u5b89\u5168\u200b\uff0c\u200b\u56e0\u6b64\u200b\u9700\u8981\u200b\u5148\u200b\u914d\u7f6e\u200b\u597d\u200bopenssl\uff0c\u200b\u5e76\u4e14\u200b\u6709\u200b\u4e00\u4e2a\u200b\u89e3\u6790\u200b\u5230\u200b\u670d\u52a1\u5668\u200b\u7684\u200b\u57df\u540d\u200b\uff0c\u200b\u5982\u200b<code>www.example.com</code>\u3002</p>"},{"location":"coding/configuration/docker/#_3","title":"\u670d\u52a1\u5668\u200b\u914d\u7f6e","text":"<p><code>dockerd</code>\u200b\u662f\u200bDocker daemon\u200b\u7684\u200b\u53ef\u6267\u884c\u7a0b\u5e8f\u200b\u3002</p> Docker DaemonDocker Desktop <p>\u200b\u5728\u200b\u670d\u52a1\u5668\u7aef\u200b\uff0c\u200b\u9700\u8981\u200b\u6267\u884c\u200b\u5982\u4e0b\u200b\u547d\u4ee4\u200b\u3002</p> <ol> <li> <p>\u200b\u751f\u6210\u200bCA\u200b\u5bc6\u94a5\u200b</p> <pre><code>openssl genrsa -aes256 -out ca-key.pem 4096\nopenssl req -new -x509 -days 365 -key ca-key.pem -sha256 -out ca.pem # (1)\n</code></pre> <ol> <li>\u200b\u4f1a\u200b\u8bf7\u6c42\u200b\u8f93\u5165\u200b\u4e00\u4e9b\u200b\u4fe1\u606f\u200b\uff0c\u200b\u5176\u4e2d\u200b<code>Common Name</code>\u200b\u4e00\u6761\u200b\u9700\u8981\u200b\u586b\u5199\u200b\u670d\u52a1\u5668\u200b\u5bf9\u5e94\u200b\u7684\u200b\u57df\u540d\u200b\u3002</li> </ol> </li> <li> <p>\u200b\u751f\u6210\u200b\u670d\u52a1\u5668\u200b\u5bc6\u94a5\u200b\u548c\u200b\u8bc1\u4e66\u200b</p> <pre><code>openssl genrsa -out server-key.pem 4096\nopenssl req -subj \"/CN=www.example.com\" -sha256 -new -key server-key.pem -out server.csr\necho subjectAltName = DNS:$HOST,IP:8.8.8.8 &gt;&gt; extfile.cnf # (1)\necho extendedKeyUsage = serverAuth &gt;&gt; extfile.cnf\nopenssl x509 -req -days 365 -sha256 -in server.csr -CA ca.pem -CAkey ca-key.pem \\\n    -CAcreateserial -out server-cert.pem -extfile extfile.cnf\n</code></pre> <ol> <li>\u200b\u5982\u679c\u200b\u4e0d\u200b\u9700\u8981\u200bIP\u200b\u5730\u5740\u200b\u8bbf\u95ee\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5220\u9664\u200b\u76f8\u5173\u200b\u6761\u76ee\u200b\u3002</li> </ol> </li> <li> <p>\u200b\u751f\u6210\u200b\u5ba2\u6237\u7aef\u200b\u5bc6\u94a5\u200b\u548c\u200b\u8bc1\u4e66\u200b</p> <pre><code>openssl genrsa -out key.pem 4096\nopenssl req -subj '/CN=client' -new -key key.pem -out client.csr\necho extendedKeyUsage = clientAuth &gt; extfile-client.cnf\nopenssl x509 -req -days 365 -sha256 -in client.csr -CA ca.pem -CAkey ca-key.pem \\\n    -CAcreateserial -out cert.pem -extfile extfile-client.cnf\n</code></pre> </li> <li> <p>\u200b\u6e05\u7406\u200b\u5176\u4f59\u200b\u6587\u4ef6\u200b\uff0c\u200b\u53ea\u200b\u7559\u4e0b\u200b<code>.pem</code>\u200b\u8bc1\u4e66\u200b\u6587\u4ef6\u200b\uff0c\u200b\u5e76\u4e14\u200b\u8c03\u6574\u200b\u8bc1\u4e66\u200b\u6587\u4ef6\u200b\u7684\u200b\u6743\u9650\u200b</p> <pre><code>rm -v client.csr server.csr extfile.cnf extfile-client.cnf\nchmod -v 0400 ca-key.pem key.pem server-key.pem\nchmod -v 0444 ca.pem server-cert.pem cert.pem\n</code></pre> </li> <li> <p>\u200b\u542f\u52a8\u200bDocker daemon</p> <pre><code>dockerd \\\n    --tlsverify \\\n    --tlscacert=ca.pem \\\n    --tlscert=server-cert.pem \\\n    --tlskey=server-key.pem \\\n    -H=0.0.0.0:2376\n</code></pre> </li> <li> <p>\u200b\u5206\u53d1\u200b\u5ba2\u6237\u7aef\u200b\u8bc1\u4e66\u200b<code>ca.pem</code>\u3001<code>cert.pem</code>\u3001<code>key.pem</code>\u3002\u200b\u5ba2\u6237\u7aef\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4ee5\u4e0b\u200b\u547d\u4ee4\u200b\u8fde\u63a5\u200b</p> <pre><code>docker --tlsverify \\\n    --tlscacert=ca.pem \\\n    --tlscert=cert.pem \\\n    --tlskey=key.pem \\\n    -H=$HOST:2376\n</code></pre> </li> </ol> <p>\u200b\u5728\u200bWindows\u200b\u4e2d\u200b\uff0cDocker\u200b\u901a\u8fc7\u200bWSL\u200b\u6216\u8005\u200bHyper-V\u200b\u865a\u62df\u5316\u200b\u5b9e\u73b0\u200bLinux\u200b\u73af\u5883\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5982\u679c\u200b\u5728\u200bWindows\u200b\u4e0b\u200b\u76f4\u63a5\u200b\u6267\u884c\u200b<code>dockerd.exe</code>\uff0c\u200b\u5f97\u5230\u200b\u7684\u200b\u662f\u200b\u8fd0\u884c\u200bWindows\u200b\u5bb9\u5668\u200b\u7684\u200bDocker\u3002\u200b\u5982\u679c\u200b\u8981\u200b\u8fdc\u7a0b\u200b\u8fde\u63a5\u200b\u8fd0\u884c\u200bLinux\u200b\u7684\u200bDocker\u200b\u73af\u5883\u200b\uff0c\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>docker-remote-api-tls</code>\u3002</p> <p>Docker Desktop Engine \u200b\u914d\u7f6e\u6587\u4ef6\u200b</p> <p>\u200b\u5728\u200bDocker Desktop\u200b\u91cc\u200b\u7684\u200bSettings - Engine\u200b\u91cc\u200b\u4fee\u6539\u200bTLS\u200b\u53c2\u6570\u200b\u914d\u7f6e\u200b\uff0c\u200b\u4f1a\u200b\u5bfc\u81f4\u200bDocker Engine\u200b\u65e0\u6cd5\u200b\u542f\u52a8\u200b\u3002</p> <ol> <li>\u200b\u6267\u884c\u200b<code>git clone https://github.com/kekru/docker-remote-api-tls.git</code>\u200b\u514b\u9686\u200b\u955c\u50cf\u200b</li> <li> <p>\u200b\u7f16\u8f91\u200b<code>docker-compose.yml</code>\uff0c\u200b\u66ff\u6362\u200b<code>&lt;password&gt;</code>\u3001<code>&lt;domain-or-ip&gt;</code>\u3001<code>&lt;client-cert-path&gt;</code>\u200b\u4e2d\u200b\u7684\u200b\u5185\u5bb9\u200b</p> <pre><code>version: \"3.4\"\nservices:\n    remote-api:\n        image: kekru/docker-remote-api-tls:v0.4.0\n        ports:\n            - 2376:443\n        environment:\n            - CREATE_CERTS_WITH_PW=&lt;password&gt;\n            - CERT_HOSTNAME=&lt;domain-or-ip&gt;\n        volumes:\n            - &lt;client-cert-path&gt;:/data/certs # (1)\n            - /var/run/docker.sock:/var/run/docker.sock:ro\n</code></pre> <ol> <li>\u200b\u4e0d\u52a0\u200b\u8fd9\u200b\u4e00\u884c\u200b\u4e5f\u200b\u884c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u4ece\u200bDocker Desktop\u200b\u91cc\u200b\u590d\u5236\u200b</li> </ol> </li> <li> <p>\u200b\u6267\u884c\u200b<code>docker-compose up -d</code>\u200b\u8fd0\u884c\u200b\u5bb9\u5668\u200b\uff0c\u200b\u6ce8\u610f\u200b\u5728\u200b\u8fdc\u7a0b\u200b\u8fde\u63a5\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u9700\u8981\u200b\u4fdd\u8bc1\u200b\u5bb9\u5668\u200b\u59cb\u7ec8\u200b\u5f00\u542f\u200b</p> </li> </ol>"},{"location":"coding/configuration/docker/#_4","title":"\u5ba2\u6237\u7aef\u200b\u914d\u7f6e","text":"<ol> <li>\u200b\u5c06\u200b<code>ca.pem</code>\u3001<code>cert.pem</code>\u3001<code>key.pem</code>\u200b\u590d\u5236\u5230\u200b\u5ba2\u6237\u7aef\u200b\u7684\u200b<code>~/.docker</code>\u200b\u76ee\u5f55\u200b\u4e0b\u200b</li> <li> <p>\u200b\u8bbe\u7f6e\u200b\u73af\u5883\u53d8\u91cf\u200b\uff08\u200b\u53ea\u200b\u9002\u7528\u200b\u4e8e\u200b\u4ece\u200b\u7ec8\u7aef\u200b\u8bbf\u95ee\u200bDocker\u200b\u7684\u200b\u60c5\u51b5\u200b\uff09</p> <pre><code>export $DOCKER_TLS_VERIFY=1\nexport $DOCKER_HOST=\"tcp://&lt;domain-or-ip&gt;:2376\"\n</code></pre> </li> <li> <p>\u200b\u6216\u8005\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200bDocker context</p> <pre><code>docker context create remote --docker \\\n    \"host=tcp://&lt;domain-or-ip&gt;:2376,ca=~/.docker/ca.pem,cert=~/.docker/cert.pem,key=~/.docker/key.pem\n</code></pre> </li> <li> <p>\u200b\u6b64\u540e\u200b\uff0cDocker\u200b\u4f1a\u200b\u9ed8\u8ba4\u200b\u4f7f\u7528\u200b\u8fdc\u7a0b\u200b\u8fde\u63a5\u200b\u3002</p> </li> </ol> <p>\u200b\u5982\u679c\u200b\u8fde\u63a5\u200b\u6210\u529f\u200b\uff0c\u200b\u6267\u884c\u200b<code>docker version</code>\u200b\u4f1a\u200b\u540c\u65f6\u200b\u8f93\u51fa\u200b\u5ba2\u6237\u7aef\u200b\u548c\u200b\u670d\u52a1\u5668\u200b\u4e24\u8005\u200b\u7684\u200b\u7248\u672c\u4fe1\u606f\u200b\uff0c\u200b\u5982\u200b</p> <pre><code>Client: Docker Engine - Community\n Version:           24.0.5\n API version:       1.43\n Go version:        go1.20.6\n Git commit:        ced0996600\n Built:             Wed Jul 19 19:44:22 2023\n OS/Arch:           darwin/arm64\n Context:           default\n\nServer: Docker Desktop 4.21.1 (114176)\n Engine:\n  Version:          24.0.2\n  API version:      1.43 (minimum version 1.12)\n  Go version:       go1.20.4\n  Git commit:       659604f\n  Built:            Thu May 25 21:52:17 2023\n  OS/Arch:          linux/amd64\n  Experimental:     false\n containerd:\n  Version:          1.6.21\n  GitCommit:        3dce8eb055cbb6872793272b4f20ed16117344f8\n runc:\n  Version:          1.1.7\n  GitCommit:        v1.1.7-0-g860f061\n docker-init:\n  Version:          0.19.0\n  GitCommit:        de40ad0\n</code></pre> <p>\u200b\u6302\u8f7d\u200b\u5377\u200b</p> <p>\u200b\u5728\u200bDocker\u200b\u4e2d\u200b\uff0c\u200b\u65e0\u8bba\u662f\u200b\u4f7f\u7528\u200b<code>-v</code>\u200b\u53c2\u6570\u200b\u8fd8\u662f\u200b<code>--mount</code>\u200b\u53c2\u6570\u200b\u6302\u8f7d\u200b\u7684\u200b\u5377\u200b\u7684\u200b\u6e90\u200b\u8def\u5f84\u200b\u90fd\u200b\u662f\u200b\u76f8\u5bf9\u200b\u4e8e\u200bDocker Daemon\u200b\u800c\u8a00\u200b\u7684\u200b\u3002</p>"},{"location":"coding/configuration/docker/#_5","title":"\u53c2\u8003\u200b\u9605\u8bfb","text":"<p>Docker registry\uff1a</p> <ul> <li>Working with the Container registry (GitHub)</li> </ul> <p>Dockerfile\uff1a</p> <ul> <li>Dockerfile reference</li> </ul> <p>\u200b\u8fdc\u7a0b\u200b\u8bbf\u95ee\u200bDocker daemon\uff1a</p> <ul> <li>Windows Docker \u200b\u4e3b\u673a\u200b\u8fdc\u7a0b\u7ba1\u7406\u200b</li> <li>Protect the Docker daemon socket</li> <li>dockerd</li> <li>Docker Remote API with TLS client authentication via container</li> <li>How can I mount a volume of files to a remote docker daemon?</li> </ul>"},{"location":"coding/configuration/nginx-rtmp/","title":"\u6e90\u7801\u200b\u7f16\u8bd1\u200bnginx\u200b\u5e76\u200b\u6dfb\u52a0\u200bRTMP\u200b\u6a21\u5757","text":""},{"location":"coding/configuration/nginx-rtmp/#_1","title":"\u5148\u51b3\u6761\u4ef6","text":"<ul> <li>nginx\u200b\u6e90\u4ee3\u7801\u200b\uff0c\u200b\u9700\u8981\u200b\u4ece\u200bGitHub Release\u200b\u4e0b\u8f7d\u200b\uff0c\u200b\u4ece\u200bnginx\u200b\u5b98\u7f51\u200b\u4e0b\u8f7d\u200b\u7684\u200b\u6e90\u4ee3\u7801\u200b\u5728\u200b\u540e\u7eed\u200b\u7f16\u8bd1\u200b\u65f6\u4f1a\u200b\u62a5\u200b\u5982\u4e0b\u200b\u9519\u8bef\u200b\uff1a   <pre><code>NMAKE : fatal error U1073: don't know how to make 'src/os/win32/ngx_win32_config.h'\n</code></pre></li> <li>openssl\u200b\u6e90\u4ee3\u7801\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4ece\u200bGitHub Release\u200b\u4e0b\u8f7d\u200b</li> <li>zlib\u200b\u6e90\u4ee3\u7801\u200b\uff0cGitHub\u200b\u4ed3\u5e93\u200b\uff1ahttps://github.com/madler/zlib</li> <li>pcre\u200b\u6e90\u4ee3\u7801\u200b\uff0c\u200b\u5b98\u7f51\u200b\u63d0\u4f9b\u200b\u5404\u200b\u7248\u672c\u200b\u7684\u200b\u4e0b\u8f7d\u200b\u94fe\u63a5\u200b\uff1ahttps://ftp.pcre.org/pub/pcre/</li> <li>RMTP\u200b\u63d2\u4ef6\u200b\uff1ahttps://github.com/arut/nginx-rtmp-module</li> <li>MSYS\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200bMinGW Installer\u200b\u8fdb\u884c\u200b\u5b89\u88c5\u200b</li> <li>Visual Studio\uff0c\u200b\u9700\u8981\u200b\u5b89\u88c5\u200bC++\u200b\u7f16\u8bd1\u5668\u200b</li> <li>NASM</li> <li>Perl&gt;=5.30</li> </ul> <p>\u200b\u672c\u6b21\u200b\u7f16\u8bd1\u200b\u4f7f\u7528\u200b\u7684\u200b\u73af\u5883\u200b\uff1a</p> <ul> <li>Windows 19042.662</li> <li>Visual Studio 2019</li> <li>Visual Studio Build Tools 2017</li> <li>Windows 10 SDK 10.0.17763.0</li> <li>nginx 1.19.5</li> <li>openssl 1.1.1g</li> <li>zlib 1.2.11</li> <li>pcre 8.44</li> <li>perl 5.32 (Strawberry)</li> <li>nasm 2.15.05</li> <li>nmake 14.27.29112.0</li> <li>MSVC(cl) 19.27.29112</li> </ul> <p>\u200b\u5c06\u200b\u5305\u542b\u200b<code>perl.exe</code>\u3001<code>nasm.exe</code>\u3001<code>sed.exe</code>\u200b\u7684\u200b\u76ee\u5f55\u200b\u6dfb\u52a0\u200b\u5230\u200bPATH\u200b\u73af\u5883\u53d8\u91cf\u200b\u3002</p>"},{"location":"coding/configuration/nginx-rtmp/#_2","title":"\u51c6\u5907\u200b\u5de5\u4f5c","text":"<p>\u200b\u542f\u52a8\u200bMSYS\uff0c<code>cd</code>\u200b\u5230\u200bnginx\u200b\u7684\u200b\u6e90\u4ee3\u7801\u200b\u76ee\u5f55\u200b\uff0c\u200b\u5207\u6362\u200b\u5230\u200bC\u200b\u76d8\u200b\u6839\u76ee\u5f55\u200b\u7684\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>cd /c/\n</code></pre> <p>\u200b\u6267\u884c\u200b<code>auto/configure</code>\uff0c\u200b\u6839\u636e\u200bnginx\u200b\u5b98\u7f51\u200b\u63d0\u4f9b\u200b\u7684\u200b\u53c2\u6570\u200b\u8fdb\u884c\u200b\uff0c\u200b\u6dfb\u52a0\u200b``\u200b\u53c2\u6570\u200b\uff0c\u200b\u5177\u4f53\u200b\u547d\u4ee4\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>auto/configure \\\n    --with-cc=cl \\\n    --with-debug \\\n    --prefix=. \\\n    --conf-path=conf/nginx.conf \\\n    --pid-path=logs/nginx.pid \\\n    --http-log-path=logs/access.log \\\n    --error-log-path=logs/error.log \\\n    --sbin-path=nginx.exe \\\n    --http-client-body-temp-path=temp/client_body_temp \\\n    --http-proxy-temp-path=temp/proxy_temp \\\n    --http-fastcgi-temp-path=temp/fastcgi_temp \\\n    --http-scgi-temp-path=temp/scgi_temp \\\n    --http-uwsgi-temp-path=temp/uwsgi_temp \\\n    --with-cc-opt=-DFD_SETSIZE=1024 \\\n    --with-pcre=pcre-8.44 \\\n    --with-zlib=zlib \\\n    --with-openssl=openssl \\\n    --with-openssl-opt=no-asm \\\n    --with-http_ssl_module \\\n    --add-module=nginx-rtmp-module\n</code></pre> <p>\u200b\u6ce8\u610f\u200b\u6b64\u5904\u200b<code>--with-pcre</code>\u3001<code>--with-zlib</code>\u3001<code>--with-openssl</code>\u200b\u53c2\u6570\u200b\u4e2d\u200b\u63d0\u4f9b\u200b\u7684\u200b\u8def\u5f84\u200b\u9700\u8981\u200b\u4e0e\u200bPCRE\u3001zlib\u3001openssl\u200b\u7684\u200b\u6e90\u4ee3\u7801\u200b\u8def\u5f84\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u3002</p> <p>\u200b\u6267\u884c\u200b\u5b8c\u6bd5\u200b\u540e\u200b\u751f\u6210\u200bMakefile\u200b\u6587\u4ef6\u200b\u300232\u200b\u4f4d\u200b\u73af\u5883\u200b\u4e0b\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u5f00\u59cb\u200b\u7f16\u8bd1\u200b\uff0c64\u200b\u4f4d\u200b\u73af\u5883\u200b\u4e0b\u200b\u9700\u8981\u200b\u5bf9\u200b<code>auto\\lib\\openssl\\makefile.msvc</code>\u200b\u6539\u4e3a\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b</p> <pre><code># Copyright (C) Igor Sysoev\n# Copyright (C) Nginx, Inc.\n\n\nall:\n    cd $(OPENSSL)\n\n    perl Configure VC-WIN64A no-shared              \\\n        --prefix=\"%cd%/openssl\"                     \\\n        --openssldir=\"%cd%/openssl/ssl\"             \\\n        $(OPENSSL_OPT)\n\n    if exist ms\\do_win64a.bat (                     \\\n        ms\\do_win64a                                \\\n        &amp;&amp; $(MAKE) -f ms\\nt.mak                     \\\n        &amp;&amp; $(MAKE) -f ms\\nt.mak install             \\\n    ) else (                                        \\\n        $(MAKE)                                     \\\n        &amp;&amp; $(MAKE) install_sw                       \\\n    )\n</code></pre> <p>\u200b\u5c06\u200b<code>VC-WIN32</code>\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>VC-WIN64A</code>\uff0c\u200b\u6240\u6709\u200b\u7684\u200b<code>ms\\do_ms.bat</code>\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>ms\\do_win64a.bat</code>\u3002</p>"},{"location":"coding/configuration/nginx-rtmp/#msvc","title":"MSVC\u200b\u7f16\u8bd1","text":"<p>\u200b\u7f16\u8f91\u200b\u5982\u4e0b\u200b\u6e90\u4ee3\u7801\u200b\u6587\u4ef6\u200b\uff1a</p> <ul> <li><code>nginx-rtmp-module/ngx_rtmp_flv_module.c</code>\uff0c\u200b\u5b9a\u4f4d\u200b\u5230\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b\u5757\u200b\uff1a</li> </ul> <p>509\u200b\u884c\u200b\uff1a</p> <pre><code>ngx_rtmp_prepare_message(s, &amp;h, ctx-&gt;msg_mask &amp; (1 &lt;&lt; h.type) ?\n                           &amp;lh : NULL, out);\n</code></pre> <p>521\u200b\u884c\u200b\uff1a</p> <pre><code>ctx-&gt;msg_mask |= (1 &lt;&lt; h.type);\n</code></pre> <p>\u200b\u5c06\u200b\u5176\u4e2d\u200b\u7684\u200b<code>(1 &lt;&lt; h.type)</code>\u200b\u6539\u4e3a\u200b<code>(unsigned long)(1 &lt;&lt; h.type)</code>\u3002</p> <ul> <li><code>nginx-rtmp-module/ngx_rtmp_core_module.c</code>\uff0c\u200b\u5b9a\u4f4d\u200b\u5230\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b\u5757\u200b\uff1a</li> </ul> <p>611\u200b\u884c\u200b\uff1a</p> <pre><code>struct sockaddr  *sa;\n</code></pre> <p>\u200b\u5c06\u200b\u6b64\u5904\u200b\u7684\u200b\u53d8\u91cf\u540d\u200b<code>sa</code>\u200b\u6539\u4e3a\u200b<code>sa2</code>\uff0c\u200b\u5e76\u200b\u5c06\u200b\u6b64\u540e\u200b\u6240\u6709\u200b\u7684\u200b<code>sa</code>\u200b\u53d8\u91cf\u200b\u6539\u4e3a\u200b<code>sa2</code>\u3002</p> <p>\u200b\u542f\u52a8\u200bx64 Native Tools Command Prompt for VS 2019\uff0c\u200b\u5207\u6362\u200b\u5230\u200b\u6e90\u4ee3\u7801\u200b\u76ee\u5f55\u200b\uff0c\u200b\u6267\u884c\u200b<code>nmake</code>\u200b\u542f\u52a8\u200b\u7f16\u8bd1\u200b\u3002\u200b\u8f93\u51fa\u200b\u6587\u4ef6\u200b\u4e3a\u200b<code>objs/nginx.exe</code></p>"},{"location":"coding/configuration/nginx-rtmp/#nginx","title":"nginx\u200b\u914d\u7f6e","text":"<p>\u200b\u5c06\u200b<code>objs/nginx.exe</code>\u200b\u590d\u5236\u200b\u4e00\u4efd\u200b\u3002\u200b\u5728\u200b\u6240\u5728\u200b\u76ee\u5f55\u200b\u4e0b\u200b\u521b\u5efa\u200b\u5982\u4e0b\u200b\u6587\u4ef6\u200b\u7ed3\u6784\u200b\uff1a</p> <pre><code>.\n\u251c\u2500 nginx.exe\n\u251c\u2500 conf/\n\u2502  \u2514\u2500 nginx.conf\n\u251c\u2500 html\n\u2502  \u2514\u2500 index.xsl\n\u251c\u2500 logs\n\u2514\u2500 temp\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff0c<code>html/index.xsl</code>\u200b\u5199\u5165\u200b\u5982\u4e0b\u200b\u5185\u5bb9\u200b</p> html/index.xsl\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b <pre><code>&lt;?xml version=\"1.0\" encoding=\"utf-8\" ?&gt;\n&lt;!--\nCopyright (C) Roman Arutyunyan\n--&gt;\n&lt;xsl:stylesheet version=\"1.0\" xmlns:xsl=\"http://www.w3.org/1999/XSL/Transform\"&gt;\n&lt;xsl:template match=\"/\"&gt;\n    &lt;html&gt;\n        &lt;head&gt;\n            &lt;title&gt;RTMP statistics&lt;/title&gt;\n        &lt;/head&gt;\n        &lt;body&gt;\n            &lt;xsl:apply-templates select=\"rtmp\"/&gt;\n            &lt;hr/&gt;\n            Generated by &lt;a href='https://github.com/arut/nginx-rtmp-module'&gt;\n            nginx-rtmp-module&lt;/a&gt;&amp;#160;&lt;xsl:value-of select=\"/rtmp/nginx_rtmp_version\"/&gt;,\n            &lt;a href=\"http://nginx.org\"&gt;nginx&lt;/a&gt;&amp;#160;&lt;xsl:value-of select=\"/rtmp/nginx_version\"/&gt;,\n            pid &lt;xsl:value-of select=\"/rtmp/pid\"/&gt;,\n            built &lt;xsl:value-of select=\"/rtmp/built\"/&gt;&amp;#160;&lt;xsl:value-of select=\"/rtmp/compiler\"/&gt;\n        &lt;/body&gt;\n    &lt;/html&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template match=\"rtmp\"&gt;\n    &lt;table cellspacing=\"1\" cellpadding=\"5\"&gt;\n        &lt;tr bgcolor=\"#999999\"&gt;\n            &lt;th&gt;RTMP&lt;/th&gt;\n            &lt;th&gt;#clients&lt;/th&gt;\n            &lt;th colspan=\"4\"&gt;Video&lt;/th&gt;\n            &lt;th colspan=\"4\"&gt;Audio&lt;/th&gt;\n            &lt;th&gt;In bytes&lt;/th&gt;\n            &lt;th&gt;Out bytes&lt;/th&gt;\n            &lt;th&gt;In bits/s&lt;/th&gt;\n            &lt;th&gt;Out bits/s&lt;/th&gt;\n            &lt;th&gt;State&lt;/th&gt;\n            &lt;th&gt;Time&lt;/th&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;\n            &lt;td colspan=\"2\"&gt;Accepted: &lt;xsl:value-of select=\"naccepted\"/&gt;&lt;/td&gt;\n            &lt;th bgcolor=\"#999999\"&gt;codec&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;bits/s&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;size&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;fps&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;codec&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;bits/s&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;freq&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;chan&lt;/th&gt;\n            &lt;td&gt;\n                &lt;xsl:call-template name=\"showsize\"&gt;\n                    &lt;xsl:with-param name=\"size\" select=\"bytes_in\"/&gt;\n                &lt;/xsl:call-template&gt;\n            &lt;/td&gt;\n            &lt;td&gt;\n                &lt;xsl:call-template name=\"showsize\"&gt;\n                    &lt;xsl:with-param name=\"size\" select=\"bytes_out\"/&gt;\n                &lt;/xsl:call-template&gt;\n            &lt;/td&gt;\n            &lt;td&gt;\n                &lt;xsl:call-template name=\"showsize\"&gt;\n                    &lt;xsl:with-param name=\"size\" select=\"bw_in\"/&gt;\n                    &lt;xsl:with-param name=\"bits\" select=\"1\"/&gt;\n                    &lt;xsl:with-param name=\"persec\" select=\"1\"/&gt;\n                &lt;/xsl:call-template&gt;\n            &lt;/td&gt;\n            &lt;td&gt;\n                &lt;xsl:call-template name=\"showsize\"&gt;\n                    &lt;xsl:with-param name=\"size\" select=\"bw_out\"/&gt;\n                    &lt;xsl:with-param name=\"bits\" select=\"1\"/&gt;\n                    &lt;xsl:with-param name=\"persec\" select=\"1\"/&gt;\n                &lt;/xsl:call-template&gt;\n            &lt;/td&gt;\n            &lt;td/&gt;\n            &lt;td&gt;\n                &lt;xsl:call-template name=\"showtime\"&gt;\n                    &lt;xsl:with-param name=\"time\" select=\"/rtmp/uptime * 1000\"/&gt;\n                &lt;/xsl:call-template&gt;\n            &lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;xsl:apply-templates select=\"server\"/&gt;\n    &lt;/table&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template match=\"server\"&gt;\n    &lt;xsl:apply-templates select=\"application\"/&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template match=\"application\"&gt;\n    &lt;tr bgcolor=\"#999999\"&gt;\n        &lt;td&gt;\n            &lt;b&gt;&lt;xsl:value-of select=\"name\"/&gt;&lt;/b&gt;\n        &lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;xsl:apply-templates select=\"live\"/&gt;\n    &lt;xsl:apply-templates select=\"play\"/&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template match=\"live\"&gt;\n    &lt;tr bgcolor=\"#aaaaaa\"&gt;\n        &lt;td&gt;\n            &lt;i&gt;live streams&lt;/i&gt;\n        &lt;/td&gt;\n        &lt;td align=\"middle\"&gt;\n            &lt;xsl:value-of select=\"nclients\"/&gt;\n        &lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;xsl:apply-templates select=\"stream\"/&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template match=\"play\"&gt;\n    &lt;tr bgcolor=\"#aaaaaa\"&gt;\n        &lt;td&gt;\n            &lt;i&gt;vod streams&lt;/i&gt;\n        &lt;/td&gt;\n        &lt;td align=\"middle\"&gt;\n            &lt;xsl:value-of select=\"nclients\"/&gt;\n        &lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;xsl:apply-templates select=\"stream\"/&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template match=\"stream\"&gt;\n    &lt;tr valign=\"top\"&gt;\n        &lt;xsl:attribute name=\"bgcolor\"&gt;\n            &lt;xsl:choose&gt;\n                &lt;xsl:when test=\"active\"&gt;#cccccc&lt;/xsl:when&gt;\n                &lt;xsl:otherwise&gt;#dddddd&lt;/xsl:otherwise&gt;\n            &lt;/xsl:choose&gt;\n        &lt;/xsl:attribute&gt;\n        &lt;td&gt;\n            &lt;a href=\"\"&gt;\n                &lt;xsl:attribute name=\"onclick\"&gt;\n                    var d=document.getElementById('&lt;xsl:value-of select=\"../../name\"/&gt;-&lt;xsl:value-of select=\"name\"/&gt;');\n                    d.style.display=d.style.display=='none'?'':'none';\n                    return false\n                &lt;/xsl:attribute&gt;\n                &lt;xsl:value-of select=\"name\"/&gt;\n                &lt;xsl:if test=\"string-length(name) = 0\"&gt;\n                    [EMPTY]\n                &lt;/xsl:if&gt;\n            &lt;/a&gt;\n        &lt;/td&gt;\n        &lt;td align=\"middle\"&gt; &lt;xsl:value-of select=\"nclients\"/&gt; &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:value-of select=\"meta/video/codec\"/&gt;&amp;#160;&lt;xsl:value-of select=\"meta/video/profile\"/&gt;&amp;#160;&lt;xsl:value-of select=\"meta/video/level\"/&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showsize\"&gt;\n                &lt;xsl:with-param name=\"size\" select=\"bw_video\"/&gt;\n                &lt;xsl:with-param name=\"bits\" select=\"1\"/&gt;\n                &lt;xsl:with-param name=\"persec\" select=\"1\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:apply-templates select=\"meta/video/width\"/&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:value-of select=\"meta/video/frame_rate\"/&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:value-of select=\"meta/audio/codec\"/&gt;&amp;#160;&lt;xsl:value-of select=\"meta/audio/profile\"/&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showsize\"&gt;\n                &lt;xsl:with-param name=\"size\" select=\"bw_audio\"/&gt;\n                &lt;xsl:with-param name=\"bits\" select=\"1\"/&gt;\n                &lt;xsl:with-param name=\"persec\" select=\"1\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:apply-templates select=\"meta/audio/sample_rate\"/&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:value-of select=\"meta/audio/channels\"/&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showsize\"&gt;\n            &lt;xsl:with-param name=\"size\" select=\"bytes_in\"/&gt;\n        &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showsize\"&gt;\n                &lt;xsl:with-param name=\"size\" select=\"bytes_out\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showsize\"&gt;\n                &lt;xsl:with-param name=\"size\" select=\"bw_in\"/&gt;\n                &lt;xsl:with-param name=\"bits\" select=\"1\"/&gt;\n                &lt;xsl:with-param name=\"persec\" select=\"1\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showsize\"&gt;\n                &lt;xsl:with-param name=\"size\" select=\"bw_out\"/&gt;\n                &lt;xsl:with-param name=\"bits\" select=\"1\"/&gt;\n                &lt;xsl:with-param name=\"persec\" select=\"1\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n        &lt;td&gt;&lt;xsl:call-template name=\"streamstate\"/&gt;&lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showtime\"&gt;\n            &lt;xsl:with-param name=\"time\" select=\"time\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr style=\"display:none\"&gt;\n        &lt;xsl:attribute name=\"id\"&gt;\n            &lt;xsl:value-of select=\"../../name\"/&gt;-&lt;xsl:value-of select=\"name\"/&gt;\n        &lt;/xsl:attribute&gt;\n        &lt;td colspan=\"16\" ngcolor=\"#eeeeee\"&gt;\n            &lt;table cellspacing=\"1\" cellpadding=\"5\"&gt;\n                &lt;tr&gt;\n                    &lt;th&gt;Id&lt;/th&gt;\n                    &lt;th&gt;State&lt;/th&gt;\n                    &lt;th&gt;Address&lt;/th&gt;\n                    &lt;th&gt;Flash version&lt;/th&gt;\n                    &lt;th&gt;Page URL&lt;/th&gt;\n                    &lt;th&gt;SWF URL&lt;/th&gt;\n                    &lt;th&gt;Dropped&lt;/th&gt;\n                    &lt;th&gt;Timestamp&lt;/th&gt;\n                    &lt;th&gt;A-V&lt;/th&gt;\n                    &lt;th&gt;Time&lt;/th&gt;\n                &lt;/tr&gt;\n                &lt;xsl:apply-templates select=\"client\"/&gt;\n            &lt;/table&gt;\n        &lt;/td&gt;\n    &lt;/tr&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template name=\"showtime\"&gt;\n    &lt;xsl:param name=\"time\"/&gt;\n\n    &lt;xsl:if test=\"$time &amp;gt; 0\"&gt;\n        &lt;xsl:variable name=\"sec\"&gt;\n            &lt;xsl:value-of select=\"floor($time div 1000)\"/&gt;\n        &lt;/xsl:variable&gt;\n\n        &lt;xsl:if test=\"$sec &amp;gt;= 86400\"&gt;\n            &lt;xsl:value-of select=\"floor($sec div 86400)\"/&gt;d\n        &lt;/xsl:if&gt;\n\n        &lt;xsl:if test=\"$sec &amp;gt;= 3600\"&gt;\n            &lt;xsl:value-of select=\"(floor($sec div 3600)) mod 24\"/&gt;h\n        &lt;/xsl:if&gt;\n\n        &lt;xsl:if test=\"$sec &amp;gt;= 60\"&gt;\n            &lt;xsl:value-of select=\"(floor($sec div 60)) mod 60\"/&gt;m\n        &lt;/xsl:if&gt;\n\n        &lt;xsl:value-of select=\"$sec mod 60\"/&gt;s\n    &lt;/xsl:if&gt;\n&lt;/xsl:template&gt;\n\n&lt;xsl:template name=\"showsize\"&gt;\n    &lt;xsl:param name=\"size\"/&gt;\n    &lt;xsl:param name=\"bits\" select=\"0\" /&gt;\n    &lt;xsl:param name=\"persec\" select=\"0\" /&gt;\n    &lt;xsl:variable name=\"sizen\"&gt;\n        &lt;xsl:value-of select=\"floor($size div 1024)\"/&gt;\n    &lt;/xsl:variable&gt;\n    &lt;xsl:choose&gt;\n        &lt;xsl:when test=\"$sizen &amp;gt;= 1073741824\"&gt;\n            &lt;xsl:value-of select=\"format-number($sizen div 1073741824,'#.###')\"/&gt; T&lt;/xsl:when&gt;\n\n        &lt;xsl:when test=\"$sizen &amp;gt;= 1048576\"&gt;\n            &lt;xsl:value-of select=\"format-number($sizen div 1048576,'#.###')\"/&gt; G&lt;/xsl:when&gt;\n\n        &lt;xsl:when test=\"$sizen &amp;gt;= 1024\"&gt;\n            &lt;xsl:value-of select=\"format-number($sizen div 1024,'#.##')\"/&gt; M&lt;/xsl:when&gt;\n        &lt;xsl:when test=\"$sizen &amp;gt;= 0\"&gt;\n            &lt;xsl:value-of select=\"$sizen\"/&gt; K&lt;/xsl:when&gt;\n    &lt;/xsl:choose&gt;\n    &lt;xsl:if test=\"string-length($size) &amp;gt; 0\"&gt;\n        &lt;xsl:choose&gt;\n            &lt;xsl:when test=\"$bits = 1\"&gt;b&lt;/xsl:when&gt;\n            &lt;xsl:otherwise&gt;B&lt;/xsl:otherwise&gt;\n        &lt;/xsl:choose&gt;\n        &lt;xsl:if test=\"$persec = 1\"&gt;/s&lt;/xsl:if&gt;\n    &lt;/xsl:if&gt;\n&lt;/xsl:template&gt;\n\n&lt;xsl:template name=\"streamstate\"&gt;\n    &lt;xsl:choose&gt;\n        &lt;xsl:when test=\"active\"&gt;active&lt;/xsl:when&gt;\n        &lt;xsl:otherwise&gt;idle&lt;/xsl:otherwise&gt;\n    &lt;/xsl:choose&gt;\n&lt;/xsl:template&gt;\n\n\n&lt;xsl:template name=\"clientstate\"&gt;\n    &lt;xsl:choose&gt;\n        &lt;xsl:when test=\"publishing\"&gt;publishing&lt;/xsl:when&gt;\n        &lt;xsl:otherwise&gt;playing&lt;/xsl:otherwise&gt;\n    &lt;/xsl:choose&gt;\n&lt;/xsl:template&gt;\n\n\n&lt;xsl:template match=\"client\"&gt;\n    &lt;tr&gt;\n        &lt;xsl:attribute name=\"bgcolor\"&gt;\n            &lt;xsl:choose&gt;\n                &lt;xsl:when test=\"publishing\"&gt;#cccccc&lt;/xsl:when&gt;\n                &lt;xsl:otherwise&gt;#eeeeee&lt;/xsl:otherwise&gt;\n            &lt;/xsl:choose&gt;\n        &lt;/xsl:attribute&gt;\n        &lt;td&gt;&lt;xsl:value-of select=\"id\"/&gt;&lt;/td&gt;\n        &lt;td&gt;&lt;xsl:call-template name=\"clientstate\"/&gt;&lt;/td&gt;\n        &lt;td&gt;\n            &lt;a target=\"_blank\"&gt;\n                &lt;xsl:attribute name=\"href\"&gt;\n                    http://apps.db.ripe.net/search/query.html&amp;#63;searchtext=&lt;xsl:value-of select=\"address\"/&gt;\n                &lt;/xsl:attribute&gt;\n                &lt;xsl:attribute name=\"title\"&gt;whois&lt;/xsl:attribute&gt;\n                &lt;xsl:value-of select=\"address\"/&gt;\n            &lt;/a&gt;\n        &lt;/td&gt;\n        &lt;td&gt;&lt;xsl:value-of select=\"flashver\"/&gt;&lt;/td&gt;\n        &lt;td&gt;\n            &lt;a target=\"_blank\"&gt;\n                &lt;xsl:attribute name=\"href\"&gt;\n                    &lt;xsl:value-of select=\"pageurl\"/&gt;\n                &lt;/xsl:attribute&gt;\n                &lt;xsl:value-of select=\"pageurl\"/&gt;\n            &lt;/a&gt;\n        &lt;/td&gt;\n        &lt;td&gt;&lt;xsl:value-of select=\"swfurl\"/&gt;&lt;/td&gt;\n        &lt;td&gt;&lt;xsl:value-of select=\"dropped\"/&gt;&lt;/td&gt;\n        &lt;td&gt;&lt;xsl:value-of select=\"timestamp\"/&gt;&lt;/td&gt;\n        &lt;td&gt;&lt;xsl:value-of select=\"avsync\"/&gt;&lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showtime\"&gt;\n            &lt;xsl:with-param name=\"time\" select=\"time\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n    &lt;/tr&gt;\n&lt;/xsl:template&gt;\n\n&lt;xsl:template match=\"publishing\"&gt;\n    publishing\n&lt;/xsl:template&gt;\n\n&lt;xsl:template match=\"active\"&gt;\n    active\n&lt;/xsl:template&gt;\n\n&lt;xsl:template match=\"width\"&gt;\n    &lt;xsl:value-of select=\".\"/&gt;x&lt;xsl:value-of select=\"../height\"/&gt;\n&lt;/xsl:template&gt;\n\n&lt;/xsl:stylesheet&gt;\n</code></pre> <p><code>conf/nginx.conf</code>\u200b\u5199\u5165\u200b\u5982\u4e0b\u200b\u5185\u5bb9\u200b\uff1a</p> <p>conf/nginx.conf\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b</p> <pre><code>worker_processes  8;\nerror_log  logs/error.log info;\nevents {\n    worker_connections  1024;\n}\n\nrtmp {\n    server {\n        listen 1935;\n\n        application live {\n            live on;\n        }\n\n        application hls {\n            live on;\n            hls on;  \n            hls_path temp/hls;  \n            hls_fragment 8s;  \n        }\n    }\n}\n\nhttp {\n    server {\n        listen      8080;\n        root html;\n        location / {\n            rtmp_stat all;\n            rtmp_stat_stylesheet index.xsl;\n        }\n\n        location /index.xsl {\n            root html;\n        }\n\n        location /hls {  \n            types{  \n                application/vnd.apple.mpegurl m3u8;  \n                video/mp2t ts;  \n            }  \n            alias temp/hls;  \n            expires -1;  \n        }  \n    }\n}\n</code></pre> <p>\u200b\u53cc\u51fb\u200b<code>nginx.exe</code>\u200b\u5373\u53ef\u200b\u542f\u52a8\u200b\u670d\u52a1\u5668\u200b\uff0c\u200b\u670d\u52a1\u5668\u200b\u540c\u65f6\u200b\u76d1\u542c\u200b1935\u200b\u7aef\u53e3\u200b\u4e0e\u200b8080\u200b\u7aef\u53e3\u200b</p> <ul> <li>\u200b\u63a8\u6d41\u200b\u4f7f\u7528\u200b\u7684\u200bRTMP\u200b\u670d\u52a1\u5668\u200b\u5730\u5740\u200b\u4e3a\u200b\uff1a<code>rtmp://localhost:1935/live</code></li> <li>\u200b\u8bbf\u95ee\u200b<code>http://localhost:8080</code>\u200b\u53ef\u4ee5\u200b\u67e5\u770b\u200b\u670d\u52a1\u5668\u7aef\u200b\u7684\u200b\u7edf\u8ba1\u200b\u4fe1\u606f\u200b</li> <li>\u200b\u5ba2\u6237\u7aef\u200b\uff08\u200b\u5982\u200bVLC\u200b\u7b49\u200b\uff09\u200b\u53ef\u4ee5\u200b\u8bbf\u95ee\u200b<code>rtmp://localhost:1935/live</code>\u200b\u63a5\u6536\u200b\u6d41\u200b\uff0c<code>localhost</code>\u200b\u53ef\u4ee5\u200b\u66f4\u200b\u6539\u4e3a\u200b\u672c\u673a\u200b\u7684\u200bIP\u200b\u5730\u5740\u200b</li> </ul> <p>\u200b\u4f7f\u7528\u200b\u547d\u4ee4\u200b<code>.\\nginx.exe -s stop</code>\u200b\u5173\u95ed\u200b\u670d\u52a1\u5668\u200b</p>"},{"location":"coding/configuration/ssh/","title":"ssh","text":""},{"location":"coding/configuration/ssh/#_1","title":"\u57fa\u672c\u200b\u4f7f\u7528","text":"<p>SSH\u200b\u53ef\u4ee5\u200b\u8fdc\u7a0b\u200b\u8fde\u63a5\u200b\u670d\u52a1\u5668\u200b\u7684\u200b\u7ec8\u7aef\u200b\u3002\u200b\u547d\u4ee4\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b\uff0c\u200b\u5176\u4e2d\u200b<code>&lt;user&gt;</code>\u200b\u4e3a\u200b\u8fdc\u7a0b\u200b\u670d\u52a1\u5668\u200b\u4e0a\u200b\u7684\u200b\u7528\u6237\u540d\u200b\uff0c<code>&lt;hostname&gt;</code>\u200b\u662f\u200b\u8fdc\u7a0b\u200b\u670d\u52a1\u5668\u200b\u7684\u200b\u7f51\u7edc\u5730\u5740\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u662f\u200bIP\u200b\u5730\u5740\u200b\u6216\u200b\u57df\u540d\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b<code>~/.ssh/config</code>\u200b\u6587\u4ef6\u200b\u4e2d\u200b\u5b9a\u4e49\u200b\u7684\u200b\u4e3b\u673a\u200b\u3002\u200b\u5982\u679c\u200b\u6307\u5b9a\u200b\u4e86\u200b<code>&lt;command&gt;</code>\u200b\u53c2\u6570\u200b\uff0c\u200b\u5219\u200bSSH\u200b\u5728\u200b\u8fde\u63a5\u200b\u6210\u529f\u200b\u540e\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u6267\u884c\u547d\u4ee4\u200b\uff0c\u200b\u5c06\u200b\u7ed3\u679c\u200b\u8f93\u51fa\u200b\u5230\u200b\u7ec8\u7aef\u200b\u5e76\u200b\u5728\u200b\u6267\u884c\u200b\u7ed3\u675f\u200b\u540e\u200b\u65ad\u5f00\u8fde\u63a5\u200b\u3002</p> <pre><code>ssh &lt;user&gt;@&lt;hostname&gt; [&lt;command&gt;]\n</code></pre> <p>\u200b\u82e5\u200b\u5ba2\u6237\u7aef\u200b\u7684\u200b\u7528\u6237\u540d\u200b\u548c\u200b\u670d\u52a1\u5668\u200b\u7684\u200b\u7528\u6237\u540d\u200b\u76f8\u540c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b<code>&lt;user&gt;</code>\u200b\u53c2\u6570\u200b\uff0c\u200b\u5982\u200b</p> <pre><code>ssh &lt;hostname&gt; [&lt;command&gt;]\n</code></pre>"},{"location":"coding/configuration/ssh/#_2","title":"\u7aef\u53e3\u200b\u8f6c\u53d1","text":"<p>SSH\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>-L</code>\u3001<code>-R</code>\u3001<code>-D</code>\u200b\u53c2\u6570\u200b\uff0c\u200b\u5206\u522b\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u672c\u5730\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u3001\u200b\u8fdc\u7a0b\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u4e0e\u200b\u52a8\u6001\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u3002\u200b\u4f7f\u7528\u200b<code>-N</code>\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u5728\u200b\u4e0d\u200b\u8fde\u63a5\u200b\u8fdc\u7a0b\u200b\u7ec8\u7aef\u200b\uff0c\u200b\u53ea\u200b\u5b9e\u73b0\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u3002\u200b\u4e3a\u200b\u6e05\u6670\u200b\u8868\u8ff0\u200b\u5404\u4e2a\u200b\u7aef\u53e3\u200b\uff0c\u200b\u672c\u8282\u200b\u4e2d\u200b\u5047\u8bbe\u200b\u5ba2\u6237\u7aef\u200bA\u3001\u200b\u670d\u52a1\u5668\u200bB\u200b\u548c\u200b\u53e6\u200b\u4e00\u200b\u4e3b\u673a\u200bC\u3002\u200b\u5ba2\u6237\u7aef\u200bA\u200b\u901a\u8fc7\u200bSSH\u200b\u8fde\u63a5\u200b\u81f3\u200b\u670d\u52a1\u5668\u200bB\u3002</p>"},{"location":"coding/configuration/ssh/#_3","title":"\u672c\u5730\u200b\u7aef\u53e3\u200b\u8f6c\u53d1","text":"<p>\u200b\u4f7f\u7528\u200b<code>-L</code>\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u6253\u5f00\u200bSSH\u200b\u7684\u200b\u672c\u5730\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u529f\u80fd\u200b\u3002\u200b\u7531\u4e8e\u200b<code>-L</code>\u200b\u53c2\u6570\u200b\u5c06\u200b\u6307\u5411\u200b\u5ba2\u6237\u7aef\u200b\u672c\u5730\u200b\u7684\u200b\u8bf7\u6c42\u200b\u8f6c\u53d1\u7ed9\u200b\u670d\u52a1\u5668\u200b\uff0c\u200b\u56e0\u6b64\u200b\u79f0\u4e3a\u200b\u672c\u5730\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u3002\u200b\u542f\u52a8\u200b\u672c\u5730\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u7684\u200b\u547d\u4ee4\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>ssh -L [&lt;bind-addr&gt;]:&lt;port-of-A&gt;:&lt;addr-of-C&gt;:&lt;port-of-C&gt; &lt;user&gt;@&lt;hostname&gt;\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u547d\u4ee4\u200b\u8868\u793a\u200b\u5c06\u200b\u5ba2\u6237\u7aef\u200bA\u200b\u4e0a\u200b\u6307\u5411\u200b<code>&lt;port-of-A&gt;</code>\u200b\u7aef\u53e3\u200b\u7684\u200b\u672c\u5730\u200b\u8bf7\u6c42\u200b\u901a\u8fc7\u200bSSH\u200b\u8f6c\u53d1\u7ed9\u200b\u670d\u52a1\u5668\u200bB\uff0c\u200b\u7531\u200b\u670d\u52a1\u5668\u200bB\u200b\u5411\u200b\u8fdc\u7a0b\u200b\u4e3b\u673a\u200b<code>&lt;addr-of-C&gt;</code>\u200b\u7684\u200b<code>&lt;port-of-C&gt;</code>\u200b\u7aef\u53e3\u200b\u53d1\u8d77\u200b\u8bf7\u6c42\u200b\u3002\u200b\u82e5\u200b\u6307\u5b9a\u200b<code>&lt;bind-addr&gt;</code>\uff0c\u200b\u5219\u200bSSH\u200b\u4f1a\u200b\u76d1\u542c\u200b<code>&lt;bind-addr&gt;</code>\u200b\u5730\u5740\u200b\uff0c\u200b\u5e76\u200b\u53ea\u200b\u8f6c\u53d1\u200b\u76ee\u6807\u200b\u4e3a\u200b<code>&lt;bind-addr&gt;</code>\u200b\u7684\u200b\u8bf7\u6c42\u200b\u3002</p>"},{"location":"coding/configuration/ssh/#_4","title":"\u8fdc\u7a0b\u200b\u7aef\u53e3\u200b\u8f6c\u53d1","text":"<p>\u200b\u4f7f\u7528\u200b<code>-R</code>\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u6253\u5f00\u200bSSH\u200b\u7684\u200b\u8fdc\u7a0b\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u529f\u80fd\u200b\u3002<code>-R</code>\u200b\u7aef\u53e3\u200b\u5c06\u200b\u6307\u5411\u200b\u8fdc\u7a0b\u200b\u670d\u52a1\u5668\u200b\u7684\u200b\u8bf7\u6c42\u200b\u8f6c\u53d1\u7ed9\u200b\u5ba2\u6237\u7aef\u200b\uff0c\u200b\u56e0\u6b64\u200b\u79f0\u4e3a\u200b\u8fdc\u7a0b\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u3002\u200b\u542f\u52a8\u200b\u8fdc\u7a0b\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u7684\u200b\u547d\u4ee4\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>ssh -R [&lt;bind-addr&gt;]:&lt;port-of-B&gt;:&lt;addr-of-C&gt;:&lt;port-of-C&gt; &lt;user&gt;@&lt;hostname&gt;\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u547d\u4ee4\u200b\u8868\u793a\u200b\u5c06\u200b\u670d\u52a1\u5668\u200bB\u200b\u4e0a\u200b\u6307\u5411\u200b<code>&lt;port-of-B&gt;</code>\u200b\u7aef\u53e3\u200b\u7684\u200b\u8bf7\u6c42\u200b\u901a\u8fc7\u200bSSH\u200b\u8f6c\u53d1\u7ed9\u200b\u5ba2\u6237\u7aef\u200bA\uff0c\u200b\u7531\u200b\u5ba2\u6237\u7aef\u200bA\u200b\u5411\u200b\u53e6\u4e00\u53f0\u200b\u4e3b\u673a\u200b<code>&lt;addr-of-C&gt;</code>\u200b\u7684\u200b<code>&lt;port-of-C&gt;</code>\u200b\u7aef\u53e3\u200b\u53d1\u8d77\u200b\u8bf7\u6c42\u200b\u3002\u200b\u82e5\u200b\u6307\u5b9a\u200b<code>&lt;bind-addr&gt;</code>\uff0c\u200b\u5219\u200bSSH\u200b\u4f1a\u200b\u76d1\u542c\u200b<code>&lt;bind-addr&gt;</code>\u200b\u5730\u5740\u200b\uff0c\u200b\u5e76\u200b\u53ea\u200b\u8f6c\u53d1\u200b\u76ee\u6807\u200b\u4e3a\u200b<code>&lt;bind-addr&gt;</code>\u200b\u7684\u200b\u8bf7\u6c42\u200b\u3002</p>"},{"location":"coding/configuration/ssh/#_5","title":"\u52a8\u6001\u200b\u7aef\u53e3\u200b\u8f6c\u53d1","text":"<p>\u200b\u52a8\u6001\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u7684\u200b\u547d\u4ee4\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>ssh -D &lt;port-of-A&gt; &lt;user&gt;@&lt;hostname&gt;\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u547d\u4ee4\u200b\u8868\u793a\u200b\u5c06\u200b\u5ba2\u6237\u7aef\u200bA\u200b\u4e0a\u200b\u6307\u5411\u200b<code>&lt;port-of-A&gt;</code>\u200b\u7684\u200b\u8bf7\u6c42\u200b\u901a\u8fc7\u200bSSH\u200b\u8f6c\u53d1\u7ed9\u200b\u670d\u52a1\u5668\u200bB\uff0c\u200b\u7531\u200bB\u200b\u4ee3\u4e3a\u200b\u53d1\u8d77\u200b\u8bf7\u6c42\u200b\uff0c\u200b\u5e76\u200b\u5c06\u200b\u8bbf\u95ee\u200b\u7ed3\u679c\u200b\u4f20\u8f93\u200b\u56de\u200b\u672c\u5730\u200b\u3002</p> <p>\u200b\u82e5\u200b\u8981\u200b\u8f6c\u53d1\u200b\u6240\u6709\u200b\u672c\u5730\u7f51\u7edc\u200b\u8bf7\u6c42\u200b\uff0c\u200b\u8bbe\u7f6e\u200bSOCKS5\u200b\u4ee3\u7406\u200b\u5c06\u200b\u672c\u5730\u200b\u8bf7\u6c42\u200b\u91cd\u5b9a\u5411\u200b\u5230\u200b<code>&lt;port-of-A&gt;</code>\u200b\u5373\u53ef\u200b\u3002</p>"},{"location":"coding/configuration/ssh/#_6","title":"\u5bc6\u94a5\u200b\u767b\u5f55","text":"<p>SSH\u200b\u9ed8\u8ba4\u200b\u4f7f\u7528\u200b\u7528\u6237\u540d\u200b-\u200b\u5bc6\u7801\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u9a8c\u8bc1\u200b\uff0c\u200b\u6bcf\u6b21\u200b\u767b\u5f55\u200b\u9700\u8981\u200b\u8f93\u5165\u200b\u5bc6\u7801\u200b\u3002\u200b\u53ef\u4ee5\u200b\u914d\u7f6e\u200bSSH\u200b\u4f7f\u7528\u200b\u5bc6\u94a5\u200b\u5bf9\u200b\u5ba2\u6237\u7aef\u200b\u8fdb\u884c\u200b\u9a8c\u8bc1\u200b\uff0c\u200b\u6b64\u540e\u200b\u4f7f\u7528\u200b\u8be5\u200b\u5ba2\u6237\u7aef\u200b\u767b\u5f55\u200b\u5c31\u200b\u4e0d\u518d\u200b\u9700\u8981\u200b\u8f93\u5165\u200b\u5bc6\u7801\u200b\u3002\u200b\u6b65\u9aa4\u200b\u5982\u4e0b\u200b</p> <ol> <li>\u200b\u5728\u200b\u5ba2\u6237\u7aef\u200b\u6267\u884c\u200b<code>ssh-keygen</code>\u200b\u547d\u4ee4\u200b\u751f\u6210\u200b\u5bc6\u94a5\u200b\u5bf9\u200b\uff0c\u200b\u6267\u884c\u200b\u5b8c\u6210\u200b\u540e\u4f1a\u200b\u5728\u200b<code>~/.ssh</code>\u200b\u76ee\u5f55\u200b\u4e0b\u200b\u751f\u6210\u200b\u79c1\u94a5\u200b\u6587\u4ef6\u200b<code>id_rsa</code>\u200b\u4e0e\u200b\u516c\u94a5\u200b\u6587\u4ef6\u200b<code>id_rsa.pub</code>\uff1b</li> <li>\u200b\u6267\u884c\u200b<code>ssh-copy-id &lt;user&gt;@&lt;hostname&gt;</code>\u200b\u547d\u4ee4\u200b\uff0c\u200b\u5c06\u200b\u516c\u94a5\u200b\u590d\u5236\u5230\u200b\u8fdc\u7a0b\u200b\u4e3b\u673a\u200b\u7684\u200b<code>~/.ssh/authorized_key</code>\u200b\u6587\u4ef6\u200b\u4e2d\u200b\uff08\u200b\u624b\u52a8\u200b\u590d\u5236\u7c98\u8d34\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\uff09\u3002</li> </ol>"},{"location":"coding/configuration/ssh/#_7","title":"\u6587\u4ef6\u4f20\u8f93","text":"<p><code>scp</code>\u200b\u53ef\u4ee5\u200b\u501f\u52a9\u200bSSH tunnel\u200b\u5b9e\u73b0\u200b\u6587\u4ef6\u4f20\u8f93\u200b\u3002\u200b\u547d\u4ee4\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>scp SOURCE DESTINATION\n</code></pre> <p>\u200b\u8868\u793a\u200b\u5c06\u200b\u6587\u4ef6\u200bSOURCE\u200b\u590d\u5236\u5230\u200bDESTINATION\u200b\u4e0b\u200b\u3002SOURCE\u200b\u548c\u200bDESTINATION\u200b\u53ef\u4ee5\u200b\u662f\u200b\uff1a</p> <ol> <li>\u200b\u672c\u5730\u200b\u6709\u6548\u200b\u8def\u5f84\u200b</li> <li>\u200b\u683c\u5f0f\u200b\u4e3a\u200b<code>&lt;user&gt;@&lt;ip&gt;:&lt;path&gt;</code>\u200b\u7684\u200b\u8fdc\u7a0b\u200b\u8def\u5f84\u200b</li> </ol>"},{"location":"coding/configuration/update-alternatives/","title":"update-alternatives","text":"<p><code>update-alternatives</code>\u200b\u662f\u200b\u7528\u4e8e\u200b\u7ba1\u7406\u200b\u8f6f\u200b\u94fe\u63a5\u200b\u7684\u200b\u547d\u4ee4\u200b\u3002</p>"},{"location":"coding/configuration/update-alternatives/#_1","title":"\u57fa\u7840\u200b\u4f7f\u7528","text":""},{"location":"coding/configuration/update-alternatives/#_2","title":"\u521b\u5efa\u200b\u94fe\u63a5","text":"<p>\u200b\u4f7f\u7528\u200b<code>--install</code>\u200b\u53c2\u6570\u200b\u521b\u5efa\u200b\u6307\u5411\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u200b\u7684\u200b\u8f6f\u200b\u94fe\u63a5\u200b\uff0c\u200b\u547d\u4ee4\u200b\u7684\u200b\u57fa\u672c\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>update-alternatives --install &lt;destination&gt; &lt;entry-name&gt; &lt;source&gt; &lt;priority&gt;\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u547d\u4ee4\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b<code>&lt;entry-name&gt;</code>\u200b\u7684\u200b\u8f6f\u200b\u94fe\u63a5\u200b\u6761\u76ee\u200b\uff0c\u200b\u5c06\u200b<code>&lt;source&gt;</code>\u200b\u6307\u5411\u200b<code>&lt;destination&gt;</code>\u3002\u200b\u5f53\u6709\u200b\u591a\u4e2a\u200b<code>&lt;source&gt;</code>\u200b\u6307\u5411\u200b\u540c\u4e00\u4e2a\u200b<code>&lt;destination&gt;</code>\u200b\u65f6\u200b\uff0c\u200b\u9ed8\u8ba4\u200b<code>&lt;priority&gt;</code>\u200b\u66f4\u5927\u200b\u7684\u200b\u751f\u6548\u200b\u3002</p>"},{"location":"coding/configuration/update-alternatives/#_3","title":"\u67e5\u8be2\u200b\u6761\u76ee","text":"<p>\u200b\u4f7f\u7528\u200b<code>--display</code>\u200b\u53c2\u6570\u200b\u67e5\u8be2\u200b<code>&lt;entry-name&gt;</code>\u200b\u4e0b\u200b\u7684\u200b\u547d\u4ee4\u200b\u6307\u5411\u200b\uff1a</p> <pre><code>update-alternatives --display &lt;entry-name&gt;\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>--get-selections</code>\u200b\u53c2\u6570\u200b\u67e5\u8be2\u200b\u6240\u6709\u200b\u7684\u200b\u6761\u76ee\u200b\u548c\u200b\u914d\u7f6e\u200b\u72b6\u6001\u200b\u3002</p> <pre><code>update-alternatives --get-selections\n</code></pre> <p>\u200b\u8f93\u51fa\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>arptables                      auto     /usr/sbin/arptables-nft\nawk                            auto     /usr/bin/gawk\nbuiltins.7.gz                  auto     /usr/share/man/man7/bash-builtins.7.gz\nc++                            auto     /usr/bin/g++\nc89                            auto     /usr/bin/c89-gcc\nc99                            auto     /usr/bin/c99-gcc\ncc                             auto     /usr/bin/gcc\ncpp                            auto     /usr/bin/cpp\ncuda                           auto     /usr/local/cuda-11.7\ncuda-11                        auto     /usr/local/cuda-11.7\n</code></pre>"},{"location":"coding/configuration/update-alternatives/#_4","title":"\u5207\u6362\u200b\u94fe\u63a5","text":"<p>\u200b\u4f7f\u7528\u200b<code>--config</code>\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u5217\u51fa\u200b\u5e76\u200b\u9009\u62e9\u200b\u9700\u8981\u200b\u7684\u200b\u8f6f\u200b\u94fe\u63a5\u200b\uff0c\u200b\u547d\u4ee4\u200b\u7684\u200b\u57fa\u672c\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>update-alternatives --config &lt;entry-name&gt;\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u547d\u4ee4\u200b\u4f1a\u200b\u5217\u51fa\u200b<code>&lt;entry-name&gt;</code>\u200b\u4e0b\u200b\u7684\u200b\u6240\u6709\u200b\u6761\u76ee\u200b\u3002</p> <pre><code>There are 2 choices for the alternative gcc (providing /usr/bin/gcc).\n\n  Selection    Path             Priority   Status\n------------------------------------------------------------\n* 0            /usr/bin/gcc-11   2         auto mode\n  1            /usr/bin/gcc-11   2         manual mode\n  2            /usr/bin/gcc-12   1         manual mode\n\nPress &lt;enter&gt; to keep the current choice[*], or type selection number:\n</code></pre> <p>\u200b\u6216\u8005\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>--set</code>\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\u94fe\u63a5\u200b\u7684\u200b\u76ee\u6807\u200b\uff0c\u200b\u683c\u5f0f\u200b\u4e3a\u200b\uff1a</p> <pre><code>update-alternatives --set &lt;entry-name&gt; &lt;source&gt;\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u547d\u4ee4\u200b\u4f7f\u7528\u200b<code>&lt;source&gt;</code>\u200b\u4f5c\u4e3a\u200b<code>&lt;entry-name&gt;</code>\u200b\u6761\u200b\u76ee\u7684\u200b\u6307\u5411\u200b\u3002</p> <p>\u200b\u4f7f\u7528\u200b<code>--auto</code>\u200b\u53c2\u6570\u200b\u6062\u590d\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u76ee\u6807\u200b\u6307\u5411\u200b\u3002</p> <pre><code>update-alternatives --auto &lt;entry-name&gt;\n</code></pre>"},{"location":"coding/configuration/update-alternatives/#_5","title":"\u5220\u9664\u200b\u94fe\u63a5","text":"<p>\u200b\u4f7f\u7528\u200b<code>--remove</code>\u200b\u53c2\u6570\u200b\u5220\u9664\u200b\u4e00\u4e2a\u200b\u6761\u76ee\u200b\u4e0b\u200b\u7684\u200b\u94fe\u63a5\u200b\u3002</p> <pre><code>update-alternatives --remove &lt;entry-name&gt; &lt;source&gt;\n</code></pre>"},{"location":"coding/configuration/vim/","title":"Vim","text":"<p>Vim\u200b\u662f\u200b\u57fa\u4e8e\u200b\u547d\u4ee4\u884c\u200b\u7684\u200b\u6587\u672c\u7f16\u8f91\u200b\u5668\u200b\u3002</p>"},{"location":"coding/configuration/vim/#_1","title":"\u547d\u4ee4\u884c","text":""},{"location":"coding/configuration/vim/#_2","title":"\u4ea4\u4e92","text":"<p>Vim\u200b\u7684\u200b\u4ea4\u4e92\u200b\u547d\u4ee4\u200b\u5982\u4e0b\u200b\u8868\u6240\u5217\u200b\uff0c\u200b\u4ea4\u4e92\u200b\u547d\u4ee4\u200b\u4ee5\u200b<code>:</code>\u200b\u5f00\u5934\u200b\u3002</p> \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b <code>v</code> \u200b\u9009\u62e9\u200b\u5b57\u7b26\u200b <code>V</code> \u200b\u9009\u62e9\u200b\u884c\u200b <code>Ctrl+V</code> \u200b\u9009\u62e9\u200b\u5757\u200b <code>i</code> \u200b\u5207\u6362\u200b\u5230\u200b\u63d2\u5165\u200b\u6a21\u5f0f\u200b \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b <code>:q</code> \u200b\u9000\u51fa\u200bVim <code>:q!</code> \u200b\u9000\u51fa\u200b\u800c\u200b\u4e0d\u200b\u4fdd\u5b58\u200b <code>:w</code> \u200b\u4fdd\u5b58\u200b <code>&lt;Esc&gt;</code> \u200b\u9000\u51fa\u200b\u63d2\u5165\u200b\u6a21\u5f0f\u200b <code>:h</code> \u200b\u67e5\u8be2\u200b\u5e2e\u52a9\u200b <code>:version</code> \u200b\u67e5\u8be2\u200b\u7248\u672c\u200b\u548c\u200b\u529f\u80fd\u200b <code>:history</code> \u200b\u67e5\u8be2\u200b\u5386\u53f2\u200b\u547d\u4ee4\u200b <code>:s</code> \u200b\u6839\u636e\u200b\u6b63\u5219\u8868\u8fbe\u5f0f\u200b\u66ff\u6362\u200b\u5b57\u7b26\u4e32\u200b <code>:edit</code> \u200b\u7f16\u8f91\u200b\u6587\u4ef6\u200b\uff08\u200b\u652f\u6301\u200bglob pattern\u200b\u5339\u914d\u200b\uff09 <code>:find</code> \u200b\u5728\u200b<code>path</code>\u200b\u4e2d\u200b\u67e5\u627e\u6587\u4ef6\u200b <code>:set</code> \u200b\u67e5\u8be2\u200b/\u200b\u8bbe\u7f6e\u200b\u73af\u5883\u53d8\u91cf\u200b <code>:Rg</code> \u200b\u5728\u200b\u6587\u4ef6\u200b\u4e2d\u200b\u67e5\u627e"},{"location":"coding/configuration/vim/#_3","title":"\u64cd\u4f5c","text":"<p>Vim\u200b\u7684\u200b\u8bed\u53e5\u200b\u7531\u200b\u64cd\u4f5c\u7b26\u200b\u548c\u200b\u52a8\u4f5c\u200b\u7ec4\u6210\u200b\uff0c\u200b\u5728\u200bVim\u200b\u754c\u9762\u200b\u4e2d\u200b\uff0c\u200b\u6309\u200b<code>:</code>\u200b\u8f93\u5165\u200b\u8bed\u53e5\u200b\u3002</p> <p>Vim\u200b\u7684\u200b\u52a8\u4f5c\u200b\u5982\u4e0b\u200b\u8868\u200b\u6240\u793a\u200b</p> \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b <code>h</code> \u200b\u5411\u200b\u5de6\u200b\u79fb\u52a8\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b <code>j</code> \u200b\u5411\u4e0b\u200b\u79fb\u52a8\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b <code>k</code> \u200b\u5411\u4e0a\u200b\u79fb\u52a8\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b <code>l</code> \u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b <code>b</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0a\u200b\u4e00\u8bcd\u200b\u5f00\u5934\u200b <code>w</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0b\u200b\u4e00\u8bcd\u200b\u5f00\u5934\u200b <code>ge</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0a\u200b\u4e00\u8bcd\u200b\u7ed3\u5c3e\u200b <code>e</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0b\u200b\u4e00\u8bcd\u200b\u7ed3\u5c3e\u200b <code>{</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0a\u200b\u4e00\u6bb5\u200b <code>}</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0b\u200b\u4e00\u6bb5\u200b <code>0</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u884c\u9996\u200b <code>$</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u884c\u5c3e\u200b <code>^</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u884c\u9996\u200b\u975e\u200b\u7a7a\u5b57\u7b26\u200b <code>g_</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u884c\u5c3e\u200b\u975e\u200b\u7a7a\u5b57\u7b26\u200b <code>n|</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u5f53\u524d\u200b\u884c\u200b\u7684\u200b\u7b2c\u200b<code>n</code>\u200b\u5217\u200b <p>Vim\u200b\u7684\u200b\u64cd\u4f5c\u7b26\u200b\u5982\u4e0b\u200b\u8868\u200b\u6240\u793a\u200b</p> \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b <code>y</code> \u200b\u590d\u5236\u200b <code>d</code> \u200b\u526a\u5207\u200b <code>c</code> \u200b\u526a\u5207\u200b\u5e76\u200b\u7f16\u8f91\u200b <code>p</code> \u200b\u7c98\u8d34\u200b <p>\u200b\u884c\u200b\u64cd\u4f5c\u200b\u6a21\u5f0f\u200b</p> <p>\u200b\u8fde\u7eed\u200b\u8f93\u5165\u200b\u4e24\u4e2a\u200b\u76f8\u540c\u200b\u64cd\u4f5c\u7b26\u200b\u5bf9\u200b\u5f53\u524d\u200b\u884c\u200b\u6267\u884c\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u5982\u200b<code>yy</code>\u200b\u590d\u5236\u200b\u5f53\u524d\u200b\u884c\u200b\u3002</p> <p>\u200b\u5728\u200b\u91cd\u590d\u200b\u7684\u200b\u64cd\u4f5c\u7b26\u200b\u524d\u9762\u200b\u52a0\u200b\u6570\u5b57\u200b<code>n</code>\uff0c\u200b\u8868\u793a\u200b\u4ece\u200b\u5f53\u524d\u200b\u884c\u200b\u5f00\u59cb\u200b\u5230\u200b\u7b2c\u200b<code>n</code>\u200b\u884c\u200b\u6267\u884c\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u5982\u200b<code>3yy</code>\u200b\u4ece\u200b\u5f53\u524d\u200b\u884c\u200b\u5f00\u59cb\u200b\u590d\u5236\u200b3\u200b\u884c\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/","title":"\u6df1\u5ea6\u200b\u5b66\u4e60","text":""},{"location":"coding/dl-from-scratch/#_2","title":"\u795e\u7ecf\u200b\u7f51\u7edc\u5c42","text":"<ul> <li>\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b</li> <li>\u200b\u635f\u5931\u200b\u51fd\u6570\u200b</li> <li>\u200b\u5f52\u4e00\u5316\u200b</li> <li>\u200b\u591a\u5c42\u200b\u611f\u77e5\u673a\u200b</li> <li>\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b</li> </ul>"},{"location":"coding/dl-from-scratch/#llm-from-scratch","title":"LLM from scratch","text":"<ol> <li>\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b</li> <li>\u200b\u7f16\u7801\u5668\u200b\u4e0e\u200b\u89e3\u7801\u5668\u200b<ul> <li>\u200b\u9a8c\u8bc1\u200btransformer\u200b\u5b9e\u73b0\u200b</li> </ul> </li> <li>\u200b\u7edd\u5bf9\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b<ul> <li>\u200b\u9a8c\u8bc1\u200bbert\u200b\u5b9e\u73b0\u200b</li> </ul> </li> <li>RoPE</li> <li>transformer\u200b\u53d8\u79cd\u200b<ul> <li>\u200b\u9a8c\u8bc1\u200bLlama-2\u200b\u5b9e\u73b0\u200b</li> </ul> </li> <li>\u200b\u5927\u200b\u6a21\u578b\u200b\u63a8\u7406\u200b</li> </ol>"},{"location":"coding/dl-from-scratch/activation/","title":"\u6fc0\u6d3b\u200b\u51fd\u6570","text":"<p>\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\u662f\u200b\u795e\u7ecf\u7f51\u7edc\u200b\u4e2d\u200b\u7528\u4e8e\u200b\u5f15\u5165\u200b\u975e\u7ebf\u6027\u200b\u56e0\u7d20\u200b\u7684\u200b\u51fd\u6570\u200b\u3002\u200b\u5e38\u7528\u200b\u7684\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\u6709\u200b\u5982\u4e0b\u200b\u51e0\u79cd\u200b\uff1a</p>"},{"location":"coding/dl-from-scratch/activation/#sigmoid","title":"Sigmoid \u200b\u51fd\u6570","text":"<p>Sigmoid\u200b\u51fd\u6570\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5c06\u200b\u8f93\u5165\u200b\u503c\u200b\u538b\u7f29\u200b\u5230\u200b0\u200b\u5230\u200b1\u200b\u4e4b\u95f4\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5e38\u7528\u200b\\(\\sigma\\)\u200b\u8868\u793a\u200b\u3002Sigmoid\u200b\u51fd\u6570\u200b\u5e38\u7528\u200b\u4e8e\u200b\u4e8c\u200b\u5206\u7c7b\u200b\u95ee\u9898\u200b\u4e2d\u200b\uff0c\u200b\u5c06\u200b\u8f93\u51fa\u200b\u503c\u200b\u6620\u5c04\u200b\u5230\u200b\u6982\u7387\u200b\u7a7a\u95f4\u200b\u3002</p> \\[ \\begin{aligned} \\sigma(x) &amp;= \\frac{1}{1 + e^{-x}} \\\\ \\sigma'(x) &amp;= \\sigma(x)(1 - \\sigma(x)) \\end{aligned} \\] <pre><code>def sigmoid(x: torch.Tensor) -&gt; torch.Tensor:\n    return 1 / (1 + torch.exp(-x))\n\nclass Sigmoid(torch.nn.Module):\n    def __init__(self):\n        super(Sigmoid, self).__init__()\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        return sigmoid(x)\n</code></pre>"},{"location":"coding/dl-from-scratch/activation/#softmax","title":"Softmax \u200b\u51fd\u6570","text":"<p>Softmax\u200b\u51fd\u6570\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5c06\u200b\u8f93\u5165\u200b\u503c\u200b\u538b\u7f29\u200b\u5230\u200b0\u200b\u5230\u200b1\u200b\u4e4b\u95f4\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5e38\u7528\u200b\u4e8e\u200b\u591a\u200b\u5206\u7c7b\u200b\u95ee\u9898\u200b\u4e2d\u200b\uff0c\u200b\u5c06\u200b\u591a\u4e2a\u200b\u5206\u7c7b\u200b\u4e0a\u200b\u8f93\u51fa\u200b\u503c\u200b\u6620\u5c04\u200b\u5230\u200b\u6982\u7387\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u6240\u6709\u200b\u7c7b\u522b\u200b\u7684\u200b\u6982\u7387\u200b\u4e4b\u200b\u548c\u200b\u4e3a\u200b1\u3002</p> \\[ \\begin{aligned} \\text{softmax}(\\bsz)_i &amp;= \\frac{e^{z_i}}{\\sum_{j=1}^n e^{z_j}} \\\\ \\text{softmax}'(\\bsz)_i &amp;= \\text{softmax}(\\bsz)_i(1 - \\text{softmax}(\\bsz)_i) \\end{aligned} \\] <pre><code>def softmax(x: torch.Tensor, dim: int = -1) -&gt; torch.Tensor:\n    return torch.exp(x) / torch.exp(x).sum(dim=dim, keepdim=True)\n\nclass Softmax(torch.nn.Module):\n    def __init__(self, dim: int = -1):\n        super(Softmax, self).__init__()\n        self.dim = dim\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        return softmax(x, dim=self.dim)\n</code></pre> <p>\u200b\u76f8\u5bf9\u200b\u5e94\u200b\u7684\u200b\uff0csoftmin\u200b\u51fd\u6570\u200b\u662f\u200bsoftmax\u200b\u51fd\u6570\u200b\u7684\u200b\u5bf9\u79f0\u200b\u51fd\u6570\u200b\uff0c\u200b\u7528\u4e8e\u200b\u6700\u5c0f\u5316\u200b\u8f93\u51fa\u200b\u503c\u200b\uff0c\\(\\text{softmin}(\\bsz) = \\text{softmax}(-\\bsz)\\)\u3002</p>"},{"location":"coding/dl-from-scratch/attention/","title":"\u6ce8\u610f\u529b\u200b\u673a\u5236","text":""},{"location":"coding/dl-from-scratch/attention/#_2","title":"\u5e8f\u5217\u200b\u5efa\u6a21","text":"<p>\u200b\u8bed\u8a00\u200b\u6a21\u578b\u200b\u662f\u200b\u4e00\u7c7b\u200b\u5e8f\u5217\u200b\u6a21\u578b\u200b\uff0c\u200b\u5176\u200b\u76ee\u6807\u200b\u662f\u200b\u5bf9\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u5e8f\u5217\u200b\u5efa\u6a21\u200b\uff0c\u200b\u5373\u200b\u5bf9\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u4f4d\u7f6e\u200b\u9884\u6d4b\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u6982\u7387\u200b\u3002\u200b\u6587\u672c\u200b\u9996\u5148\u200b\u901a\u8fc7\u200b\u5206\u8bcd\u200b\uff0c\u200b\u83b7\u5f97\u200b\u5bf9\u5e94\u200b\u7684\u200b\u79bb\u6563\u200b\u7b26\u53f7\u200b\uff08token\uff09\uff0c\u200b\u7136\u540e\u200b\u7ecf\u8fc7\u200b\u5d4c\u5165\u200b\u5c42\u200b\uff08embedding layer\uff09\u200b\u5c06\u200b\u7b26\u53f7\u200b\u6620\u5c04\u200b\u5230\u200b\\(D\\)\u200b\u7ef4\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5e8f\u5217\u200b\u5efa\u6a21\u200b\u7684\u200b\u8f93\u5165\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(N\\times L\\times D\\)\u200b\u7684\u200b\u5f20\u91cf\u200b\uff0c\u200b\u5176\u4e2d\u200b\\(N\\)\u200b\u662f\u200bbatch size\uff0c\\(L\\)\u200b\u662f\u200b\u5e8f\u5217\u200b\u957f\u5ea6\u200b\uff0c\\(D\\)\u200b\u662f\u200b\u5d4c\u5165\u200b\u7ef4\u5ea6\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/attention/#transformer","title":"Transformer","text":"<p>Transformer\u200b\u7531\u200bVaswani\u200b\u7b49\u200b\u4eba\u200b\u5728\u200b2017\u200b\u5e74\u200b\u63d0\u51fa\u200b\uff0c\u200b\u5176\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\uff0c\u200b\u5373\u200b\u901a\u8fc7\u200b\u67e5\u8be2\u200b\u3001\u200b\u952e\u200b\u7684\u200b\u5339\u914d\u200b\u5f97\u200b\u5206\u4e3a\u200b\u503c\u200b\u5206\u914d\u200b\u6743\u91cd\u200b\uff0c\u200b\u4ece\u800c\u200b\u5b9e\u73b0\u200b\u5bf9\u200b\u5e8f\u5217\u200b\u7684\u200b\u5efa\u6a21\u200b\u3002\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u7684\u200b\u6570\u5b66\u200b\u8868\u8fbe\u200b\u4e3a\u200b\uff1a</p> \\[ \\text{Attention}(Q, K, V) = \\text{softmax}\\left(\\frac{\\bsQ\\bsK^\\top}{\\sqrt{d_k}}\\right)\\bsV \\] <p>\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u6307\u200b\u7684\u200b\u662f\u200b\u67e5\u8be2\u200b\u3001\u200b\u952e\u548c\u503c\u200b\u90fd\u200b\u6765\u81ea\u200b\u540c\u4e00\u4e2a\u200b\u5e8f\u5217\u200b\u3002\u200b\u5728\u200b\u5b9e\u73b0\u200b\u4e2d\u200b\uff0c\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\\(\\bsx\\)\u200b\u5206\u522b\u200b\u7ecf\u8fc7\u200b\u4e09\u7ec4\u200b\u7ebf\u6027\u53d8\u6362\u200b\\(\\bsW^Q, \\bsW^K, \\bsW^V\\)\u200b\u5f97\u5230\u200b\u67e5\u8be2\u200b\\(\\bsQ\\)\u3001\u200b\u952e\u200b\\(\\bsK\\)\u200b\u548c\u200b\u503c\u200b\\(\\bsV\\)\uff0c\u200b\u7136\u540e\u200b\u8ba1\u7b97\u200b\u6ce8\u610f\u529b\u200b\u5f97\u5206\u200b\uff0c\u200b\u6700\u540e\u200b\u901a\u8fc7\u200b\u503c\u200b\u7684\u200b\u52a0\u6743\u200b\u6c42\u548c\u200b\u5f97\u5230\u200b\u8f93\u51fa\u200b\u3002\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u6307\u200b\u67e5\u8be2\u200b\u548c\u200b</p> <p>\u200b\u4e3a\u4ec0\u4e48\u200b\u9700\u8981\u200b\u7ebf\u6027\u53d8\u6362\u200b</p> <p>\u200b\u7ebf\u6027\u53d8\u6362\u200b\u7684\u200b\u76ee\u7684\u200b\u662f\u200b\u5c06\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u6620\u5c04\u200b\u5230\u200b\u4e0d\u540c\u200b\u7684\u200b\u8bed\u4e49\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u53e6\u5916\u200b\u4e00\u65b9\u9762\u200b\u4e5f\u200b\u4f7f\u5f97\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u4e0d\u518d\u200b\u53d7\u5230\u200b\u77e9\u9635\u200b\u5bf9\u79f0\u200b\u7684\u200b\u7ea6\u675f\u200b\uff08A\u200b\u5bf9\u200bB\u200b\u7684\u200b\u6ce8\u610f\u529b\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u7b49\u4e8e\u200bB\u200b\u5bf9\u200bA\u200b\u7684\u200b\u6ce8\u610f\u529b\u200b\uff09\u3002</p> <p>\u200b\u4e3a\u4ec0\u4e48\u200b\u8981\u200b\u9664\u4ee5\u200b\\(\\sqrt{d_k}\\)</p> <p>\u200b\u5728\u200b\\(\\bsQ\\bsK^\\top\\)\u200b\u7684\u200b\u8ba1\u7b97\u200b\u4e2d\u200b\uff0c\u200b\u7ed3\u679c\u200b\u7684\u200b\u65b9\u5dee\u200b\u4f1a\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u4e3a\u4e86\u200b\u4f7f\u200b\u7ed3\u679c\u200b\u7684\u200b\u65b9\u5dee\u200b\u4fdd\u6301\u200b\u4e0d\u53d8\u200b\uff0c\u200b\u9700\u8981\u200b\u9664\u4ee5\u200b\\(\\sqrt{d_k}\\)\uff0c\u200b\u5176\u4e2d\u200b\\(d_k\\)\u200b\u662f\u200b\u952e\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u3002\u200b\u5426\u5219\u200b\uff0c\u200b\u5f53\u200b\\(d_k\\)\u200b\u8f83\u5927\u200b\u65f6\u200b\uff0c\\(\\bsQ\\bsK^\\top\\)\u200b\u4e2d\u200b\u5143\u7d20\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5dee\u5f02\u200b\u4f1a\u200b\u53d8\u5f97\u200b\u5f88\u5927\u200b\uff0c\u200b\u5bfc\u81f4\u200bsoftmax\u200b\u51fd\u6570\u200b\u7684\u200b\u68af\u5ea6\u200b\u63a5\u8fd1\u200b\u4e8e\u200b0\u3002</p> <p>softmax\u200b\u51fd\u6570\u200b</p> <p>softmax\u200b\u51fd\u6570\u200b\u7684\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\uff1a</p> \\[ \\text{softmax}(\\bsz)_i = \\frac{e^{z_i}}{\\sum_{j=1}^n e^{z_j}} \\] <p>\u200b\u5373\u200b\u5bf9\u200b\u8f93\u5165\u200b\u5411\u91cf\u200b\\(\\bsz\\)\u200b\u4e2d\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u6307\u6570\u200b\u7684\u200b\u5f52\u4e00\u5316\u200b\u3002\u200b\u5176\u4e2d\u200b\\(\\bsz\\)\u200b\u662f\u200b\u8f93\u5165\u200b\u5411\u91cf\u200b\uff0c\\(n\\)\u200b\u662f\u200b\u5411\u91cf\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u3002</p> <p>\u200b\u4e3a\u4e86\u200b\u4ece\u200b\u591a\u4e2a\u200b\u7ef4\u5ea6\u200b\u5b66\u4e60\u200b\u5230\u200b\u5bf9\u200b\u540c\u4e00\u4e2a\u200b\u4f4d\u7f6e\u200b\u4e0d\u540c\u200b\u7279\u5f81\u200b\u7684\u200b\u8868\u5f81\u200b\uff0ctransformer\u200b\u5f15\u5165\u200b\u4e86\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u3002\u200b\u5176\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u5c06\u200b\u8f93\u5165\u200b\u7684\u200b\\(D\\)\u200b\u7ef4\u200b\u5411\u91cf\u200b\u62c6\u5206\u200b\u5230\u200b\\(h\\)\u200b\u4e2a\u5934\u200b\uff0c\u200b\u7136\u540e\u200b\u5206\u522b\u200b\u8ba1\u7b97\u200b\u6ce8\u610f\u529b\u200b\uff0c\u200b\u6700\u540e\u200b\u5c06\u200b\\(h\\)\u200b\u4e2a\u5934\u200b\u7684\u200b\u8f93\u51fa\u200b\u62fc\u63a5\u200b\u5728\u200b\u4e00\u8d77\u200b\uff0c\u200b\u518d\u200b\u7ecf\u8fc7\u200b\u4e00\u4e2a\u200b\u7ebf\u6027\u53d8\u6362\u200b\u5f97\u5230\u200b\u6700\u7ec8\u200b\u8f93\u51fa\u200b\u3002\u200b\u5728\u200b\u5b9e\u73b0\u200b\u4e2d\u200b\uff0c\u200b\u4e00\u822c\u200b\u4f7f\u7528\u200b\u7ef4\u5ea6\u200b\u53d8\u6362\u200b\u6765\u200b\u5b9e\u73b0\u200b\u8f93\u5165\u200b\u5411\u91cf\u200b\u7684\u200b\u62c6\u5206\u200b\u3002</p> <p>\u200b\u5728\u200b\u90e8\u5206\u200b\u4efb\u52a1\u200b\u4e2d\u200b\uff0c\u200b\u9700\u8981\u200b\u63a7\u5236\u200b\u5e8f\u5217\u200b\u67d0\u4e2a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u53ea\u80fd\u200b\u4f9d\u8d56\u4e8e\u200b\u8be5\u200b\u4f4d\u7f6e\u200b\u4e4b\u524d\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u800c\u200b\u4e0d\u80fd\u200b\u4f9d\u8d56\u4e8e\u200b\u4e4b\u540e\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\u200b\u6b64\u65f6\u200b\u9700\u8981\u200b\u4f7f\u7528\u200bmask\u200b\u673a\u5236\u200b\uff0c\u200b\u5373\u200b\u5728\u200b\u8ba1\u7b97\u200b\u6ce8\u610f\u529b\u200b\u5f97\u5206\u200b\u65f6\u200b\uff0c\u200b\u5c06\u200b\u4e0d\u200b\u5408\u6cd5\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5f97\u5206\u200b\u8bbe\u7f6e\u200b\u4e3a\u200b\u8d1f\u200b\u65e0\u7a77\u200b\uff0c\u200b\u4ece\u800c\u200b\u5728\u200bsoftmax\u200b\u51fd\u6570\u200b\u4e2d\u200b\u5f97\u5230\u200b0\u200b\u7684\u200b\u6982\u7387\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/attention/#_3","title":"\u4ee3\u7801\u200b\u5b9e\u73b0","text":"<p>einsum</p> <p><code>einsum</code>\u200b\u662f\u200bEinstein Summation Notation\u200b\u7684\u200b\u7f29\u5199\u200b\uff0c\u200b\u662f\u200b\u4e00\u79cd\u200b\u7528\u4e8e\u200b\u5f20\u91cf\u200b\u8fd0\u7b97\u200b\u7684\u200b\u8bb0\u53f7\u200b\u3002\u200b\u5728\u200bPyTorch\u200b\u4e2d\u200b\uff0c<code>einsum</code>\u200b\u51fd\u6570\u200b\u7684\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\uff1a</p> <pre><code>torch.einsum(equation, *operands)\n</code></pre> <p>\u200b\u5176\u4e2d\u200b<code>equation</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u63cf\u8ff0\u200b\u4e86\u200b\u5f20\u91cf\u200b\u7684\u200b\u8fd0\u7b97\u200b\u65b9\u5f0f\u200b\uff0c<code>operands</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6216\u200b\u591a\u4e2a\u200b\u5f20\u91cf\u200b\u3002\u200b\u4f8b\u5982\u200b\uff0c<code>torch.einsum(\"ij,jk-&gt;ik\", A, B)</code>\u200b\u8868\u793a\u200b\u8ba1\u7b97\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u548c\u200b\\(B\\)\u200b\u7684\u200b\u4e58\u79ef\u200b\u3002\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5176\u200b\u7406\u89e3\u200b\u4e3a\u200b\u4e58\u79ef\u200b-\u200b\u6c42\u548c\u200b\u7684\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u5373\u5c06\u200b<code>equation</code>\u200b\u4e2d\u200b\u7684\u200b\u5b57\u6bcd\u200b\u5bf9\u5e94\u200b\u5230\u200b<code>operands</code>\u200b\u4e2d\u200b\u7684\u200b\u5f20\u91cf\u200b\uff0c\u200b\u7136\u540e\u200b\u5bf9\u200b\u8fd9\u4e9b\u200b\u5f20\u91cf\u200b\u8fdb\u884c\u200b\u4e58\u79ef\u200b\u548c\u200b\u6c42\u548c\u200b\u3002\u200b\u5ffd\u7565\u200bbatch size\uff0c\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u8ba1\u7b97\u200b\\(\\bsQ\\bsK^\\top\\)\uff0c\u200b\u5176\u4e2d\u200b\\(\\bsQ\\)\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u662f\u200b\\((L, D)\\)\uff0c\\(\\bsK\\)\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u662f\u200b\\((L, D)\\)\uff0c\u200b\u8f93\u51fa\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u662f\u200b\\((L, L)\\)\uff0c\u200b\u56e0\u6b64\u200b<code>equation</code>\u200b\u4e3a\u200b<code>\"id,jd-&gt;ij\"</code>\u3002</p> <p>\u200b\u4e0b\u9762\u200b\u6211\u4eec\u200b\u901a\u8fc7\u200bPyTorch\u200b\u5b9e\u73b0\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u3002\u200b\u5728\u200b\u5b9e\u73b0\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u524d\u200b\uff0c\u200b\u9700\u8981\u200b\u5148\u200b\u5b9e\u73b0\u200b\u7ebf\u6027\u53d8\u6362\u200b\u548c\u200bsoftmax\u200b\u51fd\u6570\u200b\u3002\u200b\u4e4b\u540e\u200b\u5b9e\u73b0\u200b\u7528\u4e8e\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\u7684\u200b<code>MultiHeadSelfAttention</code>\u200b\u7c7b\u200b\uff0c\u200b\u5176\u200b\u8f93\u5165\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(N\\times L\\times D\\)\u200b\u7684\u200b\u5f20\u91cf\u200b\uff0c\u200b\u8f93\u51fa\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(N\\times L\\times D\\)\u200b\u7684\u200b\u5f20\u91cf\u200b\u3002</p> <pre><code>class MultiHeadAttention(torch.nn.Module):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadAttention, self).__init__()\n        self.d_model = d_model\n        self.num_heads = num_heads\n        if d_model % num_heads != 0:\n            raise ValueError(\"d_model must be divisible by num_heads\")\n\n        self.d_k = d_model // num_heads\n        self.sqrt_d_k = self.d_k ** 0.5\n\n        self.W_Q = torch.nn.Linear(d_model, d_model)\n        self.W_K = torch.nn.Linear(d_model, d_model)\n        self.W_V = torch.nn.Linear(d_model, d_model)\n        self.W_O = torch.nn.Linear(d_model, d_model)\n\n    def forward(\n        self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,\n        mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        N, L_Q, D = x_q.size()\n        _, L_KV, _ = x_k.size()\n        Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)\n        K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)\n        V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)\n\n        score = torch.einsum(\"nihd,njhd-&gt;nijh\", Q, K) / self.sqrt_d_k\n        score = torch.nn.functional.softmax(score, dim=2)\n        value = torch.einsum(\"nijh,njhd-&gt;nihd\", score, V).reshape(N, L_Q, self.d_model)\n        return self.W_O(value)\n\nclass MultiHeadSelfAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadSelfAttention, self).__init__(d_model, num_heads)\n\n    def forward(self, x: torch.Tensor, mask: torch.Tensor | None = None) -&gt; torch.Tensor:\n        return super().forward(x, x, x, mask)\n\nclass MultiHeadCrossAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadCrossAttention, self).__init__(d_model, num_heads)\n\n    def forward(\n        self, x_q: torch.Tensor, x_kv: torch.Tensor,\n        mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        return super().forward(x_q, x_kv, x_kv, mask)\n</code></pre> <p>\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u4e2d\u200b\u7684\u200bmask</p> <p>\u200b\u5728\u200b\u5b9e\u73b0\u200b\u4e2d\u200b\uff0c\u200b\u6709\u65f6\u200b\u9700\u8981\u200b\u5bf9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u7684\u200b\u5f97\u5206\u200b\u8fdb\u884c\u200bmask\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u4ee5\u200b\u63a7\u5236\u200b\u6a21\u578b\u200b\u7684\u200b\u884c\u4e3a\u200b\u3002mask\u200b\u901a\u5e38\u200b\u5206\u4e3a\u200b\u4e09\u79cd\u200b\uff0c\u200b\u5373\u200b</p> <ol> <li>padding mask\uff1a\u200b\u7ef4\u5ea6\u200b<code>[N, L]</code>\uff0c\u200b\u7528\u4e8e\u200b\u63a9\u76d6\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200bpadding\u200b\u5143\u7d20\u200b\uff0c\u200b\u4f7f\u200b\u5176\u200b\u4e0d\u200b\u53c2\u4e0e\u200b\u6ce8\u610f\u529b\u200b\u8ba1\u7b97\u200b\u3002</li> <li>attention mask\uff1a\u200b\u7ef4\u5ea6\u200b<code>[L, L]</code>\uff0c\u200b\u7528\u4e8e\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u4e2d\u200b\uff0c\u200b\u63a7\u5236\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u4f9d\u8d56\u200b\u5173\u7cfb\u200b\u3002\u200b\u901a\u5e38\u200b\u7528\u200bcausal mask\u200b\u6216\u200blook-ahead mask\u200b\u89c4\u5b9a\u200b\u5e8f\u5217\u200b\u67d0\u4e2a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u53ea\u80fd\u200b\u4f9d\u8d56\u4e8e\u200b\u8be5\u200b\u4f4d\u7f6e\u200b\u4e4b\u524d\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u800c\u200b\u4e0d\u80fd\u200b\u4f9d\u8d56\u4e8e\u200b\u4e4b\u540e\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</li> <li>\u200b\u5728\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u4e2d\u200b\u7684\u200bmemory mask\uff1a\u200b\u7ef4\u5ea6\u200b<code>[L_Q, L_KV]</code>\uff0c\u200b\u7528\u4e8e\u200b\u63a7\u5236\u200b\u67e5\u8be2\u200b\u548c\u952e\u200b\u4e4b\u95f4\u200b\u7684\u200b\u4f9d\u8d56\u200b\u5173\u7cfb\u200b\u3002</li> </ol> <p>\u200b\u5728\u200b\u5b9e\u73b0\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u5c06\u200bmask\u200b\u4e2d\u4e3a\u200b0\u200b\u7684\u200b\u5143\u7d20\u200b\u8bbe\u7f6e\u200b\u4e3a\u200b\u8d1f\u200b\u65e0\u7a77\u200b\uff0c\u200b\u4ece\u800c\u200b\u5728\u200bsoftmax\u200b\u51fd\u6570\u200b\u4e2d\u200b\u5f97\u5230\u200b0\u200b\u7684\u200b\u6982\u7387\u200b\u3002</p> <p>\u200b\u5b8c\u6574\u200b\u7684\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u5b9e\u73b0\u200b\u5982\u4e0b\u200b</p> <pre><code>class MultiHeadAttention(torch.nn.Module):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadAttention, self).__init__()\n        self.d_model = d_model\n        self.num_heads = num_heads\n        if d_model % num_heads != 0:\n            raise ValueError(\"d_model must be divisible by num_heads\")\n\n        self.d_k = d_model // num_heads\n        self.sqrt_d_k = self.d_k ** 0.5\n\n        self.W_Q = torch.nn.Linear(d_model, d_model)\n        self.W_K = torch.nn.Linear(d_model, d_model)\n        self.W_V = torch.nn.Linear(d_model, d_model)\n        self.W_O = torch.nn.Linear(d_model, d_model)\n\n    def forward(\n        self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        # x_q: (N, L_Q, D), x_k: (N, L_KV, D), x_v: (N, L_KV, D)\n        # padding_mask: (N, L_KV), attention_mask: (L_Q, L_KV)\n        N, L_Q, D = x_q.size()\n        _, L_KV, _ = x_k.size()\n\n        # Linear transformation -&gt; Split heads\n        Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)\n        K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)\n        V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)\n\n        # Compute attention score\n        score = torch.einsum('nihd,njhd-&gt;nijh', Q, K) / self.sqrt_d_k\n\n        # Apply attention mask\n        if attention_mask is not None:\n            score = score.masked_fill(\n                attention_mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')\n            )\n\n        # Apply padding mask\n        if padding_mask is not None:\n            score = score.masked_fill(\n                padding_mask.reshape(N, 1, L_KV, 1) == 0, float('-inf')\n            )\n\n        # Softmax -&gt; Weighted sum -&gt; Merge heads -&gt; Output transformation\n        score = torch.nn.functional.softmax(score, dim=2)\n        value = torch.einsum(\n            'nijh,njhd-&gt;nihd', score, V\n        ).reshape(N, L_Q, self.d_model)\n        return self.W_O(value)\n\n\nclass MultiHeadSelfAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadSelfAttention, self).__init__(d_model, num_heads)\n\n    def forward(\n        self, x: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        # Self attention is applied to the same input\n        return super().forward(x, x, x, padding_mask, attention_mask)\n\n\nclass MultiHeadCrossAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadCrossAttention, self).__init__(d_model, num_heads)\n\n    def forward(\n        self, x_q: torch.Tensor, x_kv: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        # Cross attention is applying query on another kv sequence\n        return super().forward(x_q, x_kv, x_kv, padding_mask, attention_mask)\n</code></pre>"},{"location":"coding/dl-from-scratch/bert/","title":"\u5b9e\u73b0\u200bBERT","text":"In\u00a0[1]: Copied! <pre>import transformers\nimport torch\n\nfrom custom_transformer import Encoder\n</pre> import transformers import torch  from custom_transformer import Encoder <p>huggingface\u200b\u4e2d\u200b\u7684\u200b\u6a21\u578b\u200b\u901a\u5e38\u200b\u5bf9\u5e94\u200b\u4e00\u7ec4\u200bconfig\uff0c\u200b\u5b58\u50a8\u200b\u6a21\u578b\u200b\u7684\u200b\u8d85\u200b\u53c2\u6570\u200b\u3002</p> In\u00a0[2]: Copied! <pre>bert_config = transformers.BertConfig.from_pretrained('bert-base-uncased', local_files_only=True)\n# Remove the pooler layer\nbert_config\n</pre> bert_config = transformers.BertConfig.from_pretrained('bert-base-uncased', local_files_only=True) # Remove the pooler layer bert_config <pre>/opt/homebrew/lib/python3.11/site-packages/huggingface_hub/file_download.py:1132: FutureWarning: `resume_download` is deprecated and will be removed in version 1.0.0. Downloads always resume when possible. If you want to force a new download, use `force_download=True`.\n  warnings.warn(\n</pre> Out[2]: <pre>BertConfig {\n  \"architectures\": [\n    \"BertForMaskedLM\"\n  ],\n  \"attention_probs_dropout_prob\": 0.1,\n  \"classifier_dropout\": null,\n  \"gradient_checkpointing\": false,\n  \"hidden_act\": \"gelu\",\n  \"hidden_dropout_prob\": 0.1,\n  \"hidden_size\": 768,\n  \"initializer_range\": 0.02,\n  \"intermediate_size\": 3072,\n  \"layer_norm_eps\": 1e-12,\n  \"max_position_embeddings\": 512,\n  \"model_type\": \"bert\",\n  \"num_attention_heads\": 12,\n  \"num_hidden_layers\": 12,\n  \"pad_token_id\": 0,\n  \"position_embedding_type\": \"absolute\",\n  \"transformers_version\": \"4.41.2\",\n  \"type_vocab_size\": 2,\n  \"use_cache\": true,\n  \"vocab_size\": 30522\n}</pre> <p>\u200b\u9996\u5148\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u52a0\u8f7d\u200b\u9884\u200b\u8bad\u7ec3\u200b\u7684\u200bbert-base-uncased\u200b\u6a21\u578b\u200b\uff0c\u200b\u7528\u4e8e\u200b\u548c\u200b\u6211\u4eec\u200b\u7684\u200b\u6a21\u578b\u200b\u8fdb\u884c\u200b\u5bf9\u6bd4\u200b\u3002</p> In\u00a0[3]: Copied! <pre>bert_tokenizer = transformers.BertTokenizer.from_pretrained(\n    'bert-base-uncased', local_files_only=True\n)\nhf_model = transformers.BertModel.from_pretrained(\n    'bert-base-uncased', config=bert_config, add_pooling_layer=False\n)\nhf_model\n</pre> bert_tokenizer = transformers.BertTokenizer.from_pretrained(     'bert-base-uncased', local_files_only=True ) hf_model = transformers.BertModel.from_pretrained(     'bert-base-uncased', config=bert_config, add_pooling_layer=False ) hf_model Out[3]: <pre>BertModel(\n  (embeddings): BertEmbeddings(\n    (word_embeddings): Embedding(30522, 768, padding_idx=0)\n    (position_embeddings): Embedding(512, 768)\n    (token_type_embeddings): Embedding(2, 768)\n    (LayerNorm): LayerNorm((768,), eps=1e-12, elementwise_affine=True)\n    (dropout): Dropout(p=0.1, inplace=False)\n  )\n  (encoder): BertEncoder(\n    (layer): ModuleList(\n      (0-11): 12 x BertLayer(\n        (attention): BertAttention(\n          (self): BertSdpaSelfAttention(\n            (query): Linear(in_features=768, out_features=768, bias=True)\n            (key): Linear(in_features=768, out_features=768, bias=True)\n            (value): Linear(in_features=768, out_features=768, bias=True)\n            (dropout): Dropout(p=0.1, inplace=False)\n          )\n          (output): BertSelfOutput(\n            (dense): Linear(in_features=768, out_features=768, bias=True)\n            (LayerNorm): LayerNorm((768,), eps=1e-12, elementwise_affine=True)\n            (dropout): Dropout(p=0.1, inplace=False)\n          )\n        )\n        (intermediate): BertIntermediate(\n          (dense): Linear(in_features=768, out_features=3072, bias=True)\n          (intermediate_act_fn): GELUActivation()\n        )\n        (output): BertOutput(\n          (dense): Linear(in_features=3072, out_features=768, bias=True)\n          (LayerNorm): LayerNorm((768,), eps=1e-12, elementwise_affine=True)\n          (dropout): Dropout(p=0.1, inplace=False)\n        )\n      )\n    )\n  )\n)</pre> <p>\u200b\u63a5\u4e0b\u6765\u200b\uff0c\u200b\u6211\u4eec\u200b\u6839\u636e\u200bconfig\u200b\u548c\u200b\u6a21\u578b\u200b\u7684\u200b\u7ed3\u6784\u200b\uff0c\u200b\u5b9e\u73b0\u200b\u4e00\u4e2a\u200bBERT\u200b\u6a21\u578b\u200b\u3002</p> In\u00a0[4]: Copied! <pre>class BertEmbeddings(torch.nn.Module):\n    def __init__(self, config: transformers.BertConfig):\n        super(BertEmbeddings, self).__init__()\n\n        self.word_embeddings = torch.nn.Embedding(\n            config.vocab_size, config.hidden_size, padding_idx=config.pad_token_id\n        )\n        self.positional_embedding = torch.nn.Embedding(\n            config.max_position_embeddings, config.hidden_size\n        )\n        self.token_type_embeddings = torch.nn.Embedding(\n            config.type_vocab_size, config.hidden_size\n        )\n        self.ln = torch.nn.LayerNorm(config.hidden_size, eps=config.layer_norm_eps)\n        self.dropout = torch.nn.Dropout(config.hidden_dropout_prob)\n\n    def forward(self, input_ids):\n        input_embeds = self.word_embeddings(input_ids)\n        pos_embeds = self.positional_embedding(\n            torch.arange(input_ids.size(1), device=input_ids.device)\n        )\n        type_embeds = self.token_type_embeddings(\n            torch.zeros_like(input_ids)\n        )\n        return self.dropout(self.ln(input_embeds + pos_embeds + type_embeds))\n\nclass CustomBertModel(torch.nn.Module):\n    def __init__(self, config: transformers.BertConfig):\n        super(CustomBertModel, self).__init__()\n        self.shared = BertEmbeddings(config)\n        self.encoder = Encoder(\n            num_layers=config.num_hidden_layers,\n            input_dim=config.hidden_size,\n            num_heads=config.num_attention_heads,\n            ffn_dim=config.intermediate_size,\n            dropout=config.hidden_dropout_prob,\n            layer_norm_eps=config.layer_norm_eps,\n            activation=config.hidden_act,\n        )\n\n    def forward(self, input_ids, padding_mask):\n        input_embeds = self.shared(input_ids)\n        return self.encoder(input_embeds, padding_mask=padding_mask)\n\ncustom_model = CustomBertModel(bert_config)\ncustom_model\n</pre> class BertEmbeddings(torch.nn.Module):     def __init__(self, config: transformers.BertConfig):         super(BertEmbeddings, self).__init__()          self.word_embeddings = torch.nn.Embedding(             config.vocab_size, config.hidden_size, padding_idx=config.pad_token_id         )         self.positional_embedding = torch.nn.Embedding(             config.max_position_embeddings, config.hidden_size         )         self.token_type_embeddings = torch.nn.Embedding(             config.type_vocab_size, config.hidden_size         )         self.ln = torch.nn.LayerNorm(config.hidden_size, eps=config.layer_norm_eps)         self.dropout = torch.nn.Dropout(config.hidden_dropout_prob)      def forward(self, input_ids):         input_embeds = self.word_embeddings(input_ids)         pos_embeds = self.positional_embedding(             torch.arange(input_ids.size(1), device=input_ids.device)         )         type_embeds = self.token_type_embeddings(             torch.zeros_like(input_ids)         )         return self.dropout(self.ln(input_embeds + pos_embeds + type_embeds))  class CustomBertModel(torch.nn.Module):     def __init__(self, config: transformers.BertConfig):         super(CustomBertModel, self).__init__()         self.shared = BertEmbeddings(config)         self.encoder = Encoder(             num_layers=config.num_hidden_layers,             input_dim=config.hidden_size,             num_heads=config.num_attention_heads,             ffn_dim=config.intermediate_size,             dropout=config.hidden_dropout_prob,             layer_norm_eps=config.layer_norm_eps,             activation=config.hidden_act,         )      def forward(self, input_ids, padding_mask):         input_embeds = self.shared(input_ids)         return self.encoder(input_embeds, padding_mask=padding_mask)  custom_model = CustomBertModel(bert_config) custom_model Out[4]: <pre>CustomBertModel(\n  (shared): BertEmbeddings(\n    (word_embeddings): Embedding(30522, 768, padding_idx=0)\n    (positional_embedding): Embedding(512, 768)\n    (token_type_embeddings): Embedding(2, 768)\n    (ln): LayerNorm((768,), eps=1e-12, elementwise_affine=True)\n    (dropout): Dropout(p=0.1, inplace=False)\n  )\n  (encoder): Encoder(\n    (layers): ModuleList(\n      (0-11): 12 x EncoderLayer(\n        (attention): MultiHeadSelfAttention(\n          (W_Q): Linear(in_features=768, out_features=768, bias=True)\n          (W_K): Linear(in_features=768, out_features=768, bias=True)\n          (W_V): Linear(in_features=768, out_features=768, bias=True)\n          (W_O): Linear(in_features=768, out_features=768, bias=True)\n        )\n        (norm1): LayerNorm((768,), eps=1e-12, elementwise_affine=True)\n        (dropout1): Dropout(p=0.1, inplace=False)\n        (ffn): FFN(\n          (fc1): Linear(in_features=768, out_features=3072, bias=True)\n          (fc2): Linear(in_features=3072, out_features=768, bias=True)\n          (act): GELU(approximate='none')\n        )\n        (norm2): LayerNorm((768,), eps=1e-12, elementwise_affine=True)\n        (dropout2): Dropout(p=0.1, inplace=False)\n      )\n    )\n  )\n)</pre> <p>\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u5c06\u200b\u9884\u200b\u8bad\u7ec3\u200b\u6a21\u578b\u200b\u7684\u200b\u6743\u91cd\u200b\u52a0\u8f7d\u200b\u5230\u200b\u6211\u4eec\u200b\u7684\u200b\u6a21\u578b\u200b\u4e2d\u200b\u3002</p> In\u00a0[5]: Copied! <pre>def copy_weights(src, dest, bias: bool = False):\n    src.weight.data.copy_(dest.weight.data)\n    if bias:\n        src.bias.data.copy_(dest.bias.data)\n\ndef load_weight_from_hf(model: CustomBertModel, hf_model: transformers.BertModel):\n    # Embeddings\n    unbiased_layer_pairs = [\n        (model.shared.word_embeddings, hf_model.embeddings.word_embeddings),\n        (model.shared.positional_embedding, hf_model.embeddings.position_embeddings),\n        (model.shared.token_type_embeddings, hf_model.embeddings.token_type_embeddings),\n    ]\n    biased_layer_pairs = [\n        (model.shared.ln, hf_model.embeddings.LayerNorm),\n    ]\n    for custom_layer, hf_layer in zip(model.encoder.layers, hf_model.encoder.layer):\n        # attn\n        biased_layer_pairs.append((custom_layer.attention.W_Q, hf_layer.attention.self.query))\n        biased_layer_pairs.append((custom_layer.attention.W_K, hf_layer.attention.self.key))\n        biased_layer_pairs.append((custom_layer.attention.W_V, hf_layer.attention.self.value))\n        biased_layer_pairs.append((custom_layer.attention.W_O, hf_layer.attention.output.dense))\n        biased_layer_pairs.append((custom_layer.norm1, hf_layer.attention.output.LayerNorm))\n        # ffn\n        biased_layer_pairs.append((custom_layer.ffn.fc1, hf_layer.intermediate.dense))\n        biased_layer_pairs.append((custom_layer.ffn.fc2, hf_layer.output.dense))\n        biased_layer_pairs.append((custom_layer.norm2, hf_layer.output.LayerNorm))\n\n    for src, dest in unbiased_layer_pairs:\n        copy_weights(src, dest)\n    for src, dest in biased_layer_pairs:\n        copy_weights(src, dest, bias=True)\n    return model\n</pre> def copy_weights(src, dest, bias: bool = False):     src.weight.data.copy_(dest.weight.data)     if bias:         src.bias.data.copy_(dest.bias.data)  def load_weight_from_hf(model: CustomBertModel, hf_model: transformers.BertModel):     # Embeddings     unbiased_layer_pairs = [         (model.shared.word_embeddings, hf_model.embeddings.word_embeddings),         (model.shared.positional_embedding, hf_model.embeddings.position_embeddings),         (model.shared.token_type_embeddings, hf_model.embeddings.token_type_embeddings),     ]     biased_layer_pairs = [         (model.shared.ln, hf_model.embeddings.LayerNorm),     ]     for custom_layer, hf_layer in zip(model.encoder.layers, hf_model.encoder.layer):         # attn         biased_layer_pairs.append((custom_layer.attention.W_Q, hf_layer.attention.self.query))         biased_layer_pairs.append((custom_layer.attention.W_K, hf_layer.attention.self.key))         biased_layer_pairs.append((custom_layer.attention.W_V, hf_layer.attention.self.value))         biased_layer_pairs.append((custom_layer.attention.W_O, hf_layer.attention.output.dense))         biased_layer_pairs.append((custom_layer.norm1, hf_layer.attention.output.LayerNorm))         # ffn         biased_layer_pairs.append((custom_layer.ffn.fc1, hf_layer.intermediate.dense))         biased_layer_pairs.append((custom_layer.ffn.fc2, hf_layer.output.dense))         biased_layer_pairs.append((custom_layer.norm2, hf_layer.output.LayerNorm))      for src, dest in unbiased_layer_pairs:         copy_weights(src, dest)     for src, dest in biased_layer_pairs:         copy_weights(src, dest, bias=True)     return model In\u00a0[6]: Copied! <pre>custom_model = load_weight_from_hf(custom_model, hf_model)\n</pre> custom_model = load_weight_from_hf(custom_model, hf_model) <p>\u200b\u4f7f\u7528\u200bBERT\u200b\u7f16\u7801\u200b\u4e00\u53e5\u200b\u8bdd\u200b\uff0c\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u5bf9\u200b\u53e5\u5b50\u200b\u5206\u8bcd\u200b\uff0c\u200b\u7136\u540e\u200b\u5c06\u200b\u5176\u200b\u8f93\u5165\u200b\u6a21\u578b\u200b\u5f97\u5230\u200b\u5bf9\u5e94\u200b\u7684\u200b\u8f93\u51fa\u200b\u3002</p> In\u00a0[7]: Copied! <pre>custom_model.eval()\nhf_model.eval()\n\nsentence = \"Hello, World!\"\ntokenized_sentence = bert_tokenizer(sentence, return_tensors='pt')\ninput_ids = tokenized_sentence['input_ids']\npadding_mask = tokenized_sentence['attention_mask']\n\ncustom_output = custom_model(input_ids, padding_mask)\nhf_output = hf_model(input_ids, attention_mask=padding_mask).last_hidden_state\ncustom_output[0, 0, :5], hf_output[0, 0, :5]\n</pre> custom_model.eval() hf_model.eval()  sentence = \"Hello, World!\" tokenized_sentence = bert_tokenizer(sentence, return_tensors='pt') input_ids = tokenized_sentence['input_ids'] padding_mask = tokenized_sentence['attention_mask']  custom_output = custom_model(input_ids, padding_mask) hf_output = hf_model(input_ids, attention_mask=padding_mask).last_hidden_state custom_output[0, 0, :5], hf_output[0, 0, :5] Out[7]: <pre>(tensor([-0.0781,  0.1587,  0.0400, -0.1986, -0.3442], grad_fn=&lt;SliceBackward0&gt;),\n tensor([-0.0781,  0.1587,  0.0400, -0.1986, -0.3442], grad_fn=&lt;SliceBackward0&gt;))</pre> In\u00a0[8]: Copied! <pre>torch.allclose(custom_output, hf_output, atol=1e-5)\n</pre> torch.allclose(custom_output, hf_output, atol=1e-5) Out[8]: <pre>True</pre>"},{"location":"coding/dl-from-scratch/custom_transformer/","title":"Custom transformer","text":"In\u00a0[\u00a0]: Copied! <pre>import torch\n</pre> import torch In\u00a0[\u00a0]: Copied! <pre>class MultiHeadAttention(torch.nn.Module):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadAttention, self).__init__()\n        self.d_model = d_model\n        self.num_heads = num_heads\n        if d_model % num_heads != 0:\n            raise ValueError(\"d_model must be divisible by num_heads\")\n\n        self.d_k = d_model // num_heads\n        self.sqrt_d_k = self.d_k ** 0.5\n\n        self.W_Q = torch.nn.Linear(d_model, d_model)\n        self.W_K = torch.nn.Linear(d_model, d_model)\n        self.W_V = torch.nn.Linear(d_model, d_model)\n        self.W_O = torch.nn.Linear(d_model, d_model)\n\n    def forward(\n        self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        # x_q: (N, L_Q, D), x_k: (N, L_KV, D), x_v: (N, L_KV, D)\n        # padding_mask: (N, L_KV), attention_mask: (L_Q, L_KV)\n        N, L_Q, D = x_q.size()\n        _, L_KV, _ = x_k.size()\n        Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)\n        K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)\n        V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)\n\n        score = torch.einsum('nihd,njhd-&gt;nijh', Q, K) / self.sqrt_d_k\n        # score: (N, L_Q, L_KV, num_heads)\n        # Apply attention mask\n        if attention_mask is not None:\n            score = score.masked_fill(\n                attention_mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')\n            )\n        # Apply padding mask\n        if padding_mask is not None:\n            score = score.masked_fill(\n                padding_mask.reshape(N, 1, L_KV, 1) == 0, float('-inf')\n            )\n        score = torch.nn.functional.softmax(score, dim=2)\n        value = torch.einsum(\n            'nijh,njhd-&gt;nihd', score, V\n        ).reshape(N, L_Q, self.d_model)\n        return self.W_O(value)\n</pre> class MultiHeadAttention(torch.nn.Module):     def __init__(self, d_model: int, num_heads: int):         super(MultiHeadAttention, self).__init__()         self.d_model = d_model         self.num_heads = num_heads         if d_model % num_heads != 0:             raise ValueError(\"d_model must be divisible by num_heads\")          self.d_k = d_model // num_heads         self.sqrt_d_k = self.d_k ** 0.5          self.W_Q = torch.nn.Linear(d_model, d_model)         self.W_K = torch.nn.Linear(d_model, d_model)         self.W_V = torch.nn.Linear(d_model, d_model)         self.W_O = torch.nn.Linear(d_model, d_model)      def forward(         self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,         padding_mask: torch.Tensor | None = None,         attention_mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         # x_q: (N, L_Q, D), x_k: (N, L_KV, D), x_v: (N, L_KV, D)         # padding_mask: (N, L_KV), attention_mask: (L_Q, L_KV)         N, L_Q, D = x_q.size()         _, L_KV, _ = x_k.size()         Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)         K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)         V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)          score = torch.einsum('nihd,njhd-&gt;nijh', Q, K) / self.sqrt_d_k         # score: (N, L_Q, L_KV, num_heads)         # Apply attention mask         if attention_mask is not None:             score = score.masked_fill(                 attention_mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')             )         # Apply padding mask         if padding_mask is not None:             score = score.masked_fill(                 padding_mask.reshape(N, 1, L_KV, 1) == 0, float('-inf')             )         score = torch.nn.functional.softmax(score, dim=2)         value = torch.einsum(             'nijh,njhd-&gt;nihd', score, V         ).reshape(N, L_Q, self.d_model)         return self.W_O(value) In\u00a0[\u00a0]: Copied! <pre>class MultiHeadSelfAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadSelfAttention, self).__init__(d_model, num_heads)\n\n    def forward(\n        self, x: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        return super().forward(x, x, x, padding_mask, attention_mask)\n</pre> class MultiHeadSelfAttention(MultiHeadAttention):     def __init__(self, d_model: int, num_heads: int):         super(MultiHeadSelfAttention, self).__init__(d_model, num_heads)      def forward(         self, x: torch.Tensor,         padding_mask: torch.Tensor | None = None,         attention_mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         return super().forward(x, x, x, padding_mask, attention_mask) In\u00a0[\u00a0]: Copied! <pre>class MultiHeadCrossAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadCrossAttention, self).__init__(d_model, num_heads)\n\n    def forward(\n        self, x_q: torch.Tensor, x_kv: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        return super().forward(x_q, x_kv, x_kv, padding_mask, attention_mask)\n</pre> class MultiHeadCrossAttention(MultiHeadAttention):     def __init__(self, d_model: int, num_heads: int):         super(MultiHeadCrossAttention, self).__init__(d_model, num_heads)      def forward(         self, x_q: torch.Tensor, x_kv: torch.Tensor,         padding_mask: torch.Tensor | None = None,         attention_mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         return super().forward(x_q, x_kv, x_kv, padding_mask, attention_mask) In\u00a0[\u00a0]: Copied! <pre>ACT2FN = {\n    'relu': torch.nn.ReLU,\n    'gelu': torch.nn.GELU,\n}\n</pre> ACT2FN = {     'relu': torch.nn.ReLU,     'gelu': torch.nn.GELU, } In\u00a0[\u00a0]: Copied! <pre>class FFN(torch.nn.Module):\n    def __init__(self, input_dim: int, hidden_dim: int, activation: str = 'relu'):\n        super(FFN, self).__init__()\n        self.fc1 = torch.nn.Linear(input_dim, hidden_dim)\n        self.fc2 = torch.nn.Linear(hidden_dim, input_dim)\n        self.act = ACT2FN[activation]()\n\n    def forward(self, x):\n        return self.fc2(self.act(self.fc1(x)))\n</pre> class FFN(torch.nn.Module):     def __init__(self, input_dim: int, hidden_dim: int, activation: str = 'relu'):         super(FFN, self).__init__()         self.fc1 = torch.nn.Linear(input_dim, hidden_dim)         self.fc2 = torch.nn.Linear(hidden_dim, input_dim)         self.act = ACT2FN[activation]()      def forward(self, x):         return self.fc2(self.act(self.fc1(x))) In\u00a0[\u00a0]: Copied! <pre>class EncoderLayer(torch.nn.Module):\n    def __init__(\n        self, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(EncoderLayer, self).__init__()\n\n        if ffn_dim is None:\n            ffn_dim = input_dim * 4\n\n        self.attention = MultiHeadSelfAttention(input_dim, num_heads)\n        self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout1 = torch.nn.Dropout(dropout)\n\n        self.ffn = FFN(input_dim, ffn_dim, activation)\n        self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout2 = torch.nn.Dropout(dropout)\n\n    def forward(\n        self, x, padding_mask: torch.Tensor | None = None,\n        src_mask: torch.Tensor | None = None\n    ):\n        x = x + self.dropout1(self.attention(x, padding_mask, src_mask))\n        x = self.norm1(x)\n\n        x = x + self.dropout2(self.ffn(x))\n        x = self.norm2(x)\n\n        return x\n</pre> class EncoderLayer(torch.nn.Module):     def __init__(         self, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1,         layer_norm_eps: float = 1e-6, activation: str = 'relu'     ):         super(EncoderLayer, self).__init__()          if ffn_dim is None:             ffn_dim = input_dim * 4          self.attention = MultiHeadSelfAttention(input_dim, num_heads)         self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout1 = torch.nn.Dropout(dropout)          self.ffn = FFN(input_dim, ffn_dim, activation)         self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout2 = torch.nn.Dropout(dropout)      def forward(         self, x, padding_mask: torch.Tensor | None = None,         src_mask: torch.Tensor | None = None     ):         x = x + self.dropout1(self.attention(x, padding_mask, src_mask))         x = self.norm1(x)          x = x + self.dropout2(self.ffn(x))         x = self.norm2(x)          return x In\u00a0[\u00a0]: Copied! <pre>class Encoder(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(Encoder, self).__init__()\n\n        self.layers = torch.nn.ModuleList([\n            EncoderLayer(\n                input_dim, num_heads, ffn_dim,\n                dropout, layer_norm_eps, activation\n            )\n            for _ in range(num_layers)\n        ])\n\n    def forward(\n        self, x, padding_mask: torch.Tensor | None = None,\n        src_mask: torch.Tensor | None = None\n    ):\n        # x: (N, L, D)\n        if src_mask is None:\n            src_mask = torch.ones(x.size(1), x.size(1)).to(x.device)\n        for layer in self.layers:\n            x = layer(x, padding_mask, src_mask)\n        return x\n</pre> class Encoder(torch.nn.Module):     def __init__(         self, num_layers: int, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1,         layer_norm_eps: float = 1e-6, activation: str = 'relu'     ):         super(Encoder, self).__init__()          self.layers = torch.nn.ModuleList([             EncoderLayer(                 input_dim, num_heads, ffn_dim,                 dropout, layer_norm_eps, activation             )             for _ in range(num_layers)         ])      def forward(         self, x, padding_mask: torch.Tensor | None = None,         src_mask: torch.Tensor | None = None     ):         # x: (N, L, D)         if src_mask is None:             src_mask = torch.ones(x.size(1), x.size(1)).to(x.device)         for layer in self.layers:             x = layer(x, padding_mask, src_mask)         return x In\u00a0[\u00a0]: Copied! <pre>class DecoderLayer(torch.nn.Module):\n    def __init__(\n        self, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(DecoderLayer, self).__init__()\n\n        if ffn_dim is None:\n            ffn_dim = input_dim * 4\n\n        self.self_attention = MultiHeadSelfAttention(input_dim, num_heads)\n        self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout1 = torch.nn.Dropout(dropout)\n\n        self.cross_attention = MultiHeadCrossAttention(input_dim, num_heads)\n        self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout2 = torch.nn.Dropout(dropout)\n\n        self.ffn = FFN(input_dim, ffn_dim, activation)\n        self.norm3 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout3 = torch.nn.Dropout(dropout)\n\n    def forward(\n        self, x: torch.Tensor, memory: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        memory_mask: torch.Tensor | None = None,\n        tgt_mask: torch.Tensor | None = None\n    ):\n\n        x = x + self.dropout1(self.self_attention(x, padding_mask, tgt_mask))\n        x = self.norm1(x)\n\n        x = x + self.dropout2(self.cross_attention(x, memory, padding_mask, memory_mask))\n        x = self.norm2(x)\n\n        x = x + self.dropout3(self.ffn(x))\n        x = self.norm3(x)\n\n        return x\n</pre> class DecoderLayer(torch.nn.Module):     def __init__(         self, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1,         layer_norm_eps: float = 1e-6, activation: str = 'relu'     ):         super(DecoderLayer, self).__init__()          if ffn_dim is None:             ffn_dim = input_dim * 4          self.self_attention = MultiHeadSelfAttention(input_dim, num_heads)         self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout1 = torch.nn.Dropout(dropout)          self.cross_attention = MultiHeadCrossAttention(input_dim, num_heads)         self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout2 = torch.nn.Dropout(dropout)          self.ffn = FFN(input_dim, ffn_dim, activation)         self.norm3 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout3 = torch.nn.Dropout(dropout)      def forward(         self, x: torch.Tensor, memory: torch.Tensor,         padding_mask: torch.Tensor | None = None,         memory_mask: torch.Tensor | None = None,         tgt_mask: torch.Tensor | None = None     ):          x = x + self.dropout1(self.self_attention(x, padding_mask, tgt_mask))         x = self.norm1(x)          x = x + self.dropout2(self.cross_attention(x, memory, padding_mask, memory_mask))         x = self.norm2(x)          x = x + self.dropout3(self.ffn(x))         x = self.norm3(x)          return x In\u00a0[\u00a0]: Copied! <pre>class Decoder(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(Decoder, self).__init__()\n\n        self.layers = torch.nn.ModuleList([\n            DecoderLayer(\n                input_dim, num_heads, ffn_dim, dropout,\n                layer_norm_eps, activation\n            )\n            for _ in range(num_layers)\n        ])\n\n    def forward(\n        self, x: torch.Tensor, memory: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        memory_mask: torch.Tensor | None = None,\n        tgt_mask: torch.Tensor | None = None\n    ):\n        # x: (N, L, D)\n        for layer in self.layers:\n            x = layer(x, memory, padding_mask, memory_mask, tgt_mask)\n        return x\n</pre> class Decoder(torch.nn.Module):     def __init__(         self, num_layers: int, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1,         layer_norm_eps: float = 1e-6, activation: str = 'relu'     ):         super(Decoder, self).__init__()          self.layers = torch.nn.ModuleList([             DecoderLayer(                 input_dim, num_heads, ffn_dim, dropout,                 layer_norm_eps, activation             )             for _ in range(num_layers)         ])      def forward(         self, x: torch.Tensor, memory: torch.Tensor,         padding_mask: torch.Tensor | None = None,         memory_mask: torch.Tensor | None = None,         tgt_mask: torch.Tensor | None = None     ):         # x: (N, L, D)         for layer in self.layers:             x = layer(x, memory, padding_mask, memory_mask, tgt_mask)         return x"},{"location":"coding/dl-from-scratch/encoder-decoder/","title":"\u7f16\u7801\u5668\u200b\u4e0e\u200b\u89e3\u7801\u5668","text":"<p>\u200b\u5728\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\uff0cVaswani\u200b\u7b49\u200b\u4eba\u200b\u63d0\u51fa\u200b\u4e86\u200b\u4e24\u79cd\u200btransformer\u200b\u67b6\u6784\u200b\uff0c\u200b\u5373\u200b\u7f16\u7801\u5668\u200b\u548c\u200b\u89e3\u7801\u5668\u200b\u3002\u200b\u7f16\u7801\u5668\u200b\u5229\u7528\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\uff0c\u200b\u5bf9\u200b\u8f93\u5165\u200b\u7684\u200b\u5e8f\u5217\u200b\u8fdb\u884c\u200b\u7f16\u7801\u200b\uff0c\u200b\u89e3\u7801\u5668\u200b\u5219\u200b\u5229\u7528\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u548c\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\uff0c\u200b\u751f\u6210\u200b\u5bf9\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u9884\u6d4b\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/encoder-decoder/#_2","title":"\u7f16\u7801\u5668","text":"<p>\u200b\u7f16\u7801\u5668\u200b\u7531\u200b\u591a\u4e2a\u200b\u76f8\u540c\u200b\u7684\u200b\u5c42\u200b\u7ec4\u6210\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5c42\u200b\u5305\u542b\u200b\u4e00\u4e2a\u200b\u6b8b\u5dee\u200b\u8fde\u63a5\u200b\u7684\u200b\u591a\u5934\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u548c\u200b\u4e00\u4e2a\u200b\u524d\u9988\u200b\u795e\u7ecf\u7f51\u7edc\u200b\u3002\u200b\u5728\u200b\u6b8b\u5dee\u200b\u8fde\u63a5\u200b\u524d\u200b\uff0c\u200b\u4f7f\u7528\u200bdropout\uff1b\u200b\u5728\u200b\u6b8b\u5dee\u200b\u8fde\u63a5\u200b\u540e\u200b\uff0c\u200b\u4f7f\u7528\u200b\u5c42\u200b\u5f52\u4e00\u5316\u200b\u6765\u200b\u9632\u6b62\u200b\u8fc7\u200b\u62df\u5408\u200b\u3002\u200b\u524d\u9988\u200b\u795e\u7ecf\u7f51\u7edc\u200b\u7531\u200b\u4e24\u4e2a\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\u7ec4\u6210\u200b\uff0c\u200b\u4e24\u4e2a\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\u4e4b\u95f4\u200b\u901a\u5e38\u200b\u4f7f\u7528\u200bReLU\u200b\u6216\u200bGeLU\u200b\u7b49\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\u3002</p> <pre><code>ACT2FN = {\n    'relu': torch.nn.ReLU,\n    'gelu': torch.nn.GELU,\n}\n\nclass FFN(torch.nn.Module):\n    def __init__(self, input_dim: int, hidden_dim: int, activation: str = 'relu'):\n        super(FFN, self).__init__()\n        self.fc1 = torch.nn.Linear(input_dim, hidden_dim)\n        self.fc2 = torch.nn.Linear(hidden_dim, input_dim)\n        self.act = ACT2FN[activation]()\n\n    def forward(self, x):\n        return self.fc2(self.act(self.fc1(x)))\n\n\nclass EncoderLayer(torch.nn.Module):\n    def __init__(\n        self, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(EncoderLayer, self).__init__()\n\n        if ffn_dim is None:\n            ffn_dim = input_dim * 4\n\n        self.attention = MultiHeadSelfAttention(input_dim, num_heads)\n        self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout1 = torch.nn.Dropout(dropout)\n\n        self.ffn = FFN(input_dim, ffn_dim, activation)\n        self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout2 = torch.nn.Dropout(dropout)\n\n    def forward(\n        self, x, padding_mask: torch.Tensor | None = None,\n        src_mask: torch.Tensor | None = None\n    ):\n        x = x + self.dropout1(self.attention(x, padding_mask, src_mask))\n        x = self.norm1(x)\n\n        x = x + self.dropout2(self.ffn(x))\n        x = self.norm2(x)\n\n        return x\n\n\nclass Encoder(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(Encoder, self).__init__()\n\n        self.layers = torch.nn.ModuleList([\n            EncoderLayer(\n                input_dim, num_heads, ffn_dim,\n                dropout, layer_norm_eps, activation\n            )\n            for _ in range(num_layers)\n        ])\n\n    def forward(\n        self, x, padding_mask: torch.Tensor | None = None,\n        src_mask: torch.Tensor | None = None\n    ):\n        # x: (N, L, D)\n        if src_mask is None:\n            src_mask = torch.ones(x.size(1), x.size(1)).to(x.device)\n        for layer in self.layers:\n            x = layer(x, padding_mask, src_mask)\n        return x\n</code></pre>"},{"location":"coding/dl-from-scratch/encoder-decoder/#_3","title":"\u89e3\u7801\u5668","text":"<p>\u200b\u89e3\u7801\u5668\u200b\u5728\u200b\u7f16\u7801\u5668\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\uff0c\u200b\u5c06\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u7531\u200b\u53cc\u5411\u200b\u6539\u4e3a\u200b\u5355\u5411\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5728\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\u548c\u200b\u524d\u9988\u200b\u795e\u7ecf\u200b\u7f51\u7edc\u5c42\u200b\u4e4b\u95f4\u200b\u589e\u52a0\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u6b8b\u5dee\u200b\u8fde\u63a5\u200b\u7684\u200b\u591a\u5934\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u3002\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u4f7f\u7528\u200b\u7f16\u7801\u5668\u200b\u7684\u200b\u8f93\u51fa\u200b\u4f5c\u4e3a\u200b\u5168\u5c40\u200b\u952e\u548c\u503c\u200b\uff0c\u200b\u89e3\u7801\u5668\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\u7684\u200b\u8f93\u51fa\u200b\u4f5c\u4e3a\u200b\u67e5\u8be2\u200b\u3002\u200b\u5728\u200b\u6b8b\u5dee\u200b\u8fde\u63a5\u200b\u524d\u540e\u200b\u4f7f\u7528\u200bdropout\u200b\u548c\u200b\u5c42\u200b\u5f52\u4e00\u5316\u200b\u3002\u200b\u6ce8\u610f\u200b\u5bf9\u4e8e\u200bdecoder-only\u200b\u7684\u200btransformer\uff0c\u200b\u6ca1\u6709\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\uff0c\u200b\u800c\u662f\u200b\u4ec5\u200b\u5305\u542b\u200b\u4e00\u4e2a\u200b\u5355\u5411\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\u3002</p> <pre><code>class DecoderLayer(torch.nn.Module):\n    def __init__(\n        self, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(DecoderLayer, self).__init__()\n\n        if ffn_dim is None:\n            ffn_dim = input_dim * 4\n\n        self.self_attention = MultiHeadSelfAttention(input_dim, num_heads)\n        self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout1 = torch.nn.Dropout(dropout)\n\n        self.cross_attention = MultiHeadCrossAttention(input_dim, num_heads)\n        self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout2 = torch.nn.Dropout(dropout)\n\n        self.ffn = FFN(input_dim, ffn_dim, activation)\n        self.norm3 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout3 = torch.nn.Dropout(dropout)\n\n    def forward(\n        self, x: torch.Tensor, memory: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        memory_mask: torch.Tensor | None = None,\n        tgt_mask: torch.Tensor | None = None\n    ):\n        if tgt_mask is None:\n            tgt_mask = torch.triu(\n                torch.ones(x.size(1), x.size(1)), diagonal=1\n            ).to(x.device)\n\n        x = x + self.dropout1(self.self_attention(x, padding_mask, tgt_mask))\n        x = self.norm1(x)\n\n        x = x + self.dropout2(self.cross_attention(x, memory, padding_mask, memory_mask))\n        x = self.norm2(x)\n\n        x = x + self.dropout3(self.ffn(x))\n        x = self.norm3(x)\n\n        return x\n\n\nclass Decoder(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(Decoder, self).__init__()\n\n        self.layers = torch.nn.ModuleList([\n            DecoderLayer(\n                input_dim, num_heads, ffn_dim, dropout,\n                layer_norm_eps, activation\n            )\n            for _ in range(num_layers)\n        ])\n\n    def forward(\n        self, x: torch.Tensor, memory: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        memory_mask: torch.Tensor | None = None,\n        tgt_mask: torch.Tensor | None = None\n    ):\n        # x: (N, L, D)\n        for layer in self.layers:\n            x = layer(x, memory, padding_mask, memory_mask, tgt_mask)\n        return x\n</code></pre>"},{"location":"coding/dl-from-scratch/encoder-decoder/#transformer","title":"transformer\u200b\u7684\u200b\u65f6\u7a7a\u200b\u590d\u6742\u5ea6","text":"<p>\u200b\u7ed9\u5b9a\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u7684\u200b\u957f\u5ea6\u200b\\(L\\)\uff0c\u200b\u8f93\u5165\u200b\u7ef4\u5ea6\u200b\\(D\\)\uff0c\u200b\u7f16\u7801\u5668\u200b\u5c42\u6570\u200b\\(N\\)\uff0c\u200b\u524d\u9988\u200b\u795e\u7ecf\u7f51\u7edc\u200b\u7684\u200b\u9690\u85cf\u200b\u5c42\u200b\u7ef4\u5ea6\u200b\\(D_{ff}\\)\uff0c\u200b\u5219\u200btransformer\u200b\u7684\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b</p> <ul> <li>\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\uff1a\u200b\u6bcf\u5c42\u200b\u5305\u542b\u200b\u56db\u4e2a\u200b\\(D \\rightarrow D\\)\u200b\u7684\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\u5305\u542b\u200b\\(D \\times D + D\\)\u200b\u4e2a\u200b\u53c2\u6570\u200b\uff08\u200b\u6743\u91cd\u200b+\u200b\u504f\u7f6e\u200b\uff09\uff0c\u200b\u5171\u200b\\(4ND(D + 1)\\)\u200b\u4e2a\u200b\u53c2\u6570\u200b\uff1b</li> <li>\u200b\u524d\u9988\u200b\u795e\u7ecf\u7f51\u7edc\u200b\uff1a\u200b\u5171\u200b\\(N\\)\u200b\u5c42\u200b\uff0c\u200b\u6bcf\u5c42\u200b\u5305\u542b\u200b\u4e00\u4e2a\u200b\\(D \\rightarrow D_{ff}\\)\u200b\u7684\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\u548c\u200b\u4e00\u4e2a\u200b\\(D_{ff} \\rightarrow D\\)\u200b\u7684\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\uff0c\u200b\u5171\u200b\\(2NDD_{ff} + ND_{ff} + ND\\)\u200b\u4e2a\u200b\u53c2\u6570\u200b\uff1b</li> <li>\u200b\u5c42\u200b\u5f52\u4e00\u5316\u200b\uff1a\u200b\u6bcf\u5c42\u200b\u7f16\u7801\u5668\u200b\u5305\u542b\u200b\u4e24\u4e2a\u200b\u5f52\u4e00\u5316\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5f52\u4e00\u5316\u200b\u5305\u542b\u200b\u4e24\u4e2a\u200b\\(D\\)\u200b\u7ef4\u200b\u7684\u200b\u7f29\u653e\u200b\u53c2\u6570\u200b\u548c\u200b\u504f\u7f6e\u200b\u53c2\u6570\u200b\uff0c\u200b\u5171\u200b\\(4ND\\)\u200b\u4e2a\u200b\u53c2\u6570\u200b\u3002</li> </ul> <p>\u200b\u56e0\u6b64\u200b\uff0ctransformer\u200b\u7684\u200b\u603b\u53c2\u200b\u6570\u91cf\u200b\u4e3a\u200b\\(4ND(D + 1) + 2NDD_{ff} + ND_{ff} + ND + 2ND\\)\u3002\u200b\u5728\u200bVaswani\u200b\u7b49\u200b\u4eba\u200b\u7684\u200b\u5b9e\u73b0\u200b\u4e2d\u200b\uff0c\\(D_{ff} = 4D\\)\uff0c\u200b\u5219\u200btransformer\u200b\u7684\u200b\u53c2\u200b\u6570\u91cf\u200b\u4e3a\u200b\\(12ND^2 + 13ND\\)\u3002\u200b\u89e3\u7801\u5668\u200b\u6bd4\u200b\u7f16\u7801\u5668\u200b\u591a\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\uff08\\(4ND(D + 1)\\)\uff09\u200b\u548c\u200b1\u200b\u4e2a\u200b\u5f52\u4e00\u5316\u200b\u5c42\u200b\uff08\\(2ND\\)\uff09\uff0c\u200b\u56e0\u6b64\u200b\u89e3\u7801\u5668\u200b\u7684\u200b\u53c2\u200b\u6570\u91cf\u200b\u4e3a\u200b\\(16ND^2 + 19ND\\)\u3002</p> <p>\u200b\u77e9\u9635\u200b\u8fd0\u7b97\u200b\\(\\bbR^{m\\times n}\\times \\bbR^{n\\times p} \\rightarrow \\bbR^{m\\times p}\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(mnp)\\)\uff0c\u200b\u9010\u6b65\u200b\u5206\u6790\u200btransformer\u200b\u7684\u200b\u8ba1\u7b97\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u5ffd\u7565\u200b\u52a0\u6cd5\u200b\u548c\u200b\u5f52\u4e00\u5316\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\uff1a</p> <ol> <li>\u200b\u7ebf\u6027\u53d8\u6362\u200b\uff1a\\(\\bbR^{L\\times D}\\times \\bbR^{D\\times D} \\rightarrow \\bbR^{L\\times D}\\)\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(LD^2)\\)\uff1b</li> <li>\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\uff1a\\(\\bbR^{L\\times D}\\times \\bbR^{D\\times L}\\rightarrow \\bbR^{L\\times L}\\)\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(L^2D)\\)\uff1b</li> <li>\u200b\u524d\u9988\u200b\u795e\u7ecf\u7f51\u7edc\u200b\uff1a\\(\\bbR^{L\\times D}\\times \\bbR^{D\\times D_{ff}}\\rightarrow \\bbR^{L\\times D_{ff}}\\)\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(LDD_{ff})\\)\u3002\u200b\u6309\u7167\u200b\\(D_{ff} = 4D\\)\uff0c\u200b\u5219\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(4LD^2) = O(LD^2)\\)\u3002</li> </ol> <p>\\(D\\)\u200b\u4e3a\u200b\u5e38\u6570\u200b\uff0c\u200b\u6a21\u578b\u200b\u5171\u200b\u53e0\u52a0\u200b\\(N\\)\u200b\u5c42\u200b\u4e32\u884c\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u56e0\u6b64\u200btransformer\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(NL^2D)\\)\u3002</p>"},{"location":"coding/dl-from-scratch/llama-2/","title":"\u5b9e\u73b0\u200bLlama-2","text":"<p>\u200b\u5728\u200b\u672c\u8282\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u5b9e\u73b0\u200b\u4e00\u4e2a\u200bLlama-2\u200b\u6a21\u578b\u200b\u3002</p> In\u00a0[1]: Copied! <pre>import functools\n\nimport torch\nimport transformers\n</pre> import functools  import torch import transformers <p>Llama\u200b\u4f7f\u7528\u200bRoPE\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u3002\u200b\u6ce8\u610f\u200b\u6b64\u5904\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u548c\u200bRoFormer\u200b\u4e2d\u200b\u5b9e\u73b0\u200b\u7684\u200b\u4e0d\u540c\u200b\uff0cRoFormer\u200b\u5c06\u200b\u76f8\u90bb\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5143\u7d20\u200b\u5206\u4e3a\u200b\u4e00\u7ec4\u200b\u8fdb\u884c\u200b\u65cb\u8f6c\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u800c\u200bLlamaRope\u200b\u662f\u200b\u5c06\u200b\u5411\u91cf\u200b\u5206\u4e3a\u200b\u524d\u540e\u200b\u4e24\u6bb5\u200b\uff0c\u200b\u524d\u540e\u200b\u4e24\u6bb5\u200b\u76f8\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u5206\u4e3a\u200b\u4e00\u7ec4\u200b\u8fdb\u884c\u200b\u65cb\u8f6c\u200b\u64cd\u4f5c\u200b\u3002</p> In\u00a0[2]: Copied! <pre>class RoPE(torch.nn.Module):\n    def __init__(self, d_model: int, theta: int | float = 10000):\n        super(RoPE, self).__init__()\n        self.d_model = d_model\n        self.theta = theta ** -(torch.arange(0, d_model, 2) / d_model)\n\n    @functools.lru_cache(maxsize=None)\n    def _forward_l(self, L: int) -&gt; torch.Tensor:\n        # Use lru_cache to avoid redundant computation for the same L\n\n        D = self.d_model\n        pos = torch.einsum(\n            'l,d-&gt;ld',\n            torch.arange(L), self.theta\n        )  # [L, D / 2]\n\n        # 0 paired with d // 2, 1 paired with d // 2 + 1, ...\n        cos = torch.cos(pos).repeat([1, 2])\n        sin = torch.sin(pos)\n\n        # Here we do not use efficient method, but construct a rotary matrix\n        result = torch.zeros(L, D, D)\n        result = torch.diagonal_scatter(result, cos, dim1=1, dim2=2)\n        result = torch.diagonal_scatter(result, sin, dim1=1, dim2=2, offset=-D // 2)\n        result = torch.diagonal_scatter(result, -sin, dim1=1, dim2=2, offset=D // 2)\n        return result\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        # x: [N, L, H, D]\n        _, L, _, D = x.size()\n        rot_matrix = self._forward_l(L).to(x.device)  # [L, D, D]\n        return torch.einsum('lde,nlhe-&gt;nlhd', rot_matrix, x)\n</pre> class RoPE(torch.nn.Module):     def __init__(self, d_model: int, theta: int | float = 10000):         super(RoPE, self).__init__()         self.d_model = d_model         self.theta = theta ** -(torch.arange(0, d_model, 2) / d_model)      @functools.lru_cache(maxsize=None)     def _forward_l(self, L: int) -&gt; torch.Tensor:         # Use lru_cache to avoid redundant computation for the same L          D = self.d_model         pos = torch.einsum(             'l,d-&gt;ld',             torch.arange(L), self.theta         )  # [L, D / 2]          # 0 paired with d // 2, 1 paired with d // 2 + 1, ...         cos = torch.cos(pos).repeat([1, 2])         sin = torch.sin(pos)          # Here we do not use efficient method, but construct a rotary matrix         result = torch.zeros(L, D, D)         result = torch.diagonal_scatter(result, cos, dim1=1, dim2=2)         result = torch.diagonal_scatter(result, sin, dim1=1, dim2=2, offset=-D // 2)         result = torch.diagonal_scatter(result, -sin, dim1=1, dim2=2, offset=D // 2)         return result      def forward(self, x: torch.Tensor) -&gt; torch.Tensor:         # x: [N, L, H, D]         _, L, _, D = x.size()         rot_matrix = self._forward_l(L).to(x.device)  # [L, D, D]         return torch.einsum('lde,nlhe-&gt;nlhd', rot_matrix, x) <p>RoPE\u200b\u662f\u200b\u5728\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u5206\u5934\u200b\u540e\u200b\u624d\u200b\u8fdb\u884c\u200b\u8ba1\u7b97\u200b\u3002\u200b\u5e76\u4e14\u200b\uff0cLlama\u200b\u7684\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u4e2d\u200b\u4e0d\u200b\u5305\u542b\u200b\u504f\u7f6e\u200b\u3002</p> In\u00a0[3]: Copied! <pre>class MultiHeadAttention(torch.nn.Module):\n    def __init__(self, d_model: int, num_heads: int, rope: RoPE):\n        super(MultiHeadAttention, self).__init__()\n        self.d_model = d_model\n        self.num_heads = num_heads\n        if d_model % num_heads != 0:\n            raise ValueError(\"d_model must be divisible by num_heads\")\n\n        self.d_k = d_model // num_heads\n        self.sqrt_d_k = self.d_k ** 0.5\n        self.rope = rope\n\n        self.W_Q = torch.nn.Linear(d_model, d_model, bias=False)\n        self.W_K = torch.nn.Linear(d_model, d_model, bias=False)\n        self.W_V = torch.nn.Linear(d_model, d_model, bias=False)\n        self.W_O = torch.nn.Linear(d_model, d_model, bias=False)\n\n    def forward(\n        self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        # x_q: (N, L_Q, D), x_k: (N, L_KV, D), x_v: (N, L_KV, D)\n        # padding_mask: (N, L_KV), attention_mask: (L_Q, L_KV)\n        N, L_Q, D = x_q.size()\n        _, L_KV, _ = x_k.size()\n        Q = self.rope(self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k))\n        K = self.rope(self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k))\n        V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)\n\n        score = torch.einsum('nihd,njhd-&gt;nijh', Q, K) / self.sqrt_d_k\n        # score: (N, L_Q, L_KV, num_heads)\n        # Apply attention mask\n        if attention_mask is not None:\n            score = score.masked_fill(\n                attention_mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')\n            )\n        # Apply padding mask\n        if padding_mask is not None:\n            score = score.masked_fill(\n                padding_mask.reshape(N, 1, L_KV, 1) == 0, float('-inf')\n            )\n        score = torch.nn.functional.softmax(score, dim=2)\n        value = torch.einsum(\n            'nijh,njhd-&gt;nihd', score, V\n        ).reshape(N, L_Q, self.d_model)\n        return self.W_O(value)\n\n\nclass MultiHeadSelfAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int, rope=RoPE):\n        super(MultiHeadSelfAttention, self).__init__(d_model, num_heads, rope)\n\n    def forward(\n        self, x: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        return super().forward(x, x, x, padding_mask, attention_mask)\n</pre> class MultiHeadAttention(torch.nn.Module):     def __init__(self, d_model: int, num_heads: int, rope: RoPE):         super(MultiHeadAttention, self).__init__()         self.d_model = d_model         self.num_heads = num_heads         if d_model % num_heads != 0:             raise ValueError(\"d_model must be divisible by num_heads\")          self.d_k = d_model // num_heads         self.sqrt_d_k = self.d_k ** 0.5         self.rope = rope          self.W_Q = torch.nn.Linear(d_model, d_model, bias=False)         self.W_K = torch.nn.Linear(d_model, d_model, bias=False)         self.W_V = torch.nn.Linear(d_model, d_model, bias=False)         self.W_O = torch.nn.Linear(d_model, d_model, bias=False)      def forward(         self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,         padding_mask: torch.Tensor | None = None,         attention_mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         # x_q: (N, L_Q, D), x_k: (N, L_KV, D), x_v: (N, L_KV, D)         # padding_mask: (N, L_KV), attention_mask: (L_Q, L_KV)         N, L_Q, D = x_q.size()         _, L_KV, _ = x_k.size()         Q = self.rope(self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k))         K = self.rope(self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k))         V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)          score = torch.einsum('nihd,njhd-&gt;nijh', Q, K) / self.sqrt_d_k         # score: (N, L_Q, L_KV, num_heads)         # Apply attention mask         if attention_mask is not None:             score = score.masked_fill(                 attention_mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')             )         # Apply padding mask         if padding_mask is not None:             score = score.masked_fill(                 padding_mask.reshape(N, 1, L_KV, 1) == 0, float('-inf')             )         score = torch.nn.functional.softmax(score, dim=2)         value = torch.einsum(             'nijh,njhd-&gt;nihd', score, V         ).reshape(N, L_Q, self.d_model)         return self.W_O(value)   class MultiHeadSelfAttention(MultiHeadAttention):     def __init__(self, d_model: int, num_heads: int, rope=RoPE):         super(MultiHeadSelfAttention, self).__init__(d_model, num_heads, rope)      def forward(         self, x: torch.Tensor,         padding_mask: torch.Tensor | None = None,         attention_mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         return super().forward(x, x, x, padding_mask, attention_mask) <p>\u200b\u5728\u200b\u524d\u9988\u200b\u7f51\u7edc\u200b\u90e8\u5206\u200b\uff0cLlama\u200b\u4f7f\u7528\u200bSwiglu\u200b\u4f5c\u4e3a\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\u3002\u200b\u6b64\u200b\u90e8\u5206\u200b\u540c\u6837\u200b\u4e0d\u200b\u4f7f\u7528\u200b\u504f\u7f6e\u200b</p> In\u00a0[4]: Copied! <pre>class FFN(torch.nn.Module):\n    def __init__(self, input_dim: int, hidden_dim: int):\n        super(FFN, self).__init__()\n        self.fc1 = torch.nn.Linear(input_dim, hidden_dim, bias=False)\n        self.fc2 = torch.nn.Linear(hidden_dim, input_dim, bias=False)\n        self.gate = torch.nn.Linear(input_dim, hidden_dim, bias=False)\n        self.act = torch.nn.SiLU()\n\n    def forward(self, x):\n        return self.fc2(self.act(self.gate(x)) * self.fc1(x))\n</pre> class FFN(torch.nn.Module):     def __init__(self, input_dim: int, hidden_dim: int):         super(FFN, self).__init__()         self.fc1 = torch.nn.Linear(input_dim, hidden_dim, bias=False)         self.fc2 = torch.nn.Linear(hidden_dim, input_dim, bias=False)         self.gate = torch.nn.Linear(input_dim, hidden_dim, bias=False)         self.act = torch.nn.SiLU()      def forward(self, x):         return self.fc2(self.act(self.gate(x)) * self.fc1(x)) <p>Llama\u200b\u4f7f\u7528\u200bRMSNorm\u200b\u4f5c\u4e3a\u200bLayerNorm\u3002</p> In\u00a0[5]: Copied! <pre>class RMSNorm(torch.nn.Module):\n    def __init__(self, dim: int, eps: float = 1e-8):\n        super(RMSNorm, self).__init__()\n        self.weight = torch.nn.Parameter(torch.ones(dim))\n        self.eps = eps\n\n    def forward(self, x: torch.Tensor):\n        return x / torch.sqrt(torch.mean(x ** 2, dim=-1, keepdim=True) + self.eps) * self.weight\n</pre> class RMSNorm(torch.nn.Module):     def __init__(self, dim: int, eps: float = 1e-8):         super(RMSNorm, self).__init__()         self.weight = torch.nn.Parameter(torch.ones(dim))         self.eps = eps      def forward(self, x: torch.Tensor):         return x / torch.sqrt(torch.mean(x ** 2, dim=-1, keepdim=True) + self.eps) * self.weight <p>Llama\u200b\u4f7f\u7528\u200bPre-norm\uff0c\u200b\u5373\u5148\u200b\u8fdb\u884c\u200bnorm\uff0c\u200b\u518d\u200b\u8fdb\u884c\u200battention/FFN\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u6700\u540e\u200b\u76f8\u52a0\u200b\u3002\u200b\u636e\u6b64\u200b\u53ef\u4ee5\u200b\u6784\u5efa\u200b\u51fa\u200bLlamaDecoder\u200b\u7684\u200b\u57fa\u672c\u200b\u7ed3\u6784\u200b\u3002</p> In\u00a0[6]: Copied! <pre>class LlamaDecoder(torch.nn.Module):\n    def __init__(\n        self, d_model: int, num_heads: int, rope: RoPE, hidden_dim: int, layernorm_eps: float\n    ):\n        super().__init__()\n        self.norm1 = RMSNorm(d_model, layernorm_eps)\n        self.attention = MultiHeadSelfAttention(d_model, num_heads, rope)\n        self.norm2 = RMSNorm(d_model, layernorm_eps)\n        self.ffn = FFN(d_model, hidden_dim)\n\n    def forward(self, x: torch.Tensor, padding_mask: torch.Tensor | None = None, attention_mask: torch.Tensor | None = None):\n        hidden = x\n        hidden += self.attention(self.norm1(hidden), padding_mask, attention_mask)\n        hidden += self.ffn(self.norm2(hidden))\n        return hidden\n</pre> class LlamaDecoder(torch.nn.Module):     def __init__(         self, d_model: int, num_heads: int, rope: RoPE, hidden_dim: int, layernorm_eps: float     ):         super().__init__()         self.norm1 = RMSNorm(d_model, layernorm_eps)         self.attention = MultiHeadSelfAttention(d_model, num_heads, rope)         self.norm2 = RMSNorm(d_model, layernorm_eps)         self.ffn = FFN(d_model, hidden_dim)      def forward(self, x: torch.Tensor, padding_mask: torch.Tensor | None = None, attention_mask: torch.Tensor | None = None):         hidden = x         hidden += self.attention(self.norm1(hidden), padding_mask, attention_mask)         hidden += self.ffn(self.norm2(hidden))         return hidden In\u00a0[7]: Copied! <pre>class LlamaModel(torch.nn.Module):\n    def __init__(\n        self, config: transformers.LlamaConfig\n    ):\n        super().__init__()\n        self.embedding = torch.nn.Embedding(config.vocab_size, config.hidden_size)\n        self.rope = RoPE(\n            config.hidden_size // config.num_attention_heads,\n            config.rope_theta\n        )\n        self.layers = torch.nn.ModuleList([\n            LlamaDecoder(\n                config.hidden_size,\n                config.num_attention_heads,\n                self.rope,\n                config.intermediate_size,\n                config.rms_norm_eps\n            )\n            for _ in range(config.num_hidden_layers)\n        ])\n        self.norm = RMSNorm(config.hidden_size)\n        self.lm_head = torch.nn.Linear(config.hidden_size, config.vocab_size, bias=False)\n\n    def forward(\n        self, input_ids: torch.Tensor,\n        padding_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        hidden = self.embedding(input_ids)\n        attention_mask = 1 - torch.triu(\n            torch.ones(hidden.size(1), hidden.size(1)), diagonal=1\n        ).to(hidden.device)\n        for layer in self.layers:\n            hidden = layer(hidden, padding_mask, attention_mask)\n        hidden = self.norm(hidden)\n        hidden = self.lm_head(hidden)\n        return hidden\n</pre> class LlamaModel(torch.nn.Module):     def __init__(         self, config: transformers.LlamaConfig     ):         super().__init__()         self.embedding = torch.nn.Embedding(config.vocab_size, config.hidden_size)         self.rope = RoPE(             config.hidden_size // config.num_attention_heads,             config.rope_theta         )         self.layers = torch.nn.ModuleList([             LlamaDecoder(                 config.hidden_size,                 config.num_attention_heads,                 self.rope,                 config.intermediate_size,                 config.rms_norm_eps             )             for _ in range(config.num_hidden_layers)         ])         self.norm = RMSNorm(config.hidden_size)         self.lm_head = torch.nn.Linear(config.hidden_size, config.vocab_size, bias=False)      def forward(         self, input_ids: torch.Tensor,         padding_mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         hidden = self.embedding(input_ids)         attention_mask = 1 - torch.triu(             torch.ones(hidden.size(1), hidden.size(1)), diagonal=1         ).to(hidden.device)         for layer in self.layers:             hidden = layer(hidden, padding_mask, attention_mask)         hidden = self.norm(hidden)         hidden = self.lm_head(hidden)         return hidden In\u00a0[8]: Copied! <pre>llama_config = transformers.AutoConfig.from_pretrained('meta-llama/Llama-2-7b-hf')\nllama = LlamaModel(llama_config)\n</pre> llama_config = transformers.AutoConfig.from_pretrained('meta-llama/Llama-2-7b-hf') llama = LlamaModel(llama_config) <pre>/usr/local/lib/python3.10/dist-packages/huggingface_hub/file_download.py:1132: FutureWarning: `resume_download` is deprecated and will be removed in version 1.0.0. Downloads always resume when possible. If you want to force a new download, use `force_download=True`.\n  warnings.warn(\n</pre> <p>\u200b\u52a0\u8f7d\u200bHuggingface\u200b\u63d0\u4f9b\u200b\u7684\u200bLlama\u200b\u6a21\u578b\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5c06\u200b\u9884\u200b\u8bad\u7ec3\u200b\u7684\u200bLlama\u200b\u53c2\u6570\u200b\u590d\u5236\u5230\u200b\u5b9e\u73b0\u200b\u7684\u200b\u6a21\u578b\u200b\u4e2d\u200b\u3002</p> In\u00a0[9]: Copied! <pre>hf_llama = transformers.LlamaForCausalLM.from_pretrained('meta-llama/Llama-2-7b-hf')\n\ndef load_params(custom_model: LlamaModel, hf_model: transformers.LlamaForCausalLM):\n    layer_pairs = [\n        (custom_model.embedding, hf_model.model.embed_tokens),\n        (custom_model.lm_head, hf_model.lm_head),\n        (custom_model.norm, hf_model.model.norm)\n    ]\n    for custom_layer, hf_layer in zip(custom_model.layers, hf_model.model.layers):\n        layer_pairs.extend([\n            (custom_layer.norm1, hf_layer.input_layernorm),\n            (custom_layer.norm2, hf_layer.post_attention_layernorm),\n            (custom_layer.attention.W_Q, hf_layer.self_attn.q_proj),\n            (custom_layer.attention.W_K, hf_layer.self_attn.k_proj),\n            (custom_layer.attention.W_V, hf_layer.self_attn.v_proj),\n            (custom_layer.attention.W_O, hf_layer.self_attn.o_proj),\n            (custom_layer.ffn.gate, hf_layer.mlp.gate_proj),\n            (custom_layer.ffn.fc1, hf_layer.mlp.up_proj),\n            (custom_layer.ffn.fc2, hf_layer.mlp.down_proj)\n        ])\n    for custom_layer, hf_layer in layer_pairs:\n        custom_layer.weight.data = hf_layer.weight.data.clone().detach()\n    return custom_model\n\nllama = load_params(llama, hf_llama)\n</pre> hf_llama = transformers.LlamaForCausalLM.from_pretrained('meta-llama/Llama-2-7b-hf')  def load_params(custom_model: LlamaModel, hf_model: transformers.LlamaForCausalLM):     layer_pairs = [         (custom_model.embedding, hf_model.model.embed_tokens),         (custom_model.lm_head, hf_model.lm_head),         (custom_model.norm, hf_model.model.norm)     ]     for custom_layer, hf_layer in zip(custom_model.layers, hf_model.model.layers):         layer_pairs.extend([             (custom_layer.norm1, hf_layer.input_layernorm),             (custom_layer.norm2, hf_layer.post_attention_layernorm),             (custom_layer.attention.W_Q, hf_layer.self_attn.q_proj),             (custom_layer.attention.W_K, hf_layer.self_attn.k_proj),             (custom_layer.attention.W_V, hf_layer.self_attn.v_proj),             (custom_layer.attention.W_O, hf_layer.self_attn.o_proj),             (custom_layer.ffn.gate, hf_layer.mlp.gate_proj),             (custom_layer.ffn.fc1, hf_layer.mlp.up_proj),             (custom_layer.ffn.fc2, hf_layer.mlp.down_proj)         ])     for custom_layer, hf_layer in layer_pairs:         custom_layer.weight.data = hf_layer.weight.data.clone().detach()     return custom_model  llama = load_params(llama, hf_llama) <pre>Loading checkpoint shards:   0%|          | 0/2 [00:00&lt;?, ?it/s]</pre> <p>\u200b\u52a0\u8f7d\u200btokenizer\u200b\u5e76\u200b\u751f\u6210\u200b\u8f93\u5165\u200b\u6570\u636e\u200b\uff0c\u200b\u6ce8\u610f\u200bLlama-2\u200b\u6ca1\u6709\u200bpadding token\uff0c\u200b\u9700\u8981\u200b\u624b\u52a8\u200b\u8bbe\u7f6e\u200b\u3002</p> In\u00a0[10]: Copied! <pre>llama_tokenizer = transformers.LlamaTokenizer.from_pretrained('meta-llama/Llama-2-7b-hf')\nllama_tokenizer.pad_token = llama_tokenizer.eos_token\ntokenized_sentence = llama_tokenizer([\n    'User: Hello Llama! How are you?\\nAssistant:',\n    'User: Hello World!\\nAssistant:'\n], return_tensors='pt', padding=True)\ninput_ids = tokenized_sentence['input_ids']\npadding_mask = tokenized_sentence['attention_mask']\n</pre> llama_tokenizer = transformers.LlamaTokenizer.from_pretrained('meta-llama/Llama-2-7b-hf') llama_tokenizer.pad_token = llama_tokenizer.eos_token tokenized_sentence = llama_tokenizer([     'User: Hello Llama! How are you?\\nAssistant:',     'User: Hello World!\\nAssistant:' ], return_tensors='pt', padding=True) input_ids = tokenized_sentence['input_ids'] padding_mask = tokenized_sentence['attention_mask'] <p>\u200b\u6839\u636e\u200b\u8f93\u5165\u200b\u6570\u636e\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u6bcf\u5c42\u200b\u6a21\u578b\u200b\u7684\u200b\u8f93\u51fa\u200b\u3002</p> In\u00a0[11]: Copied! <pre>llama_layers_output = hf_llama(input_ids, attention_mask=padding_mask, output_hidden_states=True)\n</pre> llama_layers_output = hf_llama(input_ids, attention_mask=padding_mask, output_hidden_states=True) <p>\u200b\u4ee5\u4e0b\u200b\u5bf9\u200b\u8f93\u51fa\u200b\u7ed3\u679c\u200b\u8fdb\u884c\u200b\u9a8c\u8bc1\u200b\u3002\u200b\u9996\u5148\u200b\u9a8c\u8bc1\u200bembedding\u200b\u5c42\u200b\u7684\u200b\u8f93\u51fa\u200b\u3002</p> In\u00a0[12]: Copied! <pre>llama_layers_output.hidden_states[0]\nhidden = llama.embedding(input_ids)\nattention_mask = 1 - torch.triu(\n    torch.ones(hidden.size(1), hidden.size(1)), diagonal=1\n).to(device=hidden.device)\n\n# Hidden state 1 equivalent to embedding output\ntorch.allclose(\n    llama_layers_output.hidden_states[0],\n    hidden\n)\n</pre> llama_layers_output.hidden_states[0] hidden = llama.embedding(input_ids) attention_mask = 1 - torch.triu(     torch.ones(hidden.size(1), hidden.size(1)), diagonal=1 ).to(device=hidden.device)  # Hidden state 1 equivalent to embedding output torch.allclose(     llama_layers_output.hidden_states[0],     hidden ) Out[12]: <pre>True</pre> <p>\u200b\u9a8c\u8bc1\u200b\u6bcf\u200b\u4e00\u5c42\u200bdecoder\u200b\u7684\u200b\u8f93\u51fa\u200b\uff0c\u200b\u6700\u540e\u200b\u4e00\u5c42\u200bdecoder\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u51fa\u73b0\u200b\u4e0d\u540c\u200b\u3002</p> In\u00a0[13]: Copied! <pre># Verify layer outputs\nlayer_results = []\nfor i in range(llama_config.num_hidden_layers - 1):\n    hidden = llama.layers[i](hidden, padding_mask, attention_mask)\n    layer_results.append(torch.allclose(\n        llama_layers_output.hidden_states[i + 1], hidden, atol=1e-4\n    ))\nall(layer_results), all(layer_results[:-1])\n</pre> # Verify layer outputs layer_results = [] for i in range(llama_config.num_hidden_layers - 1):     hidden = llama.layers[i](hidden, padding_mask, attention_mask)     layer_results.append(torch.allclose(         llama_layers_output.hidden_states[i + 1], hidden, atol=1e-4     )) all(layer_results), all(layer_results[:-1]) Out[13]: <pre>(False, True)</pre> <p>\u200b\u9a8c\u8bc1\u200bLM-head\u200b\u7684\u200b\u8f93\u51fa\u200b\u3002</p> In\u00a0[14]: Copied! <pre># Verify output logits\nlogits = llama(input_ids, padding_mask)\nhf_logits = llama_layers_output.logits\ntorch.allclose(logits, hf_logits, atol=1e-4)\n</pre> # Verify output logits logits = llama(input_ids, padding_mask) hf_logits = llama_layers_output.logits torch.allclose(logits, hf_logits, atol=1e-4) Out[14]: <pre>True</pre> <p>\u200b\u5c1d\u8bd5\u200b\u5bf9\u200b\u8f93\u51fa\u200b\u7684\u200b\u5355\u8bcd\u200b\u8fdb\u884c\u200b\u89e3\u7801\u200b\u3002</p> In\u00a0[15]: Copied! <pre># Decode the output tokens\nmax_prob_tokens = logits[0, -1].topk(5).indices.tolist()\nfor _ in max_prob_tokens:\n    print(llama_tokenizer.decode(_))\n</pre> # Decode the output tokens max_prob_tokens = logits[0, -1].topk(5).indices.tolist() for _ in max_prob_tokens:     print(llama_tokenizer.decode(_)) <pre>I\nHello\nFine\nHi\nOh\n</pre> In\u00a0[16]: Copied! <pre>del hf_llama\ndel llama_layers_output\ndel hf_logits\nimport gc\ngc.collect()\n</pre> del hf_llama del llama_layers_output del hf_logits import gc gc.collect() Out[16]: <pre>21</pre> In\u00a0[17]: Copied! <pre>def complete_ids(model: LlamaModel, input_ids: torch.Tensor):\n    assert input_ids.size(0) == 1\n\n    next_token = model(input_ids, torch.ones_like(input_ids))[0, -1, :].topk(1).indices[0].item()\n    return next_token\n\ndef complete_string(model: LlamaModel, tokenizer: transformers.LlamaTokenizer, prompt: str, max_length: int):\n    model.eval()\n    gc.collect()\n    tokenized_sentence = tokenizer([prompt], return_tensors='pt')['input_ids'].tolist()\n    next_token = None\n    print(tokenizer.decode(tokenized_sentence[0]))\n    num_tokens = 0\n\n    while next_token != tokenizer.eos_token_id and num_tokens &lt;= max_length:\n        num_tokens += 1\n        next_token = complete_ids(model, torch.tensor(tokenized_sentence))\n        tokenized_sentence[0].append(next_token)\n        print(tokenizer.decode(tokenized_sentence[0]))\n\n        gc.collect()\n\n    return num_tokens\n</pre> def complete_ids(model: LlamaModel, input_ids: torch.Tensor):     assert input_ids.size(0) == 1      next_token = model(input_ids, torch.ones_like(input_ids))[0, -1, :].topk(1).indices[0].item()     return next_token  def complete_string(model: LlamaModel, tokenizer: transformers.LlamaTokenizer, prompt: str, max_length: int):     model.eval()     gc.collect()     tokenized_sentence = tokenizer([prompt], return_tensors='pt')['input_ids'].tolist()     next_token = None     print(tokenizer.decode(tokenized_sentence[0]))     num_tokens = 0      while next_token != tokenizer.eos_token_id and num_tokens &lt;= max_length:         num_tokens += 1         next_token = complete_ids(model, torch.tensor(tokenized_sentence))         tokenized_sentence[0].append(next_token)         print(tokenizer.decode(tokenized_sentence[0]))          gc.collect()      return num_tokens In\u00a0[18]: Copied! <pre>import time\nquery = 'User: Given two numbers, 13.11 and 13.8, which is larger?\\nAssistant:'\n\nstart_time = time.time()\nnum_tokens = complete_string(llama, llama_tokenizer, query, 10)\nend_time = time.time()\n\ntime_span = end_time - start_time\nprint(f'Predicted {num_tokens} in {time_span:.2f} seconds, {num_tokens / time_span:.2f} tokens/s')\n</pre> import time query = 'User: Given two numbers, 13.11 and 13.8, which is larger?\\nAssistant:'  start_time = time.time() num_tokens = complete_string(llama, llama_tokenizer, query, 10) end_time = time.time()  time_span = end_time - start_time print(f'Predicted {num_tokens} in {time_span:.2f} seconds, {num_tokens / time_span:.2f} tokens/s') <pre>&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant:\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is \n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is 1\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is 13\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is 13.\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is 13.8\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is 13.8.\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is 13.8.\n\nPredicted 11 in 37.42 seconds, 0.29 tokens/s\n</pre>"},{"location":"coding/dl-from-scratch/llama-2/","title":"\u6a21\u578b\u200b\u5b9e\u73b0\u200b\u00b6","text":""},{"location":"coding/dl-from-scratch/llama-2/","title":"\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u00b6","text":""},{"location":"coding/dl-from-scratch/llama-2/","title":"\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u00b6","text":""},{"location":"coding/dl-from-scratch/llama-2/#ffn","title":"FFN\u00b6","text":""},{"location":"coding/dl-from-scratch/llama-2/#layernorm","title":"LayerNorm\u00b6","text":""},{"location":"coding/dl-from-scratch/llama-2/#decoder-layer","title":"Decoder 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of sentence\uff09\u200b\u548c\u200beos\uff08end of sentence\uff09\u200b\u7684\u200b\u9884\u6d4b\u200b\u3002\u200b\u6240\u4ee5\u200b\uff0c\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b\u57fa\u5ea7\u200b\u6a21\u578b\u200b\uff0c\u200b\u6a21\u578b\u200b\u4f1a\u200b\u4e0d\u65ad\u200b\u8f93\u51fa\u200btoken\u3002</p>"},{"location":"coding/dl-from-scratch/llm-inference/#kv-cache","title":"KV-cache","text":"<p>\u200b\u628a\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u91cd\u65b0\u200b\u8f93\u51fa\u200bLLM\uff0c\u200b\u9884\u6d4b\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5355\u8bcd\u200b\u63a8\u7406\u200b\u6d41\u7a0b\u200b\u7684\u200b\u8ba1\u7b97\u200b\u6548\u7387\u200b\u975e\u5e38\u4f4e\u200b\uff0c\u200b\u539f\u56e0\u200b\u5728\u4e8e\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u9884\u6d4b\u200b\u90fd\u200b\u9700\u8981\u200b\u5bf9\u200b\u6574\u4e2a\u200b\u5e8f\u5217\u200b\u8ba1\u7b97\u200b\u6ce8\u610f\u529b\u200b\u5f97\u5206\u200b\u3002\u200b\u7136\u800c\u200b\uff0c\u200b\u7531\u4e8e\u200b\u6bcf\u200b\u4e00\u4e2a\u200btoken\u200b\u7684\u200b\u8868\u793a\u200b\u53ea\u200b\u4f9d\u8d56\u4e8e\u200b\u524d\u9762\u200b\u7684\u200btoken\uff0c\u200b\u5728\u200b\u63a8\u7406\u200b\u9636\u6bb5\u200b\uff0c\u200b\u6211\u4eec\u200b\u4e0d\u5fc5\u200b\u91cd\u7b97\u200b\u524d\u8fb9\u200b\u7684\u200btoken\uff0c\u200b\u53ea\u200b\u9700\u8981\u200b\u5b58\u50a8\u200b\u524d\u8fb9\u200b\u7684\u200btoken\u200b\u7684\u200bKV\u200b\u8868\u793a\u200b\u5373\u53ef\u200b\u3002\u200b\u53ea\u200b\u9700\u7528\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200btoken\u200b\u7684\u200bQ\u200b\u8ba1\u7b97\u200b\u6ce8\u610f\u529b\u200b\u5373\u53ef\u200b\u5f97\u5230\u200b\u5bf9\u4e0b\u200b\u4e00\u4e2a\u200btoken\u200b\u7684\u200b\u9884\u6d4b\u200b\u3002\u200b\u8fd9\u79cd\u200b\u6280\u672f\u200b\u5373\u200b\u79f0\u4e3a\u200bKV\u200b\u7f13\u5b58\u200b\u3002\u200b\u5305\u542b\u200bKV\u200b\u7f13\u5b58\u200b\u7684\u200b\u63a8\u7406\u200b\u8fc7\u7a0b\u200b\u5206\u4e3a\u200b\u4e24\u4e2a\u200b\u9636\u6bb5\u200b\uff0c\u200b\u5373\u200b\u7f13\u5b58\u200b\u586b\u5145\u200b\uff08prefill\uff09\u200b\u9636\u6bb5\u200b\u548c\u200b\u89e3\u7801\u200b\uff08decode\uff09\u200b\u9636\u6bb5\u200b\u3002</p> <p>\u200b\u4e3a\u4f55\u200b\u53ea\u200b\u9700\u8981\u200bKV-cache\u200b\u800c\u200b\u4e0d\u200b\u9700\u8981\u200bQ-cache</p> <p>\u200b\u5047\u8bbe\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u7ecf\u8fc7\u200b\\(QKV\\)\u200b\u8ba1\u7b97\u200b\u540e\u200b\u7684\u200b\u5e8f\u5217\u200b\u4e3a\u200b\\(S^Q, S^K, S^V\\)\uff0c\u200b\u5e76\u4e14\u200b\u53ef\u4ee5\u200b\u62c6\u200b\u5206\u4e3a\u200b\u5217\u200b\u5411\u91cf\u200b\u8868\u793a\u200b\uff1a</p> \\[ \\begin{aligned} S^Q &amp;= \\begin{bmatrix} Q_1 &amp; Q_2 &amp; \\cdots &amp; Q_n \\end{bmatrix} \\\\ S^K &amp;= \\begin{bmatrix} K_1 &amp; K_2 &amp; \\cdots &amp; K_n \\end{bmatrix} \\\\ S^V &amp;= \\begin{bmatrix} V_1 &amp; V_2 &amp; \\cdots &amp; V_n \\end{bmatrix} \\] <p>\u200b\u6ce8\u610f\u529b\u200b\u5f97\u5206\u200b\u77e9\u9635\u200b\u4e3a\u200b</p> \\[ \\begin{bmatrix} Q_1K_1 &amp; Q_1K_2 &amp; \\cdots &amp; Q_1K_n \\\\ Q_2K_1 &amp; Q_2K_2 &amp; \\cdots &amp; Q_2K_n \\\\ \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\ Q_nK_1 &amp; Q_nK_2 &amp; \\cdots &amp; Q_nK_n \\] <p>\u200b\u7531\u4e8e\u200bdecoder\u200b\u7684\u200bcausal mask\u200b\u9650\u5236\u200b\uff0c\u200b\u6700\u7ec8\u200b\u8ba1\u7b97\u200b\u7684\u200b\u8868\u793a\u200b\u4e3a\u200b</p> \\[ \\begin{aligned} O_1 &amp;= \\sum_{i=1}^1 \\text{softmax}(Q_1K_i)V_i \\\\ O_2 &amp;= \\sum_{i=1}^2 \\text{softmax}(Q_2K_i)V_i \\\\ \\end{aligned} \\] <p>\u200b\u4e5f\u200b\u5373\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u8868\u793a\u200b\uff0c\u200b\u53ea\u200b\u4f9d\u8d56\u200b\u8be5\u200b\u4f4d\u7f6e\u200b\u7684\u200bQ\uff0c\u200b\u548c\u200b\u8be5\u200b\u4f4d\u7f6e\u200b\u53ca\u200b\u4e4b\u524d\u200b\u7684\u200bK\u200b\u548c\u200bV\u3002\u200b\u56e0\u6b64\u200b\u9884\u6d4b\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5355\u8bcd\u200b\u65f6\u200b\uff0c\u200b\u4e0d\u200b\u9700\u8981\u200b\u91cd\u65b0\u200b\u8ba1\u7b97\u200bKV\uff0c\u200b\u5c06\u200bKV\u200b\u5b58\u50a8\u200b\u4e0b\u6765\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u5c06\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u8f93\u5165\u200bLLM\u200b\u65f6\u200b\uff0c\u200b\u9996\u5148\u200b\u8fdb\u884c\u200b\u7f13\u5b58\u200b\u586b\u5145\u200b\uff0c\u200b\u4fdd\u5b58\u200b\u4e0b\u200b\u6240\u6709\u200b\u5c42\u200b\u7684\u200bKV-cache\uff0c\u200b\u7136\u540e\u200b\u8fdb\u884c\u200b\u89e3\u7801\u200b\uff0c\u200b\u4e0d\u65ad\u200b\u9884\u6d4b\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5355\u8bcd\u200b\uff0c\u200b\u5e76\u200b\u8bb0\u5f55\u200b\u9884\u6d4b\u200b\u5355\u8bcd\u200b\u7684\u200bKV\u200b\u8868\u793a\u200b\u3002\u200b\u5bf9\u4e8e\u200bdecoder\u200b\u7684\u200b\u6bcf\u200b\u4e00\u5c42\u200b\uff0c\u200b\u90fd\u200b\u9700\u8981\u200b\u4fdd\u5b58\u200b\u4e00\u4e2a\u200bKV-cache\uff0c\u200b\u6bcf\u200b\u4e00\u5c42\u200bcache\u200b\u7684\u200b\u5927\u5c0f\u200b\u4e3a\u200b\\(2\\times N\\times L\\times D\\)\u3002\u200b\u5bf9\u4e8e\u200b\u957f\u200b\u5e8f\u5217\u200b\uff0cKV-cache\u200b\u7684\u200b\u5927\u5c0f\u200b\u4f1a\u200b\u975e\u5e38\u200b\u5927\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/loss/","title":"\u635f\u5931\u200b\u51fd\u6570","text":"<p>\u200b\u635f\u5931\u200b\u51fd\u6570\u200b\u662f\u200b\u7528\u6765\u200b\u8bc4\u4ef7\u200b\u6a21\u578b\u200b\u9884\u6d4b\u503c\u200b\u4e0e\u200b\u771f\u5b9e\u200b\u503c\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5dee\u5f02\u200b\u7684\u200b\u51fd\u6570\u200b\u3002\u200b\u7136\u540e\u200b\u901a\u8fc7\u200b\u4f18\u5316\u200b\u7b97\u6cd5\u200b\u6765\u200b\u8c03\u6574\u200b\u6a21\u578b\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u635f\u5931\u200b\u51fd\u6570\u200b\u7684\u200b\u503c\u200b\u6700\u5c0f\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/loss/#_2","title":"\u5747\u65b9\u200b\u8bef\u5dee","text":"<p>\u200b\u5747\u65b9\u200b\u8bef\u5dee\u200b\uff08Mean Squared Error, MSE\uff09\u200b\u662f\u200b\u56de\u5f52\u200b\u4efb\u52a1\u200b\u4e2d\u200b\u6700\u200b\u5e38\u7528\u200b\u7684\u200b\u635f\u5931\u200b\u51fd\u6570\u200b\u3002\u200b\u5b83\u200b\u8ba1\u7b97\u200b\u7684\u200b\u662f\u200b\u9884\u6d4b\u503c\u200b\u4e0e\u200b\u771f\u5b9e\u200b\u503c\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5e73\u65b9\u5dee\u200b\u7684\u200b\u5747\u503c\u200b\u3002</p> \\[ \\begin{aligned} L(y, t) &amp;= \\frac{1}{2} \\sum_{i=1}^{n} (y_i - t_i)^2 \\\\ \\frac{\\partial L}{\\partial y_i} &amp;= y_i - t_i \\end{aligned} \\] <p>\u200b\u5176\u4e2d\u200b\uff0c\\(y\\) \u200b\u662f\u200b\u6a21\u578b\u200b\u7684\u200b\u9884\u6d4b\u503c\u200b\uff0c\\(t\\) \u200b\u662f\u200b\u771f\u5b9e\u200b\u503c\u200b\uff0c\\(n\\) \u200b\u662f\u200b\u6837\u672c\u200b\u6570\u91cf\u200b\u3002</p> <pre><code>def mse(y: torch.Tensor, t: torch.Tensor) -&gt; torch.Tensor:\n    return 0.5 * torch.sum((y - t) ** 2)\n\nclass MSELoss(torch.nn.Module):\n    def __init__(self):\n        super(MSELoss, self).__init__()\n\n    def forward(self, y: torch.Tensor, t: torch.Tensor) -&gt; torch.Tensor:\n        return mse(y, t)\n</code></pre>"},{"location":"coding/dl-from-scratch/loss/#_3","title":"\u4ea4\u53c9\u200b\u71b5","text":"<p>\u200b\u4ea4\u53c9\u200b\u71b5\u200b\uff08Cross Entropy\uff09\u200b\u662f\u200b\u591a\u200b\u5206\u7c7b\u200b\u4efb\u52a1\u200b\u4e2d\u200b\u6700\u200b\u5e38\u7528\u200b\u7684\u200b\u635f\u5931\u200b\u51fd\u6570\u200b\u3002\u200b\u5b83\u200b\u8ba1\u7b97\u200b\u7684\u200b\u662f\u200b\u6a21\u578b\u200b\u8f93\u51fa\u200b\u503c\u200b\u4e0e\u200b\u771f\u5b9e\u200b\u503c\u200b\u4e4b\u95f4\u200b\u7684\u200b\u4ea4\u53c9\u200b\u71b5\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200bKL\u200b\u6563\u5ea6\u200b\u63a8\u5bfc\u200b\u5f97\u5230\u200b\u3002</p> \\[ \\begin{aligned} KL(p||q) &amp;= -\\sum_{i=1}^{n} p_i (\\log q_i - \\log p_i) \\\\ &amp;= -\\sum_{i=1}^{n} p_i \\log q_i + \\sum_{i=1}^{n} p_i \\log p_i \\\\ &amp;= H(p, q) - H(p) \\end{aligned} \\] <p>\u200b\u5176\u4e2d\u200b\uff0c\\(p\\) \u200b\u662f\u200b\u771f\u5b9e\u200b\u6982\u7387\u5206\u5e03\u200b\uff0c\\(q\\) \u200b\u662f\u200b\u6a21\u578b\u200b\u8f93\u51fa\u200b\u6982\u7387\u5206\u5e03\u200b\uff0c\\(H(p)\\)\u200b\u8868\u793a\u200b\u771f\u5b9e\u200b\u5206\u5e03\u200b\\(p\\)\u200b\u7684\u200b\u71b5\u200b\uff0c\u200b\u4e3a\u200b\u5e38\u6570\u200b\u3002\u200b\u5f53\u200b\u8f93\u51fa\u200b\u5206\u5e03\u200b\\(q\\)\u200b\u4e0e\u200b\u771f\u5b9e\u200b\u5206\u5e03\u200b\\(p\\)\u200b\u76f8\u8fd1\u200b\u65f6\u200b\uff0cKL\u200b\u6563\u5ea6\u503c\u200b\u8d8a\u200b\u5c0f\u200b\uff0c\u200b\u4ea4\u53c9\u200b\u71b5\u200b\\(-\\sum_{i=1}^{n} p_i \\log q_i\\)\u200b\u503c\u200b\u4e5f\u200b\u8d8a\u200b\u5c0f\u200b\u3002</p> <p>\u200b\u9700\u8981\u200b\u6ce8\u610f\u200b\u7684\u200b\u662f\u200b\uff0c\u200b\u5728\u200bPyTorch\u200b\u4e2d\u200b\uff0c<code>torch.nn.CrossEntropyLoss</code>\u200b\u51fd\u6570\u200b\u7684\u200b\u8f93\u5165\u200b\u662f\u200b\u6a21\u578b\u200b\u7684\u200b\u8f93\u51fa\u200b\u503c\u200b\u548c\u200b\u771f\u5b9e\u200b\u6807\u7b7e\u200b\uff0c\u200b\u4e0d\u200b\u9700\u8981\u200b\u5bf9\u200b\u8f93\u51fa\u200b\u503c\u200b\u8fdb\u884c\u200bsoftmax\u200b\u64cd\u4f5c\u200b\u3002</p> <pre><code>def cross_entropy(score: torch.Tensor, label: torch.Tensor) -&gt; torch.Tensor:\n    # label: [N], score: [N, C]\n    score = torch.softmax(score, dim=-1)\n    label_logits = score[torch.arange(score.size(0)), label]\n    return -torch.sum(torch.log(label_logits))\n\nclass CrossEntropyLoss(torch.nn.Module):\n    def __init__(self):\n        super(CrossEntropyLoss, self).__init__()\n\n    def forward(self, score: torch.Tensor, label: torch.Tensor) -&gt; torch.Tensor:\n        return cross_entropy(score, label)\n</code></pre>"},{"location":"coding/dl-from-scratch/mlp/","title":"\u591a\u5c42\u200b\u611f\u77e5\u673a","text":"<p>\u200b\u591a\u5c42\u200b\u611f\u77e5\u673a\u200b\uff08Multilayer Perceptron, MLP\uff09\u200b\u662f\u200b\u6df1\u5ea6\u200b\u5b66\u4e60\u200b\u7684\u200b\u57fa\u7840\u200b\uff0c\u200b\u4e5f\u200b\u662f\u200b\u6700\u200b\u7b80\u5355\u200b\u7684\u200b\u6df1\u5ea6\u200b\u5b66\u4e60\u200b\u6a21\u578b\u200b\u4e4b\u4e00\u200b\uff0c\u200b\u7531\u200b\u7ebf\u6027\u53d8\u6362\u200b\u548c\u200b\u975e\u7ebf\u6027\u200b\u7684\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\u7ec4\u6210\u200b\u3002\u200b\u591a\u5c42\u200b\u611f\u77e5\u673a\u200b\u7684\u200b\u7f51\u7edc\u7ed3\u6784\u200b\u662f\u200b\u7531\u200b\u8f93\u5165\u200b\u5c42\u200b\u3001\u200b\u591a\u4e2a\u200b\u9690\u85cf\u200b\u5c42\u200b\u548c\u200b\u8f93\u51fa\u200b\u5c42\u200b\u7ec4\u6210\u200b\uff0c\u200b\u5176\u4e2d\u200b\u9690\u85cf\u200b\u5c42\u200b\u7684\u200b\u795e\u7ecf\u5143\u200b\u6570\u91cf\u200b\u548c\u200b\u5c42\u6570\u200b\u53ef\u4ee5\u200b\u81ea\u7531\u200b\u8bbe\u5b9a\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/mlp/#_2","title":"\u7ebf\u6027\u53d8\u6362","text":"<p>\u200b\u7ebf\u6027\u53d8\u6362\u200b\u662f\u200b\u591a\u5c42\u200b\u611f\u77e5\u673a\u200b\u7684\u200b\u57fa\u7840\u200b\uff0c\u200b\u5b83\u200b\u662f\u200b\u7531\u200b\u8f93\u5165\u200b\u5c42\u5230\u200b\u9690\u85cf\u200b\u5c42\u200b\u548c\u200b\u8f93\u51fa\u200b\u5c42\u200b\u7684\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\u7ec4\u6210\u200b\u3002\u200b\u7ebf\u6027\u53d8\u6362\u200b\u7684\u200b\u6570\u5b66\u200b\u8868\u8fbe\u5f0f\u200b\u4e3a\u200b\uff1a</p> \\[ y = Wx + b \\] <p>\u200b\u5176\u4e2d\u200b \\(W\\) \u200b\u662f\u200b\u6743\u91cd\u200b\u77e9\u9635\u200b\uff0c\\(x\\) \u200b\u662f\u200b\u8f93\u5165\u200b\u5411\u91cf\u200b\uff0c\\(b\\) \u200b\u662f\u200b\u504f\u7f6e\u200b\u5411\u91cf\u200b\uff0c\\(y\\) \u200b\u662f\u200b\u8f93\u51fa\u200b\u5411\u91cf\u200b\u3002</p> <pre><code>class Linear(torch.nn.Module):\n    def __init__(self, in_dim: int, out_dim: int):\n        super(Linear, self).__init__()\n        self.in_dim = in_dim\n        self.out_dim = out_dim\n        self.weight = torch.nn.Parameter(torch.randn(out_dim, in_dim))\n        self.bias = torch.nn.Parameter(torch.randn(out_dim))\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        return torch.einsum(\"ij,j-&gt;i\", self.weight, x) + self.bias\n</code></pre> <p>\u200b\u53cd\u5411\u200b\u4f20\u64ad\u200b\u4e2d\u200b\uff0c\u200b\u68af\u5ea6\u200b\u8ba1\u7b97\u516c\u5f0f\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned} \\frac{\\partial y}{\\partial W} &amp;= x \\\\ \\frac{\\partial y}{\\partial b} &amp;= 1 \\end{aligned} \\]"},{"location":"coding/dl-from-scratch/normalization/","title":"\u5f52\u4e00\u5316","text":"<p>\u200b\u5f52\u4e00\u5316\u200b\u662f\u200b\u6307\u200b\u5c06\u200b\u6570\u636e\u200b\u6309\u7167\u200b\u6bd4\u4f8b\u200b\u7f29\u653e\u200b\uff0c\u200b\u6539\u53d8\u200b\u5176\u200b\u5206\u5e03\u200b\u3002\u200b\u6309\u7167\u200b\u5f52\u4e00\u5316\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u4e0d\u540c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u4ee5\u4e0b\u200b\u51e0\u79cd\u200b\uff1a</p> <ul> <li>\u200b\u6837\u672c\u200b\u5f52\u4e00\u5316\u200b\uff08Batch Normalization\uff09\uff1a\u200b\u5bf9\u200b\u6bcf\u4e2a\u200b\u7279\u5f81\u200b\u8fdb\u884c\u200b\u5f52\u4e00\u5316\u200b\uff0c\u200b\u901a\u5e38\u200b\u7528\u4e8e\u200b\u56fe\u50cf\u200b\u6570\u636e\u200b\\(B \\times C \\times H \\times W\\)\uff0c\u200b\u5728\u200b\u540c\u4e00\u200b\\(C\\)\u200b\u7ef4\u5ea6\u200b\u4e0a\u200b\u8ba1\u7b97\u200b\u5747\u503c\u200b\u548c\u200b\u65b9\u5dee\u200b\u3002</li> <li>\u200b\u5c42\u200b\u5f52\u4e00\u5316\u200b\uff08Layer Normalization\uff09\uff1a\u200b\u5bf9\u200b\u6bcf\u4e2a\u200b\u6837\u672c\u200b\u8fdb\u884c\u200b\u5f52\u4e00\u5316\u200b\uff0c\u200b\u901a\u5e38\u200b\u7528\u4e8e\u200b\u5e8f\u5217\u200b\u6570\u636e\u200b\\(B\\times L \\times D\\)\uff0c\u200b\u5728\u200b\u540c\u4e00\u200b\\(B\\)\u200b\u7ef4\u5ea6\u200b\u4e0a\u200b\u8ba1\u7b97\u200b\u5747\u503c\u200b\u548c\u200b\u65b9\u5dee\u200b\u3002</li> <li>\u200b\u5b9e\u4f8b\u200b\u5f52\u4e00\u5316\u200b\uff08Instance Normalization\uff09\uff1a\u200b\u5bf9\u200b\u6bcf\u4e2a\u200b\u6837\u672c\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u7279\u5f81\u200b\u7ef4\u5ea6\u200b\u8fdb\u884c\u200b\u5f52\u4e00\u5316\u200b\uff0c\u200b\u5728\u200b\u540c\u4e00\u200b\\(B\\times C\\)\u200b\u7ef4\u5ea6\u200b\u4e0a\u200b\u8ba1\u7b97\u200b\u5747\u503c\u200b\u548c\u200b\u65b9\u5dee\u200b\u3002</li> <li>\u200b\u5206\u7ec4\u200b\u5f52\u4e00\u5316\u200b\uff08Group Normalization\uff09\uff1a\u200b\u5bf9\u200b\u6bcf\u4e2a\u200b\u6837\u672c\u200b\u4e2d\u200b\u7684\u200b\u7279\u5f81\u200b\u5206\u6210\u200b\u82e5\u5e72\u7ec4\u200b\u540e\u200b\uff0c\u200b\u5728\u200b\u540c\u200b\u4e00\u7ec4\u200b\u5185\u200b\u8fdb\u884c\u200b\u5f52\u4e00\u5316\u200b\u3002</li> </ul> <p>\u200b\u5f52\u4e00\u5316\u200b\u7684\u200b\u65b9\u5f0f\u200b\u4e3a\u200b</p> \\[ x' = \\frac{x - \\mu}{\\sigma} \\times \\gamma + \\beta \\] <p>\u200b\u5176\u4e2d\u200b\\(\\mu, \\sigma\\)\u200b\u5206\u522b\u200b\u4e3a\u200b\u6837\u672c\u5747\u503c\u200b\u548c\u200b\u6807\u51c6\u5dee\u200b\uff0c\\(\\gamma, \\beta\\)\u200b\u4e3a\u200b\u53ef\u200b\u5b66\u4e60\u200b\u7684\u200b\u91cd\u200b\u7f29\u653e\u200b\u53c2\u6570\u200b\u3002\u200b\u7ed9\u5b9a\u200b\u7279\u5f81\u200b\u7ef4\u5ea6\u200b\\(d\\)\uff0c\u200b\u5f52\u4e00\u5316\u200b\u5c42\u200b\u7684\u200b\u53c2\u200b\u6570\u91cf\u200b\u4e3a\u200b\\(2d\\)\u3002</p>"},{"location":"coding/dl-from-scratch/normalization/#rmsnorm","title":"RMSNorm","text":"<p>Zhang\u200b\u7b49\u200b\u4eba\u200b\u5728\u200bLN\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\u63d0\u51fa\u200b\u4e86\u200bRMSNorm\uff0c\u200b\u76f8\u6bd4\u200b\u4e8e\u200bLN\uff0cRMSNorm\u200b\u4e0d\u200b\u9700\u8981\u200b\u8ba1\u7b97\u200b\u5747\u503c\u200b\u548c\u200b\u65b9\u5dee\u200b\uff0c\u200b\u800c\u662f\u200b\u76f4\u63a5\u200b\u4f7f\u7528\u200b\u5747\u200b\u65b9\u6839\u200b\u4f5c\u4e3a\u200b\u5f52\u4e00\u5316\u200b\u7684\u200b\u6807\u51c6\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5728\u200b\u91cd\u200b\u7f29\u653e\u200b\u9636\u6bb5\u200b\u5220\u53bb\u200b\u4e86\u200b\u504f\u7f6e\u200b\\(\\beta\\)\u3002</p> \\[ x' = \\frac{x}{\\sqrt{\\frac{1}{N} \\sum_{i=1}^{N} x_i^2}} \\times \\gamma \\] <pre><code>class RMSNorm(torch.nn.Module):\n    def __init__(self, dim: int, eps=1e-8):\n        super(RMSNorm, self).__init__()\n        self.gamma = torch.nn.Parameter(torch.ones(dim))\n        self.eps = eps\n\n    def forward(self, x: torch.Tensor):\n        return x / torch.sqrt(torch.mean(x ** 2, dim=-1, keepdim=True) + self.eps) * self.gamma\n</code></pre>"},{"location":"coding/dl-from-scratch/positional-embedding/","title":"\u4f4d\u7f6e\u200b\u7f16\u7801","text":"<p>Attention\u200b\u673a\u5236\u200b\u867d\u7136\u200b\u80fd\u200b\u6355\u6349\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u4e0d\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u4f9d\u8d56\u200b\u5173\u7cfb\u200b\uff0c\u200b\u4f46\u662f\u200b\u65e0\u6cd5\u200b\u533a\u5206\u200b\u4e0d\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\u200b\u4e3a\u4e86\u200b\u89e3\u51b3\u200b\u8fd9\u4e2a\u200b\u95ee\u9898\u200b\uff0cTransformer\u200b\u6a21\u578b\u200b\u5f15\u5165\u200b\u4e86\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff08Positional Encoding\uff09\u3002</p>"},{"location":"coding/dl-from-scratch/positional-embedding/#_2","title":"\u7edd\u5bf9\u200b\u4f4d\u7f6e\u200b\u7f16\u7801","text":"<p>\u200b\u7edd\u5bf9\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u662f\u200bTransformer\u200b\u6a21\u578b\u200b\u4e2d\u200b\u6700\u65e9\u200b\u5f15\u5165\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u65b9\u5f0f\u200b\uff0c\u200b\u5176\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u4e3a\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u4f4d\u7f6e\u200b\u5206\u914d\u200b\u4e00\u4e2a\u200b\u552f\u4e00\u200b\u7684\u200b\u5411\u91cf\u200b\uff0c\u200b\u5c06\u200b\u8fd9\u4e2a\u200b\u5411\u91cf\u200b\u548c\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u76f8\u52a0\u200b\u5f97\u5230\u200b\u6700\u7ec8\u200b\u5e8f\u5217\u200b\u7684\u200b\u8868\u793a\u200b\u3002\u200b\u5982\u200bBERT\u200b\u6a21\u578b\u200b\uff0c\u200b\u9650\u5236\u200b\u4e86\u200b\u5e8f\u5217\u200b\u7684\u200b\u6700\u5927\u200b\u957f\u5ea6\u200b\u4e3a\u200b512\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u56fa\u5b9a\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff0c\u200b\u5c06\u200b\u6bcf\u4e2a\u200b\u4f4d\u7f6e\u200b\u6620\u5c04\u200b\u5230\u200b\u4e00\u4e2a\u200b\u53ef\u200b\u5b66\u4e60\u200b\u7684\u200b\u5411\u91cf\u200b\u3002</p> <pre><code>class AbsolutePE(torch.nn.Module):\n    def __init__(self, max_len: int, d_model: int):\n        super(AbsolutePE, self).__init__()\n        self.pe = torch.nn.Embedding(max_len, d_model)\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        # x: [N, L]\n        pos = torch.arange(x.size(1), device=x.device).unsqueeze(0)\n        return self.pe(pos) # [1, L, D], can be broadcasted to [N, L, D]\n</code></pre> <p>\u200b\u6b64\u7c7b\u200b\u65b9\u6cd5\u200b\u7684\u200b\u7f3a\u70b9\u200b\u662f\u200b\u65e0\u6cd5\u200b\u5904\u7406\u200b\u8d85\u8fc7\u200b\u6700\u5927\u200b\u957f\u5ea6\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5ffd\u89c6\u200b\u4e86\u200b\u5143\u7d20\u200b\u4e4b\u95f4\u200b\u7684\u200b\u76f8\u5bf9\u200b\u4f4d\u7f6e\u200b\u5173\u7cfb\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/positional-embedding/#_3","title":"\u6b63\u5f26\u200b\u4f4d\u7f6e\u200b\u7f16\u7801","text":"<p>Vaswani\u200b\u7b49\u200b\u4eba\u200b\u63d0\u51fa\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u5305\u542b\u200b\u4e00\u4e2a\u200b\u6b63\u5f26\u200b\u51fd\u6570\u200b\u548c\u200b\u4e00\u4e2a\u200b\u4f59\u5f26\u200b\u51fd\u6570\u200b\uff0c\u200b\u5176\u200b\u6570\u5b66\u200b\u8868\u8fbe\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned} PE_{(pos, 2i)} &amp;= \\sin(pos / 10000^{2i / d_{\\text{model}}}) \\\\ PE_{(pos, 2i + 1)} &amp;= \\cos(pos / 10000^{2i / d_{\\text{model}}}) \\\\ \\end{aligned} \\] <p>\u200b\u5176\u4e2d\u200b\uff0c\\(pos\\)\u200b\u8868\u793a\u200b\u4f4d\u7f6e\u200b\uff0c\\(i\\)\u200b\u8868\u793a\u200b\u7ef4\u5ea6\u200b\uff0c\\(d_{\\text{model}}\\)\u200b\u8868\u793a\u200b\u6a21\u578b\u200b\u7684\u200b\u7ef4\u5ea6\u200b\uff0c\u200b\u8f93\u51fa\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u76f4\u63a5\u200b\u4e0e\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u76f8\u52a0\u200b\u5f97\u5230\u200b\u6700\u7ec8\u200b\u8868\u793a\u200b\u3002\u200b\u6b63\u5f26\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u4e0d\u200b\u9700\u8981\u200b\u989d\u5916\u200b\u8fdb\u884c\u200b\u5b66\u4e60\u200b\u3002Vaswani\u200b\u7684\u200b\u5b9e\u9a8c\u200b\u8868\u660e\u200b\uff0c\u200b\u56fa\u5b9a\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u548c\u200b\u53ef\u200b\u5b66\u4e60\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u5728\u200b\u6027\u80fd\u200b\u4e0a\u200b\u6ca1\u6709\u200b\u663e\u8457\u200b\u5dee\u5f02\u200b\u3002</p> <pre><code>class SinusoidPE(torch.nn.Module):\n    def __init__(self, d_model: int):\n        super(SinusoidPE, self).__init__()\n        self.d_model = d_model\n\n    def _denominator(self, device: torch.device) -&gt; torch.Tensor:\n        return 10000 ** (torch.arange(0, self.d_model, device=device) / self.d_model)\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        # x: [N, L]\n        pos = torch.arange(x.size(1), device=x.device).unsqueeze(0) # [1, L]\n        pos = pos.unsqueeze(-1) / self._denominator(x.device) # [1, L, D]\n        pos[:, :, 0::2] = torch.sin(pos[:, :, 0::2])\n        pos[:, :, 1::2] = torch.cos(pos[:, :, 1::2])\n        return pos # [1, L, D], can be broadcasted to [N, L, D]\n</code></pre>"},{"location":"coding/dl-from-scratch/positional-embedding/#_4","title":"\u65cb\u8f6c\u200b\u4f4d\u7f6e\u200b\u7f16\u7801","text":"<p>Su\u200b\u7b49\u200b\u4eba\u200b\u63d0\u51fa\u200b\u7684\u200b\u65cb\u8f6c\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff08Rotary Positional Embedding\uff0cRoPE\uff09\u3002\u200b\u7f16\u7801\u200b\u7684\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u901a\u8fc7\u200b\u65cb\u8f6c\u200b\u77e9\u9635\u200b\u5c06\u200b\u4f4d\u7f6e\u200b\u4fe1\u606f\u200b\u5d4c\u5165\u200b\u5230\u200b\u7279\u5f81\u200b\u7a7a\u95f4\u200b\u4e2d\u200b\uff0c\u200b\u4ece\u800c\u200b\u4f7f\u200b\u6a21\u578b\u200b\u80fd\u591f\u200b\u5b66\u4e60\u200b\u5230\u200b\u4f4d\u7f6e\u200b\u4fe1\u606f\u200b\u3002</p> \\[ f_{q, k} \\bsx_m = \\bsR_{\\Theta, m}^d W_{q, k} \\bsx_m \\] <p>\u200b\u5176\u4e2d\u200b\uff0c\\(\\bsR_{\\Theta, m}\\)\u200b\u4e3a\u200b\u65cb\u8f6c\u200b\u77e9\u9635\u200b\uff0c\u200b\u5c06\u200b\u5411\u91cf\u200b\u6bcf\u200b\u4e24\u4e2a\u200b\u5206\u91cf\u200b\u8fdb\u884c\u200b\u65cb\u8f6c\u200b\u3002</p> \\[ \\begin{aligned} \\bsR^k_{\\Theta, m} &amp;= \\begin{bmatrix} \\cos(m\\theta_k) &amp; -\\sin(m\\theta_k) \\\\ \\sin(m\\theta_k) &amp; \\cos(m\\theta_k) \\end{bmatrix} \\\\ \\bsR_{\\Theta, m} &amp;= \\begin{bmatrix} \\bsR^1_{\\Theta, m} &amp; 0 &amp; \\cdots &amp; 0 \\\\ 0 &amp; \\bsR^2_{\\Theta, m} &amp; \\cdots &amp; 0 \\\\ \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\ 0 &amp; 0 &amp; \\cdots &amp; \\bsR^{d / 2}_{\\Theta, m} \\end{bmatrix}_{d\\times d} \\end{aligned} \\] <p>\u200b\u6ce8\u610f\u200b\uff0c\\(m\\)\u200b\u4e3a\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\(1, \\ldots, L\\)\uff0c\\(k\\)\u200b\u4e3a\u200b\u7ef4\u5ea6\u200b\u4e0b\u6807\u200b\uff0c\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\(1, \\ldots, d / 2\\)\u3002\u200b\u6700\u7ec8\u200b\u7684\u200b\u65cb\u8f6c\u200b\u64cd\u4f5c\u200b\u5c06\u200b\u6bcf\u4e2a\u200b\u4f4d\u7f6e\u200b\\(m\\)\u200b\u7684\u200b\u5411\u91cf\u200b\\(\\bsx_m\\)\u200b\u5e94\u7528\u200b\u65cb\u8f6c\u200b\u77e9\u9635\u200b\\(\\bsR_{\\Theta, m}\\)\uff0c\u200b\u5f97\u5230\u200b\u65b0\u200b\u7684\u200b\u5411\u91cf\u200b\u3002</p> <pre><code>import functools\n\nclass RoPE(torch.nn.Module):\n    def __init__(self, d_model: int, theta: int | float = 10000):\n        super(RoPE, self).__init__()\n        self.d_model = d_model\n        self.theta = theta ** -(torch.arange(0, d_model, 2) / d_model)\n\n    @functools.lru_cache(maxsize=None)\n    def _forward_l(self, L: int) -&gt; torch.Tensor:\n        # Use lru_cache to avoid redundant computation for the same L\n\n        D = self.d_model\n        pos = torch.einsum(\n            'l,d-&gt;ld',\n            torch.arange(L), self.theta\n        )  # [L, D / 2]\n\n        # Major diagonal is cos, cos, ..., cos; D elements\n        cos = torch.cos(pos).repeat_interleave(2)\n        # Minor diagonal is sin, 0, sin, 0, ..., sin; D - 1 elements\n        sin = torch.stack([\n            torch.sin(pos), torch.zeros_like(pos), dim=-1\n        ]).reshape(L, D)[:, :-1]\n\n        result = torch.zeros(L, D, D)\n        result = torch.diagonal_scatter(result, cos, dim1=1, dim2=2)\n        result = torch.diagonal_scatter(result, sin, dim1=1, dim2=2, offset=-1)\n        result = torch.diagonal_scatter(result, -sin, dim1=1, dim2=2, offset=1)\n        return result\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        # x: [N, L, H, D]\n        _, L, _, D = x.size()\n        rot_matrix = self._forward_l(L).to(x.device)  # [L, D, D]\n        return torch.einsum('lde,nlhe-&gt;nlhd', rot_matrix, x)\n</code></pre> <p>\u200b\u548c\u200b\u5176\u4ed6\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u65b9\u5f0f\u200b\u4e0d\u540c\u200b\uff0cRoPE\u200b\u5728\u200bQKV\u200b\u53d8\u6362\u200b\u540e\u200b\u8fdb\u884c\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff0c\u200b\u56e0\u6b64\u200b\u9700\u8981\u200b\u5bf9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u7684\u200b\u5b9e\u73b0\u200b\u8fdb\u884c\u200b\u4fee\u6539\u200b\u3002</p> <pre><code>class MultiHeadAttention(torch.nn.Module):\n    def __init__(self, d_model: int, num_heads: int, rope: RoPE | None = None):\n        ...\n\n        self.rope = rope\n\n        ...\n\n    def forward(\n        self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        ...\n\n        if self.rope is not None:\n            Q = self.rope(self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k))\n            K = self.rope(self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k))\n        else:\n            Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)\n            K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)\n        V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)\n\n        ...\n</code></pre> <p>\u200b\u4e3a\u4e86\u200b\u8ba1\u7b97\u200b\u65b9\u4fbf\u200b\uff0cLlama\u200b\u7b49\u200b\u90e8\u5206\u200b\u6a21\u578b\u200b\u7684\u200bRoPE\u200b\u5c06\u200b\u8f93\u5165\u200b\u5411\u91cf\u200b\u5206\u4e3a\u200b\u524d\u540e\u200b\u4e24\u6bb5\u200b\uff0c\u200b\u524d\u540e\u200b\u4e24\u6bb5\u200b\u76f8\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u5206\u4e3a\u200b\u4e00\u7ec4\u200b\u8fdb\u884c\u200b\u65cb\u8f6c\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u800c\u200b\u4e0d\u662f\u200b\u5c06\u200b\u6574\u4e2a\u200b\u5411\u91cf\u200b\u6309\u7167\u200b\u76f8\u90bb\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5143\u7d20\u200b\u5206\u4e3a\u200b\u4e00\u7ec4\u200b\u8fdb\u884c\u200b\u65cb\u8f6c\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/transformer-variants/","title":"Transformer\u200b\u53d8\u79cd","text":"<p>Transformer\u200b\u53d8\u79cd\u200b\u4e3b\u8981\u200b\u5305\u542b\u200b\u5982\u4e0b\u200b\u51e0\u79cd\u200b\uff1a</p> <ul> <li>\u200b\u6539\u53d8\u200btransformer\u200b\u7684\u200b\u6ce8\u610f\u529b\u200b\u8ba1\u7b97\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u5982\u200bLinformer\u3001Reformer\u200b\u7b49\u200b\uff1b</li> <li>\u200b\u6539\u53d8\u200btransformer\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u65b9\u5f0f\u200b\uff1b</li> <li>\u200b\u6539\u53d8\u200btransformer\u200b\u7684\u200b\u5f52\u4e00\u5316\u200b\u5c42\u200b\uff1b</li> <li>\u200b\u6539\u53d8\u200btransformer\u200b\u7684\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\uff1b</li> <li>\u200b\u90e8\u5206\u200b\u8bed\u8a00\u200b\u6a21\u578b\u200b\uff08\u200b\u5982\u200bLlama\uff09\u200b\u4f7f\u7528\u200b\u7684\u200b\u7ebf\u6027\u200b\u5c42\u4e0d\u5e26\u200b\u504f\u7f6e\u200b\uff1b</li> <li>\u200b\u6539\u53d8\u200btransformer\u200b\u5f52\u4e00\u5316\u200b\u5c42\u200b\u7684\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u662f\u200b\u5148\u200b\u5f52\u4e00\u5316\u200b\u518d\u200b\u52a0\u200b\u6b8b\u5dee\u200b\uff08Pre-Norm\uff09\uff0c\u200b\u8fd8\u662f\u200b\u5148\u52a0\u200b\u6b8b\u5dee\u200b\u518d\u200b\u5f52\u4e00\u5316\u200b\uff08Post-Norm\uff09\u3002</li> </ul>"},{"location":"coding/dl-from-scratch/transformer-variants/#bert","title":"BERT","text":"<p>BERT\u200b\u7684\u200b\u7ed3\u6784\u200b\u7279\u70b9\u200b\u4e3a\u200b\uff1a</p> <ul> <li>encoder-only\uff0c\u200b\u5373\u200b\u4f7f\u7528\u200bBidirectional self-attention\uff1b</li> <li>\u200b\u4f7f\u7528\u200b\u7edd\u5bf9\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff1b</li> </ul>"},{"location":"coding/dl-from-scratch/transformer-variants/#llama","title":"Llama","text":"<p>Llama\u200b\u7684\u200b\u7ed3\u6784\u200b\u7279\u70b9\u200b\u4e3a\u200b\uff1a</p> <ul> <li>decoder-only\uff0c\u200b\u5373\u200b\u4f7f\u7528\u200bCausal self-attention\uff1b</li> <li>\u200b\u5168\u90e8\u200b\u4f7f\u7528\u200b\u4e0d\u5e26\u200b\u504f\u7f6e\u200b\u7684\u200b\u7ebf\u6027\u200b\u5c42\u200b\uff1b</li> <li>\u200b\u5728\u200bFFN\u200b\u4e2d\u200b\u4f7f\u7528\u200bSiLU\u200b\u914d\u5408\u200b\u95e8\u63a7\u200b\uff1b</li> <li>\u200b\u4f7f\u7528\u200bRoPE\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff1b</li> <li>\u200b\u4f7f\u7528\u200bRMSNorm\u200b\u5f52\u4e00\u5316\u200b\u5c42\u200b\u3002\u200b\u5f52\u4e00\u5316\u200b\u65b9\u5f0f\u200b\u4e3a\u200bpre-norm\uff0c\u200b\u5373\u5148\u200b\u8fdb\u884c\u200b\u5f52\u4e00\u5316\u200b\uff0c\u200b\u518d\u200b\u8fdb\u884c\u200b\u6ce8\u610f\u529b\u200b\u6216\u8005\u200b\u5168\u200b\u8fde\u63a5\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u7136\u540e\u200b\u52a0\u200b\u6b8b\u5dee\u200b\u3002</li> </ul>"},{"location":"coding/dl-from-scratch/transformer/","title":"\u5b9e\u73b0\u200btransformer","text":"In\u00a0[1]: Copied! <pre>import torch\n</pre> import torch <p>\u200b\u9996\u5148\u200b\u5b9e\u73b0\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u3002</p> In\u00a0[2]: Copied! <pre>class MultiHeadAttention(torch.nn.Module):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadAttention, self).__init__()\n        self.d_model = d_model\n        self.num_heads = num_heads\n        if d_model % num_heads != 0:\n            raise ValueError(\"d_model must be divisible by num_heads\")\n\n        self.d_k = d_model // num_heads\n        self.sqrt_d_k = self.d_k ** 0.5\n\n        self.W_Q = torch.nn.Linear(d_model, d_model)\n        self.W_K = torch.nn.Linear(d_model, d_model)\n        self.W_V = torch.nn.Linear(d_model, d_model)\n        self.W_O = torch.nn.Linear(d_model, d_model)\n\n    def forward(\n        self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,\n        mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        N, L_Q, D = x_q.size()\n        _, L_KV, _ = x_k.size()\n        Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)\n        K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)\n        V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)\n\n        score = torch.einsum(\"nihd,njhd-&gt;nijh\", Q, K) / self.sqrt_d_k\n        if mask is not None:\n            # mask: (L, L)\n            score = score.masked_fill(\n                mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')\n            )\n        score = torch.nn.functional.softmax(score, dim=2)\n        value = torch.einsum(\"nijh,njhd-&gt;nihd\", score, V).reshape(N, L_Q, self.d_model)\n        return self.W_O(value)\n\n\nclass MultiHeadSelfAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadSelfAttention, self).__init__(d_model, num_heads)\n\n    def forward(self, x: torch.Tensor, mask: torch.Tensor | None = None) -&gt; torch.Tensor:\n        return super().forward(x, x, x, mask)\n\n\nclass MultiHeadCrossAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadCrossAttention, self).__init__(d_model, num_heads)\n\n    def forward(\n        self, x_q: torch.Tensor, x_kv: torch.Tensor,\n        mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        return super().forward(x_q, x_kv, x_kv, mask)\n</pre> class MultiHeadAttention(torch.nn.Module):     def __init__(self, d_model: int, num_heads: int):         super(MultiHeadAttention, self).__init__()         self.d_model = d_model         self.num_heads = num_heads         if d_model % num_heads != 0:             raise ValueError(\"d_model must be divisible by num_heads\")          self.d_k = d_model // num_heads         self.sqrt_d_k = self.d_k ** 0.5          self.W_Q = torch.nn.Linear(d_model, d_model)         self.W_K = torch.nn.Linear(d_model, d_model)         self.W_V = torch.nn.Linear(d_model, d_model)         self.W_O = torch.nn.Linear(d_model, d_model)      def forward(         self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,         mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         N, L_Q, D = x_q.size()         _, L_KV, _ = x_k.size()         Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)         K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)         V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)          score = torch.einsum(\"nihd,njhd-&gt;nijh\", Q, K) / self.sqrt_d_k         if mask is not None:             # mask: (L, L)             score = score.masked_fill(                 mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')             )         score = torch.nn.functional.softmax(score, dim=2)         value = torch.einsum(\"nijh,njhd-&gt;nihd\", score, V).reshape(N, L_Q, self.d_model)         return self.W_O(value)   class MultiHeadSelfAttention(MultiHeadAttention):     def __init__(self, d_model: int, num_heads: int):         super(MultiHeadSelfAttention, self).__init__(d_model, num_heads)      def forward(self, x: torch.Tensor, mask: torch.Tensor | None = None) -&gt; torch.Tensor:         return super().forward(x, x, x, mask)   class MultiHeadCrossAttention(MultiHeadAttention):     def __init__(self, d_model: int, num_heads: int):         super(MultiHeadCrossAttention, self).__init__(d_model, num_heads)      def forward(         self, x_q: torch.Tensor, x_kv: torch.Tensor,         mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         return super().forward(x_q, x_kv, x_kv, mask) <p>\u200b\u524d\u9988\u200b\u795e\u7ecf\u7f51\u7edc\u200b\u7531\u200b\u4e24\u4e2a\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\u548c\u200bReLU\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\u7ec4\u6210\u200b</p> In\u00a0[3]: Copied! <pre>class FFN(torch.nn.Module):\n    def __init__(self, input_dim: int, hidden_dim: int):\n        super(FFN, self).__init__()\n        self.fc1 = torch.nn.Linear(input_dim, hidden_dim)\n        self.fc2 = torch.nn.Linear(hidden_dim, input_dim)\n        self.relu = torch.nn.ReLU()\n\n    def forward(self, x):\n        return self.fc2(self.relu(self.fc1(x)))\n</pre> class FFN(torch.nn.Module):     def __init__(self, input_dim: int, hidden_dim: int):         super(FFN, self).__init__()         self.fc1 = torch.nn.Linear(input_dim, hidden_dim)         self.fc2 = torch.nn.Linear(hidden_dim, input_dim)         self.relu = torch.nn.ReLU()      def forward(self, x):         return self.fc2(self.relu(self.fc1(x))) <p>\u200b\u7f16\u7801\u5668\u200b\u5c42\u200b\u7531\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200bc\u200b\u5c42\u200b\u548c\u200b\u524d\u9988\u200b\u795e\u7ecf\u200b\u7f51\u7edc\u5c42\u200b\u7ec4\u6210\u200b\uff0c\u200b\u4e00\u4e2a\u200b\u7f16\u7801\u5668\u200b\u7531\u200b\u591a\u4e2a\u200b\u8fd9\u6837\u200b\u7684\u200b\u5c42\u200b\u4e32\u8054\u200b\u7ec4\u6210\u200b</p> In\u00a0[4]: Copied! <pre>class EncoderLayer(torch.nn.Module):\n    def __init__(\n        self, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6\n    ):\n        super(EncoderLayer, self).__init__()\n\n        if ffn_dim is None:\n            ffn_dim = input_dim * 4\n\n        self.attention = MultiHeadSelfAttention(input_dim, num_heads)\n        self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout1 = torch.nn.Dropout(dropout)\n\n        self.ffn = FFN(input_dim, ffn_dim)\n        self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout2 = torch.nn.Dropout(dropout)\n\n    def forward(self, x, mask):\n        x = x + self.dropout1(self.attention(x, mask))\n        x = self.norm1(x)\n\n        x = x + self.dropout2(self.ffn(x))\n        x = self.norm2(x)\n\n        return x\n\n\nclass Encoder(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6\n    ):\n        super(Encoder, self).__init__()\n\n        self.layers = torch.nn.ModuleList([\n            EncoderLayer(input_dim, num_heads, ffn_dim, dropout, layer_norm_eps)\n            for _ in range(num_layers)\n        ])\n\n    def forward(self, x, mask):\n        # x: (N, L, D)\n        for layer in self.layers:\n            x = layer(x, mask)\n        return x\n</pre> class EncoderLayer(torch.nn.Module):     def __init__(         self, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1,         layer_norm_eps: float = 1e-6     ):         super(EncoderLayer, self).__init__()          if ffn_dim is None:             ffn_dim = input_dim * 4          self.attention = MultiHeadSelfAttention(input_dim, num_heads)         self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout1 = torch.nn.Dropout(dropout)          self.ffn = FFN(input_dim, ffn_dim)         self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout2 = torch.nn.Dropout(dropout)      def forward(self, x, mask):         x = x + self.dropout1(self.attention(x, mask))         x = self.norm1(x)          x = x + self.dropout2(self.ffn(x))         x = self.norm2(x)          return x   class Encoder(torch.nn.Module):     def __init__(         self, num_layers: int, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1,         layer_norm_eps: float = 1e-6     ):         super(Encoder, self).__init__()          self.layers = torch.nn.ModuleList([             EncoderLayer(input_dim, num_heads, ffn_dim, dropout, layer_norm_eps)             for _ in range(num_layers)         ])      def forward(self, x, mask):         # x: (N, L, D)         for layer in self.layers:             x = layer(x, mask)         return x <p>\u200b\u89e3\u7801\u5668\u200b\u5c42\u200b\u7531\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\uff0c\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\u548c\u200b\u524d\u9988\u200b\u795e\u7ecf\u200b\u7f51\u7edc\u5c42\u200b\u7ec4\u6210\u200b\u3002</p> In\u00a0[5]: Copied! <pre>class DecoderLayer(torch.nn.Module):\n    def __init__(\n        self, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1\n    ):\n        super(DecoderLayer, self).__init__()\n\n        if ffn_dim is None:\n            ffn_dim = input_dim * 4\n\n        self.self_attention = MultiHeadSelfAttention(input_dim, num_heads)\n        self.norm1 = torch.nn.LayerNorm(input_dim)\n        self.dropout1 = torch.nn.Dropout(dropout)\n\n        self.cross_attention = MultiHeadCrossAttention(input_dim, num_heads)\n        self.norm2 = torch.nn.LayerNorm(input_dim)\n        self.dropout2 = torch.nn.Dropout(dropout)\n\n        self.ffn = FFN(input_dim, ffn_dim)\n        self.norm3 = torch.nn.LayerNorm(input_dim)\n        self.dropout3 = torch.nn.Dropout(dropout)\n\n    def forward(\n        self, x: torch.Tensor, memory: torch.Tensor, tgt_mask: torch.Tensor | None = None\n    ):\n\n        x = x + self.dropout1(self.self_attention(x, tgt_mask))\n        x = self.norm1(x)\n\n        x = x + self.dropout2(self.cross_attention(x, memory))\n        x = self.norm2(x)\n\n        x = x + self.dropout3(self.ffn(x))\n        x = self.norm3(x)\n\n        return x\n\n\nclass Decoder(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1\n    ):\n        super(Decoder, self).__init__()\n\n        self.layers = torch.nn.ModuleList([\n            DecoderLayer(input_dim, num_heads, ffn_dim, dropout)\n            for _ in range(num_layers)\n        ])\n\n    def forward(self, x, memory, tgt_mask):\n        # x: (N, L, D)\n        for layer in self.layers:\n            x = layer(x, memory, tgt_mask)\n        return x\n</pre> class DecoderLayer(torch.nn.Module):     def __init__(         self, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1     ):         super(DecoderLayer, self).__init__()          if ffn_dim is None:             ffn_dim = input_dim * 4          self.self_attention = MultiHeadSelfAttention(input_dim, num_heads)         self.norm1 = torch.nn.LayerNorm(input_dim)         self.dropout1 = torch.nn.Dropout(dropout)          self.cross_attention = MultiHeadCrossAttention(input_dim, num_heads)         self.norm2 = torch.nn.LayerNorm(input_dim)         self.dropout2 = torch.nn.Dropout(dropout)          self.ffn = FFN(input_dim, ffn_dim)         self.norm3 = torch.nn.LayerNorm(input_dim)         self.dropout3 = torch.nn.Dropout(dropout)      def forward(         self, x: torch.Tensor, memory: torch.Tensor, tgt_mask: torch.Tensor | None = None     ):          x = x + self.dropout1(self.self_attention(x, tgt_mask))         x = self.norm1(x)          x = x + self.dropout2(self.cross_attention(x, memory))         x = self.norm2(x)          x = x + self.dropout3(self.ffn(x))         x = self.norm3(x)          return x   class Decoder(torch.nn.Module):     def __init__(         self, num_layers: int, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1     ):         super(Decoder, self).__init__()          self.layers = torch.nn.ModuleList([             DecoderLayer(input_dim, num_heads, ffn_dim, dropout)             for _ in range(num_layers)         ])      def forward(self, x, memory, tgt_mask):         # x: (N, L, D)         for layer in self.layers:             x = layer(x, memory, tgt_mask)         return x <p>Transformer\u200b\u6a21\u578b\u200b\u7531\u200b\u7f16\u7801\u5668\u200b\u548c\u200b\u89e3\u7801\u5668\u200b\u7ec4\u6210\u200b\uff0c\u200b\u7f16\u7801\u5668\u200b\u5904\u7406\u200b\u6e90\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5c06\u200b\u7f16\u7801\u200b\u540e\u200b\u7684\u200b\u5e8f\u5217\u200b\u8f93\u5165\u200b\u5230\u200b\u89e3\u7801\u5668\u200b\u4e2d\u200b\uff0c\u200b\u89e3\u7801\u5668\u200b\u751f\u6210\u200b\u76ee\u6807\u200b\u5e8f\u5217\u200b\u3002</p> In\u00a0[6]: Copied! <pre>class CustomTransformer(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, num_decoder_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1, layer_norm_eps: float = 1e-6\n    ):\n        super(CustomTransformer, self).__init__()\n        self.encoder = Encoder(num_layers, input_dim, num_heads, ffn_dim, dropout, layer_norm_eps)\n        self.decoder = Decoder(num_decoder_layers, input_dim, num_heads, ffn_dim, dropout)\n\n    def forward(self, src, tgt, src_mask: torch.Tensor | None = None, tgt_mask: torch.Tensor | None = None):\n        memory = self.encoder(src, src_mask)\n        return self.decoder(tgt, memory, tgt_mask)\n</pre> class CustomTransformer(torch.nn.Module):     def __init__(         self, num_layers: int, num_decoder_layers: int, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1, layer_norm_eps: float = 1e-6     ):         super(CustomTransformer, self).__init__()         self.encoder = Encoder(num_layers, input_dim, num_heads, ffn_dim, dropout, layer_norm_eps)         self.decoder = Decoder(num_decoder_layers, input_dim, num_heads, ffn_dim, dropout)      def forward(self, src, tgt, src_mask: torch.Tensor | None = None, tgt_mask: torch.Tensor | None = None):         memory = self.encoder(src, src_mask)         return self.decoder(tgt, memory, tgt_mask) <p>\u200b\u5c06\u200bPyTorch\u200b\u5185\u90e8\u200b\u5b9e\u73b0\u200b\u7684\u200bTransformer\u200b\u6743\u91cd\u200b\u590d\u5236\u5230\u200b\u5b9e\u73b0\u200b\u7684\u200bTransformer\u200b\u4e2d\u200b\u3002</p> In\u00a0[7]: Copied! <pre>def _attn_load_from_torch(\n    custom_attn: MultiHeadAttention, torch_attn: torch.nn.MultiheadAttention\n):\n    embed_dim = custom_attn.d_model\n\n    def split_qkv(weight, embed_dim):\n        return weight[:embed_dim], weight[embed_dim: 2 * embed_dim], weight[2 * embed_dim:3 * embed_dim]\n    custom_attn.W_Q.weight.data, \\\n        custom_attn.W_K.weight.data, \\\n        custom_attn.W_V.weight.data = split_qkv(\n            torch_attn.in_proj_weight.data, embed_dim\n    )\n\n    custom_attn.W_Q.bias.data, \\\n        custom_attn.W_K.bias.data, \\\n        custom_attn.W_V.bias.data = split_qkv(\n            torch_attn.in_proj_bias.data, embed_dim\n    )\n\n    custom_attn.W_O.weight.data = torch_attn.out_proj.weight.data\n    custom_attn.W_O.bias.data = torch_attn.out_proj.bias.data\n\n    return custom_attn\n\n\ndef load_from_torch(\n    custom_transformer: CustomTransformer, torch_transformer: torch.nn.Transformer\n):\n    for custom_layer, torch_layer in zip(\n        [*custom_transformer.encoder.layers, *custom_transformer.decoder.layers],\n        [*torch_transformer.encoder.layers, *torch_transformer.decoder.layers],\n    ):\n\n        if hasattr(custom_layer, 'cross_attention'):\n            # Decoder\n            custom_layer.self_attention = _attn_load_from_torch(\n                custom_layer.self_attention, torch_layer.self_attn\n            )\n            custom_layer.cross_attention = _attn_load_from_torch(\n                custom_layer.cross_attention, torch_layer.multihead_attn\n            )\n        else:\n            # Encoder\n            custom_layer.attention = _attn_load_from_torch(\n                custom_layer.attention, torch_layer.self_attn\n            )\n\n        layer_pairs = [\n            (custom_layer.norm1, torch_layer.norm1),\n            (custom_layer.ffn.fc1, torch_layer.linear1),\n            (custom_layer.ffn.fc2, torch_layer.linear2),\n            (custom_layer.norm2, torch_layer.norm2)\n        ]\n        for custom, torch in layer_pairs:\n            custom.weight.data = torch.weight.data\n            custom.bias.data = torch.bias.data\n    return custom_transformer\n</pre> def _attn_load_from_torch(     custom_attn: MultiHeadAttention, torch_attn: torch.nn.MultiheadAttention ):     embed_dim = custom_attn.d_model      def split_qkv(weight, embed_dim):         return weight[:embed_dim], weight[embed_dim: 2 * embed_dim], weight[2 * embed_dim:3 * embed_dim]     custom_attn.W_Q.weight.data, \\         custom_attn.W_K.weight.data, \\         custom_attn.W_V.weight.data = split_qkv(             torch_attn.in_proj_weight.data, embed_dim     )      custom_attn.W_Q.bias.data, \\         custom_attn.W_K.bias.data, \\         custom_attn.W_V.bias.data = split_qkv(             torch_attn.in_proj_bias.data, embed_dim     )      custom_attn.W_O.weight.data = torch_attn.out_proj.weight.data     custom_attn.W_O.bias.data = torch_attn.out_proj.bias.data      return custom_attn   def load_from_torch(     custom_transformer: CustomTransformer, torch_transformer: torch.nn.Transformer ):     for custom_layer, torch_layer in zip(         [*custom_transformer.encoder.layers, *custom_transformer.decoder.layers],         [*torch_transformer.encoder.layers, *torch_transformer.decoder.layers],     ):          if hasattr(custom_layer, 'cross_attention'):             # Decoder             custom_layer.self_attention = _attn_load_from_torch(                 custom_layer.self_attention, torch_layer.self_attn             )             custom_layer.cross_attention = _attn_load_from_torch(                 custom_layer.cross_attention, torch_layer.multihead_attn             )         else:             # Encoder             custom_layer.attention = _attn_load_from_torch(                 custom_layer.attention, torch_layer.self_attn             )          layer_pairs = [             (custom_layer.norm1, torch_layer.norm1),             (custom_layer.ffn.fc1, torch_layer.linear1),             (custom_layer.ffn.fc2, torch_layer.linear2),             (custom_layer.norm2, torch_layer.norm2)         ]         for custom, torch in layer_pairs:             custom.weight.data = torch.weight.data             custom.bias.data = torch.bias.data     return custom_transformer <p>\u200b\u901a\u8fc7\u200b\u4ee3\u7801\u200b\u9a8c\u8bc1\u200b\u5b9e\u73b0\u200b\u7684\u200bTransformer\u200b\u6a21\u578b\u200b\u7684\u200b\u6b63\u786e\u6027\u200b\u3002</p> In\u00a0[8]: Copied! <pre>transformer_config = {\n    'num_layers': 6,\n    'num_decoder_layers': 6,\n    'input_dim': 512,\n    'num_heads': 8,\n    'ffn_dim': 2048,\n    'dropout': 0.1,\n    'layer_norm_eps': 1e-6\n}\n\ncustom_transformer = CustomTransformer(**transformer_config)\ntorch_transformer = torch.nn.Transformer(\n    d_model=transformer_config['input_dim'],\n    nhead=transformer_config['num_heads'],\n    num_encoder_layers=transformer_config['num_layers'],\n    num_decoder_layers=transformer_config['num_decoder_layers'],\n    dim_feedforward=transformer_config['ffn_dim'],\n    dropout=transformer_config['dropout'],\n    batch_first=True\n)\ncustom_transformer = load_from_torch(custom_transformer, torch_transformer)\ncustom_transformer.eval()\ntorch_transformer.eval()\n\nsrc = torch.randn(32, 10, transformer_config['input_dim'])\ntgt = torch.randn(32, 20, transformer_config['input_dim'])\ntgt_mask = 1 - torch.triu(torch.ones(20, 20), diagonal=1)\n\ncustom_output = custom_transformer(src, tgt, tgt_mask=tgt_mask)\ntorch_output = torch_transformer(src, tgt, tgt_mask=tgt_mask, tgt_is_causal=True)\ncustom_output[0, 0, :5], torch_output[0, 0, :5]\n</pre> transformer_config = {     'num_layers': 6,     'num_decoder_layers': 6,     'input_dim': 512,     'num_heads': 8,     'ffn_dim': 2048,     'dropout': 0.1,     'layer_norm_eps': 1e-6 }  custom_transformer = CustomTransformer(**transformer_config) torch_transformer = torch.nn.Transformer(     d_model=transformer_config['input_dim'],     nhead=transformer_config['num_heads'],     num_encoder_layers=transformer_config['num_layers'],     num_decoder_layers=transformer_config['num_decoder_layers'],     dim_feedforward=transformer_config['ffn_dim'],     dropout=transformer_config['dropout'],     batch_first=True ) custom_transformer = load_from_torch(custom_transformer, torch_transformer) custom_transformer.eval() torch_transformer.eval()  src = torch.randn(32, 10, transformer_config['input_dim']) tgt = torch.randn(32, 20, transformer_config['input_dim']) tgt_mask = 1 - torch.triu(torch.ones(20, 20), diagonal=1)  custom_output = custom_transformer(src, tgt, tgt_mask=tgt_mask) torch_output = torch_transformer(src, tgt, tgt_mask=tgt_mask, tgt_is_causal=True) custom_output[0, 0, :5], torch_output[0, 0, :5] Out[8]: <pre>(tensor([ 0.6672,  0.4102, -0.2145,  1.8782,  0.4108], grad_fn=&lt;SliceBackward0&gt;),\n tensor([ 0.6672,  0.4102, -0.2145,  1.8781,  0.4108], grad_fn=&lt;SliceBackward0&gt;))</pre> In\u00a0[9]: Copied! <pre>torch.allclose(custom_output, torch_output, atol=1e-4)\n</pre> torch.allclose(custom_output, torch_output, atol=1e-4) Out[9]: <pre>True</pre>"},{"location":"coding/dsa/","title":"\u6570\u636e\u7ed3\u6784","text":"<p>\u200b\u4ee5\u4e0b\u200b\u6574\u7406\u200b\u4e86\u200b\u9093\u4fca\u8f89\u200b\u8001\u5e08\u200b\u300a\u200b\u6570\u636e\u7ed3\u6784\u200b\u300b\u200b\u8bfe\u7a0b\u200b\u7684\u200b\u5185\u5bb9\u200b\uff1a</p> <ul> <li>\u200b\u5411\u91cf\u200b\u4e0e\u200b\u5217\u8868\u200b</li> </ul>"},{"location":"coding/dsa/vector-and-list/","title":"\u5411\u91cf\u200b\u4e0e\u200b\u5217\u8868","text":"<p>\u200b\u4e3b\u8981\u200b\u7684\u200b\u57fa\u7840\u200b\u7ebf\u6027\u200b\u6570\u636e\u7ed3\u6784\u200b\u662f\u200b\u5411\u91cf\u200b\u4e0e\u200b\u5217\u8868\u200b\u3002\u200b\u5728\u200b\u672c\u5408\u200b\u96c6\u4e2d\u200b\uff0c\u200b\u5411\u91cf\u200b\u6307\u4ee5\u200b\u6570\u7ec4\u200b\u65b9\u5f0f\u200b\u7ec4\u5408\u200b\u5e76\u200b\u5c01\u88c5\u200b\u901a\u7528\u200b\u62bd\u8c61\u200b\u63a5\u53e3\u200b\u7684\u200b\u6570\u636e\u7c7b\u578b\u200b\uff1b\u200b\u5217\u8868\u200b\u6307\u4ee5\u200b\u94fe\u8868\u200b\u65b9\u5f0f\u200b\u7ec4\u5408\u200b\u5e76\u200b\u5c01\u88c5\u200b\u901a\u7528\u200b\u62bd\u8c61\u200b\u63a5\u53e3\u200b\u7684\u200b\u6570\u636e\u7c7b\u578b\u200b\u3002</p> <p>\u200b\u8f83\u4e3a\u200b\u9ad8\u7ea7\u200b\u7684\u200b\u6570\u636e\u7ed3\u6784\u200b\u2014\u2014\u200b\u6808\u200b\u4e0e\u200b\u961f\u5217\u200b\u5efa\u7acb\u200b\u5728\u200b\u5411\u91cf\u200b\u4e0e\u200b\u5217\u8868\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\u3002\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u4e00\u4e9b\u200b\u53d7\u9650\u5236\u200b\u7684\u200b\u64cd\u4f5c\u200b\u63a5\u53e3\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_2","title":"\u7ebf\u6027\u200b\u6570\u636e\u7ed3\u6784","text":"<p>\u200b\u5bf9\u4e8e\u200b\u57fa\u7840\u200b\u7684\u200b\u7ebf\u6027\u200b\u6570\u636e\u7ed3\u6784\u200b\uff0c\u200b\u4ee5\u4e0b\u200b\u7684\u200b\u63a5\u53e3\u200b\u53ef\u4ee5\u200b\u89c6\u4e3a\u200b\u901a\u7528\u200b\uff1a</p> \u200b\u64cd\u4f5c\u200b \u200b\u63cf\u8ff0\u200b \u200b\u901a\u7528\u6027\u200b <code>size()</code> \u200b\u83b7\u53d6\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u6570\u76ee\u200b <code>get()</code> \u200b\u83b7\u53d6\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u67d0\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u503c\u200b <code>set()</code> \u200b\u4fee\u6539\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u67d0\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u503c\u200b <code>insert()</code> \u200b\u5411\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u67d0\u4e2a\u200b\u4f4d\u7f6e\u200b\u63d2\u5165\u200b\u5143\u7d20\u200b <code>remove()</code> \u200b\u79fb\u9664\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u67d0\u4e2a\u200b\u5143\u7d20\u200b <code>traverse()</code> \u200b\u904d\u5386\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\uff0c\u200b\u5e76\u200b\u6267\u884c\u200b\u67d0\u200b\u64cd\u4f5c\u200b <code>find()</code> \u200b\u5728\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u67e5\u627e\u200b\u67d0\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u4f4d\u7f6e\u200b <code>search()</code> \u200b\u5728\u200b\u6709\u5e8f\u200b\u7684\u200b\u7ed3\u6784\u200b\u4e2d\u200b\u67e5\u627e\u200b\u67d0\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u4f4d\u7f6e\u200b \u200b\u8981\u6c42\u200b\u6709\u5e8f\u200b <code>sort()</code> \u200b\u6392\u5e8f\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b \u200b\u8981\u6c42\u200b\u91cd\u8f7d\u200b<code>&lt;</code>\u200b\u548c\u200b<code>&gt;</code>\u200b\u8fd0\u7b97\u7b26"},{"location":"coding/dsa/vector-and-list/#_3","title":"\u5411\u91cf","text":"<p>\u200b\u5728\u200b\u5411\u91cf\u200b\u4e2d\u200b\uff0c\u200b\u6570\u636e\u200b\u5728\u200b\u5185\u5b58\u200b\u4e2d\u200b\u8fde\u7eed\u200b\u5b58\u50a8\u200b\uff0c\u200b\u5373\u200b\u4e0b\u6807\u200b\u76f8\u90bb\u200b\u7684\u200b\u6570\u636e\u200b\u5728\u200b\u5185\u5b58\u200b\u4e2d\u200b\u76f8\u90bb\u200b\u3002C/C++\u200b\u8bed\u8a00\u200b\u4e2d\u200b\u7684\u200b\u6570\u7ec4\u200b\u6570\u636e\u7ed3\u6784\u200b\u53ef\u4ee5\u200b\u89c6\u4e3a\u200b\u7b80\u5355\u200b\u7684\u200b\u5411\u91cf\u200b\u3002\u200b\u4f46\u200b\u6570\u7ec4\u200b\u4e0d\u200b\u63d0\u4f9b\u200b\u63d2\u5165\u200b\u3001\u200b\u5220\u9664\u200b\u7b49\u200b\u5177\u4f53\u200b\u7684\u200b\u64cd\u4f5c\u200b\u63a5\u53e3\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b\u7a0b\u5e8f\u4ee3\u7801\u200b\u4e2d\u200b\u624b\u52a8\u200b\u5b9e\u73b0\u200b\u3002\u200b\u5c06\u200b\u6570\u7ec4\u200b\u5c01\u88c5\u200b\u6210\u7c7b\u200b\u53ef\u4ee5\u200b\u63d0\u4f9b\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u64cd\u4f5c\u200b\u63a5\u53e3\u200b\uff0c\u200b\u5e76\u4e14\u200b\u4fdd\u8bc1\u6570\u636e\u200b\u88ab\u200b\u5408\u6cd5\u200b\u5730\u200b\u8bbf\u95ee\u200b\u53ca\u200b\u4fee\u6539\u200b\u3002</p> <p>\u200b\u5411\u91cf\u200b\u662f\u200b\u6570\u7ec4\u200b\u7684\u200b\u62bd\u8c61\u200b\u6cdb\u5316\u200b\u3001\u200b\u652f\u6301\u200b\u6240\u6709\u200b\u6570\u7ec4\u200b\u7684\u200b\u529f\u80fd\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u4e0d\u540c\u200b\u7c7b\u578b\u200b\u7684\u200b\u5143\u7d20\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u7edf\u4e00\u200b\u7684\u200b\u64cd\u4f5c\u200b\u63a5\u53e3\u200b\u3002</p> <p>C++\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u6a21\u677f\u200b\u7c7b\u200b\u7684\u200b\u529f\u80fd\u200b\u7528\u6765\u200b\u9488\u5bf9\u200b\u4e0d\u540c\u200b\u7c7b\u578b\u200b\u7684\u200b\u5143\u7d20\u200b\u63d0\u4f9b\u200b\u76f8\u540c\u200b\u7684\u200b\u529f\u80fd\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u67d0\u4e2a\u200bADT\uff08\u200b\u4ee5\u200b\u5411\u91cf\u200b\u4e3a\u4f8b\u200b\uff09\uff0c\u200b\u5176\u200b\u5c5e\u6027\u200b\u548c\u200b\u63a5\u53e3\u200b\u7684\u200b\u7ec4\u7ec7\u200b\u65b9\u5f0f\u200b\u53ef\u200b\u9075\u5faa\u200b\u5982\u4e0b\u200b\u89c4\u5219\u200b\uff1a</p> <pre><code>template &lt;typename T&gt; class Vector { //\u200b\u5411\u91cf\u200b\u6a21\u677f\u200b\u7c7b\u200b\nprivate: Rank _size; int _capacity; T* _elem; //\u200b\u89c4\u6a21\u200b\u3001\u200b\u5bb9\u91cf\u200b\u3001\u200b\u6570\u636e\u200b\u533a\u200b\nprotected:\n/* ... \u200b\u5185\u90e8\u200b\u51fd\u6570\u200b */\npublic:\n/* ... \u200b\u6784\u9020\u51fd\u6570\u200b */\n/* ... \u200b\u6790\u6784\u200b\u51fd\u6570\u200b */\n/* ... \u200b\u53ea\u8bfb\u200b\u63a5\u53e3\u200b */\n/* ... \u200b\u53ef\u200b\u5199\u200b\u63a5\u53e3\u200b */\n/* ... \u200b\u904d\u5386\u200b\u63a5\u53e3\u200b */\n};\n</code></pre> <p>\u200b\u901a\u8fc7\u200b\u5bf9\u200b<code>[]</code>\u200b\u8fd0\u7b97\u7b26\u200b\u8fdb\u884c\u200b\u91cd\u8f7d\u200b\uff0c\u200b\u5411\u91cf\u200b\u548c\u200b\u5217\u8868\u200b\u652f\u6301\u200b\u6570\u7ec4\u200b\u98ce\u683c\u200b\u7684\u200b\u8bbf\u95ee\u200b\u65b9\u5f0f\u200b\u3002</p> <p>\u200b\u5177\u4f53\u200b\u4ee3\u7801\u200b\u53ef\u4ee5\u200b\u53c2\u89c1\u200b\uff1a</p> \u200b\u5411\u91cf\u200b\u6a21\u677f\u200b\u7c7b\u200b\u58f0\u660e\u200b <p><pre><code>/******************************************************************************************\n* Data Structures in C++\n* ISBN: 7-302-33064-6 &amp; 7-302-33065-3 &amp; 7-302-29652-2 &amp; 7-302-26883-3\n* Junhui DENG, deng@tsinghua.edu.cn\n* Computer Science &amp; Technology, Tsinghua University\n* Copyright (c) 2003-2020. All rights reserved.\n******************************************************************************************/\n\ntypedef int Rank; //\u200b\u79e9\u200b\n#define DEFAULT_CAPACITY  3 //\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u521d\u59cb\u200b\u5bb9\u91cf\u200b\uff08\u200b\u5b9e\u9645\u200b\u5e94\u7528\u200b\u4e2d\u200b\u53ef\u200b\u8bbe\u7f6e\u200b\u4e3a\u200b\u66f4\u200b\u5927\u200b\uff09\n\ntemplate &lt;typename T&gt; class Vector { //\u200b\u5411\u91cf\u200b\u6a21\u677f\u200b\u7c7b\u200b\nprotected:\n   Rank _size; int _capacity;  T* _elem; //\u200b\u89c4\u6a21\u200b\u3001\u200b\u5bb9\u91cf\u200b\u3001\u200b\u6570\u636e\u200b\u533a\u200b\n   void copyFrom ( T const* A, Rank lo, Rank hi ); //\u200b\u590d\u5236\u200b\u6570\u7ec4\u200b\u533a\u95f4\u200bA[lo, hi)\n   void expand(); //\u200b\u7a7a\u95f4\u200b\u4e0d\u8db3\u200b\u65f6\u200b\u6269\u5bb9\u200b\n   void shrink(); //\u200b\u88c5\u586b\u200b\u56e0\u5b50\u200b\u8fc7\u200b\u5c0f\u65f6\u200b\u538b\u7f29\u200b\n   bool bubble ( Rank lo, Rank hi ); //\u200b\u626b\u63cf\u200b\u4ea4\u6362\u200b\n   void bubbleSort ( Rank lo, Rank hi ); //\u200b\u8d77\u6ce1\u200b\u6392\u5e8f\u200b\u7b97\u6cd5\u200b\n   Rank max ( Rank lo, Rank hi ); //\u200b\u9009\u53d6\u200b\u6700\u5927\u200b\u5143\u7d20\u200b\n   void selectionSort ( Rank lo, Rank hi ); //\u200b\u9009\u62e9\u200b\u6392\u5e8f\u200b\u7b97\u6cd5\u200b\n   void merge ( Rank lo, Rank mi, Rank hi ); //\u200b\u5f52\u5e76\u200b\u7b97\u6cd5\u200b\n   void mergeSort ( Rank lo, Rank hi ); //\u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b\u7b97\u6cd5\u200b\n   void heapSort ( Rank lo, Rank hi ); //\u200b\u5806\u6392\u5e8f\u200b\uff08\u200b\u7a0d\u540e\u200b\u7ed3\u5408\u200b\u5b8c\u5168\u200b\u5806\u200b\u8bb2\u89e3\u200b\uff09\n   Rank partition ( Rank lo, Rank hi ); //\u200b\u8f74\u70b9\u200b\u6784\u9020\u200b\u7b97\u6cd5\u200b\n   void quickSort ( Rank lo, Rank hi ); //\u200b\u5feb\u901f\u200b\u6392\u5e8f\u200b\u7b97\u6cd5\u200b\n   void shellSort ( Rank lo, Rank hi ); //\u200b\u5e0c\u5c14\u200b\u6392\u5e8f\u200b\u7b97\u6cd5\u200b\npublic:\n// \u200b\u6784\u9020\u51fd\u6570\u200b\n   Vector ( int c = DEFAULT_CAPACITY, int s = 0, T v = 0 ) //\u200b\u5bb9\u91cf\u200b\u4e3a\u200bc\u3001\u200b\u89c4\u6a21\u200b\u4e3a\u200bs\u3001\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u521d\u59cb\u200b\u4e3a\u200bv\n   { _elem = new T[_capacity = c]; for ( _size = 0; _size &lt; s; _elem[_size++] = v ); } //s&lt;=c\n   Vector ( T const* A, Rank n ) { copyFrom ( A, 0, n ); } //\u200b\u6570\u7ec4\u200b\u6574\u4f53\u200b\u590d\u5236\u200b\n   Vector ( T const* A, Rank lo, Rank hi ) { copyFrom ( A, lo, hi ); } //\u200b\u533a\u95f4\u200b\n   Vector ( Vector&lt;T&gt; const&amp; V ) { copyFrom ( V._elem, 0, V._size ); } //\u200b\u5411\u91cf\u200b\u6574\u4f53\u200b\u590d\u5236\u200b\n   Vector ( Vector&lt;T&gt; const&amp; V, Rank lo, Rank hi ) { copyFrom ( V._elem, lo, hi ); } //\u200b\u533a\u95f4\u200b\n// \u200b\u6790\u6784\u200b\u51fd\u6570\u200b\n   ~Vector() { delete [] _elem; } //\u200b\u91ca\u653e\u200b\u5185\u90e8\u7a7a\u95f4\u200b\n// \u200b\u53ea\u8bfb\u200b\u8bbf\u95ee\u200b\u63a5\u53e3\u200b\n   Rank size() const { return _size; } //\u200b\u89c4\u6a21\u200b\n   bool empty() const { return !_size; } //\u200b\u5224\u7a7a\u200b\n   Rank find ( T const&amp; e ) const { return find ( e, 0, _size ); } //\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u6574\u4f53\u200b\u67e5\u627e\u200b\n   Rank find ( T const&amp; e, Rank lo, Rank hi ) const; //\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u533a\u95f4\u200b\u67e5\u627e\u200b\n   Rank search ( T const&amp; e ) const //\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u6574\u4f53\u200b\u67e5\u627e\u200b\n   { return ( 0 &gt;= _size ) ? -1 : search ( e, 0, _size ); }\n   Rank search ( T const&amp; e, Rank lo, Rank hi ) const; //\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u533a\u95f4\u200b\u67e5\u627e\u200b\n// \u200b\u53ef\u200b\u5199\u8bbf\u95ee\u200b\u63a5\u53e3\u200b\n   T&amp; operator[] ( Rank r ); //\u200b\u91cd\u8f7d\u200b\u4e0b\u6807\u200b\u64cd\u4f5c\u7b26\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7c7b\u4f3c\u200b\u4e8e\u200b\u6570\u7ec4\u200b\u5f62\u5f0f\u200b\u5f15\u7528\u200b\u5404\u200b\u5143\u7d20\u200b\n   const T&amp; operator[] ( Rank r ) const; //\u200b\u4ec5\u9650\u4e8e\u200b\u505a\u53f3\u503c\u200b\u7684\u200b\u91cd\u8f7d\u200b\u7248\u672c\u200b\n   Vector&lt;T&gt; &amp; operator= ( Vector&lt;T&gt; const&amp; ); //\u200b\u91cd\u8f7d\u200b\u8d4b\u503c\u200b\u64cd\u4f5c\u7b26\u200b\uff0c\u200b\u4ee5\u4fbf\u200b\u76f4\u63a5\u200b\u514b\u9686\u200b\u5411\u91cf\u200b\n   T remove ( Rank r ); //\u200b\u5220\u9664\u200b\u79e9\u4e3a\u200br\u200b\u7684\u200b\u5143\u7d20\u200b\n   int remove ( Rank lo, Rank hi ); //\u200b\u5220\u9664\u200b\u79e9\u5728\u200b\u533a\u95f4\u200b[lo, hi)\u200b\u4e4b\u5185\u200b\u7684\u200b\u5143\u7d20\u200b\n   Rank insert ( Rank r, T const&amp; e ); //\u200b\u63d2\u5165\u200b\u5143\u7d20\u200b\n   Rank insert ( T const&amp; e ) { return insert ( _size, e ); } //\u200b\u9ed8\u8ba4\u200b\u4f5c\u4e3a\u200b\u672b\u200b\u5143\u7d20\u200b\u63d2\u5165\u200b\n   void sort ( Rank lo, Rank hi ); //\u200b\u5bf9\u200b[lo, hi)\u200b\u6392\u5e8f\u200b\n   void sort() { sort ( 0, _size ); } //\u200b\u6574\u4f53\u200b\u6392\u5e8f\u200b\n   void unsort ( Rank lo, Rank hi ); //\u200b\u5bf9\u200b[lo, hi)\u200b\u7f6e\u4e71\u200b\n   void unsort() { unsort ( 0, _size ); } //\u200b\u6574\u4f53\u200b\u7f6e\u4e71\u200b\n   int deduplicate(); //\u200b\u65e0\u5e8f\u200b\u53bb\u200b\u91cd\u200b\n   int uniquify(); //\u200b\u6709\u5e8f\u200b\u53bb\u200b\u91cd\u200b\n// \u200b\u904d\u5386\u200b\n   void traverse ( void (* ) ( T&amp; ) ); //\u200b\u904d\u5386\u200b\uff08\u200b\u4f7f\u7528\u200b\u51fd\u6570\u6307\u9488\u200b\uff0c\u200b\u53ea\u8bfb\u200b\u6216\u200b\u5c40\u90e8\u6027\u200b\u4fee\u6539\u200b\uff09\n   template &lt;typename VST&gt; void traverse ( VST&amp; ); //\u200b\u904d\u5386\u200b\uff08\u200b\u4f7f\u7528\u200b\u51fd\u6570\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u53ef\u200b\u5168\u5c40\u6027\u200b\u4fee\u6539\u200b\uff09\n}; //Vector\n</code></pre> Reference</p>"},{"location":"coding/dsa/vector-and-list/#_4","title":"\u63d2\u5165","text":"<p>\u200b\u5411\u91cf\u200b\u7684\u200b\u63d2\u5165\u200b\u64cd\u4f5c\u200b\u5206\u4e3a\u200b\u4e09\u6b65\u200b\uff1a</p> <ol> <li>\u200b\u68c0\u67e5\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u4e0e\u200b\u8f93\u5165\u200b\u4e0b\u200b\u6807\u7684\u200b\u5408\u6cd5\u6027\u200b\uff0c\u200b\u5fc5\u8981\u200b\u65f6\u200b\u6539\u53d8\u200b\u5411\u91cf\u200b\u7684\u200b\u957f\u5ea6\u200b\u4ee5\u200b\u5bb9\u7eb3\u200b\u8be5\u200b\u5143\u7d20\u200b</li> <li>\u200b\u5f53\u200b\u8f93\u5165\u200b\u4e0b\u6807\u200b\u4e0d\u662f\u200b\u5411\u91cf\u200b\u7684\u200b\u672b\u5c3e\u200b\u65f6\u200b\uff0c\u200b\u79fb\u52a8\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u5728\u200b\u5bf9\u5e94\u200b\u4f4d\u7f6e\u200b\u817e\u51fa\u200b\u7a7a\u95f4\u200b</li> <li>\u200b\u5c06\u200b\u6570\u636e\u200b\u5199\u5165\u200b\u5411\u91cf\u200b\u4e2d\u200b</li> </ol> <p>\u200b\u5728\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(n\\)\u200b\u7684\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u968f\u673a\u200b\u4f4d\u7f6e\u200b\u63d2\u5165\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u6539\u53d8\u200b\u5411\u91cf\u200b\u957f\u5ea6\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u65f6\u95f4\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)\uff0c\u200b\u79fb\u52a8\u200b\u5143\u7d20\u200b\u4ee5\u200b\u817e\u51fa\u200b\u7a7a\u95f4\u200b\u7684\u200b\u671f\u671b\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)\uff0c\u200b\u5199\u5165\u200b\u6570\u636e\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(1)\\)\uff0c\u200b\u603b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)</p>"},{"location":"coding/dsa/vector-and-list/#_5","title":"\u53ef\u200b\u6269\u5145\u200b\u5411\u91cf","text":"<p>\u200b\u5411\u91cf\u200b\u7684\u200b\u7a7a\u95f4\u200b\u662f\u200b\u6709\u9650\u200b\u7684\u200b\uff08\u200b\u5373\u200b\u4e3a\u200b\u5411\u91cf\u200b\u672c\u8eab\u200b\u7684\u200b\u957f\u5ea6\u200b\uff09\uff0c\u200b\u82e5\u5f53\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u4e0d\u8db3\u200b\u65f6\u4e3a\u200b\u5411\u91cf\u200b\u91cd\u65b0\u5206\u914d\u200b\u4e00\u5757\u200b\u66f4\u957f\u200b\u7684\u200b\u5185\u5b58\u200b\uff0c\u200b\u5c06\u200b\u539f\u200b\u6570\u636e\u200b\u590d\u5236\u5230\u200b\u65b0\u200b\u7684\u200b\u5185\u5b58\u7a7a\u95f4\u200b\u4e2d\u4ee5\u200b\u5b9e\u73b0\u200b\u5bb9\u91cf\u200b\u7684\u200b\u6269\u589e\u200b\uff0c\u200b\u5219\u200b\u5411\u91cf\u200b\u7684\u200b\u7a7a\u95f4\u200b\u53ef\u4ee5\u200b\u8fd1\u4f3c\u200b\u89c6\u4e3a\u200b\u65e0\u9650\u200b\u3002\u200b\u4e00\u822c\u200b\u5b58\u5728\u200b\u4e24\u79cd\u200b\u6269\u5145\u200b\u7b97\u6cd5\u200b\uff1a</p> <ul> <li>\u200b\u5f53\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u4e0d\u8db3\u200b\u65f6\u200b\uff0c\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u589e\u52a0\u200b\u4e00\u4e2a\u200b\u5e38\u91cf\u200b\\(I\\)</li> <li>\u200b\u5f53\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u4e0d\u8db3\u200b\u65f6\u200b\uff0c\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u4e58\u4ee5\u200b\u4e00\u5b9a\u200b\u500d\u6570\u200b\\(k\\)\uff08\u200b\u901a\u5e38\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\\(k=2\\)\uff09</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u521d\u59cb\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(0\\)\u200b\u7684\u200b\u5411\u91cf\u200b\uff0c\u200b\u8003\u5bdf\u200b\u63d2\u5165\u200b\\(n\\gg 2\\)\u200b\u4e2a\u200b\u5143\u7d20\u200b\u4e0b\u200b\u4e24\u79cd\u200b\u6269\u5145\u200b\u7b97\u6cd5\u200b\u7684\u200b\u8c03\u7528\u200b\u6b21\u6570\u200b\u3002\u200b\u9012\u589e\u200b\u7b56\u7565\u200b\u4e0b\u200b\u63d2\u5165\u200b\u7b2c\u200b\\(mI+1, (m\\in \\mathbb N)\\)\u200b\u4e2a\u200b\u5143\u7d20\u200b\u65f6\u200b\u9700\u8981\u200b\u8fdb\u884c\u200b\u6269\u5145\u200b\uff0c\u200b\u603b\u200b\u7684\u200b\u8c03\u7528\u200b\u6b21\u6570\u200b\u4e3a\u200b\\(\\frac{n}{I}\\)\uff0c\u200b\u5373\u200b\\(\\mathcal O(n)\\)\u3002\u200b\u500d\u589e\u200b\u7b56\u7565\u200b\u4e0b\u200b\u63d2\u5165\u200b\u7b2c\u200b\\(1, 2, 3, 5\\dots 2^m+1, (m\\in \\mathbb N)\\)\u200b\u4e2a\u200b\u5143\u7d20\u200b\u65f6\u200b\u9700\u8981\u200b\u8fdb\u884c\u200b\u6269\u5145\u200b\uff0c\u200b\u603b\u200b\u7684\u200b\u8c03\u7528\u200b\u6b21\u6570\u200b\u4e3a\u200b\\(\\mathcal O(\\log n)\\)\u3002</p> <p>\u200b\u590d\u5236\u200b\u6570\u636e\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)\uff0c\u200b\u56e0\u6b64\u200b\u9012\u589e\u200b\u7b56\u7565\u200b\u6bcf\u6b21\u200b\u6269\u5bb9\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u65f6\u95f4\u200b\u4e3a\u200b\\(0, I, 2I, \\dots\\)\uff0c\u200b\u5171\u200b\u9700\u8981\u200b\u6269\u5bb9\u200b\\(\\frac nI\\)\u200b\u6b21\u200b\uff0c\u200b\u6269\u5bb9\u200b\u64cd\u4f5c\u200b\u7684\u200b\u603b\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n^2)\\)\u3002\u200b\u6bcf\u6b21\u200b\u63d2\u5165\u200b\u64cd\u4f5c\u200b\u5206\u644a\u200b\\(\\mathcal O(n)\\)\u3002\u200b\u500d\u589e\u200b\u7b56\u7565\u200b\u6bcf\u6b21\u200b\u6269\u5bb9\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u65f6\u95f4\u200b\u4e3a\u200b\\(0, 1, 2, \\dots n\\)\uff0c\u200b\u5171\u200b\u9700\u8981\u200b\u6269\u5bb9\u200b\\(\\log n\\)\u200b\u6b21\u200b\u3002\u200b\u6269\u5bb9\u200b\u64cd\u4f5c\u200b\u7684\u200b\u603b\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)\uff0c\u200b\u6bcf\u6b21\u200b\u63d2\u5165\u200b\u64cd\u4f5c\u200b\u5206\u644a\u200b\\(\\mathcal O(1)\\)\u3002</p> <p>\u200b\u4f46\u200b\u500d\u589e\u200b\u7b56\u7565\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u7a7a\u95f4\u200b\u6d6a\u8d39\u200b\uff0c\u200b\u5177\u4f53\u8868\u73b0\u200b\u4e3a\u200b\u5411\u91cf\u200b\u7684\u200b\u88c5\u8f7d\u200b\u56e0\u5b50\u200b\u8f83\u200b\u4f4e\u200b\u3002\u200b\u63d2\u5165\u200b\u5927\u91cf\u200b\u5143\u7d20\u200b\u65f6\u200b\uff0c\u200b\u500d\u589e\u200b\u7b56\u7565\u200b\u7684\u200b\u88c5\u8f7d\u200b\u56e0\u5b50\u200b\u53ea\u80fd\u200b\u4fdd\u8bc1\u200b\\(&gt; 50\\)\uff0c\u200b\u800c\u200b\u9012\u589e\u200b\u7b56\u7565\u200b\u7684\u200b\u88c5\u8f7d\u200b\u56e0\u5b50\u200b\u53ef\u4ee5\u200b\u4fdd\u8bc1\u200b\u5728\u200b\\(1\\)\u200b\u5de6\u53f3\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_6","title":"\u5220\u9664","text":"<p>\u200b\u5411\u91cf\u200b\u7684\u200b\u5220\u9664\u200b\u64cd\u4f5c\u200b\u4e3b\u8981\u200b\u4e3a\u200b\u533a\u95f4\u200b\u5220\u9664\u200b\u4e0e\u200b\u5355\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u5220\u9664\u200b\u3002\u200b\u5176\u4e2d\u200b\u540e\u8005\u200b\u53ef\u4ee5\u200b\u89c6\u4e3a\u200b\u524d\u8005\u200b\u7684\u200b\u7279\u6b8a\u200b\u60c5\u51b5\u200b\u3002\u200b\u5220\u9664\u200b\u64cd\u4f5c\u200b\u4e0d\u200b\u9700\u8981\u200b\u6267\u884c\u200b\u989d\u5916\u200b\u7684\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u53ea\u200b\u9700\u8981\u200b\u5c06\u200b\u5220\u9664\u200b\u533a\u95f4\u200b\u540e\u200b\u7684\u200b\u5143\u7d20\u200b\u5411\u524d\u200b\u79fb\u52a8\u200b\uff0c\u200b\u8986\u76d6\u200b\u5220\u9664\u200b\u533a\u95f4\u200b\u5185\u200b\u7684\u200b\u6570\u636e\u200b\u5e76\u200b\u66f4\u65b0\u200b\u64cd\u4f5c\u200b\u7ed3\u675f\u200b\u540e\u200b\u5411\u91cf\u200b\u7684\u200b\u957f\u5ea6\u200b\u5373\u53ef\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u53ef\u200b\u6269\u5145\u200b\u5411\u91cf\u200b\uff0c\u200b\u5220\u9664\u200b\u540e\u200b\u5e94\u5bf9\u200b\u5411\u91cf\u200b\u7684\u200b\u88c5\u586b\u200b\u56e0\u5b50\u200b\u8fdb\u884c\u200b\u68c0\u67e5\u200b\uff0c\u200b\u5728\u200b\u5fc5\u8981\u200b\u65f6\u200b\u6536\u7f29\u200b\u5411\u91cf\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u591a\u6b21\u200b\u8c03\u7528\u200b\u5220\u9664\u200b\u5355\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u63a5\u53e3\u200b\u5b9e\u73b0\u200b\u533a\u95f4\u200b\u5220\u9664\u200b\uff0c\u200b\u7531\u4e8e\u200b\u6bcf\u6b21\u200b\u5220\u9664\u200b\u90fd\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u4e00\u6b21\u200b\u6570\u636e\u200b\u7684\u200b\u79fb\u52a8\u200b\uff0c\u200b\u603b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n^2)\\)</p> <p>\u200b\u5220\u9664\u200b\u64cd\u4f5c\u200b\u9700\u8981\u200b\u5bf9\u200b\u5220\u9664\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5408\u6cd5\u6027\u200b\u8fdb\u884c\u200b\u68c0\u9a8c\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_7","title":"\u67e5\u627e","text":"<p>\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u4e0e\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u67e5\u627e\u200b\u65b9\u6cd5\u200b\u4e0d\u540c\u200b\uff0c\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e5f\u200b\u4e0d\u540c\u200b\u3002\u200b\u5177\u4f53\u200b\u800c\u8a00\u200b\uff0c\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u987a\u5e8f\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u989d\u5916\u200b\u7684\u200b\u4fe1\u606f\u200b\uff0c\u200b\u4ece\u800c\u200b\u4f7f\u5f97\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u67e5\u627e\u200b\u64cd\u4f5c\u200b\u80fd\u591f\u200b\u5728\u200b\\(\\log n\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u5185\u200b\u5b8c\u6210\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_8","title":"\u987a\u5e8f\u200b\u67e5\u627e","text":"<p>\u200b\u987a\u5e8f\u200b\u67e5\u627e\u200b\u64cd\u4f5c\u200b\u5047\u8bbe\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u7c7b\u578b\u5b9a\u4e49\u200b\u4e86\u200b<code>==</code>\u200b\u8fd0\u7b97\u7b26\u200b\u4e0e\u200b<code>!=</code>\u200b\u8fd0\u7b97\u7b26\u200b\u3002</p> <p>\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u67e5\u627e\u200b\u53ea\u200b\u9700\u8981\u200b\u4ece\u200b\u5411\u91cf\u200b\u7684\u200b\u5f00\u5934\u200b\u4f9d\u6b21\u200b\u5bf9\u6bd4\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u4e0e\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\uff0c\u200b\u5f53\u200b\u5143\u7d20\u200b\u76f8\u540c\u200b\u65f6\u200b\u8fd4\u56de\u200b\u5373\u53ef\u200b\u3002\u200b\u5e73\u5747\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)</p>"},{"location":"coding/dsa/vector-and-list/#_9","title":"\u4e8c\u5206\u200b\u67e5\u627e","text":"<p>\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u67e5\u627e\u200b\u64cd\u4f5c\u200b\u5047\u8bbe\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u7c7b\u578b\u5b9a\u4e49\u200b\u7c7b\u200b<code>==</code>\u200b\u8fd0\u7b97\u7b26\u200b\u3001<code>!=</code>\u200b\u8fd0\u7b97\u7b26\u200b\uff0c\u200b\u56e0\u6b64\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u67e5\u627e\u200b\u65b9\u6cd5\u200b\u7ebf\u6027\u200b\u67e5\u627e\u200b\u3002\u200b\u82e5\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u5143\u7d20\u200b\u7c7b\u578b\u5b9a\u4e49\u200b\u4e86\u200b\u6bd4\u8f83\u200b\u8fd0\u7b97\u7b26\u200b<code>&lt;</code>\u200b\u4e0e\u200b<code>&gt;</code>\uff0c\u200b\u5219\u200b\u57fa\u4e8e\u200b\u6bd4\u8f83\u200b\u7684\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u964d\u200b\u81f3\u200b\\(\\mathcal O(\\log n)\\)\u3002</p> <p>\u200b\u51cf\u800c\u6cbb\u200b\u4e4b\u200b\u662f\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7684\u200b\u6838\u5fc3\u601d\u60f3\u200b\u3002\u200b\u8003\u8651\u200b\u66f4\u200b\u4e00\u822c\u200b\u7684\u200b\u901a\u7528\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(n\\)\u200b\u7684\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5728\u200b\\(\\lambda \\cdot n (0\\leq \\lambda\\leq 1)\\)\u200b\u5904\u8bbe\u200b\u4e3a\u8f74\u70b9\u200b\u3002\u200b\u6bcf\u6b21\u200b\u6bd4\u8f83\u200b\u5f53\u524d\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u7684\u200b\u8f74\u70b9\u200b\\(B\\)\u200b\u4e0e\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\\(A\\)\u3002\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u5bf9\u8f74\u70b9\u200b\u7684\u200b\u6bd4\u8f83\u200b\u6709\u200b\u4e09\u79cd\u200b\u53ef\u80fd\u200b\u7684\u200b\u7ed3\u679c\u200b\uff0c\u200b\u5047\u8bbe\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u6309\u7167\u200b\u5347\u5e8f\u200b\u6392\u5e8f\u200b\uff0c\u200b\u5219\u200b\uff1a</p> <ul> <li>\\(A&lt;B\\)\uff0c\u200b\u8868\u793a\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u53ea\u200b\u53ef\u80fd\u200b\u51fa\u73b0\u200b\u5728\u200b\u8f74\u70b9\u200b\u5de6\u4fa7\u200b\uff0c\u200b\u5c06\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u7f29\u5c0f\u200b\u81f3\u200b\u5de6\u534a\u200b\u90e8\u5206\u200b\uff0c\u200b\u7ee7\u7eed\u200b\u67e5\u627e\u200b\u8fc7\u7a0b\u200b\uff1b</li> <li>\\(A=B\\)\uff0c\u200b\u8868\u793a\u200b\u8f74\u70b9\u200b\u5904\u200b\u51fa\u73b0\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\uff0c\u200b\u76f4\u63a5\u200b\u8fd4\u56de\u200b\u4e2d\u70b9\u200b\u7684\u200b\u4e0b\u6807\u200b\u5373\u53ef\u200b\uff1b</li> <li>\\(B&lt;A\\)\uff0c\u200b\u8868\u793a\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u53ea\u200b\u53ef\u80fd\u200b\u51fa\u73b0\u200b\u5728\u200b\u8f74\u70b9\u200b\u53f3\u4fa7\u200b\uff0c\u200b\u5c06\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u7f29\u5c0f\u200b\u81f3\u200b\u53f3\u534a\u200b\u90e8\u5206\u200b\uff0c\u200b\u7ee7\u7eed\u200b\u67e5\u627e\u200b\u8fc7\u7a0b\u200b\u3002</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\uff0c\u200b\u8f74\u70b9\u200b\u4e3a\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u7684\u200b\u4e2d\u70b9\u200b\uff0c\u200b\u5373\u200b\\(\\lambda = 0.5\\)\u3002</p> <p>1234567891011</p> <p>\u200b\u4f46\u200b\u8be5\u200b\u7248\u672c\u200b\u7684\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u6709\u200b\u591a\u79cd\u200b\u610f\u5916\u200b\u60c5\u5f62\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <ol> <li>\u200b\u5411\u91cf\u200b\u4e2d\u200b\u627e\u200b\u4e0d\u5230\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\uff1b</li> <li>\u200b\u5411\u91cf\u200b\u4e2d\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u5b58\u5728\u200b\u591a\u4e2a\u200b\u3002</li> </ol> <p>\u200b\u6b64\u65f6\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u7684\u200b\u7ed3\u679c\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u552f\u4e00\u200b\u3002\u200b\u7ea6\u5b9a\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7b97\u6cd5\u200b\u8fd4\u56de\u200b\u4e0d\u200b\u5927\u4e8e\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002</p> <p>\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u901a\u8fc7\u200b\u5c3d\u53ef\u80fd\u51cf\u5c11\u200b\u9012\u5f52\u200b\u6df1\u5ea6\u200b\u7684\u200b\u65b9\u5f0f\u200b\u51cf\u5c11\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\uff0c\u200b\u7531\u6b64\u200b\uff0c\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u9012\u5f52\u200b\u7684\u200b\u4e0d\u540c\u200b\u5206\u652f\u200b\u5e94\u6709\u200b\u76f8\u540c\u200b\u7684\u200b\u671f\u671b\u200b\u65f6\u95f4\u200b\u6d88\u8017\u200b\uff08\u200b\u5373\u200b\u8f6c\u5411\u200b\u6210\u672c\u200b\uff09\u3002\u200b\u6b64\u5904\u200b\u7684\u200b\u201c\u200b\u4e8c\u5206\u200b\u201d\u200b\u53ea\u662f\u200b\u5728\u200b\u6570\u7ec4\u200b\u533a\u95f4\u200b\u7684\u200b\u610f\u4e49\u200b\u4e0a\u200b\u4e8c\u7b49\u200b\u5206\u200b\uff0c\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u8bed\u53e5\u200b\uff1a</p> <pre><code>if (A &gt; B)\n    // statement\nelse if (A &lt; B)\n    // statement\nelse\n    // statement\n</code></pre> <p>\u200b\u5de6\u53f3\u4e24\u4e2a\u200b\u8df3\u8f6c\u200b\u4f4d\u4e8e\u200b\u4e0d\u540c\u200b\u7684\u200b\u5206\u652f\u200b\uff0c\u200b\u9700\u8981\u200b\u8fdb\u884c\u200b\u6bd4\u8f83\u200b\u7684\u200b\u6b21\u6570\u200b\u4e0d\u540c\u200b\uff0c\u200b\u56e0\u6b64\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u65f6\u95f4\u200b\u5728\u200b\u4e25\u683c\u200b\u610f\u4e49\u200b\u4e0a\u200b\u662f\u200b\u4e0d\u540c\u200b\u7684\u200b\u3002\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u6539\u53d8\u200b\u8f74\u70b9\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u5c06\u200b\u540c\u4e00\u200b\u9012\u5f52\u200b\u6df1\u5ea6\u200b\u4e0b\u200b\u6bd4\u8f83\u200b\u7684\u200b\u671f\u671b\u200b\u6b21\u6570\u200b\u8c03\u6574\u200b\u4e3a\u200b\u76f8\u540c\u200b\uff0c\u200b\u4ece\u800c\u200b\u964d\u4f4e\u200b\u5e73\u5747\u200b\u67e5\u627e\u200b\u957f\u5ea6\u200b\uff0c\u200b\u5728\u200b\u5e38\u200b\u7cfb\u6570\u200b\u7a0b\u5ea6\u200b\u4e0a\u200b\u5bf9\u200b\u7b97\u6cd5\u200b\u8fdb\u884c\u200b\u4f18\u5316\u200b\u3002</p> <p>+1+2n\u03bbn(1 -\u03bb)\u00b7n</p> <p>\u200b\u8bbe\u200b\u7b97\u6cd5\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\alpha(\\lambda) \\log n\\)\uff0c\u200b\u5219\u200b\uff1a</p> <ul> <li>\u200b\u5de6\u4fa7\u200b\u5b50\u200b\u95ee\u9898\u200b\u6d88\u8017\u200b\u7684\u200b\u65f6\u95f4\u200b\u7b49\u4e8e\u200b\u5224\u65ad\u200b\u6d88\u8017\u200b\u7684\u200b\u65f6\u95f4\u200b\u53ca\u200b\u89e3\u51b3\u200b\u5b50\u200b\u95ee\u9898\u200b\u6d88\u8017\u200b\u7684\u200b\u65f6\u95f4\u200b\uff0c\u200b\u5373\u200b\uff1a\\(\\lambda (1 + \\alpha(\\lambda)  \\log (\\lambda n))\\)</li> <li>\u200b\u540c\u7406\u200b\uff0c\u200b\u53f3\u4fa7\u200b\u5b50\u200b\u95ee\u9898\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\uff1a\\((1 - \\lambda)(2+\\alpha(\\lambda)\\log((1-\\lambda)n))\\)</li> </ul> <p>\u200b\u5f97\u200b\\(\\alpha(\\lambda) = \\frac{\\ln 2(\\lambda - 2)}{\\lambda \\cdot\\ln\\lambda+(1-\\lambda)\\cdot\\ln(1-\\lambda)}\\)</p> <p>\u200b\u5f53\u200b\\(\\lambda=\\frac{\\sqrt 5-1}{2}\\approx 0.618\\)\u200b\u65f6\u200b\uff0c\\(\\alpha(\\lambda)\\)\u200b\u53d6\u200b\u6700\u5c0f\u503c\u200b\u3002</p> <p>\u200b\u6d88\u9664\u200b\u4e0d\u200b\u5bf9\u79f0\u200b\u7684\u200b\u53e6\u200b\u4e00\u79cd\u200b\u65b9\u5f0f\u200b\u662f\u200b\u5c06\u200b\u4e09\u79cd\u200b\u6bd4\u8f83\u200b\u7ed3\u679c\u200b\u53d8\u4e3a\u200b\u4e24\u79cd\u200b\uff0c\u200b\u4ece\u800c\u200b\u4e00\u6b21\u200b\u6bd4\u8f83\u200b\u5373\u53ef\u200b\u8fdb\u884c\u200b\u5212\u5206\u200b\u3002\u200b\u8bb0\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u4e3a\u200b\\(A\\)\uff0c\u200b\u8f74\u70b9\u200b\u4e3a\u200b\\(B\\)\uff0c\u200b\u5219\u200b\uff1a</p> <ul> <li>\\(A&lt;B\\)\uff1a\u200b\u5728\u200b\u8f74\u70b9\u200b\u5de6\u4fa7\u200b\u7684\u200b\u533a\u95f4\u200b\u4e2d\u200b\u67e5\u627e\u200b</li> <li>\\(A\\geq B\\)\uff1a\u200b\u5728\u200b\u8f74\u70b9\u200b\u53f3\u4fa7\u200b \uff08\u200b\u5305\u542b\u200b\u8f74\u70b9\u200b\uff09 \u200b\u7684\u200b\u533a\u95f4\u200b\u4e2d\u200b\u67e5\u627e\u200b</li> <li>\u200b\u5f53\u200b\u533a\u95f4\u200b\u957f\u5ea6\u200b\u7f29\u51cf\u200b\u4e3a\u200b1\u200b\u65f6\u200b\uff0c\u200b\u8868\u793a\u200b\u67e5\u627e\u200b\u8fc7\u7a0b\u200b\u7ed3\u675f\u200b\u3002</li> </ul> <p>\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7684\u200b\u6700\u7ec8\u200b\u5b9e\u73b0\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</p> \u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b <p><pre><code>/******************************************************************************************\n* Data Structures in C++\n* ISBN: 7-302-33064-6 &amp; 7-302-33065-3 &amp; 7-302-29652-2 &amp; 7-302-26883-3\n* Junhui DENG, deng@tsinghua.edu.cn\n* Computer Science &amp; Technology, Tsinghua University\n* Copyright (c) 2003-2020. All rights reserved.\n******************************************************************************************/\n\n// \u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7b97\u6cd5\u200b\uff08\u200b\u7248\u672c\u200bC\uff09\uff1a\u200b\u5728\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u533a\u95f4\u200b[lo, hi)\u200b\u5185\u200b\u67e5\u627e\u200b\u5143\u7d20\u200be\uff0c0 &lt;= lo &lt;= hi &lt;= _size\ntemplate &lt;typename T&gt; static Rank binSearch ( T* S, T const&amp; e, Rank lo, Rank hi ) {\n   while ( lo &lt; hi ) { //\u200b\u6bcf\u6b65\u200b\u8fed\u4ee3\u200b\u4ec5\u200b\u9700\u200b\u505a\u200b\u4e00\u6b21\u200b\u6bd4\u8f83\u200b\u5224\u65ad\u200b\uff0c\u200b\u6709\u200b\u4e24\u4e2a\u200b\u5206\u652f\u200b\n      Rank mi = ( lo + hi ) &gt;&gt; 1; //\u200b\u4ee5\u4e2d\u70b9\u200b\u4e3a\u8f74\u70b9\u200b\uff08\u200b\u533a\u95f4\u200b\u5bbd\u5ea6\u200b\u7684\u200b\u6298\u534a\u200b\uff0c\u200b\u7b49\u6548\u4e8e\u200b\u5bbd\u5ea6\u200b\u4e4b\u200b\u6570\u503c\u200b\u8868\u793a\u200b\u7684\u200b\u53f3\u79fb\u200b\uff09\n      ( e &lt; S[mi] ) ? hi = mi : lo = mi + 1; //\u200b\u7ecf\u200b\u6bd4\u8f83\u200b\u540e\u200b\u786e\u5b9a\u200b\u6df1\u5165\u200b[lo, mi)\u200b\u6216\u200b(mi, hi)\n   } //\u200b\u6210\u529f\u200b\u67e5\u627e\u200b\u4e0d\u80fd\u200b\u63d0\u524d\u200b\u7ec8\u6b62\u200b\n   return lo - 1; //\u200b\u5faa\u73af\u200b\u7ed3\u675f\u200b\u65f6\u200b\uff0clo\u200b\u4e3a\u200b\u5927\u4e8e\u200be\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u6700\u5c0f\u200b\u79e9\u200b\uff0c\u200b\u6545\u200blo - 1\u200b\u5373\u200b\u4e0d\u200b\u5927\u4e8e\u200be\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u6700\u5927\u200b\u79e9\u200b\n} //\u200b\u6709\u200b\u591a\u4e2a\u200b\u547d\u4e2d\u200b\u5143\u7d20\u200b\u65f6\u200b\uff0c\u200b\u603b\u80fd\u200b\u4fdd\u8bc1\u200b\u8fd4\u56de\u200b\u79e9\u200b\u6700\u5927\u8005\u200b\uff1b\u200b\u67e5\u627e\u200b\u5931\u8d25\u200b\u65f6\u200b\uff0c\u200b\u80fd\u591f\u200b\u8fd4\u56de\u200b\u5931\u8d25\u200b\u7684\u200b\u4f4d\u7f6e\u200b\n</code></pre> Reference</p>"},{"location":"coding/dsa/vector-and-list/#_10","title":"\u63d2\u503c\u200b\u67e5\u627e","text":"<p>\u200b\u5047\u8bbe\u200b\u533a\u95f4\u200b\u5185\u200b\u5143\u7d20\u200b\u5206\u5e03\u200b\u7684\u200b\u89c4\u5f8b\u200b\u5df2\u77e5\u200b\uff0c\u200b\u6839\u636e\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u5728\u200b\u8be5\u200b\u5206\u5e03\u200b\u4e2d\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u63a8\u65ad\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u5728\u200b\u533a\u95f4\u200b\u7684\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u4ee5\u6b64\u200b\u786e\u5b9a\u200b\u5212\u5206\u200b\u7684\u200b\u8f74\u70b9\u200b\u4f4d\u7f6e\u200b\u3002</p> <p>\u200b\u5047\u8bbe\u200b\u533a\u95f4\u200b\u5185\u200b\u5143\u7d20\u200b\u670d\u4ece\u200b\u72ec\u7acb\u200b\u7684\u200b\u5e73\u5747\u200b\u5206\u5e03\u200b\uff0c\u200b\u5219\u200b\u6bcf\u6b21\u200b\u67e5\u627e\u200b\u5e73\u5747\u200b\u4f7f\u5f97\u200b\u533a\u95f4\u200b\u7f29\u51cf\u200b\u81f3\u200b\u539f\u6765\u200b\u7684\u200b\\(\\sqrt n\\)\uff0c\u200b\u5e73\u5747\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(\\log\\log n)\\)\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_11","title":"\u53bb\u200b\u91cd","text":"<p>\u200b\u5bf9\u4e8e\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\uff0c\u200b\u53bb\u200b\u91cd\u200b\u64cd\u4f5c\u200b\u53ef\u4ee5\u200b\u5728\u200b\u7ebf\u6027\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u5185\u200b\u5b8c\u6210\u200b\uff0c\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u53bb\u200b\u91cd\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n^2)\\)</p>"},{"location":"coding/dsa/vector-and-list/#_12","title":"\u65e0\u5e8f\u200b\u5411\u91cf","text":"<p>\u200b\u4f7f\u7528\u200b\u6307\u9488\u200b\u6807\u8bb0\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u5df2\u7ecf\u200b\u5b8c\u6210\u200b\u53bb\u200b\u91cd\u200b\u7684\u200b\u90e8\u5206\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u672a\u53bb\u200b\u91cd\u200b\u90e8\u5206\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u5728\u200b\u672a\u200b\u53bb\u200b\u91cd\u200b\u90e8\u5206\u200b\u7684\u200b\u540e\u200b\u534a\u200b\u90e8\u5206\u200b\u4e2d\u200b\u67e5\u627e\u200b\u662f\u5426\u200b\u5b58\u5728\u200b\u76f8\u540c\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u5982\u679c\u200b\u627e\u5230\u200b\u76f8\u540c\u200b\u7684\u200b\u5143\u7d20\u200b\u5219\u200b\u5220\u9664\u200b\u3002\u200b\u5b8c\u6210\u200b\u67e5\u627e\u200b\u540e\u200b\u6307\u9488\u200b\u540e\u79fb\u200b\uff0c\u200b\u5c06\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u6807\u8bb0\u200b\u4e3a\u200b\u5df2\u7ecf\u200b\u5b8c\u6210\u200b\u53bb\u200b\u91cd\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_13","title":"\u6709\u5e8f\u200b\u5411\u91cf","text":"<p>\u200b\u4f7f\u7528\u200b\u53cc\u200b\u6307\u9488\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u4e00\u4e2a\u200b\u6307\u9488\u200b\\(P\\)\u200b\u6807\u8bb0\u200b\u5df2\u53bb\u200b\u91cd\u200b\u90e8\u5206\u200b\u7684\u200b\u7ed3\u5c3e\u200b\uff0c\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u6307\u9488\u200b\\(Q\\)\u200b\u6807\u8bb0\u200b\u672a\u53bb\u200b\u91cd\u200b\u90e8\u5206\u200b\u7684\u200b\u5f00\u5934\u200b\uff0c\u200b\u521d\u59cb\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\\(P\\)\u200b\u4f4d\u4e8e\u200b\u5411\u91cf\u200b\u7b2c\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u4e14\u200b\\(P+1=Q\\)\u3002\u200b\u5f53\u200b\\(P\\)\u200b\u6307\u5411\u200b\u7684\u200b\u5bf9\u8c61\u200b\u4e0e\u200b\\(Q\\)\u200b\u6307\u5411\u200b\u7684\u200b\u5bf9\u8c61\u200b\u76f8\u540c\u200b\u65f6\u200b\uff0c\\(Q\\)\u200b\u9012\u589e\u200b\u800c\u200b\\(P\\)\u200b\u4e0d\u200b\u79fb\u52a8\u200b\uff0c\u200b\u5426\u5219\u200b\u5c06\u200b\\(Q\\)\u200b\u5904\u200b\u7684\u200b\u5143\u7d20\u200b\u590d\u5236\u5230\u200b\\(P+1\\)\u200b\u5904\u200b\uff0c\u200b\u4e24\u200b\u6307\u9488\u200b\u540c\u65f6\u200b\u9012\u589e\u200b\u3002</p> <p> 33355558815333555588153355555881535555588815355558881515 </p>"},{"location":"coding/dsa/vector-and-list/#_14","title":"\u6392\u5e8f","text":"<p>\u200b\u6b64\u5904\u200b\u4ecb\u7ecd\u200b\u5bf9\u200b\u5411\u91cf\u200b\u7684\u200b\u5192\u6ce1\u6392\u5e8f\u200b\u4e0e\u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b\uff0c\u200b\u5176\u4ed6\u200b\u6392\u5e8f\u200b\u65b9\u5f0f\u200b\u5c06\u200b\u5728\u200b\u4ee5\u540e\u200b\u8fdb\u884c\u200b\u4ecb\u7ecd\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_15","title":"\u5192\u6ce1\u6392\u5e8f","text":"<p>\u200b\u5bf9\u4e8e\u200b\u6570\u5217\u200b\\(\\{a_1, \\cdots, a_n\\}\\)\uff0c\u200b\u82e5\u200b\\(a_i&gt;a_{i+1}\\)\uff0c\u200b\u79f0\u200b\\((a_i, a_{i+1})\\)\u200b\u4e3a\u200b\u4e00\u5bf9\u200b\u9006\u5e8f\u200b\u5bf9\u200b\u3002\u200b\u5426\u5219\u200b\u79f0\u4e3a\u200b\u987a\u5e8f\u200b\u5bf9\u200b\u3002\u200b\u5219\u200b\u6709\u5e8f\u200b\u6570\u5217\u200b\u4e2d\u200b\u4e0d\u200b\u5b58\u5728\u200b\u9006\u5e8f\u200b\u5bf9\u200b\uff0c\u200b\u65e0\u5e8f\u200b\u6570\u5217\u200b\u4e2d\u200b\u81f3\u5c11\u200b\u6709\u200b\u4e00\u4e2a\u200b\u9006\u5e8f\u200b\u5bf9\u200b\u3002\u200b\u5982\u679c\u200b\u80fd\u591f\u200b\u901a\u8fc7\u200b\u9006\u5e8f\u200b\u5bf9\u200b\u7684\u200b\u4ea4\u6362\u200b\u4f7f\u5f97\u200b\u6570\u5217\u200b\u4e2d\u200b\u6240\u6709\u200b\u7684\u200b\u9006\u5e8f\u200b\u5bf9\u200b\u8f6c\u4e3a\u200b\u987a\u5e8f\u200b\uff0c\u200b\u5373\u53ef\u200b\u5b8c\u6210\u200b\u5bf9\u200b\u6570\u7ec4\u200b\u7684\u200b\u6392\u5e8f\u200b\u3002\u200b\u8fd9\u662f\u200b\u5192\u6ce1\u6392\u5e8f\u200b\u7b97\u6cd5\u200b\u7684\u200b\u6838\u5fc3\u601d\u60f3\u200b\u3002</p> <ol> <li>\u200b\u8bbe\u200b\u5411\u91cf\u200b\uff08\u200b\u6570\u7ec4\u200b\uff09\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(n\\)\uff0c\u200b\u5f85\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u4e0b\u6807\u200b\u4e3a\u200b\\([0,n)\\)</li> <li>\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u626b\u63cf\u200b\u5f85\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u4e2d\u200b\u7684\u200b\u76f8\u90bb\u200b\u6570\u636e\u200b\uff0c\u200b\u82e5\u200b\u4e3a\u200b\u9006\u5e8f\u200b\u5bf9\u200b\u5219\u200b\u4ea4\u6362\u200b\u4e4b\u200b</li> <li>\u200b\u626b\u63cf\u200b\u7ed3\u675f\u200b\u540e\u200b\uff0c\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u6700\u5927\u200b\u5143\u7d20\u200b\u79fb\u52a8\u200b\u5230\u200b\u5f85\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u672b\u5c3e\u200b\uff0c\u200b\u5373\u200b\u4e3a\u200b\u5df2\u200b\u6392\u5e8f\u200b</li> <li>\u200b\u4e00\u8d9f\u200b\u626b\u63cf\u200b\u540e\u200b\u5c06\u200b\u5f85\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u672b\u5c3e\u200b\u5143\u7d20\u200b\u5212\u5165\u200b\u5df2\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u3002</li> <li>\u200b\u91cd\u590d\u200b\u626b\u63cf\u200b\uff0c\u200b\u76f4\u5230\u200b\u5f85\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u957f\u5ea6\u200b\u7f29\u5c0f\u200b\u4e3a\u200b0\uff0c\u200b\u5373\u200b\u5b8c\u6210\u200b\u6392\u5e8f\u200b\u3002</li> </ol> <p>\u200b\u63d0\u524d\u200b\u7ec8\u6b62\u200b\uff1a\u200b\u82e5\u200b\u67d0\u6b21\u200b\u626b\u63cf\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u6ca1\u6709\u200b\u53d1\u751f\u200b\u4ea4\u6362\u200b\uff0c\u200b\u8bf4\u660e\u200b\u5f85\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u5df2\u7ecf\u200b\u6709\u5e8f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u63d0\u524d\u7ed3\u675f\u200b\u7b97\u6cd5\u200b</p> <p>\u200b\u7531\u4e8e\u200b\u63d0\u524d\u200b\u7ec8\u6b62\u200b\u7684\u200b\u5b58\u5728\u200b\uff0c\u200b\u5192\u6ce1\u6392\u5e8f\u200b\u5728\u200b\u6700\u597d\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)\uff0c\u200b\u5bf9\u5e94\u200b\u8f93\u5165\u200b\u6570\u636e\u200b\u5df2\u7ecf\u200b\u6709\u5e8f\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002\u200b\u5e73\u5747\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u5192\u6ce1\u6392\u5e8f\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n^2)\\)\u3002</p> <p>\u200b\u5728\u200b\u5192\u6ce1\u6392\u5e8f\u200b\u4e2d\u200b\uff0c\u200b\u76f8\u7b49\u200b\u7684\u200b\u5143\u7d20\u200b\u4e0d\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u9006\u5e8f\u200b\u5bf9\u200b\u3002\u200b\u5982\u679c\u200b\u5c06\u200b\u76f8\u7b49\u200b\u7684\u200b\u5143\u7d20\u200b\u89c6\u4e3a\u200b\u9006\u5e8f\u200b\u5bf9\u200b\uff0c\u200b\u5192\u6ce1\u6392\u5e8f\u200b\u5c06\u200b\u5931\u53bb\u200b\u7a33\u5b9a\u6027\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_16","title":"\u5f52\u5e76\u200b\u6392\u5e8f","text":"<p>\u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b\u91c7\u7528\u200b\u5206\u800c\u6cbb\u4e4b\u200b\u7684\u200b\u65b9\u6cd5\u200b\u5bf9\u200b\u6570\u7ec4\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b\uff0c\u200b\u5176\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u9012\u5f52\u200b\u3002\u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b\u7684\u200b\u7b97\u6cd5\u200b\u8fc7\u7a0b\u200b\u4e3b\u8981\u200b\u5206\u4e3a\u200b\u4e09\u6b65\u200b\uff1a</p> <ul> <li>\u201c\u200b\u5206\u200b\u201d\uff1a\u200b\u5c06\u200b\u5411\u91cf\u200b\u5212\u5206\u200b\u4e3a\u200b\u4e24\u4e2a\u200b\uff08\u200b\u8fd1\u4f3c\u200b\uff09\u200b\u7b49\u957f\u200b\u7684\u200b\u90e8\u5206\u200b\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(1)\\)</li> <li>\u201c\u200b\u6cbb\u200b\u201d\uff1a\u200b\u5bf9\u200b\u4e24\u200b\u90e8\u5206\u200b\u5206\u522b\u200b\u8fdb\u884c\u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b\uff08\u200b\u5df2\u77e5\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(1\\)\u200b\u7684\u200b\u5411\u91cf\u200b\u662f\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\uff09\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(2\\times T(n/2)\\)</li> <li>\u201c\u200b\u5408\u200b\u201d\uff1a\u200b\u5c06\u200b\u6392\u5e8f\u200b\u540e\u200b\u7684\u200b\u4e24\u200b\u90e8\u5206\u200b\u5408\u5e76\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u6709\u5e8f\u200b\u7684\u200b\u6574\u4f53\u200b\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)</li> <li>\u200b\u7b97\u6cd5\u200b\u6574\u4f53\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n\\log n)\\)\uff0c\u200b\u5373\u4f7f\u200b\u662f\u200b\u5728\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u6709\u5e8f\u200b\u6216\u200b\u63a5\u8fd1\u200b\u6709\u5e8f\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u3002\u200b\u5f52\u5e76\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)</li> </ul> \u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b <p><pre><code>template &lt;typename T&gt; void Vector&lt;T&gt;::mergeSort( Rank lo, Rank hi ) {\n   if ( hi - lo &lt; 2 ) return; //\u200b\u5355\u200b\u5143\u7d20\u200b\u533a\u95f4\u200b\u81ea\u7136\u200b\u6709\u5e8f\u200b\uff0c\u200b\u5426\u5219\u200b...\n   int mi = (lo + hi) &gt;&gt; 1; //\u200b\u4ee5\u4e2d\u70b9\u200b\u4e3a\u754c\u200b\n   mergeSort( lo, mi ); //\u200b\u5bf9\u200b\u524d\u534a\u6bb5\u200b\u6392\u5e8f\u200b\n   mergeSort( mi, hi ); //\u200b\u5bf9\u200b\u540e\u534a\u6bb5\u200b\u6392\u5e8f\u200b\n   merge( lo, mi, hi ); //\u200b\u5f52\u5e76\u200b\n}\n</code></pre> Reference</p> <p>\u200b\u5f52\u5e76\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u6838\u5fc3\u601d\u60f3\u200b\u5373\u200b\u6309\u200b\u987a\u5e8f\u200b\u904d\u5386\u200b\u4e24\u200b\u90e8\u5206\u200b\uff0c\u200b\u5c06\u200b\u6700\u5c0f\u200b\u7684\u200b\u5199\u5165\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u3002</p> <p> </p> <p>\u200b\u5b9e\u9645\u200b\u6267\u884c\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\uff0c\u200b\u53ea\u200b\u9700\u200b\u5c06\u200b\u6570\u7ec4\u200b\u7684\u200b\u524d\u534a\u90e8\u200b\u5206\u200b\u590d\u5236\u200b\u4e00\u4efd\u200b\uff0c\u200b\u7136\u540e\u200b\u8fdb\u884c\u200b\u5408\u5e76\u200b\u5373\u53ef\u200b\u3002</p> <ul> <li>\u200b\u82e5\u200b\u524d\u534a\u90e8\u200b\u5206\u200b\u63d0\u524d\u7ed3\u675f\u200b\uff0c\u200b\u7531\u4e8e\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\u7684\u200b\u540e\u200b\u534a\u200b\u90e8\u5206\u200b\u4e0e\u200b\u539f\u200b\u6570\u7ec4\u200b\u7684\u200b\u540e\u200b\u534a\u200b\u90e8\u5206\u200b\u91cd\u5408\u200b\uff0c\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\u81ea\u52a8\u200b\u6210\u4e3a\u200b\u6709\u5e8f\u200b</li> <li>\u200b\u82e5\u540e\u200b\u534a\u200b\u90e8\u5206\u200b\u63d0\u524d\u7ed3\u675f\u200b\uff0c\u200b\u5c06\u200b\u524d\u534a\u90e8\u200b\u5206\u200b\u5269\u4f59\u200b\u7684\u200b\u6570\u636e\u200b\u590d\u5236\u5230\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\u540e\u200b\u534a\u200b\u90e8\u5206\u200b\u5bf9\u5e94\u200b\u4f4d\u7f6e\u200b\u5373\u53ef\u200b</li> </ul> <p>\u200b\u5982\u4e0b\u200b\u56fe\u200b\uff1a</p> <p> </p> <p>\u200b\u82e5\u200b\u5728\u200b\u5f52\u5e76\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u76f8\u540c\u200b\u5143\u7d20\u200b\uff0c\u200b\u5de6\u4fa7\u200b\u7684\u200b\u5143\u7d20\u200b\u4f18\u5148\u200b\u5f52\u5165\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u4fdd\u8bc1\u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b\u7684\u200b\u7a33\u5b9a\u6027\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_17","title":"\u4f4d\u200b\u56fe","text":"<p>\u200b\u4f4d\u56fe\u200b\u662f\u200b\u5176\u4e2d\u200b\u5143\u7d20\u200b\u53ea\u80fd\u200b\u53d6\u200b0/1\u200b\u7684\u200b\u5411\u91cf\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u4e00\u4e2a\u200b\u6709\u9650\u200b\u7684\u200b\u6574\u6570\u200b\u96c6\u5408\u200b\\(U\\)\uff0c\u200b\u4f4d\u200b\u56fe\u200b\u6784\u5efa\u200b\u4e86\u200b\\(U\\)\u200b\u7684\u200b\u5b50\u96c6\u200b\\(S\\)\uff0c\u200b\u5176\u4e2d\u200b\\(k\\in S\\)\u200b\u7b49\u4ef7\u200b\u4e8e\u4f4d\u200b\u56fe\u4e2d\u200b\u4e0b\u6807\u200b\u4e3a\u200b\\(k\\)\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u53d6\u503c\u200b\u4e3a\u200b\\(1\\)\u3002\u200b\u4f4d\u200b\u56fe\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>set()</code>\u3001<code>get()</code>\u3001<code>clear()</code>\u200b\u4e09\u4e2a\u200b\u63a5\u53e3\u200b\uff0c\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u4e8e\u4f4d\u200b\u56fe\u200b\u7684\u200b\u5199\u5165\u200b\u3001\u200b\u67e5\u8be2\u200b\u3001\u200b\u6e05\u9664\u200b\u64cd\u4f5c\u200b\u3002</p> <p>byte11010011</p> <p>\u200b\u57fa\u4e8e\u200b\u4f4d\u200b\u8fd0\u7b97\u200b\u7684\u200b\u5b9e\u73b0\u200b\u5982\u4e0b\u200b\uff1a</p> \u200b\u4f4d\u200b\u56fe\u200b\u65b9\u6cd5\u200b\u5b9e\u73b0\u200b <p><pre><code>bool test( int k ) { return M[ k &gt;&gt; 3 ] &amp; ( 0x80 &gt;&gt; (k &amp; 0x07) ); }\nvoid set( int k ) { expand( k ); M[ k &gt;&gt; 3 ] |= ( 0x80 &gt;&gt; (k &amp; 0x07) ); }\nvoid clear( int k ) { expand( k ); M[ k &gt;&gt; 3 ] &amp;= ~( 0x80 &gt;&gt; (k &amp; 0x07) ); }\n</code></pre> Reference</p> <p>\u200b\u8bfb\u53d6\u200b\u5b57\u8282\u200b\u4e2d\u200b\u7684\u200b\u67d0\u4e2a\u200b\u4f4d\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>&amp;</code>\u200b\u8fd0\u7b97\u7b26\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u5411\u200b\u5b57\u8282\u200b\u4e2d\u200b\u8bbe\u7f6e\u200b\u67d0\u4e2a\u200b\u4f4d\u4e3a\u200b\\(1\\)\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>|=</code>\u200b\u8fd0\u7b97\u7b26\u200b\u5b9e\u73b0\u200b\u3002</p> <p>\u200b\u4f4d\u56fe\u200b\u7684\u200b\u5e94\u7528\u200b\u573a\u666f\u200b\uff1a</p> <ol> <li>\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\u53bb\u200b\u91cd\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u501f\u52a9\u200b\u4f4d\u200b\u56fe\u200b\u53ef\u4ee5\u200b\u5728\u200b\\(O(n)\\)\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u5185\u200b\u5b8c\u6210\u200b\uff0c\u200b\u540c\u65f6\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u5b8c\u6210\u200b\u76f8\u5e94\u200b\u7684\u200b\u6392\u5e8f\u200b\u64cd\u4f5c\u200b\u3002</li> <li>\u200b\u8d28\u6570\u200b\u8ba1\u7b97\u200b\uff1aEratosthenes\u200b\u7b5b\u200b</li> </ol>"},{"location":"coding/dsa/vector-and-list/#_18","title":"\u5feb\u901f\u200b\u521d\u59cb\u5316","text":"<p>\u200b\u5728\u200b\u4e25\u683c\u200b\u610f\u4e49\u200b\u4e0a\u200b\u8bb2\u200b\uff0c\u200b\u5373\u4f7f\u200b\u8c03\u7528\u200b<code>memset()</code>\u200b\u51fd\u6570\u200b\uff0c\u200b\u4f4d\u56fe\u200b\u7684\u200b\u521d\u59cb\u5316\u200b\uff08\u200b\u6e05\u7a7a\u200b\uff09\u200b\u4e5f\u200b\u9700\u8981\u200b\\(\\mathcal O(n)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u3002\u200b\u501f\u52a9\u200b\u6821\u9a8c\u200b\u73af\u200b\u7ed3\u6784\u200b\u53ef\u4ee5\u200b\u5c06\u4f4d\u200b\u56fe\u200b\u7684\u200b\u521d\u59cb\u5316\u200b\u590d\u6742\u5ea6\u200b\u7531\u200b\\(\\mathcal O(n)\\)\u200b\u63d0\u9ad8\u200b\u5230\u200b\\(\\mathcal O(1)\\)\uff08\u200b\u4f46\u200b\u76f8\u5e94\u200b\u5730\u200b\uff0c\u200b\u4f4d\u56fe\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u200b\u4f4d\u200b\u9700\u8981\u200b\u4e24\u4e2a\u200b<code>int</code>\u200b\u7c7b\u578b\u200b\u5373\u200b64\u200b\u5b57\u8282\u200b\u8fdb\u884c\u200b\u5b58\u50a8\u200b\uff0c\u200b\u9020\u6210\u200b\u7a7a\u95f4\u200b\u7684\u200b\u5927\u91cf\u200b\u6d6a\u8d39\u200b\uff09\u3002</p> <p>\u200b\u5c06\u4f4d\u200b\u56fe\u200b\u7684\u200b\u6570\u7ec4\u200b\u62c6\u200b\u5206\u4e3a\u200b<code>A[]</code>\u3001<code>B[]</code>\u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u7b2c\u200b<code>k</code>\u200b\u4f4d\u200b\uff0c\u200b\u6ee1\u8db3\u200b\uff1a</p> <ul> <li><code>A[B[k]] = k</code></li> <li><code>B[A[k]] = k</code></li> </ul> <p>\u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u6784\u6210\u200b\u4e86\u200b\u76f8\u4e92\u200b\u6821\u9a8c\u200b\u7684\u200b\u5173\u7cfb\u200b\uff0c\u200b\u6309\u7167\u200b\u4e0b\u6807\u200b\u67e5\u8be2\u200b\uff0c\u200b\u7b26\u5408\u200b\u6821\u9a8c\u200b\u6761\u4ef6\u200b\u7684\u200b\u4e0b\u6807\u200b\u5b58\u50a8\u200b\u7684\u200b\u4f4d\u4e3a\u200b<code>1</code>\uff0c\u200b\u5426\u5219\u200b\u4e3a\u200b<code>0</code>\u3002</p> <p> 0AB0112512234567</p> <p>\u200b\u53cc\u200b\u6570\u7ec4\u200b\u6a21\u5f0f\u200b\u4e0b\u200b\uff0c\u200b\u6570\u7ec4\u200b<code>B</code>\u200b\u4e3a\u200b\u7c7b\u4f3c\u200b\u4e8e\u6808\u200b\u7684\u200b\u7ed3\u6784\u200b\uff0c\u200b\u4ece\u200b\u5f00\u5934\u200b\u9010\u6e10\u200b\u5411\u200b\u540e\u200b\u5ef6\u4f38\u200b\uff0c\u200b\u7ef4\u62a4\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b<code>top</code>\u200b\u8bb0\u5f55\u200b<code>B</code>\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u6570\u91cf\u200b\u3002\u200b\u6570\u7ec4\u200b<code>A</code>\u200b\u968f\u673a\u200b\u8bbf\u95ee\u200b\u3002</p> <ul> <li><code>set(k)</code>\uff1a\u200b\u5728\u200b\u6570\u7ec4\u200b<code>B</code>\u200b\u4e2d\u200b\u8ffd\u52a0\u200b\u4e00\u4e2a\u200b\u503c\u4e3a\u200b<code>k</code>\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u5728\u200b<code>A[k]</code>\u200b\u4e2d\u200b\u5b58\u50a8\u200b<code>B</code>\u200b\u4e2d\u200b\u5bf9\u5e94\u200b\u5143\u7d20\u200b\u7684\u200b\u4e0b\u6807\u200b\uff1b</li> <li><code>get(k)</code>\uff1a\u200b\u82e5\u200b<code>B[A[k]] == k</code>\u200b\u4e14\u200b<code>A[k]</code>\u200b\u5c0f\u4e8e\u200b\u6808\u9876\u200b\u4f4d\u7f6e\u200b\u5219\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5426\u5219\u200b\u4e3a\u200b\u5047\u200b\uff1b</li> <li><code>clear()</code>\uff1a\u200b\u6e05\u7a7a\u200b\u6570\u7ec4\u200b<code>B</code>\uff0c\u200b\u5373\u5c06\u200b<code>top</code>\u200b\u7f6e\u200b<code>0</code>\u3002</li> <li><code>remove(k)</code>\uff1a\u200b\u5148\u200b\u5c06\u200b<code>A-B</code>\u200b\u6821\u9a8c\u200b\u73af\u4e2d\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u5bf9\u200b\uff08\u200b\u5bf9\u5e94\u200b<code>B[top - 1]</code>\uff09\u200b\u590d\u5236\u5230\u200b<code>B[A[k]]</code>\u200b\u7684\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u518d\u200b<code>top--</code>\u200b\u5220\u9664\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u3002</li> </ul> <p>\u201c\u200b\u6821\u9a8c\u200b\u73af\u200b\u201d\u200b\u4f4d\u56fe\u200b\u7684\u200b\u5b9e\u73b0\u200b\uff1a</p> \u200b\u4f4d\u200b\u56fe\u200b <pre><code>void set(int k)\n{\n   A[k] = top;\n   B[top++] = A[k];\n}\nint get(int k)\n{\n   return B[A[k]] == k &amp;&amp; A[k] &lt; top;\n}\nvoid remove(int k)\n{\n   A[B[--top]] = k;\n   B[k] = B[top];\n}\nvoid clear(int k)\n{\n   top = 0;\n}\n</code></pre>"},{"location":"coding/dsa/vector-and-list/#_19","title":"\u5217\u8868","text":"<p>\u200b\u5411\u91cf\u200b\u662f\u200b\u9759\u6001\u200b\u7684\u200b\u6570\u636e\u7ed3\u6784\u200b\uff0c\u200b\u4e00\u6b21\u200b\u53ea\u80fd\u200b\u5206\u914d\u200b\u56fa\u5b9a\u200b\u957f\u5ea6\u200b\u7684\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u5411\u91cf\u200b\u7684\u200b\u6269\u589e\u200b\u9700\u8981\u200b\u5206\u914d\u200b\u4e00\u5757\u200b\u65b0\u200b\u7684\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u7136\u540e\u200b\u5c06\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u590d\u5236\u5230\u200b\u65b0\u200b\u7684\u200b\u5411\u91cf\u200b\u4e2d\u200b\u3002\u200b\u800c\u200b\u5217\u8868\u200b\u4e0e\u200b\u4e4b\u200b\u4e0d\u540c\u200b\uff0c\u200b\u5217\u8868\u200b\u7684\u200b\u5143\u7d20\u200b\u53ef\u4ee5\u200b\u52a8\u6001\u5206\u914d\u200b\uff0c\u200b\u5217\u8868\u200b\u7684\u200b\u6269\u589e\u200b\u53ea\u200b\u9700\u8981\u200b\u5206\u914d\u200b\u6269\u589e\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u7a7a\u95f4\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u5217\u8868\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u6392\u5217\u200b\u4e0e\u200b\u5217\u8868\u200b\u5143\u7d20\u200b\u5728\u200b\u5185\u5b58\u200b\u4e2d\u200b\u7684\u200b\u6392\u5217\u200b\u987a\u5e8f\u200b\u6ca1\u6709\u200b\u5fc5\u7136\u200b\u7684\u200b\u5173\u7cfb\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u5143\u7d20\u200b\u5728\u200b\u903b\u8f91\u200b\u4e0a\u200b\u5f62\u6210\u200b\u4e00\u4e2a\u200b\u5e8f\u5217\u200b\u3002\u200b\u5217\u8868\u200b\u7684\u200b\u5143\u7d20\u200b\u53c8\u79f0\u200b\u8282\u70b9\u200b\uff0c\u200b\u76f8\u90bb\u200b\u7684\u200b\u8282\u70b9\u200b\u4e92\u79f0\u200b\u524d\u9a71\u200b\u4e0e\u200b\u540e\u7ee7\u200b\u3002\u200b\u6ca1\u6709\u200b\u524d\u9a71\u200b\u7684\u200b\u8282\u70b9\u200b\u79f0\u4e3a\u200b\u9996\u200b\u8282\u70b9\u200b\uff0c\u200b\u6ca1\u6709\u200b\u540e\u7ee7\u200b\u7684\u200b\u8282\u70b9\u200b\u79f0\u4e3a\u200b\u5c3e\u200b\u8282\u70b9\u200b\u3002\u200b\u8282\u70b9\u200b\u9700\u8981\u200b\u989d\u5916\u200b\u7684\u200b\u7a7a\u95f4\u200b\u8bb0\u5f55\u200b\u76f8\u90bb\u200b\u8282\u70b9\u200b\u7684\u200b\u5730\u5740\u200b\u4ee5\u200b\u6784\u5efa\u200b\u903b\u8f91\u200b\u4e0a\u200b\u7684\u200b\u987a\u5e8f\u200b\u5173\u7cfb\u200b\u3002</p> <p>\u200b\u8fd9\u79cd\u200b\u65b9\u6cd5\u200b\u7684\u200b\u7f3a\u70b9\u200b\u5728\u4e8e\u200b\u5217\u8868\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u4e0d\u80fd\u200b\u76f4\u63a5\u200b\u6309\u7167\u200b\u4e0b\u6807\u200b\u8fdb\u884c\u200b\u8bbf\u95ee\u200b\uff0c\u200b\u53ea\u80fd\u200b\u4ece\u200b\u5217\u8868\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u5f00\u59cb\u200b\uff0c\u200b\u4f9d\u5e8f\u200b\u8bbf\u95ee\u200b\u8282\u70b9\u200b\u76f4\u5230\u200b\u8bbf\u95ee\u200b\u5230\u200b\u5bf9\u5e94\u200b\u4e0b\u200b\u6807\u7684\u200b\u8282\u70b9\u200b\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)</p>"},{"location":"coding/dsa/vector-and-list/#_20","title":"\u8282\u70b9\u200b\u7c7b\u578b","text":"<p>\u200b\u4e3a\u200b\u5b9e\u73b0\u200b\u5217\u8868\u200b\u7684\u200b\u5404\u9879\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u8282\u70b9\u200b\u7c7b\u578b\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b\u4ee5\u4e0b\u200b\u63a5\u53e3\u200b\uff1a</p> <ul> <li>\u200b\u83b7\u53d6\u200b\u4e0e\u200b\u8bbe\u7f6e\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u7684\u200b\u524d\u9a71\u200b</li> <li>\u200b\u83b7\u53d6\u200b\u4e0e\u200b\u8bbe\u7f6e\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u7684\u200b\u540e\u7ee7\u200b</li> <li>\u200b\u83b7\u53d6\u200b\u4e0e\u200b\u8bbe\u7f6e\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u7684\u200b\u53d6\u503c\u200b</li> <li>\u200b\u63d2\u5165\u200b\u524d\u9a71\u200b\u8282\u70b9\u200b</li> <li>\u200b\u63d2\u5165\u200b\u540e\u7ee7\u200b\u8282\u70b9\u200b</li> </ul> <p>\u200b\u6ce8\u610f\u200b\uff1a \u200b\u53cc\u5411\u200b\u94fe\u8868\u200b\u5728\u200b\u5b9e\u73b0\u200b\u65f6\u6709\u200b\u6f5c\u5728\u200b\u7684\u200b\u6570\u636e\u200b\u4e0d\u200b\u4e00\u81f4\u200b\u7684\u200b\u98ce\u9669\u200b\u3002\uff08\u200b\u524d\u9a71\u200b\u8282\u70b9\u200b\u7684\u200b\u540e\u7ee7\u200b\u4e0d\u200b\u7b49\u4e8e\u200b\u540e\u7ee7\u200b\u8282\u70b9\u200b\u7684\u200b\u524d\u9a71\u200b\uff09</p> <p>\u200b\u5982\u4e0b\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u7c7b\u200b\u7684\u200b\u793a\u4f8b\u200b\u63a5\u53e3\u200b\uff0c\u200b\u6ca1\u6709\u200b\u5305\u542b\u200b\u63a5\u53e3\u200b\u7684\u200b\u5177\u4f53\u200b\u5b9e\u73b0\u200b\u3002</p> \u200b\u5217\u8868\u200b\u8282\u70b9\u200b\u7c7b\u200b <p><pre><code>/******************************************************************************************\n * Data Structures in C++\n * ISBN: 7-302-33064-6 &amp; 7-302-33065-3 &amp; 7-302-29652-2 &amp; 7-302-26883-3\n * Junhui DENG, deng@tsinghua.edu.cn\n * Computer Science &amp; Technology, Tsinghua University\n * Copyright (c) 2003-2020. All rights reserved.\n ******************************************************************************************/\n\ntypedef int Rank; //\u200b\u79e9\u200b\n#define ListNodePosi(T) ListNode&lt;T&gt;* //\u200b\u5217\u8868\u200b\u8282\u70b9\u200b\u4f4d\u7f6e\u200b\n\ntemplate &lt;typename T&gt; struct ListNode { //\u200b\u5217\u8868\u200b\u8282\u70b9\u200b\u6a21\u677f\u200b\u7c7b\u200b\uff08\u200b\u4ee5\u200b\u53cc\u5411\u200b\u94fe\u8868\u200b\u5f62\u5f0f\u200b\u5b9e\u73b0\u200b\uff09\n// \u200b\u6210\u5458\u200b\n   T data; ListNodePosi(T) pred; ListNodePosi(T) succ; //\u200b\u6570\u503c\u200b\u3001\u200b\u524d\u9a71\u200b\u3001\u200b\u540e\u7ee7\u200b\n// \u200b\u6784\u9020\u51fd\u6570\u200b\n   ListNode() {} //\u200b\u9488\u5bf9\u200bheader\u200b\u548c\u200btrailer\u200b\u7684\u200b\u6784\u9020\u200b\n   ListNode ( T e, ListNodePosi(T) p = NULL, ListNodePosi(T) s = NULL )\n      : data ( e ), pred ( p ), succ ( s ) {} //\u200b\u9ed8\u8ba4\u200b\u6784\u9020\u200b\u5668\u200b\n// \u200b\u64cd\u4f5c\u200b\u63a5\u53e3\u200b\n   ListNodePosi(T) insertAsPred ( T const&amp; e ); //\u200b\u7d27\u9760\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u4e4b\u524d\u200b\u63d2\u5165\u200b\u65b0\u200b\u8282\u70b9\u200b\n   ListNodePosi(T) insertAsSucc ( T const&amp; e ); //\u200b\u7d27\u968f\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u4e4b\u540e\u200b\u63d2\u5165\u200b\u65b0\u200b\u8282\u70b9\u200b\n};\n</code></pre> Reference</p>"},{"location":"coding/dsa/vector-and-list/#_21","title":"\u63d2\u5165\u200b\u4e0e\u200b\u5220\u9664","text":"<p>\u200b\u5411\u200b\u5217\u8868\u200b\u4e2d\u200b\u63d2\u5165\u200b\u6570\u636e\u200b\u9700\u8981\u200b\u5982\u4e0b\u200b\u8fc7\u7a0b\u200b\uff1a</p> <ul> <li>\u200b\u5b9a\u4f4d\u200b\u5230\u200b\u8282\u70b9\u200b</li> <li>\u200b\u5206\u914d\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u8282\u70b9\u200b</li> <li>\u200b\u66f4\u65b0\u200b\u8282\u70b9\u200b\u4e4b\u95f4\u200b\u7684\u200b\u6307\u9488\u200b\uff0c\u200b\u5c06\u200b\u65b0\u200b\u8282\u70b9\u200b\u4e0e\u200b\u5176\u4ed6\u200b\u8282\u70b9\u200b\u76f8\u8fde\u63a5\u200b</li> </ul> <p>\u200b\u5220\u9664\u200b\u8fc7\u7a0b\u200b\u662f\u200b\u63d2\u5165\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u53cd\u5411\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u66f4\u65b0\u200b\u8282\u70b9\u200b\u4e4b\u95f4\u200b\u6307\u9488\u200b\uff0c\u200b\u5c06\u200b\u5220\u9664\u200b\u8282\u70b9\u200b\u6392\u9664\u200b\u5728\u5916\u200b\u5373\u53ef\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_22","title":"\u67e5\u627e\u200b\u4e0e\u200b\u53bb\u200b\u91cd","text":"<p>\u200b\u65e0\u5e8f\u200b\u5217\u8868\u200b\u7684\u200b\u67e5\u627e\u200b\u4e0e\u200b\u53bb\u200b\u91cd\u200b\u64cd\u4f5c\u200b\u4e0e\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u67e5\u627e\u200b\u4e0e\u200b\u53bb\u200b\u91cd\u200b\u64cd\u4f5c\u200b\u7684\u200b\u57fa\u672c\u601d\u8def\u200b\u76f8\u540c\u200b\uff0c\u200b\u6b64\u5904\u200b\u4e0d\u518d\u200b\u8d58\u8ff0\u200b\u3002</p> <p>\u200b\u6709\u200b\u5e8f\u5217\u8868\u200b\u7684\u200b\u67e5\u627e\u200b\u4e0d\u80fd\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u800c\u200b\u53ea\u80fd\u200b\u987a\u5e8f\u200b\u67e5\u627e\u200b\u3002</p> <p>\u200b\u6709\u200b\u5e8f\u5217\u8868\u200b\u7684\u200b\u53bb\u200b\u91cd\u200b\uff1a\u200b\u53cd\u590d\u200b\u8003\u5bdf\u200b\u76f8\u90bb\u200b\u7684\u200b\u8282\u70b9\u200b\uff0c\u200b\u82e5\u200b\u8282\u70b9\u200b\u76f8\u540c\u200b\uff0c\u200b\u5219\u200b\u5220\u9664\u200b\u540e\u8fb9\u200b\u7684\u200b\u8282\u70b9\u200b\uff0c\u200b\u7ee7\u7eed\u200b\u8003\u5bdf\u200b\uff0c\u200b\u5426\u5219\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u5411\u200b\u540e\u200b\u79fb\u52a8\u200b\uff0c\u200b\u91cd\u590d\u200b\u8003\u5bdf\u200b\u8fc7\u7a0b\u200b\u3002\u200b\u76f4\u5230\u200b\u5217\u8868\u200b\u5230\u8fbe\u200b\u7ed3\u5c3e\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u7ed3\u675f\u200b\u3002</p>"},{"location":"coding/leetcode/","title":"Leetcode \u200b\u9898\u76ee","text":"<p>\u200b\u6309\u200b\u6807\u7b7e\u200b\u5206\u7c7b\u200b\u7684\u200b\u9898\u76ee\u200b\u5217\u8868\u200b\u53ef\u4ee5\u200b\u53c2\u89c1\u200b\u8fd9\u91cc\u200b</p> \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 1. \u200b\u4e24\u6570\u200b\u4e4b\u200b\u548c\u200b 2. \u200b\u4e24\u200b\u6570\u200b\u76f8\u52a0\u200b 3. \u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b 7. \u200b\u6574\u6570\u200b\u53cd\u8f6c\u200b 8. \u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u200b\u6574\u6570\u200b (atoi) 9. \u200b\u56de\u6587\u200b\u6570\u200b 11. \u200b\u76db\u200b\u6700\u591a\u6c34\u200b\u7684\u200b\u5bb9\u5668\u200b 13. \u200b\u7f57\u9a6c\u6570\u5b57\u200b\u8f6c\u200b\u6574\u6570\u200b 14. \u200b\u6700\u957f\u200b\u516c\u5171\u200b\u524d\u7f00\u200b 17. \u200b\u7535\u8bdd\u53f7\u7801\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408\u200b 19. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u7684\u200b\u5012\u6570\u7b2c\u200b N \u200b\u4e2a\u200b\u7ed3\u70b9\u200b 20. \u200b\u6709\u6548\u200b\u7684\u200b\u62ec\u53f7\u200b 21. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u94fe\u8868\u200b 22. \u200b\u62ec\u53f7\u200b\u751f\u6210\u200b 23. \u200b\u5408\u5e76\u200bK\u200b\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868\u200b 24. \u200b\u4e24\u200b\u4e24\u200b\u4ea4\u6362\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9\u200b 25. K \u200b\u4e2a\u200b\u4e00\u7ec4\u200b\u7ffb\u8f6c\u200b\u94fe\u8868\u200b 26. \u200b\u5220\u9664\u200b\u6709\u5e8f\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u91cd\u590d\u200b\u9879\u200b 27. \u200b\u79fb\u9664\u200b\u5143\u7d20\u200b 28. \u200b\u5b9e\u73b0\u200b strStr() 35. \u200b\u641c\u7d22\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b 36. \u200b\u6709\u6548\u200b\u7684\u200b\u6570\u72ec\u200b 37. \u200b\u89e3\u6570\u200b\u72ec\u200b 38. \u200b\u5916\u89c2\u200b\u6570\u5217\u200b 42. \u200b\u63a5\u200b\u96e8\u6c34\u200b 43. \u200b\u5b57\u7b26\u4e32\u200b\u76f8\u4e58\u200b 46. \u200b\u5168\u200b\u6392\u5217\u200b 48. \u200b\u65cb\u8f6c\u200b\u56fe\u50cf\u200b 50. Pow(x, n) 53. \u200b\u6700\u5927\u200b\u5b50\u5e8f\u200b\u548c\u200b 55. \u200b\u8df3\u8dc3\u200b\u6e38\u620f\u200b 56. \u200b\u5408\u5e76\u200b\u533a\u95f4\u200b 59. \u200b\u87ba\u65cb\u200b\u77e9\u9635\u200b II 61. \u200b\u65cb\u8f6c\u200b\u94fe\u8868\u200b 62. \u200b\u4e0d\u540c\u200b\u8def\u5f84\u200b 64. \u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b 66. \u200b\u52a0\u4e00\u200b 69. x \u200b\u7684\u200b\u5e73\u65b9\u6839\u200b 70. \u200b\u722c\u697c\u68af\u200b 73. \u200b\u77e9\u9635\u200b\u7f6e\u200b\u96f6\u200b 74. \u200b\u641c\u7d22\u200b\u4e8c\u7ef4\u200b\u77e9\u9635\u200b 75. \u200b\u989c\u8272\u200b\u5206\u7c7b\u200b 78. \u200b\u5b50\u96c6\u200b 79. \u200b\u5355\u8bcd\u200b\u641c\u7d22\u200b 87. \u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b 88. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u6570\u7ec4\u200b 90. \u200b\u5b50\u96c6\u200b II 92. \u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b II 98. \u200b\u9a8c\u8bc1\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b 101. \u200b\u5bf9\u79f0\u200b\u4e8c\u53c9\u6811\u200b 102. \u200b\u4e8c\u53c9\u6811\u200b\u7684\u200b\u5c42\u5e8f\u200b\u904d\u5386\u200b 115. \u200b\u4e0d\u540c\u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217\u200b 118. \u200b\u6768\u8f89\u4e09\u89d2\u200b 119. \u200b\u6768\u8f89\u4e09\u89d2\u200b II 120. \u200b\u4e09\u89d2\u5f62\u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b 121. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b 122. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b II 125. \u200b\u9a8c\u8bc1\u200b\u56de\u6587\u200b\u4e32\u200b 136. \u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u7684\u200b\u6570\u5b57\u200b 150. \u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u6c42\u503c\u200b 155. \u200b\u6700\u5c0f\u200b\u6808\u200b 189. \u200b\u65cb\u8f6c\u200b\u6570\u7ec4\u200b 198. \u200b\u6253\u5bb6\u52ab\u820d\u200b 204. \u200b\u8ba1\u6570\u200b\u8d28\u6570\u200b 206. \u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b 217. \u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b 234. \u200b\u56de\u6587\u200b\u94fe\u8868\u200b 237. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9\u200b 242. \u200b\u6709\u6548\u200b\u7684\u200b\u5b57\u6bcd\u200b\u5f02\u4f4d\u200b\u8bcd\u200b 315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570\u200b 341. \u200b\u6241\u5e73\u5316\u200b\u5d4c\u5957\u200b\u5217\u8868\u200b\u8fed\u4ee3\u200b\u5668\u200b 344. \u200b\u53cd\u8f6c\u200b\u5b57\u7b26\u4e32\u200b 350. \u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u7684\u200b\u4ea4\u96c6\u200b II 368. \u200b\u6700\u5927\u200b\u6574\u9664\u200b\u5b50\u96c6\u200b 377. \u200b\u7ec4\u5408\u200b\u603b\u548c\u200b \u2163 384. \u200b\u6253\u4e71\u200b\u6570\u7ec4\u200b 387. \u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u552f\u4e00\u200b\u5b57\u7b26\u200b 403. \u200b\u9752\u86d9\u200b\u8fc7\u6cb3\u200b 433. \u200b\u6700\u5c0f\u200b\u57fa\u56e0\u200b\u53d8\u5316\u200b 488. \u200b\u7956\u739b\u200b\u6e38\u620f\u200b 509. \u200b\u6590\u6ce2\u200b\u90a3\u200b\u5951\u6570\u200b 779. \u200b\u7b2c\u200bK\u200b\u4e2a\u200b\u8bed\u6cd5\u200b\u7b26\u53f7\u200b 781. \u200b\u68ee\u6797\u200b\u4e2d\u200b\u7684\u200b\u5154\u5b50\u200b 917. \u200b\u4ec5\u4ec5\u200b\u53cd\u8f6c\u200b\u5b57\u6bcd\u200b 1006. \u200b\u7b28\u200b\u9636\u4e58\u200b 1011. \u200b\u5728\u200b D \u200b\u5929\u200b\u5185\u200b\u9001\u8fbe\u200b\u5305\u88f9\u200b\u7684\u200b\u80fd\u529b\u200b 1283. \u200b\u4f7f\u200b\u7ed3\u679c\u200b\u4e0d\u200b\u8d85\u8fc7\u200b\u9608\u503c\u200b\u7684\u200b\u6700\u5c0f\u200b\u9664\u6570\u200b 1300. \u200b\u8f6c\u53d8\u200b\u6570\u7ec4\u200b\u540e\u200b\u6700\u200b\u63a5\u8fd1\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u6570\u7ec4\u200b\u548c\u200b 1603. \u200b\u8bbe\u8ba1\u200b\u505c\u8f66\u200b\u7cfb\u7edf\u200b 1835. \u200b\u6240\u6709\u200b\u6570\u5bf9\u200b\u6309\u4f4d\u200b\u4e0e\u200b\u7ed3\u679c\u200b\u7684\u200b\u5f02\u6216\u200b\u548c"},{"location":"coding/leetcode/1/","title":"1. \u200b\u4e24\u6570\u200b\u4e4b\u200b\u548c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/1/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b<code>nums</code>\u200b\u548c\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u503c\u200b<code>target</code>\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u5728\u200b\u8be5\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u627e\u51fa\u200b\u548c\u200b\u4e3a\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u90a3\u200b\u00a0\u200b\u4e24\u4e2a\u200b\u00a0\u200b\u6574\u6570\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b\u4ed6\u4eec\u200b\u7684\u200b\u6570\u7ec4\u200b\u4e0b\u6807\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u6bcf\u79cd\u200b\u8f93\u5165\u200b\u53ea\u4f1a\u200b\u5bf9\u5e94\u200b\u4e00\u4e2a\u200b\u7b54\u6848\u200b\u3002\u200b\u4f46\u662f\u200b\uff0c\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u540c\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u4e0d\u80fd\u200b\u4f7f\u7528\u200b\u4e24\u904d\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b:</p> <pre><code>\u200b\u7ed9\u5b9a\u200b nums = [2, 7, 11, 15], target = 9\n\n\u200b\u56e0\u4e3a\u200b nums[0] + nums[1] = 2 + 7 = 9\n\u200b\u6240\u4ee5\u200b\u8fd4\u56de\u200b [0, 1]\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/1/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a </p> <p>\u200b\u5c06\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u5b58\u50a8\u200b\u5728\u200b\u54c8\u5e0c\u200b\u8868\u4e2d\u200b\uff0c\u200b\u6bcf\u6b21\u200b\u5728\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u67e5\u627e\u200b\u76ee\u6807\u503c\u200b\u4e0e\u200b\u5f53\u524d\u200b\u503c\u4e4b\u5dee\u200b\u662f\u5426\u200b\u5728\u200b\u54c8\u5e0c\u200b\u8868\u4e2d\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\n#define MAX(a, b) (a &gt; b ? a : b)\n#define MIN(a, b) (a &lt; b ? a : b)\nstruct cell\n{\n    int value;\n    int flag;\n};\nint* twoSum(int* nums, int numsSize, int target, int* returnSize){\n    *returnSize = 2;\n    int i = 0, mapSize = numsSize * 2, cur, *ret = malloc(sizeof(int) * 2), targetval, j = 0;\n    struct cell *hashMap = (struct cell *)memset(malloc(sizeof(struct cell) * mapSize), 0, sizeof(struct cell) * mapSize);\n    for (i = 0; i &lt; numsSize; i++)\n    {\n        cur = (nums[i] &gt; 0 ? nums[i] : -nums[i]) % mapSize;\n        while(hashMap[cur].flag != 0 &amp;&amp; hashMap[cur].value != nums[i])\n        {\n            cur++;\n            cur = cur == mapSize ? 0 : cur;\n        }\n        if (hashMap[cur].flag == 0)\n            hashMap[cur].value = nums[i];\n        hashMap[cur].flag++;\n    }\n    for (i = 0; i &lt; numsSize; i++)\n    {\n        targetval = target - nums[i];\n        cur = (targetval &gt; 0 ? targetval : -targetval) % mapSize;\n        while(hashMap[cur].flag != 0 &amp;&amp; hashMap[cur].value != targetval)\n        {\n            cur++;\n            cur = cur == mapSize ? 0 : cur;\n        }\n        if (hashMap[cur].value == targetval)\n        {\n            for (j = 0; j &lt; numsSize; j++)\n                if (nums[j] == targetval &amp;&amp; j != i)\n                    break;\n            if (j != numsSize)\n                break;\n        }\n    }\n    ret[0] = MIN(i, j);\n    ret[1] = MAX(i, j);\n    if (nums[i] + nums[j] == target)\n        return ret;\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/1006/","title":"1006. \u200b\u7b28\u200b\u9636\u4e58","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/1006/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u901a\u5e38\u200b\uff0c\u200b\u6b63\u6574\u6570\u200b <code>n</code> \u200b\u7684\u200b\u9636\u4e58\u200b\u662f\u200b\u6240\u6709\u200b\u5c0f\u4e8e\u200b\u6216\u200b\u7b49\u4e8e\u200b <code>n</code> \u200b\u7684\u200b\u6b63\u6574\u6570\u200b\u7684\u200b\u4e58\u79ef\u200b\u3002\u200b\u4f8b\u5982\u200b\uff0c<code>factorial(10) = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1</code>\u3002</p> <p>\u200b\u76f8\u53cd\u200b\uff0c\u200b\u6211\u4eec\u200b\u8bbe\u8ba1\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u7b28\u200b\u9636\u4e58\u200b <code>clumsy</code>\uff1a\u200b\u5728\u200b\u6574\u6570\u200b\u7684\u200b\u9012\u51cf\u200b\u5e8f\u5217\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u4ee5\u200b\u4e00\u4e2a\u200b\u56fa\u5b9a\u200b\u987a\u5e8f\u200b\u7684\u200b\u64cd\u4f5c\u7b26\u200b\u5e8f\u5217\u200b\u6765\u200b\u4f9d\u6b21\u200b\u66ff\u6362\u200b\u539f\u6709\u200b\u7684\u200b\u4e58\u6cd5\u200b\u64cd\u4f5c\u7b26\u200b\uff1a\u200b\u4e58\u6cd5\u200b(*)\uff0c\u200b\u9664\u6cd5\u200b(/)\uff0c\u200b\u52a0\u6cd5\u200b(+)\u200b\u548c\u200b\u51cf\u6cd5\u200b(-)\u3002</p> <p>\u200b\u4f8b\u5982\u200b\uff0c<code>clumsy(10) = 10 * 9 / 8 + 7 - 6 * 5 / 4 + 3 - 2 * 1</code>\u3002\u200b\u7136\u800c\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u8fd0\u7b97\u200b\u4ecd\u7136\u200b\u4f7f\u7528\u200b\u901a\u5e38\u200b\u7684\u200b\u7b97\u672f\u200b\u8fd0\u7b97\u200b\u987a\u5e8f\u200b\uff1a\u200b\u6211\u4eec\u200b\u5728\u200b\u4efb\u4f55\u200b\u52a0\u200b\u3001\u200b\u51cf\u200b\u6b65\u9aa4\u200b\u4e4b\u524d\u200b\u6267\u884c\u200b\u6240\u6709\u200b\u7684\u200b\u4e58\u6cd5\u200b\u548c\u200b\u9664\u6cd5\u200b\u6b65\u9aa4\u200b\uff0c\u200b\u5e76\u4e14\u200b\u6309\u200b\u4ece\u5de6\u5230\u53f3\u200b\u5904\u7406\u200b\u4e58\u6cd5\u200b\u548c\u200b\u9664\u6cd5\u200b\u6b65\u9aa4\u200b\u3002</p> <p>\u200b\u53e6\u5916\u200b\uff0c\u200b\u6211\u4eec\u200b\u4f7f\u7528\u200b\u7684\u200b\u9664\u6cd5\u200b\u662f\u200b\u5730\u677f\u200b\u9664\u6cd5\u200b\uff08floor division\uff09\uff0c\u200b\u6240\u4ee5\u200b\u00a0<code>10 * 9 / 8</code>\u00a0\u200b\u7b49\u4e8e\u200b\u00a0<code>11</code>\u3002\u200b\u8fd9\u200b\u4fdd\u8bc1\u200b\u7ed3\u679c\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u5b9e\u73b0\u200b\u4e0a\u9762\u200b\u5b9a\u4e49\u200b\u7684\u200b\u7b28\u200b\u51fd\u6570\u200b\uff1a\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b <code>N</code>\uff0c\u200b\u5b83\u200b\u8fd4\u56de\u200b <code>N</code> \u200b\u7684\u200b\u7b28\u200b\u9636\u4e58\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a4\n\u200b\u8f93\u51fa\u200b\uff1a7\n\u200b\u89e3\u91ca\u200b\uff1a7 = 4 * 3 / 2 + 1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a10\n\u200b\u8f93\u51fa\u200b\uff1a12\n\u200b\u89e3\u91ca\u200b\uff1a12 = 10 * 9 / 8 + 7 - 6 * 5 / 4 + 3 - 2 * 1\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= N &lt;= 10000</code></li> <li><code>-2^31 &lt;= answer &lt;= 2^31 - 1</code>  \uff08\u200b\u7b54\u6848\u200b\u4fdd\u8bc1\u200b\u7b26\u5408\u200b <code>32</code> \u200b\u4f4d\u200b\u6574\u6570\u200b\u3002\uff09</li> </ul> <p>Reference</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/1006/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5bf9\u200b\u8ba1\u7b97\u200b\u8fc7\u7a0b\u200b\u8fdb\u884c\u200b\u6a21\u62df\u200b\u5373\u53ef\u200b\uff1a</p> <pre><code>int clumsy(int N){\n    int ret = 0, i = N;\n    bool flag = true, minus = false;\n    for (i = N; i &gt; 0;)\n    {\n        if (flag)\n        {\n            if (i &gt;= 3)\n                ret += (minus ? -1 : 1) * i * (i - 1) / (i - 2);\n            else if (i == 2)\n                ret += (minus ? -1 : 1) * i * (i - 1);\n            else if (i == 1)\n                ret += (minus ? -1 : 1) * i;\n            i -= 3;\n        }\n        else\n            ret += i--;\n        flag = !flag;\n        minus = true;\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/101/","title":"101. \u200b\u5bf9\u79f0\u200b\u4e8c\u53c9\u6811","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6811","\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/101/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u4e8c\u53c9\u6811\u200b\uff0c\u200b\u68c0\u67e5\u200b\u5b83\u200b\u662f\u5426\u662f\u200b\u955c\u50cf\u200b\u5bf9\u79f0\u200b\u7684\u200b\u3002</p> <p>\u200b\u4f8b\u5982\u200b\uff0c\u200b\u4e8c\u53c9\u6811\u200b<code>[1,2,2,3,4,4,3]</code>\u200b\u662f\u200b\u5bf9\u79f0\u200b\u7684\u200b\u3002</p> <pre><code>    1\n   / \\\n  2   2\n / \\ / \\\n3  4 4  3\n</code></pre> <p>\u200b\u4f46\u662f\u200b\u4e0b\u9762\u200b\u8fd9\u4e2a\u200b<code>[1,2,2,null,3,null,3]</code>\u200b\u5219\u200b\u4e0d\u662f\u200b\u955c\u50cf\u200b\u5bf9\u79f0\u200b\u7684\u200b:</p> <pre><code>    1\n   / \\\n  2   2\n   \\   \\\n   3    3\n</code></pre> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u8fd0\u7528\u200b\u9012\u5f52\u200b\u548c\u200b\u8fed\u4ee3\u200b\u4e24\u79cd\u200b\u65b9\u6cd5\u200b\u89e3\u51b3\u200b\u8fd9\u4e2a\u200b\u95ee\u9898\u200b\u5417\u200b\uff1f</p> <p>Reference</p>","tags":["\u6811","\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/101/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u9012\u5f52\u200b\u5224\u65ad\u200b\u4e24\u4e2a\u200b\u6839\u200b\u8282\u70b9\u200b\u662f\u5426\u200b\u6210\u200b\u955c\u50cf\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for a binary tree node.\n * struct TreeNode {\n *     int val;\n *     struct TreeNode *left;\n *     struct TreeNode *right;\n * };\n */\nbool isMirrored(struct TreeNode *root1, struct TreeNode *root2)\n{\n    if (!root1 || !root2)\n        return root1 == root2;\n    if (root1-&gt;val != root2-&gt;val)\n        return false;\n    return isMirrored(root1-&gt;left, root2-&gt;right) &amp;&amp; isMirrored(root2-&gt;left, root1-&gt;right);\n}\n\nbool isSymmetric(struct TreeNode* root){\n    if (root == NULL)\n        return true;\n    return isMirrored(root-&gt;left, root-&gt;right);\n}\n</code></pre>","tags":["\u6811","\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/1011/","title":"1011. \u200b\u5728\u200b D \u200b\u5929\u200b\u5185\u200b\u9001\u8fbe\u200b\u5305\u88f9\u200b\u7684\u200b\u80fd\u529b","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/1011/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u4f20\u9001\u5e26\u200b\u4e0a\u200b\u7684\u200b\u5305\u88f9\u200b\u5fc5\u987b\u200b\u5728\u200b <code>D</code> \u200b\u5929\u5185\u200b\u4ece\u200b\u4e00\u4e2a\u200b\u6e2f\u53e3\u200b\u8fd0\u9001\u200b\u5230\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u6e2f\u53e3\u200b\u3002</p> <p>\u200b\u4f20\u9001\u5e26\u200b\u4e0a\u200b\u7684\u200b\u7b2c\u200b <code>i</code>\u00a0\u200b\u4e2a\u200b\u5305\u88f9\u200b\u7684\u200b\u91cd\u91cf\u200b\u4e3a\u200b\u00a0<code>weights[i]</code>\u3002\u200b\u6bcf\u200b\u4e00\u5929\u200b\uff0c\u200b\u6211\u4eec\u200b\u90fd\u200b\u4f1a\u200b\u6309\u200b\u7ed9\u51fa\u200b\u91cd\u91cf\u200b\u7684\u200b\u987a\u5e8f\u200b\u5f80\u200b\u4f20\u9001\u5e26\u200b\u4e0a\u200b\u88c5\u8f7d\u200b\u5305\u88f9\u200b\u3002\u200b\u6211\u4eec\u200b\u88c5\u8f7d\u200b\u7684\u200b\u91cd\u91cf\u200b\u4e0d\u4f1a\u200b\u8d85\u8fc7\u200b\u8239\u200b\u7684\u200b\u6700\u5927\u200b\u8fd0\u8f7d\u200b\u91cd\u91cf\u200b\u3002</p> <p>\u200b\u8fd4\u56de\u200b\u80fd\u200b\u5728\u200b <code>D</code> \u200b\u5929\u5185\u200b\u5c06\u200b\u4f20\u9001\u5e26\u200b\u4e0a\u200b\u7684\u200b\u6240\u6709\u200b\u5305\u88f9\u200b\u9001\u8fbe\u200b\u7684\u200b\u8239\u200b\u7684\u200b\u6700\u4f4e\u200b\u8fd0\u8f7d\u200b\u80fd\u529b\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aweights = [1,2,3,4,5,6,7,8,9,10], D = 5\n\u200b\u8f93\u51fa\u200b\uff1a15\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u8239\u8236\u200b\u6700\u4f4e\u200b\u8f7d\u91cd\u200b 15 \u200b\u5c31\u200b\u80fd\u591f\u200b\u5728\u200b 5 \u200b\u5929\u200b\u5185\u200b\u9001\u8fbe\u200b\u6240\u6709\u200b\u5305\u88f9\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a\n\u200b\u7b2c\u200b 1 \u200b\u5929\u200b\uff1a1, 2, 3, 4, 5\n\u200b\u7b2c\u200b 2 \u200b\u5929\u200b\uff1a6, 7\n\u200b\u7b2c\u200b 3 \u200b\u5929\u200b\uff1a8\n\u200b\u7b2c\u200b 4 \u200b\u5929\u200b\uff1a9\n\u200b\u7b2c\u200b 5 \u200b\u5929\u200b\uff1a10\n\n\u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u8d27\u7269\u200b\u5fc5\u987b\u200b\u6309\u7167\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u987a\u5e8f\u200b\u88c5\u8fd0\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4f7f\u7528\u200b\u8f7d\u91cd\u200b\u80fd\u529b\u200b\u4e3a\u200b 14 \u200b\u7684\u200b\u8239\u8236\u200b\u5e76\u200b\u5c06\u200b\u5305\u88c5\u200b\u5206\u6210\u200b (2, 3, 4, 5), (1, 6, 7), (8), (9), (10) \u200b\u662f\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u7684\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aweights = [3,2,2,4,1,4], D = 3\n\u200b\u8f93\u51fa\u200b\uff1a6\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u8239\u8236\u200b\u6700\u4f4e\u200b\u8f7d\u91cd\u200b 6 \u200b\u5c31\u200b\u80fd\u591f\u200b\u5728\u200b 3 \u200b\u5929\u200b\u5185\u200b\u9001\u8fbe\u200b\u6240\u6709\u200b\u5305\u88f9\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a\n\u200b\u7b2c\u200b 1 \u200b\u5929\u200b\uff1a3, 2\n\u200b\u7b2c\u200b 2 \u200b\u5929\u200b\uff1a2, 4\n\u200b\u7b2c\u200b 3 \u200b\u5929\u200b\uff1a1, 4\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aweights = [1,2,3,1,1], D = 4\n\u200b\u8f93\u51fa\u200b\uff1a3\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u7b2c\u200b 1 \u200b\u5929\u200b\uff1a1\n\u200b\u7b2c\u200b 2 \u200b\u5929\u200b\uff1a2\n\u200b\u7b2c\u200b 3 \u200b\u5929\u200b\uff1a3\n\u200b\u7b2c\u200b 4 \u200b\u5929\u200b\uff1a1, 1\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= D &lt;= weights.length &lt;= 50000</code></li> <li><code>1 &lt;= weights[i] &lt;= 500</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/1011/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u8bbe\u200b\u5217\u8868\u200b\u4e3a\u200b\\(A=\\{a_i\\}_n\\)\uff0c\u200b\u5219\u200b\u6700\u7ec8\u200b\u7ed3\u679c\u200b\u7684\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\([\\max_{i} \\{a_i\\}, \\sum_{i} a_i]\\)\u3002\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u5728\u200b\u533a\u95f4\u200b\u4e2d\u200b\u67e5\u627e\u200b\u80fd\u591f\u200b\u6ee1\u8db3\u200b\u5728\u200b\\(D\\)\u200b\u5929\u200b\u5185\u200b\u8fd0\u8f93\u200b\u5168\u90e8\u200b\u5305\u88f9\u200b\u7684\u200b\u6700\u5c0f\u200b\u8f7d\u91cd\u200b\u3002</p> <p>\u200b\u6709\u5173\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7684\u200b\u8be6\u7ec6\u200b\u8ba8\u8bba\u200b\uff0c\u200b\u8bf7\u200b\u53c2\u89c1\u200b35. \u200b\u641c\u7d22\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b</p> <pre><code>#define MAX(x, y) ((x) &gt; (y) ? (x) : (y))\n\nbool verify(int *weights, int weightsSize, int D, int limit)\n{\n    int i = 0, ub = limit;\n    for (i = 0; i &lt; weightsSize; i++)\n    {\n        if (weights[i] &gt; ub)\n        {\n            D--;\n            ub = limit;\n        }\n        ub -= weights[i];\n        if (D &lt;= 0)\n            return false;\n    }\n    return true;\n}\n\nint bSearch(int *weights, int weightsSize, int D, int lo, int hi)\n{\n    int mid;\n    while (lo &lt; hi)\n    {\n        mid = (lo + hi) &gt;&gt; 1;\n        if (verify(weights, weightsSize, D, mid))\n            hi = mid;\n        else\n            lo = mid + 1;\n    }\n    return lo;\n}\n\nint shipWithinDays(int* weights, int weightsSize, int D) {\n    int i = 0, sum = 0, max = 0;\n    printf(\"%d\", verify(weights, weightsSize, D, 6));\n    for (i = 0; i &lt; weightsSize; i++)\n    {\n        sum += weights[i];\n        max = MAX(max, weights[i]);\n    }\n    return bSearch(weights, weightsSize, D, max, sum);\n}\n</code></pre>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/102/","title":"102. \u200b\u4e8c\u53c9\u6811\u200b\u7684\u200b\u5c42\u5e8f\u200b\u904d\u5386","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6811","\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/102/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u4e8c\u53c9\u6811\u200b\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u8fd4\u56de\u200b\u5176\u200b\u6309\u200b \u200b\u5c42\u5e8f\u200b\u904d\u5386\u200b \u200b\u5f97\u5230\u200b\u7684\u200b\u8282\u70b9\u200b\u503c\u200b\u3002 \uff08\u200b\u5373\u200b\u9010\u5c42\u200b\u5730\u200b\uff0c\u200b\u4ece\u5de6\u5230\u53f3\u200b\u8bbf\u95ee\u200b\u6240\u6709\u200b\u8282\u70b9\u200b\uff09\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <p>\u200b\u4e8c\u53c9\u6811\u200b\uff1a<code>[3,9,20,null,null,15,7]</code>,</p> <pre><code>    3\n   / \\\n  9  20\n    /  \\\n   15   7\n</code></pre> <p>\u200b\u8fd4\u56de\u200b\u5176\u200b\u5c42\u6b21\u200b\u904d\u5386\u200b\u7ed3\u679c\u200b\uff1a</p> <pre><code>[\n  [3],\n  [9,20],\n  [15,7]\n]\n</code></pre> <p>Reference</p>","tags":["\u6811","\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/102/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u7b2c\u4e00\u5c42\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u5373\u6839\u200b\u8282\u70b9\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u518d\u5c42\u5e8f\u200b\u904d\u5386\u200b\u5de6\u5b50\u200b\u6811\u200b\u548c\u200b\u53f3\u5b50\u200b\u6811\u200b\uff0c\u200b\u5408\u5e76\u200b\u540c\u5c42\u200b\u7684\u200b\u7ed3\u679c\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for a binary tree node.\n * struct TreeNode {\n *     int val;\n *     struct TreeNode *left;\n *     struct TreeNode *right;\n * };\n */\n/**\n * Return an array of arrays of size *returnSize.\n * The sizes of the arrays are returned as *returnColumnSizes array.\n * Note: Both returned array and *columnSizes array must be malloced, assume caller calls free().\n */\n#define MAX(x, y) (x &gt; y ? x : y)\nint *merge(int *leftBranch, int leftSize, int *rightBranch, int rightSize, int *returnSize)\n{\n    *returnSize = leftSize + rightSize;\n    int *ret = (int *)malloc(sizeof(int) * (leftSize + rightSize));\n    if (leftSize != 0)\n    {\n        memcpy(ret, leftBranch, sizeof(int) * leftSize);\n        free(leftBranch);\n    }\n    if (rightSize != 0)\n    {\n        memcpy(ret + leftSize, rightBranch, sizeof(int) * rightSize);\n        free(rightBranch);\n    }\n    return ret;\n}\nint** levelOrder(struct TreeNode* root, int* returnSize, int** returnColumnSizes){\n    if (root == NULL)\n    {\n        *returnSize = 0;\n        *returnColumnSizes = NULL;\n        return NULL;\n    }\n    int leftBranchSize = 0, *leftColumnSizes = NULL, **leftBranch = levelOrder(root-&gt;left, &amp;leftBranchSize, &amp;leftColumnSizes),\n        rightBranchSize = 0, *rightColumnSizes = NULL, **rightBranch = levelOrder(root-&gt;right, &amp;rightBranchSize, &amp;rightColumnSizes);\n    *returnSize = 1 + MAX(leftBranchSize, rightBranchSize);\n    int **ret = (int **)malloc(sizeof(int *) * *returnSize), *returnCols = (int *)malloc(sizeof(int) * *returnSize), i = 0;\n    *returnColumnSizes = returnCols;\n    *returnCols = 1;\n    *ret = malloc(sizeof(int));\n    **ret = root-&gt;val;\n    for (i = 1; i &lt; *returnSize; i++)\n    {\n        if (i &lt;= leftBranchSize &amp;&amp; i &lt;= rightBranchSize)\n            ret[i] = merge(leftBranch[i - 1], leftColumnSizes[i - 1], rightBranch[i - 1], rightColumnSizes[i - 1], returnCols + i);\n        else if (i &lt;= leftBranchSize)\n            ret[i] = merge(leftBranch[i-1], leftColumnSizes[i-1], NULL, 0, returnCols + i);\n        else if (i &lt;= rightBranchSize)\n            ret[i] = merge(NULL, 0, rightBranch[i-1], rightColumnSizes[i-1], returnCols + i);\n    }\n    free(leftBranch);\n    free(rightBranch);\n    free(leftColumnSizes);\n    free(rightColumnSizes);\n    return ret;\n}\n</code></pre>","tags":["\u6811","\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/11/","title":"11. \u200b\u76db\u200b\u6700\u591a\u6c34\u200b\u7684\u200b\u5bb9\u5668","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u5b66","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/11/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b <code>n</code> \u200b\u4e2a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b <code>a_1\uff0ca_2\uff0c...\uff0ca_n</code>\uff0c\u200b\u6bcf\u4e2a\u200b\u6570\u200b\u4ee3\u8868\u200b\u5750\u6807\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u70b9\u200b\u00a0<code>(i,\u00a0a_i)</code> \u3002\u200b\u5728\u200b\u5750\u6807\u200b\u5185\u200b\u753b\u200b <code>n</code> \u200b\u6761\u200b\u5782\u76f4\u7ebf\u200b\uff0c\u200b\u5782\u76f4\u7ebf\u200b <code>i</code>\u00a0\u200b\u7684\u200b\u4e24\u4e2a\u200b\u7aef\u70b9\u200b\u5206\u522b\u200b\u4e3a\u200b\u00a0<code>(i,\u00a0a_i)</code> \u200b\u548c\u200b <code>(i, 0)</code> \u3002\u200b\u627e\u51fa\u200b\u5176\u4e2d\u200b\u7684\u200b\u4e24\u6761\u7ebf\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u5b83\u4eec\u200b\u4e0e\u200b\u00a0<code>x</code>\u00a0\u200b\u8f74\u200b\u5171\u540c\u200b\u6784\u6210\u200b\u7684\u200b\u5bb9\u5668\u200b\u53ef\u4ee5\u200b\u5bb9\u7eb3\u200b\u6700\u591a\u200b\u7684\u200b\u6c34\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u503e\u659c\u200b\u5bb9\u5668\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a[1,8,6,2,5,4,8,3,7]\n\u200b\u8f93\u51fa\u200b\uff1a49 \n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u56fe\u4e2d\u200b\u5782\u76f4\u7ebf\u200b\u4ee3\u8868\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b [1,8,6,2,5,4,8,3,7]\u3002\u200b\u5728\u200b\u6b64\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u5bb9\u5668\u200b\u80fd\u591f\u200b\u5bb9\u7eb3\u200b\u6c34\u200b\uff08\u200b\u8868\u793a\u200b\u4e3a\u200b\u84dd\u8272\u200b\u90e8\u5206\u200b\uff09\u200b\u7684\u200b\u6700\u5927\u503c\u200b\u4e3a\u200b\u00a049\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aheight = [1,1]\n\u200b\u8f93\u51fa\u200b\uff1a1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aheight = [4,3,2,1,4]\n\u200b\u8f93\u51fa\u200b\uff1a16\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aheight = [1,2,1]\n\u200b\u8f93\u51fa\u200b\uff1a2\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>n = height.length</code></li> <li>2 &lt;= n &lt;= 3 * 104</li> <li>0 &lt;= height[i] &lt;= 3 * 104</li> </ul> <p>Reference</p>","tags":["\u6570\u5b66","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/11/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u53cc\u200b\u6307\u9488\u200b\u6cd5\u200b\uff0c\u200b\u4e24\u4e2a\u200b\u6307\u9488\u200b\u5206\u522b\u200b\u6307\u5411\u200b\u76db\u6c34\u200b\u533a\u57df\u200b\u7684\u200b\u8fb9\u754c\u200b\u3002\u200b\u6bcf\u6b21\u200b\u5faa\u73af\u200b\u5c06\u200b\u8f83\u200b\u4f4e\u200b\u7684\u200b\u8fb9\u754c\u200b\u5411\u200b\u4e2d\u95f4\u200b\u6536\u7f29\u200b\uff0c\u200b\u5e76\u200b\u5c1d\u8bd5\u200b\u66f4\u65b0\u200b\u6700\u5927\u200b\u7684\u200b\u76db\u6c34\u200b\u533a\u57df\u200b\u3002</p> <pre><code>#define MIN(x, y) (x &lt; y ? x : y)\n#define MAX(x, y) (x &gt; y ? x : y)\nint maxArea(int* height, int heightSize){\n    int left = 0, right = heightSize - 1,\n        ret = (right - left) * MIN(height[left], height[right]), cur;\n    while (left &lt; right)\n    {\n        if (height[left] &lt; height[right])\n            left++;\n        else\n            right--;\n        cur = (right - left) * MIN(height[left], height[right]);\n        ret = MAX(cur, ret);\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u5b66","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/115/","title":"115. \u200b\u4e0d\u540c\u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u5b57\u7b26\u4e32","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/115/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b <code>s</code> \u200b\u548c\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b <code>t</code> \uff0c\u200b\u8ba1\u7b97\u200b\u5728\u200b <code>s</code> \u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217\u200b\u4e2d\u200b <code>t</code> \u200b\u51fa\u73b0\u200b\u7684\u200b\u4e2a\u6570\u200b\u3002</p> <p>\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u4e00\u4e2a\u200b \u200b\u5b50\u200b\u5e8f\u5217\u200b \u200b\u662f\u200b\u6307\u200b\uff0c\u200b\u901a\u8fc7\u200b\u5220\u9664\u200b\u4e00\u4e9b\u200b\uff08\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u5220\u9664\u200b\uff09\u200b\u5b57\u7b26\u200b\u4e14\u200b\u4e0d\u200b\u5e72\u6270\u200b\u5269\u4f59\u200b\u5b57\u7b26\u200b\u76f8\u5bf9\u200b\u4f4d\u7f6e\u200b\u6240\u200b\u7ec4\u6210\u200b\u7684\u200b\u65b0\u200b\u5b57\u7b26\u4e32\u200b\u3002\uff08\u200b\u4f8b\u5982\u200b\uff0c<code>\"ACE\"</code>\u00a0\u200b\u662f\u200b\u00a0<code>\"ABCDE\"</code>\u00a0\u200b\u7684\u200b\u4e00\u4e2a\u200b\u5b50\u200b\u5e8f\u5217\u200b\uff0c\u200b\u800c\u200b\u00a0<code>\"AEC\"</code>\u00a0\u200b\u4e0d\u662f\u200b\uff09</p> <p>\u200b\u9898\u76ee\u200b\u6570\u636e\u200b\u4fdd\u8bc1\u200b\u7b54\u6848\u200b\u7b26\u5408\u200b 32 \u200b\u4f4d\u200b\u5e26\u7b26\u53f7\u200b\u6574\u6570\u200b\u8303\u56f4\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"rabbbit\", t = \"rabbit\"\n\u200b\u8f93\u51fa\u200b\uff1a3\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u5982\u4e0b\u200b\u56fe\u200b\u6240\u793a\u200b, \u200b\u6709\u200b 3 \u200b\u79cd\u200b\u53ef\u4ee5\u200b\u4ece\u200b s \u200b\u4e2d\u200b\u5f97\u5230\u200b \"rabbit\" \u200b\u7684\u200b\u65b9\u6848\u200b\u3002\n(\u200b\u4e0a\u200b\u7bad\u5934\u200b\u7b26\u53f7\u200b ^ \u200b\u8868\u793a\u200b\u9009\u53d6\u200b\u7684\u200b\u5b57\u6bcd\u200b)\nrabbbit\n^^^^ ^^\nrabbbit\n^^ ^^^^\nrabbbit\n^^^ ^^^\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"babgbag\", t = \"bag\"\n\u200b\u8f93\u51fa\u200b\uff1a5\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u5982\u4e0b\u200b\u56fe\u200b\u6240\u793a\u200b, \u200b\u6709\u200b 5 \u200b\u79cd\u200b\u53ef\u4ee5\u200b\u4ece\u200b s \u200b\u4e2d\u200b\u5f97\u5230\u200b \"bag\" \u200b\u7684\u200b\u65b9\u6848\u200b\u3002 \n(\u200b\u4e0a\u200b\u7bad\u5934\u200b\u7b26\u53f7\u200b ^ \u200b\u8868\u793a\u200b\u9009\u53d6\u200b\u7684\u200b\u5b57\u6bcd\u200b)\nbabgbag\n^^ ^\nbabgbag\n^^    ^\nbabgbag\n^    ^^\nbabgbag\n  ^  ^^\nbabgbag\n    ^^^\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>0 &lt;= s.length, t.length &lt;= 1000</code></li> <li><code>s</code> \u200b\u548c\u200b <code>t</code> \u200b\u7531\u200b\u82f1\u6587\u5b57\u6bcd\u200b\u7ec4\u6210\u200b</li> </ul> <p>Reference</p>","tags":["\u5b57\u7b26\u4e32","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/115/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u89e3\u9898\u200b\u601d\u8def\u200b\uff1a\u200b\u8bbe\u200b\\(f(x, y)\\)\u200b\u4e3a\u200b\u5b57\u7b26\u4e32\u200b\\(x\\)\u200b\u4e2d\u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u5b57\u7b26\u4e32\u200b\\(y\\)\u200b\u51fa\u73b0\u200b\u7684\u200b\u6b21\u6570\u200b\u3002\u200b\u5e76\u4e14\u200b\u4f7f\u7528\u200b\\(x + 1\\)\u200b\u8868\u793a\u200b\u5b57\u7b26\u4e32\u200b\\(x\\)\u200b\u4ece\u200b\u7b2c\u4e8c\u4e2a\u200b\u5b57\u7b26\u200b\u5f00\u59cb\u200b\u7684\u200b\u5b50\u4e32\u200b\u3002</p> <p>\u200b\u5219\u200b\uff1a</p> <ul> <li>\u200b\u5f53\u200b\\(x, y\\)\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u5b57\u7b26\u200b\u4e0d\u200b\u540c\u65f6\u200b\uff0c\u200b\u6709\u200b\\(f(x, y) = f(x + 1, y)\\)</li> <li>\u200b\u5f53\u200b\\(x, y\\)\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u5b57\u7b26\u200b\u5339\u914d\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(f(x, y) = f(x + 1, y + 1) + f(x + 1, y)\\)</li> </ul> <p>\u200b\u7279\u6b8a\u200b\u5730\u200b\uff0c\u200b\u5f53\u200b\\(y\\)\u200b\u4e3a\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u65f6\u200b\uff0c\u200b\u8bf4\u660e\u200b\u5b57\u7b26\u4e32\u200b\u5339\u914d\u200b\u7ed3\u675f\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u4e00\u4e2a\u200b\u5339\u914d\u200b\u6210\u529f\u200b\u7684\u200b\u5b50\u4e32\u200b\uff0c\u200b\u56e0\u6b64\u200b\\(f(x, \\varnothing) = 1\\)\u3002\u200b\u5f53\u200b\\(x\\)\u200b\u4e3a\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u4e14\u200b\\(y\\)\u200b\u4e0d\u4e3a\u200b\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u65f6\u200b\uff0c\u200b\u8bf4\u660e\u200b\u5b57\u7b26\u4e32\u200b\u5339\u914d\u200b\u5931\u8d25\u200b\uff0c\\(f(\\varnothing, y) = \\left\\{\\begin{aligned}&amp;0 &amp;y\\not = \\varnothing \\\\ &amp; 1 &amp;y=\\varnothing\\end{aligned}\\right .\\)\u3002</p> <p>\u200b\u7531\u6b64\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u521d\u6b65\u200b\u5f97\u51fa\u200b\u9012\u5f52\u200b\u4ee3\u7801\u200b\uff1a</p> <pre><code>int numDistinct(char * s, char * t){\n    if (!*t)\n        return 1;\n    if (!*s)\n        return 0;\n    int ret = 0;\n    if (*s == *t)\n        ret += numDistinct(s + 1, t + 1);\n    ret += numDistinct(s + 1, t);\n    return ret;\n}\n</code></pre> <p>\u200b\u5982\u200b\u4e0a\u200b\u4ee3\u7801\u200b\u4f1a\u200b\u9020\u6210\u200b\u8d85\u65f6\u200b\uff0c\u200b\u539f\u56e0\u200b\u662f\u200b\u9012\u5f52\u200b\u65f6\u200b\u51fa\u73b0\u200b\u4e86\u200b\u5206\u652f\u200b\u5bfc\u81f4\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(2^n\\)\u3002\u200b\u4f7f\u7528\u200b\u6570\u7ec4\u200b\u5b58\u50a8\u200b\u5386\u53f2\u200b\u7ed3\u679c\u200b\u53ef\u4ee5\u200b\u907f\u514d\u200b\u9012\u5f52\u200b\u4ea7\u751f\u200b\u5206\u652f\u200b\uff0c\u200b\u4f18\u5316\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\uff1a</p> <pre><code>int cached(char *s, char *t, int x, int y, int **buffer) {\n    int ret = 0;\n    if (!*t)\n        ret = 1;\n    else if (!*s)\n        ret = 0;\n    else\n    {\n        if (buffer[x][y] &gt;= 0)\n            return buffer[x][y];\n        if (*s == *t)\n            ret += cached(s + 1, t + 1, x + 1, y + 1, buffer);\n        ret += cached(s + 1, t, x + 1, y, buffer);\n        buffer[x][y] = ret;\n    }\n    return ret;\n}\nint numDistinct(char * s, char * t){\n    int x = strlen(s), y = strlen(t);\n    int *_buffer = (int *)malloc(sizeof(int) * x * y);\n    int **buffer = (int **)malloc(sizeof(int *) * x);\n    for (int i = 0; i &lt; x; i++)\n        buffer[i] = _buffer + i * y;\n    memset(_buffer, 0xff, sizeof(int) * x * y);\n    return cached(s, t, 0, 0, buffer);\n}\n</code></pre>","tags":["\u5b57\u7b26\u4e32","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/118/","title":"118. \u200b\u6768\u8f89\u4e09\u89d2","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/118/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200bnumRows\uff0c\u200b\u751f\u6210\u200b\u6768\u8f89\u4e09\u89d2\u200b\u7684\u200b\u524d\u200bnumRows\u200b\u884c\u200b\u3002</p> <p></p> <p>\u200b\u5728\u200b\u6768\u8f89\u4e09\u89d2\u200b\u4e2d\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u6570\u662f\u200b\u5b83\u200b\u5de6\u4e0a\u65b9\u200b\u548c\u200b\u53f3\u4e0a\u65b9\u200b\u7684\u200b\u6570\u200b\u7684\u200b\u548c\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a <pre><code>\u200b\u8f93\u5165\u200b: 5\n\u200b\u8f93\u51fa\u200b:\n[\n     [1],\n    [1,1],\n   [1,2,1],\n  [1,3,3,1],\n [1,4,6,4,1]\n]\n</code></pre></p> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/118/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6839\u636e\u200b\u6768\u8f89\u4e09\u89d2\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u901a\u8fc7\u200b\u9012\u5f52\u200b\u9010\u884c\u200b\u6784\u9020\u200b\u51fa\u200b\u6bcf\u200b\u4e00\u884c\u200b\u7684\u200b\u503c\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Return an array of arrays of size *returnSize.\n * The sizes of the arrays are returned as *returnColumnSizes array.\n * Note: Both returned array and *columnSizes array must be malloced, assume caller calls free().\n */\nint **generate(int numRows, int *returnSize, int **returnColumnSizes)\n{\n    *returnSize = numRows;\n    if (numRows == 0)\n        return NULL;\n    if (numRows == 1)\n    {\n        *returnColumnSizes = (int *)malloc(sizeof(int));\n        **returnColumnSizes = 1;\n        int **ret = (int **)malloc(sizeof(int *));\n        *ret = (int *)malloc(sizeof(int));\n        **ret = 1;\n        return ret;\n    }\n    int retSize = 0,\n        *oldRetColSizes = NULL,\n        **rec = generate(numRows - 1, &amp;retSize, &amp;oldRetColSizes), \n        i = 0;\n        **ret = (int **)memcpy(malloc(sizeof(int *) * numRows), rec, (numRows - 1) * sizeof(int *));\n        *newRetColSizes = memcpy((int *)malloc(sizeof(int) * numRows), oldRetColSizes, (numRows - 1) * sizeof(int));\n        *tempArray = (int *)malloc(sizeof(int) * numRows);\n\n    free(oldRetColSizes);\n    free(rec);\n\n    newRetColSizes[numRows - 1] = numRows;\n    for (i = 1; i &lt; numRows - 1; i++)\n        tempArray[i] = *(ret[numRows - 2] + i) + *(ret[numRows - 2] + i - 1);\n    tempArray[0] = 1;\n    tempArray[numRows - 1] = 1;\n    ret[numRows - 1] = tempArray;\n\n    *returnColumnSizes = newRetColSizes;\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/119/","title":"119. \u200b\u6768\u8f89\u4e09\u89d2\u200b II","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/119/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u975e\u8d1f\u200b\u7d22\u5f15\u200b\\(k\\)\uff0c\u200b\u5176\u4e2d\u200b\\(k\\leq33\\)\uff0c\u200b\u8fd4\u56de\u200b\u6768\u8f89\u4e09\u89d2\u200b\u7684\u200b\u7b2c\u200b\\(k\\) \u200b\u884c\u200b\u3002</p> <p></p> <p>\u200b\u5728\u200b\u6768\u8f89\u4e09\u89d2\u200b\u4e2d\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u6570\u662f\u200b\u5b83\u200b\u5de6\u4e0a\u65b9\u200b\u548c\u200b\u53f3\u4e0a\u65b9\u200b\u7684\u200b\u6570\u200b\u7684\u200b\u548c\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a <pre><code>\u200b\u8f93\u5165\u200b: 3\n\u200b\u8f93\u51fa\u200b: [1,3,3,1]\n</code></pre> \u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u4f18\u5316\u200b\u4f60\u200b\u7684\u200b\u7b97\u6cd5\u200b\u5230\u200b\\(O(k)\\)\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u5417\u200b\uff1f</p> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/119/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5f53\u524d\u200b\u884c\u200b\u7684\u200b\u7ed3\u679c\u200b\u4ec5\u200b\u4f9d\u8d56\u4e8e\u200b\u4e0a\u200b\u4e00\u884c\u200b\u7684\u200b\u7ed3\u679c\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\nvoid recur(int rowIndex, int *Filling)\n{\n    if (rowIndex &lt;= 1)\n        return;\n    recur(rowIndex - 1, Filling);\n    for (int i = rowIndex - 1; i &gt; 0; i--)\n        Filling[i] += Filling[i - 1];\n}\n\nint* getRow(int rowIndex, int* returnSize){\n    *returnSize = rowIndex + 1;\n    int *ret = (int *)memset(malloc(sizeof(int) * *returnSize), 0, sizeof(int) * *returnSize);\n    ret[0] = 1;\n    recur(*returnSize, ret);\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/120/","title":"120. \u200b\u4e09\u89d2\u5f62\u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/120/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u4e09\u89d2\u5f62\u200b\uff0c\u200b\u627e\u51fa\u200b\u81ea\u9876\u5411\u4e0b\u200b\u7684\u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b\u3002\u200b\u6bcf\u200b\u4e00\u6b65\u200b\u53ea\u80fd\u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0b\u200b\u4e00\u884c\u200b\u4e2d\u200b\u76f8\u90bb\u200b\u7684\u200b\u7ed3\u70b9\u200b\u4e0a\u200b\u3002</p> <p>\u200b\u76f8\u90bb\u200b\u7684\u200b\u7ed3\u70b9\u200b \u200b\u5728\u200b\u8fd9\u91cc\u200b\u6307\u200b\u7684\u200b\u662f\u200b <code>\u200b\u4e0b\u6807\u200b</code> \u200b\u4e0e\u200b <code>\u200b\u4e0a\u200b\u4e00\u5c42\u200b\u7ed3\u70b9\u200b\u4e0b\u6807\u200b</code> \u200b\u76f8\u540c\u200b\u6216\u8005\u200b\u7b49\u4e8e\u200b <code>\u200b\u4e0a\u200b\u4e00\u5c42\u200b\u7ed3\u70b9\u200b\u4e0b\u6807\u200b + 1</code> \u200b\u7684\u200b\u4e24\u4e2a\u200b\u7ed3\u70b9\u200b\u3002</p> <p>\u200b\u4f8b\u5982\u200b\uff0c\u200b\u7ed9\u5b9a\u200b\u4e09\u89d2\u5f62\u200b\uff1a</p> <pre><code>[\n     [2],\n    [3,4],\n   [6,5,7],\n  [4,1,8,3]\n]\n</code></pre> <p>\u200b\u81ea\u9876\u5411\u4e0b\u200b\u7684\u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b\u4e3a\u200b<code>11</code>\uff08\u200b\u5373\u200b\uff0c2\u00a0+\u00a03\u00a0+\u00a05\u00a0+\u00a01\u00a0= 11\uff09\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u5982\u679c\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u53ea\u200b\u4f7f\u7528\u200b\\(O(n)\\)\u200b\u7684\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\uff08\\(n\\)\u200b\u4e3a\u200b\u4e09\u89d2\u5f62\u200b\u7684\u200b\u603b\u884c\u200b\u6570\u200b\uff09\u200b\u6765\u200b\u89e3\u51b3\u200b\u8fd9\u4e2a\u200b\u95ee\u9898\u200b\uff0c\u200b\u90a3\u4e48\u200b\u4f60\u200b\u7684\u200b\u7b97\u6cd5\u200b\u4f1a\u200b\u5f88\u200b\u52a0\u5206\u200b\u3002</p> <p>Reference</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/120/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\uff0c\u200b\u4f7f\u7528\u200b\u4e09\u89d2\u5f62\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u884c\u200b\u5b58\u653e\u200b\u6700\u5c0f\u200b\u7684\u200b\u548c\u200b\uff0c\u200b\u4ece\u200b\u4e0b\u200b\u5f80\u200b\u4e0a\u200b\u8fdb\u884c\u200b\u8ba1\u7b97\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>#define MIN(x, y) (x &lt; y ? x : y)\nint minimumTotal(int **triangle, int triangleSize, int *triangleColSize)\n{\n    if (!triangleSize)\n        return 0;\n    int *dp1 = triangle[triangleSize - 1], i = 0, j = 0;\n    for (i = triangleSize - 1; i &gt; 0; i--)\n        for (j = 0; j &lt; triangleColSize[i - 1]; j++)\n            dp1[j] = triangle[i - 1][j] + MIN(dp1[j], dp1[j + 1]);\n    return dp1[0];\n}\n</code></pre>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/121/","title":"121. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/121/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5b83\u200b\u7684\u200b\u7b2c\u200b\u00a0i \u200b\u4e2a\u200b\u5143\u7d20\u200b\u662f\u200b\u4e00\u652f\u200b\u7ed9\u5b9a\u200b\u80a1\u7968\u200b\u7b2c\u200b i \u200b\u5929\u200b\u7684\u200b\u4ef7\u683c\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u4f60\u200b\u6700\u200b\u591a\u53ea\u200b\u5141\u8bb8\u200b\u5b8c\u6210\u200b\u4e00\u7b14\u200b\u4ea4\u6613\u200b\uff08\u200b\u5373\u200b\u4e70\u5165\u200b\u548c\u200b\u5356\u51fa\u200b\u4e00\u652f\u200b\u80a1\u7968\u200b\u4e00\u6b21\u200b\uff09\uff0c\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u7b97\u6cd5\u200b\u6765\u200b\u8ba1\u7b97\u200b\u4f60\u200b\u6240\u80fd\u200b\u83b7\u53d6\u200b\u7684\u200b\u6700\u5927\u200b\u5229\u6da6\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b\uff1a\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u5728\u200b\u4e70\u5165\u200b\u80a1\u7968\u200b\u524d\u200b\u5356\u51fa\u200b\u80a1\u7968\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [7,1,5,3,6,4]\n\u200b\u8f93\u51fa\u200b: 5\n\u200b\u89e3\u91ca\u200b: \u200b\u5728\u200b\u7b2c\u200b 2 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 1\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u4e70\u5165\u200b\uff0c\u200b\u5728\u200b\u7b2c\u200b 5 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 6\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u5356\u51fa\u200b\uff0c\u200b\u6700\u5927\u200b\u5229\u6da6\u200b = 6-1 = 5 \u3002\n     \u200b\u6ce8\u610f\u200b\u5229\u6da6\u200b\u4e0d\u80fd\u200b\u662f\u200b 7-1 = 6, \u200b\u56e0\u4e3a\u200b\u5356\u51fa\u200b\u4ef7\u683c\u200b\u9700\u8981\u200b\u5927\u4e8e\u200b\u4e70\u5165\u200b\u4ef7\u683c\u200b\uff1b\u200b\u540c\u65f6\u200b\uff0c\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u5728\u200b\u4e70\u5165\u200b\u524d\u200b\u5356\u51fa\u200b\u80a1\u7968\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [7,6,4,3,1]\n\u200b\u8f93\u51fa\u200b: 0\n\u200b\u89e3\u91ca\u200b: \u200b\u5728\u200b\u8fd9\u79cd\u200b\u60c5\u51b5\u200b\u4e0b\u200b, \u200b\u6ca1\u6709\u200b\u4ea4\u6613\u200b\u5b8c\u6210\u200b, \u200b\u6240\u4ee5\u200b\u6700\u5927\u200b\u5229\u6da6\u200b\u4e3a\u200b 0\u3002\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/121/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u627e\u51fa\u200b\u76f8\u5dee\u200b\u6700\u5927\u200b\u7684\u200b\u6ee1\u8db3\u200b\\(a_i &lt; a_j\\)\u200b\u4e14\u200b\\(i &lt; j\\)\u200b\u7684\u200b\u4e24\u4e2a\u200b\u6570\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int maxProfit(int* prices, int pricesSize){\n    int max = 0, i = pricesSize - 1, value = 0, profit = 0;\n    for (; i&gt;=0; i--)\n    {\n        if (prices[i] &gt; value)\n            value = prices[i];\n        else\n            profit = value - prices[i];\n        max = max &gt; profit ? max : profit;\n    }\n    return max;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/122/","title":"122. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b II","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u8d2a\u5fc3\u7b97\u6cd5","\u6570\u7ec4"]},{"location":"coding/leetcode/122/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5b83\u200b\u7684\u200b\u7b2c\u200b\u00a0i \u200b\u4e2a\u200b\u5143\u7d20\u200b\u662f\u200b\u4e00\u652f\u200b\u7ed9\u5b9a\u200b\u80a1\u7968\u200b\u7b2c\u200b i \u200b\u5929\u200b\u7684\u200b\u4ef7\u683c\u200b\u3002</p> <p>\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u7b97\u6cd5\u200b\u6765\u200b\u8ba1\u7b97\u200b\u4f60\u200b\u6240\u80fd\u200b\u83b7\u53d6\u200b\u7684\u200b\u6700\u5927\u200b\u5229\u6da6\u200b\u3002\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5c3d\u53ef\u80fd\u200b\u5730\u200b\u5b8c\u6210\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u4ea4\u6613\u200b\uff08\u200b\u591a\u6b21\u200b\u4e70\u5356\u200b\u4e00\u652f\u200b\u80a1\u7968\u200b\uff09\u3002</p> <p>\u200b\u6ce8\u610f\u200b\uff1a\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u540c\u65f6\u200b\u53c2\u4e0e\u200b\u591a\u7b14\u200b\u4ea4\u6613\u200b\uff08\u200b\u4f60\u200b\u5fc5\u987b\u200b\u5728\u200b\u518d\u6b21\u200b\u8d2d\u4e70\u200b\u524d\u200b\u51fa\u552e\u200b\u6389\u200b\u4e4b\u524d\u200b\u7684\u200b\u80a1\u7968\u200b\uff09\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [7,1,5,3,6,4]\n\u200b\u8f93\u51fa\u200b: 7\n\u200b\u89e3\u91ca\u200b: \u200b\u5728\u200b\u7b2c\u200b 2 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 1\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u4e70\u5165\u200b\uff0c\u200b\u5728\u200b\u7b2c\u200b 3 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 5\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u5356\u51fa\u200b, \u200b\u8fd9\u7b14\u200b\u4ea4\u6613\u6240\u200b\u80fd\u200b\u83b7\u5f97\u200b\u5229\u6da6\u200b = 5-1 = 4 \u3002\n\u00a0    \u200b\u968f\u540e\u200b\uff0c\u200b\u5728\u200b\u7b2c\u200b 4 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 3\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u4e70\u5165\u200b\uff0c\u200b\u5728\u200b\u7b2c\u200b 5 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 6\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u5356\u51fa\u200b, \u200b\u8fd9\u7b14\u200b\u4ea4\u6613\u6240\u200b\u80fd\u200b\u83b7\u5f97\u200b\u5229\u6da6\u200b = 6-3 = 3 \u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,2,3,4,5]\n\u200b\u8f93\u51fa\u200b: 4\n\u200b\u89e3\u91ca\u200b: \u200b\u5728\u200b\u7b2c\u200b 1 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 1\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u4e70\u5165\u200b\uff0c\u200b\u5728\u200b\u7b2c\u200b 5 \u200b\u5929\u200b \uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 5\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u5356\u51fa\u200b, \u200b\u8fd9\u7b14\u200b\u4ea4\u6613\u6240\u200b\u80fd\u200b\u83b7\u5f97\u200b\u5229\u6da6\u200b = 5-1 = 4 \u3002\n\u00a0    \u200b\u6ce8\u610f\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u5728\u200b\u7b2c\u200b 1 \u200b\u5929\u200b\u548c\u200b\u7b2c\u200b 2 \u200b\u5929\u200b\u63a5\u8fde\u200b\u8d2d\u4e70\u200b\u80a1\u7968\u200b\uff0c\u200b\u4e4b\u540e\u200b\u518d\u200b\u5c06\u200b\u5b83\u4eec\u200b\u5356\u51fa\u200b\u3002\n\u00a0    \u200b\u56e0\u4e3a\u200b\u8fd9\u6837\u200b\u5c5e\u4e8e\u200b\u540c\u65f6\u200b\u53c2\u4e0e\u200b\u4e86\u200b\u591a\u7b14\u200b\u4ea4\u6613\u200b\uff0c\u200b\u4f60\u200b\u5fc5\u987b\u200b\u5728\u200b\u518d\u6b21\u200b\u8d2d\u4e70\u200b\u524d\u200b\u51fa\u552e\u200b\u6389\u200b\u4e4b\u524d\u200b\u7684\u200b\u80a1\u7968\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a03\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [7,6,4,3,1]\n\u200b\u8f93\u51fa\u200b: 0\n\u200b\u89e3\u91ca\u200b: \u200b\u5728\u200b\u8fd9\u79cd\u200b\u60c5\u51b5\u200b\u4e0b\u200b, \u200b\u6ca1\u6709\u200b\u4ea4\u6613\u200b\u5b8c\u6210\u200b, \u200b\u6240\u4ee5\u200b\u6700\u5927\u200b\u5229\u6da6\u200b\u4e3a\u200b 0\u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= prices.length &lt;= 3 * 10 ^ 4</code></li> <li><code>0 &lt;= prices[i]\u00a0&lt;= 10 ^ 4</code></li> </ul> <p>Reference</p>","tags":["\u8d2a\u5fc3\u7b97\u6cd5","\u6570\u7ec4"]},{"location":"coding/leetcode/122/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a </p> <ol> <li>\u200b\u5f53\u200b\u6570\u7ec4\u200b\u957f\u5ea6\u200b\u4e3a\u200b1\u200b\u65f6\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u83b7\u53d6\u200b\u4efb\u4f55\u200b\u5229\u6da6\u200b\uff0c\u200b\u8fd4\u56de\u200b<code>0</code>\uff1b</li> <li>\u200b\u4ece\u200b\u6570\u7ec4\u200b\u672b\u5c3e\u200b\u5f00\u59cb\u200b\u641c\u7d22\u200b\u7b2c\u4e00\u4e2a\u200b\u80fd\u200b\u76c8\u5229\u200b\u7684\u200b\u70b9\u200b\uff0c\u200b\u5219\u200b\u6700\u5927\u200b\u5229\u6da6\u200b\u4e3a\u200b\u672c\u6b21\u200b\u76c8\u5229\u200b\u4e0e\u200b\u4e4b\u524d\u200b\u6700\u5927\u200b\u76c8\u5229\u200b\u4e4b\u200b\u548c\u200b\u3002</li> </ol> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int maxProfit(int* prices, int pricesSize){\n    if (pricesSize &lt; 2)\n        return 0;\n    int *cur = prices + pricesSize - 1, currentVal = *cur, len = pricesSize;\n    while(cur &gt; prices)\n    {\n        if (*cur &lt; *(cur - 1))\n            break;\n        len--;\n        cur--;\n    }\n    return (currentVal - *cur) + maxProfit(prices, len - 1);\n}\n</code></pre>","tags":["\u8d2a\u5fc3\u7b97\u6cd5","\u6570\u7ec4"]},{"location":"coding/leetcode/125/","title":"125. \u200b\u9a8c\u8bc1\u200b\u56de\u6587\u200b\u4e32","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/125/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u9a8c\u8bc1\u200b\u5b83\u200b\u662f\u5426\u662f\u200b\u56de\u6587\u200b\u4e32\u200b\uff0c\u200b\u53ea\u200b\u8003\u8651\u200b\u5b57\u6bcd\u200b\u548c\u200b\u6570\u5b57\u200b\u5b57\u7b26\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5ffd\u7565\u200b\u5b57\u6bcd\u200b\u7684\u200b\u5927\u5c0f\u5199\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a \u200b\u672c\u9898\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u5c06\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\u6709\u6548\u200b\u7684\u200b\u56de\u6587\u200b\u4e32\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: \"A man, a plan, a canal: Panama\"\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: \"race a car\"\n\u200b\u8f93\u51fa\u200b: false\n</code></pre> <p>Reference</p>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/125/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4f7f\u7528\u200b\u53cc\u200b\u6307\u9488\u200b\u6cd5\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b\uff0c\u200b\u9700\u8981\u200b\u6ce8\u610f\u200b\u5982\u4e0b\u200b\u7279\u6b8a\u200b\u60c5\u51b5\u200b</p> <ol> <li>\u200b\u9700\u8981\u200b\u5ffd\u7565\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u7b26\u53f7\u200b\u800c\u200b\u53ea\u200b\u8003\u8651\u200b\u5b57\u6bcd\u200b\u548c\u200b\u6570\u5b57\u200b</li> <li>\u200b\u5927\u5c0f\u5199\u200b\u7684\u200bASCII\u200b\u6570\u503c\u200b\u76f8\u5dee\u200b32\uff0c\u200b\u4f46\u200bASCII\u200b\u6570\u503c\u200b\u76f8\u5dee\u200b32\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u5927\u5c0f\u5199\u200b\u5173\u7cfb\u200b\uff0c\u200b\u5982\u200b<code>'0'</code>(48)\u200b\u4e0e\u200b<code>'P'</code>(80)</li> </ol> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>bool isText(char s)\n{\n    return (s &gt;= 'A' &amp;&amp; s &lt;= 'Z') || (s &gt;= 'a' &amp;&amp; s &lt;= 'z') || (s &gt;= '0' &amp;&amp; s &lt;= '9');\n}\nbool isSame(char s1, char s2)\n{\n    return (s1 == s2) || (s2 &gt;= 'A' &amp;&amp; s1 - s2 == 32) || (s1 &gt;= 'A' &amp;&amp; s2 - s1 == 32);\n}\nbool isPalindrome(char * s){\n    int len = strlen(s), head = 0, tail = len - 1;\n    while (head &lt; len &amp;&amp; !isText(s[head]))\n        head++;\n    while (tail &gt;= 0 &amp;&amp; !isText(s[tail]))\n        tail--;\n    while(tail &gt; head &amp;&amp; head &lt; len &amp;&amp; tail &gt;= 0)\n    {\n        if (!isSame(s[head], s[tail]))\n            return false;\n        head++;\n        tail--;\n        while (!isText(s[head]))\n            head++;\n        while (!isText(s[tail]))\n            tail--;\n    }\n    return true;\n}\n</code></pre>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/1283/","title":"1283. \u200b\u4f7f\u200b\u7ed3\u679c\u200b\u4e0d\u200b\u8d85\u8fc7\u200b\u9608\u503c\u200b\u7684\u200b\u6700\u5c0f\u200b\u9664\u6570","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/1283/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\u00a0<code>nums</code> \u200b\u548c\u200b\u4e00\u4e2a\u200b\u6b63\u6574\u6570\u200b\u00a0<code>threshold</code>\u00a0\uff0c\u200b\u4f60\u200b\u9700\u8981\u200b\u9009\u62e9\u200b\u4e00\u4e2a\u200b\u6b63\u6574\u6570\u200b\u4f5c\u4e3a\u200b\u9664\u6570\u200b\uff0c\u200b\u7136\u540e\u200b\u5c06\u200b\u6570\u7ec4\u200b\u91cc\u200b\u6bcf\u4e2a\u200b\u6570\u90fd\u200b\u9664\u4ee5\u200b\u5b83\u200b\uff0c\u200b\u5e76\u200b\u5bf9\u200b\u9664\u6cd5\u200b\u7ed3\u679c\u200b\u6c42\u548c\u200b\u3002</p> <p>\u200b\u8bf7\u200b\u4f60\u200b\u627e\u51fa\u200b\u80fd\u591f\u200b\u4f7f\u200b\u4e0a\u8ff0\u200b\u7ed3\u679c\u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b\u9608\u503c\u200b\u00a0<code>threshold</code>\u00a0\u200b\u7684\u200b\u9664\u6570\u200b\u4e2d\u200b \u200b\u6700\u5c0f\u200b \u200b\u7684\u200b\u90a3\u4e2a\u200b\u3002</p> <p>\u200b\u6bcf\u4e2a\u200b\u6570\u200b\u9664\u4ee5\u200b\u9664\u6570\u200b\u540e\u200b\u90fd\u200b\u5411\u4e0a\u200b\u53d6\u6574\u200b\uff0c\u200b\u6bd4\u65b9\u8bf4\u200b 7/3 = 3 \uff0c 10/2 = 5 \u3002</p> <p>\u200b\u9898\u76ee\u200b\u4fdd\u8bc1\u200b\u4e00\u5b9a\u200b\u6709\u89e3\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1,2,5,9], threshold = 6\n\u200b\u8f93\u51fa\u200b\uff1a5\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5982\u679c\u200b\u9664\u6570\u200b\u4e3a\u200b 1 \uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u548c\u200b\u4e3a\u200b 17 \uff081+2+5+9\uff09\u3002\n\u200b\u5982\u679c\u200b\u9664\u6570\u200b\u4e3a\u200b 4 \uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u548c\u200b\u4e3a\u200b 7 (1+1+2+3) \u3002\u200b\u5982\u679c\u200b\u9664\u6570\u200b\u4e3a\u200b 5 \uff0c\u200b\u548c\u200b\u4e3a\u200b 5 (1+1+1+2)\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [2,3,5,7,11], threshold = 11\n\u200b\u8f93\u51fa\u200b\uff1a3\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [19], threshold = 5\n\u200b\u8f93\u51fa\u200b\uff1a4\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 5 * 10^4</code></li> <li><code>1 &lt;= nums[i] &lt;= 10^6</code></li> <li><code>nums.length &lt;=\u00a0threshold &lt;= 10^6</code></li> </ul> <p>Reference</p>","tags":["\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/1283/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u627e\u5230\u200b\u6700\u5c0f\u200b\u7684\u200b\u9664\u6570\u200b</p> <p>\u200b\u5982\u679c\u200b\u7ed3\u679c\u200b\u5927\u4e8e\u200b<code>threshold</code>\uff0c\u200b\u8bf4\u660e\u200b\u9664\u6570\u200b\u8fc7\u200b\u5c0f\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u53f3\u4fa7\u200b\u67e5\u627e\u200b\uff0c\u200b\u5426\u5219\u200b\u5728\u200b\u5de6\u4fa7\u200b\u67e5\u627e\u200b</p> <pre><code>bool test(int *nums, int numsSize, int threshold, int x)\n{\n    if (!x)\n        return INT_MAX;\n    int ret = 0, i = 0;\n    for (i = 0; i &lt; numsSize; i++)\n        ret += (nums[i] + x - 1) / x;\n    return ret &lt;= threshold;\n}\n\nint bSearch(int *nums, int numsSize, int threshold, int lo, int hi)\n{\n    if (hi - lo == 1)\n        return hi;\n    int mid = (hi + lo) &gt;&gt; 1;\n    if (!test(nums, numsSize, threshold, mid))\n        return bSearch(nums, numsSize, threshold, mid, hi);\n    else\n        return bSearch(nums, numsSize, threshold, lo, mid);\n}\n\nint smallestDivisor(int* nums, int numsSize, int threshold){\n    int max = 0, i = 0;\n    for (i = 0; i &lt; numsSize; i++)\n        if (max &lt; nums[i])\n        max = nums[i];\n    return bSearch(nums, numsSize, threshold, 0, max);\n}\n</code></pre> <pre><code>class Solution:\n    def bSearch(self, nums: List[int], threshold: int, lo: int, hi: int) -&gt; int:\n        if hi - lo == 1:\n            return hi\n        mid = (lo + hi) &gt;&gt; 1\n        return self.bSearch(nums, threshold, lo, mid) \\\n            if sum([math.ceil(_ / mid) for _ in nums]) &lt;= threshold \\\n            else self.bSearch(nums, threshold, mid, hi)\n    def smallestDivisor(self, nums: List[int], threshold: int) -&gt; int:\n        return self.bSearch(nums, threshold, 0, max(nums));\n</code></pre>","tags":["\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/13/","title":"13. \u200b\u7f57\u9a6c\u6570\u5b57\u200b\u8f6c\u200b\u6574\u6570","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/13/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7f57\u9a6c\u6570\u5b57\u200b\u5305\u542b\u200b\u4ee5\u4e0b\u200b\u4e03\u79cd\u200b\u5b57\u7b26\u200b:<code>I</code>\uff0c<code>V</code>\uff0c<code>X</code>\uff0c<code>L</code>\uff0c<code>C</code>\uff0c<code>D</code>\u200b\u548c\u200b<code>M</code>\u3002</p> <p><pre><code>\u200b\u5b57\u7b26\u200b          \u200b\u6570\u503c\u200b\nI             1\nV             5\nX             10\nL             50\nC             100\nD             500\nM             1000\n</code></pre> \u200b\u4f8b\u5982\u200b\uff0c \u200b\u7f57\u9a6c\u6570\u5b57\u200b 2 \u200b\u5199\u200b\u505a\u200b<code>II</code>\u00a0\uff0c\u200b\u5373\u200b\u4e3a\u200b\u4e24\u4e2a\u200b\u5e76\u5217\u200b\u7684\u200b 1\u300212 \u200b\u5199\u200b\u505a\u200b<code>XII</code>\u00a0\uff0c\u200b\u5373\u200b\u4e3a\u200b<code>X</code>\u00a0+\u00a0<code>II</code>\u3002 27 \u200b\u5199\u200b\u505a\u200b<code>XXVII</code>, \u200b\u5373\u200b\u4e3a\u200b\u00a0<code>XX</code>\u00a0+\u00a0<code>V</code>\u00a0+\u00a0<code>II</code>\u3002</p> <p>\u200b\u901a\u5e38\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u7f57\u9a6c\u6570\u5b57\u200b\u4e2d\u5c0f\u200b\u7684\u200b\u6570\u5b57\u200b\u5728\u200b\u5927\u200b\u7684\u200b\u6570\u5b57\u200b\u7684\u200b\u53f3\u8fb9\u200b\u3002\u200b\u4f46\u200b\u4e5f\u200b\u5b58\u5728\u200b\u7279\u4f8b\u200b\uff0c\u200b\u4f8b\u5982\u200b 4 \u200b\u4e0d\u5199\u200b\u505a\u200b<code>IIII</code>\uff0c\u200b\u800c\u662f\u200b<code>IV</code>\u3002\u200b\u6570\u5b57\u200b 1 \u200b\u5728\u200b\u6570\u5b57\u200b 5 \u200b\u7684\u200b\u5de6\u8fb9\u200b\uff0c\u200b\u6240\u200b\u8868\u793a\u200b\u7684\u200b\u6570\u200b\u7b49\u4e8e\u200b\u5927\u6570\u200b 5 \u200b\u51cf\u5c0f\u200b\u6570\u200b 1 \u200b\u5f97\u5230\u200b\u7684\u200b\u6570\u503c\u200b 4 \u3002\u200b\u540c\u6837\u200b\u5730\u200b\uff0c\u200b\u6570\u5b57\u200b 9 \u200b\u8868\u793a\u200b\u4e3a\u200b<code>IX</code>\u3002\u200b\u8fd9\u4e2a\u200b\u7279\u6b8a\u200b\u7684\u200b\u89c4\u5219\u200b\u53ea\u200b\u9002\u7528\u200b\u4e8e\u200b\u4ee5\u4e0b\u200b\u516d\u79cd\u200b\u60c5\u51b5\u200b\uff1a</p> <p><code>I</code>\u200b\u53ef\u4ee5\u200b\u653e\u5728\u200b\u00a0<code>V</code>\u00a0(5) \u200b\u548c\u200b\u00a0<code>X</code>\u00a0(10) \u200b\u7684\u200b\u5de6\u8fb9\u200b\uff0c\u200b\u6765\u200b\u8868\u793a\u200b 4 \u200b\u548c\u200b 9\u3002 <code>X</code>\u200b\u53ef\u4ee5\u200b\u653e\u5728\u200b\u00a0<code>L</code>\u00a0(50) \u200b\u548c\u200b\u00a0<code>C</code>\u00a0(100) \u200b\u7684\u200b\u5de6\u8fb9\u200b\uff0c\u200b\u6765\u200b\u8868\u793a\u200b 40 \u200b\u548c\u200b\u00a090\u3002\u00a0 <code>C</code>\u200b\u53ef\u4ee5\u200b\u653e\u5728\u200b\u00a0<code>D</code>\u00a0(500) \u200b\u548c\u200b\u00a0<code>M</code>\u00a0(1000) \u200b\u7684\u200b\u5de6\u8fb9\u200b\uff0c\u200b\u6765\u200b\u8868\u793a\u200b\u00a0400 \u200b\u548c\u200b\u00a0900\u3002 \u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u7f57\u9a6c\u6570\u5b57\u200b\uff0c\u200b\u5c06\u200b\u5176\u200b\u8f6c\u6362\u6210\u200b\u6574\u6570\u200b\u3002\u200b\u8f93\u5165\u200b\u786e\u4fdd\u200b\u5728\u200b 1\u00a0\u200b\u5230\u200b 3999 \u200b\u7684\u200b\u8303\u56f4\u200b\u5185\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\u00a0\"III\"\n\u200b\u8f93\u51fa\u200b: 3\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\u00a0\"IV\"\n\u200b\u8f93\u51fa\u200b: 4\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a03\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\u00a0\"IX\"\n\u200b\u8f93\u51fa\u200b: 9\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a04\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\u00a0\"LVIII\"\n\u200b\u8f93\u51fa\u200b: 58\n\u200b\u89e3\u91ca\u200b: L = 50, V= 5, III = 3.\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a05\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\u00a0\"MCMXCIV\"\n\u200b\u8f93\u51fa\u200b: 1994\n\u200b\u89e3\u91ca\u200b: M = 1000, CM = 900, XC = 90, IV = 4.\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <p>\u200b\u9898\u76ee\u200b\u6240\u200b\u7ed9\u200b\u6d4b\u8bd5\u7528\u4f8b\u200b\u7686\u200b\u7b26\u5408\u200b\u7f57\u9a6c\u6570\u5b57\u200b\u4e66\u5199\u200b\u89c4\u5219\u200b\uff0c\u200b\u4e0d\u4f1a\u200b\u51fa\u73b0\u200b\u8de8\u4f4d\u200b\u7b49\u200b\u60c5\u51b5\u200b\u3002</p> <p><code>IC</code>\u200b\u548c\u200b<code>IM</code>\u200b\u8fd9\u6837\u200b\u7684\u200b\u4f8b\u5b50\u200b\u5e76\u200b\u4e0d\u200b\u7b26\u5408\u200b\u9898\u76ee\u200b\u8981\u6c42\u200b\uff0c49 \u200b\u5e94\u8be5\u200b\u5199\u4f5c\u200b<code>XLIX</code>\uff0c999 \u200b\u5e94\u8be5\u200b\u5199\u4f5c\u200b <code>CMXCIX</code>\u3002</p> <p>\u200b\u5173\u4e8e\u200b\u7f57\u9a6c\u6570\u5b57\u200b\u7684\u200b\u8be6\u5c3d\u200b\u4e66\u5199\u200b\u89c4\u5219\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u53c2\u8003\u200b\u7f57\u9a6c\u6570\u5b57\u200b - Mathematics\u3002</p> <p>Reference</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/13/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u9010\u4f4d\u200b\u5904\u7406\u200b\u5373\u53ef\u200b\uff0c\u200b\u6ce8\u610f\u200b\u5c0f\u200b\u7684\u200b\u6570\u5b57\u200b\u5728\u200b\u5927\u200b\u7684\u200b\u6570\u5b57\u200b\u5de6\u8fb9\u200b\u7684\u200b\u7279\u6b8a\u200b\u60c5\u51b5\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int romanToInt(char * s){\n    char *cur = s;\n    int ret = 0;\n    while(*cur)\n    {\n        switch(*cur)\n        {\n            case 'I':\n                if (*(cur + 1) == 'V' || *(cur + 1) == 'X')\n                    ret -= 1;\n                else\n                    ret += 1;\n                break;\n            case 'V':\n                ret += 5;\n                break;\n            case 'X':\n                if (*(cur + 1) == 'L' || *(cur + 1) == 'C')\n                    ret -= 10;\n                else\n                    ret += 10;\n                break;\n            case 'L':\n                ret += 50;\n                break;\n            case 'C':\n                if (*(cur + 1) == 'D' || *(cur + 1) == 'M')\n                    ret -= 100;\n                else\n                    ret += 100;\n                break;\n            case 'M':\n                ret += 500;\n            case 'D':\n                ret += 500;\n        }\n        cur++;\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/1300/","title":"1300. \u200b\u8f6c\u53d8\u200b\u6570\u7ec4\u200b\u540e\u200b\u6700\u200b\u63a5\u8fd1\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u6570\u7ec4\u200b\u548c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/1300/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\u00a0<code>arr</code> \u200b\u548c\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u503c\u200b\u00a0<code>target</code> \uff0c\u200b\u8bf7\u200b\u4f60\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u00a0<code>value</code>\u00a0\uff0c\u200b\u4f7f\u5f97\u200b\u5c06\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u6240\u6709\u200b\u5927\u4e8e\u200b\u00a0<code>value</code> \u200b\u7684\u200b\u503c\u200b\u53d8\u6210\u200b\u00a0<code>value</code> \u200b\u540e\u200b\uff0c\u200b\u6570\u7ec4\u200b\u7684\u200b\u548c\u200b\u6700\u200b\u63a5\u8fd1\u200b <code>target</code> \uff08\u200b\u6700\u200b\u63a5\u8fd1\u200b\u8868\u793a\u200b\u4e24\u8005\u200b\u4e4b\u5dee\u200b\u7684\u200b\u7edd\u5bf9\u503c\u200b\u6700\u5c0f\u200b\uff09\u3002</p> <p>\u200b\u5982\u679c\u200b\u6709\u200b\u591a\u79cd\u200b\u4f7f\u5f97\u200b\u548c\u200b\u6700\u200b\u63a5\u8fd1\u200b\u00a0<code>target</code>\u00a0\u200b\u7684\u200b\u65b9\u6848\u200b\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u8fd4\u56de\u200b\u8fd9\u4e9b\u200b\u6574\u6570\u200b\u4e2d\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\u3002</p> <p>\u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u7b54\u6848\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u00a0<code>arr</code> \u200b\u4e2d\u200b\u7684\u200b\u6570\u5b57\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aarr = [4,9,3], target = 10\n\u200b\u8f93\u51fa\u200b\uff1a3\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5f53\u200b\u9009\u62e9\u200b value \u200b\u4e3a\u200b 3 \u200b\u65f6\u200b\uff0c\u200b\u6570\u7ec4\u200b\u4f1a\u200b\u53d8\u6210\u200b [3, 3, 3]\uff0c\u200b\u548c\u200b\u4e3a\u200b 9 \uff0c\u200b\u8fd9\u200b\u662f\u200b\u6700\u200b\u63a5\u8fd1\u200b target \u200b\u7684\u200b\u65b9\u6848\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aarr = [2,3,5], target = 10\n\u200b\u8f93\u51fa\u200b\uff1a5\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aarr = [60864,25176,27249,21296,20204], target = 56803\n\u200b\u8f93\u51fa\u200b\uff1a11361\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= arr.length &lt;= 10^4</code></li> <li><code>1 &lt;= arr[i], target &lt;= 10^5</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/1300/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5728\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\\((lo, hi)\\)\u200b\u4e2d\u200b\u786e\u5b9a\u200b\u4e09\u7b49\u5206\u200b\u70b9\u200b\\(x_1, x_2 (x_1 &lt; x_2)\\)\uff0c\u200b\u5206\u522b\u200b\u8ba1\u7b97\u200b\u4e24\u4e2a\u200b\u4e09\u7b49\u5206\u200b\u70b9\u5904\u200b\u7684\u200b\u5dee\u503c\u200b\u3002\u200b\u8bbe\u200b\\(f(x)\\)\u200b\u4e3a\u200b\u5c06\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5927\u4e8e\u200b\\(x\\)\u200b\u7684\u200b\u6570\u503c\u200b\u8f6c\u53d8\u200b\u4e3a\u200b\\(x\\)\u200b\u540e\u200b\u6570\u7ec4\u200b\u6c42\u548c\u200b\u4e0e\u200b\u76ee\u6807\u503c\u200b\u4e4b\u5dee\u200b\u7684\u200b\u7edd\u5bf9\u503c\u200b\u3002</p> <ul> <li>\u200b\u5f53\u200b\\(f(x_1) &lt; f(x_2)\\)\u200b\u65f6\u200b\uff0c\u200b\u5c06\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u7f29\u51cf\u200b\u4e3a\u200b\\((lo, x_2)\\)</li> <li>\u200b\u5f53\u200b\\(f(x_1) &gt; f(x_2)\\)\u200b\u65f6\u200b\uff0c\u200b\u5c06\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u7f29\u51cf\u200b\u4e3a\u200b\\((x_1, hi)\\)</li> </ul> <p>\\(f(x)\\)\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b+\u200b\u9884\u5904\u7406\u200b\u5728\u200b\\(\\mathcal O(\\log N)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u5185\u200b\u8ba1\u7b97\u200b\u5b8c\u6210\u200b\u3002\u200b\u9884\u5904\u7406\u200b\u9700\u8981\u200b\u82b1\u8d39\u200b\\(\\mathcal O(N)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u3002\u200b\u6392\u5e8f\u200b\u9700\u8981\u200b\u82b1\u8d39\u200b\\(\\mathcal O(N\\log N)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u3002\u200b\u9700\u8981\u200b\\(\\mathcal O(\\log N)\\)\u200b\u6b21\u200b\u4e09\u5206\u200b\u67e5\u627e\u200b\u3002\u200b\u56e0\u6b64\u200b\u603b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(N\\log N)\\)\uff08\u200b\u9884\u5904\u7406\u200b\u9636\u6bb5\u200b\uff09</p> <pre><code>#define ABS(x) ((x) &gt; 0 ? (x) : -(x))\n\nint cmp(const void *a, const void *b)\n{\n    return *((int *)a) - *((int *)b);\n}\n\nint bSearch(int *arr, int target, int lo, int hi)\n{\n    int mid;\n    while (lo &lt; hi)\n    {\n        mid = (lo + hi) &gt;&gt; 1;\n        if (arr[mid] &lt; target)\n            lo = mid + 1;\n        else\n            hi = mid;\n    }\n    return lo - 1;\n}\n\nint test(int *arr, int arrSize, int target, int val, int *presum)\n{\n    int i = bSearch(arr, val, 0, arrSize), ret = val * (arrSize - i - 1) - target;\n    if (i &gt;= 0)\n        ret += presum[i];\n    return ABS(ret);\n}\n\nint tSearch(int *arr, int arrSize, int target, int lo, int hi, int *presum)\n{\n    int left, right;\n    while (hi - lo &gt; 2)\n    {\n        left = (2 * lo + hi) / 3;\n        right = (2 * hi + lo) / 3;\n        if (test(arr, arrSize, target, left, presum) &lt;= test(arr, arrSize, target, right, presum))\n            hi = right;\n        else\n            lo = left;\n    }\n    return (2 * hi + lo) / 3;\n}\n\nint findBestValue(int* arr, int arrSize, int target){\n    int i = 0, presum[arrSize];\n    qsort(arr, arrSize, sizeof(int), cmp);\n    presum[0] = arr[0];\n    for (i = 1; i &lt; arrSize; i++)\n        presum[i] = presum[i - 1] + arr[i];\n\n    return tSearch(arr, arrSize, target, -1, arr[arrSize - 1] + 1, presum);\n}\n</code></pre>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/136/","title":"136. \u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u7684\u200b\u6570\u5b57","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u4f4d\u8fd0\u7b97","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/136/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u975e\u7a7a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\uff0c\u200b\u9664\u4e86\u200b\u67d0\u4e2a\u200b\u5143\u7d20\u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u4ee5\u5916\u200b\uff0c\u200b\u5176\u4f59\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u5747\u200b\u51fa\u73b0\u200b\u4e24\u6b21\u200b\u3002\u200b\u627e\u51fa\u200b\u90a3\u4e2a\u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e86\u200b\u4e00\u6b21\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u7684\u200b\u7b97\u6cd5\u200b\u5e94\u8be5\u200b\u5177\u6709\u200b\u7ebf\u6027\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u3002 \u200b\u4f60\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u4f7f\u7528\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\u6765\u200b\u5b9e\u73b0\u200b\u5417\u200b\uff1f</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [2,2,1]\n\u200b\u8f93\u51fa\u200b: 1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [4,1,2,1,2]\n\u200b\u8f93\u51fa\u200b: 4\n</code></pre> <p>Reference</p>","tags":["\u4f4d\u8fd0\u7b97","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/136/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5f02\u6216\u200b\u8fd0\u7b97\u200b\u5177\u6709\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ol> <li>\u200b\u4ea4\u6362\u5f8b\u200b\uff1a<code>a ^ b = b ^ a</code></li> <li>\u200b\u7ed3\u5408\u5f8b\u200b\uff1a<code>(a ^ b) ^ c = a ^ (b ^ c)</code></li> <li>\u200b\u82e5\u200b<code>a ^ b = c</code>\uff0c\u200b\u5219\u200b<code>a ^ c = b</code>\u3001<code>b ^ c = a</code></li> <li><code>a ^ a = 0</code></li> </ol> <p>\u200b\u6839\u636e\u200b\u6027\u8d28\u200b1\u30012\uff0c\u200b\u5f02\u6216\u200b\u8fd0\u7b97\u200b\u7684\u200b\u7ed3\u679c\u200b\u4e0e\u200b\u53c2\u4e0e\u200b\u8fd0\u7b97\u200b\u7684\u200b\u6240\u6709\u200b\u6570\u503c\u200b\u7684\u200b\u6392\u5217\u200b\u987a\u5e8f\u200b\u65e0\u5173\u200b\u3002\u200b\u6839\u636e\u200b\u6027\u8d28\u200b3\u30014\uff0c\u200b\u76f8\u540c\u200b\u6574\u6570\u200b\u7684\u200b\u5f02\u6216\u200b\u503c\u4e3a\u200b0\uff0c0\u200b\u4e0e\u200b\u4efb\u4f55\u200b\u6570\u200b\u5f02\u6216\u200b\u4ecd\u4e3a\u200b\u8be5\u6570\u200b\u3002\u200b\u56e0\u6b64\u200b\u5c06\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5404\u200b\u5143\u7d20\u200b\u8fdb\u884c\u200b\u6309\u4f4d\u200b\u5f02\u6216\u200b\uff0c\u200b\u5f97\u5230\u200b\u7684\u200b\u7ed3\u679c\u200b\u5373\u200b\u4e3a\u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u7684\u200b\u6570\u503c\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int singleNumber(int* nums, int numsSize){\n    int ret = 0, i = 0;\n    for (i = 0; i &lt; numsSize; i++)\n        ret ^= nums[i];\n    return ret;\n}\n</code></pre>","tags":["\u4f4d\u8fd0\u7b97","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/14/","title":"14. \u200b\u6700\u957f\u200b\u516c\u5171\u200b\u524d\u7f00","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/14/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u6765\u200b\u67e5\u627e\u200b\u5b57\u7b26\u4e32\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u6700\u957f\u200b\u516c\u5171\u200b\u524d\u7f00\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u4e0d\u200b\u5b58\u5728\u200b\u516c\u5171\u200b\u524d\u7f00\u200b\uff0c\u200b\u8fd4\u56de\u200b\u7a7a\u200b\u5b57\u7b26\u4e32\u200b<code>\"\"</code>\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [\"flower\",\"flow\",\"flight\"]\n\u200b\u8f93\u51fa\u200b: \"fl\"\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [\"dog\",\"racecar\",\"car\"]\n\u200b\u8f93\u51fa\u200b: \"\"\n\u200b\u89e3\u91ca\u200b: \u200b\u8f93\u5165\u200b\u4e0d\u200b\u5b58\u5728\u200b\u516c\u5171\u200b\u524d\u7f00\u200b\u3002\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u6240\u6709\u200b\u8f93\u5165\u200b\u53ea\u200b\u5305\u542b\u200b\u5c0f\u5199\u5b57\u6bcd\u200b<code>a-z</code>\u3002</p> <p>Reference</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/14/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5728\u200b\u6bd4\u8f83\u200b\u524d\u7f00\u200b\u65f6\u200b\u53ea\u200b\u9700\u8981\u200b\u5c06\u200b\u524d\u200b n \u200b\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u516c\u5171\u200b\u524d\u7f00\u200b\u4e0e\u200b\u7b2c\u200b n+1 \u200b\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u8fdb\u884c\u200b\u6bd4\u8f83\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>char *compareTwo(char *str1, char *str2)\n{\n    char *cur1 = str1, *cur2 = str2, *ret = NULL;\n    int len = 0;\n    while (*cur1 &amp;&amp; *cur2)\n    {\n        if (*cur1 != *cur2)\n            break;\n        else\n            len++;\n        cur1++;\n        cur2++;\n    }\n    ret = (char *)memset(malloc(sizeof(char) * (len + 1)), 0, sizeof(char) * (len + 1));\n    return memcpy(ret, str1, sizeof(char) * len);\n}\n\nchar * longestCommonPrefix(char ** strs, int strsSize){\n    if (strsSize == 0)\n        return memset(malloc(sizeof(char)), 0, sizeof(char));\n    if (strsSize == 1)\n        return strs[0];\n    if (strsSize == 2)\n        return compareTwo(strs[0], strs[1]);\n    int rec_size = (strsSize + 1) &gt;&gt; 1, i = 0;\n    char **rec = (char **)memset(malloc(sizeof(char *) * rec_size), 0, sizeof(char *) * rec_size);\n    for (i = 0; i &lt; strsSize &gt;&gt; 1; i++)\n        rec[i] = compareTwo(strs[2 * i], strs[2 * i + 1]);\n    if (strsSize &amp; 1)\n        rec[rec_size - 1] = strs[strsSize - 1];\n    return longestCommonPrefix(rec, rec_size);\n}\n</code></pre>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/150/","title":"150. \u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u6c42\u503c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6808"]},{"location":"coding/leetcode/150/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u6839\u636e\u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u793a\u6cd5\u200b\uff0c\u200b\u6c42\u200b\u8868\u8fbe\u5f0f\u200b\u7684\u200b\u503c\u200b\u3002</p> <p>\u200b\u6709\u6548\u200b\u7684\u200b\u7b97\u7b26\u200b\u5305\u62ec\u200b<code>+</code>\u3001<code>-</code>\u3001<code>*</code>\u3001<code>/</code>\u3002\u200b\u6bcf\u4e2a\u200b\u8fd0\u7b97\u200b\u5bf9\u8c61\u200b\u53ef\u4ee5\u200b\u662f\u200b\u6574\u6570\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li>\u200b\u6574\u6570\u200b\u9664\u6cd5\u200b\u53ea\u200b\u4fdd\u7559\u200b\u6574\u6570\u200b\u90e8\u5206\u200b\u3002</li> <li>\u200b\u7ed9\u5b9a\u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u603b\u662f\u200b\u6709\u6548\u200b\u7684\u200b\u3002\u200b\u6362\u53e5\u8bdd\u8bf4\u200b\uff0c\u200b\u8868\u8fbe\u5f0f\u200b\u603b\u4f1a\u200b\u5f97\u51fa\u200b\u6709\u6548\u200b\u6570\u503c\u200b\u4e14\u200b\u4e0d\u200b\u5b58\u5728\u200b\u9664\u6570\u200b\u4e3a\u200b <code>0</code> \u200b\u7684\u200b\u60c5\u51b5\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1atokens = [\"2\",\"1\",\"+\",\"3\",\"*\"]\n\u200b\u8f93\u51fa\u200b\uff1a9\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u8be5\u200b\u7b97\u5f0f\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u5e38\u89c1\u200b\u7684\u200b\u4e2d\u7f00\u200b\u7b97\u672f\u200b\u8868\u8fbe\u5f0f\u200b\u4e3a\u200b\uff1a((2 + 1) * 3) = 9\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1atokens = [\"4\",\"13\",\"5\",\"/\",\"+\"]\n\u200b\u8f93\u51fa\u200b\uff1a6\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u8be5\u200b\u7b97\u5f0f\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u5e38\u89c1\u200b\u7684\u200b\u4e2d\u7f00\u200b\u7b97\u672f\u200b\u8868\u8fbe\u5f0f\u200b\u4e3a\u200b\uff1a(4 + (13 / 5)) = 6\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a03\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1atokens = [\"10\",\"6\",\"9\",\"3\",\"+\",\"-11\",\"*\",\"/\",\"*\",\"17\",\"+\",\"5\",\"+\"]\n\u200b\u8f93\u51fa\u200b\uff1a22\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u8be5\u200b\u7b97\u5f0f\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u5e38\u89c1\u200b\u7684\u200b\u4e2d\u7f00\u200b\u7b97\u672f\u200b\u8868\u8fbe\u5f0f\u200b\u4e3a\u200b\uff1a\n  ((10 * (6 / ((9 + 3) * -11))) + 17) + 5\n= ((10 * (6 / (12 * -11))) + 17) + 5\n= ((10 * (6 / -132)) + 17) + 5\n= ((10 * 0) + 17) + 5\n= (0 + 17) + 5\n= 17 + 5\n= 22\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= tokens.length &lt;= 10^4</code></li> <li><code>tokens[i]</code> \u200b\u8981\u4e48\u200b\u662f\u200b\u4e00\u4e2a\u200b\u7b97\u7b26\u200b\uff08<code>\"+\"</code>\u3001<code>\"-\"</code>\u3001<code>\"*\"</code> \u200b\u6216\u200b <code>\"/\"</code>\uff09\uff0c\u200b\u8981\u4e48\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5728\u200b\u8303\u56f4\u200b <code>[-200, 200]</code> \u200b\u5185\u200b\u7684\u200b\u6574\u6570\u200b</li> </ul> <p>\u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\uff1a</p> <p>\u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u662f\u200b\u4e00\u79cd\u200b\u540e\u7f00\u200b\u8868\u8fbe\u5f0f\u200b\uff0c\u200b\u6240\u8c13\u200b\u540e\u7f00\u200b\u5c31\u662f\u6307\u200b\u7b97\u7b26\u200b\u5199\u200b\u5728\u200b\u540e\u9762\u200b\u3002</p> <p>\u200b\u5e73\u5e38\u200b\u4f7f\u7528\u200b\u7684\u200b\u7b97\u5f0f\u200b\u5219\u200b\u662f\u200b\u4e00\u79cd\u200b\u4e2d\u7f00\u200b\u8868\u8fbe\u5f0f\u200b\uff0c\u200b\u5982\u200b <code>( 1 + 2 ) * ( 3 + 4 )</code> \u3002 \u200b\u8be5\u200b\u7b97\u5f0f\u200b\u7684\u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u5199\u6cd5\u200b\u4e3a\u200b <code>( ( 1 2 + ) ( 3 4 + ) * )</code> \u3002 \u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u4e3b\u8981\u200b\u6709\u200b\u4ee5\u4e0b\u200b\u4e24\u4e2a\u200b\u4f18\u70b9\u200b\uff1a</p> <p>\u200b\u53bb\u6389\u200b\u62ec\u53f7\u200b\u540e\u200b\u8868\u8fbe\u5f0f\u200b\u65e0\u200b\u6b67\u4e49\u200b\uff0c\u200b\u4e0a\u200b\u5f0f\u200b\u5373\u4fbf\u200b\u5199\u6210\u200b <code>1 2 + 3 4 + *</code> \u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4f9d\u636e\u200b\u6b21\u5e8f\u200b\u8ba1\u7b97\u200b\u51fa\u200b\u6b63\u786e\u200b\u7ed3\u679c\u200b\u3002 \u200b\u9002\u5408\u200b\u7528\u6808\u200b\u64cd\u4f5c\u200b\u8fd0\u7b97\u200b\uff1a\u200b\u9047\u5230\u200b\u6570\u5b57\u200b\u5219\u200b\u5165\u6808\u200b\uff1b\u200b\u9047\u5230\u200b\u7b97\u7b26\u200b\u5219\u200b\u53d6\u51fa\u200b\u6808\u9876\u200b\u4e24\u4e2a\u200b\u6570\u5b57\u200b\u8fdb\u884c\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u5e76\u200b\u5c06\u200b\u7ed3\u679c\u200b\u538b\u5165\u200b\u6808\u200b\u4e2d\u200b\u3002</p> <p>Reference</p>","tags":["\u6808"]},{"location":"coding/leetcode/150/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u6839\u636e\u200b\u9898\u76ee\u200b\u4e2d\u200b\u7684\u200b\u63d0\u793a\u200b\u5373\u53ef\u200b\uff1a</p> <p>\u200b\u9002\u5408\u200b\u7528\u6808\u200b\u64cd\u4f5c\u200b\u8fd0\u7b97\u200b\uff1a\u200b\u9047\u5230\u200b\u6570\u5b57\u200b\u5219\u200b\u5165\u6808\u200b\uff1b\u200b\u9047\u5230\u200b\u7b97\u7b26\u200b\u5219\u200b\u53d6\u51fa\u200b\u6808\u9876\u200b\u4e24\u4e2a\u200b\u6570\u5b57\u200b\u8fdb\u884c\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u5e76\u200b\u5c06\u200b\u7ed3\u679c\u200b\u538b\u5165\u200b\u6808\u200b\u4e2d\u200b\u3002</p> <pre><code>struct ListNode *getNode(int val, struct ListNode *next)\n{\n    struct ListNode *ret = (struct ListNode *)malloc(sizeof(struct ListNode));\n    ret-&gt;val = val;\n    ret-&gt;next = next;\n    return ret;\n}\nstruct ListNode *operand(struct ListNode *Stack, char operand)\n{\n    struct ListNode *cur1 = Stack, *cur2 = Stack-&gt;next, *temp = cur2-&gt;next;\n    int ret;\n    switch (operand)\n    {\n        case '+':\n            ret = cur1-&gt;val + cur2-&gt;val;\n            break;\n        case '-':\n            ret = cur2-&gt;val - cur1-&gt;val;\n            break;\n        case '*':\n            ret = cur1-&gt;val * cur2-&gt;val;\n            break;\n        case '/':\n            ret = cur2-&gt;val / cur1-&gt;val;\n            break;\n    }\n    free(cur1);\n    free(cur2);\n    return getNode(ret, temp);\n}\nint evalRPN(char ** tokens, int tokensSize){\n    struct ListNode *Stack = NULL;\n    int i = 0;\n    char *cur;\n    for (i = 0; i &lt; tokensSize; i++)\n    {\n        cur = tokens[i];\n        if (*cur &gt;= '0' &amp;&amp; *cur &lt;= '9' || strlen(cur) &gt; 1)\n            Stack = getNode(atoi(cur), Stack);\n        else\n            Stack = operand(Stack, *cur);\n    }\n    return Stack-&gt;val;\n}\n</code></pre>","tags":["\u6808"]},{"location":"coding/leetcode/155/","title":"155. \u200b\u6700\u5c0f\u200b\u6808","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6808","\u8bbe\u8ba1"]},{"location":"coding/leetcode/155/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u652f\u6301\u200b <code>push</code>\uff0c<code>pop</code>\uff0c<code>top</code>\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u5e76\u200b\u80fd\u200b\u5728\u200b\u5e38\u6570\u200b\u65f6\u95f4\u200b\u5185\u200b\u68c0\u7d22\u200b\u5230\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b\u7684\u200b\u6808\u200b\u3002</p> <ul> <li><code>push(x)</code> \u2014\u2014 \u200b\u5c06\u200b\u5143\u7d20\u200b x \u200b\u63a8\u5165\u200b\u6808\u4e2d\u200b\u3002</li> <li><code>pop()</code>\u00a0\u2014\u2014 \u200b\u5220\u9664\u200b\u6808\u9876\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</li> <li><code>top()</code>\u00a0\u2014\u2014 \u200b\u83b7\u53d6\u200b\u6808\u9876\u200b\u5143\u7d20\u200b\u3002</li> <li><code>getMin()</code> \u2014\u2014 \u200b\u68c0\u7d22\u200b\u6808\u4e2d\u200b\u7684\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a\n[\"MinStack\",\"push\",\"push\",\"push\",\"getMin\",\"pop\",\"top\",\"getMin\"]\n[[],[-2],[0],[-3],[],[],[],[]]\n\n\u200b\u8f93\u51fa\u200b\uff1a\n[null,null,null,null,-3,null,0,-2]\n\n\u200b\u89e3\u91ca\u200b\uff1a\nMinStack minStack = new MinStack();\nminStack.push(-2);\nminStack.push(0);\nminStack.push(-3);\nminStack.getMin();   --&gt; \u200b\u8fd4\u56de\u200b -3.\nminStack.pop();\nminStack.top();      --&gt; \u200b\u8fd4\u56de\u200b 0.\nminStack.getMin();   --&gt; \u200b\u8fd4\u56de\u200b -2.\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <p><code>pop</code>\u3001<code>top</code> \u200b\u548c\u200b <code>getMin</code> \u200b\u64cd\u4f5c\u200b\u603b\u662f\u200b\u5728\u200b \u200b\u975e\u7a7a\u6808\u200b \u200b\u4e0a\u200b\u8c03\u7528\u200b\u3002</p> <p>Reference</p>","tags":["\u6808","\u8bbe\u8ba1"]},{"location":"coding/leetcode/155/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4f7f\u7528\u200b\u4e24\u4e2a\u200b\u57df\u200b\u5206\u522b\u200b\u5b58\u50a8\u200b\u5f53\u524d\u200b\u6570\u503c\u200b\u548c\u200b\u5f53\u524d\u200b\u6808\u4e2d\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>#define MIN(x, y) (x &lt; y ? x : y)\nstruct cell\n{\n    int val;\n    int curMin;\n};\ntypedef struct {\n    struct cell values[10000];\n    int top;\n} MinStack;\n/** initialize your data structure here. */\nMinStack *minStackCreate() \n{\n    return (MinStack *)memset(malloc(sizeof(MinStack)), 0, sizeof(MinStack));\n}\nvoid minStackPush(MinStack* obj, int x)\n{\n    obj-&gt;values[obj-&gt;top].val = x;\n    obj-&gt;values[obj-&gt;top].curMin = obj-&gt;top &gt; 0 ? MIN(x, obj-&gt;values[obj-&gt;top - 1].curMin) : x;\n    obj-&gt;top++;\n}\nvoid minStackPop(MinStack *obj)\n{\n    obj-&gt;top--;\n}\nint minStackTop(MinStack* obj)\n{\n    return obj-&gt;values[obj-&gt;top - 1].val;\n}\nint minStackGetMin(MinStack* obj)\n{\n    return obj-&gt;values[obj-&gt;top - 1].curMin;\n}\nvoid minStackFree(MinStack* obj)\n{\n    free(obj);\n}\n/**\n * Your MinStack struct will be instantiated and called as such:\n * MinStack* obj = minStackCreate();\n * minStackPush(obj, x);\n\n * minStackPop(obj);\n\n * int param_3 = minStackTop(obj);\n\n * int param_4 = minStackGetMin(obj);\n\n * minStackFree(obj);\n*/\n</code></pre>","tags":["\u6808","\u8bbe\u8ba1"]},{"location":"coding/leetcode/1603/","title":"1603. \u200b\u8bbe\u8ba1\u200b\u505c\u8f66\u200b\u7cfb\u7edf","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u8bbe\u8ba1"]},{"location":"coding/leetcode/1603/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u8bf7\u200b\u4f60\u200b\u7ed9\u200b\u4e00\u4e2a\u200b\u505c\u8f66\u573a\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u505c\u8f66\u200b\u7cfb\u7edf\u200b\u3002\u200b\u505c\u8f66\u573a\u200b\u603b\u5171\u200b\u6709\u200b\u4e09\u79cd\u200b\u4e0d\u540c\u200b\u5927\u5c0f\u200b\u7684\u200b\u8f66\u4f4d\u200b\uff1a\u200b\u5927\u200b\uff0c\u200b\u4e2d\u200b\u548c\u200b\u5c0f\u200b\uff0c\u200b\u6bcf\u79cd\u200b\u5c3a\u5bf8\u200b\u5206\u522b\u200b\u6709\u200b\u56fa\u5b9a\u200b\u6570\u76ee\u200b\u7684\u200b\u8f66\u4f4d\u200b\u3002</p> <p>\u200b\u8bf7\u200b\u4f60\u200b\u5b9e\u73b0\u200b\u00a0<code>ParkingSystem</code>\u00a0\u200b\u7c7b\u200b\uff1a</p> <p><code>ParkingSystem(int big, int medium, int small)``\u00a0\u200b\u521d\u59cb\u5316\u200b\u00a0ParkingSystem\u00a0\u200b\u7c7b\u200b\uff0c\u200b\u4e09\u4e2a\u200b\u53c2\u6570\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u6bcf\u79cd\u200b\u505c\u8f66\u4f4d\u200b\u7684\u200b\u6570\u76ee\u200b\u3002</code> <code>bool addCar(int carType)</code>\u00a0\u200b\u68c0\u67e5\u200b\u662f\u5426\u200b\u6709\u200b\u00a0<code>carType</code>\u00a0\u200b\u5bf9\u5e94\u200b\u7684\u200b\u505c\u8f66\u4f4d\u200b\u3002\u00a0<code>carType</code>\u00a0\u200b\u6709\u200b\u4e09\u79cd\u200b\u7c7b\u578b\u200b\uff1a\u200b\u5927\u200b\uff0c\u200b\u4e2d\u200b\uff0c\u200b\u5c0f\u200b\uff0c\u200b\u5206\u522b\u200b\u7528\u200b\u6570\u5b57\u200b\u00a0<code>1</code>\uff0c\u00a0<code>2</code>\u00a0\u200b\u548c\u200b\u00a0<code>3</code>\u00a0\u200b\u8868\u793a\u200b\u3002\u200b\u4e00\u8f86\u8f66\u200b\u53ea\u80fd\u200b\u505c\u200b\u5728\u200b\u00a0<code>carType</code>\u00a0\u200b\u5bf9\u5e94\u200b\u5c3a\u5bf8\u200b\u7684\u200b\u505c\u8f66\u4f4d\u200b\u4e2d\u200b\u3002\u200b\u5982\u679c\u200b\u6ca1\u6709\u200b\u7a7a\u8f66\u200b\u4f4d\u200b\uff0c\u200b\u8bf7\u200b\u8fd4\u56de\u200b\u00a0<code>false</code>\u00a0\uff0c\u200b\u5426\u5219\u200b\u5c06\u200b\u8be5\u8f66\u200b\u505c\u5165\u200b\u8f66\u4f4d\u200b\u5e76\u200b\u8fd4\u56de\u200b\u00a0<code>true</code>\u00a0\u3002\u00a0</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a\n[\"ParkingSystem\", \"addCar\", \"addCar\", \"addCar\", \"addCar\"]\n[[1, 1, 0], [1], [2], [3], [1]]\n\u200b\u8f93\u51fa\u200b\uff1a\n[null, true, true, false, false]\n</code></pre> <p>\u200b\u89e3\u91ca\u200b\uff1a</p> <pre><code>ParkingSystem parkingSystem = new ParkingSystem(1, 1, 0);\nparkingSystem.addCar(1); // \u200b\u8fd4\u56de\u200b true \uff0c\u200b\u56e0\u4e3a\u200b\u6709\u200b 1 \u200b\u4e2a\u7a7a\u200b\u7684\u200b\u5927\u200b\u8f66\u4f4d\u200b\nparkingSystem.addCar(2); // \u200b\u8fd4\u56de\u200b true \uff0c\u200b\u56e0\u4e3a\u200b\u6709\u200b 1 \u200b\u4e2a\u7a7a\u200b\u7684\u200b\u4e2d\u200b\u8f66\u4f4d\u200b\nparkingSystem.addCar(3); // \u200b\u8fd4\u56de\u200b false \uff0c\u200b\u56e0\u4e3a\u200b\u6ca1\u6709\u200b\u7a7a\u200b\u7684\u200b\u5c0f\u8f66\u200b\u4f4d\u200b\nparkingSystem.addCar(1); // \u200b\u8fd4\u56de\u200b false \uff0c\u200b\u56e0\u4e3a\u200b\u6ca1\u6709\u200b\u7a7a\u200b\u7684\u200b\u5927\u200b\u8f66\u4f4d\u200b\uff0c\u200b\u552f\u4e00\u200b\u4e00\u4e2a\u200b\u5927\u200b\u8f66\u4f4d\u200b\u5df2\u7ecf\u200b\u88ab\u200b\u5360\u636e\u200b\u4e86\u200b\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>0 &lt;= big, medium, small &lt;= 1000</code></li> <li><code>carType</code>\u00a0\u200b\u53d6\u503c\u200b\u4e3a\u200b\u00a0<code>1</code>\uff0c\u00a0<code>2</code>\u00a0\u200b\u6216\u200b\u00a0<code>3</code></li> <li>\u200b\u6700\u591a\u4f1a\u200b\u8c03\u7528\u200b\u00a0<code>addCar</code>\u00a0\u200b\u51fd\u6570\u200b\u00a01000\u00a0\u200b\u6b21\u200b</li> </ul> <p>Reference</p>","tags":["\u8bbe\u8ba1"]},{"location":"coding/leetcode/1603/#_2","title":"\u9898\u89e3","text":"<pre><code>typedef struct {\n    int cars[3];\n} ParkingSystem;\n\n\nParkingSystem* parkingSystemCreate(int big, int medium, int small) {\n    ParkingSystem *ret = (ParkingSystem *)malloc(sizeof(ParkingSystem));\n    ret-&gt;cars[0] = big;\n    ret-&gt;cars[1] = medium;\n    ret-&gt;cars[2] = small;\n    return ret;\n}\n\nbool parkingSystemAddCar(ParkingSystem* obj, int carType) {\n    if (obj-&gt;cars[carType - 1] == 0)\n        return false;\n    else\n    {\n        obj-&gt;cars[carType - 1]--;\n        return true;\n    }\n}\n\nvoid parkingSystemFree(ParkingSystem* obj) {\n    free(obj);\n}\n\n/**\n * Your ParkingSystem struct will be instantiated and called as such:\n * ParkingSystem* obj = parkingSystemCreate(big, medium, small);\n * bool param_1 = parkingSystemAddCar(obj, carType);\n\n * parkingSystemFree(obj);\n*/\n</code></pre>","tags":["\u8bbe\u8ba1"]},{"location":"coding/leetcode/17.21/","title":"\u9762\u8bd5\u9898\u200b 17.21. \u200b\u76f4\u65b9\u56fe\u200b\u7684\u200b\u6c34\u91cf","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>"},{"location":"coding/leetcode/17.21/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u76f4\u65b9\u56fe\u200b(\u200b\u4e5f\u200b\u79f0\u200b\u67f1\u72b6\u56fe\u200b)\uff0c\u200b\u5047\u8bbe\u200b\u6709\u4eba\u200b\u4ece\u200b\u4e0a\u9762\u200b\u6e90\u6e90\u4e0d\u65ad\u200b\u5730\u200b\u5012\u6c34\u200b\uff0c\u200b\u6700\u540e\u200b\u76f4\u65b9\u56fe\u200b\u80fd\u5b58\u200b\u591a\u5c11\u200b\u6c34\u91cf\u200b?\u200b\u76f4\u65b9\u56fe\u200b\u7684\u200b\u5bbd\u5ea6\u200b\u4e3a\u200b <code>1</code>\u3002</p> <p></p> <p>\u200b\u4e0a\u9762\u200b\u662f\u200b\u7531\u200b\u6570\u7ec4\u200b <code>[0,1,0,2,1,0,1,3,2,1,2,1]</code> \u200b\u8868\u793a\u200b\u7684\u200b\u76f4\u65b9\u56fe\u200b\uff0c\u200b\u5728\u200b\u8fd9\u79cd\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u63a5\u200b <code>6</code> \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u7684\u200b\u6c34\u200b\uff08\u200b\u84dd\u8272\u200b\u90e8\u5206\u200b\u8868\u793a\u200b\u6c34\u200b\uff09\u3002\u00a0\u200b\u611f\u8c22\u200b Marcos \u200b\u8d21\u732e\u200b\u6b64\u56fe\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b:</p> <pre><code>\u200b\u8f93\u5165\u200b: [0,1,0,2,1,0,1,3,2,1,2,1]\n\u200b\u8f93\u51fa\u200b: 6\n</code></pre> <p>Reference</p> <p>\u200b\u6ce8\u200b\uff1a\u200b\u672c\u9898\u200b\u4e0e\u200b42. \u200b\u63a5\u200b\u96e8\u6c34\u200b\u76f8\u540c\u200b\u3002</p>"},{"location":"coding/leetcode/17.21/#_2","title":"\u9898\u89e3","text":"<pre><code>#define MAX(x, y) ((x) &gt; (y) ? x : y)\n#define MIN(x, y) ((x) &lt; (y) ? x : y)\n#define ABS(x) ((x) &gt; 0 ? (x) : -(x))\nint trap(int* height, int heightSize){\n    int highest = 0, higher = 0, highestIndex = -1, higherIndex = -1, i = 0, ret = 0;\n    for (i = 0; i &lt; heightSize; i++)\n    {\n        if (height[i] &gt; highest)\n        {\n            higher = highest;\n            higherIndex = highestIndex;\n            highest = height[i];\n            highestIndex = i;\n        }\n        else if (height[i] &gt; higher)\n        {\n            higher = height[i];\n            higherIndex = i;\n        }\n    }\n    if (heightSize &lt;= 2 || higherIndex &lt; 0 || highestIndex &lt; 0)\n        return 0;\n    ret = higher * (ABS(higherIndex - highestIndex) - 1);\n    int left = MIN(higherIndex, highestIndex) + 1, right = MAX(higherIndex, highestIndex);\n    for (i = left; i &lt; right; i++)\n        ret -= height[i];\n    return ret + trap(height, left) + trap(height + right, heightSize - right);\n}\n</code></pre>"},{"location":"coding/leetcode/17/","title":"17. \u200b\u7535\u8bdd\u53f7\u7801\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u9012\u5f52","\u5b57\u7b26\u4e32","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/17/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u4ec5\u200b\u5305\u542b\u200b\u6570\u5b57\u200b\u00a0<code>2-9</code>\u00a0\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u8fd4\u56de\u200b\u6240\u6709\u200b\u5b83\u200b\u80fd\u200b\u8868\u793a\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408\u200b\u3002\u200b\u7b54\u6848\u200b\u53ef\u4ee5\u200b\u6309\u200b \u200b\u4efb\u610f\u200b\u987a\u5e8f\u200b \u200b\u8fd4\u56de\u200b\u3002</p> <p>\u200b\u7ed9\u51fa\u200b\u6570\u5b57\u200b\u5230\u200b\u5b57\u6bcd\u200b\u7684\u200b\u6620\u5c04\u200b\u5982\u4e0b\u200b\uff08\u200b\u4e0e\u200b\u7535\u8bdd\u200b\u6309\u952e\u200b\u76f8\u540c\u200b\uff09\u3002\u200b\u6ce8\u610f\u200b 1 \u200b\u4e0d\u200b\u5bf9\u5e94\u200b\u4efb\u4f55\u200b\u5b57\u6bcd\u200b\u3002</p> <p></p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1adigits = \"23\"\n\u200b\u8f93\u51fa\u200b\uff1a[\"ad\",\"ae\",\"af\",\"bd\",\"be\",\"bf\",\"cd\",\"ce\",\"cf\"]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1adigits = \"\"\n\u200b\u8f93\u51fa\u200b\uff1a[]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1adigits = \"2\"\n\u200b\u8f93\u51fa\u200b\uff1a[\"a\",\"b\",\"c\"]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>0 &lt;= digits.length &lt;= 4</li> <li>digits[i] \u200b\u662f\u200b\u8303\u56f4\u200b ['2', '9'] \u200b\u7684\u200b\u4e00\u4e2a\u200b\u6570\u5b57\u200b\u3002</li> </ul> <p>Reference</p>","tags":["\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u9012\u5f52","\u5b57\u7b26\u4e32","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/17/#_2","title":"\u9898\u89e3","text":"<pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\nvoid dfs(char *digits, char **ret, int *subscript, char *current, int end, char **map)\n{\n    if (*digits == 0)\n    {\n        strcpy(ret[(*subscript)++], current);\n        return;\n    }\n    int i = 0, target = *digits - '2';\n    char *cur = map[target];\n    while(*cur)\n    {\n        current[end] = *cur;\n        dfs(digits + 1, ret, subscript, current, end + 1, map);\n        cur++;\n    }\n}\nchar ** letterCombinations(char * digits, int* returnSize){\n    *returnSize = 1;\n    char *cur = digits, len = 0, *charMap[] = {\"abc\", \"def\", \"ghi\", \"jkl\", \"mno\", \"pqrs\", \"tuv\", \"wxyz\"};\n    while(*cur)\n    {\n        *returnSize *= (*cur == '7' || *cur == '9') ? 4 : 3;\n        cur++;\n        len++;\n    }\n    if (len == 0)\n    {\n        *returnSize = 0;\n        return NULL;\n    }\n    char **ret = (char **)malloc(sizeof(char *) * *returnSize),\n         *current = (char *)memset(malloc(sizeof(char) * (len + 1)), 0, sizeof(char) * (len + 1));\n    for (int i = 0; i &lt; *returnSize; i++)\n        ret[i] = (int *)memset(malloc(sizeof(char) * (len + 1)), 0, sizeof(char) * (len + 1));\n    int subscript = 0;\n    dfs(digits, ret, &amp;subscript, current, 0, charMap);\n    return ret;\n}\n</code></pre>","tags":["\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u9012\u5f52","\u5b57\u7b26\u4e32","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/1835/","title":"1835. \u200b\u6240\u6709\u200b\u6570\u5bf9\u200b\u6309\u4f4d\u200b\u4e0e\u200b\u7ed3\u679c\u200b\u7684\u200b\u5f02\u6216\u200b\u548c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/1835/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5217\u8868\u200b\u7684\u200b \u200b\u5f02\u6216\u200b\u548c\u200b\uff08XOR sum\uff09\u200b\u6307\u5bf9\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u8fdb\u884c\u200b\u6309\u4f4d\u200b <code>XOR</code> \u200b\u8fd0\u7b97\u200b\u7684\u200b\u7ed3\u679c\u200b\u3002\u200b\u5982\u679c\u200b\u5217\u8868\u200b\u4e2d\u4ec5\u200b\u6709\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u90a3\u4e48\u200b\u5176\u200b \u200b\u5f02\u6216\u200b\u548c\u200b \u200b\u5c31\u200b\u7b49\u4e8e\u200b\u8be5\u200b\u5143\u7d20\u200b\u3002</p> <ul> <li>\u200b\u4f8b\u5982\u200b\uff0c<code>[1,2,3,4]</code> \u200b\u7684\u200b \u200b\u5f02\u6216\u200b\u548c\u200b \u200b\u7b49\u4e8e\u200b <code>1 XOR 2 XOR 3 XOR 4 = 4</code> \uff0c\u200b\u800c\u200b <code>[3]</code> \u200b\u7684\u200b \u200b\u5f02\u6216\u200b\u548c\u200b \u200b\u7b49\u4e8e\u200b <code>3</code> \u3002</li> </ul> <p>\u200b\u7ed9\u200b\u4f60\u200b\u4e24\u4e2a\u200b\u4e0b\u6807\u200b \u200b\u4ece\u200b 0 \u200b\u5f00\u59cb\u200b \u200b\u8ba1\u6570\u200b\u7684\u200b\u6570\u7ec4\u200b <code>arr1</code> \u200b\u548c\u200b <code>arr2</code> \uff0c\u200b\u4e24\u200b\u6570\u7ec4\u200b\u5747\u200b\u7531\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u7ec4\u6210\u200b\u3002</p> <p>\u200b\u6839\u636e\u200b\u6bcf\u4e2a\u200b\u00a0<code>(i, j)</code> \u200b\u6570\u5bf9\u200b\uff0c\u200b\u6784\u9020\u200b\u4e00\u4e2a\u200b\u7531\u200b <code>arr1[i] AND arr2[j]</code>\uff08\u200b\u6309\u4f4d\u200b <code>AND</code> \u200b\u8fd0\u7b97\u200b\uff09\u200b\u7ed3\u679c\u200b\u7ec4\u6210\u200b\u7684\u200b\u5217\u8868\u200b\u3002\u200b\u5176\u4e2d\u200b <code>0 &lt;= i &lt; arr1.length</code> \u200b\u4e14\u200b <code>0 &lt;= j &lt; arr2.length</code> \u3002</p> <p>\u200b\u8fd4\u56de\u200b\u4e0a\u8ff0\u200b\u5217\u8868\u200b\u7684\u200b \u200b\u5f02\u6216\u200b\u548c\u200b \u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aarr1 = [1,2,3], arr2 = [6,5]\n\u200b\u8f93\u51fa\u200b\uff1a0\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5217\u8868\u200b = [1 AND 6, 1 AND 5, 2 AND 6, 2 AND 5, 3 AND 6, 3 AND 5] = [0,1,2,0,2,1] \uff0c\n\u200b\u5f02\u6216\u200b\u548c\u200b = 0 XOR 1 XOR 2 XOR 0 XOR 2 XOR 1 = 0 \u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aarr1 = [12], arr2 = [4]\n\u200b\u8f93\u51fa\u200b\uff1a4\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5217\u8868\u200b = [12 AND 4] = [4] \uff0c\u200b\u5f02\u6216\u200b\u548c\u200b = 4 \u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= arr1.length, arr2.length &lt;= 105</code></li> <li><code>0 &lt;= arr1[i], arr2[j] &lt;= 109</code></li> </ul> <p>Reference</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/1835/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u8bbe\u200b\u4e24\u4e2a\u200b\u6570\u5217\u200b\\(A=\\{a_i\\}_m\\)\u200b\u4e0e\u200b\\(B=\\{b_j\\}_n\\)\uff0c\u200b\u5e76\u200b\u4f7f\u7528\u200b\u8fd0\u7b97\u7b26\u200b\\(\\otimes\\)\u200b\u8868\u793a\u200b \u200b\u5f02\u6216\u200b \u200b\u64cd\u4f5c\u200b\uff0c\u200b\u4f7f\u7528\u200b\u8fd0\u7b97\u7b26\u200b\\(\\land\\)\u200b\u8868\u793a\u200b \u200b\u4e0e\u200b \u200b\u64cd\u4f5c\u200b\uff0c\u200b\u5219\u200b\u672c\u9898\u200b\u7684\u200b\u7ed3\u679c\u200b\u4e3a\u200b\uff1a</p> \\[ f(A, B) = \\bigotimes_{i=1}^{m} \\bigotimes_{j=1}^n a_i\\land b_j \\] <p>\u200b\u8003\u8651\u200b\u8868\u8fbe\u5f0f\u200b\\((a\\otimes b) \\land (a \\otimes c)\\)\uff1a</p> \\[ \\begin{aligned} &amp; (a\\otimes b) \\land (a \\otimes c) \\\\ \\Leftrightarrow &amp; ((a\\lor b)\\land (\\lnot a\\lor\\lnot b)) \\land ((a\\lor c)\\land (\\lnot a\\lor\\lnot c)) \\\\ \\Leftrightarrow &amp; (a\\lor b)\\land (\\lnot a\\lor\\lnot b) \\land (a\\lor c)\\land (\\lnot a\\lor\\lnot c) \\\\ \\Leftrightarrow &amp; (a\\land (b\\lor c)) \\land (a\\land (\\lnot b\\lor\\lnot c)) \\\\ \\Leftrightarrow &amp; a\\land (b\\otimes c) \\end{aligned} \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\\(\\otimes\\)\u200b\u8fd0\u7b97\u200b\u5bf9\u200b\\(\\land\\)\u200b\u8fd0\u7b97\u200b\u6ee1\u8db3\u200b\u5206\u914d\u5f8b\u200b\uff0c\u200b\u6709\u200b\uff1a</p> \\[ \\begin{aligned} f(A, B) &amp;= \\bigotimes_{i=1}^{m} \\bigotimes_{j=1}^n a_i\\land b_j \\\\ &amp;= \\bigotimes_{i=1}^{m} \\left(a_{i}\\land\\bigotimes_{j=1}^n b_j\\right) \\\\ &amp;= \\left(\\bigotimes_{i=1}^{m} a_i\\right)\\land\\left(\\bigotimes_{j=1}^n b_j\\right) \\end{aligned} \\] <pre><code>int getXORSum(int* arr1, int arr1Size, int* arr2, int arr2Size){\n    int a = 0, b = 0, i = 0;\n    for (i = 0; i &lt; arr1Size; i++)\n        a ^= arr1[i];\n    for (i = 0; i &lt; arr2Size; i++)\n        b ^= arr2[i];\n    return a &amp; b;\n}\n</code></pre>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/189/","title":"189. \u200b\u65cb\u8f6c\u200b\u6570\u7ec4","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/189/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5c06\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u00a0k\u00a0\u200b\u4e2a\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u5176\u4e2d\u200b\u00a0k\u00a0\u200b\u662f\u975e\u200b\u8d1f\u6570\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,2,3,4,5,6,7] \u200b\u548c\u200b k = 3\n\u200b\u8f93\u51fa\u200b: [5,6,7,1,2,3,4]\n\u200b\u89e3\u91ca\u200b:\n\u200b\u5411\u200b\u53f3\u200b\u65cb\u8f6c\u200b 1 \u200b\u6b65\u200b: [7,1,2,3,4,5,6]\n\u200b\u5411\u200b\u53f3\u200b\u65cb\u8f6c\u200b 2 \u200b\u6b65\u200b: [6,7,1,2,3,4,5]\n\u200b\u5411\u200b\u53f3\u200b\u65cb\u8f6c\u200b 3 \u200b\u6b65\u200b: [5,6,7,1,2,3,4]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [-1,-100,3,99] \u200b\u548c\u200b k = 2\n\u200b\u8f93\u51fa\u200b: [3,99,-1,-100]\n\u200b\u89e3\u91ca\u200b: \n\u200b\u5411\u200b\u53f3\u200b\u65cb\u8f6c\u200b 1 \u200b\u6b65\u200b: [99,-1,-100,3]\n\u200b\u5411\u200b\u53f3\u200b\u65cb\u8f6c\u200b 2 \u200b\u6b65\u200b: [3,99,-1,-100]\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li>\u200b\u5c3d\u53ef\u80fd\u200b\u60f3\u51fa\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u89e3\u51b3\u65b9\u6848\u200b\uff0c\u200b\u81f3\u5c11\u200b\u6709\u200b\u4e09\u79cd\u200b\u4e0d\u540c\u200b\u7684\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u89e3\u51b3\u200b\u8fd9\u4e2a\u200b\u95ee\u9898\u200b\u3002</li> <li>\u200b\u8981\u6c42\u200b\u4f7f\u7528\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(1)\\)\u200b\u7684\u200b\u539f\u5730\u200b\u7b97\u6cd5\u200b\u3002</li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/189/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5148\u200b\u53cd\u8f6c\u200b\u6574\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u518d\u200b\u4ee5\u200b\u7b2c\u200b\\(k\\)\u200b\u4e2a\u200b\u5143\u7d20\u200b\u4e3a\u754c\u200b\uff0c\u200b\u5206\u522b\u200b\u53cd\u8f6c\u200b\\(k\\)\u200b\u4e4b\u524d\u200b\u7684\u200b\u90e8\u5206\u200b\u4e0e\u200b\\(k\\)\u200b\u53ca\u200b\\(k\\)\u200b\u4e4b\u540e\u200b\u7684\u200b\u90e8\u5206\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>void Reverse(int* nums, int numsSize)\n{\n    int head = 0, tail = numsSize - 1, temp = 0;\n    while(tail &gt; head)\n    {\n        temp = nums[head];\n        nums[head] = nums[tail];\n        nums[tail] = temp;\n        tail--;\n        head++;\n    }\n}\nvoid rotate(int* nums, int numsSize, int k){\n    if (numsSize &lt; 2)\n        return;\n    k %= numsSize;\n    Reverse(nums, numsSize);\n    Reverse(nums, k);\n    Reverse(nums + k, numsSize - k);\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/19/","title":"19. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u7684\u200b\u5012\u6570\u7b2c\u200bN\u200b\u4e2a\u200b\u8282\u70b9","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/19/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b\uff0c\u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u7684\u200b\u5012\u6570\u7b2c\u200b\u00a0n\u00a0\u200b\u4e2a\u200b\u8282\u70b9\u200b\uff0c\u200b\u5e76\u4e14\u200b\u8fd4\u56de\u200b\u94fe\u8868\u200b\u7684\u200b\u5934\u200b\u7ed3\u70b9\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <p>`` \u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b: 1-&gt;2-&gt;3-&gt;4-&gt;5, \u200b\u548c\u200b n = 2.</p> <p>\u200b\u5f53\u200b\u5220\u9664\u200b\u4e86\u200b\u5012\u6570\u200b\u7b2c\u4e8c\u4e2a\u200b\u8282\u70b9\u200b\u540e\u200b\uff0c\u200b\u94fe\u8868\u200b\u53d8\u4e3a\u200b 1-&gt;2-&gt;3-&gt;5. ``</p> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u7ed9\u5b9a\u200b\u7684\u200b n\u00a0\u200b\u4fdd\u8bc1\u200b\u662f\u200b\u6709\u6548\u200b\u7684\u200b\u3002</p> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u80fd\u200b\u5c1d\u8bd5\u200b\u4f7f\u7528\u200b\u4e00\u8d9f\u200b\u626b\u63cf\u200b\u5b9e\u73b0\u200b\u5417\u200b\uff1f</p> <p>Reference</p>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/19/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5148\u200b\u627e\u5230\u200b\u5012\u6570\u7b2c\u200b n \u200b\u4e2a\u200b\u8282\u70b9\u200b\uff0c\u200b\u518d\u200b\u8fdb\u884c\u200b\u5220\u9664\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\nstruct ListNode* removeNthFromEnd(struct ListNode* head, int n){\n    struct ListNode *cur = head, *temp = NULL;\n    int length = 1, i = 0;\n    while(cur-&gt;next != NULL)\n    {\n        cur = cur-&gt;next;\n        length++;\n    }\n    cur = head;\n    for (i = 0; i &lt; length - n - 1; i++)\n        cur = cur-&gt;next;\n    if (n == length)\n        return cur-&gt;next;\n    if (cur-&gt;next)\n    {\n        cur-&gt;next = cur-&gt;next-&gt;next;\n        return head;\n    }\n    return NULL;\n}\n</code></pre>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/198/","title":"198. \u200b\u6253\u5bb6\u52ab\u820d","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/198/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u4f60\u200b\u662f\u200b\u4e00\u4e2a\u200b\u4e13\u4e1a\u200b\u7684\u200b\u5c0f\u5077\u200b\uff0c\u200b\u8ba1\u5212\u200b\u5077\u7a83\u200b\u6cbf\u8857\u200b\u7684\u200b\u623f\u5c4b\u200b\u3002\u200b\u6bcf\u95f4\u623f\u200b\u5185\u200b\u90fd\u200b\u85cf\u6709\u200b\u4e00\u5b9a\u200b\u7684\u200b\u73b0\u91d1\u200b\uff0c\u200b\u5f71\u54cd\u200b\u4f60\u200b\u5077\u7a83\u200b\u7684\u200b\u552f\u4e00\u200b\u5236\u7ea6\u200b\u56e0\u7d20\u200b\u5c31\u662f\u200b\u76f8\u90bb\u200b\u7684\u200b\u623f\u5c4b\u200b\u88c5\u6709\u200b\u76f8\u4e92\u200b\u8fde\u901a\u200b\u7684\u200b\u9632\u76d7\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u5982\u679c\u200b\u4e24\u95f4\u200b\u76f8\u90bb\u200b\u7684\u200b\u623f\u5c4b\u200b\u5728\u200b\u540c\u4e00\u200b\u665a\u4e0a\u200b\u88ab\u200b\u5c0f\u5077\u200b\u95ef\u5165\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u62a5\u8b66\u200b\u3002</p> <p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u4ee3\u8868\u200b\u6bcf\u4e2a\u200b\u623f\u5c4b\u200b\u5b58\u653e\u200b\u91d1\u989d\u200b\u7684\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u4f60\u200b \u200b\u4e0d\u200b\u89e6\u52a8\u200b\u8b66\u62a5\u200b\u88c5\u7f6e\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b \uff0c\u200b\u4e00\u591c\u200b\u4e4b\u5185\u200b\u80fd\u591f\u200b\u5077\u7a83\u200b\u5230\u200b\u7684\u200b\u6700\u9ad8\u200b\u91d1\u989d\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a[1,2,3,1]\n\u200b\u8f93\u51fa\u200b\uff1a4\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5077\u7a83\u200b 1 \u200b\u53f7\u200b\u623f\u5c4b\u200b (\u200b\u91d1\u989d\u200b = 1) \uff0c\u200b\u7136\u540e\u200b\u5077\u7a83\u200b 3 \u200b\u53f7\u200b\u623f\u5c4b\u200b (\u200b\u91d1\u989d\u200b = 3)\u3002\n\u00a0    \u200b\u5077\u7a83\u200b\u5230\u200b\u7684\u200b\u6700\u9ad8\u200b\u91d1\u989d\u200b = 1 + 3 = 4 \u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a[2,7,9,3,1]\n\u200b\u8f93\u51fa\u200b\uff1a12\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5077\u7a83\u200b 1 \u200b\u53f7\u200b\u623f\u5c4b\u200b (\u200b\u91d1\u989d\u200b = 2), \u200b\u5077\u7a83\u200b 3 \u200b\u53f7\u200b\u623f\u5c4b\u200b (\u200b\u91d1\u989d\u200b = 9)\uff0c\u200b\u63a5\u7740\u200b\u5077\u7a83\u200b 5 \u200b\u53f7\u200b\u623f\u5c4b\u200b (\u200b\u91d1\u989d\u200b = 1)\u3002\n\u00a0    \u200b\u5077\u7a83\u200b\u5230\u200b\u7684\u200b\u6700\u9ad8\u200b\u91d1\u989d\u200b = 2 + 9 + 1 = 12 \u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <pre><code>0 &lt;= nums.length &lt;= 100\n0 &lt;= nums[i] &lt;= 400\n</code></pre> <p>Reference</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/198/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\uff0c\u200b\u4ee4\u200b\\(f(i)\\)\u200b\u8868\u793a\u200b\u524d\u200b\\(i\\)\u200b\u4e2a\u200b\u623f\u95f4\u200b\u6240\u200b\u80fd\u200b\u83b7\u53d6\u200b\u7684\u200b\u6700\u5927\u200b\u91d1\u989d\u200b\uff0c\u200b\u5219\u200b\u6709\u200b\uff1a</p> \\[ f(i + 1) = \\max\\left\\{f(i), a_{i + 1} + f(i - 1)\\right\\} \\] <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>#define MAX(x, y) (x &gt; y ? x : y)\nint rob(int* nums, int numsSize){\n    if (numsSize &lt; 2)\n        return numsSize ? *nums : 0;\n    int *dp = (int *)memset(malloc(sizeof(int) * (numsSize)), 0, sizeof(int) * (numsSize)), i = 0;\n    dp[0] = *nums;\n    dp[1] = MAX(nums[0], nums[1]);\n    for (i = 2; i &lt; numsSize; i++)\n        dp[i] = MAX(dp[i-2] + nums[i], dp[i-1]);\n    return dp[numsSize-1];\n}\n</code></pre>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/2/","title":"2. \u200b\u4e24\u200b\u6570\u200b\u76f8\u52a0","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u9012\u5f52","\u94fe\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/2/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e24\u4e2a\u200b\u00a0\u200b\u975e\u7a7a\u200b \u200b\u7684\u200b\u94fe\u8868\u200b\uff0c\u200b\u8868\u793a\u200b\u4e24\u4e2a\u200b\u975e\u8d1f\u200b\u7684\u200b\u6574\u6570\u200b\u3002\u200b\u5b83\u4eec\u200b\u6bcf\u4f4d\u200b\u6570\u5b57\u200b\u90fd\u200b\u662f\u200b\u6309\u7167\u200b\u00a0\u200b\u9006\u5e8f\u200b\u00a0\u200b\u7684\u200b\u65b9\u5f0f\u200b\u5b58\u50a8\u200b\u7684\u200b\uff0c\u200b\u5e76\u4e14\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\u53ea\u80fd\u200b\u5b58\u50a8\u200b\u00a0\u200b\u4e00\u4f4d\u200b\u00a0\u200b\u6570\u5b57\u200b\u3002</p> <p>\u200b\u8bf7\u200b\u4f60\u200b\u5c06\u200b\u4e24\u4e2a\u200b\u6570\u200b\u76f8\u52a0\u200b\uff0c\u200b\u5e76\u200b\u4ee5\u200b\u76f8\u540c\u200b\u5f62\u5f0f\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u8868\u793a\u200b\u548c\u200b\u7684\u200b\u94fe\u8868\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u9664\u4e86\u200b\u6570\u5b57\u200b <code>0</code> \u200b\u4e4b\u5916\u200b\uff0c\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u6570\u90fd\u200b\u4e0d\u4f1a\u200b\u4ee5\u200b <code>0</code>\u00a0\u200b\u5f00\u5934\u200b\u3002</p> <p></p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1al1 = [2,4,3], l2 = [5,6,4]\n\u200b\u8f93\u51fa\u200b\uff1a[7,0,8]\n\u200b\u89e3\u91ca\u200b\uff1a342 + 465 = 807.\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1al1 = [0], l2 = [0]\n\u200b\u8f93\u51fa\u200b\uff1a[0]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1al1 = [9,9,9,9,9,9,9], l2 = [9,9,9,9]\n\u200b\u8f93\u51fa\u200b\uff1a[8,9,9,9,0,0,0,1]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u6bcf\u4e2a\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9\u200b\u6570\u5728\u200b\u8303\u56f4\u200b <code>[1, 100]</code> \u200b\u5185\u200b</li> <li><code>0 &lt;= Node.val &lt;= 9</code></li> <li>\u200b\u9898\u76ee\u200b\u6570\u636e\u200b\u4fdd\u8bc1\u200b\u5217\u8868\u200b\u8868\u793a\u200b\u7684\u200b\u6570\u5b57\u200b\u4e0d\u200b\u542b\u200b\u524d\u5bfc\u200b\u96f6\u200b</li> </ul>","tags":["\u9012\u5f52","\u94fe\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/2/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u9012\u5f52\u200b\u7684\u200b\u601d\u60f3\u200b\uff0c\u200b\u5c06\u200b\u8ba1\u7b97\u200b\u52a0\u6cd5\u200b\u7684\u200b\u95ee\u9898\u200b\u5206\u89e3\u200b\u4e3a\u200b\u8ba1\u7b97\u200b\u6700\u540e\u200b\u4e00\u4f4d\u200b\u76f8\u52a0\u200b\u548c\u200b\u8ba1\u7b97\u200b\u5269\u4f59\u200b\u90e8\u5206\u200b\u76f8\u52a0\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u95ee\u9898\u200b\u3002</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\nstruct ListNode *getNode(int val, struct ListNode *next)\n{\n    struct ListNode *ret = (struct ListNode *)malloc(sizeof(struct ListNode));\n    ret-&gt;val = val;\n    ret-&gt;next = next;\n    return ret;\n}\nstruct ListNode *recur(struct ListNode *l1, struct ListNode *l2, int flag)\n{\n    if (!l1)\n        return flag ? recur(getNode(1, NULL), l2, 0) : l2;\n    if (!l2)\n        return flag ? recur(getNode(1, NULL), l1, 0) : l1;\n    int ret = l1-&gt;val + l2-&gt;val + (flag ? 1 : 0);\n    return getNode(ret - (ret &gt;= 10 ? 10 : 0), recur(l1-&gt;next, l2-&gt;next, ret &gt;= 10));\n}\nstruct ListNode* addTwoNumbers(struct ListNode* l1, struct ListNode* l2){\n    return recur(l1, l2, 0);\n}\n</code></pre> <p>Reference</p>","tags":["\u9012\u5f52","\u94fe\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/20/","title":"20. \u200b\u6709\u6548\u200b\u7684\u200b\u62ec\u53f7","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6808","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/20/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u53ea\u200b\u5305\u62ec\u200b <code>'('</code>\uff0c<code>')'</code>\uff0c<code>'{'</code>\uff0c<code>'}'</code>\uff0c<code>'['</code>\uff0c<code>']'</code>\u00a0\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b <code>s</code> \uff0c\u200b\u5224\u65ad\u200b\u5b57\u7b26\u4e32\u200b\u662f\u5426\u200b\u6709\u6548\u200b\u3002</p> <p>\u200b\u6709\u6548\u200b\u5b57\u7b26\u4e32\u200b\u9700\u200b\u6ee1\u8db3\u200b\uff1a</p> <ol> <li>\u200b\u5de6\u200b\u62ec\u53f7\u200b\u5fc5\u987b\u200b\u7528\u200b\u76f8\u540c\u200b\u7c7b\u578b\u200b\u7684\u200b\u53f3\u200b\u62ec\u53f7\u200b\u95ed\u5408\u200b\u3002</li> <li>\u200b\u5de6\u200b\u62ec\u53f7\u200b\u5fc5\u987b\u200b\u4ee5\u200b\u6b63\u786e\u200b\u7684\u200b\u987a\u5e8f\u200b\u95ed\u5408\u200b\u3002</li> </ol> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"()\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"()[]{}\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a03\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"(]\"\n\u200b\u8f93\u51fa\u200b\uff1afalse\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a04\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"([)]\"\n\u200b\u8f93\u51fa\u200b\uff1afalse\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a05\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"{[]}\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>Reference</p>","tags":["\u6808","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/20/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u6808\u200b\u6a21\u62df\u200b\u5339\u914d\u200b\u8fc7\u7a0b\u200b</p> <ul> <li>\u200b\u9047\u5230\u200b\u5de6\u200b\u62ec\u53f7\u200b\u65f6\u5165\u200b\u6808\u200b\uff1b</li> <li>\u200b\u9047\u5230\u200b\u53f3\u200b\u62ec\u53f7\u200b\u65f6\u200b\uff0c\u200b\u68c0\u67e5\u200b\u6808\u9876\u200b\u62ec\u53f7\u200b\u662f\u5426\u200b\u4e0e\u200b\u53f3\u200b\u62ec\u53f7\u200b\u5339\u914d\u200b\uff0c\u200b\u82e5\u200b\u5339\u914d\u200b\u5219\u200b\u51fa\u6808\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>false</code>\uff1b</li> <li>\u200b\u5b8c\u6210\u200b\u5339\u914d\u200b\u8fc7\u7a0b\u200b\u540e\u200b\u68c0\u67e5\u200b\u6808\u200b\u662f\u5426\u200b\u4e3a\u7a7a\u200b\uff0c\u200b\u82e5\u4e3a\u200b\u7a7a\u200b\u8fd4\u56de\u200b<code>true</code>\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>false</code>\u3002</li> </ul> <pre><code>bool isValid(char * s){\n    int maxLen = strlen(s), top = 0;\n    char *holding = malloc(sizeof(char) * maxLen), *cur = s;\n    while(*cur)\n    {\n        switch(*cur)\n        {\n            case '(':\n            case '[':\n            case '{':\n                holding[top] = *cur;\n                top++;\n                break;\n            case ')':\n                if (top == 0 || holding[top - 1] != '(')\n                    return false;\n                top--;\n                break;\n            case ']':\n            case '}':\n                if (top == 0 || holding[top - 1] != (*cur) - 2)\n                    return false;\n                top--;\n                break;\n        }\n        cur++;\n    }\n    return !top;\n}\n</code></pre>","tags":["\u6808","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/204/","title":"204. \u200b\u8ba1\u6570\u200b\u8d28\u6570","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u54c8\u5e0c\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/204/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7edf\u8ba1\u200b\u6240\u6709\u200b\u5c0f\u4e8e\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b n \u200b\u7684\u200b\u8d28\u6570\u200b\u7684\u200b\u6570\u91cf\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 10\n\u200b\u8f93\u51fa\u200b: 4\n\u200b\u89e3\u91ca\u200b: \u200b\u5c0f\u4e8e\u200b 10 \u200b\u7684\u200b\u8d28\u6570\u200b\u4e00\u5171\u200b\u6709\u200b 4 \u200b\u4e2a\u200b, \u200b\u5b83\u4eec\u200b\u662f\u200b 2, 3, 5, 7 \u3002\n</code></pre> <p>Reference</p>","tags":["\u54c8\u5e0c\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/204/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4ece\u200b2\u200b\u5f00\u59cb\u200b\uff0c\u200b\u5728\u200b\u6570\u5217\u200b\\(2, 3, \\cdots, n\\)\u200b\u4e2d\u5212\u200b\u53bb\u200b\u6240\u6709\u200b2\u200b\u7684\u200b\u500d\u6570\u200b\uff0c\u200b\u5269\u4e0b\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u6ca1\u6709\u200b\u88ab\u5212\u200b\u53bb\u200b\u7684\u200b\u6570\u5b57\u200b3\u200b\u5373\u200b\u4e3a\u200b\u8d28\u6570\u200b\uff0c\u200b\u518d\u200b\u82b1\u65d7\u200b\u6240\u6709\u200b3\u200b\u7684\u200b\u500d\u6570\u200b\uff0c\u200b\u5269\u4e0b\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u6ca1\u6709\u200b\u88ab\u5212\u200b\u53bb\u200b\u7684\u200b\u6570\u5b57\u200b5\u200b\u5373\u200b\u4e3a\u200b\u8d28\u6570\u200b\u3002\u200b\u91cd\u590d\u200b\u5982\u200b\u4e0a\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u76f4\u5230\u200b\u68c0\u67e5\u200b\u5b8c\u200b\u6240\u6709\u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b\\(\\sqrt n\\)\u200b\u7684\u200b\u6574\u6570\u200b\uff0c\u200b\u5373\u200b\u5b8c\u6210\u200b\u3002</p> <p>\u200b\u6570\u5b57\u200b\u662f\u5426\u200b\u5212\u53bb\u200b\u7684\u200b\u72b6\u6001\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\u63a7\u5236\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200b\u4e00\u4e2a\u200b\u7c7b\u578b\u200b\u4e2d\u200b\u7684\u200b1\u200b\u4f4d\u200b\u8868\u793a\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int countPrimes(int n)\n{\n    unsigned char *map = memset(malloc(sizeof(unsigned char) * ((n + 1 &gt;&gt; 3) + 1)), 0x55, sizeof(unsigned char) * ((n + 1 &gt;&gt; 3) + 1));\n    int i = 0, j, ret = 0, sq = (int)sqrt(n), increment = 1, start = 2;\n    *map = 0x51;\n    for (i = 3; i &lt;= sq; i++)\n        if (!(map[i &gt;&gt; 3] &amp; (unsigned char)1 &lt;&lt; (i &amp; 7)))\n            for (j = i * 3; j &lt;= n; j += i &lt;&lt; 1)\n                map[j &gt;&gt; 3] |= (unsigned char)1 &lt;&lt; (j &amp; 7);\n    for (i = 2; i &lt; n; i++)\n        if (!(map[i &gt;&gt; 3] &amp; (unsigned char)1 &lt;&lt; (i &amp; 7)))\n            ret++;\n    return ret;\n}\n</code></pre>","tags":["\u54c8\u5e0c\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/206/","title":"206. \u200b\u53cd\u8f6c\u200b\u94fe\u8868","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/206/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u53cd\u8f6c\u200b\u4e00\u4e2a\u200b\u5355\u94fe\u200b\u8868\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a <pre><code>\u200b\u8f93\u5165\u200b: 1-&gt;2-&gt;3-&gt;4-&gt;5-&gt;NULL\n\u200b\u8f93\u51fa\u200b: 5-&gt;4-&gt;3-&gt;2-&gt;1-&gt;NULL\n</code></pre> \u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u8fed\u4ee3\u200b\u6216\u200b\u9012\u5f52\u200b\u5730\u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b\u3002\u200b\u4f60\u200b\u80fd\u5426\u200b\u7528\u200b\u4e24\u79cd\u200b\u65b9\u6cd5\u200b\u89e3\u51b3\u200b\u8fd9\u9053\u9898\u200b\uff1f</p> <p>Reference</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/206/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u53ea\u200b\u9700\u8981\u200b\u5c06\u200b\u94fe\u8868\u200b\u7ed3\u6784\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u6307\u9488\u200b\u53cd\u8f6c\u200b\uff0c\u200b\u5373\u200b\u6307\u5411\u200b\u524d\u200b\u4e00\u200b\u8282\u70b9\u200b\u7684\u200b\u6307\u9488\u200b\u73b0\u5728\u200b\u6307\u5411\u200b\u540e\u200b\u4e00\u200b\u8282\u70b9\u200b\u3002\u200b\u5b9a\u4e49\u200b\u4e24\u4e2a\u200b\u6307\u9488\u200b\u6307\u5411\u200b\u9700\u8981\u200b\u53cd\u8f6c\u200b\u90e8\u5206\u200b\u7684\u200b\u4e24\u4e2a\u200b\u8282\u70b9\u200b\uff0c\u200b\u5c06\u200b\u540e\u200b\u4e00\u200b\u6307\u9488\u200b\u7684\u200b<code>next</code>\u200b\u57df\u200b\u6307\u5411\u200b\u524d\u200b\u4e00\u200b\u6307\u9488\u200b\u3002\u200b\u6539\u200b\u53cd\u8f6c\u200b\u64cd\u4f5c\u200b\u7834\u574f\u200b\u4e86\u200b\u540e\u200b\u4e00\u200b\u6307\u9488\u200b\u539f\u6709\u200b\u7684\u200b<code>next</code>\u200b\u57df\u200b\uff0c\u200b\u9700\u8981\u200b\u4e00\u4e2a\u200b\u989d\u5916\u200b\u7684\u200b\u6307\u9488\u200b\u7528\u4e8e\u200b\u5411\u524d\u200b\u79fb\u52a8\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\n\nstruct ListNode* reverseList(struct ListNode* head){\n    if (!head || !(head-&gt;next))\n        return head;\n    struct ListNode *cur1 = head, *cur2 = head-&gt;next, *cur3 = cur2-&gt;next;\n    head-&gt;next = NULL;\n    while (cur3)\n    {\n        cur2-&gt;next = cur1;\n        cur1 = cur2;\n        cur2 = cur3;\n        cur3 = cur3-&gt;next;\n    }\n    cur2-&gt;next = cur1;\n    return cur2;\n}\n</code></pre>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/21/","title":"21. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u94fe\u8868","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u9012\u5f52","\u94fe\u8868"]},{"location":"coding/leetcode/21/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5c06\u200b\u4e24\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868\u200b\u5408\u5e76\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b \u200b\u5347\u5e8f\u200b \u200b\u94fe\u8868\u200b\u5e76\u200b\u8fd4\u56de\u200b\u3002\u200b\u65b0\u200b\u94fe\u8868\u200b\u662f\u200b\u901a\u8fc7\u200b\u62fc\u63a5\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u4e24\u4e2a\u200b\u94fe\u8868\u200b\u7684\u200b\u6240\u6709\u200b\u8282\u70b9\u200b\u7ec4\u6210\u200b\u7684\u200b\u3002\u00a0</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a1-&gt;2-&gt;4, 1-&gt;3-&gt;4\n\u200b\u8f93\u51fa\u200b\uff1a1-&gt;1-&gt;2-&gt;3-&gt;4-&gt;4\n</code></pre> <p>Reference</p>","tags":["\u9012\u5f52","\u94fe\u8868"]},{"location":"coding/leetcode/21/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6bd4\u8f83\u200b\u4e24\u4e2a\u200b\u94fe\u8868\u200b\u7684\u200b\u5934\u200b\u8282\u70b9\u200b\uff0c\u200b\u5c06\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u8282\u70b9\u200b\u5e76\u5165\u200b\u7ed3\u679c\u200b\u94fe\u8868\u200b\u4e2d\u200b\uff0c\u200b\u540c\u65f6\u200b\u94fe\u8868\u200b\u5934\u200b\u540e\u79fb\u200b\uff0c\u200b\u9012\u5f52\u200b\u5904\u7406\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\nstruct ListNode* mergeTwoLists(struct ListNode* l1, struct ListNode* l2){\n    if (!l1)\n        return l2;\n    if (!l2)\n        return l1;\n    if (l1-&gt;val &lt; l2-&gt;val)\n    {\n        l1-&gt;next = mergeTwoLists(l1-&gt;next, l2);\n        return l1;\n    }\n    else\n    {\n        l2-&gt;next = mergeTwoLists(l1, l2-&gt;next);\n        return l2;\n    }\n}\n</code></pre>","tags":["\u9012\u5f52","\u94fe\u8868"]},{"location":"coding/leetcode/217/","title":"217. \u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/217/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5224\u65ad\u200b\u662f\u5426\u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u4efb\u610f\u200b\u4e00\u503c\u200b\u5728\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u81f3\u5c11\u200b\u4e24\u6b21\u200b\uff0c\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b<code>true</code>\u3002\u200b\u5982\u679c\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u90fd\u200b\u4e0d\u200b\u76f8\u540c\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>false</code>\u3002\u00a0</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,2,3,1]\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,2,3,4]\n\u200b\u8f93\u51fa\u200b: false\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a03\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,1,1,3,3,4,3,2,4,2]\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/217/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5c06\u200b\u6570\u7ec4\u200b\u6392\u5e8f\u200b\u540e\u200b\u68c0\u67e5\u200b\u662f\u5426\u200b\u6709\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int comp(const void *a, const void *b)\n{\n    return *(int*)a - *(int*)b;\n}\nbool containsDuplicate(int* nums, int numsSize){\n    if (numsSize &lt; 2)\n        return false;\n    qsort(nums, numsSize, sizeof(int), comp);\n    int i = 0;\n    for (i = 1; i&lt;numsSize; i++)\n        if (nums[i] == nums[i - 1])\n            return true;\n    return false;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/22/","title":"22. \u200b\u62ec\u53f7\u200b\u751f\u6210","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u5b57\u7b26\u4e32","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/22/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u6570\u5b57\u200b <code>n</code>\u00a0\u200b\u4ee3\u8868\u200b\u751f\u6210\u200b\u62ec\u53f7\u200b\u7684\u200b\u5bf9\u6570\u200b\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u7528\u4e8e\u200b\u80fd\u591f\u200b\u751f\u6210\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u7684\u200b\u5e76\u4e14\u200b \u200b\u6709\u6548\u200b\u7684\u200b \u200b\u62ec\u53f7\u200b\u7ec4\u5408\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1an = 3\n\u200b\u8f93\u51fa\u200b\uff1a[\"((()))\",\"(()())\",\"(())()\",\"()(())\",\"()()()\"]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1an = 1\n\u200b\u8f93\u51fa\u200b\uff1a[\"()\"]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= n &lt;= 8</code></li> </ul> <p>Reference</p>","tags":["\u5b57\u7b26\u4e32","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/22/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u56de\u6eaf\u200b\u7b97\u6cd5\u200b</p> <ul> <li>\u200b\u5f53\u200b\u5de6\u200b\u62ec\u53f7\u200b\u6570\u91cf\u200b\u5c0f\u4e8e\u200b\u62ec\u53f7\u200b\u5bf9\u6570\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6dfb\u52a0\u200b\u5de6\u200b\u62ec\u53f7\u200b</li> <li>\u200b\u5f53\u53f3\u200b\u62ec\u53f7\u200b\u6570\u91cf\u200b\u5c0f\u4e8e\u200b\u5de6\u200b\u62ec\u53f7\u200b\u6570\u91cf\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6dfb\u52a0\u200b\u53f3\u200b\u62ec\u53f7\u200b</li> </ul> <p>\u200b\u6309\u7167\u200b\u5982\u200b\u4e0a\u200b\u65b9\u6cd5\u200b\u751f\u6210\u200b\u7684\u200b\u62ec\u53f7\u200b\u7ec4\u5408\u200b\u53ef\u4ee5\u200b\u4fdd\u8bc1\u200b\u4e00\u5b9a\u200b\u662f\u200b\u6709\u6548\u200b\u7684\u200b\u3002</p> <pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\n\nvoid dfs(char *currentStr, int n, int left, int right, char **ret, int *returnSize)\n{\n    if (left == n &amp;&amp; right == n)\n    {\n        ret[*returnSize] = (char *)malloc(sizeof(char) * (2 * n + 1));\n        strcpy(ret[*returnSize], currentStr);\n        *returnSize += 1;\n        return;\n    }\n    if (left &lt; n)\n    {\n        currentStr[left + right] = '(';\n        dfs(currentStr, n, left + 1, right, ret, returnSize);\n    }\n    if (right &lt; left)\n    {\n        currentStr[left + right] = ')';\n        dfs(currentStr, n, left, right + 1, ret, returnSize);\n    }\n}\nchar ** generateParenthesis(int n, int* returnSize){\n    const int maxLen = 10000;\n    *returnSize = 0;\n    char **ret = (char **)malloc(sizeof(char *) * maxLen),\n         *buffer = (char *)memset(malloc(sizeof(char) * (2 * n + 1)), 0, sizeof(char) * (2 * n + 1));\n    dfs(buffer, n, 0, 0, ret, returnSize);\n    return ret;\n}\n</code></pre>","tags":["\u5b57\u7b26\u4e32","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/23/","title":"23. \u200b\u5408\u5e76\u200bK\u200b\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u5806","\u94fe\u8868","\u5206\u6cbb\u7b97\u6cd5"]},{"location":"coding/leetcode/23/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b\u6570\u7ec4\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u94fe\u8868\u200b\u90fd\u200b\u5df2\u7ecf\u200b\u6309\u200b\u5347\u5e8f\u200b\u6392\u5217\u200b\u3002</p> <p>\u200b\u8bf7\u200b\u4f60\u200b\u5c06\u200b\u6240\u6709\u200b\u94fe\u8868\u200b\u5408\u5e76\u200b\u5230\u200b\u4e00\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868\u200b\u4e2d\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5408\u5e76\u200b\u540e\u200b\u7684\u200b\u94fe\u8868\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1alists = [[1,4,5],[1,3,4],[2,6]]\n\u200b\u8f93\u51fa\u200b\uff1a[1,1,2,3,4,4,5,6]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u94fe\u8868\u200b\u6570\u7ec4\u200b\u5982\u4e0b\u200b\uff1a\n[\n  1-&gt;4-&gt;5,\n  1-&gt;3-&gt;4,\n  2-&gt;6\n]\n\u200b\u5c06\u200b\u5b83\u4eec\u200b\u5408\u5e76\u200b\u5230\u200b\u4e00\u4e2a\u200b\u6709\u5e8f\u200b\u94fe\u8868\u200b\u4e2d\u200b\u5f97\u5230\u200b\u3002\n1-&gt;1-&gt;2-&gt;3-&gt;4-&gt;4-&gt;5-&gt;6\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1alists = []\n\u200b\u8f93\u51fa\u200b\uff1a[]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1alists = [[]]\n\u200b\u8f93\u51fa\u200b\uff1a[]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>k == lists.length</code></li> <li><code>0 &lt;= k &lt;= 10^4</code></li> <li><code>0 &lt;= lists[i].length &lt;= 500</code></li> <li><code>-10^4 &lt;= lists[i][j] &lt;= 10^4</code></li> <li><code>lists[i]</code> \u200b\u6309\u200b \u200b\u5347\u5e8f\u200b \u200b\u6392\u5217\u200b</li> <li><code>lists[i].length</code> \u200b\u7684\u200b\u603b\u548c\u200b\u4e0d\u200b\u8d85\u8fc7\u200b <code>10^4</code></li> </ul> <p>Reference</p>","tags":["\u5806","\u94fe\u8868","\u5206\u6cbb\u7b97\u6cd5"]},{"location":"coding/leetcode/23/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u6700\u200b\u76f4\u63a5\u200b\u7684\u200b\u5408\u5e76\u200b\u601d\u8def\u200b\u662f\u200b\u6309\u7167\u200b\u987a\u5e8f\u200b\u5bf9\u200b\u94fe\u8868\u200b\u8fdb\u884c\u200b\u5408\u5e76\u200b\u3002\u200b\u8003\u8651\u200b\u5230\u200b\u5408\u5e76\u200b\u540e\u200b\u94fe\u8868\u200b\u53d8\u200b\u957f\u200b\uff0c\u200b\u5e73\u5747\u200b\u6bcf\u6b21\u200b\u5408\u5e76\u200b\u9700\u8981\u200b\u82b1\u8d39\u200b\\(\\mathcal O(MN)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\uff0c\u200b\u5171\u9700\u200b\u5408\u5e76\u200b\\(\\mathcal O(M)\\)\u200b\u6b21\u200b\uff0c\u200b\u56e0\u6b64\u200b\u603b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(M^2N)\\)</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\n\nstruct ListNode* mergeLists(struct ListNode* l1, struct ListNode* l2);\nstruct ListNode* mergeKLists(struct ListNode** lists, int listsSize){\n    if (!listsSize)\n        return NULL;\n    struct ListNode *head = lists[0];\n    for (int i = 1; i&lt;listsSize; i++)\n    {\n        if (!head)\n        {\n            head = lists[i];\n            continue;\n        }\n        head = mergeLists(head, lists[i]);\n    }\n    return head;\n}\n\nstruct ListNode* mergeLists(struct ListNode* l1, struct ListNode* l2){\n    if (!l1)\n        return l2;\n    if (!l2)\n        return l1;\n    if (l1-&gt;val &lt; l2-&gt;val)\n    {\n        l1-&gt;next = mergeLists(l1-&gt;next, l2);\n        return l1;\n    }\n    else\n    {\n        l2-&gt;next = mergeLists(l1, l2-&gt;next);\n        return l2;\n    }\n}\n</code></pre> <p>\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b\u4e24\u200b\u4e24\u200b\u5408\u5e76\u200b\u7684\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5408\u5e76\u200b\u64cd\u4f5c\u200b\u7684\u200b\u5e73\u5747\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u964d\u4f4e\u200b\u81f3\u200b\\(\\mathcal O(N\\log M)\\)\uff0c\u200b\u603b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u53d8\u4e3a\u200b\\(\\mathcal O(MN\\log M)\\)\u3002</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\n\nstruct ListNode* mergeLists(struct ListNode* l1, struct ListNode* l2);\nstruct ListNode* mergeKLists(struct ListNode** lists, int listsSize){\n    if (!listsSize)\n        return NULL;\n    else if (listsSize == 1)\n        return lists[0];\n    else if (listsSize == 2)\n        return mergeLists(lists[0], lists[1]);\n    struct ListNode *head = lists[0];\n    int step = (listsSize + 1) &gt;&gt; 1;\n    for (int i = 0; i &lt; step; i++)\n    {\n        if (i + step &lt; listsSize)\n            lists[i] = mergeLists(lists[i], lists[i + step]);\n    }\n    return mergeKLists(lists, step);\n}\n\nstruct ListNode* mergeLists(struct ListNode* l1, struct ListNode* l2){\n    if (!l1)\n        return l2;\n    if (!l2)\n        return l1;\n    if (l1-&gt;val &lt; l2-&gt;val)\n    {\n        l1-&gt;next = mergeLists(l1-&gt;next, l2);\n        return l1;\n    }\n    else\n    {\n        l2-&gt;next = mergeLists(l1, l2-&gt;next);\n        return l2;\n    }\n}\n</code></pre>","tags":["\u5806","\u94fe\u8868","\u5206\u6cbb\u7b97\u6cd5"]},{"location":"coding/leetcode/234/","title":"234. \u200b\u56de\u6587\u200b\u94fe\u8868","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/234/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u8bf7\u200b\u5224\u65ad\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b\u662f\u5426\u200b\u4e3a\u200b\u56de\u6587\u200b\u94fe\u8868\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 1-&gt;2\n\u200b\u8f93\u51fa\u200b: false\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 1-&gt;2-&gt;2-&gt;1\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u80fd\u5426\u200b\u7528\u200b\\(O(n)\\)\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u548c\u200b\\(O(1)\\)\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u89e3\u51b3\u200b\u6b64\u9898\u200b\uff1f</p> <p>Reference</p>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/234/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u53cd\u5e8f\u200b\u6784\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u94fe\u8868\u200b\uff0c\u200b\u6bd4\u8f83\u200b\u4e24\u4e2a\u200b\u94fe\u8868\u200b\u662f\u5426\u200b\u76f8\u540c\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\nstruct ListNode *getNode(int val, struct ListNode *next)\n{\n    struct ListNode *ret = (struct ListNode *)malloc(sizeof(struct ListNode));\n    ret-&gt;val = val;\n    ret-&gt;next = next;\n    return ret;\n}\n\nbool isPalindrome(struct ListNode* head){\n    int len = 0;\n    struct ListNode *reversed = NULL, *cur1 = head, *cur2 = NULL;\n    while(cur1 != NULL)\n    {\n        reversed = getNode(cur1-&gt;val, reversed);\n        cur1 = cur1-&gt;next;\n        len++;\n    }\n    cur1 = head;\n    cur2 = reversed;\n    for (int i = 0; i &lt; len / 2; i++)\n    {\n        if (cur1-&gt;val != cur2-&gt;val)\n            return false;\n        cur1 = cur1-&gt;next;\n        cur2 = cur2-&gt;next;\n    }\n    return true;\n}\n</code></pre>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/237/","title":"237. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/237/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u8bf7\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u4f7f\u200b\u5176\u200b\u53ef\u4ee5\u200b\u5220\u9664\u200b\u67d0\u4e2a\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7ed9\u5b9a\u200b\u7684\u200b\uff08\u200b\u975e\u200b\u672b\u5c3e\u200b\uff09\u200b\u8282\u70b9\u200b\u3002\u200b\u4f20\u5165\u200b\u51fd\u6570\u200b\u7684\u200b\u552f\u4e00\u200b\u53c2\u6570\u200b\u4e3a\u200b \u200b\u8981\u200b\u88ab\u200b\u5220\u9664\u200b\u7684\u200b\u8282\u70b9\u200b \u3002</p> <p>\u200b\u73b0\u6709\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b --\u00a0head =\u00a0[4,5,1,9]\uff0c\u200b\u5b83\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b:</p> <p></p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [4,5,1,9], node = 5\n\u200b\u8f93\u51fa\u200b\uff1a[4,1,9]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u7ed9\u5b9a\u200b\u4f60\u200b\u94fe\u8868\u200b\u4e2d\u503c\u200b\u4e3a\u200b\u00a05\u00a0\u200b\u7684\u200b\u7b2c\u4e8c\u4e2a\u200b\u8282\u70b9\u200b\uff0c\u200b\u90a3\u4e48\u200b\u5728\u200b\u8c03\u7528\u200b\u4e86\u200b\u4f60\u200b\u7684\u200b\u51fd\u6570\u200b\u4e4b\u540e\u200b\uff0c\u200b\u8be5\u200b\u94fe\u8868\u200b\u5e94\u53d8\u200b\u4e3a\u200b 4 -&gt; 1 -&gt; 9.\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [4,5,1,9], node = 1\n\u200b\u8f93\u51fa\u200b\uff1a[4,5,9]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u7ed9\u5b9a\u200b\u4f60\u200b\u94fe\u8868\u200b\u4e2d\u503c\u200b\u4e3a\u200b\u00a01\u00a0\u200b\u7684\u200b\u7b2c\u4e09\u4e2a\u200b\u8282\u70b9\u200b\uff0c\u200b\u90a3\u4e48\u200b\u5728\u200b\u8c03\u7528\u200b\u4e86\u200b\u4f60\u200b\u7684\u200b\u51fd\u6570\u200b\u4e4b\u540e\u200b\uff0c\u200b\u8be5\u200b\u94fe\u8868\u200b\u5e94\u53d8\u200b\u4e3a\u200b 4 -&gt; 5 -&gt; 9.\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u94fe\u8868\u200b\u81f3\u5c11\u200b\u5305\u542b\u200b\u4e24\u4e2a\u200b\u8282\u70b9\u200b\u3002</li> <li>\u200b\u94fe\u8868\u200b\u4e2d\u200b\u6240\u6709\u200b\u8282\u70b9\u200b\u7684\u200b\u503c\u200b\u90fd\u200b\u662f\u200b\u552f\u4e00\u200b\u7684\u200b\u3002</li> <li>\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u8282\u70b9\u200b\u4e3a\u200b\u975e\u200b\u672b\u5c3e\u200b\u8282\u70b9\u200b\u5e76\u4e14\u200b\u4e00\u5b9a\u200b\u662f\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u200b\u6709\u6548\u200b\u8282\u70b9\u200b\u3002</li> <li>\u200b\u4e0d\u8981\u200b\u4ece\u200b\u4f60\u200b\u7684\u200b\u51fd\u6570\u200b\u4e2d\u200b\u8fd4\u56de\u200b\u4efb\u4f55\u200b\u7ed3\u679c\u200b\u3002</li> </ul> <p>Reference</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/237/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5c06\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u7684\u200b\u503c\u200b\u590d\u5236\u5230\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\uff0c\u200b\u7136\u540e\u200b\u5220\u53bb\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u3002\uff08\u200b\u5df2\u77e5\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u4e00\u5b9a\u200b\u4e0d\u662f\u200b\u672b\u5c3e\u200b\u8282\u70b9\u200b\uff09</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>void deleteNode(struct ListNode* node) {\n    node-&gt;val = node-&gt;next-&gt;val;\n    node-&gt;next = node-&gt;next-&gt;next;\n}\n</code></pre>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/24/","title":"24. \u200b\u4e24\u200b\u4e24\u200b\u4ea4\u6362\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u9012\u5f52","\u94fe\u8868"]},{"location":"coding/leetcode/24/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b\uff0c\u200b\u4e24\u200b\u4e24\u200b\u4ea4\u6362\u200b\u5176\u4e2d\u200b\u76f8\u90bb\u200b\u7684\u200b\u8282\u70b9\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b\u4ea4\u6362\u200b\u540e\u200b\u7684\u200b\u94fe\u8868\u200b\u3002</p> <p>\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u53ea\u662f\u200b\u5355\u7eaf\u200b\u7684\u200b\u6539\u53d8\u200b\u8282\u70b9\u200b\u5185\u90e8\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u800c\u662f\u200b\u9700\u8981\u200b\u5b9e\u9645\u200b\u7684\u200b\u8fdb\u884c\u200b\u8282\u70b9\u200b\u4ea4\u6362\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a <pre><code>\u200b\u7ed9\u5b9a\u200b 1-&gt;2-&gt;3-&gt;4, \u200b\u4f60\u200b\u5e94\u8be5\u200b\u8fd4\u56de\u200b 2-&gt;1-&gt;4-&gt;3.\n</code></pre></p> <p>Reference</p>","tags":["\u9012\u5f52","\u94fe\u8868"]},{"location":"coding/leetcode/24/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5148\u200b\u4ea4\u6362\u200b\u524d\u200b\u4e24\u4e2a\u200b\u8282\u70b9\u200b\uff0c\u200b\u518d\u200b\u9012\u5f52\u200b\u5904\u7406\u200b\u540e\u9762\u200b\u7684\u200b\u8282\u70b9\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\n\nstruct ListNode* swapPairs(struct ListNode* head){\n    if (!head || !(head-&gt;next))\n        return head;\n    struct ListNode *temp1 = head-&gt;next, *temp2 = temp1-&gt;next;\n    temp1-&gt;next = head;\n    head-&gt;next = swapPairs(temp2);\n    return temp1;\n}\n</code></pre>","tags":["\u9012\u5f52","\u94fe\u8868"]},{"location":"coding/leetcode/242/","title":"242. \u200b\u6709\u6548\u200b\u7684\u200b\u5b57\u6bcd\u200b\u5f02\u4f4d\u200b\u8bcd","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6392\u5e8f","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/242/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b s \u200b\u548c\u200b t \uff0c\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u6765\u200b\u5224\u65ad\u200b t \u200b\u662f\u5426\u662f\u200b s \u200b\u7684\u200b\u5b57\u6bcd\u200b\u5f02\u4f4d\u200b\u8bcd\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: s = \"anagram\", t = \"nagaram\"\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: s = \"rat\", t = \"car\"\n\u200b\u8f93\u51fa\u200b: false\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u5b57\u7b26\u4e32\u200b\u53ea\u200b\u5305\u542b\u200b\u5c0f\u5199\u5b57\u6bcd\u200b\u3002</p> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u5982\u679c\u200b\u8f93\u5165\u200b\u5b57\u7b26\u4e32\u200b\u5305\u542b\u200b unicode \u200b\u5b57\u7b26\u200b\u600e\u4e48\u529e\u200b\uff1f\u200b\u4f60\u200b\u80fd\u5426\u200b\u8c03\u6574\u200b\u4f60\u200b\u7684\u200b\u89e3\u6cd5\u200b\u6765\u200b\u5e94\u5bf9\u200b\u8fd9\u79cd\u200b\u60c5\u51b5\u200b\uff1f</p> <p>Reference</p>","tags":["\u6392\u5e8f","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/242/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u7edf\u8ba1\u200b\u5e76\u200b\u6bd4\u8f83\u200b\u4e24\u4e2a\u200b\u8f93\u5165\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u5404\u200b\u5b57\u7b26\u200b\u7684\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>bool isAnagram(char * s, char * t){\n    int hashMap1[26] = {0}, hashMap2[26] = {0};\n    char *cur = s;\n    while(*cur)\n    {\n        hashMap1[*cur - 'a']++;\n        cur++;\n    }\n    cur = t;\n    while(*cur)\n    {\n        hashMap2[*cur - 'a']++;\n        cur++;\n    }\n    int i = 0;\n    for (i = 0; i &lt; 26; i++)\n        if (hashMap1[i] != hashMap2[i])\n            return false;\n    return true;\n}\n</code></pre>","tags":["\u6392\u5e8f","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/25/","title":"25. K \u200b\u4e2a\u200b\u4e00\u7ec4\u200b\u7ffb\u8f6c\u200b\u94fe\u8868","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/25/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b\uff0c\u200b\u6bcf\u200b\u00a0k\u00a0\u200b\u4e2a\u200b\u8282\u70b9\u200b\u4e00\u7ec4\u200b\u8fdb\u884c\u200b\u7ffb\u8f6c\u200b\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u8fd4\u56de\u200b\u7ffb\u8f6c\u200b\u540e\u200b\u7684\u200b\u94fe\u8868\u200b\u3002</p> <p>k\u00a0\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6b63\u6574\u6570\u200b\uff0c\u200b\u5b83\u200b\u7684\u200b\u503c\u200b\u5c0f\u4e8e\u200b\u6216\u200b\u7b49\u4e8e\u200b\u94fe\u8868\u200b\u7684\u200b\u957f\u5ea6\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u8282\u70b9\u200b\u603b\u6570\u200b\u4e0d\u662f\u200b\u00a0k\u00a0\u200b\u7684\u200b\u6574\u6570\u500d\u200b\uff0c\u200b\u90a3\u4e48\u200b\u8bf7\u200b\u5c06\u200b\u6700\u540e\u200b\u5269\u4f59\u200b\u7684\u200b\u8282\u70b9\u200b\u4fdd\u6301\u200b\u539f\u6709\u200b\u987a\u5e8f\u200b\u3002</p> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <ul> <li>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u53ea\u200b\u4f7f\u7528\u200b\u5e38\u6570\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\u7684\u200b\u7b97\u6cd5\u200b\u6765\u200b\u89e3\u51b3\u200b\u6b64\u200b\u95ee\u9898\u200b\u5417\u200b\uff1f</li> <li>\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u53ea\u662f\u200b\u5355\u7eaf\u200b\u7684\u200b\u6539\u53d8\u200b\u8282\u70b9\u200b\u5185\u90e8\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u800c\u662f\u200b\u9700\u8981\u200b\u5b9e\u9645\u200b\u8fdb\u884c\u200b\u8282\u70b9\u200b\u4ea4\u6362\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [1,2,3,4,5], k = 2\n\u200b\u8f93\u51fa\u200b\uff1a[2,1,4,3,5]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [1,2,3,4,5], k = 3\n\u200b\u8f93\u51fa\u200b\uff1a[3,2,1,4,5]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [1,2,3,4,5], k = 1\n\u200b\u8f93\u51fa\u200b\uff1a[1,2,3,4,5]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [1], k = 1\n\u200b\u8f93\u51fa\u200b\uff1a[1]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u5217\u8868\u200b\u4e2d\u200b\u8282\u70b9\u200b\u7684\u200b\u6570\u91cf\u200b\u5728\u200b\u8303\u56f4\u200b <code>sz</code> \u200b\u5185\u200b</li> <li><code>1 &lt;= sz &lt;= 5000</code></li> <li><code>0 &lt;= Node.val &lt;= 1000</code></li> <li><code>1 &lt;= k &lt;= sz</code></li> </ul> <p>Reference</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/25/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u904d\u5386\u200b k \u200b\u4e2a\u200b\u5143\u7d20\u200b\u540e\u200b\u5c06\u200b\u94fe\u8868\u200b\u7684\u200b\u7ed3\u5c3e\u200b\u8bbe\u200b\u4e3a\u200b<code>NULL</code>\uff0c\u200b\u6267\u884c\u200b\u53cd\u8f6c\u200b\u64cd\u4f5c\u200b\u540e\u200b\u63a5\u4e0a\u200b\u540e\u9762\u200b\u7684\u200b\u94fe\u8868\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\\(\\mathcal O(1)\\)\u200b\u7684\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b\u3002</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\nstruct ListNode *reverse(struct ListNode *head)\n{\n    struct ListNode *prev = NULL, *cur = head, *next = head-&gt;next;\n    while (next != NULL)\n    {\n        cur-&gt;next = prev;\n        prev = cur;\n        cur = next;\n        next = next-&gt;next;\n    }\n    cur-&gt;next = prev;\n    return cur;\n}\n\nstruct ListNode* reverseKGroup(struct ListNode* head, int k){\n    int i = 0;\n    struct ListNode *ret = NULL, *tempHead = NULL, *cur = head;\n    for (i = 0; i &lt; k - 1 &amp;&amp; cur != NULL; i++)\n        cur = cur-&gt;next;\n    if (cur == NULL)\n        return head;\n    tempHead = cur-&gt;next;\n    cur-&gt;next = NULL;\n    ret = reverse(head);\n    head-&gt;next = reverseKGroup(tempHead, k);\n    return ret;\n}\n</code></pre>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/26/","title":"26. \u200b\u5220\u9664\u200b\u6392\u5e8f\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u91cd\u590d\u200b\u9879","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/26/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6392\u5e8f\u200b\u6570\u7ec4\u200b\uff0c\u200b\u4f60\u200b\u9700\u8981\u200b\u5728\u200b\u539f\u5730\u200b\u5220\u9664\u200b\u91cd\u590d\u200b\u51fa\u73b0\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\uff0c\u200b\u8fd4\u56de\u200b\u79fb\u9664\u200b\u540e\u200b\u6570\u7ec4\u200b\u7684\u200b\u65b0\u200b\u957f\u5ea6\u200b\u3002</p> <p>\u200b\u4e0d\u8981\u200b\u4f7f\u7528\u200b\u989d\u5916\u200b\u7684\u200b\u6570\u7ec4\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u4f60\u200b\u5fc5\u987b\u200b\u5728\u200b\u539f\u5730\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b \u200b\u5e76\u200b\u5728\u200b\u4f7f\u7528\u200b\\(O(1)\\)\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\u7684\u200b\u6761\u4ef6\u200b\u4e0b\u200b\u5b8c\u6210\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u7ed9\u5b9a\u200b\u6570\u7ec4\u200b nums = [1,1,2], \n\n\u200b\u51fd\u6570\u200b\u5e94\u8be5\u200b\u8fd4\u56de\u200b\u65b0\u200b\u7684\u200b\u957f\u5ea6\u200b 2, \u200b\u5e76\u4e14\u200b\u539f\u200b\u6570\u7ec4\u200b nums \u200b\u7684\u200b\u524d\u200b\u4e24\u4e2a\u200b\u5143\u7d20\u200b\u88ab\u200b\u4fee\u6539\u200b\u4e3a\u200b 1, 2\u3002 \n\n\u200b\u4f60\u200b\u4e0d\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u8d85\u51fa\u200b\u65b0\u200b\u957f\u5ea6\u200b\u540e\u9762\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u7ed9\u5b9a\u200b nums = [0,0,1,1,1,2,2,3,3,4],\n\n\u200b\u51fd\u6570\u200b\u5e94\u8be5\u200b\u8fd4\u56de\u200b\u65b0\u200b\u7684\u200b\u957f\u5ea6\u200b 5, \u200b\u5e76\u4e14\u200b\u539f\u200b\u6570\u7ec4\u200b nums \u200b\u7684\u200b\u524d\u200b\u4e94\u4e2a\u200b\u5143\u7d20\u200b\u88ab\u200b\u4fee\u6539\u200b\u4e3a\u200b 0, 1, 2, 3, 4\u3002\n\n\u200b\u4f60\u200b\u4e0d\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u8d85\u51fa\u200b\u65b0\u200b\u957f\u5ea6\u200b\u540e\u9762\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u4e3a\u4ec0\u4e48\u200b\u8fd4\u56de\u200b\u6570\u503c\u200b\u662f\u200b\u6574\u6570\u200b\uff0c\u200b\u4f46\u200b\u8f93\u51fa\u200b\u7684\u200b\u7b54\u6848\u200b\u662f\u200b\u6570\u7ec4\u200b\u5462\u200b?</p> <p>\u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u662f\u200b\u4ee5\u200b \u300c\u200b\u5f15\u7528\u200b\u300d \u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b\u7684\u200b\uff0c\u200b\u8fd9\u200b\u610f\u5473\u7740\u200b\u5728\u200b\u51fd\u6570\u200b\u91cc\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u5bf9\u4e8e\u200b\u8c03\u7528\u8005\u200b\u662f\u200b\u53ef\u89c1\u200b\u7684\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u60f3\u8c61\u200b\u5185\u90e8\u200b\u64cd\u4f5c\u200b\u5982\u4e0b\u200b:</p> <pre><code>// nums \u200b\u662f\u200b\u4ee5\u200b\u201c\u200b\u5f15\u7528\u200b\u201d\u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b\u7684\u200b\u3002\u200b\u4e5f\u5c31\u662f\u8bf4\u200b\uff0c\u200b\u4e0d\u200b\u5bf9\u200b\u5b9e\u53c2\u200b\u505a\u200b\u4efb\u4f55\u200b\u62f7\u8d1d\u200b\nint len = removeDuplicates(nums);\n\n// \u200b\u5728\u200b\u51fd\u6570\u200b\u91cc\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u5bf9\u4e8e\u200b\u8c03\u7528\u8005\u200b\u662f\u200b\u53ef\u89c1\u200b\u7684\u200b\u3002\n// \u200b\u6839\u636e\u200b\u4f60\u200b\u7684\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u7684\u200b\u957f\u5ea6\u200b, \u200b\u5b83\u4f1a\u200b\u6253\u5370\u200b\u51fa\u200b\u6570\u7ec4\u200b\u4e2d\u8be5\u200b\u957f\u5ea6\u200b\u8303\u56f4\u200b\u5185\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u3002\nfor (int i = 0; i &lt; len; i++) {\n\u00a0 \u00a0 print(nums[i]);\n}\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/26/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a </p> <ol> <li>\u200b\u5f53\u200b\u6570\u7ec4\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u65f6\u200b\uff0c\u200b\u4e0d\u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\uff1b</li> <li>\u200b\u4f7f\u7528\u200b\u4e00\u4e2a\u200b\u6307\u9488\u200b\u5206\u522b\u200b\u6307\u5411\u200b\u5220\u9664\u200b\u91cd\u590d\u200b\u503c\u540e\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u672b\u5c3e\u200b\uff0c\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u6570\u503c\u200b\u8868\u793a\u200b\u5220\u53bb\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570\u200b\u3002</li> </ol> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int removeDuplicates(int* nums, int numsSize){\n    if (numsSize &lt; 2)\n        return numsSize;\n    int steps = 0, i = 0;\n    while(i &lt; numsSize - steps - 1)\n    {\n        if (steps)\n            nums[i + 1] = nums[i + steps + 1];\n        if (nums[i] == nums[i + steps + 1])\n            steps++;\n        else\n            i++;\n    }\n    return numsSize - steps;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/27/","title":"27. \u200b\u79fb\u9664\u200b\u5143\u7d20","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/27/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b <code>nums</code>\u00a0\u200b\u548c\u200b\u4e00\u4e2a\u200b\u503c\u200b <code>val</code>\uff0c\u200b\u4f60\u200b\u9700\u8981\u200b \u200b\u539f\u5730\u200b \u200b\u79fb\u9664\u200b\u6240\u6709\u200b\u6570\u503c\u200b\u7b49\u4e8e\u200b\u00a0<code>val</code>\u00a0\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b\u79fb\u9664\u200b\u540e\u200b\u6570\u7ec4\u200b\u7684\u200b\u65b0\u200b\u957f\u5ea6\u200b\u3002</p> <p>\u200b\u4e0d\u8981\u200b\u4f7f\u7528\u200b\u989d\u5916\u200b\u7684\u200b\u6570\u7ec4\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u4f60\u200b\u5fc5\u987b\u200b\u4ec5\u200b\u4f7f\u7528\u200b <code>O(1)</code> \u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\u5e76\u200b \u200b\u539f\u5730\u200b \u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u3002</p> <p>\u200b\u5143\u7d20\u200b\u7684\u200b\u987a\u5e8f\u200b\u53ef\u4ee5\u200b\u6539\u53d8\u200b\u3002\u200b\u4f60\u200b\u4e0d\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u8d85\u51fa\u200b\u65b0\u200b\u957f\u5ea6\u200b\u540e\u9762\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b:</p> <p>\u200b\u4e3a\u4ec0\u4e48\u200b\u8fd4\u56de\u200b\u6570\u503c\u200b\u662f\u200b\u6574\u6570\u200b\uff0c\u200b\u4f46\u200b\u8f93\u51fa\u200b\u7684\u200b\u7b54\u6848\u200b\u662f\u200b\u6570\u7ec4\u200b\u5462\u200b?</p> <p>\u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u662f\u200b\u4ee5\u200b\u300c\u200b\u5f15\u7528\u200b\u300d\u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b\u7684\u200b\uff0c\u200b\u8fd9\u200b\u610f\u5473\u7740\u200b\u5728\u200b\u51fd\u6570\u200b\u91cc\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u5bf9\u4e8e\u200b\u8c03\u7528\u8005\u200b\u662f\u200b\u53ef\u89c1\u200b\u7684\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u60f3\u8c61\u200b\u5185\u90e8\u200b\u64cd\u4f5c\u200b\u5982\u4e0b\u200b:</p> <pre><code>// nums \u200b\u662f\u200b\u4ee5\u200b\u201c\u200b\u5f15\u7528\u200b\u201d\u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b\u7684\u200b\u3002\u200b\u4e5f\u5c31\u662f\u8bf4\u200b\uff0c\u200b\u4e0d\u200b\u5bf9\u200b\u5b9e\u53c2\u200b\u4f5c\u200b\u4efb\u4f55\u200b\u62f7\u8d1d\u200b\nint len = removeElement(nums, val);\n\n// \u200b\u5728\u200b\u51fd\u6570\u200b\u91cc\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u5bf9\u4e8e\u200b\u8c03\u7528\u8005\u200b\u662f\u200b\u53ef\u89c1\u200b\u7684\u200b\u3002\n// \u200b\u6839\u636e\u200b\u4f60\u200b\u7684\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u7684\u200b\u957f\u5ea6\u200b, \u200b\u5b83\u4f1a\u200b\u6253\u5370\u200b\u51fa\u200b\u6570\u7ec4\u200b\u4e2d\u200b \u200b\u8be5\u200b\u957f\u5ea6\u200b\u8303\u56f4\u200b\u5185\u200b \u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u3002\nfor (int i = 0; i &lt; len; i++) {\n\u00a0 \u00a0 print(nums[i]);\n}\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [3,2,2,3], val = 3\n\u200b\u8f93\u51fa\u200b\uff1a2, nums = [2,2]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u51fd\u6570\u200b\u5e94\u8be5\u200b\u8fd4\u56de\u200b\u65b0\u200b\u7684\u200b\u957f\u5ea6\u200b 2, \u200b\u5e76\u4e14\u200b nums \u200b\u4e2d\u200b\u7684\u200b\u524d\u200b\u4e24\u4e2a\u200b\u5143\u7d20\u200b\u5747\u200b\u4e3a\u200b 2\u3002\u200b\u4f60\u200b\u4e0d\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u8d85\u51fa\u200b\u65b0\u200b\u957f\u5ea6\u200b\u540e\u9762\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\u200b\u4f8b\u5982\u200b\uff0c\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u7684\u200b\u65b0\u200b\u957f\u5ea6\u200b\u4e3a\u200b 2 \uff0c\u200b\u800c\u200b nums = [2,2,3,3] \u200b\u6216\u200b nums = [2,2,0,0]\uff0c\u200b\u4e5f\u200b\u4f1a\u200b\u88ab\u200b\u89c6\u4f5c\u200b\u6b63\u786e\u200b\u7b54\u6848\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [0,1,2,2,3,0,4,2], val = 2\n\u200b\u8f93\u51fa\u200b\uff1a5, nums = [0,1,4,0,3]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u51fd\u6570\u200b\u5e94\u8be5\u200b\u8fd4\u56de\u200b\u65b0\u200b\u7684\u200b\u957f\u5ea6\u200b 5, \u200b\u5e76\u4e14\u200b nums \u200b\u4e2d\u200b\u7684\u200b\u524d\u200b\u4e94\u4e2a\u200b\u5143\u7d20\u200b\u4e3a\u200b 0, 1, 3, 0, 4\u3002\u200b\u6ce8\u610f\u200b\u8fd9\u200b\u4e94\u4e2a\u200b\u5143\u7d20\u200b\u53ef\u200b\u4e3a\u200b\u4efb\u610f\u200b\u987a\u5e8f\u200b\u3002\u200b\u4f60\u200b\u4e0d\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u8d85\u51fa\u200b\u65b0\u200b\u957f\u5ea6\u200b\u540e\u9762\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>0 &lt;= nums.length &lt;= 100</code></li> <li><code>0 &lt;= nums[i] &lt;= 50</code></li> <li><code>0 &lt;= val &lt;= 100</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/27/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u53cc\u200b\u6307\u9488\u200b\u6cd5\u200b\uff0c\u200b\u4e24\u4e2a\u200b\u6307\u9488\u200b\u5206\u522b\u200b\u6307\u5411\u200b\uff1a</p> <ul> <li>\u200b\u5c1a\u672a\u200b\u68c0\u67e5\u200b\u8fc7\u200b\u7684\u200b\u90e8\u5206\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u5143\u7d20\u200b</li> <li>\u200b\u5df2\u7ecf\u200b\u68c0\u67e5\u200b\u8fc7\u200b\u7684\u200b\u90e8\u5206\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b</li> </ul> <p>\u200b\u5f53\u200b\u4e24\u4e2a\u200b\u6307\u9488\u200b\u91cd\u5408\u200b\u65f6\u200b\u4e0d\u200b\u9700\u8981\u200b\u8fdb\u884c\u200b\u590d\u5236\u200b\u64cd\u4f5c\u200b\u3002</p> <pre><code>int removeElement(int* nums, int numsSize, int val){\n    int slow = 0, fast = 0, ret = numsSize;\n    if (numsSize == 0)\n        return 0;\n    do\n    {\n        if (fast &gt; slow)\n            nums[slow] = nums[fast];\n        if (nums[fast] == val)\n            ret--;\n        else\n            slow++;\n        fast++;\n    }\n    while (fast &lt; numsSize);\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/28/","title":"28. \u200b\u5b9e\u73b0\u200b strStr()","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/28/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5b9e\u73b0\u200b<code>strStr()</code>\u200b\u51fd\u6570\u200b\u3002</p> <p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b<code>haystack</code>\u200b\u5b57\u7b26\u4e32\u200b\u548c\u200b\u4e00\u4e2a\u200b<code>needle</code>\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u5728\u200b<code>haystack</code>\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u627e\u51fa\u200b needle \u200b\u5b57\u7b26\u4e32\u200b\u51fa\u73b0\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u4f4d\u7f6e\u200b (\u200b\u4ece\u200b0\u200b\u5f00\u59cb\u200b)\u3002\u200b\u5982\u679c\u200b\u4e0d\u200b\u5b58\u5728\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b\u00a0-1\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1:</p> <pre><code>\u200b\u8f93\u5165\u200b: haystack = \"hello\", needle = \"ll\"\n\u200b\u8f93\u51fa\u200b: 2\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2:</p> <pre><code>\u200b\u8f93\u5165\u200b: haystack = \"aaaaa\", needle = \"bba\"\n\u200b\u8f93\u51fa\u200b: -1\n</code></pre> <p>\u200b\u8bf4\u660e\u200b:</p> <p>\u200b\u5f53\u200b<code>needle</code>\u200b\u662f\u200b\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u65f6\u200b\uff0c\u200b\u6211\u4eec\u200b\u5e94\u5f53\u200b\u8fd4\u56de\u200b\u4ec0\u4e48\u200b\u503c\u200b\u5462\u200b\uff1f\u200b\u8fd9\u662f\u200b\u4e00\u4e2a\u200b\u5728\u200b\u9762\u8bd5\u200b\u4e2d\u200b\u5f88\u200b\u597d\u200b\u7684\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u672c\u9898\u200b\u800c\u8a00\u200b\uff0c\u200b\u5f53\u200b<code>needle</code>\u200b\u662f\u200b\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u65f6\u200b\u6211\u4eec\u200b\u5e94\u5f53\u200b\u8fd4\u56de\u200b 0 \u3002\u200b\u8fd9\u200b\u4e0e\u200bC\u200b\u8bed\u8a00\u200b\u7684\u200b<code>strstr()</code>\u200b\u4ee5\u53ca\u200bJava\u200b\u7684\u200b<code>indexOf()</code>\u200b\u5b9a\u4e49\u200b\u76f8\u7b26\u200b\u3002</p> <p>Reference</p>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/28/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u66b4\u529b\u200b\u6309\u4f4d\u200b\u5339\u914d\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int strStr(char * haystack, char * needle){\n    int len1 = strlen(haystack), len2 = strlen(needle), cur1 = 0, cur2 = 0, cur3 = 0, success = 1;\n    if (!len2)\n        return 0;\n    if (len1 &lt; len2)\n        return -1;\n    do\n    {\n        success = 1;\n        if (haystack[cur1] == needle[0])\n        {\n            cur2 = 0;\n            cur3 = 0;\n            while(cur2 &lt; len2 &amp;&amp; cur3 + cur1 &lt; len1 &amp;&amp; success)\n            {\n                if(haystack[cur1 + cur3] != needle[cur2])\n                    success = 0;\n                cur3++;\n                cur2++;\n            }\n            if (success)\n                return cur1;\n        }\n        cur1++;\n    }\n    while (haystack[cur1 + len2 - 1]);\n    return -1;\n}\n</code></pre>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/3/","title":"3. \u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u54c8\u5e0c\u8868","\u53cc\u6307\u9488","\u5b57\u7b26\u4e32","Sliding Window"]},{"location":"coding/leetcode/3/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u627e\u51fa\u200b\u5176\u4e2d\u200b\u4e0d\u200b\u542b\u6709\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u00a0\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b\u00a0\u200b\u7684\u200b\u957f\u5ea6\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01:</p> <pre><code>\u200b\u8f93\u5165\u200b: s = \"abcabcbb\"\n\u200b\u8f93\u51fa\u200b: 3 \n\u200b\u89e3\u91ca\u200b: \u200b\u56e0\u4e3a\u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b\u662f\u200b \"abc\"\uff0c\u200b\u6240\u4ee5\u200b\u5176\u200b\u957f\u5ea6\u200b\u4e3a\u200b 3\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2:</p> <pre><code>\u200b\u8f93\u5165\u200b: s = \"bbbbb\"\n\u200b\u8f93\u51fa\u200b: 1\n\u200b\u89e3\u91ca\u200b: \u200b\u56e0\u4e3a\u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b\u662f\u200b \"b\"\uff0c\u200b\u6240\u4ee5\u200b\u5176\u200b\u957f\u5ea6\u200b\u4e3a\u200b 1\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3:</p> <pre><code>\u200b\u8f93\u5165\u200b: s = \"pwwkew\"\n\u200b\u8f93\u51fa\u200b: 3\n\u200b\u89e3\u91ca\u200b: \u200b\u56e0\u4e3a\u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b\u662f\u200b\u00a0\"wke\"\uff0c\u200b\u6240\u4ee5\u200b\u5176\u200b\u957f\u5ea6\u200b\u4e3a\u200b 3\u3002\n\u00a0    \u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u4f60\u200b\u7684\u200b\u7b54\u6848\u200b\u5fc5\u987b\u200b\u662f\u200b \u200b\u5b50\u4e32\u200b \u200b\u7684\u200b\u957f\u5ea6\u200b\uff0c\"pwke\"\u00a0\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5b50\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4e0d\u662f\u200b\u5b50\u4e32\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4:</p> <pre><code>\u200b\u8f93\u5165\u200b: s = \"\"\n\u200b\u8f93\u51fa\u200b: 0\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>0 &lt;= s.length &lt;= 5 * 104</code></li> <li><code>s</code>\u00a0\u200b\u7531\u200b\u82f1\u6587\u5b57\u6bcd\u200b\u3001\u200b\u6570\u5b57\u200b\u3001\u200b\u7b26\u53f7\u200b\u548c\u200b\u7a7a\u683c\u200b\u7ec4\u6210\u200b</li> </ul>","tags":["\u54c8\u5e0c\u8868","\u53cc\u6307\u9488","\u5b57\u7b26\u4e32","Sliding Window"]},{"location":"coding/leetcode/3/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b \uff1a\u200b\u7528\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\u5b58\u50a8\u200b\u5404\u200b\u5b57\u6bcd\u200b\u51fa\u73b0\u200b\u7684\u200b\u6b21\u6570\u200b\uff08\u200b\u54c8\u5e0c\u200b\u8868\u200b\uff09\u3002\u200b\u5f53\u200b\u4e0d\u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5b57\u6bcd\u200b\u65f6\u200b\u6ed1\u52a8\u200b\u7a97\u53e3\u200b\u4e0d\u65ad\u200b\u5411\u200b\u53f3\u200b\u5ef6\u4f38\u200b\uff0c\u200b\u76f4\u5230\u200b\u51fa\u73b0\u200b\u91cd\u590d\u200b\u5b57\u6bcd\u200b\u6216\u200b\u5230\u8fbe\u200b\u5b57\u7b26\u4e32\u200b\u7ed3\u5c3e\u200b\u3002\u200b\u68c0\u6d4b\u200b\u5230\u200b\u91cd\u590d\u200b\u5b57\u6bcd\u200b\u540e\u200b\uff0c\u200b\u4ece\u200b\u5de6\u5411\u53f3\u200b\u6536\u7f29\u200b\u6ed1\u52a8\u200b\u7a97\u53e3\u200b\uff0c\u200b\u6bcf\u6b21\u200b\u6536\u7f29\u200b\u90fd\u200b\u5bf9\u200b\u54c8\u5e0c\u200b\u8868\u200b\u8fdb\u884c\u200b\u66f4\u65b0\u200b\u3002\u200b\u76f4\u5230\u200b\u4e0d\u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5b57\u6bcd\u200b\u65f6\u200b\uff0c\u200b\u6ed1\u52a8\u200b\u7a97\u53e3\u200b\u624d\u200b\u505c\u6b62\u200b\u6536\u7f29\u200b\uff0c\u200b\u7ee7\u7eed\u200b\u5411\u200b\u53f3\u200b\u5ef6\u4f38\u200b\u3002</p> <p>\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b \uff1a\\(O(N)\\)</p> <p>\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b \uff1a\\(O(1)\\)\uff0c\u200b\u4e0e\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u65e0\u5173\u200b\uff0c\u200b\u4f46\u200b\u4e0e\u200b\u5355\u4e2a\u200b\u5b57\u7b26\u200b\u7684\u200b\u5bbd\u5ea6\u200b\u6709\u5173\u200b\u3002</p> <pre><code>int lengthOfLongestSubstring(char * s){\n    int length = 0, ret = 0;\n    char hashMap[256] = {0}, *cur = s;\n    while(*cur)\n    {\n        while(hashMap[*cur])\n        {\n            hashMap[*(cur - length)] -= 1;\n            length--;\n        }\n        hashMap[*cur] = 1;\n        length++;\n        ret = ret &gt; length ? ret : length;\n        cur++;\n    }\n    return ret;\n}\n</code></pre> <p>Reference</p>","tags":["\u54c8\u5e0c\u8868","\u53cc\u6307\u9488","\u5b57\u7b26\u4e32","Sliding Window"]},{"location":"coding/leetcode/315/","title":"315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u6392\u5e8f","\u6811\u72b6\u6570\u7ec4","\u7ebf\u6bb5\u6811","\u4e8c\u5206\u67e5\u627e","\u5206\u6cbb\u7b97\u6cd5"]},{"location":"coding/leetcode/315/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b nums \uff0c\u200b\u6309\u200b\u8981\u6c42\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u6570\u7ec4\u200b\u00a0counts \u3002\u200b\u6570\u7ec4\u200b counts \u200b\u6709\u200b\u8be5\u200b\u6027\u8d28\u200b\uff1a<code>counts[i]</code>\u200b\u7684\u200b\u503c\u200b\u662f\u200b<code>nums[i]</code>\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b<code>nums[i]</code>\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u6570\u91cf\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [5,2,6,1]\n\u200b\u8f93\u51fa\u200b\uff1a[2,1,1,0] \n\u200b\u89e3\u91ca\u200b\uff1a\n5 \u200b\u7684\u200b\u53f3\u4fa7\u200b\u6709\u200b 2 \u200b\u4e2a\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u5143\u7d20\u200b (2 \u200b\u548c\u200b 1)\n2 \u200b\u7684\u200b\u53f3\u4fa7\u200b\u4ec5\u200b\u6709\u200b 1 \u200b\u4e2a\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u5143\u7d20\u200b (1)\n6 \u200b\u7684\u200b\u53f3\u4fa7\u200b\u6709\u200b 1 \u200b\u4e2a\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u5143\u7d20\u200b (1)\n1 \u200b\u7684\u200b\u53f3\u4fa7\u200b\u6709\u200b 0 \u200b\u4e2a\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u5143\u7d20\u200b\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>0 &lt;= nums.length &lt;= 10^5</code></li> <li><code>-10^4\u00a0&lt;= nums[i] &lt;= 10^4</code></li> </ul> <p>Reference</p>","tags":["\u6392\u5e8f","\u6811\u72b6\u6570\u7ec4","\u7ebf\u6bb5\u6811","\u4e8c\u5206\u67e5\u627e","\u5206\u6cbb\u7b97\u6cd5"]},{"location":"coding/leetcode/315/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4ece\u540e\u200b\u5f80\u524d\u200b\u6784\u9020\u200b\u4e00\u4e2a\u200b\u4e8c\u53c9\u200b\u67e5\u627e\u200b\u6811\u200b\uff0c\u200b\u5728\u200b\u4e8c\u53c9\u200b\u67e5\u627e\u200b\u6811\u4e2d\u200b\u7ef4\u62a4\u200b\u6811\u4e2d\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u7684\u200b\u6570\u5b57\u200b\u7684\u200b\u4e2a\u6570\u200b\u3002</p> <p>\u200b\u4e8c\u53c9\u6811\u200b\u9700\u8981\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u57df\u200b\u7528\u4e8e\u200b\u8ba1\u7b97\u200b\u5de6\u5b50\u200b\u6811\u200b\u7684\u200b\u5143\u7d20\u200b\u603b\u6570\u200b\uff0c\u200b\u5373\u200b\u5f53\u524d\u200b\u6811\u4e2d\u200b\u6570\u503c\u200b\u5c0f\u4e8e\u200b\u8be5\u200b\u8282\u70b9\u200b\u7684\u200b\u5143\u7d20\u200b\u603b\u6570\u200b\u3002\u200b\u5b9e\u73b0\u200b\u65b9\u6cd5\u200b\u662f\u200b\u5728\u200b\u5de6\u5b50\u200b\u6811\u200b\u6dfb\u52a0\u200b\u5143\u7d20\u200b\u540e\u200b\u8be5\u57df\u200b\u7684\u200b\u503c\u200b\u589e\u52a0\u200b1\u3002</p> <p>\u200b\u5728\u200b\u8282\u70b9\u200b\u7684\u200b\u53f3\u5b50\u200b\u6811\u200b\u6dfb\u52a0\u200b\u5143\u7d20\u200b\u65f6\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u548c\u200b\u8282\u70b9\u200b\u5de6\u5b50\u200b\u6811\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u8282\u70b9\u200b\u90fd\u200b\u662f\u200b\u7b26\u5408\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u8282\u70b9\u200b\u3002\u200b\u5728\u200b\u9012\u5f52\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u9700\u8981\u200b\u4f20\u9012\u200b\u4e00\u4e2a\u200b\u7528\u4f5c\u200b\u8ba1\u6570\u5668\u200b\u7684\u200b\u6307\u9488\u200b\u53c2\u6570\u200b\uff08\u200b\u8ba1\u6570\u200b\u7684\u200b\u6570\u503c\u200b\u5373\u200b\u4e3a\u200b\u6240\u6c42\u200b\u7684\u200b\u7ed3\u679c\u200b\uff09\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\nstruct TreeNode2\n{\n    struct TreeNode2 *left;\n    struct TreeNode2 *right;\n    int val;\n    int subs;\n};\n\nstruct TreeNode2 *insert(struct TreeNode2 *dest, int src, int *depth)\n{\n    if (dest == NULL)\n    {\n        struct TreeNode2 *ret = (struct TreeNode2 *)malloc(sizeof(struct TreeNode2));\n        ret-&gt;left = NULL;\n        ret-&gt;right = NULL;\n        ret-&gt;val = src;\n        ret-&gt;subs = 0;\n        return ret;\n    }\n    if (src &gt; dest-&gt;val)\n    {\n        *depth += 1 + dest-&gt;subs;\n        dest-&gt;right = insert(dest-&gt;right, src, depth);\n    }\n    else\n    {\n        dest-&gt;subs++;\n        dest-&gt;left = insert(dest-&gt;left, src, depth);\n    }\n    return dest;\n}\n\nint *countSmaller(int *nums, int numsSize, int *returnSize)\n{\n    *returnSize = numsSize;\n    int *ret = (int *)memset(malloc(sizeof(int) * numsSize), 0, sizeof(int) * numsSize), i = 0;\n    if (numsSize &lt;= 1)\n        return ret;\n    struct TreeNode2 *tempTree = NULL;\n    for (i = numsSize - 1; i &gt;= 0; i--)\n        tempTree = insert(tempTree, nums[i], ret + i);\n    return ret;\n}\n</code></pre>","tags":["\u6392\u5e8f","\u6811\u72b6\u6570\u7ec4","\u7ebf\u6bb5\u6811","\u4e8c\u5206\u67e5\u627e","\u5206\u6cbb\u7b97\u6cd5"]},{"location":"coding/leetcode/341/","title":"341. \u200b\u6241\u5e73\u5316\u200b\u5d4c\u5957\u200b\u5217\u8868\u200b\u8fed\u4ee3\u200b\u5668","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6808","\u8bbe\u8ba1"]},{"location":"coding/leetcode/341/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u5d4c\u5957\u200b\u7684\u200b\u6574\u578b\u200b\u5217\u8868\u200b\u3002\u200b\u8bf7\u200b\u4f60\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u8fed\u4ee3\u200b\u5668\u200b\uff0c\u200b\u4f7f\u200b\u5176\u200b\u80fd\u591f\u200b\u904d\u5386\u200b\u8fd9\u4e2a\u200b\u6574\u578b\u200b\u5217\u8868\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u5217\u8868\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e00\u9879\u200b\u6216\u8005\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\uff0c\u200b\u6216\u8005\u200b\u662f\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u5217\u8868\u200b\u3002\u200b\u5176\u4e2d\u200b\u5217\u8868\u200b\u7684\u200b\u5143\u7d20\u200b\u4e5f\u200b\u53ef\u80fd\u200b\u662f\u200b\u6574\u6570\u200b\u6216\u662f\u200b\u5176\u4ed6\u200b\u5217\u8868\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [[1,1],2,[1,1]]\n\u200b\u8f93\u51fa\u200b: [1,1,2,1,1]\n\u200b\u89e3\u91ca\u200b: \u200b\u901a\u8fc7\u200b\u91cd\u590d\u200b\u8c03\u7528\u200b\u00a0next \u200b\u76f4\u5230\u200b\u00a0hasNext \u200b\u8fd4\u56de\u200b false\uff0cnext\u00a0\u200b\u8fd4\u56de\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u987a\u5e8f\u200b\u5e94\u8be5\u200b\u662f\u200b: [1,1,2,1,1]\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,[4,[6]]]\n\u200b\u8f93\u51fa\u200b: [1,4,6]\n\u200b\u89e3\u91ca\u200b: \u200b\u901a\u8fc7\u200b\u91cd\u590d\u200b\u8c03\u7528\u200b\u00a0next\u00a0\u200b\u76f4\u5230\u200b\u00a0hasNext \u200b\u8fd4\u56de\u200b false\uff0cnext\u00a0\u200b\u8fd4\u56de\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u987a\u5e8f\u200b\u5e94\u8be5\u200b\u662f\u200b: [1,4,6]\u3002\n</code></pre> <p>Reference</p>","tags":["\u6808","\u8bbe\u8ba1"]},{"location":"coding/leetcode/341/#_2","title":"\u9898\u89e3","text":"<p>C\u200b\u8bed\u8a00\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5d4c\u5957\u200b\u5217\u8868\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u94fe\u8868\u200b\uff0c\u200b\u7136\u540e\u200b\u5bf9\u200b\u94fe\u8868\u200b\u8fdb\u884c\u200b\u8fed\u4ee3\u200b\uff1a</p> <pre><code>/**\n * *********************************************************************\n * // This is the interface that allows for creating nested lists.\n * // You should not implement it, or speculate about its implementation\n * *********************************************************************\n *\n * // Return true if this NestedInteger holds a single integer, rather than a nested list.\n * bool NestedIntegerIsInteger(struct NestedInteger *);\n *\n * // Return the single integer that this NestedInteger holds, if it holds a single integer\n * // The result is undefined if this NestedInteger holds a nested list\n * int NestedIntegerGetInteger(struct NestedInteger *);\n *\n * // Return the nested list that this NestedInteger holds, if it holds a nested list\n * // The result is undefined if this NestedInteger holds a single integer\n * struct NestedInteger **NestedIntegerGetList(struct NestedInteger *);\n *\n * // Return the nested list's size that this NestedInteger holds, if it holds a nested list\n * // The result is undefined if this NestedInteger holds a single integer\n * int NestedIntegerGetListSize(struct NestedInteger *);\n * };\n */\nstruct NestedIterator {\n    struct ListNode *list;\n    struct ListNode *current;\n};\n\nstruct ListNode *getNode(int val, struct ListNode *next)\n{\n    struct ListNode *ret = (struct ListNode *)malloc(sizeof(struct ListNode));\n    ret-&gt;val = val;\n    ret-&gt;next = next;\n    return ret;\n}\n\nstruct ListNode *toList(struct NestedInteger *src, struct ListNode **tail)\n{\n    if (NestedIntegerIsInteger(src))\n    {\n        struct ListNode *ret = getNode(NestedIntegerGetInteger(src), NULL);\n        if (tail)\n            *tail = ret;\n        return ret;\n    }\n    else\n    {\n        struct NestedInteger **content = NestedIntegerGetList(src);\n        struct ListNode *ret = NULL, *temp = NULL;\n        int i = 0, size = NestedIntegerGetListSize(src);\n        for (i = 0; i &lt; size; i++)\n        {\n            if (!ret)\n                ret = toList(content[i], &amp;temp);\n            else\n                (*tail)-&gt;next = toList(content[i], &amp;temp);\n            if (temp)\n                *tail = temp;\n        }\n        return ret;\n    }\n}\n\nstruct NestedIterator *nestedIterCreate(struct NestedInteger** nestedList, int nestedListSize) {\n    struct NestedIterator *ret = (struct NestedIterator *)malloc(sizeof(struct NestedIterator));\n    ret-&gt;list = NULL;\n    ret-&gt;current = NULL;\n    struct ListNode *temp = NULL;\n    for (int i = 0; i &lt; nestedListSize; i++)\n    {\n        if (!ret-&gt;list)\n            ret-&gt;list = toList(nestedList[i], &amp;temp);\n        else\n            ret-&gt;current-&gt;next = toList(nestedList[i], &amp;temp);\n        ret-&gt;current = temp;\n    }\n    ret-&gt;current = ret-&gt;list;\n    return ret;\n}\n\nbool nestedIterHasNext(struct NestedIterator *iter) {\n    return iter-&gt;current;\n}\n\nint nestedIterNext(struct NestedIterator *iter) {\n    int ret = iter-&gt;current-&gt;val;\n    iter-&gt;current = iter-&gt;current-&gt;next;\n    return ret;\n}\n\n/** Deallocates memory previously allocated for the iterator */\nvoid nestedIterFree(struct NestedIterator *iter) {\n    while (iter-&gt;list)\n    {\n        iter-&gt;current = iter-&gt;list-&gt;next;\n        free(iter-&gt;list);\n        iter-&gt;list = iter-&gt;current;\n    }\n    free(iter);\n}\n\n/**\n * Your NestedIterator will be called like this:\n * struct NestedIterator *i = nestedIterCreate(nestedList, nestedListSize);\n * while (nestedIterHasNext(i)) printf(\"%d\\n\", nestedIterNext(i));\n * nestedIterFree(i);\n */\n</code></pre> <p>Python\u200b\u8bed\u8a00\u200b\u652f\u6301\u200b<code>yield</code>\u200b\u8bed\u53e5\u200b\u4e0e\u200b<code>yield from</code>\u200b\u8bed\u53e5\u200b\uff0c\u200b\u56e0\u6b64\u200b\u76f4\u63a5\u200b\u8fed\u4ee3\u200b\u751f\u6210\u5668\u200b\u5373\u53ef\u200b</p> <pre><code># \"\"\"\n# This is the interface that allows for creating nested lists.\n# You should not implement it, or speculate about its implementation\n# \"\"\"\n#class NestedInteger:\n#    def isInteger(self) -&gt; bool:\n#        \"\"\"\n#        @return True if this NestedInteger holds a single integer, rather than a nested list.\n#        \"\"\"\n#\n#    def getInteger(self) -&gt; int:\n#        \"\"\"\n#        @return the single integer that this NestedInteger holds, if it holds a single integer\n#        Return None if this NestedInteger holds a nested list\n#        \"\"\"\n#\n#    def getList(self) -&gt; [NestedInteger]:\n#        \"\"\"\n#        @return the nested list that this NestedInteger holds, if it holds a nested list\n#        Return None if this NestedInteger holds a single integer\n#        \"\"\"\n\ndef listIter(nestNode):\n    for node in nestNode:\n        if node.isInteger():\n            yield node.getInteger()\n        else:\n            yield from listIter(node.getList())\n\nclass NestedIterator:\n    def __init__(self, nestedList):\n        self.nextvalue = 0\n        self.iterator = listIter(nestedList)\n\n    def next(self):\n        return self.nextvalue\n\n    def hasNext(self):\n        try:\n            self.nextvalue = next(self.iterator)\n        except StopIteration:\n            return False\n        return True\n\n# Your NestedIterator object will be instantiated and called as such:\n# i, v = NestedIterator(nestedList), []\n# while i.hasNext(): v.append(i.next())\n</code></pre>","tags":["\u6808","\u8bbe\u8ba1"]},{"location":"coding/leetcode/344/","title":"344. \u200b\u53cd\u8f6c\u200b\u5b57\u7b26\u4e32","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/344/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u5176\u200b\u4f5c\u7528\u200b\u662f\u200b\u5c06\u200b\u8f93\u5165\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\u53cd\u8f6c\u200b\u8fc7\u6765\u200b\u3002\u200b\u8f93\u5165\u200b\u5b57\u7b26\u4e32\u200b\u4ee5\u200b\u5b57\u7b26\u200b\u6570\u7ec4\u200b char[] \u200b\u7684\u200b\u5f62\u5f0f\u200b\u7ed9\u51fa\u200b\u3002</p> <p>\u200b\u4e0d\u8981\u200b\u7ed9\u200b\u53e6\u5916\u200b\u7684\u200b\u6570\u7ec4\u200b\u5206\u914d\u200b\u989d\u5916\u200b\u7684\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u4f60\u200b\u5fc5\u987b\u200b\u539f\u5730\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u3001\u200b\u4f7f\u7528\u200b O(1) \u200b\u7684\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\u89e3\u51b3\u200b\u8fd9\u4e00\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5b57\u7b26\u200b\u90fd\u200b\u662f\u200b ASCII \u200b\u7801\u8868\u200b\u4e2d\u200b\u7684\u200b\u53ef\u200b\u6253\u5370\u200b\u5b57\u7b26\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a <pre><code>\u200b\u8f93\u5165\u200b\uff1a[\"h\",\"e\",\"l\",\"l\",\"o\"]\n\u200b\u8f93\u51fa\u200b\uff1a[\"o\",\"l\",\"l\",\"e\",\"h\"]\n</code></pre> \u200b\u793a\u4f8b\u200b 2\uff1a <pre><code>\u200b\u8f93\u5165\u200b\uff1a[\"H\",\"a\",\"n\",\"n\",\"a\",\"h\"]\n\u200b\u8f93\u51fa\u200b\uff1a[\"h\",\"a\",\"n\",\"n\",\"a\",\"H\"]\n</code></pre></p> <p>Reference</p>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/344/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5206\u522b\u200b\u4f7f\u7528\u200b\u4e24\u4e2a\u200b\u6307\u9488\u200b\u6307\u5411\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5f00\u5934\u200b\u4e0e\u200b\u7ed3\u5c3e\u200b\uff0c\u200b\u4ea4\u6362\u200b\u4e24\u8005\u200b\u7684\u200b\u503c\u200b\u540e\u200b\u6536\u7f29\u200b\u6307\u9488\u200b\u3002\uff08\u200b\u5de6\u200b\u6307\u9488\u200b\u53f3\u79fb\u200b\uff0c\u200b\u53f3\u200b\u6307\u9488\u200b\u5de6\u79fb\u200b\uff09</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>void reverseString(char* s, int sSize)\n{\n    if (sSize &lt;= 1)\n        return;\n    char temp = 0;\n    temp = *s;\n    *s = s[sSize - 1];\n    s[sSize - 1] = temp;\n    reverseString(s + 1, sSize - 2);\n}\n</code></pre>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/35/","title":"35. \u200b\u641c\u7d22\u200b\u63d2\u5165\u200b\u4f4d\u7f6e","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/35/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6392\u5e8f\u200b\u6570\u7ec4\u200b\u548c\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u503c\u200b\uff0c\u200b\u5728\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u627e\u5230\u200b\u76ee\u6807\u503c\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5176\u200b\u7d22\u5f15\u200b\u3002\u200b\u5982\u679c\u200b\u76ee\u6807\u503c\u200b\u4e0d\u200b\u5b58\u5728\u200b\u4e8e\u200b\u6570\u7ec4\u200b\u4e2d\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5b83\u200b\u5c06\u200b\u4f1a\u200b\u88ab\u200b\u6309\u200b\u987a\u5e8f\u200b\u63d2\u5165\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u65e0\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1:</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,3,5,6], 5\n\u200b\u8f93\u51fa\u200b: 2\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02:</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,3,5,6], 2\n\u200b\u8f93\u51fa\u200b: 1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3:</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,3,5,6], 7\n\u200b\u8f93\u51fa\u200b: 4\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4:</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,3,5,6], 0\n\u200b\u8f93\u51fa\u200b: 0\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/35/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u3002</p> <p>\u200b\u8bbe\u200b\u6570\u5217\u200b\\(\\{a_i\\}\\)\u200b\u4e3a\u200b\u5347\u5e8f\u200b\u6570\u5217\u200b\uff0c\u200b\u76ee\u6807\u503c\u200b\\(b\\)\u200b\u7684\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b\\(j\\)\u200b\u6ee1\u8db3\u200b\uff1a</p> <ul> <li>\\(b \\leq a_j\\)</li> <li>\\(b &gt; a_{j - 1}\\)</li> </ul> <p>\u200b\u6ce8\u610f\u200b\u672c\u9898\u200b\u8981\u6c42\u200b\u7684\u200b\u6570\u636e\u200b\u4e0e\u200b\u666e\u901a\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u610f\u4e49\u200b\u4e0d\u540c\u200b\u3002</p> <ul> <li>\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u8fd4\u56de\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u4e0d\u200b\u5927\u4e8e\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u5143\u7d20\u200b\u4e0b\u6807\u200b\u3002</li> <li>\u200b\u672c\u9898\u200b\u8981\u6c42\u200b\u8fd4\u56de\u200b\u7b2c\u4e00\u4e2a\u200b\u4e0d\u200b\u5c0f\u4e8e\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u5143\u7d20\u200b\u4e0b\u6807\u200b\u3002</li> </ul> <pre><code>int bSearch(int *nums, int lo, int hi, int target)\n{\n    int mid;\n    while (lo &lt; hi)\n    {\n        mid = (lo + hi) &gt;&gt; 1;\n        if (nums[mid] &gt;= target)\n            hi = mid;\n        else\n            lo = mid + 1;\n    }\n    return lo;\n}\n\nint searchInsert(int* nums, int numsSize, int target){\n    return bSearch(nums, 0, numsSize, target);\n}\n</code></pre> <p>\u200b\u9644\u200b\uff1a\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u4f9b\u200b\u6bd4\u8f83\u200b</p> <pre><code>int bSearch(int *nums, int lo, int hi, int target)\n{\n    int mid;\n    while (lo &lt; hi)\n    {\n        mid = (lo + hi) &gt;&gt; 1;\n        if (nums[mid] &gt; target)\n            hi = mid;\n        else\n            lo = mid + 1;\n    }\n    return lo - 1;\n}\n</code></pre> <p>\u200b\u5728\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u4e2d\u200b\uff0c\u200b\u59cb\u7ec8\u200b\u6709\u200b\uff1a</p> <ul> <li><code>nums[lo - 1]</code> \u200b\u662f\u200b\u622a\u81f3\u200b\u5f53\u524d\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u4e0d\u200b\u5927\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5927\u200b\u5143\u7d20\u200b</li> <li><code>nums[hi]</code> \u200b\u662f\u200b\u622a\u81f3\u200b\u5f53\u524d\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u5927\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b</li> </ul> <p>\u200b\u800c\u200b\u5728\u200b\u672c\u9898\u200b\u7b97\u6cd5\u200b\u4e2d\u200b\uff0c\u200b\u59cb\u7ec8\u200b\u6709\u200b\uff1a</p> <ul> <li><code>nums[lo - 1]</code> \u200b\u662f\u200b\u622a\u81f3\u200b\u5f53\u524d\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u5c0f\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5927\u200b\u5143\u7d20\u200b</li> <li><code>nums[hi]</code> \u200b\u662f\u200b\u622a\u81f3\u200b\u5f53\u524d\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u4e0d\u200b\u5c0f\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b</li> </ul> <p>\u200b\u539f\u56e0\u200b\u5728\u4e8e\u200b\uff0c\u200b\u4e24\u200b\u7b97\u6cd5\u200b\u5bf9\u200b\u7b49\u4e8e\u200b\u7684\u200b\u60c5\u51b5\u200b\u7684\u200b\u5904\u7406\u200b\u4e0d\u540c\u200b\uff0c\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u5bf9\u200b<code>nums[mid] == target</code>\u200b\u60c5\u51b5\u200b\u7684\u200b\u5904\u7406\u200b\u4e3a\u200b <code>lo = mid + 1</code>\uff0c\u200b\u4ec5\u5f53\u200b<code>nums[mid] &gt; target</code>\u200b\u65f6\u624d\u200b\u6267\u884c\u200b<code>hi = mid</code>\u3002\u200b\u7531\u4e8e\u200b<code>[lo, hi)</code>\u200b\u662f\u200b\u5f53\u524d\u200b\u5df2\u200b \u200b\u786e\u8ba4\u200b\u7684\u200b\u533a\u95f4\u200b\uff0c\u200b\u5219\u200b<code>nums[hi]</code>\u200b\u4e3a\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u5927\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b\uff0c\u200b\u800c\u200b<code>nums[lo - 1]</code>\u200b\u4e3a\u200b\u4e0d\u200b\u5927\u4e8e\u200b <code>target</code>\u200b\u7684\u200b\u6700\u5927\u200b\u5143\u7d20\u200b\u3002\u200b\u800c\u200b\u672c\u9898\u200b\u7b97\u6cd5\u200b\u5bf9\u200b<code>nums[mid] == target</code>\u200b\u60c5\u51b5\u200b\u7684\u200b\u5904\u7406\u200b\u4e3a\u200b<code>hi = mid</code>\uff0c\u200b\u4ec5\u5f53\u200b <code>nums[mid] &lt; target</code>\u200b\u65f6\u624d\u200b\u6267\u884c\u200b<code>lo = mid + 1</code>\uff0c\u200b\u5219\u200b<code>nums[hi]</code>\u200b\u4e3a\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u4e0d\u200b\u5c0f\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5c0f\u200b \u200b\u5143\u7d20\u200b\uff0c<code>nums[lo - 1]</code>\u200b\u4e3a\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u5c0f\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5927\u200b\u5143\u7d20\u200b\u3002</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/350/","title":"350. \u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u7684\u200b\u4ea4\u96c6\u200b II","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6392\u5e8f","\u54c8\u5e0c\u8868","\u53cc\u6307\u9488","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/350/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u6765\u200b\u8ba1\u7b97\u200b\u5b83\u4eec\u200b\u7684\u200b\u4ea4\u96c6\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums1 = [1,2,2,1], nums2 = [2,2]\n\u200b\u8f93\u51fa\u200b\uff1a[2,2]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums1 = [4,9,5], nums2 = [9,4,9,8,4]\n\u200b\u8f93\u51fa\u200b\uff1a[4,9]\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li>\u200b\u8f93\u51fa\u200b\u7ed3\u679c\u200b\u4e2d\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u51fa\u73b0\u200b\u7684\u200b\u6b21\u6570\u200b\uff0c\u200b\u5e94\u200b\u4e0e\u200b\u5143\u7d20\u200b\u5728\u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\u4e00\u81f4\u200b\u3002</li> <li>\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u8003\u8651\u200b\u8f93\u51fa\u200b\u7ed3\u679c\u200b\u7684\u200b\u987a\u5e8f\u200b\u3002</li> </ul> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <ul> <li>\u200b\u5982\u679c\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u6570\u7ec4\u200b\u5df2\u7ecf\u200b\u6392\u597d\u5e8f\u200b\u5462\u200b\uff1f\u200b\u4f60\u200b\u5c06\u200b\u5982\u4f55\u200b\u4f18\u5316\u200b\u4f60\u200b\u7684\u200b\u7b97\u6cd5\u200b\uff1f</li> <li>\u200b\u5982\u679c\u200b<code>nums1</code>\u200b\u7684\u200b\u5927\u5c0f\u200b\u6bd4\u200b<code>nums2</code>\u200b\u5c0f\u200b\u5f88\u591a\u200b\uff0c\u200b\u54ea\u200b\u79cd\u200b\u65b9\u6cd5\u200b\u66f4\u4f18\u200b\uff1f</li> <li>\u200b\u5982\u679c\u200b<code>nums2</code>\u200b\u7684\u200b\u5143\u7d20\u200b\u5b58\u50a8\u200b\u5728\u200b\u78c1\u76d8\u200b\u4e0a\u200b\uff0c\u200b\u5185\u5b58\u200b\u662f\u200b\u6709\u9650\u200b\u7684\u200b\uff0c\u200b\u5e76\u4e14\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u4e00\u6b21\u200b\u52a0\u8f7d\u200b\u6240\u6709\u200b\u7684\u200b\u5143\u7d20\u200b\u5230\u200b\u5185\u5b58\u200b\u4e2d\u200b\uff0c\u200b\u4f60\u200b\u8be5\u200b\u600e\u4e48\u529e\u200b\uff1f</li> </ul> <p>Reference</p>","tags":["\u6392\u5e8f","\u54c8\u5e0c\u8868","\u53cc\u6307\u9488","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/350/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4f7f\u7528\u200b\u54c8\u5e0c\u200b\u8868\u200b\u5b58\u50a8\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5404\u200b\u5143\u7d20\u200b\u51fa\u73b0\u200b\u7684\u200b\u6b21\u6570\u200b\uff0c\u200b\u6ce8\u610f\u200b\u6570\u503c\u200b\u5728\u200b\u4ea4\u200b\u96c6\u4e2d\u200b\u53ef\u4ee5\u200b\u91cd\u590d\u200b\u51fa\u73b0\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>#define MIN(x, y) (x &gt; y ? y : x)\nstruct cell\n{\n    int value;\n    int times1;\n    int times2;\n};\nint* intersect(int* nums1, int nums1Size, int* nums2, int nums2Size, int* returnSize){\n    int i = 0, cur, mapSize = nums1Size + nums2Size, *ret = (int *)malloc(sizeof(int) * (nums1Size + nums2Size));\n    *returnSize = 0;\n    struct cell *hashMap = (struct cell *)memset(malloc(sizeof(struct cell) * mapSize), 0, sizeof(struct cell) * mapSize);\n    for (i = 0; i &lt; nums1Size; i++)\n    {\n        cur = (nums1[i] &gt; 0 ? 1 : -1) * (nums1[i] % mapSize);\n        while((hashMap[cur].times1 != 0 || hashMap[cur].times2) &amp;&amp; hashMap[cur].value != nums1[i])\n        {\n            cur++;\n            cur = cur == mapSize ? 0 : cur;\n        }\n        if (hashMap[cur].times1 == 0 &amp;&amp; hashMap[cur].times2 == 0)\n            hashMap[cur].value = nums1[i];\n        hashMap[cur].times1++;\n    }\n    for (i = 0; i &lt; nums2Size; i++)\n    {\n        cur = (nums2[i] &gt; 0 ? 1 : -1) * (nums2[i] % mapSize);\n        while((hashMap[cur].times1 != 0 || hashMap[cur].times2) &amp;&amp; hashMap[cur].value != nums2[i])\n        {\n            cur++;\n            cur = cur == mapSize ? 0 : cur;\n        }\n        if (hashMap[cur].times1 == 0 &amp;&amp; hashMap[cur].times2 == 0)\n            hashMap[cur].value = nums2[i];\n        hashMap[cur].times2++;\n    }\n    for (i = 0; i &lt; mapSize; i++)\n        while ((hashMap[i].times2--) &amp;&amp; (hashMap[i].times1--))\n            ret[(*returnSize)++] = hashMap[i].value;\n    return ret;\n}\n</code></pre>","tags":["\u6392\u5e8f","\u54c8\u5e0c\u8868","\u53cc\u6307\u9488","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/36/","title":"36. \u200b\u6709\u6548\u200b\u7684\u200b\u6570\u72ec","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/36/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5224\u65ad\u200b\u4e00\u4e2a\u200b\u00a09x9 \u200b\u7684\u200b\u6570\u72ec\u200b\u662f\u5426\u200b\u6709\u6548\u200b\u3002\u200b\u53ea\u200b\u9700\u8981\u200b\u6839\u636e\u200b\u4ee5\u4e0b\u200b\u89c4\u5219\u200b\uff0c\u200b\u9a8c\u8bc1\u200b\u5df2\u7ecf\u200b\u586b\u5165\u200b\u7684\u200b\u6570\u5b57\u200b\u662f\u5426\u200b\u6709\u6548\u200b\u5373\u53ef\u200b\u3002</p> <ol> <li>\u200b\u6570\u5b57\u200b<code>1-9</code>\u200b\u5728\u200b\u6bcf\u200b\u4e00\u884c\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002</li> <li>\u200b\u6570\u5b57\u200b<code>1-9</code>\u200b\u5728\u200b\u6bcf\u200b\u4e00\u5217\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002</li> <li>\u200b\u6570\u5b57\u200b<code>1-9</code>\u200b\u5728\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u4ee5\u200b\u7c97\u200b\u5b9e\u7ebf\u200b\u5206\u9694\u200b\u7684\u200b<code>3x3</code>\u200b\u5bab\u5185\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002</li> </ol> <p></p> <p>\u200b\u4e0a\u56fe\u200b\u662f\u200b\u4e00\u4e2a\u200b\u90e8\u5206\u200b\u586b\u5145\u200b\u7684\u200b\u6709\u6548\u200b\u7684\u200b\u6570\u72ec\u200b\u3002</p> <p>\u200b\u6570\u72ec\u200b\u90e8\u5206\u200b\u7a7a\u683c\u200b\u5185\u200b\u5df2\u200b\u586b\u5165\u200b\u4e86\u200b\u6570\u5b57\u200b\uff0c\u200b\u7a7a\u767d\u200b\u683c\u7528\u200b<code>'.'</code>\u200b\u8868\u793a\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\n[\n  [\"5\",\"3\",\".\",\".\",\"7\",\".\",\".\",\".\",\".\"],\n  [\"6\",\".\",\".\",\"1\",\"9\",\"5\",\".\",\".\",\".\"],\n  [\".\",\"9\",\"8\",\".\",\".\",\".\",\".\",\"6\",\".\"],\n  [\"8\",\".\",\".\",\".\",\"6\",\".\",\".\",\".\",\"3\"],\n  [\"4\",\".\",\".\",\"8\",\".\",\"3\",\".\",\".\",\"1\"],\n  [\"7\",\".\",\".\",\".\",\"2\",\".\",\".\",\".\",\"6\"],\n  [\".\",\"6\",\".\",\".\",\".\",\".\",\"2\",\"8\",\".\"],\n  [\".\",\".\",\".\",\"4\",\"1\",\"9\",\".\",\".\",\"5\"],\n  [\".\",\".\",\".\",\".\",\"8\",\".\",\".\",\"7\",\"9\"]\n]\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\n[\n\u00a0 [\"8\",\"3\",\".\",\".\",\"7\",\".\",\".\",\".\",\".\"],\n\u00a0 [\"6\",\".\",\".\",\"1\",\"9\",\"5\",\".\",\".\",\".\"],\n\u00a0 [\".\",\"9\",\"8\",\".\",\".\",\".\",\".\",\"6\",\".\"],\n\u00a0 [\"8\",\".\",\".\",\".\",\"6\",\".\",\".\",\".\",\"3\"],\n\u00a0 [\"4\",\".\",\".\",\"8\",\".\",\"3\",\".\",\".\",\"1\"],\n\u00a0 [\"7\",\".\",\".\",\".\",\"2\",\".\",\".\",\".\",\"6\"],\n\u00a0 [\".\",\"6\",\".\",\".\",\".\",\".\",\"2\",\"8\",\".\"],\n\u00a0 [\".\",\".\",\".\",\"4\",\"1\",\"9\",\".\",\".\",\"5\"],\n\u00a0 [\".\",\".\",\".\",\".\",\"8\",\".\",\".\",\"7\",\"9\"]\n]\n\u200b\u8f93\u51fa\u200b: false\n\u200b\u89e3\u91ca\u200b: \u200b\u9664\u4e86\u200b\u7b2c\u4e00\u884c\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u6570\u5b57\u200b\u4ece\u200b 5 \u200b\u6539\u4e3a\u200b 8 \u200b\u4ee5\u5916\u200b\uff0c\u200b\u7a7a\u683c\u200b\u5185\u200b\u5176\u4ed6\u200b\u6570\u5b57\u200b\u5747\u200b\u4e0e\u200b \u200b\u793a\u4f8b\u200b1 \u200b\u76f8\u540c\u200b\u3002\n     \u200b\u4f46\u200b\u7531\u4e8e\u200b\u4f4d\u4e8e\u200b\u5de6\u4e0a\u89d2\u200b\u7684\u200b 3x3 \u200b\u5bab\u5185\u200b\u6709\u200b\u4e24\u4e2a\u200b 8 \u200b\u5b58\u5728\u200b, \u200b\u56e0\u6b64\u200b\u8fd9\u4e2a\u200b\u6570\u72ec\u200b\u662f\u200b\u65e0\u6548\u200b\u7684\u200b\u3002\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li>\u200b\u4e00\u4e2a\u200b\u6709\u6548\u200b\u7684\u200b\u6570\u72ec\u200b\uff08\u200b\u90e8\u5206\u200b\u5df2\u200b\u88ab\u200b\u586b\u5145\u200b\uff09\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u53ef\u89e3\u200b\u7684\u200b\u3002</li> <li>\u200b\u53ea\u200b\u9700\u8981\u200b\u6839\u636e\u200b\u4ee5\u4e0a\u200b\u89c4\u5219\u200b\uff0c\u200b\u9a8c\u8bc1\u200b\u5df2\u7ecf\u200b\u586b\u5165\u200b\u7684\u200b\u6570\u5b57\u200b\u662f\u5426\u200b\u6709\u6548\u200b\u5373\u53ef\u200b\u3002</li> <li>\u200b\u7ed9\u200b\u5b9a\u6570\u200b\u72ec\u200b\u5e8f\u5217\u200b\u53ea\u200b\u5305\u542b\u200b\u6570\u5b57\u200b\u00a01-9\u00a0\u200b\u548c\u200b\u5b57\u7b26\u200b\u00a0'.'\u00a0\u3002</li> <li>\u200b\u7ed9\u200b\u5b9a\u6570\u200b\u72ec\u200b\u6c38\u8fdc\u200b\u662f\u200b\u00a09x9\u00a0\u200b\u5f62\u5f0f\u200b\u7684\u200b\u3002</li> </ul> <p>Reference</p>","tags":["\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/36/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6839\u636e\u200b\u5b9a\u4e49\u200b\u76f4\u63a5\u5224\u65ad\u200b\u8f93\u5165\u200b\u7684\u200b\u6570\u7ec4\u200b\u662f\u5426\u200b\u6ee1\u8db3\u200b\u6570\u72ec\u200b\u7684\u200b3\u200b\u4e2a\u200b\u6761\u4ef6\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>bool isValidSudoku(char** board, int boardSize, int* boardColSize){\n    int i = 0, j = 0, positions1[boardSize], positions2[boardSize], k = 0;\n    for (i = 0; i &lt; boardSize; i++)\n    {\n        memset(positions1, 0, sizeof(int) * boardSize);\n        memset(positions2, 0, sizeof(int) * boardSize);\n        for (j = 0; j &lt; boardSize; j++)\n        {\n            if (board[i][j] &gt; '0' &amp;&amp; positions1[board[i][j] - '1'])\n                return false;\n            if (board[j][i] &gt; '0' &amp;&amp; positions2[board[j][i] - '1'])\n                return false;\n            if (board[i][j] &gt; '0')\n                positions1[board[i][j] - '1'] = 1;\n            if (board[j][i] &gt; '0')\n                positions2[board[j][i] - '1'] = 1;\n        }\n    }\n    for (i = 0; i &lt; 3; i++)\n    {\n        for (j = 0; j &lt; 3; j++)\n        {\n            memset(positions1, 0, sizeof(int) * boardSize);\n            for (k = 0; k &lt; boardSize; k++)\n            {\n                if (board[i * 3 + k % 3][j * 3 + k / 3] &gt; '0' &amp;&amp; positions1[board[i * 3 + k % 3][j * 3 + k / 3] - '1'])\n                    return false;\n                if (board[i * 3 + k % 3][j * 3 + k / 3] &gt; '0')\n                    positions1[board[i * 3 + k % 3][j * 3 + k / 3] - '1'] = 1;\n            }\n        }\n    }\n    return true;\n}\n</code></pre>","tags":["\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/368/","title":"368. \u200b\u6700\u5927\u200b\u6574\u9664\u200b\u5b50\u96c6","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u5b66","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/368/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u7531\u200b \u200b\u65e0\u200b\u91cd\u590d\u200b \u200b\u6b63\u6574\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b <code>nums</code> \uff0c\u200b\u8bf7\u200b\u4f60\u200b\u627e\u51fa\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5176\u4e2d\u200b\u6700\u5927\u200b\u7684\u200b\u6574\u9664\u200b\u5b50\u96c6\u200b <code>answer</code> \uff0c\u200b\u5b50\u200b\u96c6\u4e2d\u200b\u6bcf\u4e00\u200b\u5143\u7d20\u200b\u5bf9\u200b <code>(answer[i], answer[j])</code> \u200b\u90fd\u200b\u5e94\u5f53\u200b\u6ee1\u8db3\u200b\uff1a</p> <ul> <li><code>answer[i] % answer[j] == 0</code> \uff0c\u200b\u6216\u200b</li> <li><code>answer[j] % answer[i] == 0</code></li> </ul> <p>\u200b\u5982\u679c\u200b\u5b58\u5728\u200b\u591a\u4e2a\u200b\u6709\u6548\u200b\u89e3\u200b\u5b50\u96c6\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5176\u4e2d\u200b\u4efb\u4f55\u200b\u4e00\u4e2a\u200b\u5747\u200b\u53ef\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1,2,3]\n\u200b\u8f93\u51fa\u200b\uff1a[1,2]\n\u200b\u89e3\u91ca\u200b\uff1a[1,3] \u200b\u4e5f\u200b\u4f1a\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u6b63\u786e\u200b\u7b54\u6848\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1,2,4,8]\n\u200b\u8f93\u51fa\u200b\uff1a[1,2,4,8]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 1000</code></li> <li><code>1 &lt;= nums[i] &lt;= 2 * 10^9</code></li> <li><code>nums</code> \u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u6574\u6570\u200b \u200b\u4e92\u4e0d\u200b\u76f8\u540c\u200b</li> </ul> <p>Reference</p>","tags":["\u6570\u5b66","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/368/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u6574\u6570\u200b\u4e4b\u95f4\u200b\u7684\u200b\u6574\u9664\u200b\u5173\u7cfb\u200b\u6ee1\u8db3\u200b\u81ea\u53cd\u6027\u200b\u3001\u200b\u53cd\u5bf9\u79f0\u6027\u200b\u548c\u200b\u4f20\u9012\u6027\u200b\uff0c\u200b\u4f7f\u7528\u200b\\(a|b\\)\u200b\u8868\u793a\u200b\\(a\\)\u200b\u53ef\u4ee5\u200b\u88ab\u200b\\(b\\)\u200b\u6574\u9664\u200b\uff0c\u200b\u5219\u200b\uff1a</p> <ol> <li>\u200b\u81ea\u53cd\u6027\u200b\uff1a\\(a|a = T\\)</li> <li>\u200b\u53cd\u5bf9\u79f0\u6027\u200b\uff1a\\(a|b \\overline \\lor b|a = T\\)</li> <li>\u200b\u4f20\u9012\u6027\u200b\uff1a\\(a|b \\land b|c\\Rightarrow a|c\\)</li> </ol> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u6574\u9664\u200b\u5173\u7cfb\u200b\u662f\u200b\u6b63\u6574\u6570\u200b\u96c6\u200b\\(N_+\\)\u200b\u4e0a\u200b\u7684\u200b\u76f8\u5bb9\u200b\u5173\u7cfb\u200b\u4e0e\u200b\u504f\u5e8f\u200b\u5173\u7cfb\u200b\u3002</p> <p>\u200b\u8bbe\u200b\u6b63\u6574\u6570\u200b\u96c6\u5408\u200b\u4e3a\u200b\\(S\\)\uff0c\u200b\u4ee5\u4e0b\u200b\u57fa\u4e8e\u200b\u504f\u5e8f\u200b\u5173\u7cfb\u200b\\(R\\)\u200b\u4e0e\u200b\u504f\u5e8f\u200b\u96c6\u200b\\(S\\)\u200b\u5b9a\u4e49\u200b\u94fe\u200b\u7684\u200b\u6982\u5ff5\u200b\uff1a</p> <ol> <li>\u200b\u82e5\u200b\\(a\\in S, b\\in S\\)\uff0c\u200b\u4e14\u200b\\(a|b \\lor b|a\\)\uff0c\u200b\u5219\u200b\\(a, b\\)\u200b\u662f\u200b\u53ef\u6bd4\u200b\u7684\u200b\uff1b</li> <li>\u200b\u82e5\u200b\u96c6\u5408\u200b\\(B\\subseteq S\\)\uff0c\u200b\u4e14\u200b\u5bf9\u4e8e\u200b\\(B\\)\u200b\u4e2d\u200b\u4efb\u610f\u200b\u4e00\u5bf9\u200b\u5143\u7d20\u200b\\(x, y\\)\u200b\u90fd\u200b\u662f\u200b\u53ef\u6bd4\u200b\u7684\u200b\uff0c\u200b\u5219\u200b\u79f0\u200b\\(B\\)\u200b\u4e3a\u200b\u504f\u5e8f\u200b\u96c6\u200b\\(S\\)\u200b\u4e0a\u200b\u7684\u200b\u94fe\u200b\u3002</li> <li>\u200b\u504f\u5e8f\u200b\u96c6\u200b\\(S\\)\u200b\u4e2d\u200b\u5305\u542b\u200b\u5143\u7d20\u200b\u6700\u591a\u200b\u7684\u200b\u94fe\u200b\u79f0\u4e3a\u200b\u6700\u957f\u200b\u94fe\u200b\u3002</li> </ol> <p>\u200b\u7531\u6b64\u53ef\u89c1\u200b\uff0c\u200b\u6240\u6c42\u200b\u7684\u200b\u6700\u5927\u200b\u6574\u9664\u200b\u5b50\u96c6\u200b\u5373\u200b\u4e3a\u200b\\(S\\)\u200b\u4e0a\u200b\u7684\u200b\u6700\u957f\u200b\u94fe\u200b\u3002</p> <p>\u200b\u504f\u5e8f\u200b\u96c6\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u54c8\u65af\u200b\u56fe\u200b\u8fdb\u884c\u200b\u8868\u793a\u200b\uff0c\u200b\u5982\u200b\u96c6\u5408\u200b\\(\\{2,3,4,6,8\\}\\)\u200b\u4e0a\u200b\u7684\u200b\u504f\u5e8f\u200b\u5173\u7cfb\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</p> <p></p> <p>\u200b\u54c8\u65af\u200b\u56fe\u662f\u200b\u6709\u200b\u5411\u200b\u65e0\u200b\u73af\u56fe\u200b\uff0c\u200b\u4f46\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u8fde\u901a\u200b\u3002\u200b\u94fe\u200b\u5728\u200b\u54c8\u65af\u200b\u56fe\u4e2d\u200b\u8868\u73b0\u200b\u4e3a\u200b\u4e00\u6bb5\u200b\u8def\u5f84\u200b\uff0c\u200b\u6700\u957f\u200b\u94fe\u5373\u200b\u4e3a\u200b\u54c8\u65af\u200b\u56fe\u4e2d\u200b\u7684\u200b\u6700\u957f\u200b\u8def\u5f84\u200b\uff0c\u200b\u5373\u200b\u5404\u200b\u5206\u91cf\u200b\u7684\u200b\u6700\u957f\u200b\u76f4\u5f84\u200b\uff0c\u200b\u5982\u56fe\u200b\u4e2d\u52a0\u200b\u7c97\u7ebf\u6761\u200b\u6240\u793a\u200b\u3002</p> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u56fe\u200b\u89e3\u51b3\u200b\u672c\u9898\u200b\uff0c\u200b\u7b2c\u4e00\u6b65\u200b\u6784\u9020\u200b\u504f\u5e8f\u200b\u96c6\u200b\u7684\u200b\u54c8\u65af\u200b\u56fe\u200b\uff0c\u200b\u6d41\u7a0b\u200b\u5982\u4e0b\u200b\uff1a</p> <ol> <li>\u200b\u5bf9\u200b\u96c6\u5408\u200b\u6392\u5e8f\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u96c6\u5408\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u5728\u200b\u56fe\u200b\u4e2d\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u4e0e\u200b\u4e4b\u200b\u5bf9\u5e94\u200b\uff1b</li> <li>\u200b\u4ece\u540e\u200b\u5411\u524d\u200b\u904d\u5386\u200b\u96c6\u5408\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\\(a_i\\)\uff1a<ol> <li>\u200b\u4ece\u200b\u7b2c\u200b\\(i + 1\\)\u200b\u4e2a\u200b\u5143\u7d20\u200b\u5f00\u59cb\u200b\uff0c\u200b\u5411\u200b\u540e\u200b\u904d\u5386\u200b\u96c6\u5408\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\\(a_j\\)\uff0c\u200b\u82e5\u200b\\(a_j|a_i\\)\uff0c\u200b\u5e76\u4e14\u200b\u4e0d\u200b\u5b58\u5728\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u8282\u70b9\u200b\u4e0e\u200b\\(a_i\\)\u200b\u76f8\u8fde\u200b\uff0c\u200b\u5219\u200b\u5c06\u200b\u8282\u70b9\u200b\\(i\\)\u200b\u4e0e\u200b\u8282\u70b9\u200b\\(j\\)\u200b\u76f8\u8fde\u63a5\u200b\u3002</li> </ol> </li> </ol> <p>\u200b\u4e3a\u4e86\u200b\u5feb\u901f\u200b\u6c42\u51fa\u200b\u56fe\u200b\u7684\u200b\u76f4\u5f84\u200b\uff0c\u200b\u53ef\u200b\u5728\u200b\u8282\u70b9\u200b\u4e2d\u200b\u5b58\u50a8\u200b\u4e24\u4e2a\u200b\u503c\u200b\uff0c\u200b\u5373\u200b\uff1a</p> <ul> <li>\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u8ddd\u79bb\u200b\u672b\u7aef\u200b\u8282\u70b9\u200b\uff08\u200b\u51fa\u5ea6\u200b\u4e3a\u200b\\(0\\)\u200b\u7684\u200b\u8282\u70b9\u200b\uff09\u200b\u7684\u200b\u6700\u5927\u200b\u8ddd\u79bb\u200b</li> <li>\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u8ddd\u79bb\u200b\u672b\u7aef\u200b\u8282\u70b9\u200b\u6700\u5927\u200b\u8ddd\u79bb\u200b\u6240\u200b\u5bf9\u5e94\u200b\u7684\u200b\u51fa\u8fb9\u200b</li> </ul> <p>\u200b\u7531\u4e8e\u200b\u6574\u9664\u200b\u5173\u7cfb\u200b\u8574\u542b\u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b\u5173\u7cfb\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5728\u200b\u6784\u9020\u200b\u54c8\u65af\u200b\u56fe\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u540c\u65f6\u200b\u6784\u9020\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u503c\u200b\u3002</p> <p>\u200b\u7b2c\u4e8c\u6b65\u200b\u5373\u53ef\u200b\u627e\u51fa\u200b\u6700\u957f\u200b\u76f4\u5f84\u200b\u3002\u200b\u904d\u5386\u200b\u6240\u6709\u200b\u5165\u5ea6\u200b\u4e3a\u200b\\(0\\)\u200b\u7684\u200b\u8282\u70b9\u200b\uff0c\u200b\u627e\u51fa\u200b\u5e76\u200b\u8bb0\u4e0b\u200b\u8ddd\u79bb\u200b\u672b\u7aef\u200b\u8282\u70b9\u200b\u6700\u8fdc\u200b\u7684\u200b\u8282\u70b9\u200b\u3002\u200b\u518d\u200b\u6839\u636e\u200b\u8282\u70b9\u200b\u4e2d\u200b\u5b58\u50a8\u200b\u7684\u200b\u51fa\u8fb9\u200b\u4fe1\u606f\u200b\u5373\u53ef\u200b\u5b8c\u6574\u200b\u6784\u9020\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u6700\u5927\u200b\u6574\u9664\u200b\u5b50\u96c6\u200b\u3002</p> <pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\n#define MAX(x, y) ((x) &gt; (y) ? (x) : (y))\nstruct Edge;\nstruct Vertex;\nstruct Edge\n{\n    struct Vertex *dest;\n    struct Edge *next;\n};\nstruct Vertex\n{\n    int val;\n    int status;\n    struct Edge *edges;\n    int height;\n    struct Edge *maxHeight;\n};\nstruct Edge *getEdge(struct Vertex *dest, struct Edge *next)\n{\n    struct Edge *cur = next;\n    while (cur)\n    {\n        if (dest-&gt;val % cur-&gt;dest-&gt;val == 0)\n            return next;\n        cur = cur-&gt;next;\n    }\n    struct Edge *ret = (struct Edge *)malloc(sizeof(struct Edge));\n    ret-&gt;dest = dest;\n    ret-&gt;next = next;\n    return ret;\n}\nint cmp(const void *a, const void *b)\n{\n    return *((int *)a) - *((int *)b);\n}\nvoid dfs(struct Vertex *graph)\n{\n    if (graph == NULL)\n        return;\n    graph-&gt;status = true;\n    struct Edge *edge = graph-&gt;edges;\n    while (edge)\n    {\n        dfs(edge-&gt;dest);\n        edge = edge-&gt;next;\n    }\n}\nint* largestDivisibleSubset(int* nums, int numsSize, int* returnSize){\n    struct Vertex *nodes = (struct Vertex *)malloc(sizeof(struct Vertex) * numsSize), *maxVertex = NULL;\n    int i = 0, j = 0, max = -1, *ret = NULL;\n    qsort(nums, numsSize, sizeof(int), cmp);\n    for (i = 0; i &lt; numsSize; i++)\n    {\n        nodes[i].val = nums[i];\n        nodes[i].status = 0;\n        nodes[i].edges = NULL;\n        nodes[i].height = 0;\n        nodes[i].maxHeight = NULL;\n    }\n    for (i = numsSize - 1; i &gt;= 0; i--)\n    {\n        for (j = i + 1; j &lt; numsSize; j++)\n        {\n            if (nodes[j].val % nodes[i].val)\n                continue;\n            nodes[i].edges = getEdge(nodes + j, nodes[i].edges);\n            if (nodes[i].height &lt; nodes[j].height + 1)\n            {\n                nodes[i].height = nodes[j].height + 1;\n                nodes[i].maxHeight = nodes[i].edges;\n            }\n        }\n    }\n    for (i = 0; i &lt; numsSize; i++)\n    {\n        if (nodes[i].status)\n            continue;\n        if (nodes[i].height &gt; max)\n        {\n            max = nodes[i].height;\n            maxVertex = nodes + i;\n        }\n        dfs(nodes + i);\n    }\n    *returnSize = max + 1;\n    ret = (int *)malloc(sizeof(int) * (max + 1));\n    for (i = 0; i &lt;= max; i++)\n    {\n        ret[i] = maxVertex-&gt;val;\n        if (maxVertex-&gt;maxHeight == NULL)\n            break;\n        maxVertex = maxVertex-&gt;maxHeight-&gt;dest;\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u5b66","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/37/","title":"37. \u200b\u89e3\u6570\u200b\u72ec","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u54c8\u5e0c\u8868","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/37/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u7a0b\u5e8f\u200b\uff0c\u200b\u901a\u8fc7\u200b\u586b\u5145\u200b\u7a7a\u683c\u200b\u6765\u200b\u89e3\u51b3\u200b\u6570\u72ec\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u6570\u72ec\u200b\u7684\u200b\u89e3\u6cd5\u200b\u9700\u200b \u200b\u9075\u5faa\u200b\u5982\u4e0b\u200b\u89c4\u5219\u200b\uff1a</p> <p>\u200b\u6570\u5b57\u200b\u00a0<code>1-9</code>\u00a0\u200b\u5728\u200b\u6bcf\u200b\u4e00\u884c\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002 \u200b\u6570\u5b57\u200b\u00a0<code>1-9</code>\u00a0\u200b\u5728\u200b\u6bcf\u200b\u4e00\u5217\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002 \u200b\u6570\u5b57\u200b\u00a0<code>1-9</code>\u00a0\u200b\u5728\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u4ee5\u200b\u7c97\u200b\u5b9e\u7ebf\u200b\u5206\u9694\u200b\u7684\u200b\u00a0<code>3x3</code>\u00a0\u200b\u5bab\u5185\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002\uff08\u200b\u8bf7\u200b\u53c2\u8003\u200b\u793a\u4f8b\u200b\u56fe\u200b\uff09 \u200b\u6570\u72ec\u200b\u90e8\u5206\u200b\u7a7a\u683c\u200b\u5185\u200b\u5df2\u200b\u586b\u5165\u200b\u4e86\u200b\u6570\u5b57\u200b\uff0c\u200b\u7a7a\u767d\u200b\u683c\u7528\u200b\u00a0<code>'.'</code>\u00a0\u200b\u8868\u793a\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = [\n    [\"5\",\"3\",\".\",\".\",\"7\",\".\",\".\",\".\",\".\"],\n    [\"6\",\".\",\".\",\"1\",\"9\",\"5\",\".\",\".\",\".\"],\n    [\".\",\"9\",\"8\",\".\",\".\",\".\",\".\",\"6\",\".\"],\n    [\"8\",\".\",\".\",\".\",\"6\",\".\",\".\",\".\",\"3\"],\n    [\"4\",\".\",\".\",\"8\",\".\",\"3\",\".\",\".\",\"1\"],\n    [\"7\",\".\",\".\",\".\",\"2\",\".\",\".\",\".\",\"6\"],\n    [\".\",\"6\",\".\",\".\",\".\",\".\",\"2\",\"8\",\".\"],\n    [\".\",\".\",\".\",\"4\",\"1\",\"9\",\".\",\".\",\"5\"],\n    [\".\",\".\",\".\",\".\",\"8\",\".\",\".\",\"7\",\"9\"]\n]\n\u200b\u8f93\u51fa\u200b\uff1a[\n    [\"5\",\"3\",\"4\",\"6\",\"7\",\"8\",\"9\",\"1\",\"2\"],\n    [\"6\",\"7\",\"2\",\"1\",\"9\",\"5\",\"3\",\"4\",\"8\"],\n    [\"1\",\"9\",\"8\",\"3\",\"4\",\"2\",\"5\",\"6\",\"7\"],\n    [\"8\",\"5\",\"9\",\"7\",\"6\",\"1\",\"4\",\"2\",\"3\"],\n    [\"4\",\"2\",\"6\",\"8\",\"5\",\"3\",\"7\",\"9\",\"1\"],\n    [\"7\",\"1\",\"3\",\"9\",\"2\",\"4\",\"8\",\"5\",\"6\"],\n    [\"9\",\"6\",\"1\",\"5\",\"3\",\"7\",\"2\",\"8\",\"4\"],\n    [\"2\",\"8\",\"7\",\"4\",\"1\",\"9\",\"6\",\"3\",\"5\"],\n    [\"3\",\"4\",\"5\",\"2\",\"8\",\"6\",\"1\",\"7\",\"9\"]\n]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u8f93\u5165\u200b\u7684\u200b\u6570\u72ec\u200b\u5982\u4e0a\u56fe\u200b\u6240\u793a\u200b\uff0c\u200b\u552f\u4e00\u200b\u6709\u6548\u200b\u7684\u200b\u89e3\u51b3\u65b9\u6848\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a\n</code></pre> <p></p> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>board.length == 9</code></li> <li><code>board[i].length == 9</code></li> <li><code>board[i][j]</code> \u200b\u662f\u200b\u4e00\u4f4d\u200b\u6570\u5b57\u200b\u6216\u8005\u200b <code>'.'</code></li> <li>\u200b\u9898\u76ee\u200b\u6570\u636e\u200b \u200b\u4fdd\u8bc1\u200b \u200b\u8f93\u5165\u200b\u6570\u72ec\u4ec5\u200b\u6709\u200b\u4e00\u4e2a\u200b\u89e3\u200b</li> </ul> <p>Reference</p>","tags":["\u54c8\u5e0c\u8868","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/37/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u56de\u6eaf\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u7a7a\u200b\u5355\u5143\u683c\u200b\uff0c\u200b\u68c0\u67e5\u200b\u5176\u200b\u6240\u5728\u200b\u884c\u200b\uff0c\u200b\u6240\u5728\u200b\u5217\u200b\u4e0e\u200b\u6240\u5728\u200b\u65b9\u683c\u200b\uff0c\u200b\u9010\u4e2a\u200b\u68c0\u67e5\u200b\u5176\u200b\u53ef\u4ee5\u200b\u586b\u5165\u200b\u7684\u200b\u6570\u503c\u200b\u3002\u200b\u5982\u679c\u200b \u200b\u6709\u200b\u53ef\u4ee5\u200b\u586b\u5165\u200b\u7684\u200b\u5b57\u7b26\u200b\uff0c\u200b\u5219\u200b\u4f9d\u6b21\u200b\u8fdb\u884c\u200b\u5c1d\u8bd5\u200b\u3002\u200b\u82e5\u200b\u67d0\u4e2a\u200b\u7a7a\u683c\u200b\u6ca1\u6709\u200b\u4efb\u4f55\u200b\u53ef\u4ee5\u200b\u586b\u5165\u200b\u7684\u200b\u6570\u503c\u200b\uff0c\u200b\u6216\u200b\u6240\u6709\u200b\u53ef\u200b\u586b\u5165\u200b\u7684\u200b\u503c\u200b\u90fd\u200b\u662f\u200b\u9519\u8bef\u200b\u7684\u200b \uff0c\u200b\u610f\u5473\u7740\u200b\u5f53\u524d\u200b\u641c\u7d22\u200b\u8def\u5f84\u200b\u5b58\u5728\u200b\u9519\u8bef\u200b\uff0c\u200b\u9700\u8981\u200b\u8fd4\u56de\u200b\u4e0a\u200b\u4e00\u4e2a\u200b\u72b6\u6001\u200b\u8fdb\u884c\u200b\u4e0b\u200b\u4e00\u6b65\u200b\u5c1d\u8bd5\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u65b9\u683c\u200b\u88ab\u200b\u5b8c\u5168\u200b\u586b\u6ee1\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u7acb\u5373\u200b\u8fd4\u56de\u200b\u3002\u200b\u5e76\u200b\u901a\u8fc7\u200b\u8fd4\u56de\u503c\u200b\u544a\u8bc9\u200b\u4e0a\u200b\u4e00\u6b65\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u8fd4\u56de\u200b\u7ed3\u679c\u200b\u3002</p> <pre><code>bool dfs(char **board, int boardSize, int *boardColSize)\n{\n    int x = 0, y = 0, i = 0, j = 0;\n    bool flag = true, candidate[9] = {false};\n    for (x = 0; x &lt; 9 &amp;&amp; flag; x++)\n        for (y = 0; y &lt; 9 &amp;&amp; flag; y++)\n            if (board[x][y] == '.')\n                flag = false;\n    if (flag)\n        return true;\n    x--;\n    y--;\n    for (i = 0; i &lt; 9; i++)\n    {\n        if (board[x][i] != '.')\n            candidate[board[x][i] - '1'] = true;\n        if (board[i][y] != '.')\n            candidate[board[i][y] - '1'] = true;\n        if (board[3 * (x / 3) + i / 3][3 * (y / 3) + i % 3] != '.')\n            candidate[board[3 * (x / 3) + i / 3][3 * (y / 3) + i % 3] - '1'] = true;\n    }\n    for (i = 0; i &lt; 9 &amp;&amp; !flag; i++)\n    {\n        if (!candidate[i])\n        {\n            board[x][y] = '1' + i;\n            flag = flag || dfs(board, boardSize, boardColSize);\n            if (flag)\n                return flag;\n            board[x][y] = '.';\n        }\n    }\n    return flag;\n}\n\nvoid solveSudoku(char** board, int boardSize, int* boardColSize){\n    dfs(board, boardSize, boardColSize);\n}\n</code></pre>","tags":["\u54c8\u5e0c\u8868","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/377/","title":"377. \u200b\u7ec4\u5408\u200b\u603b\u548c\u200b \u2163","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/377/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u7531\u200b \u200b\u4e0d\u540c\u200b \u200b\u6574\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u6570\u7ec4\u200b <code>nums</code> \uff0c\u200b\u548c\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u200b\u6574\u6570\u200b <code>target</code> \u3002\u200b\u8bf7\u200b\u4f60\u200b\u4ece\u200b <code>nums</code> \u200b\u4e2d\u200b\u627e\u51fa\u200b\u5e76\u200b\u8fd4\u56de\u200b\u603b\u548c\u200b\u4e3a\u200b <code>target</code> \u200b\u7684\u200b\u5143\u7d20\u200b\u7ec4\u5408\u200b\u7684\u200b\u4e2a\u6570\u200b\u3002</p> <p>\u200b\u9898\u76ee\u200b\u6570\u636e\u200b\u4fdd\u8bc1\u200b\u7b54\u6848\u200b\u7b26\u5408\u200b <code>32</code> \u200b\u4f4d\u200b\u6574\u6570\u200b\u8303\u56f4\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1,2,3], target = 4\n\u200b\u8f93\u51fa\u200b\uff1a7\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u7684\u200b\u7ec4\u5408\u200b\u4e3a\u200b\uff1a\n(1, 1, 1, 1)\n(1, 1, 2)\n(1, 2, 1)\n(1, 3)\n(2, 1, 1)\n(2, 2)\n(3, 1)\n\u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u987a\u5e8f\u200b\u4e0d\u540c\u200b\u7684\u200b\u5e8f\u5217\u200b\u88ab\u200b\u89c6\u4f5c\u200b\u4e0d\u540c\u200b\u7684\u200b\u7ec4\u5408\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [9], target = 3\n\u200b\u8f93\u51fa\u200b\uff1a0\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 200</code></li> <li><code>1 &lt;= nums[i] &lt;= 1000</code></li> <li><code>nums</code> \u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b \u200b\u4e92\u4e0d\u200b\u76f8\u540c\u200b</li> <li><code>1 &lt;= target &lt;= 1000</code></li> </ul> <p>\u200b\u8fdb\u9636\u200b\uff1a \u200b\u5982\u679c\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u542b\u6709\u200b\u8d1f\u6570\u200b\u4f1a\u200b\u53d1\u751f\u200b\u4ec0\u4e48\u200b\uff1f\u200b\u95ee\u9898\u200b\u4f1a\u200b\u4ea7\u751f\u200b\u4f55\u79cd\u200b\u53d8\u5316\u200b\uff1f\u200b\u5982\u679c\u200b\u5141\u8bb8\u200b\u8d1f\u6570\u200b\u51fa\u73b0\u200b\uff0c\u200b\u9700\u8981\u200b\u5411\u200b\u9898\u76ee\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u54ea\u4e9b\u200b\u9650\u5236\u200b\u6761\u4ef6\u200b\uff1f</p> <p>Reference</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/377/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u672c\u9898\u200b\u7684\u200b\u601d\u8def\u200b\u4e0e\u200b70. \u200b\u722c\u697c\u68af\u200b\u76f8\u540c\u200b\u3002\u200b\u6b64\u5904\u200b\u4f7f\u7528\u200b\u8bb0\u5fc6\u200b\u5316\u200b\u9012\u5f52\u200b\u89e3\u51b3\u200b\u3002</p> <p>\u200b\u8bbe\u200b\\(f(x; S)\\)\u200b\u4e3a\u200b\u7ed9\u5b9a\u200b\u6574\u6570\u200b\u96c6\u5408\u200b\\(S\\)\uff0c\u200b\u7ec4\u6210\u200b\u76ee\u6807\u503c\u200b\\(x\\)\u200b\u7684\u200b\u7ec4\u5408\u200b\u6570\u91cf\u200b\uff0c\u200b\u6211\u4eec\u200b\u9ed8\u8ba4\u200b\\(0\\in S\\)\u3002</p> <ul> <li>\u200b\u5f53\u200b\\(x &lt; 0\\)\uff0c\u200b\u7531\u4e8e\u200b\\(\\forall y\\in S, y\\geq 0\\)\uff0c\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u5f97\u5230\u200b\u5c0f\u4e8e\u200b\\(0\\)\u200b\u7684\u200b\u6570\u503c\u200b\uff0c\u200b\u6709\u200b\\(f(x; S) = 0\\)</li> <li>\\(x = 0\\)\uff0c\u200b\u7531\u4e8e\u200b\u9ed8\u8ba4\u200b\\(0\\in S\\)\uff0c\u200b\u6709\u200b\\(f(x; S) = 1\\)\u3002</li> <li>\u200b\u5f53\u200b\\(x &lt; 0\\)\uff0c\\(S\\)\u200b\u4e2d\u200b\u7684\u200b\u4efb\u610f\u200b\u5143\u7d20\u200b\\(y\\)\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u9009\u62e9\u200b\uff0c\u200b\u9009\u62e9\u200b\\(y\\)\u200b\u5f97\u5230\u200b\u7684\u200b\u7ec4\u5408\u200b\u4e2a\u6570\u200b\u4e3a\u200b\\(f(x - y; S)\\)</li> </ul> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5f97\u5230\u200b\\(f(x; S)\\)\u200b\u7684\u200b\u8868\u8fbe\u5f0f\u200b\uff1a</p> \\[ f(x; S)=\\left\\{ \\begin{aligned}     &amp; 0 &amp; x &lt; 0 \\\\     &amp; 1 &amp; x = 0 \\\\     &amp; \\sum_{y\\in S} f(x - y; S) &amp; x &gt; 0 \\end{aligned} \\right . \\] <p>\u200b\u7531\u4e8e\u200b\u8ba1\u7b97\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u4f1a\u200b\u91cd\u590d\u200b\u8ba1\u7b97\u200b\u67d0\u4e9b\u200b\\(f(x; S)\\)\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\u8bb0\u5f55\u200b\u6240\u6709\u200b\u5df2\u7ecf\u200b\u8ba1\u7b97\u200b\u8fc7\u200b\u7684\u200b\\(f(x; S)\\)\u3002</p> <pre><code>int helper(int *dp, int *nums, int numsSize, int target)\n{\n    if (target == 0)\n        return 1;\n    if (target &lt; 0)\n        return 0;\n    if (dp[target - 1] &lt; 0)\n    {\n        int ret = 0, i = 0;\n        for (i = 0; i &lt; numsSize; i++)\n            ret += helper(dp, nums, numsSize, target - nums[i]);\n        dp[target - 1] = ret;\n    }\n    return dp[target - 1];\n}\n\nint combinationSum4(int* nums, int numsSize, int target) {\n    int *dp = (int *)malloc(sizeof(int) * target);\n    memset(dp, 0xff, sizeof(int) * target);\n    return helper(dp, nums, numsSize, target);\n}\n</code></pre>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/38/","title":"38. \u200b\u5916\u89c2\u200b\u6570\u5217","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/38/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6b63\u6574\u6570\u200b\\(n\\)\uff08\\(1 \\leq\u00a0n\u00a0\\leq 30\\)\uff09\uff0c\u200b\u8f93\u51fa\u200b\u5916\u89c2\u200b\u6570\u5217\u200b\u7684\u200b\u7b2c\u200b\\(n\\)\u200b\u9879\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b\uff1a\u200b\u6574\u6570\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e00\u9879\u200b\u5c06\u200b\u8868\u793a\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u3002</p> <p>\u300c\u200b\u5916\u89c2\u200b\u6570\u5217\u200b\u300d\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4ece\u200b\u6570\u5b57\u200b 1 \u200b\u5f00\u59cb\u200b\uff0c\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e00\u9879\u200b\u90fd\u200b\u662f\u200b\u5bf9\u200b\u524d\u200b\u4e00\u9879\u200b\u7684\u200b\u63cf\u8ff0\u200b\u3002\u200b\u524d\u200b\u4e94\u9879\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>1.     1\n2.     11\n3.     21\n4.     1211\n5.     111221\n</code></pre> <p>\u200b\u7b2c\u4e00\u9879\u200b\u662f\u200b\u6570\u5b57\u200b 1</p> <p>\u200b\u63cf\u8ff0\u200b\u524d\u200b\u4e00\u9879\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u6570\u662f\u200b 1 \u200b\u5373\u200b \u201c\u200b\u4e00\u4e2a\u200b 1 \u201d\uff0c\u200b\u8bb0\u200b\u4f5c\u200b 11</p> <p>\u200b\u63cf\u8ff0\u200b\u524d\u200b\u4e00\u9879\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u6570\u662f\u200b 11 \u200b\u5373\u200b \u201c\u200b\u4e24\u4e2a\u200b 1 \u201d \uff0c\u200b\u8bb0\u200b\u4f5c\u200b 21</p> <p>\u200b\u63cf\u8ff0\u200b\u524d\u200b\u4e00\u9879\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u6570\u662f\u200b 21 \u200b\u5373\u200b \u201c\u200b\u4e00\u4e2a\u200b 2 \u200b\u4e00\u4e2a\u200b 1 \u201d \uff0c\u200b\u8bb0\u200b\u4f5c\u200b 1211</p> <p>\u200b\u63cf\u8ff0\u200b\u524d\u200b\u4e00\u9879\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u6570\u662f\u200b 1211 \u200b\u5373\u200b \u201c\u200b\u4e00\u4e2a\u200b 1 \u200b\u4e00\u4e2a\u200b 2 \u200b\u4e24\u4e2a\u200b 1 \u201d \uff0c\u200b\u8bb0\u200b\u4f5c\u200b 111221</p> <p>\u200b\u793a\u4f8b\u200b\u00a01:</p> <pre><code>\u200b\u8f93\u5165\u200b: 1\n\u200b\u8f93\u51fa\u200b: \"1\"\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u8fd9\u662f\u200b\u4e00\u4e2a\u200b\u57fa\u672c\u200b\u6837\u4f8b\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2:</p> <pre><code>\u200b\u8f93\u5165\u200b: 4\n\u200b\u8f93\u51fa\u200b: \"1211\"\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5f53\u200b n = 3 \u200b\u65f6\u200b\uff0c\u200b\u5e8f\u5217\u200b\u662f\u200b \"21\"\uff0c\u200b\u5176\u4e2d\u200b\u6211\u4eec\u200b\u6709\u200b \"2\" \u200b\u548c\u200b \"1\" \u200b\u4e24\u7ec4\u200b\uff0c\"2\" \u200b\u53ef\u4ee5\u200b\u8bfb\u4f5c\u200b \"12\"\uff0c\u200b\u4e5f\u200b\u5c31\u662f\u200b\u51fa\u73b0\u200b\u9891\u6b21\u200b = 1 \u200b\u800c\u200b \u200b\u503c\u200b = 2\uff1b\u200b\u7c7b\u4f3c\u200b \"1\" \u200b\u53ef\u4ee5\u200b\u8bfb\u4f5c\u200b \"11\"\u3002\u200b\u6240\u4ee5\u200b\u7b54\u6848\u200b\u662f\u200b \"12\" \u200b\u548c\u200b \"11\" \u200b\u7ec4\u5408\u200b\u5728\u200b\u4e00\u8d77\u200b\uff0c\u200b\u4e5f\u200b\u5c31\u662f\u200b \"1211\"\u3002\n</code></pre> <p>Reference</p>","tags":["\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/38/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6839\u636e\u200b\u5b9a\u4e49\u200b\u8fdb\u884c\u200b\u8ba1\u7b97\u200b\u5373\u53ef\u200b\uff0c\u200b\u6ce8\u610f\u200b\u6b21\u6570\u200b\u90e8\u5206\u200b\u53ef\u80fd\u200b\u8d85\u8fc7\u200b10\uff0c\u200b\u800c\u200b\u6570\u5b57\u200b\u90e8\u5206\u200b\u4e0d\u4f1a\u200b\u8d85\u8fc7\u200b10\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>char * toStr(int n)\n{\n    char *ret = (char *)memset(malloc(sizeof(char) * 10), 0, sizeof(char) * 10), temp;\n    int i = 0, head = 0, tail;\n    while (n)\n    {\n        ret[i] = n % 10 + '0';\n        n /= 10;\n        i++;\n    }\n    tail = i - 1;\n    while(head &lt; tail)\n    {\n        temp = ret[head];\n        ret[head] = ret[tail];\n        ret[tail] = temp;\n    }\n    return ret;\n}\n\nchar * countAndSay(int n){\n    if (n == 1)\n    {\n        char *ret = (char *)malloc(sizeof(char) * 2);\n        ret[0] = '1';\n        ret[1] = 0;\n        return ret;\n    }\n    char *rec = countAndSay(n - 1), *cur = rec, flag = *cur, *temp = NULL,\n         *ret = (char *)memset(malloc(sizeof(char) * (strlen(rec) * 2 + 1)), 0, sizeof(char) * (strlen(rec) * 2 + 1)), *retcur = ret;\n    int counter = 0;\n    while(*cur)\n    {\n        if (*cur == flag)\n            counter++;\n        else\n        {\n            temp = toStr(counter);\n            strcpy(retcur, temp);\n            while(*retcur)\n                retcur++;\n            *retcur = flag;\n            *retcur++;\n            flag = *cur;\n            counter = 1;\n            free(temp);\n        }\n        cur++;\n    }\n    temp = toStr(counter);\n    strcpy(retcur, temp);\n    while (*retcur)\n        retcur++;\n    *retcur = flag;\n    free(temp);\n    free(rec);\n    return ret;\n}\n</code></pre>","tags":["\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/384/","title":"384. \u200b\u6253\u4e71\u200b\u6570\u7ec4","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/384/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u6253\u4e71\u200b\u4e00\u4e2a\u200b\u6ca1\u6709\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\u7684\u200b\u6570\u7ec4\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>// \u200b\u4ee5\u200b\u6570\u5b57\u200b\u96c6\u5408\u200b 1, 2 \u200b\u548c\u200b 3 \u200b\u521d\u59cb\u5316\u200b\u6570\u7ec4\u200b\u3002\nint[] nums = {1,2,3};\nSolution solution = new Solution(nums);\n\n// \u200b\u6253\u4e71\u200b\u6570\u7ec4\u200b [1,2,3] \u200b\u5e76\u200b\u8fd4\u56de\u200b\u7ed3\u679c\u200b\u3002\u200b\u4efb\u4f55\u200b [1,2,3]\u200b\u7684\u200b\u6392\u5217\u200b\u8fd4\u56de\u200b\u7684\u200b\u6982\u7387\u200b\u5e94\u8be5\u200b\u76f8\u540c\u200b\u3002\nsolution.shuffle();\n\n// \u200b\u91cd\u8bbe\u200b\u6570\u7ec4\u200b\u5230\u200b\u5b83\u200b\u7684\u200b\u521d\u59cb\u72b6\u6001\u200b[1,2,3]\u3002\nsolution.reset();\n\n// \u200b\u968f\u673a\u200b\u8fd4\u56de\u200b\u6570\u7ec4\u200b[1,2,3]\u200b\u6253\u4e71\u200b\u540e\u200b\u7684\u200b\u7ed3\u679c\u200b\u3002\nsolution.shuffle();\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/384/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a Fisher-Yates \u200b\u6d17\u724c\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u5173\u952e\u5728\u4e8e\u200b\u5b9e\u73b0\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u65e0\u653e\u56de\u200b\u62bd\u6837\u200b\u3002</p> <p>Fisher-Yates \u200b\u6d17\u724c\u200b\u7b97\u6cd5\u200b\u8ddf\u200b\u66b4\u529b\u200b\u7b97\u6cd5\u200b\u5f88\u200b\u50cf\u200b\u3002\u200b\u5728\u200b\u6bcf\u6b21\u200b\u8fed\u4ee3\u200b\u4e2d\u200b\uff0c\u200b\u751f\u6210\u200b\u4e00\u4e2a\u200b\u8303\u56f4\u200b\u5728\u200b\u5f53\u524d\u200b\u4e0b\u6807\u200b\u5230\u200b\u6570\u7ec4\u200b\u672b\u5c3e\u200b\u5143\u7d20\u200b\u4e0b\u6807\u200b\u4e4b\u95f4\u200b\u7684\u200b\u968f\u673a\u200b\u6574\u6570\u200b\u3002\u200b\u63a5\u4e0b\u6765\u200b\uff0c\u200b\u5c06\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u548c\u200b\u968f\u673a\u200b\u9009\u51fa\u200b\u7684\u200b\u4e0b\u6807\u200b\u6240\u6307\u200b\u7684\u200b\u5143\u7d20\u200b\u4e92\u76f8\u4ea4\u6362\u200b - \u200b\u8fd9\u200b\u4e00\u6b65\u200b\u6a21\u62df\u200b\u4e86\u200b\u6bcf\u6b21\u200b\u4ece\u200b \u201c\u200b\u5e3d\u5b50\u200b\u201d \u200b\u91cc\u9762\u200b\u6478\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u5176\u4e2d\u200b\u9009\u53d6\u200b\u4e0b\u6807\u200b\u8303\u56f4\u200b\u7684\u200b\u4f9d\u636e\u200b\u5728\u4e8e\u200b\u6bcf\u4e2a\u200b\u88ab\u200b\u6478\u200b\u51fa\u200b\u7684\u200b\u5143\u7d20\u200b\u90fd\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u518d\u200b\u88ab\u200b\u6478\u51fa\u6765\u200b\u4e86\u200b\u3002\u200b\u6b64\u5916\u200b\u8fd8\u6709\u200b\u4e00\u4e2a\u200b\u9700\u8981\u200b\u6ce8\u610f\u200b\u7684\u200b\u7ec6\u8282\u200b\uff0c\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u662f\u200b\u53ef\u4ee5\u200b\u548c\u200b\u5b83\u200b\u672c\u8eab\u200b\u4e92\u76f8\u4ea4\u6362\u200b\u7684\u200b - \u200b\u5426\u5219\u200b\u751f\u6210\u200b\u6700\u540e\u200b\u7684\u200b\u6392\u5217\u7ec4\u5408\u200b\u7684\u200b\u6982\u7387\u200b\u5c31\u200b\u4e0d\u200b\u5bf9\u200b\u4e86\u200b\u3002</p> <p>Reference</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>typedef struct {\n    int size;\n    int *nums;\n} Solution;\n\n\nSolution* solutionCreate(int* nums, int numsSize) {\n    Solution *ret = (Solution *)malloc(sizeof(Solution));\n    ret-&gt;size = numsSize;\n    ret-&gt;nums = (int *)memcpy(malloc(sizeof(int) * numsSize), nums, sizeof(int) * numsSize);\n    return ret;\n}\n\n/** Resets the array to its original configuration and return it. */\nint* solutionReset(Solution* obj, int* retSize) {\n    *retSize = obj-&gt;size;\n    return obj-&gt;nums;\n}\n\n/** Returns a random shuffling of the array. */\nint* solutionShuffle(Solution* obj, int* retSize) {\n    *retSize = obj-&gt;size;\n    int *ret = (int *)memcpy(malloc(sizeof(int) * obj-&gt;size), obj-&gt;nums, sizeof(int) * obj-&gt;size), i = 0, temp, x, y;\n    unsigned int *seed = malloc(sizeof(unsigned int));\n    *seed += (unsigned int)seed;\n    srand(*seed);\n    for (i = 0; i &lt; obj-&gt;size; i++)\n    {\n        x = i;\n        y = rand() % (obj-&gt;size - i) + i;\n        temp = ret[x];\n        ret[x] = ret[y];\n        ret[y] = temp;\n    }\n    free(seed);\n    return ret;\n}\n\nvoid solutionFree(Solution* obj) {\n    free(obj-&gt;nums);\n    free(obj);\n}\n\n/**\n * Your Solution struct will be instantiated and called as such:\n * Solution* obj = solutionCreate(nums, numsSize);\n * int* param_1 = solutionReset(obj, retSize);\n\n * int* param_2 = solutionShuffle(obj, retSize);\n\n * solutionFree(obj);\n*/\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/387/","title":"387. \u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u552f\u4e00\u200b\u5b57\u7b26","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u54c8\u5e0c\u8868","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/387/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u627e\u5230\u200b\u5b83\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u4e0d\u200b\u91cd\u590d\u200b\u7684\u200b\u5b57\u7b26\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5b83\u200b\u7684\u200b\u7d22\u5f15\u200b\u3002\u200b\u5982\u679c\u200b\u4e0d\u200b\u5b58\u5728\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>-1</code>\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>s = \"leetcode\"\n\u200b\u8fd4\u56de\u200b 0\n\ns = \"loveleetcode\"\n\u200b\u8fd4\u56de\u200b 2\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a \u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u5b9a\u200b\u8be5\u200b\u5b57\u7b26\u4e32\u200b\u53ea\u200b\u5305\u542b\u200b\u5c0f\u5199\u5b57\u6bcd\u200b\u3002</p> <p>Reference</p>","tags":["\u54c8\u5e0c\u8868","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/387/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4f7f\u7528\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\u5b58\u50a8\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u5404\u200b\u5b57\u7b26\u200b\u7684\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u3002\u200b\u904d\u5386\u200b\u5b57\u7b26\u4e32\u200b\u8fdb\u884c\u200b\u7edf\u8ba1\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int firstUniqChar(char * s){\n    int hashMap[26] = {0};\n    char *cur = s;\n    while(*cur)\n    {\n        hashMap[*cur - 'a']++;\n        cur++;\n    }\n    cur = s;\n    while(*cur)\n    {\n        if (hashMap[*cur - 'a'] == 1)\n            return (int)(cur - s);\n        cur++;\n    }\n    return -1;\n}\n</code></pre>","tags":["\u54c8\u5e0c\u8868","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/403/","title":"403. \u200b\u9752\u86d9\u200b\u8fc7\u6cb3","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p> <p>\u200b\u4e00\u53ea\u200b\u9752\u86d9\u200b\u60f3\u8981\u200b\u8fc7\u6cb3\u200b\u3002 \u200b\u5047\u5b9a\u200b\u6cb3\u6d41\u200b\u88ab\u200b\u7b49\u200b\u5206\u4e3a\u200b\u82e5\u5e72\u4e2a\u200b\u5355\u5143\u683c\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5728\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u5355\u5143\u683c\u200b\u5185\u200b\u90fd\u200b\u6709\u200b\u53ef\u80fd\u200b\u653e\u200b\u6709\u200b\u4e00\u5757\u200b\u77f3\u5b50\u200b\uff08\u200b\u4e5f\u200b\u6709\u200b\u53ef\u80fd\u200b\u6ca1\u6709\u200b\uff09\u3002 \u200b\u9752\u86d9\u200b\u53ef\u4ee5\u200b\u8df3\u4e0a\u200b\u77f3\u5b50\u200b\uff0c\u200b\u4f46\u662f\u200b\u4e0d\u200b\u53ef\u4ee5\u200b\u8df3\u5165\u200b\u6c34\u4e2d\u200b\u3002</p> <p>\u200b\u7ed9\u200b\u4f60\u200b\u77f3\u5b50\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u5217\u8868\u200b <code>stones</code>\uff08\u200b\u7528\u200b\u5355\u5143\u683c\u200b\u5e8f\u53f7\u200b \u200b\u5347\u5e8f\u200b \u200b\u8868\u793a\u200b\uff09\uff0c\u00a0\u200b\u8bf7\u200b\u5224\u5b9a\u200b\u9752\u86d9\u200b\u80fd\u5426\u200b\u6210\u529f\u200b\u8fc7\u6cb3\u200b\uff08\u200b\u5373\u200b\u80fd\u5426\u200b\u5728\u200b\u6700\u540e\u200b\u4e00\u6b65\u200b\u8df3\u81f3\u200b\u6700\u540e\u200b\u4e00\u5757\u200b\u77f3\u5b50\u200b\u4e0a\u200b\uff09\u3002</p> <p>\u200b\u5f00\u59cb\u200b\u65f6\u200b\uff0c\u00a0\u200b\u9752\u86d9\u200b\u9ed8\u8ba4\u200b\u5df2\u7ad9\u200b\u5728\u200b\u7b2c\u4e00\u5757\u200b\u77f3\u5b50\u200b\u4e0a\u200b\uff0c\u200b\u5e76\u200b\u53ef\u4ee5\u200b\u5047\u5b9a\u200b\u5b83\u200b\u7b2c\u4e00\u6b65\u200b\u53ea\u80fd\u200b\u8df3\u8dc3\u200b\u4e00\u4e2a\u200b\u5355\u4f4d\u200b\uff08\u200b\u5373\u200b\u53ea\u80fd\u200b\u4ece\u200b\u5355\u5143\u683c\u200b 1 \u200b\u8df3\u81f3\u200b\u5355\u5143\u683c\u200b 2 \uff09\u3002</p> <p>\u200b\u5982\u679c\u200b\u9752\u86d9\u200b\u4e0a\u200b\u4e00\u6b65\u200b\u8df3\u8dc3\u200b\u4e86\u200b\u00a0<code>k</code>\u00a0\u200b\u4e2a\u200b\u5355\u4f4d\u200b\uff0c\u200b\u90a3\u4e48\u200b\u5b83\u200b\u63a5\u4e0b\u6765\u200b\u7684\u200b\u8df3\u8dc3\u200b\u8ddd\u79bb\u200b\u53ea\u80fd\u200b\u9009\u62e9\u200b\u4e3a\u200b\u00a0<code>k - 1</code>\u3001<code>k</code>\u00a0\u200b\u6216\u200b\u00a0<code>k + 1</code> \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u3002\u00a0\u200b\u53e6\u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u9752\u86d9\u200b\u53ea\u80fd\u200b\u5411\u200b\u524d\u65b9\u200b\uff08\u200b\u7ec8\u70b9\u200b\u7684\u200b\u65b9\u5411\u200b\uff09\u200b\u8df3\u8dc3\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1astones = [0,1,3,5,6,8,12,17]\n\u200b\u8f93\u51fa\u200b\uff1atrue\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u9752\u86d9\u200b\u53ef\u4ee5\u200b\u6210\u529f\u200b\u8fc7\u6cb3\u200b\uff0c\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u65b9\u6848\u200b\u8df3\u8dc3\u200b\uff1a\u200b\u8df3\u200b 1 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u5230\u200b\u7b2c\u200b 2 \u200b\u5757\u200b\u77f3\u5b50\u200b, \u200b\u7136\u540e\u200b\u8df3\u200b 2 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u5230\u200b\u7b2c\u200b 3 \u200b\u5757\u200b\u77f3\u5b50\u200b, \u200b\u63a5\u7740\u200b \u200b\u8df3\u200b 2 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u5230\u200b\u7b2c\u200b 4 \u200b\u5757\u200b\u77f3\u5b50\u200b, \u200b\u7136\u540e\u200b\u8df3\u200b 3 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u5230\u200b\u7b2c\u200b 6 \u200b\u5757\u200b\u77f3\u5b50\u200b, \u200b\u8df3\u200b 4 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u5230\u200b\u7b2c\u200b 7 \u200b\u5757\u200b\u77f3\u5b50\u200b, \u200b\u6700\u540e\u200b\uff0c\u200b\u8df3\u200b 5 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u5230\u200b\u7b2c\u200b 8 \u200b\u4e2a\u200b\u77f3\u5b50\u200b\uff08\u200b\u5373\u200b\u6700\u540e\u200b\u4e00\u5757\u200b\u77f3\u5b50\u200b\uff09\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1astones = [0,1,2,3,4,8,9,11]\n\u200b\u8f93\u51fa\u200b\uff1afalse\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u8fd9\u200b\u662f\u56e0\u4e3a\u200b\u7b2c\u200b 5 \u200b\u548c\u200b\u7b2c\u200b 6 \u200b\u4e2a\u200b\u77f3\u5b50\u200b\u4e4b\u95f4\u200b\u7684\u200b\u95f4\u8ddd\u200b\u592a\u200b\u5927\u200b\uff0c\u200b\u6ca1\u6709\u200b\u53ef\u9009\u200b\u7684\u200b\u65b9\u6848\u200b\u4f9b\u200b\u9752\u86d9\u200b\u8df3\u8dc3\u200b\u8fc7\u53bb\u200b\u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>2 &lt;= stones.length &lt;= 2000</code></li> <li><code>0 &lt;= stones[i] &lt;= 2^31 - 1</code></li> <li><code>stones[0] == 0</code></li> </ul> <p>Reference</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/403/#_1","title":"\u9898\u89e3","text":"<p>\u200b\u6839\u636e\u200b\u9898\u610f\u200b\uff0c\u200b\u5728\u200b\u4efb\u4f55\u200b\u72b6\u6001\u200b\uff0c\u200b\u9752\u86d9\u200b\u90fd\u200b\u6700\u200b\u591a\u200b\u53ea\u6709\u200b\u4e09\u79cd\u200b\u72b6\u6001\u200b\u53ef\u4ee5\u200b\u9009\u62e9\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6df1\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b\u67e5\u627e\u200b\u6240\u6709\u200b\u7684\u200b\u72b6\u6001\u200b\u3002\u200b\u5047\u8bbe\u200b\u9752\u86d9\u200b\u8df3\u200b\u4e86\u200b\\(k\\)\u200b\u683c\u200b\u8fbe\u5230\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b</p> <ul> <li>\u200b\u5f53\u200b\u9752\u86d9\u200b\u8fbe\u5230\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u65f6\u200b\u8fd4\u56de\u200b<code>true</code>\uff0c</li> <li>\u200b\u5f53\u200b\u9752\u86d9\u200b\u5230\u8fbe\u200b\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u65f6\u200b\uff0c\u200b\u68c0\u67e5\u200b\u524d\u9762\u200b\u7684\u200b\\(k-1, k, k+1\\)\u200b\u4e2a\u200b\u5355\u5143\u683c\u200b\u662f\u5426\u200b\u6709\u200b\u77f3\u5b50\u200b\uff0c</li> <li>\u200b\u5f53\u200b\u9752\u86d9\u200b\u65e0\u6cd5\u200b\u7ee7\u7eed\u200b\u5411\u524d\u200b\u79fb\u52a8\u200b\u65f6\u200b\u8fd4\u56de\u200b<code>false</code>\u200b\u5e76\u200b\u8fdb\u884c\u200b\u56de\u6eaf\u200b\u3002</li> </ul> <pre><code>bool dfs(int *stones, int stonesSize, int currentPos, int lastStep)\n{\n    if (currentPos == stonesSize - 1)\n        return true;\n    int i = 0, j = 0, cur_id = stones[currentPos];\n    bool ret = false;\n    for (i = -1; i &lt; 2; i++)\n    {\n        if (lastStep + i &lt;= 0)\n            continue;\n        for (j = currentPos + 1; !ret &amp;&amp; j &lt; stonesSize &amp;&amp; stones[j] &lt;= cur_id + lastStep + i; j++)\n            if (stones[j] == cur_id + lastStep + i)\n                ret = ret || dfs(stones, stonesSize, j, lastStep + i);\n    }\n    return ret;\n}\n\nbool canCross(int* stones, int stonesSize){\n    return dfs(stones, stonesSize, 0, 0);\n}\n</code></pre> <p>\u200b\u5982\u200b\u4e0a\u200bdfs\u200b\u7b97\u6cd5\u200b\u4f1a\u200b\u5bf9\u200b\u6240\u6709\u200b\u7684\u200b\u5206\u652f\u200b\u8fdb\u884c\u200b\u904d\u5386\u200b\uff0c\u200b\u76f4\u5230\u200b\u627e\u51fa\u200b\u6700\u7ec8\u200b\u7ed3\u679c\u200b\u3002\u200b\u4f1a\u200b\u9020\u6210\u200b\u5927\u91cf\u200b\u65f6\u95f4\u200b\u7684\u200b\u6d6a\u8d39\u200b\u3002\u200b\u6784\u9020\u200b\u5982\u4e0b\u200b\u8f93\u5165\u200b\u6570\u636e\u200b\u5373\u53ef\u200b\u4f7f\u8be5\u200b\u7b97\u6cd5\u200b\u8d85\u65f6\u200b\uff1a</p> <pre><code>[0,1,2,3,4,5,...,998, +infty]\n</code></pre> <p>\u200b\u5176\u4e2d\u200b<code>+infty</code>\u200b\u6307\u200b\u5145\u5206\u200b\u5927\u200b\u7684\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u52a0\u5165\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8bb0\u5fc6\u200b\u4e0d\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5230\u200b\u7ec8\u70b9\u200b\u7684\u200b\u53ef\u8fbe\u6027\u200b\uff1a</p> <pre><code>bool dfs(int *stones, int *buf, int stonesSize, int currentPos, int lastStep)\n{\n    if (buf[currentPos] != -1)\n        return buf[currentPos];\n    if (currentPos == stonesSize - 1)\n        return true;\n    int i = 0, j = 0, cur_id = stones[currentPos];\n    bool ret = false;\n    for (i = -1; i &lt; 2; i++)\n    {\n        if (lastStep + i &lt;= 0)\n            continue;\n        for (j = currentPos + 1; !ret &amp;&amp; j &lt; stonesSize &amp;&amp; stones[j] &lt;= cur_id + lastStep + i; j++)\n            if (stones[j] == cur_id + lastStep + i)\n                ret = ret || dfs(stones, buf, stonesSize, j, lastStep + i);\n    }\n    buf[currentPos] = ret;\n    return ret;\n}\n\nbool canCross(int* stones, int stonesSize){\n    int *buf = (int *)malloc(sizeof(int) * stonesSize);\n    memset(buf, 0xff, sizeof(int) * stonesSize);\n    return dfs(stones, buf, stonesSize, 0, 0);\n}\n</code></pre> <p>\u200b\u4f46\u200b\u8fd9\u4e2a\u200b\u7b97\u6cd5\u200b\u53ea\u80fd\u200b\u901a\u8fc7\u200b\u4e00\u90e8\u5206\u200b\u6d4b\u4f8b\u200b\uff0c\u200b\u90e8\u5206\u200b\u6d4b\u4f8b\u200b\u65e0\u6cd5\u200b\u901a\u8fc7\u200b\uff08\u200b\u5982\u200b\u793a\u4f8b\u200b1\uff0c<code>[0,1,3,5,6,8,12,17]</code>\uff09\u3002\u200b\u6211\u4eec\u200b\u67e5\u770b\u200b\u4e00\u4e0b\u200b\u9752\u86d9\u200b\u5728\u200b\u8282\u70b9\u200b\u95f4\u200b\u8df3\u8dc3\u200b\u7684\u200b\u60c5\u51b5\u200b\uff1a</p> <p></p> <p>\u200b\u5f53\u200b\u7b97\u6cd5\u200b\u8d70\u200b\u5230\u200b\u8282\u70b9\u200b<code>5</code>\u200b\u65f6\u200b\uff0c\u200b\u9996\u5148\u200b\u4f1a\u200b\u5c1d\u8bd5\u200b\u8def\u5f84\u200b<code>5-&gt;6-&gt;8</code>\uff0c\u200b\u4f46\u200b\u6b64\u65f6\u200b\u5728\u200b\u8282\u70b9\u200b<code>8</code>\u200b\u9752\u86d9\u200b\u6700\u200b\u591a\u200b\u53ea\u80fd\u200b\u8df3\u200b<code>3</code>\u200b\u6b65\u200b\uff0c\u200b\u4ece\u800c\u200b\u88ab\u200b\u8bb0\u5fc6\u200b\u5316\u200b\u6807\u8bb0\u200b\u4e3a\u200b\u4e0d\u53ef\u200b\u5230\u8fbe\u200b\u7ec8\u70b9\u200b\uff0c\u200b\u4f46\u200b\u5b9e\u9645\u4e0a\u200b\u5982\u679c\u200b\u76f4\u63a5\u200b\u91c7\u53d6\u200b\u8def\u5f84\u200b<code>5-&gt;8</code>\u200b\u5219\u200b\u53ef\u4ee5\u200b\u4ece\u200b\u8282\u70b9\u200b<code>8</code>\u200b\u8df3\u200b\u5230\u200b\u8282\u70b9\u200b<code>12</code>\u3002\u200b\u5373\u200b\u67d0\u4e2a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u7ec8\u70b9\u200b\u53ef\u8fbe\u6027\u200b\u53d7\u5230\u200b\u4e0a\u200b\u4e00\u6b65\u200b\u8df3\u52a8\u200b\u6b65\u6570\u200b\u7684\u200b\u5f71\u54cd\u200b\uff0c\u200b\u6211\u4eec\u200b\u5728\u200b\u8fdb\u884c\u200b\u8bb0\u5fc6\u200b\u5316\u200b\u65f6\u200b\u9700\u8981\u200b\u8bb0\u5fc6\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\u5728\u200b\u6bcf\u4e2a\u200b\u8df3\u52a8\u200b\u6b65\u6570\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u7684\u200b\u53ef\u8fbe\u6027\u200b\u3002</p> <p>\u200b\u5982\u4e0b\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u4ee3\u66ff\u200b\u904d\u5386\u200b\uff0c\u200b\u5e76\u200b\u4f7f\u7528\u200b\u62c9\u94fe\u200b\u6cd5\u200b\u5b58\u50a8\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\u7684\u200b\u53ef\u8fbe\u6027\u200b\uff0c\u200b\u94fe\u8868\u200b\u8282\u70b9\u200b\u4e2d\u200b\u5b58\u50a8\u200b\u76f8\u5e94\u200b\u7684\u200b\u8df3\u52a8\u200b\u6b65\u6570\u200b\u3002</p> <pre><code>struct BufNode\n{\n    int success;\n    int lastStep;\n    struct BufNode *next;\n};\n\nstruct BufNode *getNode(int success, int lastStep, struct BufNode *next)\n{\n    struct BufNode *ret = (struct BufNode *)malloc(sizeof(struct BufNode));\n    ret-&gt;success = success;\n    ret-&gt;lastStep = lastStep;\n    ret-&gt;next = next;\n    return ret;\n}\n\nbool check(struct BufNode *buf, int lastStep, int *success)\n{\n    if (buf == NULL)\n        return NULL;\n    if (buf-&gt;lastStep == lastStep)\n    {\n        if (success)\n            *success = buf-&gt;success;\n        return true;\n    }\n    return check(buf-&gt;next, lastStep, success);\n}\n\nint bSearch(int *stones, int lo, int hi, int target)\n{\n    int mid;\n    while (lo &lt; hi)\n    {\n        mid = (lo + hi) &gt;&gt; 1;\n        if (target &lt; stones[mid])\n            hi = mid;\n        else\n            lo = mid + 1;\n    }\n    return lo - 1;\n}\n\nbool dfs(int *stones, struct BufNode **buf, int stonesSize, int currentPos, int lastStep)\n{\n    int success = false;\n    if (check(buf[currentPos], lastStep, &amp;success))\n        return success;\n    if (currentPos == stonesSize - 1)\n        return true;\n    int i = 0, j = 0, cur_id = stones[currentPos];\n    bool ret = false;\n    for (i = -1; i &lt; 2; i++)\n    {\n        if (lastStep + i &lt;= 0)\n            continue;\n        j = bSearch(stones, currentPos + 1, stonesSize, cur_id + lastStep + i);\n        if (stones[j] == cur_id + lastStep + i)\n            ret = ret || dfs(stones, buf, stonesSize, j, lastStep + i);\n    }\n    buf[currentPos] = getNode(ret, lastStep, buf[currentPos]);\n    return ret;\n}\n\nbool canCross(int* stones, int stonesSize){\n    struct BufNode **buf = (struct BufNode **)malloc(sizeof(struct BufNode *) * stonesSize);\n    memset(buf, 0, sizeof(struct BufNode *) * stonesSize);\n    return dfs(stones, buf, stonesSize, 0, 0);\n}\n</code></pre>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/42/","title":"42. \u200b\u63a5\u200b\u96e8\u6c34","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u6808","\u6570\u7ec4","\u53cc\u6307\u9488","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/42/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u00a0<code>n</code> \u200b\u4e2a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u8868\u793a\u200b\u6bcf\u4e2a\u200b\u5bbd\u5ea6\u200b\u4e3a\u200b <code>1</code> \u200b\u7684\u200b\u67f1\u5b50\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u56fe\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u6309\u6b64\u200b\u6392\u5217\u200b\u7684\u200b\u67f1\u5b50\u200b\uff0c\u200b\u4e0b\u96e8\u200b\u4e4b\u540e\u200b\u80fd\u63a5\u200b\u591a\u5c11\u200b\u96e8\u6c34\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aheight = [0,1,0,2,1,0,1,3,2,1,2,1]\n\u200b\u8f93\u51fa\u200b\uff1a6\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u4e0a\u9762\u200b\u662f\u200b\u7531\u200b\u6570\u7ec4\u200b [0,1,0,2,1,0,1,3,2,1,2,1] \u200b\u8868\u793a\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u56fe\u200b\uff0c\u200b\u5728\u200b\u8fd9\u79cd\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u63a5\u200b 6 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u7684\u200b\u96e8\u6c34\u200b\uff08\u200b\u84dd\u8272\u200b\u90e8\u5206\u200b\u8868\u793a\u200b\u96e8\u6c34\u200b\uff09\u3002 \n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aheight = [4,2,0,3,2,5]\n\u200b\u8f93\u51fa\u200b\uff1a9\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>n == height.length</code></li> <li><code>0 &lt;= n &lt;= 3 * 10^4</code></li> <li><code>0 &lt;= height[i] &lt;= 10^5</code></li> </ul> <p>Reference</p> <p>\u200b\u6ce8\u200b\uff1a\u200b\u672c\u9898\u200b\u4e0e\u200b\u9762\u8bd5\u9898\u200b 17.21. \u200b\u76f4\u65b9\u56fe\u200b\u7684\u200b\u6c34\u91cf\u200b\u76f8\u540c\u200b\u3002</p>","tags":["\u6808","\u6570\u7ec4","\u53cc\u6307\u9488","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/42/#_2","title":"\u9898\u89e3","text":"<ol> <li> <p>\u200b\u5206\u6cbb\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u6c34\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u4e0d\u4f1a\u200b\u9ad8\u4e8e\u200b\u6700\u9ad8\u200b\u7684\u200b\u67f1\u5b50\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u5728\u200b\u6700\u9ad8\u200b\u7684\u200b\u67f1\u5b50\u200b\\(A\\)\u200b\u4e0e\u200b\u6b21\u200b\u9ad8\u200b\u7684\u200b\u67f1\u5b50\u200b\\(B\\)\u200b\u4e24\u5904\u200b\u8fdb\u884c\u200b\u5206\u5272\u200b\uff0c\u200b\u5206\u522b\u200b\u6c42\u89e3\u200b\u4e09\u6bb5\u200b\u5bb9\u7eb3\u200b\u7684\u200b\u6c34\u91cf\u200b\u3002</p> <p>\\(A, B\\)\u200b\u4e4b\u95f4\u200b\u5fc5\u7136\u200b\u53ef\u4ee5\u200b\u5bb9\u7eb3\u200b\u6c34\u200b\uff0c\u200b\u6c34\u200b\u7684\u200b\u91cf\u200b\u7b49\u4e8e\u200b\u5404\u4e2a\u200b\u67f1\u5b50\u200b\u5230\u6b21\u200b\u9ad8\u200b\u67f1\u5b50\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u4e4b\u200b\u548c\u200b\u3002\u200b\u5176\u4ed6\u200b\u533a\u95f4\u200b\u9700\u8981\u200b\u9012\u5f52\u200b\u8fdb\u884c\u200b\u6c42\u89e3\u200b\u3002\u200b\u9700\u8981\u200b\u6ce8\u610f\u200b\u5206\u5272\u200b\u65f6\u200b\u5206\u5272\u200b\u70b9\u8981\u200b\u540c\u65f6\u200b\u5305\u542b\u200b\u5728\u200b\u5de6\u53f3\u200b\u4e24\u4fa7\u200b\uff0c\u200b\u5426\u5219\u200b\u65e0\u6cd5\u200b\u5bb9\u7eb3\u200b\u6c34\u200b\u3002</p> <p>\u200b\u5f53\u200b\u67f1\u5b50\u200b\u6570\u91cf\u200b\u5c0f\u4e8e\u200b3\u200b\u65f6\u200b\uff0c\u200b\u65e0\u6cd5\u200b\u5bb9\u7eb3\u200b\u4efb\u4f55\u200b\u6c34\u200b\u3002</p> <pre><code>#define MAX(x, y) (x &gt; y ? x : y)\n#define MIN(x, y) (x &lt; y ? x : y)\n\nint trap(int* height, int heightSize){\n    int *maxLeft = (int *)malloc(sizeof(int) * heightSize),\n        *maxRight = (int *)malloc(sizeof(int) * heightSize),\n        i = 0, ret = 0;\n    if (heightSize == 0)\n        return 0;\n    maxLeft[0] = height[0];\n    maxRight[heightSize - 1] = height[heightSize - 1];\n    for (i = 1; i &lt; heightSize; i++)\n    {\n        maxLeft[i] = MAX(maxLeft[i - 1], height[i - 1]);\n        maxRight[heightSize - i - 1] = MAX(maxRight[heightSize - i], height[heightSize - i]);\n    }\n    for (i = 0; i &lt; heightSize; i++)\n        ret += MAX(0, MIN(maxLeft[i], maxRight[i]) - height[i]);\n    return ret;\n}\n</code></pre> <p>\u200b\u603b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(N)\\)\u3002</p> </li> <li> <p>\u200b\u76f4\u63a5\u200b\u601d\u8def\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u67f1\u5b50\u200b\uff0c\u200b\u5176\u200b\u5bb9\u7eb3\u200b\u6c34\u540e\u200b\u7684\u200b\u9ad8\u5ea6\u200b\uff08\u200b\u67f1\u5b50\u200b\u9ad8\u5ea6\u200b\u51cf\u53bb\u200b\u6c34\u200b\u9ad8\u5ea6\u200b\uff09\u200b\u7b49\u4e8e\u200b\u8be5\u200b\u67f1\u5b50\u200b\u5de6\u4fa7\u200b\u6700\u9ad8\u200b\u67f1\u5b50\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u4e0e\u200b\u53f3\u4fa7\u200b\u6700\u9ad8\u200b\u67f1\u5b50\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u67f1\u5b50\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7edf\u8ba1\u200b\u5728\u200b\u5176\u200b\u5de6\u4fa7\u200b\u6700\u9ad8\u200b\u67f1\u5b50\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u4e0e\u200b\u5728\u200b\u5176\u200b\u53f3\u4fa7\u200b\u6700\u9ad8\u200b\u67f1\u5b50\u200b\u7684\u200b\u9ad8\u5ea6\u200b\uff0c\u200b\u5b58\u50a8\u200b\u5728\u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u3002</p> <p>\u200b\u7ed3\u679c\u200b\u4e3a\u200b\u6bcf\u4e2a\u200b\u67f1\u5b50\u200b\u5bb9\u7eb3\u200b\u6c34\u540e\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u4e0e\u200b\u67f1\u5b50\u200b\u9ad8\u5ea6\u200b\u4e4b\u5dee\u200b\u7684\u200b\u6c42\u548c\u200b\u3002</p> <pre><code>#define MAX(x, y) (x &gt; y ? x : y)\n#define MIN(x, y) (x &lt; y ? x : y)\n\nint trap(int* height, int heightSize){\n    int *maxLeft = (int *)malloc(sizeof(int) * heightSize),\n        *maxRight = (int *)malloc(sizeof(int) * heightSize),\n        i = 0, ret = 0;\n    if (heightSize == 0)\n        return 0;\n    maxLeft[0] = height[0];\n    maxRight[heightSize - 1] = height[heightSize - 1];\n    for (i = 1; i &lt; heightSize; i++)\n    {\n        maxLeft[i] = MAX(maxLeft[i - 1], height[i - 1]);\n        maxRight[heightSize - i - 1] = MAX(maxRight[heightSize - i], height[heightSize - i]);\n    }\n    for (i = 0; i &lt; heightSize; i++)\n        ret += MAX(0, MIN(maxLeft[i], maxRight[i]) - height[i]);\n    return ret;\n}\n</code></pre> </li> </ol>","tags":["\u6808","\u6570\u7ec4","\u53cc\u6307\u9488","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/43/","title":"43. \u200b\u5b57\u7b26\u4e32\u200b\u76f8\u4e58","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/43/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e24\u4e2a\u200b\u4ee5\u200b\u5b57\u7b26\u4e32\u200b\u5f62\u5f0f\u200b\u8868\u793a\u200b\u7684\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u00a0<code>num1</code>\u00a0\u200b\u548c\u200b\u00a0<code>num2</code>\uff0c\u200b\u8fd4\u56de\u200b\u00a0<code>num1</code>\u00a0\u200b\u548c\u200b\u00a0<code>num2</code>\u00a0\u200b\u7684\u200b\u4e58\u79ef\u200b\uff0c\u200b\u5b83\u4eec\u200b\u7684\u200b\u4e58\u79ef\u200b\u4e5f\u200b\u8868\u793a\u200b\u4e3a\u200b\u5b57\u7b26\u4e32\u200b\u5f62\u5f0f\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1:</p> <pre><code>\u200b\u8f93\u5165\u200b: num1 = \"2\", num2 = \"3\"\n\u200b\u8f93\u51fa\u200b: \"6\"\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02:</p> <pre><code>\u200b\u8f93\u5165\u200b: num1 = \"123\", num2 = \"456\"\n\u200b\u8f93\u51fa\u200b: \"56088\"\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li><code>num1</code>\u00a0\u200b\u548c\u200b\u00a0<code>num2</code>\u00a0\u200b\u7684\u200b\u957f\u5ea6\u200b\u5c0f\u4e8e\u200b110\u3002</li> <li><code>num1</code> \u200b\u548c\u200b\u00a0<code>num2</code> \u200b\u53ea\u200b\u5305\u542b\u200b\u6570\u5b57\u200b\u00a0<code>0-9</code>\u3002</li> <li><code>num1</code> \u200b\u548c\u200b\u00a0<code>num2</code>\u00a0\u200b\u5747\u200b\u4e0d\u200b\u4ee5\u200b\u96f6\u200b\u5f00\u5934\u200b\uff0c\u200b\u9664\u975e\u200b\u662f\u200b\u6570\u5b57\u200b <code>0</code> \u200b\u672c\u8eab\u200b\u3002</li> <li>\u200b\u4e0d\u80fd\u200b\u4f7f\u7528\u200b\u4efb\u4f55\u200b\u6807\u51c6\u200b\u5e93\u200b\u7684\u200b\u5927\u6570\u200b\u7c7b\u578b\u200b\uff08\u200b\u6bd4\u5982\u200bBigInteger\uff09\u200b\u6216\u200b\u76f4\u63a5\u200b\u5c06\u200b\u8f93\u5165\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u6574\u6570\u200b\u6765\u200b\u5904\u7406\u200b\u3002</li> </ul> <p>Reference</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/43/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u8ba1\u7b97\u65b9\u6cd5\u200b\u4e0e\u200b\u5217\u7ad6\u5f0f\u200b\u76f8\u540c\u200b\uff0c\u200b\u6ce8\u610f\u200b\u5bf9\u200b\u7279\u6b8a\u200b\u8f93\u5165\u200b<code>0</code>\u200b\u7684\u200b\u5904\u7406\u200b\u3002\u200b\u7ed3\u679c\u200b\u7684\u200b\u957f\u5ea6\u200b\u4e0d\u4f1a\u200b\u8d85\u8fc7\u200b\u8f93\u5165\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u4e4b\u200b\u548c\u200b\u3002</p> <pre><code>char * multiply(char * num1, char * num2){\n    if (*num1 == '0' || *num2 == '0')\n        return *num1 == '0' ? num1 : num2;\n    int len1 = strlen(num1), len2 = strlen(num2), retLen = len1 + len2, i = 0, j = 0, incre, curDigit, pos;\n    char *ret = (char *)memset(malloc(sizeof(char) * (retLen + 1)), '0', sizeof(char) * (retLen + 1)),\n         *cur = ret;\n    ret[retLen] = 0;\n    for (i = len1 - 1; i &gt;= 0; i--)\n    {\n        for (j = len2 - 1; j &gt;= 0; j--)\n        {\n            incre = (num1[i] - '0') * (num2[j] - '0');\n            curDigit = incre % 10;\n            incre /= 10;\n            pos = retLen - ((len1 - i - 1) + (len2 - j - 1)) - 1;\n            while (incre != 0 || curDigit != 0)\n            {\n                ret[pos] += curDigit;\n                incre += ret[pos] &gt; '9';\n                ret[pos] -= ret[pos] &gt; '9' ? 10 : 0;\n                curDigit = incre &gt;= 10 ? incre - 10 : incre;\n                incre = incre &gt;= 10;\n                pos--;\n            }\n        }\n    }\n    while (*cur == '0')\n        cur++;\n    return cur;\n}\n</code></pre>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/433/","title":"433. \u200b\u6700\u5c0f\u200b\u57fa\u56e0\u200b\u53d8\u5316","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/433/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u4e00\u6761\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u7531\u200b\u4e00\u4e2a\u200b\u5e26\u6709\u200b8\u200b\u4e2a\u5b57\u7b26\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\u8868\u793a\u200b\uff0c\u200b\u5176\u4e2d\u200b\u6bcf\u4e2a\u200b\u5b57\u7b26\u200b\u90fd\u200b\u5c5e\u4e8e\u200b <code>\"A\"</code>, <code>\"C\"</code>, <code>\"G\"</code>, <code>\"T\"</code>\u200b\u4e2d\u200b\u7684\u200b\u4efb\u610f\u200b\u4e00\u4e2a\u200b\u3002</p> <p>\u200b\u5047\u8bbe\u200b\u6211\u4eec\u200b\u8981\u200b\u8c03\u67e5\u200b\u4e00\u4e2a\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u7684\u200b\u53d8\u5316\u200b\u3002\u200b\u4e00\u6b21\u200b\u57fa\u56e0\u200b\u53d8\u5316\u200b\u610f\u5473\u7740\u200b\u8fd9\u4e2a\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b\u53d1\u751f\u200b\u4e86\u200b\u53d8\u5316\u200b\u3002</p> <p>\u200b\u4f8b\u5982\u200b\uff0c\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u7531\u200b <code>\"AACCGGTT\"</code>\u00a0\u200b\u53d8\u5316\u200b\u81f3\u200b\u00a0<code>\"AACCGGTA\"</code>\u00a0\u200b\u5373\u200b\u53d1\u751f\u200b\u4e86\u200b\u4e00\u6b21\u200b\u57fa\u56e0\u200b\u53d8\u5316\u200b\u3002</p> <p>\u200b\u4e0e\u6b64\u540c\u65f6\u200b\uff0c\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u57fa\u56e0\u200b\u53d8\u5316\u200b\u7684\u200b\u7ed3\u679c\u200b\uff0c\u200b\u90fd\u200b\u9700\u8981\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5408\u6cd5\u200b\u7684\u200b\u57fa\u56e0\u200b\u4e32\u200b\uff0c\u200b\u5373\u8be5\u200b\u7ed3\u679c\u200b\u5c5e\u4e8e\u200b\u4e00\u4e2a\u200b\u57fa\u56e0\u5e93\u200b\u3002</p> <p>\u200b\u73b0\u5728\u200b\u7ed9\u5b9a\u200b3\u200b\u4e2a\u200b\u53c2\u6570\u200b \u2014 <code>start</code>, <code>end</code>, <code>bank</code>\uff0c\u200b\u5206\u522b\u200b\u4ee3\u8868\u200b\u8d77\u59cb\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\uff0c\u200b\u76ee\u6807\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u53ca\u200b\u57fa\u56e0\u5e93\u200b\uff0c\u200b\u8bf7\u200b\u627e\u51fa\u200b\u80fd\u591f\u200b\u4f7f\u200b\u8d77\u59cb\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u53d8\u5316\u200b\u4e3a\u200b\u76ee\u6807\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u6700\u5c11\u200b\u53d8\u5316\u200b\u6b21\u6570\u200b\u3002\u200b\u5982\u679c\u200b\u65e0\u6cd5\u200b\u5b9e\u73b0\u76ee\u6807\u200b\u53d8\u5316\u200b\uff0c\u200b\u8bf7\u200b\u8fd4\u56de\u200b <code>-1</code>\u3002</p> <p>\u200b\u6ce8\u610f\u200b\uff1a</p> <ul> <li>\u200b\u8d77\u59cb\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u9ed8\u8ba4\u200b\u662f\u200b\u5408\u6cd5\u200b\u7684\u200b\uff0c\u200b\u4f46\u662f\u200b\u5b83\u200b\u5e76\u4e0d\u4e00\u5b9a\u200b\u4f1a\u200b\u51fa\u73b0\u200b\u5728\u200b\u57fa\u56e0\u5e93\u200b\u4e2d\u200b\u3002</li> <li>\u200b\u5982\u679c\u200b\u4e00\u4e2a\u200b\u8d77\u59cb\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u9700\u8981\u200b\u591a\u6b21\u200b\u53d8\u5316\u200b\uff0c\u200b\u90a3\u4e48\u200b\u5b83\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u53d8\u5316\u200b\u4e4b\u540e\u200b\u7684\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u90fd\u200b\u5fc5\u987b\u200b\u662f\u200b\u5408\u6cd5\u200b\u7684\u200b\u3002</li> <li>\u200b\u5047\u5b9a\u200b\u8d77\u59cb\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u4e0e\u200b\u76ee\u6807\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u662f\u200b\u4e0d\u200b\u4e00\u6837\u200b\u7684\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>start: \"AACCGGTT\"\nend:   \"AACCGGTA\"\nbank: [\"AACCGGTA\"]\n\n\u200b\u8fd4\u56de\u503c\u200b: 1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>start: \"AACCGGTT\"\nend:   \"AAACGGTA\"\nbank: [\"AACCGGTA\", \"AACCGCTA\", \"AAACGGTA\"]\n\n\u200b\u8fd4\u56de\u503c\u200b: 2\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>start: \"AAAAACCC\"\nend:   \"AACCCCCC\"\nbank: [\"AAAACCCC\", \"AAACCCCC\", \"AACCCCCC\"]\n\n\u200b\u8fd4\u56de\u503c\u200b: 3\n</code></pre> <p>Reference</p>","tags":["\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/433/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u5e7f\u5ea6\u200b\u4f18\u5148\u200b\u7b97\u6cd5\u200b\u904d\u5386\u200b\u57fa\u56e0\u200b\u7684\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u6539\u52a8\u200b\u60c5\u51b5\u200b\u3002\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u957f\u5ea6\u200b\u4e3a\u200b8\uff0c\u200b\u6bcf\u6b21\u200b\u53ef\u80fd\u200b\u7684\u200b\u6539\u52a8\u200b\u60c5\u51b5\u200b\u4e3a\u200b\\(8\\times 3 = 24\\)\u200b\u79cd\u200b\uff0c\u200b\u5728\u200b\u8fd9\u200b24\u200b\u79cd\u200b\u6539\u52a8\u200b\u60c5\u51b5\u200b\u4e2d\u200b\u9009\u62e9\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u7ed3\u679c\u200b\uff1a</p> <ul> <li>\u200b\u7ed3\u679c\u200b\u662f\u200b\u5408\u6cd5\u200b\u7684\u200b\u57fa\u56e0\u200b</li> <li>\u200b\u7ed3\u679c\u200b\u672a\u200b\u88ab\u200b\u904d\u5386\u200b\u8fc7\u200b</li> </ul> <p>\u200b\u4f7f\u7528\u200b\u54c8\u5e0c\u200b\u8868\u200b\u5b58\u50a8\u200b\u57fa\u56e0\u5e93\u200b\u548c\u200b\u5df2\u7ecf\u200b\u88ab\u200b\u904d\u5386\u200b\u8fc7\u200b\u7684\u200b\u57fa\u56e0\u200b\u3002</p> <pre><code>char *genes = \"TGCATGCA\";\n\nint getCode(char *s)\n{\n    int ret = 0;\n    switch (*s)\n    {\n    case 'A':\n        ret++;\n    case 'C':\n        ret++;\n    case 'G':\n        ret++;\n    case 'T':\n        break;\n    }\n    return ret;\n}\n\nint hash(char *s)\n{\n    char *cur = s;\n    int ret = 0;\n    while (*cur)\n    {\n        ret &lt;&lt;= 2;\n        ret += getCode(cur);\n        cur++;\n    }\n    return ret;\n}\n\nchar *dehash(int o)\n{\n    char *ret = (char *)malloc(sizeof(char) * 9);\n    ret[8] = 0;\n    for (int i = 7; i &gt;= 0; i--)\n    {\n        ret[i] = genes[o &amp; 3];\n        o &gt;&gt;= 2;\n    }\n    return ret;\n}\n\nstruct Bank {\n    char *bitmap;\n};\n\nvoid deleteBank(struct Bank *dest)\n{\n    free(dest-&gt;bitmap);\n    free(dest);\n}\n\nvoid removeGene(struct Bank *dest, char *seq)\n{\n    int i = hash(seq);\n    dest-&gt;bitmap[i &gt;&gt; 3] &amp;= ~(1 &lt;&lt; (i &amp; 7));\n}\n\nvoid addGene(struct Bank *dest, char *seq)\n{\n    int i = hash(seq);\n    dest-&gt;bitmap[i &gt;&gt; 3] |= 1 &lt;&lt; (i &amp; 7);\n}\n\nbool queryGene(struct Bank *dest, char *seq)\n{\n    int i = hash(seq);\n    return dest-&gt;bitmap[i &gt;&gt; 3] &amp; (1 &lt;&lt; (i &amp; 7));\n}\n\nvoid merge(struct Bank *dest, struct Bank *src)\n{\n    int i = 0;\n    for (i = 0; i &lt; 8192; i++)\n        dest-&gt;bitmap[i] |= src-&gt;bitmap[i];\n    deleteBank(src);\n}\n\nstruct Bank *getBank(char **bank, int bankSize)\n{\n    struct Bank *ret = (struct Bank *)malloc(sizeof(struct Bank));\n    ret-&gt;bitmap = (char *)malloc(sizeof(char) * 8192);\n    memset(ret-&gt;bitmap, 0, sizeof(char) * 8192);\n    for (int i = 0; i &lt; bankSize; i++)\n        addGene(ret, bank[i]);\n    return ret;\n}\n\nstruct Bank *step(char *current, char *end, struct Bank *passed, struct Bank *fullBank, int *count)\n{\n    int startHash = hash(current);\n    struct Bank *ret = getBank(NULL, 0);\n    addGene(passed, current);\n    for (int i = 0; i &lt; 8; i++)\n    {\n        for (int j = 0; j &lt; 4; j++)\n        {\n            current[i] = genes[getCode(current + i) + 1];\n            if (startHash == hash(current))\n                continue;\n            if (!queryGene(fullBank, current))\n                continue;\n            if (queryGene(passed, current))\n                continue;\n            addGene(ret, current);\n            (*count)++;\n        }\n    }\n    return ret;\n}\n\nstruct Bank *stepBank(struct Bank *current, char *end, struct Bank *passed, struct Bank *fullBank, int* count)\n{\n    int index = 0;\n    struct Bank *ret = getBank(NULL, 0);\n    char *temp = NULL;\n    for (int i = 0; i &lt; 8192; i++)\n    {\n        if (current-&gt;bitmap[i])\n        {\n            index = i &lt;&lt; 3;\n            for (int j = 0; j &lt; 8; j++)\n            {\n                if (current-&gt;bitmap[i] &amp; (1 &lt;&lt; j))\n                {\n                    index += j;\n                    temp = dehash(index);\n                    merge(ret, step(temp, end, passed, fullBank, count));\n                    free(temp);\n                }\n            }\n        }\n    }\n    return ret;\n}\n\nint minMutation(char * start, char * end, char ** bank, int bankSize){\n    struct Bank *fullBank = getBank(bank, bankSize), *passed = getBank(NULL, 0);\n    struct Bank *temp = getBank(NULL, 0), *temp2 = NULL;\n    addGene(temp, start);\n    int ret = 0, modified = 0;\n    while (!queryGene(temp, end))\n    {\n        modified = 0;\n        temp2 = stepBank(temp, end, passed, fullBank, &amp;modified);\n        deleteBank(temp);\n        temp = temp2;\n        if (!modified)\n            return -1;\n        ret++;\n    }\n    return ret;\n}\n</code></pre>","tags":["\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/46/","title":"46. \u200b\u5168\u200b\u6392\u5217","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/46/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b \u200b\u6ca1\u6709\u200b\u91cd\u590d\u200b \u200b\u6570\u5b57\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5176\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u7684\u200b\u5168\u200b\u6392\u5217\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b:</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,2,3]\n\u200b\u8f93\u51fa\u200b:\n[\n  [1,2,3],\n  [1,3,2],\n  [2,1,3],\n  [2,3,1],\n  [3,1,2],\n  [3,2,1]\n]\n</code></pre> <p>Reference</p>","tags":["\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/46/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u56de\u6eaf\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u5e8f\u5217\u200b\uff1a</p> <ul> <li>\u200b\u82e5\u200b\u957f\u5ea6\u200b\u4e3a\u200b1\uff0c\u200b\u5219\u200b\u53ea\u6709\u200b\u4e00\u79cd\u200b\u6392\u5217\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u76f4\u63a5\u200b\u8fd4\u56de\u200b</li> <li>\u200b\u82e5\u200b\u957f\u5ea6\u200b\u5927\u4e8e\u200b1\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\uff1a<ul> <li>\u200b\u5c06\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7b2c\u4e00\u4e2a\u200b\u6570\u200b\u653e\u5728\u200b\u6392\u5217\u200b\u7684\u200b\u7b2c\u4e00\u4f4d\u200b\uff0c\u200b\u7136\u540e\u200b\u4e0e\u200b\u540e\u9762\u200b\u90e8\u5206\u200b\u7684\u200b\u5168\u200b\u6392\u5217\u200b\u5408\u5e76\u200b</li> <li>\u200b\u5c06\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7b2c\u4e8c\u4e2a\u200b\u6570\u200b\u653e\u5728\u200b\u6392\u5217\u200b\u7684\u200b\u7b2c\u4e00\u4f4d\u200b\uff0c\u200b\u7136\u540e\u200b\u4e0e\u200b\u540e\u9762\u200b\u90e8\u5206\u200b\u7684\u200b\u5168\u200b\u6392\u5217\u200b\u5408\u5e76\u200b</li> <li>\u2026\u2026</li> <li>\u200b\u5c06\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u6570\u200b\u653e\u5728\u200b\u6392\u5217\u200b\u7684\u200b\u7b2c\u4e00\u4f4d\u200b\uff0c\u200b\u7136\u540e\u200b\u4e0e\u200b\u540e\u9762\u200b\u90e8\u5206\u200b\u7684\u200b\u5168\u200b\u6392\u5217\u200b\u5408\u5e76\u200b</li> <li>\u200b\u6700\u7ec8\u200b\u7ed3\u679c\u200b\u5305\u542b\u200b\u5982\u200b\u4e0a\u200b\u6240\u6709\u200b\u7684\u200b\u6392\u5217\u200b\u3002</li> </ul> </li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(n\\)\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5171\u6709\u200b\\(n!\\)\u200b\u79cd\u200b\u6392\u5217\u200b\u3002</p> <pre><code>int fact(int x)\n{\n    if (!x)\n        return 0;\n    int i = 1, ret = 1;\n    for (i = 1; i &lt;= x; i++)\n        ret *= i;\n    return ret;\n}\nvoid swap(int *a, int *b)\n{\n    int temp = *a;\n    *a = *b;\n    *b = temp;\n}\nint** permute(int* nums, int numsSize, int* returnSize, int** returnColumnSizes){\n    if (numsSize &lt; 2)\n    {\n        *returnSize = 1;\n        *returnColumnSizes = (int *)malloc(sizeof(int) * *returnSize);\n        *returnColumnSizes[0] = numsSize;\n        if (numsSize == 0)\n            return NULL;\n        int **ret = (int **)malloc(sizeof(int *) * *returnSize);\n        *ret = (int *)malloc(sizeof(int));\n        **ret = *nums;\n        return ret;\n    }\n    *returnSize = fact(numsSize);\n    *returnColumnSizes = (int *)malloc(sizeof(int *) * *returnSize);\n    int *sub = (int *)malloc(sizeof(int) * numsSize), i = 0, **rec = NULL, recSize, *recColSize,\n        **ret = (int **)malloc(sizeof(int *) * *returnSize), retpos = 0, j = 0;\n    for (i = 0; i &lt; *returnSize; i++)\n        (*returnColumnSizes)[i] = numsSize;\n    for (i = 0; i &lt; numsSize; i++)\n    {\n        sub = memcpy(sub, nums, sizeof(int) * numsSize);\n        swap(sub + i, sub);\n        rec = permute(sub + 1, numsSize - 1, &amp;recSize, &amp;recColSize);\n        for (j = 0; j &lt; recSize; j++)\n        {\n            ret[retpos] = (int *)malloc(sizeof(int) * numsSize);\n            ret[retpos][0] = sub[0];\n            memcpy(ret[retpos] + 1, rec[j], sizeof(int) * (numsSize - 1));\n            retpos++;\n        }\n    }\n    return ret;\n}\n</code></pre>","tags":["\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/48/","title":"48. \u200b\u65cb\u8f6c\u200b\u56fe\u50cf","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/48/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b n\u00a0\u00d7\u00a0n \u200b\u7684\u200b\u4e8c\u7ef4\u200b\u77e9\u9635\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u56fe\u50cf\u200b\u3002</p> <p>\u200b\u5c06\u200b\u56fe\u50cf\u200b\u987a\u65f6\u9488\u200b\u65cb\u8f6c\u200b 90 \u200b\u5ea6\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u5fc5\u987b\u200b\u5728\u200b\u539f\u5730\u200b\u65cb\u8f6c\u200b\u56fe\u50cf\u200b\uff0c\u200b\u8fd9\u200b\u610f\u5473\u7740\u200b\u4f60\u200b\u9700\u8981\u200b\u76f4\u63a5\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u7684\u200b\u4e8c\u7ef4\u200b\u77e9\u9635\u200b\u3002\u200b\u8bf7\u200b\u4e0d\u8981\u200b\u4f7f\u7528\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u77e9\u9635\u200b\u6765\u200b\u65cb\u8f6c\u200b\u56fe\u50cf\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u7ed9\u5b9a\u200b matrix = \n[\n  [1,2,3],\n  [4,5,6],\n  [7,8,9]\n],\n\n\u200b\u539f\u5730\u200b\u65cb\u8f6c\u200b\u8f93\u5165\u200b\u77e9\u9635\u200b\uff0c\u200b\u4f7f\u200b\u5176\u200b\u53d8\u4e3a\u200b:\n[\n  [7,4,1],\n  [8,5,2],\n  [9,6,3]\n]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u7ed9\u5b9a\u200b matrix =\n[\n  [ 5, 1, 9,11],\n  [ 2, 4, 8,10],\n  [13, 3, 6, 7],\n  [15,14,12,16]\n], \n\n\u200b\u539f\u5730\u200b\u65cb\u8f6c\u200b\u8f93\u5165\u200b\u77e9\u9635\u200b\uff0c\u200b\u4f7f\u200b\u5176\u200b\u53d8\u4e3a\u200b:\n[\n  [15,13, 2, 5],\n  [14, 3, 4, 1],\n  [12, 6, 8, 9],\n  [16, 7,10,11]\n]\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/48/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5c06\u200b\u77e9\u9635\u200b\u8f6c\u7f6e\u200b\u540e\u200b\u53cd\u8f6c\u200b\u5404\u5217\u200b\u7684\u200b\u987a\u5e8f\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>void rotate(int** matrix, int matrixSize, int* matrixColSize){\n    int i = 0, j = 0, temp;\n    for (i = 0; i &lt; matrixSize; i++)\n    {\n        for (j = i; j &lt; matrixSize; j++)\n        {\n            temp = matrix[i][j];\n            matrix[i][j] = matrix[j][i];\n            matrix[j][i] = temp;\n        }\n    }\n    for (i = 0; i &lt; matrixSize; i++)\n    {\n        for (j = 0; j &lt; matrixSize / 2; j++)\n        {\n            temp = matrix[i][j];\n            matrix[i][j] = matrix[i][matrixSize - j - 1];\n            matrix[i][matrixSize - j - 1] = temp;\n        }\n    }\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/488/","title":"488. \u200b\u7956\u739b\u200b\u6e38\u620f","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u6df1\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/488/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u56de\u5fc6\u200b\u4e00\u4e0b\u200b\u7956\u739b\u200b\u6e38\u620f\u200b\u3002\u200b\u73b0\u5728\u200b\u684c\u4e0a\u200b\u6709\u200b\u4e00\u4e32\u200b\u7403\u200b\uff0c\u200b\u989c\u8272\u200b\u6709\u200b\u7ea2\u8272\u200b(R)\uff0c\u200b\u9ec4\u8272\u200b(Y)\uff0c\u200b\u84dd\u8272\u200b(B)\uff0c\u200b\u7eff\u8272\u200b(G)\uff0c\u200b\u8fd8\u6709\u200b\u767d\u8272\u200b(W)\u3002 \u200b\u73b0\u5728\u200b\u4f60\u200b\u624b\u91cc\u200b\u4e5f\u200b\u6709\u200b\u51e0\u4e2a\u200b\u7403\u200b\u3002</p> <p>\u200b\u6bcf\u200b\u4e00\u6b21\u200b\uff0c\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u4ece\u200b\u624b\u91cc\u200b\u7684\u200b\u7403\u9009\u200b\u4e00\u4e2a\u200b\uff0c\u200b\u7136\u540e\u200b\u628a\u200b\u8fd9\u4e2a\u200b\u7403\u200b\u63d2\u5165\u200b\u5230\u200b\u4e00\u4e32\u200b\u7403\u4e2d\u200b\u7684\u200b\u67d0\u4e2a\u200b\u4f4d\u7f6e\u200b\u4e0a\u200b\uff08\u200b\u5305\u62ec\u200b\u6700\u200b\u5de6\u7aef\u200b\uff0c\u200b\u6700\u200b\u53f3\u7aef\u200b\uff09\u3002\u200b\u63a5\u7740\u200b\uff0c\u200b\u5982\u679c\u200b\u6709\u200b\u51fa\u73b0\u200b\u4e09\u4e2a\u200b\u6216\u8005\u200b\u4e09\u4e2a\u200b\u4ee5\u4e0a\u200b\u989c\u8272\u200b\u76f8\u540c\u200b\u7684\u200b\u7403\u200b\u76f8\u8fde\u200b\u7684\u8bdd\u200b\uff0c\u200b\u5c31\u200b\u628a\u200b\u5b83\u4eec\u200b\u79fb\u200b\u9664\u6389\u200b\u3002\u200b\u91cd\u590d\u200b\u8fd9\u4e00\u200b\u6b65\u9aa4\u200b\u76f4\u5230\u200b\u684c\u4e0a\u200b\u6240\u6709\u200b\u7684\u200b\u7403\u200b\u90fd\u200b\u88ab\u200b\u79fb\u9664\u200b\u3002</p> <p>\u200b\u627e\u5230\u200b\u63d2\u5165\u200b\u5e76\u200b\u53ef\u4ee5\u200b\u79fb\u200b\u9664\u6389\u200b\u684c\u4e0a\u200b\u6240\u6709\u200b\u7403\u6240\u200b\u9700\u200b\u7684\u200b\u6700\u5c11\u200b\u7684\u200b\u7403\u6570\u200b\u3002\u200b\u5982\u679c\u200b\u4e0d\u80fd\u200b\u79fb\u9664\u200b\u684c\u4e0a\u200b\u6240\u6709\u200b\u7684\u200b\u7403\u200b\uff0c\u200b\u8f93\u51fa\u200b <code>-1</code> \u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = \"WRRBBW\", hand = \"RB\"\n\u200b\u8f93\u51fa\u200b\uff1a-1\n\u200b\u89e3\u91ca\u200b\uff1aWRRBBW -&gt; WRR[R]BBW -&gt; WBBW -&gt; WBB[B]W -&gt; WW\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = \"WWRRBBWW\", hand = \"WRBRW\"\n\u200b\u8f93\u51fa\u200b\uff1a2\n\u200b\u89e3\u91ca\u200b\uff1aWWRRBBWW -&gt; WWRR[R]BBWW -&gt; WWBBWW -&gt; WWBB[B]WW -&gt; WWWW -&gt; empty\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = \"G\", hand = \"GGGGG\"\n\u200b\u8f93\u51fa\u200b\uff1a2\n\u200b\u89e3\u91ca\u200b\uff1aG -&gt; G[G] -&gt; GG[G] -&gt; empty \n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = \"RBYYBBRRB\", hand = \"YRBGB\"\n\u200b\u8f93\u51fa\u200b\uff1a3\n\u200b\u89e3\u91ca\u200b\uff1aRBYYBBRRB -&gt; RBYY[Y]BBRRB -&gt; RBBBRRB -&gt; RRRB -&gt; B -&gt; B[B] -&gt; BB[B] -&gt; empty \n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u684c\u4e0a\u200b\u4e00\u200b\u5f00\u59cb\u200b\u7684\u200b\u7403\u200b\u4e2d\u200b\uff0c\u200b\u4e0d\u4f1a\u200b\u6709\u200b\u4e09\u4e2a\u200b\u53ca\u200b\u4e09\u4e2a\u200b\u4ee5\u4e0a\u200b\u989c\u8272\u200b\u76f8\u540c\u200b\u4e14\u200b\u8fde\u200b\u7740\u200b\u7684\u200b\u7403\u200b\u3002</li> <li><code>1 &lt;= board.length &lt;= 16</code></li> <li><code>1 &lt;= hand.length &lt;= 5</code></li> <li>\u200b\u8f93\u5165\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u5747\u200b\u4e3a\u200b\u975e\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u4e14\u200b\u53ea\u200b\u5305\u542b\u200b\u5b57\u7b26\u200b <code>'R','Y','B','G','W'</code>\u3002</li> </ul> <p>Reference</p>","tags":["\u6df1\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/488/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5b9a\u4e49\u200b\u5f53\u524d\u200b\u6e38\u620f\u200b\u7684\u200b\u72b6\u6001\u200b\uff0c\u200b\u4e00\u4e2a\u200b\u72b6\u6001\u200b\u7531\u200b\u684c\u4e0a\u200b\u7684\u200b\u7403\u200b\u5217\u8868\u200b\u4e0e\u200b\u624b\u4e2d\u200b\u7684\u200b\u7403\u200b\u5217\u8868\u200b\u8868\u793a\u200b\u3002\u200b\u6bcf\u6b21\u200b\u63d2\u5165\u200b\u64cd\u4f5c\u200b\u6784\u6210\u200b\u4e00\u4e2a\u200b\u72b6\u6001\u200b\u95f4\u200b\u7684\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\u8fc7\u7a0b\u200b\u3002\u200b\u4f7f\u7528\u200b\u4e24\u4e2a\u200b\u961f\u5217\u200b\u5b58\u50a8\u200b\u5f53\u524d\u200b\u72b6\u6001\u200b\uff0c\u200b\u548c\u200b\u5f53\u524d\u200b\u72b6\u6001\u200b\u7684\u200b \u200b\u6240\u6709\u200b \u200b\u53ef\u80fd\u200b\u8f6c\u79fb\u200b\u5230\u200b\u7684\u200b\u72b6\u6001\u200b\u3002\u200b\u518d\u200b\u5c06\u200b\u8f6c\u79fb\u200b\u5230\u200b\u7684\u200b\u72b6\u6001\u200b\u8bbe\u200b\u4e3a\u200b\u5f53\u524d\u200b\u72b6\u6001\u200b\uff0c\u200b\u7ee7\u7eed\u200b\u8fdb\u884c\u200b\u641c\u7d22\u200b\u8fc7\u7a0b\u200b\u3002\u200b\u672c\u9898\u200b\u4e2d\u200b\uff0c\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\u662f\u200b\u5355\u5411\u200b\u7684\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e0d\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u91cd\u590d\u200b\u7ecf\u8fc7\u200b\u4e00\u79cd\u200b\u72b6\u6001\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002</p> <p>\u200b\u7531\u4e8e\u200b\u4e00\u4e2a\u200b\u72b6\u6001\u200b\u53ef\u4ee5\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u72b6\u6001\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b\u8fd9\u4e9b\u200b\u72b6\u6001\u200b\u95f4\u200b\u8fdb\u884c\u200b\u7b5b\u9009\u200b\u3002\u200b\u7b5b\u9009\u200b\u89c4\u5219\u200b\u5982\u4e0b\u200b\uff1a</p> <ol> <li>\u200b\u5f53\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b\u4e24\u4fa7\u200b\u7684\u200b\u989c\u8272\u200b\u4e0d\u200b\u540c\u65f6\u200b\uff0c\u200b\u65b0\u7403\u200b\u7684\u200b\u989c\u8272\u200b\u9700\u8981\u200b\u4e0e\u200b\u5de6\u53f3\u200b\u4e24\u4fa7\u200b\u5176\u4e2d\u200b\u4e00\u4e2a\u200b\u7403\u200b\u7684\u200b\u989c\u8272\u200b\u76f8\u540c\u200b</li> <li>\u200b\u5f53\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b\u4e24\u4fa7\u200b\u989c\u8272\u200b\u76f8\u540c\u200b\u65f6\u200b\uff0c\u200b\u65b0\u7403\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4efb\u610f\u200b\u53ef\u80fd\u200b\u7684\u200b\u989c\u8272\u200b</li> <li>\u200b\u5f53\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b\u4f4d\u4e8e\u200b\u7aef\u70b9\u200b\u65f6\u200b\uff0c\u200b\u65b0\u7403\u200b\u7684\u200b\u989c\u8272\u200b\u9700\u8981\u200b\u4e0e\u200b\u76f8\u90bb\u200b\u7403\u200b\u7684\u200b\u989c\u8272\u200b\u76f8\u540c\u200b</li> </ol> <p>\u200b\u5982\u200b\u4e0a\u200b\u526a\u679d\u200b\u601d\u8def\u200b\u4e0d\u4f1a\u200b\u5f71\u54cd\u200b\u6d4b\u4f8b\u200b<code>\"RRYGGYYRRYGGYYRR\" \"GGBBB\"</code>\u200b\u4e0e\u200b<code>\"RRWWRRBBRR\" \"WB\"</code>\u200b\u7b49\u200b\u9700\u8981\u200b\u5c06\u200b\u8fde\u7eed\u200b\u5e8f\u5217\u200b\u5206\u5272\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u5982\u200b\u4e0a\u200b\u6761\u4ef6\u200b\u7684\u200b\u72b6\u6001\u200b\u89c6\u4e3a\u200b\u65e0\u6548\u200b\u7684\u200b\u72b6\u6001\u200b\uff0c\u200b\u76f4\u63a5\u200b\u8df3\u8fc7\u200b\u5bf9\u200b\u8fd9\u4e9b\u200b\u72b6\u6001\u200b\u7684\u200b\u641c\u7d22\u200b\u3002</p> <p>\u200b\u4f7f\u7528\u200b\u94fe\u8868\u200b\u6a21\u62df\u200b\u961f\u5217\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u94fe\u8868\u200b\u8282\u70b9\u200b\u8868\u793a\u200b\u95ee\u9898\u200b\u7684\u200b\u4e00\u4e2a\u200b\u72b6\u6001\u200b\uff0c\u200b\u7531\u4e8e\u200b\u94fe\u8868\u200b\u8282\u70b9\u200b\u9700\u8981\u200b\u591a\u6b21\u200b\u5206\u914d\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u63d0\u524d\u200b\u5206\u914d\u5185\u5b58\u200b\u6c60\u200b\u7684\u200b\u7b56\u7565\u200b\u4ee5\u200b\u51cf\u5c11\u200b<code>malloc()</code>\u200b\u7684\u200b\u8c03\u7528\u200b\u6b21\u6570\u200b\u3002</p> <pre><code>struct Queue\n{\n    char *val;\n    int colorsOnHand[5];\n    struct Queue *next;\n};\n\nstruct Queue *nodePool;\nint poolSize;\nint poolAllocated;\n\nstruct Queue *getNode(char *val, struct Queue *next, int *colorsOnHand)\n{\n    int i = 0;\n    struct Queue *ret;\n    if (!nodePool || poolAllocated &gt;= poolSize)\n    {\n        if (poolAllocated)\n            poolSize &lt;&lt;= 1;\n        else\n            poolSize = 1000;\n        poolAllocated = 0;\n        nodePool = (struct Queue *)malloc(sizeof(struct Queue) * poolSize);\n    }\n    ret = nodePool + (poolAllocated++);\n    ret-&gt;val = val;\n    ret-&gt;next = next;\n    for (i = 0; i &lt; 5; i++)\n        ret-&gt;colorsOnHand[i] = colorsOnHand[i];\n    return ret;\n}\n</code></pre> <p>\u200b\u5206\u200b\u4e24\u6b65\u200b\u8fdb\u884c\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\uff0c\u200b\u7b2c\u4e00\u6b65\u200b\u5148\u200b\u63d2\u5165\u200b\u4e00\u4e2a\u200b\u7403\u200b\uff0c\u200b\u7b2c\u4e8c\u6b65\u200b\u6d88\u9664\u200b\u8fde\u7eed\u200b\u7684\u200b\u7403\u200b\uff0c\u200b\u76f4\u5230\u200b\u65e0\u7403\u200b\u53ef\u6d88\u200b\u3002</p> <pre><code>char *strPool;\nint strSize;\nint strAllocated;\n\nchar *getStr(int len)\n{\n    if (!strPool || strAllocated + len &gt;= strSize)\n    {\n        if (strAllocated)\n            strSize &lt;&lt;= 1;\n        else\n            strSize = 32768;\n        strAllocated = 0;\n        strPool = (char *)malloc(sizeof(char) * strSize);\n    }\n    char *ret = strPool + strAllocated;\n    strAllocated += len;\n    return ret;\n}\n\nbool play(char *src)\n{\n    char *ptrfast = src, *ptrslow = src, cur = 0;\n    int i = 0;\n    bool flag = false;\n    while (*ptrfast)\n    {\n        if (cur == *ptrfast)\n            i++;\n        else\n        {\n            if (i &lt; 3)\n            {\n                while (i)\n                {\n                    *(ptrslow++) = cur;\n                    i--;\n                }\n            }\n            else\n                flag = true;\n            i = 1;\n            cur = *ptrfast;\n        }\n        ptrfast++;\n    }\n    if (i &lt; 3)\n    {\n        while (i)\n        {\n            *(ptrslow++) = cur;\n            i--;\n        }\n    }\n    else\n        flag = true;\n    *ptrslow = 0;\n    return flag;\n}\n\nchar *addBall(char *src, int len, char ball, int loc)\n{\n    char *ret = (char *)malloc(sizeof(char) * (len + 2));\n    ret[len + 1] = 0;\n    memcpy(ret, src, sizeof(char) * loc);\n    memcpy(ret + loc + 1, src + loc, sizeof(char) * (len - loc));\n    ret[loc] = ball;\n    return ret;\n}\n</code></pre> <pre><code>int findMinStep(char * board, char * hand) {\n    struct Queue *current = NULL, *temp = NULL;\n    char *cur = hand, *init = getStr(strlen(board) + 1), *newseq = NULL,\n         *colors = \"RYBGW\";\n    int i = 0, j = 0, len = 0, ret = 0, colorsOnHand[5] = {0}, flag;\n\n    nodePool = NULL;\n    poolSize = 0;\n    poolAllocated = 0;\n    strPool = NULL;\n    strSize = 0;\n    strAllocated = 0;\n\n    strcpy(init, board);\n    do\n        for (i = 0; i &lt; 5; i++)\n            if (*cur == colors[i])\n                colorsOnHand[i]++;\n    while (*(++cur));\n\n    current = getNode(init, NULL, colorsOnHand);\n    while (current)\n    {\n        while (current)\n        {\n            len = strlen(current-&gt;val);\n            if (!len)\n                return ret;\n            for (i = 0; i &lt;= len; i++)\n            {\n                flag = false;\n                for (j = 0; j &lt; 5; j++)\n                {\n                    if (current-&gt;colorsOnHand[j] == 0)\n                        continue;\n                    if (i &gt; 0)\n                        flag = flag || current-&gt;val[i - 1] == colors[j];\n                    if (i &lt; len)\n                        flag = flag || current-&gt;val[i] == colors[j];\n                    if (i &gt; 0 &amp;&amp; i &lt; len)\n                        flag = flag || current-&gt;val[i - 1] == current-&gt;val[i];\n                    if (!flag)\n                        continue;\n                    current-&gt;colorsOnHand[j]--;\n                    newseq = addBall(current-&gt;val, len, colors[j], i);\n                    while (play(newseq));\n                    temp = getNode(newseq, temp, current-&gt;colorsOnHand);\n                    current-&gt;colorsOnHand[j]++;\n                }\n            }\n            current = current-&gt;next;\n        }\n        ret++;\n        current = temp;\n        temp = NULL;\n    }\n    return -1;\n}\n</code></pre>","tags":["\u6df1\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/50/","title":"50. Pow(x, n)","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u5b66","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/50/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5b9e\u73b0\u200b\u00a0pow(x, n)\uff0c\u200b\u5373\u200b\u8ba1\u7b97\u200b x \u200b\u7684\u200b n \u200b\u6b21\u200b\u5e42\u51fd\u6570\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a <pre><code>\u200b\u8f93\u5165\u200b: 2.00000, 10\n\u200b\u8f93\u51fa\u200b: 1024.00000\n</code></pre> \u200b\u793a\u4f8b\u200b\u00a02\uff1a <pre><code>\u200b\u8f93\u5165\u200b: 2.10000, 3\n\u200b\u8f93\u51fa\u200b: 9.26100\n</code></pre> \u200b\u793a\u4f8b\u200b\u00a03\uff1a <pre><code>\u200b\u8f93\u5165\u200b: 2.00000, -2\n\u200b\u8f93\u51fa\u200b: 0.25000\n\u200b\u89e3\u91ca\u200b: 2^(-2) = (1/2)^2 = 1/4 = 0.25\n</code></pre> \u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li>-100.0 &lt;\u00a0x\u00a0&lt; 100.0</li> <li>n\u00a0\u200b\u662f\u200b 32 \u200b\u4f4d\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\uff0c\u200b\u5176\u200b\u6570\u503c\u200b\u8303\u56f4\u200b\u662f\u200b\\([\u22122^{31},\u00a02^{31}\u00a0\u2212 1]\\)\u3002</li> </ul> <p>Reference</p>","tags":["\u6570\u5b66","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/50/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5229\u7528\u200b\u5feb\u901f\u200b\u5e42\u200b\u7684\u200b\u601d\u8def\u200b\uff0c\u200b\u5b58\u50a8\u200b\u6b21\u6570\u200b\u4e3a\u200b2\u200b\u7684\u200b\u6574\u6570\u200b\u5e42\u200b\u7684\u200b\u8ba1\u7b97\u7ed3\u679c\u200b\uff0c\u200b\u7136\u540e\u200b\u7528\u200b\u8fd9\u4e9b\u200b\u7ed3\u679c\u200b\u76f8\u4e58\u200b\u5f97\u5230\u200b\u8f93\u51fa\u200b\u7ed3\u679c\u200b\u3002\u200b\u6ce8\u610f\u200b\u6307\u6570\u200b\u4e3a\u200b\u8d1f\u6570\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002\u200b\u5f53\u200b\u6307\u6570\u200b\u4e3a\u200b<code>-2147483648</code>\u200b\u65f6\u200b\u4e0d\u80fd\u200b\u53d6\u8d1f\u200b\uff0c\u200b\u6b64\u65f6\u200b\u53ef\u4ee5\u200b\u5148\u52a0\u200b\u4e00\u518d\u200b\u53d6\u8d1f\u200b\uff0c\u200b\u6700\u540e\u200b\u9664\u4ee5\u200b\u5e95\u6570\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>double myPow(double x, int n)\n{\n    if (n &lt; 0)\n        return 1 / myPow(x, -(n + 1)) / x;\n    double cache[32] = {0}, ret = 1;\n    int i = 0;\n    cache[0] = x;\n    while (n)\n    {\n        if (n &amp; 1)\n            ret *= cache[i];\n        n &gt;&gt;= 1;\n        i++;\n        cache[i] = cache[i - 1] * cache[i - 1];\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u5b66","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/509/","title":"509. \u200b\u6590\u6ce2\u200b\u90a3\u200b\u5951\u6570","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/509/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u6590\u6ce2\u200b\u90a3\u200b\u5951\u6570\u200b\uff0c\u200b\u901a\u5e38\u200b\u7528\u200b<code>F(n)</code>\u200b\u8868\u793a\u200b\uff0c\u200b\u5f62\u6210\u200b\u7684\u200b\u5e8f\u5217\u200b\u79f0\u4e3a\u200b\u6590\u6ce2\u200b\u90a3\u5951\u200b\u6570\u5217\u200b\u3002\u200b\u8be5\u200b\u6570\u5217\u200b\u7531\u200b<code>0</code>\u200b\u548c\u200b<code>1</code>\u200b\u5f00\u59cb\u200b\uff0c\u200b\u540e\u9762\u200b\u7684\u200b\u6bcf\u4e00\u9879\u200b\u6570\u5b57\u200b\u90fd\u200b\u662f\u200b\u524d\u9762\u200b\u4e24\u9879\u200b\u6570\u5b57\u200b\u7684\u200b\u548c\u200b\u3002\u200b\u4e5f\u200b\u5c31\u662f\u200b\uff1a <pre><code>F(0) = 0,\u00a0 \u00a0F(1)\u00a0= 1\nF(N) = F(N - 1) + F(N - 2), \u200b\u5176\u4e2d\u200b N &gt; 1.\n</code></pre> \u200b\u7ed9\u5b9a\u200b<code>N</code>\uff0c\u200b\u8ba1\u7b97\u200b<code>F(N)</code>\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a <pre><code>\u200b\u8f93\u5165\u200b\uff1a2\n\u200b\u8f93\u51fa\u200b\uff1a1\n\u200b\u89e3\u91ca\u200b\uff1aF(2) = F(1) + F(0) = 1 + 0 = 1.\n</code></pre></p> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a3\n\u200b\u8f93\u51fa\u200b\uff1a2\n\u200b\u89e3\u91ca\u200b\uff1aF(3) = F(2) + F(1) = 1 + 1 = 2.\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a4\n\u200b\u8f93\u51fa\u200b\uff1a3\n\u200b\u89e3\u91ca\u200b\uff1aF(4) = F(3) + F(2) = 2 + 1 = 3.\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <p>\\(0\\leq N\\leq30\\)</p> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/509/#_2","title":"\u9898\u89e3","text":"","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/509/#_3","title":"\u9012\u5f52","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4f7f\u7528\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\u5b58\u653e\u200b\u5386\u53f2\u200b\u7ed3\u679c\u200b\uff0c\u200b\u52a0\u901f\u200b\u9012\u5f52\u8ba1\u7b97\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int fib_cache(int N, int **cache, int *lenCache, int *fullCache){\n    if (!N)\n        return 0;\n    if (!(*cache))\n    {\n        *lenCache = 2;\n        *fullCache = N + 1;\n        *cache = (int *)memset(malloc(sizeof(int) * *fullCache), 0, sizeof(int) * *fullCache);\n        (*cache)[0] = 1;\n        (*cache)[1] = 1;\n    }\n    if (N &gt; *lenCache &amp;&amp; N &lt;= *fullCache)\n        (*cache)[N-1] = fib_cache(N - 1, cache, lenCache, fullCache) + fib_cache(N - 2, cache, lenCache, fullCache);\n        *lenCache = N;\n    return (*cache)[N - 1];\n}\n\nint fib(int N){\n    int *cache = NULL, lenCache = 0, fullCache = 0;\n    return fib_cache(N, &amp;cache, &amp;lenCache, &amp;fullCache);\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/509/#_4","title":"\u8fed\u4ee3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6839\u636e\u200b\u5b9a\u4e49\u200b\u63a8\u7b97\u200b\u6590\u6ce2\u200b\u90a3\u200b\u5951\u6570\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int fib(int N){\n    int ret = 0, next = 1, tmp;\n    while (N)\n    {\n        tmp = ret;\n        ret = next;\n        next = ret + tmp;\n        N--;\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/53/","title":"53. \u200b\u6700\u5927\u200b\u5b50\u5e8f\u200b\u548c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u5206\u6cbb\u7b97\u6cd5","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/53/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b <code>nums</code>\u00a0\uff0c\u200b\u627e\u5230\u200b\u4e00\u4e2a\u200b\u5177\u6709\u200b\u6700\u5927\u200b\u548c\u200b\u7684\u200b\u8fde\u7eed\u200b\u5b50\u200b\u6570\u7ec4\u200b\uff08\u200b\u5b50\u200b\u6570\u7ec4\u200b\u6700\u5c11\u200b\u5305\u542b\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\uff09\uff0c\u200b\u8fd4\u56de\u200b\u5176\u200b\u6700\u5927\u200b\u548c\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [-2,1,-3,4,-1,2,1,-5,4]\n\u200b\u8f93\u51fa\u200b\uff1a6\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u8fde\u7eed\u200b\u5b50\u200b\u6570\u7ec4\u200b\u00a0[4,-1,2,1] \u200b\u7684\u200b\u548c\u200b\u6700\u5927\u200b\uff0c\u200b\u4e3a\u200b\u00a06 \u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1]\n\u200b\u8f93\u51fa\u200b\uff1a1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [0]\n\u200b\u8f93\u51fa\u200b\uff1a0\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [-1]\n\u200b\u8f93\u51fa\u200b\uff1a-1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 5\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [-100000]\n\u200b\u8f93\u51fa\u200b\uff1a-100000\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 3 * 10^4</code></li> <li><code>-10^5 &lt;= nums[i] &lt;= 10^5</code></li> </ul> <p>\u200b\u8fdb\u9636\u200b\uff1a\u200b\u5982\u679c\u200b\u4f60\u200b\u5df2\u7ecf\u200b\u5b9e\u73b0\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b <code>O(n)</code> \u200b\u7684\u200b\u89e3\u6cd5\u200b\uff0c\u200b\u5c1d\u8bd5\u200b\u4f7f\u7528\u200b\u66f4\u4e3a\u200b\u7cbe\u5999\u200b\u7684\u200b \u200b\u5206\u6cbb\u200b\u6cd5\u200b \u200b\u6c42\u89e3\u200b\u3002</p> <p>Reference</p>","tags":["\u6570\u7ec4","\u5206\u6cbb\u7b97\u6cd5","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/53/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u4ece\u540e\u200b\u5411\u524d\u200b\u8fdb\u884c\u200b\u8003\u8651\u200b\uff0c\u200b\u8bbe\u200b\u6570\u7ec4\u200b\u4e2d\u5171\u200b\u6709\u200b\\(n\\)\u200b\u4e2a\u200b\u5143\u7d20\u200b\\(\\{a_n\\}\\)\uff0c\u200b\u5f53\u524d\u200b\u4f4d\u7f6e\u200b\\(i\\)\u200b\u4ece\u540e\u200b\u5411\u524d\u200b\u79fb\u52a8\u200b\uff0c\\(i\\)\u200b\u4f4d\u7f6e\u200b\u53ca\u200b\\(i\\)\u200b\u4f4d\u7f6e\u200b\u4e4b\u540e\u200b\u6784\u6210\u200b\u7684\u200b\u6700\u5927\u200b\u5b50\u200b\u5e8f\u5217\u200b\u6570\u503c\u200b\u4e3a\u200b\\(s_i\\)\u3002</p> <ol> <li>\u200b\u521d\u59cb\u200b\u60c5\u51b5\u200b\uff0c\\(s_n = 0\\)</li> <li>\u200b\u8ba8\u8bba\u200b\\(s_{i + 1}\\)\uff1a</li> <li>\u200b\u82e5\u200b\\(s_{i + 1} \\leq 0\\)\uff0c\u200b\u5219\u200b\\(s_{i} = a_{i}\\)</li> <li>\u200b\u82e5\u200b\\(s_{i + 1} &gt; 0\\)\uff0c\u200b\u5219\u200b\\(s_{i} = a_{i} + s_{i + 1}\\)</li> </ol> <p>\u200b\u7531\u6b64\u200b\u4e0d\u65ad\u200b\u8fdb\u884c\u200b\uff0c\u200b\u4e22\u5f03\u200b\u548c\u200b\u4e3a\u200b\u8d1f\u6570\u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217\u200b\uff0c\u200b\u6700\u7ec8\u200b\u5f97\u5230\u200b\u7684\u200b\\(s_0\\)\u200b\u5373\u200b\u4e3a\u200b\u6700\u5927\u200b\u5b50\u5e8f\u200b\u548c\u200b\u3002</p> <pre><code>#define MAX(x, y) (x &gt; y ? x : y)\nint maxSubArray(int* nums, int numsSize){\n    int ret = INT_MIN, value = 0;\n    for (int i = numsSize - 1; i &gt;= 0; i--)\n    {\n        value = MAX(value, 0) + nums[i];\n        ret = MAX(ret, value);\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u5206\u6cbb\u7b97\u6cd5","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/55/","title":"55. \u200b\u8df3\u8dc3\u200b\u6e38\u620f","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p> <p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\u00a0<code>nums</code> \uff0c\u200b\u4f60\u200b\u6700\u521d\u200b\u4f4d\u4e8e\u200b\u6570\u7ec4\u200b\u7684\u200b \u200b\u7b2c\u4e00\u4e2a\u200b\u4e0b\u6807\u200b \u3002</p> <p>\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u4ee3\u8868\u200b\u4f60\u200b\u5728\u200b\u8be5\u200b\u4f4d\u7f6e\u200b\u53ef\u4ee5\u200b\u8df3\u8dc3\u200b\u7684\u200b\u6700\u5927\u200b\u957f\u5ea6\u200b\u3002</p> <p>\u200b\u5224\u65ad\u200b\u4f60\u200b\u662f\u5426\u200b\u80fd\u591f\u200b\u5230\u8fbe\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u4e0b\u6807\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [2,3,1,1,4]\n\u200b\u8f93\u51fa\u200b\uff1atrue\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u53ef\u4ee5\u200b\u5148\u200b\u8df3\u200b 1 \u200b\u6b65\u200b\uff0c\u200b\u4ece\u200b\u4e0b\u6807\u200b 0 \u200b\u5230\u8fbe\u200b\u4e0b\u6807\u200b 1, \u200b\u7136\u540e\u200b\u518d\u200b\u4ece\u200b\u4e0b\u6807\u200b 1 \u200b\u8df3\u200b 3 \u200b\u6b65\u200b\u5230\u8fbe\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u4e0b\u6807\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [3,2,1,0,4]\n\u200b\u8f93\u51fa\u200b\uff1afalse\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u65e0\u8bba\u600e\u6837\u200b\uff0c\u200b\u603b\u4f1a\u200b\u5230\u8fbe\u200b\u4e0b\u6807\u200b\u4e3a\u200b 3 \u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002\u200b\u4f46\u200b\u8be5\u200b\u4e0b\u200b\u6807\u7684\u200b\u6700\u5927\u200b\u8df3\u8dc3\u200b\u957f\u5ea6\u200b\u662f\u200b 0 \uff0c \u200b\u6240\u4ee5\u200b\u6c38\u8fdc\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u5230\u8fbe\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u4e0b\u6807\u200b\u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 3 * 10^4</code></li> <li><code>0 &lt;= nums[i] &lt;= 10^5</code></li> </ul> <p>Reference</p>","tags":["\u8d2a\u5fc3\u7b97\u6cd5","\u6570\u7ec4"]},{"location":"coding/leetcode/55/#_1","title":"\u9898\u89e3","text":"<p>\u200b\u53cd\u5e8f\u200b\u904d\u5386\u200b\u6570\u7ec4\u200b\uff0c\u200b\u4ece\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u4f4d\u7f6e\u200b\u5f00\u59cb\u200b\uff0c\u200b\u68c0\u67e5\u200b\u5404\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u53ef\u200b\u8df3\u200b\u8ddd\u79bb\u200b\uff0c\u200b\u4ece\u800c\u200b\u6784\u9020\u200b\u4e00\u4e2a\u200b\u53ef\u4ee5\u200b\u8df3\u5230\u200b\u4e2d\u70b9\u200b\u7684\u200b\u4e0b\u6807\u200b\u533a\u95f4\u200b\u3002\u200b\u904d\u5386\u200b\u5b8c\u6210\u200b\u540e\u200b\u68c0\u67e5\u200b\u8d77\u70b9\u200b\u662f\u5426\u200b\u5728\u200b\u8be5\u200b\u4e0b\u6807\u200b\u533a\u95f4\u200b\u4e2d\u200b\u3002</p> <pre><code>bool canJump(int* nums, int numsSize) {\n\n    int distance = numsSize - 1, i = 0;\n    for (i = numsSize - 1; i &gt;= 0; i--)\n        if (i + nums[i] &gt;= distance)\n            distance = i;\n    return distance == 0;\n}\n</code></pre>","tags":["\u8d2a\u5fc3\u7b97\u6cd5","\u6570\u7ec4"]},{"location":"coding/leetcode/56/","title":"56. \u200b\u5408\u5e76\u200b\u533a\u95f4","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u6392\u5e8f"]},{"location":"coding/leetcode/56/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u4ee5\u200b\u6570\u7ec4\u200b <code>intervals</code> \u200b\u8868\u793a\u200b\u82e5\u5e72\u4e2a\u200b\u533a\u95f4\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u5176\u4e2d\u200b\u5355\u4e2a\u200b\u533a\u95f4\u200b\u4e3a\u200b <code>intervals[i] = [start_i, end_i]</code> \u3002\u200b\u8bf7\u200b\u4f60\u200b\u5408\u5e76\u200b\u6240\u6709\u200b\u91cd\u53e0\u200b\u7684\u200b\u533a\u95f4\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u4e0d\u200b\u91cd\u53e0\u200b\u7684\u200b\u533a\u95f4\u200b\u6570\u7ec4\u200b\uff0c\u200b\u8be5\u200b\u6570\u7ec4\u200b\u9700\u200b\u6070\u597d\u200b\u8986\u76d6\u200b\u8f93\u5165\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u533a\u95f4\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aintervals = [[1,3],[2,6],[8,10],[15,18]]\n\u200b\u8f93\u51fa\u200b\uff1a[[1,6],[8,10],[15,18]]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u533a\u95f4\u200b [1,3] \u200b\u548c\u200b [2,6] \u200b\u91cd\u53e0\u200b, \u200b\u5c06\u200b\u5b83\u4eec\u200b\u5408\u5e76\u200b\u4e3a\u200b [1,6].\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aintervals = [[1,4],[4,5]]\n\u200b\u8f93\u51fa\u200b\uff1a[[1,5]]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u533a\u95f4\u200b [1,4] \u200b\u548c\u200b [4,5] \u200b\u53ef\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u91cd\u53e0\u200b\u533a\u95f4\u200b\u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= intervals.length &lt;= 10^4</code></li> <li><code>intervals[i].length == 2</code></li> <li><code>0 &lt;= starti &lt;= endi &lt;= 10^4</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u6392\u5e8f"]},{"location":"coding/leetcode/56/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u9996\u5148\u200b\u5bf9\u200b\u533a\u95f4\u200b\u7684\u200b\u8d77\u70b9\u200b\u6309\u7167\u200b\u5347\u5e8f\u200b\u6392\u5e8f\u200b\uff0c\u200b\u4e4b\u540e\u200b\u904d\u5386\u200b\u533a\u95f4\u200b\u5217\u8868\u200b\uff0c\u200b\u4e0d\u65ad\u200b\u68c0\u67e5\u200b\u5df2\u200b\u5408\u5e76\u200b\u533a\u95f4\u200b\u4e0e\u200b\u5f53\u524d\u200b\u533a\u95f4\u200b\u662f\u5426\u200b\u5b58\u5728\u200b\u91cd\u53e0\u200b\uff0c\u200b\u5982\u679c\u200b\u5b58\u5728\u200b\u91cd\u53e0\u200b\uff0c\u200b\u5219\u200b\u5c06\u200b\u4e24\u200b\u533a\u95f4\u200b\u5408\u5e76\u200b\uff0c\u200b\u5426\u5219\u200b\u5c06\u200b\u5df2\u200b\u5408\u5e76\u200b\u533a\u95f4\u200b\u52a0\u5165\u200b\u7ed3\u679c\u200b\u96c6\u4e2d\u200b\uff0c\u200b\u5f53\u524d\u200b\u533a\u95f4\u200b\u8bbe\u4e3a\u200b\u5df2\u200b\u5408\u5e76\u200b\u533a\u95f4\u200b\u3002</p> <pre><code>/**\n * Return an array of arrays of size *returnSize.\n * The sizes of the arrays are returned as *returnColumnSizes array.\n * Note: Both returned array and *columnSizes array must be malloced, assume caller calls free().\n */\n#define MAX(x, y) (x &gt; y ? x : y)\nint comp(const void *x, const void *y)\n{\n    return **(int **)x - **(int **)y;\n}\nint** merge(int** intervals, int intervalsSize, int* intervalsColSize, int* returnSize, int** returnColumnSizes){\n    if (intervalsSize == 0)\n    {\n        *returnSize = 0;\n        return NULL;\n    }\n    qsort(intervals, intervalsSize, sizeof(int *), comp);\n    int **ret = (int **)malloc(sizeof(int *) * intervalsSize), i = 0;\n    ret[0] = (int *)memcpy(malloc(sizeof(int) * 2), intervals[0], sizeof(int) * 2);\n    *returnSize = 0;\n    for (i = 1; i &lt; intervalsSize; i++)\n    {\n        if (ret[*returnSize][1] &gt;= intervals[i][0])\n            ret[*returnSize][1] = MAX(intervals[i][1], ret[*returnSize][1]);\n        else\n            ret[++(*returnSize)] = (int *)memcpy(malloc(sizeof(int) * 2), intervals[i], sizeof(int) * 2);\n    }\n    (*returnSize)++;\n    *returnColumnSizes = (int *)malloc(sizeof(int) * *returnSize);\n    for (i = 0; i &lt; *returnSize; i++)\n        (*returnColumnSizes)[i] = 2;\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u6392\u5e8f"]},{"location":"coding/leetcode/59/","title":"59. \u200b\u87ba\u65cb\u200b\u77e9\u9635\u200b II","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/59/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u6b63\u6574\u6570\u200b\u00a0<code>n</code> \uff0c\u200b\u751f\u6210\u200b\u4e00\u4e2a\u200b\u5305\u542b\u200b <code>1</code> \u200b\u5230\u200b\u00a0<code>n^2</code>\u00a0\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\uff0c\u200b\u4e14\u200b\u5143\u7d20\u200b\u6309\u200b\u987a\u65f6\u9488\u200b\u987a\u5e8f\u200b\u87ba\u65cb\u200b\u6392\u5217\u200b\u7684\u200b\u00a0<code>n x n</code> \u200b\u6b63\u65b9\u5f62\u200b\u77e9\u9635\u200b <code>matrix</code> \u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1an = 3\n\u200b\u8f93\u51fa\u200b\uff1a[[1,2,3],[8,9,4],[7,6,5]]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1an = 1\n\u200b\u8f93\u51fa\u200b\uff1a[[1]]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= n &lt;= 20</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/59/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u6309\u200b\u987a\u5e8f\u200b\u586b\u5145\u200b\u77e9\u9635\u200b\uff0c\u200b\u5f53\u200b\u8d85\u51fa\u200b\u77e9\u9635\u200b\u6216\u200b\u9047\u5230\u200b\u5df2\u200b\u586b\u5145\u200b\u5143\u7d20\u200b\u65f6\u200b\u8f6c\u5411\u200b\u3002</p> <pre><code>/**\n * Return an array of arrays of size *returnSize.\n * The sizes of the arrays are returned as *returnColumnSizes array.\n * Note: Both returned array and *columnSizes array must be malloced, assume caller calls free().\n */\nvoid change(int direction, int *i, int *j, int incre)\n{\n    switch(direction)\n    {\n        case 0:\n            *j += incre;\n            break;\n        case 1:\n            *i += incre;\n            break;\n        case 2:\n            *j -= incre;\n            break;\n        case 3:\n            *i -= incre;\n            break;\n    }\n}\nint** generateMatrix(int n, int* returnSize, int** returnColumnSizes){\n    *returnSize = n;\n    *returnColumnSizes = (int *)malloc(sizeof(int) * n);\n\n    int i = 0, j = 0, m = 1, direction = 0, **ret = (int **)malloc(sizeof(int *) * n);\n    for (i = 0; i &lt; n; i++)\n    {\n        (*returnColumnSizes)[i] = n;\n        ret[i] = (int *)malloc(sizeof(int) * n);\n        memset(ret[i], 0, sizeof(int) * n);\n    }\n\n    i = 0;\n    while (m &lt; n * n)\n    {\n        ret[i][j] = m;\n        change(direction, &amp;i, &amp;j, 1);\n        m++;\n        if (i &lt; 0 || j &lt; 0 || i &gt;= n || j &gt;= n || ret[i][j])\n        {\n            change(direction, &amp;i, &amp;j, -1);\n            direction = (direction + 1) &amp; 3;\n            m--;\n        }\n    }\n    ret[i][j] = m;\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/61/","title":"61. \u200b\u65cb\u8f6c\u200b\u94fe\u8868","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/61/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b\u7684\u200b\u5934\u200b\u8282\u70b9\u200b <code>head</code> \uff0c\u200b\u65cb\u8f6c\u200b\u94fe\u8868\u200b\uff0c\u200b\u5c06\u200b\u94fe\u8868\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u00a0<code>k</code>\u00a0\u200b\u4e2a\u200b\u4f4d\u7f6e\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [1,2,3,4,5], k = 2\n\u200b\u8f93\u51fa\u200b\uff1a[4,5,1,2,3]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [0,1,2], k = 4\n\u200b\u8f93\u51fa\u200b\uff1a[2,0,1]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u94fe\u8868\u200b\u4e2d\u200b\u8282\u70b9\u200b\u7684\u200b\u6570\u76ee\u200b\u5728\u200b\u8303\u56f4\u200b <code>[0, 500]</code> \u200b\u5185\u200b</li> <li><code>-100 &lt;= Node.val &lt;= 100</code></li> <li><code>0 &lt;= k &lt;= 2 * 10^9</code></li> </ul> <p>Reference</p>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/61/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5148\u200b\u5c06\u200b\u94fe\u8868\u200b\u95ed\u73af\u200b\uff0c\u200b\u53f3\u79fb\u200b k \u200b\u4e2a\u200b\u8282\u70b9\u200b\u540e\u200b\u4ee5\u200b\u8be5\u200b\u8282\u70b9\u200b\u4f5c\u4e3a\u200b\u5934\u200b\u8282\u70b9\u200b\uff0c\u200b\u5c06\u200b\u94fe\u8868\u200b\u7684\u200b\u73af\u200b\u6253\u5f00\u200b\u3002</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\n\n\nstruct ListNode* rotateRight(struct ListNode* head, int k) {\n    if (!head)\n        return NULL;\n    struct ListNode *cur = head, *ret = NULL;\n    int len = 1;\n    while (cur-&gt;next != NULL)\n    {\n        cur = cur-&gt;next;\n        len++;\n    }\n    cur-&gt;next = head;\n    cur = head;\n    k %= len;\n    for (int i = k + 1; i &lt; len; i++)\n        cur = cur-&gt;next;\n    ret = cur-&gt;next;\n    cur-&gt;next = NULL;\n    return ret;\n}\n</code></pre>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/62/","title":"62. \u200b\u4e0d\u540c\u200b\u8def\u5f84","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/62/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u4e00\u4e2a\u200b\u673a\u5668\u4eba\u200b\u4f4d\u4e8e\u200b\u4e00\u4e2a\u200b <code>m x n</code>\u00a0\u200b\u7f51\u683c\u200b\u7684\u200b\u5de6\u4e0a\u89d2\u200b \uff08\u200b\u8d77\u59cb\u200b\u70b9\u200b\u5728\u200b\u4e0b\u56fe\u200b\u4e2d\u200b\u6807\u8bb0\u200b\u4e3a\u200b \u201cStart\u201d \uff09\u3002</p> <p>\u200b\u673a\u5668\u4eba\u200b\u6bcf\u6b21\u200b\u53ea\u80fd\u200b\u5411\u4e0b\u200b\u6216\u8005\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u4e00\u6b65\u200b\u3002\u200b\u673a\u5668\u4eba\u200b\u8bd5\u56fe\u200b\u8fbe\u5230\u200b\u7f51\u683c\u200b\u7684\u200b\u53f3\u4e0b\u89d2\u200b\uff08\u200b\u5728\u200b\u4e0b\u56fe\u200b\u4e2d\u200b\u6807\u8bb0\u200b\u4e3a\u200b \u201cFinish\u201d \uff09\u3002</p> <p>\u200b\u95ee\u200b\u603b\u5171\u200b\u6709\u200b\u591a\u5c11\u200b\u6761\u200b\u4e0d\u540c\u200b\u7684\u200b\u8def\u5f84\u200b\uff1f</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1am = 3, n = 7\n\u200b\u8f93\u51fa\u200b\uff1a28\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1am = 3, n = 2\n\u200b\u8f93\u51fa\u200b\uff1a3\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u4ece\u200b\u5de6\u4e0a\u89d2\u200b\u5f00\u59cb\u200b\uff0c\u200b\u603b\u5171\u200b\u6709\u200b 3 \u200b\u6761\u200b\u8def\u5f84\u200b\u53ef\u4ee5\u200b\u5230\u8fbe\u200b\u53f3\u4e0b\u89d2\u200b\u3002\n1. \u200b\u5411\u200b\u53f3\u200b -&gt; \u200b\u5411\u4e0b\u200b -&gt; \u200b\u5411\u4e0b\u200b\n2. \u200b\u5411\u4e0b\u200b -&gt; \u200b\u5411\u4e0b\u200b -&gt; \u200b\u5411\u200b\u53f3\u200b\n3. \u200b\u5411\u4e0b\u200b -&gt; \u200b\u5411\u200b\u53f3\u200b -&gt; \u200b\u5411\u4e0b\u200b\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1am = 7, n = 3\n\u200b\u8f93\u51fa\u200b\uff1a28\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1am = 3, n = 3\n\u200b\u8f93\u51fa\u200b\uff1a6\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= m, n &lt;= 100</code></li> <li>\u200b\u9898\u76ee\u200b\u6570\u636e\u200b\u4fdd\u8bc1\u200b\u7b54\u6848\u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b <code>2 * 10^9</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/62/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4ece\u200b\u6700\u540e\u200b\u4e00\u884c\u200b\u5f00\u59cb\u200b\u8003\u8651\u200b\uff0c\u200b\u8981\u200b\u5230\u8fbe\u200b\u53f3\u4e0b\u89d2\u200b\u7684\u200b\u65b9\u683c\u200b\uff0c\u200b\u6709\u200b\u4e24\u79cd\u200b\u65b9\u6cd5\u200b\uff1a</p> <ol> <li>\u200b\u4ece\u200b\u659c\u200b\u4e0a\u65b9\u200b\u7684\u200b\u65b9\u683c\u200b\u5f00\u59cb\u200b\uff0c\u200b\u5148\u200b\u5411\u4e0b\u200b\u540e\u200b\u5411\u200b\u53f3\u200b</li> <li>\u200b\u4ece\u200b\u659c\u200b\u4e0a\u65b9\u200b\u7684\u200b\u65b9\u683c\u200b\u5f00\u59cb\u200b\uff0c\u200b\u5148\u5411\u200b\u53f3\u540e\u200b\u5411\u4e0b\u200b</li> </ol> <p>\u200b\u7531\u6b64\u200b\u5373\u53ef\u200b\u5f97\u51fa\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\u65b9\u7a0b\u200b\u3002</p> <pre><code>int uniquePaths(int m, int n){\n    if (m &lt; n)\n        return uniquePaths(n, m);\n    int *dp = (int *)malloc(sizeof(int) * m), i = 0, j = 0;\n    for (i = 0; i &lt; m; i++)\n        dp[i] = 1;\n    for (i = 1; i &lt; n; i++)\n        for (j = 1; j &lt; m; j++)\n            dp[j] += dp[j - 1];\n    return dp[m - 1];\n}\n</code></pre> <p>\u200b\u6216\u8005\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6392\u5217\u7ec4\u5408\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u957f\u4e3a\u200b\\(m\\)\uff0c\u200b\u5bbd\u4e3a\u200b\\(n\\)\u200b\u7684\u200b\u65b9\u683c\u200b\uff0c\u200b\u9700\u8981\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\\(m - 1\\)\u200b\u6b21\u200b\uff0c\u200b\u5411\u4e0b\u200b\u79fb\u52a8\u200b\\(n - 1\\)\u200b\u6b21\u200b\uff0c\u200b\u5171\u200b\\(m + n - 2\\)\u200b\u6b21\u200b\u3002</p> <p>\u200b\u56e0\u6b64\u200b\u603b\u200b\u7684\u200b\u8def\u5f84\u200b\u4e2a\u6570\u200b\u4e3a\u200b\\(m + n - 2\\)\u200b\u4e2a\u200b\u9009\u62e9\u200b\u4e2d\u4e2d\u200b\u9009\u62e9\u200b\\(m - 1\\)\u200b\u6b21\u200b\u5411\u4e0b\u200b\u79fb\u52a8\u200b\u7684\u200b\u65b9\u6848\u200b\u603b\u6570\u200b\uff0c\u200b\u8be5\u200b\u65b9\u6848\u200b\u603b\u6570\u200b\u4e3a\u200b\uff1a</p> \\[ \\frac{(m + n - 2)!}{(m - 1)!(n - 1)!} \\]","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/64/","title":"64. \u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p> <p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5305\u542b\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u7684\u200b <code>m\u00a0x\u00a0n</code>\u00a0\u200b\u7f51\u683c\u200b\u00a0<code>grid</code> \uff0c\u200b\u8bf7\u200b\u627e\u51fa\u200b\u4e00\u6761\u200b\u4ece\u200b\u5de6\u4e0a\u89d2\u200b\u5230\u200b\u53f3\u4e0b\u89d2\u200b\u7684\u200b\u8def\u5f84\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u8def\u5f84\u200b\u4e0a\u200b\u7684\u200b\u6570\u5b57\u200b\u603b\u548c\u200b\u4e3a\u200b\u6700\u5c0f\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a\u200b\u6bcf\u6b21\u200b\u53ea\u80fd\u200b\u5411\u4e0b\u200b\u6216\u8005\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u4e00\u6b65\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1agrid = [[1,3,1],[1,5,1],[4,2,1]]\n\u200b\u8f93\u51fa\u200b\uff1a7\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u56e0\u4e3a\u200b\u8def\u5f84\u200b 1\u21923\u21921\u21921\u21921 \u200b\u7684\u200b\u603b\u548c\u200b\u6700\u5c0f\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1agrid = [[1,2,3],[4,5,6]]\n\u200b\u8f93\u51fa\u200b\uff1a12\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>m == grid.length</code></li> <li><code>n == grid[i].length</code></li> <li><code>1 &lt;= m, n &lt;= 200</code></li> <li><code>0 &lt;= grid[i][j] &lt;= 100</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/64/#_1","title":"\u9898\u89e3","text":"<p>\u200b\u8bbe\u200b\u7b2c\u200b\\(i\\)\u200b\u884c\u7b2c\u200b\\(j\\)\u200b\u5217\u200b\u7684\u200b\u5143\u7d20\u200b\\(a_{ij}\\)\u200b\u5230\u200b\u53f3\u4e0b\u89d2\u200b\u8def\u5f84\u200b\u4e0a\u200b\u7684\u200b\u6570\u5b57\u200b\u548c\u200b\u4e3a\u200b\\(f(i, j)\\)\uff0c\u200b\u7279\u522b\u200b\u5730\u200b\uff0c\u200b\u6709\u200b\\(f(m, n) = a_{mn}\\)\uff0c\u200b\u5219\u200b\uff1a</p> <p>\u200b\u5bf9\u4e8e\u200b\u6700\u540e\u200b\u4e00\u884c\u200b\u5143\u7d20\u200b\uff0c\u200b\u53ea\u80fd\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\uff0c\u200b\u6709\u200b\uff1a</p> \\[ f(m, j) = \\sum_{k=j} ^{n} a_{mk} \\] <p>\u200b\u540c\u7406\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6700\u540e\u200b\u4e00\u5217\u200b\u5143\u7d20\u200b\uff0c\u200b\u6709\u200b\uff1a</p> \\[ f(i, n) = \\sum_{k = i} ^ {m} a_{kn} \\] <p>\u200b\u8bbe\u200b\u7b2c\u200b\\(i\\)\u200b\u884c\u200b\u5230\u8fbe\u200b\u53f3\u4e0b\u89d2\u200b\u7684\u200b\u6700\u200b\u77ed\u200b\u8def\u5f84\u200b\u548c\u200b\uff08\u200b\u5373\u200b\\(f(i, j), \\forall j\\)\uff09\u200b\u4ee5\u53ca\u200b\u7b2c\u200b\\(i - 1\\)\u200b\u884c\u200b\u6700\u540e\u200b\u4e00\u5217\u200b\u5230\u8fbe\u200b\u53f3\u4e0b\u89d2\u200b\u7684\u200b\u6700\u200b\u77ed\u200b\u8def\u5f84\u200b\u548c\u200b\uff0c\u200b\u5373\u200b\\(f(i - 1, n)\\)\u200b\u5df2\u77e5\u200b\u3002\u200b\u7531\u4e8e\u200b\u6bcf\u200b\u4e00\u6b65\u200b\u53ea\u80fd\u200b\u5411\u200b\u53f3\u200b\u6216\u200b\u5411\u4e0b\u200b\u79fb\u52a8\u200b\uff0c\\(f(i - 1, j)\\)\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u5982\u4e0b\u200b\u65b9\u7a0b\u200b\u63a8\u5bfc\u200b\uff1a</p> \\[ f(i - 1, j - 1) = \\min\\{f(i - 1, j), f(i, j - 1)\\} + a_{(i - 1)(j - 1)} \\] <p>\u200b\u7531\u6b64\u200b\u5f97\u5230\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\u65b9\u7a0b\u200b\uff0c\u200b\u5373\u53ef\u200b\u4f7f\u7528\u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\u81ea\u200b\u5e95\u5411\u4e0a\u200b\u6c42\u89e3\u200b\u3002\u200b\u6c42\u89e3\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u53ea\u200b\u9700\u8981\u200b\u5b58\u50a8\u200b\u4e00\u884c\u200b\u7684\u200b\u6700\u200b\u77ed\u200b\u8def\u5f84\u200b\u548c\u200b\u5373\u53ef\u200b\u3002</p> <pre><code>#define MIN(x, y) ((x) &lt; (y) ? x : y)\nint minPathSum(int** grid, int gridSize, int* gridColSize){\n    if (gridSize == 0 || *gridColSize == 0)\n        return 0;\n    int *dp = (int *)memset(malloc(sizeof(int) * *gridColSize), 0, sizeof(int) * *gridColSize),\n        i = 0, j = 0;\n    for (i = 0; i &lt; *gridColSize; i++)\n        dp[i] = INT_MAX;\n    dp[0] = 0;\n    for (i = 0; i &lt; gridSize; i++)\n    {\n        dp[0] += grid[i][0];\n        for (j = 1; j &lt; *gridColSize; j++)\n            dp[j] = MIN(dp[j - 1], dp[j]) + grid[i][j];\n    }\n    return dp[*gridColSize - 1];\n}\n</code></pre>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/66/","title":"66. \u200b\u52a0\u4e00","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/66/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u7531\u200b\u6574\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u975e\u7a7a\u200b\u6570\u7ec4\u200b\u6240\u200b\u8868\u793a\u200b\u7684\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\uff0c\u200b\u5728\u200b\u8be5\u6570\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u52a0\u200b\u4e00\u200b\u3002</p> <p>\u200b\u6700\u9ad8\u200b\u4f4d\u200b\u6570\u5b57\u200b\u5b58\u653e\u200b\u5728\u200b\u6570\u7ec4\u200b\u7684\u200b\u9996\u4f4d\u200b\uff0c \u200b\u6570\u7ec4\u200b\u4e2d\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u53ea\u200b\u5b58\u50a8\u200b\u5355\u4e2a\u200b\u6570\u5b57\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u9664\u4e86\u200b\u6574\u6570\u200b 0 \u200b\u4e4b\u5916\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u6574\u6570\u200b\u4e0d\u4f1a\u200b\u4ee5\u200b\u96f6\u200b\u5f00\u5934\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,2,3]\n\u200b\u8f93\u51fa\u200b: [1,2,4]\n\u200b\u89e3\u91ca\u200b: \u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u8868\u793a\u200b\u6570\u5b57\u200b 123\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [4,3,2,1]\n\u200b\u8f93\u51fa\u200b: [4,3,2,2]\n\u200b\u89e3\u91ca\u200b: \u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u8868\u793a\u200b\u6570\u5b57\u200b 4321\u3002\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/66/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4ece\u200b\u6700\u540e\u200b\u4e00\u4f4d\u200b\u5f00\u59cb\u200b\u52a0\u200b\uff0c\u200b\u5411\u524d\u200b\u5904\u7406\u200b\u8fdb\u4f4d\u200b\u3002\u200b\u6ce8\u610f\u200b9999+1\u200b\u7684\u200b\u7279\u6b8a\u200b\u60c5\u51b5\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\nint* plusOne(int* digits, int digitsSize, int* returnSize){\n    int *ret = (int *)malloc(sizeof(int) * (digitsSize + 1)), i = digitsSize, add = 1;\n    *ret = 0;\n    memcpy(ret + 1, digits, sizeof(int) * digitsSize);\n    for (i = digitsSize; i &gt;= 0 &amp;&amp; add; i--)\n    {\n        if (add = ret[i] == 9)\n            ret[i] = 0;\n        else\n            ret[i]++;\n    }\n    *returnSize = (i + 1) ? digitsSize : digitsSize + 1;\n    return ret + ((i + 1) ? 1 : 0);\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/69/","title":"69. x \u200b\u7684\u200b\u5e73\u65b9\u6839","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u5b66","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/69/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5b9e\u73b0\u200b\u00a0<code>int sqrt(int x)</code>\u00a0\u200b\u51fd\u6570\u200b\u3002</p> <p>\u200b\u8ba1\u7b97\u200b\u5e76\u200b\u8fd4\u56de\u200b\u00a0<code>x</code>\u00a0\u200b\u7684\u200b\u5e73\u65b9\u6839\u200b\uff0c\u200b\u5176\u4e2d\u200b\u00a0<code>x</code> \u200b\u662f\u975e\u200b\u8d1f\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u7531\u4e8e\u200b\u8fd4\u56de\u200b\u7c7b\u578b\u200b\u662f\u200b\u6574\u6570\u200b\uff0c\u200b\u7ed3\u679c\u200b\u53ea\u200b\u4fdd\u7559\u200b\u6574\u6570\u200b\u7684\u200b\u90e8\u5206\u200b\uff0c\u200b\u5c0f\u6570\u200b\u90e8\u5206\u200b\u5c06\u200b\u88ab\u200b\u820d\u53bb\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1:</p> <pre><code>\u200b\u8f93\u5165\u200b: 4\n\u200b\u8f93\u51fa\u200b: 2\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2:</p> <pre><code>\u200b\u8f93\u5165\u200b: 8\n\u200b\u8f93\u51fa\u200b: 2\n\u200b\u8bf4\u660e\u200b: 8 \u200b\u7684\u200b\u5e73\u65b9\u6839\u200b\u662f\u200b 2.82842..., \n\u00a0    \u200b\u7531\u4e8e\u200b\u8fd4\u56de\u200b\u7c7b\u578b\u200b\u662f\u200b\u6574\u6570\u200b\uff0c\u200b\u5c0f\u6570\u200b\u90e8\u5206\u200b\u5c06\u200b\u88ab\u200b\u820d\u53bb\u200b\u3002\n</code></pre> <p>Reference</p>","tags":["\u6570\u5b66","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/69/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u5373\u53ef\u200b\uff0c\u200b\u5173\u4e8e\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7684\u200b\u8ba8\u8bba\u200b\uff0c\u200b\u8bf7\u200b\u53c2\u89c1\u200b35. \u200b\u641c\u7d22\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b</p> <pre><code>unsigned int binsearch(unsigned int target, unsigned int lo, unsigned int hi)\n{\n    int mid;\n    while (lo &lt; hi)\n    {\n        mid = (lo + hi) &gt;&gt; 1;\n        if (mid * mid &gt; target)\n            hi = mid;\n        else\n            lo = mid + 1;\n    }\n    return lo - 1;\n}\nint mySqrt(int x){\n    return binsearch(x, 0, 46341);\n}\n</code></pre>","tags":["\u6570\u5b66","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/7/","title":"7. \u200b\u6574\u6570\u200b\u53cd\u8f6c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/7/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u51fa\u200b\u4e00\u4e2a\u200b 32 \u200b\u4f4d\u200b\u7684\u200b\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\uff0c\u200b\u4f60\u200b\u9700\u8981\u200b\u5c06\u200b\u8fd9\u4e2a\u200b\u6574\u6570\u200b\u4e2d\u200b\u6bcf\u4f4d\u200b\u4e0a\u200b\u7684\u200b\u6570\u5b57\u200b\u8fdb\u884c\u200b\u53cd\u8f6c\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 123\n\u200b\u8f93\u51fa\u200b: 321\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: -123\n\u200b\u8f93\u51fa\u200b: -321\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 120\n\u200b\u8f93\u51fa\u200b: 21\n</code></pre> <p>\u200b\u6ce8\u610f\u200b\uff1a</p> <p>\u200b\u5047\u8bbe\u200b\u6211\u4eec\u200b\u7684\u200b\u73af\u5883\u200b\u53ea\u80fd\u200b\u5b58\u50a8\u200b\u5f97\u200b\u4e0b\u200b 32 \u200b\u4f4d\u200b\u7684\u200b\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\uff0c\u200b\u5219\u200b\u5176\u200b\u6570\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b \\([\u22122^{31},\u00a0 2^{31}\u00a0\u2212 1]\\) \u3002\u200b\u8bf7\u200b\u6839\u636e\u200b\u8fd9\u4e2a\u200b\u5047\u8bbe\u200b\uff0c\u200b\u5982\u679c\u200b\u53cd\u8f6c\u200b\u540e\u200b\u6574\u6570\u200b\u6ea2\u51fa\u200b\u90a3\u4e48\u200b\u5c31\u200b\u8fd4\u56de\u200b 0\u3002</p> <p>Reference</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/7/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6bcf\u200b\u4e00\u6b21\u200b\u4ece\u200b\u53f3\u4fa7\u200b\u63d0\u53d6\u200b\u51fa\u200b\u6700\u540e\u200b\u4e00\u4f4d\u200b\u52a0\u200b\u5230\u200b\u7ed3\u679c\u200b\u7684\u200b\u53f3\u4fa7\u200b\u3002\u200b\u6ce8\u610f\u200b\u8d1f\u6570\u200b<code>-2147483648</code>\u200b\u5728\u200b 32 \u200b\u4f4d\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\u73af\u5883\u200b\u4e0b\u200b\u4e0d\u80fd\u200b\u53d6\u8d1f\u200b\uff0c\u200b\u5c06\u200b\u4e2d\u95f4\u200b\u53d8\u91cf\u200b\u58f0\u660e\u200b\u4e3a\u200b<code>long</code>\u200b\u5373\u53ef\u200b\u89e3\u51b3\u200b\u3002</p> <pre><code>int reverse(int x){\n    long ret = 0, testcase = x;\n    testcase = testcase &gt; 0 ? testcase : -testcase;\n    while(testcase)\n    {\n        ret *= 10;\n        ret += testcase % 10;\n        testcase /= 10;\n    }\n    return ((ret &gt;&gt; 31) ? 0 : ret) * (x &gt; 0 ? 1 : -1);\n}\n</code></pre>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/70/","title":"70. \u200b\u722c\u697c\u68af","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/70/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5047\u8bbe\u200b\u4f60\u200b\u6b63\u5728\u200b\u722c\u697c\u68af\u200b\u3002\u200b\u9700\u8981\u200b n\u00a0\u200b\u9636\u200b\u4f60\u200b\u624d\u80fd\u200b\u5230\u8fbe\u200b\u697c\u9876\u200b\u3002</p> <p>\u200b\u6bcf\u6b21\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u722c\u200b 1 \u200b\u6216\u200b 2 \u200b\u4e2a\u200b\u53f0\u9636\u200b\u3002\u200b\u4f60\u200b\u6709\u200b\u591a\u5c11\u200b\u79cd\u200b\u4e0d\u540c\u200b\u7684\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u722c\u200b\u5230\u200b\u697c\u9876\u200b\u5462\u200b\uff1f</p> <p>\u200b\u6ce8\u610f\u200b\uff1a\u200b\u7ed9\u5b9a\u200b n \u200b\u662f\u200b\u4e00\u4e2a\u200b\u6b63\u6574\u6570\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a 2\n\u200b\u8f93\u51fa\u200b\uff1a 2\n\u200b\u89e3\u91ca\u200b\uff1a \u200b\u6709\u200b\u4e24\u79cd\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u722c\u200b\u5230\u200b\u697c\u9876\u200b\u3002\n1.  1 \u200b\u9636\u200b + 1 \u200b\u9636\u200b\n2.  2 \u200b\u9636\u200b\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a 3\n\u200b\u8f93\u51fa\u200b\uff1a 3\n\u200b\u89e3\u91ca\u200b\uff1a \u200b\u6709\u200b\u4e09\u79cd\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u722c\u200b\u5230\u200b\u697c\u9876\u200b\u3002\n1.  1 \u200b\u9636\u200b + 1 \u200b\u9636\u200b + 1 \u200b\u9636\u200b\n2.  1 \u200b\u9636\u200b + 2 \u200b\u9636\u200b\n3.  2 \u200b\u9636\u200b + 1 \u200b\u9636\u200b\n</code></pre> <p>Reference</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/70/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4f7f\u7528\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\u5b58\u50a8\u200b\u4e34\u65f6\u200b\u7ed3\u679c\u200b\uff0c\u200b\u9012\u5f52\u8ba1\u7b97\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int climbStairs_cache(int n, int **cache)\n{\n    if (n &lt;= 2)\n        return n;\n    if (*cache == NULL)\n        *cache = memset(malloc(sizeof(int) * n), 0, sizeof(int) * n);\n    if (!(*cache)[n - 1])\n        (*cache)[n - 1] = climbStairs_cache(n - 1, cache) + climbStairs_cache(n - 2, cache);\n    return (*cache)[n - 1];\n}\n\nint climbStairs(int n){\n    int *cache = NULL;\n    return climbStairs_cache(n, &amp;cache);\n}\n</code></pre>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/73/","title":"73. \u200b\u77e9\u9635\u200b\u7f6e\u200b\u96f6","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/73/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u00a0<code>m x n</code> \u200b\u7684\u200b\u77e9\u9635\u200b\uff0c\u200b\u5982\u679c\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u4e3a\u200b <code>0</code> \uff0c\u200b\u5219\u200b\u5c06\u200b\u5176\u200b\u6240\u5728\u200b\u884c\u548c\u5217\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u90fd\u200b\u8bbe\u4e3a\u200b <code>0</code> \u3002\u200b\u8bf7\u200b\u4f7f\u7528\u200b \u200b\u539f\u5730\u200b \u200b\u7b97\u6cd5\u200b\u3002</p> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <ul> <li>\u200b\u4e00\u4e2a\u200b\u76f4\u89c2\u200b\u7684\u200b\u89e3\u51b3\u65b9\u6848\u200b\u662f\u200b\u4f7f\u7528\u200b \u00a0\\(O(mn)\\)\u00a0\u200b\u7684\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u4f46\u200b\u8fd9\u200b\u5e76\u200b\u4e0d\u662f\u200b\u4e00\u4e2a\u200b\u597d\u200b\u7684\u200b\u89e3\u51b3\u65b9\u6848\u200b\u3002</li> <li>\u200b\u4e00\u4e2a\u200b\u7b80\u5355\u200b\u7684\u200b\u6539\u8fdb\u200b\u65b9\u6848\u200b\u662f\u200b\u4f7f\u7528\u200b \\(O(m\u00a0+\u00a0n)\\) \u200b\u7684\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u4f46\u200b\u8fd9\u200b\u4ecd\u7136\u200b\u4e0d\u662f\u200b\u6700\u597d\u200b\u7684\u200b\u89e3\u51b3\u65b9\u6848\u200b\u3002</li> <li>\u200b\u4f60\u200b\u80fd\u200b\u60f3\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u4ec5\u200b\u4f7f\u7528\u200b\u5e38\u91cf\u200b\u7a7a\u95f4\u200b\u7684\u200b\u89e3\u51b3\u65b9\u6848\u200b\u5417\u200b\uff1f</li> </ul> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1amatrix = [[1,1,1],[1,0,1],[1,1,1]]\n\u200b\u8f93\u51fa\u200b\uff1a[[1,0,1],[0,0,0],[1,0,1]]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1amatrix = [[0,1,2,0],[3,4,5,2],[1,3,1,5]]\n\u200b\u8f93\u51fa\u200b\uff1a[[0,0,0,0],[0,4,5,0],[0,3,1,0]]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>m == matrix.length</code></li> <li><code>n == matrix[0].length</code></li> <li><code>1 &lt;= m, n &lt;= 200</code></li> <li><code>-2^31 &lt;= matrix[i][j] &lt;= 2^31 - 1</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/73/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\\(m\\)\u200b\u4e2a\u200b\u6807\u8bb0\u53d8\u91cf\u200b\u6807\u8bb0\u200b\u884c\u4e2d\u200b\u662f\u5426\u200b\u51fa\u73b0\u200b0\uff0c\u200b\u7528\u200b\\(n\\)\u200b\u4e2a\u200b\u6807\u8bb0\u53d8\u91cf\u200b\u6807\u8bb0\u200b\u5217\u4e2d\u200b\u662f\u5426\u200b\u51fa\u73b0\u200b0\u3002</p> <pre><code>void setZeroes(int** matrix, int matrixSize, int* matrixColSize){\n    int *rows = memset(malloc(sizeof(int) * matrixSize), 0, sizeof(int) * matrixSize),\n        *cols = memset(malloc(sizeof(int) * *matrixColSize), 0, sizeof(int) * *matrixColSize),\n        i = 0, j = 0;\n    for (i = 0; i &lt; matrixSize; i++)\n    {\n        for (j = 0; j &lt; matrixColSize[i]; j++)\n        {\n            if (matrix[i][j] == 0)\n            {\n                rows[i] = 1;\n                cols[j] = 1;\n            }\n        }\n    }\n    for (i = 0; i &lt; matrixSize; i++)\n        for (j = 0; j &lt; matrixColSize[i]; j++)\n            if (rows[i] || cols[j])\n                matrix[i][j] = 0;\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/74/","title":"74. \u200b\u641c\u7d22\u200b\u4e8c\u7ef4\u200b\u77e9\u9635","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/74/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u9ad8\u6548\u200b\u7684\u200b\u7b97\u6cd5\u200b\u6765\u200b\u5224\u65ad\u200b\u00a0<code>m x n</code>\u00a0\u200b\u77e9\u9635\u200b\u4e2d\u200b\uff0c\u200b\u662f\u5426\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u503c\u200b\u3002\u200b\u8be5\u200b\u77e9\u9635\u200b\u5177\u6709\u200b\u5982\u4e0b\u200b\u7279\u6027\u200b\uff1a</p> <ul> <li>\u200b\u6bcf\u884c\u200b\u4e2d\u200b\u7684\u200b\u6574\u6570\u200b\u4ece\u5de6\u5230\u53f3\u200b\u6309\u200b\u5347\u5e8f\u200b\u6392\u5217\u200b\u3002</li> <li>\u200b\u6bcf\u884c\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u6574\u6570\u200b\u5927\u4e8e\u200b\u524d\u200b\u4e00\u884c\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1amatrix = [[1,3,5,7],[10,11,16,20],[23,30,34,60]], target = 3\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1amatrix = [[1,3,5,7],[10,11,16,20],[23,30,34,60]], target = 13\n\u200b\u8f93\u51fa\u200b\uff1afalse\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>m == matrix.length</code></li> <li><code>n == matrix[i].length</code></li> <li><code>1 &lt;= m, n &lt;= 100</code></li> <li><code>-10^4 &lt;= matrix[i][j], target &lt;= 10^4</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/74/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u6839\u636e\u200b\u77e9\u9635\u200b\u7684\u200b\u7279\u6027\u200b\u53ef\u77e5\u200b\uff0c\u200b\u5047\u8bbe\u200b\u77e9\u9635\u200b\u4e3a\u200b\\(\\{a_{ij}\\}_{m\\times n}\\)\uff0c\u200b\u76ee\u6807\u503c\u200b\\(t\\)\u200b\u4f4d\u4e8e\u200b\u5750\u6807\u200b\\((i^*, j^*)\\)\u200b\u5904\u200b\uff0c\u200b\u6709\u200b\uff1a</p> <ul> <li>\u200b\u5bf9\u4e8e\u200b\u4efb\u4f55\u200b\\(i, j\\)\uff0c\u200b\u82e5\u200b\\(i &lt; i^*\\)\uff0c\u200b\u5219\u200b\\(a_{ij} &lt; t\\)\uff0c\u200b\u5f53\u200b\\(i = i^*\\)\u200b\u65f6\u200b\uff0c\u200b\u82e5\u200b\\(j &lt; j^*\\)\uff0c\u200b\u5219\u200b\\(a_{ij} &lt; t\\)</li> <li>\u200b\u5bf9\u4e8e\u200b\u4efb\u4f55\u200b\\(i, j\\)\uff0c\u200b\u82e5\u200b\\(i &gt; i^*\\)\uff0c\u200b\u5219\u200b\\(a_{ij} &gt; t\\)\uff0c\u200b\u5f53\u200b\\(i = i^*\\)\u200b\u65f6\u200b\uff0c\u200b\u82e5\u200b\\(j &gt; j^*\\)\uff0c\u200b\u5219\u200b\\(a_{ij} &lt; t\\)</li> </ul> <p>\u200b\u5373\u200b\u76ee\u6807\u503c\u200b\u5c06\u200b\u77e9\u9635\u200b\u5206\u4e3a\u200b\u4e24\u200b\u90e8\u5206\u200b\uff1a\u200b\u53f3\u4fa7\u200b\u548c\u200b\u4e0b\u4fa7\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u5927\u4e8e\u200b\u7b49\u4e8e\u200b\u76ee\u6807\u503c\u200b\uff0c\u200b\u5de6\u4fa7\u200b\u548c\u200b\u4e0a\u4fa7\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b\u76ee\u6807\u503c\u200b\u3002</p> <p>\u200b\u4ece\u200b\u77e9\u5f62\u200b\u7684\u200b\u5de6\u4e0b\u89d2\u200b\u5f00\u59cb\u200b\uff1a</p> <ul> <li>\u200b\u82e5\u200b\u5f53\u524d\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u6570\u503c\u200b\u5c0f\u4e8e\u200b\u76ee\u6807\u503c\u200b\uff0c\u200b\u8bf4\u660e\u200b\u76ee\u6807\u503c\u200b\u4f4d\u4e8e\u200b\u5f53\u524d\u200b\u884c\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u5373\u53ef\u200b\u627e\u5230\u200b\u76ee\u6807\u503c\u200b</li> <li>\u200b\u82e5\u200b\u5f53\u524d\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u6570\u503c\u200b\u5927\u4e8e\u200b\u76ee\u6807\u503c\u200b\uff0c\u200b\u8bf4\u660e\u200b\u76ee\u6807\u503c\u200b\u4e0d\u200b\u5728\u200b\u5f53\u524d\u200b\u884c\u200b\uff0c\u200b\u9700\u8981\u200b\u5411\u4e0a\u200b\u79fb\u52a8\u200b</li> </ul> <p>\u200b\u91cd\u590d\u200b\u5982\u200b\u4e0a\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u76f4\u81f3\u200b\u627e\u5230\u200b\u76ee\u6807\u200b\u5143\u7d20\u200b\u3002</p> <pre><code>bool searchMatrix(int** matrix, int matrixSize, int* matrixColSize, int target){\n    int x = matrixSize - 1, y = 0;\n    while (x &gt;= 0 &amp;&amp; y &lt; *matrixColSize)\n    {\n        if (matrix[x][y] &lt; target)\n            y++;\n        else if (matrix[x][y] &gt; target)\n            x--;\n        else\n            return true;\n    }\n    return false;\n}\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u7b97\u6cd5\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u8fdb\u884c\u200b\u4f18\u5316\u200b\u3002</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/75/","title":"75. \u200b\u989c\u8272\u200b\u5206\u7c7b","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6392\u5e8f","\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/75/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5305\u542b\u200b\u7ea2\u8272\u200b\u3001\u200b\u767d\u8272\u200b\u548c\u200b\u84dd\u8272\u200b\uff0c\u200b\u4e00\u5171\u200b\u00a0<code>n</code> \u200b\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u6570\u7ec4\u200b\uff0c\u200b\u539f\u5730\u200b\u5bf9\u200b\u5b83\u4eec\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u76f8\u540c\u200b\u989c\u8272\u200b\u7684\u200b\u5143\u7d20\u200b\u76f8\u90bb\u200b\uff0c\u200b\u5e76\u200b\u6309\u7167\u200b\u7ea2\u8272\u200b\u3001\u200b\u767d\u8272\u200b\u3001\u200b\u84dd\u8272\u200b\u987a\u5e8f\u6392\u5217\u200b\u3002</p> <p>\u200b\u6b64\u9898\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u4f7f\u7528\u200b\u6574\u6570\u200b <code>0</code>\u3001\u00a0<code>1</code> \u200b\u548c\u200b <code>2</code> \u200b\u5206\u522b\u200b\u8868\u793a\u200b\u7ea2\u8272\u200b\u3001\u200b\u767d\u8272\u200b\u548c\u200b\u84dd\u8272\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [2,0,2,1,1,0]\n\u200b\u8f93\u51fa\u200b\uff1a[0,0,1,1,2,2]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [2,0,1]\n\u200b\u8f93\u51fa\u200b\uff1a[0,1,2]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [0]\n\u200b\u8f93\u51fa\u200b\uff1a[0]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1]\n\u200b\u8f93\u51fa\u200b\uff1a[1]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>n == nums.length</code></li> <li><code>1 &lt;= n &lt;= 300</code></li> <li><code>nums[i]</code> \u200b\u4e3a\u200b <code>0</code>\u3001<code>1</code> \u200b\u6216\u200b <code>2</code></li> </ul> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <ul> <li>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u4f7f\u7528\u200b\u4ee3\u7801\u200b\u5e93\u4e2d\u200b\u7684\u200b\u6392\u5e8f\u200b\u51fd\u6570\u200b\u6765\u200b\u89e3\u51b3\u200b\u8fd9\u9053\u9898\u200b\u5417\u200b\uff1f</li> <li>\u200b\u4f60\u200b\u80fd\u200b\u60f3\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u4ec5\u200b\u4f7f\u7528\u200b\u5e38\u6570\u200b\u7a7a\u95f4\u200b\u7684\u200b\u4e00\u8d9f\u200b\u626b\u63cf\u200b\u7b97\u6cd5\u200b\u5417\u200b\uff1f</li> </ul> <p>Reference</p>","tags":["\u6392\u5e8f","\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/75/#_2","title":"\u9898\u89e3","text":"<ol> <li>\u200b\u7edf\u8ba1\u200b\u6570\u7ec4\u200b\u4e2d\u200b<code>0</code>\u3001<code>1</code>\u200b\u548c\u200b<code>2</code>\u200b\u7684\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\uff0c\u200b\u6309\u7167\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u91cd\u65b0\u200b\u751f\u6210\u200b\u6570\u7ec4\u200b\u3002</li> <li>\u200b\u4f7f\u7528\u200b\u5feb\u901f\u200b\u6392\u5e8f\u200b\u7684\u200b\u601d\u8def\u200b\uff0c\u200b\u82e5\u200b\u51fa\u73b0\u200b<code>0</code>\u200b\u5219\u200b\u79fb\u52a8\u200b\u5230\u200b\u6570\u7ec4\u200b\u5f00\u5934\u200b\uff0c\u200b\u82e5\u200b\u51fa\u73b0\u200b<code>2</code>\u200b\u5219\u200b\u79fb\u52a8\u200b\u5230\u200b\u6570\u7ec4\u200b\u7ed3\u5c3e\u200b\u3002</li> </ol> <pre><code>void swap(int *a, int *b)\n{\n    int temp = *a;\n    *a = *b;\n    *b = temp;\n}\nvoid sortColors(int* nums, int numsSize){\n    int head = 0, tail = numsSize - 1, i = 0;\n    for (i = 0; i &lt;= tail &amp;&amp; tail &gt; head; i++)\n    {\n        if (nums[i] == 0)\n            swap(nums + head++, nums + i);\n        if (nums[i] == 2)\n            swap(nums + tail--, nums + i--);\n    }\n}\n</code></pre>","tags":["\u6392\u5e8f","\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/779/","title":"779. \u200b\u7b2c\u200bK\u200b\u4e2a\u200b\u8bed\u6cd5\u200b\u7b26\u53f7","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u9012\u5f52"]},{"location":"coding/leetcode/779/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5728\u200b\u7b2c\u4e00\u884c\u200b\u6211\u4eec\u200b\u5199\u200b\u4e0a\u200b\u4e00\u4e2a\u200b<code>0</code>\u3002\u200b\u63a5\u4e0b\u6765\u200b\u7684\u200b\u6bcf\u200b\u4e00\u884c\u200b\uff0c\u200b\u5c06\u200b\u524d\u200b\u4e00\u884c\u200b\u4e2d\u200b\u7684\u200b<code>0</code>\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>01</code>\uff0c<code>1</code>\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>10</code>\u3002</p> <p>\u200b\u7ed9\u5b9a\u200b\u884c\u200b\u6570\u200b<code>N</code>\u200b\u548c\u200b\u5e8f\u6570\u200b<code>K</code>\uff0c\u200b\u8fd4\u56de\u200b\u7b2c\u200b<code>N</code>\u200b\u884c\u4e2d\u200b\u7b2c\u200b<code>K</code>\u200b\u4e2a\u5b57\u7b26\u200b\u3002\uff08<code>K</code>\u200b\u4ece\u200b 1 \u200b\u5f00\u59cb\u200b\uff09</p> <p>\u200b\u4f8b\u5b50\u200b: <pre><code>\u200b\u8f93\u5165\u200b: N = 1, K = 1\n\u200b\u8f93\u51fa\u200b: 0\n\n\u200b\u8f93\u5165\u200b: N = 2, K = 1\n\u200b\u8f93\u51fa\u200b: 0\n\n\u200b\u8f93\u5165\u200b: N = 2, K = 2\n\u200b\u8f93\u51fa\u200b: 1\n\n\u200b\u8f93\u5165\u200b: N = 4, K = 5\n\u200b\u8f93\u51fa\u200b: 1\n\n\u200b\u89e3\u91ca\u200b:\n\u200b\u7b2c\u4e00\u884c\u200b: 0\n\u200b\u7b2c\u4e8c\u884c\u200b: 01\n\u200b\u7b2c\u4e09\u884c\u200b: 0110\n\u200b\u7b2c\u56db\u884c\u200b: 01101001\n</code></pre> \u200b\u6ce8\u610f\u200b\uff1a</p> <ul> <li><code>N</code>\u200b\u7684\u200b\u8303\u56f4\u200b<code>[1, 30]</code>.</li> <li><code>K</code>\u200b\u7684\u200b\u8303\u56f4\u200b<code>[1, 2^(N-1)]</code>.</li> </ul> <p>Reference</p>","tags":["\u9012\u5f52"]},{"location":"coding/leetcode/779/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6ce8\u610f\u200b\u5230\u200b\u6bcf\u4e2a\u200b\u7ed3\u679c\u200b\u5b57\u7b26\u4e32\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b4\u200b\u4e2a\u200b\u7b49\u200b\u957f\u200b\u7684\u200b\u90e8\u5206\u200b\uff0c\u200b\u5176\u4e2d\u200b\u7b2c\u200b1\u30014\u200b\u90e8\u5206\u200b\u76f8\u540c\u200b\uff0c2\u30013\u200b\u90e8\u5206\u200b\u76f8\u540c\u200b\uff0c\u200b\u4e24\u7ec4\u200b\u7684\u200b\u4e8c\u8fdb\u5236\u200b\u4e92\u8865\u200b\u3002\u200b\u540c\u65f6\u200b\uff0c\u200b\u4efb\u610f\u200b\u4e00\u884c\u200b\u4ee5\u4e2d\u70b9\u200b\u4e3a\u754c\u200b\uff0c\u200b\u5de6\u53f3\u200b\u4e24\u4fa7\u200b\u4e92\u8865\u200b\u3002\u200b\u56e0\u6b64\u200b\u53ea\u200b\u9700\u8981\u200b\u5728\u200b\u5de6\u4fa7\u200b\u8fdb\u884c\u200b\u67e5\u627e\u200b\uff0c\u200b\u53f3\u4fa7\u200b\u90e8\u5206\u200b\u53d6\u53cd\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int kthGrammar(int N, int K)\n{\n    if (N == 1)\n        return 0;\n    int unit = 1 &lt;&lt; (N - 2);\n    return !(K &lt;= unit) ^ kthGrammar(N - 1, ((K - 1) &amp; (unit - 1)) + 1);\n}\n</code></pre>","tags":["\u9012\u5f52"]},{"location":"coding/leetcode/78/","title":"78. \u200b\u5b50\u96c6","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u4f4d\u8fd0\u7b97","\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/78/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\u00a0<code>nums</code> \uff0c\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b \u200b\u4e92\u4e0d\u200b\u76f8\u540c\u200b \u3002\u200b\u8fd4\u56de\u200b\u8be5\u200b\u6570\u7ec4\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u7684\u200b\u5b50\u96c6\u200b\uff08\u200b\u5e42\u96c6\u200b\uff09\u3002</p> <p>\u200b\u89e3\u96c6\u200b \u200b\u4e0d\u80fd\u200b \u200b\u5305\u542b\u200b\u91cd\u590d\u200b\u7684\u200b\u5b50\u96c6\u200b\u3002\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u6309\u200b \u200b\u4efb\u610f\u200b\u987a\u5e8f\u200b \u200b\u8fd4\u56de\u200b\u89e3\u96c6\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1,2,3]\n\u200b\u8f93\u51fa\u200b\uff1a[[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3]]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [0]\n\u200b\u8f93\u51fa\u200b\uff1a[[],[0]]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 10</code></li> <li><code>-10 &lt;= nums[i] &lt;= 10</code></li> <li><code>nums</code> \u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b \u200b\u4e92\u4e0d\u200b\u76f8\u540c\u200b</li> </ul> <p>Reference</p>","tags":["\u4f4d\u8fd0\u7b97","\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/78/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5df2\u77e5\u200b\u96c6\u5408\u200b\u4e2d\u200b\u4e0d\u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6574\u6570\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u200b\u4e00\u4f4d\u200b\u8868\u793a\u200b\u96c6\u5408\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u9009\u53d6\u200b\u72b6\u6001\u200b\u3002</p> <pre><code>/**\n * Return an array of arrays of size *returnSize.\n * The sizes of the arrays are returned as *returnColumnSizes array.\n * Note: Both returned array and *columnSizes array must be malloced, assume caller calls free().\n */\nint count1(int x)\n{\n    int ret = 0;\n    while (x)\n    {\n        ret += x &amp; 1;\n        x &gt;&gt;= 1;\n    }\n    return ret;\n}\nint** subsets(int* nums, int numsSize, int* returnSize, int** returnColumnSizes){\n    *returnSize = 1 &lt;&lt; numsSize;\n    *returnColumnSizes = (int *)memset(malloc(sizeof(int) * *returnSize), 0, sizeof(int) * *returnSize);\n    int i = 0, j = 0, **ret = (int **)malloc(sizeof(int *) * *returnSize), pos;\n    for (i = 0; i &lt; *returnSize; i++)\n    {\n        (*returnColumnSizes)[i] = count1(i);\n        ret[i] = (int *)malloc(sizeof(int) * (*returnColumnSizes)[i]);\n        pos = 0;\n        for (j = 0; j &lt; numsSize; j++)\n            if (i &amp; (1 &lt;&lt; j))\n                ret[i][pos++] = nums[j];\n    }\n    return ret;\n}\n</code></pre>","tags":["\u4f4d\u8fd0\u7b97","\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/781/","title":"781. \u200b\u68ee\u6797\u200b\u4e2d\u200b\u7684\u200b\u5154\u5b50","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u54c8\u5e0c\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/781/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u68ee\u6797\u200b\u4e2d\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5154\u5b50\u200b\u90fd\u200b\u6709\u200b\u989c\u8272\u200b\u3002\u200b\u5176\u4e2d\u200b\u4e00\u4e9b\u200b\u5154\u5b50\u200b\uff08\u200b\u53ef\u80fd\u200b\u662f\u200b\u5168\u90e8\u200b\uff09\u200b\u544a\u8bc9\u200b\u4f60\u200b\u8fd8\u6709\u200b\u591a\u5c11\u200b\u5176\u4ed6\u200b\u7684\u200b\u5154\u5b50\u200b\u548c\u200b\u81ea\u5df1\u200b\u6709\u200b\u76f8\u540c\u200b\u7684\u200b\u989c\u8272\u200b\u3002\u200b\u6211\u4eec\u200b\u5c06\u200b\u8fd9\u4e9b\u200b\u56de\u7b54\u200b\u653e\u5728\u200b\u00a0<code>answers</code>\u00a0\u200b\u6570\u7ec4\u200b\u91cc\u200b\u3002</p> <p>\u200b\u8fd4\u56de\u200b\u68ee\u6797\u200b\u4e2d\u200b\u5154\u5b50\u200b\u7684\u200b\u6700\u5c11\u200b\u6570\u91cf\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b:</p> <pre><code>\u200b\u8f93\u5165\u200b: answers = [1, 1, 2]\n\u200b\u8f93\u51fa\u200b: 5\n\u200b\u89e3\u91ca\u200b:\n\u200b\u4e24\u53ea\u200b\u56de\u7b54\u200b\u4e86\u200b \"1\" \u200b\u7684\u200b\u5154\u5b50\u200b\u53ef\u80fd\u200b\u6709\u200b\u76f8\u540c\u200b\u7684\u200b\u989c\u8272\u200b\uff0c\u200b\u8bbe\u200b\u4e3a\u200b\u7ea2\u8272\u200b\u3002\n\u200b\u4e4b\u540e\u200b\u56de\u7b54\u200b\u4e86\u200b \"2\" \u200b\u7684\u200b\u5154\u5b50\u200b\u4e0d\u4f1a\u200b\u662f\u200b\u7ea2\u8272\u200b\uff0c\u200b\u5426\u5219\u200b\u4ed6\u4eec\u200b\u7684\u200b\u56de\u7b54\u200b\u4f1a\u200b\u76f8\u4e92\u200b\u77db\u76fe\u200b\u3002\n\u200b\u8bbe\u200b\u56de\u7b54\u200b\u4e86\u200b \"2\" \u200b\u7684\u200b\u5154\u5b50\u200b\u4e3a\u200b\u84dd\u8272\u200b\u3002\n\u200b\u6b64\u5916\u200b\uff0c\u200b\u68ee\u6797\u200b\u4e2d\u200b\u8fd8\u200b\u5e94\u6709\u200b\u53e6\u5916\u200b 2 \u200b\u53ea\u200b\u84dd\u8272\u200b\u5154\u5b50\u200b\u7684\u200b\u56de\u7b54\u200b\u6ca1\u6709\u200b\u5305\u542b\u200b\u5728\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u3002\n\u200b\u56e0\u6b64\u200b\u68ee\u6797\u200b\u4e2d\u200b\u5154\u5b50\u200b\u7684\u200b\u6700\u5c11\u200b\u6570\u91cf\u200b\u662f\u200b 5: 3 \u200b\u53ea\u200b\u56de\u7b54\u200b\u7684\u200b\u548c\u200b 2 \u200b\u53ea\u200b\u6ca1\u6709\u200b\u56de\u7b54\u200b\u7684\u200b\u3002\n\n\u200b\u8f93\u5165\u200b: answers = [10, 10, 10]\n\u200b\u8f93\u51fa\u200b: 11\n\n\u200b\u8f93\u5165\u200b: answers = []\n\u200b\u8f93\u51fa\u200b: 0\n</code></pre> <p>\u200b\u8bf4\u660e\u200b:</p> <ul> <li><code>answers</code>\u00a0\u200b\u7684\u200b\u957f\u5ea6\u200b\u6700\u5927\u200b\u4e3a\u200b<code>1000</code>\u3002</li> <li><code>answers[i]</code>\u00a0\u200b\u662f\u200b\u5728\u200b\u00a0<code>[0, 999]</code>\u00a0\u200b\u8303\u56f4\u200b\u5185\u200b\u7684\u200b\u6574\u6570\u200b\u3002</li> </ul> <p>Reference</p>","tags":["\u54c8\u5e0c\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/781/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5154\u5b50\u200b\u7684\u200b\u5206\u7c7b\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u60c5\u51b5\u200b\uff0c\u200b\u8bbe\u200b\u7b2c\u200b\\(i\\)\u200b\u53ea\u200b\u5154\u5b50\u200b\u7684\u200b\u62a5\u6570\u200b\u4e3a\u200b\\(a_i\\)\uff0c\u200b\u5219\u200b\uff1a</p> <ul> <li>\u200b\u82e5\u200b\\(a_i \\not = a_j\\)\uff0c\u200b\u5219\u200b\u7b2c\u200b\\(i\\)\u200b\u53ea\u200b\u5154\u5b50\u200b\u4e0e\u200b\u7b2c\u200b\\(j\\)\u200b\u53ea\u200b\u5154\u5b50\u200b\u5fc5\u7136\u200b\u4e0d\u200b\u5c5e\u4e8e\u200b\u4e00\u7c7b\u200b</li> <li>\u200b\u82e5\u200b\\(a_i = a_j\\)\uff0c\u200b\u5219\u200b\u7b2c\u200b\\(i\\)\u200b\u53ea\u200b\u5154\u5b50\u200b\u4e0e\u200b\u7b2c\u200b\\(j\\)\u200b\u53ea\u200b\u5154\u5b50\u200b\u53ef\u80fd\u200b\u5c5e\u4e8e\u200b\u4e00\u7c7b\u200b\uff0c</li> <li>\u200b\u7b2c\u200b\\(i\\)\u200b\u53ea\u200b\u5154\u5b50\u200b\u6240\u5c5e\u200b\u7684\u200b\u7c7b\u200b\u4e2d\u200b\u6700\u200b\u591a\u200b\u6709\u200b\\(a_i + 1\\)\u200b\u53ea\u200b\u5154\u5b50\u200b</li> </ul> <p>\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u5f97\u51fa\u200b\u7b97\u6cd5\u200b\uff1a\u200b\u7edf\u8ba1\u200b\u76f8\u540c\u200b\u6570\u5b57\u200b\\(a_i\\)\u200b\u51fa\u73b0\u200b\u7684\u200b\u6b21\u6570\u200b\\(b_j\\)\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u6570\u5b57\u200b\uff0c\u200b\u4e0e\u200b\u4e4b\u200b\u5bf9\u5e94\u200b\u7684\u200b\u6700\u5c11\u200b\u7684\u200b\u5154\u5b50\u200b\u6570\u91cf\u200b\u4e3a\u200b\\(b'_j\\)\uff0c\u200b\u5219\u200b\\(b'_j\\)\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\uff1a</p> <ul> <li>\\(b'_j = k \\times (a_i + 1), k\\in \\N\\)</li> <li>\\(b'_j \\geq b_j\\)</li> <li>\\(b'_j - (a_i + 1) &lt; b_j\\)</li> </ul> <p>\u200b\u5373\u200b\\(b'_j = \\left\\lceil\\frac{b_j}{a_i + 1}\\right\\rceil\\times (a_i + 1)\\)</p> <pre><code>int numRabbits(int* answers, int answersSize){\n    int hash[1000] = {0}, i = 0, ret = 0;\n    for (i = 0; i &lt; answersSize; i++)\n        hash[answers[i]]++;\n    for (i = 0; i &lt; 1000; i++)\n        ret += (hash[i] + i) / (i + 1) * (i + 1);\n    return ret;\n}\n</code></pre> <p>Python\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5185\u7f6e\u200b\u5e93\u200b<code>collections</code>\u200b\u4e2d\u200b\u7684\u200b<code>Counter</code>\u200b\u7c7b\u200b\u7edf\u8ba1\u6570\u5b57\u200b\u7684\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\uff1a</p> <pre><code>import collections\n\nclass Solution:\n    def numRabbits(self, answers: List[int]) -&gt; int:\n        return sum(map(\n            lambda args: sum(args) // (args[0] + 1) * (args[0] + 1)\n            collections.Counter(answers).items()\n        ))\n</code></pre>","tags":["\u54c8\u5e0c\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/79/","title":"79. \u200b\u5355\u8bcd\u200b\u641c\u7d22","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/79/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u00a0<code>m x n</code> \u200b\u4e8c\u7ef4\u200b\u5b57\u7b26\u200b\u7f51\u683c\u200b\u00a0<code>board</code> \u200b\u548c\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u5355\u8bcd\u200b\u00a0<code>word</code> \u3002\u200b\u5982\u679c\u200b\u00a0<code>word</code> \u200b\u5b58\u5728\u200b\u4e8e\u200b\u7f51\u683c\u200b\u4e2d\u200b\uff0c\u200b\u8fd4\u56de\u200b <code>true</code> \uff1b\u200b\u5426\u5219\u200b\uff0c\u200b\u8fd4\u56de\u200b <code>false</code> \u3002</p> <p>\u200b\u5355\u8bcd\u200b\u5fc5\u987b\u200b\u6309\u7167\u200b\u5b57\u6bcd\u200b\u987a\u5e8f\u200b\uff0c\u200b\u901a\u8fc7\u200b\u76f8\u90bb\u200b\u7684\u200b\u5355\u5143\u683c\u200b\u5185\u200b\u7684\u200b\u5b57\u6bcd\u200b\u6784\u6210\u200b\uff0c\u200b\u5176\u4e2d\u200b\u201c\u200b\u76f8\u90bb\u200b\u201d\u200b\u5355\u5143\u683c\u200b\u662f\u200b\u90a3\u4e9b\u200b\u6c34\u5e73\u200b\u76f8\u90bb\u200b\u6216\u200b\u5782\u76f4\u200b\u76f8\u90bb\u200b\u7684\u200b\u5355\u5143\u683c\u200b\u3002\u200b\u540c\u4e00\u4e2a\u200b\u5355\u5143\u683c\u200b\u5185\u200b\u7684\u200b\u5b57\u6bcd\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u88ab\u200b\u91cd\u590d\u4f7f\u7528\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = [[\"A\",\"B\",\"C\",\"E\"],[\"S\",\"F\",\"C\",\"S\"],[\"A\",\"D\",\"E\",\"E\"]], word = \"ABCCED\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = [[\"A\",\"B\",\"C\",\"E\"],[\"S\",\"F\",\"C\",\"S\"],[\"A\",\"D\",\"E\",\"E\"]], word = \"SEE\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = [[\"A\",\"B\",\"C\",\"E\"],[\"S\",\"F\",\"C\",\"S\"],[\"A\",\"D\",\"E\",\"E\"]], word = \"ABCB\"\n\u200b\u8f93\u51fa\u200b\uff1afalse\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>m == board.length</code></li> <li><code>n = board[i].length</code></li> <li><code>1 &lt;= m, n &lt;= 6</code></li> <li><code>1 &lt;= word.length &lt;= 15</code></li> <li><code>board</code> \u200b\u548c\u200b <code>word</code> \u200b\u4ec5\u200b\u7531\u200b\u5927\u5c0f\u5199\u200b\u82f1\u6587\u5b57\u6bcd\u200b\u7ec4\u6210\u200b</li> </ul> <p>\u200b\u8fdb\u9636\u200b\uff1a\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u641c\u7d22\u200b\u526a\u679d\u200b\u7684\u200b\u6280\u672f\u200b\u6765\u200b\u4f18\u5316\u200b\u89e3\u51b3\u65b9\u6848\u200b\uff0c\u200b\u4f7f\u200b\u5176\u200b\u5728\u200b <code>board</code> \u200b\u66f4\u5927\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u53ef\u4ee5\u200b\u66f4\u200b\u5feb\u200b\u89e3\u51b3\u95ee\u9898\u200b\uff1f</p> <p>Reference</p>","tags":["\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/79/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u6df1\u5ea6\u200b\u4f18\u5148\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u5728\u200b\u4e8c\u7ef4\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5bfb\u627e\u200b\u5355\u8bcd\u200b\u7b2c\u4e00\u4e2a\u200b\u5b57\u6bcd\u200b\u51fa\u73b0\u200b\u7684\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u51fa\u73b0\u200b\u4f4d\u7f6e\u200b\uff1a</p> <ul> <li>\u200b\u5c06\u200b\u5f53\u524d\u200b\u4f4d\u7f6e\u200b\u6807\u5fd7\u200b\u4e3a\u200b\u7ecf\u5386\u200b\u8fc7\u200b</li> <li>\u200b\u5bfb\u627e\u200b\u5f53\u524d\u200b\u4f4d\u7f6e\u200b\u9644\u8fd1\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u5355\u5143\u683c\u200b<ol> <li>\u200b\u5355\u5143\u683c\u200b\u4e2d\u200b\u7684\u200b\u5b57\u6bcd\u200b\u7b49\u4e8e\u200b\u5355\u8bcd\u200b\u7684\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5b57\u6bcd\u200b</li> <li>\u200b\u5355\u5143\u683c\u200b\u672a\u200b\u88ab\u200b\u904d\u5386\u200b\u8fc7\u200b</li> </ol> </li> <li>\u200b\u5982\u679c\u200b\u627e\u200b\u4e0d\u5230\u200b\u5355\u5143\u683c\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u4e8c\u7ef4\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u65e0\u6cd5\u200b\u627e\u5230\u200b\u76ee\u6807\u200b\u5355\u8bcd\u200b</li> <li>\u200b\u5982\u679c\u200b\u627e\u5230\u200b\u4e00\u4e2a\u200b\u6216\u200b\u591a\u4e2a\u200b\u5355\u5143\u683c\u200b\uff0c\u200b\u5219\u200b\u524d\u8fdb\u200b\u5230\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5355\u5143\u683c\u200b\u7ee7\u7eed\u200b\u8fdb\u884c\u200b\u5339\u914d\u200b</li> <li>\u200b\u5982\u679c\u200b\u5355\u8bcd\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u5b57\u6bcd\u200b\u6210\u529f\u200b\u5339\u914d\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u5339\u914d\u200b\u6210\u529f\u200b</li> </ul> <pre><code>bool dfs(char **board, int boardSize, int* boardColSize, char *word, int posX, int posY, int **map)\n{\n    if (*word == 0)\n        return true;\n    bool ret = false;\n    if (posX &gt; 0 &amp;&amp; board[posX - 1][posY] == *word &amp;&amp; map[posX - 1][posY] == 0)\n    {\n        map[posX - 1][posY] = 1;\n        ret = ret || dfs(board, boardSize, boardColSize, word + 1, posX - 1, posY, map);\n        map[posX - 1][posY] = 0;\n    }\n    if (posX &lt; boardSize - 1 &amp;&amp; board[posX + 1][posY] == *word &amp;&amp; map[posX + 1][posY] == 0)\n    {\n        map[posX + 1][posY] = 1;\n        ret = ret || dfs(board, boardSize, boardColSize, word + 1, posX + 1, posY, map);\n        map[posX + 1][posY] = 0;\n    }\n    if (posY &gt; 0 &amp;&amp; board[posX][posY - 1] == *word &amp;&amp; map[posX][posY - 1] == 0)\n    {\n        map[posX][posY - 1] = 1;\n        ret = ret || dfs(board, boardSize, boardColSize, word + 1, posX, posY - 1, map);\n        map[posX][posY - 1] = 0;\n    }\n    if (posY &lt; boardColSize[0] - 1 &amp;&amp; board[posX][posY + 1] == *word &amp;&amp; map[posX][posY + 1] == 0)\n    {\n        map[posX][posY + 1] = 1;\n        ret = ret || dfs(board, boardSize, boardColSize, word + 1, posX, posY + 1, map);\n        map[posX][posY + 1] = 0;\n    }\n    return ret;\n}\nbool exist(char** board, int boardSize, int* boardColSize, char * word){\n    bool ret = false;\n    int i = 0, j = 0, **map = (int **)malloc(sizeof(int *) * boardSize);\n    for (i = 0; i &lt; boardSize; i++)\n        map[i] = (int *)memset(malloc(sizeof(int) * boardColSize[i]), 0, sizeof(int) * boardColSize[i]);\n    for (i = 0; i &lt; boardSize &amp;&amp; !ret; i++)\n    {\n        for (j = 0; j &lt; boardColSize[0] &amp;&amp; !ret; j++)\n        {\n            if (board[i][j] == *word)\n            {\n                map[i][j] = 1;\n                ret = ret || dfs(board, boardSize, boardColSize, word + 1, i, j, map);\n                map[i][j] = 0;\n            }\n        }\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/8/","title":"8. \u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u200b\u6574\u6570\u200b (atoi)","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/8/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u8bf7\u200b\u4f60\u200b\u6765\u200b\u5b9e\u73b0\u200b\u4e00\u4e2a\u200b<code>atoi</code>\u200b\u51fd\u6570\u200b\uff0c\u200b\u4f7f\u200b\u5176\u80fd\u200b\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u6210\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u9996\u5148\u200b\uff0c\u200b\u8be5\u200b\u51fd\u6570\u200b\u4f1a\u200b\u6839\u636e\u200b\u9700\u8981\u200b\u4e22\u5f03\u200b\u65e0\u7528\u200b\u7684\u200b\u5f00\u5934\u200b\u7a7a\u683c\u200b\u5b57\u7b26\u200b\uff0c\u200b\u76f4\u5230\u200b\u5bfb\u627e\u200b\u5230\u200b\u7b2c\u4e00\u4e2a\u200b\u975e\u200b\u7a7a\u683c\u200b\u7684\u200b\u5b57\u7b26\u200b\u4e3a\u6b62\u200b\u3002\u200b\u63a5\u4e0b\u6765\u200b\u7684\u200b\u8f6c\u5316\u200b\u89c4\u5219\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u5982\u679c\u200b\u7b2c\u4e00\u4e2a\u200b\u975e\u200b\u7a7a\u5b57\u7b26\u200b\u4e3a\u200b\u6b63\u200b\u6216\u8005\u200b\u8d1f\u53f7\u200b\u65f6\u200b\uff0c\u200b\u5219\u200b\u5c06\u200b\u8be5\u200b\u7b26\u53f7\u200b\u4e0e\u200b\u4e4b\u200b\u540e\u9762\u200b\u5c3d\u53ef\u80fd\u200b\u591a\u200b\u7684\u200b\u8fde\u7eed\u200b\u6570\u5b57\u200b\u5b57\u7b26\u200b\u7ec4\u5408\u200b\u8d77\u6765\u200b\uff0c\u200b\u5f62\u6210\u200b\u4e00\u4e2a\u200b\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\u3002</li> <li>\u200b\u5047\u5982\u200b\u7b2c\u4e00\u4e2a\u200b\u975e\u200b\u7a7a\u5b57\u7b26\u200b\u662f\u200b\u6570\u5b57\u200b\uff0c\u200b\u5219\u200b\u76f4\u63a5\u200b\u5c06\u200b\u5176\u200b\u4e0e\u200b\u4e4b\u540e\u200b\u8fde\u7eed\u200b\u7684\u200b\u6570\u5b57\u200b\u5b57\u7b26\u200b\u7ec4\u5408\u200b\u8d77\u6765\u200b\uff0c\u200b\u5f62\u6210\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u3002</li> <li>\u200b\u8be5\u200b\u5b57\u7b26\u4e32\u200b\u5728\u200b\u6709\u6548\u200b\u7684\u200b\u6574\u6570\u200b\u90e8\u5206\u200b\u4e4b\u540e\u200b\u4e5f\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u5b58\u5728\u200b\u591a\u4f59\u200b\u7684\u200b\u5b57\u7b26\u200b\uff0c\u200b\u90a3\u4e48\u200b\u8fd9\u4e9b\u200b\u5b57\u7b26\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u5ffd\u7565\u200b\uff0c\u200b\u5b83\u4eec\u200b\u5bf9\u200b\u51fd\u6570\u200b\u4e0d\u200b\u5e94\u8be5\u200b\u9020\u6210\u200b\u5f71\u54cd\u200b\u3002</li> </ul> <p>\u200b\u6ce8\u610f\u200b\uff1a\u200b\u5047\u5982\u200b\u8be5\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u975e\u200b\u7a7a\u683c\u200b\u5b57\u7b26\u200b\u4e0d\u662f\u200b\u4e00\u4e2a\u200b\u6709\u6548\u200b\u6574\u6570\u200b\u5b57\u7b26\u200b\u3001\u200b\u5b57\u7b26\u4e32\u200b\u4e3a\u7a7a\u200b\u6216\u200b\u5b57\u7b26\u4e32\u200b\u4ec5\u200b\u5305\u542b\u200b\u7a7a\u767d\u200b\u5b57\u7b26\u200b\u65f6\u200b\uff0c\u200b\u5219\u200b\u4f60\u200b\u7684\u200b\u51fd\u6570\u200b\u4e0d\u200b\u9700\u8981\u200b\u8fdb\u884c\u200b\u8f6c\u6362\u200b\uff0c\u200b\u5373\u200b\u65e0\u6cd5\u200b\u8fdb\u884c\u200b\u6709\u6548\u200b\u8f6c\u6362\u200b\u3002</p> <p>\u200b\u5728\u200b\u4efb\u4f55\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u82e5\u200b\u51fd\u6570\u200b\u4e0d\u80fd\u200b\u8fdb\u884c\u200b\u6709\u6548\u200b\u7684\u200b\u8f6c\u6362\u200b\u65f6\u200b\uff0c\u200b\u8bf7\u200b\u8fd4\u56de\u200b 0 \u3002</p> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u672c\u9898\u200b\u4e2d\u200b\u7684\u200b\u7a7a\u767d\u200b\u5b57\u7b26\u200b\u53ea\u200b\u5305\u62ec\u200b\u7a7a\u683c\u200b\u5b57\u7b26\u200b<code>' '</code>\u3002</li> <li>\u200b\u5047\u8bbe\u200b\u6211\u4eec\u200b\u7684\u200b\u73af\u5883\u200b\u53ea\u80fd\u200b\u5b58\u50a8\u200b 32 \u200b\u4f4d\u200b\u5927\u5c0f\u200b\u7684\u200b\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\uff0c\u200b\u90a3\u4e48\u200b\u5176\u200b\u6570\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\([\u22122^{31},\u00a0 2^{31}\u00a0\u2212 1]\\)\u3002\u200b\u5982\u679c\u200b\u6570\u503c\u200b\u8d85\u8fc7\u200b\u8fd9\u4e2a\u200b\u8303\u56f4\u200b\uff0c\u200b\u8bf7\u200b\u8fd4\u56de\u200b<code>INT_MAX</code>(\\(2^{31}\u00a0\u2212 1\\))\u200b\u6216\u200b<code>INT_MIN</code>(\\(\u22122^{31}\\)) \u3002 \u00a0</li> </ul> <p>\u200b\u793a\u4f8b\u200b\u00a01:</p> <pre><code>\u200b\u8f93\u5165\u200b: \"42\"\n\u200b\u8f93\u51fa\u200b: 42\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02:</p> <pre><code>\u200b\u8f93\u5165\u200b: \"   -42\"\n\u200b\u8f93\u51fa\u200b: -42\n\u200b\u89e3\u91ca\u200b: \u200b\u7b2c\u4e00\u4e2a\u200b\u975e\u200b\u7a7a\u767d\u200b\u5b57\u7b26\u200b\u4e3a\u200b '-', \u200b\u5b83\u200b\u662f\u200b\u4e00\u4e2a\u200b\u8d1f\u53f7\u200b\u3002\n\u00a0    \u200b\u6211\u4eec\u200b\u5c3d\u53ef\u80fd\u200b\u5c06\u200b\u8d1f\u53f7\u200b\u4e0e\u200b\u540e\u9762\u200b\u6240\u6709\u200b\u8fde\u7eed\u200b\u51fa\u73b0\u200b\u7684\u200b\u6570\u5b57\u200b\u7ec4\u5408\u200b\u8d77\u6765\u200b\uff0c\u200b\u6700\u540e\u200b\u5f97\u5230\u200b -42 \u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a03:</p> <pre><code>\u200b\u8f93\u5165\u200b: \"4193 with words\"\n\u200b\u8f93\u51fa\u200b: 4193\n\u200b\u89e3\u91ca\u200b: \u200b\u8f6c\u6362\u200b\u622a\u6b62\u200b\u4e8e\u200b\u6570\u5b57\u200b '3' \uff0c\u200b\u56e0\u4e3a\u200b\u5b83\u200b\u7684\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b\u4e0d\u200b\u4e3a\u200b\u6570\u5b57\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a04:</p> <pre><code>\u200b\u8f93\u5165\u200b: \"words and 987\"\n\u200b\u8f93\u51fa\u200b: 0\n\u200b\u89e3\u91ca\u200b: \u200b\u7b2c\u4e00\u4e2a\u200b\u975e\u200b\u7a7a\u5b57\u7b26\u200b\u662f\u200b 'w', \u200b\u4f46\u200b\u5b83\u200b\u4e0d\u662f\u200b\u6570\u5b57\u200b\u6216\u200b\u6b63\u200b\u3001\u200b\u8d1f\u53f7\u200b\u3002\n     \u200b\u56e0\u6b64\u200b\u65e0\u6cd5\u200b\u6267\u884c\u200b\u6709\u6548\u200b\u7684\u200b\u8f6c\u6362\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a05:</p> <pre><code>\u200b\u8f93\u5165\u200b: \"-91283472332\"\n\u200b\u8f93\u51fa\u200b: -2147483648\n\u200b\u89e3\u91ca\u200b: \u200b\u6570\u5b57\u200b \"-91283472332\" \u200b\u8d85\u8fc7\u200b 32 \u200b\u4f4d\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\u8303\u56f4\u200b\u3002 \n\u00a0    \u200b\u56e0\u6b64\u200b\u8fd4\u56de\u200b INT_MIN (\u2212231) \u3002\n</code></pre> <p>Reference</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/8/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a </p> <ol> <li>\u200b\u6ce8\u610f\u200b\u4e00\u4e2a\u200b\u6709\u6548\u200b\u7684\u200b\u6570\u5b57\u200b\u5b57\u7b26\u4e32\u200b\u53ef\u4ee5\u200b\u4ee5\u200b<code>0-9</code>\u3001<code>'-'</code>\u3001<code>'+'</code>\u200b\u548c\u200b\u7a7a\u683c\u200b<code>' '</code>\u200b\u5f00\u5934\u200b\u3002</li> <li>\u200b\u4e3a\u4e86\u200b\u907f\u514d\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u5e94\u5f53\u200b\u4f7f\u7528\u200b<code>long</code>\u200b\u7c7b\u578b\u200b\u3002</li> </ol> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int myAtoi(char * str){\n    int flag = 0;\n    char *cur = str;\n    while (*cur &amp;&amp; *cur == ' ')\n        cur++;\n    if (*cur == '-' || *cur == '+')\n        flag = *(cur++) - '+';\n    if (*cur &lt; '0' || *cur &gt; '9')\n        return 0;\n    long ret = 0;\n    while(*cur &gt;= '0' &amp;&amp; *cur &lt;= '9')\n    {\n        ret *= 10;\n        ret += *cur - '0';\n        cur++;\n        if (ret &gt;= -(long)INT_MIN || (ret == (long)INT_MAX &amp;&amp; !flag))\n            return flag ? INT_MIN : INT_MAX;\n    }\n    return flag ? -ret : ret;\n}\n</code></pre>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/87/","title":"87. \u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u5b57\u7b26\u4e32","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/87/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u4f7f\u7528\u200b\u4e0b\u9762\u200b\u63cf\u8ff0\u200b\u7684\u200b\u7b97\u6cd5\u200b\u53ef\u4ee5\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b <code>s</code> \u200b\u5f97\u5230\u200b\u5b57\u7b26\u4e32\u200b <code>t</code> \uff1a</p> <ol> <li>\u200b\u5982\u679c\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u957f\u5ea6\u200b\u4e3a\u200b 1 \uff0c\u200b\u7b97\u6cd5\u200b\u505c\u6b62\u200b</li> <li>\u200b\u5982\u679c\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u957f\u5ea6\u200b &gt; 1 \uff0c\u200b\u6267\u884c\u200b\u4e0b\u8ff0\u200b\u6b65\u9aa4\u200b\uff1a<ul> <li>\u200b\u5728\u200b\u4e00\u4e2a\u200b\u968f\u673a\u200b\u4e0b\u6807\u200b\u5904\u200b\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u5206\u5272\u200b\u6210\u200b\u4e24\u4e2a\u200b\u975e\u7a7a\u200b\u7684\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u3002\u200b\u5373\u200b\uff0c\u200b\u5982\u679c\u200b\u5df2\u77e5\u200b\u5b57\u7b26\u4e32\u200b <code>s</code> \uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5176\u200b\u5206\u6210\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b <code>x</code> \u200b\u548c\u200b <code>y</code> \uff0c\u200b\u4e14\u200b\u6ee1\u8db3\u200b <code>s = x + y</code> \u3002</li> <li>\u200b\u968f\u673a\u200b \u200b\u51b3\u5b9a\u200b\u662f\u200b\u8981\u200b\u300c\u200b\u4ea4\u6362\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u300d\u200b\u8fd8\u662f\u200b\u8981\u200b\u300c\u200b\u4fdd\u6301\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u987a\u5e8f\u200b\u4e0d\u53d8\u200b\u300d\u3002\u200b\u5373\u200b\uff0c\u200b\u5728\u200b\u6267\u884c\u200b\u8fd9\u4e00\u200b\u6b65\u9aa4\u200b\u4e4b\u540e\u200b\uff0c<code>s</code> \u200b\u53ef\u80fd\u200b\u662f\u200b <code>s = x + y</code> \u200b\u6216\u8005\u200b <code>s = y + x</code> \u3002</li> <li>\u200b\u5728\u200b <code>x</code> \u200b\u548c\u200b <code>y</code> \u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u4e0a\u200b\u7ee7\u7eed\u200b\u4ece\u200b\u6b65\u9aa4\u200b 1 \u200b\u5f00\u59cb\u200b\u9012\u5f52\u200b\u6267\u884c\u200b\u6b64\u200b\u7b97\u6cd5\u200b\u3002</li> </ul> </li> </ol> <p>\u200b\u7ed9\u200b\u4f60\u200b\u4e24\u4e2a\u200b \u200b\u957f\u5ea6\u200b\u76f8\u7b49\u200b \u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b <code>s1</code> \u200b\u548c\u200b\u00a0<code>s2</code>\uff0c\u200b\u5224\u65ad\u200b\u00a0<code>s2</code>\u00a0\u200b\u662f\u5426\u662f\u200b\u00a0<code>s1</code>\u00a0\u200b\u7684\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b\u3002\u200b\u5982\u679c\u200b\u662f\u200b\uff0c\u200b\u8fd4\u56de\u200b <code>true</code> \uff1b\u200b\u5426\u5219\u200b\uff0c\u200b\u8fd4\u56de\u200b <code>false</code> \u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as1 = \"great\", s2 = \"rgeat\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n\u200b\u89e3\u91ca\u200b\uff1as1 \u200b\u4e0a\u200b\u53ef\u80fd\u200b\u53d1\u751f\u200b\u7684\u200b\u4e00\u79cd\u200b\u60c5\u5f62\u200b\u662f\u200b\uff1a\n\"great\" --&gt; \"gr/eat\" // \u200b\u5728\u200b\u4e00\u4e2a\u200b\u968f\u673a\u200b\u4e0b\u6807\u200b\u5904\u200b\u5206\u5272\u200b\u5f97\u5230\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\n\"gr/eat\" --&gt; \"gr/eat\" // \u200b\u968f\u673a\u200b\u51b3\u5b9a\u200b\uff1a\u300c\u200b\u4fdd\u6301\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u987a\u5e8f\u200b\u4e0d\u53d8\u200b\u300d\n\"gr/eat\" --&gt; \"g/r / e/at\" // \u200b\u5728\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u4e0a\u200b\u9012\u5f52\u200b\u6267\u884c\u200b\u6b64\u200b\u7b97\u6cd5\u200b\u3002\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u5206\u522b\u200b\u5728\u200b\u968f\u673a\u200b\u4e0b\u6807\u200b\u5904\u200b\u8fdb\u884c\u200b\u4e00\u8f6e\u200b\u5206\u5272\u200b\n\"g/r / e/at\" --&gt; \"r/g / e/at\" // \u200b\u968f\u673a\u200b\u51b3\u5b9a\u200b\uff1a\u200b\u7b2c\u4e00\u7ec4\u200b\u300c\u200b\u4ea4\u6362\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u300d\uff0c\u200b\u7b2c\u4e8c\u7ec4\u200b\u300c\u200b\u4fdd\u6301\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u987a\u5e8f\u200b\u4e0d\u53d8\u200b\u300d\n\"r/g / e/at\" --&gt; \"r/g / e/ a/t\" // \u200b\u7ee7\u7eed\u200b\u9012\u5f52\u200b\u6267\u884c\u200b\u6b64\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u5c06\u200b \"at\" \u200b\u5206\u5272\u200b\u5f97\u5230\u200b \"a/t\"\n\"r/g / e/ a/t\" --&gt; \"r/g / e/ a/t\" // \u200b\u968f\u673a\u200b\u51b3\u5b9a\u200b\uff1a\u300c\u200b\u4fdd\u6301\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u987a\u5e8f\u200b\u4e0d\u53d8\u200b\u300d\n\u200b\u7b97\u6cd5\u200b\u7ec8\u6b62\u200b\uff0c\u200b\u7ed3\u679c\u200b\u5b57\u7b26\u4e32\u200b\u548c\u200b s2 \u200b\u76f8\u540c\u200b\uff0c\u200b\u90fd\u200b\u662f\u200b \"rgeat\"\n\u200b\u8fd9\u662f\u200b\u4e00\u79cd\u200b\u80fd\u591f\u200b\u6270\u4e71\u200b s1 \u200b\u5f97\u5230\u200b s2 \u200b\u7684\u200b\u60c5\u5f62\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8ba4\u4e3a\u200b s2 \u200b\u662f\u200b s1 \u200b\u7684\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u8fd4\u56de\u200b true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as1 = \"abcde\", s2 = \"caebd\"\n\u200b\u8f93\u51fa\u200b\uff1afalse\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as1 = \"a\", s2 = \"a\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>s1.length == s2.length</code></li> <li><code>1 &lt;= s1.length &lt;= 30</code></li> <li><code>s1</code> \u200b\u548c\u200b <code>s2</code> \u200b\u7531\u200b\u5c0f\u5199\u200b\u82f1\u6587\u5b57\u6bcd\u200b\u7ec4\u6210\u200b</li> </ul> <p>Reference</p>","tags":["\u5b57\u7b26\u4e32","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/87/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u8003\u8651\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\\(s_1, s_2\\)\uff0c\u200b\u51fd\u6570\u200b\\(f(x, y)\\)\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ f(x, y) = \\left\\{ \\begin{aligned}     &amp; T &amp; y\\mathrm{\u200b\u662f\u200b}x\\mathrm{\u200b\u7684\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b} \\\\     &amp; F &amp; y\\mathrm{\u200b\u4e0d\u662f\u200b}x\\mathrm{\u200b\u7684\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b} \\end{aligned} \\right . \\] <p>\u200b\u5219\u200b\u5bf9\u4e8e\u200b\u51fd\u6570\u200b\\(f(x, y)\\)\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ul> <li>\\(f(s_1, s_2) = f(s_2, s_1)\\)</li> <li>\\(f(s_1, s_1) = T\\)</li> <li>\\(f(s_1, s_2) \\land f(s_2, s_3)\\rightarrow f(s_1, s_3)\\)</li> </ul> <p>\u200b\u5373\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b\u5173\u7cfb\u200b\u662f\u200b\u7b49\u4ef7\u5173\u7cfb\u200b\u3002</p> <p>\u200b\u5b9a\u4e49\u200b\u53e6\u200b\u4e00\u79cd\u200b\u7b49\u4ef7\u5173\u7cfb\u200b\\(g(x, y)\\)\uff1a</p> \\[ g(x, y) = \\left\\{ \\begin{aligned}     &amp; T &amp; x, y \\mathrm{\u200b\u5404\u200b\u5b57\u6bcd\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u76f8\u540c\u200b} \\\\     &amp; F &amp; x, y \\mathrm{\u200b\u5404\u200b\u5b57\u6bcd\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u4e0d\u540c\u200b} \\end{aligned} \\right . \\] <p>\u200b\u6839\u636e\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b\u64cd\u4f5c\u200b \u200b\u4e0d\u4f1a\u200b \u200b\u6539\u53d8\u200b</p> <ol> <li>\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u957f\u5ea6\u200b</li> <li>\u200b\u5b57\u7b26\u4e32\u200b\u5185\u200b\u5404\u200b\u5b57\u6bcd\u200b\u7684\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b</li> </ol> <p>\u200b\u5982\u679c\u200b\\(s_1, s_2\\)\u200b\u7684\u200b\u957f\u5ea6\u200b\u4e0d\u200b\u5339\u914d\u200b\u6216\u200b\u5b57\u6bcd\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u4e0d\u200b\u5339\u914d\u200b\uff0c\\(f(s_1, s_2)\\)\u200b\u5373\u53ef\u200b\u63d0\u524d\u200b\u8fd4\u56de\u200b\\(F\\)\u3002</p> <p>\u200b\u8003\u8651\u200b\u5206\u5272\u200b\u540e\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u5c06\u200b\\(s_1\\)\u200b\u5728\u200b\u4e0b\u6807\u200b\\(i\\)\u200b\u4e0e\u200b\u4e0b\u6807\u200b\\(i + 1\\)\u200b\u5904\u200b\u5206\u5272\u200b\u5f97\u5230\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u79f0\u4e3a\u200b \\(s_{1, i}, s'_{1, i}\\)\uff0c\\(s_{1, i}\\)\u200b\u4f4d\u4e8e\u200b\u5de6\u4fa7\u200b\uff0c\\(s'_{1, i}\\)\u200b\u4f4d\u4e8e\u200b\u53f3\u4fa7\u200b\u3002\u200b\u5bf9\u4e8e\u200b\\(s_2\\)\uff0c\u200b\u6709\u200b\u540c\u6837\u200b\u7684\u200b\u5206\u5272\u200b \u200b\u65b9\u6cd5\u200b\u5f97\u5230\u200b\\(s_{2, i}, s'_{2, i}\\)\u3002\u200b\u8bbe\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(s\\)\uff0c\u200b\u4ee4\u200b\\(-i\\)\u200b\u4e3a\u200b\u4ece\u200b\u5b57\u7b26\u4e32\u200b\u53f3\u4fa7\u200b\u5f00\u59cb\u200b\u7684\u200b\u4e0b\u6807\u200b\\(i\\)\u3002</p> <p>\u200b\u6839\u636e\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u82e5\u4efb\u200b\u4e00\u7ec4\u200b\u5206\u5272\u200b\u540e\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u6ee1\u8db3\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5173\u7cfb\u200b\uff0c\u200b\u5219\u200b\u539f\u6765\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u6ee1\u8db3\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5173\u7cfb\u200b \uff0c\u200b\u5e76\u4e14\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5173\u7cfb\u200b\u4e0e\u200b\u539f\u200b\u5b57\u7b26\u4e32\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5173\u7cfb\u200b\u65e0\u5173\u200b\uff0c\u200b\u4e0d\u4f1a\u200b\u76f8\u4e92\u5f71\u54cd\u200b\u3002\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u91c7\u7528\u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\uff0c\u200b\u5c06\u200b\u539f\u200b \u200b\u95ee\u9898\u200b\u5206\u5272\u200b\u4e3a\u200b\u4e00\u7cfb\u5217\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u5b50\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u7531\u4e8e\u200b\\(f(s_1, s_2)\\)\u200b\u5b58\u5728\u200b\u5bf9\u79f0\u6027\u200b\uff0c\u200b\u4e0d\u59a8\u200b\u56fa\u5b9a\u200b\\(s_1\\)\u200b\u7684\u200b\u5206\u5272\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u8003\u8651\u200b\\(s_2\\)\u200b\u7684\u200b\u5206\u5272\u200b\u65b9\u5f0f\u200b\u3002\\(s_2\\)\u200b\u6709\u200b\u4e24\u79cd\u200b\u5206\u5272\u200b \u200b\u65b9\u5f0f\u200b\uff0c\u200b\u5373\u200b\u201c\u200b\u4fdd\u6301\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u987a\u5e8f\u200b\u4e0d\u53d8\u200b\u201d\u200b\u6216\u200b\u201c\u200b\u4ea4\u6362\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u201d\u3002\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u4e8e\u200b\u5982\u4e0b\u200b\u60c5\u51b5\u200b\uff1a</p> <ul> <li>\\(s_{1, i}\\)\u200b\u5bf9\u5e94\u200b\u4e8e\u200b\\(s_{2, i}\\)\uff0c\\(s'_{1, i}\\)\u200b\u5bf9\u5e94\u200b\u4e8e\u200b\\(s'_{2, i}\\)\u3002</li> <li>\\(s_{1, i}\\)\u200b\u5bf9\u5e94\u200b\u4e8e\u200b\\(s'_{2, -i}\\)\uff0c\\(s'_{1, i}\\)\u200b\u5bf9\u5e94\u200b\u4e8e\u200b\\(s_{2, -i}\\)\u3002</li> </ul> <p>\u200b\u5219\u200b\\(f(s_1, s_2)\\)\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u65b9\u5f0f\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u8bbe\u200b\\(\\mathrm{len}(s_1) = \\mathrm{len}(s_2) = s\\)\u3002</p> \\[ f(s_1, s_2) = \\left\\{ \\begin{aligned}     &amp; F &amp; \\lnot g(s_1, s_2) \\\\     &amp; T &amp; s\\leq 3 \\land g(s_1, s_2) \\\\     &amp; \\left(\\bigvee_{i} f(s_{1, i}, s_{2, i})\\land f(s'_{1, i}, s'_{2, i})\\right)     \\lor     \\left(\\bigvee_{i} f(s_{1, i}, s'_{2, -i})\\land f(s'_{1, i}, s_{2, -i})\\right) &amp; s &gt; 3 \\land g(s_1, s_2)\\\\ \\end{aligned} \\right . \\] <p>\u200b\u7531\u6b64\u200b\u5373\u53ef\u200b\u5f97\u51fa\u200b\u521d\u6b65\u200b\u7684\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u4f46\u200b\u591a\u6b21\u200b\u9012\u5f52\u200b\u4f1a\u200b\u4f7f\u5f97\u200b\u7b97\u6cd5\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e0a\u5347\u200b\uff0c\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u6570\u7ec4\u200b\u4fdd\u5b58\u200b\u5b50\u200b\u95ee\u9898\u200b\u7684\u200b\u89e3\u200b\u3002\u200b\u5171\u6709\u200b \\(\\mathcal O(n^3)\\)\u200b\u89c4\u6a21\u200b\u7684\u200b\u5b50\u200b\u95ee\u9898\u200b\uff0c\u200b\u56e0\u6b64\u200b\u9700\u8981\u200b\u82b1\u8d39\u200b\\(\\mathcal O(n^3)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e0e\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u3002</p> <pre><code>inline int hash(char *s, int len)\n{\n    int i, hashMap[26] = {0};\n    unsigned long long ret = 0;\n    for (i = 0; i &lt; len; i++)\n        hashMap[s[i] - 'a']++;\n    for (i = 0; i &lt; 26; i++)\n    {\n        ret *= 31;\n        ret += hashMap[i] + 1;\n        if (ret &gt; INT_MAX)\n            ret = ret &amp; INT_MAX;\n    }\n    return (int)ret;\n}\n\nbool helper(char *s1, int offset1, char *s2, int offset2, int len, int ***dp)\n{\n    if (dp[offset1][offset2][len - 1] &gt;= 0)\n        return dp[offset1][offset2][len - 1];\n    else if (len &lt; 4)\n    {\n        dp[offset1][offset2][len - 1] = hash(s1 + offset1, len) == hash(s2 + offset2, len);\n        return dp[offset1][offset2][len - 1];\n    }\n    int flag = false, a, i;\n    dp[offset1][offset2][len - 1] = 0;\n    for (i = 1; i &lt; len; i++)\n    {\n        a = hash(s1 + offset1, i);\n        if (a == hash(s2 + offset2, i))\n            flag = flag || (\n                helper(s1, offset1, s2, offset2, i, dp) &amp;&amp; \n                helper(s1, offset1 + i, s2, offset2 + i, len - i, dp)\n            );\n        if (!flag &amp;&amp; a == hash(s2 + offset2 + len - i, i))\n            flag = flag || (\n                helper(s1, offset1, s2, offset2 + len - i, i, dp) &amp;&amp; \n                helper(s1, offset1 + i, s2, offset2, len - i, dp)\n            );\n        if (flag)\n            break;\n    }\n    dp[offset1][offset2][len - 1] = flag;\n    return dp[offset1][offset2][len - 1];\n}\n\nbool isScramble(char * s1, char * s2){\n    int len = strlen(s1), i, j,\n        ***dp = (int ***)malloc(sizeof(int **) * len),\n        **_dp = (int **)malloc(sizeof(int *) * len * len),\n        *__dp = (int *)malloc(sizeof(int) * len * len * len);\n    memset(__dp, 0xff, sizeof(int) * len * len * len);\n    for (i = 0; i &lt; len; i++)\n    {\n        dp[i] = _dp + i * len;\n        for (j = 0; j &lt; len; j++)\n            dp[i][j] = __dp + i * len * len + j * len;\n    }\n    return helper(s1, 0, s2, 0, strlen(s1), dp);\n}\n</code></pre>","tags":["\u5b57\u7b26\u4e32","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/88/","title":"88. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u6570\u7ec4","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/88/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\u00a0<code>nums1</code> \u200b\u548c\u200b <code>nums2</code>\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u5c06\u200b <code>nums2</code> \u200b\u5408\u5e76\u200b\u5230\u200b\u00a0<code>nums1</code>\u00a0\u200b\u4e2d\u200b\uff0c\u200b\u4f7f\u200b <code>nums1</code> \u200b\u6210\u4e3a\u200b\u4e00\u4e2a\u200b\u6709\u5e8f\u200b\u6570\u7ec4\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b:</p> <ul> <li>\u200b\u521d\u59cb\u5316\u200b\u00a0<code>nums1</code> \u200b\u548c\u200b <code>nums2</code> \u200b\u7684\u200b\u5143\u7d20\u200b\u6570\u91cf\u200b\u5206\u522b\u200b\u4e3a\u200b\u00a0<code>m</code> \u200b\u548c\u200b <code>n</code> \u3002</li> <li>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u00a0<code>nums1</code>\u00a0\u200b\u6709\u200b\u8db3\u591f\u200b\u7684\u200b\u7a7a\u95f4\u200b\uff08\u200b\u7a7a\u95f4\u200b\u5927\u5c0f\u200b\u5927\u4e8e\u200b\u6216\u200b\u7b49\u4e8e\u200b\u00a0<code>m + n</code>\uff09\u200b\u6765\u200b\u4fdd\u5b58\u200b <code>nums2</code> \u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\nnums1 = [1,2,3,0,0,0], m = 3\nnums2 = [2,5,6],       n = 3\n\n\u200b\u8f93\u51fa\u200b:\u00a0[1,2,2,3,5,6]\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/88/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u76f4\u63a5\u200b\u5728\u200b<code>nums1</code>\u200b\u7684\u200b\u5185\u5b58\u7a7a\u95f4\u200b\u4e0a\u200b\u4ece\u200b\u540e\u200b\u5411\u524d\u200b\u539f\u4f4d\u200b\u5408\u5e76\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>#define MAX(x, y) (x &gt; y ? x : y)\nvoid merge(int* nums1, int nums1Size, int m, int* nums2, int nums2Size, int n){\n    int x = m - 1, y = n - 1;\n    while(y &gt;= 0 &amp;&amp; x &gt;= 0)\n    {\n        nums1[x + y + 1] = MAX(nums1[x], nums2[y]);\n        if (nums1[x] &gt; nums2[y])\n            x--;\n        else\n            y--;\n    }\n    if (y &gt;= 0)\n        memcpy(nums1, nums2, sizeof(int) * (y + 1));\n}\n</code></pre>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/9/","title":"9. \u200b\u56de\u6587\u200b\u6570","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/9/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5224\u65ad\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u662f\u5426\u662f\u200b\u56de\u6587\u200b\u6570\u200b\u3002\u200b\u56de\u6587\u200b\u6570\u662f\u200b\u6307\u6b63\u200b\u5e8f\u200b\uff08\u200b\u4ece\u200b\u5de6\u5411\u53f3\u200b\uff09\u200b\u548c\u200b\u5012\u5e8f\u200b\uff08\u200b\u4ece\u53f3\u200b\u5411\u200b\u5de6\u200b\uff09\u200b\u8bfb\u200b\u90fd\u200b\u662f\u200b\u4e00\u6837\u200b\u7684\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 121\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: -121\n\u200b\u8f93\u51fa\u200b: false\n\u200b\u89e3\u91ca\u200b: \u200b\u4ece\u200b\u5de6\u5411\u53f3\u200b\u8bfb\u200b, \u200b\u4e3a\u200b -121 \u3002 \u200b\u4ece\u53f3\u200b\u5411\u200b\u5de6\u8bfb\u200b, \u200b\u4e3a\u200b 121- \u3002\u200b\u56e0\u6b64\u200b\u5b83\u200b\u4e0d\u662f\u200b\u4e00\u4e2a\u200b\u56de\u6587\u200b\u6570\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 10\n\u200b\u8f93\u51fa\u200b: false\n\u200b\u89e3\u91ca\u200b: \u200b\u4ece\u53f3\u200b\u5411\u200b\u5de6\u8bfb\u200b, \u200b\u4e3a\u200b 01 \u3002\u200b\u56e0\u6b64\u200b\u5b83\u200b\u4e0d\u662f\u200b\u4e00\u4e2a\u200b\u56de\u6587\u200b\u6570\u200b\u3002\n</code></pre> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u80fd\u200b\u4e0d\u200b\u5c06\u200b\u6574\u6570\u200b\u8f6c\u4e3a\u200b\u5b57\u7b26\u4e32\u200b\u6765\u200b\u89e3\u51b3\u200b\u8fd9\u4e2a\u200b\u95ee\u9898\u200b\u5417\u200b\uff1f</p> <p>Reference</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/9/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a </p> <ol> <li>\u200b\u5c0f\u4e8e\u200b0\u200b\u7684\u200b\u6570\u200b\u5fc5\u7136\u200b\u4e0d\u662f\u200b\u56de\u6587\u200b\u6570\u200b\uff1b</li> <li>\u200b\u5bf9\u4e8e\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\uff0c\u200b\u53ea\u200b\u9700\u8981\u200b\u5224\u65ad\u200b\u6309\u4f4d\u200b\u53cd\u8f6c\u200b\u540e\u200b\u7684\u200b\u6570\u503c\u200b\u4e0e\u200b\u539f\u200b\u6570\u503c\u200b\u662f\u5426\u200b\u76f8\u7b49\u200b\u5373\u53ef\u200b\u3002</li> </ol> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>bool isPalindrome(int x){\n    if (x &lt; 0)\n        return false;\n    long transformed = 0, original = x;\n    while (x)\n    {\n        transformed *= 10;\n        transformed += x % 10;\n        x /= 10;\n    }\n    return transformed == original;\n}\n</code></pre>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/90/","title":"90. \u200b\u5b50\u96c6\u200b II","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/90/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b <code>nums</code> \uff0c\u200b\u5176\u4e2d\u200b\u53ef\u80fd\u200b\u5305\u542b\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u8fd4\u56de\u200b\u8be5\u200b\u6570\u7ec4\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u7684\u200b\u5b50\u96c6\u200b\uff08\u200b\u5e42\u96c6\u200b\uff09\u3002</p> <p>\u200b\u89e3\u96c6\u200b \u200b\u4e0d\u80fd\u200b \u200b\u5305\u542b\u200b\u91cd\u590d\u200b\u7684\u200b\u5b50\u96c6\u200b\u3002\u200b\u8fd4\u56de\u200b\u7684\u200b\u89e3\u200b\u96c6\u4e2d\u200b\uff0c\u200b\u5b50\u96c6\u200b\u53ef\u4ee5\u200b\u6309\u200b \u200b\u4efb\u610f\u200b\u987a\u5e8f\u200b \u200b\u6392\u5217\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1,2,2]\n\u200b\u8f93\u51fa\u200b\uff1a[[],[1],[1,2],[1,2,2],[2],[2,2]]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [0]\n\u200b\u8f93\u51fa\u200b\uff1a[[],[0]]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 10</code></li> <li><code>-10 &lt;= nums[i] &lt;= 10</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/90/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5148\u200b\u7edf\u8ba1\u200b\u96c6\u5408\u200b\u4e2d\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\uff0c\u200b\u518d\u200b\u6839\u636e\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u8fdb\u884c\u200b\u56de\u6eaf\u200b\uff0c\u200b\u6700\u540e\u200b\u91cd\u65b0\u200b\u751f\u6210\u200b\u6bcf\u4e2a\u200b\u96c6\u5408\u200b\u3002</p> <pre><code>/**\n * Return an array of arrays of size *returnSize.\n * The sizes of the arrays are returned as *returnColumnSizes array.\n * Note: Both returned array and *columnSizes array must be malloced, assume caller calls free().\n */\nint *convert(int *count, int countLen, int *returnSize)\n{\n    int i = 0, j = 0, k = 0;\n    *returnSize = 0;\n    for (i = 0; i &lt; countLen; i++)\n        (*returnSize) += count[i];\n    int *ret = (int *)malloc(sizeof(int) * *returnSize);\n    for (i = 0; i &lt; countLen; i++)\n        for (j = 0; j &lt; count[i]; j++)\n            ret[k++] = i - 10;\n    return ret;\n}\n\nvoid dfs(int **ret, int *retSize, int *count, int *current, int currentPos, int maxPos, int *number)\n{\n    if (currentPos == maxPos)\n    {\n        ret[*number] = convert(current, maxPos, retSize + *number);\n        (*number)++;\n    }\n    else\n    {\n        if (count[currentPos] == 0)\n            dfs(ret, retSize, count, current, currentPos + 1, maxPos, number);\n        else\n        {\n            int i = 0;\n            for (i = 0; i &lt;= count[currentPos]; i++)\n            {\n                current[currentPos] = i;\n                dfs(ret, retSize, count, current, currentPos + 1, maxPos, number);\n                current[currentPos] = 0;\n            }\n        }\n    }\n}\n\nint** subsetsWithDup(int* nums, int numsSize, int* returnSize, int** returnColumnSizes){\n    int *count = (int *)malloc(sizeof(int) * 21), *buffer = (int *)malloc(sizeof(int) * 21),\n        i = 0, **ret = NULL;\n    memset(count, 0, sizeof(int) * 21);\n    memset(buffer, 0, sizeof(int) * 21);\n    *returnSize = 1;\n    for (i = 0; i &lt; numsSize; i++)\n        count[nums[i] + 10]++;\n    for (i = 0; i &lt; 21; i++)\n        *returnSize *= count[i] + 1;\n    *returnColumnSizes = (int *)malloc(sizeof(int) * *returnSize);\n    ret = (int **)malloc(sizeof(int *) * *returnSize);\n    i = 0;\n    dfs(ret, *returnColumnSizes, count, buffer, 0, 21, &amp;i);\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/917/","title":"917. \u200b\u4ec5\u4ec5\u200b\u53cd\u8f6c\u200b\u5b57\u6bcd","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/917/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u00a0<code>S</code>\uff0c\u200b\u8fd4\u56de\u200b\u00a0\u201c\u200b\u53cd\u8f6c\u200b\u540e\u200b\u7684\u200b\u201d\u00a0\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u5176\u4e2d\u200b\u4e0d\u662f\u200b\u5b57\u6bcd\u200b\u7684\u200b\u5b57\u7b26\u200b\u90fd\u200b\u4fdd\u7559\u200b\u5728\u200b\u539f\u5730\u200b\uff0c\u200b\u800c\u200b\u6240\u6709\u200b\u5b57\u6bcd\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u53d1\u751f\u200b\u53cd\u8f6c\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a\"ab-cd\"\n\u200b\u8f93\u51fa\u200b\uff1a\"dc-ba\"\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a\"a-bC-dEf-ghIj\"\n\u200b\u8f93\u51fa\u200b\uff1a\"j-Ih-gfE-dCba\"\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a\"Test1ng-Leet=code-Q!\"\n\u200b\u8f93\u51fa\u200b\uff1a\"Qedo1ct-eeLg=ntse-T!\"\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>S.length &lt;= 100</code></li> <li><code>33 &lt;= S[i].ASCIIcode &lt;= 122</code></li> <li><code>S</code> \u200b\u4e2d\u200b\u4e0d\u200b\u5305\u542b\u200b\u00a0<code>\\</code> or <code>\"</code></li> </ul> <p>Reference</p>","tags":["\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/917/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u548c\u200b\u53cd\u8f6c\u200b\u5b57\u7b26\u4e32\u200b\u4f7f\u7528\u200b\u76f8\u540c\u200b\u7684\u200b\u601d\u8def\u200b\uff0c\u200b\u53ea\u200b\u9700\u200b\u52a0\u4e0a\u200b\u5bf9\u200b\u5f53\u524d\u200b\u5b57\u7b26\u200b\u7684\u200b\u5224\u65ad\u200b\u5373\u53ef\u200b\u3002</p> <pre><code>bool isChar(char *s)\n{\n    return (*s &gt;= 'a' &amp;&amp; *s &lt;= 'z') || (*s &gt;= 'A' &amp;&amp; *s &lt;= 'Z');\n}\nchar * reverseOnlyLetters(char * S) {\n    int len = strlen(S);\n    char *head = S, *tail = S + len - 1, temp;\n    while (tail &gt; head)\n    {\n        while (!isChar(head) &amp;&amp; head &lt; tail)\n            head++;\n        while (!isChar(tail) &amp;&amp; head &lt; tail)\n            tail--;\n        temp = *head;\n        *head = *tail;\n        *tail = temp;\n        head++;\n        tail--;\n    }\n    return S;\n}\n</code></pre>","tags":["\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/92/","title":"92. \u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b II","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/92/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u5355\u94fe\u200b\u8868\u200b\u7684\u200b\u5934\u200b\u8282\u70b9\u200b\u00a0<code>head</code> \u200b\u548c\u200b\u4e24\u4e2a\u200b\u6574\u6570\u200b\u00a0<code>left</code> \u200b\u548c\u200b <code>right</code> \uff0c\u200b\u5176\u4e2d\u200b\u00a0<code>left &lt;= right</code> \u3002\u200b\u8bf7\u200b\u4f60\u200b\u53cd\u8f6c\u200b\u4ece\u200b\u4f4d\u7f6e\u200b <code>left</code> \u200b\u5230\u200b\u4f4d\u7f6e\u200b <code>right</code> \u200b\u7684\u200b\u94fe\u8868\u200b\u8282\u70b9\u200b\uff0c\u200b\u8fd4\u56de\u200b \u200b\u53cd\u8f6c\u200b\u540e\u200b\u7684\u200b\u94fe\u8868\u200b \u3002</p> <p></p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [1,2,3,4,5], left = 2, right = 4\n\u200b\u8f93\u51fa\u200b\uff1a[1,4,3,2,5]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [5], left = 1, right = 1\n\u200b\u8f93\u51fa\u200b\uff1a[5]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u94fe\u8868\u200b\u4e2d\u200b\u8282\u70b9\u200b\u6570\u76ee\u200b\u4e3a\u200b <code>n</code></li> <li><code>1 &lt;= n &lt;= 500</code></li> <li><code>-500 &lt;= Node.val &lt;= 500</code></li> <li><code>1 &lt;= left &lt;= right &lt;= n</code></li> </ul> <p>Reference</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/92/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5148\u200b\u8003\u8651\u200b\u53cd\u8f6c\u200b\u6574\u4e2a\u200b\u94fe\u8868\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e09\u200b\u6307\u9488\u200b\u6cd5\u5728\u200b\\(\\mathcal O(N)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u5185\u200b\u4f7f\u7528\u200b\\(\\mathcal O(1)\\)\u200b\u7684\u200b\u7a7a\u95f4\u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b</p> <p>\u200b\u94fe\u8868\u200b\u53cd\u8f6c\u200b\u7684\u200b\u4e09\u200b\u6307\u9488\u200b\u6cd5\u200b\uff1a\u200b\u5b9a\u4e49\u200b<code>cur1, cur2, cur3</code>\u200b\u4e09\u4e2a\u200b\u6307\u9488\u200b\uff0c\u200b\u5206\u522b\u200b\u6307\u5411\u200b\u94fe\u8868\u200b\u7684\u200b\u5934\u200b\u8282\u70b9\u200b\u3001\u200b\u7b2c\u4e8c\u4e2a\u200b\u8282\u70b9\u200b\u4e0e\u200b\u7b2c\u4e09\u4e2a\u200b\u8282\u70b9\u200b\u3002\u200b\u6307\u9488\u200b<code>cur1, cur2</code>\u200b\u7528\u4e8e\u200b\u4ea4\u6362\u200b\u94fe\u8868\u200b\u8282\u70b9\u200b\u7684\u200b\u8fde\u63a5\u200b\u65b9\u5411\u200b\u3002\u200b\u7531\u4e8e\u200b\u4ea4\u6362\u200b\u8282\u70b9\u200b\u64cd\u4f5c\u200b\u7834\u574f\u200b\u4e86\u200b\u539f\u6709\u200b\u7684\u200b\u8fde\u63a5\u7ed3\u6784\u200b\uff0c\u200b\u56e0\u6b64\u200b\u9700\u8981\u200b\u4e00\u4e2a\u200b\u989d\u5916\u200b\u7684\u200b\u6307\u9488\u200b<code>cur3</code>\u200b\u7528\u4e8e\u200b\u79fb\u52a8\u200b\u6307\u9488\u200b\u3002</p> <p>\u200b\u4e09\u200b\u6307\u9488\u200b\u6cd5\u200b\u7684\u200b\u4e3b\u8981\u200b\u6d41\u7a0b\u200b\u5982\u4e0b\u200b\u56fe\u200b\uff1a</p> <ol> <li>\u200b\u521d\u59cb\u5316\u200b<code>cur1, cur2, cur3</code>\uff0c\u200b\u7f6e\u200b<code>cur1-&gt;next</code>\u200b\u4e3a\u200b<code>NULL</code></li> <li>\u200b\u4ee4\u200b<code>cur2-&gt;next = cur1</code>\uff0c\u200b\u5b8c\u6210\u200b\u4ea4\u6362\u200b\u64cd\u4f5c\u200b</li> <li>\u200b\u4ee4\u200b<code>cur1 = cur2</code></li> <li>\u200b\u4ee4\u200b<code>cur2 = cur3</code></li> <li>\u200b\u5982\u679c\u200b<code>cur3</code>\u200b\u4e0d\u200b\u4e3a\u200b<code>NULL</code>\uff0c<code>cur3</code>\u200b\u5728\u200b\u94fe\u8868\u200b\u4e0a\u200b\u5411\u524d\u200b\u79fb\u52a8\u200b\uff0c\u200b\u7136\u540e\u200b\u8fd4\u56de\u200b2</li> <li>\u200b\u5982\u679c\u200b<code>cur3</code>\u200b\u4e3a\u200b<code>NULL</code>\uff0c\u200b\u7ed3\u675f\u200b</li> <li>\u200b\u8fd4\u56de\u200b<code>cur2</code></li> </ol> <p></p> <p>\u200b\u53cd\u8f6c\u200b\u7ed9\u5b9a\u200b\u8303\u56f4\u200b\u5185\u200b\u7684\u200b\u94fe\u8868\u200b\uff0c\u200b\u9700\u8981\u200b\u6ce8\u610f\u200b\u5982\u4e0b\u200b\u60c5\u51b5\u200b\uff1a</p> <ul> <li>\u200b\u5982\u679c\u200b\u53cd\u8f6c\u200b\u533a\u95f4\u200b\u4ece\u200b\u7b2c\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u5f00\u59cb\u200b\uff0c\u200b\u94fe\u8868\u200b\u7684\u200b\u5934\u200b\u8282\u70b9\u200b\u4f1a\u200b\u6539\u53d8\u200b\uff0c\u200b\u5426\u5219\u200b\u4e0d\u200b\u6539\u53d8\u200b</li> <li>\u200b\u5982\u679c\u200b\u533a\u95f4\u200b\u957f\u5ea6\u200b\u4e3a\u200b1\uff0c\u200b\u5219\u200b\u65e0\u9700\u200b\u53cd\u8f6c\u200b</li> <li>\u200b\u53cd\u8f6c\u200b\u540e\u200b\u9700\u8981\u200b\u66f4\u65b0\u200b\u53cd\u8f6c\u200b\u533a\u95f4\u200b\u524d\u540e\u200b\u7684\u200b\u8282\u70b9\u200b\u8fde\u63a5\u200b\u5173\u7cfb\u200b</li> </ul> <p>\u200b\u6574\u4f53\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\nstruct ListNode* reverseBetween(struct ListNode* head, int left, int right){\n    if (left == right)\n        return head;\n    struct ListNode *cur = head, *prev = NULL;\n    left--;\n    right--;\n    while (left)\n    {\n        left--;\n        right--;\n        prev = cur;\n        cur = cur-&gt;next;\n    }\n    struct ListNode * cur2 = cur-&gt;next, *cur3 = cur2-&gt;next, *tail = cur;\n    while (right)\n    {\n        cur2-&gt;next = cur;\n        cur = cur2;\n        cur2 = cur3;\n        if (cur3)\n            cur3 = cur3-&gt;next;\n        else\n            break;\n        right--;\n    }\n    tail-&gt;next = cur2;\n    if (prev)\n        prev-&gt;next = cur;\n    return prev ? head : cur;\n}\n</code></pre>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/98/","title":"98. \u200b\u9a8c\u8bc1\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6811","\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u9012\u5f52"]},{"location":"coding/leetcode/98/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u4e8c\u53c9\u6811\u200b\uff0c\u200b\u5224\u65ad\u200b\u5176\u200b\u662f\u5426\u662f\u200b\u4e00\u4e2a\u200b\u6709\u6548\u200b\u7684\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b\u3002</p> <p>\u200b\u5047\u8bbe\u200b\u4e00\u4e2a\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b\u5177\u6709\u200b\u5982\u4e0b\u200b\u7279\u5f81\u200b\uff1a</p> <ul> <li>\u200b\u8282\u70b9\u200b\u7684\u200b\u5de6\u5b50\u200b\u6811\u200b\u53ea\u200b\u5305\u542b\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u7684\u200b\u6570\u200b\u3002</li> <li>\u200b\u8282\u70b9\u200b\u7684\u200b\u53f3\u5b50\u200b\u6811\u200b\u53ea\u200b\u5305\u542b\u200b\u5927\u4e8e\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u7684\u200b\u6570\u200b\u3002</li> <li>\u200b\u6240\u6709\u200b\u5de6\u5b50\u200b\u6811\u200b\u548c\u200b\u53f3\u5b50\u200b\u6811\u200b\u81ea\u8eab\u200b\u5fc5\u987b\u200b\u4e5f\u200b\u662f\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\n    2\n   / \\\n  1   3\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\n    5\n   / \\\n  1   4\n\u00a0    / \\\n\u00a0   3   6\n\u200b\u8f93\u51fa\u200b: false\n\u200b\u89e3\u91ca\u200b: \u200b\u8f93\u5165\u200b\u4e3a\u200b: [5,1,4,null,null,3,6]\u3002\n\u00a0    \u200b\u6839\u200b\u8282\u70b9\u200b\u7684\u200b\u503c\u200b\u4e3a\u200b 5 \uff0c\u200b\u4f46\u662f\u200b\u5176\u53f3\u5b50\u200b\u8282\u70b9\u200b\u503c\u4e3a\u200b 4 \u3002\n</code></pre> <p>Reference</p>","tags":["\u6811","\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u9012\u5f52"]},{"location":"coding/leetcode/98/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5c06\u200b\u4e8c\u53c9\u200b\u67e5\u627e\u200b\u6811\u4e2d\u5e8f\u200b\u5c55\u5f00\u200b\u4e3a\u200b\u94fe\u8868\u200b\uff0c\u200b\u5224\u65ad\u200b\u94fe\u8868\u200b\u662f\u5426\u662f\u200b\u6709\u5e8f\u200b\u7684\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for a binary tree node.\n * struct TreeNode {\n *     int val;\n *     struct TreeNode *left;\n *     struct TreeNode *right;\n * };\n */\nstruct ListNode *insert(struct ListNode *List, int val)\n{\n    struct ListNode *ret = malloc(sizeof(struct ListNode));\n    ret-&gt;val = val;\n    ret-&gt;next = List;\n    return ret;\n}\nstruct ListNode *toList(struct TreeNode *root, struct ListNode *List)\n{\n    if (root == NULL)\n        return List;\n    List = toList(root-&gt;right, List);\n    List = insert(List, root-&gt;val);\n    List = toList(root-&gt;left, List);\n    return List;\n}\nbool isValidBST(struct TreeNode* root){\n    if (root == NULL)\n        return true;\n    struct ListNode *rec = toList(root, NULL), *cur = rec;\n    do\n    {\n        if (cur-&gt;next == NULL)\n            break;\n        if (cur-&gt;val &gt;= cur-&gt;next-&gt;val)\n            return false;\n        cur = cur-&gt;next;\n    }\n    while (true);\n    return true;\n}\n</code></pre>","tags":["\u6811","\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u9012\u5f52"]},{"location":"coding/leetcode/tags/","title":"Leetcode \u200b\u9898\u76ee","text":"<p>\u200b\u70b9\u51fb\u200b\u53f3\u4fa7\u200b\u76ee\u5f55\u200b\u53ef\u4ee5\u200b\u8df3\u8f6c\u200b\u5230\u200b\u5bf9\u5e94\u200b\u7684\u200b\u6807\u7b7e\u200b</p>"},{"location":"coding/leetcode/tags/#_1","title":"\u6570\u7ec4","text":"<p>\u200b\u6570\u7ec4\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b31\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6570\u7ec4\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 1. \u200b\u4e24\u6570\u200b\u4e4b\u200b\u548c\u200b 26. \u200b\u5220\u9664\u200b\u6392\u5e8f\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u91cd\u590d\u200b\u9879\u200b 27. \u200b\u79fb\u9664\u200b\u5143\u7d20\u200b 35. \u200b\u641c\u7d22\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b 42. \u200b\u63a5\u200b\u96e8\u6c34\u200b 48. \u200b\u65cb\u8f6c\u200b\u56fe\u50cf\u200b 53. \u200b\u6700\u5927\u200b\u5b50\u5e8f\u200b\u548c\u200b 55. \u200b\u8df3\u8dc3\u200b\u6e38\u620f\u200b 56. \u200b\u5408\u5e76\u200b\u533a\u95f4\u200b 59. \u200b\u87ba\u65cb\u200b\u77e9\u9635\u200b II 62. \u200b\u4e0d\u540c\u200b\u8def\u5f84\u200b 64. \u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b 66. \u200b\u52a0\u4e00\u200b 73. \u200b\u77e9\u9635\u200b\u7f6e\u200b\u96f6\u200b 74. \u200b\u641c\u7d22\u200b\u4e8c\u7ef4\u200b\u77e9\u9635\u200b 75. \u200b\u989c\u8272\u200b\u5206\u7c7b\u200b 78. \u200b\u5b50\u96c6\u200b 79. \u200b\u5355\u8bcd\u200b\u641c\u7d22\u200b 88. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u6570\u7ec4\u200b 90. \u200b\u5b50\u96c6\u200b II 118. \u200b\u6768\u8f89\u4e09\u89d2\u200b 119. \u200b\u6768\u8f89\u4e09\u89d2\u200b II 120. \u200b\u4e09\u89d2\u5f62\u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b 121. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b 122. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b II 189. \u200b\u65cb\u8f6c\u200b\u6570\u7ec4\u200b 217. \u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b 384. \u200b\u6253\u4e71\u200b\u6570\u7ec4\u200b 509. \u200b\u6590\u6ce2\u200b\u90a3\u200b\u5951\u6570\u200b 1011. \u200b\u5728\u200b D \u200b\u5929\u200b\u5185\u200b\u9001\u8fbe\u200b\u5305\u88f9\u200b\u7684\u200b\u80fd\u529b\u200b 1300. \u200b\u8f6c\u53d8\u200b\u6570\u7ec4\u200b\u540e\u200b\u6700\u200b\u63a5\u8fd1\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u6570\u7ec4\u200b\u548c"},{"location":"coding/leetcode/tags/#_2","title":"\u5b57\u7b26\u4e32","text":"<p>\u200b\u5b57\u7b26\u4e32\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b16\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u5b57\u7b26\u4e32\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 3. \u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b 8. \u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u200b\u6574\u6570\u200b (atoi) 13. \u200b\u7f57\u9a6c\u6570\u5b57\u200b\u8f6c\u200b\u6574\u6570\u200b 14. \u200b\u6700\u957f\u200b\u516c\u5171\u200b\u524d\u7f00\u200b 17. \u200b\u7535\u8bdd\u53f7\u7801\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408\u200b 20. \u200b\u6709\u6548\u200b\u7684\u200b\u62ec\u53f7\u200b 22. \u200b\u62ec\u53f7\u200b\u751f\u6210\u200b 28. \u200b\u5b9e\u73b0\u200b strStr() 38. \u200b\u5916\u89c2\u200b\u6570\u5217\u200b 43. \u200b\u5b57\u7b26\u4e32\u200b\u76f8\u4e58\u200b 87. \u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b 115. \u200b\u4e0d\u540c\u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217\u200b 125. \u200b\u9a8c\u8bc1\u200b\u56de\u6587\u200b\u4e32\u200b 344. \u200b\u53cd\u8f6c\u200b\u5b57\u7b26\u4e32\u200b 387. \u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u552f\u4e00\u200b\u5b57\u7b26\u200b 917. \u200b\u4ec5\u4ec5\u200b\u53cd\u8f6c\u200b\u5b57\u6bcd"},{"location":"coding/leetcode/tags/#_3","title":"\u6570\u5b66","text":"<p>\u200b\u6570\u5b66\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b15\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6570\u5b66\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 2. \u200b\u4e24\u200b\u6570\u200b\u76f8\u52a0\u200b 7. \u200b\u6574\u6570\u200b\u53cd\u8f6c\u200b 8. \u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u200b\u6574\u6570\u200b (atoi) 9. \u200b\u56de\u6587\u200b\u6570\u200b 11. \u200b\u76db\u200b\u6700\u591a\u6c34\u200b\u7684\u200b\u5bb9\u5668\u200b 13. \u200b\u7f57\u9a6c\u6570\u5b57\u200b\u8f6c\u200b\u6574\u6570\u200b 14. \u200b\u6700\u957f\u200b\u516c\u5171\u200b\u524d\u7f00\u200b 43. \u200b\u5b57\u7b26\u4e32\u200b\u76f8\u4e58\u200b 50. Pow(x, n) 69. x \u200b\u7684\u200b\u5e73\u65b9\u6839\u200b 204. \u200b\u8ba1\u6570\u200b\u8d28\u6570\u200b 368. \u200b\u6700\u5927\u200b\u6574\u9664\u200b\u5b50\u96c6\u200b 781. \u200b\u68ee\u6797\u200b\u4e2d\u200b\u7684\u200b\u5154\u5b50\u200b 1006. \u200b\u7b28\u200b\u9636\u4e58\u200b 1835. \u200b\u6240\u6709\u200b\u6570\u5bf9\u200b\u6309\u4f4d\u200b\u4e0e\u200b\u7ed3\u679c\u200b\u7684\u200b\u5f02\u6216\u200b\u548c"},{"location":"coding/leetcode/tags/#_4","title":"\u53cc\u200b\u6307\u9488","text":"<p>\u200b\u53cc\u200b\u6307\u9488\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b14\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u53cc\u200b\u6307\u9488\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 3. \u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b 11. \u200b\u76db\u200b\u6700\u591a\u6c34\u200b\u7684\u200b\u5bb9\u5668\u200b 19. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u7684\u200b\u5012\u6570\u7b2c\u200bN\u200b\u4e2a\u200b\u8282\u70b9\u200b 26. \u200b\u5220\u9664\u200b\u6392\u5e8f\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u91cd\u590d\u200b\u9879\u200b 27. \u200b\u79fb\u9664\u200b\u5143\u7d20\u200b 28. \u200b\u5b9e\u73b0\u200b strStr() 42. \u200b\u63a5\u200b\u96e8\u6c34\u200b 61. \u200b\u65cb\u8f6c\u200b\u94fe\u8868\u200b 75. \u200b\u989c\u8272\u200b\u5206\u7c7b\u200b 88. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u6570\u7ec4\u200b 125. \u200b\u9a8c\u8bc1\u200b\u56de\u6587\u200b\u4e32\u200b 234. \u200b\u56de\u6587\u200b\u94fe\u8868\u200b 344. \u200b\u53cd\u8f6c\u200b\u5b57\u7b26\u4e32\u200b 350. \u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u7684\u200b\u4ea4\u96c6\u200b II"},{"location":"coding/leetcode/tags/#_5","title":"\u52a8\u6001\u200b\u89c4\u5212","text":"<p>\u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b13\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u52a8\u6001\u200b\u89c4\u5212\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 42. \u200b\u63a5\u200b\u96e8\u6c34\u200b 53. \u200b\u6700\u5927\u200b\u5b50\u5e8f\u200b\u548c\u200b 62. \u200b\u4e0d\u540c\u200b\u8def\u5f84\u200b 64. \u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b 70. \u200b\u722c\u697c\u68af\u200b 87. \u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b 115. \u200b\u4e0d\u540c\u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217\u200b 120. \u200b\u4e09\u89d2\u5f62\u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b 121. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b 198. \u200b\u6253\u5bb6\u52ab\u820d\u200b 368. \u200b\u6700\u5927\u200b\u6574\u9664\u200b\u5b50\u96c6\u200b 377. \u200b\u7ec4\u5408\u200b\u603b\u548c\u200b \u2163 403. \u200b\u9752\u86d9\u200b\u8fc7\u6cb3"},{"location":"coding/leetcode/tags/#_6","title":"\u54c8\u5e0c\u200b\u8868","text":"<p>\u200b\u54c8\u5e0c\u200b\u8868\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b12\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u54c8\u5e0c\u200b\u8868\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 1. \u200b\u4e24\u6570\u200b\u4e4b\u200b\u548c\u200b 3. \u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b 36. \u200b\u6709\u6548\u200b\u7684\u200b\u6570\u72ec\u200b 37. \u200b\u89e3\u6570\u200b\u72ec\u200b 136. \u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u7684\u200b\u6570\u5b57\u200b 204. \u200b\u8ba1\u6570\u200b\u8d28\u6570\u200b 217. \u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b 242. \u200b\u6709\u6548\u200b\u7684\u200b\u5b57\u6bcd\u200b\u5f02\u4f4d\u200b\u8bcd\u200b 350. \u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u7684\u200b\u4ea4\u96c6\u200b II 387. \u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u552f\u4e00\u200b\u5b57\u7b26\u200b 433. \u200b\u6700\u5c0f\u200b\u57fa\u56e0\u200b\u53d8\u5316\u200b 781. \u200b\u68ee\u6797\u200b\u4e2d\u200b\u7684\u200b\u5154\u5b50"},{"location":"coding/leetcode/tags/#_7","title":"\u94fe\u8868","text":"<p>\u200b\u94fe\u8868\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b11\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u94fe\u8868\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 2. \u200b\u4e24\u200b\u6570\u200b\u76f8\u52a0\u200b 19. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u7684\u200b\u5012\u6570\u7b2c\u200bN\u200b\u4e2a\u200b\u8282\u70b9\u200b 21. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u94fe\u8868\u200b 23. \u200b\u5408\u5e76\u200bK\u200b\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868\u200b 24. \u200b\u4e24\u200b\u4e24\u200b\u4ea4\u6362\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9\u200b 25. K \u200b\u4e2a\u200b\u4e00\u7ec4\u200b\u7ffb\u8f6c\u200b\u94fe\u8868\u200b 61. \u200b\u65cb\u8f6c\u200b\u94fe\u8868\u200b 92. \u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b II 206. \u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b 234. \u200b\u56de\u6587\u200b\u94fe\u8868\u200b 237. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9"},{"location":"coding/leetcode/tags/#_8","title":"\u4e8c\u5206\u200b\u67e5\u627e","text":"<p>\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b9\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 35. \u200b\u641c\u7d22\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b 50. Pow(x, n) 69. x \u200b\u7684\u200b\u5e73\u65b9\u6839\u200b 74. \u200b\u641c\u7d22\u200b\u4e8c\u7ef4\u200b\u77e9\u9635\u200b 315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570\u200b 350. \u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u7684\u200b\u4ea4\u96c6\u200b II 1011. \u200b\u5728\u200b D \u200b\u5929\u200b\u5185\u200b\u9001\u8fbe\u200b\u5305\u88f9\u200b\u7684\u200b\u80fd\u529b\u200b 1283. \u200b\u4f7f\u200b\u7ed3\u679c\u200b\u4e0d\u200b\u8d85\u8fc7\u200b\u9608\u503c\u200b\u7684\u200b\u6700\u5c0f\u200b\u9664\u6570\u200b 1300. \u200b\u8f6c\u53d8\u200b\u6570\u7ec4\u200b\u540e\u200b\u6700\u200b\u63a5\u8fd1\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u6570\u7ec4\u200b\u548c"},{"location":"coding/leetcode/tags/#_9","title":"\u56de\u6eaf\u200b\u7b97\u6cd5","text":"<p>\u200b\u56de\u6eaf\u200b\u7b97\u6cd5\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b7\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u56de\u6eaf\u200b\u7b97\u6cd5\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 17. \u200b\u7535\u8bdd\u53f7\u7801\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408\u200b 22. \u200b\u62ec\u53f7\u200b\u751f\u6210\u200b 37. \u200b\u89e3\u6570\u200b\u72ec\u200b 46. \u200b\u5168\u200b\u6392\u5217\u200b 78. \u200b\u5b50\u96c6\u200b 79. \u200b\u5355\u8bcd\u200b\u641c\u7d22\u200b 90. \u200b\u5b50\u96c6\u200b II"},{"location":"coding/leetcode/tags/#_10","title":"\u9012\u5f52","text":"<p>\u200b\u9012\u5f52\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b6\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u9012\u5f52\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 2. \u200b\u4e24\u200b\u6570\u200b\u76f8\u52a0\u200b 17. \u200b\u7535\u8bdd\u53f7\u7801\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408\u200b 21. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u94fe\u8868\u200b 24. \u200b\u4e24\u200b\u4e24\u200b\u4ea4\u6362\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9\u200b 98. \u200b\u9a8c\u8bc1\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b 779. \u200b\u7b2c\u200bK\u200b\u4e2a\u200b\u8bed\u6cd5\u200b\u7b26\u53f7"},{"location":"coding/leetcode/tags/#_11","title":"\u6808","text":"<p>\u200b\u6808\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b5\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6808\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 20. \u200b\u6709\u6548\u200b\u7684\u200b\u62ec\u53f7\u200b 42. \u200b\u63a5\u200b\u96e8\u6c34\u200b 150. \u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u6c42\u503c\u200b 155. \u200b\u6700\u5c0f\u200b\u6808\u200b 341. \u200b\u6241\u5e73\u5316\u200b\u5d4c\u5957\u200b\u5217\u8868\u200b\u8fed\u4ee3\u200b\u5668"},{"location":"coding/leetcode/tags/#_12","title":"\u6392\u5e8f","text":"<p>\u200b\u6392\u5e8f\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b5\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6392\u5e8f\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 56. \u200b\u5408\u5e76\u200b\u533a\u95f4\u200b 75. \u200b\u989c\u8272\u200b\u5206\u7c7b\u200b 242. \u200b\u6709\u6548\u200b\u7684\u200b\u5b57\u6bcd\u200b\u5f02\u4f4d\u200b\u8bcd\u200b 315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570\u200b 350. \u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u7684\u200b\u4ea4\u96c6\u200b II"},{"location":"coding/leetcode/tags/#_13","title":"\u6df1\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22","text":"<p>\u200b\u6df1\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b4\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6df1\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 17. \u200b\u7535\u8bdd\u53f7\u7801\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408\u200b 98. \u200b\u9a8c\u8bc1\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b 101. \u200b\u5bf9\u79f0\u200b\u4e8c\u53c9\u6811\u200b 488. \u200b\u7956\u739b\u200b\u6e38\u620f"},{"location":"coding/leetcode/tags/#_14","title":"\u6811","text":"<p>\u200b\u6811\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b3\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6811\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 98. \u200b\u9a8c\u8bc1\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b 101. \u200b\u5bf9\u79f0\u200b\u4e8c\u53c9\u6811\u200b 102. \u200b\u4e8c\u53c9\u6811\u200b\u7684\u200b\u5c42\u5e8f\u200b\u904d\u5386"},{"location":"coding/leetcode/tags/#_15","title":"\u5e7f\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22","text":"<p>\u200b\u5e7f\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b3\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u5e7f\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 101. \u200b\u5bf9\u79f0\u200b\u4e8c\u53c9\u6811\u200b 102. \u200b\u4e8c\u53c9\u6811\u200b\u7684\u200b\u5c42\u5e8f\u200b\u904d\u5386\u200b 433. \u200b\u6700\u5c0f\u200b\u57fa\u56e0\u200b\u53d8\u5316"},{"location":"coding/leetcode/tags/#_16","title":"\u8bbe\u8ba1","text":"<p>\u200b\u8bbe\u8ba1\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b3\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u8bbe\u8ba1\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 155. \u200b\u6700\u5c0f\u200b\u6808\u200b 341. \u200b\u6241\u5e73\u5316\u200b\u5d4c\u5957\u200b\u5217\u8868\u200b\u8fed\u4ee3\u200b\u5668\u200b 1603. \u200b\u8bbe\u8ba1\u200b\u505c\u8f66\u200b\u7cfb\u7edf"},{"location":"coding/leetcode/tags/#_17","title":"\u5206\u6cbb\u200b\u7b97\u6cd5","text":"<p>\u200b\u5206\u6cbb\u200b\u7b97\u6cd5\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b3\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u5206\u6cbb\u200b\u7b97\u6cd5\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 23. \u200b\u5408\u5e76\u200bK\u200b\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868\u200b 53. \u200b\u6700\u5927\u200b\u5b50\u5e8f\u200b\u548c\u200b 315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570"},{"location":"coding/leetcode/tags/#_18","title":"\u8d2a\u5fc3\u200b\u7b97\u6cd5","text":"<p>\u200b\u8d2a\u5fc3\u200b\u7b97\u6cd5\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b2\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u8d2a\u5fc3\u200b\u7b97\u6cd5\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 55. \u200b\u8df3\u8dc3\u200b\u6e38\u620f\u200b 122. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b II"},{"location":"coding/leetcode/tags/#_19","title":"\u4f4d\u200b\u8fd0\u7b97","text":"<p>\u200b\u4f4d\u200b\u8fd0\u7b97\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b2\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u4f4d\u200b\u8fd0\u7b97\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 78. \u200b\u5b50\u96c6\u200b 136. \u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u7684\u200b\u6570\u5b57"},{"location":"coding/leetcode/tags/#_20","title":"\u5806","text":"<p>\u200b\u5806\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b1\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u5806\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 23. \u200b\u5408\u5e76\u200bK\u200b\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868"},{"location":"coding/leetcode/tags/#sliding-window","title":"Sliding Window","text":"<p>Sliding Window\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b1\u200b\u9053\u200b\u9898\u76ee\u200b</p> Sliding Window \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 3. \u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32"},{"location":"coding/leetcode/tags/#_21","title":"\u6811\u72b6\u200b\u6570\u7ec4","text":"<p>\u200b\u6811\u72b6\u200b\u6570\u7ec4\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b1\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6811\u72b6\u200b\u6570\u7ec4\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570"},{"location":"coding/leetcode/tags/#_22","title":"\u7ebf\u6bb5\u200b\u6811","text":"<p>\u200b\u7ebf\u6bb5\u200b\u6811\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b1\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u7ebf\u6bb5\u200b\u6811\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570"},{"location":"coding/machine-learning/","title":"\u673a\u5668\u200b\u5b66\u4e60","text":"<p>\u200b\u90e8\u5206\u200b\u673a\u5668\u200b\u5b66\u4e60\u200b\u7b97\u6cd5\u200b\u7684\u200bPython\u200b\u5b9e\u73b0\u200b</p> <ul> <li>\u200b\u7ebf\u6027\u200b\u6a21\u578b\u200b</li> <li>\u200b\u51b3\u7b56\u6811\u200b</li> <li>\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b</li> <li>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b</li> <li>\u200b\u7b97\u6cd5\u200b\u6d41\u7a0b\u200b</li> <li>\u200b\u4e3b\u9898\u200b\u6a21\u578b\u200b</li> <li>VAE-NTM</li> </ul>"},{"location":"coding/machine-learning/bayesian-optimization/","title":"\u8d1d\u53f6\u65af\u200b\u4f18\u5316","text":"In\u00a0[1]: Copied! <pre>import numpy as np\nfrom scipy.stats import norm\n\ndef objective(x):\n    return norm.pdf(x, 3, 2) * 1.5 + norm.pdf(x, 7, 1) + norm.pdf(x, 11, 2)\n</pre> import numpy as np from scipy.stats import norm  def objective(x):     return norm.pdf(x, 3, 2) * 1.5 + norm.pdf(x, 7, 1) + norm.pdf(x, 11, 2) <p>\u200b\u521d\u6b65\u200b\u5bf9\u200b\u8be5\u200b\u51fd\u6570\u200b\u8fdb\u884c\u200b\u53ef\u89c6\u5316\u200b</p> In\u00a0[2]: Copied! <pre>from matplotlib import pyplot as plt\n%config InlineBackend.figure_format = 'svg'\n\nx = np.linspace(-5, 20, 50)\nplt.plot(x, objective(x))\nplt.xlabel('x')\nplt.ylabel('f(x)')\nplt.title('Objective function')\nplt.show()\n</pre> from matplotlib import pyplot as plt %config InlineBackend.figure_format = 'svg'  x = np.linspace(-5, 20, 50) plt.plot(x, objective(x)) plt.xlabel('x') plt.ylabel('f(x)') plt.title('Objective function') plt.show() <p>\u200b\u91c7\u96c6\u200b\u51fd\u6570\u200b\u7684\u200b\u76ee\u6807\u200b\u662f\u200b\u9009\u51fa\u200b\u540e\u9a8c\u200b\u4e2d\u200b\u6700\u4f18\u200b\u7684\u200b\u70b9\u200b\u3002\u200b\u5728\u200b\u6b64\u5904\u200b\uff0c\u200b\u6211\u4eec\u200b\u4f7f\u7528\u200b\u671f\u671b\u200b\u6539\u8fdb\u200b\uff08Expected Improvement, EI\uff09\u200b\u6765\u200b\u9009\u62e9\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u91c7\u96c6\u200b\u70b9\u200b\u3002EI\u200b\u7684\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff1a</p> <p>$$ \\begin{aligned} \\text{EI}(x) &amp;= \\mathbb{E}[\\max(0, \\tilde f(x) - f(x^*) - \\xi)] \\\\ \\end{aligned} $$</p> <p>\u200b\u5176\u4e2d\u200b$\\tilde f(x)$\u200b\u662f\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u5728\u200b$x$\u200b\u5904\u200b\u7684\u200b\u9884\u6d4b\u503c\u200b\uff0c$f(x^*)$\u200b\u662f\u200b\u5f53\u524d\u200b\u6700\u4f18\u200b\u7684\u200b\u51fd\u6570\u200b\u503c\u200b\uff0c$\\xi$\u200b\u662f\u200b\u4e00\u4e2a\u200b\u63a7\u5236\u200b\u63a2\u7d22\u200b\u548c\u200b\u5f00\u53d1\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5f53\u200b$\\xi$\u200b\u8f83\u5927\u200b\u65f6\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u66f4\u200b\u503e\u5411\u200b\u4e8e\u200b\u63a2\u7d22\u200b\uff0c\u200b\u5f53\u200b$\\xi$\u200b\u8f83\u200b\u5c0f\u65f6\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u66f4\u200b\u503e\u5411\u200b\u4e8e\u200b\u5f00\u53d1\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u9ad8\u65af\u5206\u5e03\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u7ed9\u5b9a\u200b$x$, $\\tilde f(x)$\u200b\u7684\u200b\u5206\u5e03\u200b\u662f\u200b\u4e00\u4e2a\u200b\u9ad8\u65af\u5206\u5e03\u200b\uff0c\u200b\u5176\u200b\u5747\u503c\u200b\u4e3a\u200b$\\mu(x)$\uff0c\u200b\u65b9\u5dee\u200b\u4e3a\u200b$\\sigma^2(x)$\uff0c\u200b\u4ee4\u200b</p> <p>$$ z(x) = \\frac{\\mu(x) - f(x^*) - \\xi}{\\sigma(x)} $$</p> <p>\u200b\u5219\u200b\u671f\u671b\u200b\u6539\u8fdb\u200b\u4e3a\u200b</p> <p>$$ \\text{EI}(x) = \\sigma(x)[z(x)\\Phi(z(x)) + \\phi(z(x))] $$</p> In\u00a0[3]: Copied! <pre>from typing import Callable\nfrom sklearn.gaussian_process import GaussianProcessRegressor\n\nGP = GaussianProcessRegressor\nAcquisition = Callable[[np.ndarray, GP, float], np.ndarray]\n\ndef expected_improvement(x: np.ndarray, gp: GP, y_max: float, xi: float = 0.01):\n    x = np.array(x).reshape(-1, 1)\n    mu, sigma = gp.predict(x, return_std=True) # type: ignore\n    sigma = sigma.reshape(-1, 1)\n    mu = mu.reshape(-1, 1)\n\n    with np.errstate(divide='warn'):\n        Z = (mu - y_max - xi) / sigma # Normalized Z-score\n        ei = sigma * (Z * norm.cdf(Z) + norm.pdf(Z))\n\n    return ei\n</pre> from typing import Callable from sklearn.gaussian_process import GaussianProcessRegressor  GP = GaussianProcessRegressor Acquisition = Callable[[np.ndarray, GP, float], np.ndarray]  def expected_improvement(x: np.ndarray, gp: GP, y_max: float, xi: float = 0.01):     x = np.array(x).reshape(-1, 1)     mu, sigma = gp.predict(x, return_std=True) # type: ignore     sigma = sigma.reshape(-1, 1)     mu = mu.reshape(-1, 1)      with np.errstate(divide='warn'):         Z = (mu - y_max - xi) / sigma # Normalized Z-score         ei = sigma * (Z * norm.cdf(Z) + norm.pdf(Z))      return ei <p>\u200b\u6839\u636e\u200b\u91c7\u96c6\u200b\u51fd\u6570\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u5728\u200b\u8303\u56f4\u200b\u4e2d\u200b\u9009\u62e9\u200b\u4e00\u4e2a\u70b9\u200b\u4f7f\u5f97\u200bEI\u200b\u6700\u5927\u200b\uff0c\u200b\u7136\u540e\u200b\u89c2\u6d4b\u200b\u8be5\u70b9\u200b\u7684\u200b\u51fd\u6570\u200b\u503c\u200b\uff0c\u200b\u66f4\u65b0\u200b\u540e\u9a8c\u200b\uff0c\u200b\u91cd\u590d\u200b\u8fd9\u4e2a\u200b\u8fc7\u7a0b\u200b\u76f4\u5230\u200b\u8fbe\u5230\u200b\u8fed\u4ee3\u200b\u6b21\u6570\u200b\u3002</p> In\u00a0[4]: Copied! <pre>def next_point(\n    x_min: float, x_max: float, n_samples: int,\n    acquisition: Acquisition, gp: GP, y_max: float\n):\n    x = np.random.uniform(x_min, x_max, n_samples).reshape(-1, 1)\n    ei = acquisition(x, gp, y_max)\n    return x[np.argmax(ei)]\n</pre> def next_point(     x_min: float, x_max: float, n_samples: int,     acquisition: Acquisition, gp: GP, y_max: float ):     x = np.random.uniform(x_min, x_max, n_samples).reshape(-1, 1)     ei = acquisition(x, gp, y_max)     return x[np.argmax(ei)] <p>\u200b\u6b64\u540e\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u6765\u200b\u62df\u5408\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u3002<code>sklearn</code>\u200b\u4e2d\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u5b9e\u73b0\u200b\u3002</p> In\u00a0[5]: Copied! <pre>from sklearn.gaussian_process.kernels import RBF\n\nkernel = RBF(length_scale=1.0, length_scale_bounds=(1e-2, 1e2))\ngp = GP(kernel=kernel, alpha=1e-5, n_restarts_optimizer=10)\n\ndef plot_gp(ax, gp: GP, x_min: float, x_max: float, ci: float = 0.95):\n    x = np.linspace(x_min, x_max, 50).reshape(-1, 1)\n    y, sigma = gp.predict(x, return_std=True)\n\n    ci_coef = norm.ppf((1 + ci) / 2)\n\n    ax.plot(x, y, linewidth=1)\n    ax.fill_between(x.flatten(), y - ci_coef * sigma, y + ci_coef * sigma, alpha=0.2)\n    ax.set_xlabel('x')\n    ax.set_ylabel('f(x)')\n\nfig, ax = plt.subplots(1, 1)\nplot_gp(ax, gp, -5, 20)\nfig.show()\n</pre> from sklearn.gaussian_process.kernels import RBF  kernel = RBF(length_scale=1.0, length_scale_bounds=(1e-2, 1e2)) gp = GP(kernel=kernel, alpha=1e-5, n_restarts_optimizer=10)  def plot_gp(ax, gp: GP, x_min: float, x_max: float, ci: float = 0.95):     x = np.linspace(x_min, x_max, 50).reshape(-1, 1)     y, sigma = gp.predict(x, return_std=True)      ci_coef = norm.ppf((1 + ci) / 2)      ax.plot(x, y, linewidth=1)     ax.fill_between(x.flatten(), y - ci_coef * sigma, y + ci_coef * sigma, alpha=0.2)     ax.set_xlabel('x')     ax.set_ylabel('f(x)')  fig, ax = plt.subplots(1, 1) plot_gp(ax, gp, -5, 20) fig.show() <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_54343/3771838051.py:19: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <p>\u200b\u6b64\u540e\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u8fed\u4ee3\u200b\u5730\u200b\u4f18\u5316\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u3002</p> In\u00a0[6]: Copied! <pre>fig, ax = plt.subplots(4, 4, sharex=False, sharey=False, figsize=(9, 9))\nax_flatten = ax.flatten()\n\nx_min = -5\nx_max = 20\nx_samples = np.random.uniform(x_min, x_max, 3).reshape(-1, 1)\ny_samples = objective(x_samples)\ny_max = y_samples.max()\ngp = gp.fit(x_samples, y_samples)\n\nfor i, current_ax in enumerate(ax_flatten):\n    # Setup plot\n    current_ax.set_xlim(x_min, x_max)\n    current_ax.set_ylim(-1, 1)\n    if i // 4 != 3:\n        current_ax.get_xaxis().set_visible(False)\n    if i % 4 != 0:\n        current_ax.get_yaxis().set_visible(False)\n\n    # Plot prediction\n    plot_gp(current_ax, gp, x_min, x_max)\n\n    # Plot function\n    x_grid = np.linspace(x_min, x_max, 50).reshape(-1, 1)\n\n    # Plot samples\n    current_ax.plot(\n        x_grid, objective(x_grid), '--', color='gray', linewidth=0.5\n    )\n    current_ax.scatter(x_samples, y_samples, color='red', s=4, zorder=10)\n\n    # Plot optimal point\n    x_optimal = x_samples[np.argmax(y_samples)].item()\n    current_ax.scatter(x_optimal, y_max, color='green', s=10, zorder=20)\n    current_ax.text(\n        0.5, 0.9,\n        f'Best: x={x_optimal:.2f}, y={y_max:.2f}',\n        ha='center', transform=current_ax.transAxes\n    )\n\n    point = next_point(x_min, x_max, 100, expected_improvement, gp, y_max)\n\n    # Plot next point\n    current_ax.plot(\n        [point, point], [-0.8, 0.8], '--', color='red', linewidth=0.5\n    )\n    current_ax.text(\n        0.5, 0.05,\n        f'Next x: {point[0]:.2f}',\n        ha='center', transform=current_ax.transAxes\n    )\n\n    # Update samples\n    y = objective(point).item()\n    x_samples = np.vstack([x_samples, point])\n    y_samples = np.append(y_samples, y)\n\n    y_max = max(y_max, y)\n\n    # Re-fit model\n    gp = gp.fit(x_samples, y_samples)\n\nfig.show()\n</pre> fig, ax = plt.subplots(4, 4, sharex=False, sharey=False, figsize=(9, 9)) ax_flatten = ax.flatten()  x_min = -5 x_max = 20 x_samples = np.random.uniform(x_min, x_max, 3).reshape(-1, 1) y_samples = objective(x_samples) y_max = y_samples.max() gp = gp.fit(x_samples, y_samples)  for i, current_ax in enumerate(ax_flatten):     # Setup plot     current_ax.set_xlim(x_min, x_max)     current_ax.set_ylim(-1, 1)     if i // 4 != 3:         current_ax.get_xaxis().set_visible(False)     if i % 4 != 0:         current_ax.get_yaxis().set_visible(False)      # Plot prediction     plot_gp(current_ax, gp, x_min, x_max)      # Plot function     x_grid = np.linspace(x_min, x_max, 50).reshape(-1, 1)      # Plot samples     current_ax.plot(         x_grid, objective(x_grid), '--', color='gray', linewidth=0.5     )     current_ax.scatter(x_samples, y_samples, color='red', s=4, zorder=10)      # Plot optimal point     x_optimal = x_samples[np.argmax(y_samples)].item()     current_ax.scatter(x_optimal, y_max, color='green', s=10, zorder=20)     current_ax.text(         0.5, 0.9,         f'Best: x={x_optimal:.2f}, y={y_max:.2f}',         ha='center', transform=current_ax.transAxes     )      point = next_point(x_min, x_max, 100, expected_improvement, gp, y_max)      # Plot next point     current_ax.plot(         [point, point], [-0.8, 0.8], '--', color='red', linewidth=0.5     )     current_ax.text(         0.5, 0.05,         f'Next x: {point[0]:.2f}',         ha='center', transform=current_ax.transAxes     )      # Update samples     y = objective(point).item()     x_samples = np.vstack([x_samples, point])     y_samples = np.append(y_samples, y)      y_max = max(y_max, y)      # Re-fit model     gp = gp.fit(x_samples, y_samples)  fig.show() <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_54343/3753732084.py:63: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre>"},{"location":"coding/machine-learning/bayesian-optimization/","title":"\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u00b6","text":"<p>\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u662f\u200b\u4e00\u7c7b\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u4f18\u5316\u200b\u9ed1\u76d2\u200b\u51fd\u6570\u200b\u7684\u200b\u4f18\u5316\u200b\u65b9\u6cd5\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u9ed1\u76d2\u200b\u51fd\u6570\u200b\u6765\u8bf4\u200b\uff0c\u200b\u6211\u4eec\u200b\u65e0\u6cd5\u200b\u5f97\u77e5\u200b\u5176\u200b\u663e\u5f0f\u200b\u7684\u200b\u8868\u8fbe\u5f0f\u200b\uff0c\u200b\u53ea\u80fd\u200b\u901a\u8fc7\u200b\u8f93\u5165\u200b\u4e00\u4e2a\u200b\u6570\u636e\u200b\uff0c\u200b\u89c2\u6d4b\u200b\u5176\u200b\u8f93\u51fa\u200b\u7684\u200b\u65b9\u6cd5\u200b\u5bfb\u627e\u200b\u5b83\u200b\u7684\u200b\u6700\u5927\u503c\u200b\u3002\u200b\u501f\u52a9\u4e8e\u200b\u8d1d\u53f6\u65af\u200b\u7edf\u8ba1\u200b\u7684\u200b\u601d\u60f3\u200b\uff0c\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u6700\u521d\u200b\u8ba4\u4e3a\u200b\u51fd\u6570\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5b8c\u5168\u200b\u968f\u673a\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4e0d\u65ad\u200b\u5730\u200b\u89c2\u6d4b\u200b\u6765\u200b\u66f4\u65b0\u200b\u5bf9\u200b\u51fd\u6570\u200b\u7684\u200b\u4f30\u8ba1\u200b\uff08\u200b\u540e\u9a8c\u200b\uff09\u3002\u200b\u5728\u200b\u6bcf\u200b\u4e00\u6b65\u200b\u8fed\u4ee3\u200b\u4e2d\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u4f1a\u200b\u6839\u636e\u200b\u5f53\u524d\u200b\u7684\u200b\u540e\u9a8c\u200b\uff0c\u200b\u901a\u8fc7\u200b\u91c7\u96c6\u200b\u51fd\u6570\u200b\u6765\u200b\u9009\u62e9\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u89c2\u6d4b\u200b\u7684\u200b\u70b9\u200b\u3002\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u7684\u200b\u4f18\u70b9\u200b\u662f\u200b\u53ef\u4ee5\u200b\u5728\u200b\u8f83\u200b\u5c11\u200b\u7684\u200b\u8fed\u4ee3\u200b\u6b21\u6570\u200b\u5185\u200b\u627e\u5230\u200b\u5168\u5c40\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5728\u200b\u5b9e\u9645\u200b\u5e94\u7528\u200b\u4e2d\u200b\u88ab\u200b\u5e7f\u6cdb\u200b\u4f7f\u7528\u200b\u3002</p> <p>\u200b\u5047\u8bbe\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7684\u200b\u5f62\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p>"},{"location":"coding/machine-learning/decision-tree/","title":"\u51b3\u7b56\u6811","text":"In\u00a0[30]: Copied! <pre>import numpy as np\nimport pandas as pd\nimport typing as T\n</pre> import numpy as np import pandas as pd import typing as T <p>\u200b\u4ee5\u4e0b\u200b\u8003\u8651\u200b\u897f\u74dc\u200b\u6570\u636e\u200b\u96c6\u200b3.0\u200b\u4e2d\u200b\u7b2c\u4e00\u6b65\u200b\u5212\u5206\u200b\u7684\u200b\u5c5e\u6027\u200b\u9009\u62e9\u200b\u3002</p> In\u00a0[31]: Copied! <pre>df = pd.read_csv('data/melon3.0a.csv')\nattributes = ['\u200b\u8272\u6cfd\u200b', '\u200b\u6839\u8482\u200b', '\u200b\u6572\u58f0\u200b', '\u200b\u7eb9\u7406\u200b', '\u200b\u8110\u90e8\u200b', '\u200b\u89e6\u611f\u200b']\ncategory = '\u200b\u597d\u74dc\u200b'\n</pre> df = pd.read_csv('data/melon3.0a.csv') attributes = ['\u200b\u8272\u6cfd\u200b', '\u200b\u6839\u8482\u200b', '\u200b\u6572\u58f0\u200b', '\u200b\u7eb9\u7406\u200b', '\u200b\u8110\u90e8\u200b', '\u200b\u89e6\u611f\u200b'] category = '\u200b\u597d\u74dc\u200b' <p>\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b\u57fa\u4e8e\u200b\u4fe1\u606f\u71b5\u200b\uff0c\u200b\u5728\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u5212\u5206\u200b\u540e\u200b\uff0c\u200b\u96c6\u5408\u200b\u7684\u200b\u4fe1\u606f\u71b5\u200b\u4e4b\u200b\u548c\u200b\uff08\u200b\u6309\u200b\u6bd4\u4f8b\u200b\u52a0\u6743\u200b\uff09\u200b\u4f1a\u200b\u51cf\u5c0f\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b\u5373\u200b\u4e3a\u200b\u539f\u200b\u4fe1\u606f\u71b5\u200b\u4e0e\u200b\u5212\u5206\u200b\u540e\u200b\u7684\u200b\u4fe1\u606f\u71b5\u200b\u4e4b\u5dee\u200b\u3002</p> In\u00a0[32]: Copied! <pre>def entropy(df, target_col):\n    p = df[target_col].value_counts() / len(df)\n    return -sum(p * np.log2(p))\n\ndef conditional_entropy(df, discrete_col, target_col):\n    ret = 0\n    for val in df[discrete_col].unique():\n        p = len(df[df[discrete_col] == val]) / len(df) # \u200b\u6bd4\u4f8b\u200b\n        ret += p * entropy(df[df[discrete_col] == val], target_col)\n    return ret\n\ndef calc_gain(df, attributes, category) -&gt; T.Dict[str, float]:\n    entropy_original = entropy(df, category)\n    return {\n        k: entropy_original - conditional_entropy(df, k, category)\n        for k in attributes\n    }\n\ngain = pd.DataFrame([calc_gain(df, attributes, category)], index=['\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b'])\ngain\n</pre> def entropy(df, target_col):     p = df[target_col].value_counts() / len(df)     return -sum(p * np.log2(p))  def conditional_entropy(df, discrete_col, target_col):     ret = 0     for val in df[discrete_col].unique():         p = len(df[df[discrete_col] == val]) / len(df) # \u200b\u6bd4\u4f8b\u200b         ret += p * entropy(df[df[discrete_col] == val], target_col)     return ret  def calc_gain(df, attributes, category) -&gt; T.Dict[str, float]:     entropy_original = entropy(df, category)     return {         k: entropy_original - conditional_entropy(df, k, category)         for k in attributes     }  gain = pd.DataFrame([calc_gain(df, attributes, category)], index=['\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b']) gain Out[32]: \u200b\u8272\u6cfd\u200b \u200b\u6839\u8482\u200b \u200b\u6572\u58f0\u200b \u200b\u7eb9\u7406\u200b \u200b\u8110\u90e8\u200b \u200b\u89e6\u611f\u200b \u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b 0.108125 0.142675 0.140781 0.380592 0.289159 0.006046 <p>\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b\u4f1a\u200b\u66f4\u52a0\u200b\u503e\u5411\u200b\u4e8e\u200b\u9009\u62e9\u200b\u5305\u542b\u200b\u7c7b\u522b\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u5c5e\u6027\u200b\uff0c\u200b\u4f7f\u7528\u200b\u589e\u76ca\u200b\u7387\u200b\u6307\u6807\u200b\u53ef\u4ee5\u200b\u5e73\u8861\u200b\u4e0d\u540c\u200b\u7c7b\u522b\u200b\u4e4b\u95f4\u200b\u7684\u200b\u6743\u91cd\u200b\u3002\u200b\u589e\u76ca\u200b\u7387\u200b\u6307\u6807\u200b\u4e3a\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b\u4e0e\u200b\u5c5e\u6027\u200b\u56fa\u6709\u200b\u503c\u200b $\\mathrm{IV}$ \u200b\u4e4b\u200b\u6bd4\u200b\u3002</p> In\u00a0[33]: Copied! <pre>def iv(df, discrete_col):\n    p = df[discrete_col].value_counts() / len(df)\n    return -sum(p * np.log2(p))\n\ndef calc_gain_ratio(df, attributes, category) -&gt; T.Dict[str, float]:\n    entropy_original = entropy(df, category)\n    return {k: (entropy_original - conditional_entropy(df, k, category)) / iv(df, k)\n        for k in attributes\n    }\n\ngain_ratio = pd.DataFrame(\n    [calc_gain_ratio(df, attributes, category)], index=['\u200b\u589e\u76ca\u200b\u7387\u200b']\n)\ngain_ratio\n</pre> def iv(df, discrete_col):     p = df[discrete_col].value_counts() / len(df)     return -sum(p * np.log2(p))  def calc_gain_ratio(df, attributes, category) -&gt; T.Dict[str, float]:     entropy_original = entropy(df, category)     return {k: (entropy_original - conditional_entropy(df, k, category)) / iv(df, k)         for k in attributes     }  gain_ratio = pd.DataFrame(     [calc_gain_ratio(df, attributes, category)], index=['\u200b\u589e\u76ca\u200b\u7387\u200b'] ) gain_ratio Out[33]: \u200b\u8272\u6cfd\u200b \u200b\u6839\u8482\u200b \u200b\u6572\u58f0\u200b \u200b\u7eb9\u7406\u200b \u200b\u8110\u90e8\u200b \u200b\u89e6\u611f\u200b \u200b\u589e\u76ca\u200b\u7387\u200b 0.06844 0.101759 0.105627 0.263085 0.186727 0.006918 <p>\u200b\u57fa\u5c3c\u200b\u6307\u6570\u200b\u8861\u91cf\u200b\u4ece\u200b\u6570\u636e\u200b\u96c6\u4e2d\u200b\u968f\u673a\u200b\u62bd\u53d6\u200b\u4e24\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u6807\u7b7e\u200b\u53d6\u503c\u200b\u4e0d\u540c\u200b\u7684\u200b\u6982\u7387\u200b\u3002</p> In\u00a0[34]: Copied! <pre>def gini(df, target_col):\n    p = df[target_col].value_counts() / len(df)\n    return 1 - sum(p ** 2)\n\ndef gini_conditional(df, discrete_col, target_col):\n    ret = 0\n    for col in df[discrete_col].unique():\n        p = len(df[df[discrete_col] == col]) / len(df)\n        ret += p * gini(df[df[discrete_col] == col], target_col)\n    return ret\n\ndef calc_gini_index(df, attributes, category) -&gt; T.Dict[str, float]:\n    return {\n        k: sum((df[k].value_counts() / len(df)) * gini_conditional(df, k, category))\n        for k in attributes\n    }\n\ngini_index = pd.DataFrame([calc_gini_index(df, attributes, category)], index=['\u200b\u57fa\u5c3c\u200b\u6307\u6570\u200b'])\ngini_index\n</pre> def gini(df, target_col):     p = df[target_col].value_counts() / len(df)     return 1 - sum(p ** 2)  def gini_conditional(df, discrete_col, target_col):     ret = 0     for col in df[discrete_col].unique():         p = len(df[df[discrete_col] == col]) / len(df)         ret += p * gini(df[df[discrete_col] == col], target_col)     return ret  def calc_gini_index(df, attributes, category) -&gt; T.Dict[str, float]:     return {         k: sum((df[k].value_counts() / len(df)) * gini_conditional(df, k, category))         for k in attributes     }  gini_index = pd.DataFrame([calc_gini_index(df, attributes, category)], index=['\u200b\u57fa\u5c3c\u200b\u6307\u6570\u200b']) gini_index Out[34]: \u200b\u8272\u6cfd\u200b \u200b\u6839\u8482\u200b \u200b\u6572\u58f0\u200b \u200b\u7eb9\u7406\u200b \u200b\u8110\u90e8\u200b \u200b\u89e6\u611f\u200b \u200b\u57fa\u5c3c\u200b\u6307\u6570\u200b 0.427451 0.422269 0.423529 0.277124 0.344538 0.494118 In\u00a0[35]: Copied! <pre>class TreeNode():\n\n    def __init__(self, key):\n        self.key = key\n        self.mapping = {}\n\n    def add_child(self, value, child):\n        self.mapping[value] = child\n\n    def predict(self, x):\n        if self.key not in x or x[self.key] not in self.mapping:\n            return None\n        if isinstance(self.mapping[x[self.key]], TreeNode):\n            return self.mapping[x[self.key]].predict(x)\n        return self.mapping[x[self.key]]\n\ndef build_tree(df, attributes, category):\n    if len(df[category].unique()) == 1:\n        return df[category].unique()[0]\n    if len(attributes) == 0:\n        return df[category].value_counts().index[0]\n    gain = calc_gain(df, attributes, category)\n    best_attr = max(gain, key=lambda _: gain[_])\n    for attr in attributes:\n        if entropy(df, category) - conditional_entropy(df, attr, category) &gt; entropy(df, category) - conditional_entropy(df, best_attr, category):\n            best_attr = attr\n    tree = TreeNode(best_attr)\n    for value in df[best_attr].unique():\n        tree.add_child(value, build_tree(df[df[best_attr] == value], [attr for attr in attributes if attr != best_attr], category))\n    return tree\n\ntree = build_tree(df, attributes, category)\n</pre> class TreeNode():      def __init__(self, key):         self.key = key         self.mapping = {}      def add_child(self, value, child):         self.mapping[value] = child      def predict(self, x):         if self.key not in x or x[self.key] not in self.mapping:             return None         if isinstance(self.mapping[x[self.key]], TreeNode):             return self.mapping[x[self.key]].predict(x)         return self.mapping[x[self.key]]  def build_tree(df, attributes, category):     if len(df[category].unique()) == 1:         return df[category].unique()[0]     if len(attributes) == 0:         return df[category].value_counts().index[0]     gain = calc_gain(df, attributes, category)     best_attr = max(gain, key=lambda _: gain[_])     for attr in attributes:         if entropy(df, category) - conditional_entropy(df, attr, category) &gt; entropy(df, category) - conditional_entropy(df, best_attr, category):             best_attr = attr     tree = TreeNode(best_attr)     for value in df[best_attr].unique():         tree.add_child(value, build_tree(df[df[best_attr] == value], [attr for attr in attributes if attr != best_attr], category))     return tree  tree = build_tree(df, attributes, category) <p>\u200b\u51b3\u7b56\u6811\u200b\u91c7\u53d6\u200b\u81ea\u9876\u5411\u4e0b\u200b\u7684\u200bpredict\u200b\u65b9\u5f0f\u200b</p> In\u00a0[36]: Copied! <pre>assert isinstance(tree, TreeNode)\ntree.predict({\n    '\u200b\u8272\u6cfd\u200b': '\u200b\u9752\u7eff\u200b',\n    '\u200b\u6839\u8482\u200b': '\u200b\u8737\u7f29\u200b',\n    '\u200b\u6572\u58f0\u200b': '\u200b\u6d4a\u54cd\u200b',\n    '\u200b\u7eb9\u7406\u200b': '\u200b\u6e05\u6670\u200b',\n    '\u200b\u8110\u90e8\u200b': '\u200b\u51f9\u9677\u200b',\n    '\u200b\u89e6\u611f\u200b': '\u200b\u786c\u6ed1\u200b'\n})\n</pre> assert isinstance(tree, TreeNode) tree.predict({     '\u200b\u8272\u6cfd\u200b': '\u200b\u9752\u7eff\u200b',     '\u200b\u6839\u8482\u200b': '\u200b\u8737\u7f29\u200b',     '\u200b\u6572\u58f0\u200b': '\u200b\u6d4a\u54cd\u200b',     '\u200b\u7eb9\u7406\u200b': '\u200b\u6e05\u6670\u200b',     '\u200b\u8110\u90e8\u200b': '\u200b\u51f9\u9677\u200b',     '\u200b\u89e6\u611f\u200b': '\u200b\u786c\u6ed1\u200b' }) Out[36]: <pre>'\u200b\u662f\u200b'</pre> In\u00a0[119]: Copied! <pre>from sklearn.model_selection import train_test_split\n\ndef no_split(train, test, attributes, category):\n    return train[category].value_counts().index[0]\n\ndef build_pre_prune(train, test, attributes, category, tree_root=None):\n    if len(train[category].unique()) == 1:\n        return train[category].unique()[0]\n    no_split_result = no_split(train, test, attributes, category)\n    no_split_metric = np.sum(test[category] == no_split_result)\n    if len(attributes) == 0:\n        return no_split_result\n\n    # \u200b\u9009\u62e9\u200b\u5212\u5206\u200b\u5c5e\u6027\u200b\n    gain = calc_gain(train, attributes, category)\n    best_attr = max(gain, key=lambda _: gain[_])\n\n    # \u200b\u521b\u5efa\u200b\u8282\u70b9\u200b\n    tree = TreeNode(best_attr)\n    if tree_root is None:\n        tree_root = tree\n    for value in train[best_attr].unique():\n        tree.add_child(value, no_split(\n            train[train[best_attr] == value], test,\n            [attr for attr in attributes if attr != best_attr], category\n        ))\n\n    # \u200b\u8ba1\u7b97\u200b\u526a\u679d\u200b\u6548\u679c\u200b\n    result = []\n    for row in test[attributes].itertuples():\n        row_dict = row._asdict()\n        result.append(tree_root.predict(row_dict))\n    split_metric = np.sum([a == b for a, b in zip(result, test[category])])\n\n    # \u200b\u526a\u679d\u200b\n    print(f'\u200b\u526a\u679d\u200b\u524d\u200b: {no_split_metric}, \u200b\u526a\u679d\u200b\u540e\u200b: {split_metric}')\n    if split_metric &lt; no_split_metric:\n        print('\u200b\u4e0d\u200b\u526a\u679d\u200b')\n        return no_split_result\n    print('\u200b\u526a\u679d\u200b')\n    for value in train[best_attr].unique():\n        child = build_pre_prune(\n            train[train[best_attr] == value], test,\n            [attr for attr in attributes if attr != best_attr], category,\n            tree_root\n        )\n        tree.add_child(value, child)\n    return tree\n\ntrain, test = train_test_split(df, test_size=0.5)\ntree = build_pre_prune(train, test, attributes, category)\n\nresult = [\n    tree.predict(row._asdict()) if isinstance(tree, TreeNode) else tree\n    for row in test[attributes].itertuples()\n]\nsplit_metric = np.mean([a == b for a, b in zip(result, test[category])])\nsplit_metric\n</pre> from sklearn.model_selection import train_test_split  def no_split(train, test, attributes, category):     return train[category].value_counts().index[0]  def build_pre_prune(train, test, attributes, category, tree_root=None):     if len(train[category].unique()) == 1:         return train[category].unique()[0]     no_split_result = no_split(train, test, attributes, category)     no_split_metric = np.sum(test[category] == no_split_result)     if len(attributes) == 0:         return no_split_result      # \u200b\u9009\u62e9\u200b\u5212\u5206\u200b\u5c5e\u6027\u200b     gain = calc_gain(train, attributes, category)     best_attr = max(gain, key=lambda _: gain[_])      # \u200b\u521b\u5efa\u200b\u8282\u70b9\u200b     tree = TreeNode(best_attr)     if tree_root is None:         tree_root = tree     for value in train[best_attr].unique():         tree.add_child(value, no_split(             train[train[best_attr] == value], test,             [attr for attr in attributes if attr != best_attr], category         ))      # \u200b\u8ba1\u7b97\u200b\u526a\u679d\u200b\u6548\u679c\u200b     result = []     for row in test[attributes].itertuples():         row_dict = row._asdict()         result.append(tree_root.predict(row_dict))     split_metric = np.sum([a == b for a, b in zip(result, test[category])])      # \u200b\u526a\u679d\u200b     print(f'\u200b\u526a\u679d\u200b\u524d\u200b: {no_split_metric}, \u200b\u526a\u679d\u200b\u540e\u200b: {split_metric}')     if split_metric &lt; no_split_metric:         print('\u200b\u4e0d\u200b\u526a\u679d\u200b')         return no_split_result     print('\u200b\u526a\u679d\u200b')     for value in train[best_attr].unique():         child = build_pre_prune(             train[train[best_attr] == value], test,             [attr for attr in attributes if attr != best_attr], category,             tree_root         )         tree.add_child(value, child)     return tree  train, test = train_test_split(df, test_size=0.5) tree = build_pre_prune(train, test, attributes, category)  result = [     tree.predict(row._asdict()) if isinstance(tree, TreeNode) else tree     for row in test[attributes].itertuples() ] split_metric = np.mean([a == b for a, b in zip(result, test[category])]) split_metric <pre>\u200b\u526a\u679d\u200b\u524d\u200b: 4, \u200b\u526a\u679d\u200b\u540e\u200b: 4\n\u200b\u526a\u679d\u200b\n\u200b\u526a\u679d\u200b\u524d\u200b: 5, \u200b\u526a\u679d\u200b\u540e\u200b: 0\n\u200b\u4e0d\u200b\u526a\u679d\u200b\n</pre> Out[119]: <pre>0.4444444444444444</pre> In\u00a0[120]: Copied! <pre>def build_post_prune(train, test, attributes, category):\n    tree = build_tree(train, attributes, category)\n    # TODO: apply post pruning\n</pre> def build_post_prune(train, test, attributes, category):     tree = build_tree(train, attributes, category)     # TODO: apply post pruning"},{"location":"coding/machine-learning/decision-tree/","title":"\u51b3\u7b56\u6811\u200b\u00b6","text":"<p>\u200b\u51b3\u7b56\u6811\u200b\u662f\u200b\u4e00\u7c7b\u200b\u57fa\u4e8e\u200b\u89c4\u5219\u200b\u7684\u200b\u673a\u5668\u200b\u5b66\u4e60\u200b\u7b97\u6cd5\u200b</p>"},{"location":"coding/machine-learning/decision-tree/","title":"\u5212\u5206\u200b\u6307\u6807\u200b\u00b6","text":"<p>\u200b\u51b3\u7b56\u6811\u200b\u7b97\u6cd5\u200b\u4e2d\u200b\u7684\u200b\u6838\u5fc3\u200b\u662f\u200b\u6bcf\u200b\u4e00\u6b65\u200b\u5212\u5206\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u5c5e\u6027\u200b\u7684\u200b\u9009\u62e9\u200b\uff0c\u200b\u6bcf\u200b\u4e00\u6b65\u200b\u5212\u5206\u200b\u9700\u8981\u200b\u53d6\u5f97\u200b\u6700\u5927\u200b\u7684\u200b\u5212\u5206\u200b\u6548\u679c\u200b\uff0c\u200b\u5373\u200b\u4ea7\u751f\u200b\u7684\u200b\u5b50\u96c6\u200b\u201c\u200b\u7eaf\u5ea6\u200b\u201d\u200b\u6700\u9ad8\u200b\u3002\u200b\u5e38\u89c1\u200b\u7684\u200b\u4e09\u79cd\u200b\u8861\u91cf\u200b\u96c6\u5408\u200b\u201c\u200b\u7eaf\u5ea6\u200b\u201d\u200b\u7684\u200b\u6307\u6807\u200b\u4e3a\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b\u3001\u200b\u589e\u76ca\u200b\u7387\u200b\u4e0e\u200b\u57fa\u5c3c\u200b\u6307\u6570\u200b\u3002</p>"},{"location":"coding/machine-learning/decision-tree/","title":"\u51b3\u7b56\u6811\u200b\u6784\u5efa\u200b\u00b6","text":"<p>\u200b\u672c\u200b\u8282\u200b\u5b9e\u73b0\u200b\u539f\u4e66\u56fe\u200b4.2\u200b\u7684\u200b\u51b3\u7b56\u6811\u200b\u5b66\u4e60\u200b\u7b97\u6cd5\u200b\u3002</p>"},{"location":"coding/machine-learning/decision-tree/","title":"\u51b3\u7b56\u6811\u200b\u526a\u679d\u200b\u5904\u7406\u200b\u00b6","text":"<p>\u200b\u51b3\u7b56\u6811\u200b\u6709\u9884\u200b\u526a\u679d\u200b\u548c\u200b\u540e\u200b\u526a\u679d\u200b\u4e24\u4e2a\u200b\u526a\u679d\u200b\u6b65\u9aa4\u200b\uff0c\u200b\u526a\u679d\u200b\u53ef\u4ee5\u200b\u7f13\u89e3\u200b\u6a21\u578b\u200b\u51fa\u73b0\u200b\u7684\u200b\u8fc7\u200b\u62df\u5408\u200b\u73b0\u8c61\u200b\u3002</p>"},{"location":"coding/machine-learning/decision-tree/","title":"\u9884\u200b\u526a\u679d\u200b\u00b6","text":"<p>\u200b\u9884\u200b\u526a\u679d\u200b\u5728\u200b\u6a21\u578b\u200b\u7684\u200b\u8bad\u7ec3\u200b\u9636\u6bb5\u200b\u5c31\u200b\u901a\u8fc7\u200b\u9a8c\u8bc1\u200b\u96c6\u200b\u7684\u200b\u51c6\u786e\u6027\u200b\u6765\u200b\u51b3\u5b9a\u200b\u662f\u5426\u200b\u8fdb\u884c\u200b\u5212\u5206\u200b\u3002</p>"},{"location":"coding/machine-learning/decision-tree/","title":"\u540e\u200b\u526a\u679d\u200b\u00b6","text":"<p>\u200b\u540e\u200b\u526a\u679d\u200b\u5728\u200b\u51b3\u7b56\u6811\u200b\u751f\u6210\u200b\u540e\u200b\u518d\u200b\u904d\u5386\u200b\u51b3\u7b56\u6811\u200b\uff0c\u200b\u5220\u53bb\u200b\u8fc7\u591a\u200b\u7684\u200b\u8282\u70b9\u200b</p>"},{"location":"coding/machine-learning/gaussian-process/","title":"\u9ad8\u65af\u200b\u8fc7\u7a0b","text":"In\u00a0[1]: Copied! <pre>from scipy.stats import norm\nfrom matplotlib import pyplot as plt\nimport numpy as np\n\n%config InlineBackend.figure_format = 'svg'\n</pre> from scipy.stats import norm from matplotlib import pyplot as plt import numpy as np  %config InlineBackend.figure_format = 'svg' <p>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\uff08Gaussian Process\uff0cGP\uff09\u200b\u662f\u200b\u5c06\u200b\u591a\u200b\u53d8\u91cf\u200b\u9ad8\u65af\u5206\u5e03\u200b\u63a8\u5e7f\u200b\u5230\u200b\u65e0\u9650\u200b\u7ef4\u5ea6\u200b$\\mathcal X\\subseteq \\mathbb R^n$\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b\u3002\u200b\u5177\u4f53\u200b\u5730\u200b\uff0c\u200b\u5bf9\u4e8e\u200b$X = \\{x_1, \\ldots, x_n\\}\\subseteq \\mathcal X$\uff0c\u200b\u968f\u673a\u53d8\u91cf\u200b$f(x_1), \\ldots, f(x_n)$\u200b\u670d\u4ece\u200b\u591a\u5143\u200b\u9ad8\u65af\u5206\u5e03\u200b$\\mathcal N(\\mu(X), \\Sigma(X))$\u3002\u200b\u5176\u4e2d\u200b\uff0c$\\mu(X)$\u200b\u4e3a\u200b\u5747\u503c\u200b\u51fd\u6570\u200b\uff0c$\\Sigma(X)$\u200b\u4e3a\u200b\u534f\u65b9\u5dee\u200b\u51fd\u6570\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u9ad8\u65af\u5206\u5e03\u200b\u7814\u7a76\u200b\u7684\u200b\u662f\u200b\u51fd\u6570\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b\u3002</p> <p>\u200b\u591a\u5143\u200b\u9ad8\u65af\u5206\u5e03\u200b\u7684\u200b\u6761\u4ef6\u200b\u5206\u5e03\u200b\u4f9d\u7136\u200b\u662f\u200b\u9ad8\u65af\u5206\u5e03\u200b\u3002\u200b\u8bbe\u200b\u968f\u673a\u53d8\u91cf\u200b$X = (X_1, X_2)$\u200b\u670d\u4ece\u200b\u591a\u5143\u200b\u9ad8\u65af\u5206\u5e03\u200b$\\mathcal N(\\mu, \\Sigma)$\uff0c\u200b\u5176\u4e2d\u200b$\\mu = (\\mu_1, \\mu_2)$\uff0c$\\Sigma = \\begin{bmatrix} \\Sigma_{11} &amp; \\Sigma_{12} \\\\ \\Sigma_{21} &amp; \\Sigma_{22} \\end{bmatrix}$\uff0c\u200b\u5219\u200b\u7ed9\u5b9a\u200b$X_1 = x_1$\u200b\u540e\u200b\uff0c$X_2$\u200b\u7684\u200b\u6761\u4ef6\u200b\u5206\u5e03\u200b\u4e3a\u200b\uff1a</p> <p>$$ (X_2\\mid X_1 = x_1) \\sim \\mathcal N(\\mu_2 + \\Sigma_{21}\\Sigma_{11}^{-1}(x_1 - \\mu_1), \\Sigma_{22} - \\Sigma_{21}\\Sigma_{11}^{-1}\\Sigma_{12}) $$</p> <p>\u200b\u8bbe\u200b\u4e00\u7ec4\u200b\u89c2\u6d4b\u200b\u6570\u636e\u200b\u6570\u636e\u200b$\\boldsymbol X = x_1, \\ldots, x_n$\u200b\u53ca\u5176\u200b\u5bf9\u5e94\u200b\u7684\u200b\u51fd\u6570\u200b\u503c\u200b$\\boldsymbol Y = y_1, \\ldots, y_n$\u3002\u200b\u5bf9\u4e8e\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u6570\u636e\u200b\u70b9\u200b$x$\uff0c\u200b\u8981\u200b\u9884\u6d4b\u200b\u5176\u200b\u5bf9\u5e94\u200b\u7684\u200b\u51fd\u6570\u200b\u503c\u200b$y = f(x)$\u200b\u7684\u200b\u5206\u5e03\u200b\u3002\u200b\u5982\u679c\u200b\u6211\u4eec\u200b\u80fd\u200b\u8ba1\u7b97\u200b\u51fa\u200b\u89c2\u6d4b\u200b\u6570\u636e\u200b$\\boldsymbol Y$\u200b\u548c\u200b\u9884\u6d4b\u200b\u53d8\u91cf\u200b$y$\u200b\u4e4b\u95f4\u200b\u7684\u200b\u534f\u65b9\u5dee\u200b\uff0c\u200b\u4fbf\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u9ad8\u65af\u5206\u5e03\u200b\u7684\u200b\u6761\u4ef6\u200b\u5206\u5e03\u200b\u8ba1\u7b97\u200b\u5f97\u5230\u200b$f(x)$\u200b\u7684\u200b\u5747\u503c\u200b\u548c\u200b\u65b9\u5dee\u200b\u3002</p> <p>\u200b\u6b64\u5904\u200b\u4ee4\u200b$n = 1$\uff0c\u200b\u5373\u200b$X$\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\u6574\u4e2a\u200b\u5b9e\u6570\u200b\u57df\u200b\u3002</p> In\u00a0[2]: Copied! <pre>def y_posterior(x, x_obs, y_obs, kernel_func):\n    # x: (num_features)\n    # x_obs: (num_observations, num_features)\n    # y_obs: (num_observations)\n\n    sigma_21 = kernel_func(x, x_obs)  # dim: (num_observations)\n    sigma_22 = kernel_func(x, x)  # dim: scalar\n    # dim: (num_observations, num_observations)\n    sigma_11 = kernel_func(x_obs, x_obs)\n\n    inv_11 = np.linalg.inv(sigma_11 + 1e-8 * np.eye(sigma_11.shape[0]))\n    mu = np.einsum('i,ij,j-&gt;', sigma_21, inv_11, y_obs)\n    sigma = sigma_22 - np.einsum('j,jk,k-&gt;', sigma_21, inv_11, sigma_21)\n    return mu, sigma\n\ndef generate_samples(num_samples=5, target_function=None, sigma=1):\n    X = np.random.uniform(-5, 5, num_samples).reshape(-1, 1)\n    Y = np.random.normal(0, sigma, num_samples)\n    if target_function is not None:\n        Y += target_function(X)\n    return X, Y\n\ndef plot_posterior(kernel, num_samples=5, X_obs=None, Y_obs=None, ax=None, grid_size=200):\n    if X_obs is None != Y_obs is None:\n        raise ValueError('X_obs must be provided if Y_obs is provided.')\n    if (X_obs is None or Y_obs is None) and num_samples is None:\n        raise ValueError('num_samples must be provided if X_obs is provided.')\n    if X_obs is None:\n        assert Y_obs is None\n        X_obs, Y_obs = generate_samples(num_samples)\n    assert X_obs is not None and Y_obs is not None\n    if X_obs.shape[0] != Y_obs.shape[0]:\n        raise ValueError('The number of observations must be the same.')\n\n    X_min, X_max = np.min(X_obs), np.max(X_obs)\n    alpha_95 = norm.ppf(0.975)\n\n    if ax is None:\n        fig, ax = plt.subplots()\n    else:\n        fig = None\n\n    X = np.linspace(X_min, X_max, grid_size)\n    y = [y_posterior(x, X_obs, Y_obs, kernel) for x in X.reshape(-1, 1)]\n    ax.plot(X, [y[0] for y in y], linewidth=1, label='Function')\n    ax.scatter(X_obs, Y_obs, s=20, label='Observations')\n    ax.fill_between(X,\n        [y[0] - np.sqrt(y[1]) * alpha_95 for y in y],\n        [y[0] + np.sqrt(y[1]) * alpha_95 for y in y],\n        alpha=0.5, label='95% CI'\n    )\n\n    ax.set_xlabel('x')\n    ax.set_xlabel('y')\n    ax.legend()\n    if fig is not None:\n        fig.show()\n</pre> def y_posterior(x, x_obs, y_obs, kernel_func):     # x: (num_features)     # x_obs: (num_observations, num_features)     # y_obs: (num_observations)      sigma_21 = kernel_func(x, x_obs)  # dim: (num_observations)     sigma_22 = kernel_func(x, x)  # dim: scalar     # dim: (num_observations, num_observations)     sigma_11 = kernel_func(x_obs, x_obs)      inv_11 = np.linalg.inv(sigma_11 + 1e-8 * np.eye(sigma_11.shape[0]))     mu = np.einsum('i,ij,j-&gt;', sigma_21, inv_11, y_obs)     sigma = sigma_22 - np.einsum('j,jk,k-&gt;', sigma_21, inv_11, sigma_21)     return mu, sigma  def generate_samples(num_samples=5, target_function=None, sigma=1):     X = np.random.uniform(-5, 5, num_samples).reshape(-1, 1)     Y = np.random.normal(0, sigma, num_samples)     if target_function is not None:         Y += target_function(X)     return X, Y  def plot_posterior(kernel, num_samples=5, X_obs=None, Y_obs=None, ax=None, grid_size=200):     if X_obs is None != Y_obs is None:         raise ValueError('X_obs must be provided if Y_obs is provided.')     if (X_obs is None or Y_obs is None) and num_samples is None:         raise ValueError('num_samples must be provided if X_obs is provided.')     if X_obs is None:         assert Y_obs is None         X_obs, Y_obs = generate_samples(num_samples)     assert X_obs is not None and Y_obs is not None     if X_obs.shape[0] != Y_obs.shape[0]:         raise ValueError('The number of observations must be the same.')      X_min, X_max = np.min(X_obs), np.max(X_obs)     alpha_95 = norm.ppf(0.975)      if ax is None:         fig, ax = plt.subplots()     else:         fig = None      X = np.linspace(X_min, X_max, grid_size)     y = [y_posterior(x, X_obs, Y_obs, kernel) for x in X.reshape(-1, 1)]     ax.plot(X, [y[0] for y in y], linewidth=1, label='Function')     ax.scatter(X_obs, Y_obs, s=20, label='Observations')     ax.fill_between(X,         [y[0] - np.sqrt(y[1]) * alpha_95 for y in y],         [y[0] + np.sqrt(y[1]) * alpha_95 for y in y],         alpha=0.5, label='95% CI'     )      ax.set_xlabel('x')     ax.set_xlabel('y')     ax.legend()     if fig is not None:         fig.show() <p>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u5047\u8bbe\u200b\u66f4\u200b\u76f8\u4f3c\u200b\u7684\u200b$x$\u200b\u6709\u200b\u66f4\u200b\u76f8\u4f3c\u200b\u7684\u200b$y$\uff0c\u200b\u5373\u200b$\\text{Cov}(f(x), f(x')) = d_{x, x'}$\u3002\u200b\u7528\u4e8e\u200b\u63cf\u8ff0\u200b$x$\u200b\u4e4b\u95f4\u200b\u76f8\u4f3c\u200b\u5ea6\u200b\u7684\u200b\u51fd\u6570\u200b\u79f0\u4e3a\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a$d_{x, x'} = k(x, x')$\u3002\u200b\u6838\u200b\u51fd\u6570\u200b\u7684\u200b\u6570\u503c\u200b\u8d8a\u5927\u200b\uff0c\u200b\u8bf4\u660e\u200b$x$\u200b\u548c\u200b$x'$\u200b\u8d8a\u200b\u76f8\u4f3c\u200b\uff0c\u200b\u5426\u5219\u200b\u8d8a\u200b\u4e0d\u200b\u76f8\u4f3c\u200b\u3002</p> In\u00a0[3]: Copied! <pre>import functools\n\ndef kernel_wrapper(*args, **kwargs):\n    if args:\n        _kernel = args[0]\n        if kwargs:\n            _kernel = functools.partial(_kernel, **kwargs)\n\n        @functools.wraps(_kernel)\n        def new_kernel(a, b):\n            if a.shape[-1] != b.shape[-1]:\n                raise ValueError('The last dimension of a and b must be the same.')\n\n            num_a = 1 if a.ndim == 1 else a.shape[0]\n            num_b = 1 if b.ndim == 1 else b.shape[0]\n            num_hidden = a.shape[-1]\n            target_shape = (num_a, num_b, num_hidden)\n\n            x_a = np.broadcast_to(a.reshape((num_a, 1, num_hidden)), target_shape)\n            x_b = np.broadcast_to(b.reshape((1, num_b, num_hidden)), target_shape)\n\n            result = _kernel(x_a, x_b)\n            result_shape = [\n                *([] if a.ndim == 1 else [num_a]),\n                *([] if b.ndim == 1 else [num_b])\n            ]\n            result = result.reshape(result_shape)\n            return result\n\n        return new_kernel\n    else:\n        return functools.partial(kernel_wrapper, **kwargs)\n\n# Use the following method to define a kernel function\n# @kernel_wrapper - for kernels with no hyperparameters\n# @kernel_wrapper(hyperparameter=value) - for kernels with hyperparameters\n</pre> import functools  def kernel_wrapper(*args, **kwargs):     if args:         _kernel = args[0]         if kwargs:             _kernel = functools.partial(_kernel, **kwargs)          @functools.wraps(_kernel)         def new_kernel(a, b):             if a.shape[-1] != b.shape[-1]:                 raise ValueError('The last dimension of a and b must be the same.')              num_a = 1 if a.ndim == 1 else a.shape[0]             num_b = 1 if b.ndim == 1 else b.shape[0]             num_hidden = a.shape[-1]             target_shape = (num_a, num_b, num_hidden)              x_a = np.broadcast_to(a.reshape((num_a, 1, num_hidden)), target_shape)             x_b = np.broadcast_to(b.reshape((1, num_b, num_hidden)), target_shape)              result = _kernel(x_a, x_b)             result_shape = [                 *([] if a.ndim == 1 else [num_a]),                 *([] if b.ndim == 1 else [num_b])             ]             result = result.reshape(result_shape)             return result          return new_kernel     else:         return functools.partial(kernel_wrapper, **kwargs)  # Use the following method to define a kernel function # @kernel_wrapper - for kernels with no hyperparameters # @kernel_wrapper(hyperparameter=value) - for kernels with hyperparameters <p>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u5e38\u7528\u200b\u7684\u200b\u6838\u200b\u51fd\u6570\u200b\u6709\u200b\uff1a</p> <ul> <li>\u200b\u7ebf\u6027\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a$k(x, x'; v) = vx^Tx'$\uff0c\u200b\u5176\u4e2d\u200b$v$\u200b\u4e3a\u200b\u8d85\u200b\u53c2\u6570\u200b\u3002</li> </ul> In\u00a0[4]: Copied! <pre>def linear_kernel(a, b, nu=1):\n    # Input: a, b: (num_a, num_b, num_hidden)\n    return np.einsum('ijk,ijk-&gt;ij', a, b) * nu\n\nplot_posterior(kernel_wrapper(nu=1)(linear_kernel), grid_size=10)\n</pre> def linear_kernel(a, b, nu=1):     # Input: a, b: (num_a, num_b, num_hidden)     return np.einsum('ijk,ijk-&gt;ij', a, b) * nu  plot_posterior(kernel_wrapper(nu=1)(linear_kernel), grid_size=10) <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:48: RuntimeWarning: invalid value encountered in sqrt\n  [y[0] - np.sqrt(y[1]) * alpha_95 for y in y],\n/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:49: RuntimeWarning: invalid value encountered in sqrt\n  [y[0] + np.sqrt(y[1]) * alpha_95 for y in y],\n/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <ul> <li>\u200b\u5e73\u65b9\u200b\u6307\u6570\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a$k(x, x'; \\sigma, l) = \\sigma^2\\exp\\left(-\\frac{\\|x - x'\\|^2}{2l^2}\\right)$\uff0c$\\sigma$\u200b\u4e3a\u200b\u5e45\u5ea6\u200b\u53c2\u6570\u200b\uff0c\u200b\u63a7\u5236\u200b\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u968f\u673a\u200b\u6ce2\u52a8\u200b\u8303\u56f4\u200b\uff1b$l$\u200b\u4e3a\u200b\u957f\u5ea6\u200b\u53c2\u6570\u200b\uff0c\u200b\u63a7\u5236\u200b\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u968f\u673a\u200b\u6ce2\u52a8\u200b\u9891\u7387\u200b\u3002</li> </ul> In\u00a0[5]: Copied! <pre>def rbf_kernel(a, b, sigma, l):\n    # Input: a, b: (num_a, num_b, num_hidden)\n    return sigma ** 2 * np.exp(\n        -0.5 * np.linalg.norm(a - b, axis=-1) ** 2 / l ** 2\n    )\n\nplot_posterior(kernel_wrapper(sigma=1, l=1)(rbf_kernel), grid_size=80)\n</pre> def rbf_kernel(a, b, sigma, l):     # Input: a, b: (num_a, num_b, num_hidden)     return sigma ** 2 * np.exp(         -0.5 * np.linalg.norm(a - b, axis=-1) ** 2 / l ** 2     )  plot_posterior(kernel_wrapper(sigma=1, l=1)(rbf_kernel), grid_size=80) <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <ul> <li>Matern\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a$k(x, x'; \\sigma, l, \\nu) = \\frac{2^{1-\\nu}}{\\Gamma(\\nu)}\\left(\\frac{\\sqrt{2\\nu}\\|x - x'\\|}{l}\\right)^\\nu K_\\nu\\left(\\frac{\\sqrt{2\\nu}\\|x - x'\\|}{l}\\right)$\uff0c\u200b\u5176\u4e2d\u200b$\\nu$\u200b\u4e3a\u200b\u8d85\u200b\u53c2\u6570\u200b\uff0c\u200b\u7528\u4e8e\u200b\u63a7\u5236\u200b\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u5149\u6ed1\u5ea6\u200b\uff1b$K_\\nu$\u200b\u4e3a\u200b\u4fee\u6b63\u200bBessel\u200b\u51fd\u6570\u200b\u3002</li> </ul> In\u00a0[6]: Copied! <pre>from scipy.special import kn, gamma\n\ndef matern_kernel(a, b, l, nu):\n    # Input: a, b: (num_a, num_b, num_hidden)\n    x = (np.sqrt(2 * nu) * np.linalg.norm(a - b, axis=-1) / l)\n    y = 2 ** (1 - nu) / gamma(nu) * x ** nu * kn(nu, x)\n    # Replace inf with 1\n    y = np.where(x &lt; 1e-6, 1, y)\n    return y\n\nplot_posterior(kernel_wrapper(l=1, nu=2)(matern_kernel), 4, grid_size=80)\n</pre> from scipy.special import kn, gamma  def matern_kernel(a, b, l, nu):     # Input: a, b: (num_a, num_b, num_hidden)     x = (np.sqrt(2 * nu) * np.linalg.norm(a - b, axis=-1) / l)     y = 2 ** (1 - nu) / gamma(nu) * x ** nu * kn(nu, x)     # Replace inf with 1     y = np.where(x &lt; 1e-6, 1, y)     return y  plot_posterior(kernel_wrapper(l=1, nu=2)(matern_kernel), 4, grid_size=80) <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/497016935.py:6: RuntimeWarning: invalid value encountered in multiply\n  y = 2 ** (1 - nu) / gamma(nu) * x ** nu * kn(nu, x)\n/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <ul> <li>\u200b\u5468\u671f\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a$k(x, x'; \\sigma, l, p) = \\sigma^2\\exp\\left(-\\frac{2\\sin^2(\\pi\\|x - x'\\|/p)}{l^2}\\right)$\uff0c\u200b\u5176\u4e2d\u200b$p$\u200b\u4e3a\u200b\u5468\u671f\u200b\u53c2\u6570\u200b\uff0c$l$\u200b\u4e3a\u200b\u957f\u5ea6\u200b\u53c2\u6570\u200b\uff0c$\\sigma$\u200b\u4e3a\u200b\u5e45\u5ea6\u200b\u53c2\u6570\u200b\u3002</li> </ul> In\u00a0[7]: Copied! <pre>def periodic_kernel(a, b, sigma, l, p):\n    # Input: a, b: (num_a, num_b, num_hidden)\n    return sigma ** 2 * np.exp(\n        -2 * np.sin(np.pi * np.linalg.norm(a - b, axis=-1) / p) ** 2 / l ** 2\n    )\n\nplot_posterior(kernel_wrapper(sigma=1, l=1, p=2)(periodic_kernel), 4, grid_size=150)\n</pre> def periodic_kernel(a, b, sigma, l, p):     # Input: a, b: (num_a, num_b, num_hidden)     return sigma ** 2 * np.exp(         -2 * np.sin(np.pi * np.linalg.norm(a - b, axis=-1) / p) ** 2 / l ** 2     )  plot_posterior(kernel_wrapper(sigma=1, l=1, p=2)(periodic_kernel), 4, grid_size=150) <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <ul> <li><p>\u200b\u566a\u58f0\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a$k(x, x'; \\sigma, l) = \\sigma^2\\delta_{x, x'}$\uff0c\u200b\u5176\u4e2d\u200b$\\delta_{x, x'}$\u200b\u4e3a\u200bKronecker delta\u200b\u51fd\u6570\u200b\uff0c\u200b\u7528\u4e8e\u200b\u6355\u83b7\u200b\u89c2\u6d4b\u200b\u6570\u636e\u200b\u7684\u200b\u566a\u58f0\u200b\u3002</p> <p>$$   \\delta(x, x') = \\begin{cases}   1, &amp; x = x' \\\\   0, &amp; x \\neq x'   \\end{cases}   $$</p> </li> </ul> In\u00a0[8]: Copied! <pre>def noise_kernel(a, b, sigma):\n    return sigma ** 2 * np.all((a == b), axis=-1)\n\nplot_posterior(kernel_wrapper(sigma=1)(noise_kernel), 5)\n</pre> def noise_kernel(a, b, sigma):     return sigma ** 2 * np.all((a == b), axis=-1)  plot_posterior(kernel_wrapper(sigma=1)(noise_kernel), 5) <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <p>\u200b\u5728\u200b\u5b9e\u9645\u200b\u5e94\u7528\u200b\u4e2d\u200b\uff0c\u200b\u9700\u8981\u200b\u6839\u636e\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u5df2\u77e5\u200b\u7684\u200b\u7279\u6027\u200b\uff0c\u200b\u8bbe\u8ba1\u200b\u5408\u9002\u200b\u7684\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a\u200b\u5982\u679c\u200b\u51fd\u6570\u200b\u4e2d\u200b\u5305\u542b\u200b\u5468\u671f\u6027\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5468\u671f\u200b\u6838\u200b\u51fd\u6570\u200b\uff1b\u200b\u5982\u679c\u200b\u51fd\u6570\u200b\u503c\u200b\u5728\u200b\u76f8\u4f3c\u200b\u7684\u200b$x$\u200b\u4e4b\u95f4\u200b\u53d8\u5316\u200b\u8f83\u200b\u5c0f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5e73\u65b9\u200b\u6307\u6570\u200b\u6838\u200b\u51fd\u6570\u200b\u3002</p> <p>\u200b\u5982\u200b\u5bf9\u4e8e\u200b\u51fd\u6570\u200b</p> <p>$$ f(x) = \\sin(\\pi x) + 0.3 x^2 + \\varepsilon $$</p> <p>\u200b\u9700\u8981\u200b\u540c\u65f6\u200b\u5f15\u5165\u200b\u5468\u671f\u200b\u6838\u200b\u51fd\u6570\u200b\u548c\u200b\u5e73\u65b9\u200b\u6307\u6570\u200b\u6838\u200b\u51fd\u6570\u200b\uff0c\u200b\u624d\u80fd\u200b\u8f83\u200b\u597d\u200b\u5730\u200b\u9884\u6d4b\u200b\u51fd\u6570\u200b\u503c\u200b\u3002</p> In\u00a0[9]: Copied! <pre>def target_function(X):\n    x = X.reshape(-1)\n    return np.sin(x * 2 * 3.14 / 2) + x ** 2 * 0.3\n\n\ndef plot_function(func, ax, X_min=-5, X_max=5, grid_size=200, **kwargs):\n    X = np.linspace(X_min, X_max, grid_size)\n    Y = func(X.reshape(-1, 1))\n    ax.plot(X, Y, **kwargs)\n\nfig, ax = plt.subplots(figsize=(4, 4))\nplot_function(target_function, ax, grid_size=100)\nax.set_xlabel('x')\nax.set_ylabel('y')\nax.set_title('Target Function')\nfig.show()\n</pre> def target_function(X):     x = X.reshape(-1)     return np.sin(x * 2 * 3.14 / 2) + x ** 2 * 0.3   def plot_function(func, ax, X_min=-5, X_max=5, grid_size=200, **kwargs):     X = np.linspace(X_min, X_max, grid_size)     Y = func(X.reshape(-1, 1))     ax.plot(X, Y, **kwargs)  fig, ax = plt.subplots(figsize=(4, 4)) plot_function(target_function, ax, grid_size=100) ax.set_xlabel('x') ax.set_ylabel('y') ax.set_title('Target Function') fig.show() <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/4237481582.py:16: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> In\u00a0[10]: Copied! <pre>periodic = kernel_wrapper(sigma=1, l=1, p=2)(periodic_kernel)\nrbf = kernel_wrapper(sigma=1, l=1)(rbf_kernel)\nnoise = kernel_wrapper(sigma=0.1)(noise_kernel)\n\ndef added(a, b):\n    return periodic(a, b) + rbf(a, b) + noise(a, b)\n\nkernels = {\n    'Periodic': periodic,\n    'RBF': rbf,\n    'Hybrid': added\n}\n\nX_obs, Y_obs = generate_samples(10, target_function, sigma=0.5)\nX_min, X_max = np.min(X_obs), np.max(X_obs)\n\nfig, axes = plt.subplots(1, 3, figsize=(10, 3))\nfor ax, (name, kernel) in zip(axes, kernels.items()):\n    plot_posterior(kernel, X_obs=X_obs, Y_obs=Y_obs, ax=ax, grid_size=150)\n    plot_function(target_function, ax, X_min, X_max, color='red', label='Target Function', grid_size=100)\n    ax.set_title(f'{name} Kernel')\n    ax.get_legend().remove()\nfig.show()\n</pre> periodic = kernel_wrapper(sigma=1, l=1, p=2)(periodic_kernel) rbf = kernel_wrapper(sigma=1, l=1)(rbf_kernel) noise = kernel_wrapper(sigma=0.1)(noise_kernel)  def added(a, b):     return periodic(a, b) + rbf(a, b) + noise(a, b)  kernels = {     'Periodic': periodic,     'RBF': rbf,     'Hybrid': added }  X_obs, Y_obs = generate_samples(10, target_function, sigma=0.5) X_min, X_max = np.min(X_obs), np.max(X_obs)  fig, axes = plt.subplots(1, 3, figsize=(10, 3)) for ax, (name, kernel) in zip(axes, kernels.items()):     plot_posterior(kernel, X_obs=X_obs, Y_obs=Y_obs, ax=ax, grid_size=150)     plot_function(target_function, ax, X_min, X_max, color='red', label='Target Function', grid_size=100)     ax.set_title(f'{name} Kernel')     ax.get_legend().remove() fig.show() <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:48: RuntimeWarning: invalid value encountered in sqrt\n  [y[0] - np.sqrt(y[1]) * alpha_95 for y in y],\n/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:49: RuntimeWarning: invalid value encountered in sqrt\n  [y[0] + np.sqrt(y[1]) * alpha_95 for y in y],\n/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/1232705018.py:23: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <p>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u4e0d\u9002\u200b\u7528\u4e8e\u200b\u4ee5\u4e0b\u200b\u573a\u666f\u200b\uff1a</p> <ul> <li>\u200b\u6570\u636e\u91cf\u200b\u8f83\u5927\u200b\u65f6\u200b\uff0c\u200b\u7531\u4e8e\u200b\u8ba1\u7b97\u200b$m$\u200b\u7ef4\u200b\u77e9\u9635\u200b\u9006\u200b\u7684\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b$O(m^3)$\uff0c\u200b\u8ba1\u7b97\u200b\u590d\u6742\u5ea6\u200b\u8f83\u200b\u9ad8\u200b\u3002</li> <li>\u200b\u6570\u636e\u200b\u7ef4\u5ea6\u200b\u8f83\u200b\u9ad8\u65f6\u200b\uff0c\u200b\u7ef4\u5ea6\u200b\u707e\u96be\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u6838\u200b\u51fd\u6570\u200b\u5bb9\u6613\u200b\u9000\u5316\u200b\uff0c\u200b\u65e0\u6cd5\u200b\u6355\u83b7\u200b\u6570\u636e\u200b\u4e4b\u95f4\u200b\u7684\u200b\u76f8\u4f3c\u6027\u200b\u3002</li> <li>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u9884\u6d4b\u200b\u7ed3\u679c\u200b\u662f\u200b\u8fde\u7eed\u200b\u7684\u200b\uff0c\u200b\u65e0\u6cd5\u200b\u76f4\u63a5\u200b\u5904\u7406\u200b\u8fde\u7eed\u200b\u6570\u636e\u200b\u3002</li> </ul>"},{"location":"coding/machine-learning/gaussian-process/","title":"\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u00b6","text":""},{"location":"coding/machine-learning/linear-models/","title":"\u7ebf\u6027\u200b\u6a21\u578b","text":"In\u00a0[\u00a0]: Copied! <pre>import numpy as np\nimport pandas as pd\n\ndf = pd.read_csv('data/melon3.0a.csv')\nX_rcol, y_rcol = df.columns[1:-2], df.columns[-2] # columns used for regression\nX_ccol, y_ccol = df.columns[1:-1], df.columns[-1] # columns used for classification\nvalue_map = {\n    '\u200b\u8272\u6cfd\u200b': {'\u200b\u6d45\u767d\u200b': 0, '\u200b\u9752\u7eff\u200b': 1, '\u200b\u4e4c\u9ed1\u200b': 2},\n    '\u200b\u6839\u8482\u200b': {'\u200b\u8737\u7f29\u200b': 0, '\u200b\u7a0d\u200b\u8737\u200b': 1, '\u200b\u786c\u633a\u200b': 2},\n    '\u200b\u6572\u58f0\u200b': {'\u200b\u6c89\u95f7\u200b': 0, '\u200b\u6d4a\u54cd\u200b': 1, '\u200b\u6e05\u8106\u200b': 2},\n    '\u200b\u7eb9\u7406\u200b': {'\u200b\u6a21\u7cca\u200b': 0, '\u200b\u7a0d\u7cca\u200b': 1, '\u200b\u6e05\u6670\u200b': 2},\n    '\u200b\u8110\u90e8\u200b': {'\u200b\u51f9\u9677\u200b': 0, '\u200b\u7a0d\u200b\u51f9\u200b': 1, '\u200b\u5e73\u5766\u200b': 2},\n    '\u200b\u89e6\u611f\u200b': {'\u200b\u786c\u6ed1\u200b': 0, '\u200b\u8f6f\u200b\u7c98\u200b': 1},\n    '\u200b\u597d\u74dc\u200b': {'\u200b\u662f\u200b': 1, '\u200b\u5426\u200b': 0},\n}\nfor col in value_map:\n    df[col] = df[col].map(value_map[col])\n\nX = np.concatenate([df[X_rcol].values, np.ones((df.shape[0], 1))], axis=1)\ny = df[y_rcol].values.reshape(-1, 1)\n\ndef linear_regression(X, y):\n    return np.linalg.inv(X.T @ X) @ X.T @ y\n\nlinear_regression(X, y).T\n</pre> import numpy as np import pandas as pd  df = pd.read_csv('data/melon3.0a.csv') X_rcol, y_rcol = df.columns[1:-2], df.columns[-2] # columns used for regression X_ccol, y_ccol = df.columns[1:-1], df.columns[-1] # columns used for classification value_map = {     '\u200b\u8272\u6cfd\u200b': {'\u200b\u6d45\u767d\u200b': 0, '\u200b\u9752\u7eff\u200b': 1, '\u200b\u4e4c\u9ed1\u200b': 2},     '\u200b\u6839\u8482\u200b': {'\u200b\u8737\u7f29\u200b': 0, '\u200b\u7a0d\u200b\u8737\u200b': 1, '\u200b\u786c\u633a\u200b': 2},     '\u200b\u6572\u58f0\u200b': {'\u200b\u6c89\u95f7\u200b': 0, '\u200b\u6d4a\u54cd\u200b': 1, '\u200b\u6e05\u8106\u200b': 2},     '\u200b\u7eb9\u7406\u200b': {'\u200b\u6a21\u7cca\u200b': 0, '\u200b\u7a0d\u7cca\u200b': 1, '\u200b\u6e05\u6670\u200b': 2},     '\u200b\u8110\u90e8\u200b': {'\u200b\u51f9\u9677\u200b': 0, '\u200b\u7a0d\u200b\u51f9\u200b': 1, '\u200b\u5e73\u5766\u200b': 2},     '\u200b\u89e6\u611f\u200b': {'\u200b\u786c\u6ed1\u200b': 0, '\u200b\u8f6f\u200b\u7c98\u200b': 1},     '\u200b\u597d\u74dc\u200b': {'\u200b\u662f\u200b': 1, '\u200b\u5426\u200b': 0}, } for col in value_map:     df[col] = df[col].map(value_map[col])  X = np.concatenate([df[X_rcol].values, np.ones((df.shape[0], 1))], axis=1) y = df[y_rcol].values.reshape(-1, 1)  def linear_regression(X, y):     return np.linalg.inv(X.T @ X) @ X.T @ y  linear_regression(X, y).T <p>\u200b\u635f\u5931\u200b\u51fd\u6570\u200b\u4e3a\u5747\u65b9\u200b\u8bef\u5dee\u200b</p> In\u00a0[\u00a0]: Copied! <pre>def mse_loss(X, y, w):\n    return np.mean((X @ w - y) ** 2)\n\nmse_loss(X, y, linear_regression(X, y))\n</pre> def mse_loss(X, y, w):     return np.mean((X @ w - y) ** 2)  mse_loss(X, y, linear_regression(X, y)) <p>\u200b\u5f53\u200b\u6837\u672c\u200b\u77e9\u9635\u200b\u975e\u6ee1\u200b\u79e9\u65f6\u200b\uff0c\u200b\u5b58\u5728\u200b\u591a\u4e2a\u200b\u6ee1\u8db3\u200b\u8bad\u7ec3\u200b\u96c6\u200b\u7684\u200b\u6a21\u578b\u200b\uff0c\u200b\u6b64\u65f6\u200b\u53ef\u4ee5\u200b\u5728\u200b\u4f18\u5316\u200b\u76ee\u6807\u200b\u4e2d\u200b\u52a0\u5165\u200b\u6b63\u5219\u200b\u5316\u9879\u200b\uff0c\u200b\u5982\u200bL2-norm\u200b\u5373\u200b\u52a0\u5165\u200b\u6743\u91cd\u200b\u7684\u200b\u5e73\u65b9\u200b\u4e4b\u200b\u548c\u200b\u3002\u200b\u7136\u540e\u200b\u4f7f\u7528\u200b\u68af\u5ea6\u200b\u4e0b\u964d\u200b\u7b49\u200b\u6570\u503c\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u8ba1\u7b97\u200b\u3002</p> In\u00a0[\u00a0]: Copied! <pre>class SGD():\n    def __init__(self, d, lr=0.01, epochs=1000):\n        self.d = d\n        self.lr = lr\n        self.epochs = epochs\n\n    def __call__(self, X, y):\n        w = np.random.normal(0, 1, size=(X.shape[1], 1))\n        for _ in range(self.epochs):\n            w -= self.lr * self.d(X, y, w)\n        return w\n\noptim_l2 = SGD(lambda X, y, w: X.T @ (X @ w - y) + 0.1 * w)\n\nw = optim_l2(X, y)\nprint(w.T, mse_loss(X, y, w))\n</pre> class SGD():     def __init__(self, d, lr=0.01, epochs=1000):         self.d = d         self.lr = lr         self.epochs = epochs      def __call__(self, X, y):         w = np.random.normal(0, 1, size=(X.shape[1], 1))         for _ in range(self.epochs):             w -= self.lr * self.d(X, y, w)         return w  optim_l2 = SGD(lambda X, y, w: X.T @ (X @ w - y) + 0.1 * w)  w = optim_l2(X, y) print(w.T, mse_loss(X, y, w)) <p>\u200b\u5bf9\u7387\u200b\u56de\u5f52\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u7ebf\u6027\u200b\u6a21\u578b\u200b\u5e94\u7528\u200b\u5230\u200b\u4e8c\u200b\u5206\u7c7b\u200b\u95ee\u9898\u200b\u4e0a\u200b\u3002\u200b\u5bf9\u7387\u200b\u56de\u5f52\u200b\u9700\u8981\u200b\u7528\u5230\u200blogit\u200b\u51fd\u6570\u200b\u5c06\u200b\u8fde\u7eed\u200b\u7684\u200b\u56de\u5f52\u200b\u503c\u200b\u6620\u5c04\u200b\u5230\u200b $(0, 1)$ \u200b\u4e0a\u200b</p> <p>$$ f(x) = \\frac{1}{1 + e^{-x}} $$</p> In\u00a0[\u00a0]: Copied! <pre>X = df[X_ccol].values\ny = df[y_ccol].values.reshape(-1, 1)\n\ndef sigmoid(x):\n    return 1 / (1 + np.exp(-x))\n\ndef precision(X, y, w):\n    return np.mean((sigmoid(X @ w) &gt; 0.5) == y)\n</pre> X = df[X_ccol].values y = df[y_ccol].values.reshape(-1, 1)  def sigmoid(x):     return 1 / (1 + np.exp(-x))  def precision(X, y, w):     return np.mean((sigmoid(X @ w) &gt; 0.5) == y) <p>\u200b\u4f7f\u7528\u200b\u6781\u5927\u200b\u4f3c\u7136\u6cd5\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u5bf9\u7387\u200b\u56de\u5f52\u200b\u7684\u200b\u4f18\u5316\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b</p> <p>$$ l(\\boldsymbol w) = \\sum_{i=1}^m \\left(-\\boldsymbol y_i\\boldsymbol \\beta^\\top \\boldsymbol x_i + \\ln \\left(1 + e^{\\boldsymbol \\beta^\\top \\boldsymbol x_i}\\right) \\right) $$</p> In\u00a0[\u00a0]: Copied! <pre>optim_logit = SGD(lambda X, y, w: X.T @ (sigmoid(X @ w) - y))\n\nw = optim_logit(X, y)\nprint(w.T, precision(X, y, w))\n</pre> optim_logit = SGD(lambda X, y, w: X.T @ (sigmoid(X @ w) - y))  w = optim_logit(X, y) print(w.T, precision(X, y, w)) In\u00a0[\u00a0]: Copied! <pre>def cov(X, a, b):\n    return np.mean((X[:, a] - np.mean(X[:, a])) * (X[:, b] - np.mean(X[:, b])))\n\ndef cov_matrix(X):\n    return np.array([\n        [\n            cov(X, i, j)\n            for i in range(X.shape[1])\n        ]\n        for j in range(X.shape[1])\n    ])\n\nsw = cov_matrix(X[y[:, 0] == 0]) + cov_matrix(X[y[:, 0] == 1])\nmu0, mu1 = np.mean(X[y[:, 0] == 0], axis=0), np.mean(X[y[:, 0] == 1], axis=0)\nw = np.linalg.inv(sw) @ (mu0 - mu1).reshape(-1, 1)\nc0, c1 = w.T @ mu0, w.T @ mu1\n\ndef precision(X, y, w):\n    return np.mean((X @ w &lt; (c0 + c1) / 2) == y)\n\nprecision(X, y, w)\n</pre> def cov(X, a, b):     return np.mean((X[:, a] - np.mean(X[:, a])) * (X[:, b] - np.mean(X[:, b])))  def cov_matrix(X):     return np.array([         [             cov(X, i, j)             for i in range(X.shape[1])         ]         for j in range(X.shape[1])     ])  sw = cov_matrix(X[y[:, 0] == 0]) + cov_matrix(X[y[:, 0] == 1]) mu0, mu1 = np.mean(X[y[:, 0] == 0], axis=0), np.mean(X[y[:, 0] == 1], axis=0) w = np.linalg.inv(sw) @ (mu0 - mu1).reshape(-1, 1) c0, c1 = w.T @ mu0, w.T @ mu1  def precision(X, y, w):     return np.mean((X @ w &lt; (c0 + c1) / 2) == y)  precision(X, y, w)"},{"location":"coding/machine-learning/linear-models/","title":"\u7ebf\u6027\u200b\u6a21\u578b\u200b\u00b6","text":"<p>\u200b\u4ee5\u4e0b\u200b\u4f7f\u7528\u200bPython\u200b\u5b9e\u73b0\u200b\u57fa\u4e8e\u200b\u7ebf\u6027\u5173\u7cfb\u200b\u7684\u200b\u5404\u79cd\u200b\u673a\u5668\u200b\u5b66\u4e60\u200b\u6a21\u578b\u200b</p>"},{"location":"coding/machine-learning/linear-models/","title":"\u591a\u5143\u200b\u7ebf\u6027\u200b\u56de\u5f52\u200b\u00b6","text":"<p>\u200b\u591a\u5143\u200b\u7ebf\u6027\u200b\u56de\u5f52\u200b\u7684\u200b\u4e00\u822c\u200b\u5f62\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> <p>$$ \\hat{\\boldsymbol y} = f(\\boldsymbol x) = \\boldsymbol w^\\top \\boldsymbol x + b $$</p> <p>\u200b\u5f53\u200b\u6837\u672c\u200b\u77e9\u9635\u200b$\\boldsymbol X$\u200b\u6ee1\u200b\u79e9\u65f6\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u4e3a\u200b</p> <p>$$ \\newcommand{\\bmX}{\\boldsymbol X} \\hat{\\boldsymbol w}^* = (\\bmX^\\top\\bmX)^{-1}\\bmX^\\top \\boldsymbol y $$</p>"},{"location":"coding/machine-learning/linear-models/#lda","title":"LDA\u00b6","text":"<p>LDA\u200b\u662f\u200b\u7ebf\u6027\u200b\u5224\u522b\u5206\u6790\u200b\u7684\u200b\u7b80\u79f0\u200b\uff0c\u200b\u5c5e\u4e8e\u200b\u5206\u7c7b\u200b\u7b97\u6cd5\u200b\u3002\u200b\u8be5\u200b\u7b97\u6cd5\u200b\u7684\u200b\u6838\u5fc3\u200b\u601d\u8def\u200b\u4e3a\u200b\u5c06\u200b\u6837\u672c\u200b\u70b9\u200b\u6295\u5f71\u200b\u5230\u200b $n$ \u200b\u7ef4\u7a7a\u95f4\u200b\u7684\u200b\u5e73\u9762\u200b\u4e0a\u200b\uff0c\u200b\u901a\u8fc7\u200b\u9009\u62e9\u200b\u5e73\u9762\u200b\uff0c\u200b\u6700\u5c0f\u5316\u200b\u540c\u4e00\u200b\u7c7b\u522b\u200b\u5185\u200b\u6837\u672c\u200b\u70b9\u200b\u6295\u5f71\u200b\u7684\u200b\u8ddd\u79bb\u200b\uff0c\u200b\u540c\u65f6\u200b\u6700\u5927\u5316\u200b\u4e0d\u540c\u200b\u7c7b\u522b\u200b\u6837\u672c\u200b\u70b9\u200b\u6295\u5f71\u200b\u7684\u200b\u8ddd\u79bb\u200b\u3002</p>"},{"location":"coding/python/","title":"Python","text":"<ul> <li>\u200b\u5173\u95ed\u200bPython\u200b\u7ebf\u7a0b\u200b</li> <li>Python\u200b\u4e2d\u200b\u7684\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b</li> <li>\u200b\u5e8f\u5217\u200b\u7c7b\u578b\u200b</li> <li>\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b</li> <li>\u200b\u5b57\u8282\u200b\u7c7b\u578b\u200b</li> <li>\u200b\u51fd\u6570\u200b</li> </ul>"},{"location":"coding/python/bytes/","title":"Python\u200b\u4e2d\u200b\u7684\u200b\u5b57\u8282\u200b\u7c7b\u578b","text":"<p>Python\u200b\u4e2d\u200b\u7684\u200b\u5b57\u8282\u200b\u7c7b\u578b\u200b\u5305\u62ec\u200b\u5b57\u8282\u200b\u6d41\u200b\u4e0e\u200b\u5b57\u7b26\u4e32\u200b\u3002</p> <p>\u200b\u5b57\u7b26\u4e32\u200b\u662f\u200b\u7531\u200b\u5b57\u7b26\u200b\u7ec4\u6210\u200b\u7684\u200b\u5e8f\u5217\u200b\u3002\u200b\u5b57\u7b26\u200b\u662f\u200b\u7ec4\u6210\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u57fa\u672c\u200b\u5355\u4f4d\u200b\uff0c\u200b\u5bf9\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5207\u7247\u200b\u7b49\u200b\u64cd\u4f5c\u200b\u4ee5\u200b\u5b57\u7b26\u200b\u4e3a\u200b\u5355\u4f4d\u200b\u8fdb\u884c\u200b\u3002\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b\u7531\u200b\u4e24\u4e2a\u200b\u90e8\u5206\u200b\u8fdb\u884c\u200b\u5b9a\u4e49\u200b\uff1a</p> <ul> <li>\u200b\u7801\u4f4d\u200b\uff0c\u200b\u5373\u200b\u5b57\u7b26\u200b\u7684\u200b\u5b57\u8282\u200b\u6570\u503c\u200b\u3002</li> <li>\u200b\u7f16\u7801\u65b9\u5f0f\u200b\uff0c\u200b\u5373\u200b\u5b57\u8282\u200b\u6570\u503c\u200b\u4e0e\u200b\u5b57\u7b26\u200b\u7684\u200b\u5bf9\u5e94\u200b\u5173\u7cfb\u200b\u3002</li> </ul> <p>\u200b\u901a\u8fc7\u200b\u7f16\u7801\u200b\u4e0e\u200b\u89e3\u7801\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u5b57\u7b26\u4e32\u200b\u4e0e\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u4e4b\u95f4\u200b\u7684\u200b\u8f6c\u6362\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; \"abc\".encode() \nb'abc'\n&gt;&gt;&gt; b\"abc\".decode()\n'abc'\n&gt;&gt;&gt;\n</code></pre> <p>Python\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u4e24\u79cd\u200b\u5b57\u8282\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5373\u200b<code>bytes</code>\u200b\u4e0e\u200b<code>bytearray</code>\uff0c\u200b\u4e24\u8005\u200b\u90fd\u200b\u662f\u200b\u7531\u200b\u65e0\u200b\u7b26\u53f7\u200b\u5b57\u8282\u200b\uff08\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b0~255\uff09\u200b\u4e3a\u200b\u5355\u4f4d\u200b\u7ec4\u6210\u200b\u7684\u200b\u5e8f\u5217\u200b\u3002<code>bytes</code>\u200b\u662f\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\u3002</p> <pre><code>&gt;&gt;&gt; b\"\\xff\\xff\"[0]\n255\n&gt;&gt;&gt; bytes(3)[0]=1   \nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: 'bytes' object does not support item assignment\n&gt;&gt;&gt;\n</code></pre> <p>New in version 3.5</p> <p><code>bytes</code>\u200b\u5bf9\u8c61\u200b\u4e0e\u200b<code>bytearray</code>\u200b\u5bf9\u8c61\u200b\u6dfb\u52a0\u200b\u4e86\u200b<code>hex</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5b57\u8282\u200b\u7684\u200b\u5341\u516d\u8fdb\u5236\u200b\u8868\u793a\u200b\u5f62\u5f0f\u200b\u3002\u200b\u8be5\u200b\u51fd\u6570\u200b\u4e0e\u200b<code>bytes</code>\u200b\u5bf9\u8c61\u200b\u7684\u200b<code>fromhex</code>\u200b\u65b9\u6cd5\u200b\u76f8\u53cd\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import random\n&gt;&gt;&gt; seq = random.randbytes(8) # New in Python 3.9\n&gt;&gt;&gt; seq.hex() # doctest: +SKIP\n'f04be4376519e9ce'\n&gt;&gt;&gt;\n</code></pre> <p>Changed in version 3.8</p> <p><code>hex</code>\u200b\u65b9\u6cd5\u200b\u65b0\u589e\u200b\u4e86\u200b\u53ef\u9009\u200b\u7684\u200b<code>sep</code>\u200b\u53c2\u6570\u200b\u4e0e\u200b<code>bytes_per_sep</code>\u200b\u53c2\u6570\u200b\u3002</p> <ul> <li><code>sep</code>\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\u533a\u6bb5\u200b\u95f4\u200b\u7684\u200b\u8fde\u63a5\u200b\u5b57\u7b26\u4e32\u200b\uff1b</li> <li><code>bytes_per_sep</code>\u200b\u53c2\u6570\u200b\u7528\u4e8e\u200b\u5212\u5206\u200b\u8fde\u7eed\u200b\u7684\u200b\u5b57\u8282\u200b\u6bb5\u200b\uff0c\u200b\u5b57\u8282\u200b\u6bb5\u200b\u4ece\u200b\u53f3\u200b\u5f80\u200b\u5de6\u200b\u8fdb\u884c\u200b\u5212\u5206\u200b\u3002</li> </ul> <pre><code># Following the previous example\n&gt;&gt;&gt; seq.hex('-') # doctest: +SKIP\n'f0-4b-e4-37-65-19-e9-ce'\n&gt;&gt;&gt; seq.hex('-', 2) # doctest: +SKIP\n'f04b-e437-6519-e9ce'\n&gt;&gt;&gt; seq.hex('-', 3) # doctest: +SKIP\n'f04b-e43765-19e9ce'\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u5b57\u8282\u200b\u6709\u200b\u4e09\u79cd\u200b\u8868\u793a\u200b\u65b9\u5f0f\u200b\uff1a</p> <ul> <li>ASCII\u200b\u4e2d\u200b\u89c4\u5b9a\u200b\u7684\u200b\u53ef\u200b\u6253\u5370\u200b\u5b57\u7b26\u200b\uff0c\u200b\u4f7f\u7528\u200b\u8be5\u200b\u5b57\u7b26\u200b\u672c\u8eab\u200b</li> <li>\u200b\u5236\u8868\u7b26\u200b\u3001\u200b\u6362\u884c\u7b26\u200b\u3001\u200b\u56de\u8f66\u7b26\u200b\u4e0e\u200b\u53cd\u200b\u659c\u6760\u200b\u4f7f\u7528\u200b\u5bf9\u5e94\u200b\u7684\u200b\u8f6c\u4e49\u5e8f\u5217\u200b\u8868\u793a\u200b\uff0c\u200b\u5373\u200b<code>\\t, \\n, \\r, \\\\</code></li> <li>\u200b\u6240\u6709\u200b\u5b57\u8282\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5341\u516d\u8fdb\u5236\u200b\u8f6c\u4e49\u5e8f\u5217\u200b\u8868\u793a\u200b\uff0c\u200b\u5982\u200b<code>\\x00</code></li> </ul> <pre><code>&gt;&gt;&gt; b\"\\t\" == b\"\\x09\"\nTrue\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5b57\u7b26\u4e32\u200b\u4e0e\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u7684\u200b\u533a\u522b\u200b\u5728\u4e8e\u200b\uff1a\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u7d22\u5f15\u200b\u4e0e\u200b\u5207\u7247\u200b\u64cd\u4f5c\u200b\u8fd4\u56de\u200b\u7684\u200b\u5bf9\u8c61\u200b\u90fd\u200b\u662f\u200b\u5b57\u7b26\u4e32\u200b\u7c7b\u578b\u200b\uff1b\u200b\u800c\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u7684\u200b\u7d22\u5f15\u200b\u64cd\u4f5c\u200b\u8fd4\u56de\u200b<code>int</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5207\u7247\u200b\u64cd\u4f5c\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u3002</p>"},{"location":"coding/python/bytes/#_1","title":"\u6784\u9020","text":"<p>\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u6784\u9020\u200b\u975e\u5e38\u7b80\u5355\u200b\uff0c\u200b\u6b64\u5904\u200b\u4e0d\u200b\u505a\u200b\u8ba8\u8bba\u200b\u3002\u200b\u5728\u200b\u5b57\u7b26\u4e32\u200b\u524d\u52a0\u200b<code>r</code>\u200b\u53ef\u4ee5\u200b\u53d6\u6d88\u200b\u5b57\u7b26\u4e32\u200b\u5185\u90e8\u200b\u7684\u200b\u8f6c\u4e49\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; print(r\"ab\\n\") \nab\\n\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5728\u200b\u5b57\u7b26\u4e32\u200b\u524d\u52a0\u200b<code>b</code>\u200b\u53ef\u4ee5\u200b\u6784\u9020\u200b\u4e00\u4e2a\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u3002\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u53ea\u80fd\u200b\u5305\u542b\u200bASCII\u200b\u53ef\u200b\u6253\u5370\u200b\u5b57\u7b26\u200b\u3002</p> <p>\u200b\u9664\u6b64\u4e4b\u5916\u200b\uff0c\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u8fd8\u6709\u200b\u5982\u4e0b\u200b\u6784\u9020\u200b\u65b9\u5f0f\u200b\uff1a</p> <ul> <li>\u200b\u6307\u5b9a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u548c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u7f16\u7801\u65b9\u5f0f\u200b\uff0c\u200b\u5c06\u200b\u8be5\u200b\u5b57\u7b26\u4e32\u200b\u7f16\u7801\u200b\u4e3a\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b</li> <li>\u200b\u4e00\u4e2a\u200b\u4ec5\u200b\u5305\u542b\u200b<code>0~255</code>\u200b\u5185\u200b\u6570\u503c\u200b\u7684\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b</li> <li>\u200b\u4e00\u4e2a\u200b\u5b9e\u73b0\u200b\u7f13\u51b2\u200b\u534f\u8bae\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5c06\u200b\u8be5\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u590d\u5236\u5230\u200b\u65b0\u200b\u7684\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u4e2d\u200b\uff08\u200b\u53ef\u80fd\u200b\u6d89\u53ca\u200b\u7c7b\u578b\u8f6c\u6362\u200b\uff09</li> <li>\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\uff0c\u200b\u521b\u5efa\u200b\u5bf9\u5e94\u200b\u957f\u5ea6\u200b\u7684\u200b\u7a7a\u200b\u5b57\u8282\u200b\u5bf9\u8c61\u200b</li> </ul> <p>\u200b\u5982\u200b\uff0c\u200b\u4ece\u200b<code>array.array</code>\u200b\u5bf9\u8c61\u200b\u521b\u5efa\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; import array  \n&gt;&gt;&gt; import random\n&gt;&gt;&gt; a = array.array(\"H\", [51417, 45016, 65120, 9976])\n&gt;&gt;&gt; b = bytes(a)\n&gt;&gt;&gt; b\nb'\\xd9\\xc8\\xd8\\xaf`\\xfe\\xf8&amp;'\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/bytes/#_2","title":"\u7ed3\u6784\u200b\u4f53","text":"<p><code>struct</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5c06\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u4e0d\u540c\u200b\u7c7b\u578b\u200b\u5b57\u6bb5\u200b\u7684\u200b\u5143\u7ec4\u200b\uff0c\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bC\u200b\u8bed\u8a00\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7684\u200b\u529f\u80fd\u200b\u3002</p>"},{"location":"coding/python/bytes/#_3","title":"\u7ed3\u6784\u200b\u5b9a\u4e49","text":"<p>\u200b\u7ed3\u6784\u200b\u5b9a\u4e49\u200b\u5305\u542b\u200b\u4e24\u4e2a\u200b\u90e8\u5206\u200b\uff0c\u200b\u5373\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\u4e0e\u200b\u5b57\u200b\u6bb5\u200b\u3002<code>struct</code>\u200b\u6a21\u5757\u200b\u5141\u8bb8\u200b\u591a\u79cd\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b</p> \u200b\u5b57\u7b26\u200b \u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b \u200b\u5927\u5c0f\u200b \u200b\u5bf9\u9f50\u200b\u65b9\u5f0f\u200b <code>@</code> native native native <code>=</code> native standard none <code>&lt;</code> little-endian standard none <code>&gt;</code> big-endian standard none <code>!</code> network (= big-endian) standard none <p>\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\u4e3a\u200b<code>@</code>\u3002</p> <p><code>struct</code>\u200b\u6a21\u5757\u200b\u7684\u200b\u5b57\u200b\u6bb5\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff0c\u200b\u6240\u6709\u200b\u7684\u200b\u5b57\u200b\u6bb5\u200b\u5728\u200bC\u200b\u8bed\u8a00\u200b\u4e2d\u200b\u90fd\u200b\u6709\u200b\u5bf9\u5e94\u200b\u7684\u200b\u7c7b\u578b\u200b\uff1a</p> \u200b\u5b57\u7b26\u200b C \u200b\u7c7b\u578b\u200b Python \u200b\u7c7b\u578b\u200b \u200b\u5b57\u200b\u5bbd\u200b <code>x</code> \uff08\u200b\u586b\u5145\u200b\u5b57\u8282\u200b\uff09 N/A <code>c</code> <code>char</code> \u200b\u957f\u5ea6\u200b\u4e3a\u200b1\u200b\u7684\u200b\u5b57\u8282\u200b 1 <code>b</code> <code>signed char</code> <code>int</code> 1 <code>B</code> <code>unsigned char</code> <code>int</code> 1 <code>?</code> <code>_Bool</code> <code>bool</code> 1 <code>h</code> <code>short</code> <code>int</code> 2 <code>H</code> <code>unsigned short</code> <code>int</code> 2 <code>i</code> <code>int</code> <code>int</code> 4 <code>I</code> <code>unsigned int</code> <code>int</code> 4 <code>l</code> <code>long</code> <code>int</code> 4 <code>L</code> <code>unsigned long</code> <code>int</code> 4 <code>q</code> <code>long long</code> <code>int</code> 8 <code>Q</code> <code>unsigned long long</code> <code>int</code> 8 <code>n</code> <code>ssize_t</code> <code>int</code> \uff08\u200b\u4ec5\u200b\u9002\u7528\u200b\u4e8e\u200b\u9ed8\u8ba4\u200b\u6216\u200b<code>@</code>\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\uff09 <code>N</code> <code>size_t</code> <code>int</code> \uff08\u200b\u4ec5\u200b\u9002\u7528\u200b\u4e8e\u200b\u9ed8\u8ba4\u200b\u6216\u200b<code>@</code>\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\uff09 <code>e</code> \uff08\u200b\u534a\u200b\u7cbe\u5ea6\u200b\uff09 <code>float</code> 2 <code>f</code> <code>float</code> <code>float</code> 4 <code>d</code> <code>double</code> <code>float</code> 8 <code>s</code> <code>char[]</code> <code>bytes</code> \u200b\u4e0e\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u6709\u5173\u200b <code>p</code> <code>char[]</code> <code>bytes</code> \u200b\u4e0e\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u6709\u5173\u200b <code>P</code> <code>void *</code> <code>int</code> <p>\u200b\u5f53\u200b\u8bd5\u56fe\u200b\u5c06\u200b\u975e\u200b\u6574\u6570\u200b\u5bf9\u8c61\u200b\u6253\u5305\u200b\u4e3a\u200b\u6574\u6570\u200b\u7c7b\u578b\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u7684\u200b<code>__index__</code>\u200b\u65b9\u6cd5\u200b\u3002</p> <p>\u200b\u4e00\u4e2a\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7684\u200b\u5b9a\u4e49\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u7ed3\u6784\u200b\u7ec4\u7ec7\u200b\uff1a</p> <ol> <li>\u200b\u7b2c\u4e00\u4e2a\u200b\u5b57\u7b26\u200b\u8868\u793a\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b</li> <li>\u200b\u6b64\u540e\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\u8868\u793a\u200b\u7ed3\u6784\u200b\u4f53\u4e2d\u200b\u7684\u200b\u5b57\u200b\u6bb5\u200b\u7c7b\u578b\u200b</li> <li>\u200b\u9664\u200b<code>s</code>\u200b\u548c\u200b<code>p</code>\u200b\u4ee5\u5916\u200b\uff0c\u200b\u5b57\u6bcd\u200b\u524d\u200b\u7684\u200b\u6570\u5b57\u200b\u8868\u660e\u200b\u8be5\u5b57\u6bb5\u200b\u91cd\u590d\u200b\u51fa\u73b0\u200b\u7684\u200b\u6b21\u6570\u200b</li> <li><code>s</code>\u3001<code>p</code>\u200b\u524d\u200b\u7684\u200b\u6570\u5b57\u200b\u8868\u660e\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u957f\u5ea6\u200b</li> </ol> <p>\u200b\u7ed3\u6784\u200b\u4f53\u200b\u5b57\u7b26\u4e32\u200b\u53ef\u4ee5\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b<code>struct.Struct</code>\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u4e24\u8005\u200b\u5b9e\u73b0\u200b\u76f8\u540c\u200b\u7684\u200b\u529f\u80fd\u200b\u3002</p>"},{"location":"coding/python/bytes/#_4","title":"\u7ed3\u6784\u200b\u64cd\u4f5c","text":"<p>\u200b\u5bf9\u4e8e\u200b\u4e00\u4e2a\u200b\u5b9a\u4e49\u200b\u7684\u200b\u7ed3\u6784\u200b\u4f53\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u6570\u636e\u200b\u6309\u7167\u200b\u7ed3\u6784\u200b\u4f53\u200b\u8fdb\u884c\u200b\u6253\u5305\u200b\uff0c\u200b\u6216\u200b\u5c06\u200b\u7ed3\u6784\u200b\u4f53\u4e2d\u200b\u7684\u200b\u6570\u636e\u200b\u89e3\u5305\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u663e\u793a\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7684\u200b\u5927\u5c0f\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import struct   \n&gt;&gt;&gt; struct.pack('&lt;hhf', 1, 2, 3)\nb'\\x01\\x00\\x02\\x00\\x00\\x00@@'\n&gt;&gt;&gt; struct.unpack('&lt;hhf', b'\\x01\\x00\\x02\\x00\\x00\\x00@@')\n(1, 2, 3.0)\n&gt;&gt;&gt;\n</code></pre> <p>New in version 3.4</p> <p><code>struct</code>\u200b\u5bf9\u8c61\u200b\u65b0\u589e\u200b\u4e86\u200b<code>iter_unpack</code>\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u4e0d\u540c\u4e8e\u200b<code>unpack</code>\u200b\u51fd\u6570\u200b\uff0c<code>iter_unpack</code>\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002</p> <p><code>struct</code>\u200b\u6a21\u5757\u200b\u4e0d\u4f1a\u200b\u5bf9\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\u8fdb\u884c\u200b\u68c0\u6d4b\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5bf9\u4e8e\u200b\u540c\u4e00\u4e2a\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4e0d\u540c\u200b\u7684\u200b\u7ed3\u6784\u200b\u4f53\u200b\u5b9a\u4e49\u200b\u5728\u200b\u89e3\u5305\u200b\u540e\u4f1a\u200b\u6709\u200b\u4e0d\u540c\u200b\u7684\u200b\u7ed3\u679c\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; struct.unpack('&gt;hhf', b'\\x01\\x00\\x02\\x00\\x00\\x00@@') \n(256, 512, 2.304855714121459e-41)\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6bcf\u4e2a\u200b\u5b57\u6bb5\u200b\u90fd\u200b\u6709\u200b\u8303\u56f4\u200b\u9650\u5236\u200b\uff0c\u200b\u5f53\u200b\u4f20\u5165\u200b\u7684\u200b\u53c2\u6570\u200b\u8d85\u8fc7\u200b\u5b57\u6bb5\u200b\u6240\u200b\u5141\u8bb8\u200b\u7684\u200b\u8303\u56f4\u200b\uff0c\u200b\u5219\u200b\u4f1a\u200b\u629b\u51fa\u200b\u5f02\u5e38\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; struct.pack('&lt;hhf', 32768, -32769, 3)\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nstruct.error: short format requires (-32768) &lt;= number &lt;= 32767\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5f53\u200b\u89e3\u5305\u200b\u7684\u200b\u5b57\u8282\u200b\u957f\u5ea6\u200b\u4e0e\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7684\u200b\u957f\u5ea6\u200b\u4e0d\u200b\u5bf9\u5e94\u200b\u65f6\u200b\uff0c\u200b\u4e5f\u200b\u4f1a\u200b\u629b\u51fa\u200b\u5f02\u5e38\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; struct.unpack('&gt;hhf', b'\\x01\\x00\\x02\\x00\\x00\\x00@')  \nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nstruct.error: unpack requires a buffer of 8 bytes\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u4f55\u200b\u7ed3\u6784\u200b\u4f53\u200b\uff0c<code>struct</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>calcsize</code>\u200b\u65b9\u6cd5\u200b\u7528\u4e8e\u200b\u68c0\u67e5\u200b\u7ed3\u6784\u200b\u4f53\u200b\u957f\u5ea6\u200b\u3002</p>"},{"location":"coding/python/bytes/#_5","title":"\u5185\u5b58\u200b\u89c6\u56fe","text":"<p>\u200b\u5185\u5b58\u200b\u89c6\u56fe\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5728\u200b\u4e0d\u540c\u200b\u5bf9\u8c61\u200b\u95f4\u200b\u5171\u4eab\u5185\u5b58\u200b\u7684\u200b\u65b9\u5f0f\u200b\u3002</p> <p>\u200b\u6709\u5173\u200b\u5185\u5b58\u200b\u89c6\u56fe\u200b\u7684\u200b\u5185\u5bb9\u200b\uff0c\u200b\u8bf7\u200b\u53c2\u89c1\u200b\u5e8f\u5217\u200b\u7c7b\u578b\u200b\u7684\u200b\u5185\u5b58\u200b\u89c6\u56fe\u200b\u90e8\u5206\u200b\u3002</p>"},{"location":"coding/python/bytes/#_6","title":"\u5b57\u7b26\u4e32","text":"<p>\u200b\u6b64\u5904\u200b\u7740\u91cd\u200b\u8ba8\u8bba\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u76f8\u5173\u200b\u95ee\u9898\u200b</p>"},{"location":"coding/python/bytes/#_7","title":"\u7f16\u7801\u200b\u4e0e\u200b\u89e3\u7801","text":"<p>\u200b\u5982\u524d\u6240\u8ff0\u200b\uff0c<code>str.encode</code>\u200b\u65b9\u6cd5\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u4ece\u200b\u5b57\u7b26\u4e32\u200b\u5230\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u7684\u200b\u8f6c\u6362\u200b\u65b9\u5f0f\u200b\uff0c<code>encoding</code>\u200b\u53c2\u6570\u200b\u6307\u660e\u200b\u4e86\u200b\u6240\u200b\u4f7f\u7528\u200b\u7684\u200b\u7f16\u7801\u5668\u200b\u3002</p> <pre><code>&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"utf-8\") \nb'\\xe6\\xb5\\x8b\\xe8\\xaf\\x95'\n&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"utf-16\")\nb'\\xff\\xfeKm\\xd5\\x8b'\n&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"gb2312\")  \nb'\\xb2\\xe2\\xca\\xd4'\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5f53\u200b\u7f16\u7801\u200b\u8fc7\u7a0b\u200b\u51fa\u73b0\u200b\u9519\u8bef\u200b\uff0c\u200b\u5982\u200b\u7f16\u7801\u5668\u200b\u65e0\u6cd5\u200b\u8bc6\u522b\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u5b57\u7b26\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u629b\u51fa\u200b<code>UnicodeEncodeError</code>\u200b\u5f02\u5e38\u200b\u3002</p> <pre><code>&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"latin-1\")\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nUnicodeEncodeError: 'latin-1' codec can't encode characters in position 0-1: ordinal not in range(256)\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u51fa\u73b0\u200b\u9519\u8bef\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\u4ee5\u4e0b\u200b\u89e3\u51b3\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>errors</code>\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\uff1a</p> <ul> <li><code>ignore</code>\uff1a\u200b\u8df3\u8fc7\u200b\u65e0\u6cd5\u200b\u7f16\u7801\u200b\u7684\u200b\u5b57\u7b26\u200b</li> <li><code>replace</code>\uff1a\u200b\u5c06\u200b\u65e0\u6cd5\u200b\u7f16\u7801\u200b\u7684\u200b\u5b57\u7b26\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>?</code></li> <li><code>xmlcharrefreplace</code>\uff1a\u200b\u5c06\u200b\u65e0\u6cd5\u200b\u7f16\u7801\u200b\u7684\u200b\u5b57\u7b26\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>xml</code>\u200b\u5b9e\u4f53\u200b\uff08\u200b\u5373\u200bXML\u200b\u4e2d\u200b\u6240\u200b\u4f7f\u7528\u200b\u7684\u200b\u5b57\u7b26\u200b\u8f6c\u6362\u200b\u65b9\u5f0f\u200b\uff09</li> </ul> <pre><code>&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"latin-1\", errors=\"ignore\")\nb''\n&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"latin-1\", errors=\"replace\")\nb'??'\n&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"latin-1\", errors=\"xmlcharrefreplace\")\nb'&amp;#27979;&amp;#35797;'\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5bf9\u5e94\u200b\u5730\u200b\uff0c\u200b\u89e3\u7801\u5668\u200b\u65e0\u6cd5\u200b\u8bc6\u522b\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u7684\u200b\u5b57\u8282\u200b\u65f6\u4f1a\u200b\u4ea7\u751f\u200b<code>UnicodeDecodeError</code>\u200b\u5f02\u5e38\u200b\u3002\u200b\u4f46\u200b\u4e0d\u200b\u629b\u51fa\u200b\u5f02\u5e38\u200b\u4e0d\u200b\u4ee3\u8868\u200b\u89e3\u7801\u200b\u6210\u529f\u200b\uff0c\u200b\u89e3\u7801\u200b\u5f97\u5230\u200b\u7684\u200b\u6570\u636e\u200b\u53ef\u80fd\u200b\u662f\u200b\u65e0\u7528\u200b\u6570\u636e\u200b\u3002<code>errors</code>\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\u4e86\u200b\u89e3\u7801\u5668\u200b\u5728\u200b\u51fa\u9519\u200b\u65f6\u200b\u7684\u200b\u884c\u4e3a\u200b\uff0c<code>replace</code>\u200b\u5c06\u200b\u65e0\u6cd5\u200b\u7f16\u7801\u200b\u7684\u200b\u5b57\u7b26\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>\ufffd</code>\u3002</p> <p><code>chardet</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u57fa\u4e8e\u200bPython\u200b\u7684\u200b\u5b57\u7b26\u200b\u7f16\u7801\u200b\u68c0\u6d4b\u5de5\u5177\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4e8c\u8fdb\u5236\u200b\u5e8f\u5217\u200b\u5bf9\u200b\u539f\u59cb\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u7f16\u7801\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u63a8\u65ad\u200b\u3002\u200b\u4e0d\u8fc7\u200b\u63a8\u65ad\u200b\u4ec5\u200b\u9002\u7528\u200b\u4e8e\u200b\u8f83\u200b\u957f\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u56e0\u4e3a\u200b\u4efb\u4f55\u200b\u5b57\u7b26\u4e32\u200b\u4f1a\u200b\u6709\u200b\u591a\u4e2a\u200b\u7f16\u7801\u65b9\u5f0f\u200b\u9002\u7528\u200b\u4e8e\u200b\u540c\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u6240\u4ee5\u200b\u65e0\u6cd5\u200b\u5b8c\u5168\u200b\u786e\u5b9a\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u7f16\u7801\u65b9\u5f0f\u200b\u3002</p>"},{"location":"coding/python/bytes/#bom","title":"BOM","text":"<p>BOM\u200b\u662f\u200b\u5b57\u8282\u200b\u5e8f\u200b\u6807\u8bb0\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200bUnicode\u200b\u5b57\u7b26\u200b\u4e3a\u200b<code>U+FEFF</code>\uff08\u200b\u4e0d\u200b\u5b58\u5728\u200b<code>U+FFFE</code>\u200b\u5b57\u7b26\u200b\uff0c\u200b\u56e0\u6b64\u200b\u8be5\u200b\u5b57\u7b26\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u63a8\u65ad\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\uff09</p> <p>\u200b\u5728\u200bUTF-16\u200b\u7f16\u7801\u200b\u7684\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u5f00\u5934\u200b\u4f1a\u200b\u5199\u5165\u200bBOM\uff0c\u200b\u5982\u679c\u200b\u5f00\u5934\u200b\u662f\u200b<code>b'\\xff\\xfe'</code>\u200b\u4e24\u4e2a\u200b\u5b57\u8282\u200b\uff0c\u200b\u6307\u660e\u200b\u7f16\u7801\u200b\u65f6\u200b\u4f7f\u7528\u200blittle endian\u200b\u5b57\u8282\u200b\u7f16\u7801\u200b\u987a\u5e8f\u200b\u3002\u200b\u5982\u679c\u200b\u662f\u200b<code>b'\\xfe\\xff'</code>\u200b\u4e24\u4e2a\u200b\u5b57\u8282\u200b\uff0c\u200b\u8bf4\u660e\u200b\u7f16\u7801\u200b\u65f6\u200b\u4f7f\u7528\u200b\u7684\u200b\u662f\u200bbig endian\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\u3002\u200b\u5982\u679c\u200b\u6307\u660e\u200bUTF-16\u200b\u6240\u200b\u4f7f\u7528\u200b\u7684\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\uff0c\u200b\u5982\u200bUTF-16LE\u200b\u6216\u200bUTF-16BE\uff0c\u200b\u5219\u200b\u4e0d\u4f1a\u200b\u751f\u6210\u200bBOM\u3002</p> <p>BOM\u200b\u4ec5\u200b\u7528\u4e8e\u200b\u63a8\u65ad\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\u800c\u200b\u4e0d\u4f1a\u200b\u51fa\u73b0\u200b\u5728\u200b\u6700\u7ec8\u200b\u89e3\u7801\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\u3002</p>"},{"location":"coding/python/bytes/#_8","title":"\u6587\u672c\u6587\u4ef6","text":"<p>\u200b\u4f7f\u7528\u200b<code>open</code>\u200b\u51fd\u6570\u200b\u4ee5\u200b\u6587\u672c\u200b\u6a21\u5f0f\u200b\u6253\u5f00\u200b\u4e00\u4e2a\u200b\u6587\u4ef6\u200b\u65f6\u200b\uff0c\u200b\u6700\u597d\u200b\u6307\u5b9a\u200b\u6587\u4ef6\u200b\u7684\u200b\u7f16\u7801\u65b9\u5f0f\u200b\u3002</p> <p>\u200b\u4e0d\u8981\u200b\u4f7f\u7528\u200b\u4e8c\u8fdb\u5236\u200b\u65b9\u5f0f\u200b\u6253\u5f00\u200b\u6587\u672c\u6587\u4ef6\u200b\u3002</p>"},{"location":"coding/python/bytes/#unicode","title":"Unicode\u200b\u89c4\u8303\u5316","text":"<p>\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a = 'caf\u00e9'\n&gt;&gt;&gt; b = 'cafe\\u0301'             \n&gt;&gt;&gt; print(a, b)\ncaf\u00e9 cafe\u0301\n&gt;&gt;&gt; a == b\nFalse\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u76f8\u540c\u200b\u7684\u200b\u6253\u5370\u200b\u7ed3\u679c\u200b\uff0c\u200b\u5374\u200b\u5bf9\u5e94\u200b\u4e0d\u540c\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u539f\u56e0\u200b\u5728\u4e8e\u200b\u5b57\u7b26\u4e32\u200b<code>b</code>\u200b\u4f7f\u7528\u200b\u4e86\u200b<code>U+0301</code>\u200b\u5b57\u7b26\u200b\u4f5c\u4e3a\u200b\u91cd\u97f3\u200b\u6807\u8bb0\u200b\uff08\u200b\u7ec4\u5408\u200b\u5b57\u7b26\u200b\uff09\uff0c\u200b\u591a\u200b\u4f7f\u7528\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u5b57\u8282\u200b\u3002\u200b\u5bf9\u4e8e\u200bPython\u200b\u800c\u8a00\u200b\uff0c\u200b\u8fd9\u200b\u4e00\u6bb5\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u5e76\u4e0d\u76f8\u540c\u200b\uff0c\u200b\u56e0\u6b64\u200b\u8ba4\u4e3a\u200b<code>a != b</code>\u3002</p> <pre><code>&gt;&gt;&gt; a.encode(\"utf-8\")     \nb'caf\\xc3\\xa9'\n&gt;&gt;&gt; b.encode(\"utf-8\") \nb'cafe\\xcc\\x81'\n&gt;&gt;&gt;\n</code></pre> <p><code>unicodedata</code>\u200b\u6a21\u5757\u200b\u4e2d\u200b\u7684\u200b<code>normalize</code>\u200b\u51fd\u6570\u200b\u63d0\u4f9b\u200b\u4e86\u200bUnicode\u200b\u89c4\u8303\u5316\u200b\u7684\u200b\u529f\u80fd\u200b\uff0c\u200b\u8be5\u200b\u51fd\u6570\u200b\u63a5\u6536\u200b\u5982\u4e0b\u200b\u53c2\u6570\u200b\u7528\u4e8e\u200b\u786e\u5b9a\u200b\u8f6c\u6362\u200b\u6807\u51c6\u200b\uff1a</p> <ul> <li><code>'NFC'</code>\uff1a\u200b\u4f7f\u7528\u200b\u6700\u5c11\u200b\u7801\u4f4d\u200b\u6784\u6210\u200b\u7b49\u4ef7\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b</li> <li><code>'NFD'</code>\uff1a\u200b\u5c06\u200b\u7ec4\u5408\u200b\u5b57\u7b26\u200b\u5206\u89e3\u200b\u4e3a\u57fa\u200b\u5b57\u7b26\u200b\u4e0e\u200b\u5355\u72ec\u200b\u7684\u200b\u7ec4\u5408\u200b\u5b57\u7b26\u200b\uff08\u200b\u5982\u200b<code>U+0301</code>\uff09\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u53bb\u9664\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u53d8\u97f3\u200b\u7b26\u53f7\u200b</li> <li><code>NFKC</code>\u3001<code>NFKD</code>\u200b\u4f1a\u200b\u989d\u5916\u200b\u5c06\u200b\u517c\u5bb9\u200b\u5b57\u7b26\u200b\u5206\u89e3\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u6216\u200b\u591a\u4e2a\u200b\u517c\u5bb9\u200b\u5206\u89e3\u200b\uff0c\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u6570\u636e\u200b\u635f\u5931\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</li> </ul> <pre><code>&gt;&gt;&gt; from unicodedata import normalize\n&gt;&gt;&gt; normalize('NFKC', '\u337f') \n'\u200b\u682a\u5f0f\u4f1a\u793e\u200b'\n&gt;&gt;&gt;\n</code></pre> <p><code>NKFC</code>\u3001<code>NFKD</code>\u200b\u89c4\u8303\u5316\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u539f\u610f\u200b\u53d8\u5316\u200b\uff0c\u200b\u4f46\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u641c\u7d22\u5f15\u64ce\u200b\u3002</p> <p><code>str.casefold</code>\u200b\u51fd\u6570\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u53e6\u200b\u4e00\u79cd\u200b\u89c4\u8303\u5316\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u5373\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5927\u5199\u5b57\u6bcd\u200b\u8f6c\u4e3a\u200b\u5c0f\u5199\u200b\u3002\u200b\u4e0e\u200b<code>str.lower</code>\u200b\u4e0d\u540c\u200b\uff0c\u200b\u90e8\u5206\u200b\u5b57\u7b26\u200b\u4f1a\u200b\u88ab\u200b\u66ff\u6362\u6210\u200b\u65b0\u200b\u7684\u200b\u5b57\u7b26\u200b\u3002</p>"},{"location":"coding/python/bytes/#_9","title":"\u5b57\u7b26\u4e32\u200b\u6392\u5e8f","text":"<p>\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u6392\u5e8f\u200b\u4e0e\u200b\u5176\u4ed6\u200b\u6570\u636e\u200b\u7684\u200b\u6392\u5e8f\u200b\u65b9\u5f0f\u200b\u76f8\u540c\u200b\uff0c\u200b\u90fd\u200b\u662f\u200b\u6309\u7167\u200b\u7801\u4f4d\u200b\u5347\u5e8f\u200b\u6392\u5e8f\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u975e\u200b<code>ASCII</code>\u200b\u5b57\u7b26\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u4e00\u4e9b\u200b\u95ee\u9898\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; sorted(['caf\u00e9', 'cafu'])       \n['cafu', 'caf\u00e9']\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6a21\u5757\u200b<code>locale</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>strxfrm</code>\u200b\u51fd\u6570\u200b\uff0c\u200b\u7528\u4e8e\u200b\u6309\u7167\u200b\u533a\u57df\u200b\u8bbe\u7f6e\u200b\u5bf9\u200b\u5b57\u7b26\u4e32\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b\u3002\u200b\u5982\u679c\u200b\u64cd\u4f5c\u7cfb\u7edf\u200b\u652f\u6301\u200b\uff0c\u200b\u533a\u57df\u200b\u8bbe\u7f6e\u200b\u53ef\u4ee5\u200b\u5728\u200b<code>setlocale</code>\u200b\u51fd\u6570\u200b\u4e2d\u200b\u5168\u5c40\u200b\u6307\u5b9a\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import locale\n&gt;&gt;&gt; locale.setlocale(locale.LC_COLLATE, \"en_US.UTF-8\")\n'en_US.UTF-8'\n&gt;&gt;&gt; sorted(['caf\u00e9', 'cafu'], key=locale.strxfrm)\n['caf\u00e9', 'cafu']\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/bytes/#api","title":"API","text":"<p>\u200b\u90e8\u5206\u200b\u6d89\u53ca\u200b\u5230\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u4e00\u4e9b\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u8f93\u5165\u200b\u5b57\u7b26\u4e32\u200b\u6216\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u3002</p>"},{"location":"coding/python/bytes/#re","title":"re\u200b\u6a21\u5757","text":"<p>\u200b\u4e0d\u540c\u4e8e\u200b\u4f7f\u7528\u200b\u5b57\u7b26\u4e32\u200b\u6784\u9020\u200b\u7684\u200b\u6b63\u5219\u8868\u8fbe\u5f0f\u200b\uff0c\u200b\u4f7f\u7528\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u6784\u9020\u200b\u7684\u200b\u6b63\u5219\u8868\u8fbe\u5f0f\u200b\uff0c<code>\\d</code>\u200b\u4e0e\u200b<code>\\w</code>\u200b\u53ea\u80fd\u200b\u5339\u914d\u200bASCII\u200b\u5b57\u7b26\u200b\u3002</p>"},{"location":"coding/python/bytes/#os","title":"os\u200b\u6a21\u5757","text":"<p><code>os</code>\u200b\u6a21\u5757\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5b57\u7b26\u4e32\u200b\u53c2\u6570\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u66ff\u6362\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u51fd\u6570\u200b\u4f1a\u200b\u4f7f\u7528\u200b<code>sys.getfilesystemencoding()</code>\u200b\u51fd\u6570\u200b\u53d6\u5f97\u200b\u5408\u9002\u200b\u7684\u200b\u7f16\u7801\u5668\u200b\u3002\u200b\u6b64\u5916\u200b\uff0c<code>os</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b<code>fsencode</code>\u200b\u4e0e\u200b<code>fsdecode</code>\u200b\u51fd\u6570\u200b\u7528\u4e8e\u200b\u624b\u52a8\u200b\u8fdb\u884c\u200b\u7f16\u7801\u200b\u4e0e\u200b\u89e3\u7801\u200b\u64cd\u4f5c\u200b\u3002</p>"},{"location":"coding/python/decorator/","title":"\u88c5\u9970\u200b\u5668","text":"<p>\u200b\u88c5\u9970\u200b\u5668\u662f\u200b\u4e00\u4e2a\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u5e76\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u5982\u4e0b\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u88c5\u9970\u200b\u5668\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; def decorator(x):\n...     print(x)\n...     return x\n...\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u88c5\u9970\u200b\u5668\u200b\u5728\u200b\u51fd\u6570\u200b\u5b9a\u4e49\u200b\u524d\u200b\u4f7f\u7528\u200b\uff0c<code>@</code>\u200b\u6807\u8bb0\u200b\u4e00\u4e2a\u200b\u88c5\u9970\u200b\u5668\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; @decorator    \n... def target(x):\n...     return x + 1\n... \n&lt;function target at ...&gt;\n&gt;&gt;&gt;\n</code></pre> <p><code>@decorator</code>\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b\uff0c\u200b\u51fd\u6570\u200b\u88ab\u200b\u88c5\u9970\u200b\u5668\u200b\u8fd4\u56de\u200b\u7684\u200b\u51fd\u6570\u200b\u66ff\u6362\u200b</p> <pre><code>&gt;&gt;&gt; def target(x):\n...     return x + 1\n... \n&gt;&gt;&gt; target = decorator(target)\n&lt;function target at ...&gt;\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u88c5\u9970\u200b\u5668\u200b\u5728\u200b\u88ab\u200b\u4fee\u9970\u200b\u7684\u200b\u51fd\u6570\u200b\u5b9a\u4e49\u200b\u540e\u200b\u7acb\u5373\u200b\u8fd0\u884c\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5f53\u200b\u5bfc\u5165\u200b\u6a21\u5757\u200b\u65f6\u200b\uff0c\u200b\u88c5\u9970\u200b\u5668\u200b\u51fd\u6570\u200b\u4f1a\u200b\u7acb\u5373\u200b\u6267\u884c\u200b\uff0c\u200b\u800c\u200b\u88ab\u200b\u88c5\u9970\u200b\u7684\u200b\u51fd\u6570\u200b\u53ea\u4f1a\u200b\u8c03\u7528\u200b\u6267\u884c\u200b\u3002</p> <pre><code>&gt;&gt;&gt; src = \"\"\"def decorator(_):\n...     print(_)\n...     return _\n... print(\"Before A\")\n... @decorator\n... def a():\n...     return 1\n... print(\"Before B\")\n... @decorator  \n... def b():\n...     return 1\n... print(\"After B\")\n... \"\"\"\n&gt;&gt;&gt; exec(src)\nBefore A\n&lt;function a at ...&gt;\nBefore B\n&lt;function b at ...&gt;\nAfter B\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/decorator/#_2","title":"\u53d8\u91cf\u200b\u7684\u200b\u4f5c\u7528\u57df","text":"<p>Python\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u5224\u65ad\u200b\u51fd\u6570\u200b\u4ee3\u7801\u200b\u4e2d\u200b\u7684\u200b\u53d8\u91cf\u200b\u662f\u5426\u200b\u4e3a\u200b\u5c40\u90e8\u53d8\u91cf\u200b\u3002\u200b\u6240\u6709\u200b\u5728\u200b\u51fd\u6570\u200b\u4ee3\u7801\u200b\u5185\u200b\u51fa\u73b0\u200b\u7684\u200b\u53d8\u91cf\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u5c40\u90e8\u53d8\u91cf\u200b\u3002</p> <pre><code>&gt;&gt;&gt; b = 2\n&gt;&gt;&gt; def A():\n...     print(b)\n... \n&gt;&gt;&gt; def B():\n...     print(b)\n...     b = 2\n... \n&gt;&gt;&gt; A()\n2\n&gt;&gt;&gt; B()\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\n  File \"&lt;stdin&gt;\", line 2, in B\nUnboundLocalError: local variable 'b' referenced before assignment\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5982\u679c\u200b\u9700\u8981\u200b\u5728\u200b\u51fd\u6570\u200b\u5185\u90e8\u200b\u8bbf\u95ee\u200b\u5168\u5c40\u53d8\u91cf\u200b\uff0c\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>global</code>\u200b\u5173\u952e\u5b57\u200b\u58f0\u660e\u200b\u3002</p> <pre><code>&gt;&gt;&gt; def C():\n...     global b\n...     print(b)\n...     b = 3\n... \n&gt;&gt;&gt; b\n2\n&gt;&gt;&gt; C()\n2\n&gt;&gt;&gt; b\n3\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u68c0\u67e5\u200b\u7f16\u8bd1\u200b\u540e\u200b\u7684\u200b\u5b57\u8282\u200b\u7801\u200b\uff0c\u200b\u53d1\u73b0\u200b<code>global</code>\u200b\u5173\u952e\u5b57\u200b\u58f0\u660e\u200b\u53d8\u91cf\u200b\u7531\u200b<code>LOAD_FAST</code>\u200b\u4e0e\u200b<code>STORE_FAST</code>\u200b\u6539\u4e3a\u200b\u4e86\u200b<code>LOAD_GLOBAL</code>\u200b\u4e0e\u200b<code>STORE_GLOBAL</code>\uff1a</p>"},{"location":"coding/python/decorator/#_3","title":"\u95ed\u5305","text":"<p>\u200b\u95ed\u5305\u200b\u662f\u200b\u6269\u5c55\u200b\u4e86\u200b\u51fd\u6570\u200b\u4f5c\u7528\u57df\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8bbf\u95ee\u200b\u51fd\u6570\u200b\u5b9a\u4e49\u200b\u4e4b\u5916\u200b\u7684\u200b\u5c40\u90e8\u53d8\u91cf\u200b\u3002\u200b\u4ee5\u4e0b\u200b\u901a\u8fc7\u200b\u4e0d\u540c\u200b\u65b9\u5f0f\u200b\u5b9e\u73b0\u200b\u6ed1\u52a8\u200b\u5e73\u5747\u503c\u200b\u7684\u200b\u8ba1\u7b97\u200b\uff1a</p> <ul> <li>\u200b\u4f7f\u7528\u200b\u7c7b\u200b</li> </ul> <pre><code>&gt;&gt;&gt; class avg1():\n...     def __init__(self):\n...             self.values = []\n...     def __call__(self, value):\n...             self.values.append(value)\n...             return sum(self.values) / len(self.values)\n... \n&gt;&gt;&gt; avg = avg1()\n&gt;&gt;&gt; avg(10)\n10.0\n&gt;&gt;&gt; avg(11)\n10.5\n&gt;&gt;&gt;\n</code></pre> <ul> <li>\u200b\u4f7f\u7528\u200b\u9ad8\u9636\u200b\u51fd\u6570\u200b</li> </ul> <pre><code>&gt;&gt;&gt; def avg2():\n...     values = []\n...     def avg(value): \n...             values.append(value)\n...             return sum(values) / len(values)\n...     return avg\n&gt;&gt;&gt; \n&gt;&gt;&gt; avg = avg2()\n&gt;&gt;&gt; avg(10)\n10.0\n&gt;&gt;&gt; avg(11)\n10.5\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5728\u200b<code>avg2</code>\u200b\u51fd\u6570\u200b\u7ed3\u675f\u200b\u540e\u200b\uff0c\u200b\u5c40\u90e8\u53d8\u91cf\u200b<code>values</code>\u200b\u539f\u6709\u200b\u7684\u200b\u4f5c\u7528\u57df\u200b\u6d88\u5931\u200b\uff0c\u200b\u6210\u4e3a\u200b\u81ea\u7531\u200b\u53d8\u91cf\u200b\u3002\u200b\u81ea\u7531\u200b\u53d8\u91cf\u200b\u6307\u200b\u6ca1\u6709\u200b\u5728\u200b\u672c\u5730\u200b\u4f5c\u7528\u57df\u200b\u4e2d\u200b\u7ed1\u5b9a\u200b\u7684\u200b\u53d8\u91cf\u200b\u3002<code>values</code>\u200b\u53d8\u91cf\u200b\u7684\u200b\u5f15\u7528\u200b\u5b58\u50a8\u200b\u5728\u200b<code>avg</code>\u200b\u51fd\u6570\u200b\u7684\u200b<code>__closure__</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\u3002\u200b\u5373\u4f7f\u200b<code>avg2</code>\u200b\u51fd\u6570\u200b\u5df2\u7ecf\u200b\u7ed3\u675f\u200b\u6267\u884c\u200b\uff0c<code>avg</code>\u200b\u51fd\u6570\u200b\u4ecd\u7136\u200b\u80fd\u200b\u8bbf\u95ee\u200b<code>values</code>\u200b\u53d8\u91cf\u200b\u3002</p>"},{"location":"coding/python/functions/","title":"Python\u200b\u4e2d\u200b\u7684\u200b\u51fd\u6570","text":"<p>\u201c\u200b\u4e00\u7b49\u200b\u5bf9\u8c61\u200b\u201d\u200b\u6307\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff1a</p> <ul> <li>\u200b\u5728\u200b\u7a0b\u5e8f\u6267\u884c\u200b\u65f6\u200b\u521b\u5efa\u200b</li> <li>\u200b\u80fd\u591f\u200b\u8d4b\u503c\u200b\u7ed9\u200b\u53d8\u91cf\u200b\u6216\u200b\u6570\u636e\u7ed3\u6784\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b</li> <li>\u200b\u80fd\u200b\u4f5c\u4e3a\u200b\u51fd\u6570\u53c2\u6570\u200b\u4e0e\u200b\u8fd4\u56de\u503c\u200b</li> </ul> <p>Python\u200b\u4e2d\u200b\u7684\u200b\u51fd\u6570\u200b\u662f\u200b\u4e00\u7b49\u200b\u5bf9\u8c61\u200b</p>"},{"location":"coding/python/functions/#_1","title":"\u51fd\u6570\u200b\u4e0e\u200b\u9ad8\u9636\u200b\u51fd\u6570","text":"<p>\u200b\u51fd\u6570\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u901a\u5e38\u200b\u4f7f\u7528\u200b<code>def</code>\u200b\u5173\u952e\u5b57\u200b\u5b9a\u4e49\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; def constant():  \n...     return 1\n... \n&gt;&gt;&gt; type(constant)\n&lt;class 'function'&gt;\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u5728\u200b\u5176\u4ed6\u200b\u51fd\u6570\u200b\u5185\u90e8\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; def outer():\n...     def inner():\n...             return \"ok\"\n...     return inner\n... \n&gt;&gt;&gt; func = outer()\n&gt;&gt;&gt; func()\n'ok'\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u9ad8\u9636\u200b\u51fd\u6570\u200b\u662f\u200b\u63a5\u53d7\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u6216\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5982\u200b<code>map</code>\u3001<code>filter</code>\u200b\u7b49\u200b\u3002</p> <p>\u200b\u9664\u200b\u4f7f\u7528\u200b<code>def</code>\u200b\u5173\u952e\u5b57\u200b\u5b9a\u4e49\u200b\u51fd\u6570\u200b\u4ee5\u5916\u200b\uff0c<code>lambda</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u53e6\u200b\u4e00\u79cd\u200b\u521b\u5efa\u200b\u7b80\u5355\u200b\u51fd\u6570\u200b\u7684\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u8fd9\u79cd\u200b\u65b9\u6cd5\u200b\u5e38\u7528\u200b\u4e8e\u200b\u5411\u200b\u9ad8\u9636\u200b\u51fd\u6570\u200b\u4f20\u9012\u200b\u53c2\u6570\u200b\u3002<code>lambda</code>\u200b\u8868\u8fbe\u5f0f\u200b\u53ea\u80fd\u200b\u52a0\u5165\u200b\u8868\u8fbe\u5f0f\u200b\u800c\u200b\u65e0\u6cd5\u200b\u6dfb\u52a0\u200b\u63a7\u5236\u200b\u903b\u8f91\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a = lambda : \"ok\"  \n&gt;&gt;&gt; a()\n'ok'\n&gt;&gt;&gt;\n</code></pre> <p><code>lambda</code>\u200b\u8868\u8fbe\u5f0f\u200b\u53ef\u4ee5\u200b\u521b\u5efa\u200b\u4e0d\u540c\u200b\u53c2\u6570\u200b\u6570\u91cf\u200b\u7684\u200b\u51fd\u6570\u200b\uff1a</p> <ul> <li>\u200b\u6ca1\u6709\u200b\u53c2\u6570\u200b\uff0c\u200b\u5982\u200b<code>lambda : \"ok\"</code></li> <li>\u200b\u7ed9\u5b9a\u200b\u6570\u91cf\u200b\u53c2\u6570\u200b\uff0c\u200b\u5982\u200b<code>lambda x, y : x + y</code></li> <li>\u200b\u4efb\u610f\u200b\u53c2\u6570\u200b\uff0c\u200b\u5982\u200b<code>lambda *_ : sum(_)</code>\uff08\u200b\u65e0\u540d\u200b\u53c2\u6570\u200b\uff09\u200b\u6216\u200b<code>lambda **_ : sum(_)</code>\uff08\u200b\u5177\u540d\u200b\u53c2\u6570\u200b\uff09</li> </ul> <p>\u200b\u82e5\u200b<code>lambda</code>\u200b\u8868\u8fbe\u5f0f\u200b\u7684\u200b\u53f3\u4fa7\u200b\u662f\u200b\u9017\u53f7\u200b\u5206\u9694\u200b\u7684\u200b\u8868\u8fbe\u5f0f\u200b\uff0c\u200b\u751f\u6210\u200b\u7684\u200b\u662f\u200b<code>lambda</code>\u200b\u5bf9\u8c61\u200b\u7ec4\u6210\u200b\u7684\u200b\u5143\u7ec4\u200b\u3002</p> <p>\u200b\u4f7f\u7528\u200b<code>dir()</code>\u200b\u67e5\u770b\u200b\u51fd\u6570\u200b\u7684\u200b\u5185\u7f6e\u200b\u5c5e\u6027\u200b\u3002\u200b\u76f8\u6bd4\u200b\u4e8e\u200b\u4e00\u822c\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u51fd\u6570\u200b\u6709\u200b\u4ee5\u4e0b\u200b\u7684\u200b\u989d\u5916\u200b\u5c5e\u6027\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; class Obj():\n...     pass\n... \n&gt;&gt;&gt; def func():\n...     pass\n... \n&gt;&gt;&gt; sorted(list(set(dir(func)) - set(dir(Obj())))) # doctest: +SKIP\n['__annotations__', '__call__', '__closure__', '__code__', '__defaults__', '__get__', '__globals__', '__kwdefaults__', '__name__', '__qualname__']\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/functions/#_2","title":"\u5e7f\u4e49\u200b\u51fd\u6570","text":"<p>\u200b\u4efb\u4f55\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>__call__</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u5bf9\u8c61\u200b\u90fd\u200b\u662f\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u7c7b\u4f3c\u200b\u4e8e\u200b\u51fd\u6570\u8c03\u7528\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u8c03\u7528\u200b\u3002\u200b\u4ee5\u4e0b\u200b\u7c7b\u578b\u200b\u4e3a\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\uff1a</p> <ul> <li>\u200b\u51fd\u6570\u200b\u6216\u200b<code>lambda</code>\u200b\u8868\u8fbe\u5f0f\u200b</li> <li>\u200b\u5185\u7f6e\u200b\u51fd\u6570\u200b\u6216\u200b\u7c7b\u578b\u200b\u65b9\u6cd5\u200b</li> <li>\u200b\u7c7b\u578b\u200b\u65b9\u6cd5\u200b</li> <li>\u200b\u7c7b\u200b\uff08\u200b\u76f8\u5f53\u4e8e\u200b\u4e00\u4e2a\u200b\u8fd4\u56de\u200b\u7c7b\u200b\u5bf9\u8c61\u200b\u7684\u200b\u51fd\u6570\u200b\uff09</li> <li>\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>__call__</code>\u200b\u7684\u200b\u5bf9\u8c61\u200b</li> <li>\u200b\u751f\u6210\u5668\u200b\u51fd\u6570\u200b</li> </ul> <p>\u200b\u5185\u7f6e\u200b\u51fd\u6570\u200b<code>callable()</code>\u200b\u7528\u4e8e\u200b\u68c0\u6d4b\u200b\u4e00\u4e2a\u200b\u5bf9\u8c61\u200b\u662f\u5426\u662f\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u3002</p>"},{"location":"coding/python/functions/#_3","title":"\u51fd\u6570\u200b\u7684\u200b\u53c2\u6570","text":"<p>Python\u200b\u4e2d\u200b\uff0c\u200b\u51fd\u6570\u200b\u7684\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b\uff1a</p> <ul> <li>\u200b\u9017\u53f7\u200b\u5206\u9694\u200b\uff0c\u200b\u6309\u200b\u51fd\u6570\u200b\u58f0\u660e\u200b\u987a\u5e8f\u200b\u4f20\u5165\u200b\u53c2\u6570\u200b</li> <li>\u200b\u5173\u952e\u5b57\u200b\u53c2\u6570\u200b\uff0c\u200b\u5373\u200b\u53c2\u6570\u200b\u540d\u200b=\u200b\u53c2\u6570\u503c\u200b</li> <li>\u200b\u6709\u200b\u9ed8\u8ba4\u503c\u200b\u7684\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b</li> <li><code>*</code>\u200b\u89e3\u5305\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff08\u200b\u6309\u200b\u53c2\u6570\u200b\u4f4d\u7f6e\u200b\u4f20\u9012\u200b\uff09\u200b\u6216\u200b<code>**</code>\u200b\u89e3\u5305\u200b\u6620\u5c04\u200b\u5bf9\u8c61\u200b\uff08\u200b\u6309\u200b\u5173\u952e\u5b57\u200b\u4f20\u9012\u200b\uff09</li> </ul> <p>\u200b\u5bf9\u5e94\u200b\u5730\u200b\uff0c\u200b\u51fd\u6570\u200b\u5bf9\u200b\u53c2\u6570\u200b\u6709\u200b\u5982\u4e0b\u200b\u5904\u7406\u200b\u65b9\u5f0f\u200b\uff1a</p> <ul> <li>\u200b\u6309\u7167\u200b\u53c2\u6570\u200b\u4f4d\u7f6e\u200b\u4e0e\u200b\u53c2\u6570\u200b\u540d\u200b\u83b7\u53d6\u200b\u53c2\u6570\u200b</li> <li>\u200b\u5176\u4ed6\u200b\u7684\u200b\u65e0\u8bb0\u540d\u200b\u53c2\u6570\u200b\u88ab\u200b<code>*</code>\u200b\u53c2\u6570\u200b\u83b7\u53d6\u200b\uff0c\u200b\u5355\u4e2a\u200b<code>*</code>\u200b\u6216\u200b<code>*</code>\u200b\u53c2\u6570\u200b\u540e\u200b\u7684\u200b\u666e\u901a\u200b\u53c2\u6570\u200b\u53ea\u80fd\u200b\u6309\u7167\u200b\u5173\u952e\u5b57\u200b\u53c2\u6570\u200b\u7684\u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b</li> <li>\u200b\u5176\u4ed6\u200b\u7684\u200b\u8bb0\u540d\u200b\u53c2\u6570\u200b\u88ab\u200b<code>**</code>\u200b\u53c2\u6570\u200b\u83b7\u53d6\u200b</li> </ul> <p>\u200b\u51fd\u6570\u200b\u58f0\u660e\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u7684\u200b\u5355\u4e2a\u200b<code>*</code>\u200b\u4e0d\u4f1a\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u53c2\u6570\u200b\uff0c\u200b\u53ea\u200b\u7528\u4e8e\u200b\u9650\u5b9a\u200b\u6b64\u540e\u200b\u6240\u6709\u200b\u53c2\u6570\u200b\u7684\u200b\u4f20\u5165\u200b\u65b9\u5f0f\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; def a(*, a, b):\n...     return a + b\n... \n&gt;&gt;&gt; a(1, 2) \nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: a() takes 0 positional arguments but 2 were given\n&gt;&gt;&gt; a(a=1, b=2)\n3\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u51fd\u6570\u200b\u4f20\u53c2\u6709\u200b\u4e24\u79cd\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u5373\u503c\u200b\u4f20\u9012\u200b\u4e0e\u200b\u5f15\u7528\u200b\u4f20\u9012\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u53ef\u53d8\u200b\u5bf9\u8c61\u200b\u6309\u200b\u5f15\u7528\u200b\u4f20\u9012\u200b\uff0c\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5bf9\u8c61\u200b\u6309\u503c\u200b\u4f20\u9012\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; def argPass(x):\n...     x += x      \n...           \n&gt;&gt;&gt; a = \"test\"      \n&gt;&gt;&gt; b = [\"test\"]\n&gt;&gt;&gt; argPass(a)\n&gt;&gt;&gt; argPass(b)\n&gt;&gt;&gt; a\n'test'\n&gt;&gt;&gt; b\n['test', 'test']\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/functions/#_4","title":"\u51fd\u6570\u200b\u5185\u7701","text":"<p>\u200b\u51fd\u6570\u200b\u5bf9\u8c61\u200b\u7279\u6709\u200b\u7684\u200b\u5c5e\u6027\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u6709\u5173\u200b\u51fd\u6570\u200b\u7684\u200b\u4fe1\u606f\u200b\uff1a</p> \u200b\u5c5e\u6027\u200b\u540d\u200b \u200b\u5c5e\u6027\u200b\u7c7b\u578b\u200b \u200b\u5c5e\u6027\u200b\u8bf4\u660e\u200b <code>__annotations__</code> <code>dict</code> \u200b\u51fd\u6570\u53c2\u6570\u200b\u548c\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u7c7b\u578b\u200b\u6ce8\u89e3\u200b\uff0c\u200b\u952e\u4e3a\u200b\u53c2\u6570\u200b\u540d\u200b\uff0c\u200b\u503c\u4e3a\u200b\u53c2\u6570\u200b\u7c7b\u578b\u200b <code>__call__</code> <code>method-wrapper</code> \u200b\u5b9e\u73b0\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u7684\u200b\u8c03\u7528\u200b\u65b9\u6cd5\u200b <code>__closure__</code> <code>tuple</code> \u200b\u51fd\u6570\u200b\u95ed\u5305\u200b\uff0c\u200b\u53c2\u89c1\u200b\u76f8\u5173\u200b\u7ae0\u8282\u200b <code>__code__</code> <code>code</code> \u200b\u7f16\u8bd1\u6210\u200b\u4e8c\u8fdb\u5236\u200b\u683c\u5f0f\u200b\u7684\u200b\u51fd\u6570\u200b\u5143\u200b\u6570\u636e\u200b\u548c\u200b\u5b9a\u4e49\u200b\u4f53\u200b <code>__defaults__</code> <code>tuple</code> \u200b\u5f62\u53c2\u200b\u7684\u200b\u9ed8\u8ba4\u503c\u200b\u5217\u8868\u200b <code>__get__</code> <code>method-wrapper</code> \u200b\u5b9e\u73b0\u200b\u53ea\u8bfb\u200b\u63cf\u8ff0\u7b26\u200b\u534f\u8bae\u200b <code>__globals__</code> <code>dict</code> \u200b\u51fd\u6570\u200b\u6240\u5728\u200b\u6a21\u5757\u200b\u4e2d\u200b\u7684\u200b\u5168\u5c40\u53d8\u91cf\u200b <code>__kwdefaults__</code> <code>dict</code> \u200b\u9650\u5b9a\u200b\u5173\u952e\u5b57\u200b\u5f62\u5f0f\u200b\u8bbf\u95ee\u200b\u7684\u200b\u5f62\u53c2\u200b\u9ed8\u8ba4\u503c\u200b <code>__name__</code> <code>str</code> \u200b\u51fd\u6570\u200b\u540d\u79f0\u200b <code>__qualname__</code> <code>str</code> \u200b\u51fd\u6570\u200b\u7684\u200b\u9650\u5b9a\u200b\u540d\u79f0\u200b <pre><code>&gt;&gt;&gt; def dummy(a, b: int, c = 10, *args, d, e=50, **kwargs):\n...     return (a, b, c, args, d, e, kwargs) \n... \n&gt;&gt;&gt; dummy.__annotations__\n{'b': &lt;class 'int'&gt;}\n&gt;&gt;&gt; dummy.__defaults__\n(10,)\n&gt;&gt;&gt; dummy.__kwdefaults__\n{'e': 50}\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u51fd\u6570\u200b\u7684\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\u4e0e\u200b\u6570\u91cf\u200b\u5b58\u50a8\u200b\u5728\u200b<code>__code__</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; dummy.__code__.co_varnames\n('a', 'b', 'c', 'd', 'e', 'args', 'kwargs')\n&gt;&gt;&gt; dummy.__code__.co_argcount\n3\n&gt;&gt;&gt;\n</code></pre> <p><code>inspect</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u68c0\u67e5\u200b\u51fd\u6570\u53c2\u6570\u200b\u4fe1\u606f\u200b\u7684\u200b\u5de5\u5177\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import inspect\n&gt;&gt;&gt; inspect.signature(dummy)\n&lt;Signature (a, b: int, c=10, *args, d, e=50, **kwargs)&gt;\n&gt;&gt;&gt; sig = inspect.signature(dummy) \n&gt;&gt;&gt; sig.parameters\nmappingproxy(OrderedDict([('a', &lt;Parameter \"a\"&gt;), ('b', &lt;Parameter \"b: int\"&gt;), ('c', &lt;Parameter \"c=10\"&gt;), ('args', &lt;Parameter \"*args\"&gt;), ('d', &lt;Parameter \"d\"&gt;), ('e', &lt;Parameter \"e=50\"&gt;), ('kwargs', &lt;Parameter \"**kwargs\"&gt;)]))\n&gt;&gt;&gt;\n</code></pre> <p><code>parameters</code>\u200b\u5b58\u50a8\u200b\u4e86\u200b\u53c2\u6570\u200b\u540d\u200b\u4e0e\u200b\u53c2\u6570\u4fe1\u606f\u200b\u7684\u200b\u6620\u5c04\u200b\u5173\u7cfb\u200b\u3002\u200b\u6bcf\u4e2a\u200b<code>Parameter</code>\u200b\u5bf9\u8c61\u200b\u6709\u200b<code>name</code>\u3001<code>default</code>\u200b\u548c\u200b<code>kind</code>\u200b\u5c5e\u6027\u200b\u3002<code>dummy</code>\u200b\u51fd\u6570\u200b\u5404\u4e2a\u200b\u53c2\u6570\u200b\u7684\u200b\u5c5e\u6027\u200b\u4fe1\u606f\u200b\u5982\u4e0b\u200b\uff0c<code>inspect._empty</code>\u200b\u8868\u793a\u200b\u53c2\u6570\u200b\u6ca1\u6709\u200b\u9ed8\u8ba4\u503c\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; par = sig.parameters \n&gt;&gt;&gt; for parName in par:\n...     print(par[parName].name, par[parName].default, par[parName].kind)\n... \na &lt;class 'inspect._empty'&gt; POSITIONAL_OR_KEYWORD\nb &lt;class 'inspect._empty'&gt; POSITIONAL_OR_KEYWORD\nc 10 POSITIONAL_OR_KEYWORD\nargs &lt;class 'inspect._empty'&gt; VAR_POSITIONAL\nd &lt;class 'inspect._empty'&gt; KEYWORD_ONLY\ne 50 KEYWORD_ONLY\nkwargs &lt;class 'inspect._empty'&gt; VAR_KEYWORD\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u4ece\u200b\u8f93\u51fa\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u770b\u51fa\u200b\uff0c<code>dummy</code>\u200b\u5171\u6709\u200b\u56db\u79cd\u200b\u7c7b\u578b\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5373\u200b\uff1a</p> <ul> <li><code>POSITIONAL_OR_KEYWORD</code>\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u5b9a\u4f4d\u200b\u53c2\u6570\u200b\u6216\u200b\u5173\u952e\u5b57\u200b\u53c2\u6570\u200b\u4f20\u5165\u200b</li> <li><code>VAR_POSITIONAL</code>\uff0c\u200b\u591a\u4f59\u200b\u7684\u200b\u5b9a\u4f4d\u200b\u53c2\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u5143\u7ec4\u200b</li> <li><code>KEYWORD_ONLY</code>\uff0c\u200b\u53ea\u80fd\u200b\u901a\u8fc7\u200b\u5173\u952e\u5b57\u200b\u53c2\u6570\u200b\u4f20\u5165\u200b</li> <li><code>VAR_KEYWORD</code>\uff0c\u200b\u591a\u4f59\u200b\u7684\u200b\u5173\u952e\u5b57\u200b\u53c2\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u5b57\u5178\u200b</li> </ul> <p>\u200b\u53e6\u200b\u4e00\u79cd\u200b\u53c2\u6570\u200b\u7c7b\u578b\u200b\u662f\u200b\u53ea\u80fd\u200b\u901a\u8fc7\u200b\u5b9a\u4f4d\u200b\u53c2\u6570\u200b\u4f20\u5165\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5373\u200b<code>POSITIONAL_ONLY</code>\uff0cPython\u200b\u8bed\u6cd5\u200b\u4e0d\u200b\u652f\u6301\u200b\u58f0\u660e\u200b\u8fd9\u79cd\u200b\u7c7b\u578b\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5728\u200bC\u200b\u8bed\u8a00\u200bAPI\u200b\u4e2d\u200b\u53ef\u80fd\u200b\u51fa\u73b0\u200b\u3002</p> <p>Changed in version 3.8</p> <p>\u200b\u5728\u200b\u51fd\u6570\u200b\u5b9a\u4e49\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>/</code>\u200b\u5206\u9694\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\uff0c\u200b\u51fa\u73b0\u200b\u5728\u200b<code>/</code>\u200b\u4e4b\u524d\u200b\u7684\u200b\u53c2\u6570\u200b\u5fc5\u987b\u200b\u901a\u8fc7\u200b\u5b9a\u4f4d\u200b\u53c2\u6570\u200b\u4f20\u5165\u200b\u3002</p> <pre><code>&gt;&gt;&gt; def test(a, /, b, *args, c, **kwargs):\n...     pass\n...\n&gt;&gt;&gt; import inspect\n&gt;&gt;&gt; sig = inspect.signature(test)\n&gt;&gt;&gt; for val in sig.parameters.values():\n...     print(val.name, val.kind)\n...\na POSITIONAL_ONLY\nb POSITIONAL_OR_KEYWORD\nargs VAR_POSITIONAL\nc KEYWORD_ONLY\nkwargs VAR_KEYWORD\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/functions/#_5","title":"\u53c2\u6570\u200b\u7684\u200b\u7c7b\u578b\u200b\u9650\u5236","text":"<p>Python\u200b\u4e2d\u200b\u5bf9\u200b\u51fd\u6570\u53c2\u6570\u200b\u7684\u200b\u7c7b\u578b\u200b\u68c0\u67e5\u200b\u4ee5\u200b\u51fd\u6570\u200b\u6ce8\u89e3\u200b\u7684\u200b\u5f62\u5f0f\u200b\u5b9e\u73b0\u200b\u3002\u200b\u51fd\u6570\u200b\u6ce8\u89e3\u200b\u89c4\u5b9a\u200b\u4e86\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u63a5\u53d7\u200b\u7684\u200b\u53c2\u6570\u200b\u7c7b\u578b\u200b\u3002</p> <p>\u200b\u51fd\u6570\u200b\u6ce8\u89e3\u200b\u53ef\u4ee5\u200b\u4e3a\u200b\u7c7b\u578b\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u3002<code>:</code>\u200b\u7528\u4e8e\u200b\u5bf9\u200b\u53c2\u6570\u200b\u7684\u200b\u6ce8\u89e3\u200b\uff0c<code>-&gt;</code>\u200b\u8868\u793a\u200b\u5bf9\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u6ce8\u89e3\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; def another_dummy(x: int) -&gt; int:       \n...     return -x\n... \n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6ce8\u610f\u200b\uff0c\u200b\u6ce8\u89e3\u200b\u5e76\u200b\u4e0d\u4f1a\u200b\u5bfc\u81f4\u200bPython\u200b\u5bf9\u200b\u4f20\u5165\u200b\u53c2\u6570\u200b\u7684\u200b\u7c7b\u578b\u200b\u8fdb\u884c\u200b\u68c0\u67e5\u200b\u3002</p> <pre><code>&gt;&gt;&gt; another_dummy(0.5)         \n-0.5\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6240\u6709\u200b\u7684\u200b\u6ce8\u89e3\u200b\u5b58\u50a8\u200b\u5728\u200b\u51fd\u6570\u200b\u7684\u200b<code>__annotations__</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\u3002</p> <pre><code>&gt;&gt;&gt; another_dummy.__annotations__\n{'x': &lt;class 'int'&gt;, 'return': &lt;class 'int'&gt;}\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/functions/#_6","title":"\u51fd\u6570\u200b\u5f0f\u200b\u7f16\u7a0b","text":""},{"location":"coding/python/functions/#operator","title":"operator\u200b\u6a21\u5757","text":"<p><code>operator</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\u4e86\u200bPython\u200b\u4e2d\u200b\u90e8\u5206\u200b\u8fd0\u7b97\u7b26\u200b\u6216\u200b\u64cd\u4f5c\u200b\u7684\u200b\u51fd\u6570\u200b\u5f0f\u200b\u5b9e\u73b0\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import operator\n&gt;&gt;&gt; operator.add(5, 3)\n8\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6b64\u5916\u200b\uff0c<code>__getitem__</code>\u3001<code>__getattribute__</code>\u200b\u65b9\u6cd5\u200b\u5728\u200b<code>operator</code>\u200b\u6a21\u5757\u200b\u4e2d\u200b\u4e5f\u200b\u6709\u200b\u5bf9\u5e94\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u4e3a\u200b<code>itemgetter</code>\u3001<code>attrgetter</code>\u3002<code>methodcaller</code>\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u5185\u200b\u7684\u200b\u67d0\u4e2a\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u4e0e\u200b<code>operator</code>\u200b\u6a21\u5757\u200b\u4e2d\u200b\u7684\u200b\u5176\u4ed6\u200b\u51fd\u6570\u200b\u4e0d\u540c\u200b\uff0c\u200b\u8c03\u7528\u200b\u8be5\u200b\u51fd\u6570\u200b\u4f1a\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u3002</p> <pre><code>&gt;&gt;&gt; sort = operator.methodcaller('sort')\n&gt;&gt;&gt; sort_rev = operator.methodcaller('sort', reverse=True)\n&gt;&gt;&gt; a = [1, 3, 2]\n&gt;&gt;&gt; sort(a)\n&gt;&gt;&gt; a\n[1, 2, 3]\n&gt;&gt;&gt; sort_rev(a)\n&gt;&gt;&gt; a\n[3, 2, 1]\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/functions/#functools","title":"functools\u200b\u6a21\u5757","text":"<p>functools\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u4e0e\u200b\u51fd\u6570\u200b\u5f0f\u200b\u7f16\u7a0b\u200b\u76f8\u5173\u200b\u7684\u200b\u4e00\u7cfb\u5217\u200b\u51fd\u6570\u200b\u5de5\u5177\u200b\u3002</p> <p><code>partial</code>\u200b\u7528\u4e8e\u200b\u9501\u5b9a\u200b\u51fd\u6570\u200b\u7684\u200b\u4e00\u90e8\u5206\u200b\u53c2\u6570\u200b\u3002\u200b\u5176\u4f59\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u5728\u200b\u8c03\u7528\u200b\u65f6\u200b\u6307\u5b9a\u200b\uff08\u200b\u5b9a\u4f4d\u200b\u53c2\u6570\u200b\u53ea\u80fd\u200b\u6309\u200b\u987a\u5e8f\u200b\u6307\u5b9a\u200b\uff0c\u200b\u5173\u952e\u5b57\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u4efb\u610f\u200b\u6307\u5b9a\u200b\uff09\uff1a</p> <pre><code>&gt;&gt;&gt; from functools import partial\n&gt;&gt;&gt; part = partial(divmod, 3) \n&gt;&gt;&gt; part(2)\n(1, 1)\n&gt;&gt;&gt;\n</code></pre> <p><code>reduce</code>\u200b\u5c06\u200b\u63a5\u53d7\u200b\u4e24\u4e2a\u200b\u53c2\u6570\u200b\u7684\u200b\u51fd\u6570\u200b\u5e94\u7528\u200b\u5230\u200b\u4e00\u4e2a\u200b\u5e8f\u5217\u200b\u4e0a\u200b\uff0c\u200b\u5f97\u5230\u200b\u4e00\u4e2a\u200b\u7ed3\u679c\u200b\u3002\u200b\u501f\u52a9\u200b<code>operator</code>\u200b\u6a21\u5757\u200b\u4e2d\u200b\u63d0\u4f9b\u200b\u7684\u200b\u8fd0\u7b97\u7b26\u200b\uff0c<code>reduce</code>\u200b\u6a21\u5757\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u9636\u4e58\u200b\u7b49\u200b\u9700\u8981\u200b\u5faa\u73af\u200b\u624d\u80fd\u200b\u5b9e\u73b0\u200b\u7684\u200b\u8fd0\u7b97\u200b\u64cd\u4f5c\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; from functools import reduce\n&gt;&gt;&gt; def fact(x: int):\n...     return reduce(operator.mul, range(1, x + 1))\n... \n&gt;&gt;&gt; fact(5)\n120\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/killable-thread/","title":"Python\u200b\u5173\u95ed\u200b\u7ebf\u7a0b","text":"<p>\u200b\u4f7f\u7528\u200b<code>PyThreadState_SetAsyncExc</code>\u200b\u51fd\u6570\u200b\u5728\u5b50\u200b\u7ebf\u7a0b\u200b\u4e2d\u200b\u5f15\u53d1\u200b\u5f02\u5e38\u200b\uff0c\u200b\u4ece\u800c\u200b\u4f7f\u200b\u7ebf\u7a0b\u200b\u63d0\u524d\u7ed3\u675f\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>import threading \nimport ctypes \nimport time \n\nclass thread_with_exception(threading.Thread): \n    def __init__(self, name): \n        threading.Thread.__init__(self) \n        self.name = name \n\n    def run(self): \n\n        # target function of the thread class \n        try: \n            while True: \n                print('running ' + self.name) \n        finally: \n            print('ended') \n\n    def get_id(self): \n\n        # returns id of the respective thread \n        if hasattr(self, '_thread_id'): \n            return self._thread_id \n        for id, thread in threading._active.items(): \n            if thread is self: \n                return id\n\n    def raise_exception(self): \n        thread_id = self.get_id() \n        res = ctypes.pythonapi.PyThreadState_SetAsyncExc(thread_id, \n              ctypes.py_object(SystemExit)) \n        if res &gt; 1: \n            ctypes.pythonapi.PyThreadState_SetAsyncExc(thread_id, 0) \n            print('Exception raise failure') \n\nt1 = thread_with_exception('Thread 1') \nt1.start() \ntime.sleep(2) \nt1.raise_exception() \nt1.join() \n</code></pre> <p>\u200b\u628a\u200b<code>run()</code>\u200b\u4e2d\u200b\u6267\u884c\u200b\u7684\u200b\u64cd\u4f5c\u200b\u4ee5\u200b\u53c2\u6570\u200b\u7684\u200b\u5f62\u5f0f\u200b\u4f20\u9012\u200b\uff1a</p> <pre><code>class KillableThread(threading.Thread):\n    \"\"\"\n    A thread class extending threading.Thread, provides a kill() method to stop the thread and a getResult() method to get the return value of the thread.\n    \"\"\"\n\n    def __init__(self, func: Callable, *args, **kwargs):\n        super().__init__()\n        self.func: Callable = func\n        self.funcArg: dict = kwargs\n        self.funcTup: Tuple = args\n        self.result: Any = None\n        self.exception: Any = None\n\n    def run(self):\n        \"\"\"\n        Executes the function here\n        \"\"\"\n        try:\n            self.result = self.func(*self.funcTup, **self.funcArg)\n        except BaseException as e:\n            self.exception = e\n\n    def get_id(self):\n        \"\"\"\n        Get the id of the thread\n        \"\"\"\n        if hasattr(self, '_thread_id'):\n            return self._thread_id\n        for id, thread in threading._active.items():\n            if thread is self:\n                return id\n\n    def kill(self):\n        \"\"\"\n        Stops the thread\n        \"\"\"\n        thread_id = self.get_id()\n        res = ctypes.pythonapi.PyThreadState_SetAsyncExc(thread_id,\n                                                         ctypes.py_object(SystemExit))\n        if res &gt; 1:\n            ctypes.pythonapi.PyThreadState_SetAsyncExc(thread_id, 0)\n            print('Exception raise failure')\n\n    def getResult(self):\n        if self.exception is None:\n            return self.result\n        else:\n            raise self.exception\n</code></pre> <p>\u200b\u6765\u6e90\u200b\uff1ahttps://www.geeksforgeeks.org/python-different-ways-to-kill-a-thread</p>"},{"location":"coding/python/magic-functions/","title":"\u9b54\u672f\u200b\u65b9\u6cd5","text":"<p>\u200b\u672c\u6587\u200b\u4e3b\u8981\u200b\u6db5\u76d6\u200bPython\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\u3002\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\u662f\u200b\u7c7b\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u4e00\u79cd\u200b\u7279\u6b8a\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u65b9\u6cd5\u200b\u62e5\u6709\u200b\u56fa\u5b9a\u200b\u7684\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\u548c\u200b\u7279\u6b8a\u200b\u7684\u200b\u8c03\u7528\u200b\u65f6\u673a\u200b\u3002</p> <p>\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\u4ee5\u53cc\u200b\u4e0b\u5212\u7ebf\u200b<code>__</code>\u200b\u5f00\u5934\u200b\uff0c\u200b\u4ee5\u53cc\u200b\u4e0b\u5212\u7ebf\u200b\u7ed3\u5c3e\u200b\u3002</p> <p>Python\u200b\u4e2d\u200b\u7684\u200b\u8fd0\u7b97\u7b26\u200b\u91cd\u8f7d\u200b\u901a\u8fc7\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\u5206\u4e3a\u200b\u4e0e\u200b\u8fd0\u7b97\u7b26\u200b\u65e0\u5173\u200b\u7684\u200b\u7279\u6b8a\u200b\u65b9\u6cd5\u200b\u53ca\u200b\u4e0e\u200b\u8fd0\u7b97\u7b26\u200b\u6709\u5173\u200b\u7684\u200b\u7279\u6b8a\u200b\u65b9\u6cd5\u200b</p>"},{"location":"coding/python/magic-functions/#_2","title":"\u4e0e\u200b\u8fd0\u7b97\u7b26\u200b\u65e0\u5173\u200b\u7684\u200b\u9b54\u672f\u200b\u65b9\u6cd5","text":"<p>\u200b\u6d89\u53ca\u200b\u5230\u200b\u5b57\u7b26\u4e32\u200b\u4e0e\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u7684\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\uff1a</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__repr__</code> <code>self</code> \u200b\u5f53\u200b\u8c03\u7528\u200b<code>repr()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>str</code> \u200b\u8fd4\u56de\u503c\u200b\u5e94\u4e3a\u200b\u91cd\u5efa\u200b\u8be5\u200b\u5bf9\u8c61\u200b\u7684\u200bPython\u200b\u8868\u8fbe\u5f0f\u200b\uff08\u200b\u5982\u679c\u200b\u53ef\u80fd\u200b\uff09\uff0c\u200b\u6216\u200b\u5305\u542b\u200b\u5176\u4ed6\u200b\u4fe1\u606f\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b<code>&lt;...&gt;</code> <code>__str__</code> <code>self</code> \u200b\u5f53\u200b\u8c03\u7528\u200b<code>str()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>str</code> \u200b\u8c03\u7528\u200b<code>print()</code>\u200b\u7b49\u200b\u5176\u4ed6\u200b\u51fd\u6570\u200b\u65f6\u4f1a\u200b\u9690\u5f0f\u200b\u8c03\u7528\u200b<code>__str__()</code>\uff0c\u200b\u5f53\u200b\u4e0d\u200b\u5b58\u5728\u200b<code>__str__()</code>\u200b\u65f6\u4f1a\u200b\u81ea\u52a8\u200b\u8c03\u7528\u200b<code>__repr__()</code> <code>__format__</code> <code>self, format_spec</code> \u200b\u5f53\u200b\u8c03\u7528\u200b<code>format()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>str</code> <code>object.__format__(x, '')</code>\u200b\u7b49\u4ef7\u200b\u4e8e\u200b<code>str(x)</code> <code>__bytes__</code> <code>self</code> \u200b\u5f53\u200b\u8c03\u7528\u200b<code>bytes()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>bytes</code> \u200b\u8fd4\u56de\u503c\u200b\u662f\u200b\u5bf9\u8c61\u200b\u7684\u200b\u5b57\u8282\u200b\u8868\u793a\u200b <p>\u200b\u6d89\u53ca\u200b\u5230\u200b\u6570\u503c\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u7684\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\uff1a</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__bool__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>bool()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>bool</code> \u200b\u5f53\u200b<code>__bool__</code>\u200b\u51fd\u6570\u200b\u6ca1\u6709\u200b\u5b9a\u4e49\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u9690\u5f0f\u200b\u8c03\u7528\u200b<code>__len__</code>\uff0c\u200b\u5982\u679c\u200b<code>__len__</code>\u200b\u4e5f\u200b\u6ca1\u6709\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>True</code> <code>__complex__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>commplex()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>complex</code> <code>__int__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>int()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>int</code> <code>__float__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>float()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>float</code> <code>__hash__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>hash()</code> \u200b\u51fd\u6570\u200b\u65f6\u200b <code>hash()</code>\u200b\u51fd\u6570\u200b\u5c06\u200b<code>__hash__()</code>\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u622a\u65ad\u200b\u5230\u200b<code>Py_ssize_t</code>\u200b\u7684\u200b\u8303\u56f4\u200b\u5185\u200b\uff0c64\u200b\u4f4d\u200b\u7cfb\u7edf\u200b\u4e0b\u200b<code>Py_ssize_t</code>\u200b\u4e3a\u200b8\u200b\u5b57\u8282\u200b\uff1b<code>set</code>\u200b\u7b49\u200b\u7c7b\u578b\u200b\u7684\u200b\u64cd\u4f5c\u200b\u4e5f\u200b\u4f1a\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u7684\u200b<code>__hash__</code>\u200b\u51fd\u6570\u200b <code>__index__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>operator.index()</code>\u200b\u6216\u200bPython\u200b\u9700\u8981\u200b\u6574\u6570\u200b\u503c\u65f6\u200b <code>int</code> \u200b\u5f53\u200b<code>__int__</code>\u3001<code>__float__</code>\u3001<code>__complex__</code>\u200b\u6ca1\u6709\u200b\u5b9e\u73b0\u200b\u65f6\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u51fd\u6570\u200b\u9690\u5f0f\u200b\u5730\u200b\u8c03\u7528\u200b<code>__index__</code> <p>\u200b\u96c6\u5408\u200b\u76f8\u5173\u200b\u64cd\u4f5c\u200b</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__len__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>len()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>int</code> \u200b\u8fd4\u56de\u503c\u200b\u5fc5\u987b\u200b\u4e3a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\uff0c\u200b\u5728\u200bCPython\u200b\u4e2d\u200b\u8fd4\u56de\u503c\u200b\u4e0d\u80fd\u200b\u8d85\u8fc7\u200b<code>sys.maxsize</code>\u200b\u4e2d\u200b\u5b9a\u4e49\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u5426\u5219\u200b<code>len()</code>\u200b\u7b49\u200b\u51fd\u6570\u200b\u4f1a\u200b\u629b\u51fa\u200b<code>OverflowError</code>\u200b\u7b49\u200b\u5f02\u5e38\u200b <code>__getitemm__</code> <code>self, key</code> \u200b\u4f7f\u7528\u200b<code>[]</code>\u200b\u8fd0\u7b97\u7b26\u200b\u65f6\u200b\uff08\u200b\u53f3\u503c\u200b\uff09 <code>key</code>\u200b\u53ef\u4ee5\u200b\u4e3a\u200b\u5408\u6cd5\u200b\u7684\u200b\u6570\u7ec4\u200b\u4e0b\u6807\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u53cd\u5411\u200b\u7684\u200b\u4e0b\u6807\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u5e8f\u5217\u200b\u7684\u200b\u5207\u7247\u200b\uff0c\u200b\u8be5\u200b\u51fd\u6570\u200b\u5e94\u200b\u5728\u200b<code>key</code>\u200b\u7c7b\u578b\u200b\u4e0d\u200b\u5bf9\u5e94\u200b\u65f6\u200b\u629b\u51fa\u200b<code>TypeError</code>\uff0c<code>key</code>\u200b\u8d85\u51fa\u8303\u56f4\u200b\u65f6\u200b\u629b\u51fa\u200b<code>IndexError</code>\u200b\u6216\u200b<code>KeyError</code>\uff08\u200b\u5bf9\u4e8e\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\uff09 <code>__setitem__</code> <code>self, key, value</code> \u200b\u4f7f\u7528\u200b<code>[]</code>\u200b\u8fd0\u7b97\u7b26\u200b\u65f6\u200b\uff08\u200b\u5de6\u503c\u200b\uff09 <code>None</code> <code>key</code>\u200b\u53c2\u6570\u200b\u7684\u200b\u53d6\u503c\u200b\u53ca\u200b\u5f02\u5e38\u200b\u5904\u7406\u200b\u4e0e\u200b<code>__getitem__</code>\u200b\u51fd\u6570\u200b\u76f8\u540c\u200b <code>__delitem__</code> <code>self, key</code> \u200b\u4f7f\u7528\u200b<code>del</code>\u200b\u5220\u9664\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u65f6\u200b <code>None</code> \u200b\u5bf9\u4e8e\u200b\u4e0d\u53ef\u200b\u5220\u9664\u200b\u7684\u200b\u5e8f\u5217\u200b\u6216\u200b\u6620\u5c04\u200b\uff0c\u200b\u65e0\u9700\u200b\u5b9e\u73b0\u200b\u8be5\u200b\u51fd\u6570\u200b\uff0c\u200b\u5176\u4ed6\u200b\u540c\u200b<code>__getitem__</code> <code>__contains__</code> <code>self, item</code> \u200b\u4f7f\u7528\u200b<code>in</code>\u200b\u4f5c\u4e3a\u200b\u8fd0\u7b97\u7b26\u200b\u65f6\u200b <code>bool</code> \u200b\u5bf9\u4e8e\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\uff0c<code>__contains__</code>\u200b\u63a5\u6536\u200b\u7684\u200b<code>item</code>\u200b\u53c2\u6570\u200b\u5bf9\u5e94\u200b\u6620\u5c04\u200b\u7684\u200b<code>key</code>\u200b\u90e8\u5206\u200b\u800c\u200b\u4e0d\u662f\u200b<code>value</code>\u200b\u90e8\u5206\u200b\u3002\u200b\u82e5\u200b\u6ca1\u6709\u200b\u5b9a\u4e49\u200b\u8be5\u200b\u51fd\u6570\u200b\uff0cPython\u200b\u9996\u5148\u200b\u901a\u8fc7\u200b<code>__iter__</code>\u200b\u8fdb\u884c\u200b\u67e5\u627e\u200b\uff0c\u200b\u5176\u6b21\u200b\u4f7f\u7528\u200b<code>__getitem__</code>\u200b\u8fdb\u884c\u200b\u67e5\u627e\u200b <p>\u200b\u8fed\u4ee3\u200b\u5668\u200b\u4e0e\u200b\u679a\u4e3e\u200b\u7c7b\u578b\u200b</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__iter__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>iter()</code>\u200b\u51fd\u6570\u200b\u6216\u200b\u4f7f\u7528\u200b\u5faa\u73af\u200b\u65f6\u200b \u200b\u8fd4\u56de\u503c\u200b\u5e94\u4e3a\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff0c<code>__iter__</code>\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u81ea\u8eab\u200b <code>__reversed__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>reversed()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b \u200b\u8fd4\u56de\u503c\u200b\u5e94\u4e3a\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u4e14\u200b\u662f\u200b\u539f\u200b\u5e8f\u5217\u200b\u7684\u200b\u9006\u5e8f\u200b <code>__next__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>next()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b \u200b\u8fd4\u56de\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u5f53\u200b\u5230\u8fbe\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u65f6\u200b\u629b\u51fa\u200b<code>StopIteration</code> <p>\u200b\u51fd\u6570\u8c03\u7528\u200b</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__call__</code> <code>self, *args, **kwargs</code> \u200b\u5f53\u200b\u5bf9\u8c61\u200b\u4ee5\u200b\u51fd\u6570\u200b\u7684\u200b\u65b9\u5f0f\u200b\u88ab\u200b\u8c03\u7528\u200b\u65f6\u200b \u200b\u5b9a\u4e49\u200b\u4e86\u200b<code>__call__</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u5bf9\u8c61\u200b\u5c5e\u4e8e\u200b<code>typing.Callable</code>\u200b\u7c7b\u578b\u200b <p>\u200b\u4e0a\u4e0b\u6587\u200b\u7ba1\u7406\u200b\uff08<code>with</code>\uff09\u200b\u5173\u952e\u8bcd\u200b\uff1a</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__enter__</code> <code>self</code> \u200b\u4f7f\u7528\u200b<code>with</code>\u200b\u5173\u952e\u8bcd\u200b\u58f0\u660e\u200b\u4e0a\u4e0b\u6587\u200b\u65f6\u200b <code>Any</code> \u200b\u8fd4\u56de\u503c\u200b\u5206\u914d\u200b\u7ed9\u200b<code>as</code>\u200b\u5173\u952e\u8bcd\u200b\u6307\u660e\u200b\u7684\u200b\u5bf9\u8c61\u200b <code>__exit__</code> <code>self, exc_type, exc_value, traceback</code> \u200b\u79bb\u5f00\u200b<code>with</code>\u200b\u5757\u200b\u65f6\u200b <code>Any</code> \u200b\u82e5\u200b<code>with</code>\u200b\u5757\u200b\u6267\u884c\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u6ca1\u6709\u200b\u51fa\u73b0\u5f02\u5e38\u200b\uff0c\u200b\u5219\u200b<code>exc_type, exc_value, traceback</code>\u200b\u53c2\u6570\u5747\u200b\u4e3a\u200b<code>None</code>\uff0c\u200b\u5426\u5219\u200b<code>exit</code>\u200b\u5757\u200b\u53ef\u4ee5\u200b\u5bf9\u200b\u5f02\u5e38\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b <p>\u200b\u5bf9\u8c61\u200b\u7ba1\u7406\u200b</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__new__</code> <code>cls[, ...]</code> \u200b\u521b\u5efa\u200b\u7c7b\u200b\u5b9e\u4f8b\u200b\u65f6\u200b <code>Any</code> <code>__new__</code>\u200b\u4e3a\u200b\u9759\u6001\u65b9\u6cd5\u200b\u4e14\u200b\u5148\u4e8e\u200b<code>__init__</code>\u200b\u6267\u884c\u200b\uff0c<code>__new__</code>\u200b\u7684\u200b\u5176\u4f59\u200b\u53c2\u6570\u200b\u4f1a\u200b\u4f20\u9012\u200b\u7ed9\u200b<code>__init__</code>\u3002\u200b\u82e5\u200b<code>__new__</code>\u200b\u8fd4\u56de\u200b\u7684\u200b\u7c7b\u578b\u200b\u4e0e\u200b\u521b\u5efa\u200b\u7684\u200b\u7c7b\u578b\u200b\u4e0d\u200b\u5bf9\u5e94\u200b\uff0c\u200b\u5219\u200b\u4e0d\u4f1a\u200b\u6267\u884c\u200b<code>__init__</code> <code>__init__</code> <code>self, ...</code> \u200b\u521d\u59cb\u5316\u200b\u7c7b\u200b\u5b9e\u4f8b\u200b\u65f6\u200b <code>None</code> \u200b\u82e5\u200b\u57fa\u7c7b\u200b\u5b9a\u4e49\u200b\u4e86\u200b<code>__init__</code>\uff0c\u200b\u5219\u200b\u5728\u200b\u5b50\u7c7b\u200b\u7684\u200b<code>__init__</code>\u200b\u4e2d\u200b\u5fc5\u987b\u200b\u6709\u200b<code>super().__init()</code>\u200b\u7684\u200b\u663e\u5f0f\u200b\u8c03\u7528\u200b <code>__del__</code> <code>self</code> \u200b\u5bf9\u8c61\u200b\u9500\u6bc1\u200b\u65f6\u200b <code>None</code> \u200b\u5f53\u200b\u5bf9\u8c61\u200b\u7684\u200b\u5f15\u7528\u200b\u8ba1\u6570\u200b\u51cf\u200b\u4e3a\u200b0\u200b\u65f6\u200b\uff0c\u200b\u5bf9\u8c61\u200b\u624d\u200b\u53ef\u80fd\u200b\u88ab\u200b\u5220\u9664\u200b <p>\u200b\u5c5e\u6027\u200b\u7ba1\u7406\u200b</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__getattribute__</code> <code>self, name</code> \u200b\u4f7f\u7528\u200b<code>self.name</code>\u200b\u8bbf\u95ee\u200b\u5bf9\u8c61\u200b\u7684\u200b\u5c5e\u6027\u200b\u65f6\u200b <code>Any</code> \u200b\u4e3a\u200b\u907f\u514d\u200b\u65e0\u9650\u200b\u9012\u5f52\u200b\uff0c\u200b\u5b50\u7c7b\u200b\u7684\u200b<code>__getattribute__</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u5b9e\u73b0\u200b\u4e2d\u200b\u5fc5\u987b\u200b\u5305\u542b\u200b\u7236\u7c7b\u200b\u7684\u200b<code>__getattribute__</code>\u200b\u65b9\u6cd5\u200b\u8c03\u7528\u200b <code>__getattr__</code> <code>self, name</code> \u200b\u4f7f\u7528\u200b<code>self.name</code>\u200b\u8bbf\u95ee\u200b\u5bf9\u8c61\u200b\u7684\u200b\u4e0d\u200b\u5b58\u5728\u200b\u5c5e\u6027\u200b\u6216\u200b<code>__getattribute__</code>\u200b\u51fd\u6570\u200b\u629b\u51fa\u200b<code>AttributeError</code>\u200b\u5f02\u5e38\u200b <code>Any</code> \u200b\u5f53\u200b<code>__getattribute__</code>\u200b\u6b63\u5e38\u200b\u6267\u884c\u200b\u65f6\u200b<code>__getattr__</code>\u200b\u4e0d\u4f1a\u200b\u6267\u884c\u200b <code>__setattr__</code> <code>self, name, value</code> \u200b\u4f7f\u7528\u200b<code>self.name</code>\u200b\u8bbe\u7f6e\u200b\u5c5e\u6027\u200b\u7684\u200b\u503c\u65f6\u200b <code>None</code> \u200b\u5f53\u200b\u8bd5\u56fe\u200b\u901a\u8fc7\u200b<code>__setattr__</code>\u200b\u5411\u200b\u5bf9\u8c61\u200b\u8bbe\u7f6e\u200b\u5c5e\u6027\u200b\u503c\u65f6\u200b\u5fc5\u987b\u200b\u901a\u8fc7\u200b\u7c7b\u578b\u200b\u7684\u200b<code>__setattr__</code>\u200b\u65b9\u6cd5\u200b\u8fdb\u884c\u200b\u8bbe\u7f6e\u200b <code>__delattr__</code> <code>self, name</code> \u200b\u6267\u884c\u200b<code>del self.name</code>\u200b\u65f6\u200b <code>None</code> \u200b\u4ec5\u5f53\u200b<code>del self.name</code>\u200b\u8bed\u53e5\u200b\u6709\u200b\u610f\u4e49\u200b\u65f6\u200b\uff0c\u200b\u624d\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b\u8be5\u200b\u65b9\u6cd5\u200b <code>__dir__</code> <code>self</code> \u200b\u6267\u884c\u200b<code>dir()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>Sequence</code> <code>dir()</code>\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b\u5904\u7406"},{"location":"coding/python/magic-functions/#_3","title":"\u4e0e\u200b\u8fd0\u7b97\u7b26\u200b\u6709\u5173\u200b\u7684\u200b\u9b54\u672f\u200b\u65b9\u6cd5","text":"<p>\u200b\u4e00\u5143\u200b\u8fd0\u7b97\u7b26\u200b</p> \u200b\u63cf\u8ff0\u200b \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u8fd0\u7b97\u7b26\u200b \u200b\u53d6\u8d1f\u200b <code>__neg__</code> <code>-</code> \u200b\u53d6\u6b63\u200b <code>__pos__</code> <code>+</code> \u200b\u7edd\u5bf9\u503c\u200b <code>__abs__</code> <code>abs()</code> <p>\u200b\u6bd4\u8f83\u200b\u8fd0\u7b97\u7b26\u200b</p> \u200b\u63cf\u8ff0\u200b \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u8fd0\u7b97\u7b26\u200b \u200b\u5c0f\u4e8e\u200b <code>__lt__</code> <code>&lt;</code> \u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b <code>__le__</code> <code>&lt;=</code> \u200b\u7b49\u4e8e\u200b <code>__eq__</code> <code>==</code> \u200b\u4e0d\u200b\u7b49\u4e8e\u200b <code>__ne__</code> <code>!=</code> \u200b\u5927\u4e8e\u200b <code>__gt__</code> <code>&gt;</code> \u200b\u5927\u4e8e\u200b\u7b49\u4e8e\u200b <code>__ge__</code> <code>&gt;=</code> <p>\u200b\u7b97\u672f\u200b\u8fd0\u7b97\u7b26\u200b</p> \u200b\u63cf\u8ff0\u200b \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u8fd0\u7b97\u7b26\u200b \u200b\u76f8\u52a0\u200b <code>__add__</code> <code>+</code> \u200b\u76f8\u51cf\u200b <code>__sub__</code> <code>-</code> \u200b\u76f8\u4e58\u200b <code>__mul__</code> <code>*</code> \u200b\u5b9e\u6570\u200b\u9664\u200b <code>__truediv__</code> <code>/</code> \u200b\u6574\u6570\u200b\u9664\u200b <code>__floordiv__</code> <code>//</code> \u200b\u53d6\u4f59\u200b <code>__mod__</code> <code>%</code> \u200b\u6574\u6570\u200b\u9664\u200b+\u200b\u53d6\u4f59\u200b <code>__divmod__</code> <code>divmod()</code> \u200b\u4e58\u65b9\u200b <code>__pow__</code> <code>**</code> \u200b\u53d6\u6574\u200b <code>__round__</code> <code>round()</code> <p>\u200b\u53cd\u5411\u200b\u7b97\u672f\u200b\u8fd0\u7b97\u7b26\u200b\u5728\u200b\u65b9\u6cd5\u200b\u540d\u200b\u524d\u9762\u200b\u52a0\u200b<code>r</code>\uff0c\u200b\u5982\u200b<code>__radd__</code>\u3002\u200b\u589e\u91cf\u200b\u8d4b\u503c\u200b\u7b97\u672f\u200b\u8fd0\u7b97\u7b26\u200b\u5728\u200b\u65b9\u6cd5\u200b\u540d\u200b\u524d\u9762\u200b\u52a0\u200b<code>i</code>\uff0c\u200b\u5982\u200b<code>__iadd__</code>\u3002</p> <p>\u200b\u4f4d\u200b\u8fd0\u7b97\u7b26\u200b</p> \u200b\u63cf\u8ff0\u200b \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u8fd0\u7b97\u7b26\u200b \u200b\u6309\u4f4d\u200b\u53d6\u53cd\u200b <code>__invert__</code> <code>~</code> \u200b\u5de6\u79fb\u200b <code>__lshift__</code> <code>&lt;&lt;</code> \u200b\u53f3\u79fb\u200b <code>__rshift__</code> <code>&gt;&gt;</code> \u200b\u6309\u4f4d\u200b\u4e0e\u200b <code>__and__</code> <code>&amp;</code> \u200b\u6309\u4f4d\u200b\u6216\u200b <code>__or__</code> <code>|</code> \u200b\u6309\u4f4d\u200b\u5f02\u6216\u200b <code>__xor__</code> <code>^</code> <p>\u200b\u53cd\u5411\u200b\u7b97\u672f\u200b\u8fd0\u7b97\u7b26\u200b\u5728\u200b\u65b9\u6cd5\u200b\u540d\u200b\u524d\u9762\u200b\u52a0\u200b<code>r</code>\uff0c\u200b\u589e\u91cf\u200b\u8d4b\u503c\u200b\u7b97\u672f\u200b\u8fd0\u7b97\u7b26\u200b\u5728\u200b\u65b9\u6cd5\u200b\u540d\u200b\u524d\u9762\u200b\u52a0\u200b<code>i</code>\u3002</p>"},{"location":"coding/python/mapping/","title":"Python\u200b\u4e2d\u200b\u7684\u200b\u6620\u5c04\u200b\u7c7b\u578b","text":"<p>Python\u200b\u4e2d\u200b\u6700\u200b\u5e38\u89c1\u200b\u7684\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\u662f\u200b\u5b57\u5178\u200b\u3002\u200b\u672c\u6587\u200b\u5c06\u200b\u4ece\u200b\u5b57\u5178\u200b\u5165\u624b\u200b\u5bf9\u200bPython\u200b\u4e2d\u200b\u7684\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\u8fdb\u884c\u200b\u63a2\u8ba8\u200b\u3002</p>"},{"location":"coding/python/mapping/#_1","title":"\u5b57\u5178","text":"<p>\u200b\u5b57\u5178\u200b\u7c7b\u578b\u200b<code>dict</code>\u200b\u5bf9\u5e94\u200b\u7684\u200b\u62bd\u8c61\u200b\u57fa\u7c7b\u200b\u4e3a\u200b<code>MutableMapping</code>\u3002\u200b\u800c\u200b<code>MutableMapping</code>\u200b\u662f\u200b<code>Mapping</code>\u200b\u7684\u200b\u5b50\u7c7b\u200b\u3002</p> <p><code>dict</code>\u200b\u4e2d\u200b\u7684\u200b\u6570\u636e\u7ec4\u7ec7\u200b\u4e3a\u952e\u200b-\u200b\u503c\u200b\u5bf9\u200b\u7684\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u5b57\u5178\u200b\u5bf9\u503c\u200b\u7684\u200b\u7c7b\u578b\u200b\u6ca1\u6709\u200b\u8981\u6c42\u200b\uff0c\u200b\u4f46\u200b\u8981\u6c42\u200b\u952e\u200b\u7684\u200b\u7c7b\u578b\u200b\u5fc5\u987b\u200b\u53ef\u6563\u5217\u200b\uff08hashable\uff09\u3002\u200b\u7531\u6b64\u53ef\u89c1\u200b\uff0c\u200b\u5b57\u5178\u200b\u7ed3\u6784\u200b\u7684\u200b\u5e95\u5c42\u200b\u5b9e\u73b0\u200b\u7c7b\u4f3c\u200b\u6563\u200b\u5217\u8868\u200b\uff0c\u200b\u4ee5\u200b\u52a0\u5feb\u200b\u5b57\u5178\u200b\u5185\u200b\u5143\u7d20\u200b\u7684\u200b\u8bbf\u95ee\u901f\u5ea6\u200b\u3002\u200b\u5b57\u5178\u200b\u7684\u200b\u6267\u884c\u200b\u6548\u7387\u200b\u8fdc\u9ad8\u4e8e\u200b\u5217\u8868\u200b\u3002</p> <p>Python 3.6\u200b\u53ca\u200b\u4ee5\u540e\u200b\uff0c<code>dict</code>\u200b\u7c7b\u578b\u200b\u5728\u200b\u5b9e\u73b0\u200b\u4e0a\u200b\u662f\u200b\u6709\u5e8f\u200b\u7684\u200b\uff0c\u200b\u952e\u200b\u7684\u200b\u987a\u5e8f\u200b\u53d6\u51b3\u4e8e\u200b\u6dfb\u52a0\u200b\u987a\u5e8f\u200b\uff0c\u200b\u4f46\u200b\u6807\u51c6\u200b\u5e76\u200b\u6ca1\u6709\u200b\u89c4\u5b9a\u200b<code>dict</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u6709\u5e8f\u6027\u200b\u3002\u200b\u56e0\u6b64\u200b\u4e24\u4e2a\u200b\u952e\u200b\u987a\u5e8f\u200b\u4e0d\u540c\u200b\u7684\u200b\u5b57\u5178\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u76f8\u540c\u200b\u3002</p> <pre><code>&gt;&gt;&gt; a = {1: 2, 2: 1}\n&gt;&gt;&gt; b = {2: 1, 1: 2}\n&gt;&gt;&gt; a == b\nTrue\n&gt;&gt;&gt;\n</code></pre> <p>New in version 3.9</p> <p>Python 3.9\u200b\u4e2d\u200b\uff0c\u200b\u5b57\u5178\u200b\u5bf9\u8c61\u200b\u652f\u6301\u200b<code>|</code>\u200b\u8fd0\u7b97\u7b26\u200b\uff0c\u200b\u7528\u4e8e\u200b\u5c06\u200b\u4e24\u4e2a\u200b\u5b57\u5178\u200b\u5408\u5e76\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u3002\u200b\u5f53\u200b\u4e24\u4e2a\u200b\u5b57\u5178\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u91cd\u590d\u200b\u503c\u65f6\u200b\u53f3\u4fa7\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u4f18\u5148\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import sys \n&gt;&gt;&gt; ver = sys.version_info\n&gt;&gt;&gt; if ver.major == 3 and ver.minor &gt; 8:\n...     a = {1: 2}\n...     b = {3: 4}\n...     print(a | b == {1: 2, 3: 4})\n... else:\n...     print(True)\nTrue\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/mapping/#_2","title":"\u8fed\u4ee3","text":"<p><code>dict</code>\u200b\u5bf9\u8c61\u200b\u662f\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5f53\u5bf9\u200b<code>dict</code>\u200b\u5bf9\u8c61\u200b\u8fdb\u884c\u200b\u8fed\u4ee3\u200b\u65f6\u200b\uff0c\u200b\u5b9e\u9645\u4e0a\u200b\u662f\u200b\u8fed\u4ee3\u200b<code>dict</code>\u200b\u7684\u200b\u952e\u200b\u3002</p> <pre><code>&gt;&gt;&gt; for key in {1: 2, 3: 4}:\n...     print(key)\n...\n1\n3\n&gt;&gt;&gt;\n</code></pre> <ul> <li><code>dict.keys()</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b\u7531\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7684\u200b\u952e\u200b\u7ec4\u6210\u200b\u7684\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff1b</li> <li><code>dict.values()</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b\u7531\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u7ec4\u6210\u200b\u7684\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff1b</li> <li><code>dict.items()</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b\u7531\u952e\u200b-\u200b\u503c\u200b\u5143\u7ec4\u200b\u7ec4\u6210\u200b\u7684\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u3002</li> </ul> <p>\u200b\u6ce8\u610f\u200b\uff0c\u200b\u6bcf\u6b21\u200b\u8c03\u7528\u200b<code>values</code>\u200b\u65b9\u6cd5\u200b\u65f6\u200b\uff0c\u200b\u90fd\u200b\u4f1a\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5373\u4f7f\u200b\u662f\u200b\u5bf9\u200b\u540c\u4e00\u4e2a\u200b\u5b57\u5178\u200b\u8c03\u7528\u200b\u591a\u6b21\u200b<code>values</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5176\u200b\u8fd4\u56de\u503c\u200b\u4e5f\u200b\u4e0d\u60f3\u200b\u7b49\u200b\u3002</p> <p>Changed in version 3.8</p> <p>\u200b\u5185\u7f6e\u200b\u51fd\u6570\u200b<code>reversed</code>\u200b\u652f\u6301\u200b\u4f20\u9012\u200b\u5b57\u5178\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7684\u200b\u952e\u200b\u9006\u5e8f\u200b\u7ec4\u6210\u200b\u7684\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u3002</p> <p>\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\u901a\u5e38\u200b\u662f\u200b\u53ef\u200b\u4fee\u6539\u200b\u7684\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>types.MappingProxyType</code>\u200b\u4f1a\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u539f\u200b\u6620\u5c04\u200b\u7684\u200b\u89c6\u56fe\u200b\u3002\u200b\u5bf9\u539f\u200b\u6620\u5c04\u200b\u7684\u200b\u4efb\u4f55\u200b\u4fee\u6539\u200b\u90fd\u200b\u4f1a\u200b\u53cd\u6620\u200b\u5728\u200b\u89c6\u56fe\u200b\u4e2d\u200b\uff0c\u200b\u4f46\u200b\u4e0d\u80fd\u200b\u5bf9\u200b\u89c6\u56fe\u200b\u8fdb\u884c\u200b\u4efb\u4f55\u200b\u76f4\u63a5\u200b\u7684\u200b\u4fee\u6539\u200b\u3002\u200b\u6b64\u200b\u65b9\u6cd5\u200b\u95f4\u63a5\u200b\u5730\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u4e0d\u53ef\u200b\u4fee\u6539\u200b\u7684\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>&gt;&gt;&gt; from types import MappingProxyType\n&gt;&gt;&gt; a = {1: 2, 3: 4}\n&gt;&gt;&gt; b = MappingProxyType(a)\n&gt;&gt;&gt; b\nmappingproxy({1: 2, 3: 4})\n&gt;&gt;&gt; a.update({1: 5})\n&gt;&gt;&gt; b\nmappingproxy({1: 5, 3: 4})\n&gt;&gt;&gt; b[1] = 2\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: 'mappingproxy' object does not support item assignment\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/mapping/#_3","title":"\u53ef\u6563\u5217\u200b\u7c7b\u578b","text":"<p>\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u5bf9\u8c61\u200b\u79f0\u4e3a\u200b\u53ef\u6563\u5217\u200b\u5bf9\u8c61\u200b\uff1a</p> <ol> <li>\u200b\u5982\u679c\u200b\u8be5\u200b\u5bf9\u8c61\u200b\u53ef\u6563\u5217\u200b\uff0c\u200b\u5219\u200b\u5728\u200b\u5176\u200b\u751f\u547d\u5468\u671f\u200b\u4e2d\u200b\uff0c\u200b\u8be5\u6563\u200b\u5217\u503c\u200b\u4e0d\u53d8\u200b</li> <li>\u200b\u5bf9\u8c61\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>__hash__()</code>\u200b\u4e0e\u200b<code>__eq__()</code>\u200b\u65b9\u6cd5\u200b\uff08\u200b\u6ca1\u6709\u200b\u5b9e\u73b0\u200b<code>__eq__()</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u5bf9\u8c61\u200b\u4e0d\u200b\u5e94\u200b\u5b9e\u73b0\u200b<code>__hash__()</code>\u200b\u65b9\u6cd5\u200b\uff09</li> <li>\u200b\u5982\u679c\u200b\u5b83\u200b\u548c\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u5bf9\u8c61\u200b\u76f8\u7b49\u200b\uff0c\u200b\u5219\u200b\u4e24\u8005\u200b\u6709\u200b\u76f8\u540c\u200b\u7684\u200b\u6563\u5217\u503c\u200b</li> </ol> <p>\u200b\u5982\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a = \"abc\"\n&gt;&gt;&gt; b = \"\".join([\"a\", \"b\", \"c\"])\n&gt;&gt;&gt; a is b\nFalse\n&gt;&gt;&gt; hash(a) == hash(b)\nTrue\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6ce8\u200b\uff1a\u200b\u82e5\u200b\u4e00\u4e2a\u200b\u81ea\u5b9a\u4e49\u200b\u7c7b\u578b\u5b9a\u4e49\u200b\u4e86\u200b<code>__eq__</code>\uff0c\u200b\u4e14\u200b\u6ca1\u6709\u200b\u5b9a\u4e49\u200b<code>__hash__</code>\uff0c<code>__hash__</code>\u200b\u4f1a\u200b\u88ab\u200b\u8bbe\u4e3a\u200b<code>None</code>\u3002\u200b\u53ef\u6563\u5217\u200b\u7684\u200b\u81ea\u5b9a\u4e49\u200b\u5bf9\u8c61\u200b\u5fc5\u987b\u200b\u6709\u200b<code>__hash__</code>\u200b\u65b9\u6cd5\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u5185\u5bb9\u200b\u6765\u81ea\u200b\u4e8e\u200bPython\u200b\u6587\u6863\u200b </p> <p>\u200b\u6839\u636e\u200b\u5982\u200b\u4e0a\u200b\u89c4\u5219\u200b\u53ef\u4ee5\u200b\u5224\u65ad\u200b\u4e00\u4e9b\u200b\u5185\u7f6e\u200b\u6570\u636e\u7ed3\u6784\u200b\u662f\u5426\u200b\u4e3a\u200b\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b</p> <ul> <li>\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\u5728\u200b\u751f\u547d\u5468\u671f\u200b\u4e2d\u200b\u5e8f\u5217\u200b\u7684\u200b\u5185\u5bb9\u200b\u53ef\u80fd\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\uff08\u200b\u5982\u200b<code>list</code>\u3001<code>set</code>\u200b\u7b49\u200b\uff09\u200b\u4e0d\u662f\u200b\u53ef\u6563\u5217\u200b\u5bf9\u8c61\u200b\u3002</li> <li>\u200b\u539f\u5b50\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u6570\u636e\u7c7b\u578b\u200b\u6ee1\u8db3\u200b\u5982\u200b\u4e0a\u200b\u6240\u6709\u200b\u6761\u4ef6\u200b\uff0c\u200b\u662f\u200b\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b</li> <li>\u200b\u5bb9\u5668\u200b<code>frozenset</code>\u200b\u8981\u6c42\u200b\u5176\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u5747\u200b\u4e3a\u200b\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5176\u4e2d\u200b\u5185\u5bb9\u200b\u5728\u200b\u6574\u4e2a\u200b\u751f\u547d\u5468\u671f\u200b\u4e2d\u200b\u4e0d\u4f1a\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u56e0\u6b64\u200b\u662f\u200b\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b</li> <li>\u200b\u5bb9\u5668\u200b<code>tuple</code>\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u5bb9\u7eb3\u200b\u4e0d\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u56e0\u6b64\u200b<code>tuple</code>\u200b\u4e0d\u662f\u200b\u53ef\u6563\u5217\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u4f46\u200b\u4e0d\u200b\u542b\u200b\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b\u7684\u200b\u5143\u7ec4\u200b\u662f\u200b\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b\u3002</li> </ul> <pre><code>&gt;&gt;&gt; t1 = hash((1, 2))\n&gt;&gt;&gt; t2 = hash((1, [2]))\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: unhashable type: 'list'\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>hash</code>\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u53d6\u5f97\u200b\u53ef\u6563\u5217\u200b\u5bf9\u8c61\u200b\u7684\u200b\u6563\u5217\u503c\u200b\u3002\u200b\u4e3a\u4e86\u200b\u907f\u514d\u200bDOS\u200b\u653b\u51fb\u200b\uff0c\u200b\u5728\u200b\u8ba1\u7b97\u200b\u6563\u5217\u503c\u200b\u65f6\u4f1a\u200b\u52a0\u4e0a\u200b\u4e00\u4e2a\u200b\u968f\u673a\u200b\u7684\u200b\u566a\u58f0\u200b\u3002\u200b\u566a\u58f0\u200b\u7684\u200b\u503c\u200b\u4ec5\u200b\u5728\u200b\u540c\u4e00\u200bPython\u200b\u5b9e\u4f8b\u200b\u4e2d\u200b\u76f8\u540c\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\uff0c\u200b\u82e5\u200b\u5b83\u200b\u7684\u200b\u503c\u200b\u5c0f\u4e8e\u200b\u6563\u5217\u503c\u200b\u6240\u200b\u5141\u8bb8\u200b\u7684\u200b\u6700\u5927\u503c\u200b\uff08\u200b\u5373\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u653e\u8fdb\u200b\u6563\u5217\u503c\u200b\u6240\u200b\u9650\u5b9a\u200b\u7684\u200b\u7a7a\u95f4\u200b\u5185\u200b\uff09\uff0c\u200b\u5219\u200b\u6563\u5217\u200b\u7684\u200b\u503c\u200b\u5c31\u662f\u200b\u8be5\u200b\u6574\u6570\u200b\u7684\u200b\u503c\u200b\u3002</p> <p>\u200b\u6563\u200b\u5217\u8868\u200b\u7684\u200b\u4f7f\u7528\u200b\u662f\u200b\u201c\u200b\u7a7a\u95f4\u200b\u6362\u200b\u65f6\u95f4\u200b\u201d\u200b\u7b56\u7565\u200b\u7684\u200b\u5178\u578b\u200b\u4f53\u73b0\u200b\u3002\u200b\u4e3a\u4e86\u200b\u5c3d\u53ef\u80fd\u51cf\u5c11\u200b\u51b2\u7a81\u200b\uff0c\u200b\u6563\u200b\u5217\u8868\u200b\u5fc5\u987b\u200b\u6709\u200b\u8f83\u200b\u4f4e\u200b\u7684\u200b\u88c5\u586b\u200b\u56e0\u5b50\u200b\uff0c\u200b\u8fd9\u200b\u610f\u5473\u7740\u200b\u6563\u200b\u5217\u8868\u200b\u4f1a\u200b\u5360\u7528\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u5185\u5b58\u7a7a\u95f4\u200b\u3002</p>"},{"location":"coding/python/mapping/#_4","title":"\u6784\u9020","text":"<p>\u200b\u5b57\u5178\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u591a\u79cd\u200b\u6784\u9020\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6784\u9020\u200b\u5b57\u5178\u200b<code>{'a': 1, 'b': 2}</code></p> <ol> <li>\u200b\u82b1\u200b\u62ec\u53f7\u200b\u6784\u9020\u200b\uff0c\u200b\u5982\u200b<code>{'a': 1, 'b': 2}</code></li> <li>\u200b\u901a\u8fc7\u200b\u7c7b\u200b\u6784\u9020\u51fd\u6570\u200b\uff0c\u200b\u5982\u200b<code>dict(a=1, b=2)</code></li> <li>\u200b\u901a\u8fc7\u200b\u5143\u7ec4\u200b/\u200b\u5217\u8868\u200b\uff08\u200b\u53ea\u200b\u9700\u200b\u662f\u200b\u957f\u5ea6\u200b\u4e3a\u200b2\u200b\u7684\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff09\u200b\u6784\u9020\u200b\uff0c\u200b\u5982\u200b<code>dict([('a', 1), ['b', 2]])</code></li> <li>\u200b\u901a\u8fc7\u200b<code>zip</code>\u200b\u6253\u5305\u200b\u6784\u9020\u200b\uff0c\u200b\u5982\u200b<code>dict(zip(['a', 'b'], [1, 2]))</code></li> <li>\u200b\u901a\u8fc7\u200b\u5b57\u5178\u200b\u63a8\u5bfc\u200b\u6784\u9020\u200b\uff0c\u200b\u5982\u200b<code>{chr(97 + _): _ + 1 for _ in range(2)}</code></li> </ol>"},{"location":"coding/python/mapping/#_5","title":"\u53d8\u79cd","text":"<p><code>collections</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5b57\u5178\u200b\u7684\u200b\u4e24\u4e2a\u200b\u53d8\u79cd\u200b\uff0c\u200b\u5373\u200b<code>defaultdict</code>\u200b\u4e0e\u200b<code>OrderDict</code></p>"},{"location":"coding/python/mapping/#defaultdict","title":"defaultdict","text":"<p>\u200b\u5f53\u200b\u5728\u200b\u4e00\u4e2a\u200b\u5b57\u5178\u200b<code>d</code>\u200b\u4e2d\u200b\u627e\u200b\u4e0d\u5230\u200b\u952e\u200b<code>k</code>\u200b\u65f6\u200b\uff0cPython\u200b\u4f1a\u200b\u4ea7\u751f\u200b<code>KeyError</code>\u200b\u5f02\u5e38\u200b\u3002\u200b\u4f7f\u7528\u200b<code>d.get(k, default)</code>\u200b\u53ef\u4ee5\u200b\u4e3a\u200b\u627e\u200b\u4e0d\u5230\u200b\u7684\u200b\u952e\u200b\u8bbe\u7f6e\u200b\u9ed8\u8ba4\u200b\u8fd4\u56de\u503c\u200b\u3002\u200b\u4f46\u200b<code>dict.get</code>\u200b\u51fd\u6570\u200b\u4e0d\u4f1a\u200b\u5bf9\u200b\u5b57\u5178\u200b\u8fdb\u884c\u200b\u66f4\u65b0\u200b\uff0c\u200b\u63d2\u5165\u200b\u627e\u200b\u4e0d\u5230\u200b\u7684\u200b\u952e\u200b\u3002</p> <p><code>dict.setdefault</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u53e6\u200b\u4e00\u79cd\u200b\u4ece\u200b\u5b57\u5178\u200b\u4e2d\u200b\u83b7\u53d6\u200b\u503c\u200b\u7684\u200b\u65b9\u6cd5\u200b\u3002<code>setdefault</code>\u200b\u63a5\u6536\u200b\u4e24\u4e2a\u200b\u53c2\u6570\u200b\uff0c\u200b\u7b2c\u4e00\u4e2a\u200b\u53c2\u6570\u200b\u4e3a\u200b<code>key</code>\uff0c\u200b\u7b2c\u4e8c\u4e2a\u200b\u53c2\u6570\u200b\u4e3a\u200b\u53ef\u9009\u200b\u7684\u200b\u9ed8\u8ba4\u503c\u200b<code>default</code>\u3002\u200b\u5f53\u200b\u5b57\u5178\u200b\u4e2d\u200b\u627e\u5230\u200b<code>key</code>\u200b\u6240\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5143\u7d20\u200b\u65f6\u200b\uff0c<code>setdefault</code>\u200b\u76f4\u63a5\u200b\u8fd4\u56de\u200b\u8be5\u503c\u200b\uff0c\u200b\u5426\u5219\u200b<code>setdefault</code>\u200b\u4f1a\u200b\u5728\u200b\u5b57\u5178\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u952e\u503c\u200b\u5bf9\u200b<code>key: default</code>\uff0c\u200b\u7136\u540e\u200b\u8fd4\u56de\u200b<code>default</code>\u3002</p> <pre><code>&gt;&gt;&gt; a = {}\n&gt;&gt;&gt; a.setdefault(0, True)\nTrue\n&gt;&gt;&gt; a[0] = False\n&gt;&gt;&gt; a.setdefault(0, False) \nFalse\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>setdefault</code>\u200b\u4ece\u200b\u5b57\u5178\u200b\u4e2d\u200b\u83b7\u53d6\u200b\u503c\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u51cf\u5c11\u200b\u5b57\u5178\u200b\u7684\u200b\u67e5\u8be2\u200b\u6b21\u6570\u200b\uff0c\u200b\u63d0\u9ad8\u200b\u5b57\u5178\u200b\u6267\u884c\u200b\u7684\u200b\u6548\u7387\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a = {}\n&gt;&gt;&gt; try:\n...     a[0].append(True)\n... except KeyError:\n...     a[0] = []\n...     a[0].append(True)\n...        \n&gt;&gt;&gt; a\n{0: [True]}\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5982\u200b\u4e0a\u200b\u4f7f\u7528\u200b<code>try-except</code>\u200b\u7ed3\u6784\u200b\u5904\u7406\u200b\u5b57\u5178\u200b\u4e2d\u200b\u627e\u200b\u4e0d\u5230\u200b\u503c\u200b\u7684\u200b\u7ed3\u679c\u200b\uff0c\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b<code>setdefault</code>\u200b\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u6709\u6548\u200b\u51cf\u5c11\u200b\u4ee3\u7801\u200b\u957f\u5ea6\u200b\uff0c\u200b\u589e\u52a0\u200b\u4ee3\u7801\u200b\u53ef\u8bfb\u6027\u200b\u3002</p> <pre><code>&gt;&gt;&gt; a = {}\n&gt;&gt;&gt; a.setdefault(0, []).append(True)\n&gt;&gt;&gt; a\n{0: [True]}\n&gt;&gt;&gt;\n</code></pre> <p><code>defaultdict</code>\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u5904\u7406\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7f3a\u5931\u200b\u7684\u200b\u952e\u200b\u3002\u200b\u51c6\u786e\u200b\u5730\u8bf4\u200b\uff0c<code>defaultdict</code>\u200b\u5728\u200b\u521b\u5efa\u200b\u65f6\u200b\u9700\u8981\u200b\u63d0\u4f9b\u200b\u4e00\u4e2a\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\uff0c\u200b\u5b57\u5178\u200b\u4e2d\u200b\u627e\u200b\u4e0d\u5230\u200b\u952e\u65f6\u200b\uff0c<code>defaultdict</code>\u200b\u4f1a\u200b\u901a\u8fc7\u200b\u8c03\u7528\u200b\u8fd9\u4e2a\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u6765\u200b\u5f97\u5230\u200b\u8be5\u952e\u200b\u5bf9\u5e94\u200b\u7684\u200b\u9ed8\u8ba4\u503c\u200b\u3002</p> <pre><code>&gt;&gt;&gt; from collections import defaultdict\n&gt;&gt;&gt; a = defaultdict(list)\n&gt;&gt;&gt; a\ndefaultdict(&lt;class 'list'&gt;, {})\n&gt;&gt;&gt; a[False] = {}\n&gt;&gt;&gt; a[False]\n{}\n&gt;&gt;&gt; a[True]\n[]\n&gt;&gt;&gt; a\ndefaultdict(&lt;class 'list'&gt;, {False: {}, True: []})\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5982\u679c\u200b\u521b\u5efa\u200b<code>defaultdict</code>\u200b\u65f6\u200b\u6ca1\u6709\u200b\u6307\u5b9a\u200b\u53c2\u6570\u200b\uff0c\u200b\u8bbf\u95ee\u200b\u4e0d\u200b\u5b58\u5728\u200b\u7684\u200b\u952e\u200b\u65f6\u200b\u4ecd\u4f1a\u200b\u629b\u51fa\u200b<code>KeyError</code>\u3002</p> <p>\u200b\u4e8b\u5b9e\u4e0a\u200b\uff0c<code>__getitem__</code>\u200b\u901a\u8fc7\u200b\u7279\u6b8a\u200b\u65b9\u6cd5\u200b<code>__missing__</code>\u200b\u5904\u7406\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7f3a\u5931\u200b\u7684\u200b\u952e\u200b\u3002\u200b\u624b\u52a8\u200b\u8c03\u7528\u200b<code>__missing__</code>\u200b\u65b9\u6cd5\u200b\u80fd\u200b\u5b9e\u73b0\u200b\u76f8\u540c\u200b\u7684\u200b\u6548\u679c\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; from collections import defaultdict \n&gt;&gt;&gt; a = defaultdict(list)\n&gt;&gt;&gt; a \ndefaultdict(&lt;class 'list'&gt;, {})\n&gt;&gt;&gt; a.__missing__(0)  \n[]\n&gt;&gt;&gt; a\ndefaultdict(&lt;class 'list'&gt;, {0: []})\n&gt;&gt;&gt;\n</code></pre> <p><code>__missing__</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u53c2\u6570\u200b\u4e3a\u200b<code>self, key</code>\uff0c\u200b\u5176\u4e2d\u200b<code>key</code>\u200b\u4e3a\u200b\u7f3a\u5931\u200b\u7684\u200b\u952e\u200b\u3002\u200b\u5373\u4f7f\u200b<code>__missing__</code>\u200b\u6b63\u5e38\u200b\u6267\u884c\u200b\uff0c\u200b\u8fd4\u56de\u503c\u200b\u4e5f\u200b\u4e0d\u4f1a\u200b\u88ab\u200b\u6dfb\u52a0\u200b\u5230\u200b\u5b57\u5178\u200b\u4e2d\u200b\u3002</p> <pre><code>&gt;&gt;&gt; class NumDict(dict):\n...     def __init__(self, *args, **kwargs):\n...             super().__init__(*args, **kwargs)\n...     def __missing__(self, key):\n...             if isinstance(key, int):\n...                     return 0\n...             else:\n...                     raise KeyError(key)\n...\n&gt;&gt;&gt; a = NumDict([(1, 2), (\"a\", \"b\")])\n&gt;&gt;&gt; a\n{1: 2, 'a': 'b'}\n&gt;&gt;&gt; a[0]\n0\n&gt;&gt;&gt; a['b']\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\n  File \"&lt;stdin&gt;\", line 8, in __missing__\nKeyError: 'b'\n&gt;&gt;&gt; a\n{1: 2, 'a': 'b'}\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/mapping/#ordereddict","title":"OrderedDict","text":"<p><code>OrderedDict</code>\u200b\u662f\u200b\u4e25\u683c\u200b\uff08\u200b\u5728\u200b\u4efb\u4f55\u200b\u7248\u672c\u200b\u4e2d\u200b\uff09\u200b\u4fdd\u6301\u200b\u987a\u5e8f\u200b\u7684\u200b\u5b57\u5178\u200b\u3002\u200b\u5728\u200b\u8fed\u4ee3\u200b\u8fc7\u7a0b\u200b\u4e2d\u952e\u200b\u7684\u200b\u987a\u5e8f\u200b\u4fdd\u6301\u4e00\u81f4\u200b\u3002</p> <p><code>OrderedDict</code>\u200b\u65b0\u589e\u200b\u4e86\u200b<code>popitem</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u9ed8\u8ba4\u200b\u79fb\u9664\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5b57\u5178\u200b\u4e2d\u200b\u6700\u540e\u200b\u6dfb\u52a0\u200b\u7684\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u3002\u200b\u5982\u679c\u200b\u624b\u52a8\u200b\u6307\u5b9a\u200b<code>last</code>\u200b\u53c2\u6570\u200b\u4e3a\u200b<code>False</code>\uff0c\u200b\u5219\u200b\u79fb\u9664\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7b2c\u4e00\u4e2a\u200b\u6dfb\u52a0\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\u200b\u540c\u65f6\u200b\uff0c<code>move_to_end</code>\u200b\u662f\u200b\u4e0e\u200b\u4e4b\u200b\u914d\u5408\u200b\u7684\u200b\u79fb\u52a8\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7684\u200b\u952e\u503c\u200b\u5bf9\u200b\uff08\u200b\u7531\u200b<code>key</code>\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\uff09\u200b\u79fb\u52a8\u200b\u5230\u200b\u5b57\u5178\u200b\u4e00\u7aef\u200b\uff08\u200b\u7531\u200b<code>last</code>\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\uff09\u3002</p>"},{"location":"coding/python/mapping/#chainmap","title":"ChainMap","text":"<p><code>ChainMap</code>\u200b\u53ef\u4ee5\u200b\u89c6\u4e3a\u200b\u4e00\u79cd\u200b\u5bb9\u7eb3\u200b\u6620\u5c04\u200b\u7684\u200b\u5e8f\u5217\u200b\u3002\u200b\u76f8\u6bd4\u200b\u4e8e\u200b\u4f7f\u7528\u200b<code>dict.update()</code>\u200b\u6216\u200b<code>|</code>\u200b\u8fd0\u7b97\u7b26\u200b\u9010\u4e2a\u200b\u5408\u5e76\u200b\u6620\u5c04\u200b\uff0c<code>ChainMap</code>\u200b\u6709\u200b\u66f4\u200b\u9ad8\u200b\u7684\u200b\u6548\u7387\u200b\u3002</p> <p><code>ChainMap</code>\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u6620\u5c04\u200b\uff0c\u200b\u5f53\u200b\u5728\u200b<code>ChainMap</code>\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u67e5\u8be2\u200b\u65f6\u200b\uff0c<code>ChainMap</code>\u200b\u4f1a\u200b\u9010\u4e2a\u200b\u67e5\u8be2\u200b\u5176\u4e2d\u200b\u7684\u200b\u6620\u5c04\u200b\u3002\u200b\u5f53\u200b\u540c\u4e00\u4e2a\u200b\u952e\u200b\u51fa\u73b0\u200b\u5728\u200b\u4e0d\u540c\u200b\u7684\u200b\u6620\u5c04\u200b\u4e2d\u200b\u65f6\u200b\uff0c<code>ChainMap</code>\u200b\u8fd4\u56de\u200b\u67e5\u8be2\u200b\u5230\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u7ed3\u679c\u200b\u3002</p> <pre><code>&gt;&gt;&gt; from collections import ChainMap\n&gt;&gt;&gt; a = ChainMap({'a': 1, 'b': 2}, {'a': 3, 'b': 4})\n&gt;&gt;&gt; a['b']\n2\n&gt;&gt;&gt; a['a']\n1\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5b58\u50a8\u200b\u7684\u200b\u6620\u5c04\u200b\u5728\u200b<code>ChainMap.maps</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\uff0c\u200b\u5bf9\u200b\u8be5\u200b\u5c5e\u6027\u200b\u7684\u200b\u4fee\u6539\u200b\u4f1a\u200b\u540c\u65f6\u200b\u53cd\u6620\u200b\u5728\u200b<code>ChainMap</code>\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u3002<code>parents</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u901a\u8fc7\u200b\u5c5e\u6027\u200b\u65b9\u5f0f\u200b\u8bbf\u95ee\u200b\u7684\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u8fd4\u56de\u200b\u53bb\u9664\u200b\u539f\u200b<code>ChainMap</code>\u200b\u4e2d\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u6620\u5c04\u200b\u540e\u200b\u7684\u200b\u65b0\u200b<code>ChainMap</code>\uff0c\u200b\u539f\u200b<code>ChainMap</code>\u200b\u4fdd\u6301\u200b\u4e0d\u53d8\u200b\u3002</p> <p><code>new_child</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5411\u200b<code>ChainMap</code>\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u6620\u5c04\u200b\u7684\u200b\u65b9\u6cd5\u200b\u3002\u200b\u65b0\u200b\u7684\u200b\u6620\u5c04\u200b\u88ab\u200b\u6dfb\u52a0\u200b\u5230\u200b<code>ChainMap</code>\u200b\u7684\u200b\u5f00\u5934\u200b\uff0c\u200b\u5982\u679c\u200b\u6ca1\u6709\u200b\u63d0\u4f9b\u200b\u6620\u5c04\u200b\uff0c\u200b\u5219\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u7a7a\u200b\u5b57\u5178\u200b\u3002</p> <p><code>ChainMap</code>\u200b\u5e76\u200b\u6ca1\u6709\u200b\u5408\u5e76\u200b\u539f\u6709\u200b\u7684\u200b\u6620\u5c04\u200b\uff0c\u200b\u56e0\u6b64\u200b\u8fed\u4ee3\u200b<code>ChainMap</code>\u200b\u5bf9\u8c61\u200b\u6216\u200b\u5bf9\u200b<code>ChainMap</code>\u200b\u5bf9\u8c61\u200b\u8c03\u7528\u200b<code>dict()</code>\u200b\u65f6\u200b\u53ea\u4f1a\u200b\u5e94\u7528\u200b\u5230\u200b\u5176\u4e2d\u200b\u5305\u542b\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u6620\u5c04\u200b\u3002</p> <pre><code>&gt;&gt;&gt; dict(a)\n{'a': 1, 'b': 2}\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/mapping/#counter","title":"Counter","text":"<p><code>Counter</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5bf9\u200b\u4e00\u7cfb\u5217\u200b\u503c\u200b\u7684\u200b\u7edf\u8ba1\u200b\u65b9\u6cd5\u200b\u3002\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u521d\u59cb\u5316\u200b<code>Counter</code>\u200b\u7c7b\u578b\u200b\uff0c<code>Counter</code>\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u7edf\u8ba1\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u6bcf\u4e2a\u200b\u503c\u200b\u7684\u200b\u51fa\u73b0\u200b\u987a\u5e8f\u200b\u3002</p> <p><code>Counter</code>\u200b\u4e0d\u4f1a\u200b\u5bf9\u200b\u8f93\u5165\u200b\u7684\u200b\u503c\u200b\u8fdb\u884c\u200b\u68c0\u67e5\u200b\uff0c\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b\u5b57\u5178\u200b\u521d\u59cb\u5316\u200b\u5e76\u200b\u63d0\u4f9b\u200b\u975e\u200b\u6574\u6570\u200b\u7c7b\u578b\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u5728\u200b\u8c03\u7528\u200b<code>Counter.most_common()</code>\u200b\u65b9\u6cd5\u200b\u6216\u200b\u8fed\u4ee3\u200b<code>Counter.elements()</code>\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u65f6\u4f1a\u200b\u629b\u51fa\u200b<code>TypeError</code>\u200b\u5f02\u5e38\u200b\u3002 <pre><code>&gt;&gt;&gt; from collections import Counter\n&gt;&gt;&gt; a = Counter({'0': '0'})\n&gt;&gt;&gt; for i in a.elements():\n...     pass\n...\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: 'str' object cannot be interpreted as an integer\n</code></pre></p> <p>\u200b\u8fd9\u200b\u4e00\u70b9\u200b\u5728\u200bPython\u200b\u6587\u6863\u200b\u4e2d\u6709\u200b\u76f8\u5e94\u200b\u8bf4\u660e\u200b\uff1a</p> <p></p> <pre><code>&gt;&gt;&gt; from collections import Counter \n&gt;&gt;&gt; a = Counter([1, 1, 1, 2, 2, 3])          \n&gt;&gt;&gt; a\nCounter({1: 3, 2: 2, 3: 1})\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u66f4\u65b0\u200b<code>Counter</code>\u200b\u5bf9\u8c61\u200b\u65f6\u200b\uff0c\u200b\u8ba1\u6570\u5668\u200b\u4f1a\u200b\u76f8\u5e94\u200b\u5730\u200b\u66f4\u65b0\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a.update([3, 3, 3, 2, 2, 1])\n&gt;&gt;&gt; a\nCounter({1: 4, 2: 4, 3: 4})\n&gt;&gt;&gt;\n</code></pre> <p><code>Counter.elements()</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u8fed\u4ee3\u200b\u5668\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u6309\u7167\u200b\u5176\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u91cd\u590d\u200b\u3002<code>Counter.most_common</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b\u8ba1\u6570\u5668\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u6700\u591a\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</p> <p><code>Counter</code>\u200b\u7c7b\u578b\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>+</code>\u3001<code>-</code>\u3001<code>&amp;</code>\u200b\u548c\u200b<code>|</code>\u200b\u8fd0\u7b97\u7b26\u200b\uff0c\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u8ba1\u6570\u5668\u200b\u7684\u200b\u76f8\u52a0\u200b\u3001\u200b\u76f8\u51cf\u200b\u3001\u200b\u4ea4\u200b\u548c\u200b\u5e76\u200b\u64cd\u4f5c\u200b\u3002\u200b\u5f53\u200b<code>+</code>\u3001<code>-</code>\u200b\u88ab\u200b\u7528\u4f5c\u200b\u4e00\u5143\u200b\u8fd0\u7b97\u7b26\u200b\u65f6\u200b\uff0c\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u64cd\u4f5c\u6570\u200b\u9ed8\u8ba4\u200b\u4e3a\u7a7a\u200b\u8ba1\u6570\u5668\u200b\u3002</p>"},{"location":"coding/python/mapping/#_6","title":"\u6269\u5c55","text":"<p><code>collections.UserDict</code>\u200b\u7c7b\u200b\u662f\u200b\u5bf9\u200b<code>dict</code>\u200b\u7684\u200bPython\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u81ea\u5b9a\u4e49\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\u7684\u200b\u5b9e\u73b0\u200b\u3002</p> <p><code>UserDict</code>\u200b\u4e0d\u662f\u200b<code>dict</code>\u200b\u7684\u200b\u5b50\u7c7b\u200b\uff0c\u200b\u4f46\u200b<code>UserDict</code>\u200b\u4e2d\u6709\u200b\u4e00\u4e2a\u200b<code>dict</code>\u200b\u7c7b\u578b\u200b\u7684\u200b<code>data</code>\u200b\u5c5e\u6027\u200b\uff0c\u200b\u662f\u200b<code>UserDict</code>\u200b\u5b58\u50a8\u200b\u6570\u636e\u200b\u7684\u200b\u6700\u7ec8\u200b\u4f4d\u7f6e\u200b\u3002\u200b\u76f8\u6bd4\u200b\u4e8e\u200b\u76f4\u63a5\u200b\u5bf9\u200b<code>dict</code>\u200b\u7c7b\u578b\u200b\u8fdb\u884c\u200b\u6269\u5c55\u200b\uff0c<code>data</code>\u200b\u5c5e\u6027\u200b\u4fdd\u8bc1\u200b\u4e86\u200b<code>__getitem__</code>\u200b\u65b9\u6cd5\u200b\u4e0d\u4f1a\u200b\u4ea7\u751f\u200b\u5faa\u73af\u200b\u9012\u5f52\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002</p> <p><code>UserDict</code>\u200b\u7c7b\u200b\u7ee7\u627f\u200b\u81ea\u200b<code>_collections_abc.MutableMapping</code>\u200b\u4e0e\u200b<code>_collections_abc.Mapping</code>\u200b\u7c7b\u200b\u3002<code>MutableMapping</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>update()</code>\u200b\u65b9\u6cd5\u200b\u3002\u200b\u65b9\u6cd5\u200b\u7684\u200b\u5185\u90e8\u200b\u5b9e\u73b0\u200b\u8868\u660e\u200b\uff0c<code>update()</code>\u200b\u5728\u200b\u5185\u90e8\u200b\u901a\u8fc7\u200b<code>__setitem__</code>\u200b\u5b9e\u73b0\u200b\u3002<code>Mapping</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>get()</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u8be5\u200b\u65b9\u6cd5\u200b\u5728\u200b\u5185\u90e8\u200b\u901a\u8fc7\u200b<code>__getitem__</code>\u200b\u5b9e\u73b0\u200b\u3002</p> <pre><code>&gt;&gt;&gt; from collections import UserDict\n&gt;&gt;&gt; class myDict(UserDict):\n...     def __init__(self, *args, **kwargs):\n...             super().__init__(*args, **kwargs)  \n...     def __setitem__(self, index, value):\n...             print(\"Modification: \", index, \":\", value)\n...             super().__setitem__(index, value)\n... \n&gt;&gt;&gt; a = myDict()\n&gt;&gt;&gt; a.update(a=5, b=4)\nModification:  a : 5\nModification:  b : 4\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/mapping/#_7","title":"\u96c6\u5408","text":"<p>\u200b\u96c6\u5408\u200b<code>set</code>\u200b\u662f\u200b\u4e00\u79cd\u200b\u5e8f\u5217\u200b\u7c7b\u578b\u200b\uff0c\u200b\u96c6\u5408\u200b\u4e2d\u200b\u7684\u200b\u5404\u4e2a\u200b\u5143\u7d20\u200b\u6700\u200b\u591a\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002\u200b\u4efb\u4f55\u200b\u975e\u200b\u7a7a\u96c6\u5408\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5b57\u9762\u200b\u8868\u793a\u6cd5\u200b<code>{}</code>\u200b\u8fdb\u884c\u200b\u8868\u793a\u200b\uff0c\u200b\u7a7a\u96c6\u200b\u53ea\u80fd\u200b\u7528\u200b<code>set()</code>\u200b\u8868\u793a\u200b\u3002\u200b\u96c6\u5408\u200b\u540c\u6837\u200b\u9002\u7528\u200b\u4e8e\u200b\u4e0e\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u76f8\u4f3c\u200b\u7684\u200b\u63a8\u5bfc\u200b\u65b9\u5f0f\u200b\u3002</p> <p>\u200b\u96c6\u5408\u200b\u6709\u200b\u5bf9\u5e94\u200b\u7684\u200b\u53ea\u8bfb\u200b\u53d8\u79cd\u200b<code>frozenset</code>\u3002<code>set</code>\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u5b57\u9762\u200b\u8868\u793a\u6cd5\u200b\u8fdb\u884c\u200b\u6784\u9020\u200b\uff0c\u200b\u6216\u200b\u4f7f\u7528\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u8fdb\u884c\u200b\u6784\u9020\u200b\u3002<code>frozenset</code>\u200b\u53ea\u80fd\u200b\u4f7f\u7528\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u6784\u9020\u200b\u3002</p> <p>\u200b\u53ef\u4ee5\u200b\u5229\u7528\u200b<code>set</code>\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u552f\u4e00\u6027\u200b\u53bb\u9664\u200b\u5217\u8868\u200b\u4e2d\u200b\u7684\u200b\u91cd\u590d\u200b\u503c\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; src_list = [1, 1, 2, 3, 4, 2, 3]\n&gt;&gt;&gt; list(set(src_list))\n[1, 2, 3, 4]\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u53e6\u200b\u4e00\u79cd\u200b\u65b9\u6cd5\u200b\u662f\u200b\u901a\u8fc7\u200b<code>collections.Counter</code>\u200b\u53bb\u200b\u91cd\u200b\u3002</p> <pre><code>&gt;&gt;&gt; from collections import Counter\n&gt;&gt;&gt; src_list = [1, 1, 2, 3, 4, 2, 3]\n&gt;&gt;&gt; list(Counter(src_list).keys())\n[1, 2, 3, 4]\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u548c\u200b\u5b57\u5178\u200b\u7684\u200b\u952e\u200b\u4e00\u6837\u200b\uff0c\u200b\u96c6\u5408\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u5fc5\u987b\u200b\u662f\u200b\u53ef\u200b\u6563\u5217\u200b\u7684\u200b\u3002<code>set</code>\u200b\u7c7b\u578b\u200b\u672c\u8eab\u200b\u4e0d\u53ef\u200b\u6563\u5217\u200b\uff0c<code>frozenset</code>\u200b\u7c7b\u578b\u200b\u53ef\u6563\u5217\u200b\u3002</p>"},{"location":"coding/python/mapping/#_8","title":"\u8fd0\u7b97","text":"<p>\u200b\u96c6\u5408\u200b\u6709\u200b\u5982\u4e0b\u200b\u8fd0\u7b97\u200b\uff0c\u200b\u7531\u4e8e\u200b\u96c6\u5408\u200b\u7684\u200b\u5e95\u5c42\u200b\u6570\u636e\u7ed3\u6784\u200b\u662f\u200b\u6563\u200b\u5217\u8868\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u8fd0\u7b97\u200b\u65b9\u6cd5\u200b\u6709\u200b\u8f83\u200b\u9ad8\u200b\u7684\u200b\u8ba1\u7b97\u200b\u6548\u7387\u200b\u3002</p> <ul> <li>\u200b\u4ea4\u96c6\u200b\u3001\u200b\u5e76\u96c6\u200b\u3001\u200b\u5dee\u96c6\u200b\u3001\u200b\u5bf9\u79f0\u200b\u5dee\u200b\uff0c\u200b\u5373\u200b<code>&amp;</code>\u3001<code>|</code>\u3001<code>-</code>\u200b\u4e0e\u200b<code>^</code>\uff1b</li> <li>\u200b\u5b50\u96c6\u200b\u3001\u200b\u771f\u200b\u5b50\u96c6\u200b\u3001\u200b\u8d85\u96c6\u200b\u3001\u200b\u771f\u8d85\u96c6\u200b\uff0c\u200b\u5373\u200b<code>&lt;=</code>\u3001<code>&lt;</code>\u3001<code>&gt;=</code>\u200b\u4e0e\u200b<code>&gt;</code>\uff1b</li> <li>\u200b\u5c5e\u4e8e\u200b\uff0c\u200b\u5373\u200b<code>in</code></li> </ul> <p>\u200b\u4ee5\u200b<code>in</code>\u200b\u64cd\u4f5c\u200b\u4e3a\u4f8b\u200b\uff0c<code>set</code>\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u67e5\u8be2\u200b\u6563\u200b\u5217\u8868\u200b\u83b7\u53d6\u200b\u7ed3\u679c\u200b\uff0c\u200b\u4f46\u200b<code>list</code>\u200b\u5fc5\u987b\u200b\u626b\u63cf\u200b\u4e00\u904d\u200b\u5217\u8868\u200b\u624d\u80fd\u200b\u83b7\u53d6\u200b\u7ed3\u679c\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u542b\u6709\u200b\u8bb8\u591a\u200b\u5143\u7d20\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4e24\u8005\u200b\u5728\u200b\u6267\u884c\u200b\u901f\u5ea6\u200b\u4e0a\u200b\u6709\u200b\u660e\u663e\u200b\u7684\u200b\u5dee\u522b\u200b\u3002</p> <p><code>set</code>\u200b\u8fd8\u200b\u652f\u6301\u200b<code>add</code>\u3001<code>clear</code>\u3001<code>copy</code>\u3001<code>pop</code>\u3001<code>remove</code>\u200b\u7b49\u200b\u65b9\u6cd5\u200b\u3002</p>"},{"location":"coding/python/sequence/","title":"Python\u200b\u4e2d\u200b\u7684\u200b\u5e8f\u5217\u200b\u7c7b\u578b","text":"<p>\u200b\u5e8f\u5217\u200b\u6307\u200b\u53ef\u4ee5\u200b\u5b58\u50a8\u200b\u591a\u4e2a\u200b\u6570\u636e\u200b\u7684\u200b\u6570\u636e\u7ed3\u6784\u200b\u3002</p> <p>\u200b\u6309\u7167\u200b\u5b58\u50a8\u200b\u7684\u200b\u6570\u636e\u7c7b\u578b\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\uff1a</p> <ul> <li>\u200b\u5bb9\u5668\u200b\u53ef\u4ee5\u200b\u5b58\u50a8\u200b\u4e0d\u540c\u200b\u7c7b\u578b\u200b\u7684\u200b\u6570\u636e\u200b\uff0c\u200b\u8fd9\u90e8\u5206\u200b\u7c7b\u578b\u200b\u6709\u200b<code>list</code>\u3001<code>tuple</code>\u3001<code>collections.deque</code></li> <li>\u200b\u6241\u5e73\u200b\u5e8f\u5217\u200b\u53ea\u80fd\u200b\u5b58\u50a8\u200b\u67d0\u79cd\u200b\u7c7b\u578b\u200b\u7684\u200b\u6570\u636e\u200b\uff0c\u200b\u8fd9\u90e8\u5206\u200b\u7c7b\u578b\u200b\u6709\u200b<code>str</code>\u3001<code>bytes</code>\u3001<code>bytearray</code>\u3001<code>memoryview</code>\u3001<code>array.array</code></li> </ul> <p>\u200b\u6309\u7167\u200b\u5e8f\u5217\u200b\u662f\u5426\u200b\u53ef\u4ee5\u200b\u4fee\u6539\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b</p> <ul> <li>\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\uff1a<code>list</code>\u3001<code>bytearray</code>\u3001<code>array.array</code>\u3001<code>collections.deque</code>\u3001<code>memoryview</code></li> <li>\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\uff1a<code>tuple</code>\u3001<code>str</code>\u3001<code>bytes</code></li> </ul>"},{"location":"coding/python/sequence/#_1","title":"\u5217\u8868","text":"<p>\u200b\u5217\u8868\u200b\u662f\u200b\u53ef\u4ee5\u200b\u4fee\u6539\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5b58\u50a8\u200b\u591a\u79cd\u4e0d\u540c\u200b\u7c7b\u578b\u200b\u7684\u200b\u6570\u636e\u7ed3\u6784\u200b\u3002\u200b\u5982\u4e0b\u200b\u7740\u91cd\u200b\u8ba8\u8bba\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u7684\u200b\u4e66\u5199\u200b\u5f62\u5f0f\u200b\u3002</p> <p>\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u53ef\u200b\u7528\u4e8e\u200b\u4ece\u200b\u4e00\u4e2a\u200b\u5217\u8868\u200b\u751f\u6210\u200b\u65b0\u200b\u7684\u200b\u5217\u8868\u200b\uff0c\u200b\u76f8\u6bd4\u200b\u4e8e\u200bfor\u200b\u5faa\u73af\u200b\u52a0\u200b<code>append()</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u7684\u200b\u5199\u6cd5\u200b\u6709\u200b\u66f4\u597d\u200b\u7684\u200b\u53ef\u8bfb\u6027\u200b\u3002</p> <p>\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>a = [1, 2, 3]\nb = [_ ** 2 for _ in a]\n</code></pre> <p>Python 3.x\u200b\u4e2d\u200b\u7684\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u4e0d\u4f1a\u200b\u4ea7\u751f\u200b\u53d8\u91cf\u200b\u6cc4\u9732\u200b\u7684\u200b\u95ee\u9898\u200b\uff0c\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>x = 'ABC'\ny = [x for x in x]\nprint(x)\n</code></pre> <p>\u200b\u8f93\u51fa\u200b\u4e3a\u200b<code>ABC</code>\uff0c\u200b\u8bf4\u660e\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u53d8\u91cf\u200b<code>x</code>\u200b\u7684\u200b\u503c\u200b\u6ca1\u6709\u200b\u4f20\u9012\u200b\u5230\u200b\u5916\u90e8\u200b\u3002</p> <p>\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u7b49\u4ef7\u200b\u4e8e\u200b<code>map</code>\u200b\u4e0e\u200b<code>filter</code>\u200b\u7684\u200b\u7ed3\u5408\u200b\u3002<code>map</code>\u200b\u51fd\u6570\u200b\u63a5\u6536\u200b\u4e24\u4e2a\u200b\u53c2\u6570\u200b\u3002\u200b\u7b2c\u4e00\u4e2a\u200b\u53c2\u6570\u200b\u4e3a\u200b\u51fd\u6570\u200b\uff0c\u200b\u7b2c\u4e8c\u4e2a\u200b\u53c2\u6570\u200b\u4e3a\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u3002<code>map</code>\u200b\u51fd\u6570\u200b\u8fed\u4ee3\u200b\u5730\u200b\u5c06\u200b\u51fd\u6570\u200b\u4f5c\u7528\u200b\u4e8e\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u4ece\u800c\u200b\u5f97\u5230\u200b\u7ed3\u679c\u200b\u3002</p> <p>\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u8ba1\u7b97\u200b\u5217\u8868\u200b\u7684\u200b\u7b1b\u5361\u5c14\u200b\u79ef\u200b\uff0c\u200b\u5982\u200b\u4e0b\u5217\u200b\u4ee3\u7801\u200b\uff1a</p> <pre><code>[(x, y) for x in range(10) for y in range(11, 20)]\n</code></pre> <p>\u200b\u5faa\u73af\u200b\u7684\u200b\u987a\u5e8f\u200b\u4e0e\u200b<code>for</code>\u200b\u5faa\u73af\u200b\u4e00\u81f4\u200b\uff0c\u200b\u5148\u200b\u6267\u884c\u200b\u5916\u5c42\u200b\u7684\u200b\u5faa\u73af\u200b\uff0c\u200b\u518d\u200b\u6267\u884c\u200b\u5185\u5c42\u200b\u7684\u200b\u5faa\u73af\u200b\u3002</p> <p>\u200b\u5c06\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u7684\u200b\u65b9\u62ec\u53f7\u200b\u5706\u62ec\u53f7\u200b\uff0c\u200b\u5373\u200b\u4e3a\u200b\u751f\u6210\u5668\u200b\u8868\u8fbe\u5f0f\u200b\u3002\u200b\u5982\u679c\u200b\u751f\u6210\u5668\u200b\u8868\u8fbe\u5f0f\u200b\u662f\u200b\u51fd\u6570\u8c03\u7528\u200b\u7684\u200b\u552f\u4e00\u200b\u53c2\u6570\u200b\uff0c\u200b\u5706\u62ec\u53f7\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u3002</p>"},{"location":"coding/python/sequence/#_2","title":"\u5143\u7ec4","text":"<p>\u200b\u5143\u7ec4\u200b\u662f\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7406\u89e3\u200b\u4e3a\u200b\u5b57\u200b\u6bb5\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u6570\u636e\u8868\u200b\u4e2d\u200b\u7684\u200b\u67d0\u200b\u4e00\u884c\u200b\u3002\u200b\u5982\u4e0b\u200b\u7740\u91cd\u200b\u8ba8\u8bba\u200b\u5143\u7ec4\u200b\u89e3\u5305\u200b\u7684\u200b\u5185\u5bb9\u200b\u3002</p> <p>\u200b\u5e73\u884c\u200b\u8d4b\u503c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u4e00\u4e2a\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u590d\u5236\u5230\u200b\u4e00\u7ec4\u200b\u53d8\u91cf\u200b\u4e2d\u200b\u3002\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u4e0d\u200b\u8981\u6c42\u200b\u662f\u200b\u5143\u7ec4\u200b\u3002\u200b\u5982\u200b\uff1a</p> <pre><code>a, b = (1, 2)\nc, d = [3, 4]\ne, f = {1: 2, 3: 4} # \u200b\u6309\u7167\u200bkey\u200b\u7684\u200b\u53d6\u503c\u200b\u8fdb\u884c\u200b\u5206\u914d\u200b\n</code></pre> <p><code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u89e3\u5305\u200b\u5143\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u4f5c\u4e3a\u200b\u51fd\u6570\u53c2\u6570\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <pre><code>print(*(1, 2))\n</code></pre> <p><code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u4ece\u200b\u51fd\u6570\u200b\u4e2d\u200b\u83b7\u53d6\u200b\u53ef\u53d8\u200b\u6570\u91cf\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5e38\u89c1\u200b\u7528\u6cd5\u200b\u4e3a\u200b<code>*args</code>\u200b\u4e0e\u200b<code>**kwargs</code>\u3002</p> <p><code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u5e73\u884c\u200b\u8d4b\u503c\u200b\u4e2d\u200b\uff0c\u200b\u4f46\u200b\u5e73\u884c\u200b\u8d4b\u503c\u200b\u4e2d\u200b\u7684\u200b\u4e00\u7ec4\u200b\u53d8\u91cf\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u4e2a\u200b<code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u3002\u200b\u5e26\u200b<code>*</code>\u200b\u7684\u200b\u53d8\u91cf\u200b\u89e3\u5305\u200b\u540e\u200b\u6210\u4e3a\u200b\u5217\u8868\u200b\u3002</p> <pre><code>a, b, c, *d = range(5) # d = [3, 4]\na, b, *c, d = range(5) # c = [2, 3]\n</code></pre> <p>\u200b\u5f53\u5f85\u200b\u89e3\u5305\u200b\u7684\u200b\u5143\u7ec4\u200b\u672c\u8eab\u200b\u662f\u200b\u5143\u7ec4\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u62ec\u53f7\u200b\u89e3\u5305\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <pre><code>a, (b, c), d = (1, (2, 3), 4)\n</code></pre> <p>\u200b\u5143\u7ec4\u200b\u4e0d\u200b\u652f\u6301\u200b\u5217\u8868\u200b\u7684\u200b<code>append</code>\u3001<code>clear</code>\u3001<code>copy</code>\u3001<code>extend</code>\u3001<code>insert</code>\u3001<code>pop</code>\u3001<code>remove</code>\u3001<code>reverse</code>\u200b\u7b49\u200b\u65b9\u6cd5\u200b\u3002</p>"},{"location":"coding/python/sequence/#_3","title":"\u5177\u540d\u200b\u5143\u7ec4","text":"<p><code>collections.namedtuple</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u7528\u4e8e\u200b\u521b\u5efa\u200b\u5177\u540d\u200b\u5143\u7ec4\u200b\u7c7b\u578b\u200b\u7684\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5176\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>collections.namedtuple(typename, field_names, *, rename=False, defaults=None, module=None)\n</code></pre> <ul> <li><code>typename</code>\u200b\u6307\u200b\u7c7b\u578b\u200b\u7684\u200b\u540d\u79f0\u200b\uff0c\u200b\u8be5\u200b\u53d8\u91cf\u200b\u88ab\u200b\u5199\u5165\u200b<code>__repr__</code>\u200b\u65b9\u6cd5\u200b\u4e2d\u200b\u3002\u200b\u8be5\u200b\u53c2\u6570\u200b\u7684\u200b\u53d6\u503c\u200b\u4e0e\u200b\u4ee3\u7801\u200b\u4e2d\u200b\u7c7b\u578b\u200b\u88ab\u200b\u5b9e\u9645\u200b\u5206\u914d\u200b\u7684\u200b\u540d\u79f0\u200b\u65e0\u5173\u200b\u3002</li> <li><code>field_names</code>\u200b\u662f\u200b\u5143\u7ec4\u200b\u4e2d\u5b57\u200b\u6bb5\u200b\u7684\u200b\u540d\u79f0\u200b</li> <li>\u200b\u5f53\u200b<code>rename</code>\u200b\u4e3a\u200b<code>True</code>\u200b\u91cd\u590d\u200b\u6216\u200b\u4e0e\u200b\u5173\u952e\u5b57\u200b\u51b2\u7a81\u200b\u7684\u200b\u5b57\u6bb5\u540d\u200b\u79f0\u4f1a\u200b\u88ab\u200b\u81ea\u52a8\u200b\u5904\u7406\u200b\uff0c\u200b\u5426\u5219\u200b\u4f1a\u200b\u629b\u51fa\u200b<code>ValueError</code></li> <li><code>defaults</code>\u200b\u4e3a\u200b\u5b57\u200b\u6bb5\u200b\u7684\u200b\u9ed8\u8ba4\u503c\u200b\uff0c\u200b\u8be5\u200b\u5217\u8868\u200b\u4e0e\u200b\u5b57\u6bb5\u540d\u200b\u79f0\u200b\u5217\u8868\u200b\u6309\u53f3\u200b\u5bf9\u9f50\u200b\u8fdb\u884c\u200b\u5339\u914d\u200b\u3002</li> <li>\u200b\u82e5\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>module</code>\u200b\u53c2\u6570\u200b\uff0c\u200b\u8be5\u503c\u200b\u4f1a\u200b\u88ab\u200b\u5199\u5165\u200b<code>__module__</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\u3002</li> </ul> <p><code>namedtuple</code>\u200b\u8fd4\u56de\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u521b\u5efa\u5bf9\u8c61\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>&gt;&gt;&gt; from collections import namedtuple\n&gt;&gt;&gt; Point = namedtuple('Point', [\"x\", \"y\"], defaults=[1, 2]) \n&gt;&gt;&gt; Point()\nPoint(x=1, y=2)\n&gt;&gt;&gt; Point(2)\nPoint(x=2, y=2)\n&gt;&gt;&gt; Point(2, 3)\nPoint(x=2, y=3)\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/sequence/#_4","title":"\u5207\u7247","text":"<p>\u200b\u5207\u7247\u200b\u662f\u200b\u5728\u200b\u5e8f\u5217\u200b\u4e0a\u200b\u6267\u884c\u200b\u7684\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u5207\u7247\u200b\u64cd\u4f5c\u200b\u5f97\u5230\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u5e8f\u5217\u200b\u3002\u200b\u5207\u7247\u200b\u4ee5\u200b<code>s[a:b:c]</code>\u200b\u7684\u200b\u5f62\u5f0f\u200b\u6307\u5b9a\u200b\uff0c<code>a</code>\u200b\u4e3a\u200b\u8d77\u59cb\u200b\u4e0b\u6807\u200b\u3001<code>b</code>\u200b\u4e3a\u200b\u7ec8\u6b62\u200b\u4e0b\u6807\u200b\uff08\u200b\u4e0d\u200b\u5305\u542b\u200b\uff09\u3001<code>c</code>\u200b\u4e3a\u200b\u6b65\u957f\u200b\uff08\u200b\u53ef\u4ee5\u200b\u4e3a\u200b\u8d1f\u200b\uff09\u3002</p> <ul> <li><code>s[::-1]</code>\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u53cd\u8f6c\u200b<code>s</code>\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b</li> <li>\u200b\u5207\u7247\u200b\u5e76\u200b\u6ca1\u6709\u200b\u521b\u5efa\u200b\u65b0\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5bf9\u200b\u5207\u7247\u200b\u7684\u200b\u4fee\u6539\u200b\u4f1a\u200b\u53cd\u6620\u200b\u5230\u200b\u539f\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u3002\u200b\u4f46\u200b\u5bf9\u200b\u5207\u7247\u200b\u7684\u200b\u4fee\u6539\u200b\u5fc5\u987b\u200b\u6ee1\u8db3\u200b\u5143\u7d20\u200b\u6570\u91cf\u200b\u76f8\u540c\u200b\u7684\u200b\u6761\u4ef6\u200b\uff0c\u200b\u5426\u5219\u200b\u4f1a\u200b\u629b\u51fa\u200b<code>ValueError</code>\u3002</li> </ul> <p>\u200b\u53e6\u200b\u4e00\u79cd\u200b\u5207\u7247\u200b\u7684\u200b\u65b9\u5f0f\u200b\u4e3a\u200b<code>s[slice(a, b, c)]</code>\uff0c\u200b\u7701\u7565\u200b\u7684\u200b\u53c2\u6570\u200b\u4f7f\u7528\u200b<code>None</code>\u200b\u8868\u793a\u200b\u3002\u200b\u56e0\u6b64\u200b<code>s[::-1]</code>\u200b\u7b49\u4ef7\u200b\u4e8e\u200b<code>s[slice(None, None, -1)]</code>\u3002\u200b\u4e8b\u5b9e\u4e0a\u200b\uff0cPython\u200b\u5728\u200b\u5904\u7406\u200b\u5f62\u200b\u5982\u200b<code>s[a:b:c]</code>\u200b\u7684\u200b\u5207\u7247\u200b\u65f6\u4f1a\u200b\u8c03\u7528\u200b<code>s.__getitem__(slice(start, stop, step))</code></p> <p>\u200b\u5728\u200bNumPy\u200b\u6570\u7ec4\u200b\u7684\u200b\u5207\u7247\u200b\u4e2d\u200b\u6709\u65f6\u200b\u4f1a\u200b\u51fa\u73b0\u200b\u7701\u7565\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u6b64\u65f6\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>...</code>\u3002<code>...</code>\u200b\u7b26\u53f7\u200b\u4f1a\u200b\u88ab\u200bPython\u200b\u89e3\u91ca\u200b\u4e3a\u200b<code>EllipsisType</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u5e38\u91cf\u200b\u3002</p>"},{"location":"coding/python/sequence/#_5","title":"\u5e8f\u5217\u200b\u8fd0\u7b97","text":"<p>\u200b\u5e8f\u5217\u200b\u901a\u5e38\u200b\u652f\u6301\u200b<code>+</code>\u200b\u8fd0\u7b97\u200b\u4e0e\u200b<code>*</code>\u200b\u8fd0\u7b97\u200b\u3002\u200b\u4e0e\u200b\u5207\u7247\u200b\u4e0d\u540c\u200b\uff0c\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u8fd0\u7b97\u200b\u76f4\u63a5\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u5e8f\u5217\u200b\u3002</p> <pre><code>&gt;&gt;&gt; a = [1]\n&gt;&gt;&gt; b = [2] \n&gt;&gt;&gt; c = a + b\n&gt;&gt;&gt; c is a\nFalse\n&gt;&gt;&gt;\n</code></pre> <p><code>+</code>\u200b\u8fd0\u7b97\u200b\u5c40\u9650\u4e8e\u200b\u76f8\u540c\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5bf9\u4e8e\u200b<code>list</code>\u200b\u7b49\u200b\u53ef\u53d8\u200b\u7c7b\u578b\u200b\uff0c<code>a + b</code>\u200b\u7b49\u4ef7\u200b\u4e8e\u200b<code>a.extend(b)</code>\u3002\u200b\u4f46\u200b<code>a.extend(b)</code>\u200b\u65b9\u6cd5\u200b\u5bf9\u200b<code>b</code>\u200b\u7684\u200b\u7c7b\u578b\u200b\u65e0\u200b\u8981\u6c42\u200b\uff0c\u200b\u4ec5\u200b\u9700\u8981\u200b<code>b</code>\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u3002</p> <p><code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u7684\u200b\u7b2c\u4e8c\u4e2a\u200b\u64cd\u4f5c\u6570\u200b<code>y</code>\u200b\u5fc5\u987b\u200b\u4e3a\u200b\u6574\u6570\u200b\uff0c\u200b\u8868\u793a\u200b\u5c06\u200b\u5e8f\u5217\u200b\u91cd\u590d\u200b<code>y</code>\u200b\u6b21\u200b\u3002\u200b\u82e5\u200b<code>y&lt;=0</code>\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b\u7a7a\u200b\u5217\u8868\u200b</p> <p>\u200b\u6ce8\u610f\u200b\uff0c<code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u5f97\u5230\u200b\u7684\u200b\u5217\u8868\u200b\u4e2d\u200b\uff0c\u200b\u5217\u8868\u200b\u591a\u6b21\u91cd\u590d\u200b\u7684\u200b\u90e8\u5206\u200b\u4ee5\u200b\u5f15\u7528\u200b\u7684\u200b\u5f62\u5f0f\u200b\u5b58\u50a8\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; b = [[]] * 3\n&gt;&gt;&gt; b[0].append(1)\n&gt;&gt;&gt; b\n[[1], [1], [1]]\n&gt;&gt;&gt;\n</code></pre> <p><code>append</code>\u200b\u5904\u7406\u200b\u53ef\u53d8\u200b\u5bf9\u8c61\u200b\u7684\u200b\u65b9\u5f0f\u200b\u540c\u6837\u200b\u662f\u200b\u590d\u5236\u200b\u5f15\u7528\u200b\u800c\u200b\u4e0d\u662f\u200b\u590d\u5236\u200b\u503c\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e5f\u200b\u4f1a\u200b\u9020\u6210\u200b\u76f8\u540c\u200b\u7684\u200b\u95ee\u9898\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a = []\n&gt;&gt;&gt; b = []\n&gt;&gt;&gt; for i in range(3):\n...     a.append(b)\n... \n&gt;&gt;&gt; a[0].append(1)\n&gt;&gt;&gt; a\n[[1], [1], [1]]\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u56e0\u6b64\u200b\u8981\u200b\u907f\u514d\u200b\u4f7f\u7528\u200b<code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u5904\u7406\u200b\u5305\u542b\u200b\u53ef\u53d8\u200b\u5bf9\u8c61\u200b\u7684\u200b\u5217\u8868\u200b\u3002\u200b\u5efa\u8bae\u200b\u4f7f\u7528\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u521b\u5efa\u200b\u5d4c\u5957\u200b\u5217\u8868\u200b\u3002</p> <p>\u200b\u4e0e\u200b\u5e8f\u5217\u200b\u7684\u200b\u8fd0\u7b97\u200b\u4e0d\u540c\u200b\uff0c\uff08\u200b\u53ef\u53d8\u200b\uff09\u200b\u5e8f\u5217\u200b\u7684\u200b\u589e\u91cf\u200b\u8fd0\u7b97\u200b<code>+=</code>\u3001<code>*=</code>\u200b\u662f\u200b\u5728\u200b\u539f\u200b\u5e8f\u5217\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\u8fdb\u884c\u200b\u7684\u200b\uff0c\u200b\u4e0d\u4f1a\u200b\u521b\u5efa\u200b\u65b0\u200b\u7684\u200b\u5e8f\u5217\u200b\u5bf9\u8c61\u200b\u3002\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\u4e0d\u200b\u652f\u6301\u200b\u589e\u91cf\u200b\u8fd0\u7b97\u200b\u3002</p> <p>\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a = [1, 2, 3]\n&gt;&gt;&gt; a[2] += 5\n&gt;&gt;&gt; a\n[1, 2, 8]\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\u901a\u8fc7\u200b<code>__setitem__</code>\u200b\u65b9\u6cd5\u200b\u5b9e\u73b0\u200b\u4e86\u200b\u5207\u7247\u200b\u7684\u200b\u4fee\u6539\u200b\u3002\u200b\u82e5\u200b<code>a</code>\u200b\u4e3a\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\uff0c\u200b\u6267\u884c\u200b\u7ed3\u679c\u200b\u5e94\u200b\u4e3a\u4f55\u200b\uff1f</p> <p>\u200b\u5bf9\u4e8e\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5176\u200b\u672c\u8eab\u200b\u4e0d\u200b\u652f\u6301\u200b\u5bf9\u200b\u5143\u7d20\u200b\u8fdb\u884c\u200b\u7684\u200b\u4fee\u6539\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5e8f\u5217\u200b\u672c\u8eab\u200b\u7684\u200b\u503c\u200b\u4e0d\u4f1a\u200b\u6539\u53d8\u200b\u3002</p> <pre><code>&gt;&gt;&gt; a = (1, 2, 3)\n&gt;&gt;&gt; a[2] += 5\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: 'tuple' object does not support item assignment\n&gt;&gt;&gt; a\n(1, 2, 3)\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u4f46\u200b\u5982\u679c\u200b\u8981\u200b\u4fee\u6539\u200b\u7684\u200b\u5143\u7d20\u200b\u672c\u8eab\u200b\u662f\u200b\u5f15\u7528\u200b\uff0c\u200b\u5982\u200b\u4e00\u4e2a\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\uff0c\u200b\u6b64\u65f6\u200b\u5bf9\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\u7684\u200b\u4fee\u6539\u200b\u4e0d\u4f1a\u200b\u5f71\u54cd\u200b\u5230\u200b\u5f15\u7528\u200b\u672c\u8eab\u200b\uff0c\u200b\u56e0\u6b64\u200b<code>+=</code>\u200b\u8fd0\u7b97\u200b\u4ecd\u4f1a\u200b\u6b63\u5e38\u200b\u4f5c\u7528\u200b\u3002\u200b\u589e\u91cf\u200b\u64cd\u4f5c\u200b\u4e0d\u662f\u200b\u539f\u5b50\u200b\u64cd\u4f5c\u200b\u3002</p> <pre><code>&gt;&gt;&gt; t = (1, 2, [30, 40])\n&gt;&gt;&gt; t[2] += [50, 60]\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: 'tuple' object does not support item assignment\n&gt;&gt;&gt; t\n(1, 2, [30, 40, 50, 60])\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b<code>append</code>\u200b\u6216\u200b<code>extend</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u907f\u514d\u200b\u5f02\u5e38\u200b\u7684\u200b\u4ea7\u751f\u200b\u3002\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\u88ab\u200b \u200b\u6b63\u5e38\u200b\u4fee\u6539\u200b \u3002\u200b\u4e3a\u200b\u4fdd\u8bc1\u6570\u636e\u200b\u4e0d\u200b\u88ab\u200b\u9519\u8bef\u200b\u5730\u200b\u4fee\u6539\u200b\uff0c\u200b\u5e94\u200b\u907f\u514d\u200b\u5728\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u5bb9\u7eb3\u200b\u53ef\u53d8\u200b\u5bf9\u8c61\u200b\u3002</p>"},{"location":"coding/python/sequence/#_6","title":"\u6392\u5e8f","text":"<p><code>list.sort</code>\u200b\u65b9\u6cd5\u200b\u4f5c\u7528\u200b\u4e0e\u200b\u5217\u8868\u200b\u672c\u8eab\u200b\u800c\u200b\u4e0d\u200b\u8fd4\u56de\u200b\uff08\u200b\u8fd4\u56de\u503c\u200b\u4e3a\u200b<code>None</code>\uff09\u3002\u200b\u8981\u200b\u4ea7\u751f\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u5df2\u200b\u6392\u5e8f\u200b\u5217\u8868\u200b\u800c\u200b\u4e0d\u200b\u5f71\u54cd\u200b\u539f\u200b\u5217\u8868\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>sorted()</code>\u200b\u5185\u7f6e\u200b\u51fd\u6570\u200b\u3002</p> <pre><code>&gt;&gt;&gt; a = [6, 3, 2, 1]\n&gt;&gt;&gt; b = sorted(a)\n&gt;&gt;&gt; b is a\nFalse\n&gt;&gt;&gt;\n</code></pre> <p><code>sorted</code>\u200b\u4e0e\u200b<code>list.sort</code>\u200b\u652f\u6301\u200b\u6309\u7167\u200b\u5176\u4ed6\u200b\u89c4\u5219\u200b\u5bf9\u200b\u5e8f\u5217\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b\uff0c\u200b\u5373\u200b<code>key</code>\u200b\u53c2\u6570\u200b\u3002<code>key</code>\u200b\u53c2\u6570\u200b\u4e3a\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u53c2\u6570\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5982\u200b<code>key=len</code>\u200b\u5373\u200b\u4e3a\u200b\u5bf9\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u5347\u5e8f\u200b\u6392\u5e8f\u200b\uff0c<code>key=lower</code>\u200b\u5373\u200b\u4e3a\u200b\u5ffd\u7565\u200b\u5b57\u7b26\u4e32\u200b\u5927\u5c0f\u5199\u200b\u8fdb\u884c\u200b\u5347\u5e8f\u200b\u6392\u5e8f\u200b\u3002</p> <p>Python\u200b\u7684\u200b<code>bisect</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5bf9\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7684\u200b\u652f\u6301\u200b\u3002<code>bisect.bisect</code>\u200b\u4e3a\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u51fd\u6570\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5728\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5927\u4e8e\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u503c\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002<code>bisect.insort</code>\u200b\u4e3a\u200b\u63d2\u5165\u200b\u51fd\u6570\u200b\uff0c\u200b\u5728\u200b<code>bisect</code>\u200b\u51fd\u6570\u200b\u67e5\u8be2\u200b\u4f4d\u7f6e\u200b\u63d2\u5165\u200b\u5143\u7d20\u200b\u3002</p> <p><code>bisect</code>\u200b\u4e0e\u200b<code>insort</code>\u200b\u51fd\u6570\u200b\u6709\u200b\u53d8\u79cd\u200b<code>bisect_left</code>\u200b\u4e0e\u200b<code>insort_left</code>\u3002<code>bisect_left</code>\u200b\u8fd4\u56de\u200b\u5728\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u4e0d\u200b\u5927\u4e8e\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u503c\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002</p>"},{"location":"coding/python/sequence/#_7","title":"\u5176\u4ed6\u200b\u5e8f\u5217\u200b\u7c7b\u578b","text":""},{"location":"coding/python/sequence/#_8","title":"\u6570\u7ec4","text":"<p><code>array</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u6570\u7ec4\u200b\u7ed3\u6784\u200b\u7684\u200b\u652f\u6301\u200b\u3002\u200b\u6570\u7ec4\u200b\u662f\u200b\u53ea\u80fd\u200b\u5b58\u50a8\u200b\u4e00\u79cd\u200b\u7c7b\u578b\u200b\u5143\u7d20\u200b\u7684\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\uff0c\u200b\u652f\u6301\u200b<code>list</code>\u200b\u7684\u200b\u591a\u6570\u200b\u65b9\u6cd5\u200b\u3002\u200b\u540c\u65f6\u200b\u6570\u7ec4\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>fromfile, tofile</code>\u200b\u4e0e\u200b<code>frombytes, tobytes</code>\uff0c\u200b\u4ece\u800c\u200b\u652f\u6301\u200b\u76f4\u63a5\u200b\u8bfb\u5199\u200b\u6587\u4ef6\u200b\u6216\u200b\u6bd4\u7279\u6d41\u200b\u3002</p> <p>\u200b\u5728\u200b\u521b\u5efa\u200b\u6570\u7ec4\u200b\u65f6\u200b\u9700\u8981\u200b\u6307\u5b9a\u200b\u7c7b\u578b\u200b\u7801\u200b\uff0c\u200b\u5373\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5b58\u50a8\u200b\u7684\u200b\u6570\u636e\u7c7b\u578b\u200b\u3002\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u7c7b\u578b\u200b\u7801\u200b\u5bf9\u5e94\u200b\u7684\u200b\u7c7b\u578b\u200b\u53ea\u200b\u5360\u7528\u200b\u6709\u9650\u200b\u7684\u200b\u5185\u5b58\u7a7a\u95f4\u200b\u3002\u200b\u5728\u200b\u5b58\u50a8\u200b\u5927\u91cf\u200b\u6570\u636e\u200b\u65f6\u200b\u4f18\u4e8e\u200b\u5217\u8868\u200b\u7b49\u200b\u5176\u4ed6\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\u3002</p> \u200b\u7c7b\u578b\u200b\u7801\u200b C \u200b\u7c7b\u578b\u200b Python \u200b\u7c7b\u578b\u200b \u200b\u5b57\u200b\u5bbd\u200b <code>'b'</code> <code>signed char</code> <code>int</code> 1 <code>'B'</code> <code>unsigned char</code> <code>int</code> 1 <code>'u'</code> <code>wchar_t</code> Unicode character 2 <code>'h'</code> <code>signed short</code> <code>int</code> 2 <code>'H'</code> <code>unsigned short</code> <code>int</code> 2 <code>'i'</code> <code>signed int</code> <code>int</code> 2 <code>'I'</code> <code>unsigned int</code> <code>int</code> 2 <code>'l'</code> <code>signed long</code> <code>int</code> 4 <code>'L'</code> <code>unsigned long</code> <code>int</code> 4 <code>'q'</code> <code>signed long long</code> <code>int</code> 8 <code>'Q'</code> <code>unsigned long long</code> <code>int</code> 8 <code>'f'</code> <code>float</code> <code>float</code> 4 <code>'d'</code> <code>double</code> <code>float</code> 8 <p>\u200b\u6570\u7ec4\u200b\u4e0d\u200b\u652f\u6301\u200b\u5217\u8868\u200b\u7684\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li><code>clear</code></li> <li><code>copy</code></li> <li><code>sort</code></li> </ul> <p>\u200b\u4f46\u200b\u6570\u7ec4\u200b\u652f\u6301\u200b\u4ece\u200b\u5217\u8868\u200b\u6dfb\u52a0\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5373\u200b<code>fromlist</code>\u200b\u65b9\u6cd5\u200b\u3002\u200b\u5f53\u200b\u5217\u8868\u200b\u4e2d\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u5728\u200b\u6dfb\u52a0\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u51fa\u9519\u200b\uff0c\u200b\u5219\u200b\u6240\u6709\u200b\u7684\u200b\u6dfb\u52a0\u200b\u90fd\u200b\u4f1a\u200b\u88ab\u200b\u56de\u6eda\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import array\n&gt;&gt;&gt; a = array.array('f')\n&gt;&gt;&gt; b = [1.0, 2.0, 'a']\n&gt;&gt;&gt; a\narray('f')\n&gt;&gt;&gt; a.fromlist(b)\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: must be real number, not str\n&gt;&gt;&gt; a\narray('f')\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6570\u7ec4\u200b\u662f\u200bbytes-like object\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u5185\u5b58\u200b\u89c6\u56fe\u200b\u4e2d\u200b\u3002</p>"},{"location":"coding/python/sequence/#_9","title":"\u5185\u5b58\u200b\u89c6\u56fe","text":"<p><code>memoryview</code>\u200b\u5141\u8bb8\u200b\u4f7f\u7528\u200b\u4e0d\u540c\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8bfb\u5199\u200b\u540c\u4e00\u200b\u5757\u5185\u5b58\u200b\u7684\u200b\u6570\u636e\u200b\u3002</p> <p>\u200b\u5982\u4e0b\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u6620\u5c04\u200b\u5230\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u7684\u200b<code>memoryview</code>\uff0c\u200b\u5728\u200b\u4e0d\u200b\u6267\u884c\u200b\u64cd\u4f5c\u200b\u65f6\u200b\uff0c<code>memory</code>\u200b\u7684\u200b\u6570\u636e\u200b\u5185\u5bb9\u200b\u4e0e\u200b\u539f\u200b\u5e8f\u5217\u200b\u7684\u200b\u6570\u636e\u200b\u5185\u5bb9\u200b\u76f8\u540c\u200b\u3002</p> <pre><code>&gt;&gt;&gt; a = [1, 2, 3, 4]\n&gt;&gt;&gt; b = memoryview(array.array('i', a))\n&gt;&gt;&gt; b[0]\n1\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>memoryview.cast</code>\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u5c06\u200b<code>memoryview</code>\u200b\u6620\u5c04\u200b\u7684\u200b\u5185\u5b58\u200b\u533a\u57df\u200b\u4ee5\u200b\u4e0d\u540c\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u89e3\u91ca\u200b\u3002</p> <pre><code>&gt;&gt;&gt; c = b.cast('B')\n&gt;&gt;&gt; list(c)\n[1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0]\n&gt;&gt;&gt;\n</code></pre> <p><code>memoryview</code>\u200b\u7684\u200b\u53ef\u53d8\u6027\u200b\u53d6\u51b3\u4e8e\u200b\u5176\u6240\u200b\u6620\u5c04\u200b\u5185\u5b58\u200b\u533a\u57df\u200b\u7684\u200b\u6027\u8d28\u200b\uff0c\u200b\u82e5\u200b\u4f7f\u7528\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\u521b\u5efa\u200b<code>memoryview</code>\uff0c\u200b\u5219\u200b<code>memoryview</code>\u200b\u4e3a\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\u3002</p> <pre><code>&gt;&gt;&gt; d = memoryview(b\"abc\") \n&gt;&gt;&gt; d[0] = \"d\"\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: cannot modify read-only memory\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u4f5c\u4e3a\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\u7684\u200b<code>memoryview</code>\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u4fee\u6539\u200b\uff0c\u200b\u518d\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u539f\u6765\u200b\u7684\u200b\u89e3\u91ca\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u5373\u53ef\u200b\u5b9e\u73b0\u200b\u5bf9\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u67d0\u4e2a\u200b\u7279\u5b9a\u200b\u5b57\u8282\u200b\u7684\u200b\u4fee\u6539\u200b\u3002</p> <pre><code>&gt;&gt;&gt; c[1] = 1 \n&gt;&gt;&gt; b = c.cast(\"i\")\n&gt;&gt;&gt; list(b)\n[257, 2, 3, 4]\n&gt;&gt;&gt;\n</code></pre> <p><code>memoryview</code>\u200b\u7684\u200b\u5207\u7247\u200b\u662f\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b<code>memoryview</code>\u200b\u5bf9\u8c61\u200b\uff08\u200b\u57fa\u4e8e\u200b\u539f\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\uff09</p>"},{"location":"coding/python/sequence/#_10","title":"\u961f\u5217","text":"<p>\u200b\u5217\u8868\u200b\u7684\u200b<code>append</code>\u200b\u4e0e\u200b<code>pop</code>\u200b\u65b9\u6cd5\u200b\u7ec4\u5408\u200b\u4f7f\u7528\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u5bf9\u6808\u200b\u6216\u200b\u961f\u5217\u200b\u7684\u200b\u6a21\u62df\u200b\u3002<code>collection.deque</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u961f\u5217\u200b\u7684\u200b\u66f4\u200b\u9ad8\u6548\u200b\u7684\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u5e76\u4e14\u200b\u6ee1\u8db3\u200b\u7ebf\u7a0b\u200b\u5b89\u5168\u200b\u7684\u200b\u6761\u4ef6\u200b\u3002</p> <p>\u200b\u5728\u200b\u521b\u5efa\u200b\u961f\u5217\u200b\u65f6\u200b\uff0c\u200b\u53ef\u200b\u9009\u200b\u63d0\u4f9b\u200b\u961f\u5217\u200b\u7684\u200b\u6700\u5927\u200b\u957f\u5ea6\u200b<code>maxlen</code>\u3002\u200b\u8be5\u200b\u53c2\u6570\u200b\u53ea\u80fd\u200b\u5728\u200b\u6784\u9020\u51fd\u6570\u200b\u4e2d\u200b\u6307\u5b9a\u200b\uff0c\u200b\u5728\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u53ea\u8bfb\u200b\u3002</p> <p>\u200b\u961f\u5217\u200b\u63d0\u4f9b\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li><code>rotate</code>\uff0c\u200b\u5f53\u200b\u53c2\u6570\u200b<code>n &gt; 0</code>\u200b\u65f6\u200b\u5c06\u200b\u961f\u5217\u200b\u53f3\u4fa7\u200b\u7684\u200b<code>n</code>\u200b\u4e2a\u200b\u5143\u7d20\u200b\u79fb\u52a8\u200b\u5230\u200b\u961f\u5217\u200b\u5de6\u4fa7\u200b\uff0c\u200b\u53cd\u4e4b\u200b\u5c06\u200b\u961f\u5217\u200b\u5de6\u4fa7\u200b\u7684\u200b<code>n</code>\u200b\u4e2a\u200b\u5143\u7d20\u200b\u79fb\u52a8\u200b\u5230\u200b\u961f\u5217\u200b\u53f3\u4fa7\u200b</li> <li><code>appendleft</code>\uff0c\u200b\u5728\u200b\u961f\u5217\u200b\u7684\u200b\u5de6\u4fa7\u200b\u6dfb\u52a0\u200b\u5143\u7d20\u200b\uff08\u200b\u5f53\u200b\u8d85\u51fa\u200b\u961f\u5217\u200b\u957f\u5ea6\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u4ece\u200b\u961f\u5217\u200b\u53e6\u4e00\u4fa7\u200b\u5220\u9664\u200b\uff09</li> <li><code>extendleft</code>\uff0c\u200b\u5728\u200b\u961f\u5217\u200b\u7684\u200b\u5de6\u4fa7\u200b\u6dfb\u52a0\u200b\u4e00\u7ec4\u200b\u5143\u7d20\u200b\uff08\u200b\u5f53\u200b\u8d85\u51fa\u200b\u961f\u5217\u200b\u957f\u5ea6\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u4ece\u200b\u961f\u5217\u200b\u53e6\u4e00\u4fa7\u200b\u5220\u9664\u200b\uff09</li> <li>\u200b\u961f\u5217\u200b\u652f\u6301\u200b\u5217\u8868\u200b\u7684\u200b\u5927\u591a\u6570\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u9664\u200b<code>copy</code>\u3001<code>sort</code>\u3001<code>insert</code>\u3001<code>index</code>\u200b\u7b49\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u961f\u5217\u200b\u4ec5\u200b\u652f\u6301\u200b<code>+=</code>\u200b\u8fd0\u7b97\u7b26\u200b\u7684\u200b\u4f5c\u7528\u200b\u3002</li> </ul>"},{"location":"coding/reinforcement-learning/","title":"\u5f3a\u5316\u200b\u5b66\u4e60","text":"<p>\u200b\u8bb0\u5f55\u200b\u5f3a\u5316\u200b\u5b66\u4e60\u200b\u7684\u200b\u5b66\u4e60\u200b\u8fc7\u7a0b\u200b\u3002</p> <ul> <li>\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b</li> <li>\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u7b97\u6cd5\u200b</li> </ul>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/","title":"\u591a\u81c2\u200b\u8001\u864e\u673a","text":"In\u00a0[1]: Copied! <pre>import itertools\nimport random\nfrom typing import List, Optional\n\nfrom scipy.stats import norm\n\n\nclass MultiArmedBandit():\n    def __init__(self, means, stds: Optional[List[float]] = None):\n        self.means = means\n        if stds is not None:\n            if len(stds) != len(means):\n                raise ValueError(\n                    'The length of means and stds should be the same.')\n            self.stds = stds\n        else:\n            self.stds = [1.0] * len(means)\n        self.k = len(means)\n        self.dists = [\n            norm(loc=mean, scale=std) for mean, std in zip(means, self.stds)\n        ]\n\n    def __len__(self) -&gt; int:\n        return self.k\n\n    @property\n    def optimal_reward(self) -&gt; float:\n        return max(self.means)\n\n    def pull(self, action: int) -&gt; float:\n        return self.dists[action].rvs()\n\nrewards = [*range(0, 10, 2)]\nstds = [*range(1, 6)]\nargs = itertools.product(rewards, stds)\nbandit = MultiArmedBandit(*zip(*args))\nprint(bandit.pull(0))\n</pre> import itertools import random from typing import List, Optional  from scipy.stats import norm   class MultiArmedBandit():     def __init__(self, means, stds: Optional[List[float]] = None):         self.means = means         if stds is not None:             if len(stds) != len(means):                 raise ValueError(                     'The length of means and stds should be the same.')             self.stds = stds         else:             self.stds = [1.0] * len(means)         self.k = len(means)         self.dists = [             norm(loc=mean, scale=std) for mean, std in zip(means, self.stds)         ]      def __len__(self) -&gt; int:         return self.k      @property     def optimal_reward(self) -&gt; float:         return max(self.means)      def pull(self, action: int) -&gt; float:         return self.dists[action].rvs()  rewards = [*range(0, 10, 2)] stds = [*range(1, 6)] args = itertools.product(rewards, stds) bandit = MultiArmedBandit(*zip(*args)) print(bandit.pull(0)) <pre>-1.435877863923347\n</pre> In\u00a0[2]: Copied! <pre>import abc\nfrom typing import Tuple\n\nclass AbstractAgent(abc.ABC):\n    def __init__(self, T: int, k: int):\n        self.T = T\n        self.k = k\n\n        self.history: List[Tuple[int, float]] = []\n\n    @abc.abstractmethod\n    def strategy(self) -&gt; int:\n        pass\n\n    def run(self, bandit: MultiArmedBandit) -&gt; float:\n        rewards = []\n        for t in range(self.T):\n            action = self.strategy()\n            if action &gt;= self.k:\n                raise ValueError('Invalid action.')\n\n            reward = bandit.pull(action)\n            self.history.append((action, reward))\n            rewards.append(reward)\n        return self.T * bandit.optimal_reward - sum(rewards)\n</pre> import abc from typing import Tuple  class AbstractAgent(abc.ABC):     def __init__(self, T: int, k: int):         self.T = T         self.k = k          self.history: List[Tuple[int, float]] = []      @abc.abstractmethod     def strategy(self) -&gt; int:         pass      def run(self, bandit: MultiArmedBandit) -&gt; float:         rewards = []         for t in range(self.T):             action = self.strategy()             if action &gt;= self.k:                 raise ValueError('Invalid action.')              reward = bandit.pull(action)             self.history.append((action, reward))             rewards.append(reward)         return self.T * bandit.optimal_reward - sum(rewards) In\u00a0[3]: Copied! <pre>import numpy as np\n\ndef show_result(x: List[float], name: str) -&gt; str:\n    x_array = np.array(x)\n    return f'{name} - Mean: {x_array.mean():.2f} Std: {x_array.std():.2f}'\n</pre> import numpy as np  def show_result(x: List[float], name: str) -&gt; str:     x_array = np.array(x)     return f'{name} - Mean: {x_array.mean():.2f} Std: {x_array.std():.2f}' In\u00a0[4]: Copied! <pre>class RandomAgent(AbstractAgent):\n    def strategy(self) -&gt; int:\n        return random.randint(0, self.k - 1)\n\nrandom_agents = [RandomAgent(T=100, k=len(bandit)) for _ in range(1000)]\nregrets = [agent.run(bandit) for agent in random_agents]\nshow_result(regrets, 'Random Agent')\n</pre> class RandomAgent(AbstractAgent):     def strategy(self) -&gt; int:         return random.randint(0, self.k - 1)  random_agents = [RandomAgent(T=100, k=len(bandit)) for _ in range(1000)] regrets = [agent.run(bandit) for agent in random_agents] show_result(regrets, 'Random Agent') Out[4]: <pre>'Random Agent - Mean: 399.66 Std: 44.19'</pre> In\u00a0[5]: Copied! <pre>from typing import Dict\n\nclass EpsilonGreedyAgent(AbstractAgent):\n    def __init__(self, T: int, k: int, epsilon: float):\n        super().__init__(T, k)\n        self.epsilon = epsilon\n\n    def _random(self) -&gt; int:\n        return random.randint(0, self.k - 1)\n\n    def _greedy(self) -&gt; int:\n        history_dict: Dict[int, List[float]] = {}\n        for action, reward in self.history:\n            if action not in history_dict:\n                history_dict[action] = []\n            history_dict[action].append(reward)\n        return max(\n            history_dict,\n            key=lambda x: sum(history_dict[x]) / len(history_dict[x])\n        )\n\n    def strategy(self) -&gt; int:\n        if random.random() &lt; self.epsilon or not self.history:\n            return self._random()\n        return self._greedy()\n\n\nepsilon_greedy = [\n    EpsilonGreedyAgent(T=100, k=len(bandit), epsilon=0.2) for _ in range(1000)\n]\nregrets = [agent.run(bandit) for agent in epsilon_greedy]\nshow_result(regrets, 'Epsilon-Greedy Agent')\n</pre> from typing import Dict  class EpsilonGreedyAgent(AbstractAgent):     def __init__(self, T: int, k: int, epsilon: float):         super().__init__(T, k)         self.epsilon = epsilon      def _random(self) -&gt; int:         return random.randint(0, self.k - 1)      def _greedy(self) -&gt; int:         history_dict: Dict[int, List[float]] = {}         for action, reward in self.history:             if action not in history_dict:                 history_dict[action] = []             history_dict[action].append(reward)         return max(             history_dict,             key=lambda x: sum(history_dict[x]) / len(history_dict[x])         )      def strategy(self) -&gt; int:         if random.random() &lt; self.epsilon or not self.history:             return self._random()         return self._greedy()   epsilon_greedy = [     EpsilonGreedyAgent(T=100, k=len(bandit), epsilon=0.2) for _ in range(1000) ] regrets = [agent.run(bandit) for agent in epsilon_greedy] show_result(regrets, 'Epsilon-Greedy Agent') Out[5]: <pre>'Epsilon-Greedy Agent - Mean: 155.49 Std: 79.32'</pre> In\u00a0[6]: Copied! <pre>import math\n\nclass UCBAgent(AbstractAgent):\n    def __init__(self, T: int, k: int, c: float):\n        super().__init__(T, k)\n        self.c = c\n\n    def ucb(self, action: int) -&gt; float:\n        n = len([a for a, _ in self.history if a == action])\n        if n == 0:\n            return float('inf')\n        mean = sum(r for a, r in self.history if a == action) / n\n        std = math.sqrt(math.log(len(self.history)) / n)\n        return mean + self.c * std\n\n    def strategy(self) -&gt; int:\n        # If there are still unselected actions, select one of them.\n        if len(self.history) &lt; self.k:\n            return len(self.history)\n\n        return max(range(self.k), key=self.ucb)\n\nucb_agents = [UCBAgent(T=100, k=len(bandit), c=0.3) for _ in range(1000)]\nregrets = [agent.run(bandit) for agent in ucb_agents]\nshow_result(regrets, 'UCB Agent')\n</pre> import math  class UCBAgent(AbstractAgent):     def __init__(self, T: int, k: int, c: float):         super().__init__(T, k)         self.c = c      def ucb(self, action: int) -&gt; float:         n = len([a for a, _ in self.history if a == action])         if n == 0:             return float('inf')         mean = sum(r for a, r in self.history if a == action) / n         std = math.sqrt(math.log(len(self.history)) / n)         return mean + self.c * std      def strategy(self) -&gt; int:         # If there are still unselected actions, select one of them.         if len(self.history) &lt; self.k:             return len(self.history)          return max(range(self.k), key=self.ucb)  ucb_agents = [UCBAgent(T=100, k=len(bandit), c=0.3) for _ in range(1000)] regrets = [agent.run(bandit) for agent in ucb_agents] show_result(regrets, 'UCB Agent') Out[6]: <pre>'UCB Agent - Mean: 116.76 Std: 29.87'</pre> In\u00a0[7]: Copied! <pre>from scipy.stats import invgamma\n\nclass ThompsonSamplingAgent(AbstractAgent):\n    def __init__(self, T: int, k: int):\n        super().__init__(T, k)\n\n        # We know that the reward is gaussian, so we can use normal-inverse-gamma\n        self.mu = [0.0] * k\n        self.lambd = [1.0] * k\n        self.alpha = [1.0] * k\n        self.beta = [1.0] * k\n\n    def update(self, action: int, reward: float) -&gt; None:\n        self.mu[action] = (\n            self.lambd[action] * self.mu[action] + reward\n        ) / (self.lambd[action] + 1)\n        self.lambd[action] += 1\n        self.alpha[action] += 0.5\n        self.beta[action] += (\n            self.lambd[action] * (reward - self.mu[action]) ** 2\n        ) / (self.lambd[action] + 1) / 2\n\n    def strategy(self) -&gt; int:\n        if self.history:\n            self.update(*self.history[-1])\n\n        samples = [\n            norm.rvs(\n                loc=mu, scale=invgamma.rvs(a, scale=b) / math.sqrt(lambd)\n            )\n            for mu, lambd, a, b in zip(self.mu, self.lambd, self.alpha, self.beta)\n        ]\n        return max(range(self.k), key=lambda x: samples[x])\n\nthompson_sampling_agents = [\n    ThompsonSamplingAgent(T=100, k=len(bandit)) for _ in range(1000)\n]\nregrets = [agent.run(bandit) for agent in thompson_sampling_agents]\nshow_result(regrets, 'Thompson Sampling Agent')\n</pre> from scipy.stats import invgamma  class ThompsonSamplingAgent(AbstractAgent):     def __init__(self, T: int, k: int):         super().__init__(T, k)          # We know that the reward is gaussian, so we can use normal-inverse-gamma         self.mu = [0.0] * k         self.lambd = [1.0] * k         self.alpha = [1.0] * k         self.beta = [1.0] * k      def update(self, action: int, reward: float) -&gt; None:         self.mu[action] = (             self.lambd[action] * self.mu[action] + reward         ) / (self.lambd[action] + 1)         self.lambd[action] += 1         self.alpha[action] += 0.5         self.beta[action] += (             self.lambd[action] * (reward - self.mu[action]) ** 2         ) / (self.lambd[action] + 1) / 2      def strategy(self) -&gt; int:         if self.history:             self.update(*self.history[-1])          samples = [             norm.rvs(                 loc=mu, scale=invgamma.rvs(a, scale=b) / math.sqrt(lambd)             )             for mu, lambd, a, b in zip(self.mu, self.lambd, self.alpha, self.beta)         ]         return max(range(self.k), key=lambda x: samples[x])  thompson_sampling_agents = [     ThompsonSamplingAgent(T=100, k=len(bandit)) for _ in range(1000) ] regrets = [agent.run(bandit) for agent in thompson_sampling_agents] show_result(regrets, 'Thompson Sampling Agent') Out[7]: <pre>'Thompson Sampling Agent - Mean: 231.88 Std: 49.94'</pre>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/","title":"\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u00b6","text":"<p>\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\uff08Multi-armed bandit\uff09\u200b\u95ee\u9898\u200b\u6765\u81ea\u200b\u4e8e\u200b\u73b0\u5b9e\u200b\u4e2d\u200b\u7684\u200b\u8001\u864e\u673a\u200b\u95ee\u9898\u200b\u3002\u200b\u6bcf\u53f0\u200b\u8001\u864e\u673a\u200b\u90fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u62c9\u6746\u200b\uff0c\u200b\u62c9\u52a8\u200b\u62c9\u6746\u200b\u540e\u200b\uff0c\u200b\u8001\u864e\u200b\u673a\u4f1a\u200b\u968f\u673a\u200b\u7ed9\u51fa\u200b\u4e00\u4e2a\u200b\u5956\u52b1\u200b\u3002\u200b\u73a9\u5bb6\u200b\u7684\u200b\u4efb\u52a1\u200b\u662f\u200b\u5728\u200b\u6709\u9650\u200b\u7684\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u5185\u200b\u5c3d\u53ef\u80fd\u200b\u83b7\u5f97\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u5956\u52b1\u200b\u3002</p> <p>\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u95ee\u9898\u200b\u7684\u200b\u6570\u5b66\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\uff1a\u200b\u5df2\u77e5\u200b\u4e00\u7ec4\u200b$K\\in \\mathbb N^{+}$\u200b\u4e2a\u200b\u5b9a\u4e49\u200b\u5728\u200b$\\mathbb R$\u200b\u4e0a\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b$\\mathcal B = \\{D_1, \\ldots, D_K\\}$\uff0c\u200b\u5176\u4e2d\u200b$D_k$\u200b\u8868\u793a\u200b\u7b2c\u200b$k$\u200b\u4e2a\u200b\u8001\u864e\u673a\u200b\u7ed9\u51fa\u200b\u5956\u52b1\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b\uff0c$D_k$\u200b\u7684\u200b\u5747\u503c\u200b\u4e3a\u200b$\\mu_k$\u3002\u200b\u5728\u200b\u7b2c\u200b$t$\u200b\u8f6e\u4e2d\u200b\uff0c\u200b\u667a\u80fd\u200b\u4f53\u4ece\u200b$\\mathcal B$\u200b\u4e2d\u200b\u968f\u673a\u200b\u9009\u51fa\u200b\u4e00\u4e2a\u200b\u5206\u5e03\u200b$D^{(t)}\\in\\mathcal B$\uff0c\u200b\u5e76\u200b\u4ece\u4e2d\u200b\u91c7\u6837\u200b\u4e00\u4e2a\u200b\u5956\u52b1\u200b$X^{(t)}\\sim D^{(t)}$\u3002\u200b\u667a\u80fd\u200b\u4f53\u200b\u7684\u200b\u76ee\u6807\u200b\u662f\u200b\u5728\u200b\u6709\u9650\u200b\u7684\u200b\u8f6e\u6b21\u200b\u5185\u200b\u6700\u5927\u5316\u200b\u7d2f\u79ef\u200b\u5956\u52b1\u200b$\\sum_{t=1}^T X^{(t)}$\u3002\u200b\u540e\u6094\u200b\u503c\u200b$\\rho$\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\u6700\u4f18\u200b\u7b56\u7565\u200b\u7684\u200b$T$\u200b\u8f6e\u200b\u7d2f\u8ba1\u200b\u5956\u52b1\u200b\u7684\u200b\u671f\u671b\u503c\u200b\u4e0e\u200b\u667a\u80fd\u200b\u4f53\u200b\u7684\u200b$T$\u200b\u8f6e\u200b\u7d2f\u8ba1\u200b\u5956\u52b1\u200b\u4e4b\u5dee\u200b\uff0c\u200b\u5373\u200b</p> <p>$$ \\rho = T\\max\\{\\mu_k\\} - \\mathbb E\\left[\\sum_{t=1}^T X^{(t)}\\right] $$</p> <p>\u200b\u7528\u200bPython\u200b\u53ef\u4ee5\u200b\u7b80\u5355\u200b\u5b9e\u73b0\u200b\u4e00\u4e2a\u200b\u4e0e\u200b\u667a\u80fd\u200b\u4f53\u200b\u4ea4\u4e92\u200b\u7684\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u73af\u5883\u200b\uff1a</p>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/","title":"\u7b97\u6cd5\u200b\u00b6","text":"<p>\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u7b97\u6cd5\u200b\u7684\u200b\u6838\u5fc3\u200b\u662f\u200b\u5728\u200b\u63a2\u7d22\u200b\uff08Exploration\uff09\u200b\u548c\u200b\u5229\u7528\u200b\uff08Exploitation\uff09\u200b\u4e4b\u95f4\u200b\u5bfb\u627e\u200b\u5e73\u8861\u200b\u3002\u200b\u63a2\u7d22\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u627e\u5230\u200b\uff08\u200b\u76f8\u8f83\u200b\u4e8e\u200b\u5f53\u524d\u200b\u5df2\u77e5\u200b\u7684\u200b\uff09\u200b\u66f4\u597d\u200b\u7684\u200b\u5956\u52b1\u200b\uff0c\u200b\u800c\u200b\u5229\u7528\u200b\u5219\u200b\u662f\u200b\u76f4\u63a5\u200b\u9009\u62e9\u200b\u5f53\u524d\u200b\u5df2\u77e5\u200b\u7684\u200b\u6700\u597d\u200b\u7684\u200b\u5956\u52b1\u200b\u3002</p>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/","title":"\u5b8c\u5168\u200b\u968f\u673a\u200b\u7b56\u7565\u200b\u00b6","text":"<p>\u200b\u5728\u200b\u5b8c\u5168\u200b\u968f\u673a\u200b\u7b56\u7565\u200b\u4e0b\u200b\uff0c\u200b\u667a\u80fd\u200b\u4f53\u200b\u968f\u673a\u200b\u9009\u62e9\u200b\u4e00\u4e2a\u200b\u81c2\u200b\uff0c\u200b\u83b7\u53d6\u200b\u5956\u52b1\u200b\uff0c\u200b\u540e\u6094\u200b\u503c\u200b\u7684\u200b\u671f\u671b\u200b\u4e3a\u200b$\\mathbb E\\rho = \\sum_{k=1}^K (\\mu^* - \\mu_k)$\u3002</p>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/#epsilon-","title":"$\\epsilon$-\u200b\u8d2a\u5fc3\u200b\u7b97\u6cd5\u200b\u00b6","text":"<p>$\\epsilon$-\u200b\u8d2a\u5fc3\u200b\u7b97\u6cd5\u200b\u662f\u200b\u6700\u200b\u7b80\u5355\u200b\u7684\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u7b97\u6cd5\u200b\u4e4b\u4e00\u200b\u3002\u200b\u5728\u200b\u6bcf\u200b\u4e00\u8f6e\u200b\u4e2d\u200b\uff0c\u200b\u4ee5\u200b$\\epsilon$\u200b\u7684\u200b\u6982\u7387\u200b\u968f\u673a\u200b\u9009\u62e9\u200b\u4e00\u4e2a\u200b\u8001\u864e\u673a\u200b\uff0c\u200b\u4ee5\u200b$1-\\epsilon$\u200b\u7684\u200b\u6982\u7387\u200b\u9009\u62e9\u200b\u5f53\u524d\u200b\u5df2\u77e5\u200b\u7684\u200b\u6700\u597d\u200b\u7684\u200b\u8001\u864e\u673a\u200b\u3002</p>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/","title":"\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u7b97\u6cd5\u200b\u00b6","text":"<p>\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u7b97\u6cd5\u200b\uff08Upper Confidence Bound, UCB\uff09\u200b\u662f\u200b\u4e00\u79cd\u200b\u57fa\u4e8e\u200b\u7f6e\u4fe1\u533a\u95f4\u200b\u7684\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u7b97\u6cd5\u200b\u3002\u200b\u5728\u200b\u6bcf\u200b\u4e00\u8f6e\u200b\u4e2d\u200b\uff0c\u200b\u667a\u80fd\u200b\u4f53\u200b\u9009\u62e9\u200b\u4e00\u4e2a\u200b\u8001\u864e\u673a\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u8be5\u200b\u8001\u864e\u673a\u200b\u7684\u200b\u7f6e\u4fe1\u533a\u95f4\u200b\u4e0a\u200b\u754c\u200b\u6700\u5927\u200b\u3002\u200b\u7f6e\u4fe1\u533a\u95f4\u200b\u4e0a\u754c\u200b\u7684\u200b\u8ba1\u7b97\u516c\u5f0f\u200b\u4e3a\u200b</p> <p>$$ \\bar X_k + c\\cdot \\sqrt{\\frac{\\log t}{N_k}} $$</p> <p>\u200b\u5176\u4e2d\u200b$\\bar X_k$\u200b\u4e3a\u200b\u7b2c\u200b$k$\u200b\u4e2a\u200b\u8001\u864e\u673a\u200b\u7684\u200b\u5e73\u5747\u200b\u5956\u52b1\u200b\uff0c$N_k$\u200b\u4e3a\u200b\u7b2c\u200b$k$\u200b\u4e2a\u200b\u8001\u864e\u673a\u200b\u88ab\u200b\u9009\u62e9\u200b\u7684\u200b\u6b21\u6570\u200b\uff0c$c$\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u5e38\u6570\u200b\uff0c$t$\u200b\u4e3a\u200b\u5f53\u524d\u200b\u8f6e\u6b21\u200b\u3002\u200b\u5176\u200b\u601d\u60f3\u200b\u5728\u4e8e\u200b\uff0c\u200b\u667a\u80fd\u200b\u4f53\u4f1a\u200b\u9996\u5148\u200b\u904d\u5386\u200b\u6240\u6709\u200b\u81c2\u200b\uff0c\u200b\u4e4b\u540e\u200b\u4f18\u5148\u9009\u62e9\u200b\u90a3\u4e9b\u200b\u88ab\u200b\u9009\u62e9\u200b\u6b21\u6570\u200b\u8f83\u5c11\u200b\u7684\u200b\u8001\u864e\u673a\u200b\uff0c\u200b\u4ee5\u4fbf\u200b\u66f4\u597d\u200b\u5730\u200b\u4f30\u8ba1\u200b\u5176\u200b\u5956\u52b1\u200b\u3002</p>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/#thompson","title":"Thompson\u200b\u91c7\u6837\u200b\u00b6","text":"<p>Thompson\u200b\u91c7\u6837\u200b\u662f\u200b\u4e00\u79cd\u200b\u57fa\u4e8e\u200b\u8d1d\u53f6\u65af\u200b\u63a8\u65ad\u200b\u7684\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u9700\u8981\u200b\u5df2\u77e5\u200b\u6bcf\u4e2a\u200b\u8001\u864e\u673a\u200b\u7684\u200b\u5956\u52b1\u200b\u670d\u4ece\u200b\u67d0\u79cd\u200b\u5206\u5e03\u200b\u3002\u200b\u5728\u200b\u6bcf\u200b\u4e00\u8f6e\u200b\u4e2d\u200b\uff0c\u200b\u667a\u80fd\u200b\u4f53\u4ece\u200b\u6bcf\u4e2a\u200b\u8001\u864e\u673a\u200b\u7684\u200b\u5956\u52b1\u200b\u5206\u5e03\u200b\u4e2d\u200b\u91c7\u6837\u200b\u4e00\u4e2a\u200b\u5956\u52b1\u200b\uff0c\u200b\u9009\u62e9\u200b\u5956\u52b1\u200b\u6700\u5927\u200b\u7684\u200b\u8001\u864e\u673a\u200b\u3002\u200b\u6839\u636e\u200b\u73af\u5883\u200b\u7684\u200b\u53cd\u9988\u200b\uff0c\u200b\u66f4\u65b0\u200b\u6bcf\u4e2a\u200b\u8001\u864e\u673a\u200b\u7684\u200b\u540e\u9a8c\u200b\u5956\u52b1\u200b\u5206\u5e03\u200b\u3002</p>"},{"location":"coding/reinforcement-learning/upper-confidence-bound/","title":"\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u7b97\u6cd5","text":"<p>\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\uff08Upper Confidence Bound, UCB\uff09\u200b\u7b97\u6cd5\u200b\u7684\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u5c06\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u7eb3\u5165\u200b\u8003\u8651\u200b\uff0c\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u8f83\u5c11\u200b\u7684\u200b\u52a8\u4f5c\u200b\u6709\u200b\u66f4\u200b\u5927\u200b\u7684\u200b\u4e0d\u786e\u5b9a\u6027\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u4f1a\u200b\u5c3d\u53ef\u80fd\u200b\u4e50\u89c2\u200b\u5730\u200b\u8003\u8651\u200b\u8fd9\u4e9b\u200b\u6ca1\u6709\u200b\u5c1d\u8bd5\u200b\u8fc7\u200b\u7684\u200b\u52a8\u4f5c\u200b\uff0c\u200b\u5e76\u200b\u4f18\u5148\u200b\u63a2\u7d22\u200b\u8fd9\u4e9b\u200b\u52a8\u4f5c\u200b\u3002\u200b\u5728\u200b\u6bcf\u4e2a\u200b\u52a8\u4f5c\u200b\u63a2\u7d22\u200b\u6bd4\u8f83\u200b\u6df1\u5165\u200b\u540e\u200b\uff0c\u200b\u518d\u200b\u8f6c\u5411\u200b\u5229\u7528\u200b\u5956\u52b1\u200b\u8f83\u200b\u9ad8\u200b\u7684\u200b\u52a8\u4f5c\u200b\u3002</p> <p>\u200b\u5f15\u7406\u200b</p> <p>\u200b\u8bbe\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X_1, \\ldots, X_n\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u5747\u503c\u200b\u4e3a\u200b\\(0\\)\uff0c\u200b\u5bf9\u200b\u5747\u503c\u200b\u7684\u200b\u4f30\u8ba1\u200b\u4e3a\u200b</p> \\[ \\hat \\mu = \\frac{1}{n}\\sum_{i=1}^n X_i \\] <p>\u200b\u5219\u200b\u5bf9\u200b\u4efb\u610f\u200b\\(\\delta &gt; 0\\)\uff0c\u200b\u6709\u200b</p> \\[ P(\\hat\\mu &gt; \\delta) \\leq \\exp(-n\\delta^2 / 2) \\] <p>\u200b\u8be5\u200b\u5f15\u7406\u200b\u79f0\u4e3a\u200bHoeffding\u200b\u4e0d\u7b49\u5f0f\u200b\u3002</p> <p>\u200b\u4ee4\u200b\\(\\varepsilon = \\exp(-2n\\delta^2)\\)\uff0c\u200b\u5373\u200b\\(\\delta = \\sqrt{\\frac{-2\\log\\varepsilon}{n}}\\)</p> \\[ P\\left(\\hat\\mu &gt; \\sqrt{\\frac{-2\\log\\varepsilon}{n}}\\right) \\leq \\varepsilon \\] <p>\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u52a8\u4f5c\u200b\\(k\\)\uff0c\u200b\u6211\u4eec\u200b\u5c1d\u8bd5\u200b\u4e86\u200b\\(n_k\\)\u200b\u6b21\u200b\uff0c\u200b\u6bcf\u6b21\u200b\u5c1d\u8bd5\u200b\u7684\u200b\u5956\u52b1\u200b\u4e3a\u200b\\(X_1, \\ldots, X_{n_k}\\)\uff0c\u200b\u89c2\u6d4b\u200b\u5230\u200b\u7684\u200b\u5e73\u5747\u200b\u5956\u52b1\u200b\u4e3a\u200b\\(\\hat\\mu_k\\)\uff0c\u200b\u5b9a\u4e49\u200b\\(\\tilde X_i = X_i - \\hat\\mu_k\\)\uff0c\u200b\u5219\u200b\\(\\tilde X_1, \\ldots, \\tilde X_{n_k}\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u5747\u503c\u200b\u4e3a\u200b\\(0\\)\u3002</p> <p>\u200b\u4ee4\u200b\u5bf9\u200b\\(\\tilde X_1, \\ldots, \\tilde X_{n_k}\\)\u200b\u4f30\u8ba1\u200b\u7684\u200b\u5747\u503c\u200b\u4e3a\u200b\\(\\hat\\mu_0\\)\uff0c\u200b\u6211\u4eec\u200b\u6709\u200b\\(1 - \\varepsilon\\)\u200b\u7684\u200b\u6982\u7387\u200b\u4fdd\u8bc1\u200b\\(\\hat\\mu_0\\)\u200b\u4e0d\u200b\u8d85\u8fc7\u200b\\(\\sqrt{\\frac{-2\\log\\varepsilon}{n_k}}\\)\u3002</p> \\[ P\\left(\\hat\\mu_0 &gt; \\sqrt{\\frac{-2\\log\\varepsilon}{n_k}}\\right) \\leq \\varepsilon \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u6211\u4eec\u200b\u53d6\u200b\\(\\hat\\mu_k + \\sqrt{\\frac{-2\\log\\varepsilon}{n_k}}\\)\u200b\u4f5c\u4e3a\u200b\u52a8\u4f5c\u200b\\(k\\)\u200b\u7684\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u3002\u200b\u901a\u8fc7\u200b\u8c03\u6574\u200b\\(\\varepsilon\\)\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u63a7\u5236\u200b\u7f6e\u4fe1\u200b\u4e0a\u754c\u200b\u7684\u200b\u5bbd\u5ea6\u200b\u4ee5\u200b\u5728\u200b\u63a2\u7d22\u200b\u548c\u200b\u5229\u7528\u200b\u4e4b\u524d\u200b\u53d6\u5f97\u200b\u5e73\u8861\u200b\u3002</p> <p>\u200b\u901a\u5e38\u200b\u91c7\u53d6\u200b\u7684\u200b\u7b56\u7565\u200b\u662f\u200b\u53d6\u200b\\(\\varepsilon = 1/t\\)\uff0c\\(t\\)\u200b\u4e3a\u200b\u603b\u200b\u8f6e\u6b21\u200b\uff0c\u200b\u4ee5\u200b\u4fdd\u8bc1\u200b\u968f\u7740\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u7684\u200b\u589e\u52a0\u200b\uff0c\u200b\u7f6e\u4fe1\u200b\u4e0a\u754c\u200b\u7684\u200b\u5bbd\u5ea6\u200b\u9010\u6e10\u200b\u51cf\u5c0f\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u7b97\u6cd5\u200b\u9010\u6b65\u200b\u8f6c\u5411\u200b\u5229\u7528\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b</p> \\[ \\hat\\mu_k + \\sqrt{\\frac{2\\log t}{n_k}} \\] <p>\u200b\u6b64\u5916\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5176\u4ed6\u200b\u7684\u200b\u4e0a\u754c\u200b\u5bbd\u5ea6\u200b\uff0c\u200b\u5982\u200b</p> \\[ \\varepsilon = \\frac{1}{1 + t\\log^2 t} \\] <p>\u200b\u5728\u5b9e\u8df5\u4e2d\u200b\uff0c\u200b\u901a\u5e38\u200b\u8fd8\u4f1a\u200b\u5f15\u5165\u200b\u4e00\u4e2a\u200b\u53c2\u6570\u200b\\(c\\)\u200b\u63a7\u5236\u200b\u5bbd\u5ea6\u200b\uff0c\u200b\u5373\u200b</p> \\[ \\hat \\mu_k + c\\cdot\\sqrt{\\frac{\\log t}{n_k}} \\]"},{"location":"coding/rust/","title":"Rust","text":"<p>\u200b\u8bb0\u5f55\u200bRust\u200b\u7684\u200b\u5b66\u4e60\u200b\u3002</p> <ul> <li>Introduction</li> <li>Variables</li> <li>Control Flows</li> <li>Complex Objects</li> <li>Lifetimes</li> <li>Error Handling</li> <li>Supplements</li> </ul>"},{"location":"coding/rust/0/","title":"Introduction","text":"<p>Rust\u200b\u7b80\u4ecb\u200b</p> <p>Rust\u200b\u6709\u200b\u5982\u4e0b\u200b\u4e8c\u8fdb\u5236\u200b\u6587\u4ef6\u200b</p> <ul> <li><code>rustc</code>\uff1aRust\u200b\u7f16\u8bd1\u5668\u200b</li> <li><code>cargo</code>\uff1aRust\u200b\u5305\u200b\u7ba1\u7406\u5668\u200b\u548c\u200b\u6784\u5efa\u200b\u5de5\u5177\u200b</li> <li><code>rustup</code>\uff1aRust\u200b\u7248\u672c\u200b\u7ba1\u7406\u5668\u200b</li> <li><code>rustdoc</code>\uff1aRust\u200b\u6587\u6863\u200b</li> <li><code>rls</code>\uff1aRust language server</li> <li><code>rustfmt</code>\uff1aRust\u200b\u4ee3\u7801\u200b\u683c\u5f0f\u5316\u200b\u5de5\u5177\u200b</li> </ul>"},{"location":"coding/rust/0/#hello-world","title":"Hello, world!","text":"<p>  Download source code</p> <pre><code>fn main() {\n    println!(\"Hello, world!\");\n}\n</code></pre> <p><code>println!</code>\u200b\u4e2d\u200b\u7684\u200b<code>!</code>\u200b\u8868\u793a\u200b\u8fd9\u662f\u200b\u4e00\u4e2a\u200b\u5b8f\u200b\uff0c\u200b\u800c\u200b\u4e0d\u662f\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u3002\u200b\u5b8f\u662f\u200b\u4e00\u79cd\u200b\u5728\u200b\u7f16\u8bd1\u200b\u65f6\u200b\u6267\u884c\u200b\u7684\u200b\u4ee3\u7801\u200b\u3002<code>println</code>\u200b\u6709\u200bPython\u200b\u4e2d\u200b<code>str.format</code>\u200b\u7c7b\u4f3c\u200b\u7684\u200b\u529f\u80fd\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6267\u884c\u200b\u5b57\u7b26\u4e32\u200b\u683c\u5f0f\u5316\u200b\u3002<code>print!</code>\u200b\u4e0e\u200b<code>println!</code>\u200b\u7c7b\u4f3c\u200b\uff0c\u200b\u4f46\u662f\u200b\u4e0d\u4f1a\u200b\u5728\u200b\u7ed3\u5c3e\u200b\u6dfb\u52a0\u200b\u6362\u884c\u7b26\u200b\u3002</p> <pre><code>println!(\"{} days\", 31);\nprintln!(\"{0}, this is {1}. {1}, this is {0}\", \"Alice\", \"Bob\");\nprintln!(\"{greeting}, {name}!\", greeting=\"Hello\", name=\"world\");\n</code></pre>"},{"location":"coding/rust/0/#cargo","title":"Cargo","text":"<p>Cargo\u200b\u662f\u200bRust\u200b\u7684\u200b\u5305\u200b\u7ba1\u7406\u5668\u200b\u548c\u200b\u6784\u5efa\u200b\u5de5\u5177\u200b\u3002Cargo\u200b\u7684\u200b\u5de5\u4f5c\u200b\u65b9\u5f0f\u200b\u4e0e\u200bPython\u200b\u7684\u200b<code>pip</code>\u200b\u548c\u200b<code>setuptools</code>\u200b\u7c7b\u4f3c\u200b\u3002</p> <pre><code>cargo new # \u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200bRust\u200b\u9879\u76ee\u200b\ncargo init # \uff08\u200b\u5728\u200b\u5df2\u7ecf\u200b\u5b58\u5728\u200b\u7684\u200b\u76ee\u5f55\u200b\u4e2d\u200b\uff09\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200bRust\u200b\u9879\u76ee\u200b\ncargo build # \u200b\u7f16\u8bd1\u200b\u9879\u76ee\u200b\ncargo run # \u200b\u7f16\u8bd1\u200b\u5e76\u200b\u8fd0\u884c\u200b\u9879\u76ee\u200b\ncargo update # \u200b\u66f4\u65b0\u200bCargo.lock\u200b\u6587\u4ef6\u200b\u4e2d\u200b\u7684\u200b\u4f9d\u8d56\u200b\ncargo doc # \u200b\u751f\u6210\u200b\u9879\u76ee\u200b\u6587\u6863\u200b\ncargo test # \u200b\u8fd0\u884c\u200b\u9879\u76ee\u200b\u6d4b\u8bd5\u200b\n\ncargo install ...\ncargo uninstall ...\n</code></pre> <p>Rust\u200b\u7684\u200b\u4ed3\u5e93\u200b\u662f\u200b<code>crates.io</code>\uff0c\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bPython\u200b\u7684\u200b<code>PyPI</code>\u3002</p>"},{"location":"coding/rust/0/#_1","title":"\u5305\u4e0e\u200b\u6a21\u5757","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u5305\u200b\u79f0\u4e3a\u200bcrate\u3002crate\u200b\u7684\u200b\u6839\u76ee\u5f55\u200b\u4e0b\u200b\u6709\u200b\u4e00\u4e2a\u200b<code>Cargo.toml</code>\u200b\u6587\u4ef6\u200b\uff0c\u200b\u5176\u4e2d\u200b\u5305\u542b\u200bcrate\u200b\u7684\u200b\u5143\u200b\u6570\u636e\u200b\u548c\u200b\u4f9d\u8d56\u200b\u3002Rust\u200b\u9879\u76ee\u200b\u7684\u200b\u6587\u4ef6\u200b\u7ed3\u6784\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>project\n\u251c\u2500\u2500 Cargo.toml\n\u251c\u2500\u2500 Cargo.lock\n\u251c\u2500\u2500 tests\n\u2502   \u2514\u2500\u2500 ...\n\u251c\u2500\u2500 benches\n\u2502   \u2514\u2500\u2500 ...\n\u251c\u2500\u2500 examples\n\u2502   \u2514\u2500\u2500 ...\n\u2514\u2500\u2500 src\n    \u251c\u2500\u2500 bin     // \u200b\u5176\u4ed6\u200b\u7684\u200b\u4e8c\u8fdb\u5236\u200b\u7a0b\u5e8f\u200b\n    \u251c\u2500\u2500 main.rs // \u200b\u9ed8\u8ba4\u200b\u7684\u200b\u4e8c\u8fdb\u5236\u200b\u7a0b\u5e8f\u200b\n    \u2514\u2500\u2500 lib.rs  // \u200b\u5e93\u5305\u200b\uff08\u200b\u53ea\u80fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\uff09\n</code></pre> <p>Rust\u200b\u4e2d\u200b\u7684\u200b\u6a21\u5757\u200b\u7528\u200b<code>mod</code>\u200b\u5173\u952e\u5b57\u200b\u58f0\u660e\u200b\uff0c\u200b\u6a21\u5757\u200b\u53ef\u4ee5\u200b\u5d4c\u5957\u200b\u3002\u200b\u7528\u200b<code>use</code>\u200b\u5f15\u7528\u200b\u6a21\u5757\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>crate</code>\u200b\u5f00\u59cb\u200b\u7684\u200b\u7edd\u5bf9\u200b\u5f15\u7528\u200b\u8def\u5f84\u200b\u548c\u200b<code>self</code>\uff08\u200b\u672c\u200b\u6a21\u5757\u200b\uff09\u200b\u6216\u200b<code>super</code>\uff08\u200b\u7236\u200b\u6a21\u5757\u200b\uff09\u200b\u5f00\u59cb\u200b\u7684\u200b\u76f8\u5bf9\u200b\u5f15\u7528\u200b\u8def\u5f84\u200b\uff0c\u200b\u6216\u8005\u200b\u7528\u200b<code>*</code>\u200b\u5f15\u5165\u200b\u6a21\u5757\u200b\u7684\u200b\u6240\u6709\u200b\u5185\u5bb9\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u6a21\u5757\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u9879\u200b\u90fd\u200b\u662f\u200b\u79c1\u6709\u200b\u7684\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u5916\u754c\u200b\u8bbf\u95ee\u200b\uff0c\u200b\u9664\u975e\u200b\u7528\u200b<code>pub</code>\u200b\u5173\u952e\u5b57\u200b\u58f0\u660e\u200b\u4e3a\u200b\u516c\u6709\u200b\u3002\u200b\u516c\u6709\u200b\u6a21\u5757\u200b\u5185\u200b\u6240\u6709\u200b\u7684\u200b\u9879\u200b\u5fc5\u987b\u200b\u90fd\u200b\u7528\u200b<code>pub</code>\u200b\u58f0\u660e\u200b\u4e3a\u200b\u516c\u6709\u200b\u3002<code>pub</code>\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>(in &lt;module path&gt;)</code>\u200b\u8bed\u6cd5\u200b\u58f0\u660e\u200b\u5bf9\u8c61\u200b\u4ec5\u200b\u5728\u200b<code>&lt;module path&gt;</code>\u200b\u4e2d\u200b\u53ef\u89c1\u200b\u3002</p> <p>\u200b\u6bcf\u4e2a\u200b<code>.rs</code>\u200b\u6587\u4ef6\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u4e00\u4e2a\u200b\u4ee5\u200b\u6587\u4ef6\u540d\u200b\u547d\u540d\u200b\u7684\u200b\u6a21\u5757\u200b\u3002\u200b\u5982\u679c\u200b\u8981\u200b\u628a\u200b\u6587\u4ef6\u5939\u200b\u4f5c\u4e3a\u200b\u6a21\u5757\u200b\uff0c\u200b\u5219\u200b\u5fc5\u987b\u200b\u5728\u200b\u6587\u4ef6\u5939\u200b\u4e2d\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b<code>mod.rs</code>\u200b\u6587\u4ef6\u200b\u6216\u200b\u4e0e\u200b\u6587\u4ef6\u5939\u200b\u540c\u540d\u200b\u7684\u200b<code>.rs</code>\u200b\u6587\u4ef6\u200b\uff0c\u200b\u5176\u4e2d\u200b\u58f0\u660e\u200b\u5bf9\u5916\u5f00\u653e\u200b\u7684\u200b\u9879\u200b\u3002</p> <p>\u200b\u5f53\u200b\u591a\u4e2a\u200b\u6a21\u5757\u200b\u4e2d\u200b\u5b58\u5728\u200b\u76f8\u540c\u200b\u540d\u79f0\u200b\u7684\u200b\u9879\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b<code>\u200b\u6a21\u5757\u200b\u540d\u200b::\u200b\u9879\u540d\u200b</code>\u200b\u6765\u200b\u6307\u5b9a\u200b\u5177\u4f53\u200b\u7684\u200b\u9879\u200b\uff0c\u200b\u6216\u8005\u200b\u7528\u200b<code>as</code>\u200b\u58f0\u660e\u200b\u522b\u540d\u200b\u3002</p>"},{"location":"coding/rust/0/#_2","title":"\u6ce8\u91ca\u200b\u4e0e\u200b\u6587\u6863","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u6ce8\u91ca\u200b\u4e0e\u200bC/C++\u200b\u7c7b\u4f3c\u200b\uff0c\u200b\u7528\u200b<code>//</code>\u200b\u8868\u793a\u200b\u5355\u884c\u200b\u6ce8\u91ca\u200b\uff0c\u200b\u7528\u200b<code>/* */</code>\u200b\u8868\u793a\u200b\u591a\u884c\u200b\u6ce8\u91ca\u200b\u3002\u200b\u5176\u4e2d\u200b\uff0c\u200b\u5e38\u7528\u200b\u7684\u200b\u6ce8\u91ca\u200b\u65b9\u6cd5\u200b\u4e3a\u200b\uff1a</p> <ol> <li>\u200b\u5305\u200b\u6ce8\u91ca\u200b<code>//! ...</code>\u200b\u4e0e\u200b<code>/*! ... */</code>\uff1a()\u200b\u7528\u4e8e\u200b\u6574\u4e2a\u200b\u5305\u200b\u7684\u200b\u6ce8\u91ca\u200b\uff08Inner comments\uff09</li> <li>\u200b\u6587\u6863\u200b\u6ce8\u91ca\u200b<code>/// ...</code>\u200b\u4e0e\u200b<code>/** ... */</code>\uff1a\u200b\u7528\u4e8e\u200b\u6a21\u5757\u200b\u3001\u200b\u51fd\u6570\u200b\u3001\u200b\u7ed3\u6784\u200b\u4f53\u200b\u3001\u200b\u679a\u4e3e\u200b\u3001\u200b\u65b9\u6cd5\u200b\u548c\u200btrait\u200b\u7684\u200b\u6ce8\u91ca\u200b\uff08Outer comments\uff09</li> <li>\u200b\u901a\u5e38\u200b\u4e0d\u7528\u200b<code>/* */</code>\u200b\u6ce8\u91ca\u200b</li> </ol> <p>\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200bdoc attribute\u200b\u6765\u200b\u6ce8\u91ca\u200b\uff0c\u200b\u5176\u200b\u8bed\u6cd5\u200b\u4e3a\u200b<code>#![doc=...]</code>\u200b\u548c\u200b<code>#[doc=...]</code>\uff0c\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200bOuter comments\u200b\u548c\u200bInner comments\u3002</p> <p>Rust\u200b\u652f\u6301\u200b\u5728\u200b\u6587\u6863\u200b\u4e2d\u200b\u8fdb\u884c\u200b\u6d4b\u8bd5\u200b\uff0c\u200b\u88ab\u200b\u6d4b\u8bd5\u200b\u7684\u200b\u51fd\u6570\u200b\u5fc5\u987b\u200b\u662f\u200b\u516c\u6709\u200b\u7684\u200b\u3002\u200b\u4ee5\u200b<code>#</code>\u200b\u5f00\u59cb\u200b\u7684\u200b\u884c\u200b\u8868\u793a\u200b\u8be5\u884c\u200b\u4e0d\u4f1a\u200b\u5728\u200b\u6587\u6863\u200b\u4e2d\u200b\u663e\u793a\u200b\uff0c\u200b\u53ea\u4f1a\u200b\u7528\u4e8e\u200b\u6d4b\u8bd5\u200b\u3002</p> <pre><code>// Assume the name of crate is `playground`\n\n/// ```rust\n/// assert_eq!(playground::add(1, 2), 3);\n/// ```\npub fn add(a: i32, b: i32) -&gt; i32 {\n    a + b\n}\n/// ```rust, should_panic\n/// // This function calls [`panic!`] macro.\n/// playground::panic_fn();\n/// ```\npub fn panic_fn() {\n    panic!();\n}\n</code></pre>"},{"location":"coding/rust/1/","title":"Variables","text":""},{"location":"coding/rust/1/#_1","title":"\u58f0\u660e","text":"<p>Rust\u200b\u4e2d\u200b\u7528\u4e8e\u200b\u58f0\u660e\u200b\u53d8\u91cf\u200b\u7684\u200b\u5173\u952e\u5b57\u200b\u662f\u200b<code>let</code>\uff0c\u200b\u5b83\u200b\u7684\u200b\u8bed\u6cd5\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>let [mut] &lt;variable&gt;[: &lt;type&gt;] [= &lt;value&gt;];\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff0c\u200b\u7c7b\u578b\u200b\u548c\u200b\u521d\u59cb\u5316\u200b\u7684\u200b\u503c\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u3002\u200b\u5982\u679c\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\u4ece\u200b\u58f0\u660e\u200b\u5f00\u59cb\u200b\u6ca1\u6709\u200b\u88ab\u200b\u8d4b\u503c\u200b\uff0c\u200b\u5c31\u200b\u4e0d\u80fd\u200b\u4f7f\u7528\u200b\u8fd9\u4e2a\u200b\u53d8\u91cf\u200b\u3002\u200b\u5982\u679c\u200b\u6ca1\u6709\u200b\u6307\u5b9a\u200b\u7c7b\u578b\u200b\u4f46\u200b\u6307\u5b9a\u200b\u4e86\u200b\u521d\u59cb\u5316\u200b\u503c\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u5219\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u63a8\u65ad\u200b\u53d8\u91cf\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002Rust\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>{}</code>\u200b\u6765\u200b\u7528\u200b\u4e00\u7cfb\u5217\u200b\u8bed\u53e5\u200b\u6765\u200b\u521d\u59cb\u5316\u200b\u53d8\u91cf\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>()</code>\u200b\u6765\u200b\u4e00\u6b21\u6027\u200b\u58f0\u660e\u200b\u591a\u4e2a\u200b\u53d8\u91cf\u200b\u3002\u200b\u6ca1\u6709\u200b\u7528\u200b<code>mut</code>\u200b\u58f0\u660e\u200b\u7684\u200b\u53d8\u91cf\u200b\u662f\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u7684\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u4fee\u6539\u200b\u3002</p> <p>Rust\u200b\u4e2d\u200b\u7528\u4e8e\u200b\u58f0\u660e\u200b\u5e38\u91cf\u200b\u7684\u200b\u5173\u952e\u5b57\u200b\u662f\u200b<code>const</code>\uff0c\u200b\u7528\u4e8e\u200b\u58f0\u660e\u200b\u9759\u6001\u200b\u53d8\u91cf\u200b\u7684\u200b\u5173\u952e\u5b57\u200b\u662f\u200b<code>static</code>\u3002\u200b\u8bed\u6cd5\u200b\u4e0e\u200b<code>let</code>\u200b\u76f8\u4f3c\u200b\uff0c\u200b\u4f46\u200b\u9700\u8981\u200b\u540c\u65f6\u200b\u6307\u5b9a\u200b\u7c7b\u578b\u200b\u548c\u200b\u521d\u59cb\u5316\u200b\u503c\u200b\u3002\u200b\u901a\u5e38\u200b<code>const</code>\u200b\u548c\u200b<code>static</code>\u200b\u8bed\u53e5\u200b\u90fd\u200b\u662f\u200b\u5728\u200b\u5168\u5c40\u200b\u4f5c\u7528\u57df\u200b\uff08\u200b\u5373\u200b\u51fd\u6570\u200b\u5916\u200b\uff09\u200b\u4e2d\u200b\u58f0\u660e\u200b\u7684\u200b\u3002</p> <p><code>let</code>\u200b\u53ef\u4ee5\u200b\u7528\u6765\u200b\u91cd\u65b0\u200b\u58f0\u660e\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\uff0c\u200b\u9690\u85cf\u200b\u4e4b\u524d\u200b\u7684\u200b\u53d8\u91cf\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>mut</code>\u200b\u6765\u200b\u6539\u53d8\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\u7684\u200b\u53ef\u53d8\u6027\u200b\u3002</p> <p>  Download source code</p> <pre><code>fn main() {\n    let x: f64 = -20.48; // float\n    let x: i64 = x.floor() as i64; // int\n    println!(\"{}\", x); // -21\n\n    let s: &amp;str = \"hello\"; // &amp;str\n    let s: String = s.to_uppercase(); // String\n    println!(\"{}\", s) // HELLO\n}\n</code></pre>"},{"location":"coding/rust/1/#_2","title":"\u53d8\u91cf\u200b\u7c7b\u578b","text":"<p>Rust\u200b\u7684\u200b\u53d8\u91cf\u200b\u7c7b\u578b\u200b\u6709\u200b</p> <ul> <li>\u200b\u4e00\u822c\u200b\u7c7b\u578b\u200b<ul> <li>\u200b\u5e03\u5c14\u200b\uff1a<code>bool</code>\uff0c\u200b\u53d6\u503c\u200b\u4e3a\u200b<code>true</code>\u200b\u6216\u200b<code>false</code>\uff0c\u200b\u5360\u7528\u200b1\u200b\u4e2a\u200b\u5b57\u8282\u200b\u3002</li> <li>\u200b\u5b57\u7b26\u200b\uff1a<code>char</code>\uff0c\u200b\u4f7f\u7528\u200b\u5355\u5f15\u53f7\u200b\uff0c\u200b\u5360\u7528\u200b4\u200b\u4e2a\u200b\u5b57\u8282\u200b\u3002</li> <li>\u200b\u6574\u6570\u200b\uff1a<code>i8</code>\uff5e<code>i128</code>\u200b\u548c\u200b<code>isize</code>\uff0c\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\([-2^{n-1}, 2^{n-1} - 1]\\)\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>min_value()</code>\u200b\u548c\u200b<code>max_value()</code>\u200b\u51fd\u6570\u200b\u6765\u200b\u83b7\u53d6\u200b\u7c7b\u578b\u200b\u7684\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u3002<code>isize</code>\u200b\u7684\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u548c\u200b\u7cfb\u7edf\u200b\u6307\u9488\u200b\u957f\u5ea6\u200b\u76f8\u540c\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u6574\u6570\u200b\u7c7b\u578b\u200b\u662f\u200b<code>i32</code>\u3002</li> <li>\u200b\u65e0\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\uff1a<code>u8</code>\uff5e<code>u128</code>\u200b\u548c\u200b<code>usize</code>\uff0c\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\([0, 2^n - 1]\\)\u3002</li> <li>\u200b\u6d6e\u70b9\u6570\u200b\uff1a<code>f32</code>\u200b\u548c\u200b<code>f64</code>\uff0c\u200b\u5206\u522b\u200b\u4e3a\u200b\u5355\u7cbe\u5ea6\u200b\u548c\u200b\u53cc\u200b\u7cbe\u5ea6\u200b\u6d6e\u70b9\u6570\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u6d6e\u70b9\u6570\u200b\u7c7b\u578b\u200b\u662f\u200b<code>f64</code>\u3002</li> <li>\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\uff1a<code>str</code>\uff0c\u200b\u4f7f\u7528\u200b\u53cc\u5f15\u53f7\u200b\uff0c\u200b\u662f\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u7c7b\u578b\u200b\u3002\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\uff08\u200b\u53d8\u91cf\u200b\uff09\u200b\u7684\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>&amp;str</code>\uff0c\u200b\u662f\u200b\u5f15\u7528\u200b\u7c7b\u578b\u200b\u3002</li> <li>\u200b\u5b57\u7b26\u4e32\u200b\uff1a<code>String</code>\uff0c\u200b\u662f\u200b\u53ef\u53d8\u200b\u7c7b\u578b\u200b\uff0c\u200b\u4f46\u200b\u4e0d\u200b\u652f\u6301\u200b\u7d22\u5f15\u200b\u3002\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5f15\u7528\u200b\u6216\u200b\u5207\u7247\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>&amp;str</code>\u3002</li> <li>\u200b\u5355\u5143\u200b\u7c7b\u578b\u200b\uff1a<code>()</code>\uff0c\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u503c\u200b<code>()</code>\uff0c\u200b\u6ca1\u6709\u200b<code>return</code>\u200b\u8bed\u53e5\u200b\u7684\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>()</code>\uff0c\u200b\u4e0d\u200b\u5360\u7528\u200b\u5185\u5b58\u200b\u3002</li> </ul> </li> <li>\u200b\u5bb9\u5668\u200b\u53ca\u200b\u76f8\u5173\u200b\u7c7b\u578b\u200b<ul> <li>\u200b\u4f7f\u7528\u200b<code>println!</code>\u200b\u6253\u5370\u200b\u5bb9\u5668\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b<code>{:?}</code>\u200b\u6216\u200b<code>{:#?}</code>\uff0c\u200b\u524d\u8005\u200b\u4e3a\u200b\u4e00\u884c\u200b\u8f93\u51fa\u200b\uff0c\u200b\u540e\u8005\u200b\u5728\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u540e\u9762\u200b\u6dfb\u52a0\u200b\u6362\u884c\u7b26\u200b\u3002</li> <li>\u200b\u5b9a\u200b\u957f\u200b\u6570\u7ec4\u200b\uff1a<code>[T; N]</code>\uff0c\u200b\u5176\u4e2d\u200b<code>T</code>\u200b\u4e3a\u200b\u6570\u7ec4\u200b\u5143\u7d20\u200b\u7c7b\u578b\u200b\u6216\u200b\u91cd\u590d\u200b\u7684\u200b\u6570\u7ec4\u200b\uff0c<code>N</code>\u200b\u4e3a\u200b\u6570\u7ec4\u200b\u957f\u5ea6\u200b\uff0c\u200b\u5fc5\u987b\u200b\u662f\u200b\u5e38\u91cf\u200b\u3002\u200b\u53ea\u6709\u200b\u7528\u200b<code>mut</code>\u200b\u58f0\u660e\u200b\u7684\u200b\u6570\u7ec4\u200b\u624d\u80fd\u200b\u4fee\u6539\u200b\u5176\u4e2d\u200b\u5143\u7d20\u200b\u3002<ul> <li>\u200b\u6ce8\u610f\u200b\u4e0d\u200b\u652f\u6301\u200b<code>Copy</code> trait\u200b\u7684\u200b\u7c7b\u578b\u200b\u4e0d\u80fd\u200b\u7528\u200b<code>[T; N]</code>\u200b\u8bed\u6cd5\u200b\uff0c\u200b\u5fc5\u987b\u200b\u9010\u4e2a\u200b\u8d4b\u503c\u200b\u3002</li> <li>\u200b\u4e0d\u540c\u200b\u957f\u5ea6\u200b\u7684\u200b\u6570\u7ec4\u200b\u662f\u200b\u4e0d\u540c\u200b\u7684\u200b\u7c7b\u578b\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u76f8\u4e92\u200b\u8d4b\u503c\u200b\u3002</li> </ul> </li> <li>\u200b\u5b9a\u200b\u957f\u200b\u5143\u7ec4\u200b\uff1a<code>(T1, T2, ..., Tn)</code>\uff0c\u200b\u5176\u4e2d\u200b<code>T1</code>\uff5e<code>Tn</code>\u200b\u4e3a\u200b\u5143\u7ec4\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u53ea\u6709\u200b\u7528\u200b<code>mut</code>\u200b\u58f0\u660e\u200b\u7684\u200b\u5143\u7ec4\u200b\u624d\u80fd\u200b\u4fee\u6539\u200b\u5176\u4e2d\u200b\u5143\u7d20\u200b\uff08\u200b\u7c7b\u578b\u200b\u5fc5\u987b\u200b\u5339\u914d\u200b\uff09\u3002\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bPython\uff0c\u200b\u5728\u200b\u53d8\u91cf\u200b\u58f0\u660e\u200b\u9636\u6bb5\u200b\u6216\u200b\u8d4b\u503c\u200b\u9636\u6bb5\u200b\uff0c\u200b\u4ec5\u200b\u5305\u542b\u200b\uff08\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u8d4b\u503c\u200b\u7684\u200b\uff09\u200b\u53d8\u91cf\u200b\u5143\u7ec4\u200b\u53ef\u4ee5\u200b\u4f5c\u4e3a\u200b\u8d4b\u503c\u200b\u8fd0\u7b97\u200b\u7684\u200b\u5de6\u7aef\u200b\u9879\u200b\uff0c\u200b\u6b64\u65f6\u200b\u4f1a\u200b\u5206\u522b\u200b\u4e3a\u200b\u6bcf\u4e2a\u200b\u53d8\u91cf\u200b\u5206\u914d\u200b\u5bf9\u5e94\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u7701\u7565\u200b\u7684\u200b\u53d8\u91cf\u200b\u7528\u200b<code>_</code>\u200b\u8868\u793a\u200b\u3002\u200b\u5143\u7ec4\u200b\u652f\u6301\u200b\u5d4c\u5957\u200b\u3002\u200b\u5355\u200b\u5143\u7d20\u200b\u5143\u7ec4\u200b\u7684\u200b\u8bed\u6cd5\u200b\u4e3a\u200b<code>(&lt;value&gt;,)</code>\u3002</li> <li>\u200b\u6570\u7ec4\u200b\u5207\u7247\u200b\uff1a<code>&amp;&lt;parent&gt;[&lt;start&gt;..&lt;end&gt;]</code>\uff0c\u200b\u6ce8\u610f\u200b\u5207\u7247\u200b\u662f\u200b\u5bf9\u200b<code>&lt;parent&gt;</code>\u200b\u7684\u200b\u5f15\u7528\u200b\uff0c\u200b\u662f\u200b\u5f15\u7528\u200b\u7c7b\u578b\u200b\u3002\u200b\u5143\u7ec4\u200b\u4e0d\u200b\u652f\u6301\u200b\u5207\u7247\u200b\uff0c\u200b\u53ea\u80fd\u200b\u7528\u200b<code>.0</code>\u3001<code>.1</code>\u200b\u7b49\u200b\u6765\u200b\u8bbf\u95ee\u200b\u5143\u7d20\u200b\uff0c\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>[T]</code>\u3002</li> </ul> </li> <li>\u200b\u8fed\u4ee3\u200b\u5668\u200b\uff0c\u200b\u901a\u5e38\u200b\u7531\u200b\u5bb9\u5668\u200b\u5bf9\u8c61\u200b\u7684\u200b<code>iter()</code>\u200b\u65b9\u6cd5\u200b\u751f\u6210\u200b\u3002</li> <li>\u200b\u51fd\u6570\u200b\uff1a\u200b\u9700\u8981\u200b\u7528\u200b<code>fn(arg1: type1, ...) -&gt; return_type</code>\u200b\u8868\u793a\u200b\uff0c\u200b\u51fd\u6570\u200b\u4ee5\u200b\u5f15\u7528\u200b\u7684\u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b\u3002</li> </ul>"},{"location":"coding/rust/1/#_3","title":"\u6574\u6570\u200b\u7c7b\u578b","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u6574\u6570\u200b\u5b57\u9762\u200b\u91cf\u200b\u6709\u200b\u5982\u4e0b\u200b\u5199\u6cd5\u200b\uff1a</p> <ul> <li>\u200b\u5341\u8fdb\u5236\u200b\uff1a<code>123_456</code></li> <li>\u200b\u5341\u516d\u8fdb\u5236\u200b\uff1a<code>0x123</code>\uff0c<code>0x</code>\u200b\u5f00\u5934\u200b\uff0c\u200b\u6570\u5b57\u200b\u4e3a\u200b<code>0-9</code>\u200b\u548c\u200b<code>a-f</code></li> <li>\u200b\u516b\u8fdb\u5236\u200b\uff1a<code>0o123</code>\uff0c<code>0o</code>\u200b\u5f00\u5934\u200b\uff0c\u200b\u6570\u5b57\u200b\u4e3a\u200b<code>0-7</code></li> <li>\u200b\u4e8c\u8fdb\u5236\u200b\uff1a<code>0b101</code>\uff0c<code>0b</code>\u200b\u5f00\u5934\u200b\uff0c\u200b\u6570\u5b57\u200b\u4e3a\u200b<code>0</code>\u200b\u548c\u200b<code>1</code></li> <li>\u200b\u5b57\u8282\u200b\uff1a<code>b'A'</code></li> <li>\u200b\u4e0b\u5212\u7ebf\u200b<code>_</code>\u200b\u7c7b\u4f3c\u200b\u4e8e\u200b\u5343\u4f4d\u200b\u5206\u9694\u7b26\u200b\uff0c\u200b\u5b58\u5728\u200b\u4e0e\u5426\u200b\u4e0d\u200b\u5f71\u54cd\u200b\u6570\u5b57\u200b\u7684\u200b\u503c\u200b</li> </ul> <p>\u200b\u6574\u6570\u200b\u6ea2\u51fa\u200b</p> <p>\u200b\u5728\u200bDebug\u200b\u6a21\u5f0f\u200b\u4e0b\u200b\uff0cRust\u200b\u4f1a\u200b\u68c0\u67e5\u200b\u6574\u6570\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u5982\u679c\u200b\u53d1\u751f\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u7a0b\u5e8f\u200b\u4f1a\u200bpanic\u3002\u200b\u5728\u200bRelease\u200b\u6a21\u5f0f\u200b\u4e0b\u200b\uff0cRust\u200b\u4e0d\u4f1a\u200b\u68c0\u67e5\u200b\u6574\u6570\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u6ea2\u51fa\u200b\u540e\u200b\u4f1a\u200b\u6309\u7167\u200b\u4e8c\u8fdb\u5236\u200b\u8865\u7801\u200b\u8fdb\u884c\u200b\u5faa\u73af\u200b\u3002</p> <p>\u200b\u6574\u6570\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li>\u200b\u4e0b\u6587\u200b\u4e2d\u200b\u7684\u200b<code>*</code>\u200b\u6307\u200b<code>add</code>\u3001<code>sub</code>\u3001<code>mul</code>\u3001<code>div</code>\u200b\u7b49\u200b\u8fd0\u7b97\u200b</li> <li><code>wrapping_*</code>\uff1a\u200b\u5982\u679c\u200b\u8fd0\u7b97\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u5219\u200b\u53d6\u200b\u6ea2\u51fa\u200b\u540e\u200b\u7684\u200b\u503c\u200b\u3002</li> <li><code>checked_*</code>\uff1a\u200b\u5982\u679c\u200b\u8fd0\u7b97\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>None</code>\u3002</li> <li><code>saturating_*</code>\uff1a\u200b\u5982\u679c\u200b\u8fd0\u7b97\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b\u76ee\u6807\u200b\u7c7b\u578b\u200b\u7684\u200b\u6700\u5927\u503c\u200b\u6216\u200b\u6700\u5c0f\u503c\u200b\u3002</li> <li> <p><code>overflowing_*</code>\uff1a\u200b\u8fd4\u56de\u200b\u8fd0\u7b97\u200b\u7ed3\u679c\u200b\u548c\u200b\u662f\u5426\u200b\u6ea2\u51fa\u200b\u7684\u200b\u6807\u8bb0\u200b</p> <pre><code>let a: i32 = 1;\nlet b: i32 = 2;\nlet (c, overflow) = a.overflowing_add(b);\nprintln!(\"{} {}\", c, overflow); // 3 false\n</code></pre> </li> </ul>"},{"location":"coding/rust/1/#_4","title":"\u6d6e\u70b9\u6570\u200b\u7c7b\u578b","text":"<ol> <li>\u200b\u6d6e\u70b9\u6570\u200b\u6ca1\u6709\u200b\u5b9e\u73b0\u200b<code>Eq</code> trait\uff0c\u200b\u56e0\u6b64\u200b\u4e0d\u80fd\u200b\u76f4\u63a5\u200b\u7528\u200b<code>==</code>\u200b\u548c\u200b<code>!=</code>\u200b\u6765\u200b\u6bd4\u8f83\u200b\uff0c\u200b\u4e5f\u200b\u4e0d\u80fd\u200b\u4f5c\u4e3a\u200b\u5b57\u5178\u200b\u7684\u200b\u952e\u200b\u3002</li> <li>\u200b\u6570\u5b66\u200b\u4e0a\u200b\u65e0\u200b\u5b9a\u4e49\u200b\u7684\u200b\u8fd0\u7b97\u200b\uff0c\u200b\u5982\u200b<code>0.0 / 0.0</code>\uff0cRust\u200b\u4f1a\u200b\u8fd4\u56de\u200b<code>NaN</code>\uff08Not a Number\uff09\u3002\u200b\u4e0e\u200b<code>NaN</code>\u200b\u7684\u200b\u8fd0\u7b97\u200b\u7ed3\u679c\u200b\u603b\u662f\u200b<code>NaN</code>\uff0c\u200b\u5e76\u4e14\u200b<code>NaN</code>\u200b\u4e0d\u80fd\u200b\u53c2\u4e0e\u200b\u6bd4\u8f83\u200b\u8fd0\u7b97\u200b\u3002</li> <li>\u200b\u7531\u4e8e\u200b<code>NaN</code>\u200b\u7684\u200b\u5b58\u5728\u200b\uff0c\u200b\u6d6e\u70b9\u6570\u200b\u6ca1\u6709\u200b\u5b9e\u73b0\u200b<code>Ord</code> trait\uff0c\u200b\u56e0\u6b64\u200b\u4e0d\u80fd\u200b\u76f4\u63a5\u200b\u7528\u4e8e\u200b<code>sort()</code>\u200b\u7b49\u200b\u9700\u8981\u200b<code>Ord</code> trait\u200b\u7684\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b<code>partial_cmp().unwrap()</code>\u200b\u6765\u200b\u8fdb\u884c\u200b\u6bd4\u8f83\u200b\u3002</li> <li>\u200b\u7531\u4e8e\u200b\u6d6e\u70b9\u6570\u200b\u7684\u200b\u7cbe\u5ea6\u200b\u95ee\u9898\u200b\uff0c\u200b\u6d6e\u70b9\u6570\u200b\u7684\u200b\u6bd4\u8f83\u200b\u5e94\u8be5\u200b\u7528\u200b<code>abs()</code>\u200b\u548c\u200b<code>&lt;</code>\u3001<code>&gt;</code>\u200b\u6765\u200b\u8fdb\u884c\u200b\u3002</li> </ol> <p>\u200b\u6d6e\u70b9\u6570\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li><code>is_nan()</code>\uff1a\u200b\u5224\u65ad\u200b\u662f\u5426\u200b\u4e3a\u200b<code>NaN</code>\u3002</li> </ul>"},{"location":"coding/rust/1/#_5","title":"\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\u7c7b\u578b","text":"<p>\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li> <p><code>len()</code>\uff1a\u200b\u8fd4\u56de\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u3002</p> <pre><code>let s = \"hello\";\nprintln!(\"{}\", s.len()); // 5\n</code></pre> </li> <li> <p><code>lines()</code>\uff1a\u200b\u6309\u200b\u6362\u884c\u7b26\u200b<code>\\n</code>\u200b\u5206\u5272\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u8fd4\u56de\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002</p> <pre><code>let s = \"hello\\nworld\";\nfor line in s.lines() {\n    println!(\"{}\", line);\n}\n</code></pre> </li> <li> <p><code>split()</code>\uff1a\u200b\u6309\u200b\u7ed9\u5b9a\u200b\u5b57\u7b26\u200b\u5206\u5272\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u8fd4\u56de\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002</p> <pre><code>let s = \"hello world\";\nfor word in s.split(\" \") {\n    println!(\"{}\", word);\n}\n</code></pre> </li> <li> <p><code>trim()</code>\uff1a\u200b\u53bb\u9664\u200b\u5b57\u7b26\u4e32\u200b\u9996\u5c3e\u200b\u7684\u200b\u7a7a\u767d\u200b\u5b57\u7b26\u200b\u3002</p> <pre><code>let s = \" hello world \";\nprintln!(\"{}\", s.trim()); // \"hello world\"\n</code></pre> </li> <li> <p><code>parse()</code>\uff1a\u200b\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u5176\u4ed6\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5982\u200b<code>parse::&lt;i32&gt;()</code>\u200b\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u200b\u4e3a\u200b<code>i32</code>\u200b\u7c7b\u578b\u200b\u3002\u200b\u5f53\u200b\u8f6c\u4e3a\u200b\u6570\u5b57\u200b\u65f6\u200b\uff0c\u200b\u6570\u5b57\u200b\u4e0d\u80fd\u200b\u8d85\u8fc7\u200b\u7c7b\u578b\u200b\u7684\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\uff0c\u200b\u5426\u5219\u200b\u4f1a\u200b\u62a5\u9519\u200b\u3002</p> <pre><code>let s = \"123\";\nlet n: i32 = s.parse().unwrap(); // \u200b\u8fd4\u56de\u503c\u200b\u662f\u200b`Result`\u200b\u7c7b\u578b\u200b\n</code></pre> <p>\u200b\u81ea\u5b9a\u4e49\u200b\u7c7b\u578b\u200b\u5982\u679c\u200b\u9700\u8981\u200b\u901a\u8fc7\u200b<code>parse()</code>\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b<code>std::str::FromStr</code> trait\uff0c\u200b\u53c2\u89c1\u200bImpl &amp; Trait</p> </li> <li> <p><code>to_string()</code>\uff1a\u200b\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\u8f6c\u6362\u200b\u4e3a\u200b<code>String</code>\u200b\u7c7b\u578b\u200b\u3002</p> </li> <li><code>chars()</code>\uff1a\u200b\u8fd4\u56de\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5b57\u7b26\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002</li> <li><code>bytes()</code>\uff1a\u200b\u8fd4\u56de\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5b57\u8282\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002</li> </ul>"},{"location":"coding/rust/1/#_6","title":"\u5b57\u7b26\u4e32\u200b\u7c7b\u578b","text":"<p>\u200b\u5b57\u7b26\u4e32\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li><code>push()</code>\u3001<code>push_str()</code>\uff1a\u200b\u5206\u522b\u200b\u7528\u4e8e\u200b\u5728\u200b\u5b57\u7b26\u4e32\u200b\u672b\u5c3e\u200b\u6dfb\u52a0\u200b\u5b57\u7b26\u200b\u548c\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\u3002</li> <li><code>insert()</code>\u3001<code>insert_str()</code>\uff1a\u200b\u5206\u522b\u200b\u7528\u4e8e\u200b\u5728\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u6307\u5b9a\u200b\u4f4d\u7f6e\u200b\u63d2\u5165\u200b\u5b57\u7b26\u200b\u548c\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\u3002</li> <li><code>replace(A, B)</code>\u3001<code>replacen(A, B, n)</code>\uff1a\u200b\u5206\u522b\u200b\u7528\u4e8e\u200b\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b<code>A</code>\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>B</code>\uff0c<code>replacen</code>\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u6307\u5b9a\u200b\u66ff\u6362\u200b\u6b21\u6570\u200b<code>n</code>\uff0c\u200b\u8fd4\u56de\u200b\u66ff\u6362\u200b\u540e\u200b\u7684\u200b\u65b0\u200b\u5b57\u7b26\u4e32\u200b\u3002</li> <li><code>replace_range(range, A)</code>\uff1a\u200b\u7528\u4e8e\u200b\u5c06\u200b\u539f\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u6307\u5b9a\u200b\u8303\u56f4\u200b\u66ff\u6362\u200b\u4e3a\u200b\u65b0\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b<code>A</code>\uff0c\u200b\u8fd4\u56de\u200b\u66ff\u6362\u200b\u540e\u200b\u7684\u200b\u539f\u200b\u5b57\u7b26\u4e32\u200b\u3002</li> <li><code>pop()</code>\uff1a\u200b\u5220\u9664\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5b57\u7b26\u4e32\u200b\u672b\u5c3e\u200b\u7684\u200b\u5b57\u7b26\u200b\uff0c\u200b\u5982\u679c\u200b\u5b57\u7b26\u4e32\u200b\u4e3a\u200b\u7a7a\u5219\u200b\u8fd4\u56de\u200b<code>None</code>\u3002</li> <li><code>remove()</code>\uff1a\u200b\u5220\u9664\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u6307\u5b9a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5b57\u7b26\u200b\u3002\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u7d22\u5f15\u200b\u5fc5\u987b\u200b\u662f\u200b\u6709\u6548\u200b\u7684\u200b\u5b57\u7b26\u200b\u8fb9\u754c\u200b\u3002</li> <li><code>truncate()</code>\uff1a\u200b\u5220\u9664\u200b\u4ece\u200b\u6307\u5b9a\u200b\u4f4d\u7f6e\u200b\u5f00\u59cb\u200b\u5230\u200b\u5b57\u7b26\u4e32\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5168\u90e8\u200b\u5b57\u7b26\u200b\u3002</li> <li><code>clear()</code>\uff1a\u200b\u6e05\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u3002</li> <li><code>+</code>\u200b\u8fd0\u7b97\u7b26\u200b\uff1a\u200b\u7528\u4e8e\u200b\u8fde\u63a5\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u53f3\u7aef\u200b\u9879\u4e3a\u200b<code>&amp;str</code>\u200b\u7c7b\u578b\u200b\u3002</li> </ul>"},{"location":"coding/rust/1/#_7","title":"\u6570\u7ec4\u200b\u7c7b\u578b","text":"<p>\u200b\u6570\u7ec4\u200b\u8d8a\u754c\u200b</p> <p>Rust\u200b\u4f1a\u200b\u68c0\u67e5\u200b\u6570\u7ec4\u200b\u8bbf\u95ee\u200b\u8d8a\u754c\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u6570\u7ec4\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li> <p><code>std::array::from_fn()</code>\uff1a\u200b\u6839\u636e\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\u751f\u6210\u200b\u6570\u7ec4\u200b\u3002</p> <pre><code>let a: [String; 8] = std::array::from_fn(|_i| String::from(\"This is Rust\"));\n</code></pre> </li> </ul>"},{"location":"coding/rust/1/#_8","title":"\u8fed\u4ee3\u200b\u5668\u200b\u7c7b\u578b","text":"<p>\u200b\u8fed\u4ee3\u200b\u5668\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li> <p><code>map(fn)</code>\uff1a\u200b\u5bf9\u200b\u8fed\u4ee3\u200b\u5668\u4e2d\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u5e94\u7528\u200b\u51fd\u6570\u200b<code>fn</code>\uff0c\u200b\u8fd4\u56de\u200b\u65b0\u200b\u7684\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002</p> <pre><code>let v = vec![1, 2, 3];\nfor i in v.iter().map(|x| x * 2) {\n    print!(\"{}\", i); // 246\n}\n</code></pre> </li> <li> <p><code>collect()</code>\uff1a\u200b\u5c06\u200b\u8fed\u4ee3\u200b\u5668\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u5bb9\u5668\u200b\uff08\u200b\u5b9a\u200b\u957f\u200b\u6570\u7ec4\u200b\uff09\u3002</p> <pre><code>let v = vec![1, 2, 3];\nlet v2: Vec&lt;i32&gt; = v.iter().map(|x| x * 2).collect();\n</code></pre> </li> </ul>"},{"location":"coding/rust/1/#_9","title":"\u8fd0\u7b97\u7b26","text":"<p>Rust\u200b\u7684\u200b\u8fd0\u7b97\u7b26\u200b\u4e0e\u200bC/C++\u200b\u7c7b\u4f3c\u200b</p> <ul> <li>\u200b\u56db\u5219\u8fd0\u7b97\u200b\u7b26\u200b\uff1a<code>+ - * / %</code></li> <li>\u200b\u6bd4\u8f83\u200b\u8fd0\u7b97\u7b26\u200b\uff1a<code>== != &gt; &lt; &gt;= &lt;=</code></li> <li>\u200b\u903b\u8f91\u200b\u8fd0\u7b97\u7b26\u200b\uff1a<code>&amp;&amp; || !</code></li> <li>\u200b\u4f4d\u200b\u8fd0\u7b97\u7b26\u200b\uff1a<code>&amp; | ^ &lt;&lt; &gt;&gt; !</code>\uff08<code>!</code>\u200b\u4e3a\u200b\u6309\u4f4d\u200b\u53d6\u53cd\u200b\uff09</li> <li>\u200b\u539f\u4f4d\u200b\u8d4b\u503c\u200b\u8fd0\u7b97\u7b26\u200b\uff1a<code>+= -= *= /= %= &amp;= |= ^= &lt;&lt;= &gt;&gt;=</code>\u200b\u7b49\u200b</li> <li>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u8fd0\u7b97\u7b26\u200b\uff1a<code>as</code>\uff0c\u200b\u5982\u200b<code>let b = (a as f64) / 2.0;</code></li> <li>\u200b\u5f15\u7528\u200b\u4e0e\u200b\u89e3\u200b\u5f15\u7528\u200b\u8fd0\u7b97\u7b26\u200b\uff1a<code>&amp; &amp;mut *</code>\uff0c\u200b\u5176\u4e2d\u200b<code>&amp;</code>\u200b\u7528\u4e8e\u200b\u83b7\u53d6\u200b\u53d8\u91cf\u200b\u7684\u200b\u5f15\u7528\u200b\uff08borrow\uff09\uff0c<code>&amp;mut</code>\u200b\u7528\u4e8e\u200b\u83b7\u53d6\u200b\u53d8\u91cf\u200b\u7684\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\uff0c<code>*</code>\u200b\u7528\u4e8e\u200b\u89e3\u200b\u5f15\u7528\u200b\u3002</li> </ul>"},{"location":"coding/rust/2/","title":"Control Flows","text":"<p>\u200b\u672c\u200b\u8282\u200b\u4ecb\u7ecd\u200bRust\u200b\u4e2d\u200b\u7684\u200b\u51fd\u6570\u200b\u548c\u200b\u63a7\u5236\u6d41\u200b\u8bed\u53e5\u200b\u3002</p>"},{"location":"coding/rust/2/#_1","title":"\u51fd\u6570\u200b\u4e0e\u200b\u51fd\u6570\u6307\u9488","text":"<p>Rust\u200b\u4e2d\u200b\uff0c\u200b\u7528\u200b\u5173\u952e\u5b57\u200b<code>fn</code>\u200b\u58f0\u660e\u200b\u51fd\u6570\u200b\uff0c\u200b\u51fd\u6570\u200b\u7684\u200b\u683c\u5f0f\u200b\u4e3a\u200b\uff1a</p> <pre><code>fn &lt;function name&gt;([&lt;parameter name&gt;: &lt;parameter type&gt;, ...]) -&gt; [&lt;return type&gt;] {\n    &lt;function body&gt;\n}\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff0c\u200b\u51fd\u6570\u200b\u5fc5\u987b\u200b\u6307\u660e\u200b\u6240\u6709\u200b\u53c2\u6570\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u5982\u679c\u200b\u6ca1\u6709\u200b\u6307\u660e\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5219\u200b\u9ed8\u8ba4\u200b\u4e3a\u200b<code>()</code>\uff0c\u200b\u5373\u7a7a\u200b\u5143\u7ec4\u200b\u3002\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>return</code>\u200b\u5173\u952e\u5b57\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u6700\u540e\u200b\u4e00\u884c\u200b\u7684\u200b\u8868\u8fbe\u5f0f\u200b\u8fd4\u56de\u503c\u200b\u3002\u200b\u4efb\u4f55\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u5982\u679c\u200b\u901a\u8fc7\u200b\u8868\u8fbe\u5f0f\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5219\u200b\u4e0d\u80fd\u200b\u52a0\u5206\u200b\u53f7\u200b<code>;</code>\u3002</p> <p>  Download source code</p> <pre><code>fn add(a: i32, b: i32) -&gt; i32 {\n    a + b // Return by expression\n}\nfn sub(a: i32, b: i32) -&gt; i32 {\n    return a - b; // Return by return keyword\n}\n\nfn main() {\n    let (a, b) = (2, 1);\n    println!(\"{} + {} = {}\", a, b, add(a, b));\n    println!(\"{} - {} = {}\", a, b, sub(a, b));\n}\n</code></pre> <p>\u200b\u51fd\u6570\u6307\u9488\u200b\u7684\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>fn(&lt;parameter type&gt;, ...) -&gt; &lt;return type&gt;</code>\uff0c\u200b\u5c06\u200b\u51fd\u6570\u200b\u8d4b\u503c\u200b\u7ed9\u200b\u51fd\u6570\u6307\u9488\u200b\u65f6\u200b\uff0c\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\u548c\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u5fc5\u987b\u200b\u5168\u90e8\u200b\u5339\u914d\u200b\u3002</p> <p>\u200b\u5728\u200b\u51fd\u6570\u200b\u5185\u90e8\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u58f0\u660e\u200b\u51fd\u6570\u200b\uff0c\u200b\u79f0\u4e3a\u200blambda\u200b\u51fd\u6570\u200b\uff0c\u200b\u58f0\u660e\u200b\u65b9\u5f0f\u200b\u4e3a\u200b</p> <pre><code>let &lt;function name&gt; = |[&lt;parameter name&gt;: &lt;parameter type&gt;, ...]| -&gt; [&lt;return type&gt;] {\n    &lt;function body&gt;\n}\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff1a</p> <ol> <li> <p>\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>let add = |a, b| { return a + b; };\n</code></pre> </li> <li> <p>\u200b\u5982\u679c\u200b\u51fd\u6570\u200b\u4f53\u200b\u53ea\u6709\u200b\u4e00\u884c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b<code>{}</code>\uff0c\u200b\u6b64\u65f6\u200b\u4e0d\u80fd\u200b\u52a0\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>let add = |a, b| a + b;\n</code></pre> </li> <li> <p>\u200b\u53ef\u4ee5\u200b\u5728\u200b\u58f0\u660e\u200b\u7684\u200b\u65f6\u5019\u200b\u76f4\u63a5\u200b\u8c03\u7528\u200b\uff0c\u200b\u6b64\u65f6\u200b\u4e0d\u80fd\u200b\u7701\u7565\u200b<code>{}</code>\u200b\u548c\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>let result = |a, b| -&gt; i32 { a + b }(1, 2); // result = 3\n</code></pre> </li> </ol> <p>  Download source code</p> <pre><code>fn main() {\n    let (a, b) = (2, 1);\n    let add = |a, b| a + b;\n    println!(\"{} + {} = {}\", a, b, add(a, b));\n    println!(\"{} - {} = {}\", a, b, |a, b| -&gt; i32 { return a - b; }(a, b));\n}\n</code></pre>"},{"location":"coding/rust/2/#if","title":"<code>if</code>\u200b\u8bed\u53e5","text":"<p>Rust\u200b\u4e2d\u200b<code>if</code>\u200b\u8bed\u53e5\u200b\u7684\u200b\u683c\u5f0f\u200b\u4e0e\u200bC/C++\u200b\u76f8\u4f3c\u200b\uff0c\u200b\u4f46\u662f\u200b\u6761\u4ef6\u200b\u8868\u8fbe\u5f0f\u200b\u4e0d\u200b\u9700\u8981\u200b\u52a0\u200b\u62ec\u53f7\u200b\u3002</p> <pre><code>if &lt;condition&gt; {\n    &lt;expression&gt;\n} [ else if &lt;condition&gt; {\n    &lt;expression&gt;\n} ] [ else {\n    &lt;expression&gt;\n} ]\n</code></pre> <p><code>if</code>\u200b\u8bed\u53e5\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200b\u5728\u200b\u8868\u8fbe\u5f0f\u200b\u4e2d\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6bcf\u4e2a\u200b\u5206\u652f\u200b\u5757\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u884c\u200b\u4f5c\u4e3a\u200b\u8be5\u200b\u5206\u652f\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002\u200b\u6b64\u65f6\u200b\u6bcf\u4e2a\u200b\u5206\u652f\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u5fc5\u987b\u200b\u76f8\u540c\u200b\u3002</p> <p>  Download source code</p> <pre><code>fn main() {\n    let (a, b) = (5, 3);\n    println!(\"max({}, {}) = {}\", a, b, if a &gt; b { a } else { b });\n    if a &lt; b {\n        println!(\"min({}, {}) = {}\", a, b, a);\n    } else {\n        println!(\"min({}, {}) = {}\", a, b, b);\n    }\n}\n</code></pre>"},{"location":"coding/rust/2/#match","title":"<code>match</code>\u200b\u8bed\u53e5","text":"<p><code>match</code>\u200b\u8bed\u53e5\u200b\u7528\u4e8e\u200b\u5bf9\u200b\u53d8\u91cf\u200b\u7684\u200b\u503c\u200b\u8fdb\u884c\u200b\u6a21\u5f0f\u5339\u914d\u200b\uff0c\u200b\u683c\u5f0f\u200b\u4e3a\u200b\uff1a</p> <pre><code>match &lt;variable&gt; {\n    &lt;pattern&gt; =&gt; &lt;expression&gt;,\n    ...\n    _ =&gt; &lt;expression&gt;\n}\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff0c<code>&lt;pattern&gt;</code>\u200b\u53ef\u4ee5\u200b\u662f\u200b</p> <ol> <li>\u200b\u5355\u4e2a\u200b\u6570\u503c\u200b\uff0c\u200b\u5982\u200b<code>5 =&gt; \"five\"</code></li> <li>\u200b\u8303\u56f4\u200b\uff0c\u200b\u5982\u200b<code>1 ..= 5 =&gt; \"one to five\"</code></li> <li><code>|</code>\u200b\u5206\u9694\u200b\u7684\u200b\u591a\u4e2a\u200b\u6a21\u5f0f\u200b\uff0c\u200b\u5982\u200b<code>1 | 2 | 3 =&gt; \"one to three\"</code></li> <li>\u200b\u6a21\u677f\u200b\u5b9e\u4f8b\u200b\uff0c\u200b\u5982\u200b<code>Some(x) =&gt; Some(x + 1)</code></li> <li><code>_</code>\uff0c\u200b\u8868\u793a\u200b\u5339\u914d\u200b\u6240\u6709\u200b\u60c5\u51b5\u200b\u3002</li> <li>\u200b\u5982\u679c\u200b\u88ab\u200b\u5339\u914d\u200b\u7684\u200b\u53d8\u91cf\u200b\u662f\u200b\u5143\u7ec4\u200b\uff0c\u200b\u5219\u200b\u5143\u7ec4\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u662f\u200b\u4e0a\u8ff0\u200b\u6a21\u5f0f\u200b\u3002</li> <li>\u200b\u6761\u4ef6\u200b\u8bed\u53e5\u200b\uff0c\u200b\u5982\u200b<code>x if x &gt; 5 =&gt; \"greater than five\"</code>\uff0c\u200b\u6b64\u65f6\u200b\u5143\u7ec4\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6574\u4f53\u200b\u53c2\u4e0e\u200b\u6761\u4ef6\u200b\u8bed\u53e5\u200b\uff0c\u200b\u5982\u200b<code>(x, y) if x &gt; 5 &amp;&amp; y &gt; 5 =&gt; \"both greater than five\"</code></li> </ol> <p><code>&lt;expression&gt;</code>\u200b\u53ef\u4ee5\u200b\u662f\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u4ee3\u7801\u200b\u5757\u200b\uff0c\u200b\u6b64\u65f6\u200b\u4ee3\u7801\u200b\u5757\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u884c\u200b\u4f5c\u4e3a\u200b\u8fd4\u56de\u503c\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b<code>match</code>\u200b\u7684\u200b\u5206\u652f\u200b\u5fc5\u987b\u200b\u5305\u62ec\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u5426\u5219\u200b\u4f1a\u200b\u62a5\u9519\u200b\u3002</p> <p>  Download source code</p> <pre><code>fn main() {\n    let marks_paper_a: u8 = 25;\n    let marks_paper_b: u8 = 30;\n\n    let output = match (marks_paper_a, marks_paper_b) {\n        (50, 50) =&gt; \"Full marks for both papers\",\n        (50, _) =&gt; \"Full marks for paper A\",\n        (_, 50) =&gt; \"Full marks for paper B\",\n        (x, y) if x &gt; 25 &amp;&amp; y &gt; 25 =&gt; \"Good\",\n        (_, _) =&gt; \"Work hard\"\n    };\n\n    println!(\"{}\", output); // Work hard\n}\n</code></pre>"},{"location":"coding/rust/2/#if-let","title":"<code>if let</code>\u200b\u8bed\u53e5","text":"<p>\u200b\u548c\u200b<code>match</code>\u200b\u7c7b\u4f3c\u200b\uff0c<code>if let</code>\u200b\u8bed\u53e5\u200b\u7528\u4e8e\u200b\u5339\u914d\u200b\u5355\u4e2a\u200b\u6a21\u5f0f\u200b\uff0c\u200b\u683c\u5f0f\u200b\u4e3a\u200b\uff1a</p> <pre><code>if let &lt;pattern&gt; = &lt;variable&gt; {\n    &lt;expression&gt;\n}\n</code></pre> <p><code>let &lt;pattern&gt; = &lt;variable&gt;</code>\u200b\u53ef\u4ee5\u200b\u770b\u6210\u200b\u4e00\u4e2a\u200b\u6761\u4ef6\u200b\uff08\u200b\u4f46\u200b\u5b9e\u9645\u200b\u5e76\u200b\u4e0d\u662f\u200b\uff09\uff0c\u200b\u5982\u679c\u200b\u80fd\u200b\u5339\u914d\u200b\u5219\u200b\u5c06\u200b<code>&lt;variable&gt;</code>\u200b\u7684\u200b\u503c\u200b\u6309\u7167\u200b<code>&lt;pattern&gt;</code>\u200b\u8fdb\u884c\u200b\u7ed1\u5b9a\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b<code>true</code>\uff0c\u200b\u56e0\u6b64\u200b\u4e5f\u200b\u4f1a\u200b\u6709\u200b<code>while let</code>\u200b\u7b49\u200b\u5176\u4ed6\u200b\u8bed\u53e5\u200b\u3002</p>"},{"location":"coding/rust/2/#loop","title":"<code>loop</code>\u200b\u8bed\u53e5","text":"<p><code>loop</code>\u200b\u8bed\u53e5\u200b\u7528\u4e8e\u200b\u65e0\u9650\u200b\u5faa\u73af\u200b\u3002</p> <ol> <li>\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>continue</code>\u200b\u8df3\u8f6c\u200b\u5230\u200b\u4e0b\u200b\u4e00\u6b21\u200b\u5faa\u73af\u200b\uff0c\u200b\u6216\u200b<code>break</code>\u200b\u8df3\u51fa\u200b\u5faa\u73af\u200b\u3002</li> <li><code>loop</code>\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>break &lt;expression&gt;</code>\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6240\u6709\u200b<code>break</code>\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u5fc5\u987b\u200b\u76f8\u540c\u200b\u3002</li> <li>\u200b\u6bcf\u4e2a\u200b\u5faa\u73af\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>'&lt;label&gt;: loop</code>\u200b\u6807\u8bb0\u200b\uff0c\u200b\u5728\u200b<code>break</code>\u200b\u548c\u200b<code>continue</code>\u200b\u65f6\u200b\u53ef\u4ee5\u200b\u6307\u5b9a\u200b\u8df3\u51fa\u200b/\u200b\u8df3\u8fc7\u200b\u7684\u200b\u5faa\u73af\u200b\u3002</li> </ol> <p>  Download source code</p> <pre><code>fn main() {\n    let mut b1 = 1;\n\n    let (c1, c2) = 'outer_loop: loop {\n      let mut b2 = 1;\n\n      'inner_loop: loop {\n        println!(\"Current Value : [{}][{}]\", b1, b2);\n\n        if b1 == 2 &amp;&amp; b2 == 2 {\n            break 'outer_loop (b1, b2); // Leave outer_loop with return value\n        } else if b2 == 5 {\n            break; // Leave inner_loop by default\n        }\n\n        b2 += 1;\n      }\n\n      b1 += 1;\n    };\n    println!(\"b1 = {}, b2 = {}\", c1, c2);\n}\n</code></pre>"},{"location":"coding/rust/2/#while","title":"<code>while</code>\u200b\u8bed\u53e5","text":"<p><code>while</code>\u200b\u8bed\u53e5\u200b\u7528\u4e8e\u200b\u662f\u200b\u6709\u200b\u6761\u4ef6\u200b\u7684\u200b\u5faa\u73af\u200b\uff0c<code>while</code>\u200b\u8bed\u53e5\u200b\u4e0d\u80fd\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5176\u4ed6\u200b\u4e0e\u200b<code>loop</code>\u200b\u8bed\u53e5\u200b\u76f8\u540c\u200b\u3002</p> <pre><code>while &lt;condition&gt; {\n    &lt;expression&gt;\n}\n</code></pre>"},{"location":"coding/rust/2/#for","title":"<code>for</code>\u200b\u8bed\u53e5","text":"<p><code>for</code>\u200b\u8bed\u53e5\u200b\u7528\u4e8e\u200b\u904d\u5386\u200b\u8fed\u4ee3\u200b\u5668\u200b\uff0c\u200b\u683c\u5f0f\u200b\u4e3a\u200b\uff1a</p> <pre><code>for &lt;variable&gt; in &lt;iterator&gt; {\n    &lt;expression&gt;\n}\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff0c<code>&lt;iterator&gt;</code>\u200b\u53ef\u4ee5\u200b\u662f\u200b</p> <ol> <li><code>start .. end</code>\uff0c\u200b\u8868\u793a\u200b\u4ece\u200b<code>start</code>\u200b\u5230\u200b<code>end - 1</code>\u200b\u7684\u200b\u6574\u6570\u200b\u8303\u56f4\u200b\u3002</li> <li><code>start ..= end</code>\uff0c\u200b\u8868\u793a\u200b\u4ece\u200b<code>start</code>\u200b\u5230\u200b<code>end</code>\u200b\u7684\u200b\u6574\u6570\u200b\u8303\u56f4\u200b\u3002</li> <li>\u200b\u7531\u200b\u65b9\u6cd5\u200b<code>&lt;iterable&gt;.iter()</code>\u200b\u8fd4\u56de\u200b\u7684\u200b\u8fed\u4ee3\u200b\u5668\u200b\uff0c\u200b\u5982\u200b<code>[1, 2, 3].iter()</code>\u3002</li> </ol> <p><code>for</code>\u200b\u5faa\u73af\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u6dfb\u52a0\u200b\u6807\u7b7e\u200b\u3002</p> <p>  Download source code</p> <pre><code>fn main() {\n    let group : [&amp;str; 4] = [\"Mark\", \"Larry\", \"Bill\", \"Steve\"];\n\n    for n in 0..group.len() { // group.len() = 4 -&gt; 0..4 \ud83d\udc4e check group.len()on each iteration\n    println!(\"Current Person : {}\", group[n]);\n    }\n\n    for person in group.iter() { // \ud83d\udc4d group.iter() turn the array into a simple iterator\n    println!(\"Current Person : {}\", person);\n    }\n}\n</code></pre> <p>\u200b\u4e25\u683c\u200b\u610f\u4e49\u200b\u4e0a\u200b<code>for</code>\u200b\u8bed\u53e5\u200b\u4e0d\u80fd\u200b\u76f4\u63a5\u200b\u904d\u5386\u200b\u5bb9\u5668\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b<code>iter()</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002\u200b\u4f46\u200b<code>for</code>\u200b\u4f1a\u200b\u5c1d\u8bd5\u200b\u9690\u5f0f\u200b\u8c03\u7528\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e0b\u9762\u200b\u7684\u200b\u4ee3\u7801\u200b\u4e5f\u200b\u80fd\u200b\u6b63\u5e38\u200b\u6267\u884c\u200b\u3002</p> <pre><code>fn main() {\n    let a = [1, 2, 3];\n    for i in a {\n        println!(\"{}\", i);\n    }\n}\n</code></pre>"},{"location":"coding/rust/3/","title":"Complex objects","text":"<p>\u200b\u672c\u200b\u8282\u200b\u4ecb\u7ecd\u200bRust\u200b\u4e2d\u200b\u5e38\u7528\u200b\u7684\u200b\u590d\u6742\u200b\u7c7b\u578b\u200b</p>"},{"location":"coding/rust/3/#_1","title":"\u5bb9\u5668\u200b\u7c7b\u578b","text":""},{"location":"coding/rust/3/#vec","title":"<code>Vec</code>","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u5411\u91cf\u200b\u7c7b\u578b\u200b\u662f\u200b\u957f\u5ea6\u200b\u53ef\u53d8\u200b\u7684\u200b\u6570\u7ec4\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>Vec::new()</code>\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u7a7a\u200b\u7684\u200b\u5411\u91cf\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>vec![]</code>\u200b\u5b8f\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u5e26\u6709\u200b\u521d\u59cb\u503c\u200b\u7684\u200b\u5411\u91cf\u200b\u3002<code>Vec</code>\u200b\u662f\u200b\u6cdb\u578b\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>Vec&lt;type&gt;</code>\u200b\u53ef\u4ee5\u200b\u6307\u5b9a\u200b\u5411\u91cf\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u4e00\u4e2a\u200b\u5411\u91cf\u200b\u4e2d\u200b\u5b58\u50a8\u200b\u5143\u7d20\u200b\u7684\u200b\u7c7b\u578b\u200b\u5168\u90e8\u200b\u76f8\u540c\u200b\u3002\u200b\u5411\u91cf\u200b\u53ca\u5176\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6574\u4f53\u200b\uff0c\u200b\u5bf9\u200b\u5176\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u4e5f\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u5411\u91cf\u200b\u672c\u8eab\u200b\u65e0\u6cd5\u200b\u518d\u200b\u8fdb\u884c\u200b\u5f15\u7528\u200b\u3002</p> <pre><code>let v1 = Vec::new(); // Empty vector\nlet v2: Vec&lt;i32&gt; = Vec::new(); // Empty vector with type annotation\nlet v3 = vec![1, 2, 3]; // Vector with initial values\nlet v4 = vec![0; 10]; // Repeat values using colon\nlet v5 = vec![1i32, 2, 3]; // Vector with type specified to the first element\nlet v6 = Vec::with_capacity(10); // Vector with pre-allocated capacity 10\n</code></pre> <p>\u200b\u7528\u200b<code>let mut</code>\u200b\u58f0\u660e\u200b\u7684\u200b\u5411\u91cf\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u4fee\u6539\u200b\u3002<code>Vec</code>\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li><code>push()</code>\uff1a\u200b\u5411\u200b\u5411\u91cf\u200b\u7684\u200b\u5c3e\u90e8\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b</li> <li><code>pop()</code>\uff1a\u200b\u4ece\u200b\u5411\u91cf\u200b\u7684\u200b\u5c3e\u90e8\u200b\u5f39\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b</li> <li><code>extend()</code>\uff1a\u200b\u5411\u200b\u5411\u91cf\u200b\u7684\u200b\u5c3e\u90e8\u200b\u6dfb\u52a0\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u5411\u91cf\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b</li> <li><code>reserve()</code>\uff1a\u200b\u8c03\u6574\u200b\u5411\u91cf\u200b\u7684\u200b\u5bb9\u91cf\u200b</li> <li><code>len()</code>\uff1a\u200b\u8fd4\u56de\u200b\u5411\u91cf\u200b\u7684\u200b\u957f\u5ea6\u200b</li> <li><code>capacity()</code>\uff1a\u200b\u8fd4\u56de\u200b\u5411\u91cf\u200b\u7684\u200b\u5bb9\u91cf\u200b\uff08\u200b\u5f53\u200b\u8d85\u51fa\u200b\u5bb9\u91cf\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u91cd\u65b0\u5206\u914d\u200b\u5185\u5b58\u7a7a\u95f4\u200b\uff09</li> <li><code>get()</code>\uff1a\u200b\u8fd4\u56de\u200b\u5411\u91cf\u200b\u4e2d\u200b\u6307\u5b9a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u5f15\u7528\u200b\uff0c\u200b\u5982\u679c\u200b\u7d22\u5f15\u200b\u8d8a\u754c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>None</code></li> <li><code>shrink_to_fit()</code>\uff1a\u200b\u5c06\u200b\u5411\u91cf\u200b\u7684\u200b\u5bb9\u91cf\u200b\u8c03\u6574\u200b\u4e3a\u200b\u5411\u91cf\u200b\u7684\u200b\u957f\u5ea6\u200b\uff0c\u200b\u91ca\u653e\u200b\u591a\u4f59\u200b\u7684\u200b\u5185\u5b58\u7a7a\u95f4\u200b</li> <li><code>sort()</code>\u3001<code>sort_unstable()</code>\uff1a\u200b\u5bf9\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b</li> <li><code>sort_by()</code>\u3001<code>sort_unstable_by()</code>\uff1a\u200b\u5bf9\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b\uff0c\u200b\u4f7f\u7528\u200b\u81ea\u5b9a\u4e49\u200b\u7684\u200b\u6bd4\u8f83\u200b\u51fd\u6570\u200b</li> <li><code>[]</code>\u200b\u8fd0\u7b97\u7b26\u200b\uff1a\u200b\u8fd4\u56de\u200b\u5411\u91cf\u200b\u4e2d\u200b\u6307\u5b9a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\uff08\u200b\u53f3\u7aef\u200b\u9879\u200b\uff09\uff0c\u200b\u6216\u8005\u200b\u4fee\u6539\u200b\u5411\u91cf\u200b\u4e2d\u200b\u6307\u5b9a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\uff08\u200b\u5de6\u7aef\u200b\u9879\u200b\uff09\u3002</li> </ul> <p>\u200b\u5411\u91cf\u200b\u53ca\u5176\u200b\u5f15\u7528\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u7528\u200b\u5728\u200b<code>for</code>\u200b\u5faa\u73af\u200b\u4e2d\u200b\u3002</p>"},{"location":"coding/rust/3/#_2","title":"\u7ed3\u6784\u200b\u4f53","text":"<p>\u200b\u6709\u200b\u4e09\u79cd\u200b\u65b9\u6cd5\u200b\u5728\u200bRust\u200b\u4e2d\u200b\u58f0\u660e\u200b\u7ed3\u6784\u200b\u4f53\u200b\uff1a</p> <ol> <li>C\u200b\u8bed\u8a00\u200b\u98ce\u683c\u200b</li> <li>\u200b\u547d\u540d\u200b\u5143\u7ec4\u200b</li> <li>unit struct</li> </ol> <pre><code>struct CStyle {\n    x: i32,\n    y: i32,\n} // C-style struct with two fields\nstruct NamedTuple(i32, i32); // Named tuple struct with two fields\nstruct TypePattern(i32); // Type pattern struct with one field, can be used in `match` statement\nstruct UnitStruct; // Unit struct without fields\n</code></pre> <p>\u200b\u7528\u200b<code>pub</code>\u200b\u5173\u952e\u5b57\u200b\u6807\u8bb0\u200b\u7ed3\u6784\u200b\u4f53\u540e\u200b\uff0c\u200b\u5176\u4e2d\u200b\u7684\u200b\u5b57\u200b\u6bb5\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u5916\u754c\u200b\u8bbf\u95ee\u200b\u3002</p>"},{"location":"coding/rust/3/#_3","title":"\u679a\u4e3e\u200b\u7c7b\u578b","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u679a\u4e3e\u200b\u7c7b\u578b\u200b\u5173\u952e\u5b57\u200b\u4e3a\u200b<code>enum</code>\uff0c\u200b\u53ef\u4ee5\u200b\u5305\u542b\u200b\u4e0d\u540c\u200b\u7c7b\u578b\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u503c\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u6709\u200b\u4e0d\u540c\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u679a\u4e3e\u200b\u7c7b\u578b\u200b\u7684\u200b\u503c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>match</code>\u200b\u8bed\u53e5\u200b\u8fdb\u884c\u200b\u5339\u914d\u200b\u3002</p> <pre><code>enum Week {\n    Monday,\n    Tuesday,\n    Wednesday,\n    Thursday,\n    Friday,\n    Saturday,\n    Sunday,\n} // Enum type with 7 variants\n</code></pre> <p>Rust\u200b\u4e2d\u200b\u7684\u200b\u679a\u4e3e\u200b\u5143\u7d20\u200b\u53ef\u4ee5\u200b\u662f\u200b\u4e0d\u540c\u200b\u7684\u200bpattern\uff0c\u200b\u53ef\u4ee5\u200b\u662f\u200b\u7ed3\u6784\u200b\u4f53\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u5143\u7ec4\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200bunit struct\u3002</p> <pre><code>enum IpAddr {\n    V4(u8, u8, u8, u8), // Tuple\n    V6{ addr: String }, // Struct\n    LocalHost, // Unit\n}\n</code></pre> <p>\u200b\u8fd9\u4e9b\u200bpattern\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b<code>match</code>\u200b\u8bed\u53e5\u200b\u4e2d\u200b\u8fdb\u884c\u200b\u6a21\u5f0f\u5339\u914d\u200b\uff1a</p> <p>  Download source code</p> <pre><code>enum IpAddr {\n    V4(u8, u8, u8, u8), // Tuple\n    V6{ addr: String }, // Struct\n    LocalHost, // Unit\n}\n\nfn main() {\n    let v4 = IpAddr::V4(4, 4, 4, 4);\n    let v4_invalid = IpAddr::V4(0, 0, 0, 0);\n    let v6 = IpAddr::V6{ addr: String::from(\"::1\") };\n    let local = IpAddr::LocalHost;\n    print_ip(v4);\n    print_ip(v4_invalid);\n    print_ip(v6);\n    print_ip(local);\n}\n\nfn print_ip(ip: IpAddr) {\n    match ip {\n        IpAddr::V4(a, b, c, d) if a != 0 || b != 0 || c != 0 || d != 0\n            =&gt; println!(\"{}.{}.{}.{}\", a, b, c, d),\n        IpAddr::V4(a, b, c, d)\n            =&gt; println!(\"{}.{}.{}.{} is not an valid IPv4 address\", a, b, c, d),\n        IpAddr::V6{ addr } =&gt; println!(\"{}\", addr),\n        IpAddr::LocalHost =&gt; println!(\"localhost\"),\n    }\n}\n</code></pre> <p>\u200b\u7528\u200b<code>pub</code>\u200b\u5173\u952e\u5b57\u200b\u6807\u8bb0\u200b\u679a\u4e3e\u200b\u7c7b\u578b\u200b\u540e\u200b\uff0c\u200b\u5176\u4e2d\u200b\u7684\u200b\u5404\u4e2a\u200b\u5b57\u6bb5\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u5916\u754c\u200b\u8bbf\u95ee\u200b\u3002</p>"},{"location":"coding/rust/3/#_4","title":"\u6cdb\u578b","text":"<p>\u200b\u6cdb\u578b\u200b\u662f\u200b\u4e00\u79cd\u200b\u7c7b\u578b\u200b\u6a21\u677f\u200b\uff0c\u200b\u5f53\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b/\u200b\u679a\u4e3e\u200b/\u200b\u7ed3\u6784\u200b\u4f53\u200b\u9700\u8981\u200b\u5904\u7406\u200b\u591a\u79cd\u7c7b\u578b\u200b\u7684\u200b\u53c2\u6570\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6cdb\u578b\u200b\u3002\u200b\u6cdb\u578b\u200b\u7684\u200b\u7c7b\u578b\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u5728\u200b\u51fd\u6570\u200b\u540d\u200b\u540e\u9762\u200b\u7528\u200b\u5c16\u62ec\u53f7\u200b\u6307\u5b9a\u200b\uff0c\u200b\u4e4b\u540e\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\u4e2d\u200b\u7528\u200b\u5192\u53f7\u200b\u6307\u5b9a\u200b\u6cdb\u578b\u200b\u7c7b\u578b\u200b\u3002</p> <p>  Download source code</p> <pre><code>fn max&lt;T&gt; (a: T, b: T) -&gt; T\n    where T: std::cmp::PartialOrd\n{\n    if a &gt; b { a } else { b }\n}\n\nfn main() {\n    println!(\"{}\", max(1, 2)); // i32\n    println!(\"{}\", max(1.0, 2.0)); // f64\n    println!(\"{}\", max(\"a\", \"b\")); // &amp;str\n}\n</code></pre> <p>Rust\u200b\u6807\u51c6\u200b\u5e93\u4e2d\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u4e9b\u200b\u5e38\u7528\u200b\u7684\u200b\u6cdb\u578b\u200b\u7c7b\u578b\u200b\uff1a</p> <pre><code>enum Option&lt;T&gt; {\n    Some(T),\n    None,\n}\n\nenum Result&lt;T, E&gt; {\n    Ok(T),\n    Err(E),\n}\n</code></pre> <ul> <li><code>Option&lt;T&gt;</code>\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u53ef\u80fd\u200b\u5b58\u5728\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u662f\u200b<code>Some(T)</code>\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b<code>None</code>\u3002\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bPython\u200b\u4e2d\u200b\u7684\u200b<code>Optional</code>\u3002</li> <li><code>Result&lt;T, E&gt;</code>\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u53ef\u80fd\u200b\u51fa\u9519\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5982\u679c\u200b\u6210\u529f\u200b\u5219\u200b\u662f\u200b<code>Ok(T)</code>\uff0c\u200b\u5982\u679c\u200b\u5931\u8d25\u200b\u5219\u200b\u662f\u200b<code>Err(E)</code>\u3002</li> <li>\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b<code>match</code>\u200b\u8bed\u53e5\u200b\u4e2d\u200b\u8fdb\u884c\u200b\u6a21\u5f0f\u5339\u914d\u200b\u3002</li> </ul>"},{"location":"coding/rust/3/#impl-trait","title":"Impl &amp; Trait","text":"<p>\u200b\u5728\u200bRust\u200b\u4e2d\u200b\uff0c<code>impl</code>\u200b\u5173\u952e\u5b57\u200b\u7528\u4e8e\u200b\u4e3a\u200b\u7ed3\u6784\u200b\u4f53\u200b\u5b9e\u73b0\u200b\u65b9\u6cd5\u200b\uff0c<code>trait</code>\u200b\u5173\u952e\u5b57\u200b\u7528\u4e8e\u200b\u58f0\u660e\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u4e0e\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7ec4\u5408\u200b\u540e\u200b\u76f8\u5f53\u4e8e\u200bC++\u200b\u4e2d\u200b\u7684\u200b\u7c7b\u200b\u3002<code>impl</code>\u200b\u53ef\u4ee5\u200b\u8131\u79bb\u200b<code>trait</code>\u200b\u5355\u72ec\u200b\u4f7f\u7528\u200b\u3002</p> <ol> <li>\u200b\u5f53\u200b\u6ca1\u6709\u200b<code>trait</code>\u200b\u65f6\u200b\uff0c<code>impl</code>\u200b\u7684\u200b\u8bed\u6cd5\u200b\u4e3a\u200b<code>impl &lt;struct name&gt; { &lt;methods&gt;}</code>\uff0c\u200b\u5176\u4e2d\u200b<code>&lt;struct name&gt;</code>\u200b\u53ef\u4ee5\u200b\u662f\u200b\u7ed3\u6784\u200b\u4f53\u540d\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u6cdb\u578b\u200b\u53c2\u6570\u200b\u3002</li> <li>\u200b\u5f53\u6709\u200b<code>trait</code>\u200b\u65f6\u200b\uff0c<code>impl</code>\u200b\u7684\u200b\u8bed\u6cd5\u200b\u4e3a\u200b<code>impl &lt;trait name&gt; for &lt;struct name&gt; { &lt;methods&gt;}</code>\uff0c\u200b\u5176\u4e2d\u200b<code>&lt;trait name&gt;</code>\u200b\u662f\u200b\u5bf9\u5e94\u200b\u7684\u200b<code>trait</code>\u200b\u540d\u200b\uff0c<code>&lt;struct name&gt;</code>\u200b\u662f\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b\u8be5\u200b<code>trait</code>\u200b\u7684\u200b\u7ed3\u6784\u200b\u4f53\u540d\u200b\u3002</li> <li>\u200b\u7279\u6b8a\u200b\u5bf9\u8c61\u200b<code>self</code>\u200b\u8868\u793a\u200b\u5f53\u524d\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7684\u200b\u5b9e\u4f8b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4ee5\u200b\u5f15\u7528\u200b\uff08<code>&amp;self</code>\u3001<code>&amp;mut self</code>\uff09\u200b\u6216\u200b\u81ea\u8eab\u200b\u5f62\u6001\u200b\uff08<code>self</code>\uff09\u200b\u51fa\u73b0\u200b\u3002\u200b\u5982\u679c\u200b\u4e0d\u200b\u4f7f\u7528\u200b<code>self</code>\uff0c\u200b\u5219\u200b\u4e3a\u200b\u9759\u6001\u65b9\u6cd5\u200b/\u200b\u7c7b\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bPython\u200b\u4e2d\u200b\u7684\u200b<code>@staticmethod</code>\u200b\u6216\u200b<code>@classmethod</code>\u3002\u200b\u6b64\u65f6\u200b\u9700\u8981\u200b\u7528\u200b<code>&lt;struct&gt;::&lt;method&gt;</code>\u200b\u8c03\u7528\u200b\uff0c\u200b\u800c\u200b\u4e0d\u662f\u200b<code>&lt;object&gt;.&lt;method&gt;</code>\u3002</li> <li>\u200b\u7279\u6b8a\u200b\u7c7b\u578b\u200b<code>Self</code>\u200b\u8868\u793a\u200b\u5f53\u524d\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7684\u200b\u7c7b\u578b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u7684\u200b\u58f0\u660e\u200b\u3002</li> </ol> <p>\u200b\u5b64\u513f\u200b\u89c4\u5219\u200b</p> <p>\u200b\u5982\u679c\u200b\u9700\u8981\u200b\u5bf9\u200b\u7c7b\u578b\u200b<code>A</code>\u200b\u5b9e\u73b0\u200b\u7279\u5f81\u200b<code>T</code>\uff0c\u200b\u5219\u200b<code>T</code>\u200b\u6216\u200b<code>A</code>\u200b\u5176\u4e2d\u200b\u4e4b\u4e00\u200b\u5fc5\u987b\u200b\u5728\u200b\u5f53\u524d\u200b\u4f5c\u7528\u57df\u200b\u5185\u200b\u5b9a\u4e49\u200b\u3002</p> <pre><code>struct Player {\n    first_name: String,\n    last_name: String,\n}\n\ntrait GetFullName {\n    fn full_name(&amp;self) -&gt; String;\n}\n\nimpl GetFullName for Player {\n    fn full_name(&amp;self) -&gt; String {\n        format!(\"{} {}\", self.first_name, self.last_name)\n    }\n}\n\nfn main() {\n    let player_1 = Player {\n        first_name: \"Rafael\".to_string(),\n        last_name: \"Nadal\".to_string(),\n    };\n\n    println!(\"Player 01: {}\", player_1.full_name());\n}\n</code></pre> <p><code>trait</code>\u200b\u53ef\u4ee5\u200b\u548c\u200b\u6cdb\u578b\u200b\u7ed3\u5408\u200b\u4f7f\u7528\u200b\uff0c<code>impl</code>\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u5bf9\u200b\u5185\u7f6e\u200b\u7684\u200b\u7c7b\u578b\u200b\u4f7f\u7528\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <p>  Download source code</p> <pre><code>trait From&lt;T&gt; {\n    fn from_u8(x: T) -&gt; Self;\n}\nimpl From&lt;u8&gt; for u8 {\n    fn from_u8(x: u8) -&gt; Self {\n        x\n    }\n}\nimpl From&lt;u8&gt; for u16 {\n    fn from_u8(x: u8) -&gt; Self {\n        x as u16\n    }\n}\nimpl From&lt;u8&gt; for i8 {\n    fn from_u8(x: u8) -&gt; Self {\n        x as i8\n    }\n}\nimpl From&lt;u8&gt; for i16 {\n    fn from_u8(x: u8) -&gt; Self {\n        x as i16\n    }\n}\n\nfn main() {\n    println!(\"{}\", i16::from_u8(1 as u8));\n}\n</code></pre> <p><code>trait</code>\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u7ee7\u627f\u200b\u548c\u200b\u591a\u200b\u7ee7\u627f\u200b\u3002</p> <pre><code>trait Person {\n    fn full_name(&amp;self) -&gt; String;\n}\n\ntrait Employee : Person { // Employee inherits from Person\n    fn job_title(&amp;self) -&gt; String;\n}\n\ntrait ExpatEmployee : Employee + Expat { // ExpatEmployee inherits from Employee and Expat\n    fn additional_tax(&amp;self) -&gt; f64;\n}\n</code></pre> <p>\u200b\u5728\u200b<code>trait</code>\u200b\u7684\u200b\u58f0\u660e\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u7528\u200b\u51fd\u6570\u200b\u4f53\u200b\uff0c\u200b\u6b64\u65f6\u200b\u8868\u793a\u200b\u65b9\u6cd5\u200b\u7684\u200b\u9ed8\u8ba4\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u5728\u200b<code>impl</code>\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u91cd\u8f7d\u200b\u8fd9\u4e2a\u200b\u65b9\u6cd5\u200b\u3002</p> <pre><code>struct Blog {};\nstruct Tweet {};\n\npub trait Summary {\n    fn summarize(&amp;self) -&gt; String {\n        String::from(\"(Read more...)\")\n    }\n}\n\nimpl Summary for Blog {} // Use default implementation\nimpl Summary for Tweet {\n    fn summarize(&amp;self) -&gt; String { // Override default implementation\n        String::from(\"Tweet\")\n    }\n}\n</code></pre>"},{"location":"coding/rust/3/#_5","title":"\u7279\u5f81\u200b\u7ea6\u675f","text":"<p>\u200b\u7279\u5f81\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u7ea6\u675f\u200b\u51fd\u6570\u200b\u6216\u6cdb\u200b\u578b\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u5982\u200b</p> <ol> <li>\u200b\u51fd\u6570\u53c2\u6570\u200b\u7684\u200b\u9650\u5b9a\u200b\uff0c\u200b\u5982\u200b<code>fn foo(bar: impl Copy)</code>\u200b\u8868\u793a\u200b<code>bar</code>\u200b\u53c2\u6570\u200b\u5fc5\u987b\u200b\u5b9e\u73b0\u200b\u540d\u4e3a\u200b<code>Copy</code>\u200b\u7684\u200b<code>trait</code>\u3002</li> <li>\u200b\u6cdb\u578b\u200b\u53c2\u6570\u200b\u7684\u200b\u9650\u5b9a\u200b\uff0c\u200b\u5982\u200b<code>&lt;T: Copy&gt;</code>\u200b\u8868\u793a\u200b<code>T</code>\u200b\u5fc5\u987b\u200b\u5b9e\u73b0\u200b\u540d\u4e3a\u200b<code>Copy</code>\u200b\u7684\u200b<code>trait</code>\uff0c\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u66f4\u200b\u590d\u6742\u200b\u7684\u200b\u7c7b\u578b\u200b\u63a7\u5236\u200b\u3002</li> </ol> <p>\u200b\u4f7f\u7528\u200b<code>+</code>\u200b\u8fde\u63a5\u200b\u591a\u4e2a\u200b\u7279\u5f81\u200b\uff0c\u200b\u8868\u793a\u200b\u53c2\u6570\u200b\u5fc5\u987b\u200b\u540c\u65f6\u200b\u5b9e\u73b0\u200b\u591a\u4e2a\u200b\u7279\u5f81\u200b\u3002</p> <pre><code>fn foo&lt;T: Copy + Display&gt;(bar: T) {\n    println!(\"{}\", bar); // `bar` must implement `Display` and `Copy`\n}\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>where</code>\u200b\u5173\u952e\u5b57\u200b\u53ef\u4ee5\u200b\u4f7f\u200b\u7279\u5f81\u200b\u7ea6\u675f\u200b\u66f4\u52a0\u200b\u6e05\u6670\u200b\u3002</p> <pre><code>fn foo&lt;T, U&gt;(bar: T, baz: U) -&gt; U\n    where   T: Copy + Display,\n            U: Add&lt;Output = U&gt;\n{\n    println!(\"{}\", bar); // `bar` must implement `Display` and `Copy`\n    baz + 1 // `baz` must implement `Add`\n}\n</code></pre> <p>\u200b\u7279\u5f81\u200b\u7ea6\u675f\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u7528\u200b\u5728\u200b<code>impl</code>\u200b\u8bed\u53e5\u200b\u4e2d\u200b\u7528\u4e8e\u200b\u9009\u62e9\u6027\u200b\u5730\u200b\u5b9e\u73b0\u200b\u7279\u5f81\u200b</p> <pre><code>trait Increment {\n    fn increment(&amp;self) -&gt; Self;\n}\n\nimpl&lt;T&gt; Increment for T\n    where T: Add&lt;Output = T&gt; + Copy\n{\n    fn increment(&amp;self) -&gt; Self {\n        *self + 1\n    }\n}\n</code></pre> <p>\u200b\u8868\u793a\u200b<code>Increment</code>\u200b\u7279\u5f81\u200b\u53ea\u80fd\u200b\u7528\u4e8e\u200b\u540c\u65f6\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>Add&lt;Output = T&gt;</code>\u200b\u548c\u200b<code>Copy</code>\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002</p> <p>\u200b\u7279\u5f81\u200b\u7ea6\u675f\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u9650\u5b9a\u200b\uff0c\u200b\u53ea\u80fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u7279\u5f81\u200b\uff0c\u200b\u8868\u793a\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u5b9e\u73b0\u200b\u4e86\u200b\u7279\u5b9a\u200b\u7279\u5f81\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u4f46\u200b\u51fd\u6570\u200b\u4f53\u200b\u5185\u90e8\u200b\u4e0d\u80fd\u200b\u8fd4\u56de\u200b\u591a\u4e2a\u200b\u4e0d\u540c\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>fn foo(x: bool) -&gt; impl Display {\n    if x {\n        1\n    } else {\n        \"hello\" // Error, since 1 and \"hello\" have different types\n    }\n}\n</code></pre>"},{"location":"coding/rust/4/","title":"Lifetimes","text":"<p>\u200b\u5728\u200bRust\u200b\u4e2d\u200b\uff0c\u200b\u5bf9\u8c61\u200b\u751f\u547d\u5468\u671f\u200b\u548c\u200b\u6240\u6709\u6743\u200b\u662f\u200b\u7d27\u5bc6\u200b\u76f8\u5173\u200b\u7684\u200b\u3002\u200b\u6240\u6709\u6743\u200b\u89c4\u5219\u200b\u662f\u200bRust\u200b\u7684\u200b\u6838\u5fc3\u200b\u7279\u6027\u200b\uff0c\u200b\u4e5f\u200b\u662f\u200bRust\u200b\u4e0e\u200b\u5176\u4ed6\u200b\u8bed\u8a00\u200b\u6700\u5927\u200b\u7684\u200b\u4e0d\u540c\u4e4b\u5904\u200b\u3002</p>"},{"location":"coding/rust/4/#ownership","title":"\u6240\u6709\u6743\u200b\uff08Ownership\uff09","text":"<p>\u200b\u6bcf\u200b\u4e00\u5757\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u90fd\u200b\u548c\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u540d\u200b\u7ed1\u5b9a\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u53d8\u91cf\u540d\u200b\u5c31\u662f\u200b\u8fd9\u5757\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u7684\u200b\u6240\u6709\u8005\u200b\u3002\u200b\u5f53\u200b\u6240\u6709\u8005\u200b\u8d85\u51fa\u200b\u4f5c\u7528\u57df\u200b\u65f6\u200b\uff0c\u200b\u8fd9\u5757\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u5c31\u200b\u4f1a\u200b\u88ab\u200b\u91ca\u653e\u200b\u3002\u200b\u5728\u200b\u672c\u8282\u200b\u4e2d\u200b\uff0c\u201c\u200b\u53d8\u91cf\u200b\u201d\u200b\u548c\u200b\u201c\u200b\u7ed1\u5b9a\u200b\u201d\u200b\u90fd\u200b\u662f\u200b\u6307\u200b\u901a\u8fc7\u200b<code>let</code>\u200b\u5173\u952e\u5b57\u200b\u58f0\u660e\u200b\u7684\u200b\u53d8\u91cf\u200b\u3002</p> <p>\u200b\u5f53\u200b\u4f7f\u7528\u200b<code>=</code>\u200b\u5c06\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\u8d4b\u503c\u200b\u7ed9\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\uff0c\u200b\u6216\u662f\u200b\u5c06\u200b\u53d8\u91cf\u200b\u4f5c\u4e3a\u200b\u51fd\u6570\u200b\u53c2\u6570\u4f20\u9012\u200b\u65f6\u200b\uff0c\u200b\u6240\u6709\u6743\u200b\u4f1a\u200b\u53d1\u751f\u200b\u8f6c\u79fb\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b\u4e24\u79cd\u200b\u53ef\u80fd\u200b\uff1a</p> <ul> <li>\u200b\u503c\u200b\u62f7\u8d1d\u200b\uff1a\u200b\u539f\u6765\u200b\u7684\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u4f1a\u200b\u88ab\u200b\u590d\u5236\u200b\u4e00\u4efd\u200b\u7ed9\u200b\u65b0\u200b\u7684\u200b\u53d8\u91cf\u200b\uff0c\u200b\u4e24\u4e2a\u200b\u53d8\u91cf\u540d\u200b\u7ed1\u5b9a\u200b\u7684\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u4e0d\u540c\u200b\u3002</li> <li>\u200b\u503c\u200b\u79fb\u52a8\u200b\uff1a\u200b\u539f\u6765\u200b\u7684\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u4f1a\u200b\u88ab\u200b\u91cd\u65b0\u200b\u7ed1\u5b9a\u200b\u7ed9\u200b\u65b0\u200b\u7684\u200b\u53d8\u91cf\u200b\uff0c\u200b\u6b64\u65f6\u200b\u4e0d\u80fd\u200b\u518d\u200b\u901a\u8fc7\u200b\u539f\u6765\u200b\u7684\u200b\u53d8\u91cf\u540d\u200b\u8bbf\u95ee\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u3002</li> </ul> <p>\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0cRust\u200b\u4f1a\u200b\u91c7\u53d6\u200b\u503c\u200b\u79fb\u52a8\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u8d4b\u503c\u200b\uff0c\u200b\u9664\u975e\u200b\u5bf9\u8c61\u200b\u7684\u200b\u7c7b\u578b\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>Copy</code> trait\u3002\u200b\u5927\u591a\u6570\u200b\u57fa\u672c\u200b\u7c7b\u578b\u200b\u90fd\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>Copy</code> trait\u3002</p> <pre><code>fn main() {\n    let a = vec![1, 2, 3]; // vec is not a primitive type, and it does not implement Copy trait\n    let b = a;\n    println!(\"{:?} {:?}\", a, b); // Error; use of moved value: `a`\n}\n</code></pre>"},{"location":"coding/rust/4/#borrowing","title":"\u5f15\u7528\u200b\uff08Borrowing\uff09","text":"<p>\u200b\u5728\u200bRust\u200b\u4e2d\u200b\uff0c\u200b\u53d8\u91cf\u200b\u7684\u200b\u5f15\u7528\u200b\u6709\u200b\u4e24\u79cd\u200b\uff0c\u200b\u5373\u200b\u53ea\u8bfb\u200b\u5f15\u7528\u200b\uff08<code>&amp;T</code>\uff09\u200b\u548c\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\uff08<code>&amp;mut T</code>\uff09\u3002\u200b\u53ea\u8bfb\u200b\u5f15\u7528\u200b\u548c\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u90fd\u200b\u662f\u200b\u6307\u5411\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u7684\u200b\u6307\u9488\u200b\uff0c\u200b\u4f46\u200b\u53ea\u8bfb\u200b\u5f15\u7528\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u4fee\u6539\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u800c\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u5141\u8bb8\u200b\u4fee\u6539\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u7684\u200b\u503c\u200b\u3002\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u53ea\u80fd\u200b\u6307\u5411\u200b\u53ef\u53d8\u200b\u53d8\u91cf\u200b\u3002\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u89e3\u200b\u5f15\u7528\u200b\u8fd0\u7b97\u7b26\u200b<code>*</code>\u200b\u6765\u200b\u8bbf\u95ee\u200b\u5f15\u7528\u200b\u6307\u5411\u200b\u7684\u200b\u503c\u200b\u3002</p> <pre><code>fn main() {\n    let const_var = 5;\n    let mutable_ref = &amp;mut const_var;\n    // error[E0596]: cannot borrow `const_var` as mutable, as it is not declared as mutable\n}\n</code></pre> <p>\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u662f\u4ece\u200b\u5f15\u7528\u200b\u58f0\u660e\u200b\u5f00\u59cb\u200b\u5230\u200b\u5f15\u7528\u200b\u6700\u540e\u200b\u4e00\u6b21\u200b\u4f7f\u7528\u200b\u7ed3\u675f\u200b\u3002\u200b\u5173\u4e8e\u200b\u5f15\u7528\u200b\u7684\u200b\u4f7f\u7528\u200b\u6709\u200b\u5982\u4e0b\u200b\u9650\u5236\u200b\uff1a</p> <ol> <li> <p>\u200b\u5728\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u5185\u200b\uff0c\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u548c\u200b\u53ea\u8bfb\u200b\u5f15\u7528\u200b\u4e0d\u80fd\u200b\u540c\u65f6\u200b\u5b58\u5728\u200b\u3002</p> \u200b\u9519\u8bef\u200b\u6b63\u786e\u200b <pre><code>fn main() {\n    let mut owner = 5;\n    let shared_borrow = &amp;owner; // share_borrow starts here\n    let mutable_borrow = &amp;mut owner; // mutable_borrow starts here\n    println!(\"{}\", mutable_borrow); // Error\n    // mutable_borrow ends here\n    println!(\"{}\", shared_borrow);\n    // share_borrow ends here\n}\n</code></pre> <pre><code>fn main() {\n    let mut owner = 5;\n    let shared_borrow = &amp;owner; // shared_borrow starts here\n    println!(\"{}\", shared_borrow);\n    // shared_borrow ends here\n    // mutable_borrow starts here\n    let mutable_borrow = &amp;mut owner; // mutable_borrow starts here\n    println!(\"{}\", mutable_borrow);\n    // mutable_borrow ends here\n}\n</code></pre> </li> <li> <p>\u200b\u5728\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u5185\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u5bf9\u200b\u540c\u4e00\u200b\u53d8\u91cf\u200b\u58f0\u660e\u200b\u65b0\u200b\u7684\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u3002</p> \u200b\u9519\u8bef\u200b\u6b63\u786e\u200b <pre><code>fn main() {\n    let mut owner = 5;\n    let mutable_borrow = &amp;mut owner;\n    let another_mutable_borrow = &amp;mut owner;\n    println!(\"{} {}\", mutable_borrow, another_mutable_borrow);\n    // error[E0499]: cannot borrow `owner` as mutable more than once at a time\n}\n</code></pre> <pre><code>fn main() {\n    let mut owner = 5;\n    let mutable_borrow = &amp;mut owner;\n    println!(\"{}\", mutable_borrow);\n    let another_mutable_borrow = &amp;mut owner;\n    println!(\"{}\", another_mutable_borrow);\n}\n</code></pre> </li> <li> <p>\u200b\u5728\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u5185\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u8bbf\u95ee\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u6216\u200b\u5bf9\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u8fdb\u884c\u200b\u8d4b\u503c\u200b\u3002</p> \u200b\u9519\u8bef\u200b\u6b63\u786e\u200b <pre><code>fn main() {\n    let mut owner = 5;\n    let mutable_borrow = &amp;mut owner;\n    println!(\"{}\", owner + *mutable_borrow);\n    // error[E0503]: cannot use `owner` because it was mutably borrowed\n}\n</code></pre> <pre><code>fn main() {\n    let mut owner = 5;\n    let mutable_borrow = &amp;mut owner;\n    println!(\"{}\", *mutable_borrow + owner); // 10\n}\n</code></pre> </li> <li> <p>\u200b\u53ef\u4ee5\u200b\u7528\u200b\u6570\u636e\u5e93\u200b\u4e2d\u200b\u201c\u200b\u5171\u4eab\u200b\u9501\u200b\u201d\u200b\u548c\u200b\u201c\u200b\u6392\u200b\u4ed6\u200b\u9501\u200b\u201d\u200b\u7684\u200b\u6982\u5ff5\u200b\u6765\u200b\u7406\u89e3\u200b\u53ea\u8bfb\u200b\u5f15\u7528\u200b\u548c\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u3002\u200b\u53ea\u8bfb\u200b\u5f15\u7528\u200b\u76f8\u5f53\u4e8e\u200b\u5bf9\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u52a0\u200b\u5171\u4eab\u200b\u9501\u200b\uff0c\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u76f8\u5f53\u4e8e\u200b\u5bf9\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u52a0\u6392\u200b\u4ed6\u9501\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u540c\u200b\u4e00\u5757\u200b\u6570\u636e\u200b\uff0c\u200b\u5171\u4eab\u200b\u9501\u200b\u53ef\u4ee5\u200b\u540c\u65f6\u200b\u5b58\u5728\u200b\uff0c\u200b\u4f46\u6392\u200b\u4ed6\u200b\u9501\u200b\u53ea\u80fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u3002</p> </li> <li> <p>\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u4e0d\u80fd\u200b\u8d85\u8fc7\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u3002</p> E0597: ... does not live long enoughE0515: cannot return value referencing local variable ... <pre><code>struct Foo&lt;'a&gt; {\n    x: Option&lt;&amp;'a u32&gt;,\n}\n\nfn main() {\n    let mut x = Foo { x: None };\n    {\n        let y = 0;\n        x.x = Some(&amp;y);\n    } // Leaving the scope of y, y is dropped\n    println!(\"{:?}\", x.x) // error[E0597]: `y` does not live long enough\n}\n</code></pre> <pre><code>fn one(x: &amp;i32) -&gt; &amp;i32 {\n    let ret = *x + 1;\n    &amp;ret // error[E0515]: cannot return value referencing local variable `ret`\n} // Leaving the scope of ret, ret is dropped\n\nfn main() {\n    println!(\"{}\", *one(&amp;5));\n}\n</code></pre> </li> </ol>"},{"location":"coding/rust/4/#_1","title":"\u751f\u547d\u5468\u671f","text":"<p>\u200b\u5f15\u7528\u200b\u76f8\u5f53\u4e8e\u200b\u6307\u5411\u200b\u4e00\u6bb5\u200b\u5185\u5b58\u7a7a\u95f4\u200b\u7684\u200b\u6307\u9488\u200b\uff0c\u200b\u5728\u200b\u6700\u540e\u200b\u4e00\u6b21\u200b\u8bbf\u95ee\u200b\u5f15\u7528\u200b\u540e\u200b\uff0c\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u5c31\u200b\u7ed3\u675f\u200b\u4e86\u200b\uff0c\u200b\u4f46\u200b\u5728\u200b\u4e00\u4e9b\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u9700\u8981\u200b\u7ee7\u7eed\u200b\u4fdd\u6301\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\uff0c\u200b\u6bd4\u5982\u200b\u5728\u200b\u51fd\u6570\u200b\u4e2d\u200b\u8fd4\u56de\u200b\u5f15\u7528\u200b\u3002\u200b\u5728\u200b\u8fd9\u79cd\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u6765\u200b\u6307\u5b9a\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u3002</p> <pre><code>fn main() {\n    let reference_to_nothing = dangle();\n    // error[E0106]: missing lifetime specifier\n}\n\nfn dangle() -&gt; &amp;String {\n    let s = String::from(\"hello\");\n\n    &amp;s // The lifetime of s is over, so this reference is dangling\n}\n</code></pre> <p>\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u6cdb\u578b\u200b\u53c2\u6570\u200b\u3001\u200b\u51fd\u6570\u53c2\u6570\u200b\u3001\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u800c\u200b\u4e0d\u80fd\u200b\u7528\u4e8e\u200b\u5f15\u7528\u200b\u58f0\u660e\u200b\u3002\u200b\u6709\u200b\u76f8\u540c\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u7684\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u76f8\u540c\u200b\u3002<code>'static</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u7279\u6b8a\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\uff0c\u200b\u8868\u793a\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u662f\u200b\u6574\u4e2a\u200b\u7a0b\u5e8f\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\uff0c\u200b\u5373\u200b\u5f15\u7528\u200b\u6307\u5411\u200b\u7684\u200b\u5185\u5b58\u7a7a\u95f4\u200b\u5728\u200b\u7a0b\u5e8f\u200b\u7ed3\u675f\u200b\u65f6\u624d\u200b\u4f1a\u200b\u88ab\u200b\u91ca\u653e\u200b\u3002\u200b\u6240\u6709\u200b\u6307\u5411\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\u7684\u200b\u5f15\u7528\u200b\u90fd\u200b\u662f\u200b<code>'static</code>\u200b\u7c7b\u578b\u200b\u3002</p> \u200b\u51fd\u6570\u200b\u590d\u6742\u200b\u5bf9\u8c61\u200bImpl &amp; Trait\u200b\u6cdb\u578b\u200b <pre><code>fn function&lt;'a&gt;() -&gt; &amp;'a str {}\nfn function&lt;'a&gt;(x: &amp;'a str) {}\nfn function&lt;'a&gt;(x: &amp;'a str) -&gt; &amp;'a str {}\nfn function&lt;'a&gt;(x: &amp;'a str, y: &amp;'a str) -&gt; &amp;'a str {}\nfn function&lt;'a, 'b&gt;(x: &amp;'a str, y: &amp;'b str) -&gt; &amp;'a str {}\n</code></pre> <pre><code>enum Enum&lt;'a&gt; {\n    Variant1(&amp;'a str),\n    Variant2(&amp;'a str) // exist as long as parent object exists\n}\nstruct Struct&lt;'a&gt; {\n    field: &amp;'a str // exist as long as parent object exists\n}\n</code></pre> <pre><code>struct Struct&lt;'a&gt; {\n    field: &amp;'a str\n}\ntrait New&lt;'a&gt; {\n    fn new() -&gt; Self;\n}\nimpl&lt;'a&gt; New&lt;'a&gt; for Struct&lt;'a&gt; {\n    fn new() -&gt; Self {\n        Struct { field: \"hello\" }\n    }\n}\n</code></pre> <pre><code>fn function&lt;T&gt;(x: T) where for&lt;'a&gt; T: Fn(&amp;'a str) -&gt; &amp;'a str {}\nenum Enum&lt;T&gt; where for&lt;'a&gt; T: Fn(&amp;'a str) -&gt; &amp;'a str { V(T) }\nstruct Struct&lt;T&gt; where for&lt;'a&gt; T: Fn(&amp;'a str) -&gt; &amp;'a str { field: T }\nimpl&lt;T&gt; Struct&lt;T&gt; where for&lt;'a&gt; T: Fn(&amp;'a str) -&gt; &amp;'a str {\n    fn function(&amp;self) -&gt; &amp;F { &amp;self.field }\n}\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff0c<code>where for</code>\u200b\u6709\u200b\u53e6\u200b\u4e00\u79cd\u200b\u5199\u6cd5\u200b</p> <pre><code>fn function&lt;T&gt;(x: T) where T: for&lt;'a&gt; Fn(&amp;'a str) -&gt; &amp;'a str {}\n</code></pre> <p><code>for</code>\u200b\u5bf9\u200b<code>T</code>\u200b\u8fdb\u884c\u200b\u4e86\u200b\u9650\u5236\u200b\uff0c\u200b\u8868\u793a\u200b<code>T</code>\u200b\u5fc5\u987b\u200b\u662f\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u4e14\u200b\u8fd9\u4e2a\u200b\u51fd\u6570\u200b\u7684\u200b\u53c2\u6570\u200b\u548c\u200b\u751f\u547d\u5468\u671f\u200b\u5fc5\u987b\u200b\u548c\u200b\u6a21\u677f\u200b\u5339\u914d\u200b\u3002</p> <pre><code>struct Closure&lt;F&gt; {\n    data: (u8, u16),\n    func: F,\n}\n\nimpl&lt;F&gt; Closure&lt;F&gt;\n    where for&lt;'a&gt; F: Fn(&amp;'a (u8, u16)) -&gt; &amp;'a u8,\n{\n    fn call(&amp;self) -&gt; &amp;u8 {\n        (self.func)(&amp;self.data)\n    }\n}\n\nfn do_it(data: &amp;(u8, u16)) -&gt; &amp;u8 { &amp;data.0 }\n\nfn main() {\n    let clo = Closure { data: (0, 1), func: do_it };\n    println!(\"{}\", clo.call());\n}\n</code></pre> <p>\u200b\u4e0d\u540c\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u7684\u200b\u5f15\u7528\u200b\u4e0d\u80fd\u200b\u76f8\u4e92\u200b\u8d4b\u503c\u200b\uff0c\u200b\u5728\u200b\u4f5c\u4e3a\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\u65f6\u200b\uff0cRust\u200b\u4e5f\u200b\u4f1a\u200b\u68c0\u67e5\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u662f\u5426\u200b\u5339\u914d\u200b\u3002\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>'a: 'b</code>\u200b\u6765\u200b\u7ea6\u675f\u200b<code>'a</code>\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u4e0d\u77ed\u4e8e\u200b<code>'b</code>\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u3002</p> \u200b\u9519\u8bef\u200b\u6b63\u786e\u200b\u6b63\u786e\u200b <pre><code>fn two_args&lt;'a, 'b&gt;(a: &amp;'a mut i32, b: &amp;'b i32) -&gt; &amp;'b mut i32 {\n    *a += b;\n    a // error: lifetime may not live long enough\n}\nfn main() {\n    let mut a = 3;\n    let b: &amp;mut i32 = two_args(&amp;mut a, &amp;4);\n    println!(\"{}\", b);\n}\n</code></pre> <pre><code>fn two_args&lt;'a, 'b&gt;(a: &amp;'a mut i32, b: &amp;'b i32) -&gt; &amp;'a mut i32 {\n    *a += b;\n    a\n}\nfn main() {\n    let mut a = 3;\n    let b: &amp;mut i32 = two_args(&amp;mut a, &amp;4);\n    println!(\"{}\", b);\n}\n</code></pre> <pre><code>fn two_args&lt;'a: 'b, 'b&gt;(a: &amp;'a mut i32, b: &amp;'b i32) -&gt; &amp;'b mut i32 {\n    *a += b;\n    a\n}\nfn main() {\n    let mut a = 3;\n    let b: &amp;mut i32 = two_args(&amp;mut a, &amp;4);\n    println!(\"{}\", b);\n}\n</code></pre> <p>\u200b\u901a\u5e38\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u51fd\u6570\u200b\u58f0\u660e\u200b\u4e2d\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\uff0cRust\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u63a8\u65ad\u200b\u6bcf\u4e2a\u200b\u53c2\u6570\u200b\u548c\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b</p> <ol> <li> <p>\u200b\u6bcf\u4e2a\u200b\u8f93\u51fa\u200b\u53c2\u6570\u200b\u90fd\u200b\u4f1a\u200b\u5206\u914d\u200b\u4e00\u4e2a\u200b\u72ec\u7acb\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u3002</p> \u200b\u7701\u7565\u200b\u6807\u8bb0\u200b\u663e\u793a\u200b\u6807\u8bb0\u200b <pre><code>fn four_args(a: &amp;i32, b: &amp;i32, c: &amp;i32, d: &amp;i32) -&gt; i32 {\n    a + b + c + d\n}\n</code></pre> <pre><code>fn four_args&lt;'a, 'b, 'c, 'd&gt;(a: &amp;'a i32, b: &amp;'b i32, c: &amp;'c i32, d: &amp;'d i32) -&gt; i32 {\n    a + b + c + d\n}\n</code></pre> </li> <li> <p>\u200b\u5982\u679c\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u8f93\u5165\u200b\u53c2\u6570\u200b\uff0c\u200b\u90a3\u4e48\u200b\u8fd9\u4e2a\u200b\u53c2\u6570\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u5c31\u662f\u200b\uff08\u200b\u6240\u6709\u200b\uff09\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u3002</p> \u200b\u7701\u7565\u200b\u6807\u8bb0\u200b\u663e\u793a\u200b\u6807\u8bb0\u200b <pre><code>fn one_arg(a: &amp;i32) -&gt; &amp;i32 {\n    a\n}\n</code></pre> <pre><code>fn one_arg&lt;'a&gt;(a: &amp;'a i32) -&gt; &amp;'a i32 {\n    a\n}\n</code></pre> </li> <li> <p>\u200b\u5f53\u200b\u51fd\u6570\u200b\u7684\u200b\u67d0\u4e2a\u200b\u53c2\u6570\u200b\u4e3a\u200b<code>&amp;self</code>\u200b\u6216\u200b<code>&amp;mut self</code>\u200b\u65f6\u200b\uff0c\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u548c\u200b<code>&amp;self</code>\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u76f8\u540c\u200b\u3002</p> \u200b\u7701\u7565\u200b\u6807\u8bb0\u200b\u663e\u793a\u200b\u6807\u8bb0\u200b <pre><code>struct Player {\n    first_name: String,\n    last_name: String,\n}\n\nimpl Player {\n    fn full_name(&amp;self) -&gt; String {\n        format!(\"{} {}\", self.first_name, self.last_name)\n    }\n}\n</code></pre> <pre><code>struct Player {\n    first_name: String,\n    last_name: String,\n}\n\nimpl Player {\n    fn full_name&lt;'a&gt;(&amp;'a self) -&gt; String {\n        format!(\"{} {}\", self.first_name, self.last_name)\n    }\n}\n</code></pre> </li> <li> <p>\u200b\u5982\u679c\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u4e0a\u8ff0\u200b\u6761\u4ef6\u200b\uff0c\u200b\u5219\u200b\u9700\u8981\u200b\u663e\u5f0f\u200b\u6307\u5b9a\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u3002</p> \u200b\u9519\u8bef\u200b\u6b63\u786e\u200b <pre><code>fn two_args(a: &amp;mut i32, b: &amp;i32) -&gt; &amp;mut i32 {\n    *a += b;\n    a // error[E0106]: missing lifetime specifier\n}\nfn main() {\n    let mut a = 3;\n    let b: &amp;mut i32 = two_args(&amp;mut a, &amp;4);\n    println!(\"{}\", b);\n}\n</code></pre> <pre><code>fn two_args&lt;'a, 'b&gt;(a: &amp;'a mut i32, b: &amp;'b i32) -&gt; &amp;'a mut i32 {\n    *a += b;\n    a\n}\nfn main() {\n    let mut a = 3;\n    let b: &amp;mut i32 = two_args(&amp;mut a, &amp;4);\n    println!(\"{}\", b);\n}\n</code></pre> </li> </ol>"},{"location":"coding/rust/5/","title":"Error Handling","text":"<p>\u200b\u672c\u200b\u8282\u200b\u4ecb\u7ecd\u200bRust\u200b\u7684\u200b\u5f02\u5e38\u200b\u5904\u7406\u200b\u673a\u5236\u200b\u3002</p>"},{"location":"coding/rust/5/#_1","title":"\u7f16\u8bd1\u5668\u200b\u9519\u8bef","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u90e8\u5206\u200b\u9519\u8bef\u200b\u53ef\u4ee5\u200b\u5728\u200b\u7f16\u8bd1\u200b\u65f6\u200b\u88ab\u200b\u68c0\u6d4b\u200b\u51fa\u6765\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u9519\u8bef\u200b\u88ab\u200b\u79f0\u4e3a\u200b\u7f16\u8bd1\u200b\u65f6\u200b\u9519\u8bef\u200b\uff08compile-time error\uff09\u3002\u200b\u4f7f\u7528\u200b<code>rustc --explain</code>\u200b\u53ef\u4ee5\u200b\u67e5\u770b\u200b\u7f16\u8bd1\u200b\u65f6\u200b\u9519\u8bef\u200b\u7684\u200b\u8be6\u7ec6\u4fe1\u606f\u200b\u3002</p> <pre><code>rustc --explain E0382\n# A variable was used after its contents were moved elsewhere.\n#\n# Erroneous code example:\n#\n# struct MyStruct { s: u32 }\n#\n# fn main() {\n#    let mut x = MyStruct { s: 5u32 };\n#    let y = x;\n#    x.s = 6;\n#    println!(\"{}\", x.s);\n# }\n# ...\n</code></pre>"},{"location":"coding/rust/5/#panic","title":"\u5f02\u5e38\u200b\uff08Panic\uff09","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u5f02\u5e38\u200b\u88ab\u200b\u79f0\u4e3a\u200bpanic\u3002\u200b\u5f53\u200b\u7a0b\u5e8f\u200b\u53d1\u751f\u200bpanic\u200b\u65f6\u200b\uff0cRust\u200b\u4f1a\u200b\u6253\u5370\u200b\u51fa\u200b\u5f02\u5e38\u200b\u4fe1\u606f\u200b\u5e76\u200b\u9000\u51fa\u200b\u7a0b\u5e8f\u200b\u3002\u200b\u6709\u200b\u591a\u4e2a\u200b\u5b8f\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\u3002</p> <pre><code>// \u200b\u4ee5\u4e0b\u200b\u7684\u200b\u5b8f\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u63a5\u53d7\u200b\u7c7b\u4f3c\u200bprintln!\u200b\u7684\u200b\u683c\u5f0f\u5316\u200b\u5b57\u7b26\u4e32\u200b\npanic!(\"This is a panic!\");\nunimplemented!(); // \u200b\u7528\u4e8e\u200b\u6807\u8bb0\u200b\u672a\u200b\u5b9e\u73b0\u200b\u7684\u200b\u4ee3\u7801\u200b\nunreachable!(); // \u200b\u7528\u4e8e\u200b\u6807\u8bb0\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u5230\u8fbe\u200b\u7684\u200b\u5206\u652f\u200b\n\n// \u200b\u4ee5\u4e0b\u200b\u5b8f\u200b\u7528\u4e8e\u200b\u6d4b\u8bd5\u200b\nassert!(false); // \u200b\u7528\u4e8e\u200b\u6d4b\u8bd5\u200b\u8868\u8fbe\u5f0f\u200b\u662f\u5426\u200b\u4e3a\u200btrue\nassert_eq!(1, 2); // \u200b\u7528\u4e8e\u200b\u6d4b\u8bd5\u200b\u4e24\u4e2a\u200b\u8868\u8fbe\u5f0f\u200b\u7684\u200b\u503c\u200b\u662f\u5426\u200b\u76f8\u7b49\u200b\nassert_ne!(1, 1); // \u200b\u7528\u4e8e\u200b\u6d4b\u8bd5\u200b\u4e24\u4e2a\u200b\u8868\u8fbe\u5f0f\u200b\u7684\u200b\u503c\u200b\u662f\u5426\u200b\u4e0d\u200b\u76f8\u7b49\u200b\n\n// \u200b\u5728\u200bDebug\u200b\u6a21\u5f0f\u200b\u6216\u200b\u4f7f\u7528\u200b-C debug-assertions\u200b\u53c2\u6570\u200b\u65f6\u200b\uff0c\u200b\u4ee5\u4e0b\u200b\u5b8f\u624d\u200b\u4f1a\u200b\u89e6\u53d1\u200b\ndebug_assert!(false);\ndebug_assert_eq!(1, 2);\ndebug_assert_ne!(1, 1);\n</code></pre>"},{"location":"coding/rust/5/#_2","title":"\u5f02\u5e38\u200b\u5904\u7406","text":"<p>Rust\u200b\u4e2d\u200b\u5185\u7f6e\u200b\u4e86\u200b\u4e24\u4e2a\u200b\u7c7b\u578b\u200b<code>Option&lt;T&gt;</code>\u200b\u548c\u200b<code>Result&lt;T, E&gt;</code>\u200b\u7528\u4e8e\u200b\u5f02\u5e38\u200b\u5904\u7406\u200b\uff0c\u200b\u5176\u200b\u5b9a\u4e49\u200b\u4e3a\u200b</p> <pre><code>enum Option&lt;T&gt; {\n    None,\n    Some(T),\n}\nenum Result&lt;T, E&gt; {\n    Ok(T),\n    Err(E),\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\uff0c\u200b\u5747\u200b\u4ee5\u200b<code>T</code>\u200b\u4ee3\u8868\u200b<code>Option::Some</code>\u200b\u6216\u200b<code>Result::Ok</code>\u200b\u7684\u200b\u7c7b\u578b\u200b\uff0c<code>E</code>\u200b\u4ee3\u8868\u200b<code>Result::Err</code>\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002</p> <ul> <li> <p><code>Option</code>\u200b\u7c7b\u578b\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u53ef\u80fd\u200b\u4e0d\u200b\u5b58\u5728\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u5e38\u7528\u200b\u4e8e\u200b\u51fd\u6570\u53c2\u6570\u200b\u3001\u200b\u7ed3\u6784\u200b\u4f53\u5b57\u200b\u6bb5\u200b\u6216\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>match</code>\u200b\u8bed\u53e5\u200b\u53ef\u4ee5\u200b\u5339\u914d\u200b<code>Option</code>\u200b\u7c7b\u578b\u200b\u53d6\u503c\u200b\u7684\u200b\u4e0d\u540c\u200b\u60c5\u51b5\u200b\u3002</p> <pre><code>fn main() {\n    let x: Option&lt;i32&gt; = Some(5);\n    match x {\n        Some(i) =&gt; println!(\"{}\", i),\n        None =&gt; println!(\"None\"),\n    }\n}\n</code></pre> <ol> <li>\u200b\u4f7f\u7528\u200b<code>is_some</code>\u200b\u548c\u200b<code>is_none</code>\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u5feb\u901f\u200b\u5224\u65ad\u200b<code>Option</code>\u200b\u7c7b\u578b\u200b\u662f\u5426\u200b\u4e3a\u200b<code>Some</code>\u200b\u6216\u200b<code>None</code>\u3002</li> <li>\u200b\u4f7f\u7528\u200b<code>unwrap</code>\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u5c06\u200b<code>Option</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b<code>T</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5982\u679c\u200b<code>Option</code>\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>None</code>\uff0c\u200b\u5219\u200b\u4f1a\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\u3002</li> </ol> </li> <li> <p><code>Result</code>\u200b\u7c7b\u578b\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u53ef\u80fd\u200b\u51fa\u9519\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u5e38\u7528\u200b\u4e8e\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\u3002<code>Ok</code>\u200b\u8868\u793a\u200b\u51fd\u6570\u200b\u6267\u884c\u200b\u6210\u529f\u200b\uff0c<code>Err</code>\u200b\u8868\u793a\u200b\u51fd\u6570\u200b\u6267\u884c\u200b\u5931\u8d25\u200b\u3002</p> <pre><code>fn main() {\n    let x: Result&lt;i32, &amp;str&gt; = Ok(5);\n    match x {\n        Ok(i) =&gt; println!(\"{}\", i),\n        Err(e) =&gt; println!(\"{}\", e),\n    }\n}\n</code></pre> <ol> <li>\u200b\u4f7f\u7528\u200b<code>is_ok</code>\u200b\u548c\u200b<code>is_err</code>\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u5feb\u901f\u200b\u5224\u65ad\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u662f\u5426\u200b\u4e3a\u200b<code>Ok</code>\u200b\u6216\u200b<code>Err</code>\u3002</li> <li> <p>\u200b\u4f7f\u7528\u200b<code>ok</code>\u200b\u548c\u200b<code>err</code>\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u5c06\u200b<code>Result&lt;T, E&gt;</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b<code>Option&lt;T&gt;</code>\u200b\u7c7b\u578b\u200b\u3002\u200b\u5982\u679c\u200b\u8c03\u7528\u200b\u7684\u200b\u65b9\u6cd5\u200b\u548c\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u53d6\u503c\u200b\u76f8\u540c\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>Some(T)</code>\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>None</code>\u3002</p> <pre><code>fn main() {\n    let x: Result&lt;i32, &amp;str&gt; = Ok(5);\n    let y: Option&lt;i32&gt; = x.ok();\n    println!(\"{:?}\", y); // Some(5)\n    let x: Result&lt;i32, &amp;str&gt; = Err(\"error\");\n    let y: Option&lt;i32&gt; = x.ok();\n    println!(\"{:?}\", y); // None\n}\n</code></pre> </li> <li> <p>\u200b\u4f7f\u7528\u200b<code>unwrap</code>\u200b\u65b9\u6cd5\u200b\u6216\u200b<code>expect</code>\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u5c06\u200b<code>Result</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b<code>T</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5982\u679c\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>Err</code>\uff0c\u200b\u5219\u200b\u4f1a\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\uff0c\u200b\u8f93\u51fa\u200b\u7684\u200b\u5f02\u5e38\u200b\u4fe1\u606f\u200b\u5bf9\u5e94\u200b\u4e8e\u200b<code>Err</code>\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u6216\u200b<code>expect</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u53c2\u6570\u200b\u3002</p> <pre><code>fn main() {\n    let x: Result&lt;i32, &amp;str&gt; = Ok(5);\n    let y: i32 = x.unwrap();\n    println!(\"{}\", y); // 5\n    let x: Result&lt;i32, &amp;str&gt; = Err(\"error\");\n    let y: i32 = x.unwrap(); // thread 'main' panicked at 'called `Result::unwrap()` on an `Err` value: \"error\"', src/main.rs:2:26\n    println!(\"{}\", y);\n}\n</code></pre> </li> </ol> </li> </ul> <p><code>unwrap</code>\u200b\u65b9\u6cd5\u200b\u6709\u200b\u5982\u4e0b\u200b\u53d8\u79cd\u200b\uff1a</p> <ul> <li><code>expect</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\uff0c\u200b\u7528\u4e8e\u200b\u6307\u5b9a\u200b\u8df3\u51fa\u200b\u5f02\u5e38\u200b\u65f6\u200b\u7684\u200b\u63d0\u793a\u4fe1\u606f\u200b\u3002</li> <li>\uff08\u200b\u4ec5\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\uff09<code>unwrap_err</code>\u200b\u65b9\u6cd5\u200b\u548c\u200b<code>expect_err</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u4e0e\u200b<code>unwrap</code>\u200b\u76f8\u53cd\u200b\uff0c\u200b\u5f53\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>Ok</code>\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>Err</code>\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u3002</li> <li><code>unwrap_or</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b<code>T</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5728\u200b<code>unwrap</code>\u200b\u65b9\u6cd5\u200b\u4f1a\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u8fd4\u56de\u200b\u8be5\u200b\u53c2\u6570\u200b\u3002</li> <li> <p><code>unwrap_or_default</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5728\u200b<code>unwrap</code>\u200b\u65b9\u6cd5\u200b\u4f1a\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u8fd4\u56de\u200b\u8be5\u200b\u9ed8\u8ba4\u503c\u200b\u3002\u200b\u8be5\u200b\u65b9\u6cd5\u200b\u8981\u6c42\u200b<code>T</code>\u200b\u7c7b\u578b\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>Default</code>\u200b\u7279\u6027\u200b\u3002</p> <pre><code>struct MyStruct {\n    s: i32\n}\n\nfn main() {\n    let a: Option&lt;MyStruct&gt; = None;\n    println!(\"{}\", a.unwrap_or_default().s);\n    // error[E0277]: the trait bound `MyStruct: Default` is not satisfied\n}\n</code></pre> </li> <li> <p><code>unwrap_or_else</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\uff0c\u200b\u5728\u200b<code>unwrap</code>\u200b\u65b9\u6cd5\u200b\u4f1a\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u8fd4\u56de\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002\u200b\u5176\u4e2d\u200b\uff0c<code>Result</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u95ed\u5305\u200b\u53c2\u6570\u200b\u4e3a\u200b<code>E</code>\u200b\u7c7b\u578b\u200b\uff0c<code>Option</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u95ed\u5305\u200b\u53c2\u6570\u200b\u4e3a\u200b<code>()</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u9700\u8981\u200b\u548c\u200b<code>Some</code>\u200b\u6216\u200b<code>Ok</code>\u200b\u7684\u200b\u7c7b\u578b\u200b\u76f8\u540c\u200b\u3002</p> <pre><code>fn increment(x: i32) -&gt; i32 {\n    x + 1\n}\n\nfn main() {\n    let a: Result&lt;i32, i32&gt; = Err(2);\n    println!(\"{}\", a.unwrap_or_else(increment));\n    // 3\n}\n</code></pre> </li> </ul>"},{"location":"coding/rust/5/#_3","title":"\u8fd0\u7b97","text":"<p>Rust\u200b\u53ef\u4ee5\u200b\u5728\u200b\u4e0d\u200b\u53d6\u51fa\u200b<code>Option</code>\u200b\u6216\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u503c\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u5bf9\u200b\u5176\u200b\u8fdb\u884c\u200b\u8fd0\u7b97\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u4ee5\u4e0b\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u8bb0\u200b<code>A</code>\u200b\u4e3a\u200b\u8c03\u7528\u200b\u65b9\u6cd5\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff0c<code>B</code>\u200b\u4e3a\u200b\u65b9\u6cd5\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5373\u200b<code>A.method(B)</code>\u3002</p> <p><code>Option</code>\u200b\u548c\u200b<code>Result</code>\u200b\u7684\u200b\u771f\u503c\u200b</p> <p>\u200b\u4e0d\u59a8\u200b\u5c06\u200b<code>Option</code>\u200b\u548c\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u7684\u200b<code>Some</code>\u200b\u548c\u200b<code>Ok</code>\u200b\u89c6\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5176\u4e2d\u200b\u7684\u200b\u503c\u200b<code>T</code>\u200b\u79f0\u4e3a\u200b\u771f\u503c\u200b\uff0c<code>None</code>\u200b\u548c\u200b<code>Err</code>\u200b\u89c6\u4e3a\u200b\u5047\u200b\uff0c<code>Err</code>\u200b\u7684\u200b\u503c\u200b<code>E</code>\u200b\u89c6\u4e3a\u200b\u5047\u503c\u200b\u3002</p> <ol> <li>\u200b\u903b\u8f91\u8fd0\u7b97\u200b<code>A.or(B)</code>\u200b\u4e0e\u200b<code>A.and(B)</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u548c\u200b\u81ea\u8eab\u200b\u7c7b\u578b\u200b\u76f8\u540c\u200b\u7684\u200b\u53c2\u6570\u200b\u3002<ul> <li>\u200b\u5bf9\u4e8e\u200b<code>or()</code>\uff0c\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>B</code>\u200b\u503c\u200b\u3002</li> <li>\u200b\u5bf9\u4e8e\u200b<code>and()</code>\uff0c\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u5047\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u5047\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>B</code>\u3002</li> </ul> </li> <li><code>A.or_else()</code>\u200b\u4e3a\u200b<code>A.or(B)</code>\u200b\u7684\u200b\u53d8\u79cd\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002</li> <li><code>A.and_then()</code>\u200b\u4e3a\u200b<code>A.and(B)</code>\u200b\u7684\u200b\u53d8\u79cd\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b\u4e3a\u200b\u5047\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u5047\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002</li> <li>\uff08\u200b\u4ec5\u200b<code>Option</code>\u200b\u7c7b\u578b\u200b\uff09<code>A.filter()</code>\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b<code>fn (T) -&gt; bool</code>\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u4e14\u200b\u4ee5\u200b\u5176\u200b\u771f\u503c\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u540e\u200b\u8c03\u7528\u51fd\u6570\u200b\u8fd4\u56de\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>None</code>\u3002</li> <li><code>A.map()</code>\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b<code>fn (T) -&gt; U</code>\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u540e\u200b\u8c03\u7528\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u5047\u503c\u200b\u3002\u200b\u6ce8\u610f\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>Option&lt;U&gt;</code>\u200b\u6216\u200b<code>Result&lt;U, E&gt;</code>\u3002<ul> <li>\uff08\u200b\u4ec5\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\uff09<code>A.map_err()</code>\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b<code>fn (E) -&gt; F</code>\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u5047\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u5047\u503c\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u540e\u200b\u8c03\u7528\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\u3002</li> <li>\uff08\u200b\u4ec5\u200b<code>Option</code>\u200b\u7c7b\u578b\u200b\uff09<code>A.map_or()</code>\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b<code>U</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u9ed8\u8ba4\u503c\u200b\u548c\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b<code>fn (T) -&gt; U</code>\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u540e\u200b\u8c03\u7528\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b\u9ed8\u8ba4\u503c\u200b\u3002</li> <li><code>A.map_or_else()</code>\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b<code>fn () -&gt; U</code>\u200b\u548c\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b<code>fn (T) -&gt; U</code>\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u540e\u200b\u8c03\u7528\u200b\u7b2c\u4e8c\u4e2a\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b\u7b2c\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002</li> </ul> </li> <li>\uff08\u200b\u4ec5\u200b<code>Option</code>\u200b\u7c7b\u578b\u200b\uff09<code>A.ok_or()</code>\u200b\u7528\u4e8e\u200b\u5c06\u200b<code>Option</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b<code>E</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b\u53c2\u6570\u200b\u4f5c\u4e3a\u200b<code>Err</code>\u200b\u7684\u200b\u503c\u200b\u3002<ul> <li><code>A.ok_or_else()</code>\u200b\u4e3a\u200b<code>A.ok_or()</code>\u200b\u7684\u200b\u53d8\u79cd\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002</li> </ul> </li> <li><code>A.as_ref()</code>\u200b\u548c\u200b<code>A.as_mut()</code>\uff0c\u200b\u7528\u4e8e\u200b\u5c06\u200b<code>Option&lt;T&gt;</code>\u200b\u6216\u200b<code>Result&lt;T, E&gt;</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u5176\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5f15\u7528\u200b\u3002\u200b\u5176\u4e2d\u200b<code>A.as_ref()</code>\u200b\u8fd4\u56de\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5f15\u7528\u200b\uff0c\u200b\u5373\u200b<code>Option&lt;&amp;T&gt;</code>\u200b\u6216\u200b<code>Result&lt;&amp;T, &amp;E&gt;</code>\uff1b<code>A.as_mut()</code>\u200b\u8fd4\u56de\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\uff0c\u200b\u5373\u200b<code>Option&lt;&amp;mut T&gt;</code>\u200b\u6216\u200b<code>Result&lt;&amp;mut T, &amp;mut E&gt;</code>\u3002</li> </ol>"},{"location":"coding/rust/5/#_4","title":"\u5f02\u5e38\u200b\u4f20\u9012","text":"<p>Rust\u200b\u4e2d\u200b\uff0c\u200b\u6709\u200b\u4e24\u79cd\u200b\u65b9\u5f0f\u200b\u53ef\u4ee5\u200b\u5411\u4e0a\u200b\u8fd4\u56de\u200b\u5f02\u5e38\u200b\u3002</p> <ul> <li> <p><code>?</code>\u200b\u8fd0\u7b97\u7b26\u200b\uff0c\u200b\u7528\u4e8e\u200b\u5c06\u200b<code>Option</code>\u200b\u6216\u200b<code>Result</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b<code>T</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5982\u679c\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>Err</code>\u200b\u6216\u200b\uff0c\u200b\u5219\u200b\u4f1a\u200b\u76f4\u63a5\u200b\u8fd4\u56de\u200b<code>Err</code>\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u3002</p> <pre><code>fn fn_with_error&lt;'a&gt;() -&gt; Result&lt;i32, &amp;'a str&gt; {\n    Err(\"Deterministic error\")\n}\nfn use_fn&lt;'a&gt;() -&gt; Result&lt;i32, &amp;'a str&gt; {\n    let x = fn_with_error()?; // Directly return Err(\"Deterministic error\")\n    Ok(x) // unreachable\n}\nfn main() {\n    let x: &amp;str = use_fn().unwrap_err();\n    println!(\"{}\", x); // Deterministic error\n}\n</code></pre> </li> <li> <p><code>try!</code>\u200b\u5b8f\u200b\uff0c\u200b\u4e0e\u200b<code>?</code>\u200b\u7684\u200b\u6548\u679c\u200b\u7b49\u540c\u200b\uff0c\u200b\u4f7f\u7528\u200b\u65b9\u5f0f\u200b\u4e3a\u200b<code>try!(expression)</code>\u3002\u200b\u76ee\u524d\u200b\u5df2\u7ecf\u200b\u88ab\u200b\u5f03\u7528\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>?</code>\u200b\u4ee3\u66ff\u200b\u3002</p> <pre><code>fn fn_with_error&lt;'a&gt;() -&gt; Result&lt;i32, &amp;'a str&gt; {\n    Err(\"Deterministic error\")\n}\nfn use_fn&lt;'a&gt;() -&gt; Result&lt;i32, &amp;'a str&gt; {\n    let x = try!(fn_with_error()); // error: use of deprecated `try` macro\n    Ok(x) // unreachable\n}\nfn main() {\n    let x: &amp;str = use_fn().unwrap_err();\n    println!(\"{}\", x); // Deterministic error\n}\n</code></pre> </li> </ul>"},{"location":"coding/rust/5/#_5","title":"\u4e3b\u200b\u51fd\u6570","text":"<p>Rust\u200b\u7684\u200b<code>main</code>\u200b\u51fd\u6570\u200b\u53ea\u80fd\u200b\u53d6\u200b<code>()</code>\u200b\u6216\u200b<code>Result&lt;(), E&gt;</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5176\u4e2d\u200b<code>E</code>\u200b\u4e3a\u200b<code>std::error::Error</code>\u200b\u7684\u200b\u5b9e\u73b0\u200b\u7c7b\u578b\u200b\u3002\u200b\u5982\u679c\u200b<code>main</code>\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5219\u200b\u4f1a\u200b\u5c06\u200b<code>Err</code>\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u6253\u5370\u200b\u5230\u200b<code>stderr</code>\u200b\u5e76\u200b\u9000\u51fa\u200b\u7a0b\u5e8f\u200b\u3002</p> <pre><code>use std::fs::File;\n\nfn main() -&gt; std::io::Result&lt;()&gt; {\n    let _ = File::open(\"not-existing-file.txt\")?; // Result::Err\n\n    Ok(()) // Default return value\n    // Must with this line, otherwise return () which is incompatible\n}\n</code></pre> <p>\u200b\u5982\u679c\u200b<code>main</code>\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u4e0d\u662f\u200b<code>()</code>\u200b\u6216\u200b<code>Result</code>\uff0c\u200b\u5219\u200b\u65e0\u6cd5\u200b\u901a\u8fc7\u200b\u7f16\u8bd1\u200b\u3002</p> <pre><code>fn main() -&gt; i32 {\n    0\n}\n// error[E0277]: `main` has invalid return type `i32`\n</code></pre>"},{"location":"coding/rust/5/#_6","title":"\u81ea\u5b9a\u4e49\u200b\u5f02\u5e38","text":"<p>\u200b\u5728\u200bRust\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u81ea\u5b9a\u4e49\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u200b\u3002\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b<code>std::error::Error</code>\u200b\u7279\u6027\u200b\uff0c\u200b\u8be5\u200b\u7279\u6027\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4ee5\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li><code>fn source(&amp;self) -&gt; Option&lt;&amp;(dyn Error + 'static)&gt;</code>\uff0c\u200b\u7528\u4e8e\u200b\u8fd4\u56de\u200b\u5f15\u8d77\u200b\u5f02\u5e38\u200b\u7684\u200b\u539f\u56e0\u200b\uff08\u200b\u53ef\u200b\u9009\u200b\uff09\u3002</li> <li><code>fn Debug::fmt(&amp;self, f: &amp;mut Formatter&lt;'_&gt;) -&gt; Result</code>\uff0c\u200b\u7528\u4e8e\u200b\u683c\u5f0f\u5316\u200b\u5f02\u5e38\u200b\u4fe1\u606f\u200b\uff08\u200b\u7528\u4e8e\u200b\u8c03\u8bd5\u200b\uff09\u3002</li> <li><code>fn Display::fmt(&amp;self, f: &amp;mut Formatter&lt;'_&gt;) -&gt; Result</code>\uff0c\u200b\u7528\u4e8e\u200b\u683c\u5f0f\u5316\u200b\u5f02\u5e38\u200b\u4fe1\u606f\u200b\uff08\u200b\u7528\u4e8e\u200b\u8f93\u51fa\u200b\uff09\u3002</li> </ul> <pre><code>struct MyError {\n    code: i32,\n    message: String\n}\n\nimpl std::error::Error for MyError {\n    fn source(&amp;self) -&gt; Option&lt;&amp;(dyn std::error::Error + 'static)&gt; {\n        None\n    }\n}\n\nimpl std::fmt::Debug for MyError {\n    fn fmt(&amp;self, f: &amp;mut std::fmt::Formatter&lt;'_&gt;) -&gt; std::fmt::Result {\n        write!(\n            f, \"MyError {{ code: {}, message: {} }} at file {} line {}\",\n            self.code, self.message, file!(), line!()\n            // file!() and line!() are macros to get the current file and line\n        )\n    }\n}\n\nimpl std::fmt::Display for MyError {\n    fn fmt(&amp;self, f: &amp;mut std::fmt::Formatter&lt;'_&gt;) -&gt; std::fmt::Result {\n        write!(f, \"MyError {{ code: {}, message: {} }}\", self.code, self.message)\n    }\n}\n\n// Use the error\n\nfn raise_error() -&gt; Result&lt;(), MyError&gt; {\n    Err(MyError { code: 1, message: \"Error message\".to_string() })\n}\n\nfn main() -&gt; Result&lt;(), MyError&gt; {\n    let _ = raise_error()?;\n    // MyError { code: 1, message: Error message } at file src/main.rs line 16\n    Ok(())\n}\n</code></pre>"},{"location":"coding/rust/5/#_7","title":"\u5f02\u5e38\u200b\u8f6c\u6362","text":"<p>\u200b\u7531\u4e8e\u200bRust\u200b\u4e2d\u200b\u7684\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u200b\u662f\u200b\u9759\u6001\u200b\u7684\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5728\u200b\u5f02\u5e38\u200b\u4f20\u9012\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\uff0c\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u9047\u5230\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u200b\u4e0d\u200b\u5339\u914d\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002\u200b\u6b64\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>From</code>\u200b\u7279\u6027\u200b\u5c06\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b\u53e6\u200b\u4e00\u79cd\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4ee5\u200b<code>std::io::Error</code>\u200b\u4e3a\u4f8b\u200b\u3002</p> <pre><code>impl std::convert::From&lt;std::io::Error&gt; for MyError {\n    fn from(error: std::io::Error) -&gt; Self {\n        MyError { code: 2, message: error.to_string() }\n    }\n}\n</code></pre> <p>\u200b\u6b64\u540e\u200b\uff0c\u200b\u65e0\u9700\u200b\u663e\u5f0f\u200b\u8c03\u7528\u200b<code>From</code>\u200b\u7279\u6027\u200b\u7684\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5373\u53ef\u200b\u5c06\u200b<code>std::io::Error</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b<code>MyError</code>\u200b\u7c7b\u578b\u200b\u3002</p>"},{"location":"coding/rust/6/","title":"Supplements for Basics","text":""},{"location":"coding/rust/6/#_1","title":"\u5ffd\u7565\u200b\u53d8\u91cf","text":"<p>\u200b\u5728\u200b\u58f0\u660e\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\u5374\u200b\u6ca1\u6709\u200b\u4f7f\u7528\u200b\u65f6\u200b\uff0cRust\u200b\u4f1a\u200b\u7ed9\u51fa\u200b\u8b66\u544a\u200b\u3002\u200b\u5982\u679c\u200b\u4e0d\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u8fd9\u4e2a\u200b\u53d8\u91cf\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5728\u200b\u53d8\u91cf\u540d\u200b\u524d\u200b\u52a0\u4e0a\u200b\u4e0b\u5212\u7ebf\u200b\u524d\u7f00\u200b\u4f5c\u4e3a\u200b\u6807\u8bb0\u200b\u3002\u200b\u540d\u4e3a\u200b<code>_</code>\u200b\u7684\u200b\u53d8\u91cf\u200b\u662f\u200b\u4e00\u4e2a\u200b\u7279\u6b8a\u200b\u7684\u200b\u53d8\u91cf\u200b\uff0c\u200b\u5b83\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u58f0\u660e\u200b\u591a\u6b21\u200b\uff0c\u200b\u4f46\u662f\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u4f7f\u7528\u200b\u3002</p> <p>  Download source code</p> <pre><code>fn main() {\n    let _: i32 = 5;\n    println!(\"{}\", _);\n    // error: in expressions, `_` can only be used on the left-hand side of an assignment\n}\n</code></pre>"},{"location":"coding/rust/6/#_2","title":"\u6a21\u5f0f\u5339\u914d","text":""},{"location":"coding/rust-gui/","title":"Rust GUI library","text":"<p>The following pages are notes from Building a simple text editor with iced, a cross-platform GUI library for Rust using iced lib.</p> <ol> <li>Hello World</li> <li>Multi-line Input</li> <li>Theme and Cursor Indicator</li> <li>Async File Loading</li> <li>File Picker</li> <li>File Path Indicator</li> <li>New and Save</li> <li>Button prettify</li> <li>Syntax Highlighting</li> <li>Misc</li> </ol>"},{"location":"coding/rust-gui/1-hello-world/","title":"Hello World App","text":"<p>Keywords: <code>iced::Sandbox</code>, <code>iced::Settings</code>, <code>iced::widget::text</code></p>"},{"location":"coding/rust-gui/1-hello-world/#_1","title":"\u521b\u5efa\u200b\u9879\u76ee","text":"<p>\u200b\u9996\u5148\u200b\uff0c\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b Rust \u200b\u9879\u76ee\u200b\uff0c\u200b\u5e76\u200b\u6dfb\u52a0\u200b\u5bf9\u200b<code>iced</code>\u200b\u5e93\u200b\u7684\u200b\u4f9d\u8d56\u200b\u3002</p> <pre><code>cargo new text-editor\ncd text-editor\ncargo add iced\n</code></pre> <p>\u200b\u6b64\u65f6\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>fn main() {\n    println!(\"Hello, world!\");\n}\n</code></pre>"},{"location":"coding/rust-gui/1-hello-world/#gui","title":"\u521b\u5efa\u200bGUI\u200b\u5e94\u7528","text":"<p><code>iced</code>\u200b\u7684\u200bGUI\u200b\u7c7b\u6709\u200b<code>Application</code>\u200b\u548c\u200b<code>Sandbox</code>\u200b\u4e24\u79cd\u200b\u3002\u200b\u5176\u4e2d\u200b<code>Sandbox</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u7b80\u5316\u7248\u200b\u7684\u200b<code>Application</code>\uff0c\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u4e00\u4e9b\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u884c\u4e3a\u200b\u3002<code>Sandbox</code>\u200b\u7279\u6027\u200b\u5305\u542b\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <pre><code>pub trait Sandbox {\n    type Message: std::fmt::Debug + Send;\n    fn new() -&gt; Self;\n    fn title(&amp;self) -&gt; String;\n    fn update(&amp;mut self, message: Self::Message);\n    fn view(&amp;self) -&gt; Element&lt;'_, Self::Message&gt;;\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::default()\n    }\n    fn style(&amp;self) -&gt; theme::Application {\n        theme::Application::default()\n    }\n    fn scale_factor(&amp;self) -&gt; f64 {\n        1.0\n    }\n    fn run(settings: Settings&lt;()&gt;) -&gt; Result&lt;(), Error&gt;\n    where\n        Self: 'static + Sized,\n    {\n        &lt;Self as Application&gt;::run(settings)\n    }\n}\n</code></pre> <p>\u200b\u4e3a\u4e86\u200b\u4f7f\u7528\u200b<code>Sandbox</code>\uff0c\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b\u5176\u4e2d\u200b\u672a\u200b\u5b9e\u73b0\u200b\u7684\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5373\u200b<code>Message</code>\u3001<code>new</code>\u3001<code>title</code>\u3001<code>update</code>\u200b\u548c\u200b<code>view</code>\u3002</p> <ul> <li> <p><code>Message</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u679a\u4e3e\u200b\u7c7b\u578b\u200b\uff0c\u200b\u7528\u4e8e\u200b\u5b9a\u4e49\u200b\u5e94\u7528\u200b\u4f1a\u200b\u4ea7\u751f\u200b\u7684\u200b\u6d88\u606f\u200b\u7c7b\u578b\u200b\uff0c\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b<code>std::fmt::Debug</code>\u200b\u548c\u200b<code>Send</code>\u200b\u7279\u6027\u200b\u3002</p> <pre><code>#[derive(Debug)] // Inherit the Debug trait\nenum Message {} // No message required\n</code></pre> </li> <li> <p><code>new</code>\u200b\u65b9\u6cd5\u200b\u7528\u4e8e\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b<code>Sandbox</code>\u200b\u5b9e\u4f8b\u200b\uff0c\u200b\u521d\u59cb\u5316\u200b\u5b9e\u4f8b\u200b\u72b6\u6001\u200b\uff0c\u200b\u4e00\u822c\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u76f4\u63a5\u200b\u8fd4\u56de\u200b<code>Self</code>\u200b\u5373\u53ef\u200b\u3002</p> <pre><code>fn new() -&gt; Self {\n    Self\n}\n</code></pre> </li> <li> <p><code>title</code>\u200b\u65b9\u6cd5\u200b\u7528\u4e8e\u200b\u8fd4\u56de\u200bGUI\u200b\u7a97\u53e3\u200b\u7684\u200b\u6807\u9898\u200b\u3002</p> <pre><code>fn title(&amp;self) -&gt; String {\n    String::from(\"A text editor\")\n}\n</code></pre> </li> <li> <p><code>update</code>\u200b\u65b9\u6cd5\u200b\u548c\u200b<code>view</code>\u200b\u65b9\u6cd5\u200b\u5171\u540c\u200b\u7ec4\u6210\u200b\u5e94\u7528\u200b\u7684\u200b\u6d88\u606f\u200b\u5faa\u73af\u200b\uff1a<code>update</code>\u200b\u65b9\u6cd5\u200b\u7528\u4e8e\u200b\u5904\u7406\u200b\u6d88\u606f\u200b\uff0c\u200b\u66f4\u65b0\u200b\u5e94\u7528\u200b\u72b6\u6001\u200b\uff0c<code>view</code>\u200b\u65b9\u6cd5\u200b\u7528\u4e8e\u200b\u5728\u200b\u72b6\u6001\u200b\u66f4\u65b0\u200b\u540e\u200b\u66f4\u65b0\u200b\u5e94\u7528\u200b\u754c\u9762\u200b\u3002\u200b\u6b64\u5904\u200b\u6211\u4eec\u200b\u5728\u200b<code>view</code>\u200b\u65b9\u6cd5\u200b\u4e2d\u200b\u653e\u7f6e\u200b\u4e00\u4e2a\u200b\u6587\u672c\u200b\u63a7\u4ef6\u200b\u3002</p> <pre><code>fn update(&amp;mut self, message: Self::Message) {\n    match message {\n        // No message to handle\n    }\n}\n\nfn view(&amp;self) -&gt; Element&lt;'_, Self::Message&gt; {\n    text(\"Hello, world!\").into()\n}\n</code></pre> </li> </ul> <p>\u200b\u901a\u8fc7\u200b<code>Sandbox::run</code>\u200b\u65b9\u6cd5\u200b\u542f\u52a8\u200b\u5e94\u7528\u200b\uff0c\u200b\u8be5\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b<code>Result&lt;(), Error&gt;</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u4f5c\u4e3a\u200b<code>main</code>\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use iced::{Element, Sandbox, Settings};\nuse iced::widget::text;\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor;\n\n#[derive(Debug)]\nenum Message {}\n\nimpl Sandbox for Editor {\n    type Message = Message; // Define the type of messages\n    fn new() -&gt; Self {\n        Self\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) {\n        // Handle messages here\n        match message {}\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        text(\"Hello, world!\").into()\n    }\n}\n</code></pre>"},{"location":"coding/rust-gui/10-misc/","title":"Miscellaneous","text":"<p>\u200b\u672c\u200b\u8282\u200b\u6dfb\u52a0\u200b\u4e00\u4e9b\u200b\u989d\u5916\u200b\u529f\u80fd\u200b\u3002</p> <p>\u200b\u9996\u5148\u200b\uff0c\u200b\u5f53\u200b\u6587\u4ef6\u200b\u6ca1\u6709\u200b\u88ab\u200b\u4fee\u6539\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8bbe\u7f6e\u200b\u7981\u7528\u200b\u4fdd\u5b58\u200b\u6309\u94ae\u200b\u3002\u200b\u8fd9\u6837\u200b\u53ef\u4ee5\u200b\u907f\u514d\u200b\u7528\u6237\u200b\u8bef\u64cd\u4f5c\u200b\u3002\u200b\u5728\u200b<code>on_press_maybe</code>\u200b\u4e2d\u200b\u4f20\u5165\u200b<code>None</code>\u200b\u5373\u53ef\u200b\u3002\u200b\u540c\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6839\u636e\u200b\u6587\u4ef6\u200b\u7684\u200b\u4fee\u6539\u200b\u60c5\u51b5\u200b\u8bbe\u7f6e\u200b\u6309\u94ae\u200b\u7684\u200b\u6837\u5f0f\u200b\u3002</p> <pre><code>fn toolbar_button&lt;'a&gt;(description: &amp;str, callback: Option&lt;Message&gt;) -&gt; Element&lt;'a, Message&gt; {\n    let font = Font::with_name(\"editor-icon\");\n    let lower = description.to_lowercase();\n    let icon = text(match lower.as_str() {\n        \"new\" =&gt; '\\u{E800}',\n        \"open\" =&gt; '\\u{F115}',\n        \"save\" =&gt; '\\u{E801}',\n        _ =&gt; ' '\n    }).font(font);\n    let is_disabled = callback.is_none();\n    tooltip(\n        button(container(icon)\n            .width(30)  // Set the width of the button\n            .center_x() // Center the icon\n        ).on_press_maybe(callback).style(\n            if is_disabled {\n                theme::Button::Secondary\n            } else {\n                theme::Button::Primary\n            }\n        ),\n        description, tooltip::Position::FollowCursor\n    ).style(theme::Container::Box).into()\n}\n</code></pre> <p>\u200b\u540c\u65f6\u200b\u4fee\u6539\u200b<code>toolbar_button</code>\u200b\u7684\u200b\u8c03\u7528\u200b\u3002</p> <pre><code>// ... In `view` function\nlet controls = row![\n    toolbar_button(\"New\", Some(Message::NewButtonPressed)),\n    toolbar_button(\"Open\", Some(Message::OpenButtonPressed)),\n    toolbar_button(\"Save\", if self.modified { Some(Message::SaveButtonPressed) } else { None }),\n    horizontal_space(Length::Fill),\n    pick_list(highlighter::Theme::ALL, Some(self.theme), Message::ThemeChanged)\n].spacing(10);\n</code></pre> <p>\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u6dfb\u52a0\u200b\u4e0d\u540c\u200b\u7684\u200b\u5feb\u6377\u952e\u200b\uff0c\u200b\u4ee5\u200b\u65b9\u4fbf\u200b\u7528\u6237\u200b\u64cd\u4f5c\u200b\u3002</p> <pre><code>// In `impl Application for Editor`\nfn subscription(&amp;self) -&gt; Subscription&lt;Message&gt; {\n    keyboard::on_key_press(|keycode, modifier| {\n        match (keycode, modifier) {\n            (keyboard::KeyCode::S, keyboard::Modifiers::COMMAND) =&gt; {\n                Some(Message::SaveButtonPressed)\n            },\n            (keyboard::KeyCode::O, keyboard::Modifiers::COMMAND) =&gt; {\n                Some(Message::OpenButtonPressed)\n            },\n            (keyboard::KeyCode::N, keyboard::Modifiers::COMMAND) =&gt; {\n                Some(Message::NewButtonPressed)\n            },\n            _ =&gt; None\n        }\n    })\n}\n</code></pre> <p>\u200b\u8fd9\u6837\u200b\uff0c\u200b\u7528\u6237\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>Command + S</code>\u200b\u6765\u200b\u4fdd\u5b58\u200b\u6587\u4ef6\u200b\uff0c<code>Command + O</code>\u200b\u6765\u200b\u6253\u5f00\u200b\u6587\u4ef6\u200b\uff0c<code>Command + N</code>\u200b\u6765\u200b\u65b0\u5efa\u200b\u6587\u4ef6\u200b\u3002</p> <p>\u200b\u6587\u4ef6\u200b\u7684\u200b\u6807\u9898\u680f\u200b\u901a\u5e38\u200b\u663e\u793a\u6587\u4ef6\u200b\u8def\u5f84\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u548c\u200b\u5de6\u4e0b\u89d2\u200b\u7684\u200b\u72b6\u6001\u680f\u200b\u4fdd\u6301\u200b\u540c\u6b65\u200b\u3002</p> <pre><code>// ... In `impl Application for Editor`\nfn title(&amp;self) -&gt; String {\n    let path_text = match &amp;self.path {\n        None =&gt; String::from(\"New file\"),\n        Some(path) =&gt; path.to_string_lossy().to_string()\n    };\n    let suffix = if self.modified { \"*\" } else { \"\" };\n    format!(\"{}{}\", path_text, suffix)\n}\n\n// ... In `view` function\nlet path_indicator = if let Some(error) = &amp;self.error {\n    match error {\n        Error::DialogClosed =&gt; text(\"Dialog closed\"),\n        Error::IO(kind) =&gt; text(format!(\"I/O error: {:?}\", kind))\n    }\n} else {\n    text(self.title())\n};\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use std::io;\nuse std::path::{Path, PathBuf};\nuse std::sync::Arc;\n\nuse iced::{executor, keyboard, theme, Application, Command, Element, Font, Length, Settings, Subscription, Theme};\nuse iced::highlighter::{self, Highlighter};\nuse iced::widget::{button, column, container, horizontal_space, pick_list, row, text, text_editor, tooltip};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings {\n        fonts: vec![include_bytes!(\"../fonts/editor-icon.ttf\").as_slice().into()],\n        ..Default::default()\n    })\n}\n\nstruct Editor {\n    path: Option&lt;PathBuf&gt;,\n    content: text_editor::Content,\n    modified: bool,\n    error: Option&lt;Error&gt;,\n    theme: highlighter::Theme\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    FileSaved(Result&lt;PathBuf, Error&gt;),\n    ThemeChanged(highlighter::Theme),\n    NewButtonPressed,\n    OpenButtonPressed,\n    SaveButtonPressed\n}\n\nimpl Application for Editor {\n    type Message = Message; // Define the type of messages\n    type Theme = Theme;\n    type Executor = executor::Default; // Engine for running async tasks\n    type Flags = ();\n\n    fn new(_flags: Self::Flags) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                content: text_editor::Content::new(),\n                error: None,\n                modified: false,\n                path: None,\n                theme: highlighter::Theme::SolarizedDark\n            },\n            Command::perform(\n            load_file(format!(\"{}/src/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n            Message::FileOpened\n            )\n        )\n    }\n\n    fn title(&amp;self) -&gt; String {\n        let path_text = match &amp;self.path {\n            None =&gt; String::from(\"New file\"),\n            Some(path) =&gt; path.to_string_lossy().to_string()\n        };\n        let suffix = if self.modified { \"*\" } else { \"\" };\n        format!(\"{}{}\", path_text, suffix)\n    }\n\n    fn subscription(&amp;self) -&gt; Subscription&lt;Message&gt; {\n        keyboard::on_key_press(|keycode, modifier| {\n            match (keycode, modifier) {\n                (keyboard::KeyCode::S, keyboard::Modifiers::COMMAND) =&gt; {\n                    Some(Message::SaveButtonPressed)\n                },\n                (keyboard::KeyCode::O, keyboard::Modifiers::COMMAND) =&gt; {\n                    Some(Message::OpenButtonPressed)\n                },\n                (keyboard::KeyCode::N, keyboard::Modifiers::COMMAND) =&gt; {\n                    Some(Message::NewButtonPressed)\n                },\n                _ =&gt; None\n            }\n        })\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                match &amp;action {\n                    text_editor::Action::Edit(_) =&gt; self.modified = true,\n                    _ =&gt; {}\n                }\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            Message::FileSaved(Ok(path)) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                Command::none()\n            },\n            Message::FileOpened(Err(error)) | Message::FileSaved(Err(error)) =&gt; {\n                self.error = Some(error);\n                Command::none()\n            },\n            Message::ThemeChanged(theme) =&gt; {\n                self.theme = theme;\n                Command::none()\n            },\n            Message::NewButtonPressed =&gt; {\n                self.content = text_editor::Content::new();\n                self.error = None;\n                self.path = None;\n                self.modified = false;\n                Command::none()\n            },\n            Message::OpenButtonPressed =&gt; {\n                self.modified = false;\n                Command::perform(pick_file(), Message::FileOpened)\n            },\n            Message::SaveButtonPressed =&gt; {\n                let content = self.content.text();\n                match self.modified {\n                    false =&gt; Command::none(),\n                    true =&gt; Command::perform(\n                        save_file(self.path.clone(), content),\n                        Message::FileSaved\n                    )\n                }\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content)\n            .on_edit(Message::EditorEdit)\n            .highlight::&lt;Highlighter&gt;(\n                highlighter::Settings {\n                    theme: self.theme,\n                    extension: self.path\n                        .as_ref()\n                        .and_then(|path| path.extension()?.to_str())\n                        .unwrap_or(\"rs\")\n                        .to_string()\n                }, |highlighter, _theme| {\n                    highlighter.to_format()\n                }\n            )\n            .font(Font::MONOSPACE);\n        let controls = row![\n            toolbar_button(\"New\", Some(Message::NewButtonPressed)),\n            toolbar_button(\"Open\", Some(Message::OpenButtonPressed)),\n            toolbar_button(\"Save\", if self.modified { Some(Message::SaveButtonPressed) } else { None }),\n            horizontal_space(Length::Fill),\n            pick_list(highlighter::Theme::ALL, Some(self.theme), Message::ThemeChanged)\n        ].spacing(10);\n\n        // Query cursor position\n        let path_indicator = if let Some(error) = &amp;self.error {\n            match error {\n                Error::DialogClosed =&gt; text(\"Dialog closed\"),\n                Error::IO(kind) =&gt; text(format!(\"I/O error: {:?}\", kind))\n            }\n        } else {\n            text(self.title())\n        };\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![\n            path_indicator,\n            horizontal_space(Length::Fill),\n            cursor_indicator\n        ];\n\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        if self.theme.is_dark() {\n            Theme::Dark\n        } else {\n            Theme::Light\n        }\n    }\n}\n\nasync fn pick_file() -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let file_handle = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?;\n    load_file(file_handle.path()).await\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let content = tokio::fs::read_to_string(path.as_ref())\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO);\n    content.and_then(|content| Ok((path.as_ref().to_path_buf(), content)))\n}\n\nasync fn save_file(path: Option&lt;PathBuf&gt;, content: String) -&gt; Result&lt;PathBuf, Error&gt; {\n    let path = if let Some(path) = path {\n        path\n    } else {\n        rfd::AsyncFileDialog::new()\n            .set_title(\"Save the file to...\")\n            .save_file()\n            .await\n            .ok_or(Error::DialogClosed)?\n            .path()\n            .to_path_buf()\n    };\n    tokio::fs::write(&amp;path, content)\n        .await\n        .map_err(|err| err.kind())\n        .map_err(Error::IO)\n        .map(|_| path)\n}\n\nfn toolbar_button&lt;'a&gt;(description: &amp;str, callback: Option&lt;Message&gt;) -&gt; Element&lt;'a, Message&gt; {\n    let font = Font::with_name(\"editor-icon\");\n    let lower = description.to_lowercase();\n    let icon = text(match lower.as_str() {\n        \"new\" =&gt; '\\u{E800}',\n        \"open\" =&gt; '\\u{F115}',\n        \"save\" =&gt; '\\u{E801}',\n        _ =&gt; ' '\n    }).font(font);\n    let is_disabled = callback.is_none();\n    tooltip(\n        button(container(icon)\n            .width(30)  // Set the width of the button\n            .center_x() // Center the icon\n        ).on_press_maybe(callback).style(\n            if is_disabled {\n                theme::Button::Secondary\n            } else {\n                theme::Button::Primary\n            }\n        ),\n        description, tooltip::Position::FollowCursor\n    ).style(theme::Container::Box).into()\n}\n\n#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n</code></pre>"},{"location":"coding/rust-gui/2-multi-line-input/","title":"Add Multi-line Input","text":"<p>Keywords: <code>iced::widget::text_editor</code>, <code>iced::widget::container</code>.</p> <p>\u200b\u672c\u200b\u8282\u200b\u6211\u4eec\u200b\u5c06\u200b<code>text</code>\u200b\u63a7\u4ef6\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>text_editor</code>\u200b\u63a7\u4ef6\u200b\uff0c\u200b\u4ee5\u200b\u5b9e\u73b0\u200b\u591a\u884c\u200b\u6587\u672c\u200b\u7684\u200b\u8f93\u5165\u200b\u3002</p> <p>iced\u200b\u5e93\u200b\u7248\u672c\u200b</p> <p><code>text_editor</code>\u200b\u63a7\u4ef6\u200b\u5c1a\u672a\u200b\u5728\u200b\u6b63\u5f0f\u200b\u7248\u672c\u200b\u4e2d\u200b\u53d1\u5e03\u200b\uff0c\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>git</code>\u200b\u4ed3\u5e93\u200b\u4e2d\u200b\u7684\u200b\u4ee3\u7801\u200b\u3002</p> <pre><code>[dependencies]\niced = { git = \"https://github.com/iced-rs/iced.git\", rev = \"refs/tags/text-editor\" }\n</code></pre> <p>\u200b\u9996\u5148\u200b\uff0c\u200b\u5c06\u200b\u5f15\u5165\u200b\u7684\u200b<code>iced::widget::text</code>\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>iced::widget::text_editor</code>\u3002<code>text_editor</code>\u200b\u662f\u200b\u6709\u200b\u72b6\u6001\u200b\u7684\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b<code>new</code>\u200b\u4e2d\u200b\u521d\u59cb\u5316\u200b\u7a7a\u95f4\u200b\u72b6\u6001\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5728\u200b<code>view</code>\u200b\u4e2d\u200b\u6839\u636e\u200b\u72b6\u6001\u200b\u66f4\u65b0\u200b\u63a7\u4ef6\u200b\u5185\u5bb9\u200b\u3002</p> <pre><code>struct Editor {\n    content: text_editor::Content\n}\n\n// ...\n\nimpl Sandbox for Editor {\n    type Message = Message;\n\n    fn new() -&gt; Self {\n        Editor {\n            content: text_editor::Content::new() // Initialize the content\n        }\n    }\n\n    // ...\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        text_editor(&amp;self.content).into() // Link the content state\n    }\n}\n</code></pre> <p>\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c<code>into</code>\u200b\u65b9\u6cd5\u200b\u4f1a\u200b\u4f7f\u200b\u63a7\u4ef6\u200b\u5145\u6ee1\u200b\u6574\u4e2a\u200b\u7a97\u53e3\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>container</code>\u200b\u5305\u56f4\u200b<code>text_editor</code>\u200b\u63a7\u4ef6\u200b\u8f85\u52a9\u200b\u5e03\u5c40\u200b\u3002</p> <pre><code>use iced::widget::container;\n\n// ...\n\nimpl Sandbox for Editor {\n    // ...\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        let editor = text_editor(&amp;self.content);\n        container(editor).padding(10).into() // Add padding and wrap the editor\n    }\n}\n</code></pre> <p>\u200b\u6b64\u65f6\u200b\uff0c\u200b\u591a\u884c\u200b\u6587\u672c\u200b\u8f93\u5165\u6846\u200b\u7684\u200b\u5e03\u5c40\u200b\u5df2\u7ecf\u200b\u5b8c\u6210\u200b\u3002\u200b\u4f46\u200b\u7531\u4e8e\u200b\u8f93\u5165\u6846\u200b\u6ca1\u6709\u200b\u7ed1\u5b9a\u200b\u4e8b\u4ef6\u5904\u7406\u200b\uff0c\u200b\u56e0\u6b64\u200b\u76ee\u524d\u200b\u65e0\u6cd5\u200b\u8f93\u5165\u200b\u6587\u672c\u200b\u3002<code>text_editor</code>\u200b\u63a7\u4ef6\u200b\u652f\u6301\u200b\u7684\u200b\u4e8b\u4ef6\u200b\u6709\u200b</p> <pre><code>pub enum Action {\n    Move(Motion),\n    Select(Motion),\n    SelectWord,\n    SelectLine,\n    Edit(Edit),\n    Click(Point),\n    Drag(Point),\n    Scroll { lines: i32 },\n}\n</code></pre> <p>\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u5c06\u200b<code>Edit</code>\u200b\u4e8b\u4ef6\u200b\u5728\u200b<code>update</code>\u200b\u548c\u200b<code>view</code>\u200b\u4e2d\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b\uff0c\u200b\u540c\u65f6\u200b\u66f4\u65b0\u200b<code>Message</code>\u200b\u679a\u4e3e\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action)\n}\n\n// ...\n\nimpl Sandbox for Editor {\n    // ...\n    fn update(&amp;mut self, message: Message) {\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                self.content = self.content.edit(action);\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit); // Bind the edit event\n        container(editor).padding(10).into() // Add padding and wrap the editor\n    }\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use iced::{Element, Sandbox, Settings};\nuse iced::widget::{text_editor, container};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor {\n    content: text_editor::Content\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action)\n}\n\nimpl Sandbox for Editor {\n    type Message = Message; // Define the type of messages\n\n    fn new() -&gt; Self {\n        Self {\n            content: text_editor::Content::new()\n        }\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                self.content.edit(action);\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n        container(editor).padding(10).into()\n    }\n}\n</code></pre>"},{"location":"coding/rust-gui/3-theme-and-cursor-indicator/","title":"Theme and Cursor Indicator","text":"<p>Keywords: <code>iced::widget::column</code>, <code>iced::widget::row</code>, <code>iced::widget::horizontal_space</code>, <code>iced::Theme</code>, <code>iced::Length</code></p> <p>\u200b\u672c\u8282\u200b\u5c06\u200b\u7a97\u53e3\u200b\u8bbe\u7f6e\u200b\u4e3a\u200b\u6697\u8272\u200b\u8c03\u200b\uff0c\u200b\u5e76\u4e14\u200b\u901a\u8fc7\u200b<code>text</code>\u200b\u63a7\u4ef6\u200b\u663e\u793a\u200b\u5149\u6807\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002</p>"},{"location":"coding/rust-gui/3-theme-and-cursor-indicator/#_1","title":"\u8bbe\u7f6e\u200b\u7a97\u53e3\u200b\u4e3b\u9898","text":"<p>\u200b\u7a97\u53e3\u200b\u4e3b\u9898\u200b\u901a\u8fc7\u200b<code>theme</code>\u200b\u51fd\u6570\u200b\u8bbe\u7f6e\u200b\uff0c\u200b\u5b9a\u4e49\u200b\u4e3b\u9898\u200b\u98ce\u683c\u200b\u7684\u200b\u7c7b\u200b\u4e3a\u200b<code>iced::Theme</code>\uff0c<code>Light</code>\u200b\u548c\u200b<code>Dark</code>\u200b\u5206\u522b\u200b\u8868\u793a\u200b\u4eae\u8272\u200b\u548c\u200b\u6697\u8272\u200b\u8c03\u200b\u3002</p> <pre><code>use iced::Theme\n\n// ...\n\nimpl Sandbox for Editor {\n    // ...\n\n    fn theme(&amp;self) -&gt; iced::Theme {\n        iced::Theme::Dark // Set the window theme to dark\n    }\n}\n</code></pre>"},{"location":"coding/rust-gui/3-theme-and-cursor-indicator/#_2","title":"\u8bfb\u53d6\u200b\u6587\u4ef6\u200b\u5185\u5bb9","text":"<p>\u200b\u5728\u200b\u7a0b\u5e8f\u200b\u542f\u52a8\u200b\u65f6\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u8bbe\u7f6e\u200b\u4e00\u4e2a\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff0c\u200b\u6b64\u5904\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>include_str!</code>\u200b\u5b8f\u5728\u200b\u7f16\u8bd1\u200b\u65f6\u200b\u8bfb\u53d6\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\u5e76\u200b\u7528\u4e8e\u200b\u521d\u59cb\u5316\u200b\u6587\u672c\u6846\u200b\u7684\u200b\u72b6\u6001\u200b\u3002</p> <pre><code>impl Sandbox for Editor {\n    // ...\n\n    fn new() -&gt; Self {\n        Self {\n            content: text_editor::Content::with(include_str!(\"main.rs\"))\n        }\n    }\n}\n</code></pre>"},{"location":"coding/rust-gui/3-theme-and-cursor-indicator/#_3","title":"\u663e\u793a\u200b\u5149\u6807\u200b\u4f4d\u7f6e","text":"<p>\u200b\u6587\u672c\u6846\u200b\u7684\u200b\u5149\u6807\u200b\u4f4d\u7f6e\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>self.content</code>\u200b\u72b6\u6001\u200b\u7684\u200b<code>cursor_position</code>\u200b\u65b9\u6cd5\u200b\u83b7\u53d6\u200b\uff0c\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u5305\u542b\u200b\u5149\u6807\u200b\u4f4d\u7f6e\u200b\uff08\u200b\u4ece\u200b0\u200b\u5f00\u59cb\u200b\uff09\u200b\u7684\u200b\u5143\u7ec4\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6839\u636e\u200b\u5149\u6807\u200b\u4f4d\u7f6e\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b<code>text</code>\u200b\u63a7\u4ef6\u200b\u7528\u4e8e\u200b\u663e\u793a\u200b\u3002</p> <ul> <li>\u200b\u901a\u8fc7\u200b<code>column!</code>\u200b\u5b8f\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u5217\u200b\u5e03\u5c40\u200b\uff0c\u200b\u4e0a\u8fb9\u200b\u5305\u542b\u200b\u8f93\u5165\u200b\u6587\u672c\u6846\u200b\uff0c\u200b\u4e0b\u8fb9\u200b\u5305\u542b\u200b\u5149\u6807\u200b\u4f4d\u7f6e\u200b\u663e\u793a\u200b\u3002</li> <li>\u200b\u901a\u8fc7\u200b<code>row!</code>\u200b\u5b8f\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u884c\u200b\u5e03\u5c40\u200b\uff0c\u200b\u7ed3\u5408\u200b<code>Length::Fill</code>\u200b\u628a\u200b\u5149\u6807\u200b\u4f4d\u7f6e\u200b\u538b\u7f29\u200b\u5230\u200b\u53f3\u4fa7\u200b\u5bf9\u9f50\u200b\u3002</li> </ul> <pre><code>use iced::widget::{column, row, horizontal_space, text};\nuse iced::Length;\n\n// ...\n\nimpl Sandbox for Editor {\n    // ...\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        let editor = text_editor(&amp;self.content);\n        // Query cursor position\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![horizontal_space(Length::Fill), cursor_indicator];\n\n        container(column![editor, status_bar].spacing(10)).padding(10).into()\n    }\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use iced::{Theme, Element, Sandbox, Settings, Length};\nuse iced::widget::{column, container, horizontal_space, row, text, text_editor};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor {\n    content: text_editor::Content\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action)\n}\n\nimpl Sandbox for Editor {\n    type Message = Message; // Define the type of messages\n\n    fn new() -&gt; Self {\n        Self {\n            content: text_editor::Content::with(include_str!(\"main.rs\"))\n        }\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                self.content.edit(action);\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n\n        // Query cursor position\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![horizontal_space(Length::Fill), cursor_indicator];\n\n        container(column![editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::Dark\n    }\n}\n</code></pre>"},{"location":"coding/rust-gui/4-async-file-loading/","title":"Asynchronous File Loading","text":"<p>Keywords: <code>iced::Application</code>, <code>iced::Command</code>, <code>iced::executor</code>, <code>std::io</code>, <code>std::path::Path</code>, <code>std::sync::Arc</code>, <code>tokio::fs::read_to_string</code></p> <p>\u200b\u4e3a\u4e86\u200b\u5b9e\u73b0\u200b\u5f02\u6b65\u200b\u6587\u4ef6\u200b\u52a0\u8f7d\u200b\uff0c\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>tokio</code>\u200b\u5e93\u200b</p> <pre><code>[dependencies]\ntokio = { version = \"1.32\", features = [\"fs\"] }\n</code></pre> <p><code>Sandbox</code>\u200b\u4e0d\u80fd\u200b\u76f4\u63a5\u200b\u6267\u884c\u200b\u5f02\u6b65\u200b\u51fd\u6570\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b<code>Application</code>\u3002\u200b\u76f8\u6bd4\u200b\u4e8e\u200b<code>Sandbox</code>\uff0c<code>Application</code>\u200b\u9700\u8981\u200b\u989d\u5916\u200b\u5b9e\u73b0\u200b\u5982\u4e0b\u200b\u7c7b\u578b\u200b</p> <pre><code>impl Application for Editor {\n    // ...\n    type Theme; // Color theme\n    type Executor; // Engine for running async tasks\n    type Flags; // Initial state\n}\n</code></pre> <p>\u200b\u6b64\u5904\u200b\uff0c<code>Theme</code>\u200b\u548c\u200b<code>Flags</code>\u200b\u7c7b\u578b\u200b\u4e0d\u200b\u9700\u8981\u200b\u989d\u5916\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5206\u522b\u200b\u7528\u200b<code>iced::Theme</code>\u200b\u548c\u200b<code>()</code>\u200b\u4ee3\u66ff\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c<code>Executor</code>\u200b\u7c7b\u578b\u200b\u9700\u8981\u200b\u7528\u200b<code>iced::executor::Default</code>\u200b\u4ee3\u66ff\u200b\u3002</p> <p>\u200b\u5728\u200b\u4fee\u6539\u200b\u4e3a\u200b<code>Application</code>\u200b\u540e\u200b\uff0c<code>new</code>\u200b\u548c\u200b<code>update</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u6807\u7b7e\u200b\u4e5f\u200b\u9700\u8981\u200b\u4fee\u6539\u200b\uff0c<code>new</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b<code>(Self, Command&lt;Message&gt;)</code>\uff0c<code>update</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b<code>Command&lt;Message&gt;</code>\u3002\u200b\u5176\u4e2d\u200b<code>Command&lt;Message&gt;</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5f02\u6b65\u200b\u4efb\u52a1\u200b\uff0c\u200b\u5728\u200b\u6267\u884c\u200b\u5b8c\u6bd5\u200b\u540e\u200b\u4f1a\u200b\u53d1\u9001\u200b\u4e00\u4e2a\u200b\u5bf9\u5e94\u200b\u7c7b\u578b\u200b\u7684\u200b<code>Message</code>\u3002</p> <pre><code>impl Application for Editor {\n    // ...\n\n    fn new(_flags: ()) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                // ...\n            },\n            Command::none(),\n        )\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        match message {\n            // ...\n        }\n\n        Command::none()\n    }\n}\n</code></pre> <p>\u200b\u6b64\u540e\u200b\uff0c\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u7f16\u5199\u200b\u7528\u4e8e\u200b\u8bfb\u53d6\u200b\u6587\u4ef6\u200b\u7684\u200b\u51fd\u6570\u200b\u3002</p> <pre><code>use std::io;\nuse std::path::Path;\nuse std::sync::Arc;\n\nasync fn load_file&lt;T&gt;(path: T) -&gt; Result&lt;Arc&lt;String&gt;, io::ErrorKind&gt;\n    where T: AsRef&lt;Path&gt;\n{\n    tokio::fs::read_to_string(path)\n        .await\n        .map(Arc::new)\n        .map_err(|e| e.kind())\n}\n</code></pre> <p>\u200b\u5728\u200b<code>new</code>\u200b\u65b9\u6cd5\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>Command</code>\u200b\u6765\u200b\u8c03\u7528\u200b<code>load_file</code>\u200b\u51fd\u6570\u200b\u3002</p> <pre><code>enum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;Arc&lt;String&gt;, io::ErrorKind&gt;)\n}\n\nimpl Application for Editor {\n    // ...\n\n    fn new(_flags: ()) -&gt; (Self, Command&lt;Message&gt;) {\n        let path = \"path/to/file.txt\";\n        let file = load_file(path);\n\n        (\n            Self {\n                // ...\n            },\n            Command::perform(\n                load_file(format!(\"{}/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n                Message::FileLoaded,\n            ),\n        )\n    }\n}\n</code></pre> <p>\u200b\u8bfb\u53d6\u200b\u6587\u4ef6\u200b\u65f6\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u53d1\u751f\u200b\u9519\u8bef\u200b\uff0c\u200b\u9700\u8981\u200b\u5bf9\u200b\u5f02\u5e38\u200b\u6d88\u606f\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b\u3002\u200b\u5728\u200b<code>load_file</code>\u200b\u51fd\u6570\u200b\u4e2d\u200b\u5df2\u7ecf\u200b\u901a\u8fc7\u200b<code>Result</code>\u200b\u8fd4\u56de\u200b\u4e86\u200b\u5bf9\u5e94\u200b\u7684\u200b\u9519\u8bef\u200b\u7c7b\u578b\u200b\uff0c\u200b\u53ea\u200b\u9700\u8981\u200b\u5728\u200b<code>update</code>\u200b\u65b9\u6cd5\u200b\u4e2d\u200b\u5904\u7406\u200b<code>Message::FileOpened</code>\u200b\u5373\u53ef\u200b\u3002</p> <pre><code>struct Editor {\n    // ...\n    error: Option&lt;io::ErrorKind&gt;, // Use Option to store error\n}\n\n// ...\n\nimpl Application for Editor {\n    // ...\n\n    fn new(_flags: ()) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                // ...\n                error: None, // Initialize error as None\n            },\n            // ...\n        )\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // ...\n        match message {\n            // ...\n            Message::FileOpened(Error(e)) =&gt; {\n                self.error = Some(e); // Store error\n            }\n        }\n        // ...\n    }\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use iced::{Theme, Element, Sandbox, Settings, Length};\nuse iced::widget::{column, container, horizontal_space, row, text, text_editor};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor {\n    content: text_editor::Content\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action)\n}\n\nimpl Sandbox for Editor {\n    type Message = Message; // Define the type of messages\n\n    fn new() -&gt; Self {\n        Self {\n            content: text_editor::Content::with(include_str!(\"main.rs\"))\n        }\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                self.content.edit(action);\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n\n        // Query cursor position\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![horizontal_space(Length::Fill), cursor_indicator];\n\n        container(column![editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::Dark\n    }\n}\n</code></pre>"},{"location":"coding/rust-gui/5-file-picker/","title":"File Picker","text":"<p>Keywords: <code>iced::widget::button</code>, <code>rfd::AsyncFileDialog</code></p> <p>\u200b\u672c\u8282\u200b\u5728\u200b\u7a97\u53e3\u200b\u4e0a\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u6309\u94ae\u200b\uff0c\u200b\u70b9\u51fb\u200b\u540e\u5f39\u200b\u51fa\u200b\u6587\u4ef6\u200b\u9009\u62e9\u200b\u5bf9\u8bdd\u6846\u200b\uff0c\u200b\u9009\u62e9\u200b\u6587\u4ef6\u200b\u540e\u200b\u5c06\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\u663e\u793a\u200b\u5728\u200b\u7a97\u53e3\u200b\u4e0a\u200b\u3002\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u6dfb\u52a0\u200b<code>rfd</code>\u200b\u5e93\u200b\u7684\u200b\u4f9d\u8d56\u200b\uff0c\u200b\u7528\u4e8e\u200b\u6587\u4ef6\u200b\u9009\u62e9\u200b\u5bf9\u8bdd\u6846\u200b\u3002</p> <pre><code>cargo add rfd\n</code></pre> <p>\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u7f16\u5199\u200b\u7528\u4e8e\u200b\u663e\u793a\u6587\u4ef6\u200b\u9009\u62e9\u200b\u7a97\u53e3\u200b\u7684\u200b\u51fd\u6570\u200b\u3002\u200b\u6ce8\u610f\u200b<code>rfd::AsyncFileDialog</code>\u200b\u4f1a\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u5f02\u5e38\u200b\uff0c\u200b\u56e0\u6b64\u200b\u9700\u8981\u200b\u81ea\u5b9a\u4e49\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n\nasync fn pick_file() -&gt; Result&lt;Arc&lt;String&gt;, Error&gt; {\n    let handler = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?; // If error, return DialogClosed error\n    load_file(handler.path()).await\n}\n</code></pre> <p>\u200b\u8c03\u6574\u200b\u7a97\u53e3\u200b\u5e03\u5c40\u200b\uff0c\u200b\u5728\u200b\u7a97\u53e3\u200b\u4e0a\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u6309\u94ae\u200b\u3002</p> <pre><code>impl Application for Editor {\n    // ...\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        let controls = row![button(\"Open\")];\n        // ...\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n}\n</code></pre> <p>\u200b\u4e4b\u540e\u200b\u9700\u8981\u200b\u6dfb\u52a0\u200b\u6309\u94ae\u200b\u7684\u200b\u70b9\u51fb\u200b\u4e8b\u4ef6\u5904\u7406\u200b\uff0c\u200b\u8c03\u7528\u200b<code>pick_file</code>\u200b\u51fd\u6570\u200b\u3002</p> <pre><code>enum Message {\n    // ...\n    OpenButtonPressed\n}\n\nimpl Application for Editor {\n    // ...\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            // ...\n            Message::OpenButtonPressed =&gt; {\n                Command::perform(pick_file(), Message::FileOpened)\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        let controls = row![button(\"Open\").on_press(Message::OpenButtonPressed)];\n        // ...\n    }\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use std::io;\nuse std::path::Path;\nuse std::sync::Arc;\n\nuse iced::{executor, Application, Command, Element, Length, Settings, Theme};\nuse iced::widget::{button, column, container, horizontal_space, row, text, text_editor};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor {\n    content: text_editor::Content,\n    error: Option&lt;Error&gt;\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;Arc&lt;String&gt;, Error&gt;),\n    OpenButtonPressed\n}\n\nimpl Application for Editor {\n    type Message = Message; // Define the type of messages\n    type Theme = Theme;\n    type Executor = executor::Default; // Engine for running async tasks\n    type Flags = ();\n\n    fn new(_flags: Self::Flags) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                content: text_editor::Content::new(),\n                error: None\n            },\n            Command::perform(\n            load_file(format!(\"{}/src/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n            Message::FileOpened\n            )\n        )\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok(result)) =&gt; {\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            Message::FileOpened(Err(error)) =&gt; {\n                self.error = Some(error);\n                Command::none()\n            },\n            Message::OpenButtonPressed =&gt; {\n                Command::perform(pick_file(), Message::FileOpened)\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n        let controls = row![button(\"Open\").on_press(Message::OpenButtonPressed)];\n\n        // Query cursor position\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![horizontal_space(Length::Fill), cursor_indicator];\n\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::Dark\n    }\n}\n\nasync fn pick_file() -&gt; Result&lt;Arc&lt;String&gt;, Error&gt; {\n    let file_handle = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?;\n    load_file(file_handle.path()).await\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;Arc&lt;String&gt;, Error&gt; {\n    tokio::fs::read_to_string(path)\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO)\n}\n\n#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n</code></pre>"},{"location":"coding/rust-gui/6-file-path-indicator/","title":"File Path Indicator","text":"<p>\u200b\u672c\u8282\u200b\u5728\u200b\u72b6\u6001\u680f\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u663e\u793a\u6587\u4ef6\u200b\u8def\u5f84\u200b\u7684\u200b\u6587\u672c\u200b\u63a7\u4ef6\u200b\u3002\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u5b58\u50a8\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u72b6\u6001\u200b\uff0c\u200b\u5e76\u200b\u5728\u200b\u521d\u59cb\u5316\u200b\u9636\u6bb5\u200b\u8bbe\u7f6e\u200b\u4e3a\u200b<code>None</code>\u3002</p> <pre><code>struct Editor {\n    path: Option&lt;PathBuf&gt;,\n    // ...\n}\n\nimpl Application for Editor {\n    // ...\n    fn new() -&gt; (Editor, Command&lt;Message&gt;) {\n        (\n            Editor {\n                path: None,\n                // ...\n            },\n            // ...\n        )\n    }\n    // ...\n}\n</code></pre> <p>Path\u200b\u4e0e\u200bPathBuf</p> <p><code>std::path::Path</code>\u200b\u548c\u200b<code>std::path::PathBuf</code>\u200b\u7684\u200b\u5173\u7cfb\u200b\u7c7b\u4f3c\u200b\u4e8e\u200b<code>&amp;str</code>\u200b\u548c\u200b<code>String</code>\u200b\u7684\u200b\u5173\u7cfb\u200b\u3002<code>Path</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5f15\u7528\u200b\uff0c<code>PathBuf</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u53ef\u53d8\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5b58\u50a8\u200b\u8def\u5f84\u200b\u72b6\u6001\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>PathBuf</code>\uff0c\u200b\u5f15\u7528\u200b\u8def\u5f84\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>Path</code>\u3002</p> <p>\u200b\u4e4b\u524d\u200b\u7684\u200b<code>load_file</code>\u200b\u51fd\u6570\u200b\u53ea\u200b\u8fd4\u56de\u200b\u4e86\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff0c\u200b\u6b64\u5904\u200b\u9700\u8981\u200b\u5c06\u200b\u6587\u4ef6\u200b\u7684\u200b\u8def\u5f84\u200b\u4e00\u540c\u200b\u8fd4\u56de\u200b\u3002\u200b\u6ce8\u610f\u200b\u51fd\u6570\u200b\u4e0d\u80fd\u200b\u8fd4\u56de\u200b\u5f15\u7528\u200b\uff0c\u200b\u56e0\u6b64\u200b\u9700\u8981\u200b\u5c06\u200b\u8def\u5f84\u200b\u8f6c\u6362\u200b\u4e3a\u200b<code>PathBuf</code>\u200b\u7c7b\u578b\u200b\u3002<code>and_then</code>\u200b\u65b9\u6cd5\u200b\u5904\u7406\u200b<code>Result</code>\u200b\u7684\u200b<code>Ok</code>\u200b\u503c\u200b\uff0c\u200b\u5728\u200b\u6210\u529f\u200b\u8bfb\u53d6\u200b\u6587\u4ef6\u200b\u540e\u200b\u5c06\u200b<code>Path</code>\u200b\u8f6c\u6362\u200b\u4e3a\u200b<code>PathBuf</code>\u200b\u5e76\u200b\u8fd4\u56de\u200b\u3002\u200b\u8c03\u7528\u200b<code>load_file</code>\u200b\u7684\u200b<code>pick_file</code>\u200b\u51fd\u6570\u200b\u4e5f\u200b\u9700\u8981\u200b\u4e00\u5e76\u200b\u4fee\u6539\u200b\uff0c\u200b\u540c\u65f6\u200b\u8fd4\u56de\u200b\u8def\u5f84\u200b\u548c\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\u3002</p> <pre><code>async fn pick_file() -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    // ...\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let content = tokio::fs::read_to_string(path.as_ref())\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO);\n    content.and_then(|content| Ok((path.as_ref().to_path_buf(), content)))\n}\n</code></pre> <p>\u200b\u5728\u200b\u4fee\u6539\u200b\u8bfb\u53d6\u200b\u6587\u4ef6\u200b\u7684\u200b\u51fd\u6570\u200b\u540e\u200b\uff0c\u200b\u9700\u8981\u200b\u4fee\u6539\u200b\u51fd\u6570\u200b\u56de\u8c03\u200b\u4e8b\u4ef6\u200b\u7684\u200b\u7c7b\u578b\u200b\u548c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5904\u7406\u51fd\u6570\u200b</p> \u200b\u4fee\u6539\u200b\u540e\u200b\u4fee\u6539\u200b\u524d\u200b <pre><code>enum Message {\n    // ...\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n}\n\nimpl Application for Editor {\n    // ...\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        match message {\n            // ...\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n        }\n    }\n}\n</code></pre> <pre><code>enum Message {\n    // ...\n    FileOpened(Result&lt;Arc&lt;String&gt;, Error&gt;),\n}\n\nimpl Application for Editor {\n    // ...\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        match message {\n            // ...\n            Message::FileOpened(Ok(result)) =&gt; {\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n        }\n    }\n}\n</code></pre> <p>\u200b\u6700\u540e\u200b\u5728\u200b\u72b6\u6001\u680f\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u6587\u672c\u200b\u63a7\u4ef6\u200b\u663e\u793a\u6587\u4ef6\u200b\u8def\u5f84\u200b\u5373\u53ef\u200b\u3002</p> <pre><code>impl Application for Editor {\n    // ...\n    fn view(&amp;mut self) -&gt; Element&lt;Message&gt; {\n        // ...\n        let path_indicator = match &amp;self.path {\n            None =&gt; text(\"\"),\n            Some(path) =&gt; text(path.to_string_lossy())\n        };\n        let status_bar = row![\n            path_indicator, // Add path indicator here\n            horizontal_space(Length::Fill),\n            cursor_indicator\n        ];\n        // ...\n    }\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use std::io;\nuse std::path::{Path, PathBuf};\nuse std::sync::Arc;\n\nuse iced::{executor, Application, Command, Element, Length, Settings, Theme};\nuse iced::widget::{button, column, container, horizontal_space, row, text, text_editor};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor {\n    path: Option&lt;PathBuf&gt;,\n    content: text_editor::Content,\n    error: Option&lt;Error&gt;\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    OpenButtonPressed\n}\n\nimpl Application for Editor {\n    type Message = Message; // Define the type of messages\n    type Theme = Theme;\n    type Executor = executor::Default; // Engine for running async tasks\n    type Flags = ();\n\n    fn new(_flags: Self::Flags) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                content: text_editor::Content::new(),\n                error: None,\n                path: None\n            },\n            Command::perform(\n            load_file(format!(\"{}/src/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n            Message::FileOpened\n            )\n        )\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            Message::FileOpened(Err(error)) =&gt; {\n                self.error = Some(error);\n                Command::none()\n            },\n            Message::OpenButtonPressed =&gt; {\n                Command::perform(pick_file(), Message::FileOpened)\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n        let controls = row![button(\"Open\").on_press(Message::OpenButtonPressed)];\n\n        // Query cursor position\n        let path_indicator = match &amp;self.path {\n            None =&gt; text(\"\"),\n            Some(path) =&gt; text(path.to_string_lossy())\n        };\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![\n            path_indicator,\n            horizontal_space(Length::Fill),\n            cursor_indicator\n        ];\n\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::Dark\n    }\n}\n\nasync fn pick_file() -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let file_handle = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?;\n    load_file(file_handle.path()).await\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let content = tokio::fs::read_to_string(path.as_ref())\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO);\n    content.and_then(|content| Ok((path.as_ref().to_path_buf(), content)))\n}\n\n#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n</code></pre>"},{"location":"coding/rust-gui/7-new-and-save/","title":"New and Save","text":"<p>\u200b\u672c\u200b\u8282\u200b\u6dfb\u52a0\u200b\u65b0\u5efa\u200b\u548c\u200b\u4fdd\u5b58\u200b\u6587\u4ef6\u200b\u7684\u200b\u529f\u80fd\u200b\u3002</p> <p>\u200b\u9996\u5148\u200b\uff0c\u200b\u5728\u200b<code>EditorEdit</code>\u200b\u6d88\u606f\u200b\u5904\u7406\u200b\u4e2d\u200b\u68c0\u67e5\u200b\u662f\u5426\u200b\u5bf9\u200b\u6587\u4ef6\u200b\u8fdb\u884c\u200b\u4e86\u200b\u4fee\u6539\u200b\uff0c\u200b\u5e76\u4e14\u200b\u8bb0\u5f55\u200b\u6587\u4ef6\u200b\u7684\u200b\u4fee\u6539\u200b\u72b6\u6001\u200b\u3002</p> <pre><code>struct Editor {\n    // ...\n    modified: bool\n}\n// ... In `update` function\n            Message::EditorEdit(action) =&gt; {\n                match &amp;action {\n                    text_editor::Action::Edit(_) =&gt; self.modified = true,\n                    _ =&gt; {}\n                }\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            // ...\n            Message::OpenButtonPressed =&gt; {\n                self.modified = false;\n                Command::perform(pick_file(), Message::FileOpened)\n            },\n</code></pre> <p>\u200b\u5176\u6b21\u200b\uff0c\u200b\u8c03\u6574\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u63a7\u4ef6\u200b\u7684\u200b\u4e0d\u540c\u200b\u663e\u793a\u200b\u72b6\u6001\u200b\uff1a</p> <ol> <li>\u200b\u5f53\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u4e3a\u200b\u7a7a\u65f6\u200b\uff0c\u200b\u4e3a\u200b\u65b0\u200b\u6587\u4ef6\u200b\uff0c\u200b\u663e\u793a\u200b\u201cNew File\u201d\u3002</li> <li>\u200b\u5f53\u200b\u6253\u5f00\u200b\u4e00\u4e2a\u200b\u6587\u4ef6\u200b\u65f6\u200b\uff0c\u200b\u663e\u793a\u6587\u4ef6\u200b\u8def\u5f84\u200b\u3002</li> <li>\u200b\u5f53\u200b\u6587\u4ef6\u200b\u88ab\u200b\u4fee\u6539\u200b\u540e\u200b\uff0c\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u540e\u200b\u52a0\u4e0a\u200b\u201c*\u201d\u3002</li> <li>\u200b\u5f53\u200b\u6253\u5f00\u200b\u6587\u4ef6\u200b\u51fa\u9519\u200b\u65f6\u200b\uff0c\u200b\u663e\u793a\u200b\u9519\u8bef\u4fe1\u606f\u200b\u3002</li> </ol> <pre><code>let path_indicator = if let Some(error) = &amp;self.error {\n    match error {\n        Error::DialogClosed =&gt; text(\"Dialog closed\"),\n        Error::IO(kind) =&gt; text(format!(\"I/O error: {:?}\", kind))\n    }\n} else {\n    let path_text = match &amp;self.path {\n        None =&gt; String::from(\"New file\"),\n        Some(path) =&gt; path.to_string_lossy().to_string()\n    };\n    let suffix = if self.modified { \"*\" } else { \"\" };\n    text(format!(\"{}{}\", path_text, suffix))\n};\n</code></pre> <p>\u200b\u5728\u200b\u521b\u5efa\u200b\u6587\u4ef6\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u6e05\u7a7a\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u548c\u200b\u5185\u5bb9\u200b\uff0c\u200b\u4ee5\u53ca\u200b\u6e05\u7a7a\u200b\u9519\u8bef\u4fe1\u606f\u200b\u3002\u200b\u52a0\u5165\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u6d88\u606f\u200b\u7c7b\u578b\u200b<code>NewButtonPressed</code>\uff0c\u200b\u7531\u200b\u4e00\u4e2a\u200b\u6309\u94ae\u200b\u89e6\u53d1\u200b\uff0c\u200b\u5728\u200b<code>update</code>\u200b\u51fd\u6570\u200b\u4e2d\u200b\u6267\u884c\u200b\u8fd9\u4e2a\u200b\u903b\u8f91\u200b\u3002</p> \u200b\u6d88\u606f\u200b\u7c7b\u578b\u200b\u58f0\u660e\u200b\u6d88\u606f\u200b\u89e6\u53d1\u200b\u6267\u884c\u200b\u903b\u8f91\u200b <pre><code>enum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    NewButtonPressed,\n    OpenButtonPressed\n}\n</code></pre> <pre><code>// ... In `view` function\nlet controls = row![\n    button(\"New\").on_press(Message::NewButtonPressed),\n    button(\"Open\").on_press(Message::OpenButtonPressed)\n];\n</code></pre> <pre><code>// ... In matching logic in `update` function\nMessage::NewButtonPressed =&gt; {\n    self.content = text_editor::Content::new();\n    self.error = None;\n    self.path = None;\n    self.modified = false;\n    Command::none()\n},\n// ...\n</code></pre> <p>\u200b\u63a5\u4e0b\u6765\u200b\u5904\u7406\u200b\u4fdd\u5b58\u200b\u6587\u4ef6\u200b\u7684\u200b\u903b\u8f91\u200b\uff0c\u200b\u5f53\u200b\u5b58\u5728\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u65f6\u200b\uff0c\u200b\u4fdd\u5b58\u200b\u6587\u4ef6\u200b\uff0c\u200b\u5426\u5219\u200b\u6253\u5f00\u200b\u6587\u4ef6\u200b\u9009\u62e9\u200b\u5bf9\u8bdd\u6846\u200b\u3002</p> <pre><code>async fn save_file(path: Option&lt;PathBuf&gt;, content: String) -&gt; Result&lt;PathBuf, Error&gt; {\n    let path = if let Some(path) = path {\n        path\n    } else {\n        rfd::AsyncFileDialog::new()\n            .set_title(\"Save the file to...\")\n            .save_file()\n            .await\n            .ok_or(Error::DialogClosed)?\n            .path()\n            .to_path_buf()\n    };\n    tokio::fs::write(&amp;path, content)\n        .await\n        .map_err(|err| err.kind())\n        .map_err(Error::IO)\n        .map(|_| path)\n}\n</code></pre> <p>\u200b\u5728\u200b\u4fdd\u5b58\u200b\u6587\u4ef6\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b</p> <ol> <li>\u200b\u68c0\u67e5\u200b\u6587\u4ef6\u200b\u7684\u200b\u4fee\u6539\u200b\u72b6\u6001\u200b\uff0c\u200b\u5982\u679c\u200b\u6587\u4ef6\u200b\u6ca1\u6709\u200b\u4fee\u6539\u200b\uff0c\u200b\u4e0d\u200b\u6267\u884c\u200b\u4fdd\u5b58\u200b\u64cd\u4f5c\u200b\u3002</li> <li>\u200b\u68c0\u67e5\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u662f\u5426\u200b\u4e3a\u7a7a\u200b\uff0c\u200b\u5982\u679c\u200b\u4e3a\u7a7a\u200b\uff0c\u200b\u6253\u5f00\u200b\u6587\u4ef6\u200b\u9009\u62e9\u200b\u5bf9\u8bdd\u6846\u200b\uff0c\u200b\u5426\u5219\u200b\u76f4\u63a5\u200b\u4fdd\u5b58\u200b\u6587\u4ef6\u200b\u3002</li> <li>\u200b\u52a0\u5165\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u6d88\u606f\u200b\u7c7b\u578b\u200b<code>SaveButtonPressed</code>\uff0c\u200b\u7531\u200b\u4e00\u4e2a\u200b\u6309\u94ae\u200b\u89e6\u53d1\u200b\uff0c\u200b\u5728\u200b<code>update</code>\u200b\u51fd\u6570\u200b\u4e2d\u200b\u6267\u884c\u200b\u8fd9\u4e2a\u200b\u903b\u8f91\u200b\u3002</li> </ol> \u200b\u6d88\u606f\u200b\u7c7b\u578b\u200b\u58f0\u660e\u200b\u6d88\u606f\u200b\u89e6\u53d1\u200b\u6267\u884c\u200b\u903b\u8f91\u200b <pre><code>enum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    FileSaved(Result&lt;PathBuf, Error&gt;),\n    NewButtonPressed,\n    OpenButtonPressed,\n    SaveButtonPressed\n}\n</code></pre> <pre><code>// ... In `view` function\nlet controls = row![\n    button(\"New\").on_press(Message::NewButtonPressed),\n    button(\"Open\").on_press(Message::OpenButtonPressed),\n    button(\"Save\").on_press(Message::SaveButtonPressed)\n];\n</code></pre> <pre><code>// ... In matching logic in `update` function\nMessage::FileSaved(Ok(path)) =&gt; {\n    self.path = Some(path);\n    self.modified = false;\n    Command::none()\n},\nMessage::FileOpened(Err(error)) | Message::FileSaved(Err(error)) =&gt; {\n    self.error = Some(error);\n    Command::none()\n},\n// ...\nMessage::SaveButtonPressed =&gt; {\n    let content = self.content.text();\n    match self.modified {\n        false =&gt; Command::none(),\n        true =&gt; Command::perform(\n            save_file(self.path.clone(), content),\n            Message::FileSaved\n        )\n    }\n}\n</code></pre> <p>\u200b\u6700\u540e\u200b\uff0c\u200b\u8c03\u6574\u200b\u4e09\u4e2a\u200b\u6309\u94ae\u200b\u4e4b\u95f4\u200b\u7684\u200b\u95f4\u8ddd\u200b</p> <pre><code>let controls = row![\n    button(\"New\").on_press(Message::NewButtonPressed),\n    button(\"Open\").on_press(Message::OpenButtonPressed),\n    button(\"Save\").on_press(Message::SaveButtonPressed)\n].spacing(10);\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use std::io;\nuse std::path::{Path, PathBuf};\nuse std::sync::Arc;\n\nuse iced::{executor, Application, Command, Element, Length, Settings, Theme};\nuse iced::widget::{button, column, container, horizontal_space, row, text, text_editor};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor {\n    path: Option&lt;PathBuf&gt;,\n    content: text_editor::Content,\n    modified: bool,\n    error: Option&lt;Error&gt;\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    FileSaved(Result&lt;PathBuf, Error&gt;),\n    NewButtonPressed,\n    OpenButtonPressed,\n    SaveButtonPressed\n}\n\nimpl Application for Editor {\n    type Message = Message; // Define the type of messages\n    type Theme = Theme;\n    type Executor = executor::Default; // Engine for running async tasks\n    type Flags = ();\n\n    fn new(_flags: Self::Flags) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                content: text_editor::Content::new(),\n                error: None,\n                modified: false,\n                path: None\n            },\n            Command::perform(\n            load_file(format!(\"{}/src/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n            Message::FileOpened\n            )\n        )\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                match &amp;action {\n                    text_editor::Action::Edit(_) =&gt; self.modified = true,\n                    _ =&gt; {}\n                }\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            Message::FileSaved(Ok(path)) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                Command::none()\n            },\n            Message::FileOpened(Err(error)) | Message::FileSaved(Err(error)) =&gt; {\n                self.error = Some(error);\n                Command::none()\n            },\n            Message::NewButtonPressed =&gt; {\n                self.content = text_editor::Content::new();\n                self.error = None;\n                self.path = None;\n                self.modified = false;\n                Command::none()\n            },\n            Message::OpenButtonPressed =&gt; {\n                self.modified = false;\n                Command::perform(pick_file(), Message::FileOpened)\n            },\n            Message::SaveButtonPressed =&gt; {\n                let content = self.content.text();\n                match self.modified {\n                    false =&gt; Command::none(),\n                    true =&gt; Command::perform(\n                        save_file(self.path.clone(), content),\n                        Message::FileSaved\n                    )\n                }\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n        let controls = row![\n            button(\"New\").on_press(Message::NewButtonPressed),\n            button(\"Open\").on_press(Message::OpenButtonPressed),\n            button(\"Save\").on_press(Message::SaveButtonPressed)\n        ].spacing(10);\n\n        // Query cursor position\n        let path_indicator = if let Some(error) = &amp;self.error {\n            match error {\n                Error::DialogClosed =&gt; text(\"Dialog closed\"),\n                Error::IO(kind) =&gt; text(format!(\"I/O error: {:?}\", kind))\n            }\n        } else {\n            let path_text = match &amp;self.path {\n                None =&gt; String::from(\"New file\"),\n                Some(path) =&gt; path.to_string_lossy().to_string()\n            };\n            let suffix = if self.modified { \"*\" } else { \"\" };\n            text(format!(\"{}{}\", path_text, suffix))\n        };\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![\n            path_indicator,\n            horizontal_space(Length::Fill),\n            cursor_indicator\n        ];\n\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::Dark\n    }\n}\n\nasync fn pick_file() -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let file_handle = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?;\n    load_file(file_handle.path()).await\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let content = tokio::fs::read_to_string(path.as_ref())\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO);\n    content.and_then(|content| Ok((path.as_ref().to_path_buf(), content)))\n}\n\nasync fn save_file(path: Option&lt;PathBuf&gt;, content: String) -&gt; Result&lt;PathBuf, Error&gt; {\n    let path = if let Some(path) = path {\n        path\n    } else {\n        rfd::AsyncFileDialog::new()\n            .set_title(\"Save the file to...\")\n            .save_file()\n            .await\n            .ok_or(Error::DialogClosed)?\n            .path()\n            .to_path_buf()\n    };\n    tokio::fs::write(&amp;path, content)\n        .await\n        .map_err(|err| err.kind())\n        .map_err(Error::IO)\n        .map(|_| path)\n}\n\n#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n</code></pre>"},{"location":"coding/rust-gui/8-button-prettify/","title":"Button Prettify","text":"<p>\u200b\u5728\u200b\u672c\u8282\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u628a\u200b\u6309\u94ae\u200b\u7684\u200b\u6587\u672c\u200b\u66ff\u6362\u200b\u4e3a\u200b\u56fe\u6807\u200b\uff0c\u200b\u5e76\u4e14\u200b\u6dfb\u52a0\u200b\u6587\u672c\u200b\u60ac\u6d6e\u200b\u63d0\u793a\u200b\u3002</p> <p>\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u521b\u5efa\u200b\u5305\u542b\u200b\u56fe\u6807\u200b\u7684\u200b\u5b57\u4f53\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5728\u200bFontello\u200b\u4e0a\u200b\u9009\u62e9\u200b\u56fe\u6807\u200b\uff0c\u200b\u7136\u540e\u200b\u4e0b\u8f7d\u200b\u5b57\u4f53\u200b\u6587\u4ef6\u200b\u3002\u200b\u5c06\u200b<code>ttf</code>\u200b\u7248\u672c\u200b\u7684\u200b\u5b57\u4f53\u200b\u5b58\u653e\u200b\u5728\u200b\u9879\u76ee\u200b\u4e0b\u200b\u7684\u200b<code>fonts/editor-icon.ttf</code>\u200b\u4e2d\u200b\u3002</p> <p>\u200b\u7136\u540e\u200b\u5728\u200b\u4ee3\u7801\u200b\u4e2d\u200b\u52a0\u8f7d\u200b\u5b57\u4f53\u200b\u6587\u4ef6\u200b\uff0c\u200b\u5728\u200b<code>iced::Settings</code>\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u5b57\u4f53\u200b\uff1a</p> <pre><code>fn main() -&gt; iced::Result {\n    Editor::run(Settings {\n        fonts: vec![include_bytes!(\"../fonts/editor-icon.ttf\").as_slice().into()],\n        ..Default::default() // Expand the default settings\n    })\n}\n</code></pre> <p>\u200b\u52a0\u8f7d\u200b\u5b57\u4f53\u200b\u540e\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u6309\u94ae\u200b\u7684\u200b\u8f93\u5165\u200b\u6587\u672c\u200b\u66ff\u6362\u200b\u4e3a\u200b\u56fe\u6807\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>text</code>\u200b\u63a7\u4ef6\u200b\u521b\u5efa\u200b\u56fe\u6807\u200b\u3002\u200b\u5728\u200b\u7f51\u9875\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u8bfb\u53d6\u200b\u5230\u200b\u5bf9\u5e94\u200b\u65b0\u5efa\u200b\u3001\u200b\u6253\u5f00\u200b\u3001\u200b\u4fdd\u5b58\u200b\u7684\u200bUnicode\u200b\u7f16\u7801\u200b\u5206\u522b\u200b\u4e3a\u200b<code>\\u{E800}</code>\u3001<code>\\u{F115}</code>\u3001<code>\\u{E801}</code>\u3002</p> <pre><code>fn toolbar_button&lt;'a&gt;(description: &amp;str, callback: Message) -&gt; Element&lt;'a, Message&gt; {\n    let font = Font::with_name(\"editor-icon\");\n\n    let icon = text(match description {\n        \"new\" =&gt; '\\u{E800}',\n        \"open\" =&gt; '\\u{F115}',\n        \"save\" =&gt; '\\u{E801}',\n        _ =&gt; ' '\n    }).font(font);\n\n    button(container(icon)\n        .width(30)  // Set the width of the button\n        .center_x() // Center the icon\n    ).on_press(callback).into()\n}\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>button_icon</code>\u200b\u51fd\u6570\u200b\u66ff\u6362\u200b\u6309\u94ae\u200b\u539f\u672c\u200b\u7684\u200b\u8f93\u5165\u200b</p> <pre><code>// ... In `view` function\nlet controls = row![\n    toolbar_button(\"new\", Message::NewButtonPressed),\n    toolbar_button(\"open\", Message::OpenButtonPressed),\n    toolbar_button(\"save\", Message::SaveButtonPressed)\n].spacing(10);\n</code></pre> <p>\u200b\u6700\u540e\u200b\uff0c\u200b\u5b9e\u73b0\u200b\u60ac\u6d6e\u200b\u63d0\u793a\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>Tooltip</code>\u200b\u63a7\u4ef6\u200b\u5305\u88f9\u200b\u6309\u94ae\u200b\u5373\u53ef\u200b\u3002\u200b\u4e3a\u4e86\u200b\u7f8e\u89c2\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>style</code>\u200b\u65b9\u6cd5\u200b\u8bbe\u7f6e\u200b\u63d0\u793a\u6846\u200b\u7684\u200b\u6837\u5f0f\u200b\u3002</p> <pre><code>// ... In `view` function\nlet controls = row![\n    toolbar_button(\"New\", Message::NewButtonPressed),\n    toolbar_button(\"Open\", Message::OpenButtonPressed),\n    toolbar_button(\"Save\", Message::SaveButtonPressed)\n].spacing(10);\n\n// ... In the outer scope\nfn toolbar_button&lt;'a&gt;(description: &amp;str, callback: Message) -&gt; Element&lt;'a, Message&gt; {\n    let font = Font::with_name(\"editor-icon\");\n    let lower = description.to_lowercase();\n    let icon = text(match lower.as_str() {\n        \"new\" =&gt; '\\u{E800}',\n        \"open\" =&gt; '\\u{F115}',\n        \"save\" =&gt; '\\u{E801}',\n        _ =&gt; ' '\n    }).font(font);\n\n    tooltip(\n        button(container(icon)\n            .width(30)  // Set the width of the button\n            .center_x() // Center the icon\n        ).on_press(callback),\n        description, tooltip::Position::FollowCursor\n    ).style(theme::Container::Box).into() // Set the style of the tooltip\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use std::io;\nuse std::path::{Path, PathBuf};\nuse std::sync::Arc;\n\nuse iced::{executor, theme, Application, Command, Element, Font, Length, Settings, Theme};\nuse iced::widget::{button, column, container, horizontal_space, row, text, text_editor, tooltip};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings {\n        fonts: vec![include_bytes!(\"../fonts/editor-icon.ttf\").as_slice().into()],\n        ..Default::default()\n    })\n}\n\nstruct Editor {\n    path: Option&lt;PathBuf&gt;,\n    content: text_editor::Content,\n    modified: bool,\n    error: Option&lt;Error&gt;\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    FileSaved(Result&lt;PathBuf, Error&gt;),\n    NewButtonPressed,\n    OpenButtonPressed,\n    SaveButtonPressed\n}\n\nimpl Application for Editor {\n    type Message = Message; // Define the type of messages\n    type Theme = Theme;\n    type Executor = executor::Default; // Engine for running async tasks\n    type Flags = ();\n\n    fn new(_flags: Self::Flags) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                content: text_editor::Content::new(),\n                error: None,\n                modified: false,\n                path: None\n            },\n            Command::perform(\n            load_file(format!(\"{}/src/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n            Message::FileOpened\n            )\n        )\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                match &amp;action {\n                    text_editor::Action::Edit(_) =&gt; self.modified = true,\n                    _ =&gt; {}\n                }\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            Message::FileSaved(Ok(path)) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                Command::none()\n            },\n            Message::FileOpened(Err(error)) | Message::FileSaved(Err(error)) =&gt; {\n                self.error = Some(error);\n                Command::none()\n            },\n            Message::NewButtonPressed =&gt; {\n                self.content = text_editor::Content::new();\n                self.error = None;\n                self.path = None;\n                self.modified = false;\n                Command::none()\n            },\n            Message::OpenButtonPressed =&gt; {\n                self.modified = false;\n                Command::perform(pick_file(), Message::FileOpened)\n            },\n            Message::SaveButtonPressed =&gt; {\n                let content = self.content.text();\n                match self.modified {\n                    false =&gt; Command::none(),\n                    true =&gt; Command::perform(\n                        save_file(self.path.clone(), content),\n                        Message::FileSaved\n                    )\n                }\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n        let controls = row![\n            toolbar_button(\"New\", Message::NewButtonPressed),\n            toolbar_button(\"Open\", Message::OpenButtonPressed),\n            toolbar_button(\"Save\", Message::SaveButtonPressed)\n        ].spacing(10);\n\n        // Query cursor position\n        let path_indicator = if let Some(error) = &amp;self.error {\n            match error {\n                Error::DialogClosed =&gt; text(\"Dialog closed\"),\n                Error::IO(kind) =&gt; text(format!(\"I/O error: {:?}\", kind))\n            }\n        } else {\n            let path_text = match &amp;self.path {\n                None =&gt; String::from(\"New file\"),\n                Some(path) =&gt; path.to_string_lossy().to_string()\n            };\n            let suffix = if self.modified { \"*\" } else { \"\" };\n            text(format!(\"{}{}\", path_text, suffix))\n        };\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![\n            path_indicator,\n            horizontal_space(Length::Fill),\n            cursor_indicator\n        ];\n\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::Dark\n    }\n}\n\nasync fn pick_file() -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let file_handle = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?;\n    load_file(file_handle.path()).await\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let content = tokio::fs::read_to_string(path.as_ref())\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO);\n    content.and_then(|content| Ok((path.as_ref().to_path_buf(), content)))\n}\n\nasync fn save_file(path: Option&lt;PathBuf&gt;, content: String) -&gt; Result&lt;PathBuf, Error&gt; {\n    let path = if let Some(path) = path {\n        path\n    } else {\n        rfd::AsyncFileDialog::new()\n            .set_title(\"Save the file to...\")\n            .save_file()\n            .await\n            .ok_or(Error::DialogClosed)?\n            .path()\n            .to_path_buf()\n    };\n    tokio::fs::write(&amp;path, content)\n        .await\n        .map_err(|err| err.kind())\n        .map_err(Error::IO)\n        .map(|_| path)\n}\n\nfn toolbar_button&lt;'a&gt;(description: &amp;str, callback: Message) -&gt; Element&lt;'a, Message&gt; {\n    let font = Font::with_name(\"editor-icon\");\n    let lower = description.to_lowercase();\n    let icon = text(match lower.as_str() {\n        \"new\" =&gt; '\\u{E800}',\n        \"open\" =&gt; '\\u{F115}',\n        \"save\" =&gt; '\\u{E801}',\n        _ =&gt; ' '\n    }).font(font);\n\n    tooltip(\n        button(container(icon)\n            .width(30)  // Set the width of the button\n            .center_x() // Center the icon\n        ).on_press(callback),\n        description, tooltip::Position::FollowCursor\n    ).style(theme::Container::Box).into()\n}\n\n#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n</code></pre>"},{"location":"coding/rust-gui/9-syntax-highlighting/","title":"Syntax Highlighting","text":"<p>Keywords: <code>iced::widget::pick_list</code></p> <p>\u200b\u672c\u200b\u8282\u200b\u6dfb\u52a0\u200b\u4ee3\u7801\u200b\u7f16\u8f91\u5668\u200b\u7684\u200b\u8bed\u6cd5\u200b\u9ad8\u4eae\u200b\u529f\u80fd\u200b\u3002</p> <p>\u200b\u9996\u5148\u200b\uff0c\u200b\u4fee\u6539\u200b\u7f16\u8f91\u5668\u200b\u7684\u200b\u9ed8\u8ba4\u200b\u5b57\u4f53\u200b\u4e3a\u200b\u7b49\u200b\u5bbd\u200b\u5b57\u4f53\u200b\uff0c\u200b\u4ee5\u4fbf\u200b\u66f4\u597d\u200b\u5730\u200b\u663e\u793a\u200b\u4ee3\u7801\u200b\u3002</p> <pre><code>// ... In `view` function\nlet editor = text_editor(&amp;self.content)\n    .on_edit(Message::EditorEdit)\n    .font(Font::MONOSPACE);\n</code></pre> <p>\u200b\u63a5\u4e0b\u6765\u200b\uff0c\u200b\u6dfb\u52a0\u200b\u9ad8\u4eae\u200b\u5904\u7406\u200b\u7684\u200b\u903b\u8f91\u200b\u3002\u200b\u9ad8\u4eae\u200b\u529f\u80fd\u200b\u7531\u200b<code>iced</code>\u200b\u7684\u200b<code>highlighter</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b<code>Cargo.toml</code>\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u4f9d\u8d56\u200b\u3002</p> <pre><code>[dependencies]\niced = { git = \"https://github.com/iced-rs/iced.git\", rev = \"refs/tags/text-editor\", features = [ \"highlighter\" ] }\n</code></pre> <p>\u200b\u5728\u200b\u6587\u672c\u6846\u200b\u7684\u200b<code>highlight</code>\u200b\u65b9\u6cd5\u200b\u4e2d\u200b\u8bbe\u7f6e\u200b\u9ad8\u4eae\u200b\u3002<code>highlight</code>\u200b\u65b9\u6cd5\u200b\u9700\u8981\u200b\u6307\u5b9a\u200b\u914d\u8272\u200b\u65b9\u6848\u200b<code>theme</code>\u200b\u548c\u200b\u8bed\u6cd5\u200b<code>extension</code>\u3002<code>extension</code>\u200b\u53ef\u4ee5\u200b\u6839\u636e\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u7684\u200b\u540e\u7f00\u540d\u200b\u81ea\u52a8\u8bc6\u522b\u200b\uff0c\u200b\u5982\u679c\u200b\u6ca1\u6709\u200b\u540e\u7f00\u540d\u200b\uff0c\u200b\u5219\u200b\u9ed8\u8ba4\u200b\u4e3a\u200b<code>rs</code>\u3002</p> <pre><code>let editor = text_editor(&amp;self.content)\n    .on_edit(Message::EditorEdit)\n    .highlight::&lt;Highlighter&gt;( // use iced::highlighter::Highlighter\n        highlighter::Settings {\n            theme: highlighter::Theme::SolarizedDark, // Set the theme\n            extension: self.path\n                .as_ref()\n                .and_then(|path| path.extension()?.to_str())\n                .unwrap_or(\"rs\") // If extension is not found, use `rs`\n                .to_string()\n        }, |highlighter, _theme| {\n            highlighter.to_format()\n        }\n    )\n    .font(Font::MONOSPACE);\n</code></pre> <p>\u200b\u63a5\u4e0b\u6765\u200b\u6dfb\u52a0\u200b\u9ad8\u4eae\u200b\u98ce\u683c\u200b\u7684\u200b\u9009\u62e9\u200b\u529f\u80fd\u200b\u3002\u200b\u9996\u5148\u200b\u5728\u200b<code>Editor</code>\u200b\u4e2d\u200b\u6dfb\u52a0\u200b<code>theme</code>\u200b\u72b6\u6001\u200b\u4f5c\u4e3a\u200b\u5f53\u524d\u200b\u7684\u200b\u4e3b\u9898\u200b\uff0c\u200b\u5e76\u4e14\u200b\u8bbe\u7f6e\u200b\u521d\u59cb\u5316\u200b\u72b6\u6001\u200b\u3002</p> <pre><code>struct Editor {\n    // ...\n    theme: highlighter::Theme\n}\n// ... In `new` function\nSelf {\n    // ...\n    theme: highlighter::Theme::SolarizedDark\n},\n// ... In `view` function\nhighlighter::Settings {\n    theme: self.theme,\n    // ...\n}\n</code></pre> <p>\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b<code>pick_list</code>\u200b\u63a7\u4ef6\u200b\u7528\u4e8e\u200b\u9009\u62e9\u200b\u9ad8\u4eae\u200b\u98ce\u683c\u200b\uff0c<code>pick_list</code>\u200b\u9700\u8981\u200b\u8bbe\u7f6e\u200b\u9009\u9879\u200b\u3001\u200b\u5f53\u524d\u200b\u9009\u4e2d\u200b\u9879\u200b\u548c\u200b\u89e6\u53d1\u200b\u4e8b\u4ef6\u200b\u3002\u200b\u6bcf\u6b21\u200b\u66f4\u65b0\u200b\u65f6\u200b\u90fd\u200b\u4f1a\u200b\u4ee5\u200b\u9009\u4e2d\u200b\u9879\u4e3a\u200b\u53c2\u6570\u200b\u89e6\u53d1\u200b\u4e8b\u4ef6\u200b\u3002</p> <pre><code>let controls = row![\n    toolbar_button(\"New\", Message::NewButtonPressed),\n    toolbar_button(\"Open\", Message::OpenButtonPressed),\n    toolbar_button(\"Save\", Message::SaveButtonPressed),\n    horizontal_space(Length::Fill),\n    pick_list(highlighter::Theme::ALL, Some(self.theme), Message::ThemeChanged)\n].spacing(10);\n</code></pre> <p>\u200b\u540c\u65f6\u200b\u66f4\u65b0\u200b<code>ThemeChanged</code>\u200b\u4e8b\u4ef6\u200b\u7684\u200b\u5904\u7406\u200b\u903b\u8f91\u200b</p> <pre><code>// ... In `update` function\nMessage::ThemeChanged(theme) =&gt; {\n    self.theme = theme;\n    Command::none()\n}\n</code></pre> <p>\u200b\u5185\u7f6e\u200b\u7684\u200b\u90e8\u5206\u200b\u4e3b\u9898\u200b\u5e76\u200b\u4e0d\u662f\u200b\u6697\u8272\u200b\u4e3b\u9898\u200b\uff0c\u200b\u9700\u8981\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u6839\u636e\u200b\u4e3b\u9898\u200b\u7684\u200b\u4eae\u200b\u6697\u200b\u5207\u6362\u200b\u7a97\u53e3\u200b\u914d\u8272\u200b\u7684\u200b\u529f\u80fd\u200b\uff1a</p> <pre><code>// ... In `impl Application for Editor`\nfn theme(&amp;self) -&gt; Theme {\n    if self.theme.is_dark() {\n        Theme::Dark\n    } else {\n        Theme::Light\n    }\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use std::io;\nuse std::path::{Path, PathBuf};\nuse std::sync::Arc;\n\nuse iced::{executor, theme, Application, Command, Element, Font, Length, Settings, Theme};\nuse iced::highlighter::{self, Highlighter};\nuse iced::widget::{button, column, container, horizontal_space, pick_list, row, text, text_editor, tooltip};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings {\n        fonts: vec![include_bytes!(\"../fonts/editor-icon.ttf\").as_slice().into()],\n        ..Default::default()\n    })\n}\n\nstruct Editor {\n    path: Option&lt;PathBuf&gt;,\n    content: text_editor::Content,\n    modified: bool,\n    error: Option&lt;Error&gt;,\n    theme: highlighter::Theme\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    FileSaved(Result&lt;PathBuf, Error&gt;),\n    ThemeChanged(highlighter::Theme),\n    NewButtonPressed,\n    OpenButtonPressed,\n    SaveButtonPressed\n}\n\nimpl Application for Editor {\n    type Message = Message; // Define the type of messages\n    type Theme = Theme;\n    type Executor = executor::Default; // Engine for running async tasks\n    type Flags = ();\n\n    fn new(_flags: Self::Flags) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                content: text_editor::Content::new(),\n                error: None,\n                modified: false,\n                path: None,\n                theme: highlighter::Theme::SolarizedDark\n            },\n            Command::perform(\n            load_file(format!(\"{}/src/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n            Message::FileOpened\n            )\n        )\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                match &amp;action {\n                    text_editor::Action::Edit(_) =&gt; self.modified = true,\n                    _ =&gt; {}\n                }\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            Message::FileSaved(Ok(path)) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                Command::none()\n            },\n            Message::FileOpened(Err(error)) | Message::FileSaved(Err(error)) =&gt; {\n                self.error = Some(error);\n                Command::none()\n            },\n            Message::ThemeChanged(theme) =&gt; {\n                self.theme = theme;\n                Command::none()\n            },\n            Message::NewButtonPressed =&gt; {\n                self.content = text_editor::Content::new();\n                self.error = None;\n                self.path = None;\n                self.modified = false;\n                Command::none()\n            },\n            Message::OpenButtonPressed =&gt; {\n                self.modified = false;\n                Command::perform(pick_file(), Message::FileOpened)\n            },\n            Message::SaveButtonPressed =&gt; {\n                let content = self.content.text();\n                match self.modified {\n                    false =&gt; Command::none(),\n                    true =&gt; Command::perform(\n                        save_file(self.path.clone(), content),\n                        Message::FileSaved\n                    )\n                }\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content)\n            .on_edit(Message::EditorEdit)\n            .highlight::&lt;Highlighter&gt;(\n                highlighter::Settings {\n                    theme: self.theme,\n                    extension: self.path\n                        .as_ref()\n                        .and_then(|path| path.extension()?.to_str())\n                        .unwrap_or(\"rs\")\n                        .to_string()\n                }, |highlighter, _theme| {\n                    highlighter.to_format()\n                }\n            )\n            .font(Font::MONOSPACE);\n        let controls = row![\n            toolbar_button(\"New\", Message::NewButtonPressed),\n            toolbar_button(\"Open\", Message::OpenButtonPressed),\n            toolbar_button(\"Save\", Message::SaveButtonPressed),\n            horizontal_space(Length::Fill),\n            pick_list(highlighter::Theme::ALL, Some(self.theme), Message::ThemeChanged)\n        ].spacing(10);\n\n        // Query cursor position\n        let path_indicator = if let Some(error) = &amp;self.error {\n            match error {\n                Error::DialogClosed =&gt; text(\"Dialog closed\"),\n                Error::IO(kind) =&gt; text(format!(\"I/O error: {:?}\", kind))\n            }\n        } else {\n            let path_text = match &amp;self.path {\n                None =&gt; String::from(\"New file\"),\n                Some(path) =&gt; path.to_string_lossy().to_string()\n            };\n            let suffix = if self.modified { \"*\" } else { \"\" };\n            text(format!(\"{}{}\", path_text, suffix))\n        };\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![\n            path_indicator,\n            horizontal_space(Length::Fill),\n            cursor_indicator\n        ];\n\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        if self.theme.is_dark() {\n            Theme::Dark\n        } else {\n            Theme::Light\n        }\n    }\n}\n\nasync fn pick_file() -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let file_handle = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?;\n    load_file(file_handle.path()).await\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let content = tokio::fs::read_to_string(path.as_ref())\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO);\n    content.and_then(|content| Ok((path.as_ref().to_path_buf(), content)))\n}\n\nasync fn save_file(path: Option&lt;PathBuf&gt;, content: String) -&gt; Result&lt;PathBuf, Error&gt; {\n    let path = if let Some(path) = path {\n        path\n    } else {\n        rfd::AsyncFileDialog::new()\n            .set_title(\"Save the file to...\")\n            .save_file()\n            .await\n            .ok_or(Error::DialogClosed)?\n            .path()\n            .to_path_buf()\n    };\n    tokio::fs::write(&amp;path, content)\n        .await\n        .map_err(|err| err.kind())\n        .map_err(Error::IO)\n        .map(|_| path)\n}\n\nfn toolbar_button&lt;'a&gt;(description: &amp;str, callback: Message) -&gt; Element&lt;'a, Message&gt; {\n    let font = Font::with_name(\"editor-icon\");\n    let lower = description.to_lowercase();\n    let icon = text(match lower.as_str() {\n        \"new\" =&gt; '\\u{E800}',\n        \"open\" =&gt; '\\u{F115}',\n        \"save\" =&gt; '\\u{E801}',\n        _ =&gt; ' '\n    }).font(font);\n\n    tooltip(\n        button(container(icon)\n            .width(30)  // Set the width of the button\n            .center_x() // Center the icon\n        ).on_press(callback),\n        description, tooltip::Position::FollowCursor\n    ).style(theme::Container::Box).into()\n}\n\n#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n</code></pre>"},{"location":"japanese/","title":"\u65e5\u8bed","text":"<p>\u200b\u65e5\u8bed\u200b\u5b66\u4e60\u200b\u7b14\u8bb0\u200b</p> <ol> <li>\u200b\u65e5\u8bed\u200b(1) - 2023-2024\u200b\u5b66\u5e74\u200b\u79cb\u5b63\u200b\u5b66\u671f\u200b<ul> <li>\u200b\u77ed\u8bed\u200b\u4e0e\u200b\u8868\u8fbe\u200b</li> <li>\u200b\u8bed\u6cd5\u200b</li> <li>\u200b\u5355\u8bcd\u8868\u200b</li> </ul> </li> </ol>"},{"location":"japanese/japanese-1-grammar/","title":"\u65e5\u8bed\u200b(1)\u200b\u8bed\u6cd5","text":""},{"location":"japanese/japanese-1-grammar/#_1","title":"\u8bcd\u6cd5","text":"<p>\u200b\u65e5\u8bed\u200b\u4e2d\u200b\u7684\u200b\u5355\u8bcd\u200b\u8bcd\u6027\u200b\u4e3b\u8981\u200b\u6709\u200b\u540d\u8bcd\u200b\u3001\u200b\u52a8\u8bcd\u200b\u3001\u200b\u5f62\u5bb9\u8bcd\u200b\u3001\u200b\u526f\u8bcd\u200b\u3001\u200b\u52a9\u8bcd\u200b\u3001\u200b\u63a5\u7eed\u8bcd\u200b\u7b49\u200b\u3002\u200b\u5176\u4e2d\u200b\uff0c\u200b\u52a8\u8bcd\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3001III\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3001III\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3002\u200b\u5f62\u5bb9\u8bcd\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u3001II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u3002</p> <ul> <li>\u200b\u540d\u8bcd\u200b\u3001\u200b\u4ee3\u8bcd\u200b\u3001\u200b\u6570\u8bcd\u200b\u79f0\u4e3a\u200b\u300c\u200b\u4f53\u8a00\u200b\u300d\u3002</li> <li>\u200b\u52a8\u8bcd\u200b\u3001\u200b\u5f62\u5bb9\u8bcd\u200b\u3001\u200b\u5f62\u5bb9\u200b\u52a8\u8bcd\u200b\u79f0\u4e3a\u200b\u300c\u200b\u7528\u8a00\u200b\u300d\u3002</li> </ul>"},{"location":"japanese/japanese-1-grammar/#_2","title":"\u300c\u30a2\u30bd\u30b3\u30c9\u300d\u200b\u7cfb\u7edf","text":"<p>\u200b\u5728\u200b\u65e5\u8bed\u200b\u4e2d\u200b\uff0c\u300c\u30a2\u30bd\u30b3\u30c9\u300d\u200b\u548c\u200b\u300c\u308c\u300d\uff08\u200b\u4e8b\u7269\u200b\uff09\u3001\u300c\u3053\u300d\uff08\u200b\u5730\u70b9\u200b\uff09\u3001\u300c\u3061\u3089\u300d\uff08\u200b\u65b9\u4f4d\u200b\uff09\u200b\u7b49\u200b\u6307\u793a\u200b\u4ee3\u8bcd\u200b\u8fde\u7528\u200b\u65f6\u200b\uff0c\u200b\u8868\u793a\u200b\u8be5\u8bcd\u200b\u7684\u200b\u8fdc\u8fd1\u200b\u3002</p> \u200b\u524d\u7f00\u200b \u200b\u4e8b\u7269\u200b \u200b\u5730\u70b9\u200b \u200b\u65b9\u4f4d\u200b \u200b\u72b6\u6001\u200b \u200b\u8fd1\u79f0\u200b \u3053 \u3053\u308c \u3053\u3053 \u3053\u3061\u3089 \u3053\u3093\u306a \u200b\u4e2d\u79f0\u200b \u305d \u305d\u308c \u305d\u3053 \u305d\u3061\u3089 \u305d\u3093\u306a \u200b\u8fdc\u79f0\u200b \u3042 \u3042\u308c \u3042\u305d\u3053 \u3042\u3061\u3089 \u3042\u3093\u306a \u200b\u4e0d\u5b9a\u200b\u79f0\u200b \u3069 \u3069\u308c \u3069\u3053 \u3069\u3061\u3089 \u3069\u3093\u306a <p>\u200b\u4e0d\u5b9a\u200b\u79f0\u200b\u662f\u200b\u7591\u95ee\u8bcd\u200b\uff0c\u200b\u7528\u4e8e\u200b\u63d0\u95ee\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_3","title":"\u5f62\u5bb9\u8bcd","text":"<p>\u200b\u5f62\u5bb9\u8bcd\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u548c\u200bII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u3002I\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u8bcd\u5c3e\u200b\u662f\u200b\u300c\u3044\u300d\uff0c\u200b\u5176\u4ed6\u200b\u7684\u200b\u90fd\u200b\u662f\u200bII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0cII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u53c8\u200b\u79f0\u4e3a\u200b\u5f62\u5bb9\u200b\u52a8\u8bcd\u200b\uff0c\u200b\u5176\u200b\u539f\u578b\u200b\u7684\u200b\u8bcd\u5c3e\u200b\u4e3a\u200b\u300c\u3060\u300d\u3002</p> I\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b \u200b\u542b\u4e49\u200b II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b \u200b\u542b\u4e49\u200b \u3042\u305f\u3089\u3057\u3044 \u200b\u65b0\u200b\u7684\u200b \u3060\u3044\u3058\u3087\u3046\u3076 \u200b\u6ca1\u5173\u7cfb\u200b \u305f\u304b\u3044 \u200b\u9ad8\u200b\u7684\u200b \u3079\u3093\u308a \u200b\u65b9\u4fbf\u200b \u3072\u308d\u3044 \u200b\u5bbd\u200b\u7684\u200b \u3059\u304d \u200b\u559c\u6b22\u200b <p>\u200b\u4f8b\u5916\u200b\uff1a\u200b\u7dba\u9e97\u200b\uff08\u304d\u308c\u3044\uff09\u3001\u200b\u5acc\u200b\u3044\uff08\u304d\u3089\u3044\uff09\u200b\u662f\u200bII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0c\u200b\u4f46\u662f\u200b\u8bcd\u5c3e\u200b\u662f\u200b\u300c\u3044\u300d\u3002</p> <p>\u200b\u5f62\u5bb9\u8bcd\u200b\u6709\u200b\u3066\u200b\u578b\u200b\u53d8\u5f62\u200b\u3002</p> \u3066\u200b\u578b\u200b\u306a\u3044\u200b\u578b\u200b\u3063\u305f\u200b\u578b\u200b <p>\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u8fde\u7528\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8fde\u63a5\u200b\u4e24\u4e2a\u200b\u5206\u53e5\u200b\u3002</p> <ol> <li>I\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u8bcd\u5c3e\u200b\u7684\u200b\u300c\u3044\u300d\u200b\u53d8\u4e3a\u200b\u300c\u304f\u3066\u300d\u3002</li> <li>II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5728\u200b\u8bcd\u5c3e\u200b\u76f4\u63a5\u200b\u52a0\u4e0a\u200b\u300c\u3067\u300d\u3002</li> </ol> \u200b\u5f62\u5bb9\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u3066\u200b\u578b\u200b \u3042\u305f\u3089\u3057\u3044 I\u200b\u7c7b\u200b \u3042\u305f\u3089\u3057\u304f\u3066 \u305f\u304b\u3044 I\u200b\u7c7b\u200b \u305f\u304b\u304f\u3066 \u3072\u308d\u3044 I\u200b\u7c7b\u200b \u3072\u308d\u304f\u3066 \u3079\u3093\u308a II\u200b\u7c7b\u200b \u3079\u3093\u308a\u3067 \u3059\u304d II\u200b\u7c7b\u200b \u3059\u304d\u3067 <p>\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u7b80\u4f53\u200b\u5426\u5b9a\u53e5\u200b\u4e2d\u200b\u3002</p> <ol> <li>I\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u8bcd\u5c3e\u200b\u7684\u200b\u300c\u3044\u300d\u200b\u53d8\u4e3a\u200b\u300c\u304f\u306a\u3044\u300d\u3002</li> <li>II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u5728\u200b\u8bcd\u5c3e\u200b\u76f4\u63a5\u200b\u52a0\u4e0a\u200b\u300c\u3067\u306f\u306a\u3044\u300d\u3002</li> </ol> \u200b\u5f62\u5bb9\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u306a\u3044\u200b\u578b\u200b \u3042\u305f\u3089\u3057\u3044 I\u200b\u7c7b\u200b \u3042\u305f\u3089\u3057\u304f\u306a\u3044 \u305f\u304b\u3044 I\u200b\u7c7b\u200b \u305f\u304b\u304f\u306a\u3044 \u3072\u308d\u3044 I\u200b\u7c7b\u200b \u3072\u308d\u304f\u306a\u3044 \u3079\u3093\u308a II\u200b\u7c7b\u200b \u3079\u3093\u308a\u3067\u306f\u306a\u3044 \u3059\u304d II\u200b\u7c7b\u200b \u3059\u304d\u3067\u306f\u306a\u3044 <p>\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u3063\u305f\u200b\u578b\u200b\u662f\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u8fc7\u53bb\u200b\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u7b80\u4f53\u200b\u8fc7\u53bb\u200b\u53e5\u200b\u4e2d\u200b\u3002</p> <ol> <li>I\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u3063\u305f\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u8bcd\u5c3e\u200b\u7684\u200b\u300c\u3044\u300d\u200b\u53d8\u4e3a\u200b\u300c\u304b\u3063\u305f\u300d\u3002</li> <li>II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u3063\u305f\u200b\u578b\u200b\u662f\u200b\u5728\u200b\u8bcd\u5c3e\u200b\u76f4\u63a5\u200b\u52a0\u4e0a\u200b\u300c\u3060\u3063\u305f\u300d\uff08\u200b\u76f8\u5f53\u4e8e\u200b\u539f\u578b\u200b\u76f4\u63a5\u200b\u52a0\u200b\u300c\u3063\u305f\u300d\uff09\u3002</li> </ol> \u200b\u5f62\u5bb9\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u3063\u305f\u200b\u578b\u200b \u3042\u305f\u3089\u3057\u3044 I\u200b\u7c7b\u200b \u3042\u305f\u3089\u3057\u304b\u3063\u305f \u305f\u304b\u3044 I\u200b\u7c7b\u200b \u305f\u304b\u304b\u3063\u305f \u3072\u308d\u3044 I\u200b\u7c7b\u200b \u3072\u308d\u304b\u3063\u305f \u3079\u3093\u308a II\u200b\u7c7b\u200b \u3079\u3093\u308a\u3060\u3063\u305f \u3059\u304d II\u200b\u7c7b\u200b \u3059\u304d\u3060\u3063\u305f <p>\u200b\u6ce8\u610f\u200b\u300c\u306a\u3044\u300d\u200b\u578b\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u770b\u4f5c\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0c\u200b\u5176\u200b\u8fc7\u53bb\u200b\u5f62\u5f0f\u200b\u4e3a\u200b\u300c\u306a\u304b\u3063\u305f\u300d\u3002</p> <ul> <li>I\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u4fee\u9970\u200b\u4f53\u8a00\u65f6\u200b\uff0c\u200b\u76f4\u63a5\u200b\u52a0\u4f53\u8a00\u200b\u5373\u53ef\u200b\uff0c\u200b\u4fee\u9970\u200b\u7528\u8a00\u65f6\u200b\uff08\u200b\u5373\u200b\u526f\u8bcd\u200b\u5f62\u5f0f\u200b\uff09\u200b\u9700\u8981\u200b\u5c06\u200b\u300c\u3044\u300d\u200b\u53d8\u4e3a\u200b\u300c\u304f\u300d\u3002</li> <li>II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u4fee\u9970\u200b\u4f53\u8a00\u65f6\u200b\u9700\u8981\u200b\u52a0\u200b\u300c\u306a\u300d\uff0c\u200b\u4fee\u9970\u200b\u7528\u8a00\u65f6\u200b\u9700\u8981\u200b\u52a0\u200b\u300c\u306b\u300d\u3002</li> </ul>"},{"location":"japanese/japanese-1-grammar/#_4","title":"\u52a8\u8bcd","text":"<p>\u200b\u52a8\u8bcd\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3001II\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3001III\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3002</p> <ol> <li>II\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u8bcd\u5c3e\u200b\u662f\u200b\u300c\u308b\u300d\uff0c\u300c\u308b\u300d\u200b\u524d\u200b\u7684\u200b\u5047\u540d\u200b\u662f\u200b\u300c\u3044\u300d\u200b\u6bb5\u200b\u6216\u200b\u300c\u3048\u300d\u200b\u6bb5\u200b\u3002</li> <li>III\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u8bcd\u5c3e\u200b\u662f\u200b\u300c\u3059\u308b\u300d\u200b\u6216\u200b\u300c\u304f\u308b\u300d\u3002\u300c\u3059\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u52a8\u8bcd\u200b\u53c8\u200b\u79f0\u4e3a\u200b\u30b5\u200b\u53d8\u52a8\u200b\u8bcd\u200b\uff0c\u300c\u304f\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u52a8\u8bcd\u200b\u53c8\u200b\u79f0\u4e3a\u200b\u30ab\u200b\u53d8\u52a8\u200b\u8bcd\u200b\u3002</li> <li>\u200b\u5269\u4e0b\u200b\u6240\u6709\u200b\u7684\u200b\u52a8\u8bcd\u200b\u5747\u200b\u4e3a\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3002I\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u7ed3\u5c3e\u200b\u662f\u200b\u300c\u3046\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\u3002</li> </ol> <p>\u200b\u52a8\u8bcd\u200b\u6709\u200b\u307e\u3059\u200b\u578b\u200b\u3001\u306a\u3044\u200b\u578b\u200b\u3001\u305f\u200b\u578b\u200b\u3001\u3066\u200b\u578b\u200b\u7b49\u200b\u4e0d\u540c\u200b\u7684\u200b\u53d8\u5f62\u200b\u3002</p> \u307e\u3059\u200b\u578b\u200b\u306a\u3044\u200b\u578b\u200b\u305f\u200b\u578b\u200b\u3066\u200b\u578b\u200b <p>\u200b\u52a8\u8bcd\u200b\u7684\u200b\u307e\u3059\u200b\u578b\u200b\u662f\u200b\u52a8\u8bcd\u200b\u7684\u200b\u656c\u4f53\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u656c\u4f53\u200b\u53e5\u200b\u4e2d\u200b\u3002</p> <ol> <li>II\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u307e\u3059\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u307e\u3059\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3059\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u307e\u3059\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u3059\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u3057\u307e\u3059\u300d\uff0c\u200b\u4ee5\u200b\u300c\u304f\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u307e\u3059\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u304f\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u304d\u307e\u3059\u300d\u3002</li> <li>I\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u307e\u3059\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u300c\u3046\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u5bf9\u5e94\u200b\u7684\u200b\u300c\u3044\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\uff0c\u200b\u518d\u200b\u52a0\u4e0a\u200b\u300c\u307e\u3059\u300d\u3002</li> </ol> \u200b\u52a8\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u307e\u3059\u200b\u578b\u200b \u3042\u308b I\u200b\u7c7b\u200b \u3042\u308a\u307e\u3059 \u3067\u304d\u308b II\u200b\u7c7b\u200b \u3067\u304d\u307e\u3059 \u3068\u308b I\u200b\u7c7b\u200b \u3068\u308a\u307e\u3059 \u308c\u3093\u3057\u3085\u3046\u3059\u308b III\u200b\u7c7b\u200b \u308c\u3093\u3057\u3085\u3046\u3057\u307e\u3059 \u3082\u3064 I\u200b\u7c7b\u200b \u3082\u3061\u307e\u3059 <p>\u200b\u52a8\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u52a8\u8bcd\u200b\u7684\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u7b80\u4f53\u200b\u5426\u5b9a\u53e5\u200b\u4e2d\u200b\u3002</p> <ol> <li>II\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u306a\u3044\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3059\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u3059\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u3057\u306a\u3044\u300d\uff0c\u200b\u4ee5\u200b\u300c\u304f\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u304f\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u3053\u306a\u3044\u300d\u3002</li> <li>I\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u300c\u3046\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u5bf9\u5e94\u200b\u7684\u200b\u300c\u3042\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\uff0c\u200b\u518d\u200b\u52a0\u4e0a\u200b\u300c\u306a\u3044\u300d\u3002</li> </ol> \u200b\u52a8\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u306a\u3044\u200b\u578b\u200b \u3042\u308b I\u200b\u7c7b\u200b \u3042\u304b\u306a\u3044 \u3067\u304d\u308b II\u200b\u7c7b\u200b \u3067\u304d\u306a\u3044 \u3068\u308b I\u200b\u7c7b\u200b \u3068\u3089\u306a\u3044 \u308c\u3093\u3057\u3085\u3046\u3059\u308b III\u200b\u7c7b\u200b \u308c\u3093\u3057\u3085\u3046\u3057\u306a\u3044 \u3082\u3064 I\u200b\u7c7b\u200b \u3082\u305f\u306a\u3044 <p>\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u52a8\u8bcd\u200b\u7684\u200b\u8fc7\u53bb\u200b\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u7b80\u4f53\u200b\u8fc7\u53bb\u200b\u53e5\u200b\u4e2d\u200b\u3002</p> <ol> <li>II\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u305f\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3059\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u3059\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u3057\u305f\u300d\uff0c\u200b\u4ee5\u200b\u300c\u304f\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u304f\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u304d\u305f\u300d\u3002</li> <li>I\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u8bcd\u5c3e\u200b\u7684\u200b\u300c\u3046\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\u3002<ul> <li>\u200b\u4ee5\u200b\u300c\u3046\u300d\u3001\u300c\u3064\u300d\u3001\u300c\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3063\u305f\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3080\u300d\u3001\u300c\u3076\u300d\u3001\u300c\u306c\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3093\u3060\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u304f\u300d\u3001\u300c\u3050\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3044\u305f\u300d\u200b\u548c\u200b\u300c\u3044\u3060\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3059\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3057\u305f\u300d\u3002</li> </ul> </li> <li>\u200b\u7279\u4f8b\u200b\uff1a\u300c\u200b\u884c\u200b\u304f\u300d\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u300c\u3044\u3063\u305f\u300d\u3002</li> </ol> \u200b\u52a8\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u305f\u200b\u578b\u200b \u3042\u308b I\u200b\u7c7b\u200b \u3042\u3063\u305f \u3067\u304d\u308b II\u200b\u7c7b\u200b \u3067\u304d\u305f \u3068\u308b I\u200b\u7c7b\u200b \u3068\u3063\u305f \u308c\u3093\u3057\u3085\u3046\u3059\u308b III\u200b\u7c7b\u200b \u308c\u3093\u3057\u3085\u3046\u3057\u305f \u3082\u3064 I\u200b\u7c7b\u200b \u3082\u3063\u305f <p>\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u52a8\u8bcd\u200b\u7684\u200b\u8fde\u7528\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8fde\u63a5\u200b\u4e24\u4e2a\u200b\u5206\u53e5\u200b\u3002</p> <ol> <li>II\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u3066\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3059\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u3059\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u3057\u3066\u300d\uff0c\u200b\u4ee5\u200b\u300c\u304f\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u304f\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u304d\u3066\u300d\u3002</li> <li>I\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u8bcd\u5c3e\u200b\u7684\u200b\u300c\u3046\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\u3002<ul> <li>\u200b\u4ee5\u200b\u300c\u3046\u300d\u3001\u300c\u3064\u300d\u3001\u300c\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3063\u3066\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3080\u300d\u3001\u300c\u3076\u300d\u3001\u300c\u306c\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3093\u3067\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u304f\u300d\u3001\u300c\u3050\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3044\u3066\u300d\u200b\u548c\u200b\u300c\u3044\u3067\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3059\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3057\u3066\u300d\u3002</li> </ul> </li> </ol> \u200b\u52a8\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u3066\u200b\u578b\u200b \u3042\u308b I\u200b\u7c7b\u200b \u3042\u3063\u3066 \u3067\u304d\u308b II\u200b\u7c7b\u200b \u3067\u304d\u3066 \u3068\u308b I\u200b\u7c7b\u200b \u3068\u3063\u3066 \u308c\u3093\u3057\u3085\u3046\u3059\u308b III\u200b\u7c7b\u200b \u308c\u3093\u3057\u3085\u3046\u3057\u3066 \u3082\u3064 I\u200b\u7c7b\u200b \u3082\u3063\u3066"},{"location":"japanese/japanese-1-grammar/#_5","title":"\u53e5\u6cd5","text":""},{"location":"japanese/japanese-1-grammar/#_6","title":"\u8bed\u53e5\u200b\u5206\u7c7b","text":"<p>\u200b\u65e5\u8bed\u200b\u4e2d\u200b\u7684\u200b\u8bed\u53e5\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u5b58\u5728\u200b\u53e5\u200b\uff08\u3042\u308a\u307e\u3059\u3001\u3044\u307e\u3059\uff09\u3001\u200b\u5224\u65ad\u53e5\u200b\uff08\u200b\u540d\u8bcd\u200b\uff09\u3001\u200b\u53d9\u8ff0\u200b\u53e5\u200b\uff08\u200b\u4e00\u822c\u200b\u52a8\u8bcd\u200b\uff09\u3001\u200b\u63cf\u5199\u200b\u53e5\u200b\uff08\u200b\u5f62\u5bb9\u8bcd\u200b\u3001\u200b\u5f62\u5bb9\u200b\u52a8\u8bcd\u200b\uff09\u3002\u200b\u5224\u65ad\u53e5\u200b\u548c\u200b\u63cf\u5199\u200b\u53e5\u200b\u7684\u200b\u656c\u4f53\u200b\u5f62\u5f0f\u200b\u662f\u200b\u300c\u301c\u3067\u3059\u300d\uff0c\u200b\u7b80\u4f53\u200b\u5f62\u5f0f\u200b\u662f\u200b\u300c\u301c\u3060\u300d\u3002\u200b\u53d9\u8ff0\u200b\u53e5\u200b\u548c\u200b\u5b58\u5728\u200b\u53e5\u200b\u7684\u200b\u656c\u4f53\u200b\u5f62\u5f0f\u200b\u662f\u200b\u300c\u301c\u307e\u3059\u300d\uff0c\u200b\u53c8\u79f0\u200b\u307e\u3059\u200b\u578b\u200b\u3002</p> <p>\u200b\u53e5\u5b50\u200b\u7531\u200b\u4e0d\u540c\u200b\u7684\u200b\u300c\u200b\u6587\u8282\u200b\u300d\u200b\u7ec4\u6210\u200b\uff0c\u200b\u6587\u8282\u200b\u662f\u200b\u65e5\u8bed\u200b\u8bed\u6cd5\u7ed3\u6784\u200b\u4e2d\u200b\u7684\u200b\u5355\u5143\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u6587\u8282\u200b\u90fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u300c\u200b\u6838\u5fc3\u200b\u8bcd\u200b\u300d\uff0c\u200b\u6838\u5fc3\u200b\u8bcd\u200b\u7684\u200b\u8bcd\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200b\u6587\u8282\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u6587\u8282\u200b\u4e2d\u200b\u7684\u200b\u9644\u5c5e\u200b\u8bcd\u200b\uff08\u200b\u52a9\u8bcd\u200b\uff09\u200b\u7528\u4e8e\u200b\u4fee\u9970\u200b\u6838\u5fc3\u200b\u8bcd\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b\uff1a\u200b\u52a9\u8bcd\u200b\u53ea\u80fd\u200b\u7528\u4e8e\u200b\u4fee\u9970\u200b\u540d\u8bcd\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_7","title":"\u683c\u200b\u52a9\u8bcd","text":"<p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u7528\u4e8e\u200b\u540d\u8bcd\u200b\u540e\u200b\uff0c\u200b\u8868\u793a\u200b\u540d\u8bcd\u200b\u5728\u200b\u53e5\u5b50\u200b\u4e2d\u200b\u7684\u200b\u8bed\u6cd5\u200b\u6210\u5206\u200b\u3002</p> \u304c\u306e\u306b\u3067\u3092\u304b\u3089\u3078\u3088\u308a\u307b\u3069 <p>\u200b\u52a9\u8bcd\u200b\u300c\u304c\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e3b\u8bed\u200b\uff0c\u200b\u5176\u524d\u200b\u7684\u200b\u540d\u8bcd\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u6267\u884c\u8005\u200b\uff0c\u200b\u6216\u8005\u200b\u63cf\u5199\u200b\u3001\u200b\u5224\u65ad\u200b\u6216\u200b\u5b58\u5728\u200b\u7684\u200b\u5bf9\u8c61\u200b\u3002</p> <p>\u300c\u200b\u79c1\u200b\u304c\u200b\u884c\u200b\u304d\u307e\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u53bb\u200b\u3002\uff08\u200b\u53d9\u8ff0\u200b\u53e5\u200b\uff09 \u300c\u200b\u79c1\u200b\u304c\u200b\u5b66\u751f\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u662f\u200b\u5b66\u751f\u200b\u3002\uff08\u200b\u5224\u65ad\u53e5\u200b\uff09 \u300c\u200b\u79c1\u200b\u304c\u3044\u307e\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u5728\u200b\u3002\uff08\u200b\u5b58\u5728\u200b\u53e5\u200b\uff09</p> <p>\u200b\u52a9\u8bcd\u200b\u300c\u306e\u300d\u200b\u4f5c\u4e3a\u200b\u683c\u200b\u52a9\u8bcd\u200b\u65f6\u200b\u8868\u793a\u200b\u6240\u6709\u200b\u5173\u7cfb\u200b\uff0c\u200b\u5176\u524d\u200b\u7684\u200b\u540d\u8bcd\u200b\u8868\u793a\u200b\u88ab\u200b\u4fee\u9970\u200b\u7684\u200b\u5bf9\u8c61\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b\uff0c\u300c\u306e\u300d\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u540c\u4f4d\u8bed\u200b\uff0c\u200b\u56e0\u6b64\u200b\u300c\u306e\u300d\u200b\u7684\u200b\u5b9e\u9645\u200b\u542b\u4e49\u200b\u9700\u8981\u200b\u6839\u636e\u200b\u4e0a\u4e0b\u6587\u200b\u6765\u200b\u786e\u5b9a\u200b\u3002</p> <p>\u300c\u30de\u30a4\u30b1\u30eb\u3055\u3093\u306e\u200b\u5f8c\u200b\u8f29\u200b\u306e\u200b\u571f\u5c4b\u200b\u3068\u200b\u7533\u200b\u3057\u307e\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u662f\u200b\u8fc8\u514b\u5c14\u200b\u7684\u200b\u540e\u8f88\u200b\u571f\u5c4b\u200b\u3002</p> <p>\u200b\u53e5\u200b\u4e2d\u200b\u7684\u200b\u300c\u30de\u30a4\u30b1\u30eb\u3055\u3093\u306e\u200b\u5f8c\u200b\u8f29\u200b\u300d\u200b\u8868\u793a\u200b\u8fc8\u514b\u5c14\u200b\u7684\u200b\u540e\u8f88\u200b\uff0c\u200b\u662f\u200b\u6240\u6709\u200b\u5173\u7cfb\u200b\uff0c\u300c\u200b\u5f8c\u200b\u8f29\u200b\u306e\u200b\u571f\u5c4b\u200b\u300d\u200b\u8868\u793a\u200b\u571f\u5c4b\u200b\u662f\u200b\u8fc8\u514b\u5c14\u200b\u7684\u200b\u540e\u8f88\u200b\uff0c\u200b\u662f\u200b\u540c\u4f4d\u8bed\u200b\u5173\u7cfb\u200b\u3002</p> <p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u300c\u306b\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b</p> <ol> <li> <p>\u200b\u52a8\u4f5c\u200b\u53e5\u200b\u4e2d\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u6267\u884c\u200b\u65f6\u95f4\u200b\u3001\u200b\u5730\u70b9\u200b\u3001\u200b\u5bf9\u8c61\u200b\u7b49\u200b\u3002\u200b\u52a8\u8bcd\u200b\u9700\u8981\u200b\u662f\u200b\u79fb\u52a8\u200b\u52a8\u8bcd\u200b\u3002</p> <p>\u300c7\u200b\u6642\u200b\u306b\u200b\u56f3\u200b\u66f8\u9928\u200b\u306b\u200b\u884c\u200b\u304d\u307e\u3059\u3002\u300d\uff1a7\u200b\u70b9\u53bb\u200b\u56fe\u4e66\u9986\u200b\u3002</p> </li> <li> <p>\u200b\u5b58\u5728\u200b\u53e5\u200b\u4e2d\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u5b58\u5728\u200b\u7684\u200b\u5730\u70b9\u200b\u3002</p> <p>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u306b\u200b\u672c\u200b\u304c\u3042\u308a\u307e\u3059\u3002\u300d\uff1a\u200b\u56fe\u4e66\u9986\u200b\u91cc\u200b\u6709\u200b\u4e66\u200b\u3002 \u300c\u200b\u79c1\u200b\u306b\u306f\u200b\u59c9\u200b\u304c\u200b\u4e00\u4eba\u200b\u3044\u307e\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u6709\u200b\u4e00\u4e2a\u200b\u59d0\u59d0\u200b\u3002</p> </li> </ol> <p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u300c\u3067\u300d\u200b\u7528\u4e8e\u200b</p> <ol> <li> <p>\u200b\u63a5\u200b\u540d\u8bcd\u200b\u65f6\u200b\uff0c\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u6267\u884c\u200b\u624b\u6bb5\u200b\u6216\u200b\u65b9\u6cd5\u200b\u3002</p> <p>\u300c\u200b\u96fb\u8eca\u200b\u3067\u200b\u884c\u200b\u304d\u307e\u3059\u3002\u300d\uff1a\u200b\u5750\u200b\u7535\u8f66\u200b\u53bb\u200b\u3002</p> </li> <li> <p>\u200b\u63a5\u200b\u5730\u70b9\u200b\u65f6\u200b\uff0c\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u53d1\u751f\u200b\u5730\u70b9\u200b\u3002</p> <p>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u3067\u200b\u52c9\u5f37\u200b\u3057\u307e\u3059\u3002\u300d\uff1a\u200b\u5728\u200b\u56fe\u4e66\u9986\u200b\u5b66\u4e60\u200b\u3002</p> </li> </ol> <p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u300c\u3092\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff08\u200b\u4ed6\u200b\u52a8\u8bcd\u200b\uff09\u200b\u6216\u200b\u4f7f\u5f79\u200b\u6001\u200b\uff08\u200b\u81ea\u52a8\u8bcd\u200b\uff09\u3002</p> <p>\u300c\u200b\u672c\u200b\u3092\u200b\u8aad\u200b\u307f\u307e\u3059\u3002\u300d\uff1a\u200b\u770b\u4e66\u200b\u3002</p> <p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u300c\u304b\u3089\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u8d77\u70b9\u200b\u3002</p> <p>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u304b\u3089\u200b\u51fa\u200b\u767a\u200b\u3057\u307e\u3059\u3002\u300d\uff1a\u200b\u4ece\u200b\u56fe\u4e66\u9986\u200b\u51fa\u53d1\u200b\u3002</p> <p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u300c\u3078\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\uff08\u200b\u79fb\u52a8\u200b\u52a8\u8bcd\u200b\uff09\u200b\u7684\u200b\u7ec8\u70b9\u200b\u3002\uff08\u200b\u53ef\u4ee5\u200b\u4e0e\u200b\u300c\u306b\u300d\u200b\u4e92\u6362\u200b\uff09</p> <p>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u3078\u200b\u884c\u200b\u304d\u307e\u3059\u3002\u300d\uff1a\u200b\u53bb\u200b\u56fe\u4e66\u9986\u200b\u3002</p> <p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u300c\u3088\u308a\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u6bd4\u8f83\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5176\u524d\u200b\u7684\u200b\u540d\u8bcd\u200b\u7684\u200b\u7a0b\u5ea6\u200b\u6bd4\u200b\u4e3b\u8bed\u200b\u7684\u200b\u7a0b\u5ea6\u200b\u5f31\u200b\u3002</p> <p>\u300c\u200b\u6e05\u83ef\u200b\u5927\u5b66\u200b\u306f\u200b\u5317\u4eac\u5927\u5b66\u200b\u3088\u308a\u200b\u6709\u540d\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u6e05\u534e\u5927\u5b66\u200b\u6bd4\u200b\u5317\u4eac\u5927\u5b66\u200b\u6709\u540d\u200b\u3002</p> <p>\u200b\u5728\u200b\u63cf\u8ff0\u200b\u53e5\u200b\u4e2d\u683c\u200b\u52a9\u8bcd\u200b\u300c\u307b\u3069\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u7a0b\u5ea6\u200b\uff0c\u200b\u7528\u4e8e\u200b\u5426\u5b9a\u53e5\u200b\u4e2d\u200b\u8868\u793a\u200b\u5176\u524d\u200b\u7684\u200b\u540d\u8bcd\u200b\u7684\u200b\u7a0b\u5ea6\u200b\u6bd4\u200b\u4e3b\u8bed\u200b\u7684\u200b\u7a0b\u5ea6\u200b\u5f3a\u200b\u3002</p> <p>\u300c\u200b\u5317\u4eac\u5927\u5b66\u200b\u306f\u200b\u6e05\u83ef\u200b\u5927\u5b66\u200b\u307b\u3069\u200b\u6709\u540d\u200b\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u300d\uff1a\u200b\u5317\u4eac\u5927\u5b66\u200b\u6ca1\u6709\u200b\u6e05\u534e\u5927\u5b66\u200b\u6709\u540d\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_8","title":"\u526f\u200b\u52a9\u8bcd","text":"<p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u7528\u4e8e\u200b\u4fee\u9970\u8bed\u200b\u6cd5\u200b\u6210\u5206\u200b\uff0c\u200b\u4efb\u4f55\u200b\u4e0d\u662f\u200b\u683c\u200b\u52a9\u8bcd\u200b\u3001\u200b\u63a5\u7eed\u200b\u52a9\u8bcd\u200b\u3001\u200b\u7ec8\u200b\u52a9\u8bcd\u200b\u7684\u200b\u52a9\u8bcd\u200b\u90fd\u200b\u662f\u200b\u526f\u200b\u52a9\u8bcd\u200b\u3002</p> \u306f\u307e\u3067\u3082\u306e\u307f\u3060\u3051\u3068\u304b\u304f\u3089\u3044\u304b <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u306f\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u53e5\u5b50\u200b\u7684\u200b\u4e3b\u9898\u200b\uff0c\u200b\u8054\u7ed3\u200b\u524d\u200b\u4e00\u4e2a\u200b\u53e5\u5b50\u200b\u3002\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u5c06\u200b\u5176\u4ed6\u200b\u7684\u200b\u683c\u200b\u52a9\u8bcd\u200b\u4fee\u9970\u200b\u7684\u200b\u8bed\u6cd5\u200b\u6210\u5206\u200b\u4f5c\u4e3a\u200b\u4e3b\u9898\u200b\u3002\u300c\u306f\u300d\u200b\u6240\u200b\u4fee\u9970\u200b\u7684\u200b\u4e3b\u9898\u200b\u901a\u5e38\u200b\u5728\u200b\u53e5\u5b50\u200b\u7684\u200b\u5f00\u5934\u200b\u3002</p> <ul> <li> <p>\u200b\u5c06\u200b\u4e3b\u8bed\u200b\u4f5c\u4e3a\u200b\u4e3b\u9898\u200b\uff1a\u300c\u304c\u300d\u200b\u53d8\u4e3a\u200b\u300c\u306f\u300d</p> <p>\u300c\u200b\u79c1\u200b\u306f\u200b\u5b66\u751f\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u662f\u200b\u5b66\u751f\u200b\u3002</p> </li> <li> <p>\u200b\u5c06\u200b\u5730\u70b9\u200b\u4f5c\u4e3a\u200b\u4e3b\u9898\u200b\uff1a\u300c\u306b\u300d\u200b\u540e\u52a0\u200b\u300c\u306f\u300d</p> <p>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u306e\u200b\u8fd1\u200b\u304f\u306b\u306f\u30b3\u30f3\u30d3\u30cb\u304c\u3042\u308a\u307e\u3059\u3002\u300d\uff1a\u200b\u56fe\u4e66\u9986\u200b\u9644\u8fd1\u200b\u6709\u200b\u4fbf\u5229\u5e97\u200b\u3002</p> </li> </ul> <p>\u200b\u6ce8\u610f\u200b\uff0c\u200b\u7591\u95ee\u8bcd\u200b\u4e0d\u80fd\u200b\u4f5c\u4e3a\u200b\u4e3b\u9898\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e0d\u80fd\u200b\u7528\u200b\u300c\u306f\u300d\u200b\u4fee\u9970\u200b\u7591\u95ee\u8bcd\u200b\uff0c\u200b\u53ea\u80fd\u200b\u7528\u200b\u300c\u304c\u300d\u3002</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u307e\u3067\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u65f6\u95f4\u200b\u3001\u200b\u5730\u70b9\u200b\u3001\u200b\u8303\u56f4\u200b\u7b49\u200b\u7684\u200b\u7ec8\u70b9\u200b\uff0c\u200b\u8868\u793a\u200b\u201c\u200b\u76f4\u5230\u200b\u201d\u3001\u201c\u200b\u5230\u200b\u2026\u2026\u200b\u4e3a\u6b62\u200b\u201d\u3002\u200b\u53ef\u4ee5\u200b\u548c\u200b\u300c\u304b\u3089\u300d\u200b\u4e00\u8d77\u200b\u7528\u200b\uff0c\u200b\u8868\u793a\u200b\u201c\u200b\u4ece\u200b\u2026\u2026\u200b\u5230\u200b\u2026\u2026\u201d\u3002</p> <p>\u300c\u3055\u3063\u304d\u307e\u3067\u200b\u56f3\u200b\u66f8\u9928\u200b\u306b\u3044\u307e\u3057\u305f\u3002\u300d\uff1a\u200b\u521a\u624d\u200b\u4e00\u76f4\u200b\u5728\u200b\u56fe\u4e66\u9986\u200b\u3002\uff08\u200b\u65f6\u95f4\u200b\uff09 \u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u307e\u3067\u200b\u884c\u200b\u304d\u307e\u3057\u3087\u3046\u3002\u300d\uff1a\u200b\u53bb\u200b\u56fe\u4e66\u9986\u200b\u5427\u200b\u3002\uff08\u200b\u5730\u70b9\u200b\uff09</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u3082\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u201c\u200b\u4e5f\u200b\u201d\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e0e\u200b\u524d\u9762\u200b\u7684\u200b\u5185\u5bb9\u200b\u76f8\u540c\u200b\u7684\u200b\u4e8b\u7269\u200b\u3002</p> <p>\u300c\u200b\u79c1\u200b\u3082\u200b\u5b66\u751f\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u4e5f\u200b\u662f\u200b\u5b66\u751f\u200b\u3002 \u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u8fd1\u200b\u304f\u306b\u3082\u200b\u7dcf\u200b\u5408\u200b\u68df\u200b\u304c\u3042\u308a\u307e\u3059\u3002\u300d\uff1a\u200b\u56fe\u4e66\u9986\u200b\u9644\u8fd1\u200b\u4e5f\u200b\u6709\u200b\u7efc\u5408\u697c\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b\u300c\u304c\u300d\u3001\u300c\u306f\u300d\u3001\u300c\u3082\u300d\u200b\u4e0d\u80fd\u200b\u540c\u65f6\u200b\u7528\u4e8e\u200b\u540c\u4e00\u4e2a\u200b\u4e3b\u8bed\u200b\u3002</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u306e\u307f\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u201c\u200b\u53ea\u6709\u200b\u201d\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e0e\u200b\u524d\u9762\u200b\u7684\u200b\u5185\u5bb9\u200b\u4e0d\u540c\u200b\u7684\u200b\u4e8b\u7269\u200b\u3002</p> <p>\u300c\u304a\u200b\u4f1a\u8a08\u200b\u306f\u200b\u73fe\u91d1\u200b\u306e\u307f\u3067\u3059\u3002\u300d\uff1a\u200b\u53ea\u6536\u200b\u73b0\u91d1\u200b\u3002</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u3060\u3051\u300d\u200b\u7684\u200b\u542b\u4e49\u200b\u540c\u200b\u300c\u306e\u307f\u300d\uff0c\u200b\u4f46\u662f\u200b\u300c\u3060\u3051\u300d\u200b\u7684\u200b\u8bed\u6c14\u200b\u66f4\u52a0\u200b\u53e3\u8bed\u200b\u3002</p> <p>\u300c\u304a\u200b\u4f1a\u8a08\u200b\u306f\u200b\u73fe\u91d1\u200b\u3060\u3051\u3067\u3059\u3002\u300d\uff1a\u200b\u53ea\u6536\u200b\u73b0\u91d1\u200b\u3002</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u3068\u304b\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u201c\u200b\u4f8b\u5982\u200b\u201d\uff0c\u200b\u7528\u4e8e\u200b\u5217\u4e3e\u200b\u4e8b\u7269\u200b\u3002</p> <p>\u300c\u200b\u79c1\u200b\u306f\u200b\u65e5\u672c\u200b\u8a9e\u200b\u3068\u304b\u200b\u82f1\u8a9e\u200b\u3068\u304b\u200b\u4e2d\u56fd\u200b\u8a9e\u200b\u3068\u304b\u200b\u8a71\u200b\u305b\u307e\u3059\u3002\u300d\uff1a\u200b\u6211\u4f1a\u200b\u8bf4\u200b\u65e5\u8bed\u200b\u3001\u200b\u82f1\u8bed\u200b\u3001\u200b\u6c49\u8bed\u200b\u7b49\u200b\u3002</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u304f\u3089\u3044\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u201c\u200b\u5927\u7ea6\u200b\u201d\u3001\u201c\u200b\u5de6\u53f3\u200b\u201d\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u6570\u91cf\u200b\u3002\u200b\u524d\u9762\u200b\u7684\u200b\u540d\u8bcd\u200b\u4ee5\u200b\u62e8\u97f3\u200b\u7ed3\u5c3e\u200b\u65f6\u200b\uff0c\u300c\u304f\u3089\u3044\u300d\u200b\u53d8\u4e3a\u200b\u300c\u3050\u3089\u3044\u300d\u3002</p> <p>\u300c\u200b\u5168\u90e8\u200b\u3067100\u200b\u5186\u200b\u304f\u3089\u3044\u3067\u3059\u3002\u300d\uff1a\u200b\u4e00\u5171\u200b\u5927\u7ea6\u200b100\u200b\u65e5\u5143\u200b\u3002</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u304b\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e0d\u200b\u786e\u5b9a\u200b\u7684\u200b\u4e8b\u7269\u200b\u3002</p> <p>\u300c\u3044\u3064\u304b\u200b\u56f3\u200b\u66f8\u9928\u200b\u306b\u200b\u884c\u200b\u304d\u307e\u3057\u3087\u3046\u3002\u300d\uff1a\u200b\u4ec0\u4e48\u200b\u65f6\u5019\u200b\u53bb\u200b\u56fe\u4e66\u9986\u200b\u5427\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_9","title":"\u63a5\u7eed\u200b\u52a9\u8bcd","text":"<p>\u200b\u63a5\u7eed\u200b\u52a9\u8bcd\u200b\u7528\u4e8e\u200b\u8fde\u63a5\u200b\u4e24\u4e2a\u200b\u5206\u53e5\u200b\uff0c\u200b\u63a5\u200b\u5728\u200b\u7b2c\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u7684\u200b\u7ed3\u5c3e\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e24\u4e2a\u200b\u5206\u53e5\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5173\u7cfb\u200b\u3002</p> \u304c\u3051\u3069\u304b\u3089\u3068\u3082\u306e\u3067 <p>\u300c\u304c\u300d\u200b\u5728\u200b\u5206\u53e5\u200b\u7684\u200b\u7ed3\u5c3e\u200b\u65f6\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u8f7b\u5fae\u200b\u7684\u200b\u8f6c\u6298\u200b\u8bed\u6c14\u200b\u3002\u200b\u524d\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u867d\u7136\u200b\u2026\u2026\u201d\uff0c\u200b\u540e\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u4f46\u662f\u200b\u2026\u2026\u201d\u3002</p> <p>\u300c\u200b\u79c1\u200b\u3067\u3059\u304c\u3001\u200b\u4f55\u200b\u3067\u3059\u304b\u3002\u300d\uff1a\u200b\u6211\u200b\u662f\u200b\uff0c\u200b\u6709\u200b\u4ec0\u4e48\u200b\u4e8b\u200b\u5417\u200b\uff1f</p> <p>\u300c\u3051\u3069\u300d\u200b\u5728\u200b\u5206\u53e5\u200b\u7684\u200b\u7ed3\u5c3e\u200b\u65f6\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u8f6c\u6298\u200b\u8bed\u6c14\u200b\u3002\u200b\u524d\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u867d\u7136\u200b\u2026\u2026\u201d\uff0c\u200b\u540e\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u4f46\u662f\u200b\u2026\u2026\u201d\u3002\u300c\u3051\u3069\u300d\u200b\u7684\u200b\u8bed\u6c14\u200b\u6bd4\u200b\u300c\u304c\u300d\u200b\u66f4\u52a0\u200b\u53e3\u8bed\u200b\uff0c\u200b\u662f\u200b\u300c\u3051\u308c\u3069\u3082\u300d\u200b\u7684\u200b\u7f29\u5199\u200b\u3002\u200b\u9664\u4e86\u200b\u300c\u3051\u3069\u300d\uff0c\u300c\u3051\u308c\u3069\u3082\u300d\u200b\u8fd8\u6709\u200b\u300c\u3051\u308c\u3069\u300d\u3001\u300c\u3051\u3069\u3082\u300d\u200b\u7b49\u200b\u7f29\u5199\u200b\u5f62\u5f0f\u200b\u3002</p> <p>\u300c\u305d\u3046\u306a\u3093\u3067\u3059\u3051\u3069\u3001\u3084\u306f\u308a\u200b\u5927\u200b\u5909\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u867d\u7136\u200b\u662f\u200b\u8fd9\u6837\u200b\uff0c\u200b\u4f46\u662f\u200b\u8fd8\u662f\u200b\u5f88\u200b\u8f9b\u82e6\u200b\u3002</p> <p>\u300c\u304b\u3089\u300d\u200b\u5728\u200b\u5206\u53e5\u200b\u7684\u200b\u7ed3\u5c3e\u200b\u65f6\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u56e0\u679c\u5173\u7cfb\u200b\u3002\u200b\u524d\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u56e0\u4e3a\u200b\u2026\u2026\u201d\uff0c\u200b\u540e\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u6240\u4ee5\u200b\u2026\u2026\u201d\u3002\u200b\u5728\u200b\u8868\u793a\u200b\u56e0\u679c\u5173\u7cfb\u200b\u65f6\u200b\uff0c\u300c\u304b\u3089\u300d\u200b\u524d\u200b\u5fc5\u987b\u200b\u662f\u200b\u4e00\u4e2a\u200b\u53e5\u5b50\u200b\u3002\u300c\u304b\u3089\u300d\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u52a0\u200b\u540e\u8fb9\u200b\u7684\u200b\u5206\u53e5\u200b\uff0c\u200b\u7528\u4e8e\u200b\u9648\u8ff0\u200b\u539f\u56e0\u200b\u3002</p> <p>\u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u304b\u3089\u3001\u200b\u6dbc\u200b\u3057\u3044\u200b\u6240\u200b\u306b\u200b\u884c\u200b\u304d\u307e\u3057\u3087\u3046\u3002\u300d\uff1a\u200b\u56e0\u4e3a\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\uff0c\u200b\u6240\u4ee5\u200b\u53bb\u200b\u51c9\u5feb\u200b\u7684\u200b\u5730\u65b9\u200b\u5427\u200b\u3002</p> <p>\u300c\u3068\u3082\u300d\u200b\u8868\u793a\u200b\u603b\u6570\u200b\uff0c\u200b\u76f8\u5f53\u4e8e\u200b\u6c49\u8bed\u200b\u4e2d\u200b\u7684\u200b\u201c\u200b\u5171\u200b\u201d\u3001\u201c\u200b\u90fd\u200b\u201d\u3002</p> <p>\u300c\u304a\u200b\u4e8c\u4eba\u200b\u3068\u3082\u30db\u30c3\u30c8\u30b3\u30fc\u30d2\u30fc\u3067\u3088\u308d\u3057\u3044\u3067\u3059\u304b\u3002\u300d\uff1a\u200b\u4e24\u4f4d\u200b\u90fd\u200b\u8981\u200b\u559d\u200b\u70ed\u5496\u5561\u200b\u5417\u200b\uff1f</p> <p>\u300c\u306e\u3067\u300d\u200b\u5728\u200b\u5206\u53e5\u200b\u7684\u200b\u7ed3\u5c3e\u200b\u65f6\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u56e0\u679c\u5173\u7cfb\u200b\u3002\u200b\u524d\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u56e0\u4e3a\u200b\u2026\u2026\u201d\uff0c\u200b\u540e\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u6240\u4ee5\u200b\u2026\u2026\u201d\u3002\u200b\u5728\u200b\u8868\u793a\u200b\u56e0\u679c\u5173\u7cfb\u200b\u65f6\u200b\uff0c\u300c\u306e\u3067\u300d\u200b\u524d\u200b\u662f\u200b\u4e00\u4e2a\u200b\u7b80\u4f53\u200b\u53e5\u200b\u3002\u200b\u5982\u679c\u200b\u300c\u306e\u3067\u300d\u200b\u524d\u200b\u662f\u200b\u4e00\u4e2a\u200b\u540d\u8bcd\u200b\u6216\u200bII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0c\u200b\u9700\u8981\u200b\u5c06\u200b\u7b80\u4f53\u200b\u53e5\u200b\u7684\u200b\u300c\u3060\u300d\u200b\u53d8\u4e3a\u200b\u300c\u306a\u300d\u3002</p> <p>\u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u306e\u3067\u3001\u200b\u6dbc\u200b\u3057\u3044\u200b\u6240\u200b\u306b\u200b\u884c\u200b\u304d\u307e\u3057\u3087\u3046\u3002\u300d\uff1a\u200b\u56e0\u4e3a\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\uff0c\u200b\u6240\u4ee5\u200b\u53bb\u200b\u51c9\u5feb\u200b\u7684\u200b\u5730\u65b9\u200b\u5427\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_10","title":"\u7ec8\u200b\u52a9\u8bcd","text":"<p>\u200b\u65e5\u8bed\u200b\u4e2d\u200b\u7684\u200b\u7ec8\u200b\u52a9\u8bcd\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u53e5\u5b50\u200b\u7684\u200b\u8bed\u6c14\u200b\u53d8\u5316\u200b\u3002</p> \u304b\u306d\u3088\u3088\u306d <p>\u300c\u304b\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u7591\u95ee\u200b\u7684\u200b\u8bed\u6c14\u200b\u3002\u200b\u5728\u200b\u53e5\u5b50\u200b\u7684\u200b\u7ed3\u5c3e\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u7591\u95ee\u53e5\u200b\u3002</p> <p>\u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u3067\u3059\u304b\u3002\u300d\uff1a\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\u5417\u200b\uff1f</p> <p>\u300c\u306d\u300d\u200b\u53ef\u4ee5\u200b\u8868\u8fbe\u200b\u786e\u8ba4\u200b\u6216\u200b\u611f\u53f9\u200b\u7684\u200b\u8bed\u6c14\u200b\u3002\u200b\u5f53\u200b\u300c\u306d\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u786e\u8ba4\u200b\u7684\u200b\u8bed\u6c14\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u8bfb\u200b\u5347\u8c03\u200b\uff1b\u200b\u5f53\u200b\u300c\u306d\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u611f\u53f9\u200b\u7684\u200b\u8bed\u6c14\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u8bfb\u200b\u964d\u8c03\u200b\u3002</p> <p>\u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u571f\u66dc\u65e5\u200b\u3067\u3059\u306d\u3002\u300d\u200b\u4eca\u5929\u200b\u662f\u200b\u661f\u671f\u516d\u200b\u5427\u200b\uff1f\uff08\u200b\u786e\u8ba4\u200b\uff09 \u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u3067\u3059\u306d\u3002\u300d\uff1a\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\u554a\u200b\u3002\uff08\u200b\u611f\u53f9\u200b\uff09</p> <p>\u300c\u3088\u300d\u200b\u7528\u4e8e\u200b\u5f3a\u8c03\u200b\u81ea\u5df1\u200b\u7684\u200b\u89c2\u70b9\u200b\u3002</p> <p>\u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u3067\u3059\u3088\u3002\u300d\uff1a\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\u554a\u200b\u3002</p> <p>\u300c\u3088\u306d\u300d\u200b\u540c\u65f6\u200b\u5305\u542b\u200b\u4e86\u200b\u300c\u3088\u300d\u200b\u548c\u200b\u300c\u306d\u300d\u200b\u7684\u200b\u8bed\u6c14\u200b\u3002</p> <p>\u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u3067\u3059\u3088\u306d\u3002\u300d\uff1a\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\uff0c\u200b\u5bf9\u200b\u5427\u200b\uff1f</p>"},{"location":"japanese/japanese-1-grammar/#_11","title":"\u300c\u306f\u300d\u200b\u4e0e\u200b\u300c\u304c\u300d","text":"<p>\u200b\u52a9\u8bcd\u200b\u300c\u304c\u300d\u200b\u662f\u200b\u683c\u200b\u52a9\u8bcd\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e3b\u8bed\u200b\uff0c\u300c\u306f\u300d\u200b\u662f\u200b\u526f\u200b\u52a9\u8bcd\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e3b\u9898\u200b\u3002\u200b\u4e3b\u9898\u200b\u662f\u200b\u6307\u200b\u53e5\u5b50\u200b\u4e3b\u8981\u200b\u63cf\u8ff0\u200b\u7684\u200b\u8bdd\u9898\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7531\u200b\u4e3b\u8bed\u200b\u6216\u200b\u5176\u4ed6\u200b\u6210\u5206\u200b\u5145\u5f53\u200b\u3002</p> <ul> <li>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u306e\u200b\u8fd1\u200b\u304f\u306b\u306f\u30b3\u30f3\u30d3\u30cb\u304c\u3042\u308a\u307e\u3059\u3002\u300d</li> <li>\u300c\u30b3\u30f3\u30d3\u30cb\u306f\u200b\u56f3\u200b\u66f8\u9928\u200b\u306e\u200b\u8fd1\u200b\u304f\u306b\u3042\u308a\u307e\u3059\u3002\u300d</li> <li>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u306e\u3061\u304b\u304f\u306b\u30b3\u30f3\u30d3\u30cb\u304c\u3042\u308a\u307e\u3059\u3002\u300d</li> <li>\u200b\u7b2c\u200b1\u200b\u53e5\u200b\u4e2d\u200b\u4e3b\u8981\u200b\u5f3a\u8c03\u200b\u56fe\u4e66\u9986\u200b\u7684\u200b\u9644\u8fd1\u200b\u6709\u200b\u4fbf\u5229\u5e97\u200b\uff0c\u200b\u53e5\u5b50\u200b\u7684\u200b\u4e3b\u9898\u200b\u662f\u200b\u56fe\u4e66\u9986\u200b\u9644\u8fd1\u200b\u7684\u200b\u4e1c\u897f\u200b\uff1b\u200b\u7b2c\u200b2\u200b\u53e5\u200b\u4e2d\u200b\u4e3b\u8981\u200b\u5f3a\u8c03\u200b\u4fbf\u5229\u5e97\u200b\u5728\u200b\u56fe\u4e66\u9986\u200b\u9644\u8fd1\u200b\uff0c\u200b\u53e5\u5b50\u200b\u7684\u200b\u4e3b\u9898\u200b\u662f\u200b\u4fbf\u5229\u5e97\u200b\u3002</li> <li>\u200b\u7b2c\u200b3\u200b\u53e5\u200b\u4e0d\u200b\u5305\u542b\u200b\u4e3b\u9898\u200b\uff0c\u200b\u4ec5\u4ec5\u200b\u9648\u8ff0\u200b\u4e86\u200b\u56fe\u4e66\u9986\u200b\u9644\u8fd1\u200b\u6709\u200b\u4fbf\u5229\u5e97\u200b\u3002</li> </ul> <p>\u200b\u5f53\u200b\u56de\u7b54\u200b\u7591\u95ee\u53e5\u200b\u65f6\u200b\uff0c\u200b\u5982\u679c\u200b\u7591\u95ee\u53e5\u200b\u662f\u200b\u7528\u200b\u300c\u304c\u300d\u200b\u63d0\u95ee\u200b\u7684\u200b\uff0c\u200b\u5728\u200b\u56de\u7b54\u200b\u7684\u200b\u65f6\u5019\u200b\u4e5f\u200b\u9700\u8981\u200b\u7528\u200b\u300c\u304c\u300d\u3002</p> <p>\u200b\u901a\u5e38\u200b\u6765\u8bf4\u200b\uff1a</p> <ol> <li>\u300c\u306f\u300d\u200b\u901a\u5e38\u200b\u7528\u4e8e\u200b\u63d0\u793a\u200b\u5df2\u6709\u200b\u7684\u200b\u4fe1\u606f\u200b\uff0c\u300c\u304c\u300d\u200b\u7528\u4e8e\u200b\u63d0\u793a\u200b\u65b0\u200b\u7684\u200b\u4fe1\u606f\u200b\u3002</li> <li>\u300c\u306f\u300d\u200b\u5f3a\u8c03\u200b\uff08\u200b\u4f20\u8fbe\u200b\uff09\u200b\u7684\u200b\u901a\u5e38\u200b\u662f\u200b\u300c\u306f\u300d\u200b\u540e\u9762\u200b\u7684\u200b\u5185\u5bb9\u200b\uff0c\u300c\u304c\u300d\u200b\u5f3a\u8c03\u200b\uff08\u200b\u4f20\u8fbe\u200b\uff09\u200b\u7684\u200b\u901a\u5e38\u200b\u662f\u200b\u300c\u304c\u300d\u200b\u524d\u9762\u200b\u7684\u200b\u5185\u5bb9\u200b\u3002</li> <li>\u200b\u7528\u4e8e\u200b\u5f62\u5bb9\u200b\u80fd\u529b\u200b\u7684\u200bII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff08\u300c\u3058\u3087\u3046\u305a\u300d\u3001\u300c\u3078\u305f\u300d\u3001\u300c\u200b\u597d\u200b\u304d\u300d\u3001\u300c\u3068\u304f\u3044\u300d\uff09\u200b\u6216\u200b\u52a8\u8bcd\u200b\uff08\u300c\u3067\u304d\u308b\u300d\u3001\u300c\u308f\u304b\u308b\u300d\uff09\u200b\u7684\u200b\u4e3b\u8bed\u200b\u53ea\u80fd\u200b\u7528\u200b\u300c\u304c\u300d\uff0c\u200b\u4e0d\u80fd\u200b\u7528\u200b\u300c\u306f\u300d\u3002</li> </ol>"},{"location":"japanese/japanese-1-grammar/#_12","title":"\u63a5\u7eed\u8bcd","text":"<p>\u200b\u63a5\u7eed\u8bcd\u200b\u653e\u5728\u200b\u53e5\u5b50\u200b\u7684\u200b\u5f00\u5934\u200b\uff0c\u200b\u540e\u9762\u200b\u53ef\u4ee5\u200b\u63a5\u200b\u53e5\u5b50\u200b\u6216\u200b\u77ed\u8bed\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u8be5\u53e5\u200b\u548c\u200b\u524d\u200b\u4e00\u53e5\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5173\u7cfb\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_13","title":"\u3057\u304b\u3057","text":"<p>\u300c\u3057\u304b\u3057\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u8f6c\u6298\u200b\u5173\u7cfb\u200b\uff0c\u200b\u76f8\u5f53\u4e8e\u200b\u6c49\u8bed\u200b\u4e2d\u200b\u7684\u200b\u201c\u200b\u4f46\u662f\u200b\u201d\uff0c\u200b\u7528\u4e8e\u200b\u53e5\u5b50\u200b\u7684\u200b\u5f00\u5934\u200b\u3002</p> <p>\u300c\u3057\u304b\u3057\u3001\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u3067\u3059\u306d\u3002\u300d\uff1a\u200b\u4f46\u662f\u200b\uff0c\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\u5427\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_14","title":"\u3067\u3082","text":"<p>\u300c\u3067\u3082\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u8f6c\u6298\u200b\u5173\u7cfb\u200b\uff0c\u200b\u76f8\u5f53\u4e8e\u200b\u6c49\u8bed\u200b\u4e2d\u200b\u7684\u200b\u201c\u200b\u4f46\u662f\u200b\u201d\uff0c\u200b\u8bed\u6c14\u200b\u6bd4\u200b\u300c\u3057\u304b\u3057\u300d\u200b\u66f4\u52a0\u200b\u53e3\u8bed\u200b\u3002</p> <p>\u300c\u3067\u3082\u3001\u200b\u5348\u200b\u5f8c\u200b\u304b\u3089\u306f\u200b\u6dbc\u200b\u3057\u304f\u306a\u308a\u307e\u3059\u3002\u300d\uff1a\u200b\u4f46\u662f\u200b\uff0c\u200b\u4ece\u200b\u4e0b\u5348\u200b\u5f00\u59cb\u200b\u4f1a\u200b\u51c9\u5feb\u200b\u4e00\u4e9b\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_15","title":"\u305d\u308c\u304b\u3089","text":"<p>\u300c\u305d\u308c\u304b\u3089\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u987a\u63a5\u200b\u5173\u7cfb\u200b\uff0c\u200b\u76f8\u5f53\u4e8e\u200b\u6c49\u8bed\u200b\u4e2d\u200b\u7684\u200b\u201c\u200b\u7136\u540e\u200b\u201d\uff0c\u200b\u7528\u4e8e\u200b\u53e5\u5b50\u200b\u7684\u200b\u5f00\u5934\u200b\u3002</p> <p>\u300c\u305d\u308c\u304b\u3089\u3001\u200b\u56f3\u200b\u66f8\u9928\u200b\u306b\u200b\u884c\u200b\u304d\u307e\u3057\u3087\u3046\u3002\u300d\uff1a\u200b\u7136\u540e\u200b\uff0c\u200b\u53bb\u200b\u56fe\u4e66\u9986\u200b\u5427\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_16","title":"\u8bed\u6cd5","text":""},{"location":"japanese/japanese-1-grammar/#_17","title":"\u656c\u4f53\u200b\u53e5\u200b\u53ca\u200b\u65f6\u6001\u53d8\u5316","text":""},{"location":"japanese/japanese-1-grammar/#_18","title":"\u540d\u8bcd\u200b\u53e5\u200b\u4e0e\u200b\u52a8\u200b\u8bcd\u53e5","text":"<p>\u200b\u65e5\u8bed\u200b\u4e2d\u200b\u7684\u200b\u65f6\u6001\u53d8\u5316\u200b\u4e3b\u8981\u200b\u6709\u200b\u73b0\u5728\u200b\u548c\u200b\u8fc7\u53bb\u200b\u65f6\u200b\u3002\u200b\u540d\u8bcd\u200b\u656c\u4f53\u200b\u53e5\u200b\u4ee5\u200b\u300c\u3067\u3059\u300d\u200b\u7ed3\u53e5\u200b\uff0c\u200b\u5176\u200b\u5728\u200b\u4e0d\u540c\u200b\u65f6\u6001\u200b\u4e0b\u200b\u7684\u200b\u53d8\u5f62\u200b\u5982\u4e0b\u200b\uff1a</p> \u200b\u80af\u5b9a\u200b \u200b\u5426\u5b9a\u200b \u200b\u73b0\u5728\u200b \u301c\u3067\u3059\u3002 \u301c\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002 \u200b\u8fc7\u53bb\u200b \u301c\u3067\u3057\u305f\u3002 \u301c\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3067\u3057\u305f\u3002 <p>\u300c\u3067\u306f\u300d\u200b\u53ef\u4ee5\u200b\u66ff\u6362\u6210\u200b\u300c\u3058\u3083\u300d\u3002</p> <p>\u200b\u52a8\u8bcd\u200b\u656c\u4f53\u200b\u53e5\u200b\u4ee5\u200b\u300c\u307e\u3059\u300d\u200b\u7ed3\u53e5\u200b\uff0c\u200b\u5176\u200b\u5728\u200b\u4e0d\u540c\u200b\u65f6\u6001\u200b\u4e0b\u200b\u7684\u200b\u53d8\u5f62\u200b\u5982\u4e0b\u200b\uff1a</p> \u200b\u80af\u5b9a\u200b \u200b\u5426\u5b9a\u200b \u200b\u73b0\u5728\u200b \u301c\u307e\u3059\u3002 \u301c\u307e\u305b\u3093\u3002 \u200b\u8fc7\u53bb\u200b \u301c\u307e\u3057\u305f\u3002 \u301c\u307e\u305b\u3093\u3067\u3057\u305f\u3002 <p>\u200b\u5b58\u5728\u200b\u53e5\u200b\u7684\u200b\u52a8\u8bcd\u200b\u6709\u200b\u4e24\u79cd\u200b\uff0c\u200b\u5373\u200b\u300c\u3042\u308b\u300d\uff08\u3042\u308a\u307e\u3059\uff09\u200b\u548c\u200b\u300c\u3044\u308b\u300d\uff08\u3044\u307e\u3059\uff09\uff0c\u200b\u5176\u4e2d\u200b\u300c\u3042\u308b\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u65e0\u200b\u751f\u547d\u200b\u7684\u200b\u4e8b\u7269\u200b\uff0c\u300c\u3044\u308b\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u6709\u200b\u751f\u547d\u200b\u7684\u200b\u4e8b\u7269\u200b\u3002</p> <p>\u200b\u8868\u793a\u200b\u63d0\u8bae\u200b\u65f6\u200b\uff0c\u200b\u7528\u200b\u300c\u307e\u3057\u3087\u3046\u300d\u200b\u6216\u200b\u300c\u307e\u3057\u3087\u3046\u304b\u300d\u200b\u4ee3\u66ff\u200b\u300c\u307e\u3059\u300d\u3002\u200b\u8868\u793a\u200b\u63a8\u6d4b\u200b\u65f6\u200b\uff0c\u200b\u7528\u200b\u300c\u3067\u3057\u3087\u3046\u300d\u200b\u4ee3\u66ff\u200b\u300c\u3067\u3059\u300d\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_19","title":"\u63cf\u8ff0\u200b\u53e5","text":"<p>\u200b\u5bf9\u4e8e\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0c\u200b\u5176\u200b\u5728\u200b\u4e0d\u540c\u200b\u65f6\u6001\u200b\u4e0b\u200b\u7684\u200b\u53d8\u5f62\u200b\u5982\u4e0b\u200b\uff08\u201c\uff5e\u201d\u200b\u6307\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u8bcd\u5e72\u200b\uff0c\u200b\u5373\u200b\u53bb\u6389\u200b\u300c\u3044\u300d\u200b\u7684\u200b\u90e8\u5206\u200b\uff09\uff1a</p> \u200b\u80af\u5b9a\u200b \u200b\u5426\u5b9a\u200b \u200b\u73b0\u5728\u200b \u301c\u3044\u3067\u3059\u3002 \u301c\u304f\u3042\u308a\u307e\u305b\u3093\u3002 \u200b\u8fc7\u53bb\u200b \u301c\u304b\u3063\u305f\u3067\u3059\u3002 \u301c\u304f\u3042\u308a\u307e\u305b\u3093\u3067\u3057\u305f\u3002 <p>\u200b\u6ce8\u610f\u200b\u300c\u306a\u3044\u300d\u200b\u4e5f\u200b\u662f\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0c\u300c\u3042\u308a\u307e\u305b\u3093\u300d\u200b\u548c\u200b\u300c\u306a\u3044\u3067\u3059\u300d\u200b\u7b49\u4ef7\u200b\u3002\u200b\u56e0\u6b64\u200b\u8fd8\u6709\u200b\u5982\u4e0b\u200b\u53d8\u5f62\u200b\uff1a</p> \u200b\u80af\u5b9a\u200b \u200b\u5426\u5b9a\u200b \u200b\u73b0\u5728\u200b \u301c\u3044\u3067\u3059\u3002 \uff5e\u304f\u306a\u3044\u3067\u3059\u3002 \u200b\u8fc7\u53bb\u200b \u301c\u304b\u3063\u305f\u3067\u3057\u305f\u3002 \u301c\u304f\u306a\u304b\u3063\u305f\u3067\u3059 <p>\u200b\u4ee5\u4e0b\u200b\u4e24\u4e2a\u200b\u53e5\u5b50\u200b\u7b49\u4ef7\u200b\uff1a</p> <p>\u300c\u200b\u79c1\u200b\u306f\u200b\u9ad8\u200b\u304f\u306a\u3044\u3067\u3059\u3002\u300d \u300c\u200b\u79c1\u200b\u306f\u200b\u9ad8\u200b\u304f\u3042\u308a\u307e\u305b\u3093\u3002\u300d</p> <p>II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff08\u200b\u5f62\u5bb9\u200b\u52a8\u8bcd\u200b\uff09\u200b\u7684\u200b\u53d8\u5f62\u200b\u548c\u200b\u540d\u8bcd\u200b\u76f8\u540c\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_20","title":"\u52a8\u200b\u8bcd\u53e5\u200b\u7684\u200b\u8fdb\u884c\u200b\u65f6\u200b\u4e0e\u200b\u5b8c\u6210\u200b\u65f6","text":"<p>\u300c\u3066\u3044\u307e\u3059\u300d\u200b\u4e0e\u200b\u6301\u7eed\u6027\u200b\u52a8\u8bcd\u200b\u8fde\u7528\u200b\u65f6\u200b\uff0c\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u8fdb\u884c\u200b\u65f6\u200b\uff1b\u200b\u4e0e\u200b\u77ac\u95f4\u200b\u52a8\u8bcd\u200b\u8fde\u7528\u200b\u65f6\u200b\uff0c\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u5b8c\u6210\u200b\u65f6\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_21","title":"\u987a\u63a5\u200b\u5173\u7cfb","text":"<p>\u200b\u4e24\u4e2a\u200b\u5206\u53e5\u200b\u5982\u679c\u200b\u4e0d\u662f\u200b\u8868\u793a\u200b\u8f6c\u6298\u200b\u3001\u200b\u6761\u4ef6\u200b\u7b49\u200b\u5173\u7cfb\u200b\uff0c\u200b\u8868\u793a\u200b\u987a\u63a5\u200b\u5173\u7cfb\u200b\u65f6\u200b\uff0c\u200b\u6784\u6210\u200b\u987a\u63a5\u200b\u5173\u7cfb\u200b\u3002\u200b\u7b2c\u4e00\u4e2a\u200b\u4ece\u53e5\u200b\u7684\u200b\u53e5\u200b\u5c3e\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u300c\u3066\u300d\u200b\u578b\u200b\uff08\u200b\u8fde\u7528\u200b\u578b\u200b\uff09\u200b\u7ed3\u5c3e\u200b\u3002</p> <p>\u300c\u200b\u79c1\u200b\u306f\u200b\u65e5\u672c\u200b\u8a9e\u200b\u3092\u200b\u52c9\u5f37\u200b\u3057\u3066\u3001\u200b\u4e2d\u56fd\u200b\u8a9e\u200b\u3092\u200b\u52c9\u5f37\u200b\u3057\u307e\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u5b66\u4e60\u200b\u65e5\u8bed\u200b\uff0c\u200b\u4e5f\u200b\u5b66\u4e60\u200b\u6c49\u8bed\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_22","title":"\u5f62\u5f0f\u200b\u4f53\u8a00","text":"<p>\u200b\u5f62\u5f0f\u200b\u4f53\u8a00\u200b\u662f\u200b\u6307\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u62bd\u8c61\u6982\u5ff5\u200b\u7684\u200b\u540d\u8bcd\u200b\uff0c\u200b\u5982\u200b\u300c\u3053\u3068\u300d\u3001\u300c\u306e\u300d\u3001\u300c\u3082\u306e\u300d\u200b\u7b49\u200b\u3002\u200b\u5f62\u5f0f\u200b\u4f53\u8a00\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u4e3b\u8bed\u200b\u6216\u200b\u5bbe\u8bed\u200b\uff0c\u200b\u4e0e\u200b\u52a8\u8bcd\u200b\u539f\u578b\u200b\u8fde\u7528\u200b\u65f6\u200b\u76f8\u5f53\u4e8e\u200b\u52a8\u540d\u8bcd\u200b\u3002</p> \u200b\u5f62\u5f0f\u200b\u4f53\u8a00\u200b \u200b\u542b\u4e49\u200b \u3053\u3068 \u200b\u4e8b\u60c5\u200b\uff08\u200b\u4e00\u822c\u200b\u8868\u793a\u200b\u4e8b\u60c5\u200b\u3001\u200b\u4e8b\u5b9e\u200b\u3001\u200b\u5185\u5bb9\u200b\uff09 \u306e \u200b\u4e8b\u60c5\u200b\uff08\u200b\u4e00\u822c\u200b\u8868\u793a\u200b\u4eba\u200b\u3001\u200b\u4e8b\u200b\u3001\u200b\u7269\u200b\uff09 \u3068\u3053\u308d \u200b\u5730\u70b9\u200b\uff08\u200b\u7528\u6765\u200b\u505a\u200b\u2026\u2026\uff09 \u3082\u306e \u200b\u7269\u54c1\u200b\uff08\u200b\u7528\u6765\u200b\u505a\u200b\u2026\u2026\uff09 <p>\u200b\u56f3\u200b\u66f8\u9928\u200b\u306f\u200b\u52c9\u5f37\u200b\u3059\u308b\u3068\u3053\u308d\u3067\u3059\u3002\uff1a\u200b\u56fe\u4e66\u9986\u200b\u662f\u200b\u5b66\u4e60\u200b\u7684\u200b\u5730\u65b9\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_23","title":"\u6bd4\u8f83\u7ea7\u200b\u4e0e\u200b\u6700\u9ad8\u7ea7","text":"<p>\u200b\u65e5\u8bed\u200b\u4e2d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u6bd4\u8f83\u200b\u7684\u200b\u52a9\u8bcd\u200b\u4e3a\u200b\u300c\u3088\u308a\u300d\u200b\u548c\u200b\u300c\u307b\u3069\u300d\u3002\u200b\u8981\u200b\u8868\u793a\u200b\u201cA\u200b\u6bd4\u200bB\u200b\u66f4\u200b\u2026\u2026\u201d\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u7684\u200b\u8868\u8fbe\u65b9\u5f0f\u200b\uff1a</p> <ol> <li>\u300cA\u306fB\u3088\u308a\u301c\u300d\uff0c\u200b\u8868\u793a\u200b\u201cA\u200b\u6bd4\u200bB\u200b\u66f4\u200b\u2026\u2026\u201d\uff0c\u200b\u5728\u200b\u53e3\u8bed\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u300cA\u306f\u300d\u200b\u6765\u200b\u66f4\u200b\u591a\u200b\u5f3a\u8c03\u200b\u7528\u4e8e\u200b\u6bd4\u8f83\u200b\u7684\u200b\u57fa\u51c6\u200b\u3002</li> <li>\u300cB\u3088\u308aA\u306e\u200b\u65b9\u200b\u304c\u300d\u200b\u8868\u793a\u200b\u201cA\u200b\u6bd4\u200bB\u200b\u66f4\u200b\u2026\u2026\u201d\uff0c\u200b\u5728\u200b\u53e3\u8bed\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u300cB\u3088\u308a\u300d\uff0c\u200b\u66f4\u200b\u591a\u200b\u5f3a\u8c03\u200b\u4e3b\u8bed\u200bA\u3002</li> <li>\u300cB\u306fA\u307b\u3069\u301c\u3042\u308a\u307e\u305b\u3093\u300d\uff0c\u200b\u8868\u793a\u200b\u201cB\u200b\u4e0d\u5982\u200bA\u2026\u2026\u201d\u3002</li> </ol> <p>\u200b\u5728\u200b\u7591\u95ee\u53e5\u200b\u4e2d\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b\u300cA\u3068B\u3068\u3069\u3061\u3089\u306e\u200b\u65b9\u200b\u304c\u301c\u3067\u3059\u304b\u300d\u200b\u6765\u200b\u8be2\u95ee\u200bA\u200b\u548c\u200bB\u200b\u54ea\u4e2a\u200b\u66f4\u200b\u2026\u2026\u3002\u200b\u5176\u4e2d\u200b\u300c\u306e\u200b\u65b9\u200b\u300d\u200b\u8868\u793a\u200b\u201c\u2026\u2026\u200b\u7684\u200b\u7a0b\u5ea6\u200b\u201d\uff0c\u200b\u5728\u200b\u95ee\u53e5\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u3002\u200b\u6ce8\u610f\u200b\u5728\u200b\u56de\u7b54\u200b\u7591\u95ee\u53e5\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b\u300c\u304c\u300d\u3002\u200b\u5982\u679c\u200b\u4e24\u8005\u200b\u7a0b\u5ea6\u200b\u5dee\u4e0d\u591a\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b\u300c\u3069\u3061\u3089\u3082\u301c\u300d\u200b\u56de\u7b54\u200b\u3002</p> <p>\u300c\u200b\u5317\u4eac\u5927\u5b66\u200b\u3068\u200b\u6e05\u83ef\u200b\u5927\u5b66\u200b\u3068\u3069\u3061\u3089\u306e\u200b\u65b9\u200b\u304c\u200b\u6709\u540d\u200b\u3067\u3059\u304b\u3002\u300d\uff1a\u200b\u5317\u4eac\u5927\u5b66\u200b\u548c\u200b\u6e05\u534e\u5927\u5b66\u200b\u54ea\u4e2a\u200b\u66f4\u200b\u6709\u540d\u200b\uff1f \u300c\u200b\u6e05\u83ef\u200b\u5927\u5b66\u200b\u306e\u200b\u65b9\u200b\u304c\u200b\u6709\u540d\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u5317\u4eac\u5927\u5b66\u200b\u66f4\u200b\u6709\u540d\u200b\u3002 \u200b\u6216\u200b \u300c\u3069\u3061\u3089\u3082\u200b\u6709\u540d\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u4e24\u4e2a\u200b\u90fd\u200b\u6709\u540d\u200b\u3002</p> <p>\u200b\u65e5\u8bed\u200b\u4e2d\u200b\u8868\u8fbe\u200b\u6700\u9ad8\u7ea7\u200b\u7684\u200b\u65b9\u5f0f\u200b\u6709\u200b\u4e24\u79cd\u200b\uff1a</p> <ol> <li> <p>\u200b\u526f\u8bcd\u200b\u300c\u200b\u4e00\u756a\u200b\u300d\uff0c\u200b\u8868\u793a\u200b\u201c\u200b\u6700\u200b\u2026\u2026\u201d\uff0c\u200b\u5728\u200b\u53e5\u5b50\u200b\u4e2d\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u300c\u301c\u306e\u200b\u4e2d\u200b\u3067\u300d\u200b\u6765\u200b\u9650\u5236\u200b\u6bd4\u8f83\u200b\u7684\u200b\u8303\u56f4\u200b\u3002</p> <p>\u300c\u200b\u5317\u4eac\u200b\u306e\u200b\u5927\u5b66\u200b\u306e\u200b\u4e2d\u200b\u3067\u200b\u6e05\u83ef\u200b\u5927\u5b66\u200b\u304c\u200b\u4e00\u756a\u200b\u6709\u540d\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u5728\u200b\u5317\u4eac\u200b\u7684\u200b\u5927\u5b66\u200b\u4e2d\u200b\uff0c\u200b\u6e05\u534e\u5927\u5b66\u200b\u6700\u200b\u6709\u540d\u200b\u3002</p> </li> <li> <p>\u200b\u4f7f\u7528\u200b\u201c\u200b\u6ca1\u6709\u200b\u6bd4\u200b\u2026\u2026\u200b\u66f4\u200b\u2026\u2026\u201d\u200b\u7684\u200b\u53e5\u5f0f\u200b\uff0c\u200b\u5373\u200b\u300cA\u307b\u3069\u301cB\u306f\u3042\u308a\u307e\u305b\u3093\u300d\u3002\u200b\u5176\u4e2d\u200b\uff0cB\u200b\u4e3a\u200b\u5f62\u5f0f\u200b\u4f53\u8a00\u200b\uff0c\u200b\u8868\u793a\u200b\u201c\u2026\u2026\u200b\u7684\u200b\u4e1c\u897f\u200b\u201d\u3002\u200b\u6ce8\u610f\u200b\u7528\u4e8e\u200b\u6709\u200b\u751f\u547d\u200b\u7684\u200b\u4e8b\u7269\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b\u300c\u3044\u307e\u305b\u3093\u300d\u3002</p> <p>\u300c\u200b\u6e05\u83ef\u200b\u5927\u5b66\u200b\u307b\u3069\u200b\u6709\u540d\u200b\u306a\u200b\u5927\u5b66\u200b\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u300d\uff1a\u200b\u6ca1\u6709\u200b\u6bd4\u200b\u6e05\u534e\u5927\u5b66\u200b\u66f4\u200b\u6709\u540d\u200b\u7684\u200b\u5927\u5b66\u200b\u4e86\u200b\u3002</p> </li> </ol> <p>\u200b\u63d0\u95ee\u200b\u6700\u9ad8\u7ea7\u200b\u65f6\u200b\u7684\u200b\u4e00\u822c\u200b\u53e5\u5f0f\u200b\u4e3a\u200b\u300cA\u3067\u3069\u3053\u304c\u200b\u4e00\u756a\u200b\u301c\u3067\u3059\u304b\u300d\u200b\u6216\u200b\u300cA\u3067\u3069\u306e\u301c\u304c\u200b\u4e00\u756a\u200b\u301c\u3067\u3059\u304b\u300d\uff0c\u200b\u610f\u200b\u4e3a\u200bA\u200b\u4e2d\u200b\u6700\u200b\u2026\u2026\u200b\u7684\u200b\u662f\u200b\u54ea\u4e2a\u200b\u2026\u2026\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_24","title":"\u7b80\u4f53\u200b\u53e5","text":"<p>\u200b\u5f53\u200b\u4f7f\u7528\u200b\u63a5\u7eed\u200b\u52a9\u8bcd\u200b\u300c\u306e\u3067\u300d\u200b\u65f6\u200b\uff0c\u200b\u524d\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u7b80\u4f53\u200b\u53e5\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_25","title":"\u540d\u8bcd\u200b\u53e5","text":"<p>\u200b\u5bf9\u4e8e\u200b\u53e5\u5b50\u200b\u7ed3\u5c3e\u200b\u662f\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u7b80\u4f53\u200b\u53e5\u200b\u65e0\u9700\u200b\u505a\u51fa\u200b\u53d8\u5316\u200b\u3002</p> \u200b\u80af\u5b9a\u200b \u200b\u5426\u5b9a\u200b \u200b\u73b0\u5728\u200b \uff5e\u3044 \uff5e\u304f\u306a\u3044 \u200b\u8fc7\u53bb\u200b \uff5e\u304b\u3063\u305f \uff5e\u304f\u306a\u304b\u3063\u305f <p>\u200b\u5bf9\u4e8e\u200b\u53e5\u5b50\u200b\u7ed3\u5c3e\u200b\u662f\u200b\u540d\u8bcd\u200b\u548c\u200bII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u7b80\u4f53\u200b\u53e5\u200b\u662f\u200b\u5728\u200b\u53e5\u200b\u5c3e\u200b\u52a0\u4e0a\u200b\u300c\u3060\u300d\uff08II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u539f\u578b\u200b\uff09\u3002\u200b\u8fc7\u53bb\u200b\u65f6\u200b\u7b80\u4f53\u200b\u53e5\u200b\u662f\u200b\u5728\u200b\u53e5\u200b\u5c3e\u200b\u52a0\u4e0a\u200b\u300c\u3060\u3063\u305f\u300d\u3002\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b\u52a0\u4e0a\u200b\u300c\u3067\u306f\u306a\u3044\u300d\u200b\u6216\u200b\u300c\u3067\u306f\u306a\u3063\u304b\u305f\u300d\u3002</p> \u200b\u80af\u5b9a\u200b \u200b\u5426\u5b9a\u200b \u200b\u73b0\u5728\u200b \u301c\u3060 \u301c\u3067\u306f\u306a\u3044 \u200b\u8fc7\u53bb\u200b \u301c\u3060\u3063\u305f \u301c\u3067\u306f\u306a\u304b\u3063\u305f"},{"location":"japanese/japanese-1-grammar/#_26","title":"\u52a8\u200b\u8bcd\u53e5","text":"<p>\u200b\u5728\u200b\u7b80\u4f53\u200b\u53e5\u200b\u4e2d\u200b\uff0c\u200b\u52a8\u8bcd\u200b\u4f7f\u7528\u200b\u5176\u200b\u539f\u578b\u200b\u3002\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b\u4e2d\u200b\u4f7f\u7528\u200b\u5176\u200b\u306a\u3044\u200b\u578b\u200b\uff0c\u200b\u8fc7\u53bb\u200b\u65f6\u200b\u4f7f\u7528\u200b\u5176\u200b\u305f\u200b\u578b\u200b\u3002\u200b\u6ce8\u610f\u200b\u300c\u306a\u3044\u300d\u200b\u662f\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5176\u200b\u8fc7\u53bb\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b\u4e3a\u200b\u300c\u306a\u304b\u3063\u305f\u300d\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_27","title":"\u306e\u3067\u3059/\u3093\u3067\u3059\u200b\u7ed3\u6784","text":"<p>\u300c\u306e\u3067\u3059\u300d\u200b\u548c\u200b\u300c\u3093\u3067\u3059\u300d\u200b\u591a\u200b\u7528\u4e8e\u200b\u53e3\u8bed\u200b\uff0c\u200b\u8868\u793a\u200b\u89e3\u91ca\u200b\u3001\u200b\u8bf4\u660e\u200b\u3001\u200b\u8bf7\u6c42\u200b\u3001\u200b\u5efa\u8bae\u200b\u7b49\u200b\u8bed\u6c14\u200b\u3002II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u6216\u200b\u540d\u8bcd\u200b\u9700\u8981\u200b\u5728\u200b\u7b80\u4f53\u200b\u53e5\u200b\u5f62\u5f0f\u200b\u540e\u200b\u52a0\u4e0a\u200b\u300c\u306a\u300d\uff0c\u200b\u518d\u200b\u52a0\u200b\u300c\u306e\u3067\u3059\u300d\u200b\u6216\u200b\u300c\u3093\u3067\u3059\u300d\uff1bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u76f4\u63a5\u200b\u52a0\u200b\u300c\u306e\u3067\u3059\u300d\u200b\u6216\u200b\u300c\u3093\u3067\u3059\u300d\u3002</p>"},{"location":"japanese/japanese-1-pattern/","title":"\u65e5\u8bed\u200b(1)\u200b\u77ed\u8bed\u200b\u4e0e\u200b\u8868\u8fbe","text":""},{"location":"japanese/japanese-1-pattern/#1-4","title":"1-4\u200b\u8bfe","text":"<ol> <li>\u300c\u304a\u3084\u3059\u307f\u306a\u3055\u3044\u300d\uff1a\u200b\u665a\u5b89\u200b</li> <li>\u300c\u304a\u200b\u75b2\u200b\u308c\u200b\u69d8\u200b\u300d\uff1a\u200b\u8f9b\u82e6\u200b\u4e86\u200b</li> <li>\u300c\u308f\u304b\u308a\u307e\u3057\u305f\u300d\uff1a\u200b\u6211\u200b\u77e5\u9053\u200b\u4e86\u200b</li> <li>\u300c\u200b\u521d\u200b\u3081\u307e\u3057\u3066\u300d\uff1a\u200b\u521d\u6b21\u89c1\u9762\u200b</li> <li>\u300c\u304a\u200b\u5143\u200b\u6c17\u200b\u3067\u3059\u304b\u300d\uff1a\u200b\u60a8\u597d\u200b\u5417\u200b\uff1f</li> <li>\u300c\u304a\u304b\u3052\u3055\u307e\u3067\u3001\u200b\u5143\u200b\u6c17\u200b\u3067\u3059\u300d\uff1a\u200b\u8c22\u8c22\u60a8\u200b\uff0c\u200b\u6211\u200b\u5f88\u200b\u597d\u200b\u3002</li> <li>\u300c\u200b\u884c\u200b\u3063\u3066\u304d\u307e\u3059\u300d\uff1a\u200b\u6211\u200b\u8d70\u200b\u4e86\u200b</li> <li>\u300c\u200b\u884c\u200b\u3063\u3066\u3089\u3063\u3057\u3083\u3044\u300d\uff1a\u200b\u4e00\u8def\u987a\u98ce\u200b</li> <li>\u300c\u305f\u3060\u3044\u307e\u300d\uff1a\u200b\u6211\u200b\u56de\u6765\u200b\u4e86\u200b</li> <li>\u300c\u304a\u200b\u5e30\u200b\u308a\u306a\u3055\u3044\u300d\uff1a\u200b\u6b22\u8fce\u200b\u56de\u6765\u200b</li> <li>\u300c\u3055\u3088\u3046\u306a\u3089\u300d\uff1a\u200b\u518d\u89c1\u200b</li> <li>\u300c\u3042\u308a\u304c\u3068\u3046\u3054\u3056\u3044\u307e\u3059\u300d\uff1a\u200b\u975e\u5e38\u200b\u611f\u8c22\u60a8\u200b</li> <li>\u300c\u3042\u308a\u304c\u3068\u3046\u3054\u3056\u3044\u307e\u3057\u305f\u300d\uff1a\u200b\u975e\u5e38\u200b\u611f\u8c22\u60a8\u200b</li> </ol>"},{"location":"japanese/japanese-1-pattern/#5","title":"\u7b2c\u200b5\u200b\u8bfe","text":"<ol> <li>\u300c\u304a\u306f\u3088\u3046\u3054\u3056\u3044\u307e\u3059\u300d\uff1a\u200b\u65e9\u4e0a\u597d\u200b</li> <li>\u300c\u3053\u3093\u306b\u3061\u306f\u300d\uff1a\u200b\u4f60\u597d\u200b</li> <li>\u300c\u3053\u3093\u3070\u3093\u306f\u300d\uff1a\u200b\u665a\u4e0a\u200b\u597d\u200b</li> <li>\u300c\u3088\u308d\u3057\u304f\u304a\u200b\u9858\u200b\u3044\u3057\u307e\u3059\u300d\u3001\u300c\u3088\u308d\u3057\u304f\u304a\u200b\u9858\u200b\u3044\u3044\u305f\u3057\u307e\u3059\u300d\uff1a\u200b\u8bf7\u591a\u5173\u7167\u200b</li> <li>\u200b\u5173\u4e8e\u200b\u95ee\u53e5\u200b<ul> <li>\u200b\u80af\u5b9a\u200b\u56de\u590d\u200b\uff1a\u300c\u306f\u3044\u3001\u305d\u3046\u3067\u3059\u300d\uff0c\u200b\u6ce8\u610f\u200b\u300c\u306f\u3044\u300d\u200b\u4e0d\u80fd\u200b\u591a\u6b21\u91cd\u590d\u200b\u3002</li> <li>\u200b\u5426\u5b9a\u200b\u56de\u590d\u200b\uff1a\u300c\u3044\u3044\u3048\u3001\u200b\u9055\u200b\u3044\u307e\u3059\u300d</li> </ul> </li> </ol>"},{"location":"japanese/japanese-1-pattern/#6","title":"\u7b2c\u200b6\u200b\u8bfe","text":"<ol> <li>\u300c\u3048\u3048\u3068\u300d\uff1a\u200b\u55ef\u200b\u2026\u2026\uff0c\u200b\u8868\u793a\u200b\u77ed\u6682\u200b\u7684\u200b\u505c\u987f\u200b\u6216\u200b\u601d\u8003\u200b</li> <li>\u200b\u5207\u5165\u200b\u8bdd\u9898\u200b<ul> <li>\u300c\u3042\u306e\u3046\u300d\uff1a\u200b\u90a3\u4e2a\u200b\u2026\u2026</li> <li>\u300c\u3059\u307f\u307e\u305b\u3093\uff08\u304c\uff09\u300d\uff1a\u200b\u4e0d\u597d\u610f\u601d\u200b</li> </ul> </li> <li>\u200b\u7ed3\u675f\u200b\u8bdd\u9898\u200b<ul> <li>\u300c\u200b\u5931\u793c\u200b\u3057\u307e\u3059\u300d\uff1a\u200b\u5931\u966a\u200b\u4e86\u200b</li> <li>\u300c\u3042\u308a\u304c\u3068\u3046\u3054\u3056\u3044\u307e\u3057\u305f\u300d\uff1a\u200b\u4eca\u5929\u200b\u975e\u5e38\u200b\u611f\u8c22\u60a8\u200b</li> <li>\u300c\u3067\u306f\u3001\u307e\u305f\u300d\uff1a\u200b\u90a3\u4e48\u200b\uff0c\u200b\u518d\u89c1\u200b</li> </ul> </li> </ol>"},{"location":"japanese/japanese-1-pattern/#7","title":"\u7b2c\u200b7\u200b\u8bfe","text":"<ol> <li>\u300c\u200b\u5931\u793c\u200b\u3057\u307e\u3057\u305f\u300d\uff1a\u200b\u4e0d\u597d\u610f\u601d\u200b\uff0c\u200b\u7528\u4e8e\u200b\u5931\u793c\u200b\u3001\u200b\u5931\u656c\u200b\u7684\u200b\u884c\u4e3a\u200b\u3002</li> <li>\u300c\u304a\u200b\u4e16\u8a71\u200b\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u300d\uff1a\u200b\u611f\u8c22\u60a8\u200b\u4e00\u76f4\u200b\u4ee5\u6765\u200b\u7684\u200b\u7167\u987e\u200b\u3002</li> <li>\u200b\u4ecb\u7ecd\u200b\u5bb6\u5ead\u6210\u5458\u200b<ul> <li>\u300c\u200b\u5fa1\u5b85\u200b\u300d\u200b\u7528\u4e8e\u200b\u5c0a\u79f0\u200b\u5bf9\u65b9\u200b\u7684\u200b\u5bb6\u5ead\u200b\uff0c\u300c\u3046\u3061\u300d\u200b\u7528\u4e8e\u200b\u79f0\u547c\u200b\u81ea\u5df1\u200b\u7684\u200b\u5bb6\u5ead\u200b\u3002</li> <li>\u300cA\u306b\u306f\uff5e\u304c\u3044\u307e\u3059\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200bA\u200b\u6709\u200b\uff5e\u200b\u5bb6\u5ead\u6210\u5458\u200b\uff0c\u200b\u5982\u679c\u200b\u6709\u200b\u591a\u4f4d\u200b\uff0c\u200b\u5219\u200b\u5728\u200b\u300c\u304c\u300d\u200b\u540e\u52a0\u200b\u6570\u91cf\u8bcd\u200b\uff0c\u300c\u3068\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u5e76\u5217\u200b\u3002</li> </ul> </li> <li>\u300c\u305d\u3046\u306a\u3093\u3067\u3059\u300d\uff1a\u200b\u8868\u793a\u200b\u53cd\u590d\u200b\u786e\u8ba4\u200b\u540e\u200b\u7684\u200b\u5f3a\u8c03\u200b\u3002</li> </ol>"},{"location":"japanese/japanese-1-pattern/#8","title":"\u7b2c\u200b8\u200b\u8bfe","text":"<ol> <li>\u300c\u301c\u3067\u200b\u5b9c\u200b\u3057\u3044\u3067\u3059\u304b\u300d\u200b\u7528\u4e8e\u200b\u5411\u200b\u5bf9\u65b9\u200b\u786e\u8ba4\u200b\uff5e\u200b\u4e2d\u200b\u7684\u200b\u5185\u5bb9\u200b\u662f\u5426\u200b\u5408\u9002\u200b\u3002\u200b\u53ef\u4ee5\u200b\u5982\u4e0b\u200b\u56de\u7b54\u200b<ul> <li>\u300c\u306f\u3044\u3001\u200b\u5927\u4e08\u592b\u200b\u3067\u3059\u3002\u300d</li> <li>\u300c\u306f\u3044\u3001\u304a\u200b\u9858\u200b\u3044\u3057\u307e\u3059\u3002\u300d</li> </ul> </li> <li>\u200b\u8d2d\u7269\u200b<ul> <li>\u200b\u63d0\u51fa\u200b\u9700\u6c42\u200b\uff1a\u300c\u301c\u306f\u3042\u308a\u307e\u3059\u304b\u300d\u3001\u300c\u301c\u3092\u304f\u3060\u3055\u3044\u300d\u3001\u300c\u301c\u3092\u304a\u200b\u9858\u200b\u3044\u3057\u307e\u3059\u300d\u3002</li> <li>\u200b\u8be2\u95ee\u200b\u4ef7\u683c\u200b\uff1a\u300c\u301c\uff08\u304a\uff09\u3044\u304f\u3089\u3067\u3059\u304b\u300d</li> <li>\u200b\u9009\u62e9\u200b\uff1a\u300c\u301c\u306b\u3057\u307e\u3059\u300d\u3001\u300c\u301c\u3067\u300d</li> </ul> </li> <li>\u200b\u5c31\u9910\u200b<ul> <li>\u200b\u9910\u524d\u200b\uff1a\u300c\u3044\u305f\u3060\u304d\u307e\u3059\u300d\uff1a\u200b\u6211\u8981\u200b\u5f00\u59cb\u200b\u5403\u200b\u4e86\u200b\u3002</li> <li>\u200b\u9910\u540e\u200b\uff1a\u300c\u3054\u200b\u99b3\u200b\u8d70\u200b\u69d8\u200b\u3067\u3057\u305f\u300d\uff1a\u200b\u60a8\u200b\u7684\u200b\u6b3e\u5f85\u200b\u771f\u662f\u200b\u592a\u597d\u4e86\u200b\u3002</li> </ul> </li> <li>\u300c\u3054\u3061\u3085\u3046\u3082\u3093\u306f\uff1f\u300d\uff1a\u200b\u60a8\u200b\u70b9\u200b\u4ec0\u4e48\u200b\uff1f</li> </ol>"},{"location":"japanese/japanese-1-pattern/#9","title":"\u7b2c\u200b9\u200b\u8bfe","text":"<ol> <li>\u300c\u304a\u200b\u4e45\u200b\u3057\u3076\u308a\u3067\u3059\u300d\uff1a\u200b\u597d\u4e45\u4e0d\u89c1\u200b</li> <li>\u300c\uff08\u305d\u308c\u306f\uff09\u3088\u304b\u3063\u305f\u3067\u3059\u306d\u300d\uff1a\u200b\u592a\u597d\u4e86\u200b<ul> <li>\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u81ea\u5df1\u200b\uff0c\u200b\u5bf9\u200b\u81ea\u5df1\u200b\u65f6\u4e0d\u52a0\u200b\u300c\u306d\u300d</li> </ul> </li> <li>\u300c\u200b\u52c9\u5f37\u200b\u306b\u306a\u308a\u307e\u3059\u300d\uff1a\u200b\u53d7\u76ca\u532a\u6d45\u200b</li> <li>\u300c\u200b\u9811\u5f35\u200b\u3063\u3066\uff08\u304f\u3060\u3055\u3044\uff09\u300d\uff1a\u200b\u52a0\u6cb9\u200b</li> <li>\u300c\u304a\u305f\u3093\u3058\u3087\u3046\u3073\u306f\u3044\u3064\u3067\u3059\u304b\u300d\uff1a\u200b\u60a8\u200b\u7684\u200b\u751f\u65e5\u200b\u662f\u200b\u4ec0\u4e48\u200b\u65f6\u5019\u200b\uff1f<ul> <li>\u300c\u200b\u79c1\u200b\u306e\u200b\u8a95\u200b\u751f\u65e5\u200b\u306f\u306a\u306a\u304c\u3064\u306b\u3058\u3085\u3046\u308d\u304f\u306b\u3061\u3067\u3059\u300d\uff1a\u200b\u6211\u200b\u7684\u200b\u751f\u65e5\u200b\u662f\u200b7\u200b\u6708\u200b26\u200b\u65e5\u200b\u3002</li> </ul> </li> <li>\u300c\u304a\u3081\u3067\u3068\u3046\u3054\u3056\u3044\u307e\u3059\u300d\uff1a\u200b\u606d\u559c\u200b\u60a8\u200b</li> </ol>"},{"location":"japanese/japanese-1-pattern/#10","title":"\u7b2c\u200b10\u200b\u8bfe","text":"<ol> <li>\u300c\u301c\u306f\u200b\u7a7a\u200b\u3044\u3066\u3044\u307e\u3059\u300d\uff1a\uff5e\u200b\u6709\u7a7a\u200b</li> <li>\u300c\u301c\u305f\u3044\u3067\u3059\u300d\uff1a\u200b\u60f3\u8981\u200b\uff5e</li> <li>\u300c\u301c\u304c\u200b\u697d\u200b\u3057\u307f\u3067\u3059\u300d\uff1a\uff5e\u200b\u4ee4\u4eba\u200b\u671f\u5f85\u200b</li> <li>\u200b\u63d0\u51fa\u200b\u9080\u8bf7\u200b\uff1a<ul> <li>\u300c\uff08\u200b\u4e00\u7dd2\u200b\u306b\uff09\u301c\u307e\u3057\u3087\u3046\u304b\u300d\uff1a\u200b\u4e00\u8d77\u200b\uff5e\u200b\u5427\u200b\uff1f</li> <li>\u300c\u301c\u306f\u3069\u3046\u3067\u3059\u304b\u300d\uff1a\uff5e\u200b\u600e\u4e48\u6837\u200b\uff1f</li> <li>\u300c\u301c\u304c\u3044\u3044\u3067\u3059\u304b\u300d\uff1a\uff5e\u200b\u597d\u200b\u5417\u200b\uff1f\uff08\uff5e\u200b\u5fc5\u987b\u200b\u662f\u200b\u7591\u95ee\u8bcd\u200b\uff09</li> </ul> </li> <li>\u200b\u56de\u7b54\u200b\u9080\u8bf7\u200b<ul> <li>\u200b\u63a5\u53d7\u200b\u9080\u8bf7\u200b\uff1a\u300c\u306f\u3044\u3001\u3044\u3044\u3067\u3059\u306d\u300d\u3001\u300c\u305d\u3057\u307e\u3057\u3087\u3046\u300d</li> <li>\u200b\u62d2\u7edd\u200b\u9080\u8bf7\u200b\uff1a\u300c\u3059\u307f\u307e\u305b\u3093\u3001\u3061\u3087\u3063\u3068\u2026\u2026\u300d\u3001\u300c\u3044\u3044\u3048\u3001\u3051\u3063\u3053\u3046\u3067\u3059\u3002\u300d\u3001\u300c\u3044\u3044\u3067\u3059\u300d</li> </ul> </li> </ol>"},{"location":"japanese/japanese-1-pattern/#11","title":"\u7b2c\u200b11\u200b\u8bfe","text":"<ol> <li>\u300c\u200b\u5168\u7136\u200b\u301c\u300d\uff1a\u200b\u5b8c\u5168\u200b\u4e0d\u200b\uff5e\uff0c\u200b\u7528\u4e8e\u200b\u5426\u5b9a\u53e5\u200b\u3002</li> <li>\u300c\u305d\u3046\u306a\u3093\u3067\u3059\u304b\u300d\uff1a\u200b\u662f\u200b\u5417\u200b\uff1f\u200b\u8868\u793a\u200b\u66f4\u200b\u60ca\u8bb6\u200b\u7684\u200b\u8bed\u6c14\u200b\u3002</li> </ol>"},{"location":"japanese/japanese-1-pattern/#12","title":"\u7b2c\u200b12\u200b\u8bfe","text":"<ol> <li>\u300c\uff08\u3059\u307f\u307e\u305b\u3093\u3001\uff09\u301c\u3066\u304f\u3060\u3055\u3044\u300d\uff1a\u200b\u8bf7\u200b\uff5e</li> <li>\u300c\u301c\u3066\u304f\u3060\u3055\u3044\u307e\u305b\u3093\u304b\u300d\uff1a\u200b\u8bf7\u200b\uff5e\u200b\u5417\u200b\uff1f\uff08\u200b\u7528\u4e8e\u200b\u4e0b\u7ea7\u200b\u5bf9\u200b\u4e0a\u7ea7\u200b\uff09</li> </ol>"},{"location":"japanese/japanese-1-pattern/#1311","title":"\u7b2c\u200b13\u200b\u8bfe\u200b\uff08\u200b\u65b0\u4e16\u7eaa\u200b\u521d\u7ea7\u200b\u7b2c\u200b11\u200b\u8bfe\u200b\uff09","text":"<ol> <li>\u300c\u200b\u6559\u200b\u3048\u3066\u304f\u3060\u3055\u3044\u300d\uff1a\u200b\u8bf7\u200b\u544a\u8bc9\u200b\u6211\u200b</li> <li>\u300c\u200b\u5b89\u5fc3\u200b\u3057\u3066\u304f\u3060\u3055\u3044\u300d\uff1a\u200b\u8bf7\u200b\u653e\u5fc3\u200b</li> <li>\u300c\u3067\u306f\u3001\u304a\u200b\u5143\u200b\u6c17\u200b\u3067\u300d\uff1a\u200b\u90a3\u4e48\u200b\uff0c\u200b\u4fdd\u91cd\u200b</li> <li>\u300c\u3054\u200b\u5bb6\u65cf\u200b\u306e\u200b\u7686\u200b\u69d8\u200b\u306b\u3082\u3088\u308d\u3057\u304f\u300d\uff1a\u200b\u8bf7\u4ee3\u200b\u6211\u200b\u5411\u200b\u60a8\u200b\u7684\u200b\u5bb6\u4eba\u200b\u95ee\u597d\u200b</li> <li>\u300c\u3069\u3046\u3082\u300d\uff1a\u200b\u975e\u5e38\u611f\u8c22\u200b</li> </ol>"},{"location":"japanese/japanese-1-vocab/","title":"\u65e5\u8bed\u200b(1) \u200b\u5355\u8bcd\u8868","text":""},{"location":"japanese/japanese-1-vocab/#_1","title":"\u52a8\u8bcd","text":"I\u200b\u7c7b\u200b\u52a8\u8bcd\u200bII\u200b\u7c7b\u200b\u52a8\u8bcd\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b \u200b\u52a8\u8bcd\u200b \u200b\u5047\u540d\u200b \u200b\u5206\u7c7b\u200b \u200b\u542b\u4e49\u200b \u200b\u9055\u200b\u3046 \u3061\u304c\u3046 \u200b\u81ea\u200bI \u200b\u4e0d\u662f\u200b \u200b\u5408\u200b\u3046 \u3042\u3046 \u200b\u81ea\u200bI \u200b\u5408\u9002\u200b \u200b\u4ed8\u200b\u304d\u200b\u5408\u200b\u3046 \u3064\u304d\u3042\u3046 \u200b\u81ea\u200bI \u200b\u4ea4\u5f80\u200b \u3053\u305f\u308f\u308b \u200b\u81ea\u200bI \u200b\u8bb2\u7a76\u200b \u200b\u9811\u5f35\u200b\u308b \u304c\u3093\u3070\u308b \u200b\u81ea\u200bI \u200b\u52a0\u6cb9\u200b \u200b\u884c\u200b\u304f \u3044\u304f \u200b\u81ea\u200bI \u200b\u53bb\u200b \u200b\u7a7a\u200b\u304f \u3042\u304f \u200b\u81ea\u200bI \u200b\u7a7a\u95f2\u200b \u200b\u5206\u200b\u304b\u308b \u308f\u304b\u308b \u200b\u81ea\u200bI \u200b\u660e\u767d\u200b \u304b\u304b\u308b \u200b\u81ea\u200bI \u200b\u82b1\u8d39\u200b \u200b\u5e30\u200b\u308b \u304b\u3048\u308b \u200b\u81ea\u200bI \u200b\u56de\u5bb6\u200b \u200b\u8981\u200b\u308b \u3044\u308b \u200b\u81ea\u200bI \u200b\u9700\u8981\u200b \u200b\u6e07\u200b\u304f \u304b\u308f\u304f \u200b\u81ea\u200bI \u200b\u53e3\u6e34\u200b \u200b\u7b11\u200b\u3046 \u308f\u3089\u3046 \u200b\u81ea\u200bI \u200b\u7b11\u200b \u200b\u6b8b\u200b\u308b \u306e\u3053\u308b \u200b\u81ea\u200bI \u200b\u5269\u4e0b\u200b \u200b\u558b\u200b\u308b \u3057\u3083\u3079\u308b 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\u3068\u304f\u3044 \u200b\u64c5\u957f\u200b \u200b\u5fc3\u914d\u200b \u3057\u3093\u3071\u3044 \u200b\u62c5\u5fc3\u200b \u200b\u5b89\u5fc3\u200b \u3042\u3093\u3057\u3093 \u200b\u5b89\u5fc3\u200b \u200b\u5143\u200b\u6c17\u200b \u3052\u3093\u304d \u200b\u7cbe\u795e\u200b \u200b\u4e0d\u8981\u200b \u3075\u3088\u3046 \u200b\u4e0d\u8981\u200b \u200b\u7c21\u200b\u5358\u200b \u304b\u3093\u305f\u3093 \u200b\u7b80\u5355\u200b \u200b\u4e00\u751f\u200b\u61f8\u547d\u200b \u3044\u3063\u3057\u3087\u3046\u3051\u3093\u3081\u3044 \u200b\u62fc\u547d\u200b \u200b\u5927\u5207\u200b \u305f\u3044\u305b\u3064 \u200b\u91cd\u8981"},{"location":"japanese/japanese-1-vocab/#_3","title":"\u6570\u5b57\u200b\u3001\u200b\u65f6\u95f4\u200b\u3001\u200b\u65e5\u671f","text":""},{"location":"japanese/japanese-1-vocab/#_4","title":"\u6570\u5b57","text":"<p>\u200b\u6ce8\u610f\u200b\uff1a</p> <ol> <li>1\u200b\u4eba\u200b\u548c\u200b2\u200b\u4eba\u200b\u7684\u200b\u7279\u6b8a\u200b\u5199\u6cd5\u200b\u3002</li> <li>4\u200b\u4eba\u200b\u7684\u200b\u5199\u6cd5\u200b\u662f\u200b\u300c\u3088\u306b\u3093\u300d\uff0c\u200b\u800c\u200b\u4e0d\u662f\u200b\u300c\u3088\u3093\u306b\u3093\u300d\u3002</li> <li>7\u30019\u200b\u4eba\u200b\u7684\u200b\u5199\u6cd5\u200b\u6709\u200b\u4e24\u79cd\u200b\u3002</li> <li>\u300c\u3072\u3083\u304f\u300d\u200b\u548c\u200b\u300c\u305b\u3093\u300d\u200b\u53ef\u4ee5\u200b\u7528\u6765\u200b\u76f4\u63a5\u200b\u8868\u793a\u200b100\u200b\u548c\u200b1000\uff0c\u200b\u800c\u200b\u300c\u307e\u3093\u300d\u200b\u548c\u200b\u300c\u304a\u304f\u300d\u200b\u9700\u8981\u200b\u52a0\u4e0a\u200b\u300c\u3044\u3061\u300d\u200b\u6765\u200b\u8868\u793a\u200b1\u200b\u4e07\u200b\u548c\u200b1\u200b\u4ebf\u200b\u3002</li> </ol> \u200b\u6570\u5b57\u200b \u200b\u5047\u540d\u200b \u200b\u6570\u5b57\u200b \u200b\u5047\u540d\u200b 1 \u3044\u3061 1\u200b\u4eba\u200b \u3072\u3068\u308a 2 \u306b 2\u200b\u4eba\u200b \u3075\u305f\u308a 3 \u3055\u3093 3\u200b\u4eba\u200b \u3055\u3093\u306b\u3093 4 \u3088\u3093 4\u200b\u4eba\u200b \u3088\u306b\u3093 5 \u3054 5\u200b\u4eba\u200b \u3054\u306b\u3093 6 \u308d\u304f 6\u200b\u4eba\u200b \u308d\u304f\u306b\u3093 7 \u306a\u306a 7\u200b\u4eba\u200b \u3057\u3061\u306b\u3093/\u306a\u306a\u306b\u3093 8 \u306f\u3061 8\u200b\u4eba\u200b \u306f\u3061\u306b\u3093 9 \u304d\u3085\u3046 9\u200b\u4eba\u200b \u304d\u3085\u3046\u306b\u3093/\u304f\u306b\u3093 10 \u3058\u3085\u3046 10\u200b\u4eba\u200b \u3058\u3085\u3046\u306b\u3093 \u200b\u767e\u200b \u3072\u3083\u304f \u200b\u5343\u200b \u305b\u3093 \u200b\u4e07\u200b \u3044\u3061\u307e\u3093 \u200b\u5104\u200b \u304a\u304f"},{"location":"japanese/japanese-1-vocab/#_5","title":"\u91cf\u8bcd","text":"<p>\u200b\u6ce8\u610f\u200b\uff1a</p> <ol> <li>\u200b\u6570\u5b57\u200b+\u300c\u3064\u300d\u200b\u7684\u200b\u5199\u6cd5\u200b\uff0c\u200b\u9664\u4e86\u200b7\u200b\u662f\u200b\u76f4\u63a5\u200b\u52a0\u200b\u300c\u3064\u300d\u200b\u4ee5\u5916\u200b\uff0c\u200b\u5176\u4ed6\u200b\u90fd\u200b\u6709\u200b\u660e\u663e\u200b\u7684\u200b\u53d8\u5316\u200b\u3002</li> <li>\u200b\u6570\u5b57\u200b+\u300c\u200b\u500b\u200b\u3001\u200b\u672c\u200b\u3001\u200b\u518a\u200b\u300d\u200b\u7684\u200b\u5199\u6cd5\u200b\uff0c\u200b\u9664\u4e86\u200b1\u30016\u30018\u300110\u200b\u662f\u200b\u76f4\u63a5\u200b\u52a0\u200b\u300c\u200b\u500b\u200b\u300d\u200b\u4ee5\u5916\u200b\uff0c\u200b\u5176\u4ed6\u200b\u90fd\u200b\u5c06\u200b\u6570\u5b57\u200b\u8bfb\u97f3\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u5047\u540d\u200b\u8f6c\u4e3a\u200b\u501a\u97f3\u200b\u300c\u3063\u300d\uff0c\u200b\u518d\u200b\u52a0\u200b\u300c\u200b\u500b\u200b\u300d\uff0c\u200b\u5982\u200b\u300c\u3044\u3063\u3053\u300d\u3001\u300c\u308d\u3063\u3053\u300d\u3001\u300c\u306f\u3063\u3053\u300d\u3001\u300c\u3058\u3085\u3063\u3053\u300d\u3002</li> <li>\u200b\u6570\u5b57\u200b+\u300c\u200b\u672c\u200b\u300d\u200b\u7684\u200b\u5199\u6cd5\u200b\uff0c\u200b\u9700\u8981\u200b\u6ce8\u610f\u200b1\u30013\u30016\u30018\u300110\u200b\u9700\u8981\u200b\u5c06\u200b\u300c\u307b\u300d\u200b\u7684\u200b\u8bfb\u97f3\u200b\u53d8\u4e3a\u200b\u300c\u307d\u300d\u3002</li> <li>\u200b\u6570\u5b57\u200b+\u300c\u200b\u679a\u200b\u3001\u200b\u53f0\u200b\u300d\uff0c\u200b\u76f4\u63a5\u200b\u5728\u200b\u6570\u5b57\u200b\u540e\u9762\u200b\u52a0\u200b\u300c\u200b\u679a\u200b\u300d\u200b\u6216\u200b\u300c\u200b\u53f0\u200b\u300d\u200b\u5373\u53ef\u200b\u3002</li> </ol> \u200b\u6570\u91cf\u200b \u3064 \u200b\u500b\u200b \u200b\u672c\u200b \u200b\u518a\u200b \u200b\u679a\u200b \u200b\u53f0\u200b 1 \u3072\u3068\u3064 \u3044\u3063\u3053 \u3044\u3063\u307d\u3093 \u3044\u3063\u3055\u3064 \u3044\u3061\u307e\u3044 \u3044\u3061\u3060\u3044 2 \u3075\u305f\u3064 \u306b\u3053 \u306b\u307b\u3093 \u306b\u3055\u3064 \u306b\u307e\u3044 \u306b\u3060\u3044 3 \u307f\u3063\u3064 \u3055\u3093\u3053 \u3055\u3093\u307d\u3093 \u3055\u3093\u3055\u3064 \u3055\u3093\u307e\u3044 \u3055\u3093\u3060\u3044 4 \u3088\u3063\u3064 \u3088\u3093\u3053 \u3088\u3093\u307b\u3093 \u3088\u3093\u3055\u3064 \u3088\u3093\u307e\u3044 \u3088\u3093\u3060\u3044 5 \u3044\u3064\u3064 \u3054\u3053 \u3054\u307b\u3093 \u3054\u3055\u3064 \u3054\u307e\u3044 \u3054\u3060\u3044 6 \u3080\u3063\u3064 \u308d\u3063\u3053 \u308d\u3063\u307d\u3093 \u308d\u304f\u3055\u3064 \u308d\u304f\u307e\u3044 \u308d\u304f\u3060\u3044 7 \u306a\u306a\u3064 \u306a\u306a\u3053 \u306a\u306a\u307b\u3093 \u306a\u306a\u3055\u3064 \u306a\u306a\u307e\u3044 \u306a\u306a\u3060\u3044 8 \u3084\u3063\u3064 \u306f\u3063\u3053 \u306f\u3063\u307d\u3093 \u306f\u3063\u3055\u3064 \u306f\u3061\u307e\u3044 \u306f\u3061\u3060\u3044 9 \u3053\u3053\u306e\u3064 \u304d\u3085\u3046\u3053 \u304d\u3085\u3046\u307b\u3093 \u304d\u3085\u3046\u3055\u3064 \u304d\u3085\u3046\u307e\u3044 \u304d\u3085\u3046\u3060\u3044 10 \u3068\u304a \u3058\u3085\u3063\u3053 \u3058\u3085\u3063\u307d\u3093 \u3058\u3085\u3063\u3055\u3064 \u3058\u3085\u3046\u307e\u3044 \u3058\u3085\u3046\u3060\u3044"},{"location":"japanese/japanese-1-vocab/#_6","title":"\u5c0f\u65f6","text":"<p>\u200b\u534a\u5c0f\u65f6\u200b\u5728\u200b\u300c\u200b\u6642\u9593\u200b\u300d\u200b\u540e\u52a0\u200b\u300c\u200b\u534a\u200b\uff08\u306f\u3093\uff09\u300d\u200b\u5373\u53ef\u200b\u3002</p> \u200b\u65f6\u95f4\u200b \u200b\u5047\u540d\u200b \u200b\u65f6\u95f4\u200b \u200b\u5047\u540d\u200b \u200b\u65f6\u95f4\u200b \u200b\u5047\u540d\u200b 1\u200b\u6642\u9593\u200b \u3044\u3061\u3058\u304b\u3093 9\u200b\u6642\u9593\u200b \u304f\u3058\u304b\u3093 17\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u306a\u306a\u3058\u304b\u3093 2\u200b\u6642\u9593\u200b \u306b\u3058\u304b\u3093 10\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u3058\u304b\u3093 18\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u306f\u3061\u3058\u304b\u3093 3\u200b\u6642\u9593\u200b \u3055\u3093\u3058\u304b\u3093 11\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u3044\u3061\u3058\u304b\u3093 19\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u304f\u3058\u304b\u3093 4\u200b\u6642\u9593\u200b \u3088\u3058\u304b\u3093 12\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u306b\u3058\u304b\u3093 20\u200b\u6642\u9593\u200b \u306b\u3058\u3085\u3046\u3058\u304b\u3093 5\u200b\u6642\u9593\u200b \u3054\u3058\u304b\u3093 13\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u3055\u3093\u3058\u304b\u3093 21\u200b\u6642\u9593\u200b \u306b\u3058\u3085\u3046\u3044\u3061\u3058\u304b\u3093 6\u200b\u6642\u9593\u200b \u308d\u304f\u3058\u304b\u3093 14\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u3088\u3058\u304b\u3093 22\u200b\u6642\u9593\u200b \u306b\u3058\u3085\u3046\u306b\u3058\u304b\u3093 7\u200b\u6642\u9593\u200b \u3057\u3061\u3058\u304b\u3093 15\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u3054\u3058\u304b\u3093 23\u200b\u6642\u9593\u200b \u306b\u3058\u3085\u3046\u3055\u3093\u3058\u304b\u3093 8\u200b\u6642\u9593\u200b \u306f\u3061\u3058\u304b\u3093 16\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u308d\u304f\u3058\u304b\u3093 24\u200b\u6642\u9593\u200b \u306b\u3058\u3085\u3046\u3088\u3058\u304b\u3093"},{"location":"japanese/japanese-1-vocab/#_7","title":"\u5206\u949f","text":"\u65f6\u95f4\u200b \u200b\u5047\u540d\u200b \u200b\u65f6\u95f4\u200b \u200b\u5047\u540d\u200b 1\u200b\u5206\u200b \u3044\u3063\u3077\u3093 1\u200b\u5206\u534a\u200b \u3044\u3063\u3077\u3093\u306f\u3093 2\u200b\u5206\u200b \u306b\u3075\u3093 2\u200b\u5206\u534a\u200b \u306b\u3075\u3093\u306f\u3093 3\u200b\u5206\u200b \u3055\u3093\u3077\u3093 3\u200b\u5206\u534a\u200b \u3055\u3093\u3077\u3093\u306f\u3093 4\u200b\u5206\u200b \u3088\u3093\u3077\u3093 4\u200b\u5206\u534a\u200b \u3088\u3093\u3077\u3093\u306f\u3093 5\u200b\u5206\u200b \u3054\u3075\u3093 5\u200b\u5206\u534a\u200b \u3054\u3075\u3093\u306f\u3093 6\u200b\u5206\u200b \u308d\u3063\u3077\u3093 6\u200b\u5206\u534a\u200b \u308d\u3063\u3077\u3093\u306f\u3093 7\u200b\u5206\u200b \u306a\u306a\u3075\u3093 7\u200b\u5206\u534a\u200b \u306a\u306a\u3075\u3093\u306f\u3093 8\u200b\u5206\u200b \u306f\u3063\u3077\u3093 8\u200b\u5206\u534a\u200b \u306f\u3063\u3077\u3093\u306f\u3093 9\u200b\u5206\u200b \u304d\u3085\u3046\u3075\u3093 9\u200b\u5206\u534a\u200b \u304d\u3085\u3046\u3075\u3093\u306f\u3093 10\u200b\u5206\u200b \u3058\u3085\u3063\u3077\u3093 10\u200b\u5206\u534a\u200b \u3058\u3085\u3063\u3077\u3093\u306f\u3093"},{"location":"japanese/japanese-1-vocab/#_8","title":"\u65e5\u671f","text":"<p>\u200b\u6ce8\u610f\u200b\uff1a1-10\u200b\u65e5\u200b\u7684\u200b\u53d8\u5316\u200b\u6bd4\u8f83\u200b\u7279\u6b8a\u200b\uff0c\u200b\u5176\u4f59\u200b\u7684\u200b\u90fd\u200b\u662f\u200b\u6570\u5b57\u200b+\u300c\u200b\u65e5\u200b\u300d\u3002</p> \u200b\u65e5\u671f\u200b \u200b\u5047\u540d\u200b \u200b\u65e5\u671f\u200b \u200b\u5047\u540d\u200b \u200b\u65e5\u671f\u200b \u200b\u5047\u540d\u200b 1\u200b\u65e5\u200b 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7\u200b\u65e5\u200b \u306a\u306e\u304b 17\u200b\u65e5\u200b \u3058\u3085\u3046\u3057\u3061\u306b\u3061 27\u200b\u65e5\u200b \u306b\u3058\u3085\u3046\u3057\u3061\u306b\u3061 8\u200b\u65e5\u200b \u3088\u3046\u304b 18\u200b\u65e5\u200b \u3058\u3085\u3046\u306f\u3061\u306b\u3061 28\u200b\u65e5\u200b \u306b\u3058\u3085\u3046\u306f\u3061\u306b\u3061 9\u200b\u65e5\u200b \u3053\u3053\u306e\u304b 19\u200b\u65e5\u200b \u3058\u3085\u3046\u304f\u306b\u3061 29\u200b\u65e5\u200b \u306b\u3058\u3085\u3046\u304f\u306b\u3061 10\u200b\u65e5\u200b \u3068\u304a\u304b 20\u200b\u65e5\u200b \u306f\u3064\u304b 30\u200b\u65e5\u200b \u3055\u3093\u3058\u3085\u3046\u306b\u3061 31\u200b\u65e5\u200b \u3055\u3093\u3058\u3085\u3046\u3044\u3061\u306b\u3061"},{"location":"japanese/japanese-1-vocab/#_9","title":"\u661f\u671f","text":"\u661f\u671f\u200b \u200b\u661f\u671f\u200b \u200b\u661f\u671f\u4e00\u200b \u200b\u6708\u66dc\u65e5\u200b \u3052\u3064\u3088\u3046\u3073 \u200b\u661f\u671f\u4e8c\u200b 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\u200b\u53f8\u673a\u200b \u200b\u64ae\u200b\u308b \u3068\u308b \u200b\u62cd\u7167\u200b \u200b\u904b\u200b\u8ee2\u200b \u3046\u3093\u3066\u3093 \u200b\u5f00\u8f66\u200b \u200b\u672c\u5c4b\u200b \u307b\u3093\u3084 \u200b\u4e66\u5e97\u200b \u301c\u200b\u5c4b\u200b \u3084 ~\u200b\u5e97\u200b \u200b\u5929\u5a66\u200b\u7f85\u200b \u3066\u3093\u3077\u3089 \u200b\u5929\u5987\u200b\u7f57\u200b \u200b\u96e8\u200b \u3042\u3081 \u200b\u96e8\u200b \u200b\u6dbc\u200b\u3057\u3044 \u3059\u305a\u3057\u3044 \u200b\u51c9\u723d\u200b \u200b\u6674\u200b\u308c \u306f\u308c \u200b\u6674\u5929\u200b \u200b\u5546\u54c1\u200b \u3057\u3087\u3046\u3072\u3093 \u200b\u5546\u54c1\u200b \u200b\u685c\u200b \u3055\u304f\u3089 \u200b\u6a31\u82b1\u200b \u200b\u5473\u200b \u3042\u3058 \u200b\u5473\u9053\u200b \u200b\u8272\u200b \u3044\u308d \u200b\u989c\u8272\u200b \u200b\u521d\u200b\u3081\u3066 \u306f\u3058\u3081\u3066 \u200b\u7b2c\u4e00\u6b21\u200b \u200b\u7389\u5b50\u200b\u713c\u200b\u304d \u305f\u307e\u3054\u3084\u304d \u200b\u714e\u86cb\u200b \u3064\u307e\u3089\u306a\u3044 \u200b\u65e0\u804a\u200b \u200b\u697d\u200b\u5668\u200b \u304c\u3063\u304d \u200b\u4e50\u5668\u200b \u200b\u9eba\u200b \u3081\u3093 \u200b\u9762\u6761\u200b \u200b\u9055\u200b\u3046 \u3061\u304c\u3046 \u200b\u4e0d\u200b\u4e00\u6837"},{"location":"japanese/japanese-1-vocab/#1311","title":"\u7b2c\u200b13\u200b\u8bfe\u200b\uff08\u200b\u65b0\u4e16\u7eaa\u200b\u521d\u7ea7\u200b\u7b2c\u200b11\u200b\u8bfe\u200b\uff09","text":"\u5355\u8bcd\u200b \u200b\u5047\u540d\u200b \u200b\u542b\u4e49\u200b \u200b\u5355\u8bcd\u200b \u200b\u5047\u540d\u200b \u200b\u542b\u4e49\u200b \u200b\u7a7a\u6e2f\u200b \u304f\u3046\u3053\u3046 \u200b\u673a\u573a\u200b \u200b\u6c7a\u5b9a\u200b \u3051\u3063\u3066\u3044 \u200b\u51b3\u5b9a\u200b \u200b\u6765\u200b\u65e5\u200b \u3089\u3044\u306b\u3061 \u200b\u6765\u200b\u65e5\u672c\u200b \u200b\u5230\u200b\u7740\u200b \u3068\u3046\u3061\u3083\u304f \u200b\u5230\u8fbe\u200b \u200b\u4f55\u6642\u200b \u306a\u3093\u3058 \u200b\u51e0\u70b9\u200b \u200b\u4e88\u5b9a\u200b \u3088\u3066\u3044 \u200b\u9884\u5b9a\u200b \u200b\u8fce\u200b\u3048\u308b \u3080\u304b\u3048\u308b \u200b\u8fce\u63a5\u200b \u200b\u5e30\u200b\u308b \u304b\u3048\u308b \u200b\u56de\u5bb6\u200b \u200b\u5c31\u200b\u8077\u200b \u3057\u3085\u3046\u3057\u3087\u304f \u200b\u5c31\u804c\u200b \u200b\u8cbf\u6613\u200b \u307c\u3046\u3048\u304d \u200b\u8d38\u6613\u200b \u200b\u4f7f\u200b\u3046 \u3064\u304b\u3046 \u200b\u4f7f\u7528\u200b \u200b\u6b8b\u696d\u200b \u3056\u3093\u304e\u3087\u3046 \u200b\u52a0\u73ed\u200b \u200b\u65e5\u200b\u3005\u200b \u3072\u3073 \u200b\u65e5\u5b50\u200b \u200b\u61d0\u200b\u304b\u3057\u3044 \u306a\u3064\u304b\u3057\u3044 \u200b\u6000\u5ff5\u200b \u200b\u4f4f\u5c45\u200b \u3058\u3085\u3046\u304d\u3087 \u200b\u4f4f\u6240\u200b \u200b\u6c7a\u200b\u3057\u3066 \u3051\u3063\u3057\u3066 \u200b\u7edd\u4e0d\u200b \u200b\u5fc3\u914d\u200b \u3057\u3093\u3071\u3044 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\u304a\u3066\u3042\u3089\u3044 \u200b\u6d17\u624b\u95f4\u200b \u307e\u3064 \u200b\u7b49"},{"location":"japanese/japanese-1-vocab/#1412","title":"\u7b2c\u200b14\u200b\u8bfe\u200b\uff08\u200b\u65b0\u4e16\u7eaa\u200b\u521d\u7ea7\u200b\u7b2c\u200b12\u200b\u8bfe\u200b\uff09","text":"\u5355\u8bcd\u200b \u200b\u5047\u540d\u200b \u200b\u542b\u4e49\u200b \u200b\u5355\u8bcd\u200b \u200b\u5047\u540d\u200b \u200b\u542b\u4e49\u200b \u200b\u6d77\u5916\u200b \u304b\u3044\u304c\u3044 \u200b\u6d77\u5916\u200b \u200b\u65c5\u884c\u200b \u308a\u3087\u3053\u3046 \u200b\u65c5\u884c\u200b \u200b\u73cd\u200b\u3057\u3044 \u3081\u305a\u3089\u3057\u3044 \u200b\u73cd\u8d35\u200b \u3053\u3068 \u200b\u4e8b\u60c5\u200b \u200b\u4f8b\u200b\u3048\u3070 \u305f\u3068\u3048\u3070 \u200b\u4f8b\u5982\u200b \u200b\u65b0\u5a5a\u200b \u3057\u3093\u3053\u3093 \u200b\u65b0\u5a5a\u200b \u200b\u56fd\u5185\u200b \u3053\u304f\u306a\u3044 \u200b\u56fd\u5185\u200b \u200b\u9078\u200b\u3076 \u3048\u3089\u3076 \u200b\u9009\u62e9\u200b 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\u304f\u308b \u200b\u6765\u200b \u200b\u4e07\u91cc\u200b\u9577\u200b\u57ce\u200b \u3070\u3093\u308a\u3061\u3087\u3046\u3058\u3087\u3046 \u200b\u4e07\u91cc\u957f\u57ce\u200b \u200b\u6848\u5185\u200b \u3042\u3093\u306a\u3044 \u200b\u5bfc\u6e38\u200b \u305d\u3046 \u200b\u8fd9\u6837\u200b \u200b\u826f\u200b\u3044 \u3088\u3044 \u200b\u597d"},{"location":"latex/","title":"LaTeX","text":"<p>\u200b\u8bb0\u5f55\u200bLaTeX\u200b\u7684\u200b\u4f7f\u7528\u200b\u3002</p>"},{"location":"latex/#_1","title":"\u57fa\u672c\u200b\u6392\u7248","text":"<ul> <li>\u200b\u5b57\u4f53\u200b\u4e0e\u200b\u5b57\u53f7\u200b</li> <li>\u200b\u5217\u8868\u200b</li> </ul>"},{"location":"latex/#_2","title":"\u6570\u5b66\u516c\u5f0f\u200b\u6392\u7248","text":"<ul> <li>\u200b\u6570\u5b66\u200b\u7b26\u53f7\u200b</li> <li>\u200b\u7b26\u53f7\u200b\u5806\u53e0\u200b</li> <li>\u200b\u7279\u6b8a\u200b\u6280\u5de7\u200b</li> </ul>"},{"location":"latex/#_3","title":"\u56fe\u8868\u200b\u6392\u7248","text":""},{"location":"latex/#tikz","title":"tikz\u200b\u7ed8\u56fe","text":"<ul> <li>pgfplots\u200b\u56fe\u8868\u200b</li> </ul>"},{"location":"latex/#_4","title":"\u8868\u683c\u200b\u6392\u7248","text":"<ul> <li>tabular\u200b\u73af\u5883\u200b</li> </ul>"},{"location":"latex/#_5","title":"\u5176\u4ed6","text":"<ul> <li>musixtex</li> <li>\u200b\u4f7f\u7528\u200bLaTeX\u200b\u521b\u5efa\u200b\u5e7b\u706f\u7247\u200b</li> <li>\u200b\u9012\u5f52\u200b</li> </ul>"},{"location":"latex/advanced/recursion/","title":"\u9012\u5f52","text":"<p>LaTeX\u200b\u4e2d\u200b\u7684\u200b\u9012\u5f52\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>\\def</code>\u200b\u547d\u4ee4\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u5728\u200b\u5b9a\u4e49\u200b\u7684\u200b\u5b8f\u4e2d\u200b\u5305\u542b\u200b\u81ea\u8eab\u200b\u5373\u53ef\u200b\u3002</p> <pre><code>\\def\\recursion#1{\n    \\ifx#1\\relax\n        \\relax\n    \\else\n        do something\n        \\expandafter\\recursion\n    \\fi\n}\n</code></pre> <p>\u200b\u5728\u200b\u5b8f\u200b\u7b2c\u4e00\u6b21\u200b\u5c55\u5f00\u200b\u540e\u200b\uff0c\u200b\u5982\u679c\u200b<code>#1</code>\u200b\u4e0d\u662f\u200b<code>\\relax</code>\uff0c\u200b\u5219\u200b\u4f1a\u200b\u8f6c\u5165\u200b<code>\\else</code>\u200b\u8bed\u53e5\u200b\u8fdb\u884c\u200b\u7b2c\u4e8c\u6b21\u200b\u5c55\u5f00\u200b\u3002</p> <ul> <li>\u200b\u5b8f\u200b\u5b9a\u4e49\u200b\u4e2d\u200b\u5f15\u7528\u200b\u7684\u200b\u5b8f\u200b\u540d\u79f0\u200b\u524d\u200b\u5fc5\u987b\u200b\u52a0\u200b<code>\\expandafter</code>\uff0c\u200b\u8868\u793a\u200b\u5728\u200b\u5c55\u5f00\u200b\u5f53\u524d\u200b\u5b8f\u540e\u200b\u624d\u200b\u5c55\u5f00\u200b<code>\\recursion</code>\uff1b</li> <li><code>\\relax</code>\u200b\u662f\u200b\u5b8f\u200b\u7684\u200b\u7ec8\u6b62\u200b\u6761\u4ef6\u200b\uff0c\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u9700\u8981\u200b\u4ee5\u200b<code>\\relax</code>\u200b\u7ed3\u5c3e\u200b\u3002</li> </ul>"},{"location":"latex/basics/fonts/","title":"LaTeX\u200b\u4e2d\u200b\u7684\u200b\u5b57\u4f53\u200b\u4e0e\u200b\u5b57\u53f7","text":""},{"location":"latex/basics/fonts/#_1","title":"\u5b57\u53f7","text":"<p>\\(\\LaTeX\\)\u200b\u4e2d\u5171\u200b\u6709\u200b<code>\\tiny</code>\u3001<code>\\scriptsize</code>\u3001<code>\\footnotesize</code>\u3001<code>\\small</code>\u3001<code>\\normalsize</code>\u3001<code>\\large</code>\u3001<code>\\Large</code>\u3001<code>\\LARGE</code>\u3001<code>\\huge</code>\u3001<code>\\Huge</code>\u200b\u7b49\u200b10\u200b\u79cd\u200b\u76f8\u5bf9\u200b\u5c3a\u5bf8\u200b\u7684\u200b\u5b57\u53f7\u200b\u8bbe\u7f6e\u200b\u3002\u200b\u6bcf\u6b21\u200b\u4fee\u6539\u200b\u5b57\u53f7\u200b\u540e\u200b\uff0c\u200b\u4f1a\u200b\u5bf9\u200b\u547d\u4ee4\u200b\u540e\u200b\u6240\u6709\u200b\u7684\u200b\u6587\u672c\u200b\u751f\u6548\u200b\u3002\u200b\u5982\u679c\u200b\u4ec5\u200b\u9700\u8981\u200b\u4fee\u6539\u200b\u4e00\u90e8\u5206\u200b\u6587\u5b57\u200b\u7684\u200b\u5b57\u53f7\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u9700\u8981\u200b\u8c03\u6574\u200b\u5b57\u53f7\u200b\u7684\u200b\u90e8\u5206\u200b\u4f7f\u7528\u200b\u82b1\u200b\u62ec\u53f7\u200b<code>{}</code>\u200b\u5305\u56f4\u200b\u3002</p> <p>LaTeX\u200b\u4e2d\u200b\u7684\u200b\u5b57\u53f7\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\centering\n\\tiny tiny \\\\\n\\scriptsize scriptsize \\\\\n\\footnotesize footnotesize \\\\\n\\small small \\\\\n\\normalsize normalsize \\\\\n\\large large \\\\\n\\Large Large \\\\\n\\LARGE LARGE \\\\\n\\huge huge \\\\\n\\Huge Huge \\\\\n</code></pre> <p></p>"},{"location":"latex/basics/lists/","title":"\u5217\u8868","text":"<p>\u200b\u5728\u200bLaTeX\u200b\u4e2d\u200b\uff0c<code>enumerate</code>\u200b\u73af\u5883\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u7f16\u53f7\u200b\u5217\u8868\u200b\uff0c<code>itemize</code>\u200b\u73af\u5883\u200b\u4e0e\u200b<code>description</code>\u200b\u73af\u5883\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u7b26\u53f7\u200b\u5217\u8868\u200b\u3002\u200b\u6240\u6709\u200b\u7684\u200b\u5217\u8868\u200b\u4f7f\u7528\u200b<code>\\item</code>\u200b\u547d\u4ee4\u200b\u5b9a\u4e49\u200b\u5217\u8868\u200b\u9879\u200b\u3002\u200b\u5217\u8868\u200b\u53ef\u4ee5\u200b\u81ea\u5b9a\u4e49\u200b\u4f7f\u7528\u200b\u7684\u200b\u7f16\u53f7\u200b\u4e0e\u200b\u7b26\u53f7\u200b\u3002</p> <p>\u200b\u5f53\u200b\u5217\u8868\u200b\u88ab\u200b\u5d4c\u5957\u200b\u65f6\u200b\uff0c\u200b\u4e0d\u540c\u200b\u5c42\u6b21\u200b\u7684\u200b\u5217\u8868\u200b\u4f1a\u200b\u4f7f\u7528\u200b\u4e0d\u540c\u200b\u7684\u200b\u7f16\u53f7\u200b\u4ee5\u200b\u907f\u514d\u200b\u6df7\u6dc6\u200b\u3002</p>"},{"location":"latex/basics/lists/#_2","title":"\u57fa\u672c\u200b\u4f7f\u7528","text":""},{"location":"latex/basics/lists/#enumerate","title":"<code>enumerate</code>\u200b\u73af\u5883","text":"<p><code>enumerate</code>\u200b\u73af\u5883\u200b\u7684\u200b\u4f7f\u7528\u200b\u65b9\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>\\begin{enumerate}\n    \\item This is an item.\n    \\item This is an item.\n    \\item This is an item.\n\\end{enumerate}\n</code></pre> <p>\u200b\u6e32\u67d3\u200b\u540e\u200b\u7684\u200b\u6548\u679c\u200b\uff1a</p> <ol> <li>This is an item.</li> <li>This is an item.</li> <li>This is an item.</li> </ol> <p><code>enumerate</code>\u200b\u73af\u5883\u200b\u9ed8\u8ba4\u200b\u4f7f\u7528\u200b<code>1. 2. 3. ...</code>\u200b\u8fdb\u884c\u200b\u7f16\u53f7\u200b\uff0c\u200b\u81ea\u5b9a\u4e49\u200b\u7f16\u53f7\u200b\u901a\u8fc7\u200b\u5728\u200b<code>\\begin{enumerate}</code>\u200b\u540e\u200b\u6dfb\u52a0\u200b\u53c2\u6570\u200b\u5b9e\u73b0\u200b\u3002</p> <pre><code>\\begin{enumerate}[A.]\n    \\item This is an item.\n    \\item This is an item.\n    \\item This is an item.\n\\end{enumerate}\n</code></pre> <p>\u200b\u6e32\u67d3\u200b\u540e\u200b\u7684\u200b\u6548\u679c\u200b\uff1a</p> <p>A. This is an item. B. This is an item. C. This is an item.</p> <p>\u200b\u5728\u200b<code>enumerate</code>\u200b\u4e0e\u200b<code>item</code>\u200b\u73af\u5883\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>\\setlength</code>\u200b\u8c03\u8282\u200b\u5217\u8868\u200b\u9879\u200b\u4e4b\u95f4\u200b\u7684\u200b\u8ddd\u79bb\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</p> <pre><code>\\setlength{\\itemsep}{0pt}\n\\setlength{\\parskip}{0pt}\n\\setlength{\\itemindent}{1em}\n</code></pre> <p>\u200b\u81ea\u5b9a\u4e49\u200b\u7f16\u53f7\u200b\u5217\u8868\u200b\u9700\u8981\u200b\u5f15\u5165\u200b<code>enumerate</code>\u200b\u5305\u200b\uff1a<code>\\usepackage{enumerate}</code></p>"},{"location":"latex/basics/lists/#itemize","title":"<code>itemize</code>\u200b\u73af\u5883","text":"<p><code>itemize</code>\u200b\u73af\u5883\u200b\u7528\u4e8e\u200b\u751f\u6210\u200b\u7b26\u53f7\u200b\u5217\u8868\u200b\uff0c\u200b\u4f7f\u7528\u200b\u65b9\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>\\begin{itemize}\n    \\item This is an item.\n    \\item This is an item.\n    \\item This is an item.\n\\end{itemize}\n</code></pre> <p>\u200b\u6e32\u67d3\u200b\u540e\u200b\u7684\u200b\u6548\u679c\u200b\uff1a</p> <ul> <li>This is an item.</li> <li>This is an item.</li> <li>This is an item.</li> </ul> <p><code>\\item</code>\u200b\u547d\u4ee4\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e2d\u62ec\u53f7\u200b\u6307\u5b9a\u200b\u5f53\u524d\u200b\u9879\u200b\u4f7f\u7528\u200b\u7684\u200b\u7b26\u53f7\u200b\uff0c\u200b\u5982\u200b<code>\\item[-]</code>\u200b\u4f7f\u7528\u200b<code>-</code>\u200b\u4f5c\u4e3a\u200b\u7b26\u53f7\u200b</p>"},{"location":"latex/floats/table/","title":"tabular\u200b\u73af\u5883","text":"<p><code>tabular</code>\u200b\u73af\u5883\u200b\u662f\u200b\\(\\LaTeX{}\\)\u200b\u7528\u4e8e\u200b\u521b\u5efa\u8868\u683c\u200b\u7684\u200b\u73af\u5883\u200b\u3002</p>"},{"location":"latex/floats/table/#_1","title":"\u57fa\u672c\u200b\u4f7f\u7528","text":"<p>\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b\u6f14\u793a\u200b\u4e86\u200b<code>tabular</code>\u200b\u8bed\u53e5\u200b\u7684\u200b\u4f7f\u7528\u200b\u65b9\u6cd5\u200b,\u200b\u751f\u6210\u200b\u4e00\u4e2a\u200b3\u200b\u884c\u200b3\u200b\u5217\u200b\u7684\u200b\u8868\u683c\u200b\u3002</p> <p>tabular\u200b\u57fa\u672c\u200b\u4f7f\u7528\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{tabular}{|c|c|c|}\n    \\hline 2    &amp; 9     &amp; 4     \\\\\n    \\hline 7    &amp; 5     &amp; 3     \\\\\n    \\hline 6    &amp; 1     &amp; 8     \\\\\n    \\hline\n\\end{tabular}\n</code></pre> <p></p>"},{"location":"latex/floats/table/#_2","title":"\u5bfc\u8a00\u200b\u533a","text":"<p><code>\\begin{tabular}</code>\u200b\u5176\u540e\u200b\u7d27\u8ddf\u200b\u7684\u200b\u4e00\u5bf9\u200b\u62ec\u53f7\u200b\u5185\u200b\u662f\u200b\u5bfc\u8a00\u200b\u533a\u200b\uff0c\u200b\u5bfc\u8a00\u200b\u533a\u200b\u89c4\u5b9a\u200b\u4e86\u200b\u5355\u5143\u683c\u200b\u7684\u200b\u7eb5\u5411\u200b\u8fb9\u6846\u200b\u548c\u200b\u5355\u5143\u683c\u200b\u5185\u200b\u5143\u7d20\u200b\u7684\u200b\u5bf9\u9f50\u200b\u65b9\u5f0f\u200b\u3002</p> <ul> <li><code>l</code>\u3001<code>c</code>\u3001<code>r</code>\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u5de6\u200b\u5bf9\u9f50\u200b\uff0c\u200b\u5c45\u4e2d\u200b\uff0c\u200b\u53f3\u200b\u5bf9\u9f50\u200b</li> <li><code>|</code>\u3001<code>||</code>\u200b\u5bf9\u5e94\u200b\u8868\u683c\u200b\u7eb5\u5411\u200b\u6846\u7ebf\u200b\u7c7b\u578b\u200b\uff08\u200b\u5355\u7ebf\u200b\u6216\u200b\u53cc\u7ebf\u200b\uff09</li> <li><code>@{exp}</code>\u200b\u6307\u5b9a\u200b\u63d2\u5165\u200b\u5230\u200b\u5217\u4e2d\u200b\u7684\u200b\u6587\u672c\u200b</li> <li><code>*{n}{pre}</code>\u200b\u6307\u200b\u91cd\u590d\u200b<code>pre</code>\u200b\u5185\u5bb9\u200b<code>n</code>\u200b\u6b21\u200b</li> <li><code>p{len}</code>\u200b\u5c06\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5217\u200b\u653e\u5165\u200b\u4e00\u4e2a\u200bparbox\u200b\u4e2d\u200b</li> </ul> <p>\u200b\u6839\u636e\u200b\u5982\u200b\u4e0a\u200b\u8868\u8ff0\u200b\uff0c\u200b\u5bfc\u8a00\u200b\u533a\u200b<code>{|c|c|c|}</code>\u200b\u7b49\u4ef7\u200b\u4e8e\u200b<code>{*{3}{|c}|}</code></p> <p>\u200b\u5bfc\u8a00\u200b\u533a\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{tabular}{||l|c|r||}\n    \\hline 2    &amp; 91    &amp; 4     \\\\\n    \\hline 71   &amp; 5     &amp; 32    \\\\\n    \\hline 652  &amp; 251   &amp; 89    \\\\\n    \\hline\n\\end{tabular}\n</code></pre> <p></p>"},{"location":"latex/floats/table/#_3","title":"\u5408\u5e76\u200b\u5355\u5143\u683c","text":"<p>\u200b\u6a2a\u5411\u200b\u5408\u5e76\u200b\u5355\u5143\u683c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>\\multicolumn</code>\u200b\u547d\u4ee4\u200b\uff0c\u200b\u7eb5\u5411\u200b\u5408\u5e76\u200b\u5355\u5143\u683c\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>multirow</code>\u200b\u5b8f\u5305\u200b\u4e2d\u200b\u7684\u200b<code>\\multirow</code>\u200b\u547d\u4ee4\u200b\uff1a</p> <ul> <li><code>\\multicolumn</code>\u200b\u547d\u4ee4\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\u5408\u5e76\u200b\u5217\u200b\u7684\u200b\u6570\u91cf\u200b\uff0c\u200b\u7b2c\u4e8c\u4e2a\u200b\u53c2\u6570\u200b\u662f\u200b\u5408\u5e76\u200b\u540e\u200b\u5355\u5143\u683c\u200b\u7684\u200b\u5bfc\u8a00\u200b\u5217\u200b</li> <li><code>\\multirow</code>\u200b\u547d\u4ee4\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\u5408\u5e76\u200b\u884c\u200b\u7684\u200b\u6570\u91cf\u200b\uff0c\u200b\u7b2c\u4e8c\u4e2a\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200bLaTeX\u200b\u81ea\u884c\u200b\u8bbe\u7f6e\u200b\u5bbd\u5ea6\u200b\uff0c\u200b\u7b2c\u4e09\u4e2a\u200b\u53c2\u6570\u200b\u4e3a\u200b\u5355\u5143\u683c\u200b\u5185\u5bb9\u200b</li> </ul> <p>\u200b\u5408\u5e76\u200b\u5355\u5143\u683c\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{tabular}{|ccc|}\n    \\hline\n    2   &amp; 9     &amp; 4\\\\\n    7   &amp; \\multicolumn{2}{c|} {\\multirow{2}*{{?}}} \\\\\n    6   &amp;       &amp;\\\\\n    \\hline\n\\end{tabular}\n</code></pre> <p></p>"},{"location":"latex/floats/table/#_4","title":"\u4e09\u7ebf\u200b\u8868","text":"<p>LaTeX\u200b\u4e2d\u200b\u4f7f\u7528\u200b\u4e09\u7ebf\u200b\u8868\u200b\u9700\u8981\u200b\u7528\u5230\u200b<code>booktabs</code>\u200b\u5b8f\u5305\u200b\uff0c\u200b\u52a0\u5165\u200b\u5b8f\u5305\u200b\u540e\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>\\toprule</code>\u3001<code>\\midrule</code>\u200b\u4e0e\u200b<code>\\bottomrule</code>\u200b\u753b\u7ebf\u200b\uff08\u200b\u4e0d\u5e94\u200b\u518d\u200b\u4f7f\u7528\u200b<code>\\hline</code>\u200b\u547d\u4ee4\u200b\uff09\u3002</p> <p>\u200b\u4e09\u7ebf\u200b\u8868\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{tabular}{ccc}\n    \\toprule\n    2   &amp; 91    &amp; 4     \\\\\n    \\midrule\n    71  &amp; 5     &amp; 32    \\\\\n    652 &amp; 251   &amp; 89    \\\\\n    \\bottomrule\n\\end{tabular}\n</code></pre> <p></p>"},{"location":"latex/floats/table/#_5","title":"\u989c\u8272\u200b\u586b\u5145","text":"<p>\u200b\u989c\u8272\u200b\u586b\u5145\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>colortbl</code>\u200b\u5b8f\u5305\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>\\rowcolor</code>\u200b\u547d\u4ee4\u200b\u6307\u5b9a\u200b\u884c\u200b\u7684\u200b\u586b\u5145\u200b\u989c\u8272\u200b\uff0c\u200b\u6216\u200b<code>\\cellcolor</code>\u200b\u547d\u4ee4\u200b\u6307\u5b9a\u200b\u5355\u5143\u683c\u200b\u7684\u200b\u586b\u5145\u200b\u989c\u8272\u200b\u3002\u200b\u586b\u5145\u200b\u989c\u8272\u200b\u53ef\u4ee5\u200b\u9009\u62e9\u200b\u7070\u5ea6\u200b<code>[gray]</code>\u200b\u6216\u200b\u5f69\u8272\u200b<code>[rgb]</code>\u3002</p> <p>\u200b\u989c\u8272\u200b\u586b\u5145\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{tabular}{ccc}\n    \\rowcolor[gray]{0.6} 2    &amp; 9     &amp; 4     \\\\\n    \\rowcolor[gray]{0.7} 7    &amp; 5     &amp; 3     \\\\\n    \\rowcolor[gray]{0.8} 6    &amp; 1     &amp; 8     \\\\\n\\end{tabular}\n</code></pre> <p></p>"},{"location":"latex/floats/table/#_6","title":"\u659c\u7ebf\u200b\u8868\u5934","text":"<p>\u200b\u659c\u7ebf\u200b\u8868\u5934\u200b\u7531\u200b<code>diagbox</code>\u200b\u5b8f\u5305\u200b\u63d0\u4f9b\u200b\uff0c\u200b\u4f7f\u7528\u200b\u65b9\u6cd5\u200b\u5982\u4e0b\u200b\uff1a</p> <p>\u200b\u659c\u7ebf\u200b\u8868\u5934\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{tabular}{|l|ccc|}\n    \\hline\n    \\diagbox{Time}{Room}{Day} &amp; Mon &amp; Tue &amp; Wed \\\\\n    \\hline\n    Morning &amp; used &amp; used &amp; \\\\\n    Afternoon &amp; &amp; used &amp; used \\\\\n    \\hline\n\\end{tabular}\n</code></pre> <p></p>"},{"location":"latex/math/stack/","title":"LaTeX \u200b\u7b26\u53f7\u200b\u5806\u53e0","text":""},{"location":"latex/math/stack/#_1","title":"\u7b26\u53f7\u200b\u4e0a\u4e0b\u200b\u52a0\u200b\u6587\u672c","text":"<p>\u200b\u90e8\u5206\u200b\u5927\u578b\u200b\u8fd0\u7b97\u7b26\u200b\uff08\u200b\u53c2\u89c1\u200b\u6570\u5b66\u200b\u7b26\u53f7\u200b\uff09\u200b\u53ef\u4ee5\u200b\u52a0\u5165\u200b\u4e0a\u4e0b\u200b\u6807\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u884c\u95f4\u200b\u516c\u5f0f\u200b\uff0c\u200b\u8fd0\u7b97\u7b26\u200b\u7684\u200b\u4e0a\u4e0b\u200b\u6807\u6709\u200b\u4e24\u79cd\u200b\u4f4d\u7f6e\u200b\u9009\u62e9\u200b\uff1a</p> <ul> <li>\u200b\u4e0a\u4e0b\u200b\u6807\u5728\u200b\u7b26\u53f7\u200b\u4e0a\u4e0b\u200b\u4e24\u4fa7\u200b\uff0c\u200b\u5982\u200b   $$   \\sum_{i=0}^\\infty   $$</li> <li>\u200b\u4e0a\u4e0b\u200b\u6807\u5728\u200b\u7b26\u53f7\u200b\u53f3\u4fa7\u200b\uff0c\u200b\u5982\u200b   $$   \\int_0^\\infty   $$</li> </ul> <p>\u200b\u884c\u5185\u200b\u516c\u5f0f\u200b\u9ed8\u8ba4\u200b\u6240\u6709\u200b\u7684\u200b\u4e0a\u4e0b\u200b\u6807\u90fd\u200b\u5728\u200b\u7b26\u53f7\u200b\u53f3\u4fa7\u200b\uff0c\u200b\u5982\u200b\\(\\sum_{i=0}^\\infty\\)\u3002\u200b\u4f7f\u7528\u200b<code>\\limits</code>\u200b\u53ef\u4ee5\u200b\u5f3a\u5236\u200b\u4e0a\u4e0b\u200b\u6807\u51fa\u200b\u73b0\u5728\u200b\u8fd0\u7b97\u7b26\u200b\u4e0a\u4e0b\u200b\u4e24\u4fa7\u200b\uff0c\u200b\u5982\u200b</p> <p><code>\\limits</code>\u200b\u547d\u4ee4\u200b</p> LaTeX\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>$\\sum\\limits_{i=0}^\\infty$\n</code></pre> <p>\\(\\sum\\limits_{i=0}^\\infty\\)</p> <p><code>\\overset</code>\u3001<code>\\underset</code>\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u5728\u200b\u7b26\u53f7\u200b\u4e0a\u4e0b\u200b\u63d2\u5165\u200b\u7b26\u53f7\u200b\u3002<code>\\stackrel</code>\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u4e0e\u200b<code>\\overset</code>\u200b\u76f8\u540c\u200b\u7684\u200b\u529f\u80fd\u200b\u3002</p> <p><code>\\overset</code>\u200b\u547d\u4ee4\u200b\u4e0e\u200b<code>\\underset</code>\u200b\u547d\u4ee4\u200b</p> LaTeX\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{equation*}\n    \\begin{aligned}\n        &amp;\\overset{above}{\\rightarrow} \\\\\n        &amp;\\underset{below}{\\rightarrow} \\\\\n    \\end{aligned}\n\\end{\\equation*}\n</code></pre> \\[ \\begin{aligned}     &amp;\\overset{above}{\\rightarrow} \\\\     &amp;\\underset{below}{\\rightarrow} \\\\ \\end{aligned} \\] <p>\u200b\u5982\u679c\u200b\u9700\u8981\u200b\u6362\u884c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>\\substack</code>\u200b\u547d\u4ee4\u200b\uff1a</p> <p><code>\\substack</code>\u200b\u547d\u4ee4\u200b</p> LaTeX\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{equation*}\n    \\begin{aligned}\n        &amp;\\overset{\\substack{above1 \\\\ above2}}{\\rightarrow} \\\\\n        &amp;\\underset{\\substack{below1 \\\\ below2}}{\\rightarrow} \\\\\n    \\end{aligned}\n\\end{\\equation*}\n</code></pre> \\[ \\begin{aligned}     &amp;\\overset{\\substack{above1 \\\\ above2}}{\\rightarrow} \\\\     &amp;\\underset{\\substack{below1 \\\\ below2}}{\\rightarrow} \\\\ \\end{aligned} \\]"},{"location":"latex/math/stack/#_2","title":"\u7bad\u5934\u200b\u4e0a\u4e0b\u200b\u52a0\u200b\u6587\u672c","text":"<p><code>\\overset</code>\u3001<code>\\stackrel</code>\u200b\u4e0e\u200b<code>\\underset</code>\u200b\u53ea\u80fd\u200b\u5b9e\u73b0\u200b\u7bad\u5934\u200b\u4e0a\u4e0b\u200b\u52a0\u200b\u6587\u672c\u200b\uff0c\u200b\u7bad\u5934\u200b\u7684\u200b\u957f\u5ea6\u200b\u4e0d\u80fd\u200b\u968f\u200b\u6587\u672c\u200b\u957f\u5ea6\u200b\u81ea\u52a8\u200b\u8c03\u6574\u200b\u3002<code>amsmath</code>\u200b\u5b8f\u5305\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>\\xleftarrow</code>\u200b\u4e0e\u200b<code>\\xrightarrow</code>\uff0c\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u7bad\u5934\u200b\u4e0a\u4e0b\u200b\u52a0\u200b\u6587\u672c\u200b\uff0c\u200b\u540c\u65f6\u200b\u7bad\u5934\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u9002\u5e94\u200b\u6587\u672c\u200b\u7684\u200b\u957f\u5ea6\u200b\u3002<code>extarrows</code>\u200b\u5b8f\u5305\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>\\xlongequal</code>\u200b\u547d\u4ee4\u200b\uff0c\u200b\u5b9e\u73b0\u200b\u7b49\u200b\u53f7\u200b\u4e0a\u4e0b\u200b\u52a0\u200b\u6587\u672c\u200b\uff0c\u200b\u793a\u4f8b\u200b\u5982\u4e0b\u200b\uff1a</p> <p><code>\\xrightarrow</code>\u200b\u547d\u4ee4\u200b</p> LaTeX\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{itemize}\n    \\item $\\xleftarrow[\\text{below}]{\\text{above}}$\n    \\item $\\xrightarrow[\\text{very very long below}]{\\text{very very long above}}$\n    \\item $\\xlongequal[\\text{very very long below}]{\\text{very very long above}}$\n\\end{itemize}\n</code></pre> <p></p> <p>\u200b\u5b9a\u4e49\u200b\u7b26\u53f7\u200b\\(\\triangleq\\)\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u4f7f\u7528\u200b<code>\\triangleq</code>\u200b\u8f93\u5165\u200b\u3002</p>"},{"location":"latex/math/symbols/","title":"LaTeX \u200b\u6570\u5b66\u200b\u7b26\u53f7","text":"<p>\u200b\u6536\u96c6\u200b\\(\\LaTeX\\)\u200b\u5e38\u7528\u200b\u7684\u200b\u6570\u5b66\u200b\u7b26\u53f7\u200b\u547d\u4ee4\u200b</p>"},{"location":"latex/math/symbols/#_1","title":"\u5b57\u6bcd\u200b\u53d8\u4f53","text":"<p>\u200b\u5e38\u7528\u547d\u4ee4\u200b\u6709\u200b\uff1a<code>\\mathcal</code>\u3001<code>\\mathbb</code>\u3001<code>\\mathfrak</code>\u3001<code>\\mathsf</code>\u3001<code>\\mathbf</code>\u3001<code>\\boldsymbol</code>\u3001<code>\\mathit</code>\u3001<code>\\mathrm</code>\u200b\u7b49\u200b\u3002\u200b\u5176\u4e2d\u200b<code>\\mathcal</code>\u3001<code>\\mathbb</code>\u200b\u4ec5\u200b\u9002\u7528\u200b\u4e8e\u200b\u5927\u5199\u5b57\u6bcd\u200b\uff0c\u200b\u5176\u4f59\u200b\u547d\u4ee4\u200b\u5927\u5199\u200b\u3001\u200b\u5c0f\u5199\u5b57\u6bcd\u200b\u901a\u7528\u200b</p> <ul> <li><code>\\mathcal{A}</code>\uff1a\\(\\mathcal A\\)</li> <li><code>\\mathbb{A}</code>\uff1a\\(\\mathbb A\\)</li> <li><code>\\mathfrak{A}</code>\uff1a\\(\\mathfrak A\\)</li> <li><code>\\mathsf{A}</code>\uff1a\\(\\mathsf A\\)</li> <li><code>\\mathbf{A}</code>\uff1a\\(\\mathbf A\\)</li> <li><code>\\boldsymbol{A}</code>\uff1a\\(\\boldsymbol A\\)</li> <li><code>\\mathit{A}</code>\uff1a\\(\\mathit A\\)\uff08\u200b\u65e0\u200b\u6548\u679c\u200b\uff09</li> <li><code>\\mathrm{A}</code>\uff1a\\(\\mathrm A\\)</li> <li><code>\\mathscr{A}</code>\uff1a\\(\\mathscr A\\)\uff08\u200b\u9700\u8981\u200b\u5f15\u5165\u200b\u5b8f\u5305\u200b<code>mathrsfs</code>\uff09</li> </ul>"},{"location":"latex/math/symbols/#_2","title":"\u5e0c\u814a\u5b57\u6bcd","text":"\u5c0f\u5199\u200b \u200b\u547d\u4ee4\u200b \u200b\u5927\u5199\u200b \u200b\u547d\u4ee4\u200b \u200b\u53d8\u4f53\u200b \u200b\u547d\u4ee4\u200b \\(\\alpha\\) <code>\\alpha</code> \\(A\\) <code>A</code> \\(\\beta\\) <code>\\beta</code> \\(B\\) <code>B</code> \\(\\gamma\\) <code>\\gamma</code> \\(\\Gamma\\) <code>\\Gamma</code> \\(\\varGamma\\) <code>\\varGamma</code> \\(\\delta\\) <code>\\delta</code> \\(\\Delta\\) <code>\\Delta</code> \\(\\varDelta\\) <code>\\varDelta</code> \\(\\epsilon\\) <code>\\epsilonn</code> \\(E\\) <code>E</code> \\(\\varepsilon\\) <code>\\varepsilon</code> \\(\\zeta\\) <code>\\zeta</code> \\(Z\\) <code>Z</code> \\(\\eta\\) <code>\\eta</code> \\(H\\) <code>H</code> \\(\\theta\\) <code>\\theta</code> \\(\\Theta\\) <code>\\Theta</code> \\(\\vartheta,\\varTheta\\) <code>\\vartheta,\\varTheta</code> \\(\\iota\\) <code>\\iota</code> \\(I\\) <code>I</code> \\(\\kappa\\) <code>\\kappa</code> \\(K\\) <code>K</code> \\(\\varkappa\\) <code>\\varkappa</code> \\(\\lambda\\) <code>\\lambda</code> \\(\\Lambda\\) <code>\\Lambda</code> \\(\\varLambda\\) <code>\\varLambda</code> \\(\\mu\\) <code>\\mu</code> \\(M\\) <code>M</code> \\(\\nu\\) <code>\\nu</code> \\(N\\) <code>N</code> \\(\\xi\\) <code>\\xi</code> \\(\\Xi\\) <code>\\Xi</code> \\(\\varXi\\) <code>\\varXi</code> \\(o\\) <code>o</code> \\(O\\) <code>O</code> \\(\\pi\\) <code>\\pi</code> \\(\\Pi\\) <code>\\Pi</code> \\(\\varpi, \\varPi\\) <code>\\varpi,\\varPi</code> \\(\\rho\\) <code>\\rho</code> \\(P\\) <code>P</code> \\(\\varrho\\) <code>\\varrho</code> \\(\\sigma\\) <code>\\sigma</code> \\(\\Sigma\\) <code>\\Sigma</code> \\(\\varsigma, \\varSigma\\) <code>\\varsigma,\\varSigma</code> \\(\\tau\\) <code>\\tau</code> \\(T\\) <code>T</code> \\(\\upsilon\\) <code>\\upsilon</code> \\(\\Upsilon\\) <code>\\Upsilon</code> \\(\\varUpsilon\\) <code>\\varUpsilon</code> \\(\\phi\\) <code>\\phi</code> \\(\\Phi\\) <code>\\Phi</code> \\(\\varphi, \\varPhi\\) <code>\\varphi,\\varPhi</code> \\(\\chi\\) <code>\\chi</code> \\(X\\) <code>X</code> \\(\\psi\\) <code>\\psi</code> \\(\\Psi\\) <code>\\Psi</code> \\(\\varPsi\\) <code>\\varPsi</code> \\(\\omega\\) <code>\\omega</code> \\(\\Omega\\) <code>\\Omega</code> \\(\\varOmega\\) <code>\\varOmega</code>"},{"location":"latex/math/symbols/#_3","title":"\u8fd0\u7b97\u7b26","text":"<p>\u200b\u5e38\u89c1\u200b\u8fd0\u7b97\u7b26\u200b\u5217\u4e8e\u200b\u4e0b\u8868\u200b\uff1a</p>"},{"location":"latex/math/symbols/#_4","title":"\u6570\u5b66\u200b\u8fd0\u7b97","text":"\u8fd0\u7b97\u7b26\u200b \u200b\u547d\u4ee4\u200b \\(\\times\\) <code>\\times</code> \\(\\div\\) <code>\\div</code> \\(\\odot\\) <code>\\odot</code> \\(\\oplus\\) <code>\\oplus</code> \\(\\otimes\\) <code>\\otimes</code>"},{"location":"latex/math/symbols/#_5","title":"\u96c6\u5408\u200b\u8fd0\u7b97","text":"\u8fd0\u7b97\u7b26\u200b \u200b\u547d\u4ee4\u200b \\(\\in\\) <code>\\in</code> \\(\\ni\\) <code>\\ni</code> \\(\\subset\\) <code>\\subset</code> \\(\\subseteq\\) <code>\\subseteq</code> \\(\\subseteqq\\) <code>\\oplsubseteqqu</code> \\(\\subsetneq\\) <code>\\subsetneq</code> \\(\\subsetneqq\\) <code>\\subsetneqq</code> \\(\\supset\\) <code>\\subset</code> \\(\\supseteq\\) <code>\\subseteq</code> \\(\\supseteqq\\) <code>\\oplsubseteqqu</code> \\(\\supsetneq\\) <code>\\subsetneq</code> \\(\\supsetneqq\\) <code>\\subsetneqq</code> \\(\\cap\\) <code>\\cap</code> \\(\\cup\\) <code>\\cup</code>"},{"location":"latex/math/symbols/#_6","title":"\u5173\u7cfb\u200b\u8fd0\u7b97\u7b26","text":"\u8fd0\u7b97\u7b26\u200b \u200b\u547d\u4ee4\u200b \\(\\geq\\) <code>\\geq</code> \\(\\geqq\\) <code>\\geqq</code> \\(\\geqslant\\) <code>\\geqslant</code> \\(\\gg\\) <code>\\gg</code> \\(\\ggg\\) <code>\\ggg</code> \\(\\leq\\) <code>\\leq</code> \\(\\leqq\\) <code>\\leqq</code> \\(\\leqslant\\) <code>\\leqslant</code> \\(\\ll\\) <code>\\ll</code> \\(\\lll\\) <code>\\lll</code> \\(\\sim\\) <code>\\sim</code> \\(\\approx\\) <code>\\approx</code>"},{"location":"latex/math/symbols/#_7","title":"\u903b\u8f91\u200b\u8fd0\u7b97\u7b26","text":"\u8fd0\u7b97\u7b26\u200b \u200b\u547d\u4ee4\u200b \\(\\lnot\\) <code>\\lnot</code> \\(\\land\\) <code>\\land</code> \\(\\lor\\) <code>\\lor</code> \\(\\forall\\) <code>\\forall</code> \\(\\exists\\) <code>\\exists</code>"},{"location":"latex/math/symbols/#_8","title":"\u51fd\u6570\u200b\u4e0e\u200b\u590d\u6742\u200b\u8fd0\u7b97","text":"\u8fd0\u7b97\u7b26\u200b \u200b\u547d\u4ee4\u200b \\(\\frac ab\\) <code>\\frac{a}{b}</code> \\(\\sum_i^n f\\) <code>\\sum_{i}^{n} f</code> \\(\\prod_i^n f\\) <code>\\prod_{i}^{n} f</code> \\(\\sqrt[x]{y}\\) <code>\\sqrt[x]{y}</code> \\(\\int_a^bf(x)\\mathrm dx\\) <code>\\int_a^bf(x)\\mathrm dx</code> \\(\\oint_Df(x)\\mathrm dx\\) <code>\\oint_Df(x)\\mathrm dx</code> \\(\\iint_Df(x)\\mathrm dx\\) <code>\\iint_Df(x)\\mathrm dx</code> \\(\\iiint_Df(x)\\mathrm dx\\) <code>\\iiint_Df(x)\\mathrm dx</code> \\(\\partial\\) <code>\\partial</code> \\(\\log\\) <code>\\log</code> \\(\\ln\\) <code>\\ln</code> \\(\\lg\\) <code>\\lg</code> \\(\\exp\\) <code>\\exp</code> \\(a\\mod b\\) <code>a\\mod b</code> \\(\\max\\) <code>\\max</code> \\(\\min\\) <code>\\min</code> \\(\\arg\\) <code>\\arg</code> \\(\\sin\\) <code>\\sin</code> \\(\\cos\\) <code>\\cos</code> \\(\\tan\\) <code>\\tan</code> \\(\\cot\\) <code>\\cot</code> \\(\\sinh\\) <code>\\sinh</code> \\(\\cosh\\) <code>\\cosh</code> \\(\\tanh\\) <code>\\tanh</code> \\(\\coth\\) <code>\\coth</code> \\(\\arcsin\\) <code>\\arcsin</code> \\(\\arccos\\) <code>\\arccos</code> \\(\\arctan\\) <code>\\arctan</code> \\(\\ker\\) <code>\\ker</code> \\(\\dim\\) <code>\\dim</code> \\(\\det\\) <code>\\det</code> \\(\\lim\\) <code>\\lim</code> \\(\\inf\\) <code>\\inf</code>"},{"location":"latex/math/symbols/#_9","title":"\u62ec\u53f7","text":"\u8fd0\u7b97\u7b26\u200b \u200b\u547d\u4ee4\u200b \\(\\{\\) <code>\\{</code> \\(\\}\\) <code>\\}</code> \\(\\langle\\) <code>\\langle</code> \\(\\rangle\\) <code>\\rangle</code> \\(\\lceil\\) <code>\\lceil</code> \\(\\rceil\\) <code>\\rceil</code> \\(\\lfloor\\) <code>\\lfloor</code> \\(\\rfloor\\) <code>\\rfloor</code> \\(\\|\\) <code>\\|</code>"},{"location":"latex/math/tricks/","title":"\u7279\u6b8a\u200b\u6280\u5de7","text":""},{"location":"latex/math/tricks/#_2","title":"\u516c\u5f0f\u200b\u5185\u200b\u4e2d\u6587","text":"<p>\u200b\u4f7f\u7528\u200b\u547d\u4ee4\u200b<code>\\text</code>\u200b\u6216\u200b<code>\\mbox</code>\u200b\u53ef\u4ee5\u200b\u5728\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u52a0\u5165\u200b\u4e2d\u6587\u200b\u6587\u672c\u200b\u3002\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b<code>\\mathrm</code>\uff0c\u200b\u5219\u200b\u4e0d\u4f1a\u200b\u663e\u793a\u200b\u4e2d\u6587\u200b\u3002</p>"},{"location":"latex/math/tricks/#_3","title":"\u9690\u5f62\u200b\u5b57\u7b26","text":"<p><code>\\phantom</code>\u200b\u547d\u4ee4\u200b\u53ef\u4ee5\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u5360\u636e\u200b\u4e00\u5b9a\u200b\u7a7a\u95f4\u200b\u7684\u200b\u7a7a\u767d\u200b\uff0c\u200b\u7a7a\u767d\u200b\u7684\u200b\u957f\u5bbd\u200b\u7531\u200b\u53c2\u6570\u200b\u7684\u200b\u5c3a\u5bf8\u200b\u51b3\u5b9a\u200b\uff1a</p> \\[ \\begin{aligned}     \\bbox[border: 2px solid magenta]{\\phantom{\\int_0^1 x\\mathrm dx}} \\; &amp; \\bbox[border: 2px solid magenta]{\\int_0^1 x\\mathrm dx} \\\\     \\bbox[border: 2px solid magenta]{\\int_0^1 x\\mathrm dx} \\; &amp; \\end{aligned} \\] <p>\u200b\u6b64\u5916\u200b\uff0c<code>\\phantom</code>\u200b\u6709\u200b\u4e24\u4e2a\u200b\u53d8\u79cd\u200b<code>\\vphantom</code>\u200b\u4e0e\u200b<code>\\hphantom</code>\uff0c\u200b\u524d\u8005\u200b\u521b\u9020\u200b\u4e00\u4e2a\u200b\u5bbd\u5ea6\u200b\u4e3a\u200b0\uff0c\u200b\u9ad8\u5ea6\u200b\u4e3a\u200b\u6307\u5b9a\u200b\u9ad8\u5ea6\u200b\u7684\u200b\u7a7a\u767d\u200b\uff1b\u200b\u540e\u8005\u200b\u521b\u9020\u200b\u4e00\u4e2a\u200b\u9ad8\u5ea6\u200b\u4e3a\u200b0\uff0c\u200b\u5bbd\u5ea6\u200b\u4e3a\u200b\u6307\u5b9a\u200b\u5bbd\u5ea6\u200b\u7684\u200b\u7a7a\u767d\u200b\u3002\u200b\u5728\u200b\u5c3a\u5bf8\u200b\u6839\u636e\u200b\u5185\u5bb9\u200b\u800c\u53d8\u200b\u7684\u200b\u5bb9\u5668\u200b\u4e2d\u200b\uff0c<code>\\phantom</code>\u200b\u7b49\u200b\u4e00\u7cfb\u5217\u200b\u547d\u4ee4\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u63a7\u5236\u200b\u5bb9\u5668\u200b\u7684\u200b\u5927\u5c0f\u200b\u3002</p> <p><code>\\vphantom</code>\u200b\u4f7f\u7528\u200b\u573a\u666f\u200b</p> <p>\u200b\u5728\u200b<code>\\left</code>\u3001<code>\\right</code>\u200b\u7b49\u200b\u547d\u4ee4\u200b\u53ea\u80fd\u200b\u5728\u200b\u4e00\u884c\u200b\u4e0a\u200b\u63a7\u5236\u200b\u5de6\u53f3\u200b\u4e24\u4fa7\u200b\u62ec\u53f7\u200b\u7684\u200b\u5927\u5c0f\u200b\uff0c\u200b\u800c\u200b\u5f53\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u8de8\u884c\u200b\uff0c<code>\\left</code>\u3001<code>\\right</code>\u200b\u5219\u200b\u65e0\u6cd5\u200b\u6b63\u786e\u200b\u5339\u914d\u200b\u3002\u200b\u6b64\u65f6\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>\\vphantom</code>\u200b\u547d\u4ee4\u200b\u5c06\u200b\u62ec\u53f7\u200b\u6491\u5f00\u200b\u3002</p> \u200b\u4e0d\u200b\u4f7f\u7528\u200b<code>\\vphantom</code>\u200b\u4f7f\u7528\u200b<code>\\vphantom</code> <pre><code>\\begin{equation}\n    \\begin{aligned}\n        \\frac{\\partial ^2\\Pi(x, t)}{\\partial x\\partial t} =\n        &amp; -(p-s)\\mathbb E_{Q, B, L}\\left[D(t)|Q(t)\\leq x\\right]f_{Q(t)}(x) \\\\\n        &amp; + h\\mathbb P(t &lt; B + L) - h\\mathbb E_{B, L} \\\\\n        &amp; \\times \\left[\\int_{t}^\\mathbb{B+L}\\mathbb E_Q\\left[D(t)|Q(t) - Q(\\tau)\\leq x, B\\leq t, L\\right]\\right. \\\\\n        &amp; \\color{magenta} \\times \\left. f_{Q(t) - Q(\\tau) | B\\leq t}(x)\\mathbb P(B\\leq t)\\mathrm d \\tau\\right]\n    \\end{aligned}\n\\end{equation}\n</code></pre> \\[ \\begin{aligned}     \\frac{\\partial ^2\\Pi(x, t)}{\\partial x\\partial t} =     &amp; -(p-s)\\mathbb E_{Q, B, L}\\left[D(t)|Q(t)\\leq x\\right]f_{Q(t)}(x) \\\\     &amp; + h\\mathbb P(t &lt; B + L) - h\\mathbb E_{B, L} \\\\     &amp; \\times \\left[\\int_{t}^\\mathbb{B+L}\\mathbb E_Q\\left[D(t)|Q(t) - Q(\\tau)\\leq x, B\\leq t, L\\right]\\right. \\\\     &amp; \\color{magenta} \\times \\left. f_{Q(t) - Q(\\tau) | B\\leq t}(x)\\mathbb P(B\\leq t)\\mathrm d \\tau\\right] \\end{aligned} \\] <pre><code>\\begin{equation}\n    \\begin{aligned}\n        \\frac{\\partial ^2\\Pi(x, t)}{\\partial x\\partial t} =\n        &amp; -(p-s)\\mathbb E_{Q, B, L}\\left[D(t)|Q(t)\\leq x\\right]f_{Q(t)}(x) \\\\\n        &amp; + h\\mathbb P(t &lt; B + L) - h\\mathbb E_{B, L} \\\\\n        &amp; \\times \\left[\\int_{t}^\\mathbb{B+L}\\mathbb E_Q\\left[D(t)|Q(t) - Q(\\tau)\\leq x, B\\leq t, L\\right]\\right. \\\\\n        &amp; \\color{magenta} \\times \\left.\\vphantom{\\int_t^{B+L}} f_{Q(t) - Q(\\tau) | B\\leq t}(x)\\mathbb P(B\\leq t)\\mathrm d \\tau\\right]\n    \\end{aligned}\n\\end{equation}\n</code></pre> \\[ \\begin{aligned}     \\frac{\\partial ^2\\Pi(x, t)}{\\partial x\\partial t} =     &amp; -(p-s)\\mathbb E_{Q, B, L}\\left[D(t)|Q(t)\\leq x\\right]f_{Q(t)}(x) \\\\     &amp; + h\\mathbb P(t &lt; B + L) - h\\mathbb E_{B, L} \\\\     &amp; \\times \\left[\\int_{t}^\\mathbb{B+L}\\mathbb E_Q\\left[D(t)|Q(t) - Q(\\tau)\\leq x, B\\leq t, L\\right]\\right. \\\\     &amp; \\color{magenta} \\times \\left.\\vphantom{\\int_t^{B+L}} f_{Q(t) - Q(\\tau) | B\\leq t}(x)\\mathbb P(B\\leq t)\\mathrm d \\tau\\right] \\end{aligned} \\]"},{"location":"latex/misc/beamer/","title":"\u4f7f\u7528\u200bLaTeX\u200b\u521b\u5efa\u200b\u5e7b\u706f\u7247","text":"<p>\u200b\u9996\u5148\u200b\uff0c\u200b\u9700\u8981\u200b\u6307\u5b9a\u200b<code>documentclass</code>\u200b\u4e3a\u200b<code>beamer</code>\u200b\u7c7b\u578b\u200b\uff1a</p> <pre><code>\\documentclass{beamer}\n</code></pre> <p>\u200b\u6bcf\u200b\u4e00\u5f20\u200b\u5e7b\u706f\u7247\u200b\u4f7f\u7528\u200b<code>frame</code>\u200b\u73af\u5883\u200b\uff0c\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u7a7a\u767d\u200b\u5e7b\u706f\u7247\u200b\uff1a</p> <pre><code>\\begin{frame}\n\\end{frame}\n</code></pre> <p>\u200b\u7b2c\u4e00\u9875\u200bPPT\u200b\u901a\u5e38\u200b\u663e\u793a\u200bPPT\u200b\u7684\u200b\u6807\u9898\u200b\u3001\u200b\u4f5c\u8005\u200b\u7684\u200b\u4e2a\u4eba\u4fe1\u606f\u200b\u7b49\u200b\u5185\u5bb9\u200b\uff1a</p> <pre><code>\\title{\u200b\u6807\u9898\u200b}\n\\subtitle{\u200b\u526f\u6807\u9898\u200b}\n\\author{\u200b\u4f5c\u8005\u200b}\n\\institute{\u200b\u7ec4\u7ec7\u200b}\n\\date{\\today}  % \u200b\u663e\u793a\u200b\u65e5\u671f\u200b\n\\titlepage\n</code></pre>"},{"location":"latex/misc/musixtex/","title":"\u4e50\u8c31\u200b\u6392\u7248","text":"<p><code>musixtex</code>\u200b\u662f\u200bLaTeX\u200b\u4e2d\u200b\u7528\u6765\u200b\u6392\u7248\u200b\u4e50\u8c31\u200b\u7684\u200b\u5b8f\u5305\u200b\u3002\u200b\u4ee5\u4e0b\u5185\u5bb9\u200b\u9488\u5bf9\u200b\u9f13\u8c31\u200b\u6392\u7248\u200b\u3002</p> <pre><code>\\usepackage{musixtex}\n\\input{musixper} % \u200b\u52a0\u5165\u200b\u4e86\u200b\u9f13\u8c31\u200b\u6392\u7248\u200b\u7684\u200b\u4e00\u4e9b\u200b\u5185\u5bb9\u200b\n</code></pre>"},{"location":"latex/misc/musixtex/#_2","title":"\u97f3\u9ad8\u200b\u5bf9\u7167\u8868","text":"<p>\u200b\u4ece\u200b\u5de6\u5411\u53f3\u200b\u4f9d\u6b21\u200b\u4e3a\u200bA-Q</p> <p></p>"},{"location":"latex/misc/musixtex/#_3","title":"\u57fa\u672c\u200b\u4f7f\u7528","text":"<p><code>musixtex</code>\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u4e86\u200b<code>music</code>\u200b\u73af\u5883\u200b\uff0c\u200b\u5728\u200b<code>music</code>\u200b\u73af\u5883\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u5bf9\u200b\u4e50\u8c31\u200b\u8fdb\u884c\u200b\u8bbe\u7f6e\u200b\u3002\u200b\u7136\u540e\u200b\u5728\u200b\u547d\u4ee4\u200b<code>\\startextract</code>\u200b\u548c\u200b<code>\\zendextract</code>\u200b\u4e4b\u95f4\u200b\u8f93\u5165\u200b\u97f3\u7b26\u200b\u3002</p> <pre><code>\\begin{music}\n    \\instrumentnumber{1}        % \u200b\u4e50\u5668\u200b\n    \\setclefsymbol1\\empty       % \u200b\u8c31\u53f7\u200b\n    \\nobarnumbers               % \u200b\u53d6\u6d88\u200b\u5c0f\u8282\u200b\u7f16\u53f7\u200b\n    \\generalmeter{\\meterfrac44} % \u200b\u8282\u62cd\u200b\n    \\startextract\n        % Enter notes here\n    \\zendextract\n\\end{music}\n</code></pre> <p>\u200b\u5728\u200b\u672c\u200b\u5c0f\u8282\u200b\u4e2d\u200b\uff0c<code>p</code>\u200b\u8868\u793a\u200b\u97f3\u7b26\u200b\u5bf9\u5e94\u200b\u7684\u200b\u97f3\u9ad8\u200b\u3002</p>"},{"location":"latex/misc/musixtex/#_4","title":"\u786e\u5b9a\u200b\u97f3\u7b26\u200b\u5206\u5e03\u200b\u95f4\u8ddd","text":"<p>\u200b\u4f7f\u7528\u200b<code>\\notes</code>\u200b\u7684\u200b\u4e0d\u540c\u200b\u5927\u5c0f\u5199\u200b\u5f62\u5f0f\u200b\u6765\u200b\u786e\u5b9a\u200b\u97f3\u7b26\u200b\u7684\u200b\u5206\u5e03\u200b\u95f4\u8ddd\u200b\u3002</p> <pre><code>\\notes ... \\en % Suitable for sixteenth notes\n\\Notes ... \\en % Suitable for eighth notes\n\\NOtes ... \\en % Suitable for quarter notes\n</code></pre> <code>\\notes</code><code>\\Notes</code><code>\\NOtes</code> <p></p> <p></p> <p></p>"},{"location":"latex/misc/musixtex/#_5","title":"\u97f3\u7b26","text":"<p>\u200b\u97f3\u7b26\u200b\u547d\u4ee4\u200b\u7531\u200b\u4e24\u200b\u90e8\u5206\u200b\u7ec4\u6210\u200b\uff1a\u200b\u65f6\u503c\u200b+\u200b\u65b9\u5411\u200b\u3002\u200b\u4e0d\u540c\u200b\u97f3\u7b26\u200b\u7684\u200b\u65f6\u503c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u547d\u4ee4\u200b\u5982\u4e0b\u200b\u8868\u200b\u6240\u793a\u200b</p> \u200b\u65f6\u503c\u200b \u200b\u5168\u97f3\u7b26\u200b \u200b\u4e8c\u5206\u200b \u200b\u56db\u5206\u200b \u200b\u516b\u5206\u200b \u200b\u5341\u516d\u5206\u200b \u200b\u547d\u4ee4\u200b <code>w</code> <code>h</code> <code>q</code> <code>c</code> <code>cc</code> <p><code>u</code>\u200b\u8868\u793a\u200b\u5411\u4e0a\u200b\u7684\u200b\u97f3\u7b26\u200b\uff0c<code>l</code>\u200b\u8868\u793a\u200b\u5411\u4e0b\u200b\u7684\u200b\u97f3\u7b26\u200b\uff0c<code>a</code>\u200b\u8868\u793a\u200b\u81ea\u52a8\u200b\u8bbe\u7f6e\u200b\u97f3\u7b26\u200b\u65b9\u5411\u200b\u3002\u200b\u5982\u200b<code>\\qu{p}</code>\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u5411\u4e0a\u200b\u7684\u200b\u56db\u5206\u97f3\u7b26\u200b\u3002\u200b\u5982\u679c\u200b\u4e0d\u200b\u52a0\u200b\u65b9\u5411\u200b\uff0c\u200b\u5219\u200b\u6ca1\u6709\u200b\u97f3\u7b26\u200b\u7684\u200b\u201c\u200b\u6746\u200b\u201d\u3002</p> <p>\u200b\u5728\u200b\u65b9\u5411\u200b\u540e\u9762\u200b\u52a0\u4e0a\u200b<code>p</code>\u200b\u8868\u793a\u200b\u9644\u70b9\u200b\uff0c<code>pp</code>\u200b\u8868\u793a\u200b\u53cc\u200b\u9644\u70b9\u200b\u3002</p> <p>\u200b\u6bcf\u200b\u8f93\u5165\u200b\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\uff0c\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u547d\u4ee4\u200b\u8f93\u5165\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u4fbf\u4f1a\u200b\u5411\u200b\u540e\u200b\u79fb\u52a8\u200b\u3002\u200b\u5982\u679c\u200b\u8981\u200b\u8f93\u5165\u200b\u548c\u5f26\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\u4e4b\u524d\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>z</code>+\u200b\u65f6\u503c\u200b\u8868\u793a\u200b\u8fd9\u4e2a\u200b\u97f3\u7b26\u200b\u4e0d\u5411\u200b\u540e\u200b\u79fb\u52a8\u200b\u8f93\u5165\u200b\u4f4d\u7f6e\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u4e24\u4e2a\u200b\u97f3\u7b26\u200b\u79bb\u5f97\u200b\u8fc7\u8fd1\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>l</code>\u200b\u5c06\u200b\u97f3\u7b26\u200b\u5411\u200b\u5de6\u200b\u79fb\u52a8\u200b\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\u7684\u200b\u4f4d\u7f6e\u200b\uff0c<code>r</code>\u200b\u5c06\u200b\u97f3\u7b26\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002</p> <pre><code>\\NOtes \\zq d\\qu g\\rq g\\qu f\\zq e\\qu h\\zq f\\qu k \\en\n</code></pre> <p></p>"},{"location":"latex/misc/musixtex/#_6","title":"\u7b26\u6760","text":"<p>\u200b\u5982\u679c\u200b\u516b\u5206\u200b\u6216\u200b\u66f4\u200b\u77ed\u200b\u7684\u200b\u97f3\u7b26\u200b\u8fde\u7eed\u200b\u51fa\u73b0\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u97f3\u7b26\u200b\u4e4b\u95f4\u200b\u4f7f\u7528\u200b\u7b26\u6760\u200b\u76f8\u8fde\u200b\uff0c\u200b\u901a\u5e38\u200b\u6bcf\u200b\u56db\u4e2a\u200b\u97f3\u7b26\u200b\u8fde\u6210\u200b\u4e00\u7ec4\u200b\u3002\u200b\u7b26\u6760\u200b\u7684\u200b\u5b9a\u4e49\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>% \u200b\u4e0a\u65b9\u200b\n\\ibu{n}{p}{s} &lt;notes-under-beam&gt; \\tbu{n} &lt;the-last-note&gt;\n% \u200b\u4e0b\u65b9\u200b\n\\ibl{n}{p}{s} &lt;notes-under-beam&gt; \\tbl{n} &lt;the-last-note&gt;\n</code></pre> <p>\u200b\u5176\u4e2d\u200b<code>n</code>\u200b\u4e3a\u200b\u7f16\u53f7\u200b\uff0c<code>p</code>\u200b\u4e3a\u200b\u5f00\u59cb\u200b\u7684\u200b\u97f3\u7b26\u200b\u9ad8\u5ea6\u200b\uff0c<code>s</code>\u200b\u4e3a\u200b\u659c\u5ea6\u200b\u3002\u200b\u547d\u4ee4\u200b\u91cc\u200b<code>b</code>\u200b\u7684\u200b\u4e2a\u6570\u200b\u7b49\u4e8e\u200b\u663e\u793a\u200b\u7684\u200b\u7b26\u200b\u6760\u6570\u200b\u3002\u200b\u4e0e\u200b\u7b26\u6760\u200b\u76f8\u8fde\u200b\u7684\u200b\u97f3\u7b26\u200b\u6807\u8bb0\u200b\u4e3a\u200b\u65f6\u503c\u200b+<code>b</code>+\u200b\u7f16\u53f7\u200b\uff0c\u200b\u5982\u679c\u200b\u540c\u4e00\u200b\u4f4d\u7f6e\u200b\u6709\u200b\u591a\u4e2a\u200b\u97f3\u7b26\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b\u65f6\u503c\u200b\u524d\u9762\u200b\u52a0\u200b<code>z</code>\u3002\u200b\u6ce8\u610f\u200b\u97f3\u7b26\u200b\u4e4b\u95f4\u200b\u4e0d\u8981\u200b\u52a0\u200b\u7a7a\u683c\u200b\u3002</p> <pre><code>\\Notes\\ibu0j0\\qb0j\\qb0j\\qb0j\\tbu0\\qb0j\\en\n\\notes\\ibbu0j0\\qb0j\\qb0j\\qb0j\\tbu0\\qb0j\\en\n\\NOtes\\qu j\\en\n</code></pre> <p></p> <p><code>\\tbu{n}\\qb{n}{p}</code>\u200b\u53ef\u4ee5\u200b\u7f29\u5199\u200b\u4e3a\u200b<code>\\tqu{n}{p}</code>\uff0c<code>\\tbl{n}{p}\\qb{n}{p}</code>\u200b\u53ef\u4ee5\u200b\u7f29\u5199\u200b\u4e3a\u200b<code>\\tql{n}{p}</code>\uff0c<code>\\tbu{n}\\zqb{n}{p}</code>\u200b\u53ef\u4ee5\u200b\u7f29\u5199\u200b\u4e3a\u200b<code>\\zqu{n}{p}</code>\uff0c<code>\\tbl{n}{p}\\zqb{n}{p}</code>\u200b\u53ef\u4ee5\u200b\u7f29\u5199\u200b\u4e3a\u200b<code>\\zql{n}{p}</code></p> <p><code>\\ibu</code>\u200b\u4e2d\u200b\u9700\u8981\u200b\u624b\u52a8\u200b\u786e\u5b9a\u200b\u659c\u7387\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b\u5bf9\u5e94\u200b\u7684\u200b<code>\\Ibu</code>\u200b\u547d\u4ee4\u200b\u81ea\u52a8\u200b\u786e\u5b9a\u200b\uff1a</p> <pre><code>\\Ibu{n}{p1}{p2}{np}\n</code></pre> <p>\u200b\u5176\u4e2d\u200b<code>{n}</code>\u200b\u4e3a\u200b\u7f16\u53f7\u200b\uff0c<code>p1</code>\u200b\u4e3a\u200b\u7b2c\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\u9ad8\u5ea6\u200b\uff0c<code>p2</code>\u200b\u4e3a\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\u9ad8\u5ea6\u200b\uff0c<code>np</code>\u200b\u4e3a\u200b\u97f3\u7b26\u200b\u4e2a\u6570\u200b\u3002</p> <pre><code>\\Notes\\Ibu0dk4\\qb0d\\qb0h\\qb0f\\tbu0\\qb0k\\en\n\\Notes\\Ibu0kd4\\qb0k\\qb0f\\qb0h\\tbu0\\qb0d\\en\n</code></pre> <p></p> <p>\u200b\u5f53\u200b\u4e0d\u200b\u5305\u542b\u200b\u548c\u5f26\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>D</code>\u3001<code>T</code>\u3001<code>Q</code>\u200b\u5feb\u901f\u200b\u8f93\u5165\u200b\u5305\u542b\u200b2\u30013\u30014\u200b\u4e2a\u200b\u97f3\u7b26\u200b\u7684\u200b\u7b26\u6760\u200b\u3002</p> <pre><code>\\Notes\\Qqbu dhfk\\en\n\\notes\\Qqbbu kfhd\\Qqbbl dhfk\\en\n</code></pre> <p></p> <p>\u200b\u5f53\u200b\u5728\u200b\u7b26\u6760\u200b\u4e2d\u200b\u9700\u8981\u200b\u6539\u53d8\u200b\u7b49\u7ea7\u200b\u65f6\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>\\nbu</code>\u200b\u5c06\u200b\u7b49\u7ea7\u200b\u589e\u52a0\u200b\u5230\u200b<code>b</code>\u200b\u7684\u200b\u4e2a\u6570\u200b\u5bf9\u5e94\u200b\u7684\u200b\u7ea7\u6570\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>\\ibu</code>\u200b\u5c06\u200b\u7b49\u7ea7\u200b\u51cf\u5c11\u200b\u5230\u200b<code>b</code>\u200b\u7684\u200b\u4e2a\u6570\u200b\u51cf\u4e00\u200b\u5bf9\u5e94\u200b\u7684\u200b\u7ea7\u6570\u200b\u3002</p> <pre><code>\\Notes\\ibu0k0\\qb0j\\en\\notes\\nbbu0\\qb0j\\tbu0\\qb0j\\en\n\\notes\\ibbu0j0\\qb0j\\tbbu0\\qb0j\\en\\Notes\\tbu0\\qb0j\\en\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>\\roff{}</code>\u200b\u5728\u200b\u5f53\u524d\u200b\u97f3\u7b26\u200b\u53f3\u4fa7\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u5bf9\u5e94\u200b\u7b49\u7ea7\u200b\u7684\u200b\u77ed\u6760\u200b\uff0c\u200b\u5f53\u200b<code>\\tbbu</code>\u200b\u914d\u5408\u200b\u66f4\u200b\u9ad8\u7b49\u7ea7\u200b\u8f93\u5165\u200b\u65f6\u200b\uff0c\u200b\u5728\u200b\u5f53\u524d\u200b\u97f3\u7b26\u200b\u5de6\u4fa7\u200b\u4f4d\u7f6e\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u5bf9\u5e94\u200b\u7b49\u7ea7\u200b\u77ed\u6760\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u6765\u200b\u8f93\u5165\u200b\u5207\u5206\u97f3\u200b\u3002</p> <pre><code>\\notes\\ibbu0j0\\qb0j\\tbbu0\\qb0j\\en\n\\Notes\\tbu0\\qb0j\\en\n\n\\notes\\ibbu0j0\\roff{\\tbbu0}\\qb0j\\en\n\\Notes\\qb0j\\tbbu0\\en\n\\notes\\tbu0\\qb0j\\en\n</code></pre> <p></p>"},{"location":"latex/misc/musixtex/#_7","title":"\u4f11\u6b62\u7b26","text":"<p>\u200b\u4f11\u6b62\u7b26\u200b\u7684\u200b\u547d\u4ee4\u200b\u4e3a\u200b</p> \u200b\u65f6\u503c\u200b \u200b\u5168\u97f3\u7b26\u200b \u200b\u4e8c\u5206\u200b \u200b\u56db\u5206\u200b \u200b\u516b\u5206\u200b \u200b\u5341\u516d\u5206\u200b \u200b\u547d\u4ee4\u200b <code>\\pause</code> <code>\\hp</code> <code>\\qp</code> <code>\\ds</code> <code>\\qs</code> <p>\u200b\u5728\u200b\u547d\u4ee4\u200b\u524d\u200b\u4f7f\u7528\u200b<code>\\raise</code>\u200b\u8c03\u6574\u200b\u4f11\u6b62\u7b26\u200b\u9ad8\u5ea6\u200b\uff0c<code>\\Interligne</code>\u200b\u8868\u793a\u200b\u7ebf\u200b\u95f4\u8ddd\u200b\uff1b\u200b\u4f7f\u7528\u200b<code>\\rlap</code>\u200b\u4f7f\u4e0b\u200b\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\u505c\u7559\u200b\u5728\u200b\u539f\u4f4d\u200b\u3002</p> <pre><code>\\NOtes\\qp\\en\n\\Notes\\ds\\ds\\en\n\\NOtes\\raise2\\Interligne\\rlap\\qp\\raise-2\\Interligne\\qp\\qp \\en \\doublebar\n</code></pre> <p></p>"},{"location":"latex/misc/musixtex/#_8","title":"\u6587\u672c","text":"<p>\u200b\u4f7f\u7528\u200b<code>charnote{p}{text}</code>\u200b\u524d\u9762\u200b\u52a0\u200b<code>\\z</code>\u3001<code>\\c</code>\u3001<code>\\l</code>\u200b\u5728\u200b\u8c31\u200b\u4e0a\u200b\u6dfb\u52a0\u200b\u6587\u5b57\u200b\uff0c<code>\\z</code>\u3001<code>\\c</code>\u3001<code>\\l</code>\u200b\u5206\u522b\u200b\u8868\u793a\u200b\u5de6\u200b\u3001\u200b\u4e2d\u200b\u3001\u200b\u53f3\u200b\u5bf9\u9f50\u200b\u5f53\u524d\u200b\u97f3\u7b26\u200b\u3002<code>char{num}{text}</code>\u200b\u548c\u200b<code>charnote</code>\u200b\u7528\u6cd5\u200b\u76f8\u540c\u200b\uff0c\u200b\u4f46\u200b\u4f7f\u7528\u200b\u6570\u5b57\u200b\u6765\u200b\u8868\u793a\u200b\u4f4d\u7f6e\u200b\uff1a\u200b\u4ece\u200b\u6700\u200b\u4e0b\u65b9\u200b\u7684\u200b\u7ebf\u200b\u5f00\u59cb\u200b\uff0c\u200b\u6587\u672c\u200b\u57fa\u7ebf\u200b\u5bf9\u9f50\u200b\u6570\u5b57\u200b\u8868\u793a\u200b\u7684\u200b\u7ebf\u200b\u3002</p> <p></p>"},{"location":"latex/misc/musixtex/#_9","title":"\u8282\u62cd","text":"<p>\u200b\u4f7f\u7528\u200b<code>\\metron{\\qu}{60}</code>\u200b\u6765\u200b\u8bbe\u7f6e\u200b\u8282\u62cd\u200b\uff0c\u200b\u8282\u62cd\u200b\u4e5f\u200b\u662f\u200b\u6587\u672c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b\u76f8\u5173\u200b\u547d\u4ee4\u200b\u653e\u5728\u200b\u8c31\u200b\u4e0a\u200b\u3002\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>\\smallnotesize</code>\u3001<code>\\tinynotesize</code>\u200b\u8c03\u8282\u200b\u97f3\u7b26\u200b\u5927\u5c0f\u200b\u3002</p> <p></p>"},{"location":"latex/misc/musixtex/#_10","title":"\u9f13\u8c31","text":"<p>\u200b\u9f13\u8c31\u200b\u4e2d\u200b\u4ee3\u8868\u200b\u51fb\u6253\u200b\u9572\u200b\u7247\u200b\u7684\u200b\u97f3\u7b26\u200b\u7531\u200b<code>musixper</code>\u200b\u6269\u5c55\u200b\u5305\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>\\input</code>\u200b\u547d\u4ee4\u200b\u5f15\u5165\u200b\u3002\u200b\u5982\u4e0b\u200b\u7b26\u53f7\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b<code>\\dc, \\dh, \\do, \\y, \\x, \\ox, \\ro, k</code>\uff0c\u200b\u52a0\u200b\u5728\u200b\u5bf9\u5e94\u200b\u7684\u200b\u97f3\u7b26\u200b\u65f6\u503c\u200b\u524d\u200b\u3002</p> <pre><code>\\Notes\\ibu0o0\\dcqb0o\\dhqb0o\\doqb0o\\tbu0\\yqb0o\\en\n\\Notes\\ibu0o0\\xqb0o\\oxqb0o\\roqb0o\\tbu0\\kqb0o\\en\n</code></pre> <p></p> <p>\u200b\u6253\u51fb\u4e50\u200b\u4f7f\u7528\u200b\u7684\u200b\u8c31\u53f7\u200b\u7528\u200b<code>\\setclefsymbol1{\\drumclef}</code>\u200b\u547d\u4ee4\u200b\u5f15\u5165\u200b\u3002\u200b\u97f3\u9ad8\u200b\u8bb0\u53f7\u200b\u548c\u200b\u9ad8\u97f3\u200b\u8c31\u53f7\u200b\u4e00\u81f4\u200b\uff0c\u200b\u7235\u58eb\u9f13\u200b\u4e2d\u200b\u7684\u200b\u4e0d\u540c\u200b\u58f0\u90e8\u200b\u5bf9\u5e94\u200b\u4e94\u7ebf\u8c31\u200b\u7684\u200b\u4e0d\u540c\u200b\u97f3\u9ad8\u200b\u5982\u4e0b\u200b\u56fe\u200b\u6240\u793a\u200b</p> <p></p>"},{"location":"latex/tikz/pgfplots/","title":"pgfplots","text":"<p>\u200b\u501f\u52a9\u200b\u5b8f\u5305\u200b<code>pgfplots</code>\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200bTikZ\u200b\u7ed8\u5236\u200b\u6298\u7ebf\u56fe\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\u3002</p> <p>pgfplots\u200b\u7ed8\u5236\u200b\u6298\u7ebf\u56fe\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{figure}[htbp]\n    \\begin{tikzpicture}\n        \\begin{axis}[sharp plot]\n            \\addplot coordinates{(0, 5)(1, 9)(2, 7)(3, 4)(4, 10)};\n        \\end{axis}\n    \\end{tikzpicture}\n\\end{figure}\n</code></pre> <p></p> <p><code>axis</code>\u200b\u63a7\u5236\u200b\u4e00\u4e2a\u200b\u56fe\u50cf\u200b\uff0c<code>\\begin{axis}</code>\u200b\u540e\u200b\u53ef\u4ee5\u200b\u52a0\u200b\u4e00\u5bf9\u200b\u65b9\u62ec\u53f7\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8bbe\u5b9a\u200b\u56fe\u7247\u200b\u7684\u200b\u53c2\u6570\u200b\u3002<code>addplot</code>\u200b\u63a7\u5236\u200b\u56fe\u50cf\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5176\u540e\u200b\u540c\u6837\u200b\u53ef\u4ee5\u200b\u52a0\u200b\u4e00\u5bf9\u200b\u65b9\u62ec\u53f7\u200b\u7528\u4e8e\u200b\u63a7\u5236\u53c2\u6570\u200b\u3002</p>"},{"location":"latex/tikz/pgfplots/#axis","title":"<code>axis</code>\u200b\u73af\u5883","text":"<p><code>axis</code>\u200b\u73af\u5883\u200b\u53ef\u7528\u200b\u7684\u200b\u53c2\u6570\u200b\u5982\u4e0b\u200b\u6240\u200b\u5217\u200b\uff1a</p> <ul> <li><code>[sharp plot, smooth]</code>\u200b\u7b49\u200b\uff0c\u200b\u8868\u660e\u200b\u6298\u7ebf\u56fe\u200b\u7684\u200b\u66f2\u7ebf\u200b\u662f\u5426\u200b\u7ecf\u8fc7\u200b\u5e73\u6ed1\u200b</li> <li><code>xlabel</code>\u200b\u4e0e\u200b<code>ylabel</code>\uff0c\u200b\u63a7\u5236\u200b\u5750\u6807\u8f74\u200b\u7684\u200b\u6807\u6ce8\u200b\u3002</li> <li><code>title</code>\u200b\u63a7\u5236\u200b\u56fe\u50cf\u200b\u7684\u200b\u6807\u9898\u200b</li> <li><code>xmode</code>\u200b\u4e0e\u200b<code>ymode</code>\u200b\u63a7\u5236\u200b\u5750\u6807\u8f74\u200b\u662f\u200b\u7ebf\u6027\u200b\u5750\u6807\u8f74\u200b\uff08<code>normal</code>\uff09\u200b\u8fd8\u662f\u200b\u5bf9\u6570\u200b\u5750\u6807\u8f74\u200b\uff08<code>log</code>\uff09</li> <li><code>xtick distance</code>\u200b\u4e0e\u200b<code>ytick distance</code>\u200b\u63a7\u5236\u200b\u5750\u6807\u8f74\u200b\u523b\u5ea6\u200b\u5927\u5c0f\u200b</li> <li><code>xmin, xmax</code>\u200b\u4e0e\u200b<code>ymin, ymax</code>\u200b\u63a7\u5236\u200b\u5750\u6807\u8f74\u200b\u7684\u200b\u8303\u56f4\u200b<ul> <li>\u200b\u5f53\u200b\u6307\u5b9a\u200b\u5750\u6807\u8f74\u200b\u8303\u56f4\u200b\u65f6\u200b\uff0c\u200b\u5982\u679c\u200b<code>enlargelimits</code>\u200b\u6307\u5b9a\u200b\u4e3a\u200b<code>true</code>\uff0c\u200b\u5219\u200b\u5750\u6807\u8f74\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u653e\u5927\u200b\u4ee5\u200b\u7b26\u5408\u200b\u8f93\u5165\u200b\u6570\u636e\u200b</li> </ul> </li> </ul> <p>\u200b\u5982\u4e0b\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u7a7a\u200b\u7684\u200b\u56fe\u50cf\u200b\uff0cx\u200b\u5750\u6807\u8f74\u200b\u4e3a\u200b\u7ebf\u6027\u200b\uff0cy\u200b\u5750\u6807\u8f74\u200b\u4e3a\u200b\u5bf9\u6570\u200b\uff1a</p> <p>axis\u200b\u53c2\u6570\u200b\u793a\u4f8b\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{figure}[htbp]\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xlabel=Year,\n            ylabel=Inventory,\n            xmin=2019, xmax=2021,\n            ymin=100, ymax=1000,\n            xtick distance=1,\n            title=Inventory Graph,\n            xmode=normal, ymode=log\n        ]\n        \\end{axis}\n    \\end{tikzpicture}\n\\end{figure}\n</code></pre> <p></p> <p>\u200b\u6709\u5173\u200b\u66f4\u200b\u591a\u200b\u53c2\u6570\u200b\uff0c\u200b\u8bf7\u200b\u53c2\u89c1\u200bpgfplots \u200b\u624b\u518c\u200b</p>"},{"location":"latex/tikz/pgfplots/#addplot","title":"<code>\\addplot</code>","text":"<p><code>\\addplot</code>\u200b\u547d\u4ee4\u200b\u53ef\u4ee5\u200b\u5411\u200b\u56fe\u50cf\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u6298\u7ebf\u56fe\u200b\u6216\u200b\u51fd\u6570\u200b\u3002</p> <ol> <li><code>\\addplot coordinates{}</code> \u200b\u7528\u4e8e\u200b\u7ed8\u5236\u200b\u6298\u7ebf\u56fe\u200b\uff0c\u200b\u5176\u4e2d\u200b<code>coordinates</code>\u200b\u4e2d\u200b\u7684\u200b\u5750\u6807\u200b\u7528\u200b\u5706\u62ec\u53f7\u200b\u62ec\u8d77\u200b</li> <li><code>\\addplot [smooth] {f(x)}</code> \u200b\u7528\u4e8e\u200b\u7ed8\u5236\u200b\u5e73\u6ed1\u200b\u7684\u200b\u6298\u7ebf\u56fe\u200b\uff0c\u200b\u5176\u4e2d\u200b<code>f(x)</code>\u200b\u4e3a\u200b\u51fd\u6570\u200b\u8868\u8fbe\u5f0f\u200b</li> </ol> <p>axis\u200b\u53c2\u6570\u200b\u793a\u4f8b\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{figure}[htbp]\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xlabel=$x$ variable,\n            ylabel=$y$ variable,\n            xmin=0, xmax=7,\n            title=Regression Example,\n        ]\n            \\addplot coordinates{(0, 5)(1, 9)(2, 7)(3, 4)(4, 10)};\n            \\addlegendentry{\\small Data}\n            \\addplot [smooth, red]{0.5*x + 6};\n            \\addlegendentry{\\small Regressor}\n        \\end{axis}\n    \\end{tikzpicture}\n\\end{figure}\n</code></pre> <p></p> <p><code>\\addplot</code>\u200b\u6709\u200b\u5982\u4e0b\u200b\u53c2\u6570\u200b\uff1a</p> <ul> <li><code>smooth</code>\u200b\u5982\u679c\u200b\u5b58\u5728\u200b\uff0c\u200b\u7ed8\u5236\u200b\u4e0d\u200b\u5305\u542b\u200b\u70b9\u200b\u5750\u6807\u200b\u7684\u200b\u5e73\u6ed1\u200b\u56fe\u50cf\u200b</li> <li><code>[color]</code>\u200b\u8bbe\u7f6e\u200b\u989c\u8272\u200b\u3002</li> </ul>"},{"location":"math/","title":"\u8bfe\u7a0b\u200b\u7b14\u8bb0","text":"<p>\u200b\u4ee5\u4e0b\u200b\u6574\u7406\u200b\u4e86\u200b\u8bfe\u4e0a\u200b\u7684\u200b\u5185\u5bb9\u200b</p> <ol> <li>\u200b\u79bb\u6563\u6570\u5b66\u200b</li> <li>\u200b\u8fd0\u7b79\u5b66\u200b\u4e0e\u200b\u7ebf\u6027\u89c4\u5212\u200b</li> <li>\u200b\u5bf9\u7b56\u200b\u3001\u200b\u51b3\u7b56\u200b\u4e0e\u200b\u6392\u961f\u200b\u8bba\u200b</li> <li>\u200b\u8fd0\u4f5c\u200b\u7ba1\u7406\u200b</li> <li>\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b</li> <li>\u200b\u51f8\u200b\u4f18\u5316\u200b</li> <li>\u200b\u56e0\u679c\u200b\u63a8\u65ad\u200b</li> </ol>"},{"location":"math/#_2","title":"\u6269\u5c55\u200b\u9605\u8bfb","text":""},{"location":"math/#_3","title":"\u6570\u5b66\u5206\u6790","text":"<ul> <li>\u200b\u521d\u4e2d\u200b\u7ade\u8d5b\u200b:\\(x+y+z=1\\)\uff0c\u200b\u6c42\u200b\\(xy+2yz+3zx\\)\u200b\u7684\u200b\u6700\u5927\u503c\u200b? - \u200b\u4e88\u4e00\u4eba\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u628a\u200b\\([-0.5,0.5]\\)\u200b\u5185\u200b\u6240\u6709\u200b\u5b9e\u6570\u200b\u76f8\u52a0\u200b\uff0c\u200b\u5f97\u51fa\u200b\u7684\u200b\u7ed3\u679c\u200b\u662f\u200b\\(0\\)\u200b\u5417\u200b\uff1f - Be2SiO4\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u5982\u4f55\u200b\u6c42\u200b\u5fae\u5206\u65b9\u7a0b\u200b \\(y'' + y = 0\\) \u200b\u7684\u200b\u901a\u89e3\u200b\uff1f - \u200b\u4e88\u4e00\u4eba\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u4e3a\u4ec0\u4e48\u200b \\(\\ln x\\) \u200b\u6c42\u5bfc\u200b\u662f\u200b \\(1/x\\)\uff1f - \u200b\u8881\u8d85\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\\(\\ln x (x&gt;1, x\\in \\mathbb Z)\\) \u200b\u53ef\u80fd\u200b\u662f\u200b\u6709\u7406\u6570\u200b\u5417\u200b\uff1f - \u200b\u4e88\u4e00\u4eba\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u79ef\u5206\u200b\u7b26\u53f7\u200b\u5185\u53d6\u200b\u5fae\u5206\u200b\u662f\u200b\u4e00\u79cd\u200b\u4ec0\u4e48\u200b\u65b9\u6cd5\u200b\uff1f - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u4e00\u6761\u200b\u4ece\u200b\\((0,0)\\)\u200b\u51fa\u53d1\u200b\u7684\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(1\\)\u200b\u7684\u200b\u66f2\u7ebf\u200b\uff0c\u200b\u56f4\u7ed5\u200b\\(x\\)\u200b\u8f74\u200b\u65cb\u8f6c\u200b\u4ea7\u751f\u200b\u7684\u200b\u65cb\u8f6c\u4f53\u200b\u7684\u200b\u6700\u5927\u200b\u4f53\u79ef\u200b\u662f\u200b\u591a\u5c11\u200b\uff1f - \u200b\u865a\u200b\u8c03\u5b50\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> </ul>"},{"location":"math/#_4","title":"\u6982\u7387\u8bba\u200b\u4e0e\u200b\u968f\u673a\u200b\u8fc7\u7a0b","text":"<ul> <li>\u200b\u6570\u5b66\u200b\u671f\u671b\u200b\u5927\u4e8e\u200b\u96f6\u200b\u7684\u200b\u6e38\u620f\u200b\u4e00\u5b9a\u200b\u80fd\u200b\u8d5a\u94b1\u200b\u5417\u200b - \u200b\u9648\u58a8\u77b3\u200bnono\u200b\u7684\u200b\u6587\u7ae0\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u7532\u4e59\u200b\u4e24\u4eba\u200b\u4e0b\u56f4\u68cb\u200b\uff0c\u200b\u7532\u80dc\u200b\u7684\u200b\u6982\u7387\u200b\u4e3a\u200b\\(a\\)\uff0c\u200b\u4e59\u80dc\u200b\u7684\u200b\u6982\u7387\u200b\u4e3a\u200b\\(b\\)\uff0c\\(a+b=1\\)\uff0c\u200b\u6bd4\u200b\u5bf9\u65b9\u200b\u591a\u200b\u8d62\u200b\u4e24\u5c40\u200b\u8005\u200b\u83b7\u80dc\u200b\uff0c\u200b\u6c42\u7532\u200b\u8d62\u200b\u7684\u200b\u6982\u7387\u200b? - Dylaaan\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u5f3a\u5927\u200b\u6570\u5f8b\u200b\uff08SLLN) - \u200b\u4f55\u5bb6\u5fd7\u200b\u7684\u200b\u6587\u7ae0\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u5982\u679c\u200b\u6211\u200b\u6709\u200b \\(2^{-n}\\) \u200b\u6982\u7387\u200b\u83b7\u5f97\u200b \\(2^n\\) \u200b\u5143\u200b\uff0c\u200b\u90a3\u200b\u83b7\u5f97\u200b\u94b1\u200b\u7684\u200b\u6570\u5b66\u200b\u671f\u671b\u503c\u200b\u5e94\u8be5\u200b\u662f\u200b\u65e0\u7a77\u200b\uff0c\u200b\u4f46\u200b\u5b9e\u9645\u200b\u4e0d\u662f\u200b\u8fd9\u6837\u200b\uff0c\u200b\u4e3a\u4ec0\u4e48\u200b\uff1f - ClefRybak\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> </ul>"},{"location":"math/#_5","title":"\u7ebf\u6027\u4ee3\u6570","text":"<ul> <li>\u200b\u5982\u4f55\u200b\u5224\u5b9a\u200b\u4e0d\u53ef\u200b\u7ea6\u200b\u591a\u9879\u5f0f\u200b\uff1f - \u200b\u5982\u8564\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u8ba1\u7b97\u673a\u200b\u56fe\u5f62\u5b66\u200b\u4e00\u200b\uff1a\u200b\u57fa\u7840\u200b\u53d8\u6362\u200b\u77e9\u9635\u200b\u603b\u7ed3\u200b(\u200b\u7f29\u653e\u200b\uff0c\u200b\u65cb\u8f6c\u200b\uff0c\u200b\u4f4d\u79fb\u200b)</li> </ul>"},{"location":"math/#_6","title":"\u79bb\u6563\u6570\u5b66\u200b\u4e0e\u200b\u6570\u8bba","text":"<ul> <li>\u200b\u4e00\u822c\u200b\u4e00\u5143\u200b\u4e94\u6b21\u200b\u65b9\u7a0b\u200b\u6ca1\u6709\u200b\u6839\u5f0f\u200b\u89e3\u200b\u7684\u200b\u79d8\u5bc6\u200b\u7a76\u7adf\u200b\u5728\u200b\u54ea\u91cc\u200b\uff1f\u200b\u82b1\u8d39\u200b\u4eba\u7c7b\u200b\u4e09\u767e\u5e74\u200b.? - Geometrie\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u7ec4\u5408\u200b\u6570\u5b66\u200b\u5b66\u4e60\u200b\u7b14\u8bb0\u200b\uff08\u200b\u5341\u200b\uff09 - sola\u200b\u7684\u200b\u6587\u7ae0\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>ACM\u2014\u2014\u200b\u5e38\u89c1\u200b\u7684\u200b\u51e0\u79cd\u200b\u5206\u89e3\u200b\u8d28\u200b\u56e0\u5b50\u200b\u7684\u200b\u65b9\u6cd5\u200b - \u200b\u4e25\u683c\u200b\u9e3d\u200b\u7684\u200b\u6587\u7ae0\u200b - \u200b\u77e5\u4e4e\u200b</li> </ul>"},{"location":"math/#_7","title":"\u5176\u4ed6","text":"<ul> <li>\u200b\u4e94\u4e2a\u200b\u56da\u72af\u200b\u5148\u540e\u200b\u4ece\u200b100\u200b\u9897\u200b\u7eff\u8c46\u200b\u4e2d\u200b\u6293\u200b\u7eff\u8c46\u200b\u3002\u200b\u6293\u5f97\u200b\u6700\u200b\u591a\u200b\u548c\u200b\u6700\u5c11\u200b\u7684\u200b\u4eba\u200b\u5c06\u200b\u88ab\u200b\u5904\u6b7b\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u4ea4\u6d41\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6478\u200b\u51fa\u200b\u5269\u4e0b\u200b\u7eff\u8c46\u200b\u7684\u200b\u6570\u91cf\u200b\uff0c\u200b\u8c01\u200b\u7684\u200b\u5b58\u6d3b\u200b\u51e0\u7387\u200b\u6700\u5927\u200b\uff1f - \u200b\u66fe\u52a0\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>1000\u200b\u6876\u200b\u6c34\u200b\uff0c\u200b\u5176\u4e2d\u200b\u4e00\u6876\u200b\u6709\u6bd2\u200b\uff0c\u200b\u732a\u200b\u559d\u200b\u6bd2\u6c34\u200b\u540e\u4f1a\u200b\u5728\u200b15\u200b\u5206\u949f\u200b\u5185\u200b\u6b7b\u53bb\u200b\uff0c\u200b\u60f3\u200b\u7528\u200b\u4e00\u4e2a\u200b\u5c0f\u65f6\u200b\u627e\u5230\u200b\u8fd9\u6876\u200b\u6bd2\u6c34\u200b\uff0c\u200b\u81f3\u5c11\u200b\u9700\u8981\u200b\u51e0\u5934\u200b\u732a\u200b\uff1f - \u200b\u82d7\u534e\u680b\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u9664\u4e86\u200b 3\uff0c4\uff0c5 \u200b\u4ee5\u5916\u200b\u662f\u5426\u200b\u8fd8\u6709\u200b\u522b\u7684\u200b\u4e09\u89d2\u5f62\u200b\uff0c\u200b\u5b83\u200b\u7684\u200b\u4e09\u6761\u200b\u8fb9\u200b\u662f\u200b\u8fde\u7eed\u200b\u81ea\u7136\u6570\u200b\uff0c\u200b\u5b83\u200b\u7684\u200b\u9762\u79ef\u200b\u4e5f\u200b\u662f\u200b\u81ea\u7136\u6570\u200b\uff1f - Pandora Eartha\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u72b6\u6001\u200b\u7a7a\u95f4\u200b\u8868\u8fbe\u5f0f\u200b\u7684\u200b\u89e3\u200b - Anonymity67\u200b\u7684\u200b\u6587\u7ae0\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u5982\u4f55\u200b\u6c42\u200b\u6570\u5217\u200b\\(a_{n+1} = a_n^2 - 2a_n\\)? - Reuental\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> </ul>"},{"location":"math/causal-inference/","title":"\u56e0\u679c\u200b\u63a8\u65ad","text":"<p>According to Causal Inference: What If.</p> <ol> <li>A definition of causal effect</li> <li>Randomized experiments</li> <li>Observational studies</li> <li>Effect modification</li> <li>Interaction</li> <li>Graphical representation of causal effects</li> <li>Confounding</li> <li>Selection bias</li> <li>Measurement bias and \"noncausal\" diagrams</li> </ol>"},{"location":"math/causal-inference/causal-inference-stale/","title":"\u56e0\u679c\u200b\u63a8\u65ad","text":"<p>Casual inference is about studying the consequence of doing one thing vs another.</p> <p>Simpson\u2019s Paradox</p> <p>\u200b\u67d0\u4e2a\u200b\u7edf\u8ba1\u200b\u5173\u8054\u200b\uff0c\u200b\u53ef\u80fd\u200b\u5728\u200b\u6240\u6709\u200b\u5b50\u200b\u7fa4\u4f53\u200b\u4e0a\u200b\u7684\u200b\u6548\u679c\u200b\u90fd\u200b\u662f\u200b\u76f8\u53cd\u200b\u7684\u200b\u3002</p> <p>A statistical association that holds for the entire population can be reversed in every subpopulation.</p> <p></p> <p>Berkson\u2019s Paradox</p> <p>Variables that have no relationships can appear to have strong associations in a selected subpopulation.</p> <p>\u200b\u56e0\u679c\u200b\u63a8\u65ad\u200b\u6709\u200b\u4e24\u4e2a\u200b\u4e3b\u8981\u200b\u7684\u200b\u7814\u7a76\u200b\u6846\u67b6\u200b\uff0c\u200b\u5373\u200b Potential outcome framework \u200b\u548c\u200b Causal graph \u3002</p>"},{"location":"math/causal-inference/causal-inference-stale/#potential-outcome-framework","title":"Potential Outcome Framework","text":""},{"location":"math/causal-inference/causal-inference-stale/#_2","title":"\u6846\u67b6\u200b\u8bbe\u5b9a","text":"<ul> <li>Unit of analysis\uff1a\u200b\u4e0d\u540c\u200b\u65f6\u95f4\u200b\u4e0b\u200b\u7684\u200b\u540c\u4e00\u4e2a\u200b\u4e2a\u4f53\u200b\u5e94\u5f53\u200b\u770b\u4f5c\u200b\u4e0d\u540c\u200b\u7684\u200b\u4e2a\u4f53\u200b</li> <li>Treatment \\(D\\)\uff1a\u200b\u901a\u5e38\u200b\u5212\u5206\u200b\u4e3a\u200b\u5b9e\u9a8c\u7ec4\u200b\uff08\\(D=1\\)\uff09\u200b\u4e0e\u200b\u5bf9\u7167\u7ec4\u200b\uff08\\(D=0\\)\uff09\uff0c\u200b\u4f46\u200b\\(D\\)\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u53d6\u591a\u503c\u200b\u6216\u200b\u8fde\u7eed\u200b</li> <li>Potential outcome \\(Y(d)\\)\uff1a\u200b\u5f53\u200b\u5e72\u9884\u200b\u6c34\u5e73\u200b\u4e3a\u200b\\(d\\)\u200b\u65f6\u200b\u5e94\u5f53\u200b\u51fa\u73b0\u200b\u7684\u200b\u7ed3\u679c\u200b\u3002</li> <li>Counterfactual outcome\uff1a\u200b\u5f53\u200b\u5e72\u9884\u200b\u6c34\u5e73\u200b\u4e3a\u200b\\(D\\)\u200b\u65f6\u200b\u6240\u6709\u200b\\(d\\not = D\\)\u200b\u7684\u200b\u7ed3\u679c\u200b\u2014\u2014\u200b\u5b58\u5728\u200b\u4f46\u200b\u65e0\u6cd5\u200b\u89c2\u6d4b\u200b\u5230\u200b</li> <li>Casual effect\uff1a\u200b\u6bd4\u8f83\u200b\u4e24\u4e2a\u200b\u5e73\u884c\u200b\u7ed3\u679c\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5dee\u5f02\u200b\uff0c\\(Y(1) - Y(0)\\)</li> </ul> <p>\u200b\u5f53\u200b\u4e00\u4e2a\u200b\u4e2a\u4f53\u200b\u63a5\u6536\u200b\u5230\u200b\\(d=D\\)\u200b\u4e4b\u540e\u200b\uff0c\u200b\u5176\u200b\u5e72\u9884\u200b\u72b6\u6001\u200b\u5c31\u200b\u4e0d\u4f1a\u200b\u518d\u200b\u6539\u53d8\u200b\u3002\u200b\u56e0\u6b64\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u89c2\u6d4b\u200b\u5230\u200b\u4e00\u4e2a\u200b\u4e2a\u4f53\u6240\u6709\u200b\u53ef\u80fd\u200b\u51fa\u73b0\u200b\u7684\u200b\u5e72\u9884\u200b\u7ed3\u679c\u200b\u3002</p> <p>SUTVA assumption</p> <p>\u200b\u5047\u8bbe\u200b\u4e2a\u4f53\u200b\\(i\\)\u200b\u7684\u200b\u5e72\u9884\u200b\u72b6\u6001\u200b\\(D_i\\)\u3001\u200b\u5e72\u9884\u200b\u548c\u200b\u4e0d\u200b\u5e72\u9884\u200b\u7684\u200b\u6548\u679c\u200b\\(Y_i(0), Y_i(1)\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u5373\u200b</p> \\[ \\{D_i, Y_i(0), Y_i(1)\\}\\sim\\{D, Y(0), Y(1)\\} \\] <p>\u200b\u540c\u65f6\u200b\u5047\u8bbe\u200b</p> <ol> <li>\u200b\u6bcf\u4e2a\u200b\u4e2a\u4f53\u200b\u7684\u200b\u7ed3\u679c\u200b\u4e0d\u200b\u53d7\u5230\u200b\u5176\u4ed6\u200b\u4e2a\u4f53\u200b\u5e72\u9884\u200b\u6c34\u5e73\u200b\u7684\u200b\u5f71\u54cd\u200b\uff08\u200b\u4e0d\u200b\u5b58\u5728\u200b\u4e2a\u4f53\u200b\u4e4b\u95f4\u200b\u7684\u200b\u76f8\u4e92\u5f71\u54cd\u200b\uff0c\u200b\u4e2a\u4f53\u200b\u7684\u200b\u7ed3\u679c\u200b\u53ea\u53d7\u200b\u5e72\u9884\u200b\u6c34\u5e73\u200b\u7684\u200b\u5f71\u54cd\u200b\uff09</li> <li>\u200b\u4e0d\u200b\u5b58\u5728\u200b\u5e72\u9884\u200b\u6c34\u5e73\u200b\u7684\u200b\u9690\u85cf\u200b\u72b6\u6001\u200b\uff08\u200b\u5373\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5e72\u9884\u200b\u72b6\u6001\u200b\u90fd\u200b\u662f\u200b\u539f\u5b50\u200b\u7684\u200b\uff0c\u200b\u4e0d\u200b\u53ef\u4ee5\u200b\u8fdb\u4e00\u6b65\u200b\u5212\u5206\u200b\uff09</li> </ol> <p>\u200b\u8003\u8651\u200b\u603b\u4f53\u200b\u7684\u200b\u671f\u671b\u200b\u5e72\u9884\u200b\u6548\u679c\u200b\uff08 Average Treatment Effect, ATE\uff09</p> \\[ \\tau = \\mathbb E[Y(1) - Y(0)] \\] <p>\u200b\u7531\u4e8e\u200b\u5bf9\u4e8e\u200b\u4e2a\u4f53\u200b\\(i\\)\uff0c\u200b\u65e0\u6cd5\u200b\u540c\u65f6\u200b\u89c2\u6d4b\u200b\\(Y_i(1), Y_i(0)\\)\uff0c\u200b\u56e0\u6b64\u200b\\(\\tau\\)\u200b\u7684\u200b\u786e\u5207\u200b\u6570\u503c\u200b\u540c\u6837\u200b\u65e0\u6cd5\u200b\u89c2\u6d4b\u200b\u3002</p> \\[ \\begin{aligned}     \\hat\\tau &amp;= \\frac{1}{n_1} \\sum_{i;D_i = 1}Y_i - \\frac{1}{n_0} \\sum_{i;D_i = 0}Y_i \\\\     &amp;= \\mathbb E[Y(1) | D = 1] - \\mathbb E[Y(0) | D = 0] \\end{aligned} \\] <p>\u200b\u9700\u8981\u200b\u8003\u8651\u200b\\(D\\)\u200b\u7684\u200b\u5206\u5e03\u200b\uff0c\u200b\u5373\u200b\u5b9e\u9a8c\u7ec4\u200b\u4e0e\u200b\u5bf9\u7167\u7ec4\u200b\u5982\u4f55\u200b\u4ea7\u751f\u200b\uff0c\u200b\u6216\u200b\\(Y, D\\)\u200b\u7684\u200b\u72ec\u7acb\u6027\u200b\u3002</p> <p>\u200b\u5f53\u200b\\(Y, D\\)\u200b\u6ee1\u8db3\u200b</p> \\[ (Y(0), Y(1))\\perp D \\label{1} \\] <p>\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(\\hat\\tau = \\tau\\)</p>"},{"location":"math/causal-inference/causal-inference-stale/#_3","title":"\u6df7\u6dc6\u200b\u56e0\u5b50","text":"<p>\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u573a\u666f\u200b\uff1a</p> <ol> <li>\u200b\u4e00\u7fa4\u200b\u88ab\u8bd5\u200b\u968f\u673a\u200b\u63a5\u53d7\u200b\u836f\u7269\u200b\u8bd5\u9a8c\u200b</li> <li>\u200b\u88ab\u200b\u8bd5\u4e2d\u200b\u75c7\u72b6\u200b\u6bd4\u8f83\u4e25\u91cd\u200b\u7684\u200b\u4e2a\u4f53\u200b\u63a5\u53d7\u200b\u836f\u7269\u200b\u5b9e\u9a8c\u200b</li> </ol> <ol> <li>\u200b\u968f\u673a\u200b\u8bd5\u9a8c\u200b\u6307\u200b\u5206\u914d\u673a\u5236\u200b\u7531\u200b\u7814\u7a76\u8005\u200b\u53ef\u63a7\u200b\u3002\u200b\u5982\u200b\u6309\u7167\u200b\\(p = 0.5\\)\u200b\u7684\u200b\u6982\u7387\u200b\u8fdb\u884c\u200b\u4f2f\u52aa\u5229\u200b\u5b9e\u9a8c\u200b\u968f\u673a\u200b\u5206\u914d\u200b\uff0c\u200b\u5219\u200b\\((Y(1)\uff0c Y(0))\\perp D\\)\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b\\(\\hat\\tau = \\tau\\)</li> <li>\u200b\u89c2\u6d4b\u200b\u7814\u7a76\u200b\u6307\u200b\u5206\u914d\u673a\u5236\u200b\u4e0d\u7531\u200b\u7814\u7a76\u8005\u200b\u63a7\u5236\u200b\u3002\u200b\u5982\u200b\u88ab\u200b\u8bd5\u200b\u6839\u636e\u200b\u4e2a\u4f53\u200b\u611f\u53d7\u200b\u81ea\u884c\u51b3\u5b9a\u200b\u5403\u836f\u200b\u3002\u200b\u6b64\u65f6\u200b\u5b58\u5728\u200b\uff08\u200b\u53ef\u4ee5\u200b\u89c2\u6d4b\u200b\u5230\u200b\u7684\u200b\uff09\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\\(X\\)</li> </ol> <p>\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b</p> <p>\u200b\u53d8\u91cf\u200b\\(X\\)\u200b\u662f\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\uff0c\u200b\u5982\u679c\u200b\uff1a</p> <ol> <li>\\(X\\)\u200b\u4f1a\u200b\u5f71\u54cd\u200b\u5206\u914d\u200b\uff0c\u200b\u5373\u200b\\(X\\not \\perp D\\)\uff0c\u200b\u6216\u200b</li> <li>\\(X\\)\u200b\u4f1a\u200b\u5f71\u54cd\u200b\u7ed3\u679c\u200b\uff0c\u200b\u5373\u200b\\(X\\not \\perp (Y(1)\uff0c Y(0))\\)</li> </ol> <p>\u200b\u901a\u5e38\u200b\u5c06\u200b\u53ef\u200b\u89c2\u6d4b\u200b\u5230\u200b\u7684\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u8bb0\u200b\u4e3a\u200b\\(X\\)\uff0c\u200b\u4e0d\u80fd\u200b\u89c2\u6d4b\u200b\u5230\u200b\u7684\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u8bb0\u200b\u4e3a\u200b\\(U\\)\u3002</p> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5f53\u200b\u5b58\u5728\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u65f6\u200b\uff0c\\(\\eqref{1}\\)\u200b\u4e0d\u518d\u200b\u6210\u7acb\u200b\uff0c\u200b\u5176\u4e2d\u200b\u7684\u200b\\(D\\)\u200b\u53d8\u4e3a\u200b\\(D|X\\)</p> <p>\u200b\u5f53\u200b\u6240\u6709\u200b\u7684\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u90fd\u200b\u88ab\u200b\u89c2\u6d4b\u200b\u5230\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u9002\u7528\u200b unconfoundedness assumption</p> <p>Unconfoundedness assumption</p> <p>\u200b\u5047\u8bbe\u200b\u6210\u7acb\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b</p> \\[ (Y(1), Y(0)) \\perp D|X \\] <p>\u200b\u63a7\u5236\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b</p> \\[\\begin{aligned}     &amp; \\mathbb E [\\mathbb E[Y | D = 1, X] - \\mathbb E[Y | D = 0, X]] \\\\     =&amp; \\mathbb E [\\mathbb E[Y(1) | D = 1, X] - \\mathbb E[Y(0) | D = 0, X]] \\\\     =&amp; \\mathbb E [\\mathbb E[Y(1) | X] - \\mathbb E[Y(0) | X]] \\\\     =&amp; \\mathbb E [\\mathbb E[Y(1) - Y(0) | X]] \\\\     =&amp; \\mathbb E[Y(1) - Y(0)] \\end{aligned} \\] <p>\u200b\u4f46\u200b\u5b9e\u9645\u4e0a\u200b\uff0c\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u89c2\u6d4b\u200b\u5230\u200b\u6240\u6709\u200b\u7684\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u3002\u200b\u6b64\u65f6\u200b</p> \\[ \\begin{aligned}     &amp; \\bbE [Y | D = 1] - \\bbE [Y| D = 0] \\\\     =&amp; \\bbE[\\bbE[Y | D = 1, X] | D = 1] - \\bbE[\\bbE[Y | D = 0, X] | D = 0] \\end{aligned} \\] <p>\u200b\u4f46\u200b\u5b9e\u9645\u4e0a\u200b\uff0c\\(\\bbP(X|D = 1) \\not = \\bbP(X|D = 0)\\)\uff0c\u200b\u56e0\u6b64\u200b\u9020\u6210\u200b\u504f\u5dee\u200b\u3002</p> <p>\u200b\u5e72\u9884\u200b\u4e0e\u200b\u6761\u4ef6\u200b</p> <ul> <li>\\(\\bbE(Y|D = 0)\\)\u200b\u53cd\u6620\u200b\u6761\u4ef6\u200b\uff0c\u200b\u4f5c\u7528\u200b\u5728\u200b\u7fa4\u4f53\u200b\\(D = 0\\)\u200b\u4e0a\u200b\u3002</li> <li>\\(\\bbE(Y(0))\\)\u200b\u53cd\u6620\u200b\u5e72\u9884\u200b\uff0c\u200b\u4f5c\u7528\u200b\u5728\u200b\u5168\u4f53\u200b\u4e0a\u200b\u3002</li> </ul> <p>\u200b\u5f53\u200b\\((Y(1), Y(0))\\perp D | X\\)\u200b\u6210\u7acb\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\bbE(Y(1) | X) = \\bbE(Y | D = 1, X) \\] <p>\u200b\u524d\u63d0\u200b\u662f\u200b\uff0c overlap assumption \u200b\u6210\u7acb\u200b\u3002</p> <p>Overlap assumption</p> <p>\u200b\u5bf9\u4e8e\u200b\u6240\u6709\u200b\u7684\u200b\\(X\\)\uff0c\\(Y(0), Y(1)\\)\u200b\u5fc5\u987b\u200b\u90fd\u200b\u5b58\u5728\u200b\uff0c\u200b\u5373\u200b</p> \\[ \\bbP(D=1|X)\\in (0, 1) \\] <p>\u200b\u5b9a\u4e49\u200b\\(\\bbP(D=1|X)\\in (0, 1)\\)\u200b\u4e3a\u200bpropensity score\u3002</p>"},{"location":"math/causal-inference/causal-inference-stale/#_4","title":"\u5b9e\u8bc1\u200b\u7814\u7a76","text":"<p>\u200b\u5b9e\u9645\u4e0a\u200b\uff0c unconfoundedness \u200b\u548c\u200b overlap \u200b\u5047\u8bbe\u200b\u5747\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u6210\u7acb\u200b\uff0c\u200b\u56e0\u4e3a\u200b\u6709\u200b\u8bb8\u591a\u200b\u9690\u85cf\u200b\u7684\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u524d\u200b\u6587\u4e2d\u200b\u7684\u200b Simpson\u2019s Paradox \uff0c\u200b\u51fa\u73b0\u200b\u53cd\u200b\u76f4\u89c9\u200b\u73b0\u8c61\u200b\u7684\u200b\u539f\u56e0\u200b\u5728\u4e8e\u200b\uff0c\u200b\u6027\u522b\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\uff1a</p> <ol> <li>\u200b\u5973\u6027\u200b\u7684\u200b\u88ab\u200b\u8bd5\u200b\u6bd4\u4f8b\u200b\u66f4\u9ad8\u200b\uff08\\(D\\not\\perp X\\)\uff09</li> <li>\u200b\u5973\u6027\u200b\u6574\u4f53\u200b\u7684\u200b\u5eb7\u590d\u200b\u7387\u200b\u66f4\u200b\u4f4e\u200b\uff08\\(Y\\not\\perp X\\)\uff09</li> </ol> <p>\u200b\u5728\u200b\u5b9e\u8bc1\u200b\u7814\u7a76\u200b\u4e2d\u200b\uff0c\u200b\u5f80\u5f80\u200b\u8fd8\u6709\u200b\u7814\u7a76\u8005\u200b\u672a\u80fd\u200b\u89c2\u6d4b\u200b\u5230\u200b\u7684\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u3002</p> <p>\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u65e2\u200b\u662f\u200b\u969c\u788d\u200b\uff0c\u200b\u4e5f\u200b\u662f\u200b\u5de5\u5177\u200b\u3002\u200b\u5728\u200b\u7406\u60f3\u5316\u200b\u7684\u200b\u7edf\u8ba1\u200b\u63a8\u65ad\u200b\u4e2d\u200b\uff0c\u200b\u76f8\u5173\u6027\u200b\u4e0d\u80fd\u200b\u63a8\u51fa\u200b\u56e0\u679c\u6027\u200b\u3002\u200b\u4f46\u200b\u5728\u200b\u5b9e\u8bc1\u200b\u7814\u7a76\u200b\u4e2d\u200b\uff0c\u200b\u7531\u4e8e\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u7684\u200b\u5b58\u5728\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u533a\u5206\u200b\u76f8\u5173\u6027\u200b\u4e2d\u200b\u7684\u200b\u56e0\u679c\u5173\u7cfb\u200b\u3002</p> <p>\u200b\u4e00\u4e9b\u200b\u8861\u91cf\u200b\u5e72\u9884\u200b\u6548\u679c\u200b\u7684\u200b\u6307\u6807\u200b\uff08\u200b\u53c2\u6570\u200b\uff09</p> <p>ATE</p> \\[ \\tau = \\bbE[Y(1) - Y(0)] \\] <p>\u200b\u88ab\u8bd5\u200b\u7684\u200b ATE</p> \\[ \\tau_1 = \\bbE[Y(1) - Y(0) | D = 1] \\] <p>\u200b\u6761\u4ef6\u200b ATE\uff08 CATE \uff09</p> \\[ \\tau_C = \\bbE[Y(1) - Y(0) | (D, X)\\in \\text{subgroup}] \\] <p>QTE\uff08 Quantile treatment effect \uff09</p> \\[ \\begin{gathered}     \\text{QTE} = F_1^{-1}(\\gamma) - F_0^{-1}(\\gamma) \\\\     F_d^{-1}(\\gamma) = \\inf\\{y|\\bbP(Y(d)\\leq y)\\geq \\gamma\\} \\end{gathered} \\] <p>Causal risk ratio\uff08\u200b\u53ea\u200b\u9002\u7528\u200b\u4e8e\u200b\u7ed3\u679c\u200b\u4e8c\u5206\u200b\u7684\u200b\u60c5\u51b5\u200b\uff09</p> \\[ \\text{CRR} = \\bbP(Y(1) = 1) / \\bbP(Y(0) = 1) \\] <p>\u200b\u4ee5\u4e0a\u200b\u8fd9\u4e9b\u200b\u53c2\u6570\u200b\u90fd\u200b\u53d7\u5230\u200b\u6df7\u6dc6\u200b\u53d8\u91cf\u200b\uff08\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u80fd\u200b\u88ab\u200b\u89c2\u6d4b\u200b\u5230\u200b\uff09\u200b\u7684\u200b\u5f71\u54cd\u200b\u3002\u200b\u76f8\u5bf9\u200b\u5e94\u200b\u7684\u200b\uff0c\u200b\u4ece\u200b\u5b9e\u9a8c\u200b\u7ed3\u679c\u200b\u7684\u200b\u7edf\u8ba1\u200b\u6307\u6807\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u8fd9\u4e9b\u200b\u6307\u6807\u200b\u7684\u200b\u4e00\u4e2a\u200b\u4f30\u8ba1\u200b\u3002</p> <p>\u200b\u5982\u200b\u5bf9\u4e8e\u200b ATE \u200b\u6307\u6807\u200b\\(\\tau = \\bbE[Y(1) - Y(0)]\\)\uff0c\\(\\tilde \\tau = \\bbE[Y | D = 1] - \\bbE[Y | D = 0]\\)\u200b\u5c31\u662f\u200b\\(\\tau\\)\u200b\u7684\u200b\u4e00\u4e2a\u200b\u4f30\u8ba1\u200b\u3002</p> <p>\\(\\tau\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\u56e0\u679c\u200b\u53c2\u6570\u200b\uff0c\u200b\u53d7\u5230\u200b\\(X, D, Y(0), Y(1)\\)\u200b\u7684\u200b\u5171\u540c\u200b\u5f71\u54cd\u200b\u3002\\(\\tilde\\tau\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\u53ef\u4ee5\u200b\u4ece\u200b\u7edf\u8ba1\u6570\u636e\u200b\u4e2d\u200b\u89c2\u6d4b\u200b\u5230\u200b\u7684\u200b\u6307\u6807\u200b\uff0c\u200b\u53d7\u5230\u200b\\(X, D, Y\\)\u200b\u7684\u200b\u5171\u540c\u200b\u5f71\u54cd\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u67d0\u4e2a\u200b\u53c2\u6570\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4ece\u200b\u7edf\u8ba1\u6570\u636e\u200b\u4e2d\u200b\u89c2\u6d4b\u200b\u5230\u200b\u4e00\u4e2a\u200b\u552f\u4e00\u200b\u7684\u200b\u65e0\u504f\u200b\u4f30\u8ba1\u200b\uff0c\u200b\u5219\u200b\u79f0\u4e3a\u200b identifiable \u3002\u200b\u7531\u4e8e\u200b\u53cd\u200b\u4e8b\u5b9e\u200b\u7684\u200b\u5b58\u5728\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ea\u80fd\u200b\u89c2\u6d4b\u200b\u5230\u200b\\(Y = Y(D)\\)\uff0c\u200b\u56e0\u6b64\u200b\u5373\u4f7f\u200b\u6709\u200b\u65e0\u9650\u200b\u591a\u200b\u7684\u200b\u6570\u636e\u200b\u96c6\u200b\uff0c\u200b\u4e5f\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u80fd\u200b\u5b9e\u73b0\u200b identification \u3002\u200b\u5728\u200b unconfoundedness \u200b\u5047\u8bbe\u200b\u6210\u7acb\u200b\u7684\u200b\u6761\u4ef6\u200b\u4e0b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b identification \u3002</p> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u4e3a\u4e86\u200b\u9a8c\u8bc1\u200b\u4e00\u4e2a\u200b\u56e0\u679c\u5173\u7cfb\u200b\u662f\u5426\u200b\u6210\u7acb\u200b\uff0c\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u4e24\u200b\u65b9\u9762\u200b\u7684\u200b\u4fe1\u606f\u200b</p> <ol> <li>\u200b\u5145\u8db3\u200b\u7684\u200b\u6570\u636e\u200b\u96c6\u4ee5\u200b\u53cd\u6620\u200b\\(X, D, Y\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u3002</li> <li>\u200b\u5173\u4e8e\u200b\\(X, D, Y(0), Y(1)\\)\u200b\u7684\u200b\u4e00\u4e9b\u200b\u5047\u8bbe\u200b\uff0c\u200b\u5982\u200b\u968f\u673a\u200b\u8bd5\u9a8c\u200b\u5047\u8bbe\u200b\u6216\u200b unconfoundedness \u200b\u5047\u8bbe\u200b\u3002</li> </ol>"},{"location":"math/causal-inference/causal-inference-stale/#causal-graph","title":"Causal Graph","text":"<p>\u200b\u4e00\u7cfb\u5217\u200b\u53d8\u91cf\u200b\u7684\u200b\u56e0\u679c\u5173\u7cfb\u200b\u53ef\u4ee5\u200b\u6784\u6210\u200b\u56e0\u679c\u200b\u56fe\u200b\u3002\u200b\u56e0\u679c\u200b\u56fe\u662f\u200b\u4e00\u4e2a\u200b\u6709\u200b\u5411\u200b\u65e0\u200b\u73af\u56fe\u200b\uff08 DAG \uff09\uff0c\u200b\u5176\u4e2d\u200b\u7684\u200b\u8282\u70b9\u200b\u8868\u793a\u200b\u53d8\u91cf\u200b\uff0c\u200b\u6bcf\u6761\u200b\u8fb9\u200b\u4ece\u7236\u200b\u53d8\u91cf\u200b\uff08\u200b\u8282\u70b9\u200b\uff09\u200b\u51fa\u53d1\u200b\uff0c\u200b\u6307\u5411\u200b\u5b50\u200b\u53d8\u91cf\u200b\uff08\u200b\u8282\u70b9\u200b\uff09\u3002</p> <ul> <li>\u200b\u76f8\u90bb\u200b\uff1a\u200b\u5982\u679c\u200b\u4e24\u4e2a\u200b\u8282\u70b9\u200b\u4e4b\u95f4\u200b\u6709\u200b\u4e00\u6761\u200b\u8fb9\u200b\uff0c\u200b\u79f0\u200b\u4e24\u4e2a\u200b\u8282\u70b9\u200b\u76f8\u90bb\u200b</li> <li>\uff08\u200b\u56e0\u679c\u200b\uff09\u200b\u8def\u5f84\u200b\uff1a\u200b\u4e00\u7cfb\u5217\u200b\u76f8\u90bb\u200b\u7684\u200b\u8282\u70b9\u200b\u7ec4\u6210\u200b\u4e00\u6761\u200b\u8def\u5f84\u200b\uff08\u200b\u65e0\u89c6\u200b\u8fb9\u200b\u7684\u200b\u65b9\u5411\u200b\uff09</li> <li>\u200b\u6709\u200b\u5411\u200b\u8def\u5f84\u200b\uff1a\u200b\u65b9\u5411\u200b\u76f8\u540c\u200b\u7684\u200b\u4e00\u7cfb\u5217\u200b\u8def\u5f84\u200b\u6240\u200b\u8fde\u63a5\u200b\u7684\u200b\u8282\u70b9\u200b\u5f62\u6210\u200b\u7684\u200b\u8def\u5f84\u200b</li> </ul> <p>\u200b\u8def\u5f84\u200b\u6709\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\uff1a</p> <ul> <li>\u200b\u94fe\u200b\uff1a\\(A\\ra B\\ra C\\)</li> <li>\u200b\u6c47\u805a\u200b\uff1a\\(A\\ra B\\la C\\)</li> <li>\u200b\u5206\u652f\u200b\uff1a\\(A\\la B\\ra C\\)</li> </ul> <p>\u200b\u5728\u200b\u56e0\u679c\u200b\u56fe\u4e2d\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\u90fd\u200b\u662f\u200b\u5176\u5b50\u200b\u8282\u70b9\u200b\uff08\u200b\u540e\u4ee3\u200b\uff09\u200b\u7684\u200b\u76f4\u63a5\u200b\u539f\u56e0\u200b\uff08\u200b\u7956\u5148\u200b\uff09\uff08\u200b\u5373\u200b\uff0c\u200b\u4e0d\u200b\u5b58\u5728\u200b\u4e2d\u95f4\u200b\u7684\u200b\u53d8\u91cf\u200b\u5f71\u54cd\u200b\u5b50\u200b\u8282\u70b9\u200b\uff09\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u5982\u4e0b\u200b\u56fe\u200b\u6240\u793a\u200b\u7684\u200b\u56e0\u679c\u200b\u56fe\u200b</p> <p></p> <ol> <li>\\(X_1\\)\u200b\u5f71\u54cd\u200b\u53d8\u91cf\u200b\\(X_4\\)\uff0c\u200b\u4f46\u200b\u4e0d\u662f\u200b\u76f4\u63a5\u200b\u5f71\u54cd\u200b\u3002</li> <li> <p>\u200b\u5982\u679c\u200b\u56fa\u5b9a\u200b\\(X_3\\)\uff0c\u200b\u5219\u200b\\(X_1\\)\u200b\u7684\u200b\u53d8\u5316\u200b\u4e0d\u4f1a\u200b\u5f71\u54cd\u200b\\(X_4\\)\uff0c\u200b\u5373\u200b</p> \\[ X_4\\perp (X_1, X_2) | X_3 \\] </li> <li> <p>\u200b\u4f46\u200b\\(X_4\\)\u200b\u4ecd\u7136\u200b\u4e0e\u200b\\(X_5\\)\u200b\u6709\u5173\u200b\u3002</p> </li> </ol> <p>\u200b\u8fdb\u4e00\u6b65\u200b\u5730\u200b\uff0c\u200b\u6211\u4eec\u200b\u4e0d\u59a8\u200b\u5047\u8bbe\u200b\uff0c\u200b\u5982\u679c\u200b\u63a7\u5236\u200b\u4e86\u200b\u67d0\u4e2a\u200b\u53d8\u91cf\u200b\\(X_i\\)\u200b\u5168\u90e8\u200b\u7684\u200b\u76f4\u63a5\u200b\u7956\u5148\u200b\uff0c\u200b\u5219\u200b\\(X_i\\)\u200b\u53ea\u200b\u4e0e\u5176\u200b\u540e\u4ee3\u200b\u6709\u5173\u200b\uff08\u200b\u56e0\u679c\u200bMarkov\u200b\u5047\u8bbe\u200b\uff09\u3002</p> <p>Markov\u200b\u56e0\u5b50\u200b\u5206\u89e3\u200b</p> <p>\u200b\u5f53\u200bMarkov\u200b\u5047\u8bbe\u200b\u6210\u7acb\u200b\u65f6\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\\(n\\)\u200b\u4e2a\u200b\u53d8\u91cf\u200b\\(X_1, \\ldots, X_n\\)\uff0c\u200b\u6709\u200b</p> \\[ \\bbP(X_1, \\ldots, X_n) = \\prod_{i=1}^n \\bbP(X_i | A_i) \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\uff0c\\(A_i\\)\u200b\u4e3a\u200b\\(X_i\\)\u200b\u5168\u90e8\u200b\u7684\u200b\u7956\u5148\u200b\u3002</p>"},{"location":"math/causal-inference/causal-inference-stale/#d-","title":"d-\u200b\u5206\u5272","text":"<p>\u200b\u5bf9\u4e8e\u200b\u6709\u200b\u4e24\u4e2a\u200b\u8282\u70b9\u200b\u7684\u200b\u56e0\u679c\u200b\u56fe\u200b\uff0c\\(X_1\\la X_2\\)\u200b\u548c\u200b\\(X_1\\ra X_2\\)\u200b\u662f\u200b\u4e24\u4e2a\u200b\u4e0d\u540c\u200b\u7684\u200b\u56e0\u679c\u5173\u7cfb\u200b\uff0c\u200b\u4f46\u200b\u5728\u200b\u76f8\u5173\u6027\u200b\u4e0a\u200b\u6709\u200b\u76f8\u540c\u200b\u7684\u200b\u8868\u73b0\u200b\uff0c\u200b\u65e0\u6cd5\u200b\u533a\u5206\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u6709\u200b\u4e09\u4e2a\u200b\u8282\u70b9\u200b\u7684\u200b\u56e0\u679c\u200b\u56fe\u200b\uff1a</p> <ul> <li>\\(X_1\\ra X_2\\ra X_3\\)\u200b\u548c\u200b\\(X_1\\la X_2\\ra X_3\\)\u200b\u662f\u200b\u4e24\u4e2a\u200b\u4e0d\u540c\u200b\u7684\u200b\u56e0\u679c\u5173\u7cfb\u200b\uff0c\u200b\u4f46\u200b\u5b83\u4eec\u200b\u90fd\u200b\u6ee1\u8db3\u200b\\(X_1\\perp X_3 | X_2, X_1\\not\\perp X_3\\)\uff0c\u200b\u4ecd\u7136\u200b\u65e0\u6cd5\u200b\u533a\u5206\u200b\u3002\u200b\u5e76\u4e14\u200b\u7531\u4e8e\u200b\u53d8\u91cf\u200b\\(X_2\\)\u200b\u7684\u200b\u5b58\u5728\u200b\uff0c\u200b\u5373\u4f7f\u200b\\(X_1, X_3\\)\u200b\u5b58\u5728\u200b\u76f8\u5173\u200b\u5173\u7cfb\u200b\uff0c\u200b\u4e5f\u200b\u4e0d\u80fd\u200b\u8bf4\u660e\u200b\u5176\u200b\u5b58\u5728\u200b\u76f4\u63a5\u200b\u6216\u200b\u95f4\u63a5\u200b\u7684\u200b\u56e0\u679c\u5173\u7cfb\u200b\u3002\u200b\u63a7\u5236\u200b\\(X_2\\)\u200b\u53ef\u4ee5\u200b\u963b\u65ad\u200b\\(X_1, X_3\\)\u200b\u7684\u200b\u56e0\u679c\u200b\u94fe\u200b\u3002</li> <li>\u200b\u5bf9\u4e8e\u200b\\(\\X_1\\ra X_2\\la X_3\\)\uff0c\u200b\u5982\u679c\u200b\u63a7\u5236\u200b\\(X_2\\)\uff0c\u200b\u5219\u200b\u6709\u200b\\(X_1\\not \\perp X_3 | X_2\\)\uff0c\u200b\u4f46\u200b\\(X_1\\perp X_3\\)\u3002\\(X_2\\)\u200b\u963b\u65ad\u200b\u4e86\u200b\\(X_1, X_3\\)\u200b\u7684\u200b\u56e0\u679c\u200b\u94fe\u200b\u3002<ul> <li>\u200b\u5982\u679c\u200b\u6709\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\\(X_4\\)\u200b\u6ee1\u8db3\u200b\\(X_2\\ra X_4\\)\uff0c\u200b\u5219\u200b\u63a7\u5236\u200b\\(X_4\\)\u200b\u4e5f\u200b\u4f1a\u200b\u4f7f\u200b\\(X_1, X_3\\)\u200b\u8868\u73b0\u200b\u51fa\u200b\u76f8\u5173\u200b\uff0c\u200b\u5373\u200b\\(X_1\\not \\perp X_3 | X_4\\)\u3002</li> </ul> </li> </ul> <p>\u200b\u963b\u65ad\u200b\u56e0\u679c\u200b\u94fe\u200b</p> <p>\u200b\u4e24\u4e2a\u200b\u53d8\u91cf\u200b\u4e4b\u95f4\u200b\u7684\u200b\u8def\u5f84\u200b\u5982\u679c\u200b\u6ee1\u8db3\u200b\u4e0b\u5217\u200b\u4e4b\u4e00\u200b\uff0c\u200b\u5219\u200b\u56e0\u679c\u5173\u7cfb\u200b\u4e0d\u80fd\u200b\u901a\u8fc7\u200b\u8fd9\u200b\u6761\u200b\u8def\u5f84\u200b\uff08\u200b\u88ab\u200b\u963b\u65ad\u200b\uff09\u3002</p> <ol> <li>\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u6ca1\u6709\u200b\u88ab\u200b\u63a7\u5236\u200b\u7684\u200b\u6c47\u805a\u200b\u8282\u70b9\u200b\uff1b</li> <li>\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u88ab\u200b\u63a7\u5236\u200b\u7684\u200b\u975e\u200b\u6c47\u805a\u200b\u8282\u70b9\u200b\u3002</li> </ol> <p>\u200b\u5982\u679c\u200b\u4e24\u4e2a\u200b\u53d8\u91cf\u200b\u4e4b\u95f4\u200b\u7684\u200b\u6240\u6709\u200b\u8def\u5f84\u200b\u90fd\u200b\u88ab\u200b\u963b\u65ad\u200b\uff0c\u200b\u5219\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u53d8\u91cf\u200b\u88ab\u200b\u5206\u5272\u200b\u3002</p> <p>\u200b\u4e00\u822c\u200b\u5b9a\u4e49\u200b\u88ab\u200b\u63a7\u5236\u200b\u7684\u200b\u53d8\u91cf\u200b\u96c6\u5408\u200b\u4e3a\u200b\\(Z\\)\u3002\u200b\u5982\u679c\u200b\u56e0\u679c\u200b\u56fe\u200b\\(\\calG\\)\u200b\u4e2d\u200b\uff0c\u200b\u63a7\u5236\u53d8\u91cf\u200b\\(Z\\)\u200b\u5c06\u200b\u56e0\u679c\u200b\u56fe\u200b\u5206\u4e3a\u200b\u4e86\u200b\u4e24\u4e2a\u200b\u90e8\u5206\u200b\uff0c\u200b\u5219\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u90e8\u5206\u200b\\(X, Y\\)\u200b\u6784\u6210\u200b\u4e86\u200b\u56e0\u679c\u200b\u56fe\u200b\u7684\u200b\u4e00\u4e2a\u200bd-\u200b\u5206\u5272\u200b\uff0c\u200b\u5373\u200b\\(X\\perp_\\calG Y | Z\\)\u3002\u200b\u8fdb\u4e00\u6b65\u200b\u5730\u200b\uff0c\u200b\u5982\u679c\u200b\u6ee1\u8db3\u200b Markov \u200b\u5047\u8bbe\u200b\uff0c\u200b\u5219\u200b\\(X\\)\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u53d8\u91cf\u200b\u548c\u200b\\(Y\\)\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u53d8\u91cf\u200b\u72ec\u7acb\u200b\u3002</p> <p>\u200b\u4f46\u662f\u200b\u53d8\u91cf\u200b\u72ec\u7acb\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u80fd\u200b\u5bfc\u51fa\u200bd-\u200b\u5206\u5272\u200b\uff0c\u200b\u7531\u6b64\u200b\u5f97\u5230\u200b faithfulness assumption</p> <p>faithfulness assumption</p> <p>\u200b\u5bf9\u4e8e\u200b\u56e0\u679c\u200b\u56fe\u200b\\(\\calG\\)\uff0c\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\u4e92\u4e0d\u200b\u76f8\u4ea4\u200b\u7684\u200b\\(X, Y, Z\\)\u200b\u4e14\u200b\\(X\\perp Y | Z\\)\uff0c\u200b\u5219\u200b\\(X, Y\\)\u200b\u6784\u6210\u200b\u4e00\u4e2a\u200bd-\u200b\u5206\u5272\u200b\u3002</p> <p>\u200b\u5728\u200b Markov \u200b\u5047\u8bbe\u200b\u548c\u200b faithfulness \u200b\u5047\u8bbe\u200b\u6210\u7acb\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0cd-\u200b\u5206\u5272\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u53d8\u91cf\u200b\u72ec\u7acb\u200b\u3002</p>"},{"location":"math/causal-inference/causal-inference-stale/#_5","title":"\u76f8\u5173\u6027\u200b\u4e0e\u200b\u56e0\u679c\u6027\u200b\u7684\u200b\u8def\u5f84","text":"<p>\u200b\u5728\u200b\u56e0\u679c\u200b\u56fe\u4e0a\u200b\uff1a</p> <ul> <li>\u200b\u76f8\u5173\u6027\u200b\u6cbf\u200b\u8def\u5f84\u200b\u6d41\u52a8\u200b</li> <li>\u200b\u56e0\u679c\u6027\u200b\u6cbf\u6709\u200b\u5411\u200b\u8def\u5f84\u200b\u6d41\u52a8\u200b</li> </ul> <p>\u200b\u5982\u679c\u200b\u963b\u65ad\u200b\u4e86\u200b\u6240\u6709\u200b\u975e\u200b\u56e0\u679c\u6027\u200b\u7684\u200b\u8def\u5f84\u200b\uff0c\u200b\u5219\u200b\u4e24\u4e2a\u200b\u53d8\u91cf\u200b\u4e4b\u95f4\u200b\u5b58\u5728\u200b\u56e0\u679c\u5173\u7cfb\u200b\u3002</p> <p>do\u200b\u7b97\u5b50\u200b</p> <p>\u200b\u5728\u200b\u56e0\u679c\u200b\u56fe\u4e2d\u200b\uff0c\u200b\u5bf9\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\u7684\u200b\u5e72\u9884\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u963b\u65ad\u200b\u6240\u6709\u200b\u6307\u5411\u200b\u8be5\u200b\u53d8\u91cf\u200b\u7684\u200b\u8fb9\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(\\mathrm {do}(\\cdot)\\)</p> <p>\u200b\u975e\u200b\u56e0\u679c\u200b\u7684\u200b\u8def\u5f84\u200b\u6709\u200b\u4e24\u4e2a\u200b\u6765\u6e90\u200b\uff1a</p> <ul> <li>\u200b\u6ca1\u6709\u200b\u63a7\u5236\u200b\u5206\u652f\u200b\u53d8\u91cf\u200b\uff0c\u200b\u5373\u200b confounding bias</li> <li>\u200b\u63a7\u5236\u200b\u4e86\u200b\u6c47\u805a\u200b\u53d8\u91cf\u200b\uff0c\u200b\u5373\u200b selection bias</li> </ul>"},{"location":"math/causal-inference/causal-inference-stale/#confounding","title":"Confounding \u200b\u4e0e\u200b\u540e\u95e8\u200b\u8c03\u6574","text":"<p>Confounding \u200b\u6307\u200b\u56e0\u679c\u200b\u8def\u5f84\u200b\u7684\u200b\u8d77\u59cb\u200b\u548c\u200b\u7ec8\u7ed3\u200b\u53d8\u91cf\u200b\u90fd\u200b\u53d7\u5230\u200b\u540c\u4e00\u4e2a\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u7684\u200b\u5f71\u54cd\u200b\u3002\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u540e\u95e8\u200b\u8c03\u6574\u200b\u6d88\u9664\u200b\u5f71\u54cd\u200b\u3002</p>"},{"location":"math/causal-inference/chapter-1/","title":"A Definition of Causal Effect","text":"<p>Key concepts: treatment, outcome, average causal effect, potential outcome, consistency, causation-association difference, identifiability</p>"},{"location":"math/causal-inference/chapter-1/#treatment-and-outcome","title":"Treatment and Outcome","text":"<p>We want to compare the result when a specified action is or is not applied to an individual. The action is called a treatment, and the result is called an outcome, which are denoted by \\(A\\) and \\(Y\\) respectively. Usually we take \\(A\\) and \\(Y\\) to be both binary.</p> <p>The outcome under a specified treatment \\(A = a\\) is denoted by \\(Y^{A = a}\\). In reality, we can only observe one value of \\(Y^{A = 1}\\) or \\(Y^{A = 0}\\) for each individual. Each \\(Y^{A = a}\\) is called a potential outcome. If \\(A = a\\) is observed, \\(Y^{A = a}\\) is factual, and \\(Y^{A = 1 - a}\\) is counterfactual. The probability of potential outcome \\(Y = 1\\) is called risk.</p> <p>Consistency assumption</p> <p>If \\(A = a\\) is observed, then \\(Y = Y^{A = a}\\).</p>"},{"location":"math/causal-inference/chapter-1/#causal-effects","title":"Causal Effects","text":"<p>Identifiability</p> <p>A quantity is identifiable if it can be expressed (unbiasedly) as a function of the distribution of the observed data.</p> <p>Before we clarify the definition of causal effect, we need to make some assumptions on individual outcome.</p> <p>No interference assumption</p> <p>The potential outcome under \\(A = a\\) does not depend on the treatment of other individuals (i.e. no interference).</p>"},{"location":"math/causal-inference/chapter-1/#individual-causal-effect","title":"Individual Causal Effect","text":"<p>If for individual \\(i\\), \\(Y_i^{A = 1} \\not = Y_i^{A = 0}\\), then we say that the treatment has a causal effect for individual \\(i\\).</p>"},{"location":"math/causal-inference/chapter-1/#average-causal-effect","title":"Average Causal Effect","text":"<p>The average causal effect is defined as</p> \\[ P(Y^{A = 1} = 1) - P(Y^{A = 0} = 1) \\] <p>or</p> \\[ E(Y^{A = 1}) - E(Y^{A = 0}) \\] <p>The two forms of average causal effect are equivalent. We can test average causal effect by testing the causal null hypothesis \\(H_0: P(Y^{A = 1} = 1) = P(Y^{A = 0} = 1)\\). However, the absence of average causal effect does not imply the absence of individual causal effect. We call the null hypothesis \\(H_0: \\forall i, Y_i^{a = 1} = Y_i^{a = 0}\\) as sharp causal null hypothesis.</p>"},{"location":"math/causal-inference/chapter-1/#functional-causal-effect","title":"Functional Causal Effect","text":"<p>We can measure the outcome \\(Y\\) over a function \\(f(y)\\). For example \\(V(Y^{a = 1}) - V(Y^{a = 0})\\) as the causal effect on sample variance. Notice that \\(V(Y^{a = 1}) - V(Y^{a = 0})\\) usually does not equal to \\(V(Y^{a = 1} - Y^{a = 0})\\).</p>"},{"location":"math/causal-inference/chapter-1/#measuring-causal-effects","title":"Measuring Causal Effects","text":"<p>The causal null hypothesis can be written as</p> <ol> <li>Difference: \\(H_0: P(Y^{a = 1} = 1) - P(Y^{a = 0} = 1) = 0\\)</li> <li>Ratio: \\(H_0: \\frac{P(Y^{a = 1} = 1)}{P(Y^{a = 0} = 1)} = 1\\)</li> <li>Odds ratio: \\(H_0: \\frac{P(Y^{a = 1} = 1) / P(Y^{a = 1} = 0)}{P(Y^{a = 0} = 1) / P(Y^{a = 0} = 0)} = 1\\)</li> </ol> <p>When \\(H_0\\) is rejected, we can use number needed to treat (NNT) to measure the strength of causal effect. NNT is defined as the number of individuals needed to be treated to prevent one additional bad outcome. NNT is given by:</p> \\[ \\text{NNT} = \\frac{-1}{P(Y^{a = 1} = 1) - P(Y^{a = 0} = 1)} \\]"},{"location":"math/causal-inference/chapter-1/#random-variability","title":"Random Variability","text":"<p>In reality, we can only collect a sample of individuals in the population, and our estimate on causal effect suffers from random variability.</p> <ol> <li>Sampling variablility: We use \\(\\hat P(Y^{a = 1} = 1) = N(Y^{a = 1}) / N(Y)\\) as an estimator of \\(P(Y^{a = 1} = 1)\\). The estimator suffers from random variability.</li> <li>Potential outcomes are usually not deterministic, which makes another source of random variability.</li> </ol>"},{"location":"math/causal-inference/chapter-1/#causation-and-association","title":"Causation and Association","text":"<p>In the real world, we can only observe one potential outcome for each individual. Therefore, we cannot directly measure the outcome \\(P(Y^{a = 1} = 1)\\) and \\(P(Y^{a = 0} = 1)\\). Instead, we can only use \\(P(Y = 1 | A = 1)\\) and \\(P(Y = 1 | A = 0)\\) as estimators.</p> <p>Like the causal null hypothesis, we can also define the independent (\\(A\\perp Y\\)) null hypothesis in three forms:</p> <ol> <li>Risk difference: \\(H_0: P(Y = 1 | A = 1) - P(Y = 1 | A = 0) = 0\\)</li> <li>Risk ratio: \\(H_0: \\frac{P(Y = 1 | A = 1)}{P(Y = 1 | A = 0)} = 1\\)</li> <li>Odds ratio: \\(H_0: \\frac{P(Y = 1 | A = 1) / P(Y = 0 | A = 1)}{P(Y = 1 | A = 0) / P(Y = 0 | A = 0)} = 1\\)</li> </ol> <p>Notice that any terms in the form of conditional probability or expectation are associations. Only marginal probability or expectation can be causal effects, since the marginal probability takes all other factors into consideration.</p> <p></p>"},{"location":"math/causal-inference/chapter-2/","title":"Randomized Experiments","text":"<p>Key concepts: random experiment, exchangeability, effect modification</p>"},{"location":"math/causal-inference/chapter-2/#randomized-experiments_1","title":"Randomized Experiments","text":"<p>Consider the following treatment assignment:</p> <p></p> <p>Assume that the treatment assignment is determined by a random procedure, the experiment is called a random experiment. Then exchanging the individual in treated group with those in control group will not affect the association measure.</p> <p>If a random experiment satisfies the following conditions, it is called an ideal randomized experiment:</p> <ol> <li>Full adherence: All individuals follow the treatment assignment.</li> <li>No loss to follow-up: All individuals are followed up and their outcomes are observed.</li> <li>Single treatment: Each individual is assigned to one treatment only.</li> <li>Double-blind assignment: The treatment assignment is unknown to both the individuals and the researchers.</li> </ol> <p>If all the possible values of treatment variable \\(A = a\\) are sequentially assigned to individuals, the experiment is called crossover experiment. Individual causal effects can be estimated using crossover experiments if the following conditions are met:</p> <ol> <li>No carryover effect: The treatment effect of \\(a\\) does not depend on the previous treatments.</li> <li>The individual causal effect \\(\\alpha_i = Y^{a_t = 1}_{it} - Y^{a_t = 0}_{it}\\) does not depend on time.</li> <li>The potential outcome under no treatment does not depend on time.</li> </ol> <p>By assigning the treatment value sequence randomly, we can remove the required conditiona 3. In such case:</p> \\[ (Y_{i1} - Y_{i0})A_{i1} - (Y_{i0} - Y_{i1})A_{i0} \\] <p>estimates the individual causal effect \\(E(\\alpha_i)\\) under condition 2. If condition 2 is not met, the estimator estimates \\(E(\\alpha_{i0} + \\alpha_{i1}) / 2\\).</p>"},{"location":"math/causal-inference/chapter-2/#exchangeability","title":"Exchangeability","text":"<p>Exchangeability</p> <ul> <li>(Full) exchangeability indicates that swapping the treatment assignment of two individuals does not affect the association measure, or \\(Y^a \\perp A\\).</li> <li>Mean exchangeability indicates that \\(\\forall a', E(Y^a | A = a') = E(Y^a)\\), or the expectation of switched potential outcome is the same as the original one.</li> </ul> <p>When \\(Y\\) and \\(A\\) are binary, full exchangeability is equivalent to mean exchangeability. Mean exchangeability is sufficient for deriving \\(E(Y^a) = E(Y | A = a)\\).</p> <p>Exchangeability indicates that all the potential outcome and the treatment assignment are independent, i.e. \\(\\forall a, Y^{A = a} \\perp A\\), or the treatment assignment is exogenous. Under exchangeability, \\(P(Y = 1 | A = a)\\) is an unbiased estimator of \\(P(Y^{a} = 1)\\).</p> <p>\\(Y^a\\perp A\\) and \\(Y\\perp A\\)</p> <p>Notice that \\(Y^a \\perp A\\) is not equivalent to \\(Y \\perp A\\). The latter indicates that treatment \\(A\\) does not have any causal effect on the outcome \\(Y\\). If \\(\\forall a, E(Y^a) = E(Y)\\), then \\(Y\\perp A\\).</p> <p>Notice that a randomized experiment could be unexchangeable even if infinite samples are collected.</p>"},{"location":"math/causal-inference/chapter-2/#conditional-randomization","title":"Conditional Randomization","text":"<p>Consider an additional factor \\(L\\) affects the assignment of treatments, i.e., when \\(L\\) takes different values, the distribution of \\(A\\) is different. Such a randomized experiment is called conditional randomized experiment. In a conditional randomized experiment, the corresponding randomized experiment using the overall distribution of \\(A\\) is called marginally randomized experiment.</p> <p>Although the marginally randomized experiment is exchangeable, the conditional randomized experiment is not exchangeable in general. Only conditional exchangeability holds in such randomized experiments, i.e. \\(\\forall a, Y^a \\perp A | L\\).</p> <p>We can measure the causal effects on sub-groups \\(P(Y^{a = 1} = 1 | L = l) - P(Y^{a = 0} = 1 | L = l)\\) using conditional exchangeability (stratification) or directly measure the overall causal effect \\(P(Y^{a = 1} = 1) - P(Y^{a = 0} = 1)\\) on the population using marginal exchangeability. For different sub-groups, the causal effects may be different, which is called effect modification or treatment effect heterogeneity.</p> <p>Stratification</p> <p>Stratification is to split the population into mutually exclusive sub-groups in which the treatment assignment is exchangeable. The causal effect can be estimated in each sub-group.</p>"},{"location":"math/causal-inference/chapter-2/#standardization","title":"Standardization","text":"<p>We can view the marginal risk as a weighted average of the conditional risks:</p> \\[ P(Y^a = 1) = \\sum_{l\\in \\calL} P(Y^a = 1 | L = l) P(L = l) \\] <p>With conditional exchangeability holds, we can derive that \\(P(Y^a = 1) = \\sum_{l\\in \\calL} P(Y = 1 | A = a, L = l) P(L = l)\\), which is called standardization, making the unobserved potential outcome \\(P(Y^a = 1)\\) to be identifiable.</p>"},{"location":"math/causal-inference/chapter-2/#inverse-probability-weighting","title":"Inverse Probability Weighting","text":"<p>Notice that</p> \\[ P(A = a | L = l) = \\frac{P(A = a, L = l)}{P(L = l)} \\] <p>We can assign the weight \\(w_l = \\frac{1}{P(A = a | L = l)}\\) to each individual with \\(L = l\\) to construct a pseudo-population. The pseudo-population satisfies exchangeability, and we can use the pseudo-population to estimate the causal effect. The IP weighted mean of outcome \\(Y\\) is \\(E\\left( I(A = a) Y / P(A = a | L)\\right)\\), where \\(I(A = a)\\) is the indicator function.</p> <p>Assuming that \\(\\forall l, P(A = a | L = l) &gt; 0\\), we can derive that</p> \\[ \\begin{aligned}     &amp; E\\left( \\frac{I(A = a) Y}{P(A = a | L)}\\right) \\\\     =&amp; \\sum_{l} \\frac{E\\left( I(A = a) Y | L = l\\right) P(L = l)}{P(A = a | L = l)} \\\\     =&amp; \\sum_{l} \\frac{E\\left( Y | A = a, L = l\\right) P(A = a | L = l)P(L = l)}{P(A = a | L = l)} \\\\     =&amp; \\sum_{l} E\\left( Y | A = a, L = l\\right) P(L = l) \\\\     =&amp; \\sum_{l} E(Y^a | L = l) P(L = l) \\\\     =&amp; E(Y^a) \\end{aligned} \\] <p>Therefore, under positivity assumption, IP weighting is equivalent to standardization.</p> <p>Positivity</p> <p>For any individual in any sub-group \\(l\\) has a non-zero probability of receiving treatment \\(a\\), i.e. \\(\\forall l, \\forall a, P(A = a | L = l) &gt; 0\\).</p>"},{"location":"math/causal-inference/chapter-3/","title":"Observational Studies","text":"<p>In an observational study, the variables are neither assigned nor controlled by the researcher. The researcher only observes the values of the variables. Compared with randomized experiments, observational studies are more limited to the researchers.</p>"},{"location":"math/causal-inference/chapter-3/#identifiability-conditions","title":"Identifiability Conditions","text":"<p>If the following identifiability conditions are met, the observational study can be regarded as a conditionally randomized experiment.</p> <p>Identifiability condition</p> <ol> <li>Consistency: The values of treatment under comparison correspond to well-defined interventions that, in turn, correspond to the versions of treatment in the data.</li> <li>Exchangeability: The conditional probability of receiving any value of treatment depends only on observed covariates \\(L\\).</li> <li>Positivity: The probability of receiving any value of treatment is strictly positive for all combinations of covariates \\(L\\).</li> </ol> <p>If the identifiability conditions are not met, we can use other methods including instrumental variable to establish another group of identifiability condition.</p>"},{"location":"math/causal-inference/chapter-3/#exchangeability","title":"Exchangeability","text":"<p>In observational studies, the exchangeablitiy condition may or may not hold, as the observed \\(L\\) may not be the only predictor of the treatment. The true predictor may even be unobserved or unobservable.</p> <p>Exchangeability is given by</p> \\[ Y^a \\perp A \\mid L \\Longleftrightarrow \\forall a, P(Y^a  = 1 \\mid A = a, L = l) = P(Y^a = 1\\mid A \\not = a, L = l) \\] <p>However, in observational studies, \\(P(Y^a = 1\\mid A\\not = a, L = l)\\) is unidentifiable. Thus, the exchangeablitiy cannot be empirically verified.</p>"},{"location":"math/causal-inference/chapter-3/#positivity","title":"Positivity","text":"<p>To measure causal effects, we need to measure the distribution of outcome under treatment and under no treatment. That is, in the whole sample or any observed subgroup, the probability of receiving any value of treatment should be strictly positive, which is called positivity. Notice that positivity is only required for these observed covariates that requires exchangeability.</p> <p>Standardization and IP weighting are meaningful only if positivity holds.</p> <p>IP weighting without positivity</p> <p>If positivity does not hold, IP weighting is undefined since the denominator in</p> \\[ E\\left[\\frac{I(A = a)Y}{P(A = a | L)}\\right] \\] <p>is zero. However, the IP weighted mean is still defined.</p> \\[ E\\left[\\frac{I(A = a)Y}{N(A = a, L) / N(L)}\\right] \\] <p>Notice that \\(N(A = a, L) / N(L)\\) is defined on observed data. Let \\(Q(a) = \\{l | N(A = a, L) / N(L) &gt; 0\\}\\). Now, \\(E\\left[\\frac{I(A = a)Y}{N(a, L) / N(L)}\\right] = E[Y^a | L \\in Q(a)]P(L\\in Q(a))\\) is a biased estimator of \\(E[Y^a]\\).</p> <p>When positivity does not hold, then \\(Q(0) \\not = Q(1)\\), under this condition, the difference between IP weighted means (\\(E\\left[\\frac{I(A = 1)Y}{N(A = 1, L) / N(L)}\\right] - E\\left[\\frac{I(A = 0)Y}{N(A = 0, L) / N(L)}\\right]\\)) cannot be interpreted causally.</p>"},{"location":"math/causal-inference/chapter-3/#consistency","title":"Consistency","text":"<p>Consider the following examples in heart transplant studies:</p> <ol> <li>Two patients received heart transplants performed by different surgeons.</li> <li>Two patients received heart transplants performed under different procedures.</li> </ol> <p>Due to the complexity of real-world situations, we always need to aggregate the different treatments into a single treatment value. The map between discrete values of the treatment variable and real-world intervention is called treatment protocol. The first step for investigating the causal effect is to precisely specify the treatment protocol, so that any variation regarded as treatment will lead to the same outcome, or treatment variation irrelevance. Under this condition can we define the potential outcome under the specified treatment value. In addition, the treatment protocol of interest and the treatment protocol of the observed data should be the same.</p> <p>Under the treatment variation irrelevance, we can define the potential outcome \\(Y^a\\).</p> <p>Treatment variation irrelevance</p> <p>Suppose that receiving heart transplant is denoted as \\(a\\), the heart transplant is performed by Alex and Bob. But the treatment protocol is observed as the heart transplant is performed by Alex. Then, the treatment variation irrelevance is violated and consistency assumption \\(Y = Y^{a = 1}\\) does not hold.</p>"},{"location":"math/causal-inference/chapter-3/#target-trial","title":"Target Trial","text":"<p>Under the identifiability conditions, we can regard the observational study as a conditionally randomized experiment. The equivalent hypothetical, randomized experiment is called target trial.</p>"},{"location":"math/causal-inference/chapter-4/","title":"Effect Modification","text":"<p>A null average causal effect does not imply a null average causal effect for a subset of the population. A variable \\(V\\) is an effect modifier if the average causal effect of \\(A\\) on \\(Y\\) differs across levels of \\(V\\).</p> <p>An effect modifier does need to directly have causal relationship with the different causal effect. If the difference in causal effects can be interpreted causally, the variable is a causal effect modifier, otherwise it is a surrogate effect modifier.</p> <p>Effects in the treated</p> <p>Notice the treatment variable itself can be regarded as a subgroup. The average causal effect in the treated is measured on subgroup \\(A = 1\\).</p> \\[ P(Y^{a = 1} = 1 | A = 1) - P(Y^{a = 0} = 1 | A = 1) \\] <ol> <li>If the distribution of individual causal effects varies between the treated and untreated, the average causal effect in the treated is not equal to the average causal effect in the population.</li> <li>\\(P(Y^{a = 0} = 1 | A = 1)\\) is unidentifiable. Estimating \\(P(Y^{a = 0} = 1 | A = 1)\\) requires partial exchangeability \\(Y^{a = 0} \\perp A \\mid L\\).</li> <li>Following the definition of effects in the treated, we can define effects in the untreated, which is defined on subgroup \\(a = 0\\). Effects in the untreated requires partial exchangeability \\(Y^{a = 1} \\perp A \\mid L\\).</li> </ol> <p>The counterfactual \\(P(Y^{a} = 1 | A = a')\\) can be estimated using standardization or IP weighting.</p> <ol> <li>Standardization: \\(P(Y^{a} = 1 | A = a') = \\sum_{l} P(Y = 1 | A = a, L = l) P(L = l | A = a')\\)</li> <li>IP weighting: \\(P(Y^{a} = 1 | A = a') = E\\left[\\frac{I(A = a)Y}{f(A | L)P(A = a' | L)}\\right]/E\\left[\\frac{I(A = a)}{f(A | L)P(A = a' | L)}\\right]\\)</li> </ol> <p>The extrapolation of computed causal effects across populations are called transportability. Transportability is an unverifiable assumption. If the effect modification measure, version of treatment and the in-group interference are similar, we can assume transportability.</p> <p>From effect modification, we can identify the subgroup that benefits most from the treatment or intervention. Moreover, effect modification helps us to learn underlying mechanisms of the outcome.</p>"},{"location":"math/causal-inference/chapter-4/#effect-modification-vs-prognostic-factor","title":"Effect Modification v.s. Prognostic Factor","text":"<p>When marginal randomization does not hold, there exists a prognostic factor \\(L\\) is a variable that affect the treatment assignment, while an effect modifier \\(V\\) affects the effect. To estimate causal effects in such conditions, extra adjustments are required</p>"},{"location":"math/causal-inference/chapter-4/#stratification","title":"Stratification","text":"<p>Stratification can be used to adjust the marginal causal effect of \\(A\\) on \\(V\\). However, it requires to take all the combinations of \\(V\\) and \\(L\\) into account. And stratification requires positivity condition to be met on \\(L\\).</p> <p>Using stratification, we can yield the conditional causal effect of \\(A\\) on \\(Y\\) for each stratum. The per-stratum causal effect can be further weighted to get a pooled causal effect.</p> <p>Notice that stratification will introduce sampling variability when the stratification is getting finer.</p>"},{"location":"math/causal-inference/chapter-4/#matching","title":"Matching","text":"<p>Matching strategy is similar but inversed approach to IP weighting.</p> <p>The goal of matching is to construct a subset of the population in which the distribution of \\(L\\) is similar between the treated and untreated by sampling in the treated part or untreated part. Stratas with only treated or untreated individuals are excluded.</p> <p>Usually, the group with more individuals between the treated and untreated is sampled to match the other group. The subgroup being matched is used to calculate the average effect in the treated or untreated.</p> <p>However, as matching changes the distribution of \\(L\\), the distribution of causal modifier \\(V\\) will also be changed.</p>"},{"location":"math/causal-inference/chapter-4/#overview-of-adjustment-methods","title":"Overview of Adjustment Methods","text":"<p>There are 4 methods to adjust the causal effect of \\(A\\) on \\(Y\\), including standardization, IP weighting, matching and stratification.</p> <ol> <li>Standardization and IP weighting can be used to compute marginal or conditional causal effects.</li> <li>Matching and stratification can be used to compute conditional causal effects in subgroups.</li> <li>With effect modification, the causal effect of different adjustment methods may be different.</li> </ol>"},{"location":"math/causal-inference/chapter-5/","title":"Interaction","text":"<p>This chapter considers a system that consists multiple treatment variables.</p>"},{"location":"math/causal-inference/chapter-5/#from-joint-intervention-to-interaction","title":"From Joint Intervention to Interaction","text":"<p>Intervention is to manually set the value of treatment variable ignoring the original distribution of the variable. Intervention on multiple variables are called joint intervention.</p> <p>Let the treatment variable be \\(A\\) and \\(B\\). If the causal effect (measured in risk difference) of treatment \\(A\\) is different with different values of \\(B\\), then \\(B\\) has an additive interaction on \\(A\\). Notice that \\(B\\) has an interaction on \\(A\\) is equivalent to \\(A\\) has an interaction on \\(B\\).</p> \\[ P(Y^{a = 1, b = 1} = 1) - P(Y^{a = 0, b = 1} = 1) \\not = P(Y^{a = 1, b = 0} = 1) - P(Y^{a = 0, b = 0} = 1) \\] <p>Superadditive and subadditive</p> <p>Rewrite the above inequality as</p> \\[ \\begin{aligned}     P(Y^{a = 1, b = 1} = 1) - P(Y^{a = 0, b = 0} = 1) &amp;\\not = [P(Y^{a = 1, b = 1} = 1) - P(Y^{a = 0, b = 1} = 1)] \\\\     &amp;+ [P(Y^{a = 1, b = 0} = 1) - P(Y^{a = 0, b = 0} = 1)] \\end{aligned} \\] <p>If the \\(\\not =\\) can be replaced by \\(&gt;\\) or \\(&lt;\\), then the interaction is called superadditive or subadditive, respectively.</p> <p>Similarly, if the causal effect (measured in risk ratio) of treatment \\(A\\) is different with different values of \\(B\\), or vice versa, then there is a multiplicative interaction between \\(A\\) and \\(B\\). If the \\(\\not =\\) can be replaced by \\(&gt;\\) or \\(&lt;\\), then the interaction is called supermultiplicative or submultiplicative, respectively.</p> \\[ \\frac{P(Y^{a = 1, b = 1} = 1)}{P(Y^{a = 0, b = 0} = 1)} \\not = \\frac{P(Y^{a = 1, b = 0} = 1)}{P(Y^{a = 0, b = 0} = 1)} \\times \\frac{P(Y^{a = 1, b = 1} = 1)}{P(Y^{a = 0, b = 1} = 1)} \\]"},{"location":"math/causal-inference/chapter-5/#idenfication-of-interactions","title":"Idenfication of Interactions","text":"<p>To identify the interaction between \\(A\\) and \\(B\\), we need make sure that consistency, exchangeability, positivity for \\(A\\) is satisfied for all values of both \\(A\\) and \\(B\\).</p> <p>Notice the difference between effect modification and interaction. If \\(B\\) is randomly assigned, then \\(B\\) interacts with \\(A\\) is equivalent to that \\(B\\) serves as an effect modifier for \\(A\\).</p> <p>Effect modification v.s. interaction</p> <p>In an interaction, the two variables are of equal status, we can intervene on either or both of them. In effect modification, one variable is the treatment and the other is the modifier, we can only intervene on the treatment variable.</p> <p>For treatment variables, we need consistency, exchangeability, positivity. But such conditions are not required for modifier variables.</p>"},{"location":"math/causal-inference/chapter-5/#response-type","title":"Response Type","text":"<p>For each individual, the potential outcome pattern for receiving different levels of treatment is called response type.</p> <p></p> <p>For binary outcome and one binary treatment, there are \\(2^{2^1} = 4\\) response types. For binary outcome and two binary treatments, there are \\(2^{2 ^ 2} = 16\\) response types.</p> <p>Monotonicity</p> <p>For a response type, when treatment variables except \\(A\\) are fixed, if \\(Y^{a = 1} \\geq Y^{a = 0}\\), then the causal effect of \\(A\\) on \\(Y\\) is monotonic.</p>"},{"location":"math/causal-inference/chapter-5/#sufficient-cause-interaction","title":"Sufficient Cause Interaction","text":"<p>Consider the mechanism of yielding certain outcome by controlling one binary treatment variable \\(A\\). There are three types of mechanisms to inevitably yield the outcome.</p> <ol> <li>With treatment \\(A\\).</li> <li>Without treatment \\(A\\).</li> <li>\\(A\\) is irrelevant to the outcome.</li> </ol> <p>Since the outcome is stochastic, there exists some unknown factors \\(U_1, U_2, U_0\\) that leads to the outcome, respectively. For example, we can say \\(\\{A = 1, U_1 = 1\\}, \\{A = 0, U_2 = 1\\}\\) or \\(\\{U_3 = 1\\}\\) are sufficient-component causes for the outcome.</p> <p>For models with \\(k\\) binary treatment variables, there are \\(3^k\\) possible mechanisms. If any individual \\(U_i = 1\\) corresponds to multiple variables exists, then there exists sufficient cause interaction among these variables.</p> <p>Let \\(k = 2\\) and the treatment variables be \\(A, B\\), respectively. The sufficient cause interaction is synergistic if \\(A = 1, B = 1\\) is a sufficient-component cause for the outcome. The sufficient cause interaction is antagonistic if \\(A = 1, B = 0\\) or \\(A = 0, B = 1\\) are sufficient-component causes for the outcome.</p> <p>When monotonicity holds, some sufficient causes are not possible.</p>"},{"location":"math/causal-inference/chapter-5/#summary","title":"Summary","text":"<p>Sufficient-component-cause framework and the counterfactual framework focus on different interaction questions:</p> <ol> <li>Sufficient-component-cause framework focuses on the causal mechanism (reason) of the outcome.</li> <li>Counterfactual framework focuses on the causal effect of the treatment.</li> </ol>"},{"location":"math/causal-inference/chapter-6/","title":"Graphical Representation of Causal Effects","text":"<p>A causal graph consists the following elements:</p> <ul> <li>Nodes in causal graphs represent random variables.</li> <li>Edges in causal graphs represent direct causal effects.<ul> <li>A trail is a sequence of edges that does not meet the same node twice.</li> <li>A path is a trail that all the edges are pointing in the same direction.</li> </ul> </li> <li>Conventionally, time flows from left to right in causal graphs.</li> <li>Causal graphs follow the properties of Bayesian networks (acyclic, markov factorization).</li> </ul> <p>Markov Factorization</p> <p>The joint distribution of a set of random variables can be factorized into a product of conditional distributions of each variable given its parents in the graph.</p> \\[ P(X_1, \\dots, X_n) = \\prod_{i=1}^n P(X_i \\mid \\text{Pa}(X_i)) \\] <p>where \\(\\text{Pa}(X_i)\\) is the set of parents of \\(X_i\\).</p> <p>Markov factorization is equivalent to the following local independence property:</p> \\[ X_i \\perp \\text{ND}(X_i) \\mid \\text{Pa}(X_i) \\] <p>where \\(\\text{ND}(X_i)\\) is the set of non-descendants of \\(X_i\\).</p> <p>A causal graph contains the following building blocks: fork, chain, collider.</p> <p></p>"},{"location":"math/causal-inference/chapter-6/#d-seperation-and-independence","title":"D-seperation and Independence","text":"<p>Conditional independence relations are encoded in the graph structure.</p>"},{"location":"math/causal-inference/chapter-6/#blocked-trails-and-d-seperation","title":"Blocked Trails and D-seperation","text":"<p>Trails in causal graphs are regarded as blocked or open according to the following rules:</p> <ol> <li>If there are no nodes being observed, a path is blocked if and only if there exists one or more colliders on the path.</li> <li>If the trail contains a non-collider that is observed, the trail is blocked.</li> <li>A collider which is observed does not block the trail.</li> <li>A collider which any of its descendants is observed does not block the trail.</li> </ol> <p>A trail is blocked if it satisfies any of the following conditions:</p> <ol> <li>There exists a collider on the trail that itself or all of its descendants are not observed.</li> <li>There exists a non-collider on the trail that is observed.</li> </ol> <p>If all possible trails between two nodes \\(X, Y\\) are blocked \\(Z\\), the two nodes are d-separated. D-separated nodes are conditionally independent given the observed nodes, i.e., \\(X \\perp Y \\mid Z\\).</p> <p>Faithfulness assumption</p> <p>Faithfulness assumptions states that the observed conditional independence relations are consistent with the graph structure. That is: observed conditional independence \\(X \\perp Y \\mid Z\\) implies that \\(X\\) and \\(Y\\) are d-separated given \\(Z\\) in the graph.</p>"},{"location":"math/causal-inference/chapter-6/#flows-of-association-and-causation","title":"Flows of Association and Causation","text":"<p>If two nodes are not d-separated, there exists at least one open trail between the nodes.</p> <ul> <li>Association flows along open trails.</li> <li>Causation flows along open paths.</li> </ul> <p>There may be multiple trails and paths between two nodes, each trail provides a flow of association or causation. The overall observed association is affected by all the flows of association and causation. Existence of undesired flows of association may lead to lack of exchangeability and biased estimation of causal effects. If there are no flow of association between two nodes, the association is causal.</p> <p>There two types of flows of association:</p> <ol> <li> <p>Common cause: two nodes are associated because they are both affected by a common cause.</p> <p></p> </li> <li> <p>Condition on common effect: two nodes are associated because they are both affecting an observed common effect.</p> <p></p> </li> </ol> <p>Blocking a trail can be regarded as blocking the flow of association or causation on the graph.</p>"},{"location":"math/causal-inference/chapter-6/#hidden-properties","title":"Hidden Properties","text":"<p>Causal graphs only encode conditional independence relations and flows of causation or association. Some properties or assumptions cannot be represent in causal graphs.</p> <ul> <li>Positivity: Positivity in causal graphs can be represented that all of the edges in the graph are not deterministic.</li> <li>Consistency: Consistency is implicitly assumed in causal graphs.</li> </ul> <p>In a causal graph, the treatment node should lead to well-defined treatment.\\</p>"},{"location":"math/causal-inference/chapter-7/","title":"Confounding","text":"<p>Confounding is a situation in which the effect of an exposure on an outcome is distorted by the presence of another variable that is associated with both the exposure and the outcome. The confounding effect is a non-causal association that leads to biased estimation of causal effect.</p> <p>In a causal graph, the confounding effect is caused by an open backdoor path. A backdoor path is a non-path trail from treatment variable \\(A\\) to outcome variable \\(Y\\) that points to \\(A\\). The simplest structure of confounding contains a variable \\(L\\) pointing to treatment \\(A\\) and outcome \\(Y\\).</p> <p></p> <p>Suppose the causal graph is the true causal graph and satisfies faithfulness.</p>"},{"location":"math/causal-inference/chapter-7/#exchangeability","title":"Exchangeability","text":"<p>Following the definition of backdoor path, the definition of confounding can be alternatively expressed as</p> <p>Alternative definition of confounding</p> <p>Confounding is any systematic bias that can be removed by randomized assignment of treatment \\(A\\) (eliminating the backdoor path).</p> <p>When backdoor criterion is satisfied, exchangeability stands. The backdoor criterion requires on of the following:</p> <ol> <li>No confounding: no common cause of treatment \\(A\\) and outcome \\(Y\\).</li> <li>No unmeasured confounding: common cause of \\(A\\) and \\(Y\\) exists but a subset \\(L\\) of measured (observed) non-descendants of \\(A\\) blocks all the backdoor paths. The corresponding set \\(L\\) is called a sufficient set for confounding adjustment. Variable \\(L\\) is also called a confounder.</li> </ol> <p>Unconditional causal effect can be measured when unconditional exchangeability is satisfied. Conditional causal effect can be measured when conditional exchangeability is satisfied. The following table shows causal graph structures holding different exchangeability. The conditioned variable is \\(L\\).</p> Causal identifiability table Unconditional exchangeable Not unconditional exchangeable Conditional exchangeable Not conditional exchangeable <p>To identify conditional or unconditional causal effect, we can apply standardization on measured variables when conditional or unconditional exchangeability is satisfied. Take the following graph as an example.</p> <p></p> <p>When either \\(L_2\\) or \\(L\\) is observed, neither unconditional nor conditional exchangeability is satisfied. However, when both \\(L_2\\) and \\(L\\) are observed, both unconditional and conditional exchangeability is satisfied. We can identify the following causal effect based on observation.</p> <ol> <li>Conditional causal effect within strata of \\(L\\) and \\(L_2\\): \\(E[Y | A = a, L = l, L_2 = l_2]\\)</li> <li>Unconditional causal effect: \\(E[Y | A = a, L = l, L_2 = l_2]P(L_2 = l_2, L = l)\\)</li> <li>Conditional causal effect within strata of \\(L_2\\): \\(E[Y | A = a, L_2 = l_2, L = l]P(L = l | L_2 = l_2)\\)</li> <li>Conditional causal effect within strata of \\(L\\): \\(E[Y | A = a, L = l, L_2 = l_2]P(L_2 = l_2 | L = l)\\)</li> </ol>"},{"location":"math/causal-inference/chapter-7/#confounders","title":"Confounders","text":"<p>The sufficient set for identifying confounder depends on variables observed or measured. For a causal graph, there may also exist multiple sufficient sets for confounding adjustment.</p>"},{"location":"math/causal-inference/chapter-7/#traditional-confounder","title":"Traditional Confounder","text":"<p>A confounder satisfies the following conditions, which is the definition of a traditional confounder.</p> <ol> <li>The variable associates with the treatment.</li> <li>Condition on the treatment, the variable associates with the outcome.</li> <li>The variable is not on the causal path from the treatment to the outcome.</li> </ol> <p>However, a variable that satisfies the above conditions is not necessarily a confounder. Variable \\(L\\) in the following graph is a traditional confounder. When \\(L\\) is not adjusted, unconditional causal effect can be directly identified, adjusting for \\(L\\) will introduce extra selection bias.</p> <p></p>"},{"location":"math/causal-inference/chapter-7/#surrogate-confounder","title":"Surrogate Confounder","text":"<p>A surrogate confounder is a variable that is the result of an unobserved common cause of the treatment and the outcome. The surrogate confounder itself does not lie on a backdoor path. The following graph shows a surrogate confounder \\(L\\).</p> <p></p> <p>Although \\(L\\) does not lie on the backdoor path, observing distribution of \\(L\\) may provide information about the unobserved common cause \\(U\\). In this case, adjusting for \\(L\\) can partially reduce the confounding bias.</p>"},{"location":"math/causal-inference/chapter-7/#single-world-intervention-graph","title":"Single-world Intervention Graph","text":"<p>A single-world intervention graph (SWIG) is a causal graph that explicitly include the conterfactual variables on the graph. Unlike the causal graph that represents the real world, the SWIG represents a counterfactual world created by an intervention.</p> Causal graph Single-world intervention graph <p>In a SWIG, the original treatment variable \\(A\\) is now regarded as two distinct nodes \\(A\\) and \\(a\\). \\(A\\) encodes the possible value of \\(A\\) under no intervention, \\(a\\) is the value of intervened treatment variable. Any descendant \\(X\\) of \\(A\\) is replaced by \\(X^a\\), which is the value of \\(X\\) under intervention \\(A = a\\). From the SWIG, we can see that after conditioning on \\(L\\), the trail \\(A\\leftarrow L\\leftarrow U\\rightarrow Y^a\\) is blocked, indicating that exchangeability \\(A \\perp Y^a | L\\) holds.</p> <p>Confounder cannot be descendants of treatment</p> <p>A confounder cannot be a descendant of the treatment variable. Suppose \\(L\\) is a descendant of \\(A\\), and \\(L\\) connects to \\(Y\\) through some trail. If the trail is a path from \\(L\\) to \\(Y\\), then the whole path through \\(L\\) is a causal path, adjustment on \\(L\\) will block such path. Otherwise, there should be a collider on the path. If the collider is \\(L\\), adjustment on \\(L\\) will open the non-causal trail, introducing extra selection bias. If the collider is not \\(L\\), adjustment on \\(L\\) have no effect.</p>"},{"location":"math/causal-inference/chapter-7/#confounding-adjustment","title":"Confounding Adjustment","text":"<p>Methods adjust for confounder \\(L\\) can be classified into two categories:</p> <ol> <li>G-methods: methods that adjust for \\(L\\) by conditioning on \\(L\\) - IP weighting, G-estimation, standardization.</li> <li>Stratification methods: methods that adjust for \\(L\\) by stratifying on \\(L\\) - stratification, matching.</li> <li>Other methods: difference-in-difference, instrumental variable, front-door adjustment. These methods does not rely on conditional exchangeability.</li> </ol>"},{"location":"math/causal-inference/chapter-7/#difference-in-difference","title":"Difference-in-Difference","text":"<p>Consider the following causal graph. We want to measure causal effect on treated \\(E(Y^1 | A = 1) - E(Y^0 | A = 1)\\). But there exists some unknown factors \\(U\\) that affects both treatment \\(A\\) and outcome \\(Y\\). For example, \\(U\\) can be the severity of disease, which affects both the treatment decision and the outcome.</p> <p></p> <p>Notice that</p> \\[ \\begin{aligned} &amp;&amp;&amp; \\underbrace{E(Y | A = 1) - E(Y | A = 0)}_{\\text{Measured association}} \\\\ &amp;=&amp;&amp; E(Y^1 | A = 1) - E(Y^0 | A = 0) \\\\ &amp;=&amp;&amp; \\underbrace{(E(Y^1 | A = 1) - E(Y^0 | A = 1))}_{\\text{Causal effect on treated}} + \\underbrace{(E(Y^0 | A = 1) - E(Y^0 | A = 0))}_{\\text{Confounding effect}} \\end{aligned} \\] <p>To estimate confounding effect \\(E(Y^0 | A = 1) - E(Y^0 | A = 0)\\) we can measure a pre-treatment outcome \\(C\\) before the treatment is applied. There is no direct causal relationship between pre-treatment outcome \\(C\\) and treatment \\(A\\), but \\(C\\) and \\(A\\) show some association since backdoor \\(C\\leftarrow U\\rightarrow A\\) presents, that is, \\(E(C | A = 1) - E(C | A = 0) \\neq 0\\). The metric \\(E(C | A = 1) - E(C | A = 0) \\neq 0\\) measures the strength of confounding effect of \\(A\\) on \\(C\\). If the confounding effect of \\(A\\) on \\(Y\\) caused by \\(Y\\leftarrow U\\rightarrow A\\) is the same as the confounding effect of \\(A\\) on \\(C\\) (additive equi-confounding assumption), then the confounding effect of \\(A\\) on \\(Y\\) can be removed by difference-in-difference.</p> \\[ E(Y^0 | A = 1) - E(Y^0 | A = 0) = E(C | A = 1) - E(C | A = 0) \\] <p>The causal effect can be measured in</p> \\[ \\begin{aligned} &amp;&amp;&amp; E(Y^1 | A = 1) - E(Y^0 | A = 1) \\\\ &amp;=&amp;&amp; (E(Y | A = 1) - E(Y | A = 0)) - (E(C | A = 1) - E(C | A = 0)) \\end{aligned} \\]"},{"location":"math/causal-inference/chapter-7/#front-door-criterion","title":"Front-door Criterion","text":"<p>Consider a causal graph with a treatment \\(A\\), a mediator \\(M\\) and an outcome \\(Y\\). The causal effect of \\(A\\) on \\(Y\\) is confounded by an unobserved variable \\(U\\). The mediator fully mediates the effect of \\(A\\) on \\(Y\\), i.e. all the causal path from \\(A\\) to \\(Y\\) passes through \\(M\\).</p> <p></p> <p>First we have</p> \\[ P(Y^a = 1) = \\sum_{m} P(Y^a = 1 | M^a = m) P(M^a = m) \\] <p>\\(M\\) is not confounded, so</p> \\[ P(M^a = m) = P(M = m | A = a) \\] <p>Notice that</p> \\[ P(Y^a = 1 | M^a = m) = P(Y^m = 1) = P(Y = 1 | M = m) \\] <p>and</p> \\[ P(Y = 1 | M = m) = \\sum_{a'} P(Y^m = 1 | M = m, A = a')P(A = a') \\] <p>by conditional exchangeability \\(Y^m \\perp M \\mid A\\). Under this case, \\(P(Y^a)\\) can be identified by</p> \\[ P(Y^a = 1) = \\sum_{m} P(M = m | A = a)\\sum_{a'} P(Y = 1 | M = m, A = a')P(A = a') \\]"},{"location":"math/causal-inference/chapter-8/","title":"Selection bias","text":"<p>Selection bias is caused by extra association caused by only part of the population is selected for analysis. Selection bias is caused by conditioning on a common effect of treatment and outcome, even if the treatment actually has no individual causal effect on the outcome.</p> <p>On the causal graph, selection bias is caused by either condition on a collider or descendant of a collider. In reality, selection bias can both appear in observational studies or randomized experiments since participants may be removed from the study before the outcome is observed. If participants are removed not in random, selection bias is introduced.</p> <p></p> \\[ \\frac{P(Y = 1 | A = 1)}{P(Y = 1 | A = 0)} = \\frac{P(Y^{a = 1} = 1)}{P(Y^{a = 0} = 1)} \\not = \\frac{P(Y = 1 | A = 1, L = 0)}{P(Y = 0 | A = 1, L = 0)} \\] <p>Selection bias and hazard ratio</p> <p>Hazards is defined as the probability of a participant to die at a certain time. Following the definition of risk, the hazard ratio is the same as risk ratio. Consider the following causal graph.</p> <p></p> <p>In the graph, treatment \\(A\\) denote the heart transplant. The outcome \\(Y_1\\) and \\(Y_2\\) denote the death of the patient. Unmeasured variable \\(U\\) affect the overall death rate of the patient. For each time, we can define the associational hazard ratio as</p> \\[ \\begin{aligned}     aRR_{AY_1} &amp;= \\frac{P(Y_1 = 1 | A = 1)}{P(Y_1 = 1 | A = 0)} \\\\     aRR_{AY_2} &amp;= \\frac{P(Y_2 = 1 | A = 1)}{P(Y_2 = 1 | A = 0)} \\end{aligned} \\] <p>However, we can only measure the hazard ratio among the patients who are still alive at that time, that is:</p> \\[ aRR_{AY_2 \\mid Y_1 = 0} = \\frac{P(Y_1 = 1 | A = 1, Y_1 = 0)}{P(Y_1 = 1 | A = 0, Y_1 = 0)} \\] <p>However, condition on \\(Y_1\\) opens a trail \\(A \\ra Y_1 \\la U\\ra Y_2\\). Therefore, unless \\(U\\) is measured, from the data collected we cannot distinguish the existence of the path \\(A \\ra Y_2\\).</p>"},{"location":"math/causal-inference/chapter-8/#selection-without-bias","title":"Selection without Bias","text":"<p>Selection will cause bias within the study, but in some cases such bias can be restricted to some strata of the study. Consider the following causal graph. \\(Y = 0\\) if and only if \\(Y_A = Y_E = Y_O = 0\\).</p> <p></p> <ol> <li>\\(Y = 0\\) is equivalent to \\(Y_A = 0\\) and \\(Y_E = 0\\). In such case, \\(A\\) is independent of \\(E\\).</li> <li>Consider the case when \\(Y = 1\\) and \\(Y_O = 0\\), then \\(Y_A = 0\\) indicates \\(Y_E = 1\\) and vice versa. In such case, \\(A\\) is dependent of \\(E\\).</li> </ol>"},{"location":"math/causal-inference/chapter-8/#adjustment-for-selection-bias","title":"Adjustment for Selection Bias","text":"<p>Assume that positivity holds for \\(C = 0\\) and consistency holds for the analysis. Selection bias arises when the participants are not randomly removed from the study, causing the distribution of remaining participants to be different from the original population, i.e. the joint distribution \\(P'(A, L) = P(A, L | C = 0)\\) is no longer identical to \\(P(A, L)\\).</p> <p>Selection bias is often unavoidable. IP weighting and standardization can be used to adjust for selection bias. The inverse probability weight \\(W^C\\) is defined as</p> \\[ W^C = \\frac{1}{P(C = 0 | \\cdot)} \\] <p>where \\(\\cdot\\) denote all the variable that directly affects \\(C\\). Since we can only observed variables for uncensored (\\(C = 0\\)) individuals, the IP weight only uses \\(C = 0\\). IP weighting assigns different weight to the probability distribution of each pair of \\((A, L)\\), so that the distribution of the weighted sample is identical to the original population.</p> \\[ \\begin{aligned} &amp;&amp;&amp; \\frac{P(A = a, L = l, C = 0)}{P(C = 0 | A = a, L = l)} \\\\ &amp;=&amp;&amp; \\frac{P(A = a, L = l, C = 0)}{P(A = a, L = l, C = 0) / P(A = a, L = l)} \\\\ &amp;=&amp;&amp; P(A = a, L = l) \\end{aligned} \\] <p>Difference in confounding bias and selection bias</p> <p>In confounding bias, IP weighting is applied on the treatment variable \\(A\\), while in selection bias, IP weighting is applied on the censoring variable \\(C\\).</p> <p>When there are measured variable \\(L\\) on the trail through \\(C\\) that is able to block the trail causing selection bias, we can use stratification to adjust for selection bias by conditioning on \\(L\\).</p>"},{"location":"math/causal-inference/chapter-9/","title":"Measurement bias and \"noncausal\" diagrams","text":""},{"location":"math/causal-inference/chapter-9/#measurement-bias","title":"Measurement Bias","text":"<p>Measurement bias is caused by errors in measuring values of variables, i.e. \\(A^* \\not = A\\). Measurement error is defined as the difference between the measured value and the true value of a variable, \\(e_A = A^* - A\\). Taken measurement error into consideration, the causal diagram is modified as follows:</p> <p></p> <p>Measurement error follow two properties:</p> <ol> <li>Indepedence: \\(e_A \\perp e_Y\\).     </li> <li>Nondifferentiality: \\(e_A \\perp Y\\) and \\(e_Y \\perp A\\).     </li> </ol> <p>Lack of either property will bring extra association and lead to bias.</p> <ul> <li>Edge \\(Y\\ra U_A\\) will introduce recall bias.</li> <li>Edge \\(A\\ra U_Y\\) will introduce reverse causation bias.</li> <li>Edge \\(U_A\\la U_{AY}\\ra U_Y\\) will introduce independent measurement error.</li> </ul> <p>Correcting for measurement error usually requires additional validated non-biased samples.</p>"},{"location":"math/causal-inference/chapter-9/#noncausal-diagrams","title":"\"Noncausal\" Diagrams","text":"<p>A causal graph requires that all of the edges in the graph can be interpreted causally, together with well-defined intervention. For graphs with non-causal edges, adjustments might fail to remove bias, as the adjusted variable is not on the true causal path.</p> <p></p>"},{"location":"math/convex-optimization/","title":"\u51f8\u200b\u4f18\u5316","text":""},{"location":"math/convex-optimization/#_2","title":"\u76ee\u5f55","text":"<ul> <li>\u200b\u51f8\u96c6\u200b</li> <li>\u200b\u51f8\u51fd\u6570\u200b</li> </ul>"},{"location":"math/convex-optimization/#notations-and-definitions","title":"Notations and Definitions","text":"<p>\u200b\u672c\u200b\u8282\u200b\u5217\u4e3e\u200b\u51f8\u200b\u4f18\u5316\u200b\u4e2d\u200b\u6240\u7528\u200b\u7b26\u53f7\u200b\u7684\u200b\u8868\u8ff0\u200b\uff0c\u200b\u5e76\u200b\u501f\u6b64\u200b\u56de\u987e\u200b\u76f8\u5173\u200b\u7684\u200b\u6570\u5b66\u200b\u6982\u5ff5\u200b</p>"},{"location":"math/convex-optimization/#_3","title":"\u6570\u57df","text":"<ul> <li>\\(\\mathbb R, \\mathbb N, \\mathbb Z, \\mathbb C\\)\uff1a\u200b\u5b9e\u6570\u200b\u57df\u200b\u3001\u200b\u81ea\u7136\u6570\u200b\u57df\u200b\u3001\u200b\u6574\u6570\u200b\u57df\u200b\u3001\u200b\u590d\u6570\u200b\u57df\u200b</li> <li>\u200b\u4e0b\u6807\u200b \\({}_\\plus, {}_{\\plus\\plus}\\) \u200b\u5728\u200b\u5b9e\u6570\u200b\u57df\u548c\u5b9e\u200b\u5411\u91cf\u200b\u57df\u200b\u4e0a\u200b\u5206\u522b\u200b\u8868\u793a\u200b\u975e\u8d1f\u200b\u3001\u200b\u4e25\u683c\u200b\u6b63\u200b\uff0c\u200b\u5728\u200b\u5b9e\u77e9\u9635\u200b\u57df\u200b\u4e0a\u200b\u8868\u793a\u200b\u534a\u200b\u6b63\u5b9a\u200b\u3001\u200b\u6b63\u5b9a\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/#_4","title":"\u77e9\u9635\u200b\u4e0e\u200b\u5411\u91cf","text":"<p>\\(A\\in \\mathbb R^{m\\times n}\\)\u200b\u8868\u793a\u200b\\(A\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(m\\)\u200b\u884c\u200b\\(n\\)\u200b\u5217\u200b\u7684\u200b\u5b9e\u77e9\u9635\u200b\uff0c\u200b\u5176\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u4e3a\u200b\\(a_{ij}\\)\uff0c\u200b\u67d0\u5217\u200b\uff08\u200b\u6216\u200b\u67d0\u884c\u200b\uff09\u200b\u7684\u200b\u5207\u7247\u200b\u4e3a\u200b\\(A_{\\cdot j}\\)</p> <p>\\(\\boldsymbol x\\in \\mathbb R^m\\)\u200b\u8868\u793a\u200b\\(\\boldsymbol x\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(m\\)\u200b\u7ef4\u7684\u5b9e\u200b\u5411\u91cf\u200b\uff0c\\(\\boldsymbol 1\\)\u200b\u8868\u793a\u200b\u6240\u6709\u200b\u5206\u91cf\u200b\u5747\u200b\u4e3a\u200b\\(1\\)\u200b\u7684\u200b\u5411\u91cf\u200b\u3002</p> <p>\u200b\u65b9\u9635\u200b\u662f\u200b\u884c\u6570\u200b\u4e0e\u200b\u5217\u6570\u200b\u76f8\u7b49\u200b\u7684\u200b\u77e9\u9635\u200b\uff0c\\(A\\in \\mathbb R^{n\\times n}\\)\u3002\u200b\u5bf9\u79f0\u200b\u77e9\u9635\u200b\u6ee1\u8db3\u200b\\(A^\\top = A\\)\uff0c\u200b\u53cd\u5bf9\u200b\u79f0\u200b\u77e9\u9635\u200b\u6ee1\u8db3\u200b\\(A^\\top = - A\\)\u3002\u200b\u6b63\u5b9a\u200b\u77e9\u9635\u200b\u6ee1\u8db3\u200b\\(\\forall x\\in \\mathbb R^n, x^\\top Ax &gt; 0\\)\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(A\\succ 0\\)\uff1b\u200b\u534a\u200b\u6b63\u5b9a\u200b\u77e9\u9635\u200b\u6ee1\u8db3\u200b\\(\\forall x\\in \\mathbb R^n, x^\\top Ax \\geq 0\\)\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(A\\succeq 0\\)\u3002\u200b\u5bf9\u4e8e\u200b\u590d\u200b\u65b9\u9635\u200b\\(A\\in \\mathbb C^{n\\times n}\\)\uff0c\u200b\u5b9a\u4e49\u200b\u5171\u8f6d\u200b\u8f6c\u7f6e\u200b\uff08Hermitian\uff09\u200b\u77e9\u9635\u200b\u4e3a\u200b\\(A^* = \\overline {A^\\top}\\)\u3002\u200b\u5168\u4f53\u200b\\(k\\)\u200b\u9636\u200b\u5bf9\u79f0\u200b\u77e9\u9635\u200b\u7684\u200b\u96c6\u5408\u200b\u4e3a\u200b\\(\\mathbb S^k\\)\u3002</p> <p>\u200b\u77e9\u9635\u7684\u8ff9\u200b\u662f\u200b\u5bf9\u89d2\u7ebf\u200b\u4e0a\u200b\u5404\u4e2a\u200b\u5143\u7d20\u200b\u4e4b\u200b\u548c\u200b</p> \\[ \\mathrm {tr}(A) = \\sum_{i=1}^n a_{ii} \\] \u200b\u8ff9\u200b\u7684\u200b\u6027\u8d28\u200b <ol> <li>\\(\\mathrm {tr}(A) = \\mathrm {tr}(A^\\top)\\)</li> <li>\\(\\mathrm {tr}(A+B) = \\mathrm {tr}(A) + \\mathrm {tr}(B)\\)</li> <li>\\(\\mathrm {tr}(t\\cdot A) = t\\cdot \\mathrm {tr}(A)\\)</li> <li>\\(\\mathrm {tr}(AB) = \\mathrm {tr}(BA)\\)\uff08\u200b\u4e0d\u200b\u8981\u6c42\u200b\\(A, B, \\cdots\\)\u200b\u4e3a\u200b\u65b9\u9635\u200b\uff0c\u200b\u53ea\u200b\u9700\u8981\u200b\u6ee1\u8db3\u200b\u8fd0\u7b97\u200b\u7ed3\u679c\u200b\u4e3a\u200b\u65b9\u9635\u200b\u5373\u53ef\u200b\uff1b\u200b\u5bf9\u4e8e\u200b\u591a\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u4e58\u6cd5\u200b\u987a\u5e8f\u200b\u9700\u8981\u200b\u6ee1\u8db3\u200b\u8f6e\u6362\u200b\u5bf9\u79f0\u6027\u200b\u624d\u80fd\u200b\u4fdd\u8bc1\u200b\u7b49\u5f0f\u200b\u6210\u7acb\u200b\uff09</li> </ol> <p>\u200b\u5b9a\u4e49\u200b\u4e24\u4e2a\u200b\u77e9\u9635\u200b\\(X, Y\\)\u200b\u7684\u200b\u5185\u79ef\u200b\u4e3a\u200b\u5176\u200b\u4e58\u79ef\u200b\u7684\u200b\u8ff9\u200b\\(\\mathrm{tr}(XY)\\)</p>"},{"location":"math/convex-optimization/#_5","title":"\u8303\u6570","text":"<p>\u200b\u8303\u6570\u200b\u662f\u200b\u6ee1\u8db3\u200b\u4ee5\u4e0b\u200b\u6027\u8d28\u200b\u7684\u200b\u51fd\u6570\u200b\\(f\\)</p> <ol> <li>\u200b\u975e\u200b\u8d1f\u6027\u200b\uff1a\\(f(\\boldsymbol{x}) \\geq 0\\)</li> <li>\u200b\u89c4\u8303\u6027\u200b\uff1a\\(f(\\boldsymbol{x}) = 0 \\Leftrightarrow \\boldsymbol x = 0\\)</li> <li>\u200b\u9f50\u6b21\u200b\u6027\u200b\uff1a\\(f(t\\cdot \\boldsymbol{x}) = t\\cdot f(\\boldsymbol{x})\\)</li> <li>\u200b\u4e09\u89d2\u200b\u4e0d\u7b49\u5f0f\u200b\uff1a\\(f(\\boldsymbol x + \\boldsymbol y)\\leq f(\\boldsymbol x) + f(\\boldsymbol y)\\)</li> </ol> \u200b\u5e38\u89c1\u200b\u7684\u200b\u8303\u6570\u200b <ol> <li>1-\u200b\u8303\u6570\u200b\uff1a\\(f(\\boldsymbol{x}) = \\sum_{i=1}^n \\left|x_i\\right|\\)</li> <li>2-\u200b\u8303\u6570\u200b\uff1a\\(f(\\boldsymbol{x}) = \\sqrt{\\sum_{i=1}^n \\left|x_i^2\\right|}\\)</li> <li>\\(\\infty\\)-\u200b\u8303\u6570\u200b\uff1a\\(f(\\boldsymbol{x}) = \\sqrt[\\infty]{\\sum_{i=1}^n \\left|x_i^\\infty\\right|}\\)</li> <li>0-\u200b\u8303\u6570\u200b\uff08\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u5b9a\u4e49\u200b3\uff0c\u200b\u56e0\u6b64\u200b\u4e0d\u662f\u200b\u8303\u6570\u200b\uff09\uff1a\\(f(\\boldsymbol{x}) = \\sum_{i=1}^n \\boldsymbol{1}_{\\{x_i \\not = 0\\}}\\)</li> <li>\\(l_p\\)-\u200b\u8303\u6570\u200b\uff1a\\(f(\\boldsymbol{x}) = \\sqrt[p]{\\sum_{i=1}^n \\left|x_i^p\\right|}\\)</li> <li>Frobenius\u200b\u8303\u6570\u200b\uff1a\\(f(A) = \\sqrt{\\sum_i\\sum_j a_{ij}^2} = \\sqrt{\\mathrm{tr}(A^\\top A)}\\)</li> </ol> <p>\u200b\u8303\u6570\u200b\u7b49\u4ef7\u200b\u5b9a\u7406\u200b\uff1a\u200b\u5bf9\u4e8e\u200b\u4e24\u4e2a\u200b\u8303\u6570\u200b\\(f, g\\)\uff0c\u200b\u5b58\u5728\u200b\u6b63\u5e38\u200b\u6570\u200b\\(c\\)\u200b\u4f7f\u5f97\u200b\\(\\forall \\boldsymbol{x}, f(\\boldsymbol{x}) \\leq cg(\\boldsymbol{x})\\)</p> <p>\u200b\u8bbe\u200b\\(f\\)\u200b\u4e3a\u200b\\(\\mathbb R^n\\)\u200b\u4e0a\u200b\u7684\u200b\u8303\u6570\u200b\uff0c\u200b\u5bf9\u5076\u200b\u8303\u6570\u200b\\(f_*\\)\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b</p> \\[ f_*(\\boldsymbol{z}) = \\sup_{\\boldsymbol{x}} \\{\\boldsymbol{z^\\top x} | f(\\boldsymbol{x}) \\leq 1\\} \\] \u200b\u5bf9\u5076\u200b\u8303\u6570\u200b <ul> <li>\u200b\u5bf9\u5076\u200b\u8303\u6570\u200b\u7684\u200b\u5bf9\u5076\u200b\u8303\u6570\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u539f\u200b\u8303\u6570\u200b\u3002</li> <li> <p>\\(p\\)-\u200b\u8303\u6570\u200b\u7684\u200b\u5bf9\u5076\u200b\u8303\u6570\u200b\u662f\u200b\\(q\\)-\u200b\u8303\u6570\u200b\uff0c\u200b\u6ee1\u8db3\u200b</p> \\[ \\frac{1}{p} + \\frac{1}{q} = 1\\qquad p, q\\geq 1 \\] </li> </ul> <p>\u200b\u6839\u636e\u200b\u8303\u6570\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u7403\u200b\u7684\u200b\u6982\u5ff5\u200b</p> \u200b\u7403\u200b <p>\u200b\u7ed9\u5b9a\u200b\\(\\mathbb R^n\\)\u200b\u7a7a\u95f4\u200b\u4e2d\u200b\u7684\u200b\u8303\u6570\u200b\\(\\Vert\\cdot\\Vert\\)\uff0c\\(B(x, r)\\)\u200b\u8868\u793a\u200b\u8303\u6570\u200b\\(\\Vert \\cdot\\Vert\\)\u200b\u4e2d\u200b\u7684\u200b\uff08\u200b\u5f00\u200b\uff09\u200b\u7403\u200b\uff0c\u200b\u5373\u200b</p> \\[ B(x, r) = \\{y | \\Vert y - x\\Vert &lt; r\\} \\] <p>\\(\\bar B(x, r)\\)\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u95ed\u7403\u200b</p> \\[ \\bar B(x, r) = \\{y | \\Vert y - x\\Vert \\leq r\\} \\]"},{"location":"math/convex-optimization/#_6","title":"\u96c6\u5408","text":"<p>\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(C\\subseteq \\mathbb R^n, x\\in C\\)\uff0c\u200b\u4f7f\u7528\u200b2-\u200b\u8303\u6570\u200b\u5ea6\u91cf\u200b\u8ddd\u79bb\u200b\u3002</p> <ul> <li>\u200b\u82e5\u200b\\(\\exists r \\in \\mathbb R_{\\plus\\plus}, B(x, r)\\subseteq C\\)\uff0c\u200b\u79f0\u200b\\(x\\)\u200b\u4e3a\u200b\u5185\u70b9\u200b\u3002\u200b\u5168\u4f53\u200b\u5185\u70b9\u200b\u6784\u6210\u200b\u96c6\u5408\u200b\u7684\u200b\u5185\u90e8\u200b\\(C^\\circ\\)\uff0c\u200b\u663e\u7136\u200b\\(C^\\circ \\subseteq C\\)\u3002</li> <li>\u200b\u82e5\u200b\\(\\forall r\\in \\mathbb R_{+}, B(x, r)\\cap C \\not = \\varnothing, B(x, r) \\cap C^C \\not = \\varnothing\\)\uff0c\u200b\u79f0\u200b\\(x\\)\u200b\u4e3a\u200b\u8fb9\u754c\u70b9\u200b\u3002\u200b\u5168\u4f53\u200b\u8fb9\u754c\u70b9\u200b\u6784\u6210\u200b\u96c6\u5408\u200b\u7684\u200b\u8fb9\u754c\u200b\\(\\partial C\\)\u3002<ul> <li>\u200b\u6839\u636e\u200b\u8fb9\u754c\u70b9\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u6709\u200b\\(\\partial C = \\partial C^C\\)\u3002</li> </ul> </li> <li>\u200b\u82e5\u200b\\(x\\)\u200b\u4e0d\u662f\u200b\u5185\u70b9\u200b\uff0c\u200b\u5219\u200b\\(x\\)\u200b\u4e3a\u200b\u8fb9\u754c\u70b9\u200b\uff0c\u200b\u5373\u200b\\(\\partial C \\cup C^\\circ = C, \\partial C\\cap C^\\circ = \\varnothing\\)\u3002</li> </ul> <p>\u200b\u82e5\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u70b9\u200b\u90fd\u200b\u662f\u200b\u5185\u70b9\u200b\uff0c\u200b\u5373\u200b\\(C^\\circ = C\\)\uff0c\u200b\u79f0\u200b\\(C\\)\u200b\u4e3a\u200b\u5f00\u96c6\u200b\u3002\u200b\u82e5\u200b\\(C\\)\u200b\u4e0d\u662f\u200b\u5f00\u96c6\u200b\uff0c\u200b\u5219\u200b\\(C\\)\u200b\u4e3a\u200b\u95ed\u96c6\u200b\u3002\u200b\u5f00\u96c6\u200b\u7684\u200b\u8865\u96c6\u200b\u662f\u200b\u95ed\u96c6\u200b\u3002</p> <p>\u200b\u5b9a\u4e49\u200b\u95ed\u5305\u200b\u4e3a\u200b</p> \\[ \\mathbf{cl}C = \\{x \\in \\mathbb R^n| \\forall r\\in \\mathbb R_{++}, B(x, r)\\cap C\\not = \\varnothing\\} \\] <p>\u200b\u76f4\u89c2\u200b\u4e0a\u200b\uff0c\u200b\u95ed\u5305\u200b\u53ef\u4ee5\u200b\u7406\u89e3\u200b\u4e3a\u200b\\(C\\cup \\partial C^C\\)\uff0c\u200b\u5373\u200b\u96c6\u5408\u200b\u4e0e\u5176\u200b\u8865\u96c6\u200b\u7684\u200b\u8fb9\u754c\u200b\u7684\u200b\u5e76\u200b\u3002</p> <p>\\(\\varnothing, \\mathbb R^n\\)\u200b\u65e2\u200b\u662f\u200b\u5f00\u96c6\u200b\u4e5f\u200b\u662f\u200b\u95ed\u96c6\u200b\uff1a</p> <ul> <li>\\(\\varnothing\\)\u200b\u7684\u200b\u95ed\u5305\u200b\u4ecd\u200b\u4e3a\u200b\\(\\varnothing\\)\uff0c\u200b\u6240\u4ee5\u200b\u662f\u200b\u95ed\u96c6\u200b\uff1b\u200b\u7136\u800c\u200b\\(\\varnothing^\\circ = \\varnothing\\)\uff0c\u200b\u6240\u4ee5\u200b\u662f\u200b\u5f00\u96c6\u200b\u3002</li> <li>\\(\\varnothing^C = \\mathbb R^n\\)\uff0c\u200b\u6240\u4ee5\u200b\\(\\mathbb R^n\\)\u200b\u65e2\u200b\u662f\u200b\u5f00\u96c6\u200b\u4e5f\u200b\u662f\u200b\u95ed\u96c6\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-function/","title":"\u51f8\u51fd\u6570\u200b\u53ca\u5176\u200b\u6027\u8d28","text":"<p>\u200b\u51f8\u51fd\u6570\u200b\u662f\u200b\u4e00\u7c7b\u200b\u51fd\u6570\u200b\u4e0a\u65b9\u200b\uff08\u200b\u4e0a\u5883\u200b\u56fe\u200b\uff0cepigraph\uff09\u200b\u662f\u200b\u51f8\u96c6\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5373\u200b\u51fd\u6570\u200b\u66f2\u7ebf\u200b\u4e0a\u200b\u4efb\u610f\u200b\u4e24\u70b9\u200b\u8fde\u6210\u200b\u7684\u200b\u7ebf\u6bb5\u200b\u90fd\u200b\u5728\u200b\u51fd\u6570\u200b\u66f2\u7ebf\u200b\u7684\u200b\u4e0a\u65b9\u200b\u3002</p> <p>\u200b\u82e5\u200b\u51fd\u6570\u200b\\(f: \\mathbb R^n\\rightarrow \\mathbb R\\)\u200b\u6ee1\u8db3\u200b\u7434\u751f\u200b\u4e0d\u7b49\u5f0f\u200b\uff1a</p> \\[ f(\\theta x + (1 - \\theta)y)\\leq \\theta f(x) + (1 - \\theta)f(y), 0\\leq \\theta\\leq 1 \\] <p>\uff0c\u200b\u5219\u200b\u79f0\u200b\\(f\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\u3002</p> \u200b\u7434\u751f\u200b\u4e0d\u7b49\u5f0f\u200b\u7684\u200b\u6269\u5c55\u200b <p>\u200b\u7434\u751f\u200b\u4e0d\u7b49\u5f0f\u200b\u53ef\u4ee5\u200b\u6269\u5c55\u200b\u5230\u200b\u66f4\u200b\u591a\u70b9\u200b\u7684\u200b\u51f8\u200b\u7ec4\u5408\u200b\u4e0a\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u51f8\u51fd\u6570\u200b\\(f\\)\uff0c\u200b\u8bbe\u200b\\(\\theta_1, \\ldots, \\theta_k\\geq 0, \\theta_1 + \\cdots + \\theta_k = 1\\)\uff0c\u200b\u5219\u200b</p> \\[ f(\\theta_1 x_1 + \\cdots + \\theta_k x_k)\\leq \\theta_1 f(x_1) + \\cdots + \\theta_k f(x_k) \\] <p>\u200b\u8be5\u5f0f\u200b\u53ef\u4ee5\u200b\u8fdb\u4e00\u6b65\u200b\u62d3\u5c55\u200b\u81f3\u200b\u7ea7\u6570\u200b\u3001\u200b\u79ef\u5206\u200b\u3001\u200b\u671f\u671b\u200b\u7b49\u200b\u5f62\u5f0f\u200b\u3002</p> <p>\u200b\u82e5\u200b\\(f(\\theta x + (1 - \\theta)y)&lt; \\theta f(x) + (1 - \\theta)f(y), 0\\leq \\theta\\leq 1\\)\uff0c\u200b\u5219\u200b\u4e3a\u200b\u4e25\u683c\u200b\u51f8\u51fd\u6570\u200b\u3002\u200b\u51f8\u51fd\u6570\u200b\u7684\u200b\u4e0a\u200b\u5883\u56fe\u200b\uff08\\(\\mathbf{epi}f\\)\uff09\u200b\u662f\u200b\u51f8\u96c6\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\\(-f\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\u51f9\u200b\u51fd\u6570\u200b\uff0c\u200b\u82e5\u200b\\(-f\\)\u200b\u662f\u200b\u4e25\u683c\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\u4e25\u683c\u200b\u51f9\u200b\u51fd\u6570\u200b\u3002\u200b\u51f9\u200b\u51fd\u6570\u200b\u7684\u200b\u4e9a\u56fe\u200b\uff08\\(\\mathbf{hypo}f\\)\uff09\u200b\u662f\u200b\u51f8\u96c6\u200b\u3002</p> <p>\u200b\u6240\u6709\u200b\u7684\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\u90fd\u200b\u65e2\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u4e5f\u200b\u662f\u200b\u51f9\u200b\u51fd\u6570\u200b\u3002</p> <p>\\(f\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u5728\u200b\u5b9a\u4e49\u57df\u200b\u5185\u200b\u4efb\u4f55\u200b\u76f4\u7ebf\u200b\u4e0a\u200b\u90fd\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5373\u200b</p> \\[ \\forall v,\\forall x, g(t; v, x) = f(x + tv) \\] <p>\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002</p> <p>\u200b\u8bb0\u200b\\(f\\)\u200b\u7684\u200b\u5ef6\u4f38\u200b\\(\\tilde f\\)\u200b\u4e3a\u200b\u5206\u6bb5\u200b\u51fd\u6570\u200b</p> \\[ \\tilde f(x) = \\left\\{\\begin{aligned}     &amp; f(x) &amp; x\\in \\mathbf{dom}f \\\\     &amp; \\infty &amp; x\\not\\in \\mathbf{dom}f \\end{aligned}\\right. \\] <p>\u200b\u7c7b\u4f3c\u200b\u5730\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u51f9\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5176\u200b\u5b9a\u4e49\u57df\u200b\u5916\u200b\u7684\u200b\u90e8\u5206\u200b\u5ef6\u4f38\u200b\u81f3\u200b\\(-\\infty\\)\u3002 \u200b\u5728\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e2d\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u793a\u6027\u200b\u51fd\u6570\u200b\\(\\tilde I_C(x) = 0, x\\in C\\)\u200b\u6765\u200b\u5c06\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u89e3\u200b\u9650\u5b9a\u200b\u5230\u200b\u51f8\u96c6\u200b\\(C\\)\u200b\u4e0a\u200b\u3002\\(\\tilde I_C(x)\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002</p> <p>\u200b\u5176\u4ed6\u200b\u7684\u200b\u51f8\u51fd\u6570\u200b\u8fd8\u6709\u200b</p> <ul> <li>\u200b\u6307\u6570\u51fd\u6570\u200b\\(f: \\mathbb R\\rightarrow\\mathbb R = e^{ax}, a\\in \\mathbb R\\)</li> <li>\u200b\u5e42\u51fd\u6570\u200b\\(f: \\mathbb R_{ + + }\\rightarrow\\mathbb R = x^a, a\\geq 1\\)\u200b\u6216\u200b\\(a\\leq 0\\)</li> <li>\u200b\u7edd\u5bf9\u503c\u200b\u5e42\u51fd\u6570\u200b\\(f: \\mathbb R\\rightarrow\\mathbb R = |x|^p, p\\geq 1\\)</li> <li>\u200b\u8d1f\u200b\u5bf9\u6570\u51fd\u6570\u200b\\(f: \\mathbb R_{\\plus\\plus}\\rightarrow \\mathbb R = -\\log x\\)</li> <li>\u200b\u8d1f\u71b5\u200b\\(f: \\mathbb R_{\\plus\\plus}\\rightarrow \\mathbb R = x\\log x\\)</li> <li>\u200b\u8303\u6570\u200b\uff08\u200b\u4e09\u89d2\u200b\u4e0d\u7b49\u5f0f\u200b\uff09<ul> <li>\u200b\u6700\u5927\u503c\u200b\u51fd\u6570\u200b\u76f8\u5f53\u4e8e\u200b\u65e0\u7a77\u200b\u9636\u200b\u8303\u6570\u200b\\(\\Vert\\cdot\\Vert_\\infty\\)\uff0c\u200b\u56e0\u6b64\u200b\u4e5f\u200b\u662f\u200b\u51f8\u200b\u7684\u200b\u3002</li> </ul> </li> <li>\u200b\u6307\u6570\u200b\u548c\u200b\u7684\u200b\u5bf9\u6570\u200b\\(f: \\mathbb R^n\\rightarrow \\mathbb R = \\log \\sum_{i=1}^n e^{x_i}\\)</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(\\alpha\\)\uff0c\u200b\u51f8\u51fd\u6570\u200b\u7684\u200b\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u200b\\(\\{x\\in \\mathbf{dom} f | f(x) \\leq \\alpha\\}\\)\u200b\u662f\u200b\u51f8\u96c6\u200b\u3002\u200b\u6216\u8005\u8bf4\u200b\uff0c\u200b\u51f8\u51fd\u6570\u200b\u7684\u200b\u7b49\u9ad8\u7ebf\u200b\u662f\u200b\u51f8\u96c6\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-function/#_2","title":"\u4e00\u9636\u200b\u6761\u4ef6\u200b\u4e0e\u200b\u4e8c\u9636\u200b\u6761\u4ef6","text":""},{"location":"math/convex-optimization/convex-function/#_3","title":"\u4e00\u9636\u200b\u6761\u4ef6","text":"<p>\u200b\u82e5\u200b\\(f\\)\u200b\u7684\u200b\u5b9a\u4e49\u57df\u200b\u4e3a\u200b\u51f8\u200b\u96c6\u4e14\u200b\u53ef\u5fae\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u7b49\u4ef7\u200b\u4e8e\u200b</p> \\[ f(y) \\geq f(x) + (\\nabla f(x))^\\top (y - x) \\] <p>\u200b\u51f8\u51fd\u6570\u200b\u7684\u200b\u4e00\u9636\u200b\u8fd1\u4f3c\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u5168\u5c40\u200b\u4e0b\u200b\u4f30\u8ba1\u200b\u3002</p> <p></p> <p>\u200b\u4e00\u9636\u200b\u6761\u4ef6\u200b\u540c\u6837\u200b\u4e5f\u200b\u80fd\u200b\u7528\u6765\u200b\u53cd\u6620\u200b\u4e25\u683c\u200b\u51f8\u51fd\u6570\u200b\u3001\u200b\u51f9\u200b\u51fd\u6570\u200b\u548c\u200b\u4e25\u683c\u200b\u51f9\u200b\u51fd\u6570\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-function/#_4","title":"\u4e8c\u9636\u200b\u6761\u4ef6","text":"<p>\u200b\u82e5\u200b\\(f\\)\u200b\u7684\u200b\u5b9a\u4e49\u57df\u200b\u4e3a\u200b\u51f8\u200b\u96c6\u4e14\u200b\u4e8c\u9636\u200b\u53ef\u5fae\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u7b49\u4ef7\u200b\u4e8e\u200b</p> \\[ \\nabla ^2f(x)\\succeq 0 \\] <p>\u200b\u4e8c\u9636\u200b\u6761\u4ef6\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u6765\u200b\u53cd\u6620\u200b\u4e25\u683c\u200b\u51f8\u51fd\u6570\u200b\u3001\u200b\u51f9\u200b\u51fd\u6570\u200b\u548c\u200b\u4e25\u683c\u200b\u51f9\u200b\u51fd\u6570\u200b\u3002\u200b\u6ce8\u610f\u200b\u4e25\u683c\u200b\u51f8\u51fd\u6570\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u6ee1\u8db3\u200b\\(\\nabla^2 f(x)\\succ 0\\)\uff0c\u200b\u4e25\u683c\u200b\u51f9\u200b\u51fd\u6570\u200b\u540c\u7406\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-function/#_5","title":"\u4fdd\u51f8\u200b\u8fd0\u7b97","text":"<p>\u200b\u5bf9\u4e8e\u200b\u51fd\u6570\u200b\\(f, g, f_1, \\ldots f_k: \\bbR^n\\rightarrow \\bbR\\)\uff08\u200b\u5982\u672a\u200b\u6307\u660e\u200b\uff0c\u200b\u5219\u200b\u5047\u5b9a\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff09\uff0c\u200b\u4ee5\u4e0b\u200b\u51fd\u6570\u200b\u4ecd\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff1a</p> <ul> <li> <p>\u200b\u51f8\u51fd\u6570\u200b\u7684\u200b\u7ebf\u6027\u7ec4\u5408\u200b\uff1a\\(f'(x) = \\sum_{i=1}^k \\theta_i f_i(x)\\)</p> \u200b\u6269\u5c55\u200b\u5230\u200b\u79ef\u5206\u200b\u5f62\u5f0f\u200b <p>\u200b\u8bbe\u200b\\(f(x, y)\\)\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(y\\)\uff0c\u200b\u5173\u4e8e\u200b\\(x\\)\u200b\u90fd\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u51fd\u6570\u200b\\(w(y)\\)\u200b\u6ee1\u8db3\u200b\\(w(y)\\geq 0\\)\uff0c\u200b\u5219\u200b</p> \\[ g(x) = \\int_{\\mathcal A} w(y)f(x, y)\\dd y \\] <p>\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\u3002</p> </li> <li> <p>\u200b\u4eff\u5c04\u53d8\u6362\u200b\uff1a\\(f'(x): \\bbR^m\\rightarrow \\bbR = f(Ax + b)\\)\uff0c\u200b\u5176\u4e2d\u200b\\(A\\in \\bbR^{m\\times n}, b\\in \\bbR^n\\)</p> </li> <li> <p>\u200b\u9010\u70b9\u200b\u6700\u5927\u200b\uff1a\\(f'(x) = \\max\\{f_1(x), \\ldots, f_k(x)\\}\\)</p> <p>\u200b\u9010\u70b9\u200b\u6700\u5927\u200b\u53ef\u4ee5\u200b\u7406\u89e3\u200b\u4e3a\u200b\u4e00\u7cfb\u5217\u200b\u51fd\u6570\u200b\u4e0a\u5883\u200b\u56fe\u200b\u7684\u200b\u4ea4\u96c6\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\\(f'\\)\uff0c\u200b\u6709\u200b</p> \\[ \\mathbf{epi} f' = \\bigcap_{i=1}^k \\mathbf{epi} f_k \\] <p>\u200b\u6839\u636e\u200b\u51f8\u96c6\u200b\u7684\u200b\u4fdd\u51f8\u200b\u6027\u8d28\u200b\u53ef\u77e5\u200b\uff0c\\(\\mathbf{epi} f'\\)\u200b\u662f\u200b\u51f8\u96c6\u200b\u3002</p> \u200b\u6269\u5c55\u200b\u5230\u200b\u65e0\u9650\u200b\u4e2a\u200b\u51fd\u6570\u200b\u7684\u200b\u5f62\u5f0f\u200b <p>\u200b\u8bbe\u200b\\(f(x, y)\\)\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(y\\)\uff0c\u200b\u5173\u4e8e\u200b\\(x\\)\u200b\u90fd\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b</p> \\[ g(x) = \\sup_{y\\in \\calA} f(x, y) \\] <p>\u200b\u4e5f\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\u3002</p> </li> <li> <p>\u200b\u590d\u5408\u200b\u51fd\u6570\u200b\uff1a\\(g\\circ f\\)</p> <ul> <li>\u200b\u5f53\u200b\\(g: \\bbR\\rightarrow \\bbR\\)\u200b\u65f6\u200b\uff0c\\(f' = g\\circ f = g(f(x))\\)\u200b\u6ee1\u8db3\u200b<ol> <li>\u200b\u82e5\u200b\\(g\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u4e14\u200b\u975e\u51cf\u200b\uff0c\\(f\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f'\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b</li> <li>\u200b\u82e5\u200b\\(g\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u4e14\u200b\u975e\u589e\u200b\uff0c\\(f\\)\u200b\u4e3a\u200b\u51f9\u200b\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f'\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b</li> </ol> </li> <li>\u200b\u5f53\u200b\\(g: \\bbR^k\\rightarrow \\bbR\\)\u200b\u65f6\u200b\uff0c\\(f' = g\\circ f = g(f_1(x), \\ldots, f_k(x))\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002<ol> <li>\u200b\u82e5\u200b\\(g\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u4e14\u200b\u5728\u200b\u5404\u4e2a\u200b\u5206\u91cf\u200b\u4e0a\u200b\u975e\u51cf\u200b\uff0c\\(f_i\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f'\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b</li> <li>\u200b\u82e5\u200b\\(g\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u4e14\u200b\u5728\u200b\u5404\u4e2a\u200b\u5206\u91cf\u200b\u4e0a\u200b\u975e\u589e\u200b\uff0c\\(f_i\\)\u200b\u4e3a\u200b\u51f9\u200b\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f'\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b</li> </ol> </li> </ul> </li> <li> <p>\u200b\u6700\u5c0f\u5316\u200b\uff1a\u200b\u82e5\u200b\\(f(x, y): \\bbR^m\\times \\bbR^n\\rightarrow \\bbR\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\u5bf9\u4e8e\u200b\u975e\u7a7a\u200b\u51f8\u96c6\u200b\\(C\\)\uff0c\\(\\inf_{y\\in C}f(x, y)\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\u3002</p> </li> <li>\u200b\u900f\u89c6\u200b\u51fd\u6570\u200b\uff1a\\(g(x, t) = tf(x / t)\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-function/#_6","title":"\u5171\u8f6d\u200b\u51fd\u6570","text":"<p>\u200b\u51fd\u6570\u200b\\(f: \\bbR^n\\rightarrow \\bbR\\)\u200b\u7684\u200b\u5171\u8f6d\u200b\u51fd\u6570\u200b\\(f^*: \\bbR^n\\rightarrow\\bbR\\)\u200b\u4e3a\u200b</p> \\[ f^*(y)\\triangleq \\sup_{x\\in \\mathbf{dom}f} (y^\\top x - f(x)) \\] <p>\u200b\u5171\u8f6d\u200b\u51fd\u6570\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002\\(f^{**}\\)\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u548c\u200b\\(f\\)\u200b\u76f8\u7b49\u200b\uff0c\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\u65f6\u200b\uff0c\\(f^{**} = f\\)\u3002</p> \u200b\u5171\u8f6d\u200b\u51fd\u6570\u200b\u7684\u200b\u51f8\u6027\u200b <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\u51fd\u6570\u200b\\(f: \\bbR^n\\rightarrow \\bbR\\)\u200b\u548c\u200b\\(x\\in \\bbR^n\\)\uff0c\\(g(y, x) = y^\\top x - f(x)\\)\u200b\u4e3a\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\uff08\u200b\u4e5f\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff09</p> <p>\\(f^*(y)\\)\u200b\u4e3a\u200b\\(g(y, x)\\)\u200b\u5173\u4e8e\u200b\\(x\\)\u200b\u7684\u200b\u9010\u70b9\u200b\u4e0a\u200b\u786e\u754c\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\u3002</p> <p>\u200b\u6839\u636e\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u51fd\u6570\u200b\\(f\\)\u200b\u53ca\u5176\u200b\u5171\u8f6d\u200b\u6ee1\u8db3\u200b</p> \\[ f(x) + f^*(y)\\geq x^\\top y \\] \u200b\u4e8c\u6b21\u200b\u51fd\u6570\u200b\u7684\u200b\u5171\u8f6d\u200b <p>\u200b\u5bf9\u4e8e\u200b\u4e8c\u6b21\u200b\u51fd\u6570\u200b\\(f(x) = \\frac{1}{2} x^\\top Qx, Q\\in \\bbS_{\\plus\\plus}^n\\)\uff0c\u200b\u5176\u200b\u5171\u8f6d\u200b\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ f^*(y) = \\frac{1}{2} y^\\top Q^{-1}y \\] \u200b\u8303\u6570\u200b\u7684\u200b\u5171\u8f6d\u200b <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\u8303\u6570\u200b\\(\\Vert\\cdot\\Vert\\)\uff0c\u200b\u5176\u200b\u5bf9\u5076\u200b\u8303\u6570\u200b\u4e3a\u200b\\(\\Vert\\cdot\\Vert_*\\)\uff0c\u200b\u5219\u200b\\(f(x) = \\Vert x\\Vert\\)\u200b\u7684\u200b\u5171\u8f6d\u200b\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ f^*(y) = \\left\\{ \\begin{aligned}     &amp; 0 &amp; \\Vert y\\Vert_* \\leq 1 \\\\     &amp; \\infty &amp; \\text{otherwise} \\end{aligned} \\right. \\] <p>\u200b\u51fd\u6570\u200b\u548c\u200b\u7684\u200b\u5171\u8f6d\u200b\u7b49\u4e8e\u200b\u5176\u200b\u5171\u8f6d\u200b\u7684\u200b\u548c\u200b</p> \\[ (f + g)^* = f^* + g^* \\] <p>\u200b\u5bf9\u4e8e\u200b\u53ef\u5fae\u200b\u51fd\u6570\u200b\\(f\\)\uff0c\u200b\u6709\u200b</p> \\[ \\begin{gather*}     f(y) = z^\\top \\nabla f(z) - f(z) \\\\     \\nabla f(z) = y \\end{gather*} \\] \u200b\u7ebf\u6027\u53d8\u6362\u200b\u7684\u200b\u5171\u8f6d\u200b <p>\u200b\u8bbe\u200b\\(g(x) = f(Ax + b)\\)\uff0c\u200b\u5176\u4e2d\u200b\\(A\\in \\bbR^{n\\times n}\\)\u200b\u4e14\u200b\u6ee1\u200b\u79e9\u200b\uff0c\u200b\u5219\u200b</p> \\[ \\begin{aligned} g^*(y) &amp;= \\sup_{x} (x^\\top y - f(Ax + b)) \\\\ &amp;= \\sup_{x} [A^{-1}[(Ax + b) - b]]^\\top y - f(Ax + b) \\\\ &amp;= \\sup_x [(A^{-1}(Ax + b))^\\top y - f(Ax + b)] - b^\\top A^{-\\top}y \\\\ &amp;= \\sup_x [(Ax+b)^\\top (A^{-\\top}y) - f(Ax + b)] - b^\\top A^{-\\top}y \\\\ &amp;= f^*(A^{-\\top}y) - b^\\top A^{-\\top}y \\end{aligned} \\] <p>\u200b\u8bbe\u200b\\(h(x) = af(x) + b\\)\uff0c\u200b\u5219\u200b\uff1a</p> \\[ \\begin{aligned} h^*(y) &amp;= \\sup_x y^\\top x - af(x) - b \\\\ &amp;= \\sup_{x} a(a^{-1}y^\\top x - f(x)) - b \\\\ &amp;= af^*(a^{-1}y) - b \\end{aligned} \\]"},{"location":"math/convex-optimization/convex-function/#_7","title":"\u62df\u51f8\u51fd\u6570","text":"<p>\u200b\u82e5\u200b\u51fd\u6570\u200b\\(f\\)\u200b\u7684\u200b\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u200b\\(\\{x|f(x)\\leq \\alpha\\}\\)\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(\\alpha\\)\u200b\u90fd\u200b\u4e3a\u200b\u51f8\u96c6\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\u62df\u51f8\u51fd\u6570\u200b\u3002\u200b\u82e5\u200b\\(-f\\)\u200b\u4e3a\u200b\u62df\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\u62df\u51f9\u51fd\u6570\u200b\u3002\u200b\u82e5\u200b\\(f\\)\u200b\u65e2\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\uff0c\u200b\u53c8\u200b\u662f\u200b\u62df\u51f9\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\u62df\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\u3002</p> <p>\u200b\u6240\u6709\u200b\u7684\u200b\u51f8\u51fd\u6570\u200b\u90fd\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\u3002\u200b\u4f46\u200b\u62df\u51f8\u51fd\u6570\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff0c\\(\\nabla f(x_0) = 0\\not \\Rightarrow f(x)\\geq f(x_0)\\)</p>"},{"location":"math/convex-optimization/convex-function/#_8","title":"\u6027\u8d28","text":"<p>\u200b\u62df\u51f8\u51fd\u6570\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ul> <li> <p>\u200b\u51fd\u6570\u200b\\(f\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\mathbf{dom} f\\)\u200b\u662f\u200b\u51f8\u96c6\u200b\uff0c\u200b\u4e14\u200b\\(f\\)\u200b\u6ee1\u8db3\u200b</p> \\[ f(\\theta x + (1 - \\theta)y)\\leq \\max\\{f(x), f(y)\\} \\] </li> <li> <p>\u200b\u5bf9\u4e8e\u200b\u5b9a\u4e49\u200b\u5728\u200b\\(\\bbR\\)\u200b\u4e0a\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\\(f\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u5176\u4e2d\u200b\u4e4b\u4e00\u200b</p> <ul> <li>\\(f\\)\u200b\u662f\u200b\u5355\u8c03\u200b\u51fd\u6570\u200b</li> <li>\\(\\exists x_0\\)\uff0c\u200b\u5bf9\u4e8e\u200b\\(x\\geq x_0, f\\)\u200b\u975e\u51cf\u200b\uff0c\\(x\\leq x_0, f\\)\u200b\u975e\u589e\u200b</li> </ul> </li> <li>\u200b\u5bf9\u4e8e\u200b\\(\\bbR\\rightarrow \\bbR\\)\u200b\u4e0a\u200b\u7684\u200b\u4efb\u610f\u200b\u5355\u8c03\u200b\u51fd\u6570\u200b\\(f\\)\uff0c\\(-f\\)\u200b\u4e5f\u200b\u662f\u200b\u5355\u8c03\u200b\u51fd\u6570\u200b\uff0c\u200b\u56e0\u6b64\u200b\\(f\\)\u200b\u662f\u200b\u62df\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-function/#_9","title":"\u4e00\u9636\u200b\u6761\u4ef6\u200b\u4e0e\u200b\u4e8c\u9636\u200b\u6761\u4ef6","text":"<p>\uff08\u200b\u4e00\u9636\u200b\u6761\u4ef6\u200b\uff09\u200b\u5bf9\u4e8e\u200b\u5b9a\u4e49\u200b\u5728\u200b\u51f8\u96c6\u200b\u4e0a\u200b\u7684\u200b\u53ef\u5fae\u200b\u51fd\u6570\u200b\\(f: \\bbR^n\\rightarrow \\bbR, f\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u6ee1\u8db3\u200b</p> \\[ \\forall x, y\\in \\mathbf{dom}f, f(y) \\leq f(x)\\Longrightarrow \\nabla f(x)^\\top (y - x)\\leq 0 \\] \u200b\u4e00\u9636\u200b\u6761\u4ef6\u200b\u7684\u200b\u51e0\u4f55\u200b\u89e3\u91ca\u200b <p>\u200b\u6839\u636e\u200b\u62df\u51f8\u51fd\u6570\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(x\\)\uff0c\u200b\u5176\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u200b\\(S_x = \\{y | f(y)\\leq f(x)\\}\\)\u200b\u5747\u200b\u4e3a\u200b\u51f8\u96c6\u200b\u3002</p> <p>\u200b\u6839\u636e\u200b\u51f8\u96c6\u200b\u7684\u200b\u6027\u8d28\u200b\uff0c\u200b\u4efb\u4f55\u200b\u51f8\u96c6\u200b\u90fd\u200b\u80fd\u200b\u8868\u793a\u200b\u6210\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u5bf9\u5e94\u200b\u534a\u200b\u7a7a\u95f4\u200b\u7684\u200b\u4ea4\u96c6\u200b\u3002\u200b\u96c6\u5408\u200b\\(\\{y | \\nabla f(x)^\\top (y - x)\\leq 0\\}\\)\u200b\u662f\u200b\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u200b\\(S_x\\)\u200b\u5728\u200b\\(x\\)\u200b\u5904\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u5bf9\u5e94\u200b\u7684\u200b\u534a\u200b\u7a7a\u95f4\u200b\u3002</p> <p>\uff08\u200b\u4e8c\u9636\u200b\u6761\u4ef6\u200b\uff09\u200b\u5bf9\u4e8e\u200b\u5b9a\u4e49\u200b\u5728\u200b\u51f8\u96c6\u200b\u4e0a\u200b\u7684\u200b\u4e8c\u6b21\u200b\u53ef\u5fae\u200b\u51fd\u6570\u200b\\(f: \\bbR^n\\rightarrow \\bbR, f\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u6ee1\u8db3\u200b</p> \\[ \\forall x\\in \\mathbf{dom} f, \\nabla f(x) = 0\\Longrightarrow \\nabla^2 f(x)\\succeq 0 \\] <p>\u200b\u5373\u200b\\(f\\)\u200b\u5728\u200b\u68af\u5ea6\u200b\u4e3a\u200b\\(0\\)\u200b\u7684\u200b\u5730\u65b9\u200b\uff0c\u200b\u4e8c\u9636\u200b\u68af\u5ea6\u200b\u6b63\u5b9a\u200b\uff08\u200b\u975e\u8d1f\u200b\uff09\u3002</p> <p>\u200b\u62df\u51f8\u51fd\u6570\u200b\u7684\u200b\u4e8c\u9636\u200b\u68af\u5ea6\u200b\u6700\u200b\u591a\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u8d1f\u200b\u7279\u5f81\u503c\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-function/#_10","title":"\u4fdd\u62df\u200b\u51f8\u200b\u8fd0\u7b97","text":"<p>\u200b\u8bbe\u200b\\(f, g, f_1,\\ldots, f_k\\)\u200b\u4e3a\u200b\u62df\u51f8\u51fd\u6570\u200b\u3002\u200b\u5982\u672a\u200b\u6307\u660e\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u51fd\u6570\u200b\u5b9a\u4e49\u200b\u5728\u200b\\(\\bbR^n \\rightarrow \\bbR\\)\u200b\u4e0a\u200b\u3002\u200b\u4ee5\u4e0b\u200b\u51fd\u6570\u200b\u4e3a\u200b\u62df\u51f8\u51fd\u6570\u200b\uff1a</p> <ul> <li>\u200b\u975e\u8d1f\u200b\u52a0\u6743\u200b\u6700\u5927\u503c\u200b\uff1a\u200b\u8bbe\u200b\\(w_1, \\ldots, w_k\\geq 0, f' = \\max\\{w_1f_1(x), \\ldots, w_kf_k(x)\\}\\)\u200b\u4e3a\u200b\u62df\u51f8\u51fd\u6570\u200b\u3002</li> <li>\u200b\u590d\u5408\u200b\u51fd\u6570\u200b\uff1a\\(f' = g\\circ f\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5176\u4e2d\u200b\\(g: \\bbR\\rightarrow \\bbR\\)\u200b\u5355\u8c03\u200b\u975e\u51cf\u200b\u3002</li> <li>\u200b\u6700\u5c0f\u503c\u200b\uff1a\\(f(x, y): \\bbR^m\\times \\bbR^n\\rightarrow \\bbR\\)\u200b\u5173\u4e8e\u200b\\(x, y\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\u51f8\u96c6\u200b\\(C, g(x) = \\inf_{y\\in C}f(x, y)\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-function/#-","title":"\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570","text":"<p>\u200b\u5b9a\u4e49\u200b\\(\\log 0 = -\\infty\\)\u3002\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u51fd\u6570\u200b\\(f: \\bbR^n\\rightarrow \\bbR_{\\plus}\\)\uff0c\u200b\u6709\u200b\\(\\log f\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\u79f0\u200b\\(f\\)\u200b\u4e3a\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\u3002\u200b\u82e5\u200b\\(1 / f\\)\u200b\u4e3a\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\u3002</p> <p>\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff08\u200b\u6307\u6570\u51fd\u6570\u200b\\(e^x\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff09\uff0c\u200b\u975e\u8d1f\u200b\u7684\u200b\u51f9\u200b\u51fd\u6570\u200b\u662f\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\uff0c\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\u662f\u200b\u62df\u51f9\u51fd\u6570\u200b\u3002</p> \u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\u7684\u200b\u5145\u8981\u6761\u4ef6\u200b <p>\\(f: \\bbR^n \\rightarrow \\bbR\\)\u200b\u662f\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b</p> \\[ f(\\theta x + (1 - \\theta)y)\\geq f^\\theta(x)f^{1 - \\theta}(y) \\] <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(x, y\\in \\mathbf{dom} f, 0\\leq\\theta\\leq 1\\)\u200b\u6210\u7acb\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u4e8c\u6b21\u200b\u53ef\u5fae\u200b\u7684\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\\(f\\)\uff0c\u200b\u6709\u200b</p> \\[ f(x)\\nabla^2 f(x)\\preceq \\nabla f(x)^\\top \\nabla f(x) \\] <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(x\\in \\mathbf{dom} f\\)\u200b\u6210\u7acb\u200b\uff0c\u200b\u53cd\u4e4b\u4ea6\u7136\u200b\u3002</p> <p>\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\u4e4b\u200b\u548c\u200b\u4ecd\u200b\u662f\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\uff08\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\u4e4b\u200b\u548c\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\uff09\uff0c\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u200b\uff08\u200b\u51f9\u200b\uff09\u200b\u51fd\u6570\u200b\u7684\u200b\u4e58\u79ef\u200b\u662f\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u200b\uff08\u200b\u51f9\u200b\uff09\u200b\u51fd\u6570\u200b\u3002</p> \u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\u4e4b\u200b\u548c\u200b\u6027\u8d28\u200b\u7684\u200b\u62d3\u5c55\u200b <p>\u200b\u8bbe\u200b\\(f(x, y): \\bbR^n\\times \\bbR^m\\rightarrow \\bbR\\)\u200b\u6ee1\u8db3\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(y\\)\uff0c\\(f(x, y)\\)\u200b\u5173\u4e8e\u200b\\(x\\)\u200b\u662f\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b</p> \\[ \\int_{y} f(x, y)\\dd y \\] <p>\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-function/#_11","title":"\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u5b9a\u4e49\u200b\u7684\u200b\u51f8\u6027","text":"<p>\u200b\u8bbe\u200b\\(K\\)\u200b\u4e3a\u200b\u6b63\u5e38\u200b\u9525\u200b\uff0c\u200b\u5b9a\u4e49\u200b\\(f(x)\\)\u200b\u7684\u200b\u589e\u51cf\u200b\u6027\u200b\u5982\u4e0b\u200b\u3002</p> <ul> <li>\\(f\\)\u200b\u4e3a\u200b\\(K\\)-\u200b\u589e\u200b\uff0c\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u6ee1\u8db3\u200b\\(x\\succeq_K y, x\\not = y\\Longrightarrow f(x)\\succ_K f(y)\\)\uff0c\u200b\u82e5\u200b\\(-f\\)\u200b\u4e3a\u200b\\(K\\)-\u200b\u589e\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\\(K\\)-\u200b\u975e\u200b\u589e\u51fd\u6570\u200b\u3002</li> <li>\\(f\\)\u200b\u4e3a\u200b\\(K\\)-\u200b\u975e\u51cf\u200b\uff0c\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u6ee1\u8db3\u200b\\(x\\succeq_K y\\Longrightarrow f(x)\\succeq_K f(y)\\)</li> </ul> <p>\\(f\\)\u200b\u662f\u200b\\(K\\)-\u200b\u975e\u51cf\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\nabla f\\succeq_{K^*} 0\\)\uff0c\\(f\\)\u200b\u662f\u200b\\(K\\)-\u200b\u589e\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\nabla f\\succ_{K^*} 0\\)\u3002</p> <p>\\(f\\)\u200b\u662f\u200b\\(K\\)-\u200b\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\forall x, y, \\theta f(x) + (1 - \\theta) f(y)\\succeq_K f(\\theta x + (1 - \\theta) y)\\)</p> <p>\\(f\\)\u200b\u662f\u200b\u4e25\u683c\u200b\\(K\\)-\u200b\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\forall x\\not = y, \\theta f(x) + (1 - \\theta) f(y)\\succ_K f(\\theta x + (1 - \\theta) y)\\)</p> <p>\\(f\\)\u200b\u662f\u200b\\(K\\)-\u200b\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\forall w\\succeq_{K^*} 0, w^\\top f(x)\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff1b\\(f\\)\u200b\u662f\u200b\u4e25\u683c\u200b\\(K\\)-\u200b\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\forall w\\succeq_{K^*} 0, w\\not = 0, w^\\top f\\)\u200b\u662f\u200b\u4e25\u683c\u200b\u51f8\u51fd\u6570\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/","title":"\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898","text":"<p>\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u662f\u200b\u6307\u200b\u5728\u200b\u4e00\u5b9a\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff08\u200b\u7b49\u5f0f\u200b\u6216\u200b\u4e0d\u7b49\u5f0f\u200b\uff09\u200b\u4e0b\u200b\u4f7f\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u53d6\u5f97\u200b\u6700\u5927\u200b\uff08\u200b\u6700\u5c0f\u200b\uff09\u200b\u503c\u200b\u7684\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_2","title":"\u4e00\u822c\u200b\u4f18\u5316\u200b\u95ee\u9898","text":"<p>\\(\\eqref{1}\\)\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p> \\[ \\label{1} \\optim{\\min}{f_0(x)}{\\cases{     \\begin{aligned}     &amp; f_i(x)\\leq 0 &amp; i = \\oneto m\\\\     &amp; h_i(x) = 0 &amp; i = \\oneto p     \\end{aligned} }} \\] <p>\u200b\u5f0f\u200b\\(\\eqref{1}\\)\u200b\u4e2d\u200b\uff0c\\(x\\in \\bbR^n\\)\u200b\u4e3a\u200b\u4f18\u5316\u200b\u53d8\u91cf\u200b\uff0c\\(f_0(x): \\bbR^n \\ra \\bbR\\)\u200b\u4e3a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\uff0c\u200b\u4e0d\u7b49\u5f0f\u200b\\(f_i(x)\\leq 0\\)\u200b\u79f0\u4e3a\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\uff0c\\(h_0(x) = 0\\)\u200b\u79f0\u4e3a\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u3002\\(m = p = 0\\)\u200b\u7684\u200b\u95ee\u9898\u200b\u79f0\u4e3a\u200b\u65e0\u7ea6\u675f\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u548c\u200b\u5168\u4f53\u200b\u7ea6\u675f\u200b\u51fd\u6570\u200b\u7684\u200b\u5b9a\u4e49\u57df\u200b\u4ea4\u96c6\u200b\u662f\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u5b9a\u4e49\u57df\u200b\\(\\calD\\)\uff0c\u200b\u5f53\u200b\u5b58\u5728\u200b\\(x\\)\u200b\u6ee1\u8db3\u200b\u6240\u6709\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u65f6\u200b\uff0c\u200b\u79f0\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u662f\u200b\u53ef\u884c\u200b\u7684\u200b\uff0c\u200b\u5426\u5219\u200b\u662f\u200b\u4e0d\u200b\u53ef\u884c\u200b\u7684\u200b\u3002\u200b\u5b9a\u4e49\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u53ef\u884c\u200b\u57df\u200b\u4e3a\u200b\u5168\u4f53\u200b\u53ef\u884c\u200b\u70b9\u200b\u7684\u200b\u96c6\u5408\u200b\u3002</p> \\[ X = \\{x | f_i(x)\\leq 0, h_j(x) = 0, i = \\oneto m, j = \\oneto p\\} \\] <p>\u200b\u5bf9\u4e8e\u200b\u6700\u5c0f\u5316\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u5176\u200b\u6700\u4f18\u200b\u503c\u4e3a\u200b</p> \\[ p^\\ast = \\inf\\{f_0(x) | x\\in X\\} \\] <p>\u200b\u53ef\u884c\u200b\u57df\u200b\u4e3a\u200b\u7a7a\u96c6\u200b\u7684\u200b\u60c5\u51b5\u200b</p> <p>\u200b\u6211\u4eec\u200b\u5141\u8bb8\u200b\\(p^*\\)\u200b\u53d6\u200b\\(\\pm\\infty\\)\uff0c\u200b\u5e76\u4e14\u200b\u5b9a\u4e49\u200b</p> \\[ \\begin{aligned} \\inf\\varnothing &amp;= \\infty \\\\ \\sup\\varnothing &amp;= -\\infty \\end{aligned} \\] <p>\u200b\u5982\u679c\u200b\u5b58\u5728\u200b\u4e00\u7ec4\u200b\u53ef\u884c\u200b\u89e3\u200b\\(x_1, \\ldots, x_k\\)\u200b\u6ee1\u8db3\u200b\\(k\\ra\\infty\\)\u200b\u65f6\u200b\\(f_0(x_k)\\ra -\\infty\\)\uff0c\u200b\u79f0\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u65e0\u200b\u4e0b\u754c\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u5b58\u5728\u200b\\(x^*\\in X, f_0(x^*) = p^*\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\\(x^*\\)\u200b\u4e3a\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u5168\u4f53\u200b\u6700\u4f18\u200b\u89e3\u200b\u7684\u200b\u96c6\u5408\u200b\u4e3a\u200b\u6700\u4f18\u200b\u96c6\u200b\u3002\u200b\u82e5\u200b\u6700\u4f18\u200b\u96c6\u975e\u7a7a\u200b\uff0c\u200b\u79f0\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u53ef\u89e3\u200b\u3002\u200b\u5f53\u200b\u95ee\u9898\u200b\u4e0d\u53ef\u200b\u89e3\u65f6\u200b\uff0c\u200b\u96c6\u5408\u200b\\(\\{x | x\\in X, f_0(x) \\leq p^* + \\varepsilon\\}\\)\u200b\u79f0\u4e3a\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\\(\\varepsilon\\)-\u200b\u6b21\u4f18\u200b\u96c6\u200b\u3002</p> <p>\u200b\u6700\u4f18\u200b\u503c\u200b\u4e0d\u53ef\u200b\u8fbe\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b</p> <p>\u200b\u8bbe\u200b\u7ea6\u675f\u200b\u95ee\u9898\u200b\u4e3a\u200b\\(\\eqref{2}\\)\uff0c\u200b\u4ee5\u4e0b\u200b\u95ee\u9898\u200b\u90fd\u200b\u662f\u200b\u6700\u4f18\u200b\u503c\u200b\u4e0d\u53ef\u200b\u8fbe\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\optim{\\min}{f_0(x)}{x &gt; 0} \\label{2} \\] <ul> <li>\u200b\u5f53\u200b\\(f_0(x) = 1/x\\)\u200b\u65f6\u200b\uff0c\\(p^* = 0\\)\uff0c\u200b\u4f46\u200b\u6700\u4f18\u200b\u96c6\u4e3a\u200b\u7a7a\u200b\u3002</li> <li>\u200b\u5f53\u200b\\(f_0(x) = -\\log x\\)\u200b\u65f6\u200b\uff0c\\(p^* = -\\infty\\)\uff0c\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u65e0\u200b\u4e0b\u754c\u200b\u3002</li> </ul> <p>\u200b\u5982\u679c\u200b\\(x_0\\)\u200b\u6ee1\u8db3\u200b\\(\\eqref{3}\\)\uff0c\u200b\u79f0\u4e3a\u200b\u5c40\u90e8\u200b\u6700\u4f18\u200b\u3002</p> \\[ \\begin{gathered}     &amp; x_0\\in X \\\\     &amp; \\forall z\\in X, \\Vert z - x_0\\Vert_2\\leq R, f_0(z)\\geq f_0(x_0) \\end{gathered} \\label{3} \\] <p>\u200b\u82e5\u200b\\(f_i(x^*) = 0\\)\uff0c\u200b\u79f0\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\\(f_i\\)\u200b\u5728\u200b\u6700\u4f18\u200b\u503c\u5904\u200b\u8d77\u200b\u4f5c\u7528\u200b\uff0c\u200b\u5426\u5219\u200b\u4e0d\u8d77\u4f5c\u7528\u200b\u3002\u200b\u5982\u679c\u200b\u4ece\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e2d\u200b\u5220\u9664\u200b\u67d0\u4e2a\u200b\u7ea6\u675f\u200b\u4e0d\u4f1a\u200b\u5f71\u54cd\u200b\u53ef\u884c\u200b\u57df\u200b\uff0c\u200b\u79f0\u200b\u7ea6\u675f\u200b\u5197\u4f59\u200b\u3002</p> <p>\u200b\u53ef\u884c\u6027\u200b\u95ee\u9898\u200b</p> <p>\u200b\u5f53\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\\(f_0(x) = 0\\)\u200b\u65f6\u200b\uff0c\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u662f\u200b\u4e00\u4e2a\u200b\u53ef\u884c\u6027\u200b\u95ee\u9898\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b</p> \\[ \\begin{gathered} p^* = -\\infty \\Lolra X = \\varnothing \\\\ p^* = 0 \\Lolra X \\not = \\varnothing \\end{gathered} \\] <p>\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\uff1a</p> <ul> <li>\u200b\u7ea6\u675f\u200b\u7684\u200b\u53f3\u7aef\u200b\u9879\u4e3a\u200b\u96f6\u200b</li> <li>\u200b\u4f18\u5316\u200b\u76ee\u6807\u200b\u4e3a\u200b\u6781\u5c0f\u200b\u5316\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b</li> <li>\u200b\u9664\u4e86\u200b\u975e\u8d1f\u200b\u7ea6\u675f\u200b\u4ee5\u5916\u200b\uff0c\u200b\u4e0d\u200b\u5305\u542b\u200b\\(\\geq\\)\u200b\u7ea6\u675f\u200b</li> </ul>"},{"location":"math/convex-optimization/convex-optimization/#_3","title":"\u95ee\u9898\u200b\u7684\u200b\u7b49\u4ef7\u200b\u53d8\u6362","text":"<p>\u200b\u4e00\u4e9b\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u6027\u8d28\u200b\u53ef\u80fd\u200b\u4e0d\u591f\u200b\u5f3a\u200b\u5bfc\u81f4\u200b\u96be\u4ee5\u200b\u8fdb\u884c\u200b\u4f18\u5316\u200b\uff0c\u200b\u6b64\u65f6\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4e00\u4e9b\u200b\u7b49\u4ef7\u200b\u53d8\u6362\u200b\uff0c\u200b\u5c06\u200b\u539f\u672c\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u8f6c\u53d8\u200b\u4e3a\u200b\u6027\u8d28\u200b\u8f83\u5f3a\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_4","title":"\u53d8\u91cf\u200b\u4ee3\u6362","text":"<p>\u200b\u8bbe\u200b\u51fd\u6570\u200b\\(\\phi: \\bbR^n\\ra\\bbR^n\\)\u200b\u662f\u200b\u53cc\u5c04\u200b\uff0c\u200b\u4e14\u200b\u51fd\u6570\u200b\u7684\u200b\u503c\u57df\u200b\u5305\u542b\u200b\u95ee\u9898\u200b\u7684\u200b\u5b9a\u4e49\u57df\u200b\uff0c\u200b\u5219\u200b\u5982\u4e0b\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e0e\u200b\u539f\u200b\u95ee\u9898\u200b\\(\\eqref{1}\\)\u200b\u7b49\u4ef7\u200b\uff1a</p> \\[ \\optim{\\min}{f_0(\\phi(z))}{\\cases{     \\begin{aligned}         &amp; f_i(\\phi(z))\\leq 0 &amp; i = \\oneto m \\\\         &amp; h_i(\\phi(z)) = 0 &amp; i = \\oneto p     \\end{aligned} }} \\]"},{"location":"math/convex-optimization/convex-optimization/#_5","title":"\u51fd\u6570\u200b\u4ee3\u6362","text":"<p>\u200b\u8bbe\u200b\u51fd\u6570\u200b\\(\\psi_0: \\bbR\\ra\\bbR\\)\u200b\u5355\u8c03\u200b\u9012\u589e\u200b\uff0c\\(\\psi_{1}, \\ldots, \\psi_{m}\\)\u200b\u6ee1\u8db3\u200b\\(u\\leq 0\\Lora \\psi_{i}(u) = 0\\)\uff0c\\(\\phi_{1}, \\ldots, \\psi_{p}\\)\u200b\u6ee1\u8db3\u200b\\(u = 0\\Lora \\phi_i(u) = 0\\)\uff0c\u200b\u5219\u200b\u5982\u4e0b\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e0e\u200b\u539f\u200b\u95ee\u9898\u200b\\(\\eqref{1}\\)\u200b\u7b49\u4ef7\u200b\uff1a</p> \\[ \\optim{\\min}{\\psi_0(f_0(x))}{\\cases{     \\begin{aligned}         &amp; \\psi_i(f_i(z))\\leq 0 &amp; i = \\oneto m \\\\         &amp; \\phi_i(h_i(z)) = 0 &amp; i = \\oneto p     \\end{aligned} }} \\]"},{"location":"math/convex-optimization/convex-optimization/#_6","title":"\u6d88\u9664\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f","text":"<p>\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u52a0\u5165\u200b\u975e\u8d1f\u200b\u53d8\u91cf\u200b\u7684\u200b\u65b9\u5f0f\u200b\u5c06\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u548c\u200b\u975e\u8d1f\u200b\u7ea6\u675f\u200b\u3002</p> \\[ \\optim{\\min}{f_0(x)}{\\cases{     \\begin{aligned}         &amp; s_i\\geq 0 &amp; i = \\oneto m \\\\         &amp; f_i(x) + s_i 0 &amp; i = \\oneto m \\\\         &amp; h_i(x) = 0 &amp; i = \\oneto p     \\end{aligned} }} \\] <p>\u200b\u79f0\u200b\\(s_i\\)\u200b\u4e3a\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b\uff0c\u200b\u6b64\u65f6\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u5171\u6709\u200b\\(n + m\\)\u200b\u4e2a\u200b\u4f18\u5316\u200b\u53d8\u91cf\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_7","title":"\u6d88\u9664\u200b\u7b49\u5f0f\u200b\u7ea6\u675f","text":"<p>\u200b\u8bbe\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u7684\u200b\u89e3\u96c6\u200b\u53ef\u4ee5\u200b\u7528\u200b\u53cc\u5c04\u200b\\(\\phi: \\bbR^k\\ra\\bbR^n\\)\u200b\u8868\u793a\u200b\uff0c\u200b\u5219\u200b\u539f\u200b\u95ee\u9898\u200b\\(\\eqref{1}\\)\u200b\u4e0e\u200b\u5982\u4e0b\u200b\u95ee\u9898\u200b\u7b49\u4ef7\u200b\uff1a</p> \\[ \\optim{\\min}{f_0(\\phi(z))}{f_i(\\phi(z))\\leq 0} \\] <p>\u200b\u5bf9\u4e8e\u200b\u7ebf\u6027\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\\(Ax - b = 0\\)\uff0c\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\\(\\phi(z) = Fz + x_0\\)\uff0c\u200b\u5176\u4e2d\u200b\\(x_0\\)\u200b\u4e3a\u200b\\(Ax - b = 0\\)\u200b\u7684\u200b\u4e00\u4e2a\u200b\u7279\u89e3\u200b\uff0c\\(F\\)\u200b\u4e3a\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u7684\u200b\u96f6\u7a7a\u95f4\u200b\u7684\u200b\u4e00\u7ec4\u200b\u57fa\u200b\u6784\u6210\u200b\u7684\u200b\u77e9\u9635\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_8","title":"\u4f18\u5316\u200b\u90e8\u5206\u200b\u53d8\u91cf","text":"<p>\u200b\u6ce8\u610f\u200b\u5230\u200b\\(\\inf_{x}\\inf{y} f(x, y) = \\inf_{(x, y)}f(x, y) = \\inf_{y}\\inf_x f(x, y)\\)\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u5148\u200b\u4f18\u5316\u200b\u4e00\u90e8\u5206\u200b\u53d8\u91cf\u200b\uff0c\u200b\u518d\u200b\u4f18\u5316\u200b\u53e6\u200b\u4e00\u90e8\u5206\u200b\u53d8\u91cf\u200b\uff0c\u200b\u7531\u6b64\u200b\u5f97\u5230\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e0e\u200b\u539f\u200b\u95ee\u9898\u200b\\(\\eqref{1}\\)\u200b\u7b49\u4ef7\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_9","title":"\u4e0a\u5883\u200b\u56fe\u200b\u95ee\u9898","text":"<p>\u200b\u5982\u4e0b\u200b\u95ee\u9898\u200b\u4e0e\u200b\u539f\u200b\u95ee\u9898\u200b\\(\\eqref{1}\\)\u200b\u7b49\u4ef7\u200b\uff1a</p> \\[ \\optim{\\min}{t}{\\cases{     \\begin{aligned}         &amp; f_0(x) - t\\leq 0 \\\\         &amp; f_i(x)\\leq 0 &amp; i = \\oneto m \\\\         &amp; h_i(x) = 0 &amp; i= \\oneto p     \\end{aligned} }} \\]"},{"location":"math/convex-optimization/convex-optimization/#_10","title":"\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u53ca\u5176\u200b\u6027\u8d28","text":"<p>\\(\\eqref{4}\\)\u200b\u662f\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\optim{\\min}{f_0(x)}{\\cases{     \\begin{aligned}         &amp; f_i(x)\\leq 0 &amp; i = \\oneto m \\\\         &amp; A^\\top x = b     \\end{aligned} }} \\label{4} \\] <p>\u200b\u5e76\u4e14\u200b\uff0c\u200b\u6709\u200b</p> <ul> <li>\\(A\\in \\bbR^{p\\times n}, b\\in \\bbR^p\\)</li> <li>\\(f_i, i\\in 0, \\ldots, m\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002</li> </ul> <p>\u200b\u5982\u679c\u200b\\(f_0(x)\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\u4e3a\u200b\u62df\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_11","title":"\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u6027\u8d28","text":"<p>\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ul> <li>\u200b\u53ef\u884c\u200b\u57df\u200b\u662f\u200b\u51f8\u96c6\u200b</li> <li>\u200b\u4efb\u610f\u200b\u5c40\u90e8\u200b\u6700\u4f18\u200b\u89e3\u90fd\u200b\u662f\u200b\u5168\u5c40\u200b\u6700\u4f18\u200b\u89e3\u200b</li> <li>\u200b\u5982\u679c\u200b\\(f_0\\)\u200b\u53ef\u5fae\u200b\uff0c\u200b\u5219\u200b\\(\\forall x\\in X, \\nabla f_0(x)^\\top (x - x^*)\\geq 0\\)<ul> <li>\u200b\u5bf9\u4e8e\u200b\u65e0\u7ea6\u675f\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u65b9\u7a0b\u200b\\(\\nabla f_0(x) = 0\\)\u200b\u7684\u200b\u89e3\u5373\u200b\u4e3a\u200b\u6700\u4f18\u200b</li> <li>\u200b\u5bf9\u4e8e\u200b\u4ec5\u200b\u5305\u542b\u200b\u7ebf\u6027\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\\(Ax - b = 0\\)\u200b\u7684\u200b\u95ee\u9898\u200b\uff0c\u200b\u65b9\u7a0b\u200b\\(\\nabla f_0(x)^\\top v = 0, v\\in \\ker A\\)\u200b\u7684\u200b\u89e3\u4e3a\u200b\u6700\u4f18\u200b</li> <li>\u200b\u5bf9\u4e8e\u200b\u4ec5\u200b\u5305\u542b\u200b\u7b26\u53f7\u200b\u7ea6\u675f\u200b\\(x\\succeq 0\\)\u200b\u7684\u200b\u95ee\u9898\u200b\uff0c\\(\\nabla f_0(x)\\succeq 0, \\nabla f_0(x)^\\top x = 0\\)\u200b\u7684\u200b\u89e3\u5373\u200b\u4e3a\u200b\u6700\u4f18\u200b\uff08\u200b\u89e3\u200b\u548c\u200b\u6b64\u5904\u200b\u7684\u200b\u68af\u5ea6\u200b\u81f3\u5c11\u200b\u6709\u200b\u4e00\u4e2a\u200b\u5206\u91cf\u200b\u4e3a\u200b\u96f6\u200b\uff09\u3002</li> </ul> </li> </ul>"},{"location":"math/convex-optimization/convex-optimization/#_12","title":"\u4fdd\u51f8\u200b\u8fd0\u7b97","text":"<p>\u200b\u5bf9\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u8fdb\u884c\u200b\u5982\u4e0b\u200b\u53d8\u6362\u200b\uff0c\u200b\u5f97\u5230\u200b\u7684\u200b\u95ee\u9898\u200b\u4ecd\u7136\u200b\u662f\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b</p> <ul> <li>\u200b\u6d88\u9664\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b</li> <li>\u200b\u5f15\u5165\u200b\uff08\u200b\u7ebf\u6027\u200b\uff09\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b</li> <li>\u200b\u5f15\u5165\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b</li> <li>\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u4e0a\u200b\u5883\u56fe\u200b\u95ee\u9898\u200b</li> <li>\u200b\u5206\u6b65\u200b\u4f18\u5316\u200b\u53d8\u91cf\u200b</li> </ul>"},{"location":"math/convex-optimization/convex-optimization/#_13","title":"\u62df\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u6027\u8d28","text":"<p>\u200b\u62df\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u5b58\u5728\u200b\uff08\u200b\u4e0d\u662f\u200b\u5168\u5c40\u200b\u6700\u4f18\u200b\u7684\u200b\uff09\u200b\u5c40\u90e8\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u53ef\u5fae\u200b\u7684\u200b\u62df\u51f8\u51fd\u6570\u200b\\(f_0\\)\uff0c\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\u4ecd\u7136\u200b\u6210\u7acb\u200b\uff1a</p> \\[ \\forall x\\in X, \\nabla f_0(x)^\\top (x - x^*)\\geq 0 \\] <p>\u200b\u4f46\u200b\u4f7f\u5f97\u200b\\(\\forall x\\in X, \\nabla f_0(x)^\\top (x - y)\\geq 0\\)\u200b\u6210\u7acb\u200b\u7684\u200b\\(y\\)\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u5168\u5c40\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002\u200b\u62df\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u89e3\u200b\u7684\u200b\u5145\u8981\u6761\u4ef6\u200b\u4e3a\u200b</p> \\[ \\nabla f_0(x)^\\top (y - x) &gt; 0, y\\in X - \\{x\\} \\] <p>\u200b\u62df\u51f8\u51fd\u6570\u200b\u7684\u200b\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u662f\u200b\u51f8\u96c6\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e00\u7ec4\u200b\u51f8\u51fd\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u4e0d\u7b49\u5f0f\u200b\u6761\u4ef6\u200b\u8868\u793a\u200b\u5176\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u200b\u3002\u200b\u7531\u6b64\u200b\uff0c\u200b\u62df\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u53ef\u4ee5\u200b\u5bf9\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u8fdb\u884c\u200b\u4e8c\u5206\u200b\u4ee5\u200b\u6784\u9020\u200b\u76f8\u5e94\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u6c42\u89e3\u200b\u6700\u4f18\u200b\u503c\u200b\\(p^*\\)\u3002\u200b\u5bf9\u4e8e\u200b\u5f62\u200b\u5982\u200b\\(\\eqref{4}\\)\u200b\u7684\u200b\u62df\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u6784\u9020\u200b\u53ef\u884c\u6027\u200b\u95ee\u9898\u200b\\(\\eqref{5}\\)\uff1a</p> \\[ \\optim{\\min}{0}{\\cases{     \\begin{aligned}         &amp; \\phi(x; t)\\leq 0 \\\\         &amp; f_i(x)\\leq 0 &amp; i = \\oneto m \\\\         &amp; Ax=b     \\end{aligned} }} \\label{5} \\] <p>\u200b\u5176\u4e2d\u200b\\(\\phi(x, t): \\bbR^n\\times \\bbR\\ra\\bbR\\)\u200b\u662f\u200b\u6ee1\u8db3\u200b\\(f_0(x) \\leq t\\Lolra \\phi(x, t)\\leq 0\\)\u200b\u7684\u200b\u51f8\u51fd\u6570\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u5bf9\u4e8e\u200b\u67d0\u4e2a\u200b\\(t\\)\uff0c\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\\(\\eqref{5}\\)\u200b\u53ef\u884c\u200b\uff0c\u200b\u5219\u200b\u5728\u200b\\(f_0(x) \\geq t\\)\u200b\u7684\u200b\u533a\u95f4\u200b\u5185\u200b\u5bfb\u627e\u200b\u6700\u4f18\u200b\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u5728\u200b\\(f_0(x) &lt; t\\)\u200b\u7684\u200b\u533a\u95f4\u200b\u5185\u200b\u5bfb\u627e\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#-","title":"\u51e0\u4f55\u200b\u89c4\u5212\u200b - \u200b\u8f6c\u5316\u200b\u4e3a\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898","text":"<p>\u200b\u5f62\u200b\u5982\u200b\\(\\eqref{5}\\)\u200b\u7684\u200b\u51fd\u6570\u200b\u79f0\u4e3a\u200b\u5355\u9879\u5f0f\u200b\uff0c\u200b\u5355\u9879\u5f0f\u200b\u7684\u200b\u548c\u200b\u79f0\u4e3a\u200b\u6b63\u9879\u200b\u5f0f\u200b\u3002</p> \\[ f(x) = c\\prod_{i=1}^n x_i^{a_i} \\label{5} \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\uff0c\\(x\\in \\bbR^n_{\\plus\\plus}, c &gt; 0\\)\u3002</p> <p>\u200b\u5f62\u200b\u5982\u200b\\(\\eqref{6}\\)\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e3a\u200b\u51e0\u4f55\u200b\u89c4\u5212\u200b\uff08GP\uff09</p> \\[ \\optim{\\min}{f_0(x)}{\\cases{     \\begin{aligned}         &amp; f_i(x)\\leq 1 &amp; i = \\oneto m \\\\         &amp; h_i(x) = 1 &amp; i = \\oneto p     \\end{aligned} }} \\label{6} \\] <p>\u200b\u5176\u4e2d\u200b\\(f_i\\)\u200b\u4e3a\u200b\u6b63\u200b\u9879\u5f0f\u200b\uff0c\\(h_i\\)\u200b\u4e3a\u200b\u5355\u9879\u5f0f\u200b\u3002</p> <p>\u200b\u51e0\u4f55\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u4e0d\u662f\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u4f46\u662f\u200b\u53ef\u4ee5\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u7b49\u4ef7\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_14","title":"\u4e00\u4e9b\u200b\u5e38\u89c1\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898","text":""},{"location":"math/convex-optimization/convex-optimization/#_15","title":"\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898","text":"<p>\u200b\u5982\u679c\u200b\\(\\eqref{4}\\)\u200b\u4e2d\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u548c\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u90fd\u200b\u662f\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\u65f6\u200b\uff0c\u200b\u6b64\u65f6\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e3a\u200b\u7ebf\u6027\u89c4\u5212\u200b\uff08LP\uff09\u3002</p> \\[ \\optim{\\min}{c^\\top x + d}{\\cases{     \\begin{aligned}         &amp; Gx\\preceq h \\\\         &amp; Ax = b     \\end{aligned} }} \\] <p>\u200b\u6d88\u9664\u200b\u7ebf\u6027\u89c4\u5212\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e2d\u200b\u7684\u200b\u5e38\u6570\u200b\u9879\u200b\u3001\u200b\u5f15\u5165\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b\u4ee5\u200b\u6d88\u9664\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u3001\u200b\u4f7f\u7528\u200b\u975e\u8d1f\u200b\u53d8\u91cf\u200b\u8868\u793a\u200b\u65e0\u7ea6\u675f\u200b\u53d8\u91cf\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\uff1a</p> \\[ \\optim{\\min}{c^\\top x}{\\cases{     \\begin{aligned}         &amp; A^\\top x = b \\\\         &amp; x\\succeq 0     \\end{aligned} }} \\label{7} \\] <p>\u200b\u5982\u679c\u200b\\(\\eqref{4}\\)\u200b\u4e2d\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u662f\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\u4e4b\u200b\u6bd4\u200b\uff0c\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u65f6\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\uff0c\u200b\u6b64\u65f6\u200b\u5f97\u5230\u200b\u7ebf\u6027\u200b\u5206\u5f0f\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u3002</p> \\[ \\optim{\\min}{\\frac{c^\\top x + d}{e^\\top x + f}}{\\cases{     \\begin{aligned}         &amp; Gx\\preceq h \\\\         &amp; Ax = b     \\end{aligned} }} \\] <p>\u200b\u7ebf\u6027\u200b\u5206\u5f0f\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u662f\u200b\u62df\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8f6c\u5316\u200b\u5982\u4e0b\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b</p> \\[ \\optim{\\min}{c^\\top y + dz}{\\cases{     \\begin{aligned}         &amp; Gy - hz\\preceq 0 \\\\         &amp; Ay - bz = 0 \\\\         &amp; e^\\top y + fz = 1 \\\\         &amp; z\\geq 0     \\end{aligned} }} \\] <p>\u200b\u7531\u4e8e\u200b\u4e00\u7ec4\u200b\u62df\u51f8\u51fd\u6570\u200b\u7684\u200b\u6700\u5927\u503c\u200b\u4ecd\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u7ebf\u6027\u200b\u5206\u5f0f\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u66ff\u6362\u200b\u4e3a\u200b\u4e00\u200b\u7ec4\u5206\u200b\u5f0f\u200b\u51fd\u6570\u200b\u7684\u200b\u6700\u5927\u503c\u200b\uff0c\u200b\u6b64\u65f6\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4ecd\u7136\u200b\u662f\u200b\u62df\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_16","title":"\u4e8c\u6b21\u200b\u4f18\u5316\u200b\u95ee\u9898","text":"<p>\u200b\u5982\u679c\u200b\\(\\eqref{4}\\)\u200b\u4e2d\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u548c\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u90fd\u200b\u662f\u200b\u51f8\u200b\u4e8c\u6b21\u200b\u578b\u200b\uff0c\u200b\u6b64\u65f6\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e3a\u200b\u4e8c\u6b21\u200b\u7ea6\u675f\u200b\u4e8c\u6b21\u200b\u89c4\u5212\u200b\uff08QCQP\uff09\u200b\u95ee\u9898\u200b\u3002</p> \\[ \\optim{\\min}{\\frac 12x^\\top P_0x + q_0^\\top x + r}{\\cases{     \\begin{aligned}         &amp; \\frac 12x^\\top P_ix + q_i^\\top x + r_i\\leq 0 &amp; i = \\oneto m \\\\         &amp; Ax = b     \\end{aligned} }} \\label{8} \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\\(P_i\\in \\bbS^n_\\plus, q_i\\in \\bbR^n, A\\in \\bbR^{p\\times n}\\)\u3002</p> <p>\u200b\u5982\u679c\u200b\\(\\eqref{8}\\)\u200b\u4e2d\u200b\u7684\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u662f\u200b\u7ebf\u6027\u200b\u7684\u200b\uff0c\u200b\u5219\u200b\u79f0\u4e3a\u200b\u4e8c\u6b21\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\uff08QP\uff09\u3002</p> \\[ \\optim{\\min}{\\frac 12x^\\top Px + q^\\top x + r}{\\cases{     \\begin{aligned}         &amp; Gx\\preceq h \\\\         &amp; Ax = b     \\end{aligned} }} \\label{9} \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\\(G\\in \\bbR^{m\\times n}\\)</p>"},{"location":"math/convex-optimization/convex-optimization/#_17","title":"\u4e8c\u9636\u200b\u9525\u200b\u89c4\u5212","text":"<p>\u200b\u5f53\u200b\\(\\eqref{4}\\)\u200b\u4e2d\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e3a\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\uff0c\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u6784\u6210\u200b\u4e8c\u9636\u200b\u51f8\u9525\u200b\uff0c\u200b\u6b64\u65f6\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u662f\u200b\u4e8c\u9636\u200b\u9525\u200b\u89c4\u5212\u200b\uff08SOCP\uff09\u3002</p> \\[ \\optim{\\min}{f^\\top x}{\\cases{     \\begin{aligned}         &amp; \\Vert A_ix + b\\Vert_2\\leq c_i^\\top x + d_i &amp; i = \\oneto m \\\\         &amp; Fx = g     \\end{aligned} }} \\label{10} \\] <p>\u200b\u79f0\u200b\\(\\Vert A_ix + b\\Vert_2\\leq c_i^\\top x + d_i\\)\u200b\u4e3a\u200b\u4e8c\u9636\u200b\u9525\u200b\u7ea6\u675f\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_18","title":"\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u4e0b\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898","text":"<p>\u200b\u672c\u200b\u8282\u200b\u8ba8\u8bba\u200b\\(f_i: \\bbR^n\\ra\\bbR^{k_i}\\)\u200b\u5728\u200b\\(\\preceq_{K_i}\\)\u200b\u60c5\u5f62\u200b\u4e0b\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_19","title":"\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f","text":"<p>\u200b\u5f62\u200b\u5982\u4e0b\u200b\u5f0f\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u79f0\u4e3a\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p> \\[ \\optim{\\min}{f_0(x)}{\\cases{     \\begin{aligned}         &amp; f_i\\preceq_{K_i} 0 &amp; i = \\oneto m \\\\         &amp; Ax = b     \\end{aligned} }} \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\\(f_0: \\bbR^n\\ra\\bbR\\)\uff0c\\(K_i\\subseteq \\bbR^{k_i}\\)\u200b\u4e3a\u200b\u6b63\u5e38\u200b\u9525\u200b\uff0c\\(f_i: \\bbR^n\\ra\\bbR^{k_i}\\)\u200b\u4e3a\u200b\\(K_i\\)-\u200b\u51f8\u51fd\u6570\u200b\u3002\u200b\u5927\u591a\u6570\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u7ed3\u8bba\u200b\u5728\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u4e0b\u200b\u4ecd\u7136\u200b\u6210\u7acb\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_20","title":"\u9525\u200b\u89c4\u5212","text":"<p>\u200b\u9525\u200b\u89c4\u5212\u200b\u662f\u200b\u5177\u6709\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b</p> \\[ \\optim{\\min}{c^\\top x}{\\cases{     \\begin{aligned}         &amp; Fx + g\\preceq_K 0 \\\\         &amp; Ax = b     \\end{aligned} }} \\] <p>\u200b\u5f53\u200b\\(K = \\bbR^n_{\\plus\\plus}\\)\u200b\u65f6\u200b\uff0c\u200b\u9525\u200b\u89c4\u5212\u200b\u9000\u5316\u200b\u4e3a\u200bLP\u3002\u200b\u4eff\u7167\u200bLP\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\u7684\u200b\u9525\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u3002</p> \\[ \\optim{\\min}{c^\\top x}{\\cases{     \\begin{aligned}         &amp; x\\succeq_K 0 \\\\         &amp; Ax = b     \\end{aligned} }} \\]"},{"location":"math/convex-optimization/convex-optimization/#_21","title":"\u534a\u5b9a\u200b\u89c4\u5212","text":"<p>\u200b\u5f53\u200b\\(K\\in \\bbS^k_{\\plus}\\)\u200b\u65f6\u200b\uff0c\u200b\u76f8\u5e94\u200b\u7684\u200b\u9525\u200b\u89c4\u5212\u200b\u4e3a\u200b\u534a\u5b9a\u200b\u89c4\u5212\u200b\uff08SDP\uff09\u3002</p> \\[ \\optim{\\min}{c^\\top x}{\\cases{     \\begin{aligned}         &amp; \\sum_{i=1}^n x_iF_i + G\\preceq 0 \\\\         &amp; Ax = b     \\end{aligned} }} \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\\(G, F_i\\in \\bbS^k\\)\u3002\u200b\u82e5\u200b\u8fd9\u4e9b\u200b\u77e9\u9635\u200b\u90fd\u200b\u662f\u200b\u5bf9\u89d2\u200b\u77e9\u9635\u200b\uff0cSDP\u200b\u9000\u5316\u200b\u4e3a\u200bLP\u3002\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\\(\\sum_{i=1}^n x_iF_i\\preceq G\\)\u200b\u79f0\u4e3a\u200b\u7ebf\u6027\u200b\u77e9\u9635\u200b\u4e0d\u7b49\u5f0f\u200b\uff08LMI\uff09\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u5206\u5757\u200b\u5bf9\u89d2\u200b\u77e9\u9635\u200b\u5c06\u200b\u591a\u4e2a\u200bLMI\u200b\u7ec4\u5408\u200b\u5728\u200b\u4e00\u8d77\u200b\u3002</p> <p>\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\u7684\u200b\u534a\u5b9a\u200b\u89c4\u5212\u200b\u4e3a\u200b</p> \\[ \\optim{\\min}{\\mathbf{tr}(CX)}{\\cases{     \\begin{aligned}         &amp; \\mathbf{tr(A_iX)} = b_i &amp; i = \\oneto p \\\\         &amp; X\\succeq 0     \\end{aligned} }} \\]"},{"location":"math/convex-optimization/convex-optimization/#_22","title":"\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u5305\u542b\u200b\u5173\u7cfb","text":"<ul> <li>\u200b\u5f53\u200bQP\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e2d\u200b\u7684\u200b\\(P = 0\\)\u200b\u65f6\u200b\uff0cQP\u200b\u9000\u5316\u200b\u4e3a\u200bLP\u200b\u95ee\u9898\u200b</li> <li>\u200b\u5f53\u200bQCQP\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u4e2d\u200b\u7684\u200b\\(P_i = 0\\)\u200b\u65f6\u200b\uff0cQCQP\u200b\u9000\u5316\u200b\u4e3a\u200bQP\u200b\u95ee\u9898\u200b</li> <li>\u200b\u5f53\u200bSOCP\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u4e2d\u200b\u7684\u200b\\(c_i = 0\\)\u200b\u65f6\u200b\uff0cSOCP\u200b\u9000\u5316\u200b\u4e3a\u200bQCQP</li> <li>\u200b\u5f62\u200b\u5982\u200b\\(\\eqref{10}\\)\u200b\u7684\u200bSOCP\u200b\u4e0e\u200b\u5982\u4e0b\u200bSDP\u200b\u95ee\u9898\u200b\u7b49\u4ef7\u200b</li> </ul> \\[ \\optim{\\min}{f^\\top x}{\\cases{     \\begin{bmatrix}         (c_i^\\top x + d_i)I &amp; A_ix + b_i \\\\         (A_ix + b_i)I &amp; (c_i^\\top + d_i)     \\end{bmatrix}\\succeq_{\\bbS_\\plus^n} 0 }} \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\text{LP}\\subseteq \\text{QP}\\subseteq \\text{QCQP}\\subseteq \\text{SOCP}\\subseteq \\text{SDP} \\]"},{"location":"math/convex-optimization/convex-set/","title":"\u51f8\u96c6\u200b\u53ca\u5176\u200b\u6027\u8d28","text":"<p>\u200b\u4ece\u200b\u76f4\u89c2\u200b\u610f\u4e49\u200b\u4e0a\u200b\u8bb2\u200b\uff0c\u200b\u51f8\u96c6\u200b\u5c31\u662f\u200b\u6ca1\u6709\u200b\u51f9\u200b\u8fdb\u53bb\u200b\u7684\u200b\u90e8\u5206\u200b\u7684\u200b\u96c6\u5408\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_2","title":"\u5b9a\u4e49","text":""},{"location":"math/convex-optimization/convex-set/#_3","title":"\u76f4\u7ebf","text":"<p>\u200b\u8bbe\u200b\\(x_1 \\not = x_2\\)\u200b\u4e3a\u200b\\(\\mathbb R^n\\)\u200b\u4e0a\u200b\u7684\u200b\u4e24\u4e2a\u200b\u70b9\u200b\uff0c\u200b\u5219\u200b</p> <ul> <li>\\(y = \\theta x_1 + (1 - \\theta) x_2\\) \u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u6761\u200b\u76f4\u7ebf\u200b\u3002</li> <li>\\(y = \\theta x_1 + (1 - \\theta) x_2, 0\\leq\\theta\\leq 1\\) \u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u6761\u200b\u7ebf\u6bb5\u200b\u3002</li> </ul> <p>\u200b\u76f4\u7ebf\u200b\\(y = \\theta x_1 + (1 - \\theta) x_2\\)\u200b\u662f\u200b\u4e00\u6761\u200b\u4ece\u200b\\(x_2\\)\u200b\u51fa\u53d1\u200b\uff0c\u200b\u6cbf\u200b\u5411\u91cf\u200b\\(x_1 - x_2\\)\u200b\u65b9\u5411\u200b\u5ef6\u4f38\u200b\u7684\u200b\u76f4\u7ebf\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_4","title":"\u4eff\u5c04\u200b\u96c6\u5408","text":"<p>\u200b\u5bf9\u4e8e\u200b\\(\\mathbb R^n\\)\u200b\u4e2d\u200b\u7684\u200b\\(m\\)\u200b\u4e2a\u70b9\u200b\\(x_1, \\ldots, x_m\\)\uff0c\u200b\u82e5\u200b\\(\\sum_{i = 1}^m \\theta_i = 1\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\\(\\sum_{i=1}^m \\theta_ix_i\\)\u200b\u4e3a\u200b\\(x_1, \\ldots, x_m\\)\u200b\u7684\u200b\u4eff\u5c04\u200b\u7ec4\u5408\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u4e00\u7ec4\u200b\u5411\u91cf\u200b\\(x_1, \\ldots, x_m\\)\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u5411\u91cf\u200b\u90fd\u200b\u4e0d\u80fd\u200b\u7531\u200b\u5176\u4ed6\u200b\u5411\u91cf\u200b\u7ecf\u200b\u4eff\u5c04\u200b\u7ec4\u5408\u200b\u5f97\u5230\u200b\uff0c\u200b\u79f0\u200b\u8fd9\u7ec4\u200b\u5411\u91cf\u200b\u4eff\u5c04\u200b\u65e0\u5173\u200b\u3002</p> \u200b\u4eff\u5c04\u200b\u65e0\u5173\u200b\u4e0e\u200b\u7ebf\u6027\u200b\u65e0\u5173\u200b <p>\u200b\u4e00\u7ec4\u200b\u5411\u91cf\u200b\\(x_0, x_1, \\ldots, x_m\\)\u200b\u4eff\u5c04\u200b\u65e0\u200b\uff08\u200b\u76f8\u200b\uff09\u200b\u5173\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u5411\u91cf\u200b\\(x_1 - x_0, \\ldots, x_m - x_0\\)\u200b\u7ebf\u6027\u200b\u65e0\u200b\uff08\u200b\u76f8\u200b\uff09\u200b\u5173\u200b\u3002</p> <p>\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(C \\subseteq \\mathbb R^n\\)\uff0c\u200b\u82e5\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u4e2d\u200b\u7684\u200b\u4efb\u610f\u200b\u4e24\u4e2a\u200b\u4e0d\u540c\u70b9\u200b\u6784\u6210\u200b\u7684\u200b\u76f4\u7ebf\u200b\uff08\u200b\u5373\u200b\u5b83\u4eec\u200b\u7684\u200b\u4eff\u5c04\u200b\u7ec4\u5408\u200b\uff09\u200b\u4ecd\u7136\u200b\u4f4d\u4e8e\u200b\u96c6\u5408\u200b\u4e2d\u200b\uff0c\u200b\u5219\u200b\\(C\\)\u200b\u4e3a\u200b\u4eff\u5c04\u200b\u96c6\u5408\u200b\u3002</p> \\[ \\forall x_1, x_2 \\in C, x_1 \\not = x_2, \\forall \\theta\\in \\mathbb R, y = \\theta x_1 + (1 - \\theta) x_2\\in C \\] <p>\u200b\u4eff\u5c04\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ul> <li>\\(C\\)\u200b\u4e2d\u200b\u5305\u542b\u200b\u5176\u4e2d\u200b\u4efb\u610f\u200b\u4e2a\u70b9\u200b\u7684\u200b\u4eff\u5c04\u200b\u7ec4\u5408\u200b\u3002</li> <li>\u200b\u96c6\u5408\u200b\\(V = \\{x - x_0 | x \\in C\\}\\)\u200b\u4e3a\u200b\\(\\mathbb R^n\\)\u200b\u7684\u200b\u5b50\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u4e14\u200b\u5176\u7ef4\u6570\u200b\u4e0e\u200b\u504f\u79fb\u200b\\(x_0\\)\u200b\u65e0\u5173\u200b\u3002\u200b\u5b9a\u4e49\u200b\u4eff\u5c04\u200b\u96c6\u5408\u200b\u7684\u200b\u7ef4\u6570\u200b\u4e3a\u200b\\(V\\)\u200b\u7684\u200b\u7ef4\u6570\u200b\u3002</li> </ul> <p>\u200b\u8bbe\u200b\\(C\\)\u200b\u4e3a\u200b\u4efb\u610f\u200b\u96c6\u5408\u200b\uff0c\u200b\u5219\u200b\\(C\\)\u200b\u4e2d\u200b\u6240\u6709\u200b\u70b9\u200b\u7684\u200b\u4eff\u5c04\u200b\u7ec4\u5408\u200b\u79f0\u4e3a\u200b\\(C\\)\u200b\u7684\u200b\u4eff\u5c04\u200b\u5305\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(\\mathbf{aff} C\\)\uff0c\u200b\u4eff\u5c04\u200b\u5305\u662f\u200b\u5305\u542b\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u7684\u200b\u6700\u5c0f\u200b\uff08\u200b\u5373\u7ef4\u6570\u200b\u6700\u4f4e\u200b\u7684\u200b\uff09\u200b\u4eff\u5c04\u200b\u96c6\u5408\u200b\uff0c\u200b\u5176\u7ef4\u6570\u200b\u79f0\u4e3a\u200b\u4eff\u5c04\u200b\u7ef4\u6570\u200b\u3002</p> <p>\u200b\u5f53\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u7684\u200b\u4eff\u5c04\u200b\u7ef4\u6570\u200b\\(\\dim \\mathbf{aff} C &lt; n\\)\u200b\u65f6\u200b\uff0c\\(C^\\circ = \\varnothing\\)\u3002\u200b\u6b64\u65f6\u200b\u9700\u8981\u200b\u5b9a\u4e49\u200b\u76f8\u5bf9\u200b\u5185\u90e8\u200b\u4e0e\u200b\u76f8\u5bf9\u200b\u8fb9\u754c\u200b\u3002</p> <ul> <li>\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u7684\u200b\u76f8\u5bf9\u200b\u5185\u90e8\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\\(\\mathbf{relint}C = \\{x\\in C | \\exists r\\in \\mathbb R_{ + + }, B(x, r)\\cap \\mathbf{aff} C \\subseteq C\\}\\)</li> <li>\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u7684\u200b\u76f8\u5bf9\u200b\u8fb9\u754c\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\\(\\mathbf{cl} C - \\mathbf{relint}C\\)\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-set/#_5","title":"\u51f8\u96c6","text":"<p>\u200b\u5bf9\u4e8e\u200b\u96c6\u5408\u200b\\(C\\subseteq \\mathbb R^n\\)\u200b\u4e2d\u200b\u7684\u200b\\(m\\)\u200b\u4e2a\u70b9\u200b\\(x_1, \\ldots, x_m\\)\uff0c\u200b\u82e5\u200b\\(\\sum_{i=1}^m \\theta_i = 1, \\theta_i \\geq 0\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\\(\\sum_{i=1}^m \\theta_i x_i\\)\u200b\u4e3a\u200b\\(x_1, \\ldots, x_m\\)\u200b\u7684\u200b\u51f8\u200b\u7ec4\u5408\u200b\u3002</p> <p>\u200b\u82e5\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\u5185\u200b\u4efb\u610f\u200b\u4e24\u70b9\u200b\u7684\u200b\u51f8\u200b\u7ec4\u5408\u200b\u90fd\u200b\u5728\u200b\u8be5\u200b\u96c6\u5408\u200b\u4e2d\u200b\uff0c\u200b\u79f0\u8be5\u200b\u96c6\u5408\u200b\u4e3a\u200b\u51f8\u96c6\u200b\u3002</p> \\[ \\forall x_1, x_2\\in C, \\forall 0\\leq \\theta\\leq 1, \\theta x_1 + (1 - \\theta) x_2\\in C \\] <p>\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u7684\u200b\u6240\u6709\u200b\u70b9\u200b\u7684\u200b\u51f8\u200b\u7ec4\u5408\u200b\u79f0\u4e3a\u200b\u51f8\u5305\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(\\mathbf{conv}C\\)\uff0c\\(C\\)\u200b\u7684\u200b\u51f8\u200b\u5305\u662f\u200b\u5305\u542b\u200b\\(C\\)\u200b\u7684\u200b\u6700\u5c0f\u200b\u51f8\u96c6\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_6","title":"\u9525","text":"<p>\u200b\u5bf9\u4e8e\u200b\u96c6\u5408\u200b\\(C\\)\uff0c\u200b\u82e5\u200b\\(\\forall x\\in C, \\forall \\theta\\geq 0, \\theta x\\in C\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\\(C\\)\u200b\u4e3a\u200b\u9525\u200b\u3002\u200b\u82e5\u200b\\(C\\)\u200b\u8fd8\u662f\u200b\u51f8\u96c6\u200b\uff0c\u200b\u5219\u200b\u79f0\u4e3a\u200b\u51f8\u9525\u200b\u3002\u200b\u51f8\u9525\u200b\u6ee1\u8db3\u200b</p> \\[ \\forall x_1, x_2\\in C, \\forall \\theta_1, \\theta_2 \\geq 0, \\theta_1x_1 + \\theta_2x_2 \\in C \\] <p>\u200b\u5bf9\u4e8e\u200b\u96c6\u5408\u200b\\(C\\subseteq \\mathbb R^n\\)\u200b\u4e2d\u200b\u7684\u200b\\(m\\)\u200b\u4e2a\u70b9\u200b\\(x_1, \\ldots, x_m\\)\uff0c\u200b\u82e5\u200b\\(\\theta_i \\geq 0\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\\(\\sum_{i=1}^m \\theta_i x_i\\)\u200b\u4e3a\u200b\\(x_1, \\ldots, x_m\\)\u200b\u7684\u200b\u9525\u200b\u7ec4\u5408\u200b\u3002</p> <p>\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u4e2d\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u7ec4\u6210\u200b\u7684\u200b\u6240\u6709\u200b\u9525\u200b\u7ec4\u5408\u200b\u79f0\u4e3a\u200b\u9525\u5305\u200b\uff0c\\(C\\)\u200b\u7684\u200b\u9525\u200b\u5305\u662f\u200b\u5305\u542b\u200b\\(C\\)\u200b\u7684\u200b\u6700\u5c0f\u200b\u51f8\u9525\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_7","title":"\u5e38\u89c1\u200b\u7684\u200b\u51f8\u96c6","text":"<p>\u200b\u4ee5\u4e0b\u200b\u662f\u200b\u4e00\u4e9b\u200b\u5e73\u51e1\u200b\u51f8\u96c6\u200b\uff1a</p> <ul> <li>\u200b\u7a7a\u96c6\u200b\\(\\varnothing\\)\u3001\u200b\u5355\u70b9\u200b\u96c6\u200b\\(\\{x_0\\}\\)\u3001\u200b\u5168\u7a7a\u95f4\u200b\\(\\mathbb R^n\\)\u200b\u53ca\u5176\u200b\u5b50\u200b\u7a7a\u95f4\u200b\u90fd\u200b\u662f\u200b\u4eff\u5c04\u200b\u96c6\u200b\uff08\u200b\u4e5f\u200b\u662f\u200b\u51f8\u96c6\u200b\uff09\u3002</li> <li>\u200b\u4efb\u610f\u200b\u76f4\u7ebf\u200b\u90fd\u200b\u662f\u200b\u51f8\u96c6\u200b\uff1b\u200b\u8fc7\u200b\u539f\u70b9\u200b\u7684\u200b\u76f4\u7ebf\u200b\u662f\u200b\u4eff\u5c04\u200b\u96c6\u200b\uff1b\u200b\u7ebf\u6bb5\u200b\u662f\u200b\u51f8\u96c6\u200b\uff1b\u200b\u5c04\u7ebf\u200b\u662f\u200b\u51f8\u96c6\u200b\uff1b\u200b\u8d77\u70b9\u200b\u5728\u200b\u539f\u70b9\u200b\u7684\u200b\u5c04\u7ebf\u200b\u662f\u200b\u51f8\u200b\u9525\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-set/#_8","title":"\u8d85\u5e73\u9762\u200b\u4e0e\u200b\u534a\u200b\u7a7a\u95f4","text":"<p>\u200b\u8bbe\u200b\\(a\\in \\mathbb R^n, a \\not = 0, b\\in \\mathbb R\\)\uff0c\u200b\u65b9\u7a0b\u200b\\(a^\\top x = b\\)\u200b\u786e\u5b9a\u200b\u4e86\u200b\u7ebf\u6027\u200b\u7a7a\u95f4\u200b\\(\\mathbb R^n\\)\u200b\u5185\u200b\u7684\u200b\u4e00\u4e2a\u200b\u8d85\u5e73\u9762\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u8d85\u5e73\u9762\u200b\u5c06\u200b\u7a7a\u95f4\u200b\u5206\u4e3a\u200b\u4e24\u200b\u90e8\u5206\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u5f00\u534a\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u6bcf\u200b\u4e00\u90e8\u5206\u200b\u52a0\u4e0a\u200b\u8d85\u5e73\u9762\u200b\u79f0\u4e3a\u200b\u534a\u200b\u7a7a\u95f4\u200b\u3002</p> <ul> <li>\u200b\u8d85\u5e73\u9762\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b\\(\\{x | a^\\top x = b\\}\\)</li> <li>\u200b\u534a\u200b\u7a7a\u95f4\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b\\(\\{x | a^\\top x\\leq b\\}\\)</li> </ul> <p>\u200b\u5411\u91cf\u200b\\(a\\)\u200b\u4e3a\u200b\u8d85\u5e73\u9762\u200b\u7684\u200b\u6cd5\u7ebf\u200b\u65b9\u5411\u200b\uff0c\u200b\u8d85\u5e73\u9762\u200b\u4e0a\u200b\u7684\u200b\u6240\u6709\u200b\u70b9\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u7531\u200b\u4e00\u4e2a\u200b\u8d77\u70b9\u200b\\(x_0\\)\u200b\u52a0\u4e0a\u200b\u4e0e\u200b\\(a\\)\u200b\u6b63\u4ea4\u200b\u7684\u200b\u6240\u6709\u200b\u5411\u91cf\u200b\u5f97\u5230\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_9","title":"\u7403","text":"<p>\\(\\mathbb R^n\\)\u200b\u4e2d\u200b\u7684\u200b\u6b27\u6c0f\u200b\u7403\u4e3a\u200b2-\u200b\u8303\u6570\u200b\u4e0b\u200b\u7684\u200b\u7403\u200b\uff0c\u200b\u5373\u200b</p> \\[ B(x, r) = \\{y | \\Vert y - x\\Vert_2\\leq r\\} \\] <p>\u200b\u5176\u4e2d\u200b\\(x\\in \\mathbb R^n\\)\u200b\u4e3a\u200b\u6b27\u6c0f\u200b\u7403\u200b\u7684\u200b\u7403\u5fc3\u200b\uff0c\\(r\\in \\mathbb R_}\\)\u200b\u4e3a\u200b\u5176\u200b\u534a\u5f84\u200b\u3002 <p>\u200b\u692d\u7403\u200b\u662f\u200b\u5404\u4e2a\u200b\u65b9\u5411\u200b\u5ef6\u4f38\u200b\u7a0b\u5ea6\u200b\u4e0d\u540c\u200b\u7684\u200b\u6b27\u6c0f\u200b\u7403\u200b\uff0c\u200b\u5ef6\u4f38\u200b\u7a0b\u5ea6\u200b\u7531\u200b\u77e9\u9635\u200b\\(P\\in \\mathbb S_{\\)\u200b\u786e\u5b9a\u200b\uff0c\u200b\u534a\u8f74\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(\\sqrt{\\lambda_i}\\)\u3002</p> \\[ \\mathcal E = \\{y | (x - y)^\\top P^{-1}(x - y)\\leq 1\\} \\] <p>\u200b\u5f53\u200b\\(P = r^2I\\)\u200b\u65f6\u200b\uff0c\\(\\mathcal E = B(x, r)\\)\u3002</p> <p>\u200b\u5c06\u200b\u6b27\u6c0f\u200b\u7403\u4e2d\u200b\u7684\u200b2-\u200b\u8303\u6570\u200b\u6362\u6210\u200b\u5176\u4ed6\u200b\u7684\u200b\u8303\u6570\u200b\uff0c\u200b\u5c31\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u8303\u6570\u200b\u7403\u200b\uff1a</p> \\[ B_*(x, r) = \\{y | \\Vert y - x\\Vert_* \\leq r\\} \\]"},{"location":"math/convex-optimization/convex-set/#_10","title":"\u9525","text":"<p>\u200b\u65b9\u7a0b\u200b\\(\\Vert x\\Vert_2 \\leq t\\)\u200b\u5728\u200b\\(R^{n+1}\\)\u200b\u4e0a\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u9525\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u4e8c\u9636\u200b\u9525\u200b\u3002\u200b\u9525\u200b\u53ef\u4ee5\u200b\u89c6\u4e3a\u200b\u7531\u200b\u4f4e\u200b\u4e00\u7ef4\u200b\u7684\u200b\u7403\u200b\u5806\u79ef\u200b\u5f97\u5230\u200b\u3002</p> <p>\u200b\u5c06\u200b2-\u200b\u8303\u6570\u200b\u66ff\u6362\u200b\u4e3a\u200b\u5176\u4ed6\u200b\u8303\u6570\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u8303\u6570\u200b\u9525\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_11","title":"\u591a\u9762\u4f53","text":"<p>\u200b\u6709\u9650\u200b\u4e2a\u200b\u7ebf\u6027\u200b\u7b49\u5f0f\u200b\u548c\u200b\u4e0d\u7b49\u5f0f\u200b\u7684\u200b\u89e3\u96c6\u200b\u6784\u6210\u200b\u591a\u9762\u4f53\u200b\u3002</p> \\[ \\begin{aligned}     \\mathcal P &amp;= \\{x | a_i^\\top x\\leq b_i, i = 1, \\ldots, m, c_j^\\top x = d_j, j = 1, \\ldots, p\\} \\\\     &amp;= \\{x | Ax\\preceq b, Cx = d\\} \\end{aligned} \\] <p>\u200b\u591a\u9762\u4f53\u200b\u53ef\u4ee5\u200b\u662f\u200b\u9525\u200b\uff0c\u200b\u5982\u975e\u200b\u8d1f\u200b\u8c61\u9650\u200b\u3002</p> <p>\u200b\u5355\u7eaf\u5f62\u200b\u662f\u200b\u4e00\u79cd\u200b\u7279\u6b8a\u200b\u7684\u200b\u591a\u9762\u4f53\u200b\uff0c\u200b\u4e00\u7ec4\u200b\u4eff\u5c04\u200b\u65e0\u5173\u200b\u7684\u200b\u70b9\u200b\\(x_0, x_1, \\ldots, x_m\\)\u200b\u7684\u200b\u51f8\u200b\u7ec4\u5408\u200b\u786e\u5b9a\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u5355\u7eaf\u5f62\u200b\uff0c\u200b\u7528\u200b\\(\\mathbf{conv}\\{x_0, \\ldots, x_m\\}\\)\u200b\u8868\u793a\u200b\uff0c\u200b\u5176\u200b\u4eff\u5c04\u200b\u7ef4\u5ea6\u200b\u4e3a\u200b\\(m\\)\u3002</p> \\[ \\mathbf{conv}\\{x_0, \\ldots, x_m\\} = \\left\\{\\theta_0x_0 + \\theta_1x_1 + \\cdots + \\theta_mx_m \\middle| \\theta_i \\geq 0, \\sum_{i=1}^m \\theta_i = 1\\right\\} \\] <ul> <li>\u200b\u5355\u4f4d\u200b\u5355\u7eaf\u5f62\u200b\u662f\u200b\u96f6\u200b\u5411\u91cf\u200b\u548c\u200b\u5355\u4f4d\u5411\u91cf\u200b\u786e\u5b9a\u200b\u7684\u200b\\(n\\)\u200b\u7ef4\u200b\u5355\u7eaf\u5f62\u200b\uff0c\u200b\u5373\u200b\\(\\{x | x\\geq 0, \\mathbf{1}^\\top x \\leq 1\\}\\)\u3002</li> <li>\u200b\u6982\u7387\u200b\u5355\u7eaf\u5f62\u200b\u662f\u200b\u5355\u4f4d\u5411\u91cf\u200b\u786e\u5b9a\u200b\u7684\u200b\\(n-1\\)\u200b\u7ef4\u200b\u5355\u7eaf\u5f62\u200b\uff0c\u200b\u5373\u200b\\(\\{x | \\mathbf 1^\\top x = 1\\}\\)\u3002</li> </ul> \u200b\u5355\u7eaf\u5f62\u200b\u4e0e\u200b\u591a\u9762\u4f53\u200b <p>\u200b\u5355\u7eaf\u5f62\u200b\u662f\u200b\u4e00\u79cd\u200b\u7279\u6b8a\u200b\u7684\u200b\u591a\u9762\u4f53\u200b\u3002\u200b\u8bbe\u200b\u5355\u7eaf\u5f62\u200b\\(C\\)\u200b\u7531\u200b\u4eff\u5c04\u200b\u65e0\u5173\u200b\u7684\u200b\u5411\u91cf\u200b\\(x_0, \\ldots, x_m\\)\u200b\u786e\u5b9a\u200b\uff0c\u200b\u5219\u200b</p> \\[ B = \\begin{bmatrix}     x_1 - x_0 &amp; \\cdots &amp; x_m - x_0 \\end{bmatrix} \\] <p>\\(B\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(n\\times m\\)\u200b\u7ef4\u200b\u7684\u200b\u6ee1\u200b\u79e9\u200b\u77e9\u9635\u200b\u3002\u200b\u4ee4\u200b\\(\\theta = \\begin{pmatrix}\\theta_0 &amp; \\theta_1 &amp; \\cdots &amp; \\theta_m\\end{pmatrix}, \\theta' = \\begin{pmatrix}\\theta_1 &amp; \\cdots &amp; \\theta_m\\end{pmatrix}\\)\u3002\u200b\u5f53\u200b\u7a7a\u95f4\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u70b9\u200b\\(x\\)\u200b\u6ee1\u8db3\u200b\\(x\\in C\\)\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\exists \\theta\\succeq 0, \\mathbf 1^\\top \\theta = 1, x = \\sum_{i=0}^m \\theta_i x_i \\Longleftrightarrow \\exists \\theta'\\succeq 0, \\mathbf 1^\\top \\theta' \\leq 1, x = x_0 + B\\theta' \\] <p>\u200b\u800c\u200b\\(B\\)\u200b\u662f\u200b\u6ee1\u200b\u79e9\u200b\u77e9\u9635\u200b\uff0c\u200b\u5219\u200b\u5b58\u5728\u200b\u6ee1\u200b\u79e9\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u4f7f\u5f97\u200b</p> \\[ AB = \\begin{bmatrix}     A_1 \\\\ A_2 \\end{bmatrix}B = \\begin{bmatrix}     I \\\\ 0 \\end{bmatrix} \\] <p>\u200b\u5219\u200b</p> \\[ Ax = \\begin{bmatrix}     A_1 \\\\ A_2 \\end{bmatrix}x = \\begin{bmatrix}     A_1x \\\\ A_2x \\end{bmatrix} = \\begin{bmatrix}     A_1x_0 + \\theta' \\\\ A_2x_0 \\end{bmatrix} \\] <p>\u200b\u8be5\u200b\u5355\u7eaf\u5f62\u200b\u7684\u200b\u591a\u9762\u4f53\u200b\u8868\u793a\u200b\u4e3a\u200b\uff1a</p> \\[ C = \\{x | A_1x\\succeq A_1x_0, A_2x = A_2x_0, \\mathbf 1^\\top A_1x - \\mathbf 1^\\top A_1x_0\\leq 1\\} \\]"},{"location":"math/convex-optimization/convex-set/#_12","title":"\u534a\u200b\u6b63\u5b9a\u200b\u9525","text":"<p>\u200b\u8bbe\u200b\\(\\mathbb S^n_{+}\\)\u200b\u8868\u793a\u200b\u5168\u4f53\u200b\\(n\\)\u200b\u9636\u200b\u5bf9\u79f0\u200b\u534a\u200b\u6b63\u5b9a\u200b\u77e9\u9635\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u5219\u200b\\(\\mathbb S^n_+\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\u51f8\u9525\u200b\u3002\uff08\u200b\u6ce8\u610f\u200b\u6b64\u5904\u200b\u5bf9\u200b\u9525\u200b\u7684\u200b\u7406\u89e3\u200b\u548c\u200b\u8303\u6570\u200b\u9525\u200b\u4e0d\u540c\u200b\uff09\u3002</p> \\[ A\\succeq 0, B\\succeq 0\\Longrightarrow x^\\top (\\theta_1 A + \\theta_2 B)x \\geq 0\\Longrightarrow (\\theta_1A + \\theta_2B)\\succeq 0 \\]"},{"location":"math/convex-optimization/convex-set/#_13","title":"\u4fdd\u51f8\u200b\u8fd0\u7b97","text":"<p>\u200b\u4fdd\u51f8\u200b\u8fd0\u7b97\u200b\u662f\u200b\u5728\u200b\u8fd0\u7b97\u200b\u524d\u540e\u200b\u4fdd\u6301\u200b\u96c6\u5408\u200b\u51f8\u6027\u200b\u7684\u200b\u8fd0\u7b97\u200b\u3002</p> <ol> <li>\u200b\u90e8\u5206\u200b\u548c\u200b\\(\\{(x, y_1 + y_2) | (x, y_1) \\in A, (x, y_2)\\in B\\}\\)\uff0c\u200b\u6709\u200b\u4e24\u79cd\u200b\u6781\u7aef\u200b\u60c5\u51b5\u200b</li> <li>\\(\\dim y_1 = \\dim y_2 = 0\\)\u200b\u65f6\u200b\u9000\u5316\u200b\u4e3a\u4ea4\u200b\u8fd0\u7b97\u200b\\(\\cap\\)\uff1a\u200b\u4e00\u4e2a\u200b\u51f8\u96c6\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b\u4e00\u7cfb\u5217\u200b\u534a\u200b\u7a7a\u95f4\u200b\u7684\u200b\u4ea4\u96c6\u200b\u3002</li> <li>\\(\\dim x = 0\\)\u200b\u65f6\u200b\u9000\u5316\u200b\u4e3a\u200b\u548c\u200b\\(+: A + B = \\{x + y | x\\in A, y\\in B\\}\\)\u3002</li> <li>\u200b\u7ebf\u6027\u53d8\u6362\u200b\\(f: \\mathbb R^n\\rightarrow \\mathbb R^m = Ax + b\\)\u3002</li> <li>\u200b\u76f4\u79ef\u200b\\(\\times, A\\times B = \\{(x, y) | x\\in A, y\\in B\\}\\)\u3002</li> <li>\u200b\u6295\u5f71\u200b\u51fd\u6570\u200b\\(P: \\mathbb R^n\\times \\mathbb R_}\\rightarrow \\mathbb R^n\\)\uff0c\\(P(z, t) = z / t\\)\u3002\u200b\u6295\u5f71\u200b\u51fd\u6570\u200b\u5c06\u200b\\(\\mathbb R^{n}\\times \\mathbb R_{\\)\u200b\u7a7a\u95f4\u200b\u4e2d\u200b\u7684\u200b\u70b9\u200b\u6295\u5f71\u200b\u5230\u200b\u5e73\u9762\u200b\\(x_{n+1} = 1\\)\u200b\u4e0a\u200b\u3002 <li>\u200b\u7ebf\u6027\u53d8\u6362\u200b\u548c\u200b\u6295\u5f71\u200b\u51fd\u6570\u200b\u7684\u200b\u7ec4\u5408\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u7ebf\u6027\u200b\u5206\u5f0f\u200b\u51fd\u6570\u200b\u3002</li>"},{"location":"math/convex-optimization/convex-set/#_14","title":"\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f","text":"<p>\u200b\u5b9a\u4e49\u200b\u6b63\u5e38\u200b\u9525\u4e3a\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u9525\u200b\uff1a</p> <ul> <li>\u200b\u662f\u200b\u51f8\u200b\u9525\u200b\u3002</li> <li>\u200b\u662f\u200b\u95ed\u96c6\u200b\uff08\u200b\u5b58\u5728\u200b\u8fb9\u754c\u200b\uff09</li> <li>\u200b\u5b58\u5728\u200b\u5185\u90e8\u200b\u3002</li> <li>\u200b\u4e0d\u200b\u5305\u542b\u200b\u76f4\u7ebf\u200b\uff08\\(x\\in K, -x\\in K\\Longleftrightarrow x = 0\\)\uff09</li> </ul> <p>\u200b\u6b63\u5e38\u200b\u9525\u200b\\(K\\)\u200b\u53ef\u4ee5\u200b\u5728\u200b\u7a7a\u95f4\u200b\\(\\mathbb R^n\\)\u200b\u4e0a\u200b\u5b9a\u4e49\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\uff1a</p> \\[ \\begin{gathered}     x \\preceq_K y \\triangleq y - x\\in K \\\\     x \\prec_K y \\triangleq y - x\\in K^\\circ \\end{gathered} \\] <ul> <li>\u200b\u82e5\u200b\\(K = \\mathbb R_+\\)\uff0c\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u5b9e\u6570\u200b\u57df\u200b\u4e0a\u200b\u7684\u200b\u4e0d\u7b49\u5f0f\u200b\u3002</li> <li>\u200b\u82e5\u200b\\(K = \\mathbb R^n_+\\)\uff0c\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\\(\\mathbb R^n\\)\u200b\u7a7a\u95f4\u200b\u4e2d\u200b\u5411\u91cf\u200b\u7684\u200b\u5206\u91cf\u200b\u4e0d\u7b49\u5f0f\u200b\u3002</li> <li>\u200b\u82e5\u200b\\(K = \\mathbb S^n_+\\)\uff0c\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u77e9\u9635\u200b\u4e0d\u7b49\u5f0f\u200b\u3002</li> </ul> <p>\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ul> <li>\\(x\\preceq_K y, u\\preceq_K v\\Longrightarrow x + u\\preceq_K y + v\\)\uff0c\u200b\u5bf9\u200b\\(\\prec_K\\)\u200b\u4e5f\u200b\u6210\u7acb\u200b\u3002</li> <li>\\(x\\preceq_K y, y\\preceq_K z\\Longrightarrow x\\preceq_K z\\)\uff0c\u200b\u5bf9\u200b\\(\\prec_K\\)\u200b\u4e5f\u200b\u6210\u7acb\u200b\u3002</li> <li>\\(x\\preceq_K y, \\alpha\\geq 0\\Longrightarrow \\alpha x\\preceq_K \\alpha y\\)\uff0c\u200b\u5f53\u200b\\(\\alpha &gt; 0\\)\u200b\u65f6\u200b\u5bf9\u200b\\(\\prec_K\\)\u200b\u4e5f\u200b\u6210\u7acb\u200b\u3002</li> <li>\\(x\\prec_K y\\Longrightarrow x\\preceq_K y\\)</li> <li>\\(x\\preceq_K x, x\\not\\prec_K x\\)</li> <li>\\(x\\preceq_K y, y\\preceq_K x\\Longrightarrow x = y\\)</li> <li>\\(\\forall i\\in \\mathbb N, x_i\\preceq_K y_i, \\lim_{i\\rightarrow\\infty} x_i = x, \\lim_{i\\rightarrow\\infty} y_i = y\\Longrightarrow x\\preceq_K y\\)</li> <li>\\(x\\preceq_K y\\Longrightarrow \\exists u, \\exists v, x + u\\prec_K y+ v\\)</li> </ul> <p>\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u4e0d\u662f\u200b\u7ebf\u6027\u200b\u7684\u200b\uff0c\u200b\u5373\u200b\\(\\exists x, \\exists y, x\\not \\preceq_K y, y\\not \\preceq_K x\\)\uff0c\u200b\u6b64\u65f6\u200b\u79f0\u200b\\(x, y\\)\u200b\u4e0d\u53ef\u200b\u6bd4\u200b\u3002</p> <p>\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u504f\u5e8f\u200b\u5173\u7cfb\u200b\u540e\u200b\u5c31\u200b\u53ef\u4ee5\u200b\u5728\u200b\u6b64\u57fa\u7840\u200b\u4e0a\u200b\u5b9a\u4e49\u200b\u6781\u5c0f\u503c\u200b\u4e0e\u200b\u6700\u5c0f\u503c\u200b\u3002</p> <ul> <li>\\(x\\in S\\)\u200b\u662f\u200b\u6700\u5c0f\u200b\u5143\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(S\\subseteq \\{y | y\\succeq_K x\\}\\)\uff0c\u200b\u5373\u200b\\(S\\)\u200b\u4e2d\u200b\u5143\u7d20\u200b\u5168\u90e8\u200b\u4e0e\u200b\\(x\\)\u200b\u53ef\u6bd4\u200b\u4e14\u200b\u5927\u4e8e\u200b\u7b49\u4e8e\u200b\\(x\\)\u3002</li> <li>\\(x\\in S\\)\u200b\u662f\u200b\u6781\u5c0f\u200b\u5143\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(S\\cap \\{y | y\\preceq_K x\\} = \\{x\\}\\)\uff0c\u200b\u5373\u200b\\(S\\)\u200b\u4e2d\u200b\u4e0d\u200b\u5b58\u5728\u200b\u548c\u200b\\(x\\)\u200b\u53ef\u6bd4\u200b\u4e14\u200b\u6bd4\u200b\\(x\\)\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-set/#_15","title":"\u8d85\u5e73\u9762\u200b\u4e0e\u200b\u51f8\u96c6","text":""},{"location":"math/convex-optimization/convex-set/#_16","title":"\u5206\u79bb\u200b\u8d85\u5e73\u9762","text":"<p>\u200b\u7ed9\u5b9a\u200b\u51f8\u96c6\u200b\\(A, B\\)\u200b\u6ee1\u8db3\u200b\\(A\\cap B = \\varnothing\\)\uff0c\u200b\u5219\u200b\u5b58\u5728\u200b\\(a\\in \\mathbb R^n, b\\in \\mathbb R, a\\not = 0\\)\u200b\u4f7f\u5f97\u200b</p> \\[ \\forall x\\in A, \\forall y\\in B, a^\\top x - b\\leq 0, a^\\top y - b\\geq 0 \\] <p>\u200b\u79f0\u200b\\(\\{x | a^\\top x = b\\}\\)\u200b\u4e3a\u200b\u96c6\u5408\u200b\\(A, B\\)\u200b\u7684\u200b\u5206\u79bb\u200b\u8d85\u5e73\u9762\u200b\u3002</p> <p>\u200b\u82e5\u200b\u8d85\u5e73\u9762\u200b\\(\\{x | a^\\top x = b\\}\\)\u200b\u6ee1\u8db3\u200b</p> \\[ \\forall x\\in A, \\forall y\\in B, a^\\top x - b &lt; 0, a^\\top y - b &gt; 0 \\] <p>\u200b\u79f0\u200b\\(\\{x | a^\\top x = b\\}\\)\u200b\u4e25\u683c\u200b\u5206\u79bb\u200b\\(A, B\\)\uff0c\u200b\u4e25\u683c\u200b\u5206\u79bb\u200b\u7684\u200b\u8d85\u5e73\u9762\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u5b58\u5728\u200b\u3002</p> \u200b\u4e0d\u200b\u5b58\u5728\u200b\u4e25\u683c\u200b\u5206\u79bb\u200b\u7684\u200b\u4f8b\u5b50\u200b <p>\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(A = \\{(x, y) | xy\\geq 1\\}, B = \\{(x, y) | x\\leq 0, y\\geq 0\\}\\)\uff0c\u200b\u5219\u200b\u96c6\u5408\u200b\\(A, B\\)\u200b\u88ab\u200b\u8d85\u5e73\u9762\u200b\\(\\{(x, y) | x = 0\\}\\)\u200b\u548c\u200b\\(\\{(x, y) | y = 0\\}\\)\u200b\u5206\u79bb\u200b\uff0c\u200b\u4f46\u200b\u4e0d\u200b\u5b58\u5728\u200b\u4e25\u683c\u200b\u5206\u79bb\u200b\u4e24\u4e2a\u200b\u96c6\u5408\u200b\u7684\u200b\u8d85\u5e73\u9762\u200b\u3002</p> <p></p> <p>\u200b\u540c\u7406\u200b\uff0c\\(\\{(x, y) | y\\leq \\ln x\\}\\)\u200b\u548c\u200b\\(\\{(x, y) | x\\leq 0\\}\\)\u200b\u4e4b\u95f4\u200b\u4e5f\u200b\u4e0d\u200b\u5b58\u5728\u200b\u4e25\u683c\u200b\u5206\u79bb\u200b\u7684\u200b\u8d85\u5e73\u9762\u200b</p> <p>\u200b\u8d85\u5e73\u9762\u200b\u5206\u79bb\u200b\u7684\u200b\u9006\u5b9a\u7406\u200b\u4e0d\u200b\u6210\u7acb\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_17","title":"\u652f\u6491\u200b\u8d85\u5e73\u9762","text":"<p>\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(C\\subseteq \\mathbb R^n\\)\uff0c\u200b\u5bf9\u4e8e\u200b\\(x_0\\in \\partial C\\)\uff0c\u200b\u82e5\u200b\u5b58\u5728\u200b\\(a\\not = 0\\)\u200b\u6ee1\u8db3\u200b</p> \\[ \\forall x\\in C, a^\\top x\\leq a^\\top x_0 \\] <p>\u200b\u79f0\u200b\\(\\{x| a^\\top x = a^\\top x_0\\}\\)\u200b\u4e3a\u200b\\(C\\)\u200b\u5728\u200b\\(x_0\\)\u200b\u5904\u200b\u7684\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u8fb9\u754c\u200b\u4e0a\u200b\u7684\u200b\u67d0\u4e2a\u200b\u70b9\u200b\uff0c\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u552f\u4e00\u200b\u3002</p> <p>\u200b\u82e5\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u662f\u200b\u51f8\u96c6\u200b\uff0c\u200b\u5219\u200b\u5176\u200b\u8fb9\u754c\u200b\u4e0a\u200b\u6240\u6709\u200b\u70b9\u200b\u90fd\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u3002\u200b\u7ed9\u5b9a\u200b\\(C\\)\u200b\u662f\u200b\u95ed\u96c6\u4e14\u200b\\(C^\\circ\\not =\\varnothing\\)\uff0c\u200b\u5219\u200b\u9006\u5b9a\u7406\u200b\u4e5f\u200b\u6210\u7acb\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_18","title":"\u5bf9\u5076\u200b\u9525","text":"<p>\u200b\u7ed9\u5b9a\u200b\u9525\u200b\\(K\\)\uff0c\u200b\u82e5\u200b\\(K^*\\)\u200b\u6ee1\u8db3\u200b</p> \\[ y\\in K^* \\Longleftrightarrow \\forall x\\in K, x^\\top y\\geq 0 \\] <p>\u200b\u79f0\u200b\\(K^*\\)\u200b\u4e3a\u200b\\(K\\)\u200b\u7684\u200b\u5bf9\u5076\u200b\u9525\u200b\u3002\u200b\u5bf9\u5076\u200b\u9525\u200b\u5fc5\u5b9a\u200b\u662f\u200b\u51f8\u200b\u9525\u200b\u3002</p> \u200b\u5bf9\u5076\u200b\u9525\u200b\u7684\u200b\u51f8\u6027\u200b <p>\u200b\u7ed9\u5b9a\u200b\u9525\u200b\\(K\\)\uff0c\u200b\u6709\u200b\\(\\forall x\\in K, \\forall \\theta\\geq 0, \\theta x \\in K\\)\uff0c\u200b\u5219\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(y\\in K^*\\)</p> \\[ \\theta x^\\top y = x^\\top (\\theta y)\\geq 0\\Longrightarrow \\theta y\\in K^* \\] <p>\u200b\u56e0\u6b64\u200b\\(K^*\\)\u200b\u662f\u200b\u9525\u200b\u3002\u200b\u8bbe\u200b\\(y_1, y_2\\in K^*, x\\in K, 0\\leq \\theta\\leq 1\\)\uff0c\u200b\u5219\u200b</p> \\[ x^\\top (\\theta y_1)\\geq 0, x^\\top(1 - \\theta)y_2\\geq 0\\Longrightarrow x^\\top (\\theta y_1 + (1 - \\theta)y_2)\\geq 0 \\] <p>\u200b\u56e0\u6b64\u200b\\(K^*\\)\u200b\u662f\u200b\u51f8\u200b\u7684\u200b\u3002</p> <p>\u200b\u5bf9\u5076\u200b\u9525\u200b\u53ef\u80fd\u200b\u548c\u200b\u539f\u9525\u200b\u76f8\u7b49\u200b\uff0c\u200b\u5982\u200b\\(\\mathbb R^n_{+}, \\mathbb S^n_{+}\\)\u200b\u7684\u200b\u5bf9\u5076\u200b\u5206\u522b\u200b\u662f\u200b\u5176\u200b\u672c\u8eab\u200b\u3002</p> <ul> <li>\\(K^*\\)\u200b\u662f\u200b\u95ed\u51f8\u200b\u9525\u200b\u3002</li> <li>\\(K_1\\subseteq K_2\\Longrightarrow K_2^*\\subseteq K_1^*\\)</li> <li>\u200b\u82e5\u200b\\(K^\\circ\\not = \\varnothing\\)\uff0c\u200b\u5219\u200b\\(K^*\\)\u200b\u662f\u200b\u5c16\u200b\u7684\u200b\u3002</li> <li>\u200b\u82e5\u200b\\(\\mathbf{cl}K\\)\u200b\u662f\u200b\u5c16\u200b\u7684\u200b\uff0c\u200b\u5219\u200b\\((K^*)^\\circ\\not = \\varnothing\\)</li> <li>\\(K^{**} = \\mathbf{cl}K\\)</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u6b63\u5e38\u200b\u9525\u200b\\(K\\)\uff0c\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u5bf9\u5076\u200b\u9525\u200b\\(K^*\\)\uff0c\u200b\u4e24\u8005\u200b\u90fd\u200b\u80fd\u200b\u7528\u4e8e\u200b\u5b9a\u4e49\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\\(\\preceq_K\\)\uff0c\u200b\u79f0\u200b\\(\\preceq_{K^*}\\)\u200b\u4e3a\u200b\\(\\preceq_K\\)\u200b\u7684\u200b\u5bf9\u5076\u200b\u3002</p> <p>\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7684\u200b\u5bf9\u5076\u200b\\(\\preceq_{K^*}\\)\u200b\u6ee1\u8db3\u200b</p> <ul> <li>\\(x\\preceq_K y\\Longleftrightarrow \\forall \\lambda\\succeq_{K^*}0, \\lambda^\\top x\\leq \\lambda^\\top y\\)</li> <li>\\(x\\prec_K y\\Longleftrightarrow \\forall \\lambda\\succeq_{K^*}0, \\lambda\\not = 0, \\lambda^\\top x &lt; \\lambda^\\top y\\)</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u96c6\u5408\u200b\\(S\\)\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\\(x\\in S\\)\uff1a</p> <ul> <li>\\(x\\)\u200b\u5173\u4e8e\u200b\\(\\preceq_K\\)\u200b\u662f\u200b\u6700\u5c0f\u200b\u5143\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\\(\\forall \\lambda \\succeq_{K^\\ast}0\\)\uff0c\\(x\\)\u200b\u662f\u200b\u6781\u5c0f\u200b\u5316\u200b\\(\\lambda^\\top z, z\\in S\\)\u200b\u7684\u200b\u552f\u4e00\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u5373\u200b\\(\\forall \\lambda \\succeq_{K^\\ast}0, \\{z | \\lambda^\\top x = \\lambda^\\top z\\}\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\u4e25\u683c\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u3002</li> <li>\u200b\u82e5\u200b\\(x\\)\u200b\u5173\u4e8e\u200b\\(\\preceq_K\\)\u200b\u662f\u200b\u6781\u5c0f\u200b\u5143\u200b\uff0c\u200b\u5219\u200b\u5b58\u5728\u200b\\(\\lambda\\succ_{K^*} 0\\)\uff0c\\(\\forall z\\in S, \\lambda^\\top z\\geq \\lambda^\\top x\\)\uff0c\u200b\u4e5f\u200b\u5373\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u4e25\u683c\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u3002\u200b\u5f53\u200b\\(S\\)\u200b\u4e3a\u200b\u51f8\u200b\u96c6\u65f6\u200b\uff0c\u200b\u9006\u5b9a\u7406\u200b\u5bf9\u4e8e\u200b\u5b58\u5728\u200b\\(\\lambda\\succeq_{K^*}0\\)\u200b\u6210\u7acb\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/duality/","title":"\u5bf9\u5076","text":""},{"location":"math/convex-optimization/duality/#_2","title":"\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u200b\u5076\u51fd\u6570","text":"\\[ \\optim{\\min}{f_0(x)}{\\cases{\\begin{aligned}     &amp; f_i(x)\\leq 0 &amp; i = \\oneto m \\\\     &amp; h_i(x) = 0 &amp; i = \\oneto p \\end{aligned}}} \\label{1} \\] <p>\u200b\u5bf9\u4e8e\u200b\u4e00\u4e2a\u200b\u5f62\u200b\u5982\u200b\\(\\eqref{1}\\)\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff08\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff09\uff0c\u200b\u5176\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(L: \\bbR^n\\times \\bbR^m\\times \\bbR^p\\)\uff0c\u200b\u5176\u4e2d\u200b\\(\\lambda, \\nu\\)\u200b\u4e3a\u200b\u5411\u91cf\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u4e58\u5b50\u200b\u5411\u91cf\u200b\u3002</p> \\[ L(x, \\lambda, \\nu) = f_0(x) + \\sum_{i=1}^m \\lambda_if_i(x) + \\sum_{i=1}^p\\nu_i h_i(x) \\label{2} \\] <p>\\(\\eqref{2}\\)\u200b\u5173\u4e8e\u200b\\(x\\)\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\\(g(\\lambda, \\nu)\\)\u200b\u79f0\u4e3a\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u200b\u5076\u51fd\u6570\u200b\u3002</p> \\[ g(\\lambda, \\nu) = \\inf_{x\\in \\calD}\\left(f_0(x) + \\sum_{i=1}^m \\lambda_if_i(x) + \\sum_{i=1}^p\\nu_i h_i(x)\\right) \\label{3} \\] <p>\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u200b\u5076\u51fd\u6570\u200b\u662f\u200b\u539f\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u503c\u200b\u7684\u200b\u4e0b\u754c\u200b\uff1a\\(\\forall \\lambda \\succeq 0, \\forall \\nu, g(\\lambda, \\nu)\\leq p^*\\)\uff0c\u200b\u5e76\u4e14\u200b\u662f\u200b\u51f9\u200b\u51fd\u6570\u200b\u3002\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u200b\u5076\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u7531\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\\(f_0\\)\u200b\u7684\u200b\u5171\u8f6d\u200b\u51fd\u6570\u200b\u8868\u793a\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_3","title":"\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u5076\u200b\u95ee\u9898","text":"<p>\u200b\u8bb0\u200b\\(\\eqref{1}\\)\u200b\u7684\u200b\uff08\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\uff09\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e3a\u200b\uff1a</p> \\[ \\optim{\\max}{g(\\lambda, \\nu)}{\\lambda \\succeq 0} \\] <p>\u200b\u5176\u200b\u6700\u4f18\u200b\u89e3\u200b\\((\\lambda^*, \\nu^*)\\)\u200b\u79f0\u4e3a\u200b\u5176\u200b\u5bf9\u5076\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u662f\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u6807\u51c6\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b</p> <p>\u200b\u8003\u8651\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b</p> \\[ \\optim{\\min}{c^\\top x}{\\cases{\\begin{aligned}     &amp; Ax = b \\\\     &amp; x\\succeq 0 \\end{aligned}}} \\] <p>\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u200b\u5076\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ \\begin{aligned}     g(\\lambda, \\nu) &amp;= \\inf_{x}(c^\\top x - \\lambda^\\top x + \\nu^\\top (Ax - b)) \\\\     &amp;= \\inf_{x}(-\\nu^\\top b + (c - \\lambda + A^\\top \\nu)^\\top x) \\\\     &amp;= \\cases{\\begin{aligned}         &amp; -\\nu^\\top b &amp; c - \\lambda + A^\\top\\nu = 0 \\\\         &amp; -\\infty &amp; \\otherwise     \\end{aligned}} \\end{aligned} \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e3a\u200b</p> \\[ \\optim{\\max}{\\nu^\\top b}{\\cases{\\begin{aligned}     &amp; c - \\lambda + A^\\top \\nu = 0 \\\\     &amp; \\lambda \\succeq 0 \\end{aligned}}} \\] <p>\u200b\u5373\u200b</p> \\[ \\optim{\\max}{\\nu^\\top b}{     c + A^\\top \\nu \\succeq 0 } \\] <p>\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u5bf9\u5076\u200b\u7b49\u4ef7\u200b\u4e8e\u539f\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_4","title":"\u5f31\u200b\u5bf9\u5076\u6027","text":"<p>\u200b\u8bb0\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u4e3a\u200b\\(d^*\\)\u3002\u200b\u6839\u636e\u200b\u5bf9\u200b\u5076\u51fd\u6570\u200b\u7684\u200b\u6027\u8d28\u200b\uff0c\u200b\u6709\u200b\\(d^*\\leq p^*\\)\uff0c\u200b\u6b64\u4e3a\u200b\u5f31\u200b\u5bf9\u5076\u6027\u200b\u3002</p> <ul> <li>\u200b\u82e5\u539f\u200b\u95ee\u9898\u200b\u65e0\u200b\u4e0b\u754c\u200b\uff0c\u200b\u5219\u200b\\(p^* = -\\infty\\)\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b\\(d^* = -\\infty\\)\uff0c\u200b\u5373\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e0d\u200b\u53ef\u884c\u200b\u3002</li> <li>\u200b\u82e5\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u65e0\u4e0a\u754c\u200b\uff0c\u200b\u5219\u200b\\(d^* = \\infty\\)\uff0c\u200b\u6b64\u65f6\u200b\\(p^* = \\infty\\)\uff0c\u200b\u5373\u539f\u200b\u95ee\u9898\u200b\u4e0d\u200b\u53ef\u884c\u200b\u3002</li> </ul> <p>\u200b\u8bb0\u200b\\(p^* - d^*\\geq 0\\)\u200b\u4e3a\u200b\u5bf9\u5076\u200b\u6700\u4f18\u200b\u95f4\u9699\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_5","title":"\u5f3a\u200b\u5bf9\u5076\u6027","text":"<p>\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u5373\u200b\\(p^* - d^* = 0\\)\u3002\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u5e76\u200b\u4e0d\u662f\u200b\u603b\u200b\u6210\u7acb\u200b\uff0c\u200b\u9700\u8981\u200b\u5bf9\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u65bd\u52a0\u200b\u7ea6\u675f\u200b\u51c6\u5219\u200b\u3002</p> <p>\u200b\u82e5\u200b\uff08\u200b\u4efb\u610f\u200b\uff09\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u6ee1\u8db3\u200bSlater\u200b\u6761\u4ef6\u200b\uff0c\u200b\u5373\u200b\\(\\exists x\\in \\mathbf{relint} \\calD\\)\u200b\u4f7f\u5f97\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u4e25\u683c\u200b\u6210\u7acb\u200b\uff0c\u200b\u5373\u200b</p> \\[ \\cases{\\begin{aligned}     &amp; f_i(x)\\leq 0 &amp; i = \\oneto m \\\\     &amp; Ax = b \\end{aligned}} \\] <p>\u200b\u65f6\u200b\uff0c\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u539f\u200b\u95ee\u9898\u200b\u4e2d\u200b\u7684\u200b\u67d0\u4e2a\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u662f\u200b\u7ebf\u6027\u200b\u7684\u200b\uff0c\u200b\u5219\u200b\u8fd9\u4e2a\u200b\u7ea6\u675f\u200b\u4e0d\u200b\u9700\u8981\u200b\u6ee1\u8db3\u200b\u4e25\u683c\u200b\u6210\u7acb\u200b\u7684\u200b\u6761\u4ef6\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_6","title":"\u51e0\u4f55\u200b\u89e3\u91ca","text":"<p>\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(\\calG = \\{(f_1(x), \\ldots, f_m(x), h_1(x), \\ldots, h_p(x), f_0(x))\\in \\bbR^m\\times \\bbR^p\\times\\bbR | x\\in \\calD\\}\\)\uff0c\u200b\u5219\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u4e3a\u200b</p> \\[ p^* = \\inf_t\\{(u, v, t)\\in \\calG | u\\preceq 0, v = 0\\} \\] <p>\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u200b\u5076\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b\\(g(\\lambda, \\nu) = \\inf\\{ (\\lambda, \\nu, 1)^\\top (u, v, t) | (u, v, t)\\in \\calG\\}\\leq (\\lambda, \\nu, 1)^\\top (u, v, t)\\)\uff0c\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u8d85\u5e73\u9762\u200b\uff0c\u200b\u5176\u6cd5\u200b\u5411\u91cf\u200b\u4e3a\u200b\\((\\lambda, \\nu, 1)\\)\uff0c\u200b\u4e0e\u200b\u76f4\u7ebf\u200b\\(u=v=0\\)\u200b\u76f8\u4ea4\u200b\u4e8e\u200b\\((0, 0, g(\\lambda, \\nu))\\)\u3002\u200b\u4e5f\u200b\u5373\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u89e3\u200b\u90fd\u200b\u662f\u200b\u96c6\u5408\u200b\\(\\calG\\)\u200b\u7684\u200b\u4e00\u4e2a\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u3002</p> <p>\u200b\u5b9a\u4e49\u200b\\(\\calG\\)\u200b\u7684\u200b\u4e0a\u200b\u5883\u56fe\u200b\\(\\calA = \\calG + \\bbR_\\plus^m\\times\\{0\\}^p\\times \\bbR_\\plus\\)\uff0c\u200b\u5373\u200b</p> \\[ \\calA = \\{(u_0 + x, v_0, t_0 + z) | (u_0, v_0, t_0)\\in \\calG, x\\in \\bbR_\\plus^m, z\\in \\bbR_\\plus\\} \\] <p>\u200b\u5219\u200b\u5bf9\u5e94\u200b\u7684\u200b\u539f\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u503c\u4e3a\u200b</p> \\[ p^* = \\inf\\{t | (0, 0, t)\\in \\calA\\} \\] <p>\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\calA\\)\u200b\u4e2d\u200b\u5b58\u5728\u200b\u4e00\u7ec4\u200b\\((u, v, t)\\in \\mathbf{bd}\\calA\\)\u200b\u4f7f\u5f97\u200b\u5728\u200b\u8be5\u5904\u200b\u7684\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u4e0e\u200b\u5750\u6807\u8f74\u200b\\(u=v=0\\)\u200b\u76f8\u4ea4\u200b\u4e8e\u200b\\((0, 0, p^*)\\)\uff0c\u200b\u4e5f\u200b\u5373\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\calA\\)\u200b\u662f\u200b\u51f8\u96c6\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_7","title":"\u6700\u5927\u200b\u6700\u5c0f\u200b\u89e3\u91ca","text":"<p>\u200b\u8bbe\u200b\\(p = 0\\)\u3002\u200b\u5219\u200b\u539f\u200b\u95ee\u9898\u200b\u53ef\u4ee5\u200b\u5199\u4f5c\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b\u65e0\u7ea6\u675f\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\optim{\\min}{\\sup_{\\lambda\\succeq 0}L(x, \\lambda)}{x\\in \\bbR^n} \\] <p>\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u53ef\u4ee5\u200b\u5199\u4f5c\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\optim{\\max}{\\inf_{x} L(x, \\lambda)}{\\lambda\\succeq 0} \\] <p>\u200b\u56e0\u6b64\u200b\u5f31\u200b\u5bf9\u5076\u6027\u200b\u53ef\u4ee5\u200b\u5199\u4f5c\u200b</p> \\[ \\sup_{\\lambda\\succeq 0}\\inf_{x}L(x, \\lambda) \\leq \\inf_{x}\\sup_{\\lambda\\succeq 0}L(x, \\lambda) \\] <p>\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u53ef\u4ee5\u200b\u5199\u4f5c\u200b</p> \\[ \\sup_{\\lambda\\succeq 0}\\inf_{x}L(x, \\lambda) = \\inf_{x}\\sup_{\\lambda\\succeq 0}L(x, \\lambda) \\] <p>\u200b\u5373\u200b\\(L(x, \\lambda)\\)\u200b\u6ee1\u8db3\u200b\u978d\u70b9\u200b\u6027\u8d28\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_8","title":"\u6700\u4f18\u6027\u200b\u6761\u4ef6","text":"<p>\u200b\u5982\u679c\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u53ef\u884c\u200b\u89e3\u200b\\((\\lambda, \\nu)\\)\uff0c\u200b\u5219\u200b\u8bf4\u660e\u200b\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u200b\u6ee1\u8db3\u200b\\(p^*\\geq g(\\lambda, \\nu)\\)\u3002\u200b\u6b64\u65f6\u200b\u5bf9\u4e8e\u200b\u4e00\u4e2a\u200b\u53ef\u884c\u200b\u89e3\u200b\\(x\\)\uff0c\\(x\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(\\varepsilon = f_0(x) - g(\\lambda, \\nu)\\)-\u200b\u6b21\u4f18\u200b\u89e3\u200b\u3002\u200b\u79f0\u200b\u6b64\u5904\u200b\u7684\u200b\\(f_0(x) - g(\\lambda, \\nu)\\)\u200b\u4e3a\u200b\u5bf9\u5076\u200b\u95f4\u9699\u200b\u3002</p> <p>\u200b\u539f\u200b\u95ee\u9898\u200b\u548c\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u200b\u6ee1\u8db3\u200b</p> \\[ p^*\\in [g(\\lambda, \\nu), f_0(x)]\\qquad d^*\\in [g(\\lambda, \\nu), f_0(x)] \\] <p>\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u5f97\u51fa\u200b\u63a8\u8bba\u200b\uff1a\u200b\u82e5\u200b\\(f_0(x) = g(\\lambda, \\nu)\\)\uff0c\u200b\u5219\u200b\u6709\u200b\\(p^*=d^*=f_0(x)\\)</p>"},{"location":"math/convex-optimization/duality/#_9","title":"\u4e92\u8865\u200b\u677e\u5f1b\u200b\u6027","text":"<p>\u200b\u5f53\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\u65f6\u200b\uff0c\u200b\u4e92\u8865\u200b\u677e\u5f1b\u200b\u6027\u6307\u200b\u5bf9\u4e8e\u200b\u539f\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u89e3\u200b\\(x^*\\)\u200b\u53ca\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u89e3\u200b\\(\\lambda^* = (\\lambda_1^*, \\ldots, \\lambda_m^*)\\)\uff0c\u200b\u6ee1\u8db3\u200b</p> \\[ \\lambda_i^* &gt; 0 \\Lora f_i(x^*) = 0 \\] <p>\u200b\u5373\u200b\u5bf9\u5076\u200b\u6700\u4f18\u200b\u89e3\u4e2d\u200b\u7684\u200b\u975e\u200b\u96f6\u200b\u5206\u91cf\u200b\u5bf9\u5e94\u200b\u539f\u200b\u95ee\u9898\u200b\u4e2d\u200b\u7684\u200b\u7d27\u200b\u7ea6\u675f\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#kkt","title":"KKT\u200b\u6700\u4f18\u200b\u6761\u4ef6","text":"<p>\u200b\u8bbe\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u548c\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u5747\u200b\u53ef\u5fae\u200b\uff0c\\(x^*\\)\u200b\u4e3a\u539f\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\\(\\lambda^*, \\nu^*\\)\u200b\u4e3a\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u4e14\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\uff0c\u200b\u5219\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u51fd\u6570\u200b\\(L(x, \\lambda^*, \\nu^*)\\)\u200b\u5728\u200b\\(x^*\\)\u200b\u5904\u200b\u53d6\u5f97\u200b\u6700\u5c0f\u503c\u200b\uff0c\u200b\u5219\u200b</p> \\[ \\nabla L(x^*, \\lambda^*, \\nu^*) = 0 \\] <p>\u200b\u7efc\u5408\u200b\u5176\u5b83\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200bKKT\u200b\u6700\u4f18\u200b\u6761\u4ef6\u200b\u3002</p> \\[ \\cases{ \\begin{aligned}     f_i(x^*) &amp; \\leq 0 &amp; \\text{\uff08\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\uff09} \\\\     h_i(x^*) &amp; = 0 &amp; \\text{\uff08\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\uff09} \\\\     \\lambda_i^* &amp; \\geq 0 &amp; \\text{\uff08\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7ea6\u675f\u200b\uff09} \\\\     \\lambda_i^* f_i(x^*) &amp;= 0 &amp; \\text{\uff08\u200b\u4e92\u8865\u200b\u677e\u5f1b\u200b\u6027\u200b\uff09} \\\\     \\nabla L(x^*, \\lambda^*, \\nu^*) &amp;= 0 &amp; \\text{\uff08\u200b\u6700\u4f18\u200b\u6761\u4ef6\u200b\uff09}\\\\ \\end{aligned} } \\] <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u82e5\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\uff0c\u200b\u5219\u200b\u6700\u4f18\u200b\u89e3\u200b\u5fc5\u987b\u200b\u6ee1\u8db3\u200bKKT\u200b\u6761\u4ef6\u200b\uff08\u200b\u53cd\u8fc7\u6765\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u6210\u7acb\u200b\uff09\u3002\u200b\u5bf9\u4e8e\u200b\u6ee1\u8db3\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u6ee1\u8db3\u200bKKT\u200b\u6761\u4ef6\u200b\u7684\u200b\u70b9\u200b\u662f\u200b\u539f\u200b\u95ee\u9898\u200b\u548c\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_10","title":"\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u7684\u200b\u5bf9\u5076\u200b\u95ee\u9898","text":"<p>\u200b\u8bbe\u539f\u200b\u95ee\u9898\u200b\u4e3a\u200b</p> \\[ \\optim{\\min}{f_0(x)}{\\cases{     \\begin{aligned}         &amp; f_i(x)\\preceq_{K_i} 0 \\\\         &amp; h_i(x) = 0     \\end{aligned} }} \\] <p>\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ L(x, \\lambda, \\nu) = f_0(x) + \\sum_{i=1}^m \\lambda_i^\\top f_i(x) + \\sum_{j=1}^p \\nu_j^\\top h_j(x) \\] <p>\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e3a\u200b</p> \\[ \\optim{\\max}{g(\\lambda, \\nu) = \\inf_{x}L(x, \\lambda, \\nu)}{\\lambda_i\\succeq_{K_i^*}0} \\] <p>\u200b\u5728\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u4e0b\u200b\uff0c\u200b\u5f31\u200b\u5bf9\u5076\u6027\u200b\u603b\u662f\u200b\u6210\u7acb\u200b\u3002\u200b\u6ee1\u8db3\u200b\uff08\u200b\u5e7f\u4e49\u200b\uff09Slater\u200b\u6761\u4ef6\u200b\\(\\eqnref{4}\\)\u200b\u65f6\u200b\uff0c\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\u3002</p> \\[ \\forall i=1,\\ldots, m,\\exists x\\in \\mathbf{relint}\\calD, Ax=b, f_i(x)\\prec_{K_i}0 \\label{4} \\] <p>\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\u65f6\u200b\uff0c\u200b\u4e92\u8865\u200b\u677e\u5f1b\u200b\u6027\u200b\u548c\u200bKKT\u200b\u6761\u4ef6\u200b\u6210\u7acb\u200b\u3002</p>"},{"location":"math/convex-optimization/unconstrained-optimization/","title":"\u65e0\u7ea6\u675f\u200b\u4f18\u5316","text":"<p>\u200b\u65e0\u7ea6\u675f\u200b\u4f18\u5316\u200b\u662f\u200b\u6307\u200b\u5728\u200b\\(\\bbR^n\\)\u200b\u4e0a\u200b\u6700\u5c0f\u5316\u200b\\(f(x)\\)\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002\u200b\u4e00\u822c\u6765\u8bf4\u200b\uff0c\\(f: \\bbR^n\\ra \\bbR\\)\u200b\u662f\u200b\u4e8c\u6b21\u200b\u53ef\u5fae\u200b\u7684\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u4e14\u200b\u5728\u200b\\(\\mathbf{dom} f\\)\u200b\u4e0a\u200b\u5b58\u5728\u200b\u6700\u5c0f\u503c\u200b\\(p^*\\)\uff0c\u200b\u4e14\u200b\u552f\u4e00\u200b\u5bf9\u5e94\u200b\u4e00\u4e2a\u200b\u89e3\u4e3a\u200b\\(x^*\\)\u3002\u200b\u6839\u636e\u200b\u6700\u4f18\u6027\u200b\u6761\u4ef6\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\nabla f(x^*) = 0 \\label{1} \\] <p>\\(\\eqref{1}\\)\u200b\u6709\u200b\u4e24\u79cd\u200b\u6c42\u89e3\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5373\u200b\u8fed\u4ee3\u6cd5\u200b\u6c42\u200b\u6570\u503c\u200b\u89e3\u200b\u6216\u200b\u76f4\u63a5\u200b\u6c42\u89e3\u200b\u65b9\u7a0b\u200b\u7684\u200b\u89e3\u6790\u200b\u89e3\u200b\u3002\u200b\u5728\u200b\u90e8\u5206\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\\(\\nabla f\\)\u200b\u53ef\u80fd\u200b\u96be\u4ee5\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u6b64\u65f6\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u8fed\u4ee3\u6cd5\u200b\\(\\eqref{2}\\)\u3002\\(x^{(0)}, x^{(1)}, \\ldots\\)\u200b\u79f0\u4e3a\u200b\\(\\eqref{1}\\)\u200b\u7684\u200b\u6781\u5c0f\u200b\u5316\u70b9\u5217\u200b\u3002</p> \\[ x^{(0)}, x^{(1)}, \\ldots \\qquad \\lim_{k\\ra \\infty} f\\left(x^{(k)}\\right) = p^* \\label{2} \\] <p>\u200b\u5bf9\u4e8e\u200b\u8fed\u4ee3\u6cd5\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u4e0d\u4f1a\u200b\u5728\u200b\u6709\u9650\u200b\u65f6\u95f4\u200b\u5185\u200b\u7ed3\u675f\u200b\u3002\u200b\u4f46\u200b\u5f53\u200b\\(f\\left(x^{(k)}\\right) - p^*\\leq \\varepsilon\\)\u200b\u65f6\u200b\uff0c\u200b\u5373\u53ef\u200b\u8ba4\u4e3a\u200b\u8fbe\u5230\u200b\u6700\u4f18\u200b\u503c\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u7ec8\u6b62\u200b\u3002\u200b\u8fed\u4ee3\u200b\u7b97\u6cd5\u200b\u5bf9\u200b\u521d\u59cb\u200b\u70b9\u200b\\(x_0\\)\u200b\u6709\u200b\u4e00\u5b9a\u200b\u8981\u6c42\u200b\uff0c\u200b\u5373\u200b\\(f\\)\u200b\u5728\u200b\\(x_0\\)\u200b\u5904\u200b\u7684\u200b\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u662f\u200b\u95ed\u96c6\u200b\u3002</p>"},{"location":"math/convex-optimization/unconstrained-optimization/#_2","title":"\u5f3a\u51f8\u6027","text":"<p>\u200b\u51fd\u6570\u200b\\(f\\)\u200b\u5982\u679c\u200b\u6ee1\u8db3\u200b\u4e00\u9636\u200b\u6761\u4ef6\u200b</p> \\[ f(y)\\geq f(x) + \\nabla f(x)^\\top (y - x) + \\frac{\\gamma}{2} \\Vert y - x\\Vert^2 \\label{3} \\] <p>\u200b\u6216\u200b\u4e8c\u9636\u200b\u6761\u4ef6\u200b</p> \\[ \\nabla^2 f(x)\\succeq \\gamma I \\label{4} \\] <p>\u200b\u5219\u200b\u79f0\u200b\\(f\\)\u200b\u4e3a\u200b\\(\\gamma\\)-\u200b\u5f3a\u200b\u51f8\u51fd\u6570\u200b\u3002\u200b\u5f3a\u200b\u51f8\u51fd\u6570\u200b\u76f8\u8f83\u200b\u4e8e\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5728\u200b\u6700\u4f18\u200b\u503c\u5904\u200b\u6709\u200b\u66f4\u597d\u200b\u7684\u200b\u6027\u8d28\u200b\u3002</p> \u200b\u6b21\u4f18\u200b\u6027\u200b\u6761\u4ef6\u200b <p>\u200b\u5bf9\u4e8e\u200b\u7ed9\u5b9a\u200b\\(x\\)\uff0c\\(\\eqref{3}\\)\u200b\u662f\u200b\\(y\\)\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5bf9\u200b\u5176\u6c42\u200b\u68af\u5ea6\u200b\u5f97\u5230\u200b</p> \\[ \\nabla f(y) = \\nabla f(x) + \\gamma(y - x) \\] <p>\u200b\u4ee4\u200b\\(\\nabla f(\\tilde y) = 0\\)\uff0c\u200b\u5219\u200b\\(\\tilde y = x - \\nabla f(x) / \\gamma\\)\u3002\u200b\u6b64\u65f6\u200b\uff0c\u200b\u6709\u200b</p> \\[ f(y)\\geq f(\\tilde y)\\geq f(x) + \\nabla f(x)^\\top (\\tilde y - x) + \\frac{\\gamma}{2} \\Vert\\tilde y - x\\Vert^2 = f(x) - \\frac{1}{2\\gamma}\\Vert \\nabla f(x)\\Vert^2 \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\\(f(x) - p^*\\leq \\frac{1}{2\\gamma}\\Vert \\nabla f(x)\\Vert^2\\)\u3002\u200b\u5373\u200b\u4efb\u4f55\u200b\u68af\u5ea6\u200b\u8db3\u591f\u200b\u5c0f\u200b\u7684\u200b\u70b9\u200b\u90fd\u200b\u662f\u200b\u8fd1\u4f3c\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002</p> \\[ \\Vert\\nabla f(x)\\Vert\\leq \\sqrt{2\\gamma\\varepsilon}\\Lora f(x) - p^*\\leq \\varepsilon \\] <p>\u200b\u6216\u200b</p> \\[ \\Vert x - x^*\\Vert\\leq \\frac{2}{m}\\Vert\\nabla f(x)\\Vert \\] <p>\u200b\u663e\u7136\u200b\uff0c\\(x^*\\)\u200b\u662f\u200b\u552f\u4e00\u200b\u7684\u200b\u3002</p> <p>\u200b\u5728\u200b\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u200b\\(S\\)\u200b\u4e0a\u200b\uff0c\u200b\u4e8c\u9636\u200b\u68af\u5ea6\u200b\\(\\nabla^2 f\\)\u200b\u5b58\u5728\u200b\u4e0a\u200b\u754c\u200b\\(\\Gamma I\\)\uff0c\u200b\u5b9a\u4e49\u200b\\(\\kappa = \\Gamma / \\gamma\\)\u200b\u4e3a\u200b\\(\\nabla^2 f(x)\\)\u200b\u7684\u200b\u6761\u4ef6\u200b\u6570\u200b\u3002</p>"},{"location":"math/convex-optimization/unconstrained-optimization/#_3","title":"\u4e0b\u964d\u200b\u65b9\u6cd5","text":"<p>\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\\(\\eqref{5}\\)\u200b\u7684\u200b\u65b9\u6cd5\u200b\u6784\u9020\u200b\u70b9\u5217\u200b\uff1a</p> \\[ x^{(k+1)} = x^{(k)} + t^{(k)}\\Delta x^{(k)} \\label{5} \\] <p>\u200b\u5176\u4e2d\u200b\u6b65\u957f\u200b\\(t^{(k)} &gt; 0\\)\uff0c\\(\\Delta x\\)\u200b\u4e3a\u200b\u524d\u8fdb\u65b9\u5411\u200b\u3002\u200b\u82e5\u200b\u6ee1\u8db3\u200b\\(f{(k+1)} &lt; f{(k)}\\)\uff0c\u200b\u5219\u200b\u70b9\u5217\u200b\u6784\u6210\u200b\u4e00\u79cd\u200b\u4e0b\u964d\u200b\u65b9\u6cd5\u200b\u3002\u200b\u6839\u636e\u200b\uff08\u200b\u5f3a\u200b\uff09\u200b\u51f8\u51fd\u6570\u200b\u7684\u200b\u6027\u8d28\u200b\uff0c\\(\\Delta x^{(k)}\\)\u200b\u5fc5\u987b\u200b\u6ee1\u8db3\u200b</p> \\[ \\nabla f(x^{(k)})^\\top \\Delta x^{(k)} &lt; 0 \\] <p>\u200b\u5373\u200b\u524d\u8fdb\u65b9\u5411\u200b\u5fc5\u987b\u200b\u80cc\u5bf9\u200b\u68af\u5ea6\u65b9\u5411\u200b\u3002</p>"},{"location":"math/convex-optimization/unconstrained-optimization/#_4","title":"\u7b97\u6cd5\u200b\u8868\u8ff0","text":""},{"location":"math/discrete-mathematics/","title":"\u79bb\u6563\u6570\u5b66","text":"<p>\u200b\u6839\u636e\u200b\u300a\u200b\u6570\u7406\u903b\u8f91\u200b\u4e0e\u200b\u96c6\u5408\u8bba\u200b\u300b\uff0c\u200b\u77f3\u200b\u7eaf\u4e00\u200b\u7b49\u200b\uff0c\u200b\u6e05\u534e\u5927\u5b66\u51fa\u7248\u793e\u200b\u4e0e\u200b\u6e05\u534e\u5927\u5b66\u200b\u8ba1\u7b97\u673a\u7cfb\u200b\u201c\u200b\u79bb\u6563\u6570\u5b66\u200b\u201d\u200b\u8bfe\u7a0b\u200b\u6574\u7406\u200b\u3002</p> <ul> <li>\u200b\u7b2c\u4e00\u7ae0\u200b \u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u57fa\u672c\u6982\u5ff5\u200b</li> <li>\u200b\u7b2c\u4e8c\u7ae0\u200b \u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u7b49\u503c\u200b\u548c\u200b\u63a8\u7406\u200b\u6f14\u7b97\u200b</li> <li>\u200b\u7b2c\u4e09\u7ae0\u200b \u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u5316\u200b</li> <li>\u200b\u7b2c\u56db\u7ae0\u200b \u200b\u8c13\u8bcd\u200b\u903b\u8f91\u200b\u7684\u200b\u57fa\u672c\u6982\u5ff5\u200b</li> <li>\u200b\u7b2c\u4e94\u7ae0\u200b \u200b\u8c13\u8bcd\u200b\u903b\u8f91\u200b\u7684\u200b\u7b49\u503c\u200b\u548c\u200b\u63a8\u7406\u200b\u6f14\u7b97\u200b</li> <li>\u200b\u7b2c\u4e5d\u7ae0\u200b \u200b\u96c6\u5408\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-1/","title":"\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u57fa\u672c\u6982\u5ff5","text":"<p>\u200b\u547d\u9898\u200b\u662f\u200b\u4e00\u4e2a\u200b\u975e\u771f\u200b\u5373\u200b\u5047\u200b\u7684\u200b\u9648\u8ff0\u53e5\u200b\u3002</p> <ul> <li>\u200b\u547d\u9898\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u975e\u771f\u200b\u975e\u5047\u200b\uff0c\u200b\u4e5f\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u65e2\u200b\u771f\u200b\u53c8\u200b\u5047\u200b</li> <li>\u200b\u547d\u9898\u200b\u7684\u200b\u771f\u5047\u200b\u6027\u200b\u53ef\u4ee5\u200b\u5f85\u5b9a\u200b</li> </ul> <p>\u200b\u547d\u9898\u200b\u7684\u200b\u771f\u5047\u200b\u53ef\u4ee5\u200b\u7528\u200bTrue\u200b\u6216\u200bFalse\u200b\u8868\u793a\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200b1\u200b\u6216\u200b0\u200b\u8868\u793a\u200b</p> <p>\u200b\u547d\u9898\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4e0e\u200b\u3001\u200b\u6216\u200b\u3001\u200b\u975e\u200b\u7b49\u200b\u903b\u8f91\u200b\u8054\u7ed3\u8bcd\u200b\u8fde\u63a5\u200b\uff0c\u200b\u5f97\u5230\u200b\u590d\u5408\u200b\u547d\u9898\u200b</p> <p>\u200b\u5f53\u200b\u4e0d\u200b\u6307\u5b9a\u200b\u547d\u9898\u200b\u7684\u200b\u5177\u4f53\u5185\u5bb9\u200b\u65f6\u200b\uff0c\u200b\u79f0\u8be5\u200b\u547d\u9898\u200b\u4e3a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5177\u4f53\u200b\u547d\u9898\u200b\u79f0\u4e3a\u200b\u8be5\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u7684\u200b\u4e00\u4e2a\u200b\u89e3\u91ca\u200b\u3002\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u7684\u200b\u89e3\u91ca\u200b\u53ef\u4ee5\u200b\u662f\u200b\u4efb\u610f\u200b\u547d\u9898\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-1/#_2","title":"\u903b\u8f91\u200b\u8054\u7ed3\u8bcd","text":"<p>\u200b\u547d\u9898\u903b\u8f91\u200b\u4e2d\u200b\u5e38\u7528\u200b\u7684\u200b\u903b\u8f91\u200b\u8054\u7ed3\u8bcd\u200b\u4e3a\u200b\u4e0e\u200b\uff08\\(\\land\\)\uff0c\u200b\u5408\u53d6\u200b\uff09\u3001\u200b\u6216\u200b\uff08\\(\\lor\\)\uff0c\u200b\u6790\u53d6\u200b\uff09\u3001\u200b\u975e\u200b\uff08\\(\\lnot\\)\uff0c\u200b\u5426\u5b9a\u200b\uff09\u3001\u200b\u8574\u542b\u200b\uff08\\(\\rightarrow\\)\uff09\u3001\u200b\u53cc\u200b\u8574\u542b\u200b\uff08\\(\\leftrightarrow\\)\uff09\u200b\u7b49\u200b\u3002\u200b\u9664\u200b\\(\\lnot\\)\u200b\u4e3a\u200b\u4e00\u5143\u200b\u8fd0\u7b97\u7b26\u200b\u4ee5\u5916\u200b\uff0c\u200b\u5176\u4f59\u200b\u903b\u8f91\u200b\u8fd0\u7b97\u7b26\u200b\u5747\u200b\u4e3a\u200b\u4e8c\u5143\u200b\u8fd0\u7b97\u7b26\u200b\u3002\u200b\u8fd9\u4e9b\u200b\u8fd0\u7b97\u7b26\u200b\u7684\u200b\u771f\u503c\u8868\u200b\u5982\u4e0b\u200b\u6240\u200b\u5217\u200b\uff1a</p> <ul> <li>\u200b\u4e0e\u200b</li> </ul> \\(P\\) \\(Q\\) \\(P\\land Q\\) 0 0 0 0 1 0 1 0 0 1 1 1 <ul> <li>\u200b\u6216\u200b</li> </ul> \\(P\\) \\(Q\\) \\(P\\lor Q\\) 0 0 0 0 1 1 1 0 1 1 1 1 <ul> <li>\u200b\u975e\u200b</li> </ul> \\(P\\) \\(\\lnot P\\) 0 1 1 0 <ul> <li>\u200b\u8574\u542b\u200b</li> </ul> \\(P\\) \\(Q\\) \\(P\\rightarrow Q\\) 0 0 1 0 1 1 1 0 0 1 1 1 <ul> <li>\u200b\u53cc\u200b\u8574\u542b\u200b</li> </ul> \\(P\\) \\(Q\\) \\(P\\leftrightarrow Q\\) 0 0 1 0 1 0 1 0 0 1 1 1 <p>\u200b\u6b64\u5916\u200b\uff0c\u200b\u5982\u4e0b\u200b\u903b\u8f91\u200b\u8054\u7ed3\u8bcd\u200b\u4e5f\u200b\u8f83\u4e3a\u200b\u5e38\u7528\u200b\uff1a</p> <ul> <li>\u200b\u4e0e\u975e\u200b\uff1a\\(A\\uparrow B = \\lnot (A\\land B)\\)</li> <li>\u200b\u6216\u975e\u200b\uff1a\\(A\\downarrow B = \\lnot (A\\lor B)\\)</li> <li>\u200b\u5f02\u6216\u200b\uff1a\\(A\\overline \\lor B = (A\\lor B)\\land (\\lnot A\\lor \\lnot B)\\)</li> </ul>"},{"location":"math/discrete-mathematics/chapter-1/#_3","title":"\u5408\u5f0f\u516c\u5f0f","text":"<p>\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff08\u200b\u7b80\u79f0\u200b\u516c\u5f0f\u200b\uff09\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u89c4\u5219\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u7b80\u5355\u200b\u547d\u9898\u200b\u662f\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u82e5\u200b\\(A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\\(\\lnot A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u82e5\u200b\\(A, B\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\\(A\\land B, A\\lor B, A\\rightarrow B, A\\leftrightarrow B\\)\u200b\u5747\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\u7ecf\u8fc7\u200b\u6709\u9650\u200b\u6b21\u200b\u5e94\u7528\u200b\u524d\u200b\u4e09\u6761\u200b\u89c4\u5219\u200b\u6240\u200b\u5f97\u5230\u200b\u7684\u200b\u516c\u5f0f\u200b\u79f0\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> </ul> <p>\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4e2d\u200b\u8fd0\u7b97\u7b26\u200b\u7684\u200b\u4f18\u5148\u7ea7\u200b\u6392\u5217\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ \\lnot &gt; \\land &gt; \\lor &gt; \\rightarrow &gt; \\leftrightarrow \\] <p>\u200b\u5982\u679c\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u5728\u200b\u4efb\u610f\u200b\u89e3\u91ca\u200b\u4e0b\u200b\u90fd\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u79f0\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\u6216\u6c38\u771f\u5f0f\u200b\u3002\u200b\u5982\u679c\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u5728\u200b\u67d0\u4e2a\u200b\u89e3\u91ca\u200b\u4e0b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u79f0\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u53ef\u200b\u6ee1\u8db3\u200b\u3002\u200b\u5982\u679c\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u5728\u200b\u4efb\u610f\u200b\u89e3\u91ca\u200b\u4e0b\u200b\u90fd\u200b\u4e3a\u200b\u5047\u200b\uff0c\u200b\u79f0\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4e3a\u200b\u77db\u76fe\u200b\u5f0f\u200b\u6216\u200b\u6c38\u5047\u5f0f\u200b\u3002\u200b\u4e24\u4e2a\u200b\u91cd\u8a00\u5f0f\u200b\u4f7f\u7528\u200b\\(\\land, \\lor, \\rightarrow, \\leftrightarrow\\)\u200b\u8fde\u63a5\u200b\u540e\u200b\u5f97\u5230\u200b\u7684\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4ecd\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-1/#_4","title":"\u91cd\u8a00\u5f0f\u200b\u7684\u200b\u4ee3\u5165\u200b\u89c4\u5219","text":"<p>\u200b\u82e5\u200b\\(A\\)\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\uff0c\u200b\u5c06\u200b\\(A\\)\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\\(C\\)\u200b\u66ff\u6362\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\\(D\\)\uff0c\u200b\u5219\u200b\u66ff\u6362\u200b\u540e\u200b\u7684\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4ecd\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\u3002\u200b\u8bb0\u200b\u4f5c\u200b\\(\\frac CD\\)</p> <ul> <li>\u200b\u4ee3\u5165\u200b\u89c4\u5219\u200b\u53ea\u80fd\u200b\u5bf9\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u8fdb\u884c\u200b\u66ff\u4ee3\u200b\uff0c\u200b\u800c\u200b\u4e0d\u80fd\u200b\u66ff\u4ee3\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u200b\u590d\u5408\u200b\u547d\u9898\u200b</li> <li>\u200b\u5bf9\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u67d0\u4e2a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u8fdb\u884c\u200b\u4ee3\u5165\u200b\u65f6\u200b\uff0c\u200b\u5fc5\u987b\u200b\u4ee3\u5165\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u7684\u200b\u6240\u6709\u200b\u547d\u9898\u200b\u53d8\u9879\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-2/","title":"\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u7b49\u503c\u200b\u548c\u200b\u63a8\u7406\u200b\u6f14\u7b97","text":"<p>\u200b\u7b49\u503c\u200b\uff1a\u200b\u8bbe\u200b\\(A, B\\)\u200b\u4e3a\u200b\u516c\u5f0f\u200b\uff0c\\(P_1, \\dots, P_n\\)\u200b\u662f\u200b\u51fa\u73b0\u200b\u5728\u200b\\(A, B\\)\u200b\u4e2d\u200b\u6240\u6709\u200b\u7684\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\uff0c\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\u7684\u200b\u4e00\u7ec4\u200b\\(\\left\\langle P_1, \\dots, P_n\\right\\rangle\\)\uff0c\u200b\u90fd\u200b\u6709\u200b\\(A=B\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u4e0e\u200b\u516c\u5f0f\u200b\\(B\\)\u200b\u7b49\u503c\u200b\uff08\u200b\u7b49\u4ef7\u200b\uff09\uff0c\u200b\u8bb0\u200b\u4e3a\u200b\\(A\\Leftrightarrow B\\)\u200b\u6216\u200b\\(A=B\\)\u3002</p> <p>\u200b\u7b49\u503c\u200b\u5173\u7cfb\u200b\u6ee1\u8db3\u200b\u81ea\u53cd\u6027\u200b\u3001\u200b\u5bf9\u79f0\u6027\u200b\u4e0e\u200b\u4f20\u9012\u6027\u200b\uff1a</p> <ul> <li>\\(A=A\\)</li> <li>\u200b\u82e5\u200b\\(A=B\\)\uff0c\u200b\u5219\u200b\\(B=A\\)</li> <li>\u200b\u82e5\u200b\\(A=B, B=C\\)\uff0c\u200b\u5219\u200b\\(A=C\\)</li> </ul> <p>\\(A=B\\)\u200b\u7684\u200b\u5145\u5206\u200b\u5fc5\u8981\u6761\u4ef6\u200b\u662f\u200b\\(A\\leftrightarrow B\\)\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-2/#_2","title":"\u7b49\u503c\u200b\u516c\u5f0f","text":"<p>\u200b\u57fa\u672c\u200b\u7684\u200b\u7b49\u503c\u200b\u516c\u5f0f\u200b\u5217\u4e3e\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u53cc\u91cd\u200b\u5426\u5b9a\u200b\u5f8b\u200b</li> <li>\\(P = \\lnot \\lnot P\\)</li> <li>\u200b\u7ed3\u5408\u5f8b\u200b</li> <li>\\((P\\lor Q)\\lor R = P \\lor (Q\\lor R)\\)</li> <li>\\((P\\land Q)\\land R = P \\land (Q\\land R)\\)</li> <li>\\((P\\leftrightarrow Q)\\leftrightarrow R = P \\leftrightarrow (Q\\leftrightarrow R)\\)</li> <li>\u200b\u8fd0\u7b97\u7b26\u200b\\(\\rightarrow\\)\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u7ed3\u5408\u5f8b\u200b</li> <li>\u200b\u4ea4\u6362\u5f8b\u200b</li> <li>\\(P\\lor Q = Q\\lor P\\)</li> <li>\\(P\\land Q = Q\\land P\\)</li> <li>\\(P\\leftrightarrow Q = Q\\leftrightarrow P\\)</li> <li>\u200b\u8fd0\u7b97\u7b26\u200b\\(\\rightarrow\\)\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u4ea4\u6362\u5f8b\u200b</li> <li>\u200b\u5206\u914d\u5f8b\u200b</li> <li>\\(P\\lor (Q\\land R) = (P\\lor Q)\\land (P\\lor R)\\)</li> <li>\\(P\\land (Q\\lor R) = (P\\land Q)\\lor (P\\land R)\\)</li> <li>\\(P\\rightarrow (Q\\rightarrow R) = (P\\rightarrow Q)\\rightarrow (P\\rightarrow R)\\)</li> <li>\u200b\u8fd0\u7b97\u7b26\u200b\\(\\leftrightarrow\\)\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u5206\u914d\u5f8b\u200b</li> <li>\u200b\u6052\u7b49\u200b\u5f8b\u200b</li> <li>\\(P\\lor P = P\\)</li> <li>\\(P\\land P = P\\)</li> <li>\\(P\\rightarrow P = T\\)</li> <li>\\(P\\leftrightarrow P = T\\)</li> <li>\u200b\u5438\u6536\u7387\u200b</li> <li>\\(P\\lor (P\\land Q) = P\\)</li> <li>\\(P\\land (P\\lor Q) = P\\)</li> <li>\u200b\u6469\u6839\u200b\u5f8b\u200b</li> <li>\\(\\lnot (P\\lor Q) = \\lnot P\\land \\lnot Q\\)</li> <li>\\(\\lnot (P\\land Q) = \\lnot P\\lor \\lnot Q\\)</li> <li>\\(\\lnot (P\\rightarrow Q) = P\\land \\lnot Q\\)</li> <li>\\(\\lnot (P\\leftrightarrow Q) = (P\\land \\lnot Q) \\lor (\\lnot P\\land Q)\\)</li> <li>\u200b\u540c\u4e00\u5f8b\u200b</li> <li>\\(P\\lor F = P\\)</li> <li>\\(P\\land P = T\\)</li> <li>\\(T\\rightarrow P = P\\)</li> <li>\\(T\\leftrightarrow P = P\\)</li> <li>\\(P\\rightarrow F = \\lnot P\\)</li> <li>\\(P\\leftrightarrow F = \\lnot P\\)</li> <li>\u200b\u96f6\u5f8b\u200b</li> <li>\\(P\\lor T = T\\)</li> <li>\\(P\\land F = F\\)</li> <li>\\(P\\rightarrow T = T\\)</li> <li>\\(F\\rightarrow P = T\\)</li> <li>\u200b\u8865\u4f59\u5f8b\u200b</li> <li>\\(P\\lor \\lnot P = T\\)</li> <li>\\(P\\land \\lnot P = F\\)</li> <li>\\(P\\rightarrow \\lnot P = \\lnot P\\)</li> <li>\\(P\\leftrightarrow \\lnot P = F\\)</li> <li>\u200b\u5176\u4ed6\u200b\u5e38\u7528\u200b\u7684\u200b\u7b49\u503c\u200b\u516c\u5f0f\u200b</li> <li>\\(P\\rightarrow Q = \\lnot Q\\rightarrow \\lnot P\\)\uff08\u200b\u9006\u5426\u547d\u9898\u200b\uff09</li> <li>\\(P\\rightarrow (Q\\rightarrow R) = (P\\land Q)\\rightarrow R\\)</li> <li>\\((P\\rightarrow R)\\land (Q\\rightarrow R) = (P\\lor Q) \\rightarrow R\\)</li> </ul> <p>\u200b\u7531\u200b\u516c\u5f0f\u200b2\u200b\u7684\u200b\u76f4\u63a5\u200b\u63a8\u8bba\u200b\uff1a\\(P\\rightarrow (Q\\rightarrow R) = Q\\rightarrow (P\\rightarrow R)\\)</p> <p>\u200b\u7f6e\u6362\u200b\uff1a\u200b\u5bf9\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u7684\u200b\u5b50\u200b\u516c\u5f0f\u200b\\(B\\)\uff0c\u200b\u4f7f\u7528\u200b\u4e0e\u200b\u4e4b\u200b\u7b49\u503c\u200b\u7684\u200b\u516c\u5f0f\u200b\u8fdb\u884c\u200b\u4ee3\u6362\u200b\u79f0\u4e3a\u200b\u7f6e\u6362\u200b</p> <p>\u200b\u8bbe\u7f6e\u200b\u6362\u540e\u200b\u7684\u200b\u516c\u5f0f\u200b\u4e3a\u200b\\(C\\)\uff0c\u200b\u5219\u200b\u5fc5\u6709\u200b\\(A=C\\)</p>"},{"location":"math/discrete-mathematics/chapter-2/#_3","title":"\u4ece\u200b\u771f\u503c\u8868\u200b\u8ba1\u7b97\u200b\u751f\u6210\u200b\u547d\u9898\u200b\u516c\u5f0f","text":"<p>\u200b\u4ece\u200b\u771f\u503c\u8868\u200b\u751f\u6210\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u6709\u200b\u4e24\u79cd\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u65e2\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u503c\u4e3a\u200b\\(T\\)\u200b\u7684\u200b\u884c\u200b\u751f\u6210\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u503c\u4e3a\u200b\\(F\\)\u200b\u7684\u200b\u884c\u200b\u751f\u6210\u200b</p>"},{"location":"math/discrete-mathematics/chapter-2/#t","title":"\u4ece\u503c\u200b\u4e3a\u200bT\u200b\u7684\u200b\u884c\u200b\u751f\u6210","text":"<p>\u200b\u5bf9\u4e8e\u200b\u5305\u542b\u200b\\(n\\)\u200b\u4e2a\u200b\u547d\u9898\u200b\u53d8\u5143\u200b\\(P_1, \\dots, P_n\\)\u200b\u7684\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5199\u51fa\u200b\u5982\u4e0b\u200b\u5408\u53d6\u200b\u5f0f\u200b\uff1a</p> \\[ P_1\\land P_2\\land \\dots \\land P_n \\] <p>\u200b\u5bf9\u4e8e\u200b\u771f\u503c\u8868\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u53d6\u503c\u200b\u4e3a\u200b\\(T\\)\u200b\u7684\u200b\u884c\u200b\uff0c\u200b\u82e5\u200b\u5bf9\u5e94\u200b\u7684\u200b\\(P_i\\)\u200b\u4e3a\u200b\\(F\\)\uff0c\u200b\u5219\u200b\u5c06\u200b\u5408\u53d6\u200b\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u5bf9\u5e94\u200b\\(P_i\\)\u200b\u66ff\u6362\u200b\u4e3a\u200b\\(\\lnot P_i\\)\u3002\u200b\u5c06\u200b\u6240\u5f97\u200b\u7684\u200b\u516c\u5f0f\u200b\u4f7f\u7528\u200b\u6790\u53d6\u200b\u8fd0\u7b97\u7b26\u200b\u8fde\u63a5\u200b\u5373\u5f97\u200b\u7b26\u5408\u200b\u771f\u503c\u8868\u200b\u7684\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-2/#f","title":"\u4ece\u503c\u200b\u4e3a\u200bF\u200b\u7684\u200b\u884c\u200b\u751f\u6210","text":"<p>\u200b\u5bf9\u4e8e\u200b\u5305\u542b\u200b\\(n\\)\u200b\u4e2a\u200b\u547d\u9898\u200b\u53d8\u5143\u200b\\(P_1, \\dots, P_n\\)\u200b\u7684\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5199\u51fa\u200b\u5982\u4e0b\u200b\u6790\u53d6\u200b\u5f0f\u200b\uff1a</p> \\[ P_1\\lor P_2\\lor \\dots \\lor P_n \\] <p>\u200b\u5bf9\u4e8e\u200b\u771f\u503c\u8868\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u53d6\u503c\u200b\u4e3a\u200b\\(F\\)\u200b\u7684\u200b\u884c\u200b\uff0c\u200b\u82e5\u200b\u5bf9\u5e94\u200b\u7684\u200b\\(P_i\\)\u200b\u4e3a\u200b\\(T\\)\uff0c\u200b\u5219\u200b\u5c06\u200b\u6790\u53d6\u200b\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u5bf9\u5e94\u200b\\(P_i\\)\u200b\u66ff\u6362\u200b\u4e3a\u200b\\(\\lnot P_i\\)\u3002\u200b\u5c06\u200b\u6240\u5f97\u200b\u7684\u200b\u516c\u5f0f\u200b\u4f7f\u7528\u200b\u5408\u53d6\u200b\u8fd0\u7b97\u7b26\u200b\u8fde\u63a5\u200b\u5373\u5f97\u200b\u7b26\u5408\u200b\u771f\u503c\u8868\u200b\u7684\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-2/#_4","title":"\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6","text":"<p>\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(A\\)\u200b\u4e3a\u200b\u7531\u200b\u8054\u7ed3\u8bcd\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u82e5\u200b\u4efb\u610f\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u90fd\u200b\u6709\u200b\u901a\u8fc7\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u4f7f\u7528\u200b\\(A\\)\u200b\u4e2d\u200b\u7684\u200b\u8054\u7ed3\u8bcd\u200b\u7ec4\u5408\u800c\u6210\u200b\u7684\u200b\u516c\u5f0f\u200b\u4e0e\u200b\u4e4b\u200b\u7b49\u503c\u200b\uff0c\u200b\u5219\u200b\\(A\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b</p> <p>\u200b\u901a\u8fc7\u200b\u5982\u4e0a\u200b\u5bf9\u200b\u771f\u503c\u8868\u200b\u7684\u200b\u5206\u6790\u200b\uff0c\u200b\u4efb\u4f55\u200b\u771f\u503c\u8868\u200b\u90fd\u200b\u80fd\u200b\u4f7f\u7528\u200b\\(\\{\\land, \\lor, \\lnot\\}\\)\u200b\u4e09\u4e2a\u200b\u8fd0\u7b97\u7b26\u200b\u8868\u793a\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u3002\u200b\u5047\u8bbe\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u516c\u5f0f\u200b\\(P\\)\uff0c\u200b\u4e0d\u200b\u5b58\u5728\u200b\\(\\{\\land, \\lor, \\lnot\\}\\)\u200b\u7ec4\u6210\u200b\u7684\u200b\u516c\u5f0f\u200b\u4e0e\u200b\u4e4b\u200b\u7b49\u503c\u200b\u3002\u200b\u4f46\u200b\u901a\u8fc7\u200b\\(P\\)\u200b\u7684\u200b\u771f\u503c\u8868\u200b\u53ef\u4ee5\u200b\u6784\u9020\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\\(Q\\)\uff0c\u200b\u5219\u200b\\(Q\\)\u200b\u4e0e\u200b\\(P\\)\u200b\u7684\u200b\u771f\u503c\u8868\u200b\u76f8\u540c\u200b\uff0c\\(P=Q\\)\uff0c\u200b\u77db\u76fe\u200b\u3002\u200b\u56e0\u6b64\u200b\\(\\{\\land, \\lor, \\lnot\\}\\)\u200b\u662f\u200b\u5b8c\u5907\u200b\u7684\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u5217\u51fa\u200b\u4e86\u200b\u4e00\u90e8\u5206\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u6700\u5c0f\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff1a</p> <ul> <li>\\(\\{\\lor, \\lnot\\}\\)</li> <li>\\(\\{\\land, \\lnot\\}\\)</li> <li>\\(\\{\\uparrow\\}\\)</li> <li>\\(\\{\\downarrow\\}\\)</li> <li>\\(\\{\\lnot, \\rightarrow\\}\\)</li> </ul> <p>\u200b\u5982\u679c\u200b\\(A\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff0c\u200b\u96c6\u5408\u200b\\(B\\)\u200b\u4e2d\u200b\u7684\u200b\u8054\u7ed3\u8bcd\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u7ec4\u5408\u200b\u5f97\u5230\u200b\u4e0e\u200b\\(A\\)\u200b\u4e2d\u200b\u8054\u7ed3\u8bcd\u200b\u7b49\u4ef7\u200b\u7684\u200b\u8054\u7ed3\u8bcd\u200b\uff0c\u200b\u5219\u200b\\(B\\)\u200b\u4e5f\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\u3002</p> <p>\u200b\u8bc1\u660e\u200b\uff1a\\(\\{\\lor, \\lnot\\}\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff1a</p> \\[ P\\land Q = \\lnot (\\lnot P\\lor \\lnot Q) \\] <p>\u200b\u8bc1\u660e\u200b\uff1a\\(\\{\\land, \\lnot\\}\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff1a</p> \\[ P\\lor Q = \\lnot (\\lnot P\\land \\lnot Q) \\] <p>\u200b\u8bc1\u660e\u200b\uff1a\\(\\{\\uparrow\\}\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff1a</p> <ul> <li>\u200b\u8bc1\u660e\u200b\\(\\lnot P\\)\u200b\u53ef\u4ee5\u200b\u4ec5\u200b\u4f7f\u7528\u200b\\(\\uparrow\\)\u200b\u8fd0\u7b97\u7b26\u200b\u8868\u793a\u200b\uff1a</li> </ul> <p>$$   \\begin{aligned}   \\lnot P &amp;= \\lnot (P\\land P) \\   &amp;= P\\uparrow P   \\end{aligned}   $$</p> <ul> <li>\u200b\u8bc1\u660e\u200b\\(P\\land Q\\)\u200b\u53ef\u4ee5\u200b\u4ec5\u200b\u4f7f\u7528\u200b\\(\\uparrow\\)\u200b\u8fd0\u7b97\u7b26\u200b\u8868\u793a\u200b\uff1a</li> </ul> <p>$$   \\begin{aligned}   P\\land Q&amp;= \\lnot \\lnot (P\\land Q) \\   &amp;= \\lnot (P\\uparrow Q) \\   &amp;= (P\\uparrow Q) \\uparrow (P\\uparrow Q)   \\end{aligned}   $$</p> <p>\u200b\u8bc1\u660e\u200b\uff1a\\(\\{\\downarrow\\}\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff1a</p> <ul> <li>\u200b\u8bc1\u660e\u200b\\(\\lnot P\\)\u200b\u53ef\u4ee5\u200b\u4ec5\u200b\u4f7f\u7528\u200b\\(\\downarrow\\)\u200b\u8fd0\u7b97\u7b26\u200b\u8868\u793a\u200b\uff1a</li> </ul> <p>$$   \\begin{aligned}   \\lnot P &amp;= \\lnot (P\\lor P) \\   &amp;= P\\downarrow P   \\end{aligned}   $$</p> <ul> <li>\u200b\u8bc1\u660e\u200b\\(P\\lor Q\\)\u200b\u53ef\u4ee5\u200b\u4ec5\u200b\u4f7f\u7528\u200b\\(\\downarrow\\)\u200b\u8fd0\u7b97\u7b26\u200b\u8868\u793a\u200b\uff1a</li> </ul> <p>$$   \\begin{aligned}   P\\lor Q&amp;= \\lnot \\lnot (P\\lor Q) \\   &amp;= \\lnot (P\\downarrow Q) \\   &amp;= (P\\downarrow Q) \\downarrow (P\\downarrow Q)   \\end{aligned}   $$</p> <p>\u200b\u8bc1\u660e\u200b\uff1a\\(\\{\\rightarrow, \\lnot\\}\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff1a</p> <p>\\(P\\lor Q = \\lnot \\lnot P\\lor Q = \\lnot P\\rightarrow Q\\)</p>"},{"location":"math/discrete-mathematics/chapter-2/#_5","title":"\u5bf9\u5076\u200b\u5f0f","text":"<p>\u200b\u8bbe\u200b\\(A\\)\u200b\u4e3a\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5c06\u200b\\(A\\)\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u7684\u200b\\(\\lor, \\land, T, F\\)\u200b\u5206\u522b\u200b\u4f7f\u7528\u200b\\(\\land, \\lor, F, T\\)\u200b\u8fdb\u884c\u200b\u66ff\u6362\u200b\uff0c\u200b\u6240\u200b\u5f97\u5230\u200b\u7684\u200b\u516c\u5f0f\u200b\u79f0\u4e3a\u200b\\(A\\)\u200b\u7684\u200b\u5bf9\u5076\u200b\u5f0f\u200b\uff0c\u200b\u8bb0\u200b\u4e3a\u200b\\(A^*\\)</p> <p>\u200b\u8bbe\u200b\\(A\\)\u200b\u4e3a\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5c06\u200b\\(A\\)\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u7684\u200b\u6240\u6709\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\\(P_1,\\dots, P_n\\)\u200b\u66ff\u6362\u200b\u4e3a\u200b\\(\\lnot P_1, \\dots, \\lnot P_n\\)\uff0c\u200b\u6240\u200b\u5f97\u5230\u200b\u7684\u200b\u516c\u5f0f\u200b\u79f0\u4e3a\u200b\\(A\\)\u200b\u7684\u200b\u5426\u200b\u547d\u9898\u200b\uff0c\u200b\u8bb0\u200b\u4e3a\u200b\\(A^-\\)</p> <p>\u200b\u5219\u200b\u53d6\u200b\u5bf9\u5076\u200b\u5f0f\u200b\u8fd0\u7b97\u200b\u3001\u200b\u53d6\u5426\u200b\u547d\u9898\u200b\u8fd0\u7b97\u200b\u6709\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ol> <li>\\(\\lnot (A^*) = (\\lnot A)^*, \\lnot (A^-) = (\\lnot A)^-\\)</li> <li>\\((A^*)^*=A, (A^-)^- =A\\)</li> <li>\\(A^{*-}=\\lnot A\\)</li> <li>\\(A=B\\Rightarrow A^*=B^*\\)</li> <li>\u200b\u5bf9\u200b\\(A\\rightarrow B\\)\u200b\u53d6\u200b\u5bf9\u5076\u200b\u5f0f\u200b\u53ef\u5f97\u200b\\(A\\rightarrow B=T\\Rightarrow B^*\\rightarrow A^*=T\\)</li> </ol> <p>\u200b\u6839\u636e\u200b\u5bf9\u5076\u200b\u5f0f\u200b\u548c\u200b\u5426\u200b\u547d\u9898\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff0c2\u200b\u5f0f\u200b\u81ea\u7136\u200b\u6210\u7acb\u200b\u30021\u200b\u5f0f\u200b\u548c\u200b3\u200b\u5f0f\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u6570\u5b66\u200b\u5f52\u7eb3\u6cd5\u200b\u8fdb\u884c\u200b\u8bc1\u660e\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-2/#_6","title":"\u8303\u5f0f","text":"<p>\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\u6982\u5ff5\u200b\uff1a</p> <ul> <li>\u200b\u6587\u5b57\u200b\uff1a\u200b\u7b80\u5355\u200b\u547d\u9898\u200b\\(P\\)\u200b\u53ca\u5176\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b</li> <li>\u200b\u5408\u53d6\u200b\u5f0f\u200b\uff1a\u200b\u4e00\u4e9b\u200b\u6587\u5b57\u200b\u7684\u200b\u5408\u53d6\u200b\uff0c\u200b\u5f62\u200b\u5982\u200b\\(P_1\\land \\dots \\land P_n\\)</li> <li>\u200b\u6790\u53d6\u200b\u5f0f\u200b\uff08\u200b\u5b50\u53e5\u200b\uff09\uff1a\u200b\u4e00\u4e9b\u200b\u6587\u5b57\u200b\u7684\u200b\u6790\u53d6\u200b\uff0c\u200b\u5f62\u200b\u5982\u200b\\(P_1\\lor \\dots\\lor P_n\\)</li> <li>\u200b\u4e92\u8865\u200b\u5bf9\u200b\uff1a\u200b\u6587\u5b57\u200b\u4e0e\u5176\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b</li> <li>\u200b\u6790\u53d6\u8303\u5f0f\u200b\uff1a\u200b\u5408\u53d6\u200b\u5f0f\u200b\u901a\u8fc7\u200b\u6790\u53d6\u200b\u8054\u7ed3\u8bcd\u200b\u8fde\u63a5\u200b\u7684\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5f62\u200b\u5982\u200b\\(A_1\\lor \\dots\\lor A_n\\)\uff0c\u200b\u5176\u4e2d\u200b\\(A_i\\)\u200b\u4e3a\u200b\u5408\u53d6\u200b\u5f0f\u200b</li> <li>\u200b\u5408\u53d6\u8303\u5f0f\u200b\uff1a\u200b\u6790\u53d6\u200b\u5f0f\u200b\u901a\u8fc7\u200b\u5408\u53d6\u200b\u8054\u7ed3\u8bcd\u200b\u8fde\u63a5\u200b\u7684\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5f62\u200b\u5982\u200b\\(A_1\\land \\dots \\land A_n\\)\uff0c\u200b\u5176\u4e2d\u200b\\(A_i\\)\u200b\u4e3a\u200b\u6790\u53d6\u200b\u5f0f\u200b</li> </ul> <p>\u200b\u7531\u6b64\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5bfc\u51fa\u200b\u8303\u5f0f\u200b\u5b9a\u7406\u200b\uff0c\u200b\u5373\u200b\u4efb\u4e00\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u90fd\u200b\u5b58\u5728\u200b\u6709\u200b\u4e0e\u200b\u4e4b\u200b\u7b49\u503c\u200b\u7684\u200b\u5408\u53d6\u8303\u5f0f\u200b\u4e0e\u200b\u6790\u53d6\u8303\u5f0f\u200b\u3002\u200b\u8303\u5f0f\u200b\u5b9a\u7406\u200b\u7684\u200b\u6b63\u786e\u6027\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u5217\u5199\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u7684\u200b\u771f\u503c\u8868\u200b\u4e0e\u200b\u4ece\u200b\u771f\u503c\u8868\u200b\u5217\u5199\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u8bc1\u660e\u200b\u3002</p> <p>\u200b\u8ba1\u7b97\u200b\u5408\u53d6\u8303\u5f0f\u200b\u4e0e\u200b\u6790\u53d6\u8303\u5f0f\u200b\uff1a</p> <ol> <li>\u200b\u6d88\u200b\u53bb\u200b\u903b\u8f91\u200b\u8054\u7ed3\u8bcd\u200b\\(A\\rightarrow B\\)\u200b\u4e0e\u200b\\(A\\leftrightarrow B\\)\uff0c\u200b\u66ff\u6362\u200b\u4e3a\u200b\\(\\lnot A\\lor B\\)\u200b\u4e0e\u200b\\((\\lnot A\\lor B)\\land (A\\lor \\lnot B)\\)</li> <li>\u200b\u4f7f\u7528\u200b\u6469\u6839\u200b\u5f8b\u200b\u4e0e\u200b\u53cc\u91cd\u200b\u5426\u5b9a\u200b\u5f8b\u200b\u6d88\u9664\u200b\\(\\lor\\)\u200b\u6216\u200b\\(\\land\\)\uff0c\u200b\u5e76\u200b\u6d88\u9664\u200b\u591a\u4f59\u200b\u7684\u200b\\(\\lnot\\)</li> <li>\u200b\u5c06\u200b\u6240\u6709\u200b\u7684\u200b\\(\\lnot\\)\u200b\u79fb\u52a8\u200b\u5230\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\\(P_i\\)\u200b\u524d\u200b</li> <li>\u200b\u4f7f\u7528\u200b\u5206\u914d\u5f8b\u200b\u5c06\u200b\u516c\u5f0f\u5316\u200b\u4e3a\u200b\u8303\u5f0f\u200b</li> </ol>"},{"location":"math/discrete-mathematics/chapter-2/#_7","title":"\u4e3b\u200b\u8303\u5f0f","text":"<p>\u200b\u5bf9\u4e8e\u200b\u516c\u5f0f\u200b\\(Q = Q_1\\land \\dots \\land Q_n\\)\uff0c\u200b\u5f0f\u200b\u4e2d\u200b\\(Q_i=P_i\\)\u200b\u6216\u200b\\(Q_i=\\lnot P_i\\)\u3002\u200b\u5219\u200b\\(Q\\)\u200b\u4e3a\u200b\u6781\u5c0f\u200b\u9879\u200b\uff0c\u200b\u8bb0\u200b\u4e3a\u200b\\(m_i\\)\u3002</p> <ul> <li>\u200b\u6781\u5c0f\u200b\u9879\u4e2d\u200b\u5fc5\u987b\u200b\u51fa\u73b0\u200b\u6240\u6709\u200b\u7684\u200b\u547d\u9898\u200b\u53d8\u9879\u200b</li> <li>\\(n\\)\u200b\u4e2a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u53ef\u4ee5\u200b\u7ec4\u6210\u200b\\(2^n\\)\u200b\u4e2a\u200b\u6781\u5c0f\u200b\u9879\u200b\uff0c\u200b\u4f9d\u6b21\u200b\u8868\u793a\u200b\u4e3a\u200b\\(m_1, \\dots m_{2^n}\\)</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u516c\u5f0f\u200b\\(R = R_1\\lor \\dots \\lor R_n\\)\uff0c\u200b\u5f0f\u200b\u4e2d\u200b\\(R_i = P_i\\)\u200b\u6216\u200b\\(R_i = \\lnot P_i\\)\u3002\u200b\u5219\u200b\\(R\\)\u200b\u4e3a\u200b\u6781\u5927\u200b\u9879\u200b\uff0c\u200b\u8bb0\u200b\u4e3a\u200b\\(M_i\\)</p> <p>\u200b\u4e0e\u200b\u6781\u5c0f\u200b\u9879\u200b\u76f8\u540c\u200b\uff0c\u200b\u6240\u6709\u200b\u7684\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u5fc5\u987b\u200b\u90fd\u200b\u51fa\u73b0\u200b\u5728\u200b\u6781\u5927\u200b\u9879\u4e2d\u200b\uff0c\u200b\u4e14\u200b\\(n\\)\u200b\u4e2a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u53ef\u4ee5\u200b\u7ec4\u6210\u200b\\(2^n\\)\u200b\u4e2a\u200b\u6781\u5927\u200b\u9879\u200b\u3002</p> <p>\u200b\u4efb\u610f\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u7684\u200b\u8303\u5f0f\u200b\u4e2d\u200b\uff0c\u200b\u4ec5\u200b\u7531\u200b\u6781\u5c0f\u200b\u9879\u200b\u6784\u6210\u200b\u7684\u200b\u6790\u53d6\u8303\u5f0f\u200b\u4e3a\u4e3b\u200b\u6790\u53d6\u8303\u5f0f\u200b\uff0c\u200b\u4ec5\u200b\u7531\u200b\u6781\u5927\u200b\u9879\u200b\u6784\u6210\u200b\u7684\u200b\u5408\u53d6\u8303\u5f0f\u200b\u4e3a\u4e3b\u200b\u5408\u53d6\u8303\u5f0f\u200b\u3002\u200b\u4e3b\u200b\u6790\u53d6\u8303\u5f0f\u200b\u4e0e\u200b\u4e3b\u200b\u5408\u53d6\u8303\u5f0f\u200b\u90fd\u200b\u662f\u200b\u552f\u4e00\u200b\u7684\u200b\u3002</p> <p>\u200b\u6781\u5927\u200b\u9879\u200b\u3001\u200b\u6781\u5c0f\u200b\u9879\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ol> <li>\u200b\u6781\u5927\u200b\u9879\u200b\u3001\u200b\u6781\u5c0f\u200b\u9879\u200b\u7684\u200b\u6570\u91cf\u200b\u7b49\u4e8e\u200b\u516c\u5f0f\u200b\u53ef\u80fd\u200b\u7684\u200b\u89e3\u91ca\u200b\u6570\u91cf\u200b\uff0c\u200b\u5373\u200b\\(2^n\\)</li> <li>\u200b\u6bcf\u4e2a\u200b\u6781\u5c0f\u200b\u9879\u4ec5\u200b\u5728\u200b\u516c\u5f0f\u200b\u7684\u200b\u4e00\u4e2a\u200b\u89e3\u91ca\u200b\u4e0b\u200b\u4e3a\u200b\\(T\\)\uff0c\u200b\u6bcf\u4e2a\u200b\u6781\u5927\u200b\u9879\u4ec5\u200b\u5728\u200b\u516c\u5f0f\u200b\u7684\u200b\u4e00\u4e2a\u200b\u89e3\u91ca\u200b\u4e0b\u200b\u4e3a\u200b\\(F\\)</li> <li>\u200b\u6781\u5c0f\u200b\u9879\u200b\u3001\u200b\u6781\u5927\u200b\u9879\u200b\u4e24\u200b\u4e24\u200b\u4e0d\u200b\u7b49\u503c\u200b\uff0c\u200b\u6709\u200b\\(m_i\\land m_j\\leftrightarrow i=j, M_i\\lor M_j\\leftrightarrow i\\not= j\\)\u3002</li> <li>\u200b\u4efb\u4e00\u200b\u542b\u6709\u200b\\(n\\)\u200b\u4e2a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u7684\u200b\u516c\u5f0f\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u7528\u200b\\(k\\leq n\\)\u200b\u4e2a\u200b\u6781\u5c0f\u200b\u9879\u200b\u6216\u200b\\(k\\leq n\\)\u200b\u4e2a\u200b\u6781\u5927\u200b\u9879\u200b\u8868\u793a\u200b\u3002</li> <li>\\(\\bigwedge _{i=0} ^{2^n-1} M_i=F, \\bigvee _{i=0}^{2^n-1} m_i = T\\)</li> </ol>"},{"location":"math/discrete-mathematics/chapter-2/#_8","title":"\u63a8\u7406\u200b\u5f62\u5f0f","text":"<p>\u200b\u5c06\u200b\u81ea\u7136\u200b\u8bed\u53e5\u200b\u63cf\u8ff0\u200b\u7684\u200b\u63a8\u7406\u200b\u5173\u7cfb\u200b\u7ecf\u200b\u62bd\u8c61\u5316\u200b\u540e\u200b\u4f7f\u7528\u200b\u6761\u4ef6\u200b\u5f0f\u200b\u7684\u200b\u8868\u793a\u200b</p> <p>\u200b\u82e5\u200b\\(A\\rightarrow B=T\\)\uff0c\u200b\u79f0\u200b\\(A\\)\u200b\u91cd\u8a00\u200b\u8574\u542b\u200b\\(B\\)\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(A\\Rightarrow B\\)\u3002\u200b\u91cd\u8a00\u200b\u8574\u542b\u200b\u5173\u7cfb\u200b\u5177\u6709\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ol> <li>\u200b\u82e5\u200b\\(A=T\\)\uff0c\u200b\u4e14\u200b\\(A\\Rightarrow B\\)\uff0c\u200b\u5219\u200b\\(B=T\\)</li> <li>\u200b\u82e5\u200b\\(A\\Rightarrow B\\land B\\Rightarrow A\\)\uff0c\u200b\u5219\u200b\\(A=B\\)</li> <li>\u200b\u82e5\u200b\\(A\\Rightarrow B\\land B\\Rightarrow C\\)\uff0c\u200b\u5219\u200b\\(A\\Rightarrow C\\)</li> <li>\u200b\u82e5\u200b\\(A\\Rightarrow B\\land A\\Rightarrow C\\)\uff0c\u200b\u5219\u200b\\(A\\Rightarrow (B\\land C)\\)</li> <li>\u200b\u82e5\u200b\\(A\\Rightarrow C\\land B\\Rightarrow C\\)\uff0c\u200b\u5219\u200b\\((A\\lor B)\\Rightarrow C\\)</li> </ol> <p>\u200b\u8981\u200b\u8bc1\u660e\u200b\u4e00\u4e2a\u200b\u5f0f\u5b50\u200b\u4e3a\u91cd\u200b\u8a00\u200b\u8574\u542b\u200b\u5f0f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8bc1\u660e\u200b\\(A\\rightarrow B\\)\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\u6216\u200b\\(\\lnot A\\land B\\)\u200b\u4e3a\u200b\u77db\u76fe\u200b\u5f0f\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-2/#_9","title":"\u57fa\u672c\u200b\u7684\u200b\u63a8\u7406\u200b\u516c\u5f0f","text":"<p>\u200b\u5982\u4e0b\u200b\u5217\u51fa\u200b\u4e00\u4e9b\u200b\u57fa\u672c\u200b\u7684\u200b\u63a8\u7406\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5f0f\u200b\u4e2d\u200b\\(P,Q,R,S\\)\u200b\u5747\u200b\u4e3a\u200b\u4efb\u610f\u200b\u547d\u9898\u200b\u53d8\u9879\u200b</p> <ul> <li>\\(P\\land Q\\Rightarrow P\\)</li> <li>\\(\\lnot(P\\rightarrow Q)\\Rightarrow P\\)</li> <li>\\(\\lnot(P\\rightarrow Q)\\Rightarrow \\lnot Q\\)</li> <li>\\(P\\Rightarrow P\\lor Q\\)</li> <li>\\(\\lnot P\\Rightarrow P\\rightarrow Q\\)</li> <li>\\(Q\\Rightarrow P\\rightarrow Q\\)</li> <li>\\(\\lnot P \\land(P\\lor Q)\\Rightarrow Q\\)</li> <li>\\(P\\land (P\\rightarrow Q) \\Rightarrow Q\\)\uff0c\u200b\u5206\u79bb\u200b\u89c4\u5219\u200b</li> <li>\\(\\lnot Q\\land (P\\rightarrow Q)\\Rightarrow \\lnot P\\)</li> <li>\\((P\\rightarrow Q)\\land (Q\\rightarrow R)\\Rightarrow P\\rightarrow R\\)\uff0c\u200b\u4e09\u6bb5\u8bba\u200b</li> <li>\\((P\\leftrightarrow Q)\\land (Q\\leftrightarrow R)\\Rightarrow P\\leftrightarrow R\\)</li> <li>\\((P\\rightarrow R)\\land (Q\\rightarrow R)\\land (P\\lor Q)\\Rightarrow R\\)</li> <li>\\((R\\rightarrow Q)\\land (R\\rightarrow S)\\land (P\\lor R)\\Rightarrow Q\\lor S\\)</li> <li>\\((P\\rightarrow Q)\\land (R\\rightarrow S)\\land (\\lnot Q\\lor \\lnot S)\\Rightarrow (\\lnot P\\lor \\lnot R)\\)</li> <li>\\(Q\\rightarrow R\\Rightarrow ((P\\lor Q)\\rightarrow (P\\lor R))\\)</li> <li>\\(Q\\rightarrow R\\Rightarrow ((P\\rightarrow Q)\\rightarrow (P\\rightarrow R))\\)</li> </ul>"},{"location":"math/discrete-mathematics/chapter-2/#_10","title":"\u63a8\u7406\u200b\u6f14\u7b97","text":"<p>\u200b\u63a8\u7406\u200b\u6f14\u7b97\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u89c4\u5219\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\uff1a</p> <ul> <li>\u200b\u524d\u63d0\u200b\u5f15\u5165\u200b\u89c4\u5219\u200b\uff1a\u200b\u63a8\u7406\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u968f\u65f6\u200b\u5f15\u5165\u200b\u524d\u63d0\u200b</li> <li>\u200b\u7ed3\u8bba\u200b\u5f15\u7528\u200b\u89c4\u5219\u200b\uff1a\u200b\u63a8\u7406\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u7684\u200b\u4e2d\u95f4\u200b\u7ed3\u8bba\u200b\u53ef\u4ee5\u200b\u4f5c\u4e3a\u200b\u540e\u7eed\u200b\u63a8\u7406\u200b\u7684\u200b\u524d\u63d0\u200b</li> <li>\u200b\u4ee3\u5165\u200b\u89c4\u5219\u200b\uff1a\u200b\u63a8\u7406\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u5bf9\u200b\u91cd\u8a00\u5f0f\u200b\u7684\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4ee3\u5165\u200b\u89c4\u5219\u200b</li> <li>\u200b\u7f6e\u6362\u200b\u89c4\u5219\u200b\uff1a\u200b\u63a8\u7406\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u4efb\u4f55\u200b\u90e8\u5206\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u7528\u200b\u4e0e\u200b\u4e4b\u200b\u7b49\u503c\u200b\u7684\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u8fdb\u884c\u200b\u66ff\u6362\u200b</li> <li>\u200b\u5206\u79bb\u200b\u89c4\u5219\u200b\uff1a\u200b\u82e5\u200b\\(A\\)\u200b\u4e14\u200b\\(A\\rightarrow B\\)\u200b\u5219\u200b\\(B\\)</li> <li>\u200b\u6761\u4ef6\u200b\u8bc1\u660e\u200b\u89c4\u5219\u200b\uff1a\\(A_1\\land A_2\\Rightarrow B \\Leftrightarrow A_1\\Rightarrow A_2\\rightarrow B\\)</li> </ul>"},{"location":"math/discrete-mathematics/chapter-2/#_11","title":"\u5f52\u7ed3\u200b\u63a8\u7406","text":"<p>\u200b\u601d\u8def\u200b\uff1a\u200b\u8981\u200b\u8bc1\u660e\u200b\\(A\\Rightarrow B\\)\uff0c\u200b\u53ef\u200b\u8bc1\u660e\u200b\\(A\\land \\lnot B=F\\)\u3002</p> <ul> <li>\u200b\u5c06\u200b\\(C = A\\land \\lnot B\\)\u200b\u5316\u4e3a\u200b\u5408\u53d6\u8303\u5f0f\u200b\\(Q_1\\land \\dots \\land Q_n\\)\uff0c\u200b\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u6790\u53d6\u200b\u5f0f\u200b\u6784\u6210\u200b\u5b50\u53e5\u200b\u96c6\u5408\u200b\\(\\{Q_1,\\dots, Q_n\\}\\)</li> <li>\u200b\u6d88\u200b\u53bb\u200b\u5b50\u53e5\u200b\u96c6\u5408\u200b\u4e2d\u200b\u7684\u200b\u4e92\u8865\u200b\u5bf9\u200b\uff0c\u200b\u5982\u200b\\(P\\lor R\\)\u200b\u4e0e\u200b\\(\\lnot P\\lor \\lnot Q\\)\u200b\u5f52\u7ed3\u200b\u5f97\u5230\u200b\\(R\\lor \\lnot Q\\)\u3002\u200b\u4e0d\u65ad\u200b\u91cd\u590d\u200b\u6d88\u200b\u53bb\u200b\u8fc7\u7a0b\u200b\u3002</li> <li>\u200b\u76f4\u5230\u200b\u5b50\u53e5\u200b\u96c6\u5408\u200b\u4e3a\u7a7a\u200b\uff0c\u200b\u5f97\u5230\u200b\u7a7a\u5b50\u200b\u53e5\u200b\\(\\square\\)\uff0c\u200b\u8bc1\u660e\u200b\\(A\\land \\lnot B\\)\u200b\u662f\u200b\u77db\u76fe\u200b\u5f0f\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-3/","title":"\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u5316","text":"<p>\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u5305\u542b\u200b\u5982\u4e0b\u200b\u7ed3\u6784\u200b\uff1a</p> <ul> <li>\u200b\u521d\u59cb\u200b\u7b26\u53f7\u200b\uff1a\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u5141\u8bb8\u200b\u51fa\u73b0\u200b\u7684\u200b\u7b26\u53f7\u200b\u96c6\u5408\u200b</li> <li>\u200b\u5f62\u6210\u200b\u89c4\u5219\u200b\uff1a\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u7b26\u53f7\u200b\u5e8f\u5217\u200b\u7684\u200b\u5408\u6cd5\u6027\u200b</li> <li>\u200b\u516c\u7406\u200b\uff1a\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u6700\u200b\u57fa\u672c\u200b\u7684\u200b\u91cd\u8a00\u5f0f\u200b\uff0c\u200b\u662f\u200b\u63a8\u6f14\u200b\u5176\u4ed6\u200b\u6240\u6709\u200b\u91cd\u8a00\u5f0f\u200b\u7684\u200b\u4f9d\u636e\u200b</li> <li>\u200b\u53d8\u5f62\u200b\u89c4\u5219\u200b\uff1a\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u6240\u200b\u89c4\u5b9a\u200b\u7684\u200b\u63a8\u6f14\u200b\u89c4\u5219\u200b\uff0c\u200b\u516c\u7406\u200b\u901a\u8fc7\u200b\u53d8\u5f62\u200b\u89c4\u5219\u200b\u5f97\u5230\u200b\u7684\u200b\u516c\u5f0f\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u5b9a\u7406\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-3/#_2","title":"\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u200b\u7cfb\u7edf","text":"<p>\u200b\u4f9d\u636e\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u7684\u200b\u7ed3\u6784\u200b\u5efa\u7acb\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u3002\u200b\u5e94\u5f53\u200b\u6ce8\u610f\u200b\u5230\u200b\uff0c\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u4e0d\u662f\u200b\u552f\u4e00\u200b\u7684\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-3/#_3","title":"\u521d\u59cb\u200b\u7b26\u53f7","text":"<p>\u200b\u5927\u5199\u200b\u82f1\u6587\u5b57\u6bcd\u200b\\(A,\\dots ,Z\\)\u200b\u8868\u793a\u200b\u547d\u9898\u200b \\(\\lnot, \\lor\\)\u200b\u8868\u793a\u200b\u8054\u7ed3\u8bcd\u200b \\((, )\\)\u200b\u4e00\u5bf9\u200b\u62ec\u53f7\u200b\u89c4\u5b9a\u200b\u4e86\u200b\u8054\u7ed3\u8bcd\u200b\u8fd0\u7b97\u200b\u7684\u200b\u4f18\u5148\u200b\u987a\u5e8f\u200b \\(\\vdash\\)\u200b\u8868\u660e\u200b\u4e00\u4e2a\u200b\u516c\u5f0f\u200b\u662f\u200b\u6c38\u771f\u5f0f\u200b</p>"},{"location":"math/discrete-mathematics/chapter-3/#_4","title":"\u5f62\u6210\u200b\u89c4\u5219","text":"<p>\u200b\u79f0\u200b\u521d\u59cb\u200b\u7b26\u53f7\u200b\u6784\u6210\u200b\u7684\u200b\u7b26\u53f7\u200b\u5e8f\u5217\u200b\u4e3a\u200b\u516c\u5f0f\u200b\uff0c\u200b\u53ea\u6709\u200b\u7b26\u5408\u200b\u4ee5\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u7b26\u53f7\u200b\u5e8f\u5217\u200b\u662f\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff1a</p> <ol> <li>\u200b\u5355\u4e2a\u200b\u7b26\u53f7\u200b\\(\\pi\\)\uff0c\\(\\pi\\)\u200b\u662f\u200b\u5927\u5199\u200b\u82f1\u6587\u5b57\u6bcd\u200b</li> <li>\u200b\u82e5\u200b\\(A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\\(\\lnot A\\)\u200b\u662f\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u82e5\u200b\\(A, B\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\\((A\\lor B)\\)\u200b\u662f\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> </ol> <p>\u200b\u4e3a\u200b\u7b80\u5316\u200b\u5176\u4ed6\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u7684\u200b\u8868\u8fbe\u200b\uff0c\u200b\u5f15\u5165\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\u7684\u200b\u516c\u5f0f\u200b\uff1a</p> <ul> <li>\\((A\\rightarrow B):= (\\lnot A\\lor B)\\)</li> <li>\\((A\\land B):= \\lnot (\\lnot A\\lor \\lnot B)\\)</li> <li>\\((A\\leftrightarrow B):= ((A\\rightarrow B)\\land (B\\rightarrow A))\\)</li> </ul>"},{"location":"math/discrete-mathematics/chapter-3/#_5","title":"\u516c\u7406","text":"<p>\u200b\u5728\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u5f15\u5165\u200b\u5982\u4e0b\u200b\u516c\u7406\u200b\uff1a</p> <ul> <li>\\(\\vdash ((P\\lor P)\\rightarrow P)\\)</li> <li>\\(\\vdash (P\\rightarrow (P\\lor Q))\\)</li> <li>\\(\\vdash ((P\\lor Q) \\rightarrow (Q\\lor P))\\)</li> <li>\\(\\vdash ((Q\\rightarrow R)\\rightarrow ((P\\lor Q)\\rightarrow (P\\lor R)))\\)</li> </ul>"},{"location":"math/discrete-mathematics/chapter-3/#_6","title":"\u63a8\u7406\u200b\u89c4\u5219","text":"<p>\u200b\u4ee3\u5165\u200b\u89c4\u5219\u200b\u3001\u200b\u5206\u79bb\u200b\u89c4\u5219\u200b\u3001\u200b\u7f6e\u6362\u200b\u89c4\u5219\u200b\u662f\u200b\u547d\u9898\u903b\u8f91\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u7684\u200b\u63a8\u7406\u200b\u89c4\u5219\u200b</p>"},{"location":"math/discrete-mathematics/chapter-3/#_7","title":"\u738b\u6d69\u200b\u7b97\u6cd5","text":""},{"location":"math/discrete-mathematics/chapter-3/#_8","title":"\u521d\u59cb\u200b\u7b26\u53f7","text":"<p>\u200b\u5927\u5199\u5b57\u6bcd\u200b\\(A, \\dots, Z\\)\u200b\u8868\u793a\u200b\u547d\u9898\u200b \\(\\lnot, \\land, \\lor, \\rightarrow, \\leftrightarrow\\)\u200b\u8868\u793a\u200b\u8054\u7ed3\u8bcd\u200b \\((, ), ,\\)\u200b\u8868\u793a\u200b\u5706\u62ec\u53f7\u200b\u4e0e\u200b\u9017\u53f7\u200b \\(\\alpha, \\beta, \\dots, \\omega\\)\u200b\u8868\u793a\u200b\u516c\u5f0f\u200b\u4e32\u200b</p>"},{"location":"math/discrete-mathematics/chapter-3/#_9","title":"\u5f62\u6210\u200b\u89c4\u5219","text":"<p>\u200b\u7b26\u5408\u200b\u4ee5\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u7b26\u53f7\u200b\u5e8f\u5217\u200b\u79f0\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff1a</p> <ol> <li>\u200b\u5355\u4e2a\u200b\u7b26\u53f7\u200b\\(\\pi\\)\uff0c\\(\\pi\\)\u200b\u662f\u200b\u5927\u5199\u200b\u82f1\u6587\u5b57\u6bcd\u200b</li> <li>\u200b\u82e5\u200b\\(A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\u200b\u5219\u200b\\(\\lnot A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u82e5\u200b\\(A, B\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\u200b\u5219\u200b\\((A\\lnot B), (A\\lor B), (A\\rightarrow B), (A\\leftrightarrow B)\\)\u200b\u662f\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> </ol> <p>\u200b\u6ee1\u8db3\u200b\u4ee5\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u79f0\u4e3a\u200b\u516c\u5f0f\u200b\u4e32\u200b\uff1a</p> <ol> <li>\u200b\u7a7a\u200b\u7b26\u53f7\u4e32\u200b\u662f\u200b\u516c\u5f0f\u200b\u4e32\u200b</li> <li>\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u662f\u200b\u516c\u5f0f\u200b\u4e32\u200b</li> <li>\u200b\u5982\u679c\u200b\\(\\alpha\\)\u200b\u548c\u200b\\(\\beta\\)\u200b\u662f\u200b\u516c\u5f0f\u200b\u4e32\u200b\uff0c\u200b\u5219\u200b\\(\\alpha, \\beta\\)\u200b\u662f\u200b\u516c\u5f0f\u200b\u4e32\u200b</li> </ol> <p>\u200b\u516c\u5f0f\u200b\u4e32\u200b\u4e0e\u200b\u7ec4\u6210\u200b\u516c\u5f0f\u200b\u4e32\u200b\u7684\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u7684\u200b\u987a\u5e8f\u200b\u65e0\u5173\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-3/#_10","title":"\u5b9a\u4e49","text":"<ol> <li>\u200b\u5b9a\u4e49\u200b\u76f8\u7ee7\u200b\u5f0f\u200b\u4e3a\u200b\\(\\alpha\\stackrel s\\rightarrow\\beta\\)\uff0c\u200b\u5176\u4e2d\u200b\\(\\alpha, \\beta\\)\u200b\u90fd\u200b\u662f\u200b\u516c\u5f0f\u200b\u4e32\u200b</li> <li>\u200b\u524d\u4ef6\u200b\u4e2d\u200b\u7684\u200b\\(,\\)\u200b\u89e3\u91ca\u200b\u4e3a\u200b\\(\\land\\)\uff0c\u200b\u540e\u4ef6\u200b\u4e2d\u200b\u7684\u200b\\(,\\)\u200b\u89e3\u91ca\u200b\u4e3a\u200b\\(\\lor\\)</li> <li>\u200b\u82e5\u200b\\(\\alpha\\stackrel s\\rightarrow \\beta\\)\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8bb0\u200b\u4f5c\u200b\\(\\alpha\\stackrel s\\Rightarrow \\beta\\)</li> </ol>"},{"location":"math/discrete-mathematics/chapter-3/#_11","title":"\u516c\u7406","text":"<p>\u200b\u82e5\u200b\\(\\alpha\\)\u200b\u548c\u200b\\(\\beta\\)\u200b\u7684\u200b\u516c\u5f0f\u200b\u4ec5\u4e3a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\uff0c\u200b\u4e0d\u200b\u5305\u542b\u200b\u8054\u7ed3\u8bcd\u200b\uff0c\u200b\u5219\u200b\\(\\alpha\\stackrel s\\rightarrow \\beta\\)\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\\(\\alpha\\)\u200b\u4e0e\u200b\\(\\beta\\)\u200b\u4e2d\u200b\u81f3\u5c11\u200b\u542b\u6709\u200b\u4e00\u4e2a\u200b\u76f8\u540c\u200b\u7684\u200b\u547d\u9898\u200b\u53d8\u9879\u200b</p>"},{"location":"math/discrete-mathematics/chapter-3/#_12","title":"\u63a8\u7406\u200b\u89c4\u5219","text":"<p>\u200b\u5bf9\u4e8e\u200b\u5b58\u5728\u200b\u4e8e\u200b\u524d\u4ef6\u200b\u4e2d\u200b\u7684\u200b\u8fd0\u7b97\u7b26\u200b\uff0c\u200b\u63a8\u7406\u200b\u89c4\u5219\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\\(\\lnot\\)\uff1a\u200b\u82e5\u200b\\(\\alpha, \\beta \\stackrel s\\Rightarrow X, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha, \\lnot X, \\beta \\stackrel x\\Rightarrow \\gamma\\)</li> <li>\\(\\land\\)\uff1a\u200b\u82e5\u200b\\(X, Y, \\alpha, \\beta\\stackrel s\\Rightarrow \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha, X\\land Y, \\beta\\stackrel s\\Rightarrow Y\\)</li> <li>\\(\\lor\\)\uff1a\u200b\u82e5\u200b\\(X, \\alpha, \\beta \\stackrel s\\Rightarrow \\gamma\\)\u200b\u4e14\u200b\\(Y, \\alpha, \\beta \\stackrel s\\Rightarrow X, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha, X\\lor Y, \\beta\\stackrel s\\Rightarrow \\gamma\\)</li> <li>\\(\\rightarrow\\)\uff1a\u200b\u82e5\u200b\\(Y, \\alpha, \\beta\\stackrel s\\Rightarrow \\gamma\\)\u200b\u4e14\u200b\\(\\alpha, \\beta\\stackrel s\\Rightarrow X, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha, X\\rightarrow Y, \\beta\\stackrel s\\Rightarrow \\gamma\\)</li> <li>\\(\\leftrightarrow\\)\uff1a\u200b\u82e5\u200b\\(X, Y, \\alpha, \\beta\\stackrel s\\Rightarrow \\gamma\\)\u200b\u4e14\u200b\\(\\alpha, \\beta\\stackrel s\\Rightarrow X, Y, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha, X\\leftrightarrow Y, \\beta\\stackrel s\\Rightarrow \\gamma\\)</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u5b58\u5728\u200b\u4e8e\u200b\u540e\u4ef6\u200b\u4e2d\u200b\u7684\u200b\u8fd0\u7b97\u7b26\u200b\uff0c\u200b\u63a8\u7406\u200b\u89c4\u5219\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\\(\\lnot\\)\uff1a\u200b\u82e5\u200b\\(X, \\alpha\\stackrel s\\Rightarrow \\beta, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha\\stackrel s\\Rightarrow\\beta, \\lnot X, \\gamma\\)</li> <li>\\(\\land\\)\uff1a\u200b\u82e5\u200b\\(\\alpha\\stackrel s\\Rightarrow X, \\beta, \\gamma\\)\u200b\u4e14\u200b\\(\\alpha\\stackrel s\\Rightarrow\\)</li> <li>\\(\\land\\)\uff1a\u200b\u82e5\u200b\\(\\alpha\\stackrel s\\Rightarrow X, \\beta, \\gamma\\)\u200b\u4e14\u200b\\(\\alpha\\stackrel s\\Rightarrow Y, \\beta, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha \\stackrel s\\Rightarrow\\beta, X\\land Y, \\gamma\\)</li> <li>\\(\\lor\\)\uff1a\u200b\u82e5\u200b\\(\\alpha\\stackrel s\\Rightarrow X, Y, \\beta, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha\\stackrel s\\Rightarrow \\beta, X\\lor Y, \\gamma\\)</li> <li>\\(\\rightarrow\\)\uff1a\u200b\u82e5\u200b\\(X, \\alpha\\stackrel s\\Rightarrow Y, \\beta, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha\\stackrel s\\Rightarrow \\beta, X\\rightarrow Y, \\gamma\\)</li> <li>\\(\\leftrightarrow\\)\uff1a\u200b\u82e5\u200b\\(X, \\alpha\\stackrel s\\Rightarrow Y, \\beta, \\gamma\\)\u200b\u4e14\u200b\\(Y, \\alpha\\stackrel s\\Rightarrow X, \\beta, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha \\stackrel s\\Rightarrow \\beta, X\\leftrightarrow Y, \\gamma\\)</li> </ul> <p>\u200b\u5b9a\u7406\u200b\u7684\u200b\u63a8\u6f14\u200b\u8fc7\u7a0b\u200b\uff1a</p> <ol> <li>\u200b\u5c06\u8981\u200b\u8bc1\u660e\u200b\u7684\u200b\u5b9a\u7406\u200b\u5199\u6210\u200b\u76f8\u7ee7\u200b\u5f0f\u200b\u5f62\u5f0f\u200b</li> <li>\u200b\u4ece\u200b\u76f8\u7ee7\u200b\u5f0f\u200b\u51fa\u53d1\u200b\u4f7f\u7528\u200b\u53d8\u5f62\u200b\u89c4\u5219\u200b\u6d88\u200b\u53bb\u200b\u76f8\u7ee7\u200b\u5f0f\u200b\u4e24\u4fa7\u200b\u7684\u200b\u6240\u6709\u200b\u903b\u8f91\u200b\u8054\u7ed3\u8bcd\u200b</li> <li>\u200b\u82e5\u200b\u5f97\u5230\u200b\u7684\u200b\u6240\u6709\u200b\u76f8\u7ee7\u200b\u5f0f\u200b\u90fd\u200b\u662f\u200b\u516c\u7406\u200b\uff0c\u200b\u5219\u200b\u5b9a\u7406\u200b\u6210\u7acb\u200b</li> </ol>"},{"location":"math/discrete-mathematics/chapter-3/#_13","title":"\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u81ea\u7136\u200b\u6f14\u7ece\u200b\u7cfb\u7edf","text":""},{"location":"math/discrete-mathematics/chapter-3/#_14","title":"\u521d\u59cb\u200b\u7b26\u53f7","text":"<p>\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u81ea\u7136\u200b\u6f14\u7ece\u200b\u7cfb\u7edf\u200b\u7684\u200b\u4e00\u90e8\u5206\u200b\u521d\u59cb\u200b\u7b26\u53f7\u200b\u7ee7\u627f\u200b\u81ea\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u3002\u200b\u6b64\u5916\u200b\uff0c\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u81ea\u7136\u200b\u6f14\u7ece\u200b\u7cfb\u7edf\u200b\u8fd8\u200b\u5305\u542b\u200b\u5982\u4e0b\u200b\u521d\u59cb\u200b\u7b26\u53f7\u200b\uff1a</p> \\[ \\varGamma = \\{A_1, \\dots, A_n\\}=A_1, \\dots, A_n \\] <p>\u200b\u8868\u793a\u200b\u6709\u9650\u200b\u4e2a\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u96c6\u5408\u200b\uff0c\u200b\u5b9a\u4e49\u200b\\(\\varGamma \\vdash A\\)\u200b\u8868\u793a\u200b\\(\\varGamma, A\\)\u200b\u4e4b\u95f4\u200b\u6709\u200b\u5f62\u5f0f\u200b\u63a8\u7406\u200b\u5173\u7cfb\u200b\uff0c\\(\\varGamma\\)\u200b\u4e3a\u200b\u5f62\u5f0f\u200b\u524d\u63d0\u200b\uff0c\\(A\\)\u200b\u4e3a\u200b\u5f62\u5f0f\u200b\u7ed3\u8bba\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-3/#_15","title":"\u5f62\u6210\u200b\u89c4\u5219","text":"<p>\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u81ea\u7136\u200b\u6f14\u7ece\u200b\u7cfb\u7edf\u200b\u7684\u200b\u5f62\u6210\u200b\u89c4\u5219\u200b\u4e0e\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u7684\u200b\u5f62\u6210\u200b\u89c4\u5219\u200b\u76f8\u540c\u200b</p>"},{"location":"math/discrete-mathematics/chapter-3/#_16","title":"\u53d8\u5f62\u200b\u89c4\u5219","text":"<p>\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u81ea\u7136\u200b\u6f14\u7ece\u200b\u7cfb\u7edf\u200b\u6709\u200b\u5982\u4e0b\u200b\u53d8\u5f62\u200b\u89c4\u5219\u200b\uff1a</p> <ol> <li>\\(A_1, \\dots, A_n\\vdash A_i(i=1, \\dots, n)\\)\uff0c\u200b\u5373\u200b\u524d\u63d0\u200b\u4e2d\u200b\u7684\u200b\u4efb\u4f55\u200b\u547d\u9898\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u4f5c\u4e3a\u200b\u7ed3\u8bba\u200b\u51fa\u73b0\u200b</li> <li>\u200b\u4f20\u9012\u200b\u5f8b\u200b\uff1a\u200b\u82e5\u200b\\(\\varGamma \\vdash A\\)\u200b\u4e14\u200b\\(A\\vdash B\\)\uff0c\u200b\u5219\u200b\\(\\varGamma\\vdash B\\)</li> <li>\u200b\u53cd\u8bc1\u200b\u5f8b\u200b\uff1a\u200b\u82e5\u200b\\(\\varGamma, \\lnot A\\vdash B\\)\u200b\u4e14\u200b\\(\\varGamma, \\lnot A\\vdash \\lnot B\\)\uff0c\u200b\u5219\u200b\\(\\varGamma \\vdash A\\)\uff08\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u8bc1\u660e\u200b\\(\\varGamma \\land \\lnot A=F\\)\uff09</li> <li>\u200b\u5206\u79bb\u200b\u89c4\u5219\u200b\uff1a\\(A, A\\rightarrow B\\vdash B\\)</li> <li>\u200b\u8574\u542b\u200b\u8bcd\u200b\u5f15\u5165\u200b\uff1a\u200b\u82e5\u200b\\(\\varGamma, A\\vdash B\\)\uff0c\u200b\u5219\u200b\\(\\varGamma\\vdash A\\rightarrow B\\)</li> </ol>"},{"location":"math/discrete-mathematics/chapter-3/#_17","title":"\u5b9a\u7406","text":"<p>\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u81ea\u7136\u200b\u6f14\u7ece\u200b\u7cfb\u7edf\u200b\u5305\u542b\u200b\u7684\u200b\u5b9a\u7406\u200b\u4e0e\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u5305\u542b\u200b\u7684\u200b\u5b9a\u7406\u200b\u76f8\u540c\u200b</p>"},{"location":"math/discrete-mathematics/chapter-3/#_18","title":"\u975e\u6807\u51c6\u200b\u903b\u8f91","text":"<ol> <li>\u200b\u591a\u503c\u903b\u8f91\u200b\uff0c\u200b\u82e5\u200b\\(P\\)\u200b\u53d6\u503c\u200b\u4e8e\u200b\\([0,1]\\)\uff0c\u200b\u53ef\u4ee5\u200b\u89e3\u91ca\u200b\u4e3a\u200b<ul> <li>\\(P=0\\)\u200b\u8868\u793a\u200b\\(P\\)\u200b\u771f\u200b</li> <li>\\(P=1\\)\u200b\u8868\u793a\u200b\\(P\\)\u200b\u5047\u200b</li> <li>\\(0&lt;P&lt;1\\)\u200b\u8868\u793a\u200b\\(P\\)\u200b\u4e3a\u200b\u771f\u7684\u200b\u6982\u7387\u200b\u4e3a\u200b\\(1-P\\) </li> </ul> </li> <li>\u200b\u4e09\u503c\u200b\u903b\u8f91\u200b<ul> <li>Kleene\u200b\u903b\u8f91\u200b\uff1a\\(T, F, U\\)\uff0c\\(U\\)\u200b\u8868\u793a\u200b\u672a\u77e5\u200b</li> <li>Lukasiewicz\u200b\u903b\u8f91\u200b\uff1a\\(T, F, I\\)\uff0c\\(I\\)\u200b\u8868\u793a\u200b\u5c06\u6765\u200b\u53ef\u80fd\u200b\uff0c\u200b\u76ee\u524d\u200b\u4e0d\u200b\u5177\u5907\u200b\u771f\u503c\u200b</li> <li>Bochvar\u200b\u903b\u8f91\u200b\uff1a\\(T, F, M\\)\uff0c\\(M\\)\u200b\u8868\u793a\u200b\u6096\u8bba\u200b</li> </ul> </li> </ol>"},{"location":"math/discrete-mathematics/chapter-4/","title":"\u8c13\u8bcd\u200b\u903b\u8f91\u200b\u7684\u200b\u57fa\u672c\u6982\u5ff5","text":"<p>\u200b\u8c13\u8bcd\u200b\u8868\u793a\u200b\u4e86\u200b\u591a\u4e2a\u200b\u4e2a\u4f53\u200b\u8bcd\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5173\u7cfb\u200b\u3002\u200b\u8fdb\u4e00\u6b65\u200b\u5730\u200b\u8bb2\u200b\u662f\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u4e2a\u4f53\u200b\u8bcd\u5230\u200b\u96c6\u5408\u200b\\(\\{T, F\\}\\)\u200b\u4e0a\u200b\u7684\u200b\u4e00\u4e2a\u200b\u6620\u5c04\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5176\u200b\u5199\u200b\u4e3a\u200b\u51fd\u6570\u200b\u7684\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u5982\u200b\\(P(x, y)\\)</p> <p>\u200b\u51fd\u6570\u200b\u8868\u793a\u200b\u4ece\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\u5230\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\u7684\u200b\u6620\u5c04\u200b\uff0c\u200b\u901a\u5e38\u200b\u4f7f\u7528\u200b\u5c0f\u5199\u5b57\u6bcd\u200b\u8fdb\u884c\u200b\u8868\u793a\u200b\u3002</p> <p>\u200b\u6709\u200b\u4e24\u79cd\u200b\u91cf\u8bcd\u200b\uff0c\u200b\u5373\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u4e0e\u200b\u5b58\u5728\u200b\u91cf\u8bcd\u200b\u3002\u200b\u53d7\u5230\u200b\u91cf\u8bcd\u200b\u7ea6\u675f\u200b\u7684\u200b\u53d8\u5143\u200b\u79f0\u4e3a\u200b\u7ea6\u675f\u200b\u53d8\u5143\u200b\uff0c\u200b\u5426\u5219\u200b\u79f0\u4e3a\u200b\u81ea\u7531\u200b\u53d8\u5143\u200b\u3002\u200b\u91cf\u8bcd\u200b\u53ea\u80fd\u200b\u4f5c\u7528\u200b\u4e8e\u200b\u53d8\u5143\u200b\u800c\u200b\u4e0d\u80fd\u200b\u4f5c\u7528\u200b\u4e8e\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u4e0e\u200b\u8c13\u8bcd\u200b\u53d8\u9879\u200b\u3002\u200b\u53d8\u5143\u200b\u4f7f\u7528\u200b\u5c0f\u5199\u5b57\u6bcd\u200b\u8868\u793a\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-4/#_2","title":"\u5408\u5f0f\u516c\u5f0f","text":"<p>\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff1a</p> <ol> <li>\u200b\u547d\u9898\u200b\u5e38\u9879\u200b\u3001\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u548c\u200b\u539f\u5b50\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u90fd\u200b\u662f\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u82e5\u200b\\(A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\u200b\u5219\u200b\\(\\lnot A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u82e5\u200b\\(A, B\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\u200b\u4e14\u200b\\(A, B\\)\u200b\u4e2d\u65e0\u53d8\u200b\u5143\u200b\u5728\u200b\u4e00\u4e2a\u200b\u4e2d\u200b\u7ea6\u675f\u200b\u800c\u200b\u5728\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u4e2d\u200b\u81ea\u7531\u200b\uff0c\u200b\u5219\u200b\\((A\\land B), (A\\lor B), (A\\rightarrow B), (A\\leftrightarrow B)\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u3002</li> <li>\u200b\u82e5\u200b\\(A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\\(x\\)\u200b\u5728\u200b\\(A\\)\u200b\u4e2d\u4e3a\u200b\u81ea\u7531\u200b\u53d8\u5143\u200b\uff0c\u200b\u5219\u200b\\((\\forall x)A, (\\exists x)A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> </ol> <p>\u200b\u5728\u200b\u6709\u9650\u200b\u8bba\u57df\u200b\u4e0b\u200b\uff0c\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u4e0e\u200b\u7279\u79f0\u200b\u91cf\u8bcd\u200b\u53ef\u200b\u4e0e\u200bi\u200b\u8868\u793a\u200b\u4e3a\u200b\u547d\u9898\u903b\u8f91\u200b\u4e0b\u200b\u7684\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff1a</p> <ul> <li>\\((\\forall x)P(x) = P(1)\\land P(2)\\land \\dots \\land P(k)\\)</li> <li>\\((\\exists x)P(x) = P(1)\\lor P(2)\\lor \\dots \\lor P(k)\\)\\</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\uff0c\u200b\u82e5\u200b\u5728\u200b\u4efb\u4f55\u200b\u89e3\u91ca\u200b\u4e0b\u200b\u771f\u503c\u200b\u90fd\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u79f0\u8c13\u200b\u8bcd\u200b\u516c\u5f0f\u200b\u4e3a\u200b\u666e\u904d\u200b\u6709\u6548\u200b\u7684\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\uff1b\u200b\u82e5\u200b\u5728\u200b\u67d0\u4e2a\u200b\u89e3\u91ca\u200b\u4e0b\u200b\u771f\u503c\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u79f0\u8c13\u200b\u8bcd\u200b\u516c\u5f0f\u200b\u4e3a\u200b\u53ef\u200b\u6ee1\u8db3\u200b\u7684\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5426\u5219\u200b\u79f0\u4e3a\u200b\u4e0d\u53ef\u200b\u6ee1\u8db3\u200b\u7684\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u3002</p> <ul> <li>\u200b\u8c13\u8bcd\u200b\u903b\u8f91\u200b\u662f\u200b\u4e0d\u53ef\u5224\u5b9a\u200b\u7684\u200b</li> <li>\u200b\u53ea\u200b\u542b\u6709\u200b\u4e00\u5143\u200b\u8c13\u8bcd\u200b\u53d8\u9879\u200b\u7684\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u662f\u200b\u53ef\u200b\u5224\u5b9a\u200b\u7684\u200b</li> <li>\u200b\u4e2a\u4f53\u200b\u57df\u200b\u6709\u9650\u200b\u7684\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u662f\u200b\u53ef\u200b\u5224\u5b9a\u200b\u7684\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-5/","title":"\u8c13\u8bcd\u200b\u903b\u8f91\u200b\u7684\u200b\u7b49\u503c\u200b\u548c\u200b\u63a8\u7406\u200b\u6f14\u7b97","text":"<p>\u200b\u5bf9\u4e8e\u200b\u4e24\u4e2a\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\\(A, B\\)\uff0c\u200b\u82e5\u200b\u5728\u200b\\(A, B\\)\u200b\u7684\u200b\u4efb\u4e00\u200b\u89e3\u91ca\u200b\u4e0b\u200b\uff0c\\(A, B\\)\u200b\u90fd\u200b\u6709\u200b\u76f8\u540c\u200b\u7684\u200b\u771f\u503c\u200b\uff0c\u200b\u5373\u200b\\(A\\leftrightarrow B\\)\uff0c\u200b\u79f0\u200b\\(A, B\\)\u200b\u7b49\u503c\u200b\u3002\u200b\u8bb0\u200b\u4e3a\u200b\\(A=B\\)\u200b\u6216\u200b\\(A\\Leftrightarrow B\\)\u3002</p>"},{"location":"math/discrete-mathematics/chapter-5/#_2","title":"\u7b49\u503c\u200b\u5f0f","text":"<ol> <li>\u200b\u5c06\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u4ee3\u5165\u200b\u547d\u9898\u903b\u8f91\u200b\u4e0b\u200b\u7684\u200b\u7b49\u503c\u200b\u516c\u5f0f\u200b\u5373\u53ef\u200b\u5f97\u5230\u200b\u8c13\u8bcd\u200b\u903b\u8f91\u200b\u7684\u200b\u7b49\u503c\u200b\u516c\u5f0f\u200b</li> <li>\u200b\u5426\u5b9a\u200b\u578b\u200b\u7b49\u503c\u200b\u5f0f\u200b</li> <li>\\(\\lnot (\\forall x)P(x) = (\\exists x)\\lnot P(x)\\)</li> <li>\\(\\lnot (\\exists x)P(x) = (\\forall x)\\lnot P(x)\\)</li> <li>\u200b\u91cf\u8bcd\u200b\u5206\u914d\u200b\u7b49\u503c\u200b\u5f0f\u200b\uff08\\(q\\)\u200b\u4e3a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\uff0c\\(x\\)\u200b\u4e3a\u200b\u53d8\u5143\u200b\uff09</li> <li>\\((\\forall x)(P(x)\\lor q) = (\\forall x)P(x)\\lor q\\)</li> <li>\\((\\exists x)(P(x)\\lor q) = (\\exists x)P(x)\\lor q\\)</li> <li>\\((\\forall x)(P(x)\\land q) = (\\forall x)P(x)\\land q\\)</li> <li>\\((\\exists x)(P(x)\\land q) = (\\forall x)P(x)\\land q\\)</li> <li>\u200b\u5bf9\u200b\\(\\rightarrow\\)\u200b\u7684\u200b\u5206\u914d\u5f8b\u200b</li> <li>\\((\\forall x)(P(x)\\rightarrow q) = (\\exists x)P(x)\\rightarrow q\\)</li> <li>\\((\\exists x)(P(x)\\rightarrow q) = (\\forall x)P(x)\\rightarrow q\\)</li> <li>\\((\\forall x)(p\\rightarrow Q(x)) = p\\rightarrow (\\forall x)Q(x)\\)</li> <li>\\((\\exists x)(p\\rightarrow Q(x)) = p\\rightarrow (\\exists x)Q(x)\\)</li> <li>\u200b\u91cf\u8bcd\u200b\\(\\forall\\)\u200b\u5bf9\u200b\\(\\land\\)\u3001\u200b\u91cf\u8bcd\u200b\\(\\exists\\)\u200b\u5bf9\u200b\\(\\lor\\)\u200b\u7684\u200b\u5206\u914d\u5f8b\u200b</li> <li>\\((\\forall x)(P(x)\\land Q(x)) = (\\forall x)P(x) \\land (\\forall x)Q(x)\\)</li> <li>\\((\\exists x)(P(x)\\lor Q(x)) = (\\exists x)P(x) \\lor (\\exists x)Q(x)\\)</li> <li>\u200b\u53d8\u5143\u200b\u6613\u540d\u200b\u7684\u200b\u5206\u914d\u5f8b\u200b</li> <li>\\((\\forall x)(\\forall y)(P(x)\\lor Q(y)) = (\\forall x)P(x) \\lor (\\forall x)Q(x)\\)</li> <li>\\((\\exists x)(\\exists y)(P(x)\\land Q(y)) = (\\forall x)P(x) \\land (\\forall x)Q(x)\\)</li> </ol>"},{"location":"math/discrete-mathematics/chapter-5/#_3","title":"\u8303\u5f0f","text":"<p>\u200b\u82e5\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u4e2d\u200b\u7684\u200b\u4e00\u5207\u200b\u91cf\u8bcd\u200b\u90fd\u200b\u4f4d\u4e8e\u200b\u516c\u5f0f\u200b\u7684\u200b\u6700\u200b\u5de6\u4fa7\u200b\uff0c\u200b\u4e14\u200b\u91cf\u8bcd\u200b\u7684\u200b\u4f5c\u7528\u57df\u200b\u5ef6\u4f38\u200b\u5230\u200b\u516c\u5f0f\u200b\u7684\u200b\u672b\u7aef\u200b\uff0c\u200b\u5219\u200b\u79f0\u8c13\u200b\u8bcd\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u4e3a\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\u3002\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\u7684\u200b\u4e00\u822c\u200b\u5f62\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ (Q_1x_1)\\dots (Q_nx_n)M(x_1, \\dots, x_n) \\] <p>\u200b\u5176\u4e2d\u200b\\(Q_i\\)\u200b\u4e3a\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\\(\\forall\\)\u200b\u6216\u200b\u7279\u79f0\u200b\u91cf\u8bcd\u200b\\(\\exists\\)\uff0c\\(M\\)\u200b\u79f0\u4e3a\u200b\u6bcd\u5f0f\u200b\uff0c\u200b\u4e0d\u200b\u542b\u6709\u200b\u91cf\u8bcd\u200b\u3002</p> <p>\u200b\u4efb\u4f55\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u90fd\u200b\u6709\u200b\u4e0e\u200b\u4e4b\u200b\u7b49\u503c\u200b\u7684\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\uff0c\u200b\u4f46\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\u4e0d\u200b\u552f\u4e00\u200b\u3002\u200b\u901a\u8fc7\u200b\u5982\u4e0b\u200b\u6d41\u7a0b\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u5316\u200b\u4e3a\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\uff1a</p> <ul> <li>\u200b\u5c06\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\\(\\rightarrow\\)\u200b\u8054\u7ed3\u8bcd\u200b\u4e0e\u200b\\(\\leftrightarrow\\)\u200b\u8054\u7ed3\u8bcd\u200b\u6d88\u200b\u53bb\u200b</li> <li>\u200b\u4f7f\u7528\u200b\u6469\u6839\u200b\u5f8b\u200b\uff0c\u200b\u5c06\u200b\\(\\lnot\\)\u200b\u5185\u79fb\u200b\u5230\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u6216\u200b\u8c13\u8bcd\u200b\u4e4b\u524d\u200b</li> <li>\u200b\u4f7f\u7528\u200b\u91cf\u8bcd\u200b\u5206\u914d\u200b\u7b49\u503c\u200b\u5f0f\u200b\u5c06\u200b\u91cf\u8bcd\u200b\u79fb\u52a8\u200b\u5230\u200b\u5de6\u4fa7\u200b</li> <li>\u200b\u4f7f\u7528\u200b\u53d8\u5143\u200b\u6613\u540d\u200b\u5206\u914d\u200b\u7b49\u503c\u200b\u5f0f\u200b\u5408\u5e76\u200b\u8c13\u8bcd\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-5/#skolem","title":"Skolem\u200b\u6807\u51c6\u578b","text":"<p>\u200b\u79f0\u200b\u4e00\u4e2a\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u4e3a\u200b\\(\\exists\\)\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u5b58\u5728\u200b\u91cf\u8bcd\u200b\u90fd\u200b\u5728\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u7684\u200b\u5de6\u8fb9\u200b\uff0c\u200b\u5e76\u4e14\u200b\u53ef\u4ee5\u200b\u4fdd\u6301\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u81f3\u5c11\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u5b58\u5728\u200b\u91cf\u8bcd\u200b\u3002\u200b\u6bcd\u5f0f\u200b\u4e2d\u200b\u4e0d\u200b\u542b\u6709\u200b\u4efb\u4f55\u200b\u91cf\u8bcd\u200b\uff0c\u200b\u4e5f\u200b\u4e0d\u200b\u542b\u200b\u81ea\u7531\u200b\u4e2a\u4f53\u200b\u53d8\u9879\u200b\u3002 \u200b\u79f0\u200b\u4e00\u4e2a\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u4e3a\u200b\u4ec5\u200b\u4fdd\u7559\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u7684\u200b\u524d\u675f\u200b\u578b\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u53ea\u200b\u5305\u542b\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u800c\u200b\u4e0d\u200b\u5305\u542b\u200b\u5b58\u5728\u200b\u91cf\u8bcd\u200b\u3002\u200b\u6bcd\u5f0f\u200b\u4e2d\u200b\u4e0d\u200b\u542b\u6709\u200b\u4efb\u4f55\u200b\u91cf\u8bcd\u200b\uff0c\u200b\u4e5f\u200b\u4e0d\u200b\u542b\u200b\u81ea\u7531\u200b\u4e2a\u4f53\u200b\u53d8\u9879\u200b\u3002</p> <ul> <li>\u200b\u4e00\u4e2a\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u53ef\u4ee5\u200b\u5316\u6210\u200b\u76f8\u5e94\u200b\u7684\u200b\\(\\exists\\)\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\uff0c\\(A\\)\u200b\u666e\u904d\u200b\u6709\u6548\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\exists\\)\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\u666e\u904d\u200b\u6709\u6548\u200b</li> <li>\u200b\u4e00\u4e2a\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u53ef\u4ee5\u200b\u5316\u6210\u200b\u76f8\u5e94\u200b\u7684\u200b\u4ec5\u200b\u4fdd\u7559\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u7684\u200b\u524d\u675f\u200b\u578b\u200b\uff0c\\(A\\)\u200b\u4e0d\u53ef\u200b\u6ee1\u8db3\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u4ec5\u200b\u4fdd\u7559\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u7684\u200b\u524d\u675f\u200b\u578b\u200b\u4e0d\u53ef\u200b\u6ee1\u8db3\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-9/","title":"\u96c6\u5408","text":"<p>\u200b\u96c6\u5408\u200b\u662f\u200b\u4e00\u4e9b\u200b\u786e\u5b9a\u200b\u7684\u200b\u3001\u200b\u53ef\u4ee5\u200b\u533a\u5206\u200b\u7684\u200b\u4e8b\u7269\u200b\u6c47\u805a\u200b\u5728\u200b\u4e00\u8d77\u200b\u7ec4\u6210\u200b\u7684\u200b\u4e00\u4e2a\u200b\u6574\u4f53\u200b\uff0c\u200b\u7ec4\u6210\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u4e8b\u7269\u200b\u79f0\u4e3a\u200b\u8be5\u200b\u96c6\u5408\u200b\u7684\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u3002</p> <ul> <li>\u200b\u96c6\u5408\u200b\u7684\u200b\u5143\u7d20\u200b\u53ef\u4ee5\u200b\u662f\u200b\u4efb\u4f55\u4e8b\u7269\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\uff0c\u200b\u4f46\u200b\u4e0d\u80fd\u200b\u662f\u200b\u81ea\u8eab\u200b</li> <li>\u200b\u96c6\u5408\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u6ee1\u8db3\u200b\u552f\u4e00\u6027\u200b\uff0c\u200b\u4e0d\u4f1a\u200b\u51fa\u73b0\u200b\u76f8\u540c\u200b\u7684\u200b\u5143\u7d20\u200b</li> <li>\u200b\u96c6\u5408\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u6ca1\u6709\u200b\u987a\u5e8f\u200b\u5173\u7cfb\u200b</li> <li>\u200b\u5143\u7d20\u200b\u4e0e\u200b\u96c6\u5408\u200b\u7684\u200b\u5173\u7cfb\u200b\u662f\u200b\u786e\u5b9a\u200b\u7684\u200b\uff0c\u200b\u5373\u200b\u5c5e\u4e8e\u200b\uff08\\(\\in\\)\uff09\u200b\u4e0e\u200b\u4e0d\u200b\u5c5e\u4e8e\u200b\uff08\\(\\not\\in\\)\uff09</li> </ul>"},{"location":"math/discrete-mathematics/chapter-9/#_2","title":"\u96c6\u5408\u200b\u7684\u200b\u8868\u793a","text":"<p>\u200b\u4ee5\u4e0b\u200b\u5217\u4e3e\u200b\u51fa\u200b\u5e38\u89c1\u200b\u7684\u200b\u96c6\u5408\u200b\uff1a</p> <ul> <li>\\(\\mathbf N\\)\u200b\u8868\u793a\u200b\u5168\u4f53\u200b\u81ea\u7136\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b</li> <li>\\(\\mathbf Z\\)\u200b\u8868\u793a\u200b\u5168\u4f53\u200b\u6574\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b</li> <li>\\(\\mathbf Q\\)\u200b\u8868\u793a\u200b\u5168\u4f53\u200b\u6709\u7406\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b</li> <li>\\(\\mathbf R\\)\u200b\u8868\u793a\u200b\u5168\u4f53\u200b\u5b9e\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b</li> <li>\\(\\mathbf C\\)\u200b\u8868\u793a\u200b\u5168\u4f53\u200b\u590d\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b</li> </ul> <p>\u200b\u53ef\u4ee5\u200b\u7528\u200b\u5916\u5ef6\u200b\u8868\u793a\u6cd5\u200b\u6216\u200b\u5185\u6db5\u200b\u8868\u793a\u6cd5\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\uff1a</p> <ul> <li>\u200b\u5916\u5ef6\u200b\u8868\u793a\u6cd5\u200b\uff1a\u200b\u4e00\u4e00\u5217\u4e3e\u200b\u51fa\u200b\u96c6\u5408\u200b\u7684\u200b\u5168\u4f53\u200b\u5143\u7d20\u200b</li> <li>\u200b\u5185\u6db5\u200b\u8868\u793a\u6cd5\u200b\uff1a\u200b\u4f7f\u7528\u200b\u8c13\u8bcd\u200b\u8868\u8ff0\u200b\u96c6\u5408\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u6027\u8d28\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-9/#_3","title":"\u96c6\u5408\u200b\u7684\u200b\u8fd0\u7b97","text":""},{"location":"math/discrete-mathematics/chapter-9/#_4","title":"\u96c6\u5408\u200b\u7684\u200b\u5173\u7cfb\u200b\u8fd0\u7b97","text":"<p>\u200b\u96c6\u5408\u200b\u4e0e\u200b\u5143\u7d20\u200b\u5b58\u5728\u200b\u5c5e\u4e8e\u200b\u4e0e\u200b\u4e0d\u200b\u5c5e\u4e8e\u200b\u5173\u7cfb\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(\\in, \\not\\in\\)</p> <p>\u200b\u96c6\u5408\u200b\u4e4b\u95f4\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u5173\u7cfb\u200b\\(=, \\subseteq, \\subset, \\supseteq, \\supset\\)\u3002</p> <ul> <li>\\(A=B\\Leftrightarrow (\\forall x)(x\\in A\\leftrightarrow x\\in B)\\)</li> <li>\\(A\\subseteq B\\Leftrightarrow (\\forall x)(x\\in A\\rightarrow x\\in B)\\)</li> <li>\\(A\\subset B\\Leftrightarrow (\\forall x)(x\\in A\\rightarrow x\\in B)\\land A\\not =B\\)</li> <li>\\(A\\supseteq B\\Leftrightarrow B\\subseteq A\\)</li> <li>\\(A\\supset B\\Leftrightarrow B\\subset A\\)</li> </ul> <p>\\(\\subseteq, \\supseteq, =\\)\u200b\u5173\u7cfb\u200b\u6ee1\u8db3\u200b\u81ea\u53cd\u6027\u200b\u3001\u200b\u4f20\u9012\u6027\u200b\uff0c\\(=\\)\u200b\u5173\u7cfb\u200b\u6ee1\u8db3\u200b\u5bf9\u79f0\u6027\u200b\u3002</p> <p>\\(A\\subseteq B\\land B\\subseteq A\\Leftrightarrow A=B\\)</p> <p>\u200b\u5b9a\u4e49\u200b\u7a7a\u96c6\u200b\u4e3a\u200b\u4e0d\u200b\u542b\u200b\u4efb\u4f55\u200b\u5143\u7d20\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(\\varnothing\\)\uff0c\u200b\u5176\u200b\u5185\u6db5\u200b\u8868\u793a\u200b\u4e3a\u200b\\(\\varnothing = \\{x|x\\not= x\\}\\)\uff0c\u200b\u7a7a\u96c6\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ul> <li>\u200b\u7a7a\u96c6\u200b\u662f\u200b\u4efb\u4f55\u200b\u96c6\u5408\u200b\u7684\u200b\u5b50\u96c6\u200b</li> <li>\u200b\u7a7a\u96c6\u200b\u662f\u200b\u552f\u4e00\u200b\u7684\u200b</li> </ul> <p>\u200b\u5b9a\u4e49\u200b\u5168\u96c6\u200b\u4e3a\u200b\u6240\u6709\u200b\u4e8b\u7269\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(E\\)\uff0c\u200b\u5176\u200b\u5185\u6db5\u200b\u8868\u793a\u200b\u4e3a\u200b\\(E=\\{x|x=x\\}\\)</p>"},{"location":"math/discrete-mathematics/chapter-9/#_5","title":"\u96c6\u5408\u200b\u7684\u200b\u57fa\u672c\u200b\u8fd0\u7b97","text":"<p>\u200b\u96c6\u5408\u200b\u5305\u542b\u200b\u5982\u4e0b\u200b\u57fa\u672c\u200b\u8fd0\u7b97\u200b\uff1a</p> <ul> <li>\u200b\u5e76\u96c6\u200b\uff1a\\(A\\cup B = \\{x\\in A\\lor x\\in B\\}\\)</li> <li>\u200b\u4ea4\u96c6\u200b\uff1a\\(A\\cap B = \\{x\\in A\\land x\\in B\\}\\)</li> <li>\u200b\u5dee\u96c6\u200b\uff1a\\(A-B = \\{x\\in A\\land x\\not \\in B\\}\\)</li> <li>\u200b\u8865\u96c6\u200b\uff1a\\(-A=E-A=\\{x|x\\not\\in A\\}\\)</li> <li>\u200b\u5bf9\u79f0\u200b\u5dee\u200b\uff1a\\(A\\oplus B=(A-B)\\cup (B-A) = \\{x| x\\in A\\overline\\lor x\\in B\\}\\)</li> </ul> <p>\u200b\u57fa\u4e8e\u200b\u5e76\u96c6\u200b\u4e0e\u200b\u4ea4\u96c6\u200b\u8fd0\u7b97\u200b\u53ef\u4ee5\u200b\u62d3\u5c55\u200b\u51fa\u200b\u5e7f\u4e49\u200b\u5e76\u200b\u548c\u200b\u5e7f\u4e49\u200b\u4ea4\u200b\u7684\u200b\u6982\u5ff5\u200b\uff0c\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(A\\)\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u90fd\u200b\u662f\u200b\u96c6\u5408\u200b\uff0c\u200b\u5b9a\u4e49\u200b$A\u200b\u7684\u200b\u5e7f\u4e49\u200b\u5e76\u200b\u3001\u200b\u5e7f\u4e49\u200b\u4ea4\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u5e7f\u4e49\u200b\u5e76\u200b\uff1a\\(\\cup A=\\{x| (\\exists z)(z\\in A\\land x\\in z)\\}\\)</li> <li>\u200b\u5e7f\u4e49\u200b\u4ea4\u200b\uff1a\\(\\cap A=\\{x| (\\forall z)(z\\in A\\rightarrow x\\in z)\\}\\)</li> </ul> <p>\u200b\u5373\u200b\uff1a\u200b\u96c6\u5408\u200b\\(A\\)\u200b\u7684\u200b\u5e7f\u4e49\u200b\u5e76\u200b\u662f\u200b\u96c6\u5408\u200b\u4e2d\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u7684\u200b\u5e76\u200b\u96c6\u200b\uff0c\u200b\u96c6\u5408\u200b\\(A\\)\u200b\u7684\u200b\u5e7f\u4e49\u200b\u4ea4\u662f\u200b\u96c6\u5408\u200b\u4e2d\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u7684\u200b\u4ea4\u96c6\u200b\u3002</p> <p>\u200b\u5b9a\u4e49\u200b\u5e42\u96c6\u200b\u4e3a\u200b\u96c6\u5408\u200b\u6240\u6709\u200b\u5b50\u96c6\u200b\uff08\u200b\u5305\u62ec\u200b\u81ea\u8eab\u200b\uff09\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(P(A)\\)</p> <p>\u200b\u5b9a\u4e49\u200b\u7b1b\u5361\u5c14\u200b\u79ef\u200b\\(A\\times B\\)\u200b\u4e3a\u200b\u4e8c\u5143\u200b\u7ec4\u200b\uff08\u200b\u6709\u5e8f\u200b\u5bf9\u200b\uff09\\(\\langle x, y\\rangle\\)\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u5176\u4e2d\u200b\\(x\\in A, y\\in B\\)\u3002\u200b\u6709\u5e8f\u200b\u5bf9\u200b\u7684\u200b\u5b9a\u4e49\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u552f\u4e00\u200b\uff0c\u200b\u5982\u200b\u96c6\u5408\u200b\\(\\{x, \\{x, y\\}\\}\\)\u200b\u5373\u53ef\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u6709\u5e8f\u200b\u5bf9\u200b\u3002\u200b\u6709\u5e8f\u200b\u5bf9\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ul> <li>\\(x\\not = y\\Rightarrow \\langle x,y\\rangle \\not =\\langle y,x\\rangle\\)</li> <li>\\(\\langle x,y\\rangle = \\langle u,v\\rangle\\Leftrightarrow x=u\\land y=v\\)</li> </ul> <p>\u200b\u53ef\u4ee5\u200b\u5728\u200b\u4e8c\u5143\u200b\u7ec4\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\u62d3\u5c55\u200b\\(n\\)\u200b\u5143\u7ec4\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff1a\\(\\langle x_1, \\dots, x_n\\rangle =\\langle\\dots\\langle x_1, x_2\\rangle, \\dots, x_n\\rangle\\)\u3002\u200b\u7531\u6b64\u200b\\(n\\)\u200b\u9636\u200b\u7b1b\u5361\u5c14\u200b\u79ef\u200b\\(A_1\\times A_2\\times\\dots\\times A_n\\)\u200b\u5373\u200b\u4e3a\u200b\\(\\{\\langle x_1, \\dots, x_n\\rangle|x_1\\in A_1\\land \\dots\\land x_n\\in A_n\\}\\)</p>"},{"location":"math/discrete-mathematics/chapter-9/#_6","title":"\u96c6\u5408\u200b\u8fd0\u7b97\u200b\u7684\u200b\u4f18\u5148\u7ea7","text":"<p>\u200b\u96c6\u5408\u200b\u8fd0\u7b97\u200b\u7684\u200b\u4f18\u5148\u7ea7\u200b\u5217\u4e3e\u5982\u4e0b\u200b\uff1a</p> \\[ \\begin{aligned}     &amp;-A, P(A), \\cap A, \\cup A \\\\     &amp;-, \\cap, \\cup, \\oplus, \\times \\\\     &amp;=, \\subseteq , \\subset, \\in \\\\     &amp;\\lnot \\\\     &amp;\\land, \\lor, \\rightarrow, \\leftrightarrow \\\\     &amp;\\Leftrightarrow, \\Rightarrow \\end{aligned} \\]"},{"location":"math/discrete-mathematics/chapter-9/#_7","title":"\u96c6\u5408\u200b\u8fd0\u7b97\u200b\u7684\u200b\u6027\u8d28","text":"<p>\u200b\u96c6\u5408\u200b\u7684\u200b\u57fa\u672c\u200b\u8fd0\u7b97\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ol> <li>\u200b\u4ea4\u6362\u5f8b\u200b</li> <li>\\(A\\cap B=B\\cap A\\)</li> <li>\\(A\\cup B=B\\cup A\\)</li> <li>\u200b\u7ed3\u5408\u5f8b\u200b</li> <li>\\((A\\cup B)\\cup C = A\\cup (B\\cup C)\\)</li> <li>\\((A\\cap B)\\cap C = A\\cap (B\\cap C)\\)</li> </ol>"},{"location":"math/discrete-mathematics/chapter-9/#_8","title":"\u96c6\u5408\u200b\u7684\u200b\u56fe\u793a\u200b\u6cd5","text":"<ul> <li>\u200b\u5bf9\u4e8e\u200b\u96c6\u5408\u200b\u7684\u200b\u57fa\u672c\u200b\u8fd0\u7b97\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6587\u6c0f\u56fe\u200b\u8868\u793a\u200b</li> <li>\u200b\u5bf9\u4e8e\u200b\u5e42\u96c6\u200b\u8fd0\u7b97\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u7f51\u7edc\u56fe\u200b\u8868\u793a\u200b</li> <li>\u200b\u5bf9\u4e8e\u200b\u7b1b\u5361\u5c14\u200b\u79ef\u200b\u8fd0\u7b97\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e8c\u7ef4\u200b\u5750\u6807\u7cfb\u200b\u8868\u793a\u200b</li> </ul>"},{"location":"math/operating-research/","title":"\u8fd0\u7b79\u5b66","text":"<ul> <li>\u200b\u7ebf\u6027\u89c4\u5212\u200b</li> <li>\u200b\u8fd0\u8f93\u200b\u95ee\u9898\u200b</li> <li>\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b</li> </ul>"},{"location":"math/operating-research/chapter-1/","title":"\u7ebf\u6027\u89c4\u5212","text":"<p>\u200b\u5b9a\u4e49\u200b</p> <p>\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\uff1a\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\\(f(x_1, \\dots, x_n)\\)\u200b\u548c\u200b\u4e00\u7cfb\u5217\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff0c\u200b\u6c42\u200b\u6ee1\u8db3\u200b\u6240\u6709\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u6539\u53d8\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\\(x_1, x_2, \\cdots, x_n\\)\u200b\u4f7f\u5f97\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u6240\u200b\u80fd\u200b\u8fbe\u5230\u200b\u7684\u200b\u6700\u5927\u503c\u200b\u6216\u200b\u6700\u5c0f\u503c\u200b\u3002\u200b\u5f53\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u53d6\u200b\u6700\u5927\u200b\uff08\u200b\u6700\u5c0f\u200b\uff09\u200b\u503c\u65f6\u200b\uff0c\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u7684\u200b\u503c\u79f0\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u7684\u200b\u89e3\u200b\u3002</p> <p>\u200b\u7ebf\u6027\u89c4\u5212\u200b\uff1a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u548c\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u5747\u200b\u4e3a\u200b \u200b\u7ebf\u6027\u200b \u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b</p> <p>\u200b\u7ebf\u6027\u89c4\u5212\u200b\u6a21\u578b\u200b\u7684\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned} &amp;\\max z = c_1x_1 + c_2x_2 + \\cdots + c_nx_n \\\\ s.t. &amp; \\left\\{     \\begin{aligned}         &amp; a_{11}x_1 + a_{12}x_2 + \\cdots + a_{1n}x_n = b_1 \\\\         &amp; a_{21}x_1 + a_{22}x_2 + \\cdots + a_{2n}x_n = b_2 \\\\         &amp; \\vdots \\\\         &amp; a_{m1}x_1 + a_{m2}x_2 + \\cdots + a_{mn}x_n = b_m \\\\         &amp; x_j \\in \\mathbb R \\land x_j \\geq 0, j = 1, \\cdots, n     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5176\u4e2d\u200b\\(a_{ij}, b_i, c_i\\)\u200b\u4e3a\u200b\u5e38\u6570\u200b\uff0c\u200b\u5206\u522b\u200b\u4f7f\u7528\u200b\u77e9\u9635\u200b\\(A, b, c^T\\)\u200b\u8fdb\u884c\u200b\u8868\u793a\u200b\u3002\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\\((x_1, x_2, \\cdots, x_n)^T\\)\u200b\u4f7f\u7528\u200b\u5411\u91cf\u200b\\(x\\)\u200b\u8fdb\u884c\u200b\u8868\u793a\u200b\uff0c\u200b\u5219\u200b\u6807\u51c6\u578b\u200b\u53ef\u4ee5\u200b\u5199\u4f5c\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max z = CX \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; Ax = b \\\\         &amp; x \\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5982\u679c\u200b\\(\\mathrm r(A) &lt; m\\)\uff0c\u200b\u5219\u200b\u90e8\u5206\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u53ef\u4ee5\u200b\u76f8\u4e92\u200b\u62b5\u6d88\u200b\uff0c\u200b\u4ee5\u4e0a\u200b\u53ea\u200b\u8ba8\u8bba\u200b\\(\\mathrm r(A) = m\\)\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u7ea6\u675f\u200b\u4e2d\u200b\u542b\u6709\u200b\u4e0d\u7b49\u5f0f\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u6dfb\u52a0\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b\u4e0e\u200b\u5269\u4f59\u200b\u53d8\u91cf\u200b\u8f6c\u53d8\u200b\u4e3a\u200b\u7b49\u5f0f\u200b</p> <ul> <li>\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b\uff1a\\(a_{i1}x_1 + a_{i2}x_2 + \\cdots + a_{in}x_n \\leq b_i\\)\u200b\u52a0\u5165\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b\\(x_{n+1}\\)\u200b\u540e\u200b\u8f6c\u4e3a\u200b\\(a_{i1}x_1 + a_{i2}x_2 + \\cdots + a_{in}x_n + x_{n+1}= b_i\\)</li> <li>\u200b\u5269\u4f59\u200b\u53d8\u91cf\u200b\uff1a\\(a_{j1}x_1 + a_{j2}x_2 + \\cdots + a_{jn}x_n \\geq b_j\\)\u200b\u52a0\u5165\u200b\u5269\u4f59\u200b\u53d8\u91cf\u200b\\(x_{n+1}\\)\u200b\u540e\u200b\u8f6c\u4e3a\u200b\\(a_{j1}x_1 + a_{j2}x_2 + \\cdots + a_{jn}x_n - x_{n+1}= b_j\\)</li> </ul> <p>\u200b\u82e5\u200b\u67d0\u4e2a\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\\(x_i\\)\u200b\u7684\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\(\\leq\\)\uff0c\u200b\u5219\u200b\u5c06\u200b\\(x_i\\)\u200b\u66ff\u6362\u200b\u4e3a\u200b\\(x_i' = -x_i\\)\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u7ebf\u6027\u89c4\u5212\u200b\u4e2d\u200b\u7684\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\uff0c\u200b\u82e5\u200b\u5728\u200b\u6700\u4f18\u200b\u65f6\u8be5\u200b\u7ea6\u675f\u200b\u4e3a\u200b\u7b49\u5f0f\u200b\uff0c\u200b\u5219\u200b\u79f0\u4e3a\u200b\u7d27\u200b\u7ea6\u675f\u200b\uff0c\u200b\u5426\u5219\u200b\u79f0\u4e3a\u200b\u975e\u7d27\u200b\u7ea6\u675f\u200b\u3002</p>"},{"location":"math/operating-research/chapter-1/#_2","title":"\u6c42\u89e3\u200b\u7ebf\u6027\u89c4\u5212","text":"<p>\u200b\u7ebf\u6027\u89c4\u5212\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u56fe\u5f62\u200b\u6cd5\u200b\u3001\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u7b49\u200b\u65b9\u6cd5\u200b\u6c42\u89e3\u200b</p>"},{"location":"math/operating-research/chapter-1/#_3","title":"\u56fe\u5f62\u200b\u6cd5","text":"<p>\u200b\u56fe\u5f62\u200b\u6cd5\u200b\u53ea\u80fd\u200b\u7528\u4e8e\u200b\u6c42\u89e3\u200b\u5305\u542b\u200b\u4e24\u4e2a\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\u3002\u200b\u6b65\u9aa4\u200b\u53ef\u4ee5\u200b\u5217\u4e3e\u5982\u4e0b\u200b\uff1a</p> <ol> <li>\u200b\u5728\u200b\u5750\u6807\u8f74\u200b\u4e0a\u200b\u7ed8\u5236\u200b\u51fa\u200b\u6240\u6709\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff0c\u200b\u786e\u5b9a\u200b\u53ef\u884c\u200b\u57df\u200b</li> <li>\u200b\u5c06\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\\(z = c_1x_1 + c_2x_2\\)\u200b\u6539\u5199\u200b\u4e3a\u200b\\(x_2 = \\frac{z - c_1x_1}{c_2}\\)\uff0c\u200b\u5176\u200b\u659c\u7387\u200b\u4e3a\u200b\\(-\\frac{c_1}{c_2}\\)</li> <li>\u200b\u5728\u200b\u53ef\u884c\u200b\u57df\u200b\u4e2d\u200b\u5bfb\u627e\u200b\u4f7f\u5f97\u200b\u622a\u8ddd\u200b\u6700\u5927\u200b\uff08\u200b\u6700\u5c0f\u200b\uff09\u200b\u4e14\u200b\u659c\u7387\u200b\u4e3a\u200b\\(-\\frac{c_1}{c_2}\\)\u200b\u7684\u200b\u76f4\u7ebf\u200b\u3002\u200b\u8be5\u200b\u622a\u8ddd\u200b\u4e3a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7684\u200b\\(1/c_2\\)<ul> <li>\u200b\u82e5\u200b\u53ef\u884c\u200b\u57df\u200b\u4e3a\u200b\u7a7a\u200b\uff0c\u200b\u5219\u200b\u7ebf\u6027\u89c4\u5212\u200b\u65e0\u200b\u89e3\u200b</li> <li>\u200b\u82e5\u200b\u76f4\u7ebf\u200b\u53ef\u4ee5\u200b\u65e0\u9650\u200b\u5411\u200b\u4f18\u5316\u200b\u65b9\u5411\u200b\u5e73\u79fb\u200b\uff0c\u200b\u5219\u200b\u7ebf\u6027\u89c4\u5212\u200b\u65e0\u200b\u754c\u200b</li> <li>\u200b\u82e5\u200b\u6700\u4f18\u200b\u65f6\u200b\u76f4\u7ebf\u200b\u4e0e\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u7684\u200b\u659c\u7387\u200b\u76f8\u540c\u200b\uff0c\u200b\u5219\u200b\u7ebf\u6027\u89c4\u5212\u200b\u6709\u200b\u65e0\u6570\u200b\u89e3\u200b</li> <li>\u200b\u5426\u5219\u200b\u7ebf\u6027\u89c4\u5212\u200b\u6709\u200b\u552f\u4e00\u200b\u6700\u4f18\u200b\u89e3\u200b</li> </ul> </li> </ol>"},{"location":"math/operating-research/chapter-1/#_4","title":"\u5355\u7eaf\u5f62\u6cd5","text":"<p>\u200b\u5df2\u77e5\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u6ee1\u200b\u79e9\u200b\uff0c\u200b\u6784\u6210\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u7684\u200b\u5217\u200b\u5411\u91cf\u200b\u7684\u200b\u6781\u5927\u200b\u7ebf\u6027\u200b\u65e0\u5173\u200b\u7ec4\u200b\u5305\u542b\u200b\\(m\\)\u200b\u4e2a\u5217\u200b\u5411\u91cf\u200b\u3002\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u4f5c\u200b\u5982\u4e0b\u200b\u5212\u5206\u200b\uff1a</p> \\[ A = [B, N] \\] <p>\u200b\u76f8\u4f3c\u200b\u5730\u200b\uff0c\u200b\u5411\u91cf\u200b\\(x\\)\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4f5c\u200b\u5982\u4e0b\u200b\u5212\u5206\u200b\uff1a</p> \\[ x = \\begin{bmatrix}     x_B \\\\ x_N \\end{bmatrix} \\] <p>\u200b\u5176\u4e2d\u200b\\(B\\)\u200b\u4e3a\u200b\u53ef\u9006\u200b\u77e9\u9635\u200b\u3002\u200b\u82e5\u200b\u4e0d\u200b\u5b58\u5728\u200b\u8fd9\u6837\u200b\u7684\u200b\u77e9\u9635\u200b\uff0c\u200b\u5c06\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u7684\u200b\u5404\u5217\u200b\u8fdb\u884c\u200b\u4ea4\u6362\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u57fa\u200b\u3001\u200b\u57fa\u200b\u5411\u91cf\u200b\u4e0e\u200b\u57fa\u200b\u53d8\u91cf\u200b</p> <p>\u200b\u77e9\u9635\u200b\\(B\\)\u200b\u79f0\u4e3a\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b \u200b\u57fa\u200b \u3002 \u200b\u57fa\u200b \u200b\u4e2d\u200b\u7684\u200b\u5404\u4e2a\u200b\u5217\u200b\u5411\u91cf\u200b\\(P_j\\)\u200b\u4e3a\u200b \u200b\u57fa\u200b\u5411\u91cf\u200b \uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u79f0\u4e3a\u200b \u200b\u57fa\u200b\u53d8\u91cf\u200b \u3002\u200b\u5206\u5272\u200b\u540e\u975e\u200b\u9f50\u6b21\u200b\u7ebf\u6027\u65b9\u7a0b\u7ec4\u200b\\(Ax=b\\)\u200b\u8f6c\u5316\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{bmatrix}     B &amp; N \\end{bmatrix}\\begin{bmatrix}     x_B \\\\ x_N \\end{bmatrix} = Bx_B + Nx_N = b \\] <p>\u200b\u7531\u4e8e\u200b\u65b9\u9635\u200b\\(B\\)\u200b\u6ee1\u200b\u79e9\u200b\uff0c\u200b\u6709\u200b\\(x_B = B^{-1}(b - Nx_N)\\)\u3002\u200b\u5176\u4e2d\u200b\\(x_N\\)\u200b\u4e3a\u200b\u81ea\u7531\u200b\u53d8\u91cf\u200b\u3002\\(-B^{-1}N\\)\u200b\u4e2d\u200b\u7684\u200b\u5217\u200b\u5411\u91cf\u200b\u4e3a\u200b\u65b9\u7a0b\u200b\\(Ax=b\\)\u200b\u7684\u200b\u57fa\u7840\u200b\u89e3\u7cfb\u200b\u3002\u200b\u4e0d\u59a8\u200b\u8bbe\u200b\\(x_N = 0\\)\uff0c\u200b\u5219\u200b\\(x_B = B^{-1}b\\)\u200b\u4e3a\u200b\u7ebf\u6027\u65b9\u7a0b\u7ec4\u200b\u7684\u200b\u4e00\u4e2a\u200b\u89e3\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u57fa\u89e3\u200b\u3002</p> <p>\u200b\u53ef\u884c\u200b\u89e3\u4e0e\u200b\u53ef\u884c\u200b\u57fa\u200b</p> <p>\u200b\u82e5\u200b\\(x_B\\geq 0\\)\uff0c\u200b\u5219\u200b\u89e3\u200b\\(x = \\begin{bmatrix} x_B \\\\ x_N \\end{bmatrix}\\)\u200b\u6ee1\u8db3\u200b\u975e\u8d1f\u200b\u7ea6\u675f\u200b\uff0c\u200b\u4e3a\u200b \u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b \uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u57fa\u200b\u79f0\u4e3a\u200b \u200b\u53ef\u884c\u200b\u57fa\u200b \u3002</p> <p>\u200b\u57fa\u89e3\u200b\u4e0e\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ul> <li>\u200b\u7531\u4e8e\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u5171\u200b\\(n\\)\u200b\u5217\u200b\uff0c\u200b\u4ece\u200b\u5176\u4e2d\u200b\u9009\u53d6\u200b\\(m\\)\u200b\u5217\u200b\uff0c\u200b\u6709\u200b\\(\\binom{n}{m}\\)\u200b\u79cd\u200b\u9009\u53d6\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u800c\u200b\u53ea\u6709\u200b\u7ec4\u6210\u200b\u7ebf\u6027\u200b\u65e0\u5173\u200b\u7684\u200b\\(m\\)\u200b\u4e2a\u200b\u5411\u91cf\u200b\u624d\u80fd\u200b\u6784\u6210\u200b\u4e00\u7ec4\u200b\u57fa\u200b\uff0c\u200b\u56e0\u6b64\u200b\u57fa\u89e3\u200b\u7684\u200b\u6570\u76ee\u200b\u4e0d\u4f1a\u200b\u8d85\u8fc7\u200b\\(\\binom nm\\)\u3002</li> <li>\\(X\\)\u200b\u4e3a\u57fa\u200b\u53ef\u884c\u200b\u89e3\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(X\\)\u200b\u7684\u200b\u975e\u200b\u96f6\u200b\u5206\u91cf\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5217\u200b\u5411\u91cf\u200b\u7ebf\u6027\u200b\u65e0\u5173\u200b\u3002\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u662f\u200b\u53ef\u884c\u200b\u57df\u200b\u7684\u200b\u9876\u70b9\u200b\u3002</li> </ul>"},{"location":"math/operating-research/chapter-1/#_5","title":"\u539f\u7406","text":"<p>\u200b\u5982\u4e0b\u200b\u9488\u5bf9\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u6781\u5927\u200b\u5316\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u6781\u5c0f\u200b\u5316\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u53d6\u8d1f\u200b\uff0c\u200b\u5373\u200b\u5f97\u5230\u200b\u9700\u8981\u200b\u6781\u5927\u200b\u5316\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u3002</p> <p>\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u7684\u200b\u601d\u8def\u200b\uff1a</p> <ol> <li>\u200b\u5148\u200b\u627e\u51fa\u200b\u4e00\u4e2a\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b</li> <li>\u200b\u5224\u65ad\u200b\u5f53\u524d\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\u662f\u5426\u200b\u6700\u4f18\u200b<ol> <li>\u200b\u82e5\u200b\u6700\u4f18\u200b\uff0c\u200b\u7ed3\u675f\u200b</li> <li>\u200b\u82e5\u200b\u4e0d\u662f\u200b\u6700\u4f18\u200b\uff0c\u200b\u5219\u200b\u8f6c\u5411\u200b\u76f8\u90bb\u200b\u7684\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b</li> </ol> </li> </ol>"},{"location":"math/operating-research/chapter-1/#_6","title":"\u72b6\u6001\u200b\u8f6c\u79fb","text":"<p>\u200b\u8bbe\u200b\u521d\u59cb\u200b\u57fa\u4e3a\u200b\\(B\\)\u3002\u200b\u5c06\u200b\u65b9\u7a0b\u200b\\(Ax=b\\)\u200b\u5de6\u53f3\u200b\u4e24\u8fb9\u200b\u540c\u65f6\u200b\u5de6\u4e58\u200b\\(B^{-1}\\)\uff0c\u200b\u6709\u200b\\((B^{-1}A)x = B^{-1}b\\)\uff0c\u200b\u5373\u200b\uff1a</p> \\[ \\begin{bmatrix}     I &amp; B^{-1}N \\end{bmatrix}\\begin{bmatrix}     x_B \\\\ x_N \\end{bmatrix} = B^{-1}b \\] <p>\u200b\u8bbe\u200b\\(B^{-1}N = \\{d_{ij}\\}_{m\\times(n-m)}\\)\u3002\u200b\u6839\u636e\u200b\\(x_B = B^{-1}b - B^{-1}Nx_N\\)\uff0c\u200b\u5bf9\u4e8e\u200b\u7b2c\u200b\\(i\\)\u200b\u4e2a\u200b\u5206\u91cf\u200b\uff0c\u200b\u6709\u200b\uff1a</p> \\[ x_i = b_i - \\sum_{j = 1}^{(n - m)}d_{ij}x_j, 1\\leq j\\leq (n-m) \\]"},{"location":"math/operating-research/chapter-1/#_7","title":"\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c","text":"<p>\u200b\u5bf9\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u8fdb\u884c\u200b\u53ef\u9006\u200b\u7ebf\u6027\u53d8\u6362\u200b\uff08\u200b\u53f3\u4e58\u200b\u53ef\u9006\u200b\u77e9\u9635\u200b\uff09\u200b\u4e0d\u4f1a\u200b\u6539\u53d8\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u89e3\u200b\u3002\u200b\u4e3a\u200b\u7b80\u5316\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u4e0d\u59a8\u200b\u5047\u8bbe\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u7684\u200b\u4e00\u4e2a\u200b\u57fa\u200b\u77e9\u9635\u200b\u4e3a\u200b\u5355\u4f4d\u77e9\u9635\u200b\uff0c\u200b\u5373\u200b\uff1a</p> \\[ A = \\begin{bmatrix}     1 &amp; 0 &amp; \\cdots &amp; 0 &amp; a_{1(m + 1)} &amp; a_{1(m + 2)} &amp; \\cdots &amp;  a_{1n} \\\\     0 &amp; 1 &amp; \\cdots &amp; 0 &amp; a_{2(m + 1)} &amp; a_{2(m + 2)} &amp; \\cdots &amp;  a_{2n} \\\\     \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots &amp; \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\     0 &amp; 0 &amp; \\cdots &amp; 1 &amp; a_{m(m + 1)} &amp; a_{m(m + 2)} &amp; \\cdots &amp;  a_{mn} \\\\ \\end{bmatrix} \\] <p>\u200b\u663e\u7136\u200b\uff0c\u200b\u5355\u4f4d\u77e9\u9635\u200b\\(I\\)\u200b\u662f\u200b\\(A\\)\u200b\u7684\u200b\u57fa\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u57fa\u89e3\u200b\\(x_B=b - Nx_N\\)\u3002\u200b\u5c06\u200b\u5411\u91cf\u200b\\(c\\)\u200b\u5206\u4e3a\u200b\\(c_B\\)\u200b\u4e0e\u200b\\(c_N\\)\uff0c\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\\(x_B\\)\u200b\u4e0e\u200b\\(x_N\\)\uff0c\u200b\u6b64\u65f6\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u8ba1\u7b97\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned}     z &amp;= c^Tx \\\\     &amp;= c_B^Tx_B + c_N^T x_N \\\\     &amp;= c_B^Tb + c_N^Tx_N - c_B^TNx_N \\\\     &amp;= c_B^Tb + (c_N - c_B^TN)x_N \\end{aligned} \\] <p>\u200b\u8bbe\u200b\u5411\u91cf\u200b\\(\\lambda = c_N - c_B^TN\\)\uff0c\u200b\u5219\u200b\\(\\frac{\\partial z}{\\partial x_N} = \\lambda\\)\u3002</p> <p>\u200b\u5b9a\u7406\u200b</p> <ul> <li>\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u67d0\u4e2a\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\\(x\\)\uff0c\u200b\u6709\u200b\\(\\lambda &lt; 0\\)\uff0c\u200b\u8bf4\u660e\u200b\\(x\\)\u200b\u4e3a\u200b\u6700\u4f18\u200b\u89e3\u200b</li> <li>\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u67d0\u4e2a\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\\(x\\)\uff0c\u200b\u6709\u200b\\(\\lambda \\leq 0\\)\u200b\u4e14\u200b\\(\\lambda\\)\u200b\u5b58\u5728\u200b\u4e3a\u200b\u96f6\u200b\u7684\u200b\u5206\u91cf\u200b\uff0c\u200b\u8bf4\u660e\u200b\u539f\u200b\u95ee\u9898\u200b\u6709\u200b\u65e0\u6570\u4e2a\u200b\u6700\u4f18\u200b\u89e3\u200b</li> <li>\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u67d0\u4e2a\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\\(x\\)\uff0c\u200b\u5411\u91cf\u200b\\(\\lambda\\)\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\\(\\lambda_{m + i} &gt; 0\\)\uff0c\u200b\u4e14\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5217\u200b\u5411\u91cf\u200b\\(P_{m + i}\\leq 0\\)\uff0c\u200b\u5219\u200b\u539f\u200b\u95ee\u9898\u200b\u65e0\u200b\u754c\u200b</li> </ul>"},{"location":"math/operating-research/chapter-1/#_8","title":"\u57fa\u7684\u200b\u8fed\u4ee3","text":"<p>\u200b\u5728\u200b\u8fed\u4ee3\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u9700\u8981\u200b\u51b3\u5b9a\u200b\u6362\u5165\u200b\u7684\u200b\u57fa\u200b\u4e0e\u200b\u6362\u200b\u51fa\u200b\u7684\u200b\u57fa\u200b\uff1a</p> <ul> <li>\u200b\u4f7f\u7528\u200b\u8d2a\u5fc3\u200b\u7b97\u6cd5\u200b\u9009\u62e9\u200b\u6362\u5165\u200b\u7684\u200b\u57fa\u200b\uff0c\u200b\u9009\u62e9\u200b\u68af\u5ea6\u200b\u6700\u5927\u200b\uff08\u200b\u4e0e\u200b\\(\\lambda\\)\u200b\u6700\u5927\u200b\u6b63\u200b\u5206\u91cf\u200b\u5bf9\u5e94\u200b\uff09\u200b\u7684\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u6362\u5165\u200b\uff0c\\(i = \\arg\\max \\lambda\\)</li> <li>\u200b\u5bf9\u4e8e\u200b\u6362\u5165\u200b\u53d8\u91cf\u200b\\(x_i\\)\uff0c\u200b\u6362\u51fa\u200b\u7684\u200b\u57fa\u200b\\(x_j\\)\u200b\u5e94\u5f53\u200b\u5c3d\u53ef\u80fd\u200b\u4fdd\u8bc1\u200b\u662f\u200b\u53ef\u884c\u200b\u89e3\u200b\u3002\u200b\u6839\u636e\u200b\\(x_B = b - Nx_N\\geq 0\\)\uff0c\u200b\u5f97\u200b\\(Nx_N \\leq b - x_B\\)\uff0c\u200b\u6709\u200b\\(j = \\arg\\min _{j} \\frac{b_j}{a_{ij}}\\)</li> </ul> <p>\u200b\u4e3e\u4f8b\u200b</p> <p>\u200b\u5bf9\u4e8e\u200b\u5982\u4e0b\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max z = 3x_1 - 3x_2 + 5x_4 - x_5 \\\\     &amp;s.t. \\left\\{     \\begin{aligned}         &amp;x_1 &amp; &amp; -2x_3 &amp; +2x_4 &amp; &amp;=12 \\\\         &amp;&amp; x_2 &amp;-2x_3 &amp; &amp; &amp;=1 \\\\         &amp;&amp;&amp; -4x_3 &amp;+3x_4 &amp;+x_5 &amp;=27     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5217\u51fa\u200b\u5355\u7eaf\u5f62\u200b\u8868\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</p> <p></p> <p>\u200b\u5c06\u200b\u53d8\u91cf\u200b\\(x_4\\)\u200b\u8f6c\u5165\u200b\u57fa\u200b\u53d8\u91cf\u200b\uff0c\\(x_1\\)\u200b\u8f6c\u51fa\u200b\u57fa\u200b\u53d8\u91cf\u200b\uff0c\u200b\u5f97\u5230\u200b\u5982\u4e0b\u200b\u5355\u7eaf\u5f62\u200b\u8868\u200b\uff1a</p> <p></p>"},{"location":"math/operating-research/chapter-1/#_9","title":"\u4e24\u200b\u9636\u6bb5\u200b\u6cd5","text":"<p>\u200b\u5f53\u200b\\(A\\)\u200b\u4e0d\u200b\u4fbf\u4e8e\u200b\u8ba1\u7b97\u200b\u51fa\u57fa\u200b\\(B\\)\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5728\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u4e2d\u200b\u5f15\u5165\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u3002\u200b\u7b2c\u4e00\u9636\u6bb5\u200b\u7684\u200b\u5f00\u59cb\u200b\u72b6\u6001\u200b\uff0c\u200b\u6240\u6709\u200b\u7684\u200b\u57fa\u200b\u53d8\u91cf\u200b\u90fd\u200b\u662f\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\uff0c\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e3a\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u4e4b\u200b\u548c\u200b\uff0c\u200b\u5411\u200b\u6700\u5c0f\u503c\u200b\u65b9\u5411\u200b\u4f18\u5316\u200b\u3002\u200b\u901a\u8fc7\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u9010\u6b65\u200b\u5c06\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u6362\u200b\u51fa\u57fa\u200b\u53d8\u91cf\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b</p> <p>\u200b\u5982\u679c\u200b\u67d0\u4e2a\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u5728\u200b\u57fa\u200b\u53d8\u91cf\u200b\u4e2d\u200b\uff0c\u200b\u4f46\u200b\\(\\lambda \\leq 0\\)\uff0c\u200b\u5219\u200b\u7ebf\u6027\u89c4\u5212\u200b\u65e0\u200b\u53ef\u884c\u200b\u89e3\u200b\u3002</p> <p>\u200b\u5c06\u200b\u6240\u6709\u200b\u7684\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u6362\u200b\u51fa\u57fa\u200b\u53d8\u91cf\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6240\u6709\u200b\u7684\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u53d6\u200b\\(0\\)\u3002\u200b\u5220\u53bb\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u5e76\u200b\u5c06\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u66ff\u6362\u200b\u4e3a\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\uff0c\u200b\u5f97\u5230\u200b\u7b2c\u4e8c\u9636\u6bb5\u200b\u5f00\u59cb\u200b\u72b6\u6001\u200b\u7684\u200b\u5355\u7eaf\u5f62\u200b\u8868\u200b\u3002</p>"},{"location":"math/operating-research/chapter-1/#_10","title":"\u7ebf\u6027\u89c4\u5212\u200b\u89e3\u200b\u7684\u200b\u6027\u8d28","text":"<p>\u200b\u6839\u636e\u200b\u56fe\u89e3\u6cd5\u200b\uff0c\u200b\u6709\u5982\u200b\u4e0b\u7ed3\u8bba\u200b\uff1a</p> <ul> <li>\u200b\u53ef\u884c\u200b\u57df\u200b\u662f\u200b\u4e00\u4e2a\u200b\u51f8\u96c6\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u53ef\u884c\u200b\u57df\u200b\\(X\\)\u200b\u4e2d\u200b\u7684\u200b\u4efb\u610f\u200b\u4e24\u70b9\u200b\\(x = (x_1, x_2, \\cdots, x_n)\\in S\\)\u200b\u4e0e\u200b\\(y = (y_1, y_2, \\cdots, y_n)\\in S\\)\uff0c\u200b\u6709\u200b\\(\\alpha x + (1 - \\alpha) y \\in S, 0\\leq \\alpha \\leq 1\\)<ul> <li>\u200b\u51f8\u200b\u96c6\u4e2d\u200b\u7684\u200b\u4efb\u4f55\u200b\u4e00\u4e2a\u70b9\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b\u51f8\u96c6\u200b\u5404\u4e2a\u200b\u9876\u70b9\u200b\u7684\u200b\u51f8\u200b\u7ec4\u5408\u200b</li> </ul> </li> <li>\u200b\u6700\u4f18\u200b\u89e3\u200b\uff08\u200b\u5982\u679c\u200b\u5b58\u5728\u200b\uff09\u200b\u5728\u200b\u53ef\u884c\u200b\u57df\u200b\u7684\u200b\u8fb9\u754c\u200b\u4e0a\u200b\uff0c\u200b\u4e14\u200b\u53ef\u884c\u200b\u57df\u200b\u7684\u200b\u81f3\u5c11\u200b\u4e00\u4e2a\u200b\u9876\u70b9\u200b\u662f\u200b\u6700\u4f18\u200b\u89e3\u200b</li> </ul>"},{"location":"math/operating-research/chapter-1/#_11","title":"\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u5bf9\u5076\u200b\u95ee\u9898","text":"<p>\u200b\u6bcf\u4e2a\u200b\u539f\u200b\u95ee\u9898\u200b\u5bf9\u5e94\u200b\u4e14\u200b\u4ec5\u200b\u5bf9\u5e94\u200b\u4e00\u4e2a\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\uff0c\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u662f\u200b\u539f\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b\u539f\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max z = c_1x_1 + c_2x_2 + c_3x_3 \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; a_{11}x_1 + a_{12}x_2 + a_{13}x_3 \\leq b_1 \\\\         &amp; a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = b_2 \\\\         &amp; a_{31}x_1 + a_{32}x_2 + a_{33}x_3 \\geq b_3 \\\\         &amp; x_1 \\leq 0 \\\\         &amp; x_3 \\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5176\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\min w = b_1y_1 + b_2y_2 + b_3y_3 \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; a_{11}y_1 + a_{21}y_2 + a_{31}y_3 \\leq c_1 \\\\         &amp; a_{12}y_1 + a_{22}y_2 + a_{32}y_3 = c_2 \\\\         &amp; a_{13}y_1 + a_{23}y_2 + a_{33}y_3 \\geq c_3 \\\\         &amp; y_1 \\geq 0 \\\\         &amp; y_3 \\leq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5047\u8bbe\u200b\u539f\u200b\u95ee\u9898\u200b\u5411\u200b\u6781\u5927\u503c\u200b\u4f18\u5316\u200b\uff0c\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u6709\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ul> <li>\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u5411\u200b\u6781\u5c0f\u503c\u200b\u4f18\u5316\u200b</li> <li>\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7684\u200b\u7cfb\u6570\u200b\u662f\u200b\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u7cfb\u6570\u200b</li> <li>\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u7cfb\u6570\u200b\u662f\u200b\u539f\u200b\u95ee\u9898\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7684\u200b\u7cfb\u6570\u200b</li> <li>\u200b\u539f\u200b\u95ee\u9898\u200b\u4e2d\u200b\\(\\geq 0\\)\u3001\\(\\leq 0\\)\u200b\u7684\u200b\u53d8\u91cf\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e2d\u200b\\(\\geq\\)\u3001\\(\\leq\\)\u200b\u7c7b\u578b\u200b\u7684\u200b\u7ea6\u675f\u200b</li> <li>\u200b\u539f\u200b\u95ee\u9898\u200b\u4e2d\u200b\\(\\geq\\)\u3001\\(\\leq\\)\u200b\u7c7b\u578b\u200b\u7684\u200b\u7ea6\u675f\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e2d\u200b\\(\\leq 0\\)\u3001\\(\\geq 0\\)\u200b\u7684\u200b\u53d8\u91cf\u200b</li> <li>\u200b\u4efb\u4f55\u200b\\(=\\)\u200b\u7c7b\u578b\u200b\u7684\u200b\u7ea6\u675f\u200b\u5bf9\u5e94\u200b\u81ea\u7531\u200b\u53d8\u91cf\u200b\uff0c\u200b\u53cd\u4e4b\u4ea6\u7136\u200b\u3002</li> </ul> <p>\u200b\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7b49\u4e8e\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u3002\u200b\u4e14\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u89e3\u4f1a\u200b\u4ee5\u200b\u8d1f\u503c\u200b\u7684\u200b\u5f62\u5f0f\u200b\u51fa\u73b0\u200b\u5728\u200b\u539f\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u5355\u7eaf\u5f62\u200b\u8868\u200b\u7684\u200b\\(\\lambda\\)\u200b\u90e8\u5206\u200b\u3002\u200b\u5373\u200b\uff0c\\(\\lambda = -Y^*\\)</p>"},{"location":"math/operating-research/chapter-1/#_12","title":"\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u5bf9\u5076\u6027","text":"<p>\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b\u539f\u200b\u95ee\u9898\u200b\\((P)\\)\u200b\u4e0e\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\\((D)\\)</p> \\[ \\begin{aligned}     &amp; (P) &amp; \\begin{aligned}         &amp; \\max z=c^Tx \\\\         &amp; s.t. \\left\\{         \\begin{aligned}             &amp; AX \\leq b \\\\             &amp; X\\geq 0         \\end{aligned}         \\right .     \\end{aligned} \\\\     &amp; (D) &amp; \\begin{aligned}         &amp; \\min w=b^Ty \\\\         &amp; s.t. \\left\\{         \\begin{aligned}             &amp; A^Ty \\geq c \\\\             &amp; y\\geq 0         \\end{aligned}         \\right .     \\end{aligned} \\end{aligned} \\]"},{"location":"math/operating-research/chapter-1/#_13","title":"\u5bf9\u5076\u200b\u5b9a\u7406","text":""},{"location":"math/operating-research/chapter-1/#_14","title":"\u5f31\u200b\u5bf9\u5076\u200b\u5b9a\u7406","text":"<p>\u200b\u8bbe\u200b\\((P), (D)\\)\u200b\u7684\u200b\u4e00\u7ec4\u200b\u53ef\u884c\u200b\u89e3\u200b\u5206\u522b\u200b\u4e3a\u200b\\(x, y\\)\uff0c\u200b\u5219\u200b\u6709\u200b\\(c^Tx \\leq b^Ty\\)</p> <p>\u200b\u8bc1\u660e\u200b</p> <p>\u200b\u8003\u8651\u200b\\(y^TAx\\)\uff0c\u200b\u6709\u200b\uff1a</p> <ol> <li>\\(y^TAx = (A^Ty)^TX \\geq c^TX\\)</li> <li>\\(y^TAx \\leq y^Tb = b^T y\\)</li> </ol> <p>\u200b\u6839\u636e\u200b\u5f31\u200b\u5bf9\u5076\u200b\u5b9a\u7406\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5f97\u51fa\u200b\u5982\u4e0b\u200b\u7ed3\u8bba\u200b\uff1a</p> <ol> <li>\u200b\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u662f\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u4e0b\u754c\u200b\uff0c\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u662f\u200b\u539f\u200b\u95ee\u9898\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u4e0a\u754c\u200b</li> <li>\u200b\u82e5\u539f\u200b\u95ee\u9898\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u65e0\u200b\u4e0a\u754c\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u65e0\u200b\u89e3\u200b\uff0c\u200b\u82e5\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u65e0\u200b\u4e0b\u754c\u200b\u7b49\u4ef7\u200b\u4e8e\u539f\u200b\u95ee\u9898\u200b\u65e0\u200b\u89e3\u200b</li> </ol>"},{"location":"math/operating-research/chapter-1/#_15","title":"\u6700\u4f18\u6027\u200b\u5b9a\u7406","text":"<p>\u200b\u8bbe\u200b\\((P), (D)\\)\u200b\u7684\u200b\u4e00\u7ec4\u200b\u53ef\u884c\u200b\u89e3\u200b\u5206\u522b\u200b\u4e3a\u200b\\(x^*, y^*\\)\uff0c\u200b\u82e5\u200b\\(c^Tx^* = b^Ty^*\\)\uff0c\u200b\u5219\u200b\\(x^*, y^*\\)\u200b\u5206\u522b\u200b\u4e3a\u200b\u5404\u81ea\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002</p> <p>\u200b\u8bc1\u660e\u200b</p> <p>\u200b\u6839\u636e\u200b\u5f31\u200b\u5bf9\u5076\u200b\u5b9a\u7406\u200b\uff0c\u200b\u6709\u200b\\(c^Tx\\leq b^Ty^* = c^Tx^*\\leq b^Ty\\)</p>"},{"location":"math/operating-research/chapter-1/#_16","title":"\u6700\u4f18\u200b\u89e3\u200b\u7684\u200b\u5bf9\u5076\u6027","text":"<p>\u200b\u82e5\u200b\\(B\\)\u200b\u4e3a\u200b\\((P)\\)\u200b\u7684\u200b\u6700\u4f18\u200b\u57fa\u200b\uff0c\u200b\u5219\u200b\\((D)\\)\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u4e3a\u200b\\(y^* = c_BB^{-1}\\)\u3002\u200b\u82e5\u200b\\((P)\\)\u200b\u548c\u200b\\((D)\\)\u200b\u5747\u200b\u6709\u200b\u53ef\u884c\u200b\u89e3\u200b\uff0c\u200b\u5219\u200b\u4e24\u8005\u200b\u5747\u200b\u6709\u200b\u6709\u754c\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u5e76\u4e14\u200b\u4e24\u8005\u200b\u7684\u200b\u6700\u4f18\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u76f8\u7b49\u200b\u3002</p>"},{"location":"math/operating-research/chapter-1/#_17","title":"\u4e92\u8865\u200b\u677e\u5f1b\u200b\u5b9a\u7406","text":"<p>\u200b\u5982\u679c\u200b\u539f\u200b\u95ee\u9898\u200b\\((P)\\)\u200b\u4e2d\u200b\u67d0\u200b\u4e00\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5bf9\u5076\u200b\u53d8\u91cf\u503c\u200b\u5927\u4e8e\u200b\\(0\\)\uff0c\u200b\u5219\u200b\u4e0e\u200b\u8be5\u200b\u5bf9\u5076\u200b\u53d8\u91cf\u200b\u5bf9\u5e94\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u53d6\u200b\u7b49\u5f0f\u200b\u3002\u200b\u76f8\u53cd\u200b\uff0c\u200b\u5982\u679c\u200b\u67d0\u4e00\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u53d6\u200b\u4e0d\u7b49\u5f0f\u200b\uff0c\u200b\u5219\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5bf9\u5076\u200b\u53d8\u91cf\u503c\u200b\u4e3a\u200b\\(0\\)\u3002</p> <p>\u200b\u6ce8\u610f\u200b</p> <p>\u200b\u8be5\u200b\u5b9a\u7406\u200b\u7684\u200b\u9006\u547d\u9898\u200b\u4e0d\u200b\u6210\u7acb\u200b\uff0c\u200b\u5373\u200b\u5f53\u539f\u200b\u95ee\u9898\u200b\u4e2d\u200b\u67d0\u200b\u4e00\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u53d6\u200b\u7b49\u5f0f\u200b\u65f6\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5bf9\u5076\u200b\u53d8\u91cf\u503c\u200b\u65e0\u6cd5\u200b\u786e\u5b9a\u200b\u662f\u5426\u200b\u4e3a\u200b\\(0\\)</p> <p>\u200b\u5229\u7528\u200b\u4e92\u8865\u200b\u677e\u5f1b\u200b\u6027\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u6c42\u89e3\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\u4ece\u800c\u200b\u7b80\u5316\u200b\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u6c42\u89e3\u200b</p>"},{"location":"math/operating-research/chapter-1/#_18","title":"\u5f71\u5b50\u200b\u4ef7\u683c","text":"<p>\u200b\u8003\u8651\u200b\u5f62\u200b\u5982\u200b\\((P)\\)\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\uff0c\u200b\u5f53\u200b\u67d0\u4e2a\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u7684\u200b\u6570\u503c\u200b\u53d8\u52a8\u200b\u4e00\u4e2a\u200b\u5355\u4f4d\u200b\u800c\u200b\u5176\u4ed6\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u4fdd\u6301\u200b\u4e0d\u53d8\u200b\u65f6\u200b\uff0c\u200b\u8be5\u200b\u53d8\u52a8\u200b\u5bf9\u200b\u6700\u4f18\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u9020\u6210\u200b\u7684\u200b\u5f71\u54cd\u200b\u3002\u200b\u5bf9\u5e94\u200b\u7ecf\u6d4e\u5b66\u200b\u4e2d\u200b\u8fb9\u9645\u200b\u6210\u672c\u200b\u7684\u200b\u6982\u5ff5\u200b\u3002</p> <ul> <li>\u200b\u6bcf\u4e2a\u200b\u7ea6\u675f\u200b\u5bf9\u5e94\u200b\u4e00\u4e2a\u200b\u5f71\u5b50\u200b\u4ef7\u683c\u200b</li> <li>\\(y = \\frac{\\partial z}{\\partial b}\\)</li> </ul> <p>\u200b\u5f71\u5b50\u200b\u4ef7\u683c\u200b\u53cd\u6620\u200b\u8d44\u6e90\u200b\u5728\u200b\u7cfb\u7edf\u200b\u5185\u200b\u7684\u200b\u7a00\u7f3a\u200b\u7a0b\u5ea6\u200b\uff0c\u200b\u5f53\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u4e3a\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u65f6\u200b\uff0c\u200b\u8be5\u200b\u8d44\u6e90\u200b\u7684\u200b\u91cf\u200b\u5728\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u5b9e\u9645\u4e0a\u200b\u4e0d\u200b\u53d7\u7ea6\u675f\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5f71\u5b50\u200b\u4ef7\u683c\u200b\u4e3a\u200b\\(0\\)\u3002</p>"},{"location":"math/operating-research/chapter-1/#_19","title":"\u5bf9\u5076\u200b\u5355\u7eaf\u5f62\u6cd5","text":"<p>\u200b\u6c42\u89e3\u200b\u5f62\u200b\u5982\u200b\\((P)\\)\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\u7684\u200b\u601d\u8def\u200b\u4e3a\u200b\u63a7\u5236\u200b\\(B^{-1}b \\geq 0\\)\u200b\u5bfb\u627e\u200b\u4f7f\u5f97\u200b\\(C-C_{B}B^{-1}A\\leq 0\\)\u200b\u7684\u200b\u57fa\u200b\u3002\u200b\u5bf9\u5076\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u4e0e\u200b\u4e4b\u200b\u76f8\u53cd\u200b\uff0c\u200b\u4fdd\u6301\u200b\\(C-C_{B}B^{-1}A\\leq 0\\)\uff0c\u200b\u5bfb\u627e\u200b\u4f7f\u5f97\u200b\\(B^{-1}b \\geq 0\\)\u200b\u7684\u200b\u4e00\u7ec4\u200b\u57fa\u200b\u3002</p> <p>\u200b\u6362\u5165\u200b\u53d8\u91cf\u200b\u7684\u200b\u786e\u5b9a\u200b\u4e0e\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u76f8\u540c\u200b</p>"},{"location":"math/operating-research/chapter-1/#_20","title":"\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u7075\u654f\u5ea6","text":"<p>\u200b\u7075\u654f\u5ea6\u200b\u8ba8\u8bba\u200b\u7ebf\u6027\u89c4\u5212\u200b\u4e2d\u200b\u53c2\u6570\u200b\u53d8\u5316\u200b\u5bf9\u200b\u7ebf\u6027\u89c4\u5212\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u5f71\u54cd\u200b\uff0c\u200b\u5e94\u5f53\u200b\u6ce8\u610f\u200b\uff0c\u200b\u5982\u679c\u200b\u53c2\u6570\u200b\u53d8\u52a8\u200b\u8fc7\u5927\u200b\uff0c\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u7ebf\u6027\u89c4\u5212\u200b\u89e3\u200b\u53d1\u751f\u53d8\u5316\u200b\u3002</p> <p>\u200b\u8003\u8651\u200b\u5f62\u200b\u5982\u4e0b\u200b\u5f0f\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max Z = 5x_1 + 8x_2 + 6x_3 \\\\     s.t. &amp; \\left\\{     \\begin{aligned}         x_1 &amp; + &amp; x_2 + &amp; x_3 + &amp; x_4 &amp; &amp;= &amp; 12 \\\\         x_1 &amp; + &amp; 2x_2 + &amp; 2x_3 + &amp; &amp; x_5 &amp;= &amp; 20 \\\\         x_i &amp;\\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5176\u200b\u6700\u4f18\u200b\u5355\u7eaf\u5f62\u200b\u8868\u4e3a\u200b\uff1a</p> <p> </p> <p>\u200b\u4efb\u4f55\u200b\u7cfb\u6570\u200b\u7684\u200b\u53d8\u5316\u200b\u90fd\u200b\u4f1a\u200b\u9020\u6210\u200b\u4e24\u79cd\u200b\u60c5\u51b5\u200b\uff1a</p> <ol> <li>\u200b\u6700\u4f18\u200b\u89e3\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b</li> <li>\u200b\u6700\u4f18\u200b\u89e3\u200b\u53d1\u751f\u53d8\u5316\u200b</li> </ol>"},{"location":"math/operating-research/chapter-1/#_21","title":"\u76ee\u6807\u200b\u51fd\u6570\u200b\u7cfb\u6570\u200b\u7684\u200b\u7075\u654f\u5ea6","text":"<p>\u200b\u5f53\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7cfb\u6570\u200b\u53d1\u751f\u53d8\u5316\u200b\u65f6\u200b\uff0c\u200b\u5982\u679c\u200b\u6700\u4f18\u200b\u89e3\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u5219\u200b\\(\\frac{\\partial Z}{\\partial C} = X\\)\u3002</p> <ul> <li>\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7cfb\u6570\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\u7684\u200b\u5145\u8981\u6761\u4ef6\u200b\u662f\u200b\u53d8\u5316\u200b\u540e\u200b\u7684\u200b\u68c0\u9a8c\u200b\u6570\u200b\u4ecd\u200b\u4e0d\u200b\u8d85\u8fc7\u200b\\(0\\)\u3002</li> <li>\u200b\u591a\u4e2a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7cfb\u6570\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\u7684\u200b\u5145\u5206\u6761\u4ef6\u200b\u662f\u200b\u5404\u4e2a\u200b\u7cfb\u6570\u200b\u53d8\u5316\u200b\u7684\u200b\u5e45\u5ea6\u200b\u6ee1\u8db3\u200b\\(100\\%\\)\u200b\u6cd5\u5219\u200b\u3002</li> </ul> <p>100%\u200b\u6cd5\u5219\u200b</p> <p>\u200b\u5b9a\u4e49\u200b\\(r_i = |\\frac{\\Delta c_i}{u_i}|\\)\uff0c\u200b\u5176\u4e2d\u200b\\(u_i\\)\u200b\u4e3a\u200b\u7cfb\u6570\u200b\\(c_i\\)\u200b\u5141\u8bb8\u200b\u53d8\u5316\u200b\u7684\u200b\u6700\u200b\u5927\u5e45\u5ea6\u200b\u3002\u200b\u82e5\u200b\\(\\sum r_i &lt; 1\\)\uff0c\u200b\u5219\u200b\u6700\u4f18\u200b\u89e3\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\u3002</p>"},{"location":"math/operating-research/chapter-1/#_22","title":"\u7ea6\u675f\u6761\u4ef6\u200b\u7684\u200b\u7075\u654f\u5ea6","text":"<p>\u200b\u5f53\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u53d1\u751f\u53d8\u5316\u200b\u65f6\u200b\uff0c\u200b\u5982\u679c\u200b\u5355\u7eaf\u5f62\u200b\u8868\u4e2d\u200b\u7684\u200b\\(b' = B^{-1}b\\)\u200b\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u5219\u200b\u6700\u4f18\u200b\u57fa\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u5426\u5219\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5bf9\u5076\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u6c42\u200b\u51fa\u65b0\u200b\u7684\u200b\u6700\u4f18\u200b\u57fa\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b</p> <p>\u200b\u5728\u200b\u67d0\u4e9b\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u5373\u4f7f\u200b\u6700\u4f18\u200b\u57fa\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u200b\u4e5f\u200b\u4f1a\u200b\u53d1\u751f\u53d8\u5316\u200b\u3002</p> <p>\u200b\u8bbe\u539f\u200b\u95ee\u9898\u200b\u4e2d\u200b\\(b_1\\)\u200b\u5141\u8bb8\u200b\u53d8\u5316\u200b\u7684\u200b\u8303\u56f4\u200b\u4e3a\u200b\\(\\lambda\\)\uff0c\u200b\u5219\u200b\uff1a</p> \\[ b' = B^{-1}b = \\begin{bmatrix}     1 &amp; 1 \\\\ 1 &amp; 2 \\end{bmatrix} ^{-1} \\begin{bmatrix}     \\lambda \\\\ 0 \\end{bmatrix} = \\begin{bmatrix}     4 + 2\\lambda \\\\ 8 - \\lambda \\end{bmatrix} \\] <p>\u200b\u5982\u679c\u200b\u6700\u4f18\u200b\u57fa\u200b\u4e0d\u53d8\u200b\uff0c\u200b\u5219\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\u4e0d\u53d8\u200b\uff0c\u200b\u5373\u200b\u5f71\u5b50\u200b\u4ef7\u683c\u200b\u4e0d\u53d8\u200b\u3002\u200b\u5f71\u5b50\u200b\u4ef7\u683c\u200b\u7684\u200b\u53d8\u5316\u89c4\u5f8b\u200b\u4ecd\u200b\u6ee1\u8db3\u200b\\(100\\%\\)\u200b\u6cd5\u5219\u200b\u3002</p>"},{"location":"math/operating-research/chapter-1/#_23","title":"\u6dfb\u52a0\u200b\u65b0\u200b\u53d8\u91cf\u200b\u6216\u200b\u65b0\u200b\u7ea6\u675f\u200b\u7684\u200b\u7075\u654f\u5ea6","text":"<p>\u200b\u6dfb\u52a0\u200b\u65b0\u200b\u53d8\u91cf\u200b\u65f6\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u65b0\u200b\u53d8\u91cf\u200b\u7684\u200b\u68c0\u9a8c\u200b\u6570\u200b\uff0c\u200b\u5982\u679c\u200b\u68c0\u9a8c\u200b\u6570\u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b\\(0\\)\uff0c\u200b\u5219\u200b\u65b0\u200b\u52a0\u5165\u200b\u7684\u200b\u53d8\u91cf\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u4e3a\u200b\\(0\\)\uff0c\u200b\u4e0d\u4f1a\u200b\u5bf9\u539f\u200b\u6700\u4f18\u200b\u89e3\u200b\u4ea7\u751f\u200b\u5f71\u54cd\u200b\u3002</p> <p>\u200b\u6dfb\u52a0\u200b\u65b0\u200b\u7ea6\u675f\u200b\u65f6\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u7ea6\u675f\u200b\u5728\u200b\u5f53\u524d\u200b\u7ed3\u679c\u200b\u4e0b\u200b\u662f\u5426\u200b\u6ee1\u8db3\u200b\uff0c\u200b\u5982\u679c\u200b\u7ea6\u675f\u200b\u6ee1\u8db3\u200b\u5219\u200b\u6700\u4f18\u200b\u89e3\u200b\u4e0d\u53d8\u200b\uff0c\u200b\u5426\u5219\u200b\u6700\u4f18\u200b\u89e3\u200b\u6539\u53d8\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5bf9\u5076\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u8fdb\u884c\u200b\u6c42\u89e3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-1/#_24","title":"\u6539\u53d8\u200b\u7cfb\u6570\u200b\u77e9\u9635\u200b\u7684\u200b\u7075\u654f\u5ea6","text":"<p>\u200b\u8ba1\u7b97\u200b\\(A' = B^{-1}A\\)\uff0c\u200b\u7136\u540e\u200b\u4f7f\u7528\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u6216\u200b\u5bf9\u5076\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u8fdb\u884c\u200b\u6c42\u89e3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/","title":"\u8fd0\u8f93\u200b\u95ee\u9898","text":"<p>\u200b\u8fd0\u8f93\u200b\u95ee\u9898\u200b\u662f\u200b\u4e00\u79cd\u200b\u7279\u6b8a\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\uff0c\u200b\u5177\u6709\u200b\u5982\u4e0b\u200b\u7684\u200b\u6570\u5b66\u200b\u5f62\u5f0f\u200b\uff1a</p> <ul> <li>\u200b\u7269\u8d44\u200b\u7684\u200b\u4ea7\u5730\u200b\u4e3a\u200b\\(A_i,\\; i=1, 2, \\dots m\\)\uff0c\u200b\u4ea7\u91cf\u200b\u4e3a\u200b\\(a_i\\)</li> <li>\u200b\u7269\u8d44\u200b\u7684\u200b\u9500\u5730\u200b\u4e3a\u200b\\(B_j,\\; j=1, 2, \\dots n\\)\uff0c\u200b\u9500\u91cf\u200b\u4e3a\u200b\\(b_j\\)</li> <li>\u200b\u4ece\u200b\\(i\\)\u200b\u5230\u200b\\(j\\)\u200b\u7684\u200b\u5355\u4f4d\u200b\u8fd0\u4ef7\u200b\\(c_{ij}\\)</li> <li>\u200b\u4f18\u5316\u200b\u8fd0\u8f93\u200b\u5b89\u6392\u200b\u4f7f\u5f97\u200b\u603b\u200b\u8fd0\u4ef7\u200b\u6700\u5c0f\u200b</li> </ul> <p>\u200b\u5176\u200b\u7ebf\u6027\u89c4\u5212\u200b\u5f62\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\min Z = \\sum_{i=1}^m \\sum_{i=1}^n c_{ij}x_{ij} \\\\     s.t. &amp; \\left\\{     \\begin{aligned}         &amp; \\sum_{i = 1}^m x_{ij} = b_{j} \\\\         &amp; \\sum_{j = 1}^n x_{ij} = a_{i} \\\\         &amp; x_{ij} \\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max W = \\sum_{i=1}^m a_iu_i + \\sum_{j=1}^n b_jv_j \\\\     s.t. &amp; \\left\\{         \\begin{aligned}             &amp; u_j + v_j \\leq c_{ij} \\\\             &amp; u_i \\geq 0 \\\\             &amp; v_j \\geq 0 \\\\             &amp; i = 1, 2, \\dots m \\\\             &amp; j = 1, 2, \\dots n         \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5982\u679c\u200b\u8fd0\u8f93\u200b\u95ee\u9898\u200b\u7684\u200b\u4ea7\u9500\u200b\u5e73\u8861\u200b\uff0c\u200b\u5373\u200b\\(\\sum a_i = \\sum b_j\\)\uff0c\u200b\u5219\u200b\u8fd0\u8f93\u200b\u95ee\u9898\u200b\u5fc5\u6709\u200b\u6709\u9650\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u4e2d\u57fa\u200b\u53d8\u91cf\u200b\u7684\u200b\u4e2a\u6570\u200b\u4e3a\u200b\\(m + n - 1\\)</p>"},{"location":"math/operating-research/chapter-2/#_2","title":"\u8868\u4e0a\u200b\u4f5c\u4e1a\u200b\u6cd5","text":"<p>\u200b\u8868\u683c\u200b\u7684\u200b\u901a\u5e38\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> <p> </p> <ol> <li>\u200b\u8ba1\u7b97\u200b\u521d\u59cb\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\uff0c\u200b\u5373\u200b\u5728\u200b\u8868\u683c\u200b\u4e0a\u200b\u586b\u5165\u200b\\(m + n - 1\\)\u200b\u4e2a\u200b\u6570\u5b57\u200b</li> <li>\u200b\u8ba1\u7b97\u200b\u8868\u4e2d\u200b\u5404\u200b\u7a7a\u683c\u200b\uff08\u200b\u975e\u57fa\u200b\u53d8\u91cf\u200b\uff09\u200b\u7684\u200b\u68c0\u9a8c\u200b\u6570\u200b\uff0c\u200b\u5224\u65ad\u200b\u5f53\u524d\u200b\u89e3\u200b\u662f\u5426\u200b\u6700\u4f18\u200b\u89e3\u200b</li> <li>\u200b\u82e5\u200b\u5f53\u524d\u200b\u89e3\u200b\u4e0d\u662f\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u5219\u200b\u8fed\u4ee3\u200b\u5230\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u57fa\u200b</li> </ol> <p>\u200b\u6ce8\u610f\u200b</p> <p>\u200b\u57fa\u200b\u53d8\u91cf\u200b\u4e0d\u80fd\u200b\u6784\u6210\u200b\u95ed\u200b\u56de\u8def\u200b\uff0c\u200b\u6784\u6210\u200b\u4e00\u7ec4\u200b\u95ed\u200b\u56de\u8def\u200b\u7684\u200b\u53d8\u91cf\u200b\u4e2d\u200b\u5fc5\u7136\u200b\u5305\u542b\u200b\u81f3\u5c11\u200b\u4e00\u4e2a\u200b\u975e\u57fa\u200b\u53d8\u91cf\u200b\u3002</p> <p>\u200b\u95ed\u200b\u56de\u8def\u200b\u793a\u610f\u56fe\u200b\u5982\u4e0b\u200b\uff1a</p> <p> </p> <p>\u200b\u4e00\u4e2a\u200b\u95ed\u200b\u56de\u8def\u200b\u4e2d\u200b\u542b\u6709\u200b\u5076\u6570\u200b\u4e2a\u200b\u5355\u5143\u683c\u200b\u3002</p> <p>\u200b\u5982\u4e0b\u200b\u4f7f\u7528\u200b \u200b\u7a7a\u683c\u200b \u200b\u8868\u793a\u200b\\(x_{ij} = 0\\)\u200b\u7684\u200b\u5355\u5143\u683c\u200b\uff0c\u200b\u4f7f\u7528\u200b \u200b\u6570\u5b57\u200b\u683c\u200b \u200b\u8868\u793a\u200b\\(x_{ij} &gt; 0\\)\u200b\u7684\u200b\u5355\u5143\u683c\u200b</p>"},{"location":"math/operating-research/chapter-2/#_3","title":"\u786e\u5b9a\u200b\u521d\u59cb\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3","text":"<p>\u200b\u6709\u200b\u4e09\u79cd\u200b\u65b9\u6cd5\u200b\u7528\u4e8e\u200b\u786e\u5b9a\u200b\u521d\u59cb\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\uff1a</p> <ol> <li>\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b\u6cd5\u200b</li> <li>\u200b\u897f\u5317\u89d2\u200b\u6cd5\u200b</li> <li>Vogol\u200b\u6cd5\u200b</li> </ol>"},{"location":"math/operating-research/chapter-2/#_4","title":"\u6700\u5c0f\u200b\u5143\u7d20\u200b\u6cd5","text":"<p>\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b\u6cd5\u200b\u4f18\u5148\u200b\u5bfb\u627e\u200b\u8fd0\u4ef7\u200b\u8f83\u200b\u4f4e\u200b\u7684\u200b\u53d8\u91cf\u200b\u4f5c\u4e3a\u200b\u57fa\u200b\u53d8\u91cf\u200b\u3002\u200b\u5982\u679c\u200b\u9009\u53d6\u200b\u51fa\u200b\u7684\u200b\\(x_{ij}\\)\u200b\u6b63\u200b\u5206\u91cf\u200b\u4e0d\u200b\u5305\u542b\u200b\u95ed\u200b\u56de\u8def\u200b\uff0c\u200b\u5219\u200b\\(\\{x_{ij}\\}\\)\u200b\u4e3a\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/#_5","title":"\u897f\u5317\u89d2\u200b\u6cd5","text":"<p>\u200b\u897f\u5317\u89d2\u200b\u6cd5\u4ece\u200b\u5de6\u4e0a\u200b\u5f00\u59cb\u200b\uff0c\u200b\u4f18\u5148\u200b\u6700\u5927\u5316\u200b\u6ee1\u8db3\u200b\u5de6\u4e0a\u89d2\u200b\u7684\u200b\u8fd0\u8f93\u200b\u9700\u6c42\u200b\u3002\u200b\u5982\u679c\u200b\u4f9b\u7ed9\u65b9\u200b\u9700\u6c42\u200b\u5df2\u7ecf\u200b\u6ee1\u8db3\u200b\uff0c\u200b\u5219\u200b\u5212\u53bb\u200b\u4f9b\u7ed9\u65b9\u200b\u6240\u5728\u200b\u7684\u200b\u884c\u200b\uff0c\u200b\u5426\u5219\u200b\u5212\u200b\u53bb\u200b\u9700\u6c42\u65b9\u200b\u6240\u5728\u200b\u7684\u200b\u5217\u200b\u3002\u200b\u91cd\u590d\u200b\u4ee5\u4e0a\u200b\u8fc7\u7a0b\u200b\u76f4\u5230\u200b\u53f3\u4e0b\u89d2\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/#vogol","title":"Vogol\u200b\u6cd5","text":"<p>Vogol\u200b\u6cd5\u200b\u8ba1\u7b97\u200b\u5404\u884c\u200b\u5404\u5217\u200b\u4e2d\u200b\u6700\u5c0f\u200b\\(c_{ij}\\)\u200b\u4e0e\u200b\u6b21\u200b\u5c0f\u200b\\(c_{ij}\\)\u200b\u7684\u200b\u5dee\u200b\u4f5c\u4e3a\u200b\u7f5a\u6570\u200b\uff0c\u200b\u4f18\u5148\u200b\u5b89\u6392\u200b\u7f5a\u6570\u200b\u6bd4\u8f83\u200b\u5927\u200b\u7684\u200b\u4ea7\u5730\u200b-\u200b\u9500\u5730\u200b\u3002\u200b\u9009\u53d6\u200b\u7f5a\u6570\u200b\u6700\u5927\u200b\u7684\u200b\u884c\u200b\u6216\u200b\u5217\u200b\uff0c\u200b\u5e76\u200b\u586b\u5165\u200b\u6700\u5c0f\u200b\u7684\u200b\\(c_{ij}\\)\uff0c\u200b\u7136\u540e\u200b\u5212\u200b\u53bb\u200b\u8be5\u884c\u200b\u6216\u5217\u200b\uff0c\u200b\u91cd\u65b0\u200b\u8ba1\u7b97\u200b\u7f5a\u6570\u200b\u3002</p> <p>Vogol\u200b\u6cd5\u200b\u901a\u5e38\u200b\u80fd\u591f\u200b\u5f97\u5230\u200b\u66f4\u4f18\u200b\u7684\u200b\u521d\u59cb\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/#_6","title":"\u6700\u4f18\u6027\u200b\u68c0\u9a8c","text":""},{"location":"math/operating-research/chapter-2/#_7","title":"\u95ed\u200b\u56de\u8def\u200b\u6cd5","text":"<p>\u200b\u5b9a\u7406\u200b</p> <p>\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u7a7a\u683c\u200b\uff0c\u200b\u90fd\u200b\u5b58\u5728\u200b\u4e00\u6761\u200b\u95ed\u200b\u56de\u8def\u200b\uff1a</p> <ol> <li>\u200b\u7a7a\u683c\u200b\u4f4d\u4e8e\u200b\u8be5\u95ed\u200b\u56de\u8def\u200b\u7684\u200b\u62d0\u70b9\u200b\u4e0a\u200b</li> <li>\u200b\u95ed\u200b\u56de\u8def\u200b\u7684\u200b\u5176\u4ed6\u200b\u62d0\u70b9\u200b\u662f\u200b\u6570\u5b57\u200b\u683c\u200b</li> </ol> <p>\u200b\u5c06\u200b\u4ece\u200b\u7a7a\u683c\u200b\u51fa\u53d1\u200b\uff0c\u200b\u5076\u6570\u200b\u5355\u5143\u683c\u200b\u7684\u200b\u8fd0\u4ef7\u200b\u548c\u200b\u4e0e\u200b\u5947\u6570\u200b\u5355\u5143\u683c\u200b\u7684\u200b\u8fd0\u4ef7\u200b\u548c\u200b\u4e4b\u200b\u5dee\u200b\u8bb0\u4e3a\u200b\\(\\lambda_{ij}\\)\uff0c\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u6240\u6709\u200b\u7a7a\u683c\u200b\uff0c\u200b\u90fd\u200b\u6709\u200b\\(\\lambda_{ij} \\leq 0\\)\uff0c\u200b\u5219\u200b\u5f53\u524d\u200b\u7684\u200b\\(\\{x_{ij}\\}\\)\u200b\u4e3a\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/#_8","title":"\u4f4d\u52bf\u200b\u6cd5","text":"<p>\u200b\u8bbe\u200b\\((u_{i}, v_{j})\\)\u200b\u4e3a\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u53ef\u884c\u200b\u89e3\u200b\uff0c\\(\\{x_{ij}\\}\\)\u200b\u4e3a\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u53ef\u884c\u200b\u89e3\u200b\u3002\u200b\u6839\u636e\u200b\u4e92\u8865\u200b\u677e\u5f1b\u200b\u6027\u200b\uff0c\u200b\u82e5\u200b\\(x_{ij}(c_{ij} - u_i - v_j) = 0\\)\uff0c\u200b\u5219\u200b\u4e24\u7ec4\u200b\u89e3\u200b\u5206\u522b\u200b\u4e3a\u200b\u5bf9\u5e94\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002</p> <p>\u200b\u5df2\u77e5\u200b\u5f53\u524d\u200b\u4e00\u7ec4\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\\(\\{x_{ij}\\}\\)\uff1a</p> <ol> <li>\u200b\u5bf9\u4e8e\u200b\u57fa\u200b\u53d8\u91cf\u200b\\(\\{x_{ij}\\}\\)\uff0c\u200b\u5217\u51fa\u200b\u65b9\u7a0b\u200b\\(u_i + v_j = c_{ij}\\)\uff0c\u200b\u7ec4\u5408\u6210\u200b\u65b9\u7a0b\u7ec4\u200b</li> <li>\u200b\u8be5\u200b\u65b9\u7a0b\u7ec4\u200b\u6709\u200b\\(m+n\\)\u200b\u4e2a\u200b\u53d8\u91cf\u200b\u4f46\u200b\u53ea\u6709\u200b\\(m + n - 1\\)\u200b\u4e2a\u200b\u65b9\u7a0b\u200b\uff0c\u200b\u56e0\u6b64\u200b\u6709\u200b\u65e0\u6570\u200b\u89e3\u200b\uff0c\u200b\u4efb\u9009\u200b\u4e00\u7ec4\u200b\u89e3\u200b\u5373\u53ef\u200b</li> <li>\u200b\u5bf9\u4e8e\u200b\u975e\u57fa\u200b\u53d8\u91cf\u200b\\(x_{ij}\\)\uff0c\u200b\u8ba1\u7b97\u200b\\(\\lambda_{ij} = c_{ij} - u_i - v_j\\)</li> <li>\u200b\u82e5\u200b\\(\\lambda _{ij}\\)\u200b\u5168\u90e8\u200b\\(\\geq 0\\)\uff0c\u200b\u5219\u200b\u5f53\u524d\u200b\u89e3\u4e3a\u200b\u6700\u4f18\u200b\u89e3\u200b</li> <li>\u200b\u5426\u5219\u200b\u8f6c\u200b\u51fa\u200b\\((i, j) = \\arg\\max_{(i, j)} -\\lambda_{ij}\\)\uff0c\u200b\u8c03\u6574\u200b\u8be5\u200b\u7a7a\u683c\u200b\u6240\u5728\u200b\u7684\u200b\u95ed\u200b\u56de\u8def\u200b</li> </ol>"},{"location":"math/operating-research/chapter-2/#_9","title":"\u8fd0\u8f93\u200b\u95ee\u9898\u200b\u7684\u200b\u53d8\u79cd","text":""},{"location":"math/operating-research/chapter-2/#_10","title":"\u4e24\u5730\u200b\u65e0\u200b\u8fde\u63a5","text":"<p>\u200b\u5982\u679c\u200b\u6a21\u578b\u200b\u4e2d\u200b\u5b58\u5728\u200b\u4e24\u4e2a\u200b\u5730\u70b9\u200b\u4e4b\u95f4\u200b\u4e0d\u80fd\u200b\u8fd0\u8f93\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u4e24\u5730\u200b\u95f4\u200b\u7684\u200b\u8fd0\u4ef7\u200b\u8bbe\u200b\u4e3a\u200b\u8db3\u591f\u200b\u9ad8\u200b\u7684\u200b\u6b63\u6570\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/#_11","title":"\u4ea7\u9500\u200b\u4e0d\u200b\u5e73\u8861","text":"<p>\u200b\u5f53\u200b\u95ee\u9898\u200b\u4e2d\u200b\u4ea7\u91cf\u200b\u4e0e\u200b\u9500\u91cf\u200b\u4e0d\u200b\u76f8\u7b49\u200b\u65f6\u200b\uff1a</p> <ul> <li>\u200b\u5982\u679c\u200b\u4ea7\u91cf\u200b\u5927\u4e8e\u200b\u9500\u552e\u91cf\u200b\uff0c\u200b\u5219\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u865a\u62df\u200b\u4e70\u65b9\u200b\uff0c\u200b\u589e\u52a0\u200b\\(m\\)\u200b\u4e2a\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b</li> <li>\u200b\u5982\u679c\u200b\u9500\u552e\u91cf\u200b\u5927\u4e8e\u200b\u4ea7\u91cf\u200b\uff0c\u200b\u5219\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u865a\u62df\u200b\u5356\u65b9\u200b\uff0c\u200b\u589e\u52a0\u200b\\(n\\)\u200b\u4e2a\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b</li> </ul>"},{"location":"math/operating-research/chapter-2/#_12","title":"\u9700\u6c42\u200b\u8303\u56f4","text":"<p>\u200b\u5982\u679c\u200b\u4e70\u65b9\u200b\u7684\u200b\u9700\u6c42\u200b\u5728\u200b\u4e00\u4e2a\u200b\u8303\u56f4\u200b\u4e2d\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5b58\u5728\u200b\u8303\u56f4\u200b\u9700\u6c42\u200b\u7684\u200b\u4e70\u65b9\u200b\u62c6\u200b\u5206\u4e3a\u200b\u4e24\u4e2a\u200b\u90e8\u5206\u200b\uff0c\u200b\u5373\u200b\u5fc5\u987b\u200b\u6ee1\u8db3\u200b\u7684\u200b\u9700\u6c42\u200b\u90e8\u5206\u200b\u4e0e\u200b\u53ef\u9009\u200b\u6ee1\u8db3\u200b\u7684\u200b\u9700\u6c42\u200b\u90e8\u5206\u200b\uff0c\u200b\u4e24\u200b\u90e8\u5206\u200b\u5230\u200b\u5356\u65b9\u200b\u7684\u200b\u8fd0\u4ef7\u200b\u76f8\u540c\u200b\u3002\u200b\u540c\u65f6\u200b\u52a0\u5165\u200b\u4e00\u4e2a\u200b\u865a\u62df\u200b\u5356\u65b9\u200b\uff0c\u200b\u865a\u62df\u200b\u5356\u65b9\u200b\u5230\u200b\u5fc5\u987b\u200b\u6ee1\u8db3\u200b\u7684\u200b\u865a\u62df\u200b\u4e70\u5bb6\u200b\u7684\u200b\u8fd0\u4ef7\u200b\u4e3a\u200b\u8db3\u591f\u200b\u5927\u200b\u7684\u200b\u6b63\u6570\u200b\uff0c\u200b\u865a\u62df\u200b\u5356\u65b9\u200b\u5230\u200b\u53ef\u9009\u200b\u6ee1\u8db3\u200b\u7684\u200b\u865a\u62df\u200b\u4e70\u5bb6\u200b\u7684\u200b\u8fd0\u4ef7\u200b\u4e3a\u200b\\(0\\)\u3002</p>"},{"location":"math/operating-research/chapter-2/#_13","title":"\u4f9b\u5e94\u200b\u95ee\u9898","text":"<p>\u200b\u8003\u8651\u200b\u6309\u200b\u6708\u200b\u751f\u4ea7\u200b\u7684\u200b\u5356\u65b9\u200b\u4e0e\u200b\u6309\u200b\u6708\u200b\u4ea7\u751f\u200b\u9700\u6c42\u200b\u7684\u200b\u4e70\u65b9\u200b\uff0c\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u5ef6\u671f\u200b\u4ea4\u8d27\u200b\u3002\u200b\u5356\u65b9\u200b\u751f\u4ea7\u200b\u7684\u200b\u4ea7\u54c1\u200b\u5f53\u6708\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u9500\u552e\u200b\u3002\u200b\u5219\u200b\u8be5\u200b\u95ee\u9898\u200b\u53ef\u4ee5\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u8fd0\u8f93\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/#_14","title":"\u591a\u7ef4\u200b\u8fd0\u8f93\u200b\u95ee\u9898","text":"<p>\u200b\u5047\u8bbe\u200b\u4e24\u5730\u200b\u4e4b\u95f4\u200b\u5b58\u5728\u200b\\(k\\)\u200b\u4e2a\u200b\u4e2d\u95f4\u200b\u8282\u70b9\u200b\\(c_k\\)\uff0c\u200b\u4ece\u200b\\(A\\)\u200b\u5230\u200b\\(B\\)\u200b\u7684\u200b\u6240\u6709\u200b\u8def\u5f84\u200b\u90fd\u200b\u5fc5\u987b\u200b\u7ecf\u8fc7\u200b\u8282\u70b9\u200b\\(C\\)\uff0c\u200b\u6c42\u200b\u6700\u4f18\u200b\u8fd0\u8f93\u200b\u65b9\u6848\u200b\u3002</p>"},{"location":"math/operating-research/chapter-3/","title":"\u76ee\u6807\u200b\u89c4\u5212","text":"<p>\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u89e3\u51b3\u200b\u73b0\u5b9e\u751f\u6d3b\u200b\u4e2d\u200b\u7684\u200b\u591a\u200b\u76ee\u6807\u200b\u51b3\u7b56\u95ee\u9898\u200b\u3002</p> <p>\u200b\u8bbe\u200b\u7ebf\u6027\u89c4\u5212\u200b\\((P)\\)\uff1a</p> \\[ \\begin{aligned}     &amp; \\max z=c^Tx \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; AX \\leq b \\\\         &amp; X\\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5047\u8bbe\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u76ee\u6807\u200b\u8c03\u6574\u200b\u4e3a\u200b\uff1a</p> <p>\u200b\u5bfb\u627e\u200b\u4f7f\u5f97\u200b\\(z\\geq z_0\\)\u200b\u7684\u200b\\(X\\)\uff0c\u200b\u5176\u4e2d\u200b\\(z_0\\)\u200b\u4e3a\u200b\u6307\u5b9a\u200b\u7684\u200b\u76ee\u6807\u200b\u3002\u200b\u5141\u8bb8\u200b\\(z &gt; z_0\\)\u200b\u4f46\u200b\u4e0d\u200b\u5141\u8bb8\u200b\\(z &lt; z_0\\)</p> <p>\u200b\u6b64\u65f6\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u5373\u200b\u4e3a\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\uff0c\u200b\u5f15\u5165\u200b\u4e24\u4e2a\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b\\(d^+, d^-\\)\uff0c\u200b\u5206\u522b\u200b\u8868\u793a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u8d85\u51fa\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u90e8\u5206\u200b\u4e0e\u200b\u4e0d\u8db3\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u90e8\u5206\u200b\uff0c\u200b\u4e3a\u200b\u4fdd\u8bc1\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u8d85\u8fc7\u200b\u76ee\u6807\u503c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u65b9\u5f0f\u200b\u5efa\u6a21\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max w = d^- \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; c^Tx - d^+ + d^- = z_0 \\\\         &amp; AX \\leq b \\\\         &amp; X\\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u7531\u6b64\u53ef\u89c1\u200b\uff0c\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u76f8\u5f53\u4e8e\u200b\u5728\u200b\u7ebf\u6027\u89c4\u5212\u200b\u4e2d\u200b\u5f15\u5165\u200b\u4e00\u79cd\u200b\u65b0\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff0c\u200b\u5373\u200b\u76ee\u6807\u200b\u7ea6\u675f\u200b\u3002\u200b\u6bcf\u4e2a\u200b\u76ee\u6807\u200b\u7ea6\u675f\u200b\u90fd\u200b\u9700\u8981\u200b\u5f15\u5165\u200b\u4e00\u5bf9\u200b\u53d8\u91cf\u200b\\(d^+\\)\u200b\u4e0e\u200b\\(d^-\\)\u3002\u200b\u5982\u679c\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e2d\u200b\u5b58\u5728\u200b\u591a\u4e2a\u200b\u76ee\u6807\u200b\u4e14\u200b\u4e4b\u95f4\u200b\u5b58\u5728\u200b\u4f18\u5148\u7ea7\u200b\u5173\u7cfb\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\\(P_i\\)\u200b\u4f5c\u4e3a\u200b\u9700\u8981\u200b\u4f18\u5316\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7cfb\u6570\u200b\u3002\\(P_i\\)\u200b\u6ee1\u8db3\u200b\\(P_1 \\gg P_2 \\gg \\dots \\gg P_n\\)\u200b\u5e76\u4e14\u200b\u5404\u200b\\(P_i\\)\u200b\u4e3a\u200b\u8db3\u591f\u200b\u5927\u200b\u7684\u200b\u6b63\u6570\u200b\u3002</p> <p>\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e2d\u200b\u53ea\u6709\u200b\u504f\u5dee\u200b\u53d8\u91cf\u200b\uff0c\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u7684\u200b\u4f18\u5316\u200b\u76ee\u6807\u200b\u662f\u200b\u6700\u5c0f\u5316\u200b\u504f\u5dee\u200b\u53d8\u91cf\u200b\u3002\u200b\u5982\u679c\u200b\u6700\u4f18\u200b\u6761\u4ef6\u200b\u4e0b\u200b\\(Z=0\\)\uff0c\u200b\u8bf4\u660e\u200b\u6240\u6709\u200b\u76ee\u6807\u200b\u53ef\u4ee5\u200b\u540c\u65f6\u200b\u6ee1\u8db3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-3/#_2","title":"\u76ee\u6807\u200b\u89c4\u5212\u200b\u7684\u200b\u6c42\u89e3\u200b\u65b9\u6cd5","text":"<p>\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u56fe\u89e3\u6cd5\u200b\u6216\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u5bf9\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u8fdb\u884c\u200b\u6c42\u89e3\u200b</p>"},{"location":"math/operating-research/chapter-3/#_3","title":"\u56fe\u89e3\u6cd5","text":"<p>\u200b\u4f7f\u7528\u200b\u56fe\u89e3\u6cd5\u200b\u6c42\u89e3\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u65f6\u200b\uff0c\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u753b\u51fa\u200b\u76ee\u6807\u200b\u7ea6\u675f\u200b\u4e2d\u200b\u7684\u200b\u786c\u200b\u7ea6\u675f\u200b\uff0c\u200b\u518d\u200b\u6309\u7167\u200b\u76ee\u6807\u200b\u7ea6\u675f\u200b\u7684\u200b\u4f18\u5148\u7ea7\u200b\u753b\u51fa\u200b\u8f6f\u200b\u7ea6\u675f\u200b\uff0c\u200b\u5e76\u200b\u6ce8\u660e\u200b\u7ea6\u675f\u200b\u504f\u5dee\u200b\u65b9\u5411\u200b\u3002\u200b\u6309\u7167\u200b\u7ea6\u675f\u200b\u7684\u200b\u4f18\u5148\u7ea7\u200b\u4f9d\u6b21\u200b\u5212\u5206\u200b\u80fd\u591f\u200b\u6ee1\u8db3\u200b\u7ea6\u675f\u200b\u7684\u200b\u53ef\u884c\u200b\u57df\u200b\u3002\u200b\u76f4\u5230\u200b\u53ef\u884c\u200b\u57df\u200b\u4e3a\u200b\u7a7a\u200b\u6216\u200b\u7ea6\u675f\u200b\u5168\u90e8\u200b\u6ee1\u8db3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-3/#_4","title":"\u5355\u7eaf\u5f62\u6cd5","text":"<p>\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u5c06\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u4e2d\u200b\u7684\u200b\u786c\u200b\u7ea6\u675f\u200b\u5316\u4e3a\u200b\u6807\u51c6\u578b\u200b\u3002\u200b\u5217\u51fa\u200b\u5355\u7eaf\u5f62\u200b\u8868\u540e\u200b\uff0c\u200b\u9010\u6b65\u200b\u5c06\u200b\u5305\u542b\u200b\u5728\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e2d\u200b\u7684\u200b\u504f\u5dee\u200b\u4ece\u200b\u57fa\u4e2d\u6362\u200b\u51fa\u200b\u3002\u200b\u5e76\u4e14\u200b\u5c3d\u53ef\u80fd\u200b\u4f7f\u5f97\u200b\u975e\u57fa\u200b\u53d8\u91cf\u200b\u7684\u200b\u68c0\u9a8c\u200b\u6570\u4e3a\u200b\u6b63\u200b\uff08\u200b\u56e0\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u7684\u200b\u4f18\u5316\u200b\u65b9\u5411\u200b\u4e3a\u200b\u6700\u5c0f\u5316\u200b\uff09\u3002</p>"},{"location":"math/operating-research/chapter-4/","title":"\u6574\u6570\u200b\u89c4\u5212","text":"<p>\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u662f\u200b\u90e8\u5206\u200b\uff08\u200b\u6216\u200b\u5168\u90e8\u200b\uff09\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u5747\u53d6\u200b\u6574\u6570\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u3002\u200b\u5168\u90e8\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u5747\u53d6\u200b\u6574\u6570\u200b\u7684\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u79f0\u4e3a\u200b\u7eaf\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u3002\u200b\u5982\u679c\u200b\u6574\u6570\u200b\u53d8\u91cf\u200b\u53ea\u80fd\u200b\u53d6\u200b0-1\u200b\u4e24\u4e2a\u200b\u503c\u200b\uff0c\u200b\u5219\u200b\u79f0\u4e3a\u200b0-1\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u3002</p>"},{"location":"math/operating-research/chapter-4/#_2","title":"\u6574\u6570\u200b\u89c4\u5212\u200b\u7684\u200b\u6c42\u89e3\u200b\u65b9\u6cd5","text":"<p>\u200b\u6c42\u89e3\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u76ee\u524d\u200b\u6709\u200b\u5982\u4e0b\u200b\u7b97\u6cd5\u200b\uff1a</p> <ol> <li>\u200b\u5272\u200b\u5e73\u9762\u200b\u6cd5\u200b</li> <li>\u200b\u5206\u652f\u200b\u5b9a\u754c\u200b\u6cd5\u200b</li> <li>\u200b\u9690\u200b\u679a\u4e3e\u6cd5\u200b\uff08\u200b\u9002\u7528\u200b\u4e8e\u200b\u6c42\u89e3\u200b0-1\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\uff09</li> <li>\u200b\u542f\u53d1\u5f0f\u200b\u7b97\u6cd5\u200b</li> </ol> <p>\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b\u7eaf\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max z=c^Tx \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; AX \\leq b \\\\         &amp; X\\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5176\u4e2d\u200b\\(A, b, x\\)\u200b\u5747\u200b\u4e3a\u200b\u6574\u6570\u200b\u3002</p>"},{"location":"math/operating-research/chapter-4/#_3","title":"\u5272\u200b\u5e73\u9762\u200b\u6cd5","text":"<p>\u200b\u79f0\u200b\u6d88\u9664\u200b\u6574\u6570\u200b\u7ea6\u675f\u200b\u7684\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u4e3a\u200b\u677e\u5f1b\u200b\u95ee\u9898\u200b\uff0c\u200b\u5272\u200b\u5e73\u9762\u200b\u6cd5\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u6c42\u89e3\u200b\u677e\u5f1b\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u8bbe\u200b\u4e3a\u200b\\(X^*\\)\u3002\u200b\u901a\u5e38\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\\(X^*\\)\u200b\u4e0d\u662f\u200b\u6574\u6570\u200b\u6216\u200b\u4e0d\u5168\u662f\u200b\u6574\u6570\u200b\u3002\u200b\u5355\u7eaf\u5f62\u200b\u8868\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</p> <p> </p> <p>\u200b\u8bbe\u200b\u5355\u7eaf\u5f62\u200b\u8868\u4e2d\u200b\u7684\u200b\u7b2c\u200b\\(i\\)\u200b\u884c\u200b\u5bf9\u5e94\u200b\u7684\u200b\\(b'_i\\)\u200b\u4e0d\u200b\u4e3a\u200b\u6574\u6570\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u8fd9\u200b\u4e00\u884c\u200b\uff0c\u200b\u6709\u200b\uff1a</p> \\[ x_i + \\sum_{j=m+1}^n a'_{ij}x_j = b'_i \\] <p>\u200b\u5c06\u200b\\(a'_{ij}, b'_i\\)\u200b\u5212\u5206\u200b\u4e3a\u200b\u6574\u6570\u200b\u90e8\u5206\u200b\u4e0e\u200b\u5c0f\u6570\u200b\u90e8\u5206\u200b\uff0c\u200b\u5373\u200b\uff1a</p> \\[ \\begin{aligned}     a'_{ij} &amp;= \\tilde a_{ij} + \\alpha_{ij}, \\; \\tilde a_{ij} = \\lfloor a'_{ij}\\rfloor \\\\     b'_{i} &amp;= \\tilde b_{i} + \\beta_{i}, \\; \\tilde b_{ij} = \\lfloor b'_{i}\\rfloor \\end{aligned} \\] <p>\u200b\u5c06\u200b\u8fd9\u200b\u4e00\u884c\u200b\u7b49\u5f0f\u200b\u53d8\u5f62\u200b\uff0c\u200b\u5f97\u5230\u200b\uff1a</p> \\[ \\beta_i - \\sum_{j=m+1}^n\\alpha_{ij}y_j = x_i - \\tilde b_i + \\sum_{j=m+1}^n \\tilde \\alpha_{ij}y_j \\] <p>\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u7684\u200b\u6761\u4ef6\u200b\u8981\u6c42\u200b\u7b49\u5f0f\u200b\u53f3\u4fa7\u200b\u4e3a\u200b\u6574\u6570\u200b\uff0c\u200b\u56e0\u6b64\u200b\u7b49\u5f0f\u200b\u5de6\u4fa7\u200b\u540c\u6837\u200b\u9700\u8981\u200b\u4e3a\u200b\u6574\u6570\u200b\uff0c\u200b\u5219\u200b\u6709\u200b\\(\\beta_i - \\sum_{j=m+1}^n\\alpha_{ij}y_j + s_i = 0\\)\uff0c\u200b\u5176\u4e2d\u200b\\(s_i\\)\u200b\u4e3a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u79f0\u200b\u65b9\u7a0b\u200b</p> \\[ \\beta_i - \\sum_{j=m+1}^n\\alpha_{ij}y_j + s_i = 0 \\] <p>\u200b\u4e3a\u200b\u5272\u200b\u5e73\u9762\u200b\u65b9\u7a0b\u200b\uff0c\u200b\u5272\u200b\u5e73\u9762\u200b\u65b9\u7a0b\u200b\u4f5c\u4e3a\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff0c\u200b\u6d88\u53bb\u200b\u4e86\u200b\u975e\u200b\u6574\u6570\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u7559\u4e0b\u200b\u6574\u6570\u200b\u89e3\u200b\u3002\u200b\u5411\u200b\u677e\u5f1b\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u5355\u7eaf\u5f62\u200b\u8868\u4e2d\u200b\u52a0\u5165\u200b\u5272\u200b\u5e73\u9762\u200b\u65b9\u7a0b\u200b\u4f5c\u4e3a\u200b\u65b0\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff0c\u200b\u7ee7\u7eed\u200b\u6c42\u89e3\u200b\u7ebf\u6027\u89c4\u5212\u200b\u5373\u53ef\u200b\u5f97\u5230\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-4/#_4","title":"\u5206\u652f\u200b\u5b9a\u754c\u200b\u6cd5","text":"<p>\u200b\u5148\u200b\u6c42\u89e3\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u5bf9\u5e94\u200b\u7684\u200b\u677e\u5f1b\u200b\u95ee\u9898\u200b\uff0c\u200b\u5e76\u200b\u4f30\u8ba1\u200b\u4e00\u4e2a\u200b\u975e\u200b\u6700\u4f18\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u4f5c\u4e3a\u200b\u8bb0\u5f55\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u6700\u4f18\u200b\u89e3\u4e2d\u200b\u7684\u200b\u975e\u200b\u6574\u6570\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\uff0c\u200b\u9009\u62e9\u200b\u5176\u4e2d\u200b\u4e00\u4e2a\u200b\\(x_i\\)\u200b\u8fdb\u884c\u200b\u5206\u652f\u200b\uff1a</p> <ol> <li>\u200b\u6dfb\u52a0\u200b\u7ea6\u675f\u200b\\(x'_i \\leq \\lfloor x_i \\rfloor\\)</li> <li>\u200b\u6dfb\u52a0\u200b\u7ea6\u675f\u200b\\(x'_i \\geq \\lceil x_i \\rceil\\)</li> </ol> <p>\u200b\u6dfb\u52a0\u200b\u7ea6\u675f\u200b\u540e\u200b\u91cd\u65b0\u200b\u6c42\u89e3\u200b\u7ebf\u6027\u89c4\u5212\u200b\uff0c\u200b\u5982\u679c\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u89e3\u200b\u6ee1\u8db3\u200b\u6574\u6570\u200b\u7ea6\u675f\u200b\uff0c\u200b\u5219\u200b\u5c06\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u4e0e\u200b\u8bb0\u5f55\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u76f8\u200b\u6bd4\u8f83\u200b\u3002\u200b\u5982\u679c\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u66f4\u4f18\u200b\uff0c\u200b\u5219\u200b\u66f4\u65b0\u200b\u6700\u4f18\u200b\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u505c\u6b62\u200b\u5411\u4e0b\u200b\u63a2\u7d22\u200b\uff08\u200b\u526a\u679d\u200b\uff09\u3002</p> <p>\u200b\u5206\u652f\u200b\u53ef\u4ee5\u200b\u91c7\u53d6\u200b\u6df1\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b\u6216\u200b\u5e7f\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b\u3002</p>"},{"location":"math/operating-research/chapter-4/#_5","title":"\u9690\u200b\u679a\u4e3e\u6cd5","text":"<p>\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b0-1\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max z=c^Tx \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; AX \\leq b \\\\         &amp; x_i \\in \\{0, 1\\}     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u9996\u5148\u200b\u8bd5\u63a2\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u53ef\u884c\u200b\u89e3\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u4f5c\u4e3a\u200b\u8bb0\u5f55\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u200b\\(Z_0\\)\uff0c\u200b\u518d\u200b\u679a\u4e3e\u200b\u6240\u6709\u200b\u7684\u200b\u53ef\u884c\u200b\u89e3\u200b\uff0c\u200b\u82e5\u89e3\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u8d85\u8fc7\u200b\u6700\u4f18\u200b\u503c\u5219\u200b\u66f4\u65b0\u200b\u6700\u4f18\u200b\u503c\u200b\u3002</p>"},{"location":"math/operating-research/chapter-4/#_6","title":"\u6307\u6d3e\u200b\u95ee\u9898","text":"<p>\u200b\u6307\u6d3e\u200b\u95ee\u9898\u200b\u662f\u200b\u7279\u6b8a\u200b\u7684\u200b0-1\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\uff0c\u200b\u5176\u200b\u76ee\u6807\u200b\u662f\u200b\u5c06\u200b\\(n\\)\u200b\u4e2a\u200b\u5de5\u4eba\u200b\u5728\u200b\\(n\\)\u200b\u4e2a\u200b\u5de5\u4f5c\u200b\u4e2d\u200b\u5206\u914d\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5de5\u4eba\u200b\\(i\\)\u200b\u5728\u200b\u8fdb\u884c\u200b\u5de5\u4f5c\u200b\\(j\\)\u200b\u65f6\u200b\uff0c\u200b\u90fd\u200b\u4f1a\u200b\u4ea7\u751f\u200b\u6210\u672c\u200b\\(c_{ij}\\)\uff0c\u200b\u51b3\u7b56\u95ee\u9898\u200b\u7684\u200b\u76ee\u6807\u200b\u662f\u200b\u6700\u5c0f\u5316\u200b\u603b\u6210\u672c\u200b\u3002\u200b\u8bb0\u200b\u6210\u672c\u200b\u77e9\u9635\u200b\u4e3a\u200b</p> \\[ A = \\begin{bmatrix}     c_{11} &amp; c_{12} &amp; \\cdots &amp; c_{1n} \\\\     c_{21} &amp; c_{22} &amp; \\cdots &amp; c_{2n} \\\\     \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\     c_{n1} &amp; c_{n2} &amp; \\cdots &amp; c_{nn} \\end{bmatrix} \\] <p>\u200b\u6027\u8d28\u200b</p> <p>\u200b\u82e5\u200b\u7cfb\u6570\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u7684\u200b\u67d0\u200b\u4e00\u884c\u200b\u6216\u200b\u67d0\u200b\u4e00\u5217\u200b\u51cf\u53bb\u200b\u5206\u522b\u200b\u51cf\u53bb\u200b\u4e00\u4e2a\u200b\u5e38\u6570\u200b\\(k\\)\uff0c\u200b\u4e0d\u200b\u5f71\u54cd\u200b\u6307\u6d3e\u200b\u95ee\u9898\u200b\u7684\u200b\u89e3\u200b\u3002</p> <p>\u200b\u4f7f\u7528\u200b\u5308\u7259\u5229\u200b\u7b97\u6cd5\u200b\u6c42\u89e3\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u3002\u200b\u9996\u5148\u200b\u5bf9\u200b\u7cfb\u6570\u200b\u77e9\u9635\u200b\u5148\u200b\u51cf\u53bb\u200b\u6bcf\u884c\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\uff0c\u200b\u518d\u200b\u51cf\u53bb\u200b\u6bcf\u5217\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\u3002\u200b\u4ece\u800c\u200b\u6bcf\u884c\u200b\u3001\u200b\u6bcf\u5217\u200b\u4e2d\u200b\u81f3\u5c11\u200b\u51fa\u73b0\u200b\u4e00\u4e2a\u96f6\u200b\u3002\u200b\u5b9a\u4e49\u200b\u201c\u200b\u72ec\u7acb\u200b\u96f6\u200b\u201d\u200b\u4e3a\u200b\u7cfb\u6570\u200b\u77e9\u9635\u200b\u4e2d\u200b\u65e2\u200b\u4e0d\u200b\u5904\u4e8e\u200b\u540c\u4e00\u200b\u884c\u200b\u3001\u200b\u4e5f\u200b\u4e0d\u200b\u5904\u4e8e\u200b\u540c\u4e00\u200b\u5217\u200b\u7684\u200b\\(0\\)\u200b\u7684\u200b\u6700\u5927\u200b\u4e2a\u6570\u200b\u3002</p> <ol> <li>\u200b\u82e5\u200b\u201c\u200b\u72ec\u7acb\u200b\u96f6\u200b\u201d\u200b\u7684\u200b\u4e2a\u6570\u200b\u7b49\u4e8e\u200b\u65b9\u9635\u200b\u7684\u200b\u7ef4\u5ea6\u200b\uff0c\u200b\u5219\u200b\u5f97\u5230\u200b\u6307\u6d3e\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u200b\u7684\u200b\u6307\u6d3e\u200b\u65b9\u6848\u200b\u4e0e\u200b\u72ec\u7acb\u200b\u96f6\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u5206\u5e03\u200b\u76f8\u540c\u200b\u3002</li> <li>\u200b\u82e5\u200b\u201c\u200b\u72ec\u7acb\u200b\u96f6\u200b\u201d\u200b\u7684\u200b\u4e2a\u6570\u200b\u5c0f\u4e8e\u200b\u65b9\u9635\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u3002\u200b\u9996\u5148\u200b\u627e\u51fa\u200b\u65b9\u9635\u200b\u4e2d\u200b\u80fd\u591f\u200b\u8986\u76d6\u200b\u6240\u6709\u200b\u96f6\u200b\u5143\u7d20\u200b\u7684\u200b\u6700\u5c11\u200b\u76f4\u7ebf\u200b\u6570\u200b\u3002\u200b\u5728\u200b\u4e0d\u200b\u88ab\u200b\u76f4\u7ebf\u200b\u8986\u76d6\u200b\u7684\u200b\u5143\u7d20\u200b\u4e2d\u200b\u5bfb\u627e\u200b\u6700\u5c0f\u503c\u200b\uff0c\u200b\u5c06\u200b\u6240\u5728\u200b\u884c\u6216\u5217\u200b\u51cf\u53bb\u200b\u8fd9\u4e2a\u200b\u6700\u5c0f\u503c\u200b\u3002\u200b\u7136\u540e\u200b\u5bf9\u200b\u4ea7\u751f\u200b\u7684\u200b\u8d1f\u503c\u200b\u8fdb\u884c\u200b\u8c03\u6574\u200b\uff0c\u200b\u76f4\u5230\u200b\u77e9\u9635\u200b\u4e2d\u200b\u4e0d\u200b\u5b58\u5728\u200b\u8d1f\u503c\u200b\u3002\u200b\u7ed3\u675f\u200b\u540e\u200b\u7ee7\u7eed\u200b\u68c0\u67e5\u200b\u72ec\u7acb\u200b\u96f6\u200b\u7684\u200b\u4e2a\u6570\u200b\u3002</li> </ol>"},{"location":"math/operations-mamagement/","title":"\u9ad8\u7b49\u200b\u8fd0\u4f5c\u200b\u7ba1\u7406","text":"<ul> <li>\u200b\u9884\u6d4b\u200b\u6280\u672f\u200b</li> <li>\u200b\u9884\u6d4b\u200b\u4e0e\u200b\u9700\u6c42\u200b\u5efa\u6a21\u200b</li> <li>\u200b\u7efc\u5408\u200b\u751f\u4ea7\u200b\u8ba1\u5212\u200b</li> <li>\u200b\u786e\u5b9a\u200b\u9700\u6c42\u200b\u4e0b\u200b\u7684\u200b\u5e93\u5b58\u200b\u63a7\u5236\u200b</li> <li>\u200b\u968f\u673a\u200b\u9700\u6c42\u200b\u4e0b\u200b\u7684\u200b\u5e93\u5b58\u200b\u63a7\u5236\u200b</li> </ul>"},{"location":"math/operations-mamagement/#todo","title":"TODO","text":"<ul> <li> \u200b\u5468\u671f\u200b\u68c0\u67e5\u200b\u6a21\u578b\u200b\u2014\u2014\\((s, S)\\)\u200b\u7b56\u7565\u200b</li> <li> \u200b\u591a\u7ea7\u200b\u7cfb\u7edf\u200b\u4e0e\u200b\u534f\u8c03\u200b\u8ba2\u8d27\u200b</li> </ul>"},{"location":"math/operations-mamagement/aggregrate-planning/","title":"\u7efc\u5408\u200b\u751f\u4ea7\u200b\u8ba1\u5212","text":""},{"location":"math/operations-mamagement/aggregrate-planning/#_2","title":"\u6a21\u578b\u200b\u8bbe\u5b9a","text":"<p>\u200b\u6a21\u578b\u200b\u4e2d\u200b\u5305\u542b\u200b\u4e00\u4e2a\u200b\u4f01\u4e1a\u200b\uff0c\u200b\u4f01\u4e1a\u200b\u901a\u8fc7\u200b\u96c7\u4f63\u5de5\u4eba\u200b\u751f\u4ea7\u200b\u5e76\u200b\u9500\u552e\u200b\u4ea7\u54c1\u200b\u4f5c\u4e3a\u200b\u6536\u5165\u200b\u3002\u200b\u76f8\u5173\u200b\u7684\u200b\u53c2\u6570\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u6210\u672c\u200b\u53c2\u6570\u200b</li> <li>\u200b\u5e93\u5b58\u200b\u76f8\u5173\u200b\uff1a\u200b\u539f\u6750\u6599\u200b\u6210\u672c\u200b\u3001\u200b\u6301\u6709\u200b\u6210\u672c\u200b\u3001\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b</li> <li>\u200b\u4eba\u5458\u200b\u76f8\u5173\u200b\uff1a\u200b\u96c7\u4f63\u200b\u6210\u672c\u200b\u3001\u200b\u89e3\u96c7\u200b\u6210\u672c\u200b\u3001\u200b\u5de5\u65f6\u200b\u6210\u672c\u200b\u3001\u200b\u95f2\u7f6e\u200b\u6210\u672c\u200b\u4e0e\u200b\u52a0\u73ed\u200b\u6210\u672c\u200b</li> <li>\u200b\u751f\u4ea7\u529b\u200b\u76f8\u5173\u200b\uff1a\u200b\u8f6c\u5305\u200b\u6210\u672c\u200b</li> <li>\u200b\u6536\u76ca\u200b\u53c2\u6570\u200b\uff1a\u200b\u5355\u4f4d\u200b\u4ea7\u54c1\u200b\u552e\u4ef7\u200b</li> <li>\u200b\u751f\u4ea7\u200b\u53c2\u6570\u200b</li> <li>\u200b\u5355\u4f4d\u200b\u4ea7\u54c1\u200b\u751f\u4ea7\u200b\u5de5\u65f6\u200b</li> <li>\u200b\u65e5\u5e38\u200b\u5de5\u4f5c\u200b\u5de5\u65f6\u200b</li> <li>\u200b\u5e93\u5b58\u200b\u53c2\u6570\u200b</li> <li>\u200b\u671f\u521d\u200b\u5b58\u91cf\u200b\u4e0e\u200b\u671f\u672b\u200b\u5b58\u91cf\u200b</li> <li>\u200b\u9700\u6c42\u91cf\u200b</li> </ul> \u200b\u53d8\u91cf\u200b \u200b\u8bf4\u660e\u200b \u200b\u53d8\u91cf\u200b \u200b\u8bf4\u660e\u200b \\(c_F\\) \u200b\u89e3\u96c7\u200b\u6210\u672c\u200b \\(c_H\\) \u200b\u96c7\u4f63\u200b\u6210\u672c\u200b \\(c_I\\) \u200b\u6301\u6709\u200b\u6210\u672c\u200b \\(c_P\\) \u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b \\(c_R\\) \u200b\u65e5\u5e38\u200b\u751f\u4ea7\u6210\u672c\u200b \\(c_O\\) \u200b\u52a0\u73ed\u200b\u751f\u4ea7\u6210\u672c\u200b \\(c_S\\) \u200b\u8f6c\u5305\u200b\u6210\u672c\u200b \\(c_U\\) \u200b\u751f\u4ea7\u529b\u200b\u7684\u200b\u95f2\u7f6e\u200b\u6210\u672c\u200b \\(D_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u9700\u6c42\u200b \\(n_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u5de5\u200b\u65f6\u6570\u200b \\(F_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u89e3\u96c7\u200b\u4eba\u6570\u200b \\(H_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u96c7\u4f63\u200b\u4eba\u6570\u200b \\(K\\) \u200b\u5355\u4f4d\u200b\u5de5\u65f6\u200b\u4ea7\u91cf\u200b \\(W_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u5458\u5de5\u200b\u4eba\u6570\u200b \\(P_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u603b\u5de5\u200b\u65f6\u200b \\(O_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u603b\u200b\u52a0\u73ed\u200b\u5de5\u65f6\u200b \\(U_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u603b\u200b\u95f2\u7f6e\u200b\u5de5\u65f6\u200b \\(S_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u5916\u5305\u200b\u4ea7\u91cf\u200b \\(I_t^+\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u591a\u4f59\u200b\u5e93\u5b58\u200b \\(I_t^-\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u7f3a\u8d27\u200b\u6c34\u5e73\u200b <p>\u200b\u4e0a\u8868\u200b\u4e2d\u200b\\(I_t^\\pm, W_t\\)\u200b\u5206\u522b\u200b\u6307\u7b2c\u200b\\(t\\)\u200b\u4e2a\u200b\u5468\u671f\u200b\u7ed3\u675f\u200b\u65f6\u200b\u7684\u200b\u6570\u503c\u200b\u3002\u200b\u7279\u522b\u200b\u5730\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u521d\u59cb\u72b6\u6001\u200b\uff0c\u200b\u6709\u200b</p> <ul> <li>\\(I_0\\)\u200b\u4e3a\u200b\u521d\u59cb\u200b\u5e93\u5b58\u200b</li> <li>\\(W_0\\)\u200b\u4e3a\u200b\u521d\u59cb\u200b\u5458\u5de5\u200b\u6570\u200b</li> </ul>"},{"location":"math/operations-mamagement/aggregrate-planning/#_3","title":"\u7b56\u7565\u200b\u5206\u7c7b","text":"<p>\u200b\u6839\u636e\u200b\u4e0d\u540c\u200b\u7684\u200b\u6210\u672c\u200b\u7b56\u7565\u200b\u4e0e\u200b\u98ce\u9669\u200b\u504f\u597d\u200b\uff0c\u200b\u7efc\u5408\u200b\u8ba1\u5212\u200b\u7c7b\u578b\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u591a\u79cd\u4e0d\u540c\u200b\u7684\u200b\u8ba1\u5212\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <ol> <li>\u200b\u8ffd\u9010\u200b\u7b56\u7565\u200b\uff08\u200b\u6700\u5c0f\u5316\u200b\u5e93\u5b58\u200b\uff09</li> <li>\u200b\u5e73\u7a33\u200b\u7b56\u7565\u200b\uff08\u200b\u6700\u5c0f\u5316\u200b\u5458\u5de5\u200b\u7684\u200b\u53d8\u52a8\u200b\uff09</li> <li>\u200b\u5f39\u6027\u200b\u5de5\u4f5c\u200b\u65f6\u95f4\u200b\u7b56\u7565\u200b</li> <li>\u200b\u6df7\u5408\u7b56\u7565\u200b</li> </ol>"},{"location":"math/operations-mamagement/aggregrate-planning/#_4","title":"\u6c42\u89e3\u200b\uff1a\u200b\u7ebf\u6027\u89c4\u5212","text":"<p>\u200b\u5bf9\u4e8e\u200b\u5e93\u5b58\u200b\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u7ea6\u675f\u200b\uff1a</p> \\[ \\begin{aligned}     I_t &amp;= I_t^+ - I_t^- \\\\     I_t &amp;= I_{t-1} + P_t + S_t - D_t \\end{aligned} \\] <p>\u200b\u5bf9\u4e8e\u200b\u5458\u5de5\u200b\u7684\u200b\u96c7\u4f63\u200b\u4e0e\u200b\u89e3\u96c7\u200b\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u7ea6\u675f\u200b\uff1a</p> \\[ \\begin{aligned}     W_t &amp;= W_{t-1} + H_t - F_t \\\\     0 &amp;= H_t\\cdot F_t \\end{aligned} \\] <p>\u200b\u5bf9\u4e8e\u200b\u5458\u5de5\u200b\u7684\u200b\u751f\u4ea7\u200b\u5de5\u65f6\u200b\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u7ea6\u675f\u200b\uff1a</p> \\[ \\begin{aligned}     P_t &amp;= Kn_t W_t + O_t - U_t \\\\     0 &amp;= O_t \\cdot U_t \\end{aligned} \\] <p>\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e3a\u200b\u6700\u5c0f\u5316\u200b\u5404\u9879\u200b\u6210\u672c\u200b\u3002</p>"},{"location":"math/operations-mamagement/fdm/","title":"\u9884\u6d4b\u200b\u4e0e\u200b\u9700\u6c42\u200b\u5efa\u6a21","text":"<p>\u200b\u7ecf\u5178\u200b\u6a21\u578b\u200b\uff08\u200b\u5982\u200b\u6ed1\u52a8\u200b\u5e73\u5747\u200b\u3001\u200b\u6307\u6570\u200b\u5e73\u6ed1\u200b\u7b49\u200b\uff09\u200b\u7684\u200b\u7f3a\u9677\u200b\u5728\u4e8e\u200b\uff1a</p> <ul> <li>\u200b\u7ecf\u5178\u200b\u6a21\u578b\u200b\u6301\u7eed\u200b\u4f9d\u8d56\u4e8e\u200b\u5386\u53f2\u200b\u89c2\u6d4b\u200b\u6570\u636e\u200b</li> <li>\u200b\u90e8\u5206\u200b\u6a21\u578b\u200b\uff08\u200b\u5982\u200b\u6ed1\u52a8\u200b\u5e73\u5747\u200b\uff09\u200b\u4ec5\u200b\u9002\u7528\u200b\u4e8e\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b</li> <li>\u200b\u4ea7\u54c1\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u4e0d\u540c\u200b\uff0c\u200b\u5982\u679c\u200b\u4e00\u4e2a\u200b\u4ea7\u54c1\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u660e\u663e\u200b\u8f83\u200b\u77ed\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u6ca1\u6709\u200b\u5145\u8db3\u200b\u7684\u200b\u5386\u53f2\u6570\u636e\u200b\u5bf9\u200b\u8be5\u200b\u4ea7\u54c1\u200b\u8fdb\u884c\u200b\u9884\u6d4b\u200b\u3002</li> </ul> <p>\u200b\u672c\u7ae0\u200b\u8ba8\u8bba\u200b\u5982\u4e0b\u200b\u6a21\u578b\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u6a21\u578b\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u9884\u6d4b\u200b\u751f\u547d\u5468\u671f\u200b\u8f83\u200b\u77ed\u200b\u7684\u200b\u4ea7\u54c1\u200b\u3002</p> <ul> <li>Bass\u200b\u6269\u6563\u200b\u6a21\u578b\u200b</li> <li>\u200b\u9886\u5148\u200b\u6307\u6807\u200b\u6a21\u578b\u200b</li> <li>\u200b\u9009\u62e9\u200b\u6a21\u578b\u200b</li> </ul>"},{"location":"math/operations-mamagement/fdm/#bass","title":"Bass\u200b\u6269\u6563\u200b\u6a21\u578b","text":"<p>Bass\u200b\u6a21\u578b\u200b\u63d0\u51fa\u200b\u4e86\u200b\u6269\u6563\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u6982\u5ff5\u200b\u3002\uff08\u200b\u67d0\u200b\u4e00\u7c7b\u200b\uff09\u200b\u65b0\u200b\u4ea7\u54c1\u200b\u9996\u6b21\u200b\u4e0a\u5e02\u200b\u540e\u200b\uff0c\u200b\u8be5\u200b\u4ea7\u54c1\u200b\u7684\u200b\u8bbe\u8ba1\u200b\u7406\u5ff5\u200b\u7b49\u200b\u5185\u5bb9\u200b\u4f1a\u200b\u5728\u200b\u5e02\u573a\u200b\u4e0a\u200b\u6269\u6563\u200b\uff08\u200b\u88ab\u200b\u6a21\u4eff\u200b\uff09\uff0c\u200b\u9020\u6210\u200b\u540c\u7c7b\u4ea7\u54c1\u200b\u7684\u200b\u4ea7\u91cf\u200b\u4e0a\u5347\u200b\u3002</p> <p>\u200b\u524d\u63d0\u200b\u5047\u8bbe\u200b\uff1a</p> <ul> <li>\u200b\u5e02\u573a\u200b\u4e0a\u200b\u7684\u200b\u7528\u6237\u6570\u91cf\u200b\u4e0d\u53d8\u200b</li> <li>\u200b\u7528\u6237\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u7b80\u5355\u200b\u5206\u4e3a\u200b\u521b\u65b0\u8005\u200b\u4e0e\u200b\u6a21\u4eff\u8005\u200b</li> <li>\u200b\u521b\u65b0\u8005\u200b\u8d2d\u4e70\u200b\u4ea7\u54c1\u200b\u7684\u200b\u51b3\u7b56\u200b\u4e0d\u200b\u53d7\u5230\u200b\u5176\u4ed6\u200b\u4e2a\u4f53\u200b\u7684\u200b\u5f71\u54cd\u200b</li> <li>\u200b\u6a21\u4eff\u8005\u200b\u8d2d\u4e70\u200b\u4ea7\u54c1\u200b\u7684\u200b\u51b3\u7b56\u200b\u53d7\u5230\u200b\u5176\u4ed6\u200b\u4e2a\u4f53\u200b\u7684\u200b\u5f71\u54cd\u200b\uff0c\u200b\u8fd9\u79cd\u200b\u5f71\u54cd\u200b\u65b9\u5f0f\u200b\u901a\u8fc7\u200b\u4e2a\u4f53\u200b\u95f4\u200b\u7684\u200b\u4ea4\u6d41\u200b\u5b9e\u73b0\u200b</li> </ul> <p>\u200b\u5b9a\u4e49\u200b\uff1a\\(P(t)\\)\u200b\u4e3a\u200b\u5e02\u573a\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u200b\u987e\u5ba2\u200b\u5728\u200b\\(t\\)\u200b\u65f6\u523b\u200b\u4f5c\u51fa\u200b\u8d2d\u4e70\u51b3\u7b56\u200b\u7684\u200b\u6982\u7387\u200b</p> \\[ P(t) = p + \\frac{q}{m} D(t) \\] <ul> <li>\\(p\\)\u200b\u4e3a\u200b\u521b\u65b0\u200b\u7cfb\u6570\u200b</li> <li>\\(q\\)\u200b\u4e3a\u200b\u6a21\u4eff\u200b\u7cfb\u6570\u200b\uff0c\\(p\\ll q\\)</li> <li>\\(m\\)\u200b\u4e3a\u200b\u6d88\u8d39\u8005\u200b\u7684\u200b\u603b\u91cf\u200b\uff0c\u200b\u4e5f\u200b\u5373\u200b\u5e02\u573a\u200b\u89c4\u6a21\u200b\uff0c\u200b\u5e38\u6570\u200b</li> <li>\\(D(t)\\)\u200b\u4e3a\u200b\u65f6\u95f4\u200b\u4ece\u200b\\(0\\)\u200b\u5230\u200b\\(t\\)\u200b\u7684\u200b\u7d2f\u79ef\u200b\u8d2d\u4e70\u91cf\u200b</li> </ul> <p>\u200b\u5373\u200b\uff0c\u200b\u5e02\u573a\u200b\u4e0a\u200b\u7684\u200b\u7d2f\u8ba1\u200b\u9700\u6c42\u200b\\(D(t)\\)\u200b\u8d8a\u5927\u200b\uff0c\u200b\u6d88\u8d39\u8005\u200b\u8d8a\u200b\u5bb9\u6613\u200b\u4f5c\u51fa\u200b\u8d2d\u4e70\u51b3\u7b56\u200b\u3002</p> <p>\u200b\u4e0d\u59a8\u200b\u8bbe\u200b\\(d(t) = \\frac{\\mathrm dD}{\\mathrm dt}\\)\uff0c\u200b\u5373\u200b\\(t\\)\u200b\u65f6\u523b\u200b\u5e02\u573a\u200b\u4e0a\u200b\u8d2d\u4e70\u200b\u4eba\u6570\u200b\uff0c\u200b\u6839\u636e\u200b\\(P(t)\\)\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u6709\u200b\uff1a</p> \\[ P(t) = \\frac{d(t)}{m - D(t)} \\] <p>\u200b\u8054\u7acb\u200b\u4e8c\u200b\u5f0f\u200b\uff0c\u200b\u5f97\u5230\u200b\uff1a</p> \\[ \\begin{aligned} p + \\frac{q}{m} D(t) &amp;= \\frac{d(t)}{m - D(t)} \\\\ d(t) &amp;= (m-D(t))\\left(p + \\frac qmD(t)\\right) \\\\ d(t) &amp;= -\\frac qmD^2(t) + (q-p)D(t) + pm \\end{aligned} \\] <p>\u200b\u89e3\u200b\u5fae\u5206\u65b9\u7a0b\u200b\\(\\frac{\\mathrm dD}{\\mathrm dt} = -\\frac qmD^2 + (q-p)D + pm\\)\uff0c\u200b\u5f97\u200b\\(D(t)\\)\uff1a</p> \\[ \\boxed{\\begin{aligned}D(t) &amp;= m\\frac{1-e^{-(p+q)t}}{1+\\frac qp e^{-(p+q)t}} \\\\ d(t) &amp;= m\\frac{p(p+q)^2e^{-(p+q)t}}{\\left(p+qe^{-(p+q)t}\\right)^2}\\end{aligned}} \\] <p>\u200b\u5bf9\u200b\\(d(t)\\)\u200b\u8fdb\u4e00\u6b65\u200b\u6c42\u5bfc\u200b\uff0c\u200b\u5f97\u5f53\u200b\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(t = \\frac{1}{p+q}\\log \\frac{q}{p}\\)\u200b\u65f6\u200b\uff0c\\(d(t)\\)\u200b\u53d6\u5f97\u200b\u6700\u5927\u503c\u200b\u3002</p> <p>\u200b\u6b64\u65f6\u200b\uff1a</p> \\[ \\boxed{\\begin{aligned}d(t) &amp;= \\frac{m(p+q)^2}{4q} \\\\ D(t) &amp;= \\frac{m(q-p)}{2q}\\end{aligned}} \\] <p>\u200b\u7ed3\u679c\u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li>\u200b\u5f53\u200b\\(p\\)\u200b\u5f88\u200b\u5c0f\u65f6\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u9500\u552e\u91cf\u200b\u7684\u200b\u589e\u957f\u200b\u7f13\u6162\u200b</li> <li>\u200b\u5f53\u200b\\(p, q\\)\u200b\u5f88\u5927\u200b\u65f6\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u9500\u552e\u91cf\u200b\u7684\u200b\u589e\u957f\u200b\u4e0e\u200b\u4e0b\u8dcc\u200b\u5747\u200b\u8fc5\u901f\u200b</li> <li>\u200b\u4ec5\u5f53\u200b\\(q&gt;p\\)\u200b\u65f6\u200b\uff0c\u200b\u4e0a\u5f0f\u200b\u624d\u200b\u6709\u200b\u610f\u4e49\u200b\uff0c\u200b\u5426\u5219\u200b\u610f\u5473\u7740\u200b\u9500\u552e\u91cf\u200b\u6301\u7eed\u200b\u4e0b\u8dcc\u200b</li> </ul>"},{"location":"math/operations-mamagement/forecast/","title":"\u9884\u6d4b\u200b\u6280\u672f","text":"<p>\u200b\u5e38\u89c1\u200b\u7684\u200b\u5ba2\u89c2\u200b\u9884\u6d4b\u200b\u65b9\u6cd5\u200b\u6709\u200b\u56e0\u679c\u200b\u6a21\u578b\u200b\u4e0e\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u65b9\u6cd5\u200b\u3002\u200b\u8fd9\u4e9b\u200b\u65b9\u6cd5\u200b\u7684\u200b\u524d\u63d0\u200b\u5047\u8bbe\u200b\u662f\u200b\u8fc7\u53bb\u200b\u7684\u200b\u53d8\u91cf\u200b\u53ca\u5176\u200b\u76f8\u4e92\u4f5c\u7528\u200b\u5173\u7cfb\u200b\u5728\u200b\u672a\u6765\u200b\u5c06\u200b\u7ee7\u7eed\u200b\u5b58\u5728\u200b\u5e76\u200b\u53d1\u6325\u4f5c\u7528\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#_2","title":"\u56e0\u679c\u200b\u6a21\u578b","text":"<p>\u200b\u56e0\u679c\u200b\u6a21\u578b\u200b\u5373\u200b\u627e\u5230\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u4f7f\u7528\u200b\u4e0e\u200b\u9884\u6d4b\u200b\u53d8\u91cf\u200b\u76f8\u5173\u200b\u7684\u200b\u5176\u4ed6\u200b\u53d8\u91cf\u200b\u8fdb\u884c\u200b\u9884\u6d4b\u200b\u3002</p> \\[ y = f(X_1, \\dots, X_n) \\] <p>\u200b\u5176\u4e2d\u200b\\(X_1, \\dots, X_n\\)\u200b\u662f\u200b\u4e0e\u200b\\(y\\)\u200b\u76f8\u5173\u200b\u7684\u200b\u5176\u4ed6\u200b\u53d8\u91cf\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u7ebf\u6027\u200b\u56e0\u679c\u200b\u6a21\u578b\u200b\uff0c\u200b\u51fd\u6570\u200b\\(f\\)\u200b\u662f\u200b\u5173\u4e8e\u200b\\(X_1, \\dots, X_n\\)\u200b\u7684\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\uff0c\u200b\u5373\u200b\uff1a</p> <p>\\(f(X_1, \\dots, X_n) = \\alpha_0 + \\alpha_1X_1 + \\dots + \\alpha_nX_n\\)</p>"},{"location":"math/operations-mamagement/forecast/#_3","title":"\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u65b9\u6cd5","text":"<p>\u200b\u6b64\u5904\u200b\u7684\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u6307\u5f80\u671f\u200b\u5bf9\u200b\u67d0\u200b\u53d8\u91cf\u200b\u89c2\u6d4b\u200b\u503c\u200b\u7684\u200b\u96c6\u5408\u200b\u3002\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u65b9\u6cd5\u200b\u4ec5\u4ec5\u200b\u4f9d\u9760\u200b\u8fd9\u4e9b\u200b\u89c2\u6d4b\u200b\u503c\u200b\u5bf9\u200b\u8be5\u200b\u53d8\u91cf\u200b\u7684\u200b\u672a\u6765\u200b\u8fdb\u884c\u200b\u9884\u6d4b\u200b\u3002</p> <p>\u200b\u5728\u200b\u73b0\u5b9e\u200b\u7684\u200b\u751f\u4ea7\u200b\u8fd0\u4f5c\u200b\u4e2d\u200b\uff0c\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u901a\u5e38\u200b\u4f1a\u200b\u8868\u73b0\u200b\u51fa\u200b\u4ee5\u4e0b\u200b\u6a21\u5f0f\u200b\uff1a</p> <ul> <li>\u200b\u8d8b\u52bf\u200b\uff0c\u200b\u5373\u200b\u89c2\u6d4b\u200b\u503c\u200b\u5728\u200b\u6574\u4e2a\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u6240\u200b\u8868\u73b0\u200b\u51fa\u200b\u7684\u200b\u8d8b\u52bf\u200b</li> <li>\u200b\u5b63\u8282\u200b\uff0c\u200b\u5373\u200b\u89c2\u6d4b\u200b\u503c\u200b\u5728\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u4e00\u90e8\u5206\u200b\u6240\u200b\u8868\u73b0\u200b\u51fa\u200b\u7684\u200b\u8d8b\u52bf\u200b</li> <li>\u200b\u5468\u671f\u200b\uff0c\u200b\u5373\u200b\u89c2\u6d4b\u200b\u503c\u200b\u5728\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u91cd\u590d\u200b\u591a\u6b21\u200b\u51fa\u73b0\u200b\u7684\u200b\u8d8b\u52bf\u200b</li> <li>\u200b\u968f\u673a\u200b\uff0c\u200b\u6307\u200b\u89c2\u6d4b\u200b\u503c\u200b\u65e0\u89c4\u5219\u200b\u7684\u200b\u53d8\u52a8\u200b</li> </ul> <p>\u200b\u5b9e\u9645\u200b\u7684\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u53ef\u80fd\u200b\u540c\u65f6\u200b\u5305\u542b\u200b\u4ee5\u4e0a\u200b\u6240\u6709\u200b\u7279\u5f81\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#_4","title":"\u89c2\u6d4b\u200b\u4e0e\u200b\u9884\u6d4b","text":"<ol> <li>\\(\\{D_t\\}\\)\u200b\u4e3a\u200b\u89c2\u6d4b\u200b\u53d8\u91cf\u200b\u5728\u200b\\(1\\sim t\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u5185\u200b\u7684\u200b\u89c2\u6d4b\u200b\u503c\u200b\u7ec4\u6210\u200b\u7684\u200b\u5e8f\u5217\u200b\uff1b</li> <li>\\(F_{t-\\tau, t}\\)\u200b\u8868\u793a\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u5728\u200b\\(t-\\tau\\)\u200b\u5468\u671f\u200b\u5185\u200b\u5bf9\u200b\\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u89c2\u6d4b\u200b\u503c\u200b\u4f5c\u51fa\u200b\u7684\u200b\u9884\u6d4b\u200b\u3002\u200b\u7279\u6b8a\u200b\u5730\u200b\uff0c\\(F_t=F_{t-1, t}\\)\uff0c\u200b\u5373\u200b\u4e3a\u200b\u63d0\u524d\u200b\u4e00\u4e2a\u200b\u5468\u671f\u200b\u7684\u200b\u9884\u6d4b\u200b\u3002</li> </ol> <p>\u200b\u5982\u4e0b\u200b\u6307\u6807\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u8861\u91cf\u200b\u9884\u6d4b\u200b\u6548\u679c\u200b\uff0c\u200b\u5b9a\u4e49\u200b\u504f\u5dee\u200b\\(e_t = D_t - F_t\\)</p> <ul> <li>\u200b\u5e73\u5747\u200b\u7edd\u5bf9\u200b\u504f\u5dee\u200b\uff08MAD\uff09\uff1a\\(\\frac{1}{n}\\sum_{i=1}^n|e_i|\\)</li> <li>\u200b\u5747\u65b9\u200b\u8bef\u5dee\u200b\uff08MSE\uff09\uff1a\\(\\frac{1}{n}\\sum_{i=1}^ne_i^2\\)</li> <li>\u200b\u5e73\u5747\u200b\u7edd\u5bf9\u200b\u767e\u5206\u200b\u8bef\u5dee\u200b\uff08MAPE\uff09\uff1a\\(\\frac{1}{n}\\sum_{i=1}^n|e_i/D_i|\\times 100\\%\\)</li> </ul>"},{"location":"math/operations-mamagement/forecast/#_5","title":"\u5e73\u7a33\u200b\u5e8f\u5217\u200b\u7684\u200b\u9884\u6d4b","text":"<p>\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b\u662f\u200b\u8054\u5408\u200b\u6982\u7387\u5206\u5e03\u200b\u4e0d\u968f\u200b\u65f6\u95f4\u200b\u6539\u53d8\u200b\uff0c\u200b\u5f62\u200b\u5982\u200b\\(D_t=\\mu + \\epsilon_t\\)\u200b\u7684\u200b\u5e8f\u5217\u200b\u3002</p> <p>\u200b\u5bf9\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5e38\u7528\u200b\u4e24\u79cd\u200b\u9884\u6d4b\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5373\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\u4e0e\u200b\u6307\u6570\u200b\u5e73\u6ed1\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#_6","title":"\u79fb\u52a8\u200b\u5e73\u5747","text":"<p>\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\uff0c\\(MA(N)\\)\uff1a</p> \\[ F_t = \\frac{1}{N}\\sum_{i=t-N}^{t-1}\\alpha_{i-t+N+1}D_i \\] <p>\u200b\u5f53\u200b\\(\\alpha_i=1\\)\u200b\u65f6\u200b\uff0c\u200b\u79f0\u8be5\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\u4e3a\u200b\u7b80\u5355\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\uff0c\u200b\u5426\u5219\u200b\u4e3a\u200b\u52a0\u6743\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u7b80\u5355\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\uff0c\u200b\u6839\u636e\u200b\u524d\u200b\u4e00\u200b\u5468\u671f\u200b\u7684\u200b\u9884\u6d4b\u503c\u200b\u548c\u200b\u65b0\u200b\u7684\u200b\u89c2\u6d4b\u200b\u503c\u200b\u53ef\u4ee5\u200b\u8ba1\u7b97\u200b\u4e0b\u200b\u4e00\u200b\u5468\u671f\u200b\u7684\u200b\u9884\u6d4b\u503c\u200b\uff1a</p> \\[ \\begin{aligned} F_{t+1} &amp;= \\frac{1}{N}\\sum_{i=t-N+1}^{t}D_i \\\\ &amp;= \\frac{1}{N}\\sum_{i=t-N}^{t-1} - D_{t-N} + D_t \\\\ &amp;= F_{t} + \\frac 1N \\left(D_t - D_{t-N}\\right) \\end{aligned} \\] <p>\u200b\u5bf9\u4e8e\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b\uff0c\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\u7684\u200b\u9884\u6d4b\u200b\u901a\u5e38\u200b\u6ede\u540e\u200b\u4e8e\u200b\u8d8b\u52bf\u200b\u7684\u200b\u53d1\u5c55\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#_7","title":"\u6307\u6570\u200b\u5e73\u6ed1","text":"<p>\u200b\u5f53\u524d\u200b\u7684\u200b\u9884\u6d4b\u503c\u200b\u7b49\u4e8e\u200b\u4e0a\u200b\u4e00\u6b21\u200b\u9884\u6d4b\u503c\u200b\u4e0e\u200b\u5f53\u524d\u200b\u9700\u6c42\u200b\u5b9e\u9645\u200b\u503c\u200b\u7684\u200b\u52a0\u6743\u200b\u5e73\u5747\u200b\uff0c\u200b\u5373\u200b\uff1a</p> \\[ \\begin{aligned} F_t &amp;= \\alpha D_{t-1} + (1-\\alpha)F_{t-1} \\\\ &amp;= F_{t-1} + \\alpha e_{t-1} \\end{aligned} \\] <p>\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7406\u89e3\u200b\u4e3a\u200b\u5bf9\u200b\u4e0a\u200b\u4e00\u6b21\u200b\u9884\u6d4b\u200b\u7684\u200b\u504f\u5dee\u200b\u4f5c\u51fa\u200b\u7cfb\u6570\u200b\u4e3a\u200b\\(\\alpha\\)\u200b\u7684\u200b\u4fee\u6b63\u200b\u3002</p> <p>\u200b\u5df2\u77e5\u200b\\(t=0\\)\u200b\u65f6\u200b\u5bf9\u200b\\(D_1\\)\u200b\u7684\u200b\u9884\u6d4b\u200b\u4e3a\u200b\\(F_1\\)\uff0c\u200b\u5219\u200b\uff1a</p> \\[ \\begin{aligned}     F_2 &amp;= \\alpha D_1 + (1-\\alpha) F_1 \\\\     F_3 &amp;= \\alpha D_2 + (1-\\alpha) F_2 =\\alpha D_2 + (1-\\alpha)\\alpha D_1 + (1-\\alpha) ^ 2 F_1 \\\\     \\vdots \\\\     F_n &amp;= \\sum_{i=1} ^{n-1} \\alpha(1-\\alpha)^{i-1}D_{n-i} + (1-\\alpha)^{n-1}F_1 \\end{aligned} \\]"},{"location":"math/operations-mamagement/forecast/#_8","title":"\u8d8b\u52bf\u200b\u5e8f\u5217","text":"<p>\u200b\u5bf9\u200b\u8d8b\u52bf\u200b\u5e8f\u5217\u200b\u7684\u200b\u9884\u6d4b\u200b\u901a\u5e38\u200b\u4f7f\u7528\u200b\u56de\u5f52\u200b\u5206\u6790\u200b\uff0c\u200b\u5373\u200b\u5047\u8bbe\u200b\u89c2\u6d4b\u200b\u53d8\u91cf\u200b\u4e0e\u200b\u9884\u6d4b\u200b\u53d8\u91cf\u200b\u95f4\u200b\u5b58\u5728\u200b\u51fd\u6570\u200b\u5173\u7cfb\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u6839\u636e\u200b\u89c2\u6d4b\u200b\u53d8\u91cf\u200b\u548c\u200b\u9884\u6d4b\u200b\u53d8\u91cf\u200b\u7684\u200b\u89c2\u6d4b\u200b\u503c\u200b\u5bf9\u200b\u9884\u6d4b\u200b\u53d8\u91cf\u200b\u8fdb\u884c\u200b\u9884\u6d4b\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#_9","title":"\u7ebf\u6027\u200b\u56de\u5f52","text":"<p>\u200b\u5f53\u200b\u56de\u5f52\u200b\u4f7f\u7528\u200b\u5f62\u200b\u5982\u200b\\(\\hat Y=a+bX\\)\u200b\u7684\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\u65f6\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u7ebf\u6027\u200b\u56de\u5f52\u200b\u3002\u200b\u7531\u6b64\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6700\u5c0f\u200b\u4e8c\u200b\u4e58\u6cd5\u200b\u8fdb\u884c\u200b\u62df\u5408\u200b\u3002</p> <p>\u200b\u6700\u5c0f\u200b\u4e8c\u200b\u4e58\u6cd5\u200b\u4f7f\u5f97\u200b\\(\\sum|Y-\\hat Y|^2\\)\u200b\u6700\u5c0f\u200b\uff0c\u200b\u5373\u200b\u9884\u6d4b\u200b\u70b9\u200b\u548c\u200b\u89c2\u6d4b\u70b9\u200b\u7684\u200b\u8ddd\u79bb\u200b\u5e73\u65b9\u548c\u200b\u6700\u5c0f\u200b\u3002\u200b\u5047\u8bbe\u200b\u89c2\u6d4b\u200b\u53d8\u91cf\u200b\u4e3a\u200b\\(X_1, \\dots, X_m\\)\uff0c\u200b\u9884\u6d4b\u200b\u53d8\u91cf\u200b\u4e3a\u200b\\(y\\)\uff0c\u200b\u5171\u200b\\(n\\)\u200b\u7ec4\u200b\u6570\u636e\u200b\u70b9\u200b\uff0c\u200b\u5219\u200b\u7ebf\u6027\u200b\u56de\u5f52\u200b\u6a21\u578b\u200b\u4e3a\u200b\uff1a</p> \\[ \\hat y = X\\vec a \\] <p>\u200b\u5176\u4e2d\u200b\\(\\vec a, , \\hat y\\)\u200b\u4e3a\u200b\\(m\\)\u200b\u7ef4\u5217\u200b\u5411\u91cf\u200b\uff0c\\(X\\)\u200b\u4e3a\u200b\\(n\\times m\\)\u200b\u77e9\u9635\u200b\u3002</p> <p>\u200b\u5219\u200b\u6b8b\u5dee\u200b\u5e73\u65b9\u548c\u200b\u5373\u200b\u4e3a\u200b\\(f(a; X, y) = (y - \\hat y)^T(y - \\hat y)=(y - Xa)^T(y - Xa)\\)\u3002</p> <p>\u200b\u5bf9\u200b\\(\\vec a, \\vec b\\)\u200b\u6c42\u5bfc\u200b\uff0c\u200b\u5f97\u200b\uff1a</p> \\[     \\frac{\\partial f}{\\partial \\vec a} = 0 \\] <p>\\(\\frac{\\partial f}{\\partial \\vec a} = 0\\)\u200b\u5373\u200b\uff1a</p> \\[ \\begin{aligned}     \\frac{\\partial f}{\\partial \\vec a} &amp;= 2X^T(X\\vec a - y) \\\\     &amp;= 2X^TX\\vec a + 2X^Ty \\\\     &amp;= \\vec 0 \\end{aligned} \\] <p>\u200b\u5f97\u200b\\(\\vec a = (X^TX)^{-1}X^Ty\\)</p>"},{"location":"math/operations-mamagement/forecast/#_10","title":"\u53cc\u91cd\u200b\u6307\u6570\u200b\u5e73\u6ed1","text":"<p>\u200b\u53cc\u91cd\u200b\u6307\u6570\u200b\u5e73\u6ed1\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u9884\u6d4b\u200b\u5177\u6709\u200b\u7ebf\u6027\u200b\u8d8b\u52bf\u200b\u7684\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5176\u4e2d\u200b\u4f7f\u7528\u200b\u4e24\u4e2a\u200b\u5e73\u6ed1\u200b\u53c2\u6570\u200b\u5206\u522b\u200b\u9884\u6d4b\u200b\u5e8f\u5217\u200b\u503c\u200b\\(S_t\\)\u200b\u548c\u200b\u8d8b\u52bf\u200b\u503c\u200b\\(G_t\\)</p> <ul> <li>\u200b\u5e8f\u5217\u200b\u503c\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u66f2\u7ebf\u200b\u7684\u200b\u622a\u8ddd\u200b   $$   S_t = \\alpha D_t + (1 - \\alpha)(S_{t-1} + G_{t-1})   $$</li> <li>\u200b\u8d8b\u52bf\u200b\u503c\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u66f2\u7ebf\u200b\u7684\u200b\u659c\u7387\u200b   $$   G_t = \\beta(S_t - S_{t-1})+(1-\\beta)G_{t-1}   $$</li> <li>\u200b\u5728\u200b\\(t\\)\u200b\u65f6\u523b\u200b\u63d0\u524d\u200b\\(\\tau\\)\u200b\u6b65\u200b\u4f5c\u51fa\u200b\u7684\u200b\u9884\u6d4b\u200b\\(F_{t, t+\\tau} = S_t+\\tau G_t\\)</li> </ul>"},{"location":"math/operations-mamagement/forecast/#_11","title":"\u5b63\u8282\u6027\u200b\u5e8f\u5217","text":"<p>\u200b\u5b63\u8282\u6027\u200b\u5e8f\u5217\u200b\u662f\u200b\u6bcf\u200b\\(N\\)\u200b\u4e2a\u200b\u5468\u671f\u200b\u91cd\u590d\u200b\u4e00\u6b21\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5176\u4e2d\u200b\\(N\\)\u200b\u88ab\u200b\u79f0\u4e3a\u200b\u5b63\u200b\u7684\u200b\u957f\u5ea6\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u5b63\u8282\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u5468\u671f\u200b\uff0c\u200b\u90fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\\(c_t\\)\uff0c\u200b\u6307\u200b\u5e73\u5747\u200b\u610f\u4e49\u200b\u4e0b\u200b\u7b2c\u200b\\(t\\)\u200b\u4e2a\u200b\u5468\u671f\u200b\u7684\u200b\u9700\u6c42\u200b\u4e0e\u200b\u6574\u4f53\u200b\u5e73\u5747\u200b\u9700\u6c42\u200b\u7684\u200b\u6bd4\u503c\u200b\uff0c\u200b\u6709\u200b\\(\\sum c_t = N\\)</p> <ul> <li>\u200b\u8ba1\u7b97\u200b\u6240\u6709\u200b\u6570\u636e\u200b\u7684\u200b\u6837\u672c\u5747\u503c\u200b</li> <li>\u200b\u5c06\u200b\u6bcf\u4e2a\u200b\u89c2\u6d4b\u200b\u503c\u200b\u9664\u4ee5\u200b\u6837\u672c\u5747\u503c\u200b\uff0c\u200b\u5373\u200b\u4e3a\u200b\u6bcf\u4e2a\u200b\u89c2\u6d4b\u200b\u5468\u671f\u200b\u7684\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b</li> <li>\u200b\u5c06\u200b\u6bcf\u5b63\u200b\u4e2d\u200b\u76f8\u540c\u200b\u5468\u671f\u200b\u7684\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\u53d6\u200b\u5747\u503c\u200b\uff0c\u200b\u5f97\u5230\u200b\\(N\\)\u200b\u4e2a\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u540c\u65f6\u200b\u5177\u6709\u200b\u5b63\u8282\u6027\u200b\u6ce2\u52a8\u200b\u548c\u200b\u8d8b\u52bf\u200b\u7684\u200b\u5b63\u8282\u6027\u200b\u5e8f\u5217\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u9884\u6d4b\u200b\uff1a</p> <ul> <li>\u200b\u8ba1\u7b97\u200b\u5e8f\u5217\u200b\u7684\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b</li> <li>\u200b\u5c06\u200b\u89c2\u6d4b\u200b\u503c\u200b\u9664\u4ee5\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\uff0c\u200b\u5f97\u5230\u200b\u6d88\u9664\u200b\u5b63\u8282\u6027\u200b\u6ce2\u52a8\u200b\u7684\u200b\u5e8f\u5217\u200b</li> <li>\u200b\u7528\u200b\u8d8b\u52bf\u200b\u5e8f\u5217\u200b\u7684\u200b\u9884\u6d4b\u200b\u65b9\u6cd5\u200b\u8fdb\u884c\u200b\u9884\u6d4b\u200b</li> <li>\u200b\u9884\u6d4b\u200b\u7ed3\u679c\u200b\u4e58\u4ee5\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\uff0c\u200b\u5f97\u5230\u200b\u6700\u7ec8\u200b\u7684\u200b\u9884\u6d4b\u503c\u200b</li> </ul> <p>\u200b\u7f3a\u70b9\u200b\uff1a\u200b\u5728\u200b\u83b7\u5f97\u200b\u65b0\u200b\u7684\u200b\u89c2\u6d4b\u200b\u503c\u540e\u200b\uff0c\u200b\u9700\u8981\u200b\u91cd\u65b0\u200b\u8ba1\u7b97\u200b\u6240\u6709\u200b\u7684\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#_12","title":"\u4e09\u91cd\u200b\u6307\u6570\u200b\u5e73\u6ed1","text":"<p>\u200b\u5047\u8bbe\u200b\u9700\u6c42\u200b\u4e3a\u200b\\(D_t = (\\mu +G_t)c_t + \\epsilon_t\\)\uff08\u200b\u4e58\u79ef\u200b\u578b\u200b\uff09\u3002\u200b\u8be5\u200b\u5f62\u5f0f\u200b\u4f7f\u7528\u200b\u4e09\u4e2a\u200b\u6307\u6570\u200b\u5e73\u6ed1\u200b\u5bf9\u200b\u5e8f\u5217\u200b\uff08\u200b\u622a\u8ddd\u200b\uff09\u3001\u200b\u8d8b\u52bf\u200b\uff08\u200b\u659c\u7387\u200b\uff09\u200b\u548c\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\uff08\u200b\u5468\u671f\u200b\uff09\u200b\u5206\u522b\u200b\u8fdb\u884c\u200b\u62df\u5408\u200b\u3002</p> <ul> <li>\u200b\u5e8f\u5217\u200b\uff1a\\(S_t = \\alpha \\frac{D_t}{c_{t-N}} + (1-\\alpha) (S_{t-1} + G_{t-1})\\)\uff0c\\(c_{t-N}\\)\u200b\u7528\u4e8e\u200b\u6d88\u9664\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\u7684\u200b\u5f71\u54cd\u200b\u3002</li> <li>\u200b\u8d8b\u52bf\u200b\uff1a\\(G_t = \\beta(S_t - S_{t-1}) + (1-\\beta) G_{t-1}\\)</li> <li>\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\uff1a\\(c_t = \\gamma\\frac{D_t}{S_t} + (1-\\gamma) c_{t-N}\\)</li> </ul>"},{"location":"math/operations-mamagement/forecast/#box-jenkinsarima","title":"Box-Jenkins\u200b\u6a21\u578b\u200b\uff1aARIMA","text":"<p>ARIMA\u200b\u6a21\u578b\u200b\u662f\u200b\u5229\u7528\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u7684\u200b\u81ea\u200b\u76f8\u5173\u200b\u7ed3\u6784\u200b\uff1a</p> <p>Auto Regressive Integrated Moving Average</p> <p>\u200b\u6784\u5efa\u200b\u6b65\u9aa4\u200b\uff1a</p> <ol> <li>\u200b\u6570\u636e\u200b\u8f6c\u6362\u200b</li> <li>\u200b\u6a21\u578b\u200b\u8bc6\u522b\u200b</li> <li>\u200b\u53c2\u6570\u4f30\u8ba1\u200b</li> <li>\u200b\u9884\u6d4b\u200b</li> <li>\u200b\u8bc4\u4ef7\u200b</li> </ol>"},{"location":"math/operations-mamagement/forecast/#arp","title":"\u81ea\u200b\u56de\u5f52\u200b\u8fc7\u7a0b\u200b\uff1aAR(p)","text":"<p>\u200b\u5bf9\u4e8e\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b\uff1a</p> <ol> <li>\u200b\u95f4\u9694\u200b\u76f8\u540c\u200b\u7684\u200b\u4efb\u610f\u200b\u4e00\u7ec4\u200b\u9700\u6c42\u200b\u7684\u200b\u8054\u5408\u200b\u8fb9\u9645\u200b\u5206\u5e03\u200b\u76f8\u540c\u200b-&gt;\u200b\u534f\u65b9\u5dee\u200b\u76f8\u540c\u200b</li> <li>\u200b\u4efb\u610f\u200b\u4e24\u4e2a\u200b\u89c2\u6d4b\u200b\u7684\u200b\u534f\u65b9\u5dee\u200b\u4ec5\u200b\u53d6\u51b3\u4e8e\u200b\u76f8\u9694\u200b\u7684\u200b\u5468\u671f\u200b\u6570\u200b</li> </ol> \\[ \\mathrm{Cov}(D_{t+m}, D_{t+m+k}) = E(D_{t+m}D_{t+m+k}) - E(D_{t+m})E(D_{t+m+k}) \\] <p>\u200b\u76f8\u5173\u7cfb\u6570\u200b\uff1a\u200b\u6ede\u540e\u200b\\(k\\)\u200b\u671f\u200b\u7684\u200b\u81ea\u200b\u76f8\u5173\u7cfb\u6570\u200b</p> \\[ \\rho_k = \\frac{\\mathrm {Cov}(D_{t+m}, D_{t+m+k})}{\\sqrt{\\mathrm {Var}(D_{t+m})}\\sqrt{\\mathrm {Var}(D_{t+m+k})}} \\] <p>\u200b\u6837\u672c\u200b\u7684\u200b\u81ea\u200b\u76f8\u5173\u200b\u51fd\u6570\u200b\uff1a</p> \\[ r_k = \\frac{\\sum_{t=k+1} ^ n(D_t - \\overline D)(D_{t-k} - \\overline D)}{\\sum_{t=1} ^ n(D_t - \\overline D)^2} \\] \\[ D_t = a_0 + a_1D_{t-1} + \\dots + a_pD_{t-p} + \\epsilon_t \\] <p>\\(p\\)\u200b\u4e3a\u200b\u81ea\u200b\u56de\u5f52\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u9636\u6570\u200b\uff0c\u200b\u4e00\u9636\u81ea\u200b\u56de\u5f52\u200b\u6a21\u578b\u200b\\(AR(1)\\)\uff1a</p> \\[ D_t = a_0 + a_1 D_{t-1} + \\epsilon_t \\] <p>\u200b\u81ea\u200b\u76f8\u5173\u200b\u51fd\u6570\u200b\u4e3a\u200b\uff1a\\(\\rho_j = a_1^j\\)</p> <p>\u200b\u5c06\u200b\u6837\u672c\u200b\u7684\u200b\u81ea\u200b\u76f8\u5173\u200b\u51fd\u6570\u200b\u4e0e\u200b\u7406\u8bba\u200b\u7684\u200b\u81ea\u200b\u76f8\u5173\u200b\u51fd\u6570\u200b\u8fdb\u884c\u200b\u6bd4\u8f83\u200b\uff0c\u200b\u5224\u65ad\u200bARIMA\u200b\u7684\u200b\u62df\u5408\u200b\u6548\u679c\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#maq","title":"\u79fb\u52a8\u200b\u5e73\u5747\u200b\u8fc7\u7a0b\u200b\uff1aMA(q)","text":"\\[ D_t = b_0 - b_1\\epsilon_{t-1} -\\dots - b_q\\epsilon{t-q} +\\epsilon_t \\] <p>\\(AR(1)\\)\u200b\u8fc7\u7a0b\u200b\u672c\u8d28\u200b\u4e0a\u200b\u662f\u200b\\(MA(\\infty)\\)\u200b\u8fc7\u7a0b\u200b\u3002</p> <p>\\(MA(1)\\)\u200b\u8fc7\u7a0b\u200b\uff1a\\(D_t = b_0 - b_1\\epsilon_{t-1} + \\epsilon_t\\)</p> <p>\u200b\u5bf9\u5e94\u200b\u4e8e\u200b\\(AR(1)\\)\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u81ea\u200b\u76f8\u5173\u200b\u51fd\u6570\u200b\uff1a</p> \\[ \\left\\{\\begin{aligned}   \\rho_1 &amp;= \\frac{-b_1}{1+b_1^2} \\\\   \\rho_2 &amp;= \\rho_3 = \\dots = 0 \\end{aligned}\\right. \\]"},{"location":"math/operations-mamagement/forecast/#arma","title":"ARMA\u200b\u6a21\u578b","text":"<p>\\(ARMA(p, q)\\)\u200b\u6a21\u578b\u200b\u662f\u200b\\(AR(p)\\)\u200b\u6a21\u578b\u200b\u4e0e\u200b\\(MA(q)\\)\u200b\u6a21\u578b\u200b\u7684\u200b\u7ed3\u5408\u200b\uff0c\u200b\u5982\u200b\\(ARMA(1, 1)\\)\uff1a</p> \\[ D_t = c+a_1D_{t-1} - b_1\\epsilon_{t-1} + \\epsilon_t \\] <p>\u200b\u5982\u679c\u200b\u5e8f\u5217\u200b\u4e0d\u662f\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b\uff0c\u200b\u800c\u200b\u5b58\u5728\u200b\u8d8b\u52bf\u6027\u200b\u548c\u200b/\u200b\u6216\u200b\u5b63\u8282\u6027\u200b\uff0c\u200b\u9700\u8981\u200b\u5c06\u200b\u975e\u200b\u5e73\u7a33\u200b\u8fc7\u7a0b\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u5e73\u7a33\u200b\u8fc7\u7a0b\u200b\u3002</p> <ul> <li>\u200b\u5f53\u200b\\(D_t\\)\u200b\u5b58\u5728\u200b\u7ebf\u6027\u200b\u8d8b\u52bf\u200b\u65f6\u200b\uff0c\u200b\u4f7f\u7528\u200b\u5dee\u5206\u200b\u65b9\u6cd5\u200b\u6d88\u9664\u200b\u8d8b\u52bf\u6027\u200b\uff1a\\(U_t = D_t - D_{t-1}\\)\uff0c\\(U_t\\)\u200b\u662f\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b\u3002</li> <li>\u200b\u5f53\u200b\\(D_t\\)\u200b\u5b58\u5728\u200b\u5b63\u8282\u6027\u200b\u8d8b\u52bf\u200b\u65f6\u200b\uff0c\u200b\u4f7f\u7528\u200b\u5dee\u5206\u200b\u65b9\u6cd5\u200b\u6d88\u9664\u200b\u8d8b\u52bf\u6027\u200b\uff1a\\(U_t = D_t - D_{t-N}\\)</li> <li>\u200b\u5bf9\u200b\\(U_t\\)\u200b\u8fdb\u884c\u200b\u52a0\u200b\u548c\u200b\uff0c\u200b\u5f97\u5230\u200b\u521d\u59cb\u200b\u7684\u200b\\(D_t\\)</li> </ul>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/","title":"\u786e\u5b9a\u200b\u9700\u6c42\u200b\u4e0b\u200b\u7684\u200b\u5e93\u5b58\u200b\u63a7\u5236","text":"<p>\u200b\u5728\u200b\u6301\u6709\u200b\u6210\u672c\u200b\u4e0e\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u4e4b\u95f4\u200b\u5e73\u8861\u200b\u2014\u2014\u200b\u5355\u6b21\u200b\u8ba2\u8d27\u91cf\u200b\\(\\bsQ\\)</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#eoq","title":"EOQ\u200b\u6a21\u578b","text":""},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_2","title":"\u8bb0\u53f7","text":"<ul> <li>\u200b\u9700\u6c42\u200b\u7387\u200b\\(\\lambda\\)\uff0c\u200b\u5468\u671f\u200b\u957f\u5ea6\u200b\\(T\\)</li> <li>\u200b\u6210\u672c\u200b\uff1a\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(K\\)\u3001\u200b\u8fb9\u9645\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(c\\)\u3001\u200b\u8fb9\u9645\u200b\u6301\u6709\u200b\u6210\u672c\u200b\\(h\\)</li> </ul> <p>\u200b\u8fb9\u9645\u200b\u6301\u6709\u200b\u6210\u672c\u200b\\(h\\)\u200b\u901a\u5e38\u200b\u4ee5\u200b\u5229\u7387\u200b\\(I\\)\u200b\u7684\u200b\u5f62\u5f0f\u200b\u8868\u793a\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b\\(h = Ic\\)</p> <ul> <li>\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u200b\\(\\tau = 0\\)</li> <li>\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u7f3a\u8d27\u200b</li> </ul>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_3","title":"\u63a8\u5bfc","text":"<p>\u200b\u8003\u8651\u200b\u4e00\u4e2a\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\uff0c\u200b\u7531\u4e8e\u200b\u4e0d\u200b\u8003\u8651\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u91cf\u200b\uff0c\u200b\u8ba2\u8d27\u200b\u540e\u200b\u4ea7\u54c1\u200b\u4f1a\u200b\u7acb\u5373\u200b\u5230\u8fbe\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e3a\u200b\u964d\u4f4e\u200b\u6301\u6709\u200b\u6210\u672c\u200b\uff0c\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u200b\u65b9\u5f0f\u200b\u4e3a\u200b\u5e93\u5b58\u200b\u4e3a\u200b\u96f6\u65f6\u200b\u91cd\u65b0\u200b\u8ba2\u8d27\u200b\u3002\u200b\u56e0\u6b64\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\u6ee1\u8db3\u200b\\(Q = \\lambda T\\)\u3002\u200b\u4e00\u4e2a\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\u4e2d\u200b\u7684\u200b\u6210\u672c\u200b\u4e3a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u548c\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u4e4b\u200b\u548c\u200b\uff0c\u200b\u5373\u200b\\(C = K + cQ + hQT/2\\)\u3002</p> <p></p> <p>\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u7684\u200b\u6210\u672c\u200b</p> \\[ C(Q) = \\frac{C}{T} = \\frac{K\\lambda}{Q} + \\lambda c + \\frac{hQ}{2} \\] <p>\u200b\u5bf9\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u6210\u672c\u200b\u6c42\u5bfc\u200b\uff0c\u200b\u5f97\u5230\u200b</p> \\[ C'(Q) = \\frac{h}{2} - \\frac{K\\lambda}{Q^2} \\] <p>\u200b\u5f53\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u6210\u672c\u200b\u6700\u200b\u5c0f\u65f6\u200b\uff0c\\(C'(Q^\\ast) = 0\\)\uff0c\u200b\u89e3\u5f97\u200b</p> \\[ Q^\\ast = \\sqrt{\\frac{2K\\lambda}{h}} \\] <p>\u200b\u4e0e\u200b\u6b64\u5bf9\u5e94\u200b\u7684\u200b\u6700\u5c0f\u200b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ C(Q^\\ast) = \\frac{K\\lambda}{Q^\\ast} + \\lambda c + \\frac{hQ^\\ast}{2} = \\lambda c + \\sqrt{2K\\lambda h} \\]"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_4","title":"\u6a21\u578b\u200b\u6269\u5145","text":""},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_5","title":"\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f","text":"<p>\u200b\u8bbe\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u4e3a\u200b\\(\\tau &gt; 0\\)\uff0c\u200b\u9996\u5148\u200b\u8003\u8651\u200b\\(\\tau \\leq T\\)\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u7531\u4e8e\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u7f3a\u8d27\u200b\uff0c\u200b\u5728\u200b\u5e93\u5b58\u200b\u63d0\u524d\u200b\u671f\u5185\u200b\u7684\u200b\u6700\u4f4e\u200b\u5e93\u5b58\u200b\u5e94\u5f53\u200b\u4e3a\u200b\u96f6\u200b\uff0c\u200b\u5373\u200b\\(R = \\lambda\\tau\\)\u200b\u65f6\u200b\u5e94\u5f53\u200b\u91cd\u65b0\u200b\u8ba2\u8d27\u200b\uff0c\\(R\\)\u200b\u79f0\u4e3a\u200b\u91cd\u200b\u8ba2\u8d27\u200b\u70b9\u200b\u3002</p> <p>\u200b\u5f53\u200b\\(\\tau &gt; T\\)\u200b\u65f6\u200b\uff0c\u200b\u7531\u4e8e\u200b\u6a21\u578b\u200b\u9762\u5bf9\u200b\u5468\u671f\u6027\u200b\u7684\u200b\u9700\u6c42\u200b\uff0c\u200b\u53ea\u200b\u9700\u200b\u6839\u636e\u200b\u63d0\u524d\u200b\u671f\u200b\\(\\tau' = \\tau\\mod T\\)\u200b\u8ba1\u7b97\u200b\u91cd\u200b\u8ba2\u8d27\u200b\u70b9\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_6","title":"\u654f\u611f\u6027\u200b\u5206\u6790","text":"<p>\u200b\u654f\u611f\u6027\u200b\u5206\u6790\u200b\u6307\u200b\\(Q^*\\)\u200b\u7684\u200b\u4f30\u8ba1\u200b\u504f\u5dee\u200b\u5bf9\u200b\u603b\u6210\u672c\u200b\u7684\u200b\u5f71\u54cd\u200b\u3002\u200b\u504f\u5dee\u200b\u4ee5\u200b\\(C(Q) / C(Q^\\ast)\\)\u200b\u8861\u91cf\u200b\uff0c\u200b\u4e3a\u200b\u4fbf\u4e8e\u200b\u5206\u6790\u200b\uff0c\u200b\u8bbe\u200b\u8fb9\u9645\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(c = 0\\)\uff0c\u200b\u5219\u200b\u603b\u6210\u672c\u200b\u7684\u200b\u504f\u5dee\u200b\u91cf\u200b\\(C(Q) / C(Q^\\ast)\\)\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned}     \\frac{C(Q)}{C(Q^\\ast)} &amp;= \\frac{\\frac{K\\lambda}{Q}  + \\frac{hQ}{2}}{\\sqrt{2K\\lambda h}} \\\\     &amp;= \\sqrt{\\frac{K\\lambda}{2h}}\\frac{1}{Q} + \\sqrt{\\frac{h}{2K\\lambda}}Q \\\\     &amp;= \\frac{1}{2}\\left(\\frac{Q}{Q^\\ast} + \\frac{Q^\\ast}{Q}\\right) \\end{aligned} \\]"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_7","title":"\u5141\u8bb8\u200b\u7f3a\u8d27","text":"<p>\u200b\u5f53\u200b\u5141\u8bb8\u200b\u7f3a\u8d27\u200b\u65f6\u200b\uff0c\u200b\u8bbe\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\u4e3a\u200b\\(p\\)\uff0c\u200b\u6700\u5927\u200b\u7f3a\u8d27\u91cf\u200b\\(xQ\\)\uff0c\u200b\u4e00\u4e2a\u200b\u5468\u671f\u200b\u5185\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u53d8\u5316\u200b\u5982\u4e0b\u200b\u56fe\u200b\u6240\u793a\u200b\uff1a</p> <p></p> <p>\u200b\u6210\u672c\u200b\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ C(Q, x) = \\frac{C}{T} = \\frac{(1-x)^2 Qh + x^2Qp}{2} + \\frac{K\\lambda}{Q} + \\lambda c \\] <p>\u200b\u5206\u522b\u200b\u6c42\u200b\\(C\\)\u200b\u5173\u4e8e\u200b\\(Q, x\\)\u200b\u7684\u200b\u504f\u200b\u5bfc\u6570\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\begin{aligned}   \\frac{\\partial C}{\\partial Q} &amp;= \\frac{(1-x)^2h + x^2p}{2} - \\frac{K\\lambda}{Q^2} \\\\   \\frac{\\partial C}{\\partial x} &amp;= xpQ + (x-1)hQ \\end{aligned} \\] <p>\u200b\u89e3\u65b9\u7a0b\u200b\\(\\partial C/\\partial x =0\\)\u200b\u5f97\u5230\u200b\\(x^* = h/(h+p)\\)\uff0c\u200b\u4ee3\u5165\u200b\u65b9\u7a0b\u200b\\(\\partial C/\\partial Q =0\\)\uff0c\u200b\u89e3\u5f97\u200b\uff1a</p> \\[ Q^* = \\sqrt{\\frac{2K\\lambda}{(1-x)^2h+x^2p}} = \\sqrt{\\frac{2K\\lambda(h+p)}{hp}} \\] <p>\u200b\u82e5\u200b\u5b9a\u4e49\u200b\\(h' = hp/(h+p)\\)\u3002\u200b\u5219\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bEOQ\u200b\u7684\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u5373\u200b\\(\\sqrt{\\frac{2K\\lambda}{h'}}\\)</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_8","title":"\u6709\u9650\u200b\u751f\u4ea7\u7387\u200b\u6a21\u578b","text":"<p>\u200b\u5728\u200b\u6709\u9650\u200b\u751f\u4ea7\u7387\u200b\u6a21\u578b\u200b\u4e2d\u200b\uff0c\u200b\u4ea7\u54c1\u200b\u901a\u8fc7\u200b\u751f\u4ea7\u200b\u800c\u200b\u4e0d\u662f\u200b\u8ba2\u8d2d\u200b\u83b7\u5f97\u200b\u3002\u200b\u4ea7\u54c1\u200b\u7684\u200b\u751f\u4ea7\u7387\u200b\u4e3a\u200b\\(P &gt; \\lambda\\)\uff0c\u200b\u751f\u4ea7\u7ebf\u200b\u7684\u200b\u542f\u52a8\u200b\u6210\u672c\u200b\u4e3a\u200b\\(K\\)\uff0c\u200b\u4ea7\u54c1\u200b\u7684\u200b\u5355\u4f4d\u200b\u751f\u4ea7\u6210\u672c\u200b\u4e3a\u200b\\(c\\)\u3002</p> <p>\u200b\u751f\u4ea7\u6210\u672c\u200b\uff08\u200b\u539f\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\uff09\u200b\u90e8\u5206\u200b\u7684\u200b\u5904\u7406\u200b\u4e0e\u200bEOQ\u200b\u6a21\u578b\u200b\u76f8\u540c\u200b\u3002\u200b\u4f46\u200b\u6709\u9650\u200b\u751f\u4ea7\u7387\u200b\u6a21\u578b\u200b\u7684\u200b\u5e73\u5747\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u4e0e\u200bEOQ\u200b\u6a21\u578b\u200b\u4e0d\u540c\u200b\u3002</p> <ul> <li>\u200b\u4e00\u4e2a\u200b\u5468\u671f\u200b\u4e2d\u200bEOQ\u200b\u6a21\u578b\u200b\u7684\u200b\u5e73\u5747\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\uff1a\\(Q/2\\)</li> <li>\u200b\u4e00\u4e2a\u200b\u5468\u671f\u200b\u4e2d\u200b\u6709\u9650\u200b\u751f\u4ea7\u7387\u200b\u6a21\u578b\u200b\u7684\u200b\u5e73\u5747\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\uff1a\\(Q(P-\\lambda)/2P\\)</li> </ul> <p></p> <p>\u200b\u56e0\u6b64\u200b\u6709\u9650\u200b\u751f\u4ea7\u7387\u200b\u6a21\u578b\u200b\u7684\u200b\u6210\u672c\u200b\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ C(Q) = \\frac{C}{T} = \\frac{K\\lambda}{Q} + \\frac{hQ(P-\\lambda)}{2P} + \\lambda c \\] <p>\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\u4e3a\u200b</p> \\[ Q^\\ast = \\sqrt{\\frac{2PK\\lambda}{h(P-\\lambda)}} \\] <p>\u200b\u82e5\u200b\u5b9a\u4e49\u200b\\(h' = (P-\\lambda) / P\\)\uff0c\u200b\u5219\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bEOQ\u200b\u7684\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u5373\u200b\\(\\sqrt{\\frac{2K\\lambda}{h'}}\\)</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_9","title":"\u6570\u91cf\u200b\u6298\u6263\u200b\u6a21\u578b","text":"<p>\u200b\u5728\u200b\u6570\u91cf\u200b\u6298\u6263\u200b\u6a21\u578b\u200b\u4e2d\u200b\uff0c\u200b\u5f53\u200b\u6bcf\u4e2a\u200b\u5468\u671f\u200b\u7684\u200b\u8ba2\u8d27\u91cf\u200b\u8fbe\u5230\u200b\u4e00\u5b9a\u200b\u6c34\u5e73\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u83b7\u5f97\u200b\u8fb9\u9645\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u7684\u200b\u6298\u6263\u200b\uff0c\u200b\u6298\u6263\u200b\u5206\u4e3a\u200b\u4e24\u79cd\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5373\u200b\u5168\u91cf\u200b\u6298\u6263\u200b\u4e0e\u200b\u589e\u91cf\u200b\u6298\u6263\u200b\u3002</p> <ul> <li>\u200b\u5168\u91cf\u200b\u6298\u6263\u200b\u6307\u5f53\u200b\u8ba2\u8d2d\u200b\u91cf\u200b\u8d85\u8fc7\u200b\u4e34\u754c\u503c\u200b\u65f6\u200b\uff0c\u200b\u5c06\u200b\u6298\u6263\u200b\u5e94\u7528\u200b\u4e8e\u200b\u5168\u90e8\u200b\u7684\u200b\u8ba2\u8d2d\u200b\u6570\u91cf\u200b\u4e0a\u200b\u3002</li> <li>\u200b\u589e\u91cf\u200b\u6298\u6263\u200b\u6307\u5f53\u200b\u8ba2\u8d2d\u200b\u91cf\u200b\u8d85\u8fc7\u200b\u4e34\u754c\u503c\u200b\u65f6\u200b\uff0c\u200b\u5c06\u200b\u6298\u6263\u200b\u5e94\u7528\u200b\u4e8e\u200b\u8d85\u8fc7\u200b\u4e34\u754c\u503c\u200b\u7684\u200b\u90e8\u5206\u200b\u4e0a\u200b\u3002</li> </ul> <p>\u200b\u6570\u91cf\u200b\u6298\u6263\u200b\u6a21\u578b\u200b\u9488\u5bf9\u200b\u7684\u200b\u662f\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u7684\u200b\u6539\u53d8\u200b\uff0c\u200b\u8fb9\u9645\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u7684\u200b\u6539\u53d8\u200b\u4e0d\u4f1a\u200b\u5f71\u54cd\u200bEOQ\u200b\u6a21\u578b\u200b\u7684\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\u3002</p> <p>\u200b\u901a\u8fc7\u200b\u6570\u91cf\u200b\u6298\u6263\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8bf1\u5bfc\u200b\u96f6\u552e\u5546\u200b\u63d0\u9ad8\u200b\u8ba2\u8d27\u200b\u6279\u91cf\u200b\uff0c\u200b\u4ece\u800c\u200b\u5b9e\u73b0\u200b\u8d85\u91cf\u200b\u5e93\u5b58\u200b\u7684\u200b\u8f6c\u79fb\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_10","title":"\u5168\u91cf\u200b\u6298\u6263\u200b\u7684\u200b\u6c42\u89e3\u200b\u8fc7\u7a0b","text":"<p>\u200b\u6839\u636e\u200bEOQ\u200b\u6a21\u578b\u200b\u8ba1\u7b97\u200b\u51fa\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u6298\u6263\u200b\u4ef7\u683c\u200b\\(h_i\\)\u200b\u4e0b\u200b\u5bf9\u5e94\u200b\u7684\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\\(Q^*_i\\)\uff0c\u200b\u786e\u5b9a\u200b\u6700\u5927\u200b\u53ef\u884c\u200b\u7684\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\\(Q^\\ast = \\max Q^\\ast_i\\)\u3002\u200b\u53d6\u8be5\u200b\u6298\u6263\u200b\u70b9\u200b\u4ee5\u53ca\u200b\u9ad8\u4e8e\u200b\u8be5\u200b\u6298\u6263\u200b\u70b9\u200b\u7684\u200b\u6240\u6709\u200b\u6298\u6263\u200b\u70b9\u4e3a\u200b\u5019\u9009\u200b\uff0c\u200b\u5373\u200b\\(I = \\{i | i\\geq \\arg\\max_i Q^\\ast_i\\}\\)\u3002\u200b\u5206\u522b\u200b\u6bd4\u8f83\u200b\u6bcf\u4e2a\u200b\u6298\u6263\u200b\u70b9\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5e73\u5747\u200b\u6210\u672c\u200b\uff0c\u200b\u6210\u672c\u200b\u6700\u4f4e\u200b\u7684\u200b\\(Q\\)\u200b\u5373\u200b\u4e3a\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u200b\u6279\u91cf\u200b\u3002</p> <p></p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_11","title":"\u589e\u91cf\u200b\u6298\u6263\u200b\u7684\u200b\u6c42\u89e3\u200b\u8fc7\u7a0b","text":"<p>\u200b\u6839\u636e\u200bEOQ\u200b\u6a21\u578b\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u51fa\u200b\u6240\u6709\u200b\u6210\u672c\u200b\u66f2\u7ebf\u200b\u6700\u4f4e\u70b9\u200b\u5bf9\u5e94\u200b\u7684\u200b\\(Q\\)\u200b\u503c\u200b\uff0c\u200b\u5e76\u200b\u7b5b\u9009\u200b\u51fa\u200b\u6ee1\u8db3\u200b\u589e\u91cf\u200b\u6298\u6263\u200b\u7ea6\u675f\u200b\u7684\u200b\u8ba2\u8d27\u91cf\u200b\uff0c\u200b\u5e73\u5747\u200b\u6210\u672c\u200b\u6700\u4f4e\u200b\u7684\u200b\\(Q\\)\u200b\u5373\u200b\u4e3a\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u200b\u6279\u91cf\u200b\u3002</p> <p></p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_12","title":"\u77ed\u671f\u200b\u6298\u6263","text":"<p>\u200b\u77ed\u671f\u200b\u6298\u6263\u200b\u5373\u200b\u751f\u4ea7\u5546\u200b\u53ea\u200b\u5728\u200b\u4e00\u4e2a\u200b\u5468\u671f\u200b\u4e2d\u200b\u63d0\u4f9b\u200b\u6298\u6263\u200b\u3002\u200b\u4e3a\u200b\u7b80\u5316\u200b\u5904\u7406\u200b\uff0c\u200b\u6298\u6263\u200b\u65f6\u200b\u96f6\u552e\u5546\u200b\u7684\u200b\u8d2d\u4e70\u91cf\u200b\\(Q_d\\)\u200b\u4e3a\u200bEOQ\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\\(Q^*\\)\u200b\u7684\u200b\u6574\u6570\u200b\\(k\\)\u200b\u500d\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#eoq_1","title":"\u8d44\u6e90\u200b\u7ea6\u675f\u200b\u4e0b\u200b\u7684\u200bEOQ\u200b\u6a21\u578b","text":"<p>EOQ\u200b\u6a21\u578b\u200b\u4ec5\u200b\u9002\u7528\u200b\u4e8e\u200b\u4e00\u79cd\u200b\u4ea7\u54c1\u200b\u7684\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u200b\u60c5\u5f62\u200b\uff0c\u200b\u65e0\u6cd5\u200b\u5904\u7406\u200b\u591a\u79cd\u4ea7\u54c1\u200b\u9700\u8981\u200b\u534f\u8c03\u200b\u8ba2\u8d2d\u200b\u7684\u200b\u95ee\u9898\u200b\u3002\u200b\u5728\u200b\u591a\u79cd\u4ea7\u54c1\u200b\u7684\u200b\u8ba2\u8d2d\u200b\u573a\u666f\u200b\u4e0b\u200b\uff0c\u200b\u4e0d\u540c\u200b\u7684\u200b\u4ea7\u54c1\u200b\u6709\u200b\u4e0d\u540c\u200b\u7684\u200b\u9700\u6c42\u200b\u7387\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b\u4e0d\u540c\u200b\u4ea7\u54c1\u200b\u7684\u200bEOQ\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\u8fdb\u884c\u200b\u5e73\u8861\u200b\u3002</p> <p>\u200b\u6a21\u578b\u200b\u5047\u8bbe\u200b\uff1a\u200b\u90e8\u5206\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u4e0e\u200b\u4ea7\u54c1\u79cd\u7c7b\u200b\u6709\u5173\u200b\u2014\u2014\\(K_i\\)\uff0c\u200b\u90e8\u5206\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u4e0e\u200b\u4ea7\u54c1\u79cd\u7c7b\u200b\u65e0\u5173\u200b\u2014\u2014\\(K\\)\u3002</p> <p>\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u5206\u4e3a\u200b\u8d44\u91d1\u200b\u7ea6\u675f\u200b\u4e0e\u200b\u5e93\u5b58\u200b\u7a7a\u95f4\u200b\u7ea6\u675f\u200b\u3002\u200b\u5982\u200b</p> \\[ \\left\\{ \\begin{aligned}   C&amp;\\geq \\sum_{i=1}^n c_iQ_i \\\\   Q_i^{\\mathrm{EOQ}} &amp;= \\sqrt{\\frac{2K_i\\lambda_i}{h_i}} \\end{aligned} \\right. \\] <p>\u200b\u5bf9\u4e8e\u200b\u7d27\u200b\u7ea6\u675f\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u5c06\u200b\u7ea6\u675f\u200b\u5199\u4f5c\u200b\\(\\sum_{i=1}^n c_iQ_i = C\\)\uff0c\u200b\u5f15\u5165\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u7b97\u5b50\u200b\\(\\theta\\)\uff0c\u200b\u5c06\u200b\u6b8b\u5dee\u200b\u9879\u200b\u4e0e\u200b\u4e58\u200b\u5b50\u9879\u200b\u76f8\u52a0\u200b\uff0c\u200b\u5f97\u5230\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u4f18\u5316\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\\(C(Q_1, \\cdots, Q_n, \\theta)\\)</p> \\[ \\min C(Q_1, \\cdots, Q_n, \\theta) = \\sum_{i=1}^n \\left(\\frac{h_iQ_i}{2} + \\frac{K_i\\lambda_i}{Q_i}\\right) + \\theta\\left(\\sum_{i=1}^n c_iQ_i - C\\right) \\] <p>\u200b\u6700\u4f18\u200b\u6761\u4ef6\u200b\u4e0b\u200b\uff0c\u200b\u6709\u200b\\(\\partial C/\\partial \\theta = \\partial C/\\partial Q_i = 0\\)\uff0c\u200b\u5373\u200b</p> \\[ \\begin{aligned}   \\sum_{i=1}^n c_iQ_i^* - C &amp;= 0 \\\\   \\frac{h_i}{2} - \\frac{K_i\\lambda_i}{ {Q_i^*}^2} + \\theta c_i &amp;= 0 \\end{aligned} \\] <p>\u200b\u89e3\u5f97\u200b\\(Q_i^*\\)\uff0c\u200b\u4ee3\u5165\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u5373\u53ef\u200b\u89e3\u5f97\u200b\u6700\u4f18\u200b\u7684\u200b\\(\\theta\\)</p> \\[ \\sum_{i=q}^n c_i\\sqrt{\\frac{2K_i\\lambda_i}{h_i + 2\\theta c_i}} = C \\] <p>\u200b\u4e0d\u59a8\u200b\u8bbe\u200b\\(h_i / c_i = I, \\forall i=1, \\cdots, n\\)\u3002\u200b\u5219\u200b\\(Q_i^* = \\sqrt{\\frac{2K_i\\lambda_i}{h_i}}\\sqrt{\\frac{1}{1 + 2\\theta / I}}\\)\uff0c\u200b\u6ce8\u610f\u200b\u5230\u200b\\(Q_{i}^{\\mathrm{EOQ}} = \\sqrt{\\frac{2K_i\\lambda_i}{h_i}}\\)\uff0c\u200b\u5219\u200b</p> \\[ \\left\\{ \\begin{aligned}   Q_i^* &amp;= \\sqrt{\\frac{1}{1 + 2\\theta / I}}Q_{i}^{\\mathrm{EOQ}} \\\\   C &amp;= \\sum_{i=1}^n c_iQ_i^* \\end{aligned} \\right. \\] <p>\u200b\u89e3\u5f97\u200b</p> \\[ \\theta = \\frac{\\sum_{i=1}^n \\left(c_iQ_i^{\\mathrm{EOQ}}\\right) - c}{2c / I} \\]"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#eoq_2","title":"\u591a\u200b\u4ea7\u54c1\u200bEOQ\u200b\u6a21\u578b","text":"<p>\u200b\u5728\u200b\u591a\u200b\u4ea7\u54c1\u200bEOQ\u200b\u6a21\u578b\u200b\u4e2d\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u9700\u8981\u200b\u9762\u5bf9\u200b\\(U = \\{1, \\cdots, n\\}\\)\u200b\u4e2a\u200b\u4ea7\u54c1\u200b\u7684\u200b\u9700\u6c42\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u4ea7\u54c1\u200b\\(i\\)\u200b\u6709\u200b\u5982\u4e0b\u200b\u7684\u200b\u53c2\u6570\u200b\uff1a</p> <ul> <li>\u200b\u9644\u52a0\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(K_i\\)\u3001\u200b\u8fb9\u9645\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(c_i\\)</li> <li>\u200b\u4ee5\u200b\u5229\u7387\u200b\u5f62\u5f0f\u200b\u8868\u793a\u200b\u7684\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\\(h_i = I_ic_i\\)</li> <li>\u200b\u9700\u6c42\u200b\u7387\u200b\\(\\lambda_i\\)</li> <li>\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u7f3a\u8d27\u200b</li> </ul> <p>\u200b\u5411\u200b\u91cf\u5316\u200b\u8868\u793a\u200b\u7684\u200b\u5404\u4e2a\u200b\u53c2\u6570\u200b\u5206\u522b\u200b\u4e3a\u200b\\(\\boldsymbol{K}, \\boldsymbol{c}, \\boldsymbol{h}, \\boldsymbol{I}, \\boldsymbol{\\lambda}\\)\u3002\u200b\u53e6\u5916\u200b\uff0c\u200b\u6709\u200b\u5168\u5c40\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(K\\)\u3002\u200b\u5728\u200b\u591a\u200b\u4ea7\u54c1\u200bEOQ\u200b\u6a21\u578b\u200b\u4e2d\u200b\uff0c\u200b\u5047\u8bbe\u200b\u67d0\u6b21\u200b\u7684\u200b\u8ba2\u8d27\u91cf\u200b\u4e3a\u200b\\(\\boldsymbol Q = (Q_i)_{n}\\)\uff0c\u200b\u5219\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ \\begin{aligned}   G(\\boldsymbol Q) =&amp; K + \\sum_{i=1}^n K_i\\boldsymbol{1}_{\\{Q_i &gt; 0\\}} &amp; \\text{Setup cost} \\\\   &amp;+\\sum_{i=1}^n c_iQ_i &amp; \\text{Marginal cost} \\end{aligned} \\] <p>\u200b\u73b0\u6709\u200b\u4e09\u79cd\u200b\u8ba2\u8d27\u200b\u65b9\u6cd5\u200b</p> <ol> <li>\u200b\u72ec\u7acb\u200b\u8ba2\u8d27\u200b\u2014\u2014\u200b\u6bcf\u6b21\u200b\u8ba2\u8d2d\u200b\u90fd\u200b\u9700\u8981\u200b\u591a\u200b\u82b1\u8d39\u200b\\(K\\)\u200b\u7684\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b</li> <li>\u200b\u5b8c\u6574\u200b\u8ba2\u8d27\u200b\u2014\u2014\u200b\u5bf9\u4e8e\u200b\u9700\u6c42\u200b\u7387\u4e0d\u9ad8\u200b\u7684\u200b\u4ea7\u54c1\u200b\uff0c\u200b\u8ba2\u8d27\u200b\u8fc7\u4e8e\u200b\u9891\u7e41\u200b\u5bfc\u81f4\u200b\u989d\u5916\u200b\u82b1\u8d39\u200b\u4e86\u200b\\(K_i\\)\u200b\u7684\u200b\u9644\u52a0\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b</li> <li>\u200b\u8054\u5408\u200b\u8ba2\u8d27\u200b\uff1a\u200b\u4e0d\u200b\u9700\u8981\u200b\u8ba2\u8d2d\u200b\u6240\u6709\u200b\u7c7b\u578b\u200b\u7684\u200b\u4ea7\u54c1\u200b</li> </ol> <p>\u200b\u4ee5\u4e0b\u200b\u5bf9\u200b\u4e09\u79cd\u200b\u8ba2\u8d27\u200b\u65b9\u6cd5\u200b\u8fdb\u884c\u200b\u8ba8\u8bba\u200b</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_13","title":"\u72ec\u7acb\u200b\u8ba2\u8d27","text":"<p>\u200b\u72ec\u7acb\u200b\u8ba2\u8d27\u200b\u4e0b\u200b\uff0c\u200b\u6bcf\u79cd\u200b\u4ea7\u54c1\u200b\u6309\u200b\u5176\u200b\u5bf9\u5e94\u200b\u7684\u200bEOQ\u200b\u8ba2\u8d27\u91cf\u200b\u8fdb\u884c\u200b\u8ba2\u8d27\u200b\uff0c\u200b\u5219\u200b\u5e73\u5747\u200b\u6210\u672c\u200b</p> \\[ G^* = \\sum_{i=1}^nG_i^* = \\sum_{i=1}^n\\sqrt{2{K + K_i}\\lambda_i h_i} \\]"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_14","title":"\u5b8c\u6574\u200b\u8ba2\u8d27","text":"<p>\u200b\u5b8c\u6574\u200b\u8ba2\u8d27\u200b\u4e0b\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5468\u671f\u200b\u5185\u200b\u90fd\u200b\u9700\u8981\u200b\u8ba2\u8d2d\u200b\u6240\u6709\u200b\u4ea7\u54c1\u200b\uff0c\u200b\u8bbe\u200b\u8ba2\u8d27\u91cf\u200b\u4e3a\u200b\\(\\boldsymbol{Q} = T\\boldsymbol\\lambda\\)\uff0c\u200b\u7c7b\u6bd4\u200b\u4e8e\u200bEOQ\u200b\u6a21\u578b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u603b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ G(Q) = \\frac{K + \\sum_{i} K_i}{T} + \\boldsymbol{c}^\\top\\boldsymbol{\\lambda} + \\frac{T\\boldsymbol{h}^\\top\\boldsymbol \\lambda}{2} \\] <p>\u200b\u8f6c\u5316\u200b\u4e3a\u5355\u200b\u4ea7\u54c1\u200bEOQ\u200b\u95ee\u9898\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u4e3a\u200b</p> \\[ \\begin{aligned}   T^* &amp;= \\sqrt{\\frac{2\\left(K + \\sum_{i} K_i\\right)}{\\boldsymbol{h}^\\top \\boldsymbol{\\lambda}}} \\\\   G^* &amp;= \\sqrt{2\\left(K + \\sum_{i} K_i\\right)\\boldsymbol{h}^\\top \\boldsymbol{\\lambda}} \\\\ \\end{aligned} \\]"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_15","title":"\u8054\u5408\u200b\u8ba2\u8d27","text":"<p>\u200b\u8054\u5408\u200b\u8ba2\u8d27\u200b\u6761\u4ef6\u200b\u4e0b\u200b\uff0c\u200b\u6bcf\u6b21\u200b\u8ba2\u8d27\u200b\u4e0d\u9700\u200b\u5305\u542b\u200b\u6240\u6709\u200b\u7c7b\u578b\u200b\u7684\u200b\u4ea7\u54c1\u200b\uff0c\u200b\u4f46\u200b\u9700\u6c42\u91cf\u200b\u6700\u9ad8\u200b\u7684\u200b\u4ea7\u54c1\u200b\u5fc5\u7136\u200b\u5728\u200b\u6bcf\u4e2a\u200b\u8ba2\u5355\u200b\u4e2d\u200b\u90fd\u200b\u51fa\u73b0\u200b\uff0c\u200b\u56e0\u6b64\u200b\uff0c\u200b\u9996\u5148\u200b\u8003\u8651\u200bEOQ\u200b\u6a21\u578b\u200b\u4e0b\u200b\u4ea7\u54c1\u200b\u7684\u200b\u8ba2\u8d27\u200b\u9891\u7387\u200b\uff1a</p> \\[ n_i = \\frac{1}{T_i} = \\sqrt{\\frac{h_i\\lambda_i}{(K + K_i)}} \\] <p>\u200b\u4ee5\u200b\u8ba2\u8d27\u200b\u6700\u200b\u9891\u7e41\u200b\u7684\u200b\u4ea7\u54c1\u200b\\(i^\\ast = \\arg\\max\\limits_{i} n_i\\)\u200b\u5bf9\u5e94\u200b\u7684\u200b\u8ba2\u8d27\u200b\u9891\u7387\u200b\\(\\bar n = n_{i^\\ast}\\)\u3001\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\\(\\bar T = T_{i^\\ast}\\)\u200b\u4e3a\u200b\u57fa\u51c6\u200b\uff0c\u200b\u8003\u8651\u200b\u5176\u4ed6\u200b\u7c7b\u578b\u200b\u4ea7\u54c1\u200b\u7684\u200b\u8ba2\u8d27\u200b\u9891\u7387\u200b\uff1a</p> \\[ m_i = \\left\\lceil\\frac{\\bar n}{n_i}\\right\\rceil \\] <p>\\(m_i\\)\u200b\u8868\u793a\u200b\u4ea7\u54c1\u200b\\(i\\)\u200b\u7684\u200b\u8ba2\u8d2d\u200b\u5468\u671f\u200b\u4e3a\u200b\\(m_i\\bar T_i\\)\uff0c\u200b\u5373\u200b\u6700\u200b\u9891\u7e41\u200b\u8ba2\u5355\u200b\u5468\u671f\u200b\u7684\u200b\\(m_i\\)\u200b\u500d\u200b\uff0c\u200b\u5219\u200b\u4e00\u4e2a\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\u5185\u200b\u7684\u200b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ \\begin{aligned}   G(\\boldsymbol Q) =&amp; \\frac{1}{T}\\left(K + \\sum_i\\frac{K_i}{m_i}\\right) &amp; \\text{Setup cost} \\\\   &amp;+\\frac{h}{2}\\sum_{i}\\lambda_i m_i &amp; \\text{Marginal cost} \\end{aligned} \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\u4e0e\u200b\u6700\u4f18\u200b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ \\begin{aligned}   T^* &amp;= \\sqrt{\\frac{\\sum_{i} \\lambda_i m_i h}{2\\left(K + \\sum_i\\frac{K_i}{m_i}\\right)}} \\\\   G^* &amp;= \\sqrt{2\\left(\\sum_{i} \\lambda_i m_i h\\right)\\left(K + \\sum_i\\frac{K_i}{m_i}\\right)} \\end{aligned} \\]"},{"location":"math/operations-mamagement/inventory-control-random-demand/","title":"\u968f\u673a\u200b\u9700\u6c42\u200b\u4e0b\u200b\u7684\u200b\u5e93\u5b58\u200b\u63a7\u5236","text":"<p>\u200b\u4f18\u5316\u200b\u76ee\u6807\u200b\uff1a\u200b\u6700\u5c0f\u5316\u200b\u6210\u672c\u200b\u7684\u200b\u6570\u5b66\u200b\u671f\u671b\u200b</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_2","title":"\u5355\u200b\u5468\u671f\u200b\u5355\u7ea7\u200b\u2014\u2014\u200b\u62a5\u7ae5\u200b\u6a21\u578b","text":"<p>\u200b\u62a5\u7ae5\u200b\u6a21\u578b\u200b\u662f\u200b\u5355\u200b\u5468\u671f\u200b\u4e0b\u200b\u7684\u200b\u5e93\u5b58\u200b\u63a7\u5236\u200b\u95ee\u9898\u200b</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_3","title":"\u8bb0\u53f7","text":"<ul> <li>\\(c_o\\)\u200b\u8868\u793a\u200b\u8d85\u989d\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\uff0c\\(c_u\\)\u200b\u8868\u793a\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b</li> <li>\\(f(\\cdot), F(\\cdot)\\)\u200b\u5206\u522b\u200b\u4e3a\u200b\u9700\u6c42\u200b\u7684\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\u4e0e\u200b\u6982\u7387\u5206\u5e03\u200b\u51fd\u6570\u200b</li> <li>\\(Q\\)\u200b\u4e3a\u200b\u8ba2\u8d27\u91cf\u200b</li> <li>\\(x^+ = \\max\\{x, 0\\}\\)</li> </ul>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_4","title":"\u63a8\u5bfc","text":"<p>\u200b\u62a5\u7ae5\u200b\u6a21\u578b\u200b\u7684\u200b\u6210\u672c\u200b\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ G(Q, D) = c_o\\max\\{0, Q-D\\} + c_u\\max\\{0, D-Q\\} \\] <p>\\(G(Q, D)\\)\u200b\u662f\u200b\u968f\u673a\u53d8\u91cf\u200b\\(D\\)\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e5f\u200b\u662f\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u5bf9\u200b\\(D\\)\u200b\u53d6\u200b\u671f\u671b\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\begin{aligned}     C(Q) &amp;= E(G(Q, D) | Q) \\\\     &amp;= \\int_{0}^\\infty G(Q, D)f(x)\\mathrm dx \\\\     &amp;= \\int_{0}^Q c_o(Q-x)f(x)\\mathrm dx + \\int_Q^\\infty c_u(x-Q)f(x)\\mathrm dx \\\\ \\end{aligned} \\] <p>\u200b\u5f53\u200b\\(C(Q)\\)\u200b\u53d6\u200b\u6781\u5c0f\u503c\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(\\frac{\\mathrm dC}{\\mathrm dD} = 0\\)\uff0c\u200b\u5373\u200b</p> \\[ \\begin{aligned}     \\frac{\\mathrm dC}{\\mathrm dQ} &amp;= c_o\\int_0^Q f(x)\\mathrm dx + c_u\\int_Q^\\infty -f(x)\\mathrm dx \\\\     &amp;= c_oF(Q) - c_u(1-F(Q)) \\\\     &amp;= 0 \\end{aligned} \\] <p>\u200b\u9a8c\u8bc1\u200b\uff1a\\(\\frac{\\mathrm d^2 C}{\\mathrm dQ^2} = (c_o + c_u)f(Q) &gt; 0\\)\uff0c\u200b\u56e0\u6b64\u200b\u4e3a\u200b\u6781\u5c0f\u503c\u200b\u70b9\u200b\u3002\u200b\u6b64\u65f6\u200b\\(F(Q^*) = c_u / (c_o + c_u)\\)\u3002</p> <p>\u200b\u79f0\u200b\\(c_u / (c_o + c_u)\\)\u200b\u4e3a\u200b\u5173\u952e\u200b\u6bd4\u4f8b\u200b\uff08crucial ratio\uff09</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_5","title":"\u5355\u200b\u5468\u671f\u200b\u6a21\u578b\u200b\u6269\u5c55","text":""},{"location":"math/operations-mamagement/inventory-control-random-demand/#_6","title":"\u79bb\u6563\u200b\u9700\u6c42\u200b\u7684\u200b\u6700\u4f18\u200b\u7b56\u7565","text":"<p>\u200b\u786e\u5b9a\u200b\\(\\max_{Q_0}, \\min_{Q_1}\\)\u200b\u6ee1\u8db3\u200b\\(Q_1 &gt; Q_0\\)\u200b\u4e14\u200b\\(F(Q_0) &lt; c_u / (c_u + c_o) &lt; F(Q_1)\\)\uff0c\u200b\u6b64\u65f6\u200b\\(Q_1\\)\u200b\u5373\u200b\u4e3a\u200b\u6700\u4f18\u200b\u7684\u200b\\(Q^*\\)\u3002\u200b\u5373\u200b\u9ad8\u4e8e\u200b\u6700\u4f18\u200b\u5173\u952e\u200b\u6bd4\u4f8b\u200b\u7684\u200b\u6700\u5c0f\u200b\u5173\u952e\u200b\u6bd4\u4f8b\u200b\u5bf9\u5e94\u200b\u7684\u200b\\(Q\\)</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#u","title":"\u5b58\u5728\u200b\u521d\u59cb\u200b\u5e93\u5b58\u200b\\(u\\)","text":"<p>\\(u\\)\u200b\u7684\u200b\u5b58\u5728\u200b\u4e0d\u200b\u5f71\u54cd\u200b\\(Q^*\\)\uff0c\u200b\u56e0\u6b64\u200b\uff0c\u200b\u8ba2\u8d27\u91cf\u200b\\(Q'\\)\u200b\u6ee1\u8db3\u200b</p> \\[ Q^{*\\prime} = \\max\\{Q^*-u, 0\\} \\]"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_7","title":"\u591a\u7ea7\u200b\u7cfb\u7edf","text":"<p>\u200b\u591a\u7ea7\u200b\u5e93\u5b58\u200b\u7cfb\u7edf\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u4e32\u884c\u200b\u7cfb\u7edf\u200b\u3001\u200b\u88c5\u914d\u200b\u7cfb\u7edf\u200b\u548c\u200b\u914d\u9001\u200b\u7cfb\u7edf\u200b\u3002</p> <p>\u200b\u8003\u8651\u200b\u4e00\u4e2a\u200b\u4e32\u884c\u200b\u7684\u200b\u4f9b\u5e94\u94fe\u200b\uff0c\u200b\u4f9b\u5e94\u94fe\u200b\u4e0b\u6e38\u200b\u5411\u200b\u4f9b\u5e94\u94fe\u200b\u4e0a\u6e38\u200b\u91c7\u8d2d\u200b\u4ea7\u54c1\u200b\u3002\u200b\u901a\u5e38\u200b\u91c7\u7528\u200b\u5d4c\u5957\u200b\u7b56\u7565\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u8ba2\u8d27\u200b\uff0c\u200b\u5373\u5f53\u200b\u4e0b\u6e38\u200b\u8ba2\u8d27\u200b\u65f6\u200b\uff0c\u200b\u4e0a\u6e38\u200b\u624d\u200b\u53ef\u80fd\u200b\u8fdb\u884c\u200b\u8ba2\u8d27\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_8","title":"\u5468\u671f\u200b\u68c0\u67e5\u200b\u6a21\u578b","text":"<p>\u200b\u5728\u200b\u5468\u671f\u200b\u68c0\u67e5\u200b\u6a21\u578b\u200b\u4e2d\u200b\uff0c\u200b\u9996\u5148\u200b\u4e0d\u200b\u8003\u8651\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(K\\)\u200b\u7684\u200b\u5f71\u54cd\u200b\u3002\u200b\u9700\u6c42\u200b\u5206\u4e3a\u200b\u591a\u4e2a\u200b\u5468\u671f\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5468\u671f\u200b\u7684\u200b\u9700\u6c42\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_9","title":"\u8bb0\u53f7","text":"<ul> <li>\u200b\u5269\u4f59\u200b\u5468\u671f\u200b\u6570\u4e3a\u200b\\(n\\)</li> <li>\u200b\u6bcf\u4e2a\u200b\u5468\u671f\u200b\u7684\u200b\u9700\u6c42\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u9700\u6c42\u200b\u7684\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\u4e3a\u200b\\(f(\\cdot)\\)\uff0c\u200b\u6982\u7387\u5206\u5e03\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(F(\\cdot)\\)\u3002</li> <li>\u200b\u5468\u671f\u200b\u4e4b\u95f4\u200b\u7684\u200b\u8d34\u73b0\u7387\u200b\\(\\alpha\\)\u200b\u8868\u793a\u200b</li> <li>\u200b\u4e0d\u200b\u8003\u8651\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u200b\uff0c\\(L=0\\)</li> <li>\\(y_i\\)\u200b\u8868\u793a\u200b\u7b2c\u200b\\(i\\)\u200b\u671f\u200b\u7ed3\u675f\u200b\u65f6\u200b\u7684\u200b\u5e93\u5b58\u200b\uff0c\\(y_0\\)\u200b\u8868\u793a\u200b\u671f\u521d\u200b\u5e93\u5b58\u200b</li> <li>\\(C_n(y_0)\\)\u200b\u8868\u793a\u200b\u7b2c\u200b\\(n\\)\u200b\u671f\u200b\u603b\u200b\u671f\u671b\u200b\u8d34\u73b0\u200b\u6210\u672c\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b</li> </ul> \\[ C_n(y_0) = \\min_{y\\geq y_0}\\left\\{L(y) - cy_0 + \\alpha\\int_{0}^\\infty C_{n-1}[t(y, x)]f(x)\\mathrm dx\\right\\} \\]"},{"location":"math/operations-mamagement/inventory-control-random-demand/#q-r","title":"\u8fde\u7eed\u200b\u68c0\u67e5\u200b\u6a21\u578b\u200b\uff1a(Q, R)\u200b\u6a21\u578b","text":"<p>\u200b\u5728\u200b\u8fde\u7eed\u200b\u68c0\u67e5\u200b\u6a21\u578b\u200b\u4e2d\u200b\uff0c\u200b\u9700\u8981\u200b\u533a\u5206\u200b\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b\u4e0e\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u7684\u200b\u6982\u5ff5\u200b</p> <ul> <li>\u200b\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b\uff08Inventory Position\uff09\uff0c\u200b\u6307\u200b\u76ee\u524d\u200b\u6301\u6709\u200b\u7684\u200b\u5e93\u5b58\u200b\u603b\u91cf\u200b\u52a0\u4e0a\u200b\u5904\u4e8e\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u200b\u7684\u200b\u5e93\u5b58\u91cf\u200b</li> <li>\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\uff08Inventory Level\uff09\uff0c\u200b\u6307\u200b\u76ee\u524d\u200b\u6301\u6709\u200b\u7684\u200b\u5e93\u5b58\u200b\u603b\u91cf\u200b</li> </ul>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_10","title":"\u6a21\u578b\u200b\u8bbe\u5b9a\u200b\u53ca\u200b\u63a8\u5bfc","text":"<p>\\((Q, R)\\)\u200b\u6a21\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u8bbe\u5b9a\u200b</p> <ol> <li>\u200b\u6bcf\u6b21\u200b\u8ba2\u8d27\u200b\u7684\u200b\u56fa\u5b9a\u6210\u672c\u200b\u4e3a\u200b\\(K\\)\uff0c\u200b\u8fb9\u9645\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u4e3a\u200b\\(c\\)\uff0c\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u4e3a\u200b\\(\\tau\\)\uff1b</li> <li>\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u5185\u200b\u7684\u200b\u9700\u6c42\u200b\\(D\\)\u200b\u5728\u200b\u65f6\u95f4\u200b\u4e0a\u200b\u8fde\u7eed\u200b\uff0c\u200b\u5728\u200b\u6570\u91cf\u200b\u4e0a\u200b\u670d\u4ece\u200b\u5747\u503c\u200b\u4e3a\u200b\\(\\mu\\)\u3001\u200b\u6807\u51c6\u5dee\u200b\u4e3a\u200b\\(\\sigma\\)\u200b\u7684\u200b\u67d0\u4e2a\u200b\u6982\u7387\u5206\u5e03\u200b\\(f(\\cdot), F(\\cdot)\\)\uff1b\\(\\lambda\\)\u200b\u4e3a\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u7684\u200b\u671f\u671b\u200b\u9700\u6c42\u200b\u7387\u200b\uff1b</li> <li>\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u7684\u200b\u8fb9\u9645\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u4e3a\u200b\\(h = Ic\\)\uff08\u200b\u5f0f\u200b\u4e2d\u200b\\(I\\)\u200b\u4e3a\u200b\u5229\u7387\u200b\uff09</li> <li>\u200b\u5141\u8bb8\u200b\u7f3a\u8d27\u200b\uff0c\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\\(p\\)\uff08backorder\uff09\uff1b</li> <li>\u200b\u6a21\u578b\u200b\u7684\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u4e3a\u91cd\u200b\u8ba2\u8d27\u200b\u70b9\u200b\\(R\\)\u3001\u200b\u4e0e\u200b\u8ba2\u8d27\u91cf\u200b\\(Q\\)\uff1a\u200b\u5f53\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u964d\u4f4e\u200b\u81f3\u200b\\(R\\)\u200b\u65f6\u200b\uff0c\u200b\u8ba2\u8d2d\u200b\\(Q\\)\u200b\u5355\u4f4d\u200b\u7684\u200b\u5e93\u5b58\u200b\u3002\u200b\u53d7\u200b\u8ba2\u8d27\u91cf\u200b\\(Q\\)\u200b\u7684\u200b\u5f71\u54cd\u200b\uff0c\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\\(T\\)\u200b\u6ee1\u8db3\u200b\\(Q = \\lambda T\\)</li> <li> <p>\u200b\u6a21\u578b\u200b\u4e2d\u200b\u4f7f\u7528\u200b\u7684\u200b\u5176\u4ed6\u200b\u8bb0\u53f7\u200b\u6709\u200b\uff1a</p> <ul> <li>\\(\\mathrm{IP}\\)\uff1a\u200b\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b</li> <li>\\(\\mathrm{IL}\\)\uff1a\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b</li> <li>\\(\\mathrm{SS} = R - \\mu\\)\uff1a\u200b\u5b89\u5168\u200b\u5e93\u5b58\u200b</li> </ul> </li> </ol> <p>\\((Q, R)\\)\u200b\u6a21\u578b\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u6765\u6e90\u200b\u7684\u200b\u6210\u672c\u200b\uff1a</p> <ol> <li> <p>\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\uff1a\u200b\u6bcf\u6b21\u200b\u8ba2\u8d2d\u200b\\(Q\\)\u200b\u5355\u4f4d\u200b\u7684\u200b\u5e93\u5b58\u200b\uff0c\u200b\u603b\u6210\u672c\u200b\u4e3a\u200b\\(K + cQ\\)\uff1b</p> \\[ \\frac{K + cQ}{T} = \\frac{K\\lambda}{Q} + c\\lambda \\] </li> <li> <p>\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\uff1a\u200b\u9996\u5148\u200b\u8ba1\u7b97\u200b\u6bcf\u4e2a\u200b\u5468\u671f\u200b\u5185\u200b\u7684\u200b\u671f\u671b\u200b\u7f3a\u8d27\u91cf\u200b\\(n(R)\\)\uff0c\u200b\u6709\u200b</p> \\[ n(R) = E(\\max(D - R, 0)) = \\int_R^\\infty (x-R)f(x)\\mathrm dx \\] <p>\u200b\u6839\u636e\u200b\\(n(R)\\)\uff0c\u200b\u5373\u53ef\u200b\u5f97\u5230\u200b\u4e00\u4e2a\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\\(T\\)\u200b\u65f6\u95f4\u200b\u7684\u200b\u5e73\u5747\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\\(\\lambda n(R)p/Q\\)</p> </li> <li> <p>\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\uff1a\u200b\u4e00\u4e2a\u200b\u5468\u671f\u200b\u5185\u200b\u671f\u671b\u200b\u7684\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u53d8\u5316\u200b\u4e3a\u200b\u7ebf\u6027\u200b\uff0c\u200b\u5f53\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u964d\u200b\u81f3\u200b\\(R\\)\u200b\u540e\u200b\u91cd\u65b0\u200b\u8ba2\u8d27\u200b\uff0c\u200b\u671f\u671b\u200b\u5e93\u5b58\u200b\u7ee7\u7eed\u200b\u4e0b\u964d\u200b\uff0c\u200b\u76f4\u81f3\u200b\u964d\u4f4e\u200b\u81f3\u200b\\(R - \\lambda\\tau\\)\u200b\u65f6\u200b\u8ba2\u5355\u200b\u5230\u8fbe\u200b\uff0c\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u5185\u200b\u7684\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ h\\left(\\frac{Q}{2} + R - \\lambda\\tau\\right) \\] </li> </ol> <p>\u200b\u6839\u636e\u200b\u5982\u200b\u4e0a\u200b\u5206\u6790\u200b\uff0c\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u5185\u200b\u7684\u200b\u671f\u671b\u200b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ \\begin{aligned} G(Q, R) &amp;= h\\left(\\frac{Q}{2} + R - \\lambda\\tau\\right) + \\frac{K\\lambda}{Q} + \\frac{p\\lambda n(R)}{Q} \\\\ &amp;= \\frac{hQ}{2} + \\lambda(K + pn(R))\\frac{1}{Q} + hR - h\\lambda\\tau \\end{aligned} \\] <p>\\((Q, R)\\)\u200b\u6a21\u578b\u200b\u7684\u200b\u76ee\u6807\u200b\u5373\u200b\u4e3a\u200b\u6c42\u89e3\u200b\u51fa\u200b\u4f7f\u5f97\u200b\\(G(Q, R)\\)\u200b\u6700\u5c0f\u200b\u7684\u200b\\(Q, R\\)\u3002\u200b\u82e5\u200b\u8868\u793a\u200b\u4e3a\u200b\\((s, S)\\)\u200b\u7b56\u7565\u200b\u7684\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u5373\u200b\u6709\u200b\\(s = R, S = Q + R\\)</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_11","title":"\u6c42\u89e3","text":"<p>\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u8fed\u4ee3\u200b\u7684\u200b\u65b9\u5f0f\u200b\u903c\u8fd1\u200b\\((Q, R)\\)\u200b\u6a21\u578b\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u5206\u522b\u200b\u5bf9\u200b\\(G\\)\u200b\u6c42\u200b\\(Q, R\\)\u200b\u7684\u200b\u504f\u200b\u5bfc\u6570\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\begin{aligned}     \\frac{\\partial G}{\\partial Q} &amp;= \\frac{h}{2} - \\lambda (K + pn(R))\\frac{1}{Q^2} \\\\     \\frac{\\partial G}{\\partial R} &amp;= h + \\frac{\\lambda p n'(R)}{Q} \\\\     &amp;= h - \\frac{\\lambda p(1-F(R))}{Q} \\end{aligned} \\] <p>\u200b\u8fed\u4ee3\u200b\u8fc7\u7a0b\u200b\u4ece\u200bEOQ\u200b\u6a21\u578b\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\\(Q_{\\mathrm{EOQ}}^* = \\sqrt{\\frac{2K\\lambda}{h}}\\)\u200b\u5f00\u59cb\u200b\uff0c\u200b\u6c42\u200b\u51fa\u200b\u4f7f\u5f97\u200b\\(G\\)\u200b\u6700\u4f18\u200b\u7684\u200b\\(R\\)\u200b\u503c\u200b\uff0c\u200b\u518d\u200b\u5c06\u200b\\(R\\)\u200b\u4ee3\u5165\u200b\u6c42\u200b\u51fa\u200b\u6700\u4f18\u200b\u7684\u200b\\(Q\\)\u200b\u503c\u200b\uff0c\u200b\u5982\u6b64\u200b\u5faa\u73af\u200b\uff0c\\(Q, R\\)\u200b\u5373\u200b\u903c\u8fd1\u200b\u6700\u4f18\u200b\u7684\u200b\\(Q^*, R^*\\)\u200b\u53d6\u503c\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u65b9\u7a0b\u200b\uff1a</p> \\[ \\begin{aligned}     Q &amp;= \\sqrt{\\frac{2\\lambda (K + pn(R))}{h}} \\\\     F(R) &amp;= 1 - \\frac{hQ}{\\lambda p} \\end{aligned} \\]"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_12","title":"\u670d\u52a1\u6c34\u5e73","text":"<p>\u200b\u7c7b\u4f3c\u200b\u4e8e\u200b\u5468\u671f\u200b\u68c0\u67e5\u200b\u6a21\u578b\u200b\uff0c\u200b\u5728\u200b\u8fde\u7eed\u200b\u68c0\u67e5\u200b\u6a21\u578b\u200b\u4e2d\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u7b2c\u4e00\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\u4e0e\u200b\u7b2c\u4e8c\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\u7b49\u200b\u6307\u6807\u200b\u3002\u200b\u82e5\u200b\u5047\u8bbe\u200b\u6bcf\u4e2a\u200b\u63d0\u524d\u200b\u671f\u5185\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u5728\u200b\u9014\u200b\u8ba2\u5355\u200b\uff1a</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_13","title":"\u7b2c\u4e00\u7c7b\u200b\u670d\u52a1\u6c34\u5e73","text":"<p>\u200b\u7b2c\u4e00\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\u6307\u5728\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u4e2d\u200b\u4e0d\u200b\u53d1\u751f\u200b\u7f3a\u8d27\u200b\u7684\u200b\u6982\u7387\u200b\uff0c\u200b\u7528\u200b\\(\\alpha\\)\u200b\u8868\u793a\u200b\u3002\u200b\u5219\u200b\u6709\u200b\\(\\alpha = F(R)\\)\u3002\u200b\u6c42\u89e3\u200b\u6ee1\u8db3\u200b\u7b2c\u4e00\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\u6761\u4ef6\u200b\u7684\u200b\\((Q, R)\\)\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u6b65\u9aa4\u200b\u8fdb\u884c\u200b</p> <ol> <li>\u200b\u6c42\u89e3\u200b\u6ee1\u8db3\u200b\\(F(R) = \\alpha\\)\u200b\u7684\u200b\\(R\\)\u200b\u503c\u200b</li> <li>\\(Q\\)\u200b\u503c\u53d6\u200bEOQ\u200b\u6700\u4f18\u200b\u503c\u200b\\(\\sqrt{\\frac{2K\\lambda}{h}}\\)\u200b\u5373\u53ef\u200b</li> </ol>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#fill-rate","title":"\u7b2c\u4e8c\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\uff1aFill rate","text":"<p>\u200b\u7b2c\u4e8c\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\u6307\u200b\u6ee1\u8db3\u200b\u7684\u200b\u9700\u6c42\u200b\u5360\u200b\u603b\u9700\u6c42\u200b\u7684\u200b\u6bd4\u4f8b\u200b\u6bd4\u4f8b\u200b\uff0c\u200b\u7528\u200b\\(\\beta\\)\u200b\u8868\u793a\u200b\uff0c\u200b\u7531\u4e8e\u200b\\(n(R)\\)\u200b\u4e3a\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u4e2d\u200b\u7684\u200b\u671f\u671b\u200b\u7f3a\u8d27\u91cf\u200b\uff0c\u200b\u5219\u200b\u6709\u200b\\(n(R)/Q = 1 - \\beta\\)\u3002\u200b\u6839\u636e\u200b\\(F(R) = 1 - \\frac{hQ}{\\lambda p}\\)\uff0c\u200b\u53ef\u200b\u5f97\u200b</p> \\[ p = \\frac{Qh}{\\lambda(1 - F(R))} \\] <p>\u200b\u5c06\u200b\u5176\u200b\u4ee3\u5165\u200b\\(Q = \\sqrt{\\frac{2\\lambda (K + pn(R))}{h}}\\)\uff0c\u200b\u6574\u7406\u200b\u5f97\u5230\u200b</p> \\[ hQ^2 - \\frac{2hn(R)}{1-F(R)}Q - 2K\\lambda = 0 \\] <p>\u200b\u89e3\u5f97\u200b</p> \\[ Q = \\frac{n(R)}{1 - F(R)} + \\sqrt{\\frac{2K\\lambda}{h} + \\left(\\frac{n(R)}{1-F(R)}\\right)^2} \\] <p>\u200b\u53c8\u200b\u6839\u636e\u200b\u7b2c\u4e8c\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\uff0c\u200b\u6709\u200b</p> \\[n(R) = (1-\\beta)Q\\] <p>\u200b\u8054\u200b\u7acb\u5373\u200b\u5f97\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u200b\u4f7f\u7528\u200b\u8fed\u4ee3\u200b\u65b9\u5f0f\u200b\u6c42\u89e3\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_14","title":"\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b\u4e0e\u200b\u5e93\u5b58\u200b\u6c34\u5e73","text":"<p>\u200b\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b\u4e0e\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u7684\u200b\u5173\u7cfb\u200b\uff1a</p> \\[ \\mathrm{IL}(t+L) = \\mathrm{IP}(t) - D(t, t+L) \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\\(D(t, t+L)\\)\u200b\u4e3a\u200b\\([t, t+L]\\)\u200b\u65f6\u95f4\u200b\u5185\u200b\u7684\u200b\u9700\u6c42\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u82e5\u200b\u5df2\u77e5\u200b\\(\\mathrm{IP}(t), D(t, t+L)\\)\u200b\u7684\u200b\u5206\u5e03\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u636e\u6b64\u200b\u8ba1\u7b97\u200b\\(\\mathrm{IL}(t+L)\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u7a33\u6001\u200b\uff0c\u200b\u5728\u200b\u4efb\u610f\u200b\u65f6\u523b\u200b\\(t\\)\uff0c\u200b\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b\\(\\mathrm{IP}(t), \\mathrm{IL}(t+L), D(t, t+L)\\)\u200b\u5206\u522b\u200b\u5bf9\u200b\u4e0d\u540c\u200b\u7684\u200b\\(t\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\u3002\u200b\u7531\u6b64\u200b\uff0c\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u968f\u673a\u53d8\u91cf\u200b\\(\\mathrm{IP}, \\mathrm{IL}, D(L)\\)\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5206\u5e03\u200b\u5173\u7cfb\u200b\u3002\u200b\u5df2\u77e5\u200b\\(\\mathrm{IP}\\)\u200b\u4e3a\u200b\u5747\u5300\u5206\u5e03\u200b\uff0c\u200b\u8bbe\u200b\\(D(L)\\)\u200b\u7684\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\u4e3a\u200b\\(f(x;L)\\)\uff0c\u200b\u5219\u200b</p> \\[ \\begin{aligned}     F_{\\mathrm{IL}}(x) &amp;= P(\\mathrm{IL}\\leq x) \\\\     &amp;= \\int_{R}^{R+Q}P(\\mathrm{IP} = u)P(\\mathrm{IP} - \\mathrm{IL} \\geq u-x) \\mathrm du \\\\     &amp;= \\frac{1}{Q}\\int_{R}^{R+Q} P(D \\geq u-x) \\mathrm du \\\\     &amp;= \\frac{1}{Q}\\int_{R}^{R+Q} (1 - F(u-x;L)) \\mathrm du \\end{aligned} \\]"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_15","title":"\u6a21\u578b\u200b\u6269\u5c55","text":""},{"location":"math/operations-mamagement/inventory-control-random-demand/#_16","title":"\u6b63\u6001\u5206\u5e03\u200b\u9700\u6c42","text":"<p>\u200b\u5f53\u200b\u63d0\u524d\u200b\u671f\u5185\u200b\u9700\u6c42\u200b\\(D\\)\u200b\u670d\u4ece\u200b\u6b63\u6001\u5206\u5e03\u200b\\(N(\\mu, \\sigma)\\)\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6b63\u6001\u5206\u5e03\u200b\u7684\u200b\u6807\u51c6\u200b\u635f\u5931\u200b\u51fd\u6570\u200b\\(L(z)\\)\u200b\u7b80\u5316\u200b\\(n(R)\\)\u200b\u7684\u200b\u8ba1\u7b97\u200b</p> \\[ n(R) = \\sigma L\\left(\\frac{R-\\mu}{\\sigma}\\right) = \\sigma L(z) \\] <p>\\(L(z)\\)\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\\(\\phi(z), \\varPhi(z)\\)\u200b\u8ba1\u7b97\u200b\u5f97\u5230\u200b\uff1a</p> \\[ L(z) = \\phi(z) - z + z\\varPhi(z) \\] <p>\u200b\u5c06\u200b\\(D(L) \\sim N(\\mu, \\sigma)\\)\u200b\u4ee3\u5165\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u7684\u200b\u5206\u5e03\u200b\uff0c\u200b\u5f97\u5230\u200b</p> \\[ F_{\\mathrm{IL}}(x) = \\frac{1}{Q}\\int_{R}^{R+Q} \\left[1 - \\varPhi\\left(\\frac{u-x-\\mu}{\\sigma}\\right)\\right] \\mathrm du \\] <p>\u200b\u7531\u6b64\u200b\u53ef\u200b\u5f97\u200b\uff0c\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b\\(f_{\\mathrm{IL}}(x)\\)\u200b\u4e3a\u200b</p> \\[ f_{\\mathrm{IL}}(x) = \\frac{1}{Q}\\left[\\varPhi\\left(\\frac{R+Q-x-\\mu}{\\sigma}\\right) - \\varPhi\\left(\\frac{R-x-\\mu}{\\sigma}\\right)\\right] \\]"},{"location":"math/operations-mamagement/inventory-control-random-demand/#lost-sales","title":"Lost-sales","text":"<p>\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u7f3a\u8d27\u200b\uff0c\u200b\u5219\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u4e0d\u200b\u4f4e\u4e8e\u200b0\uff0c\u200b\u6b64\u65f6\u200b\u5e73\u5747\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u53d8\u4e3a\u200b</p> \\[ \\frac{Q}{2} + \\max(R-\\mu, 0) \\] <p>\u200b\u5176\u4f59\u200b\u6210\u672c\u200b\u4e0d\u53d8\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_17","title":"\u968f\u673a\u200b\u63d0\u524d\u200b\u671f","text":""},{"location":"math/operations-mamagement/inventory-control-random-demand/#one-for-one-policy","title":"One-for-one policy","text":"<p>\u200b\u5f53\u200b\u8ba2\u8d27\u200b\u6279\u91cf\u200b\\(Q = 1\\)\u200b\u65f6\u200b\uff0c\u200b\u79f0\u4e3a\u200bOne-for-one policy\uff0c\u200b\u901a\u5e38\u200b\u9002\u7528\u200b\u4e8e\u200b\u987e\u5ba2\u200b\u9700\u6c42\u200b\u7387\u200b\u8f83\u200b\u4f4e\u200b\u800c\u200b\u4ea7\u54c1\u200b\u4ef7\u503c\u200b\u8f83\u200b\u9ad8\u200b\uff0c\u200b\u5982\u200b\u6c7d\u8f66\u200b\u7b49\u200b\u5927\u5b97\u200b\u5546\u54c1\u200b\u3002\u200b\u8be5\u200b\u7b56\u7565\u200b\u4e00\u200b\u51fa\u73b0\u200b\u9700\u6c42\u200b\u5c31\u200b\u8ba2\u8d27\u200b\uff0c\u200b\u4ece\u800c\u200b\u4f7f\u5f97\u200b\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b\u59cb\u7ec8\u200b\u5904\u4e8e\u200b\\(S = Q + R\\)\u3002\u200b\u4ee5\u4e0b\u200b\u8003\u8651\u200b\u9700\u6c42\u200b\u79bb\u6563\u200b\u7684\u200b\u60c5\u51b5\u200b\uff1a</p> <ol> <li>\u200b\u9700\u6c42\u200b\u670d\u4ece\u200b\u53c2\u6570\u200b\u4e3a\u200b\\(\\lambda\\)\u200b\u7684\u200b\u6cca\u677e\u200b\u5206\u5e03\u200b</li> <li>\u200b\u5141\u8bb8\u200b\u5ef6\u671f\u200b\u4ea4\u8d27\u200b</li> <li>\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u7684\u200b\u5355\u4f4d\u200b\u6301\u6709\u200b\u6210\u672c\u200b\u4e0e\u200b\u5355\u4f4d\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\u4e3a\u200b\\(h, p\\)</li> </ol> <p>\u200b\u5f53\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u4e3a\u200b\u5e38\u6570\u200b\\(\\tau\\)\u200b\u65f6\u200b\uff0c\u200b\u5206\u6790\u200b\u6301\u6709\u200b\u6210\u672c\u200b\u4e0e\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\u3002\u200b\u9996\u5148\u200b\u8003\u8651\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u5904\u4e8e\u200b\u4e0d\u540c\u200b\u72b6\u6001\u200b\u7684\u200b\u6982\u7387\u200b\uff1a</p> \\[ P(\\mathrm{IL} = i) = P(D = S - i) = \\frac{e^{-\\lambda t}(\\lambda t)^{S-i}}{(S-i)!} \\] <p>\u200b\u5df2\u77e5\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u4e3a\u200b\\(i\\)\uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5b58\u8d27\u91cf\u200b\u4e3a\u200b\\(\\max\\{S - D, 0\\}\\)\uff0c\u200b\u7f3a\u8d27\u91cf\u200b\u4e3a\u200b\\(\\max\\{D-S, 0\\}\\)\uff0c\u200b\u56e0\u6b64\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u4e0e\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\u5206\u522b\u200b\u4e3a\u200b</p> \\[ \\begin{aligned}     c_h &amp;= h\\sum_{D=0}^S (S-D)\\frac{e^{-\\lambda t}(\\lambda t)^{D}}{D!} \\\\     c_u &amp;= p\\sum_{D=S}^\\infty (D-S)\\frac{e^{-\\lambda t}(\\lambda t)^{D}}{D!} \\end{aligned} \\] <p>\u200b\u5f53\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u200b\u968f\u673a\u200b\uff0c\u200b\u4e0d\u540c\u200b\u8ba2\u5355\u200b\u7684\u200b\u63d0\u524d\u200b\u671f\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\u65f6\u200b\uff0c\u200b\u4e3a\u200b\u7b80\u5316\u200b\u5904\u7406\u200b\uff0c\u200b\u8bbe\u200b\u63d0\u524d\u200b\u671f\u200b\u670d\u4ece\u200b\u5747\u503c\u200b\u4e3a\u200b\\(\\tau\\)\u200b\u7684\u200b\u6307\u6570\u5206\u5e03\u200b\uff0c\u200b\u5219\u200b\u7cfb\u7edf\u200b\u7684\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\u56fe\u4e3a\u200b</p> <p></p> <p>\u200b\u8be5\u200b\u7cfb\u7edf\u200b\u4e3a\u751f\u200b\u706d\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u72b6\u6001\u200b\\(i &lt; S\\)\uff0c\u200b\u6709\u200b</p> \\[ \\left(\\lambda + \\frac{S-i}{\\tau}\\right)P(\\mathrm{IL} = i) = \\lambda P(\\mathrm{IL} = i+1) + \\frac{S-i+1}{\\tau} P(\\mathrm{IL} = i-1) \\] <p>\u200b\u5f53\u200b\\(i = S\\)\u200b\u65f6\u200b\uff0c\\(\\lambda P(\\mathrm{IL}  = S) = \\frac{1}{\\tau}P(\\mathrm{IL} = S-1)\\)\u3002\u200b\u7531\u6b64\u200b\u53ef\u4ee5\u200b\u63a5\u200b\u7684\u200b</p> \\[ \\begin{aligned}     P(\\mathrm{IL} = S) &amp;= e^{-\\lambda\\tau} \\\\     P(\\mathrm{IL} = i) &amp;= \\frac{(\\lambda\\tau)^{S-i}}{(S-i)!}P(\\mathrm{IL} = S) \\\\     &amp;= \\frac{(\\lambda\\tau)^{S-i}e^{-\\lambda\\tau}}{(S-i)!} \\end{aligned} \\] <p>\u200b\u5f53\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u5ef6\u671f\u200b\u4ea4\u8d27\u200b\u65f6\u200b\uff0c\\(\\mathrm{IL}\\geq 0\\)\uff0c\u200b\u60c5\u51b5\u200b\u4e0e\u200b\u5141\u8bb8\u200b\u5ef6\u671f\u200b\u4ea4\u8d27\u200b\u65f6\u200b\u53c8\u200b\u6709\u200b\u4e0d\u540c\u200b\uff0c\u200b\u4ece\u200b\u65e0\u9650\u200b\u72b6\u6001\u200b\u9a6c\u6c0f\u94fe\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u6709\u9650\u200b\u72b6\u6001\u200b\u9a6c\u6c0f\u94fe\u200b\u3002</p>"},{"location":"math/queue-theory/","title":"\u5bf9\u7b56\u200b\u3001\u200b\u51b3\u7b56\u200b\u4e0e\u200b\u6392\u961f\u200b\u8bba","text":"<p>\u200b\u8bfe\u7a0b\u200b\u76ee\u5f55\u200b\uff1a</p> <ol> <li>Chapter 1\uff0c</li> <li>Chapter 2\uff0c</li> <li>Chapter 3\uff0c</li> <li>Chapter 4\uff0c</li> <li>Chapter 5\uff0c</li> <li>Chapter 6\uff0c</li> <li>Chapter 7\uff0c</li> <li>Chapter 8\uff0c</li> <li>Chapter 9\uff0c</li> <li>Chapter 10\uff0c</li> <li>Chapter 11\uff0c</li> <li>Chapter 12\uff0c\u200b\u5e38\u89c1\u200b\u7684\u200b\u6392\u961f\u6a21\u578b\u200b</li> </ol>"},{"location":"math/queue-theory/chapter-12/","title":"\u5e38\u89c1\u200b\u6392\u961f\u6a21\u578b","text":"<p>\u200b\u4e00\u4e2a\u200b\u6392\u961f\u6a21\u578b\u200b\u4e3b\u8981\u200b\u7531\u200b\u4ee5\u4e0b\u200b\u90e8\u5206\u200b\u7ec4\u6210\u200b</p> <ul> <li>\u200b\u987e\u5ba2\u200b\uff08\u200b\u5b9e\u4f53\u200b\uff09\u200b\u5904\u7406\u200b</li> <li>\u200b\u5b9e\u4f53\u200b\u6392\u961f\u200b\u7b49\u5f85\u200b\u5904\u7406\u200b\uff08\u200b\u7cfb\u7edf\u200b\u7684\u200b\u6392\u961f\u200b\u7b56\u7565\u200b\u53ef\u80fd\u200b\u4e0d\u540c\u200b\uff0c\u200b\u635f\u5931\u200b\u5236\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u6392\u961f\u200b\u3001\u200b\u6df7\u5408\u5236\u200b\u5141\u8bb8\u200b\u6392\u200b\u6709\u9650\u200b\u957f\u5ea6\u200b\u7684\u200b\u961f\u5217\u200b\uff09</li> <li>\u200b\u5b9e\u4f53\u200b\u5904\u7406\u5b8c\u6bd5\u200b\uff0c\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b</li> </ul>"},{"location":"math/queue-theory/chapter-12/#_2","title":"\u6392\u961f\u6a21\u578b\u200b\u7684\u200b\u63cf\u8ff0","text":""},{"location":"math/queue-theory/chapter-12/#_3","title":"\u8fdb\u5165\u200b\u7cfb\u7edf","text":"<p>\u200b\u5047\u8bbe\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u5b9e\u4f53\u200b\u6570\u91cf\u200b\u65e0\u9650\u200b\u3002\u200b\u8003\u5bdf\u200b\u76f8\u90bb\u200b\u4e24\u4e2a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u65f6\u95f4\u5dee\u200b\uff1a</p> <ol> <li>\u200b\u5bf9\u4e8e\u200b\u5b9a\u200b\u957f\u200b\u5206\u5e03\u200b\uff0c\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u65f6\u95f4\u5dee\u200b\u4e3a\u200b\u5e38\u91cf\u200b\\(t\\)\uff0c\\(t\\)\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\u4e3a\u200b\uff1a</li> </ol> \\[ f(t; \\alpha)=\\left\\{ \\begin{aligned}     &amp; 1 &amp; t=\\alpha \\\\     &amp; 0 &amp; t\\not =\\alpha \\end{aligned} \\right . \\] <ol> <li>Poisson \u200b\u6d41\u200b\uff0c\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u65f6\u95f4\u5dee\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u670d\u4ece\u200b\u53c2\u6570\u200b\u4e3a\u200b\\(\\lambda\\)\u200b\u7684\u200b\u6307\u6570\u5206\u5e03\u200b</li> </ol> \\[ f_{\\xi_i}(t; \\lambda)=\\left\\{ \\begin{aligned}     &amp; \\lambda e^{-\\lambda t} &amp; t\\geq 0 \\\\     &amp; 0 &amp; t &lt; 0 \\end{aligned} \\right . \\] <ol> <li>Erlang \u200b\u5206\u5e03\u200b\uff0c\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u65f6\u95f4\u5dee\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u670d\u4ece\u200b\u5982\u4e0b\u200b\u516c\u5f0f\u200b\u6240\u793a\u200b\u7684\u200b\u5206\u5e03\u200b\u3002\u200b\u82e5\u200b\u67d0\u4e2a\u200b\u7cfb\u7edf\u200b\u4e2d\u6709\u200b\\(K\\)\u200b\u4e2a\u200b\u5e76\u8054\u200b\u7684\u200b\u670d\u52a1\u53f0\u200b\uff0c\u200b\u8f93\u5165\u200b\u5b9e\u4f53\u200b\u4e3a\u200b Poisson \u200b\u6d41\u200b\uff0c\u200b\u5219\u200b\u7b2c\u200b\\(K\\)\u200b\u4e2a\u200b\u670d\u52a1\u53f0\u200b\u7684\u200b\u987e\u5ba2\u200b\u6d41\u4e3a\u200b Erlang \u200b\u6d41\u200b\uff0c\u200b\u524d\u9762\u200b\u6240\u6709\u200b\\(K - 1\\)\u200b\u4e2a\u200b\u670d\u52a1\u53f0\u200b\u5728\u200b\u7b2c\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u5230\u8fbe\u200b\u4ee5\u540e\u200b\u7684\u200b\u8f93\u5165\u200b\u6d41\u200b\u540c\u6837\u200b\u4e3a\u200b Erlang \u200b\u6d41\u200b\u3002</li> </ol> \\[ f(t; \\lambda, K)=\\left\\{ \\begin{aligned}     &amp; \\frac{\\lambda(\\lambda t)^{K-1}}{(K-1)!} &amp; t\\geq 0 \\\\     &amp; 0 &amp; t &lt; 0 \\end{aligned} \\right . \\]"},{"location":"math/queue-theory/chapter-12/#_4","title":"\u5b9e\u4f53\u200b\u6392\u961f\u200b\u4e0e\u200b\u5904\u7406","text":"<p>\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u53ef\u80fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u6216\u200b\u591a\u4e2a\u200b\u670d\u52a1\u53f0\u200b\u3002\u200b\u5728\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u670d\u52a1\u53f0\u200b\u90fd\u200b\u5904\u4e8e\u200b\u8fd0\u884c\u200b\u72b6\u6001\u200b\u65f6\u200b\uff0c\u200b\u65b0\u200b\u8fdb\u5165\u200b\u7684\u200b\u5b9e\u4f53\u200b\u65e0\u6cd5\u200b\u7acb\u5373\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b\uff0c\u200b\u800c\u662f\u200b\u8fdb\u5165\u200b\u961f\u5217\u200b\u6216\u200b\u76f4\u63a5\u200b\u79bb\u5f00\u200b\uff0c\u200b\u53d6\u51b3\u4e8e\u200b\u7cfb\u7edf\u200b\u7684\u200b\u6392\u961f\u200b\u7b56\u7565\u200b\u3002\u200b\u6392\u961f\u200b\u7b56\u7565\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u635f\u5931\u200b\u5236\u200b\u3001\u200b\u7b49\u5f85\u200b\u5236\u200b\u4e0e\u200b\u6df7\u5408\u5236\u200b\u3002  </p> <p>\u200b\u5728\u200b\u635f\u5931\u200b\u5236\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\uff0c\u200b\u5982\u679c\u200b\u65b0\u200b\u8fdb\u5165\u200b\u7684\u200b\u5b9e\u4f53\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u7acb\u5373\u200b\u5904\u7406\u200b\uff0c\u200b\u5219\u200b\u4f1a\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b\u3002 \u200b\u5728\u200b\u7b49\u5f85\u200b\u5236\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\uff0c\u200b\u5982\u679c\u200b\u65b0\u200b\u8fdb\u5165\u200b\u7684\u200b\u5b9e\u4f53\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u7acb\u5373\u200b\u5904\u7406\u200b\uff0c\u200b\u5219\u200b\u4f1a\u200b\u5728\u200b\u961f\u5217\u200b\u4e2d\u200b\u7b49\u5f85\u200b\uff0c\u200b\u76f4\u5230\u200b\u88ab\u200b\u5904\u7406\u200b\u3002 \u200b\u5728\u200b\u6df7\u5408\u5236\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\uff0c\u200b\u5982\u679c\u200b\u65b0\u200b\u8fdb\u5165\u200b\u7684\u200b\u5b9e\u4f53\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u7acb\u5373\u200b\u5904\u7406\u200b\u4e14\u200b\u6b64\u65f6\u200b\u7684\u200b\u961f\u5217\u200b\u957f\u5ea6\u200b\u5927\u4e8e\u200b\u67d0\u4e00\u5e38\u200b\u6570\u503c\u200b\uff0c\u200b\u5219\u200b\u4f1a\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u5426\u5219\u200b\u4f1a\u200b\u8fdb\u5165\u200b\u961f\u5217\u200b\u7b49\u5f85\u200b\u3002</p> <p>\u200b\u5b9e\u4f53\u200b\u5728\u200b\u6392\u961f\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u7684\u200b\u884c\u4e3a\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u5982\u4e0b\u200b\u7c7b\u578b\u200b\uff1a</p> <ul> <li>\u200b\u79bb\u5f00\u200b\uff1a\u200b\u5f53\u200b\u5b9e\u4f53\u200b\u5f97\u77e5\u200b\u9700\u8981\u200b\u7b49\u5f85\u200b\u6216\u200b\u6ca1\u6709\u200b\u8010\u5fc3\u200b\u7ee7\u7eed\u200b\u7b49\u5f85\u200b\u65f6\u200b\uff0c\u200b\u5b9e\u4f53\u200b\u53ef\u80fd\u200b\u9000\u51fa\u200b\u961f\u5217\u200b</li> <li>\u200b\u53d8\u6362\u200b\uff1a\u200b\u5373\u200b\u5b9e\u4f53\u200b\u4e3a\u200b\u51cf\u5c11\u200b\u6392\u961f\u200b\u65f6\u95f4\u200b\u800c\u200b\u5728\u200b\u4e0d\u540c\u200b\u670d\u52a1\u53f0\u200b\u7684\u200b\u961f\u5217\u200b\u4e4b\u95f4\u200b\u79fb\u52a8\u200b</li> </ul> <p>\u200b\u670d\u52a1\u53f0\u200b\u4f1a\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u53ef\u80fd\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u670d\u52a1\u200b\uff1a</p> <ul> <li>FCFS\uff1a\u200b\u5148\u200b\u8fdb\u5165\u200b\u961f\u5217\u200b\u7684\u200b\u5148\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b</li> <li>LCFS\uff1a\u200b\u540e\u200b\u8fdb\u5165\u200b\u961f\u5217\u200b\u7684\u200b\u5148\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b</li> <li>\u200b\u968f\u673a\u200b\u5904\u7406\u200b</li> <li>PS\uff1a\u200b\u4f18\u5148\u7ea7\u200b\u9ad8\u200b\u7684\u200b\u5b9e\u4f53\u200b\u5148\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b</li> <li>\u200b\u65ad\u7eed\u200b\u5904\u7406\u200b\uff1a\u200b\u5728\u200b\u5904\u7406\u200b\u67d0\u4e2a\u200b\u5b9e\u4f53\u200b\u4e00\u6bb5\u65f6\u95f4\u200b\u540e\u200b\uff0c\u200b\u8f6c\u800c\u200b\u5904\u7406\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\uff0c\u200b\u4e0d\u65ad\u200b\u5faa\u73af\u200b\u76f4\u5230\u200b\u5b9e\u4f53\u200b\u5904\u7406\u200b\u5b8c\u6210\u200b</li> </ul> <p>\u200b\u5728\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u7684\u200b\u670d\u52a1\u53f0\u200b\u7684\u200b\u670d\u52a1\u200b\u65f6\u95f4\u200b\u662f\u200b\u4e00\u4e2a\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u6982\u7387\u5206\u5e03\u200b\u4e3a\u200b\u5e38\u6570\u200b\u3001\u200b\u6307\u6570\u5206\u5e03\u200b\u3001K\u200b\u7ea7\u200b Erlang \u200b\u5206\u5e03\u200b\u3001\u200b\u4e00\u822c\u200b\u5206\u5e03\u200b\u7b49\u200b\u3002</p>"},{"location":"math/queue-theory/chapter-12/#_5","title":"\u7cfb\u7edf\u200b\u7684\u200b\u7b26\u53f7\u200b\u8868\u793a","text":"<p>Kendall\u200b\u8bb0\u53f7\u200b\uff0c\u200b\u683c\u5f0f\u200b\u4e3a\u200b\\(X/Y/Z/A/B/C\\)</p> <ul> <li>\\(X\\)\u200b\u8868\u793a\u200b\u5b9e\u4f53\u200b\u5230\u8fbe\u200b\u95f4\u9694\u65f6\u95f4\u200b\u7684\u200b\u5206\u5e03\u200b<ul> <li>\\(M\\)\u200b\u8868\u793a\u200b\u5230\u8fbe\u200b\u8fc7\u7a0b\u200b\u4e3a\u200b Poisson \u200b\u8fc7\u7a0b\u200b\uff08\u200b\u95f4\u9694\u65f6\u95f4\u200b\u670d\u4ece\u200b\u6307\u6570\u5206\u5e03\u200b\uff09\u200b\u6216\u8d1f\u200b\u6307\u6570\u5206\u5e03\u200b</li> <li>\\(D\\)\u200b\u8868\u793a\u200b\u95f4\u9694\u65f6\u95f4\u200b\u4e3a\u200b\u5e38\u6570\u200b</li> <li>\\(E_k\\)\u200b\u8868\u793a\u200b\u95f4\u9694\u65f6\u95f4\u200b\u670d\u4ece\u200b\\(k\\)\u200b\u9636\u200b Erlang \u200b\u5206\u5e03\u200b</li> <li>\\(G\\)\u200b\u8868\u793a\u200b\u95f4\u9694\u65f6\u95f4\u200b\u670d\u4ece\u200b\u4e00\u822c\u200b\u3001\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\u7684\u200b\u968f\u673a\u200b\u5206\u5e03\u200b</li> </ul> </li> <li>\\(Y\\)\u200b\u8868\u793a\u200b\u670d\u52a1\u53f0\u200b\u5355\u6b21\u200b\u670d\u52a1\u200b\u65f6\u95f4\u200b\u7684\u200b\u5206\u5e03\u200b\uff0c\u200b\u8bb0\u53f7\u200b\u4e0e\u200b\\(X\\)\u200b\u7684\u200b\u542b\u4e49\u200b\u76f8\u540c\u200b</li> <li>\\(Z\\)\u200b\u8868\u793a\u200b\u670d\u52a1\u53f0\u200b\u4e2a\u6570\u200b</li> <li>\\(A\\)\u200b\u8868\u793a\u200b\u7cfb\u7edf\u200b\u7684\u200b\u7b49\u5f85\u200b\u7a7a\u95f4\u200b\u5bb9\u91cf\u200b\uff0c\\(0\\)\u200b\u4e3a\u200b\u635f\u5931\u200b\u5236\u200b\u7cfb\u7edf\u200b\uff0c\\(\\infty\\)\u200b\u4e3a\u200b\u7b49\u5f85\u200b\u5236\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u5426\u5219\u200b\u4e3a\u200b\u6df7\u5408\u5236\u200b\u7cfb\u7edf\u200b\u3002</li> <li>\\(B\\)\u200b\u8868\u793a\u200b\u5b9e\u4f53\u200b\u7684\u200b\u603b\u200b\u6570\u91cf\u200b\uff0c\\(\\infty\\)\u200b\u8868\u793a\u200b\u5b9e\u4f53\u200b\u6e90\u200b\u65e0\u9650\u200b</li> <li>\\(C\\)\u200b\u8868\u793a\u200b\u670d\u52a1\u200b\u89c4\u5219\u200b</li> </ul> <p>\\(A, B, C\\)\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u3002\u200b\u5206\u522b\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\\(\\infty, \\infty, \\mathit{FCFS}\\)</p>"},{"location":"math/queue-theory/chapter-12/#_6","title":"\u7cfb\u7edf\u200b\u7684\u200b\u89c2\u6d4b\u200b\u6307\u6807","text":"<p>\u200b\u8861\u91cf\u200b\u7cfb\u7edf\u200b\u6027\u80fd\u200b\u7684\u200b\u5e38\u7528\u200b\u89c2\u6d4b\u200b\u6307\u6807\u200b\u4e3a\u200b\uff1a</p> <ul> <li>\\(N(L)\\)\uff1a\u200b\u7a33\u6001\u200b\u7cfb\u7edf\u200b\u7684\u200b\u961f\u957f\u200b\uff08\u200b\u6307\u200b\u7cfb\u7edf\u200b\u5185\u200b\u6240\u6709\u200b\u7684\u200b\u5b9e\u4f53\u200b\u6570\u91cf\u200b\uff09</li> <li>\\(N_q(L_q)\\)\u200b\u7a33\u6001\u200b\u7cfb\u7edf\u200b\u7684\u200b\u6392\u961f\u200b\u957f\u200b</li> <li>\\(T(W)\\)\uff1a\u200b\u987e\u5ba2\u200b\u5728\u200b\u7a33\u6001\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u7684\u200b\u505c\u7559\u65f6\u95f4\u200b\uff08\u200b\u5904\u7406\u200b+\u200b\u7b49\u5f85\u200b\uff09</li> <li>\\(T_q(W_q)\\)\uff1a\u200b\u987e\u5ba2\u200b\u5728\u200b\u7a33\u6001\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u7684\u200b\u7b49\u5f85\u65f6\u95f4\u200b</li> <li>\\(p_n\\)\uff1a\u200b\u7a33\u6001\u200b\u7cfb\u7edf\u200b\u4efb\u610f\u200b\u65f6\u523b\u200b\u72b6\u6001\u200b\u4e3a\u200b\\(n\\)\u200b\u7684\u200b\u6982\u7387\u200b\uff0c\\(p_n=P(N=n)=\\lim_{t\\rightarrow \\infty}P(N(t) = n)\\)</li> <li>\\(\\rho\\)\uff1a\u200b\u5229\u7528\u7387\u200b</li> <li>\\(p_D\\)\uff1a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u65f6\u200b\u9700\u8981\u200b\u7b49\u5f85\u200b\u7684\u200b\u6982\u7387\u200b</li> </ul> <p>\u200b\u82e5\u200b\u5c06\u200b\u7cfb\u7edf\u200b\u89c6\u4e3a\u200b\u7ba1\u9053\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u5b9e\u4f53\u200b\u7684\u200b\u6570\u91cf\u200b\u7b49\u4e8e\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u901f\u5ea6\u200b\u4e0e\u200b\u7cfb\u7edf\u200b\u5e73\u5747\u200b\u54cd\u5e94\u200b\uff08\u200b\u5904\u7406\u200b\uff09\u200b\u65f6\u95f4\u200b\u4e4b\u79ef\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\\(M/M/s\\)\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u4ee4\u200b\\(\\lambda\\)\u200b\u4e3a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u901f\u5ea6\u200b\uff0c\\(\\mu\\)\u200b\u4e3a\u200b\u670d\u52a1\u53f0\u200b\u5904\u7406\u200b\u5b9e\u4f53\u200b\u7684\u200b\u901f\u5ea6\u200b\uff0c\\(s\\)\u200b\u4e3a\u200b\u670d\u52a1\u53f0\u200b\u7684\u200b\u6570\u91cf\u200b\uff0c\u200b\u5219\u200b\uff1a</p> \\[ \\begin{aligned} L &amp;= \\lambda_e W \\\\ L_q &amp;= \\lambda_e W_q \\end{aligned} \\]"},{"location":"math/queue-theory/chapter-12/#_7","title":"\u968f\u673a\u200b\u8fc7\u7a0b","text":"<p>\u200b\u5e38\u89c1\u200b\u7684\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b\u6709\u200b Markov \u200b\u8fc7\u7a0b\u200b\u3001\u200b\u751f\u706d\u200b\u8fc7\u7a0b\u200b\u3001 Poisson \u200b\u8fc7\u7a0b\u200b\u7b49\u200b\u3002\u200b\u6240\u6709\u200b\u7684\u200b Poisson \u200b\u8fc7\u7a0b\u200b\u90fd\u200b\u662f\u200b\u751f\u706d\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u6240\u6709\u200b\u7684\u200b\u751f\u706d\u200b\u8fc7\u7a0b\u200b\u90fd\u200b\u662f\u200b Markov \u200b\u8fc7\u7a0b\u200b\u3002</p>"},{"location":"math/queue-theory/chapter-12/#markov","title":"Markov \u200b\u8fc7\u7a0b","text":"<p>\u200b\u7279\u70b9\u200b\uff1a\\(t_0\\)\u200b\u65f6\u523b\u200b\u7684\u200b\u72b6\u6001\u200b\u4e0e\u200b\u4efb\u610f\u200b\\(t&lt;t_0\\)\u200b\u65f6\u200b\u7684\u200b\u72b6\u6001\u200b\u65e0\u5173\u200b</p> <p>Markov \u200b\u94fe\u200b\u7531\u200b\u9a6c\u5c14\u53ef\u592b\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u7684\u200b\u79bb\u6563\u200b\u72b6\u6001\u200b\u7ec4\u5408\u200b\u3002\u200b\u5bf9\u4e8e\u200b Markov \u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u65e0\u200b\u8bb0\u5fc6\u6027\u200b\uff0c\u200b\u5f53\u524d\u200b\u7684\u200b\u72b6\u6001\u200b\u4e0d\u4f1a\u200b\u5bf9\u200b\u540e\u9762\u200b\u7684\u200b\u72b6\u6001\u200b\u4ea7\u751f\u200b\u5f71\u54cd\u200b\u3002</p> <p>\\(M/M/s\\)\u200b\u7cfb\u7edf\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b Markov \u200b\u8fc7\u7a0b\u200b\u5efa\u6a21\u200b\u3002\u200b\u961f\u5217\u200b\u4e2d\u200b\u5b9e\u4f53\u200b\u7684\u200b\u6570\u91cf\u200b\u662f\u200b\u4e00\u4e2a\u200b Markov \u200b\u94fe\u200b\u3002</p>"},{"location":"math/queue-theory/chapter-12/#_8","title":"\u751f\u706d\u200b\u8fc7\u7a0b","text":"<p>\u200b\u751f\u706d\u200b\u8fc7\u7a0b\u200b\u6309\u200b\u5982\u4e0b\u200b\u56fe\u200b\u8fdb\u884c\u200b\u63cf\u8ff0\u200b\uff08\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\u56fe\u200b\uff09\uff1a</p> <p></p> <p>\u200b\u5047\u8bbe\u200b\u5b9e\u4f53\u200b\u79bb\u6563\u200b\u8fdb\u5165\u200b\u7684\u200b\\(M/M/s\\)\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u961f\u5217\u200b\u4e2d\u200b\u7684\u200b\u5b9e\u4f53\u200b\u6570\u91cf\u200b\u670d\u4ece\u200b\u751f\u706d\u200b\u8fc7\u7a0b\u200b</p>"},{"location":"math/queue-theory/chapter-12/#poisson","title":"Poisson \u200b\u8fc7\u7a0b","text":"<p>\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7684\u200b\u65f6\u95f4\u5dee\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\u4e14\u200b\u670d\u4ece\u200b\u540c\u4e00\u200b\u6307\u6570\u5206\u5e03\u200b\u7684\u200b\u8fc7\u7a0b\u200b</p>"},{"location":"math/queue-theory/chapter-12/#_9","title":"\u751f\u706d\u200b\u8fc7\u7a0b","text":"<p>\u200b\u5bf9\u4e8e\u200b\u751f\u706d\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u5047\u8bbe\u200b</p> <ul> <li>\u200b\u6bcf\u4e2a\u200b\u65f6\u523b\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b</li> <li>\u200b\u5b9a\u4e49\u200b\u7cfb\u7edf\u200b\u7684\u200b\u72b6\u6001\u200b\u4e3a\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u7684\u200b\u5b9e\u4f53\u200b\u6570\u91cf\u200b\\(n\\)</li> <li>\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u7684\u200b\u72b6\u6001\u200b\u7531\u200b\\(n-1\\)\u200b\u8f6c\u53d8\u200b\u4e3a\u200b\\(n\\)</li> <li>\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u7684\u200b\u72b6\u6001\u200b\u7531\u200b\\(n\\)\u200b\u8f6c\u53d8\u200b\u4e3a\u200b\\(n+1\\)</li> <li>\u200b\u7cfb\u7edf\u200b\u5904\u4e8e\u200b\u72b6\u6001\u200b\\(n\\)\u200b\u65f6\u200b\uff0c\u200b\u5b9a\u4e49\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7684\u200b\u901f\u5ea6\u200b\u4e3a\u200b\\(\\lambda _n\\)\uff0c\u200b\u7cfb\u7edf\u200b\u7684\u200b\u5904\u7406\u901f\u5ea6\u200b\u4e3a\u200b\\(\\mu_n\\)\u3002\u200b\u5bf9\u4e8e\u200b\\(\\lambda_n\\)\uff0c\\(n\\geq 0\\)\uff1b\u200b\u5bf9\u4e8e\u200b\\(\\mu_n\\)\uff0c\\(n&gt;0\\)</li> </ul> <p>\u200b\u95ee\u9898\u200b\uff1a\u200b\u7cfb\u7edf\u200b\u5904\u4e8e\u200b\u7a33\u5b9a\u200b\u72b6\u6001\u200b\u65f6\u200b\uff0c\u200b\u751f\u706d\u200b\u7cfb\u7edf\u200b\u5904\u4e8e\u200b\u72b6\u6001\u200b\\(n\\)\u200b\u7684\u200b\u6982\u7387\u200b\\(p_n\\)\uff1f</p> <p>\u200b\u5bf9\u4e8e\u200b\u5904\u4e8e\u200b\u72b6\u6001\u200b\\(j\\)\u200b\u7684\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u8003\u5bdf\u200b\u4e00\u4e2a\u200b\u6781\u77ed\u200b\u7684\u200b\u65f6\u95f4\u200b\u95f4\u9694\u200b\\(\\Delta t\\)\uff0c\u200b\u7531\u4e8e\u200b\\(\\Delta t\\)\u200b\u6781\u77ed\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5ffd\u7565\u200b\\(\\Delta t\\)\u200b\u65f6\u95f4\u200b\u5185\u200b\u4e24\u4e2a\u200b\u5b9e\u4f53\u200b\u540c\u65f6\u200b\u8fdb\u5165\u200b/\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b\u6216\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u3001\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002\\(\\Delta t\\)\u200b\u65f6\u95f4\u200b\u5185\u200b\u7cfb\u7edf\u200b\u53d1\u751f\u53d8\u5316\u200b\u7684\u200b\u6982\u7387\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\uff1a\\(P(n(t+\\Delta t)=j+1 | n(t) = j)=\\lambda_j\\Delta t\\)</li> <li>\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b\uff1a\\(P(n(t+\\Delta t)=j-1| n(t) = j) = \\mu_j\\Delta t\\)</li> <li>\u200b\u7cfb\u7edf\u200b\u5728\u200b\\(\\Delta t\\)\u200b\u65f6\u95f4\u200b\u5185\u200b\u6ca1\u6709\u200b\u53d1\u751f\u200b\u72b6\u6001\u200b\u6539\u53d8\u200b\uff1a\\(P(n(t+\\Delta t) = j | n(t)=j) = 1 - (\\lambda_j+\\mu_j)\\Delta t\\)</li> </ul> <p>\u200b\u6839\u636e\u200b\u5047\u8bbe\u200b\uff0c\u200b\u5904\u4e8e\u200b\u72b6\u6001\u200b\\(i\\)\u200b\u7684\u200b\u7cfb\u7edf\u200b\u5728\u200b\\(t+\\Delta t\\)\u200b\u65f6\u200b\u53ea\u200b\u53ef\u80fd\u200b\u5904\u4e8e\u200b\\(i-1, i, i+1\\)\u200b\u4e09\u79cd\u200b\u72b6\u6001\u200b\u4e2d\u200b\u7684\u200b\u4e00\u79cd\u200b\u3002\u200b\u4e5f\u200b\u5373\u200b\uff0c\u200b\u5904\u4e8e\u200b\u72b6\u6001\u200b\\(i\\)\u200b\u7684\u200b\u7cfb\u7edf\u200b\u5728\u200b\\(t-\\Delta t\\)\u200b\u65f6\u200b\u53ea\u200b\u53ef\u80fd\u200b\u5904\u4e8e\u200b\\(i-1, i, i+1\\)\u200b\u4e09\u79cd\u200b\u72b6\u6001\u200b\u4e2d\u200b\u7684\u200b\u4e00\u79cd\u200b\u3002\u200b\u7531\u6b64\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5217\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ \\begin{aligned}   p_0(t+\\Delta t) &amp;= (1-\\lambda_0\\Delta t)p_0(t) + \\mu_1\\Delta tp_1(t) \\\\   p_j(t+\\Delta t) &amp;= \\lambda_{j-1}\\Delta tp_{j-1}(t)+(1-\\mu_j\\Delta t-\\lambda_j\\Delta t)p_j(t)+\\mu_{j+1}\\Delta tp_{j+1}(t) \\end{aligned} \\] <p>\u200b\u6574\u7406\u200b\u540e\u53d6\u200b\u6781\u9650\u200b\\(\\Delta t\\rightarrow 0\\)\uff0c\u200b\u4e0a\u200b\u5f0f\u200b\u53d8\u4e3a\u200b\u5bfc\u6570\u200b\u5f62\u5f0f\u200b\uff1a</p> \\[ \\begin{aligned}   &amp;\\lim_{\\Delta t\\rightarrow 0} \\frac{p_j(t+\\Delta t)-p_j(t)}{\\Delta t}=\\lambda_{j-1}p_{j-1}(t)+\\mu_{j+1}p_{j+1}(t) - (\\mu_j+\\lambda_j)p_j(t) \\\\   \\Rightarrow &amp;\\frac{\\mathrm dp_j(t)}{\\mathrm dt}=\\lambda_{j-1}p_{j-1}(t)+\\mu_{j+1}p_{j+1}(t) - (\\mu_j+\\lambda_j)p_j(t) \\end{aligned} \\] <p>\u200b\u5f53\u200b\\(t\\rightarrow \\infty\\)\u200b\u65f6\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u8d8b\u4e8e\u200b\u7a33\u6001\u200b\uff0c\u200b\u5219\u200b\u6709\u200b\\(\\lim_{t\\rightarrow \\infty}p_j(t) = p_j\\)\uff0c\u200b\u6536\u655b\u200b\u7684\u200b\u5145\u5206\u200b\u5fc5\u8981\u6761\u4ef6\u200b\u4e3a\u200b\\(\\lim_{t\\rightarrow\\infty} \\frac{\\mathrm dp_j(t)}{\\mathrm dt}=0\\)\u3002\u200b\u5373\u200b\\(\\lambda_{j-1}p_{j-1}(t)+\\mu_{j+1}p_{j+1}(t) - (\\mu_j+\\lambda_j)p_j(t)=0\\)\u3002</p> <p>\u200b\u89e3\u200b\\(p_{j+1}\\)\uff0c\u200b\u89e3\u5f97\u200b\uff1a</p> \\[ p_{j+1} = \\left(\\frac{\\mu_j+\\lambda_j}{\\mu_{j+1}}\\right)p_j-\\frac{\\lambda_{j-1}}{\\mu_{j+1}}p_{j-1} \\] <p>\u200b\u5bf9\u4e8e\u200b\\(p_1\\)\uff0c\u200b\u6709\u200b\\(p_1=\\frac{\\lambda_0}{\\mu_1}p_0\\)\u3002</p> <p>\u200b\u6839\u636e\u200b\u9012\u63a8\u200b\u516c\u5f0f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b\\(p_0\\)\u200b\u8868\u793a\u200b\u51fa\u200b\\(p_j\\)\uff1a</p> \\[ \\begin{aligned}   p_2&amp;=\\left(\\frac{\\mu_1+\\lambda_1}{\\mu_2}\\right)p_1-\\frac{\\lambda _0}{\\mu_2}p_0 = \\frac{\\lambda_0(\\mu_1+\\lambda_1)}{\\mu_1\\mu_2}p_0-\\frac{\\lambda_0}{\\mu_2}p_0=\\frac{\\lambda_0\\lambda_1}{\\mu_1\\mu_2}p_0 \\\\   p_3&amp;=\\left(\\frac{\\mu_2+\\lambda_2}{\\mu_3}\\right)p_2-\\frac{\\lambda_1}{\\mu_3}p_1=\\frac{\\lambda_0\\lambda_1(\\mu_2+\\lambda_2)}{\\mu_1\\mu_2\\mu_3}p_0-\\frac{\\lambda_0\\lambda_1}{\\mu_1\\mu_3}p_0=\\frac{\\lambda_0\\lambda_1\\lambda_2}{\\mu_1\\mu_2\\mu_3}p_0 \\\\   &amp;\\vdots \\\\   &amp;p_j = \\frac{\\lambda_0\\lambda_1\\cdots\\lambda_{j-1}}{\\mu_1\\mu_2\\cdots\\mu_j}p_0=p_0\\prod_{i = 1}^{j}\\frac{\\lambda_{i-1}}{\\mu_i} \\end{aligned} \\] <p>\u200b\u7531\u4e8e\u200b\u7cfb\u7edf\u200b\u7684\u200b\u72b6\u6001\u200b\u53ea\u200b\u53ef\u80fd\u200b\u53d6\u200b\\(0\\sim\\infty\\)\u200b\u4e4b\u95f4\u200b\u7684\u200b\u6574\u6570\u200b\u503c\u200b\uff0c\u200b\u6982\u7387\u200b\u4e4b\u200b\u548c\u200b\u4e3a\u200b\\(1\\)\uff0c\u200b\u53ef\u4ee5\u200b\u8ba1\u7b97\u200b\u51fa\u200b\\(p_0\\)\uff1a</p> \\[ p_0=\\frac{1}{1+\\sum_{n=1}^\\infty \\prod_{i=0}^{n}\\frac{\\lambda_{i-1}}{\\mu_i}} \\]"},{"location":"math/queue-theory/chapter-12/#mm1","title":"M/M/1\u200b\u961f\u5217","text":"<p>\u200b\u5bf9\u4e8e\u200b\\(M/M/1\\)\u200b\u7cfb\u7edf\u200b\uff0c\\(\\lambda_0=\\lambda_1=\\cdots=\\lambda_n=\\lambda, \\mu_1=\\cdots=\\mu_n=\\mu\\)\u3002</p> <p>\u200b\u7531\u6b64\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u51fa\u200b\\(M/M/1\\)\u200b\u7cfb\u7edf\u200b\u5728\u200b\u7a33\u6001\u200b\u65f6\u200b\u5904\u4e8e\u200b\\(p_0\\)\u200b\u7684\u200b\u6982\u7387\u200b\uff1a</p> \\[ p_0=\\frac{1}{1+\\sum_{n=1}^\\infty \\prod_{i=0}^{n}\\frac{\\lambda}{\\mu}} = \\frac{1}{1+\\sum_{n=1}^\\infty \\rho^n}=\\frac{1}{\\sum_{n=0}^\\infty \\rho^n}=1-\\rho \\] <p>\u200b\u6982\u7387\u5206\u5e03\u200b\u4e3a\u200b\uff1a\\(P(N=n)=\\rho^n(1-\\rho)\\)</p>"},{"location":"math/queue-theory/chapter-12/#mms","title":"M/M/s\u200b\u961f\u5217","text":"<p>\u200b\u5bf9\u4e8e\u200b\\(M/M/s\\)\u200b\u7cfb\u7edf\u200b\uff0c\\(\\lambda_i, \\mu_i\\)\u200b\u670d\u4ece\u200b\u5982\u4e0b\u200b\u89c4\u5f8b\u200b\uff1a</p> <ul> <li>\\(\\lambda_i = \\lambda \\qquad i=0, 1\\cdots, \\infty\\)</li> <li>\\(\\mu_i = \\max\\{i, s\\}\\mu \\qquad i=1, 2, \\cdots, \\infty\\)</li> </ul> <p>\u200b\u8ba1\u7b97\u200b\u51fa\u200b\\(M/M/s\\)\u200b\u7cfb\u7edf\u200b\u5728\u200b\u7a33\u6001\u200b\u65f6\u200b\u5904\u4e8e\u200b\\(p_0\\)\u200b\u7684\u200b\u6982\u7387\u200b\uff0c\u200b\u4ee4\u200b\\(\\rho_0=\\frac{\\lambda}{\\mu}\\)\uff1a</p> \\[ \\begin{aligned}   p_0&amp;=\\frac{1}{1+\\sum_{n=1}^\\infty \\prod_{i=0}^{n}\\frac{\\lambda}{\\max\\{i, s\\}\\mu}} \\\\   &amp;= \\frac{1}{\\sum_{i=0}^{s-1}\\frac{\\rho_0^i}{i!}+\\sum_{i=s}^\\infty \\frac{\\rho_0^i}{s!s^{i-s}}} \\\\   &amp;=\\frac{1}{\\sum_{i=0}^{s-1}\\frac{\\rho_0^i}{i!}+\\frac{\\rho_0^s}{s!}\\sum_{i=0}^\\infty \\frac{\\rho_0^i}{s^i}} \\\\   &amp;=\\frac{1}{\\sum_{i=0}^{s-1}\\frac{\\rho_0^i}{i!}+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}} \\end{aligned} \\] <p>\u200b\u8bbe\u200b\\(\\sum_{i=0}^{s-1}\\frac{\\rho_0^i}{i!}=T\\)\uff0c\\(p_i\\)\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\\(p_0\\)\u200b\u63a8\u5bfc\u200b\u5f97\u200b\uff1a</p> \\[ p_i=\\frac{\\lambda_0\\lambda_1\\cdots\\lambda_{i-1}}{\\mu_1\\mu_2\\cdots\\mu_i}p_0=\\left\\{ \\begin{aligned} &amp; \\frac{\\rho_0^i}{i!}p_0=\\frac{\\rho_0^i}{i!\\left(T+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}\\right)} &amp; i\\leq s \\\\ &amp; \\frac{\\rho_0^i}{s!s^{i-s}}p_0=\\frac{\\rho_0^i}{s!s^{i-s}\\left(T+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}\\right)} &amp; i &gt; s \\end{aligned} \\right. \\] <p>\u200b\u5f53\u200b\\(n \\geq s\\)\u200b\u65f6\u200b\uff0c\u200b\u987e\u5ba2\u200b\u5230\u8fbe\u200b\u7cfb\u7edf\u200b\u65f6\u200b\u9700\u8981\u200b\u7b49\u5f85\u200b\uff0c\u200b\u5219\u200b\u6709\u200b\uff1a</p> \\[ p_D=\\sum_{i=s}^\\infty p_i= \\sum_{i=s}^\\infty \\frac{\\rho^i_0}{s!s^{i-s}}p_0 = \\frac{\\rho^s_0p_0}{(s-1)!(s-\\rho_0)}=\\frac{\\rho^s_0}{(s-1)!(s-\\rho_0)T+\\rho_0^s} \\] <p>\u200b\u5f53\u200b\\(N\\leq s\\)\u200b\u65f6\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u6ca1\u6709\u200b\u961f\u5217\u200b\uff0c\\(L_q=0\\)\uff0c\u200b\u5f53\u200b\\(N&gt;s\\)\u200b\u65f6\u200b\uff0c\u200b\u961f\u5217\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(N-s\\)\uff0c\u200b\u56e0\u6b64\u200b\u7a33\u6001\u200b\u65f6\u200b\u7684\u200b\u5e73\u5747\u200b\u6392\u961f\u200b\u957f\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned}   L_q&amp;=\\sum_{i=s + 1}^\\infty (i - s)p_i \\\\   &amp;=\\sum_{i=s}^\\infty (i-s)\\frac{\\rho^i_0}{s!s^{i-s}}p_0 \\\\   &amp;= \\frac{\\rho^s_0p_0}{s!}\\sum_{i=0}^\\infty \\frac{i\\rho_0^i}{s^i} \\\\   &amp;= \\frac{\\rho^s_0p_0}{s!}\\sum_{i=1}^\\infty \\left(\\frac{\\rho_0}{s}\\right)^i\\frac{s}{s-\\rho_0} \\\\   &amp;= \\frac{\\rho^s_0p_0}{(s-1)!(s-\\rho)}\\frac{\\rho_0/s}{1-\\rho_0/s} \\\\   &amp;= \\frac{\\rho^{s+1}_0p_0}{(s-1)!(s-\\rho_0)^2} \\end{aligned} \\] <p>\u200b\u72b6\u6001\u200b\u4e3a\u200b\\(n\\)\u200b\u7684\u200b\u7a33\u6001\u200b\u65f6\u200b\u7cfb\u7edf\u200b\u63a5\u53d7\u200b\u670d\u52a1\u200b\u7684\u200b\u5b9e\u4f53\u200b\u6570\u91cf\u200b\u4e3a\u200b\uff1a</p> \\[ \\left\\{ \\begin{aligned} &amp;n &amp; n\\leq s \\\\ &amp;s &amp; n&gt; s \\end{aligned} \\right . \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u7a33\u6001\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u63a5\u53d7\u200b\u670d\u52a1\u200b\u7684\u200b\u5b9e\u4f53\u200b\u5e73\u5747\u200b\u6570\u91cf\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned}   L-L_q&amp;=\\sum_{i=0}^s ip_i+\\sum_{i=s+1}^\\infty sp_i \\\\   &amp;=\\sum_{i=0}^s \\frac{\\rho_0^i}{(i-1)!}p_0+s\\sum_{i=s+1}^\\infty \\frac{\\rho_0^i}{s!s^{i-s}}p_0 \\\\   &amp;= \\sum_{i=0}^s \\frac{\\rho_0^i}{(i-1)!}p_0 + \\frac{\\rho_0^sp_0}{(s-1)!}\\sum_{i=1}^\\infty \\frac{\\rho_0^i}{s^i} \\\\   &amp;= \\sum_{i=0}^s \\frac{\\rho_0^i}{(i-1)!}p_0 + \\frac{\\rho_0^{s+1}p_0}{(s-1)!(s-\\rho_0)} \\\\   &amp;= \\frac{\\sum_{i=0}^s \\frac{\\rho_0^i}{(i-1)!}}{\\left(T+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}\\right)} + \\frac{\\rho_0^{s+1}p_0}{(s-1)!(s-\\rho_0)} \\\\   &amp;= \\frac{\\rho_0 T+1}{T+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}} + \\frac{\\rho_0^{s+1}}{(s-1)!(s-\\rho_0)T+\\rho_0^s} \\\\   &amp;= \\frac{(\\rho_0 T+1)(s-1)!(s-\\rho_0) + \\rho_0^{s+1}}{(s-1)!(s-\\rho_0)T+\\rho_0^s}\\\\   &amp;= \\rho_0 + \\frac{(s-1)!(s-\\rho_0)}{(s-1)!(s-\\rho_0)T+\\rho_0^s} \\end{aligned} \\] <p>\u200b\u987e\u5ba2\u200b\u7684\u200b\u5e73\u5747\u200b\u9017\u7559\u200b\u65f6\u95f4\u200b\\(W=\\frac{L}{\\lambda}\\)\uff1a</p> \\[ \\begin{aligned}   W&amp;=\\frac{L}{\\lambda} \\\\   &amp;= \\frac{\\sum_{i=0}^\\infty ip_i}{\\lambda} \\\\   &amp;= \\frac{\\sum_{i=1}^s \\left(\\frac{\\rho_0^i}{(i-1)!\\left(T+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}\\right)}\\right) + \\sum_{i=s+1}^\\infty \\left(i\\frac{\\rho_0^i}{s!s^{i-s}\\left(T+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}\\right)}\\right) }{\\lambda} \\\\   &amp;= \\frac{\\frac{T(s-1)!(s-\\rho_0)}{T(s-1)!(s-\\rho_0) + \\rho_0^s} + \\frac{\\rho_0^{s+1}(s-\\rho_0+1)/(s-\\rho_0)}{T(s-1)!(s-\\rho_0) + \\rho_0^s}}{\\lambda} \\\\   &amp;= \\frac{T(s-1)!(s-\\rho_0) + \\rho_0^{s+1}(s-\\rho_0+1)/(s-\\rho_0)}{\\lambda(T(s-1)!(s-\\rho_0) + \\rho_0^s)} \\end{aligned} \\]"},{"location":"math/random-process/","title":"\u968f\u673a\u200b\u8fc7\u7a0b","text":"<ul> <li>\u200b\u9884\u5907\u200b\u77e5\u8bc6\u200b</li> <li>\u200b\u6cca\u677e\u200b\u8fc7\u7a0b\u200b</li> <li>\u200b\u8bb0\u53f7\u200b</li> </ul>"},{"location":"math/random-process/basics/","title":"\u9884\u5907\u200b\u77e5\u8bc6","text":""},{"location":"math/random-process/basics/#_2","title":"\u53ef\u6d4b\u200b\u7a7a\u95f4\u200b\u3001\u200b\u6982\u7387\u200b\u7a7a\u95f4\u200b\u4e0e\u200b\u6982\u7387\u200b\u6d4b\u5ea6","text":"<p>\\(\\renewcommand{\\geq}{\\geqslant}\\renewcommand{\\leq}{\\leqslant}\\) \u200b\u5f53\u200b\u4e00\u4e2a\u200b\u8bd5\u9a8c\u200b\u7684\u200b\u7ed3\u679c\u200b\u65e0\u6cd5\u200b\u9884\u5148\u786e\u5b9a\u200b\u65f6\u200b\uff0c\u200b\u79f0\u8be5\u200b\u5b9e\u9a8c\u200b\u4e3a\u200b\u968f\u673a\u200b\u8bd5\u9a8c\u200b\u3002\u200b\u968f\u673a\u200b\u8bd5\u9a8c\u200b\u53ef\u80fd\u200b\u51fa\u73b0\u200b\u7684\u200b\u6240\u6709\u200b\u7ed3\u679c\u200b\u96c6\u5408\u200b\u6784\u6210\u200b\u6837\u672c\u7a7a\u95f4\u200b\\(\\Omega\\)\uff0c\u200b\u6bcf\u4e2a\u200b\u53ef\u80fd\u200b\u7684\u200b\u7ed3\u679c\u200b\u79f0\u4e3a\u200b\u6837\u672c\u200b\u70b9\u200b\\(\\omega\\)\u3002\\(\\Omega\\)\u200b\u7684\u200b\u5b50\u96c6\u200b\u6784\u6210\u200b\u7684\u200b\u96c6\u5408\u200b\u79f0\u4e3a\u200b\u96c6\u7c7b\u200b\\(\\mathcal F\\)\u3002</p> <p>\u200b\u53ef\u6d4b\u200b\u7a7a\u95f4\u200b</p> <p>\u200b\u8bbe\u200b\\(\\mathcal F\\)\u200b\u4e3a\u200b\u7531\u200b\\(\\Omega\\)\u200b\u7684\u200b\u67d0\u4e9b\u200b\u5b50\u96c6\u200b\u6784\u6210\u200b\u7684\u200b\u975e\u200b\u7a7a\u96c6\u200b\u7c7b\u200b\uff0c\u200b\u82e5\u200b\u6ee1\u8db3\u200b\u4ee5\u4e0b\u200b\u6761\u4ef6\u200b\uff1a</p> <ol> <li>\u200b\u82e5\u200b\\(A\\in \\mathcal F\\)\uff0c\u200b\u5219\u200b\\(A^C = \\Omega - A\\in \\mathcal F\\)\u3002</li> <li>\u200b\u82e5\u200b\\(A_n\\in \\mathcal F\\)\uff0c\u200b\u5219\u200b\\(\\bigcup\\limits_{n=1}^\\infty A_n \\in \\mathcal F\\)</li> </ol> <p>\u200b\u79f0\u200b\\(\\mathcal F\\)\u200b\u4e3a\u200b\\(\\sigma\\)\u200b\u57df\u200b\uff0c\\((\\Omega, \\mathcal F)\\)\u200b\u4e3a\u200b\u53ef\u6d4b\u200b\u7a7a\u95f4\u200b\u3002</p> <p>\\(\\sigma\\)\u200b\u57df\u200b\\(\\mathcal F\\)\u200b\u5bf9\u200b\\(\\cap, \\cup, -\\)\u200b\u8fd0\u7b97\u200b\u5c01\u95ed\u200b\uff0c\u200b\u4efb\u4f55\u200b\u5143\u7d20\u200b\u7ecf\u8fc7\u200b\u53ef\u5217\u6b21\u200b\u8fd0\u7b97\u200b\u540e\u200b\u4ecd\u200b\u5c5e\u4e8e\u200b\\(\\mathcal F\\)\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u96c6\u7c7b\u200b\\(\\mathcal A\\)\uff0c\u200b\u5305\u542b\u200b\\(\\mathcal A\\)\u200b\u7684\u200b\\(\\sigma\\)\u200b\u57df\u200b\u7684\u200b\u4ea4\u200b\u79f0\u4e3a\u200b\\(\\mathcal A\\)\u200b\u751f\u6210\u200b\u7684\u200b\\(\\sigma\\)\u200b\u57df\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(\\sigma(\\mathcal A)\\)\u3002\u200b\u5982\u200b\uff1a\\(\\sigma(\\{\\varnothing, A, \\Omega\\}) = \\{\\varnothing, A, A^C, \\Omega\\}\\)\u3002\u200b\u7279\u6b8a\u200b\u5730\u200b\uff0c\u200b\u8bb0\u200b\\(\\mathcal B = \\sigma(\\{(-\\infty, \\alpha], \\forall \\alpha \\in \\mathbb R\\})\\)\u200b\u4e3a\u200bBorel\u200b\u57df\u200b\u3002Borel\u200b\u57df\u200b\u89e3\u51b3\u200b\u4e86\u200b\u6837\u672c\u7a7a\u95f4\u200b\u5728\u200b\\(\\mathbb R\\)\u200b\u4e0a\u200b\u8fde\u7eed\u200b\u7684\u200b\u95ee\u9898\u200b\u3002\u200b\u53ef\u4ee5\u200b\u8bc1\u660e\u200b\uff0c\\(\\forall a &lt; b, [a, b]\\in \\mathcal B, (a, b]\\in \\mathcal B, [a, b)\\in \\mathcal B, (a, b)\\in \\mathcal B\\)\u3002\u200b\u5b9a\u4e49\u200b\\(\\mathcal B[a, b]\\)\u200b\u4e3a\u200b\u9650\u5236\u200b\u5728\u200b\\([a, b]\\)\u200b\u4e0a\u200b\u7684\u200bBorel\u200b\u57df\u200b\u3002</p> <p>\u200b\u6982\u7387\u200b\u6d4b\u5ea6\u200b\u4e0e\u200b\u6982\u7387\u200b\u7a7a\u95f4\u200b</p> <p>\u200b\u8bbe\u200b\\((\\Omega, \\mathcal F)\\)\u200b\u4e3a\u200b\u53ef\u6d4b\u200b\u7a7a\u95f4\u200b\uff0c\\(P: \\mathcal F\\rightarrow [0, 1]\\)\u200b\u4e3a\u200b\u5b9a\u4e49\u200b\u5728\u200b\\(\\mathcal F\\)\u200b\u4e0a\u200b\u7684\u200b\u96c6\u200b\u51fd\u6570\u200b\u3002\u200b\u4e14\u200b\\(P\\)\u200b\u6ee1\u8db3\u200b</p> <ol> <li>\u200b\u975e\u200b\u8d1f\u6027\u200b\uff1a\\(\\forall A\\in \\mathcal F, P(A)\\geq 0\\)</li> <li>\u200b\u89c4\u4e00\u6027\u200b\uff1a\\(P(\\Omega) = 1\\)</li> <li> <p>\u200b\u53ef\u5217\u200b\u53ef\u52a0\u6027\u200b\uff1a\u200b\u82e5\u200b\\(\\forall i\\in \\mathbb N, A_i\\in \\mathcal F\\)\uff0c\u200b\u4e14\u200b\\(\\forall i\\not = j, A_i\\cap A_j=\\varnothing\\)\uff0c\u200b\u5219\u200b</p> \\[ P\\left(\\bigcup_{n=0}^\\infty A_n\\right) = \\sum_{n=0}^\\infty P(A_n) \\] </li> </ol> <p>\u200b\u79f0\u200b\\(P\\)\u200b\u4e3a\u200b\u53ef\u6d4b\u200b\u7a7a\u95f4\u200b\\((\\Omega, \\mathcal F)\\)\u200b\u4e0a\u200b\u7684\u200b\u6982\u7387\u200b\u6d4b\u5ea6\u200b\uff0c\\((\\Omega, \\mathcal F, P)\\)\u200b\u4e3a\u200b\u6982\u7387\u200b\u7a7a\u95f4\u200b\u3002\\(\\mathcal F\\)\u200b\u4e3a\u200b\u4e8b\u4ef6\u200b\u57df\u200b\uff0c\\(A\\in \\mathcal F\\)\u200b\u4e3a\u200b\uff08\u200b\u968f\u673a\u200b\uff09\u200b\u4e8b\u4ef6\u200b</p> <p>\u200b\u6982\u7387\u200b\u6d4b\u5ea6\u200b\\(P\\)\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ol> <li>\u200b\u6709\u9650\u200b\u53ef\u52a0\u6027\u200b\uff1a\u200b\u53ef\u4ee5\u200b\u4ee4\u200b\\(A_{n+1} = A_{n+2} = \\cdots = \\varnothing\\)\uff0c\u200b\u7ed3\u5408\u200b\u53ef\u5217\u200b\u53ef\u52a0\u6027\u200b\u63a8\u51fa\u200b</li> <li>\\(P(A^C) = 1 - P(A)\\)<ul> <li>\\(P(\\varnothing) = 0\\)</li> </ul> </li> <li>\u200b\u96c6\u5408\u200b\u7684\u200b\u5305\u542b\u200b\u5173\u7cfb\u200b\uff1a\\(A\\subset B\\Rightarrow P(A)\\leq P(B)\\)</li> <li> <p>\u200b\u5bb9\u65a5\u200b\u539f\u7406\u200b\uff1a</p> \\[ P\\left(\\bigcup_{i=1}^n A_i\\right) = \\sum_{i=1}^n P(A_i) - \\sum_{1\\leq i &lt; j\\leq n}P(A_i\\cap A_j) + \\cdots + (-1)^{n+1}P(A_1\\cap \\cdots \\cap A_n) \\] <ul> <li>\\(P(A\\cup B) = P(A) + P(B) - P(A\\cap B)\\)</li> <li>\\(P(A - B) = P(A\\cup B) - P(B) = P(A) - P(A\\cap B)\\)</li> <li>\\(P\\left(\\bigcup_{i=1}^n A_i\\right)\\leq \\sum_{i=1}^n P(A_i)\\)</li> </ul> </li> </ol> <p>\u200b\u6ee1\u8db3\u200b\\(A_n\\subset A_{n+1}\\)\u200b\u7684\u200b\u4e8b\u4ef6\u200b\u5217\u200b\\(\\{A_n, n\\geq 1\\}\\)\u200b\u79f0\u4e3a\u200b\u5355\u8c03\u200b\u589e\u200b\u5e8f\u5217\u200b\uff0c\u200b\u6ee1\u8db3\u200b\\(A_n\\supset A_{n+1}\\)\u200b\u7684\u200b\u4e8b\u4ef6\u200b\u5217\u200b\\(\\{A_n, n\\geq 1\\}\\)\u200b\u79f0\u4e3a\u200b\u5355\u8c03\u200b\u51cf\u200b\u5e8f\u5217\u200b\u3002\u200b\u7531\u6b64\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u4e8b\u4ef6\u200b\u5217\u200b\u7684\u200b\u6781\u9650\u200b\uff1a</p> <ol> <li>\u200b\u82e5\u200b\\(\\{A_n, n\\geq 1\\}\\)\u200b\u4e3a\u200b\u5355\u8c03\u200b\u589e\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5219\u200b\\(\\lim\\limits_{n\\rightarrow \\infty}A_n = \\bigcup_{i=1}^\\infty A_i\\)</li> <li>\u200b\u82e5\u200b\\(\\{A_n, n\\geq 1\\}\\)\u200b\u4e3a\u200b\u5355\u8c03\u200b\u51cf\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5219\u200b\\(\\lim\\limits_{n\\rightarrow \\infty}A_n = \\bigcap_{i=1}^\\infty A_i\\)</li> </ol> <p>\u200b\u4e8b\u4ef6\u200b\u5217\u200b\u7684\u200b\u6781\u9650\u200b\u6ee1\u8db3\u200b</p> \\[ \\lim_{n\\rightarrow\\infty}P(A_n) = P\\left(\\lim_{n\\rightarrow\\infty} A_n\\right) \\] <p>\u200b\u53ef\u4ee5\u200b\u8bc1\u660e\u200b\uff0c\u200b\u5355\u70b9\u200b\u96c6\u200b\\(\\{a\\}\\)\u200b\u4e3a\u200b\u4e8b\u4ef6\u200b\u5217\u200b\\(\\{[a, a+1/n], n &gt; 0\\}\\)\u200b\u7684\u200b\u6781\u9650\u200b\u3002</p> <p>Borel-Cantelli\u200b\u5f15\u7406\u200b</p> <p>\u200b\u8bbe\u200b\\(\\{A_n, n\\geq 1\\}\\)\u200b\u4e3a\u200b\u4e8b\u4ef6\u200b\u5e8f\u5217\u200b\uff0c\u200b\u6ee1\u8db3\u200b\\(\\sum_{i=1}^\\infty P(A_i) &lt; \\infty\\)\uff0c\u200b\u5219\u200b</p> \\[ P\\left(\\lim_{i\\rightarrow\\infty}\\sup A_i\\right) \\triangleq P\\left(\\bigcap_{n=1}^\\infty \\bigcup_{i=n}^\\infty A_i\\right) = 0 \\] <p>\u200b\u5b9a\u4e49\u200b\u6761\u4ef6\u200b\u6982\u7387\u200b</p> \\[ P(A|B) = \\frac{P(A\\cap B)}{P(B)} \\] <p>\u200b\u82e5\u200b\u4e8b\u4ef6\u200b\\(A, B\\)\u200b\u6ee1\u8db3\u200b\\(P(A\\cap B) = P(A)P(B)\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\u4e8b\u4ef6\u200b\\(A, B\\)\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\uff0c\u200b\u4e14\u200b\\(P(A|B) = P(A)\\)\u3002\u200b\u540c\u7406\u200b\u53ef\u4ee5\u200b\u63a8\u5e7f\u200b\u81f3\u200b\\(n\\)\u200b\u4e2a\u200b\u4e8b\u4ef6\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002\u200b\u8bbe\u200b\\(A_1, \\cdots, A_n\\in \\mathcal F\\)\uff0c\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u5176\u4e2d\u200b\u4efb\u610f\u200b\\(k\\)\u200b\u4e2a\u200b\u4e8b\u4ef6\u200b\uff0c\u200b\u90fd\u200b\u6709\u200b</p> \\[ P(A_{i_1}\\cap\\cdots\\cap A_{i_k}) = P(A_{i_1})\\cdots P(A_{i_k}) \\] <p>\u200b\u79f0\u200b\\(A_1, \\cdots, A_n\\)\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\u3002</p> <p>\u200b\u82e5\u200b\\(\\{A_n, n\\geq 1\\}\\)\u200b\u4e3a\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\u7684\u200b\u4e8b\u4ef6\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4e14\u200b\\(\\sum_{n=1}^\\infty P(A_n) = \\infty\\)\uff0c\u200b\u5219\u200b</p> \\[ P\\left(\\lim_{i\\rightarrow\\infty}\\sup A_i\\right) \\triangleq P\\left(\\bigcap_{n=1}^\\infty \\bigcup_{i=n}^\\infty A_i\\right) = 1 \\]"},{"location":"math/random-process/basics/#_3","title":"\u968f\u673a\u53d8\u91cf\u200b\u4e0e\u200b\u5206\u5e03\u200b\u51fd\u6570","text":"<p>\u200b\u8bbe\u200b\u6837\u672c\u7a7a\u95f4\u200b\u4e3a\u200b\\(\\Omega\\)\u3002</p>"},{"location":"math/random-process/basics/#_4","title":"\u968f\u673a\u53d8\u91cf","text":"<p>\u200b\u8bbe\u200b\\((\\Omega, \\mathcal F, P)\\)\u200b\u4e3a\u200b\u6982\u7387\u200b\u7a7a\u95f4\u200b\uff0c\\(X(\\omega)\\)\u200b\u4e3a\u200b\u5b9a\u4e49\u200b\u5728\u200b\\(\\Omega\\)\u200b\u4e0a\u200b\u7684\u200b\u5355\u503c\u200b\u5b9e\u200b\u51fd\u6570\u200b\uff0c\u200b\u5373\u200b\\(X: \\Omega\\rightarrow \\mathbb R\\)\uff0c\u200b\u82e5\u200b\\(\\forall a\\in \\mathbb R\\)\uff0c\u200b\u6709\u200b\\(\\{\\omega:  X(\\omega) \\leq a\\}\\in \\mathcal F\\)\uff0c\u200b\u79f0\u200b\\(X(\\omega)\\)\u200b\u4e3a\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u7b80\u8bb0\u200b\u4e3a\u200b\\(X\\)\u3002\u200b\u5b9a\u4e49\u200b\\(F(x) = P(X\\leq x) = P(X\\in (-\\infty, X])\\)\u200b\u4e3a\u200b\\(X\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u51fd\u6570\u200b\u3002</p> <ul> <li>\u200b\u79bb\u6563\u200b\u578b\u200b\u968f\u673a\u53d8\u91cf\u200b\uff1a\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X\\)\u200b\u7684\u200b\u53ef\u80fd\u200b\u53d6\u503c\u200b\u7684\u200b\u5168\u4f53\u200b\u662f\u200b\u53ef\u5217\u96c6\u200b\u6216\u8005\u200b\u6709\u9650\u200b\u96c6\u200b\u3002</li> <li> <p>\u200b\u8fde\u7eed\u578b\u200b\u968f\u673a\u53d8\u91cf\u200b\uff1a\u200b\u82e5\u200b\\(\\forall B\\in \\mathcal F\\)\uff0c\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\\(f(x)\\)\u200b\u6ee1\u8db3\u200b</p> \\[ P(X\\in B) = \\int_B f(x)\\mathrm dx \\] <p>\u200b\u79f0\u200b\\(X\\)\u200b\u4e3a\u200b\u8fde\u7eed\u578b\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\\(f(x)\\)\u200b\u4e3a\u200b\\(X\\)\u200b\u7684\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\u3002</p> </li> </ul> <p>\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\u4e0e\u200b\u6982\u7387\u5206\u5e03\u200b\u51fd\u6570\u200b</p> <p>\u200b\u6839\u636e\u200b\u6982\u7387\u5206\u5e03\u200b\u51fd\u6570\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u6709\u200b</p> \\[ P(x &lt; X\\leq x + h) = F(x + h) - F(x) = \\int_{x}^{x + h} f(x)\\mathrm dx = f(x)h + o(h) \\] <p>\u200b\u4ee4\u200b\\(h\\rightarrow 0\\)\u200b\u5e76\u53d6\u200b\u6781\u9650\u200b\uff0c\u200b\u5f97\u5230\u200b</p> \\[ \\frac{\\mathrm dF(x)}{\\mathrm dx} = \\lim_{h\\rightarrow 0} \\frac{F(x + h) - F(x)}{h} = f(x) \\] <p>\u200b\u5bf9\u4e8e\u200b\u4e8c\u7ef4\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X, Y\\)\uff0c\u200b\u5b9a\u4e49\u200b\u8054\u5408\u200b\u5206\u5e03\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(F(x, y) = P(X\\leq x, Y\\leq y)\\)\uff0c\u200b\u8fb9\u7f18\u200b\u5206\u5e03\u200b\u4e3a\u200b\\(F_X(x) = P(X\\leq x), F_Y = P(Y\\leq y)\\)\u3002\u200b\u540c\u7406\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\\(f(x, y)\\)\uff1a</p> \\[ F(x, y) = \\int_{-\\infty}^x\\int_{-\\infty}^y f(v, u)\\mathrm du\\mathrm dv \\] <p>\u200b\u82e5\u200b\\(F(x, y) = F_X(x)F_Y(y)\\)\uff0c\u200b\u79f0\u200b\\(X\\)\u200b\u4e0e\u200b\\(Y\\)\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\u3002</p>"},{"location":"math/random-process/basics/#_5","title":"\u6570\u5b57\u200b\u7279\u5f81","text":"<p>\u200b\u8bbe\u200b\\(X\\)\u200b\u4e3a\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u5206\u5e03\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(F(x)\\)\uff0c\u200b\u82e5\u200b\\(\\int_{-\\infty}^\\infty |x|\\mathrm dF(x)\\)\u200b\u5b58\u5728\u200b\uff0c\u200b\u5219\u200b\u5b9a\u4e49\u200b\\(X\\)\u200b\u7684\u200b\u671f\u671b\u200b\\(E(X)\\)\u200b\u4e3a\u200b</p> \\[ E(X) = \\int_{-\\infty}^\\infty x\\mathrm dF(x) \\] <p>\u200b\u6570\u5b66\u200b\u671f\u671b\u200b\\(E(X)\\)\u200b\u6ee1\u8db3\u200b</p> <ul> <li> <p>\u200b\u53ef\u52a0\u6027\u200b</p> \\[ E\\left(\\sum_{i=1}^n c_iX_i\\right) = \\sum_{i=1}^n c_iE(X_i) \\] </li> <li> <p>\u200b\u51fd\u6570\u200b</p> \\[ E(g(x)) = \\int_{-\\infty}^\\infty g(x)\\mathrm dF(x) \\] </li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u79bb\u6563\u200b\u578b\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u6709\u200b</p> \\[ E(X) = \\sum_{i=1}^\\infty x_nP(X = x_n) \\] <p>\u200b\u5bf9\u4e8e\u200b\u8fde\u7eed\u578b\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u6709\u200b</p> \\[ E(X) = \\int_{-\\infty}^\\infty xf(x)\\mathrm dx \\] <p>\u200b\u82e5\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X, Y\\)\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\uff0c\u200b\u5219\u200b\u6709\u200b</p> \\[ \\begin{aligned}     E(XY) &amp;= \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty xyf(x, y)\\mathrm dx\\mathrm dy \\\\     &amp;= \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty xyf_X(x)f_Y(y)\\mathrm dx\\mathrm dy \\\\     &amp;= \\int_{-\\infty}^\\infty xf_X(x)\\mathrm dx\\int_{-\\infty}^\\infty yf_Y(y)\\mathrm dy \\\\     &amp;= E(X)E(Y) \\end{aligned} \\] <p>\u200b\u5b9a\u4e49\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X\\)\u200b\u7684\u200b\u65b9\u5dee\u200b\u4e3a\u200b\\(\\sigma_X^2 = D(X) = E(X - E(X)) = E(X^2) - E^2(X)\\)\uff0c\u200b\u5b9a\u4e49\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X, Y\\)\u200b\u7684\u200b\u534f\u65b9\u5dee\u200b\u4e3a\u200b\\(\\mathrm{cov}(X, Y) = E(XY) - E(X)E(Y)\\)\uff0c\u200b\u76f8\u5173\u7cfb\u6570\u200b\u4e3a\u200b\\(\\rho = \\mathrm{cov}(X, Y) / (\\sigma_X\\sigma_Y)\\)\uff0c\\(k\\)\u200b\u9636\u77e9\u200b\u4e3a\u200b\\(E(X^k)\\)</p> <p>\u200b\u5b9a\u4e49\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X\\)\u200b\u7684\u200b\u77e9\u6bcd\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(\\psi(t) = E\\left(e^{tX}\\right)\\)\uff0c\u200b\u7279\u5f81\u51fd\u6570\u200b\u4e3a\u200b\\(\\phi(t) = E\\left(e^{\\mathbf itX}\\right)\\)</p> <p>\u200b\u77e9\u6bcd\u200b\u51fd\u6570\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ol> <li>\\(E\\left(X^k\\right) = \\psi^{(k)}(0)\\)</li> <li>\u200b\u8bbe\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X, Y\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(F_X(t), F_Y(t)\\)\uff0c\u200b\u77e9\u6bcd\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(\\psi_X(t), \\psi_Y(t)\\)\uff0c\u200b\u5219\u200b\\(\\psi_X(t) = \\psi_Y(t) \\Leftrightarrow F_X(t) = F_Y(t)\\)</li> </ol> <p>\u200b\u7279\u5f81\u51fd\u6570\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ol> <li>\u200b\u8bbe\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X, Y\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(F_X(t), F_Y(t)\\)\uff0c\u200b\u7279\u5f81\u51fd\u6570\u200b\u4e3a\u200b\\(\\phi_X(t), \\phi_Y(t)\\)\uff0c\u200b\u5219\u200b\\(\\phi_X(t) = \\phi_Y(t) \\Leftrightarrow F_X(t) = F_Y(t)\\)</li> <li>\\(i^kE(X^k) = \\phi^{(k)}(0)\\)</li> <li>\u200b\u82e5\u200b\\(X, Y\\)\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\uff0c\u200b\u6709\u200b\\(\\phi_{X+Y}(t) = \\phi_{X}(t) + \\phi_{Y}(t)\\)</li> </ol>"},{"location":"math/random-process/basics/#_6","title":"\u5e38\u89c1\u200b\u968f\u673a\u53d8\u91cf\u200b\u7684\u200b\u5206\u5e03","text":"<ol> <li> <p>\u200b\u4e8c\u9879\u5206\u5e03\u200b\uff1a\\(X\\sim B(n, p)\\)</p> \\[ P(X = k) = \\binom{n}{k} p^k(1 - p)^{n - k} \\] </li> <li> <p>\u200b\u6cca\u677e\u200b\u5206\u5e03\u200b\uff1a\\(X\\sim P(\\lambda)\\)</p> \\[ P(X = k) = \\frac{\\lambda^k}{k!}e^{-\\lambda} \\] </li> <li> <p>\u200b\u51e0\u4f55\u200b\u5206\u5e03\u200b\uff1a\\(X\\sim G(p)\\)</p> \\[ P(X = k) = (1 - p)^{k - 1}p \\] </li> <li> <p>\u200b\u5747\u5300\u5206\u5e03\u200b\uff1a\\(X\\sim U(a, b)\\)</p> \\[ f(x) = \\left\\{\\begin{aligned}     &amp; \\frac{1}{b-a} &amp; a &lt; x &lt; b \\\\     &amp; 0 &amp; \\text{otherwise} \\end{aligned}\\right. \\] </li> <li> <p>\u200b\u6b63\u6001\u5206\u5e03\u200b\uff1a\\(X\\sim N(\\mu, \\sigma^2)\\)</p> \\[ f(x) = \\frac{1}{\\sqrt{2\\pi}\\sigma} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}} \\] </li> <li> <p>\u200b\u6307\u6570\u5206\u5e03\u200b\uff1a\\(X\\sim E(\\lambda)\\)</p> \\[ f(x) = \\left\\{\\begin{aligned}     &amp; \\lambda e^{-\\lambda x} &amp; x\\geq 0 \\\\     &amp; 0 &amp; x &lt; 0 \\end{aligned}\\right. \\] </li> </ol>"},{"location":"math/random-process/basics/#_7","title":"\u6761\u4ef6\u200b\u6570\u5b66\u200b\u671f\u671b","text":"<p>\u200b\u5bf9\u4e8e\u200b\u8fde\u7eed\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u6761\u4ef6\u200b\u6570\u5b66\u200b\u671f\u671b\u200b\\(E(X|Y = y)\\)\u200b\u7684\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ E(X|Y = y) = \\int_{-\\infty}^\\infty x\\frac{f(x, y)}{f_Y(y)}\\mathrm dx \\] <p>\u200b\u5b9a\u4e49\u200b\\(E(X|Y)\\)\u200b\u4e3a\u200b\\(X\\)\u200b\u5173\u4e8e\u200b\\(Y\\)\u200b\u7684\u200b\u6761\u4ef6\u200b\u6570\u5b66\u200b\u671f\u671b\u200b</p> \\[ E(X|Y) = \\sum_j \\boldsymbol 1_{\\{Y = y_j\\}}(\\omega)E(X|Y=y_j) \\] <p>\u200b\u5bf9\u4e8e\u200b\u79bb\u6563\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u6761\u4ef6\u200b\u6570\u5b66\u200b\u671f\u671b\u200b\\(E(X|Y = y)\\)\u200b\u7684\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ E(X|Y = y) = \\sum_i x_iP(X=x_i | Y=y) \\] <p>\u200b\u663e\u7136\u200b\uff0c\\(E(X|Y = y)\\)\u200b\u662f\u200b\\(y\\)\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5b9a\u4e49\u200b\u968f\u673a\u53d8\u91cf\u200b\\(E(X|Y)\\)\u200b\u4e3a\u200b\\(X\\)\u200b\u5173\u4e8e\u200b\\(Y\\)\u200b\u7684\u200b\u6761\u4ef6\u671f\u671b\u200b\uff0c\u200b\u82e5\u200b\\(E(X|Y)\\)\u200b\u6ee1\u8db3\u200b</p> <ol> <li>\\(E(X|Y)\\)\u200b\u662f\u200b\\(Y\\)\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5f53\u200b\\(Y = y\\)\u200b\u65f6\u200b\uff0c\\(E(X|Y) = E(X|Y = y)\\)</li> <li> <p>\\(\\forall D\\in \\mathcal B\\)\uff0c\u200b\u6709\u200b</p> \\[ E[E(X|Y)|Y\\in D] = E(X|Y\\in D) \\] </li> </ol> <p>\u200b\u6761\u4ef6\u200b\u6570\u5b66\u200b\u671f\u671b\u200b\u7684\u200b\u6027\u8d28\u200b</p> <p>\u200b\u6761\u4ef6\u200b\u6570\u5b66\u200b\u671f\u671b\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ol> <li>\\(E(E(X|Y)) = EX\\)</li> <li>\u200b\u82e5\u200b\\(X, Y\\)\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\uff0c\u200b\u5219\u200b\\(E(X|Y) = EX\\)</li> <li>\\(E[g(X)h(Y)|Y] = h(Y)E(g(X)|Y)\\quad \\text{a.s.}\\)</li> </ol>"},{"location":"math/random-process/basics/#_8","title":"\u968f\u673a\u200b\u8fc7\u7a0b","text":"<p>\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b\u662f\u200b\u4e00\u65cf\u200b\u65e0\u7a77\u200b\u591a\u4e2a\u200b\uff0c\u200b\u76f8\u4e92\u200b\u6709\u5173\u200b\u7684\u200b\u968f\u673a\u53d8\u91cf\u200b\u3002\u200b\u8bbe\u200b\\(X(t, \\omega)\\)\u200b\u4e3a\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u5176\u4e2d\u200b\\(t\\in T\\subset \\mathbb R\\)\u200b\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u79f0\u200b\\(X_T = \\{X(t, \\omega), t\\in T\\}\\)\u200b\u4e3a\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b\u3002</p> <ol> <li>\\(X(t, \\omega)\\)\u200b\u53ef\u4ee5\u200b\u7b80\u8bb0\u200b\u4e3a\u200b\\(X(t)\\)\uff1b</li> <li>\u200b\u5f53\u200b\\(T\\)\u200b\u4e3a\u200b\u53ef\u5217\u200b\u96c6\u65f6\u200b\uff0c\u200b\u79f0\u200b\\(X_T\\)\u200b\u4e3a\u200b\u968f\u673a\u200b\u5e8f\u5217\u200b\uff1b</li> <li>\\(X_T\\)\u200b\u7684\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u79f0\u4e3a\u200b\u72b6\u6001\u200b\u7a7a\u95f4\u200b\\(S\\)\uff1b</li> <li>\\(X(\\cdot, \\omega), \\omega\\in \\Omega\\)\u200b\u662f\u200b\u5173\u4e8e\u200b\\(t\\)\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u8f68\u9053\u200b\u3002</li> </ol>"},{"location":"math/random-process/notations/","title":"\u8bb0\u53f7","text":"<p>\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u5e38\u7528\u200b\u7684\u200b\u8bb0\u53f7\u200b\u8868\u793a\u200b</p>"},{"location":"math/random-process/notations/#_2","title":"\u9884\u5907\u200b\u77e5\u8bc6","text":"<p>\u200b\u53ef\u5217\u96c6\u200b</p> <p>\u200b\u82e5\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\\(A\\)\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u53ef\u4ee5\u200b\u4ee5\u200b\u5e8f\u5217\u200b\\(\\{a_n\\}\\)\u200b\u7684\u200b\u5f62\u5f0f\u200b\u8868\u793a\u200b\uff0c\u200b\u5373\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\\(\\mathbb N_+\\rightarrow A\\)\u200b\u7684\u200b\u53cc\u5c04\u200b\uff0c\u200b\u79f0\u200b\u96c6\u5408\u200b\\(A\\)\u200b\u4e3a\u200b\u53ef\u5217\u96c6\u200b\u3002</p> <p>\u200b\u51e0\u4e4e\u200b\u5904\u5904\u200b</p> <p>\u200b\u8bbe\u200b\\(Z_1, Z_2\\)\u200b\u4e3a\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u4e14\u200b\\(P(Z_1 = Z_2) = 1\\)\uff0c\u200b\u79f0\u200b\\(Z_1, Z_2\\)\u200b\u51e0\u4e4e\u200b\u5904\u5904\u200b\u76f8\u7b49\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(Z_1 = Z_2\\quad \\text{a.s.}\\)</p>"},{"location":"math/random-process/notations/#_3","title":"\u6cca\u677e\u200b\u8fc7\u7a0b","text":"<p>\u200b\u8ba1\u6570\u200b\u8fc7\u7a0b\u200b</p> <p>\u200b\u82e5\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b\\(\\{N(t), t\\geq 0\\}\\)\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\uff0c\u200b\u79f0\u200b\\(N(t)\\)\u200b\u4e3a\u200b\u8ba1\u6570\u200b\u8fc7\u7a0b\u200b\u3002</p> <ol> <li>\\(N(t)\\in \\mathbb N\\)\uff0c\u200b\u5373\u200b\\(N(t)\\)\u200b\u53d6\u503c\u200b\u4e3a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b</li> <li>\\(\\forall t &gt; s, N(t)\\geq N(s)\\)</li> </ol>"},{"location":"math/random-process/poisson/","title":"\u6cca\u677e\u200b\u8fc7\u7a0b","text":"<p>\\(\\renewcommand{\\geq}{\\geqslant}\\renewcommand{\\leq}{\\leqslant}\\) \u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b\\(\\{N(t), t\\geq 0\\}\\)\u200b\u4e3a\u200b\u65f6\u200b\u9f50\u6cca\u677e\u200b\u8fc7\u7a0b\u200b\u3002</p> <ol> <li>\\(\\{N(t), t\\geq 0\\}\\)\u200b\u4e3a\u200b\u8ba1\u6570\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u4e14\u200b\\(N(0) = 0\\)\uff1b</li> <li>\\(\\{N(t), t\\geq 0\\}\\)\u200b\u4e3a\u200b\u72ec\u7acb\u200b\u589e\u91cf\u200b\u8fc7\u7a0b\u200b\uff1b</li> <li>\\(\\{N(t), t\\geq 0\\}\\)\u200b\u5177\u5907\u200b\u589e\u91cf\u200b\u5e73\u7a33\u6027\u200b\uff0c\u200b\u5373\u200b\\(N(t) - N(s)\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u4ec5\u200b\u4e0e\u200b\\(t - s\\)\u200b\u6709\u5173\u200b</li> <li> <p>\u200b\u5bf9\u4e8e\u200b\u5145\u5206\u200b\u5c0f\u200b\u7684\u200b\\(\\Delta t &gt; 0\\)\uff0c\u200b\u6709\u200b</p> \\[ \\left\\{ \\begin{aligned}     P(N(t + \\Delta t) - N(t) = 1) &amp;= \\lambda \\Delta t + o(\\Delta t) \\\\     P(N(t + \\Delta t) - N(t) \\geq 2) &amp;= o(\\Delta t) \\end{aligned} \\right. \\] </li> </ol>"},{"location":"math/random-process/poisson/#_2","title":"\u589e\u91cf\u200b\u7684\u200b\u6982\u7387\u5206\u5e03","text":"<p>\u200b\u6cca\u677e\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u589e\u91cf\u200b\\(N(t + s) - N(s)\\)\u200b\u670d\u4ece\u200b\u53c2\u6570\u200b\u4e3a\u200b\\(\\lambda t\\)\u200b\u7684\u200b\u6cca\u677e\u200b\u5206\u5e03\u200b\uff0c\u200b\u5373\u200b</p> \\[ P[N(t + s) - N(s) = k] = \\frac{(\\lambda t)^k}{k!}e^{-\\lambda t} \\] \u200b\u8bc1\u660e\u200b <p>\u200b\u6839\u636e\u200b\u6cca\u677e\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u589e\u91cf\u200b\u5e73\u7a33\u6027\u200b\uff0c\u200b\u6709\u200b</p> \\[ P(N(s + t) - N(s) = n) = P(N(t) = n) \\] <ol> <li> <p>\\(n = 0\\)\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(\\{N(h + t) = 0, h &gt; 0\\} = \\{N(h+ t) - N(t) = 0, N(t) = 0, h &gt; 0\\}\\)\uff0c\u200b\u6839\u636e\u200b\u589e\u91cf\u200b\u72ec\u7acb\u6027\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\begin{aligned}     P(N(h + t) = 0) &amp;= P(N(h + t) - N(t) = 0) P(N(t) = 0) \\\\     &amp;= P(N(h) = 0) P(N(t) = 0) \\end{aligned} \\] <p>\u200b\u6839\u636e\u200b\u6cca\u677e\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u6027\u8d28\u200b\\(P(N(t + \\Delta t) - N(t) = 1) = \\lambda \\Delta t + o(\\Delta t)\\)\uff0c\u200b\u6709\u200b\uff1a</p> \\[ \\begin{aligned}     \\lim_{h\\rightarrow 0} \\frac{P(t + h) - P(t)}{h} &amp;= \\lim_{h\\rightarrow 0} \\frac{P(t)(1 - \\lambda h - o(h)) - P(t)}{h} \\\\     &amp;= -\\lambda P(t) - \\lim_{h\\rightarrow 0} \\frac{o(h)}{h} \\\\     &amp;= -\\lambda P(t) \\end{aligned} \\] <p>\u200b\u89e3\u200b\u5fae\u5206\u65b9\u7a0b\u200b\\(P_0'(t) = -\\lambda P_0(t)\\)\uff0c\u200b\u89e3\u5f97\u200b\\(P_0(t) = e^{-\\lambda t + C}\\)\uff0c\u200b\u53c8\u200b\u56e0\u4e3a\u200b\\(N(0) = 0, P_0(N(0) = 0) = 1\\)\uff0c\u200b\u5f97\u200b\\(C = 0\\)\uff0c\u200b\u4ece\u800c\u200b\u6709\u200b</p> \\[ P_0(t) = e^{-\\lambda t} \\] </li> <li> <p>\\(n &gt; 0\\)\u200b\u65f6\u200b\uff0c\\(\\{N(t + h) = n\\} = \\bigcup_{k = 0} ^ n \\{N(t) = k, N(t + h) - N(t) = (n - k)\\}\\)\uff0c\u200b\u800c\u200b\\(n - k \\geq 2\\)\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(P(N(t + h) - N(t) = n - k) = o(h)\\)\uff0c\u200b\u7531\u200b\u589e\u91cf\u200b\u72ec\u7acb\u6027\u200b</p> \\[ \\begin{aligned}     P(N(t + h) = n) &amp;= \\sum_{k = 0} ^ n P(N(t) = k) P(N(t + h) - N(t) = n - k) \\\\     &amp;= P(N(t) = n)(1 - \\lambda t - o(h)) + P(N(t) = n - 1)(\\lambda t + o(h)) \\\\     &amp;+ \\sum_{k = 2}^n P(N(t) = n - k)o(h) \\end{aligned} \\] <p>\u200b\u7531\u6b64\u200b</p> \\[ \\begin{aligned}     &amp;\\lim_{h\\rightarrow 0} \\frac{P(N(t + h) = n) - P(N(t) = n)}{h} \\\\     =&amp; \\lim_{h\\rightarrow 0}\\frac{P(N(t) = n)(\\sum_{k = 2}^n P(N(t) = n - k)o(h) - P(N(t) = n)}{h} \\\\     +&amp; \\lim_{h\\rightarrow 0}\\frac{\\sum_{k = 2}^n P(N(t) = n - k)o(h) - P(N(t) = n)}{h} \\\\     =&amp; \\lim_{h\\rightarrow 0}\\frac{P(N(t) = n)(- \\lambda t) + P(N(t) = n - 1)(\\lambda t)}{h} \\\\ \\end{aligned} \\] <p>\u200b\u5f97\u5230\u200b</p> \\[ P_n'(t) = -\\lambda P_n(t) + \\lambda P_{n-1}(t) \\] <p>\u200b\u8003\u8651\u200b\u7ea6\u675f\u6761\u4ef6\u200b\\(P_n(0) = 0\\)\uff0c\u200b\u89e3\u5f97\u200b</p> \\[ \\frac{\\mathrm d}{\\mathrm dt}[e^{\\lambda t} P_n(t)] = \\lambda e^{\\lambda t}P_{n - 1}(t) \\] <ol> <li>\\(n = 0\\)\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(P_n(t) = e^{-\\lambda t} = \\frac{(\\lambda t)^0}{0!}e^{-\\lambda t}\\)</li> <li> <p>\u200b\u8bbe\u200b\u5bf9\u4e8e\u200b\\(n &lt; k\\)\u200b\u65f6\u200b\u547d\u9898\u200b\u6210\u7acb\u200b\uff0c\u200b\u53ea\u200b\u9700\u200b\u8bc1\u660e\u200b\\(n = k\\)\u200b\u65f6\u200b\u547d\u9898\u200b\u6210\u7acb\u200b\u5373\u53ef\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\begin{aligned}     P_k(t) &amp;= e^{-\\lambda t}\\int \\frac{\\mathrm d}{\\mathrm dt}[e^{\\lambda t} P_k(t)] \\mathrm dt \\\\     &amp;= e^{-\\lambda t}\\int \\lambda e^{\\lambda t}P_{k - 1}(t) \\mathrm dt \\\\     &amp;= e^{-\\lambda t}\\int \\lambda \\frac{(\\lambda t)^{k-1}}{(k-1)!} \\mathrm dt \\\\     &amp;= \\frac{(\\lambda t)^k}{k!}e^{-\\lambda t} \\end{aligned} \\] </li> </ol> </li> </ol>"},{"location":"math/random-process/poisson/#_3","title":"\u76f8\u90bb\u200b\u4e8b\u4ef6\u200b\u65f6\u95f4\u200b\u95f4\u9694\u200b\u7684\u200b\u6982\u7387\u5206\u5e03","text":"<p>\u200b\u8bbe\u200b\\(\\{N(t), t \\geq 0\\}\\)\u200b\u4e3a\u200b\u8ba1\u6570\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u8bbe\u200b\\(S_n\\)\u200b\u8868\u793a\u200b\u7b2c\u200b\\(n\\)\u200b\u4e2a\u200b\u65f6\u95f4\u200b\u53d1\u751f\u200b\u7684\u200b\u65f6\u523b\u200b\uff0c\u200b\u5373\u200b\\(S_n = \\inf\\{t, t &gt; S_{n - 1}, N(t) = n\\}\\)\uff0c\u200b\u7279\u522b\u200b\u5730\u200b\uff0c\u200b\u4ee4\u200b\\(S_0 = 0\\)\u3002\\(S_n, N(t)\\)\u200b\u7684\u200b\u4e8b\u4ef6\u200b\u6ee1\u8db3\u200b</p> \\[ \\begin{aligned}     \\{S_n \\leq t\\} &amp;\\equiv \\{N(t) \\geq n\\} \\\\     \\{N(t) = n\\} &amp;\\equiv \\{S_n\\leq t &lt; S_{n + 1}\\} \\equiv \\{S_n\\leq t\\} - \\{S_{n+1}\\leq t\\} \\end{aligned} \\] <p>\u200b\u7531\u4e8e\u200b\\(\\{S_n \\leq t\\} \\equiv \\{N(t) \\geq n\\}\\)\uff0c\u200b\u53ef\u4ee5\u200b\u8ba1\u7b97\u200b\\(S_n\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u51fd\u6570\u200b</p> \\[ P(S_n\\leq t) = P(N(t)\\geq n) = 1 - e^{-\\lambda t}\\sum_{k=0}^{n-1} \\frac{(\\lambda t)^{k}}{k!} \\] <p>\u200b\u7279\u522b\u200b\u5730\u200b\uff0c\u200b\u5f53\u200b\\(n = 1\\)\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(P(X_1\\leq t) = P(S_1\\leq t) = (1 - e^{-\\lambda t}) I_{\\{t\\geq 0\\}}\\)</p> <p>\u200b\u8ba1\u6570\u200b\u8fc7\u7a0b\u200b\\(\\{N(t), t\\geq 0\\}\\)\u200b\u662f\u200b\u6cca\u677e\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u5145\u8981\u6761\u4ef6\u200b\u662f\u200b\\(\\{X_n = S_{n} - S_{n-1}, n\\geq 1\\}\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u670d\u4ece\u200b\u7684\u200b\u5206\u5e03\u200b\u4e3a\u200b\u53c2\u6570\u200b\\(\\lambda\\)\u200b\u7684\u200b\u6307\u6570\u5206\u5e03\u200b\u3002</p> \u200b\u8bc1\u660e\u200b <ol> <li> <p>\\(\\Leftarrow\\)\uff1a\u200b\u5fc5\u8981\u6027\u200b\u8bc1\u660e\u200b\uff0c\u200b\u5148\u6c42\u200b\\(S_1, \\cdots, S_n\\)\u200b\u7684\u200b\u8054\u5408\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\uff0c\u200b\u8bbe\u200b\\(t_1 &lt; t_2 &lt; \\cdots &lt; t_n\\)\uff0c\u200b\u8bbe\u200b\\(h\\)\u200b\u4e3a\u200b\u5145\u5206\u200b\u5c0f\u200b\u7684\u200b\u6b63\u6570\u200b\uff0c\u200b\u6ee1\u8db3\u200b</p> \\[ t_1 + \\frac h2 &lt; t_2 - \\frac h2 &lt; t_2 + \\frac h2 &lt; \\cdots &lt; t_n - \\frac h2 \\] <p>\u200b\u8003\u8651\u200b\u4e8b\u4ef6\u200b\\(A\\)</p> \\[ \\begin{aligned}     A&amp; \\\\     &amp;\\equiv \\left\\{t_1 - \\frac h2 &lt; S_1 \\leq t_1 + \\frac h2, t_2 - \\frac h2 &lt; S_2 \\leq t_2 + \\frac h2, \\cdots, t_n - \\frac h2 &lt; S_n \\leq t_n + \\frac h2\\right\\} \\\\     \\equiv &amp; \\left\\{N\\left(t_1 - \\frac h2\\right) = 0, N\\left(t_{i+1} - \\frac h2\\right) - N\\left(t_i + \\frac h2\\right) = 0, N\\left(t_{i} + \\frac h2\\right) - N\\left(t_{i} - \\frac h2\\right) = 1\\right\\} \\\\     &amp; 0\\leq i\\leq n - 1 \\end{aligned} \\] <p>\u200b\u800c\u200b\u6839\u636e\u200b\u6cca\u677e\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u6027\u8d28\u200b</p> \\[ \\begin{aligned}     P\\left(N\\left(t_n + \\frac h2\\right) - N\\left(t_n - \\frac h2\\right)= 1\\right) &amp;= \\lambda h + o(h) \\\\     P\\left(N\\left(t_n + \\frac h2\\right) - N\\left(t_n - \\frac h2\\right)\\geq 2\\right) &amp;= o(h) \\\\ \\end{aligned} \\] <p>\u200b\u6709\u200b</p> \\[ P(A) = (\\lambda h)^ne^{-\\lambda (t_n + h/2)} + o\\left(h^n\\right) = \\lambda ^nh^ne^{-\\lambda t_n} + o\\left(h^n\\right) \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\\(\\left(S_1, \\cdots, S_n\\right)\\)\u200b\u7684\u200b\u6982\u7387\u5bc6\u5ea6\u200b\u4e3a\u200b</p> \\[ f(t_1, \\cdots, t_n) = \\left\\{ \\begin{aligned}     &amp; \\lambda^n e^{-\\lambda t_n} &amp; 0 &lt; t_1 &lt; \\cdots &lt; t_n \\\\     &amp; 0 &amp; \\mathrm{otherwise} \\end{aligned} \\right. \\] <p>\u200b\u800c\u200b\\(X_n = S_n - S_{n-1}\\)\uff0c\u200b\u8bbe\u200b\\(x_n = t_n - t_{n - 1}\\)\uff0c\u200b\u5219\u200b\\(t_n = \\sum_{i=1}^n x_n\\)\uff0c\u200b\u4ece\u800c\u200b\u53d8\u6362\u200b\u7684\u200bJacob\u200b\u77e9\u9635\u200b\u4e3a\u200b</p> \\[ J = \\frac{\\partial (t_1, \\cdots, t_n)}{\\partial (x_1, \\cdots, x_n)} = \\begin{vmatrix}     1 &amp; 0 &amp; \\cdots &amp; 0 \\\\     1 &amp; 1 &amp; \\cdots &amp; 0 \\\\     \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\     1 &amp; 1 &amp; \\cdots &amp; 1 \\end{vmatrix} \\] <p>\u200b\u56e0\u6b64\u200b\\(X_1, \\cdots, X_n\\)\u200b\u7684\u200b\u6982\u7387\u5bc6\u5ea6\u200b\u4e3a\u200b</p> \\[ g(x_1, \\cdots, x_n) = \\lambda^n e^{-\\lambda (x_1 + \\cdots + x_n)} = \\prod_{i=1}^n \\left(\\lambda e^{-\\lambda x_i}\\right), x_i \\geq 0 \\] <p>\u200b\u56e0\u6b64\u200b\\(X_1, \\cdots, X_n\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u6307\u6570\u5206\u5e03\u200b\u3002</p> </li> <li> <p>\\(\\Rightarrow\\)\uff1a\u200b\u5145\u5206\u6027\u200b\u8bc1\u660e\u200b\uff0c\u200b\u8bbe\u200b\\(\\{X_k, k\\geq 1\\}\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u6307\u6570\u5206\u5e03\u200b\uff0c\u200b\u8bbe\u200b\\(S_0 = 0, S_n = S_0 + \\sum_{i = 1}^n X_i\\)\u3002\u200b\u5b9a\u4e49\u200b</p> \\[ N(t) = \\sum_{n=1}^\\infty I_{\\{S_n\\leq t\\}} \\] <ul> <li>\u200b\u7531\u4e8e\u200b\\(I_{\\{S_n\\leq t\\}}\\in \\mathbb N\\)\uff0c\u200b\u5219\u200b\\(N(t)\\in \\mathbb N\\)\u3002</li> <li> <p>\u200b\u5bf9\u4e8e\u200b\\(t &gt; s\\)\uff0c\u200b\u6709\u200b</p> \\[ N(t) - N(s) = \\sum_{n = 1}^\\infty I_{\\{S_n\\leq t\\}}- I_{\\{S_n\\leq s\\}} \\] <p>\u200b\u800c\u200b\\(S_n - S_{n-1} = X_n &gt; 0\\)\uff0c\u200b\u56e0\u6b64\u200b\\(S_n\\)\u200b\u4e3a\u200b\u5355\u8c03\u200b\u589e\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4ece\u800c\u200b\\(\\{S_n\\leq s\\}\\subset \\{S_n\\leq t\\}\\)\uff0c\u200b\u5373\u200b\\(I_{\\{S_n\\leq t\\}}- I_{\\{S_n\\leq s\\}} \\geq 0\\)\uff0c\u200b\u56e0\u6b64\u200b\\(N(t) - N(s)\\)</p> </li> </ul> <p>\u200b\u53ef\u200b\u5f97\u200b\\(N(t)\\)\u200b\u4e3a\u200b\u8ba1\u6570\u200b\u8fc7\u7a0b\u200b\u3002\u200b\u4ee5\u4e0b\u200b\u8ba1\u7b97\u200b\\(S_n\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u3002\u200b\u5df2\u77e5\u200b\\(X_1, \\cdots, X_n\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u6307\u6570\u5206\u5e03\u200b\uff0c\u200b\u8bbe\u200b\\(t_n = \\sum_{i=1}^n x_i\\)\uff0c\u200b\u5219\u200b\\(x_n = t_n - t_{n-1}\\)\uff0c\u200b\u4ece\u800c\u200bJacob\u200b\u77e9\u9635\u200b\u4e3a\u200b</p> \\[ J = \\frac{\\partial (x_1, \\cdots, x_n)}{\\partial (t_1, \\cdots, t_n)} = \\begin{vmatrix}     1 &amp; 0 &amp; 0 &amp; \\cdots &amp; 0 \\\\     -1 &amp; 1 &amp; 0 &amp; \\cdots &amp; 0 \\\\     0 &amp; -1 &amp; 1 &amp; \\cdots &amp; 0 \\\\     \\vdots &amp; \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\     0 &amp; 0 &amp; 0 &amp; \\cdots &amp; 1 \\end{vmatrix} \\] <p>\u200b\u800c\u200b\\(X_1, \\cdots, X_n\\)\u200b\u7684\u200b\u8054\u5408\u200b\u6982\u7387\u5bc6\u5ea6\u200b\u4e3a\u200b</p> \\[ g(x_1, \\cdots, x_n) = \\lambda^n e^{-\\lambda (x_1 + \\cdots + x_n)} = \\prod_{i=1}^n \\left(\\lambda e^{-\\lambda x_i}\\right), x_i \\geq 0 \\] <p>\u200b\u56e0\u6b64\u200b\\(S_1, \\cdots, S_n\\)\u200b\u7684\u200b\u8054\u5408\u200b\u6982\u7387\u5bc6\u5ea6\u200b\u4e3a\u200b</p> \\[ f(t_1, \\cdots, t_n) = \\left\\{ \\begin{aligned}     &amp; \\lambda^n e^{-\\lambda t_n} &amp; 0 &lt; t_1 &lt; \\cdots &lt; t_n \\\\     &amp; 0 &amp; \\mathrm{otherwise} \\end{aligned} \\right. \\] </li> </ol>"},{"location":"notes/2020/","title":"\u4e00\u4e9b\u200b\u5f88\u200b\u559c\u6b22\u200b\u7684\u200b\u63a8\u9001","text":"<p>\u300a\u200b\u6628\u665a\u200b\u300b by \u200b\u8001\u9f20\u200b\u4ec0\u4e48\u200b\u90fd\u200b\u77e5\u9053\u200b</p> <p>\u300a\u200b\u6740\u6b7b\u200b\u65e0\u804a\u200b\u7684\u200b\u540c\u65f6\u200b\uff0c\u200b\u4f60\u200b\u4e5f\u200b\u5728\u200b\u6740\u6b7b\u200b\u81ea\u5df1\u200b\u300b by \u200b\u4eba\u7269\u200b</p> <p>\u300a\u200b\u75db\u82e6\u200b\u7684\u200b\u4e00\u767e\u5f20\u200b\u5e95\u7247\u200b\u300b by \u200b\u4eba\u7269\u200b</p> <p>\u300a\u200b\u603b\u200b\u4e0d\u80fd\u200b\u4e27\u6c14\u200b\u592a\u4e45\u200b\u5427\u200b\u3002\u300b by \u200b\u6982\u7387\u8bba\u200b</p> <p>\u300a\u200b\u6211\u4eec\u200b\u628a\u200b\u8fd9\u573a\u200b\u521d\u96ea\u200b\u505a\u6210\u200b\u4e86\u200b\u7535\u5f71\u6d77\u62a5\u200b\u300b by \u200b\u6e05\u534e\u5927\u5b66\u200b\u6e05\u65b0\u200b\u65f6\u62a5\u200b</p> <p>\u300a\u200b\u56f0\u4e8e\u200b\u4eba\u8138\u8bc6\u522b\u200b\u300b by \u200b\u4eba\u7269\u200b</p> <p>\u300a\u200b\u65b0\u200b\u7684\u200b\u5341\u5e74\u200b\uff0c\u200b\u4eba\u751f\u200b\u7684\u200b\u5947\u5e7b\u200b\u6f02\u6d41\u200b\u300b by \u200b\u4eba\u7269\u200b</p> <p>\u300a\u200b\u6587\u827a\u200b\u662f\u200b\u4e00\u573a\u200b\u9a97\u5c40\u200b\u5417\u200b\uff1f\u300b by \u200b\u4eba\u7269\u200b</p> <p>\u300a\u200b\u4e00\u7897\u200b\u7c73\u200b\u8c46\u8150\u200b\uff0c\u200b\u7a7f\u8d8a\u200b\u5c71\u6d77\u200b\u300b by \u200b\u4eba\u7269\u200b</p> <p>\u300a\u200b\u4fdd\u7814\u200b\u5931\u8d25\u200b\u4ee5\u540e\u200b\uff0c\u200b\u6211\u200b\u53c8\u200b\u8003\u200b\u4e86\u200b\u6b21\u200b\u6e05\u534e\u200b\u300b by \u200b\u6e05\u534e\u200b\u7814\u8bfb\u200b\u95f4\u200b</p> <p>\u300a\u200b\u4eca\u5929\u200b\u662f\u200b2020\u200b\u5e74\u200b\u4f60\u200b\u5728\u200b\u6e05\u534e\u200b\u7206\u809d\u200b\u7684\u200b\u7b2c\u200b366\u200b\u5929\u200b\u300b by \u200b\u6e05\u534e\u5927\u5b66\u200b\u6e05\u65b0\u200b\u65f6\u62a5\u200b</p>"},{"location":"notes/2021/","title":"\u4e00\u4e9b\u200b\u5f88\u200b\u559c\u6b22\u200b\u7684\u200b\u63a8\u9001","text":"<p>\u300a2020 \u200b\u6700\u200b\u820d\u4e0d\u5f97\u200b\u5220\u200b\u7684\u200b\u7167\u7247\u200b\u300b by \u200b\u6e05\u534e\u5927\u5b66\u200b</p> <p>\u200b\u665a\u5b89\u200b\u6d77\u62a5\u200b by \u200b\u90d1\u5dde\u200b\u53d1\u5e03\u200b</p> <p>\u200b\u77e5\u4e4e\u200b\u5b87\u5b99\u200b\u91cc\u200b\u7684\u200b\u90a3\u4e9b\u200b\u8d22\u5bcc\u200b\u5bc6\u7801\u200b by \u200b\u4e3a\u200b\u4f60\u200b\u5199\u200b\u4e00\u4e2a\u200b\u6545\u4e8b\u200b</p> <p>\u200b\u540c\u5b66\u200b\uff0c\u200b\u8fd8\u200b\u8bb0\u5f97\u200b\u5f53\u5e74\u200b\u90a3\u9053\u200b\u9ad8\u8003\u9898\u200b\u5417\u200b\uff1f by \u200b\u6e05\u534e\u5927\u5b66\u200b</p> <p>\u200b\u7ea2\u200b\u4e86\u200b\u7ea2\u200b\u4e86\u200b\uff0c\u200b\u8fd9\u6b21\u200b\u771f\u7684\u200b\u7ea2\u200b\u4e86\u200b by GQ\u200b\u5b9e\u9a8c\u5ba4\u200b</p> <p>\u200b\u751f\u6d3b\u200b\u5c0f\u7a8d\u95e8\u200b \u200b\u4e4b\u200b \u200b\u5982\u4f55\u200b\u6577\u884d\u200b\u5973\u751f\u8282\u200b by \u200b\u55e8\u200b\u76ae\u200b\u8fd0\u8425\u90e8\u200b</p> <p>\u200b\u6c99\u5c18\u66b4\u200b\u5bf9\u200b\u6e05\u534e\u56ed\u200b\u8bf4\u200b\uff0c\u200b\u4eca\u5929\u200b\u7ed9\u200b\u4f60\u200b\u70b9\u200b\u989c\u8272\u200b\u770b\u770b\u200b by \u200b\u5bf9\u200b\u725b\u200b\u8c08\u6e05\u200b</p>"},{"location":"notes/2022/","title":"\u4e00\u4e9b\u200b\u5f88\u200b\u559c\u6b22\u200b\u7684\u200b\u63a8\u9001","text":"<p>\u200b\u665a\u5b89\u200b\u6d77\u62a5\u200b by \u200b\u90d1\u5dde\u200b\u53d1\u5e03\u200b</p> <p>\u200b\u6e05\u534e\u200b\u5bf9\u200b\u4f60\u200b\u8bf4\u200b\u665a\u5b89\u200b\uff01 by \u200b\u6e05\u534e\u5927\u5b66\u200b</p> <p>\u200b\u5c0f\u533a\u200b\u88ab\u200b\u5c01\u200b\u4e94\u5929\u200b\u540e\u200b\uff0c\u200b\u6211\u200b\u6e10\u6e10\u200b\u61c2\u200b\u4e86\u200b by \u200b\u96f7\u53d4\u200b\u5199\u200b\u6545\u4e8b\u200b</p> <p>\u201c\u200b\u6211\u4eec\u200b\u7ec8\u200b\u4f1a\u200b\u627e\u5230\u200b\u610f\u4e49\u200b\u3002\u201d by \u200b\u6e05\u534e\u5927\u5b66\u200b</p> <p>40\u200b\u4e2a\u200b\u975e\u5fc5\u8981\u200b\u751f\u6d3b\u200b\u4e0b\u200b\u7684\u200b\u5e78\u798f\u200b\u77ac\u95f4\u200b by \u200b\u5168\u5143\u200b\u5149\u6ed1\u200b</p> <p>\u200b\u7ba1\u200b\u4ed6\u200b\u51e0\u5c81\u200b\uff0c\u200b\u9752\u6625\u200b\u4e07\u5c81\u200b by \u200b\u5c0f\u6797\u200b</p> <p>\u300c\u200b\u65f6\u4ee3\u200b\u4e4b\u97f3\u200b\u300d\u200b\u53bb\u200b\u54ea\u513f\u200b\u4e86\u200b\uff1f by \u200b\u4eba\u7269\u200b</p> <p>\u200b\u5e94\u200b \u200b\u6da6\u200b \u200b\u5c3d\u200b \u200b\u6da6\u200b by \u200b\u5168\u5143\u200b\u5149\u6ed1\u200b</p> <p>\u200b\u6bd5\u4e1a\u200b\u7279\u520a\u200b | \u200b\u634c\u200b\u6765\u200b\u65e0\u6059\u200b\uff0c\u200b\u540e\u4f1a\u6709\u671f\u200b by \u200b\u6e05\u534e\u5927\u5b66\u200b\u6e05\u65b0\u200b\u65f6\u62a5\u200b</p> <p>\u200b\u6bd5\u4e1a\u200b\u65f6\u200b\uff0c\u200b\u4f60\u200b\u6700\u200b\u60f3\u200b\u5e26\u8d70\u200b\u7684\u200b\u4e00\u4ef6\u200b\u4e1c\u897f\u200b\u662f\u200b\uff1f by \u200b\u6e05\u534e\u5927\u5b66\u200b</p> <p>\u200b\u6bd5\u4e1a\u200b\u5feb\u4e50\u200b | \u200b\u7247\u523b\u200b\u6210\u4e3a\u200b\u6c38\u6052\u200b by \u200b\u957f\u6b4c\u200b\u695a\u5929\u200b\u78a7\u200b</p> <p>\u200b\u4fdd\u7814\u200b\u5931\u8d25\u200b\u4e86\u200b\uff0c\u200b\u82df\u4f4f\u200b\u5c31\u200b\u597d\u200b by\u200b\u4e28\u200bJust Lose It\u200b\u4e28\u200b</p> <p>10\u200b\u6708\u200b\uff0c\u200b\u5728\u200b\u51ac\u5929\u200b\u5230\u6765\u200b\u4e4b\u524d\u200b\uff0c\u200b\u7559\u4e0b\u200b\u70b9\u200b\u4ec0\u4e48\u200b by \u200b\u4eba\u7269\u200b</p>"}]}
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+{"config":{"lang":["en"],"separator":"[\\s\\u200b\\-]","pipeline":["stemmer"]},"docs":[{"location":"","title":"Welcome to HuangFuSL's blog","text":""},{"location":"#table-of-contents","title":"Table of contents","text":"<ul> <li>\u200b\u63a8\u9001\u200b</li> <li>LaTeX</li> <li>\u200b\u6570\u5b66\u200b</li> <li>Coding</li> <li>\u200b\u65e5\u8bed\u200b</li> <li>\u200b\u5173\u4e8e\u200b\u6211\u200b</li> </ul>"},{"location":"#blog-timeline","title":"Blog timeline","text":"<p>Jupyter</p> 2024-12-16<p>Rust\u200b\u7684\u200bevcxr_jupyter\u200b\u5305\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5bf9\u200bjupyter notebook\u200b\u7684\u200b\u652f\u6301\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5728\u200bnotebook\u200b\u4e2d\u200b\u8fd0\u884c\u200bRust\u200b\u4ee3\u7801\u200b\u3002\u200b\u4f7f\u7528\u200b\u5982\u4e0b\u200b\u547d\u4ee4\u200b\u5b89\u88c5\u200b\uff1a</p> <p>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b</p> 2024-12-16<p>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\uff08Gaussian Process\uff0cGP\uff09\u200b\u662f\u200b\u5c06\u200b\u591a\u200b\u53d8\u91cf\u200b\u9ad8\u65af\u5206\u5e03\u200b\u63a8\u5e7f\u200b\u5230\u200b\u65e0\u9650\u200b\u7ef4\u5ea6\u200b\\(\\mathcal X\\subseteq \\mathbb R^n\\)\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b\u3002\u200b\u5177\u4f53\u200b\u5730\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\\(X = \\{x_1, \\ldots, x_n\\}\\subseteq \\mathcal X\\)\uff0c\u200b\u968f\u673a\u53d8\u91cf\u200b\\(f(x_1), \\ldots, f(x_n)\\)\u200b\u670d\u4ece\u200b\u591a\u5143\u200b\u9ad8\u65af\u5206\u5e03\u200b\\(\\mathcal N(\\mu(X), \\Sigma(X))\\)\u3002\u200b\u5176\u4e2d\u200b\uff0c\\(\\mu(X)\\)\u200b\u4e3a\u200b\u5747\u503c\u200b\u51fd\u6570\u200b\uff0c\\(\\Sigma(X)\\)\u200b\u4e3a\u200b\u534f\u65b9\u5dee\u200b\u51fd\u6570\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u9ad8\u65af\u5206\u5e03\u200b\u7814\u7a76\u200b\u7684\u200b\u662f\u200b\u51fd\u6570\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b\u3002</p> <p>\u200b\u7b97\u6cd5\u200b</p> 2024-12-09<p>\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u662f\u200b\u4e00\u7c7b\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u4f18\u5316\u200b\u9ed1\u76d2\u200b\u51fd\u6570\u200b\u7684\u200b\u4f18\u5316\u200b\u65b9\u6cd5\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u9ed1\u76d2\u200b\u51fd\u6570\u200b\u6765\u8bf4\u200b\uff0c\u200b\u6211\u4eec\u200b\u65e0\u6cd5\u200b\u5f97\u77e5\u200b\u5176\u200b\u663e\u5f0f\u200b\u7684\u200b\u8868\u8fbe\u5f0f\u200b\uff0c\u200b\u53ea\u80fd\u200b\u901a\u8fc7\u200b\u8f93\u5165\u200b\u4e00\u4e2a\u200b\u6570\u636e\u200b\uff0c\u200b\u89c2\u6d4b\u200b\u5176\u200b\u8f93\u51fa\u200b\u7684\u200b\u65b9\u6cd5\u200b\u5bfb\u627e\u200b\u5b83\u200b\u7684\u200b\u6700\u5927\u503c\u200b\u3002\u200b\u501f\u52a9\u4e8e\u200b\u8d1d\u53f6\u65af\u200b\u7edf\u8ba1\u200b\u7684\u200b\u601d\u60f3\u200b\uff0c\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u6700\u521d\u200b\u8ba4\u4e3a\u200b\u51fd\u6570\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5b8c\u5168\u200b\u968f\u673a\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4e0d\u65ad\u200b\u5730\u200b\u89c2\u6d4b\u200b\u6765\u200b\u66f4\u65b0\u200b\u5bf9\u200b\u51fd\u6570\u200b\u7684\u200b\u4f30\u8ba1\u200b\uff08\u200b\u540e\u9a8c\u200b\uff09\u3002\u200b\u5728\u200b\u6bcf\u200b\u4e00\u6b65\u200b\u8fed\u4ee3\u200b\u4e2d\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u4f1a\u200b\u6839\u636e\u200b\u5f53\u524d\u200b\u7684\u200b\u540e\u9a8c\u200b\uff0c\u200b\u901a\u8fc7\u200b\u91c7\u96c6\u200b\u51fd\u6570\u200b\u6765\u200b\u9009\u62e9\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u89c2\u6d4b\u200b\u7684\u200b\u70b9\u200b\u3002\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u7684\u200b\u4f18\u70b9\u200b\u662f\u200b\u53ef\u4ee5\u200b\u5728\u200b\u8f83\u200b\u5c11\u200b\u7684\u200b\u8fed\u4ee3\u200b\u6b21\u6570\u200b\u5185\u200b\u627e\u5230\u200b\u5168\u5c40\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5728\u200b\u5b9e\u9645\u200b\u5e94\u7528\u200b\u4e2d\u200b\u88ab\u200b\u5e7f\u6cdb\u200b\u4f7f\u7528\u200b\u3002</p> <p>\u200b\u5f3a\u5316\u200b\u5b66\u4e60\u200b</p> 2024-12-04<p>\u200b\u8bb0\u5f55\u200b\u5f3a\u5316\u200b\u5b66\u4e60\u200b\u7684\u200b\u5b66\u4e60\u200b\u8fc7\u7a0b\u200b\u3002</p> <p>\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b</p> 2024-12-04<p>\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\uff08Multi-armed bandit\uff09\u200b\u95ee\u9898\u200b\u6765\u81ea\u200b\u4e8e\u200b\u73b0\u5b9e\u200b\u4e2d\u200b\u7684\u200b\u8001\u864e\u673a\u200b\u95ee\u9898\u200b\u3002\u200b\u6bcf\u53f0\u200b\u8001\u864e\u673a\u200b\u90fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u62c9\u6746\u200b\uff0c\u200b\u62c9\u52a8\u200b\u62c9\u6746\u200b\u540e\u200b\uff0c\u200b\u8001\u864e\u200b\u673a\u4f1a\u200b\u968f\u673a\u200b\u7ed9\u51fa\u200b\u4e00\u4e2a\u200b\u5956\u52b1\u200b\u3002\u200b\u73a9\u5bb6\u200b\u7684\u200b\u4efb\u52a1\u200b\u662f\u200b\u5728\u200b\u6709\u9650\u200b\u7684\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u5185\u200b\u5c3d\u53ef\u80fd\u200b\u83b7\u5f97\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u5956\u52b1\u200b\u3002</p> <p>\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u7b97\u6cd5\u200b</p> 2024-12-04<p>\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\uff08Upper Confidence Bound, UCB\uff09\u200b\u7b97\u6cd5\u200b\u7684\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u5c06\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u7eb3\u5165\u200b\u8003\u8651\u200b\uff0c\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u8f83\u5c11\u200b\u7684\u200b\u52a8\u4f5c\u200b\u6709\u200b\u66f4\u200b\u5927\u200b\u7684\u200b\u4e0d\u786e\u5b9a\u6027\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u4f1a\u200b\u5c3d\u53ef\u80fd\u200b\u4e50\u89c2\u200b\u5730\u200b\u8003\u8651\u200b\u8fd9\u4e9b\u200b\u6ca1\u6709\u200b\u5c1d\u8bd5\u200b\u8fc7\u200b\u7684\u200b\u52a8\u4f5c\u200b\uff0c\u200b\u5e76\u200b\u4f18\u5148\u200b\u63a2\u7d22\u200b\u8fd9\u4e9b\u200b\u52a8\u4f5c\u200b\u3002\u200b\u5728\u200b\u6bcf\u4e2a\u200b\u52a8\u4f5c\u200b\u63a2\u7d22\u200b\u6bd4\u8f83\u200b\u6df1\u5165\u200b\u540e\u200b\uff0c\u200b\u518d\u200b\u8f6c\u5411\u200b\u5229\u7528\u200b\u5956\u52b1\u200b\u8f83\u200b\u9ad8\u200b\u7684\u200b\u52a8\u4f5c\u200b\u3002</p> <p>\u200b\u6a21\u578b\u200b\u63a8\u7406\u200b</p> 2024-11-25<p>\u200b\u5728\u200b\u63a8\u7406\u200b\u9636\u6bb5\u200b\uff0c\u200b\u5927\u200b\u6a21\u578b\u200b\u6839\u636e\u200b\u8f93\u5165\u200b\u7684\u200b\u6587\u672c\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4e0d\u65ad\u200b\u9884\u6d4b\u200b\u8f93\u5165\u200b\u6587\u672c\u200b\u7684\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u8bcd\u200b\uff0c\u200b\u76f4\u5230\u200b\u9884\u6d4b\u51fa\u200b\u7ed3\u675f\u200b\u7b26\u53f7\u200b\u4e3a\u6b62\u200b\u3002\u200b\u4e0d\u8fc7\u200b\uff0c\u200b\u5728\u200b\u9884\u200b\u8bad\u7ec3\u200b\u9636\u6bb5\u200b\u4e2d\u200b\uff0c\u200b\u4e0d\u200b\u5305\u542b\u200b\u5bf9\u200bbos\uff08begin of sentence\uff09\u200b\u548c\u200beos\uff08end of sentence\uff09\u200b\u7684\u200b\u9884\u6d4b\u200b\u3002\u200b\u6240\u4ee5\u200b\uff0c\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b\u57fa\u5ea7\u200b\u6a21\u578b\u200b\uff0c\u200b\u6a21\u578b\u200b\u4f1a\u200b\u4e0d\u65ad\u200b\u8f93\u51fa\u200btoken\u3002</p> <p>\u200b\u5b9e\u73b0\u200bLlama-2</p> 2024-09-07<p>\u200b\u5728\u200b\u672c\u8282\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u5b9e\u73b0\u200b\u4e00\u4e2a\u200bLlama-2\u200b\u6a21\u578b\u200b\u3002</p> <p>\u200b\u5f52\u4e00\u5316\u200b</p> 2024-09-07<p>\u200b\u5f52\u4e00\u5316\u200b\u662f\u200b\u6307\u200b\u5c06\u200b\u6570\u636e\u200b\u6309\u7167\u200b\u6bd4\u4f8b\u200b\u7f29\u653e\u200b\uff0c\u200b\u6539\u53d8\u200b\u5176\u200b\u5206\u5e03\u200b\u3002\u200b\u6309\u7167\u200b\u5f52\u4e00\u5316\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u4e0d\u540c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u4ee5\u4e0b\u200b\u51e0\u79cd\u200b\uff1a</p> <p>\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b</p> 2024-09-07<p>Attention\u200b\u673a\u5236\u200b\u867d\u7136\u200b\u80fd\u200b\u6355\u6349\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u4e0d\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u4f9d\u8d56\u200b\u5173\u7cfb\u200b\uff0c\u200b\u4f46\u662f\u200b\u65e0\u6cd5\u200b\u533a\u5206\u200b\u4e0d\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\u200b\u4e3a\u4e86\u200b\u89e3\u51b3\u200b\u8fd9\u4e2a\u200b\u95ee\u9898\u200b\uff0cTransformer\u200b\u6a21\u578b\u200b\u5f15\u5165\u200b\u4e86\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff08Positional Encoding\uff09\u3002</p>"},{"location":"#recent-updates","title":"Recent updates","text":"<ul> <li>2024-12-16\uff1aJupyter\uff0c\u200b\u673a\u5668\u200b\u5b66\u4e60\u200b\uff0c\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b</li> <li>2024-12-09\uff1a\u200b\u7b97\u6cd5\u200b</li> <li>2024-12-04\uff1a\u200b\u6df1\u5ea6\u200b\u5b66\u4e60\u200b\uff0c\u200b\u6570\u636e\u7ed3\u6784\u200b\uff0cPython\uff0c\u200b\u5f3a\u5316\u200b\u5b66\u4e60\u200b\uff0c\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\uff0c\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u7b97\u6cd5\u200b\uff0cRust</li> <li>2024-11-25\uff1a\u200b\u5b9e\u73b0\u200bLlama-2\uff0c\u200b\u6a21\u578b\u200b\u63a8\u7406\u200b</li> <li>2024-09-07\uff1aCustom transformer\uff0c\u200b\u5f52\u4e00\u5316\u200b\uff0c\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff0ctransformer\u200b\u53d8\u79cd\u200b\uff0c\u200b\u5b9e\u73b0\u200btransformer</li> </ul>"},{"location":"#currently-working-on","title":"Currently working on","text":"<ul> <li>Vim</li> </ul>"},{"location":"#customization","title":"Customization","text":"<p>Click on the buttons to change the primary color.</p> Red Pink Purple Deep Purple Indigo Blue Light Blue Cyan Teal Green Light Green Lime Yellow Amber Orange Deep Orange Brown Grey Blue Grey White <p>Click on the buttons to change the accent color.</p> Red Pink Purple Deep Purple Indigo Blue Light Blue Cyan Teal Green Light Green Lime Yellow Amber Orange Deep Orange <p>However, if you try to switch from dark mode to light mode or reversed, changes to the primary color and accent color will lose.</p>"},{"location":"#building-documentation","title":"Building documentation","text":"<p>Run <code>git clone https://github.com/HuangFuSL/HuangFuSL.github.io.git</code> to get the source code.</p>"},{"location":"#bootstrap-icon-installation","title":"Bootstrap icon installation","text":"<p>The site uses bootstrap icons, which are added as submodules in <code>third_party/icons</code>. You have to manually initialize the submodule.</p> <pre><code>git submodule update --recursive --remote\n</code></pre>"},{"location":"#latex-support","title":"LaTeX support","text":"<p>The site uses <code>xelatex</code> and <code>dvisvgm</code> to render tex document to SVG images embedded in the markdown files. However, as the SVG images are ignored by <code>.gitignore</code>, you have to manually perform the conversion.</p> <p>For GitHub repository clones:</p> <ul> <li>Run <code>git submodule update --recursive --remote</code> to receive the template.</li> <li>Make sure you have installed and correctly configured <code>xelatex</code> and <code>dvisvgm</code>.</li> <li>Add <code>./template</code> directory to <code>$TEXINPUTS</code> environmental variable.</li> <li>Execute <code>ci/convert.py</code> in the root directory of the repository.</li> <li>Run <code>mkdocs serve</code> to view the images.</li> </ul> <p>The template is located at HuangFuSL/latex-template</p>"},{"location":"#github-workflow","title":"GitHub workflow","text":"<p>You need to install the dependencies stored in <code>requirements.txt</code> before you can  start building the site:</p> <pre><code>pip install -r requirements.txt\n</code></pre> <p>There are cross-links in the site which require metadata defined in the page, so the project should be built before <code>mkdocs serve</code> is executed. The exported metadata is saved in <code>meta.json</code> after a build is successfully executed. To build the site, execute the following command:</p> <pre><code>mkdocs build -d build\n</code></pre> <p>Execute <code>mkdocs serve</code>, the built site will appear at http://127.0.0.1:8000</p>"},{"location":"#acknowledgements","title":"Acknowledgements","text":"<p>The blog relies on the following open-source projects:</p> <ul> <li>mkdocs</li> <li>Python Markdown</li> <li>Python Markdown Extension</li> <li>jupyter</li> </ul> <p>The blog uses the following mkdocs plugins to function correctly.</p> <ul> <li>Neoteroi/mkdocs-plugins</li> <li>lukasgeiter/mkdocs-awesome-pages-plugin</li> <li>timvink/mkdocs-git-revision-date-localized-plugin</li> <li>zhaoterryy/mkdocs-git-revision-date-plugin</li> <li>squidfunk/mkdocs-material</li> <li>facelessuser/mkdocs-material-extensions</li> <li>fralau/mkdocs_macros_plugin</li> <li>danielfrg/mkdocs-jupyter</li> <li>prcr/mkdocs-meta-descriptions-plugin</li> </ul> <p>Unless noted, content in this blog are shared under CC-BY-NC-SA 4.0 license.</p>"},{"location":"#version-information","title":"Version information","text":"<pre><code>commit 3634d38c7e95f633ed80e816b6dbfa4e3edce5ac\nMerge: 4b7d0e34 98aca170\nAuthor: github-actions[bot] &lt;41898282+github-actions[bot]@users.noreply.github.com&gt;\nDate:   Mon Dec 16 20:57:31 2024 +0800\n\n    Merge pull request #422 from HuangFuSL/rust-jupyter\n\n    Update: Rust Jupyter\n</code></pre>"},{"location":"about/","title":"\u5173\u4e8e\u200b\u6211","text":"<p>Guess who is here?</p> <p></p>"},{"location":"coding/","title":"Coding","text":"<p>\u200b\u7f16\u7a0b\u200b\u76f8\u5173\u200b</p>"},{"location":"coding/#_1","title":"\u7f16\u7a0b\u8bed\u8a00","text":"<ul> <li>Python</li> <li>Rust<ul> <li>GUI</li> <li>Jupyter</li> </ul> </li> </ul>"},{"location":"coding/#_2","title":"\u7b97\u6cd5","text":"<ul> <li>Leetcode</li> <li>\u200b\u6570\u636e\u7ed3\u6784\u200b</li> </ul>"},{"location":"coding/#_3","title":"\u673a\u5668\u200b\u5b66\u4e60","text":"<ul> <li>\u200b\u673a\u5668\u200b\u5b66\u4e60\u200b</li> <li>\u200b\u6df1\u5ea6\u200b\u5b66\u4e60\u200b</li> <li>\u200b\u5f3a\u5316\u200b\u5b66\u4e60\u200b</li> </ul>"},{"location":"coding/#_4","title":"\u5176\u4ed6","text":"<ul> <li>\u200b\u7cfb\u7edf\u914d\u7f6e\u200b</li> <li>SVG</li> </ul>"},{"location":"coding/rust-jupyter/","title":"Jupyter for Rust","text":"In\u00a0[2]: Copied! <pre>fn add(a: i32, b: i32) -&gt; i32 {\n    a + b\n}\n\nadd(1, 2) // The last expression is evaluated and printed\n</pre> fn add(a: i32, b: i32) -&gt; i32 {     a + b }  add(1, 2) // The last expression is evaluated and printed Out[2]: <pre>3</pre> <p>\u200b\u4f7f\u7528\u200b<code>:dep &lt;package&gt;</code>\u200b\u6765\u200b\u5bfc\u5165\u200b\u5305\u200b\u3002<code>evcxr_input</code>\u200b\u5305\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>get_string</code>\u200b\u548c\u200b<code>get_password</code>\u200b\u51fd\u6570\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5728\u200bnotebook\u200b\u4e2d\u200b\u8f93\u5165\u200b\u5b57\u7b26\u4e32\u200b\u548c\u200b\u5bc6\u7801\u200b\u3002</p> In\u00a0[3]: Copied! <pre>:dep evcxr_input\nlet name = evcxr_input::get_string(\"Name?\");\nlet password = evcxr_input::get_password(\"Password?\");\n\nprintln!(\"Hello, {name}! Your password is {password}\", name=name, password=password);\n</pre> :dep evcxr_input let name = evcxr_input::get_string(\"Name?\"); let password = evcxr_input::get_password(\"Password?\");  println!(\"Hello, {name}! Your password is {password}\", name=name, password=password); <pre>Hello, Alice! Your password is alice123\n</pre> <p>\u200b\u4f7f\u7528\u200b\u5982\u4e0b\u200b\u547d\u4ee4\u200b\u5378\u8f7d\u200bRust</p> <pre>evcxr_jupyter --uninstall\ncargo uninstall evcxr_jupyter\n</pre>"},{"location":"coding/rust-jupyter/#jupyter-for-rust","title":"Jupyter for Rust\u00b6","text":"<p>Rust\u200b\u7684\u200b<code>evcxr_jupyter</code>\u200b\u5305\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5bf9\u200bjupyter notebook\u200b\u7684\u200b\u652f\u6301\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5728\u200bnotebook\u200b\u4e2d\u200b\u8fd0\u884c\u200bRust\u200b\u4ee3\u7801\u200b\u3002\u200b\u4f7f\u7528\u200b\u5982\u4e0b\u200b\u547d\u4ee4\u200b\u5b89\u88c5\u200b\uff1a</p> <pre>cargo install --locked evcxr_jupyter\nevcxr_jupyter --install\n</pre> <p>\u200b\u53e6\u5916\u200b\uff0c\u200b\u8fd8\u200b\u9700\u8981\u200b\u5b89\u88c5\u200bRust\u200b\u6807\u51c6\u200b\u5e93\u200b\u7684\u200b\u6e90\u4ee3\u7801\u200b\uff1a</p> <pre>rustup component add rust-src\n</pre> <p>\u200b\u4e4b\u540e\u200b\uff0c\u200b\u5c31\u200b\u53ef\u4ee5\u200b\u5728\u200bnotebook\u200b\u4e2d\u200b\u8fd0\u884c\u200bRust\u200b\u4ee3\u7801\u200b\u4e86\u200b\u3002</p> <pre>jupyter notebook\n</pre>"},{"location":"coding/svg/","title":"SVG\u200b\u52a8\u753b","text":"<p>SVG\u200b\u7684\u200b\u52a8\u753b\u200b\u4f7f\u7528\u200b<code>&lt;animate&gt;</code>\u200b\u6807\u7b7e\u200b\uff0c\u200b\u5728\u200b\u4f7f\u7528\u200b\u65f6\u200b\uff0c<code>&lt;animate&gt;</code>\u200b\u6807\u7b7e\u200b\u9700\u8981\u200b\u653e\u5728\u200b\u52a8\u753b\u200b\u64cd\u4f5c\u200b\u5bf9\u8c61\u200b\u7684\u200b\u5185\u90e8\u200b\u3002</p> <p>\u200b\u4f5c\u4e3a\u200b\u5bf9\u6bd4\u200b\uff0c\u200b\u6211\u4eec\u200b\u5148\u200b\u6784\u9020\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u77e9\u5f62\u200b\uff0c\u200b\u5728\u200b\u77e9\u5f62\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\u6dfb\u52a0\u200b\u52a8\u753b\u200b\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#_1","title":"\u7b80\u5355\u200b\u52a8\u753b","text":"<p>\u200b\u5728\u200b<code>&lt;rect&gt;</code>\u200b\u6807\u7b7e\u200b\u4e2d\u200b\u6dfb\u52a0\u200b<code>&lt;animate&gt;</code>\u200b\u6807\u7b7e\u200b\uff0c\u200b\u6307\u5b9a\u200b\u52a8\u753b\u200b\u7684\u200b\u7ed1\u5b9a\u200b\u5c5e\u6027\u200b\u3001\u200b\u5f00\u59cb\u200b\u7ed3\u675f\u200b\u7684\u200b\u53d6\u503c\u200b\u548c\u200b\u65f6\u957f\u200b\u5373\u53ef\u200b\u3002</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" from=\"20\" to=\"150\" begin=\"click\" dur=\"1.5s\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre> <p>\u200b\u901a\u8fc7\u200b<code>attributeName</code>\u200b\u5c5e\u6027\u200b\u8bbe\u7f6e\u200b\u4ea4\u4e92\u200b\u7ed1\u5b9a\u200b\u7684\u200b\u5c5e\u6027\u200b\uff0c\u200b\u5141\u8bb8\u200b\u7684\u200b\u5c5e\u6027\u200b\u5217\u4e8e\u200b\u4e0b\u8868\u200b\uff1a</p> \u200b\u6807\u7b7e\u200b \u200b\u5c5e\u6027\u200b\u540d\u200b \u200b\u5907\u6ce8\u200b <code>&lt;animate&gt;</code> <code>x</code> <code>&lt;animate&gt;</code> <code>y</code> <code>&lt;animate&gt;</code> <code>width</code> <code>&lt;animate&gt;</code> <code>height</code> <code>&lt;animate&gt;</code> <code>r</code> <code>cx</code>, <code>cy</code> <code>&lt;animate&gt;</code> <code>opacity</code> <code>&lt;animate&gt;</code> <code>d</code> <code>&lt;animate&gt;</code> <code>points</code> <code>&lt;animate&gt;</code> <code>stroke</code> <code>stroke-width</code>, <code>stroke-linecap</code>, <code>stroke-dashoffset</code> <code>&lt;animate&gt;</code> <code>fill</code> <code>&lt;set&gt;</code> <code>visibility</code> <code>&lt;animateTransform&gt;</code> <code>translate</code> <code>&lt;animateTransform&gt;</code> <code>transform</code> <code>&lt;animateTransform&gt;</code> <code>scale</code> <code>&lt;animateTransform&gt;</code> <code>rotate</code> <code>&lt;animateTransform&gt;</code> <code>skewX</code> <code>&lt;animateTransform&gt;</code> <code>skewY</code> <code>&lt;animateMotion&gt;</code> <code>path</code> <code>&lt;animateMotion&gt;</code> <code>rotate</code> <code>&lt;animateMotion&gt;</code> <code>keypoints</code> <code>&lt;animateMotion&gt;</code> <code>mpath</code>"},{"location":"coding/svg/#_2","title":"\u81ea\u5b9a\u4e49\u200b\u52a8\u753b","text":"<p><code>&lt;animate&gt;</code>\u200b\u6807\u7b7e\u200b\u652f\u6301\u200b\u591a\u79cd\u200b\u81ea\u5b9a\u4e49\u200b\u5c5e\u6027\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u5c5e\u6027\u200b\u5141\u8bb8\u200b\u5728\u200b\u52a8\u753b\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u5173\u952e\u5e27\u200b\u3001\u200b\u8ba9\u200b\u52a8\u753b\u200b\u54cd\u5e94\u200b\u7528\u6237\u200b\u4ea4\u4e92\u200b\u7b49\u200b\u9ad8\u7ea7\u200b\u529f\u80fd\u200b\u7684\u200b\u5b9e\u73b0\u200b</p>"},{"location":"coding/svg/#fill","title":"<code>fill</code>\u200b\u5c5e\u6027","text":"<p><code>fill</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u52a8\u753b\u200b\u7ed3\u675f\u200b\u65f6\u200b\u52a8\u753b\u200b\u5bf9\u8c61\u200b\u7684\u200b\u72b6\u6001\u200b\uff0c\u200b\u53d6\u503c\u200b\u53ef\u4ee5\u200b\u4e3a\u200b<code>remove</code>\u200b\u6216\u200b<code>freeze</code>\uff0c\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c<code>fill</code>\u200b\u7684\u200b\u53d6\u503c\u200b\u4e3a\u200b<code>remove</code>\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b<code>fill</code>\u200b\u53d6\u503c\u200b\u4e3a\u200b<code>freeze</code>\u200b\u7684\u200bSVG\u200b\u52a8\u753b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u770b\u51fa\u200b\uff0c\u200b\u52a8\u753b\u200b\u7ed3\u675f\u200b\u540e\u200b\u77e9\u5f62\u200b\u6ca1\u6709\u200b\u56de\u5230\u200b\u5f00\u59cb\u200b\u7684\u200b\u72b6\u6001\u200b\u3002</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" from=\"20\" to=\"150\" begin=\"click\" dur=\"1.5s\" fill=\"freeze\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#repeatcount","title":"<code>repeatCount</code>\u200b\u5c5e\u6027","text":"<p><code>repeatcount</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200b\u52a8\u753b\u200b\u5728\u200b\u5f00\u59cb\u200b\u4ee5\u540e\u200b\u91cd\u590d\u200b\u7684\u200b\u6b21\u6570\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4e3a\u200b\u6574\u6570\u200b\u503c\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4e3a\u200b<code>indefinite</code>\u200b\u8868\u793a\u200b\u65e0\u9650\u200b\u6b21\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4f7f\u200b\u77e9\u5f62\u200b\u79fb\u52a8\u200b\u7684\u200b\u52a8\u4f5c\u200b\u91cd\u590d\u200b\u4e09\u6b21\u200b\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" from=\"20\" to=\"150\" begin=\"click\" dur=\"1.5s\" repeatcount=\"3\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#values","title":"<code>values</code>\u200b\u5c5e\u6027","text":"<p><code>values</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200b\u52a8\u753b\u200b\u7684\u200b\u5c5e\u6027\u200b\u5173\u952e\u5e27\u200b\uff0c\u200b\u5c5e\u6027\u200b\u503c\u4e3a\u200b\u5404\u200b\u5173\u952e\u5e27\u200b\u7684\u200b\u5c5e\u6027\u200b\u53d6\u503c\u200b\uff0c\u200b\u4ee5\u5206\u200b\u53f7\u200b\u5206\u9694\u200b\u3002<code>values</code>\u200b\u5c5e\u6027\u200b\u53ef\u4ee5\u200b\u540c\u65f6\u200b\u66ff\u6362\u200b<code>from</code>\u200b\u5c5e\u6027\u200b\u4e0e\u200b<code>to</code>\u200b\u5c5e\u6027\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4f7f\u7528\u200b<code>values</code>\u200b\u5c5e\u6027\u200b\uff0c\u200b\u4ee4\u200b\u77e9\u5f62\u200b\u5728\u200b\u79fb\u52a8\u200b\u540e\u200b\u6298\u8fd4\u200b\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" values=\"20;150;20\" begin=\"click\" dur=\"3s\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#keytimes","title":"<code>keyTimes</code>\u200b\u5c5e\u6027","text":"<p><code>keyTimes</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200b\u5173\u952e\u5e27\u200b\u7684\u200b\u65f6\u95f4\u200b\u5206\u914d\u200b\uff0c\u200b\u5f53\u200b<code>calcMode</code>\u200b\u5c5e\u6027\u200b\u4e3a\u200b<code>linear</code>\u200b\u6216\u200b<code>spline</code>\u200b\u65f6\u200b\uff0c\u200b\u8be5\u200b\u5c5e\u6027\u200b\u6709\u6548\u200b\uff0c\u200b\u5c5e\u6027\u200b\u7684\u200b\u503c\u200b\u4e3a\u200b\u5404\u4e2a\u200b\u5173\u952e\u5e27\u200b\u5728\u200b\u6574\u4e2a\u200b\u52a8\u753b\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u7684\u200b\u8fdb\u5ea6\u200b\uff08\u200b\u53d6\u503c\u200b\u5728\u200b0-1\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5c0f\u6570\u200b\uff09\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4f7f\u7528\u200b<code>keyTimes</code>\u200b\u5c5e\u6027\u200b\uff0c\u200b\u4ee4\u200b\u77e9\u5f62\u200b\u5728\u200b\u53f3\u79fb\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u7684\u200b\u901f\u5ea6\u200b\u662f\u200b\u5de6\u79fb\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u901f\u5ea6\u200b\u7684\u200b\u4e09\u500d\u200b\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" values=\"20;150;20\" keyTimes=\"0;0.25;1\" begin=\"click\" dur=\"3s\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#calcmodekeysplines","title":"<code>calcMode</code>\u200b\u5c5e\u6027\u200b\u4e0e\u200b<code>keySplines</code>\u200b\u5c5e\u6027","text":"<p><code>calcMode</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200b\u5173\u952e\u5e27\u200b\u63d2\u503c\u200b\u7684\u200b\u8ba1\u7b97\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u8fdb\u4e00\u6b65\u200b\u5730\u200b\uff0c<code>keySplines</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200b\u5173\u952e\u5e27\u200b\u63d2\u503c\u200b\u6240\u7528\u200b\u7684\u200b\u51fd\u6570\u200b\uff08\u200b\u8d1d\u585e\u5c14\u200b\u66f2\u7ebf\u200b\uff09\u3002<code>calcMode</code>\u200b\u5c5e\u6027\u200b\u5141\u8bb8\u200b\u7684\u200b\u53d6\u503c\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li><code>paced</code>\uff1a\u200b\u5ffd\u7565\u200b<code>keyTimes</code>\u200b\u5c5e\u6027\u200b\uff0c\u200b\u5c5e\u6027\u200b\u503c\u200b\u6309\u7167\u200b<code>values</code>\u200b\u4e2d\u200b\u5b9a\u4e49\u200b\u7684\u200b\u5173\u952e\u5e27\u200b\u7ebf\u6027\u200b\u53d8\u5316\u200b</li> <li><code>linear</code>\uff1a\u200b\u9ed8\u8ba4\u200b\u53d6\u503c\u200b\uff0c\u200b\u5c5e\u6027\u200b\u503c\u200b\u6309\u7167\u200b<code>values</code>\u200b\u4e2d\u200b\u5b9a\u4e49\u200b\u7684\u200b\u5173\u952e\u5e27\u200b\u5728\u200b<code>keyTimes</code>\u200b\u65f6\u95f4\u200b\u5185\u200b\u7ebf\u6027\u200b\u53d8\u5316\u200b</li> <li><code>discrete</code>\uff1a\u200b\u5c5e\u6027\u200b\u503c\u200b\u9636\u8dc3\u200b\u53d8\u5316\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8ba4\u4e3a\u200b\u6ca1\u6709\u200b\u63d2\u503c\u200b</li> <li><code>spline</code>\uff1a\u200b\u5c5e\u6027\u200b\u503c\u200b\u6309\u7167\u200b<code>keySplines</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\u5b9a\u4e49\u200b\u7684\u200b\u8d1d\u585e\u5c14\u200b\u66f2\u7ebf\u200b\u53d8\u5316\u200b</li> </ul> <p>\u200b\u4ee5\u4e0b\u200b\u5206\u522b\u200b\u6f14\u793a\u200b<code>calcMode</code>\u200b\u5c5e\u6027\u200b\u56db\u79cd\u200b\u53d6\u503c\u200b\u7684\u200b\u6548\u679c\u200b\uff0c<code>keySplines</code>\u200b\u5c5e\u6027\u200b\u503c\u200b\u8bbe\u5b9a\u200b\u4e3a\u200b<code>1 0 1 1; 0 0 0 1</code>\uff1a</p> SVG\u200b\u56fe\u50cf\u200b pacedlineardiscretespline <p> </p> <p> </p> <p> </p> <p> </p>"},{"location":"coding/svg/#beginend","title":"<code>begin</code>\u200b\u5c5e\u6027\u200b\u4e0e\u200b<code>end</code>\u200b\u5c5e\u6027","text":"<p><code>begin</code>\u200b\u5c5e\u6027\u200b\u4e0e\u200b<code>end</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200bSVG\u200b\u52a8\u753b\u200b\u5728\u200b\u4f55\u65f6\u200b\u5f00\u59cb\u200b\uff0c\u200b\u6b64\u65f6\u200b\u7684\u200b\u65f6\u95f4\u200b\u96f6\u70b9\u200b\u4e3a\u200b\u9875\u9762\u200b\u6e32\u67d3\u200b\u5b8c\u6210\u200b\u7684\u200b\u65f6\u95f4\u200b\uff0c\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5982\u679c\u200b\u4e0d\u200b\u5728\u200b<code>begin</code>\u200b\u4e2d\u200b\u6307\u5b9a\u200b\u7528\u6237\u200b\u4ea4\u4e92\u200b\u4e8b\u4ef6\u200b\uff0c\u200b\u5219\u200b\u52a8\u753b\u200b\u5728\u200b\u6e32\u67d3\u200b\u5b8c\u6210\u200b\u540e\u200b\u5373\u200b\u5f00\u59cb\u200b\u8ba1\u65f6\u200b\u3002<code>end</code>\u200b\u5c5e\u6027\u200b\u53ef\u4ee5\u200b\u63a7\u5236\u200b\u52a8\u753b\u200b\u7684\u200b\u5f3a\u5236\u200b\u7ed3\u675f\u200b\u65f6\u95f4\u200b\uff0c\u200b\u5982\u679c\u200b\u52a8\u753b\u200b\u5728\u200b\u8be5\u200b\u65f6\u95f4\u200b\u5185\u200b\u6ca1\u6709\u200b\u7ed3\u675f\u200b\uff0c\u200b\u5219\u200b\u7acb\u5373\u200b\u7ed3\u675f\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4f7f\u7528\u200b<code>begin</code>\u200b\u5c5e\u6027\u200b\u63a7\u5236\u200b\u52a8\u753b\u200b\u5ef6\u8fdf\u200b\u5f00\u59cb\u200b2s\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" from=\"20\" to=\"150\" begin=\"click+2s\" dur=\"1.5s\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#additive","title":"<code>additive</code>\u200b\u5c5e\u6027","text":"<p><code>additive</code>\u200b\u5c5e\u6027\u200b\u51b3\u5b9a\u200b<code>values</code>\u200b\u53ca\u200b<code>from</code>\u200b\u4e0e\u200b<code>to</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\u7684\u200b\u5c5e\u6027\u200b\u503c\u200b\u662f\u200b\u7edd\u5bf9\u503c\u200b\u8fd8\u662f\u200b\u76f8\u5bf9\u503c\u200b\u3002\u200b\u5141\u8bb8\u200b\u7684\u200b\u53d6\u503c\u200b\u4e3a\u200b<code>sum</code>\u200b\u4e0e\u200b<code>replace</code>\uff0c\u200b\u5206\u522b\u200b\u8868\u793a\u200b\u76f8\u5bf9\u503c\u200b\u4e0e\u200b\u7edd\u5bf9\u503c\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c<code>additive</code>\u200b\u5c5e\u6027\u200b\u53d6\u200b<code>replace</code>\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4f7f\u7528\u200b\u76f8\u5bf9\u503c\u200b\u63a7\u5236\u200b\u77e9\u5f62\u200b\u7684\u200b\u79fb\u52a8\u200b\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100%\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" from=\"0\" to=\"130\" additive=\"sum\" begin=\"click\" dur=\"1.5s\" /&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#accumulate","title":"<code>accumulate</code>\u200b\u5c5e\u6027","text":"<p><code>accumulate</code>\u200b\u5c5e\u6027\u200b\u5f53\u200b\u52a8\u753b\u200b\u91cd\u590d\u200b\u65f6\u200b\uff0c\u200b\u5c5e\u6027\u200b\u503c\u200b\u662f\u5426\u200b\u7d2f\u52a0\u200b\u3002\u200b\u5141\u8bb8\u200b\u7684\u200b\u53d6\u503c\u200b\u4e3a\u200b<code>sum</code>\u200b\u4e0e\u200b<code>none</code>\u3002\u200b\u5206\u522b\u200b\u8868\u793a\u200b\u4e0a\u200b\u4e00\u6b21\u200b\u52a8\u753b\u200b\u7ed3\u675f\u200b\u7684\u200b\u503c\u200b\u4e0e\u200b\u52a8\u753b\u200b\u539f\u672c\u200b\u7684\u200b\u5c5e\u6027\u200b\u503c\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c<code>accumulate</code>\u200b\u5c5e\u6027\u200b\u53d6\u200b<code>none</code>\u3002\u200b\u5f53\u200b\u5c5e\u6027\u200b\u503c\u200b\u4e0d\u200b\u652f\u6301\u200b\u76f8\u52a0\u200b\u6216\u200b\u52a8\u753b\u200b\u4e0d\u200b\u91cd\u590d\u200b\u65f6\u200b\uff0c\u200b\u8be5\u200b\u5c5e\u6027\u200b\u88ab\u200b\u5ffd\u7565\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u5b9e\u73b0\u200b\u6bcf\u6b21\u200b\u70b9\u51fb\u200b\u540e\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u77e9\u5f62\u200b\u4e24\u6b21\u200b\uff0c\u200b\u6bcf\u6b21\u200b65\u200b\u5355\u4f4d\u200b\uff1a</p> SVG\u200b\u56fe\u50cf\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p> </p> <pre><code>&lt;svg width=\"100%\" height=\"100\"&gt;\n    &lt;rect x=\"20\" y=\"20\" fill=\"#69EC69\" width=\"100\" height=\"60\"&gt;\n        &lt;animate attributeName=\"x\" values=\"0;65\" begin=\"click\" keyTimes=\"0;1\" keySplines=\"1 0 1 1\" calcMode=\"spline\" additive=\"sum\" accumulate=\"sum\" dur=\"1.5s\" fill=\"freeze\" repeatCount=\"2\"/&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/svg/#_3","title":"\u4ea4\u4e92\u200b\u52a8\u753b","text":"<p><code>begin</code>\u200b\u5c5e\u6027\u200b\u7684\u200b\u53d6\u503c\u200b\u53ef\u4ee5\u200b\u4e3a\u200b<code>click</code>\u200b\u6216\u200b<code>touch</code>\u200b\u4ee5\u200b\u54cd\u5e94\u200b\u7528\u6237\u200b\u8f93\u5165\u200b\u3002<code>click</code>\u200b\u65e2\u200b\u53ef\u4ee5\u200b\u54cd\u5e94\u200b\u9f20\u6807\u200b\u70b9\u51fb\u200b\u4e8b\u4ef6\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u54cd\u5e94\u200b\u7528\u6237\u200b\u89e6\u6478\u200b\u3002<code>touch</code>\u200b\u53ea\u80fd\u200b\u54cd\u5e94\u200b\u7528\u6237\u200b\u89e6\u6478\u200b\u3002</p>"},{"location":"coding/svg/#svg_1","title":"\u5e38\u89c1\u200b\u7684\u200bSVG\u200b\u52a8\u753b\u200b\u6548\u679c","text":"<ol> <li>\u200b\u4f4d\u79fb\u200b</li> <li>\u200b\u53d8\u6362\u200b</li> <li>\u200b\u6de1\u5165\u6de1\u51fa\u200b</li> <li>\u200b\u63cf\u8fb9\u200b</li> <li>\u200b\u8def\u5f84\u200b\u53d8\u5316\u200b</li> <li>\u200b\u989c\u8272\u200b\u53d8\u5316\u200b</li> </ol>"},{"location":"coding/svg/#_4","title":"\u4f4d\u79fb","text":"<p>\u200b\u4f4d\u79fb\u200b\u64cd\u4f5c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4f7f\u7528\u200b<code>&lt;animate&gt;</code>\u200b\u63a7\u5236\u200b\u5bf9\u8c61\u200b\u7684\u200b<code>x</code>\u3001<code>y</code>\u200b\u5c5e\u6027\u200b\u4e0e\u200b<code>width</code>\u3001<code>height</code>\u200b\u5c5e\u6027\u200b\u5b9e\u73b0\u200b\u3002\u200b\u53ef\u80fd\u200b\u7684\u200b\u5e94\u7528\u200b\u573a\u666f\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u8fdb\u5ea6\u6761\u200b\u3001\u200b\u76f4\u65b9\u56fe\u200b\u7b49\u200b\u7528\u4e8e\u200b\u4ea4\u4e92\u200b\u7684\u200b\u6570\u636e\u200b\u56fe\u8868\u200b</li> <li>\u200b\u4ea4\u901a\u5de5\u5177\u200b\u6216\u200b\u5176\u4ed6\u200b\u5bf9\u8c61\u200b\u7684\u200b\u79fb\u52a8\u200b</li> </ul> <p>\u200b\u8fdb\u5ea6\u6761\u200b\u793a\u4f8b\u200b\uff1a</p> \u200b\u8fdb\u5ea6\u6761\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p></p> <pre><code>&lt;svg id=\"\u200b\u56fe\u5c42\u200b_1\" data-name=\"\u200b\u56fe\u5c42\u200b 1\" xmlns=\"http://www.w3.org/2000/svg\" width=\"700\" height=\"195\" viewBox=\"0 0 700 195\"&gt;\n    &lt;rect x=\"0.5\" y=\"0.5\" width=\"699\" height=\"58\" fill=\"#fff\"/&gt;&lt;path d=\"M699,1V58H1V1H699m1-1H0V59H700V0Z\"/&gt;\n    &lt;rect x=\"1\" y=\"1\" width=\"446\" height=\"57\" fill=\"red\"&gt;\n        &lt;animate attributeName=\"width\" values=\"0;698\" keytimes=\"0;1\" calcmode=\"spline\" keysplines=\"0.44 0.27 0.21 0.99\" begin=\"click\" dur=\"1s\" fill=\"freeze\" repeatcount=\"3\"/&gt;\n    &lt;/rect&gt;\n    &lt;rect x=\"0.5\" y=\"68.5\" width=\"699\" height=\"58\" fill=\"#fff\"/&gt;\n    &lt;path d=\"M699,69v57H1V69H699m1-1H0v59H700V68Z\"/&gt;\n    &lt;rect x=\"1\" y=\"69\" width=\"446\" height=\"57\" fill=\"#dd2\"&gt;\n        &lt;animate attributeName=\"width\" values=\"0;698\" keytimes=\"0;1\" calcmode=\"spline\" keysplines=\"0 0 0.5 1\" begin=\"click\" dur=\"1s\" fill=\"freeze\" repeatcount=\"3\"/&gt;\n    &lt;/rect&gt;\n    &lt;rect x=\"0.5\" y=\"136.5\" width=\"699\" height=\"58\" fill=\"#fff\"/&gt;\n    &lt;path d=\"M699,137v57H1V137H699m1-1H0v59H700V136Z\"/&gt;\n    &lt;rect x=\"1\" y=\"137\" width=\"446\" height=\"57\" fill=\"lime\"&gt;\n        &lt;animate attributeName=\"width\" values=\"0;698\" keytimes=\"0;1\" calcmode=\"spline\" keysplines=\"0.5 0 1 1\" begin=\"click\" dur=\"1s\" fill=\"freeze\" repeatcount=\"3\"/&gt;\n    &lt;/rect&gt;\n&lt;/svg&gt;\n</code></pre> <p>\u200b\u4e00\u4e2a\u200b\u79fb\u52a8\u200b\u7684\u200b\u5706\u70b9\u200b\uff1a</p> \u200b\u5706\u70b9\u200b\u52a8\u753b\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p></p> <pre><code>&lt;svg id=\"\u200b\u56fe\u5c42\u200b_1\" data-name=\"\u200b\u56fe\u5c42\u200b 1\" xmlns=\"http://www.w3.org/2000/svg\" width=\"150\" viewBox=\"0 0 700 195\"&gt;\n    &lt;rect width=\"700\" height=\"195\" fill=\"none\"/&gt;\n    &lt;circle cx=\"279.5\" cy=\"97.5\" r=\"25.5\"&gt;\n        &lt;animate attributeName=\"cx\" values=\"0;90;0\" keytimes=\"0;0.5;1\" calcmode=\"spline\" keysplines=\"0.5 0 0.5 1; 0.5 0 0.5 1\" begin=\"0s\" dur=\"2s\" fill=\"freeze\" repeatcount=\"indefinite\" additive=\"sum\"/&gt;\n    &lt;/circle&gt;\n&lt;/svg&gt;\n</code></pre> <p>\u200b\u6b64\u5916\u200b\uff0c\u200b\u90e8\u5206\u200b\u4f4d\u79fb\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>&lt;animateMotion&gt;</code>\u200b\u6807\u7b7e\u200b\u8fdb\u884c\u200b\u63a7\u5236\u200b\uff0c<code>&lt;animateMotion&gt;</code>\u200b\u6807\u7b7e\u200b\u652f\u6301\u200b\u5bf9\u8c61\u200b\u6309\u7167\u200b\u8def\u5f84\u200b\u8fdb\u884c\u200b\u4f4d\u79fb\u200b\u3002</p> <p>\u200b\u590d\u6742\u200b\u7684\u200b\u4f4d\u79fb\u200b\u9700\u8981\u200b\u63a7\u5236\u200b\u5bf9\u8c61\u200b\u7684\u200b<code>translate</code>\u200b\u5c5e\u6027\u200b\u3002</p>"},{"location":"coding/svg/#_5","title":"\u53d8\u6362","text":"<p>\u200b\u4ee5\u4e0b\u200b\u5c5e\u6027\u200b\u5f71\u54cd\u200b\u56fe\u5f62\u200b\u7684\u200b\u53d8\u6362\u200b\uff1a</p> <ul> <li><code>r</code>\uff1a\u200b\u534a\u5f84\u200b\uff08\u200b\u5bf9\u4e8e\u200b\u5706\u200b\uff09</li> <li><code>rotate</code>\uff1a\u200b\u65cb\u8f6c\u200b\uff08\u200b\u4ee5\u200b\u4e2d\u5fc3\u70b9\u200b\uff09</li> <li><code>skewX</code>\u3001<code>skewY</code>\uff1a\u200b\u503e\u659c\u200b</li> <li><code>transform</code>\uff1a\u200b\u590d\u6742\u200b\u53d8\u6362\u200b</li> </ul> <p>\u200b\u4ee5\u4e0b\u200b\u7ed9\u51fa\u200b<code>r</code>\u200b\u53d8\u6362\u200b\u7684\u200b\u793a\u4f8b\u200b\uff1a</p> r\u200b\u53d8\u6362\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p>     &gt;&gt;&gt; \u200b\u70b9\u51fb\u200b\u5706\u70b9\u200b \u200b\u67e5\u770b\u200b\u9690\u85cf\u200b\u5185\u5bb9\u200b &lt;&lt;&lt;      Hello              World!      <pre><code>&lt;svg id=\"\u200b\u56fe\u5c42\u200b_1\" data-name=\"\u200b\u56fe\u5c42\u200b 1\" xmlns=\"http://www.w3.org/2000/svg\" width=\"350\" viewBox=\"0 0 749 1334\"&gt;\n    &lt;rect width=\"749\" height=\"1334\" fill=\"#fff\"/&gt;\n    &lt;text transform=\"translate(221.4 1070.91)\" font-size=\"21\" font-family=\"PingFang-SC-Regular, PingFang SC\"&gt;\n        &amp;gt;&amp;gt;&amp;gt; \u200b\u70b9\u51fb\u200b\u5706\u70b9\u200b \u200b\u67e5\u770b\u200b\u9690\u85cf\u200b\u5185\u5bb9\u200b &amp;lt;&amp;lt;&amp;lt;\n    &lt;/text&gt;\n    &lt;circle cx=\"375\" cy=\"992\" r=\"29\" fill=\"#ed1e79\"&gt;\n        &lt;animate attributeName=\"r\" values=\"0;1100\" keytimes=\"0;1\" calcmode=\"spline\" keysplines=\"0 0 0.5 1\" begin=\"click\" dur=\"0.75s\" fill=\"freeze\" additive=\"sum\" restart=\"never\"&gt;\n    &lt;/circle&gt;\n    &lt;text transform=\"translate(0 295.12)\" font-size=\"192\" fill=\"#fff\" font-family=\"PingFang-SC-Heavy, PingFang SC\" font-weight=\"800\"&gt;\n        Hello\n        &lt;tspan x=\"0\" y=\"230.4\"&gt;\n            World!\n        &lt;/tspan&gt;\n    &lt;/text&gt;\n&lt;/svg&gt;\n</code></pre> <p><code>transform</code>\u200b\u5c5e\u6027\u200b\u5728\u200b<code>animateTransform</code>\u200b\u6807\u7b7e\u200b\u4e2d\u200b\u66f4\u6539\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u8f83\u4e3a\u200b\u590d\u6742\u200b\u7684\u200b\u53d8\u6362\u200b\uff0c\u200b\u5982\u7ed5\u200b\u67d0\u200b\u4e00\u4e2a\u200b\u7279\u5b9a\u200b\u70b9\u200b\u8fdb\u884c\u200b\u65cb\u8f6c\u200b\uff1a</p> transform\u200b\u53d8\u6362\u200b \u200b\u6e32\u67d3\u200b\u56fe\u50cf\u200bSVG\u200b\u4ee3\u7801\u200b <p></p> <pre><code>&lt;svg id=\"\u200b\u56fe\u5c42\u200b_1\" data-name=\"\u200b\u56fe\u5c42\u200b 1\" xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 749 1340\" width=\"350\"&gt;\n    &lt;rect y=\"6\" width=\"749\" height=\"1334\" fill=\"#fff\"/&gt;\n    &lt;g&gt;\n        &lt;animateTransform attributeName=\"transform\" type=\"rotate\" values=\"0 375 0; -10 375 0; 10 375 0; 0 375 0\" keytimes=\"0;0.25;0.75;1\" calcmode=\"spline\" keysplines=\"0 0 0.5 1; 0.5 0 0.5 1; 0.5 0 1 1\" start=\"0s\" dur=\"2s\" repeatcount=\"indefinite\" /&gt;\n        &lt;line x1=\"375\" y1=\"6\" x2=\"375\" y2=\"969.5\" fill=\"none\" stroke=\"#000\" stroke-linecap=\"round\" stroke-miterlimit=\"10\" stroke-width=\"12\"/&gt;\n        &lt;circle cx=\"375\" cy=\"969\" r=\"45\" stroke=\"#000\" stroke-linecap=\"round\" stroke-miterlimit=\"10\" stroke-width=\"12\"/&gt;\n    &lt;/g&gt;\n&lt;/svg&gt;\n</code></pre>"},{"location":"coding/configuration/","title":"\u7cfb\u7edf\u914d\u7f6e","text":""},{"location":"coding/configuration/#_2","title":"\u547d\u4ee4\u884c","text":"<ul> <li>apt</li> <li>Docker</li> <li>ssh</li> <li>Vim</li> <li>update-alternatives</li> </ul>"},{"location":"coding/configuration/#_3","title":"\u8f6f\u4ef6\u200b\u7f16\u8bd1\u200b\u6d41\u7a0b","text":"<ul> <li>\u200b\u6e90\u7801\u200b\u7f16\u8bd1\u200bnginx\u200b\u5e76\u200b\u6dfb\u52a0\u200bRTMP\u200b\u6a21\u5757\u200b</li> </ul>"},{"location":"coding/configuration/apt-get/","title":"apt\uff1aUbuntu \u200b\u5305\u200b\u7ba1\u7406\u5de5\u5177","text":"<p>Ubuntu\u200b\u4e0b\u200b\u7684\u200b\u5305\u200b\u7ba1\u7406\u5de5\u5177\u200b\u662f\u200b<code>dpkg</code>\uff0c\u200b\u6709\u200b\u4e24\u4e2a\u200b\u524d\u7aef\u200b<code>apt</code>\u200b\u548c\u200b<code>apt-get</code>\u3002</p>"},{"location":"coding/configuration/apt-get/#_1","title":"\u5b89\u88c5\u200b\u4e2d\u65ad\u200b\u540e\u200b\u7684\u200b\u6062\u590d","text":"<p>\u200b\u5728\u200b\u67d0\u4e2a\u200b\u8f6f\u4ef6\u200b\u5b89\u88c5\u200b\u4e2d\u65ad\u200b\uff08\u200b\u5982\u200bCtrl+C\u200b\u6216\u200bkill\uff09\u200b\u540e\u200b\uff0c\u200b\u8f6f\u4ef6\u200b\u7684\u200b\u5b89\u88c5\u200b\u5e76\u200b\u6ca1\u6709\u200b\u5b9e\u9645\u200b\u5b8c\u6210\u200b\uff0c\u200b\u9700\u8981\u200b\u7ee7\u7eed\u6267\u884c\u200b\u5b89\u88c5\u64cd\u4f5c\u200b\u6216\u200b\u56de\u6eda\u200b\u5230\u200b\u5b89\u88c5\u200b\u4e4b\u524d\u200b\u7684\u200b\u72b6\u6001\u200b\u3002</p> <p>\u200b\u82e5\u200b\u8981\u200b\u7ee7\u7eed\u200b\u5b89\u88c5\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6267\u884c\u200b\u5982\u4e0b\u200b\u547d\u4ee4\u200b\uff0c<code>dpkg</code>\u200b\u4f1a\u200b\u5c1d\u8bd5\u200b\u5bf9\u200b\u5df2\u7ecf\u200b\u89e3\u538b\u200b\u672a\u200b\u5b8c\u6210\u200b\u5b89\u88c5\u200b\u7684\u200b\u8f6f\u4ef6\u5305\u200b\u8fdb\u884c\u200b\u6062\u590d\u200b\u3002</p> <pre><code>sudo dpkg --configure -a\n</code></pre> <p>\u200b\u82e5\u8981\u200b\u56de\u6eda\u200b\u5b89\u88c5\u200b\uff0c\u200b\u9700\u8981\u200b\u5148\u200b\u5728\u200b<code>dpkg</code>\u200b\u7684\u200b\u5b89\u88c5\u200b\u8bb0\u5f55\u200b\u4e0b\u200b\u627e\u5230\u200b\u672a\u200b\u5b8c\u5168\u5b89\u88c5\u200b\u7684\u200b\u5305\u200b\uff0c\u200b\u5c06\u200b\u5176\u200b\u5220\u9664\u200b\u540e\u200b\u91cd\u65b0\u914d\u7f6e\u200b<code>apt</code>\u3002\u200b\u6b64\u5904\u200b\u4ee5\u200b<code>nvidia-cudnn</code>\u200b\u4e3a\u4f8b\u200b\uff1a</p> <pre><code>sudo ls -l /var/lib/dpkg/info | grep -i nvidia-cudnn\nsudo mv /var/lib/dpkg/info/nvidia-cudnn.* /tmp\nsudo dpkg --configure -a\nsudo apt-get autoremove\n</code></pre>"},{"location":"coding/configuration/docker/","title":"Docker","text":"<p><code>docker</code>\u200b\u662f\u200b\u7528\u4e8e\u200b\u63d0\u4f9b\u200b\u5bb9\u5668\u200b\u955c\u50cf\u200b\u7684\u200b\u8f6f\u4ef6\u200b\u3002\u200b\u955c\u50cf\u200b\u662f\u200b\u4e00\u5957\u200b\u914d\u7f6e\u200b\u597d\u200b\u7684\u200b\u8fd0\u884c\u200b\u73af\u5883\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5feb\u901f\u200b\u90e8\u7f72\u200b\u4e3a\u200b\u5e94\u7528\u7a0b\u5e8f\u200b\u3002</p>"},{"location":"coding/configuration/docker/#docker-registry","title":"Docker registry","text":"<p>Docker registry\u200b\u662f\u200bdocker\u200b\u7684\u200b\u955c\u50cf\u200b\u6e90\u200b\u3002</p>"},{"location":"coding/configuration/docker/#_1","title":"\u767b\u5f55","text":"<p>\u200b\u767b\u5f55\u200b\u7ec8\u7aef\u200b\u540e\u200b\uff0c\u200b\u624d\u80fd\u200b\u4ece\u200b\u7f51\u7edc\u200b\u4e0a\u200bpush\u200b\u548c\u200bpull\u200b\u955c\u50cf\u200b\u3002\u200b\u767b\u5f55\u200b\u7684\u200b\u547d\u4ee4\u200b\u4e3a\u200b</p> <pre><code>docker login &lt;registry-url&gt;\n</code></pre> <p>\u200b\u5982\u679c\u200b\u6ca1\u6709\u200b\u6307\u5b9a\u200b<code>&lt;registry-url&gt;</code>\uff0c\u200b\u5219\u200b\u9ed8\u8ba4\u200b\u4f7f\u7528\u200bDocker registry <code>docker.io</code>\u3002</p> <p>\u200b\u767b\u5f55\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u4f1a\u200b\u8bf7\u6c42\u200b\u7528\u6237\u200b\u8f93\u5165\u200b\u7528\u6237\u540d\u200b\u548c\u200b\u5bc6\u7801\u200b\uff08\u200b\u6216\u200bAccess Token\uff09\u3002\u200b\u5982\u679c\u200b\u8d26\u6237\u200b\u914d\u7f6e\u200b\u4e86\u200b2FA\uff0c\u200b\u5219\u200b\u53ea\u80fd\u200b\u901a\u8fc7\u200bAccess Token\u200b\u7684\u200b\u65b9\u5f0f\u200b\u767b\u5f55\u200b\u3002\u200b\u767b\u5f55\u200b\u4fe1\u606f\u200b\u4fdd\u5b58\u200b\u5728\u200b<code>~/.docker/config.json</code>\u200b\u6216\u200b\u7cfb\u7edf\u200b\u7684\u200b\u94a5\u5319\u200b\u4e32\u4e2d\u200b\u3002\u200b\u5982\u679c\u200b\u52a0\u4e0a\u200b<code>--password-stdin</code>\u200b\u53c2\u6570\u200b\uff0cdocker\u200b\u4f1a\u200b\u4ece\u200b\u547d\u4ee4\u884c\u200b\u4e2d\u200b\u8bfb\u53d6\u200b\u5bc6\u7801\u200b\u3002</p> <p>\u200b\u914d\u7f6e\u200bdocker\u200b\u4f7f\u7528\u200bmacOS\u200b\u94a5\u5319\u200b\u4e32\u200b</p> <ol> <li>\u200b\u6267\u884c\u200b<code>docker logout</code>\uff0c\u200b\u5220\u9664\u200b\u539f\u5148\u200b\u7684\u200b\u767b\u5f55\u200b\u51ed\u8bc1\u200b\uff1b</li> <li>\u200b\u4ece\u200bHomebrew\u200b\u5b89\u88c5\u200b<code>docker-credential-helper</code>\uff1b</li> <li>\u200b\u5728\u200b<code>~/.docker/config.json</code>\u200b\u4e2d\u200b\u52a0\u5165\u200b\u5982\u4e0b\u200b\u4e00\u884c\u200b\uff1a</li> </ol> <pre><code>{\n    \"credsStore\": \"osxkeychain\"\n}\n</code></pre> <ol> <li>\u200b\u91cd\u65b0\u200b\u6267\u884c\u200b<code>docker login</code>\u200b\u767b\u5f55\u200b\uff1b</li> <li>\u200b\u786e\u8ba4\u200b<code>~/.docker/config.json</code>\u200b\u91cc\u200b\u6ca1\u6709\u200b\u767b\u5f55\u200b\u51ed\u8bc1\u200b\uff1a</li> </ol> <pre><code>{\n    \"auths\": {\n            \"https://index.docker.io/v1/\": {}\n    }\n}\n</code></pre>"},{"location":"coding/configuration/docker/#_2","title":"\u7b2c\u4e09\u65b9\u200b\u955c\u50cf\u200b\u6e90\u200b\u5217\u8868","text":"<p>ghcr.io</p> <p><code>ghcr.io</code>\u200b\u662f\u200bGitHub\u200b\u63d0\u4f9b\u200b\u7684\u200b\u955c\u50cf\u200b\u6e90\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u65e0\u7f1d\u200b\u96c6\u6210\u200b\u81f3\u200bGitHub Actions\u3002</p> <p>\u200b\u9a8c\u8bc1\u200b\u65b9\u5f0f\u200b\uff1a</p> <ol> <li>\u200b\u4f7f\u7528\u200bGitHub Access Token\u200b\u9a8c\u8bc1\u200b\uff0cAccess Token\u200b\u5fc5\u987b\u200b\u81f3\u5c11\u200b\u5177\u5907\u200b<code>read:packages</code>\u200b\u6743\u9650\u200b\uff0c\u200b\u5982\u679c\u200b\u8981\u200b\u63a8\u9001\u200b\u955c\u50cf\u200b\uff0c\u200b\u5219\u200b\u8fd8\u200b\u9700\u8981\u200b<code>write:packages</code>\u200b\u6743\u9650\u200b</li> <li>\u200b\u5728\u200bGitHub Actions\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>GITHUB_TOKEN</code>\u200b\u9a8c\u8bc1\u200b\u3002</li> </ol>"},{"location":"coding/configuration/docker/#dockerfile","title":"Dockerfile","text":"<p><code>dockerfile</code>\u200b\u6587\u4ef6\u200b\u7528\u6765\u200b\u5c06\u200b\u4e00\u5957\u200b\u73af\u5883\u200b\u6253\u5305\u200b\u6210\u200b\u955c\u50cf\u200b\u3002\u200b\u6bcf\u200b\u4e00\u6761\u200b\u6307\u4ee4\u200b\u90fd\u200b\u5bf9\u5e94\u200b\u4e86\u200b\u5bb9\u5668\u200b\u955c\u50cf\u200b\u7684\u200b\u4e00\u5c42\u200b\u3002</p>"},{"location":"coding/configuration/docker/#docker_1","title":"Docker\u200b\u955c\u50cf\u200b\u57fa\u7840","text":"<p>Docker\u200b\u955c\u50cf\u200b\u7684\u200b\u57fa\u672c\u200b\u683c\u5f0f\u200b\u4e3a\u200b<code>[&lt;registry-name&gt;]/&lt;namespace&gt;/&lt;image-name&gt;:[&lt;tag&gt;]</code></p> <ul> <li><code>&lt;registry-name&gt;</code>\uff1a\u200b\u53ef\u200b\u9009\u200b\uff0c\u200b\u51b3\u5b9a\u200bDocker\u200b\u955c\u50cf\u200b\u53d1\u5e03\u200b\u5728\u200b\u54ea\u4e2a\u200bRegistry\u200b\u4e0a\u200b\uff1b</li> <li><code>&lt;namespace&gt;</code>\uff1a\u200b\u4e00\u822c\u200b\u662f\u200b\u7528\u6237\u540d\u200b\uff1b</li> <li><code>&lt;image-name&gt;</code>\uff1a\u200b\u955c\u50cf\u200b\u540d\u79f0\u200b\uff1b</li> <li><code>&lt;tag&gt;</code>\uff1a\u200b\u53ef\u200b\u9009\u200b\uff0c\u200b\u955c\u50cf\u200b\u7248\u672c\u200b\uff0c\u200b\u9ed8\u8ba4\u200b\u4e3a\u200b<code>latest</code></li> </ul> <p>\u200b\u4f7f\u7528\u200b<code>docker pull</code>\u200b\u547d\u4ee4\u200b\u4ece\u200bregistry\u200b\u4e0b\u8f7d\u200b\u955c\u50cf\u200b\u3002</p> <p>\u200b\u6267\u884c\u200b<code>docker build &lt;path&gt;</code>\u200b\u6765\u200b\u4ece\u200bDockerfile\u200b\u6784\u5efa\u200b\u955c\u50cf\u200b\u3002\u200b\u7528\u200b<code>-t</code>\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u6307\u5b9a\u200b\u955c\u50cf\u200b\u7684\u200b\u540d\u79f0\u200b\u548c\u200b\u7248\u672c\u200b\u3002</p> <p>\u200b\u5728\u200b\u955c\u50cf\u200b\u6784\u5efa\u200b\u5b8c\u6210\u200b\u540e\u200b\uff0c\u200b\u6267\u884c\u200b<code>docker push &lt;image-name&gt;</code>\u200b\u5c06\u200b\u955c\u50cf\u200b\u63a8\u9001\u200b\u5230\u200b\u5bf9\u5e94\u200b\u7684\u200bregistry\u200b\u4e0a\u200b\u3002</p>"},{"location":"coding/configuration/docker/#add","title":"<code>ADD</code>","text":"<p><code>ADD</code>\uff08\u200b\u6216\u200b<code>COPY</code>\uff09\u200b\u547d\u4ee4\u200b\u5c06\u200b\u672c\u5730\u200b\u6216\u200b\u4e92\u8054\u7f51\u200b\u4e0a\u200b\u7684\u200b\u6587\u4ef6\u200b\u590d\u5236\u5230\u200b\u5bb9\u5668\u200b\u955c\u50cf\u200b\u4e2d\u200b\u3002</p> <pre><code>ADD &lt;local-path&gt; &lt;image-path&gt;\n</code></pre> <ul> <li>\u200b\u5982\u679c\u200b\u8def\u5f84\u200b\u4e2d\u200b\u5305\u542b\u200b\u7a7a\u683c\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b\u53cc\u5f15\u53f7\u200b\u628a\u200b\u8def\u5f84\u200b\u62ec\u200b\u8d77\u6765\u200b\uff1b</li> <li><code>&lt;local-path&gt;</code>\u200b\u652f\u6301\u200b\u901a\u914d\u7b26\u200b<code>*?</code>\uff1b</li> </ul>"},{"location":"coding/configuration/docker/#from","title":"<code>FROM</code>","text":"<p><code>FROM</code>\u200b\u547d\u4ee4\u200b\u6307\u660e\u200b\u5f53\u524d\u200b\u7684\u200b<code>dockerfile</code>\u200b\u662f\u200b\u5728\u200b\u54ea\u200b\u4e00\u4e2a\u200b\u955c\u50cf\u200b\u4e0a\u200b\u6267\u884c\u200b\u7684\u200b\u3002\u200b\u6bcf\u4e2a\u200bDockerfile\u200b\u5fc5\u987b\u200b\u6709\u200b\u4e00\u6761\u200b<code>FROM</code>\u200b\u8bed\u53e5\u200b\u3002</p> <p>\u200b\u7a7a\u767d\u200bDocker\u200b\u955c\u50cf\u200b</p> <p><code>scratch</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u9884\u5b9a\u200b\u4e49\u200b\u7684\u200b\u7a7a\u767d\u200bDocker\u200b\u955c\u50cf\u200b\uff0c\u200b\u4e5f\u200b\u662f\u200b\u6240\u6709\u200bDocker\u200b\u955c\u50cf\u200b\u7684\u200b\u8d77\u59cb\u200b\u3002</p> <pre><code>FROM scratch\n</code></pre>"},{"location":"coding/configuration/docker/#run","title":"<code>RUN</code>","text":"<p><code>RUN</code>\u200b\u6267\u884c\u200b\u4e00\u4e2a\u200b\u547d\u4ee4\u200b\uff0c\u200b\u5982\u679c\u200b\u8981\u200b\u6267\u884c\u200b\u591a\u4e2a\u200b\u547d\u4ee4\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>&amp;&amp;</code>\u200b\u8fde\u63a5\u200b</p> <pre><code>RUN command-1 &amp;&amp; \\\n    command-2 &amp;&amp; \\\n    ...\n</code></pre>"},{"location":"coding/configuration/docker/#tls","title":"\u914d\u7f6e\u200bTLS\u200b\u8fdc\u7a0b\u200b\u8bbf\u95ee","text":"<p>Docker\u200b\u662f\u200bc/s\u200b\u67b6\u6784\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u53ef\u4ee5\u200b\u4ece\u200b\u8fdc\u7a0b\u200b\u8bbf\u95ee\u200b\u4e3b\u673a\u200b\u7684\u200bDocker daemon\uff0c\u200b\u4ece\u200b\u8fdc\u7a0b\u200b\u76f4\u63a5\u200b\u64cd\u4f5c\u200bDocker\u200b\u73af\u5883\u200b\uff0c\u200b\u5bb9\u5668\u200b\u548c\u200b\u955c\u50cf\u200b\u5747\u200b\u4fdd\u5b58\u200b\u5728\u200b\u8fdc\u7a0b\u200b\u4e3b\u673a\u200b\u4e0a\u200b\u3002\u200b\u4f46\u200b\u5728\u200b\u7f51\u7edc\u200b\u4e0a\u200b\u9700\u8981\u200bTLS\u200b\u52a0\u5bc6\u4f20\u8f93\u200b\u4ee5\u200b\u4fdd\u8bc1\u200b\u5b89\u5168\u200b\uff0c\u200b\u56e0\u6b64\u200b\u9700\u8981\u200b\u5148\u200b\u914d\u7f6e\u200b\u597d\u200bopenssl\uff0c\u200b\u5e76\u4e14\u200b\u6709\u200b\u4e00\u4e2a\u200b\u89e3\u6790\u200b\u5230\u200b\u670d\u52a1\u5668\u200b\u7684\u200b\u57df\u540d\u200b\uff0c\u200b\u5982\u200b<code>www.example.com</code>\u3002</p>"},{"location":"coding/configuration/docker/#_3","title":"\u670d\u52a1\u5668\u200b\u914d\u7f6e","text":"<p><code>dockerd</code>\u200b\u662f\u200bDocker daemon\u200b\u7684\u200b\u53ef\u6267\u884c\u7a0b\u5e8f\u200b\u3002</p> Docker DaemonDocker Desktop <p>\u200b\u5728\u200b\u670d\u52a1\u5668\u7aef\u200b\uff0c\u200b\u9700\u8981\u200b\u6267\u884c\u200b\u5982\u4e0b\u200b\u547d\u4ee4\u200b\u3002</p> <ol> <li> <p>\u200b\u751f\u6210\u200bCA\u200b\u5bc6\u94a5\u200b</p> <pre><code>openssl genrsa -aes256 -out ca-key.pem 4096\nopenssl req -new -x509 -days 365 -key ca-key.pem -sha256 -out ca.pem # (1)\n</code></pre> <ol> <li>\u200b\u4f1a\u200b\u8bf7\u6c42\u200b\u8f93\u5165\u200b\u4e00\u4e9b\u200b\u4fe1\u606f\u200b\uff0c\u200b\u5176\u4e2d\u200b<code>Common Name</code>\u200b\u4e00\u6761\u200b\u9700\u8981\u200b\u586b\u5199\u200b\u670d\u52a1\u5668\u200b\u5bf9\u5e94\u200b\u7684\u200b\u57df\u540d\u200b\u3002</li> </ol> </li> <li> <p>\u200b\u751f\u6210\u200b\u670d\u52a1\u5668\u200b\u5bc6\u94a5\u200b\u548c\u200b\u8bc1\u4e66\u200b</p> <pre><code>openssl genrsa -out server-key.pem 4096\nopenssl req -subj \"/CN=www.example.com\" -sha256 -new -key server-key.pem -out server.csr\necho subjectAltName = DNS:$HOST,IP:8.8.8.8 &gt;&gt; extfile.cnf # (1)\necho extendedKeyUsage = serverAuth &gt;&gt; extfile.cnf\nopenssl x509 -req -days 365 -sha256 -in server.csr -CA ca.pem -CAkey ca-key.pem \\\n    -CAcreateserial -out server-cert.pem -extfile extfile.cnf\n</code></pre> <ol> <li>\u200b\u5982\u679c\u200b\u4e0d\u200b\u9700\u8981\u200bIP\u200b\u5730\u5740\u200b\u8bbf\u95ee\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5220\u9664\u200b\u76f8\u5173\u200b\u6761\u76ee\u200b\u3002</li> </ol> </li> <li> <p>\u200b\u751f\u6210\u200b\u5ba2\u6237\u7aef\u200b\u5bc6\u94a5\u200b\u548c\u200b\u8bc1\u4e66\u200b</p> <pre><code>openssl genrsa -out key.pem 4096\nopenssl req -subj '/CN=client' -new -key key.pem -out client.csr\necho extendedKeyUsage = clientAuth &gt; extfile-client.cnf\nopenssl x509 -req -days 365 -sha256 -in client.csr -CA ca.pem -CAkey ca-key.pem \\\n    -CAcreateserial -out cert.pem -extfile extfile-client.cnf\n</code></pre> </li> <li> <p>\u200b\u6e05\u7406\u200b\u5176\u4f59\u200b\u6587\u4ef6\u200b\uff0c\u200b\u53ea\u200b\u7559\u4e0b\u200b<code>.pem</code>\u200b\u8bc1\u4e66\u200b\u6587\u4ef6\u200b\uff0c\u200b\u5e76\u4e14\u200b\u8c03\u6574\u200b\u8bc1\u4e66\u200b\u6587\u4ef6\u200b\u7684\u200b\u6743\u9650\u200b</p> <pre><code>rm -v client.csr server.csr extfile.cnf extfile-client.cnf\nchmod -v 0400 ca-key.pem key.pem server-key.pem\nchmod -v 0444 ca.pem server-cert.pem cert.pem\n</code></pre> </li> <li> <p>\u200b\u542f\u52a8\u200bDocker daemon</p> <pre><code>dockerd \\\n    --tlsverify \\\n    --tlscacert=ca.pem \\\n    --tlscert=server-cert.pem \\\n    --tlskey=server-key.pem \\\n    -H=0.0.0.0:2376\n</code></pre> </li> <li> <p>\u200b\u5206\u53d1\u200b\u5ba2\u6237\u7aef\u200b\u8bc1\u4e66\u200b<code>ca.pem</code>\u3001<code>cert.pem</code>\u3001<code>key.pem</code>\u3002\u200b\u5ba2\u6237\u7aef\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4ee5\u4e0b\u200b\u547d\u4ee4\u200b\u8fde\u63a5\u200b</p> <pre><code>docker --tlsverify \\\n    --tlscacert=ca.pem \\\n    --tlscert=cert.pem \\\n    --tlskey=key.pem \\\n    -H=$HOST:2376\n</code></pre> </li> </ol> <p>\u200b\u5728\u200bWindows\u200b\u4e2d\u200b\uff0cDocker\u200b\u901a\u8fc7\u200bWSL\u200b\u6216\u8005\u200bHyper-V\u200b\u865a\u62df\u5316\u200b\u5b9e\u73b0\u200bLinux\u200b\u73af\u5883\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5982\u679c\u200b\u5728\u200bWindows\u200b\u4e0b\u200b\u76f4\u63a5\u200b\u6267\u884c\u200b<code>dockerd.exe</code>\uff0c\u200b\u5f97\u5230\u200b\u7684\u200b\u662f\u200b\u8fd0\u884c\u200bWindows\u200b\u5bb9\u5668\u200b\u7684\u200bDocker\u3002\u200b\u5982\u679c\u200b\u8981\u200b\u8fdc\u7a0b\u200b\u8fde\u63a5\u200b\u8fd0\u884c\u200bLinux\u200b\u7684\u200bDocker\u200b\u73af\u5883\u200b\uff0c\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>docker-remote-api-tls</code>\u3002</p> <p>Docker Desktop Engine \u200b\u914d\u7f6e\u6587\u4ef6\u200b</p> <p>\u200b\u5728\u200bDocker Desktop\u200b\u91cc\u200b\u7684\u200bSettings - Engine\u200b\u91cc\u200b\u4fee\u6539\u200bTLS\u200b\u53c2\u6570\u200b\u914d\u7f6e\u200b\uff0c\u200b\u4f1a\u200b\u5bfc\u81f4\u200bDocker Engine\u200b\u65e0\u6cd5\u200b\u542f\u52a8\u200b\u3002</p> <ol> <li>\u200b\u6267\u884c\u200b<code>git clone https://github.com/kekru/docker-remote-api-tls.git</code>\u200b\u514b\u9686\u200b\u955c\u50cf\u200b</li> <li> <p>\u200b\u7f16\u8f91\u200b<code>docker-compose.yml</code>\uff0c\u200b\u66ff\u6362\u200b<code>&lt;password&gt;</code>\u3001<code>&lt;domain-or-ip&gt;</code>\u3001<code>&lt;client-cert-path&gt;</code>\u200b\u4e2d\u200b\u7684\u200b\u5185\u5bb9\u200b</p> <pre><code>version: \"3.4\"\nservices:\n    remote-api:\n        image: kekru/docker-remote-api-tls:v0.4.0\n        ports:\n            - 2376:443\n        environment:\n            - CREATE_CERTS_WITH_PW=&lt;password&gt;\n            - CERT_HOSTNAME=&lt;domain-or-ip&gt;\n        volumes:\n            - &lt;client-cert-path&gt;:/data/certs # (1)\n            - /var/run/docker.sock:/var/run/docker.sock:ro\n</code></pre> <ol> <li>\u200b\u4e0d\u52a0\u200b\u8fd9\u200b\u4e00\u884c\u200b\u4e5f\u200b\u884c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u4ece\u200bDocker Desktop\u200b\u91cc\u200b\u590d\u5236\u200b</li> </ol> </li> <li> <p>\u200b\u6267\u884c\u200b<code>docker-compose up -d</code>\u200b\u8fd0\u884c\u200b\u5bb9\u5668\u200b\uff0c\u200b\u6ce8\u610f\u200b\u5728\u200b\u8fdc\u7a0b\u200b\u8fde\u63a5\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u9700\u8981\u200b\u4fdd\u8bc1\u200b\u5bb9\u5668\u200b\u59cb\u7ec8\u200b\u5f00\u542f\u200b</p> </li> </ol>"},{"location":"coding/configuration/docker/#_4","title":"\u5ba2\u6237\u7aef\u200b\u914d\u7f6e","text":"<ol> <li>\u200b\u5c06\u200b<code>ca.pem</code>\u3001<code>cert.pem</code>\u3001<code>key.pem</code>\u200b\u590d\u5236\u5230\u200b\u5ba2\u6237\u7aef\u200b\u7684\u200b<code>~/.docker</code>\u200b\u76ee\u5f55\u200b\u4e0b\u200b</li> <li> <p>\u200b\u8bbe\u7f6e\u200b\u73af\u5883\u53d8\u91cf\u200b\uff08\u200b\u53ea\u200b\u9002\u7528\u200b\u4e8e\u200b\u4ece\u200b\u7ec8\u7aef\u200b\u8bbf\u95ee\u200bDocker\u200b\u7684\u200b\u60c5\u51b5\u200b\uff09</p> <pre><code>export $DOCKER_TLS_VERIFY=1\nexport $DOCKER_HOST=\"tcp://&lt;domain-or-ip&gt;:2376\"\n</code></pre> </li> <li> <p>\u200b\u6216\u8005\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200bDocker context</p> <pre><code>docker context create remote --docker \\\n    \"host=tcp://&lt;domain-or-ip&gt;:2376,ca=~/.docker/ca.pem,cert=~/.docker/cert.pem,key=~/.docker/key.pem\n</code></pre> </li> <li> <p>\u200b\u6b64\u540e\u200b\uff0cDocker\u200b\u4f1a\u200b\u9ed8\u8ba4\u200b\u4f7f\u7528\u200b\u8fdc\u7a0b\u200b\u8fde\u63a5\u200b\u3002</p> </li> </ol> <p>\u200b\u5982\u679c\u200b\u8fde\u63a5\u200b\u6210\u529f\u200b\uff0c\u200b\u6267\u884c\u200b<code>docker version</code>\u200b\u4f1a\u200b\u540c\u65f6\u200b\u8f93\u51fa\u200b\u5ba2\u6237\u7aef\u200b\u548c\u200b\u670d\u52a1\u5668\u200b\u4e24\u8005\u200b\u7684\u200b\u7248\u672c\u4fe1\u606f\u200b\uff0c\u200b\u5982\u200b</p> <pre><code>Client: Docker Engine - Community\n Version:           24.0.5\n API version:       1.43\n Go version:        go1.20.6\n Git commit:        ced0996600\n Built:             Wed Jul 19 19:44:22 2023\n OS/Arch:           darwin/arm64\n Context:           default\n\nServer: Docker Desktop 4.21.1 (114176)\n Engine:\n  Version:          24.0.2\n  API version:      1.43 (minimum version 1.12)\n  Go version:       go1.20.4\n  Git commit:       659604f\n  Built:            Thu May 25 21:52:17 2023\n  OS/Arch:          linux/amd64\n  Experimental:     false\n containerd:\n  Version:          1.6.21\n  GitCommit:        3dce8eb055cbb6872793272b4f20ed16117344f8\n runc:\n  Version:          1.1.7\n  GitCommit:        v1.1.7-0-g860f061\n docker-init:\n  Version:          0.19.0\n  GitCommit:        de40ad0\n</code></pre> <p>\u200b\u6302\u8f7d\u200b\u5377\u200b</p> <p>\u200b\u5728\u200bDocker\u200b\u4e2d\u200b\uff0c\u200b\u65e0\u8bba\u662f\u200b\u4f7f\u7528\u200b<code>-v</code>\u200b\u53c2\u6570\u200b\u8fd8\u662f\u200b<code>--mount</code>\u200b\u53c2\u6570\u200b\u6302\u8f7d\u200b\u7684\u200b\u5377\u200b\u7684\u200b\u6e90\u200b\u8def\u5f84\u200b\u90fd\u200b\u662f\u200b\u76f8\u5bf9\u200b\u4e8e\u200bDocker Daemon\u200b\u800c\u8a00\u200b\u7684\u200b\u3002</p>"},{"location":"coding/configuration/docker/#_5","title":"\u53c2\u8003\u200b\u9605\u8bfb","text":"<p>Docker registry\uff1a</p> <ul> <li>Working with the Container registry (GitHub)</li> </ul> <p>Dockerfile\uff1a</p> <ul> <li>Dockerfile reference</li> </ul> <p>\u200b\u8fdc\u7a0b\u200b\u8bbf\u95ee\u200bDocker daemon\uff1a</p> <ul> <li>Windows Docker \u200b\u4e3b\u673a\u200b\u8fdc\u7a0b\u7ba1\u7406\u200b</li> <li>Protect the Docker daemon socket</li> <li>dockerd</li> <li>Docker Remote API with TLS client authentication via container</li> <li>How can I mount a volume of files to a remote docker daemon?</li> </ul>"},{"location":"coding/configuration/nginx-rtmp/","title":"\u6e90\u7801\u200b\u7f16\u8bd1\u200bnginx\u200b\u5e76\u200b\u6dfb\u52a0\u200bRTMP\u200b\u6a21\u5757","text":""},{"location":"coding/configuration/nginx-rtmp/#_1","title":"\u5148\u51b3\u6761\u4ef6","text":"<ul> <li>nginx\u200b\u6e90\u4ee3\u7801\u200b\uff0c\u200b\u9700\u8981\u200b\u4ece\u200bGitHub Release\u200b\u4e0b\u8f7d\u200b\uff0c\u200b\u4ece\u200bnginx\u200b\u5b98\u7f51\u200b\u4e0b\u8f7d\u200b\u7684\u200b\u6e90\u4ee3\u7801\u200b\u5728\u200b\u540e\u7eed\u200b\u7f16\u8bd1\u200b\u65f6\u4f1a\u200b\u62a5\u200b\u5982\u4e0b\u200b\u9519\u8bef\u200b\uff1a   <pre><code>NMAKE : fatal error U1073: don't know how to make 'src/os/win32/ngx_win32_config.h'\n</code></pre></li> <li>openssl\u200b\u6e90\u4ee3\u7801\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4ece\u200bGitHub Release\u200b\u4e0b\u8f7d\u200b</li> <li>zlib\u200b\u6e90\u4ee3\u7801\u200b\uff0cGitHub\u200b\u4ed3\u5e93\u200b\uff1ahttps://github.com/madler/zlib</li> <li>pcre\u200b\u6e90\u4ee3\u7801\u200b\uff0c\u200b\u5b98\u7f51\u200b\u63d0\u4f9b\u200b\u5404\u200b\u7248\u672c\u200b\u7684\u200b\u4e0b\u8f7d\u200b\u94fe\u63a5\u200b\uff1ahttps://ftp.pcre.org/pub/pcre/</li> <li>RMTP\u200b\u63d2\u4ef6\u200b\uff1ahttps://github.com/arut/nginx-rtmp-module</li> <li>MSYS\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200bMinGW Installer\u200b\u8fdb\u884c\u200b\u5b89\u88c5\u200b</li> <li>Visual Studio\uff0c\u200b\u9700\u8981\u200b\u5b89\u88c5\u200bC++\u200b\u7f16\u8bd1\u5668\u200b</li> <li>NASM</li> <li>Perl&gt;=5.30</li> </ul> <p>\u200b\u672c\u6b21\u200b\u7f16\u8bd1\u200b\u4f7f\u7528\u200b\u7684\u200b\u73af\u5883\u200b\uff1a</p> <ul> <li>Windows 19042.662</li> <li>Visual Studio 2019</li> <li>Visual Studio Build Tools 2017</li> <li>Windows 10 SDK 10.0.17763.0</li> <li>nginx 1.19.5</li> <li>openssl 1.1.1g</li> <li>zlib 1.2.11</li> <li>pcre 8.44</li> <li>perl 5.32 (Strawberry)</li> <li>nasm 2.15.05</li> <li>nmake 14.27.29112.0</li> <li>MSVC(cl) 19.27.29112</li> </ul> <p>\u200b\u5c06\u200b\u5305\u542b\u200b<code>perl.exe</code>\u3001<code>nasm.exe</code>\u3001<code>sed.exe</code>\u200b\u7684\u200b\u76ee\u5f55\u200b\u6dfb\u52a0\u200b\u5230\u200bPATH\u200b\u73af\u5883\u53d8\u91cf\u200b\u3002</p>"},{"location":"coding/configuration/nginx-rtmp/#_2","title":"\u51c6\u5907\u200b\u5de5\u4f5c","text":"<p>\u200b\u542f\u52a8\u200bMSYS\uff0c<code>cd</code>\u200b\u5230\u200bnginx\u200b\u7684\u200b\u6e90\u4ee3\u7801\u200b\u76ee\u5f55\u200b\uff0c\u200b\u5207\u6362\u200b\u5230\u200bC\u200b\u76d8\u200b\u6839\u76ee\u5f55\u200b\u7684\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>cd /c/\n</code></pre> <p>\u200b\u6267\u884c\u200b<code>auto/configure</code>\uff0c\u200b\u6839\u636e\u200bnginx\u200b\u5b98\u7f51\u200b\u63d0\u4f9b\u200b\u7684\u200b\u53c2\u6570\u200b\u8fdb\u884c\u200b\uff0c\u200b\u6dfb\u52a0\u200b``\u200b\u53c2\u6570\u200b\uff0c\u200b\u5177\u4f53\u200b\u547d\u4ee4\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>auto/configure \\\n    --with-cc=cl \\\n    --with-debug \\\n    --prefix=. \\\n    --conf-path=conf/nginx.conf \\\n    --pid-path=logs/nginx.pid \\\n    --http-log-path=logs/access.log \\\n    --error-log-path=logs/error.log \\\n    --sbin-path=nginx.exe \\\n    --http-client-body-temp-path=temp/client_body_temp \\\n    --http-proxy-temp-path=temp/proxy_temp \\\n    --http-fastcgi-temp-path=temp/fastcgi_temp \\\n    --http-scgi-temp-path=temp/scgi_temp \\\n    --http-uwsgi-temp-path=temp/uwsgi_temp \\\n    --with-cc-opt=-DFD_SETSIZE=1024 \\\n    --with-pcre=pcre-8.44 \\\n    --with-zlib=zlib \\\n    --with-openssl=openssl \\\n    --with-openssl-opt=no-asm \\\n    --with-http_ssl_module \\\n    --add-module=nginx-rtmp-module\n</code></pre> <p>\u200b\u6ce8\u610f\u200b\u6b64\u5904\u200b<code>--with-pcre</code>\u3001<code>--with-zlib</code>\u3001<code>--with-openssl</code>\u200b\u53c2\u6570\u200b\u4e2d\u200b\u63d0\u4f9b\u200b\u7684\u200b\u8def\u5f84\u200b\u9700\u8981\u200b\u4e0e\u200bPCRE\u3001zlib\u3001openssl\u200b\u7684\u200b\u6e90\u4ee3\u7801\u200b\u8def\u5f84\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u3002</p> <p>\u200b\u6267\u884c\u200b\u5b8c\u6bd5\u200b\u540e\u200b\u751f\u6210\u200bMakefile\u200b\u6587\u4ef6\u200b\u300232\u200b\u4f4d\u200b\u73af\u5883\u200b\u4e0b\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u5f00\u59cb\u200b\u7f16\u8bd1\u200b\uff0c64\u200b\u4f4d\u200b\u73af\u5883\u200b\u4e0b\u200b\u9700\u8981\u200b\u5bf9\u200b<code>auto\\lib\\openssl\\makefile.msvc</code>\u200b\u6539\u4e3a\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b</p> <pre><code># Copyright (C) Igor Sysoev\n# Copyright (C) Nginx, Inc.\n\n\nall:\n    cd $(OPENSSL)\n\n    perl Configure VC-WIN64A no-shared              \\\n        --prefix=\"%cd%/openssl\"                     \\\n        --openssldir=\"%cd%/openssl/ssl\"             \\\n        $(OPENSSL_OPT)\n\n    if exist ms\\do_win64a.bat (                     \\\n        ms\\do_win64a                                \\\n        &amp;&amp; $(MAKE) -f ms\\nt.mak                     \\\n        &amp;&amp; $(MAKE) -f ms\\nt.mak install             \\\n    ) else (                                        \\\n        $(MAKE)                                     \\\n        &amp;&amp; $(MAKE) install_sw                       \\\n    )\n</code></pre> <p>\u200b\u5c06\u200b<code>VC-WIN32</code>\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>VC-WIN64A</code>\uff0c\u200b\u6240\u6709\u200b\u7684\u200b<code>ms\\do_ms.bat</code>\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>ms\\do_win64a.bat</code>\u3002</p>"},{"location":"coding/configuration/nginx-rtmp/#msvc","title":"MSVC\u200b\u7f16\u8bd1","text":"<p>\u200b\u7f16\u8f91\u200b\u5982\u4e0b\u200b\u6e90\u4ee3\u7801\u200b\u6587\u4ef6\u200b\uff1a</p> <ul> <li><code>nginx-rtmp-module/ngx_rtmp_flv_module.c</code>\uff0c\u200b\u5b9a\u4f4d\u200b\u5230\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b\u5757\u200b\uff1a</li> </ul> <p>509\u200b\u884c\u200b\uff1a</p> <pre><code>ngx_rtmp_prepare_message(s, &amp;h, ctx-&gt;msg_mask &amp; (1 &lt;&lt; h.type) ?\n                           &amp;lh : NULL, out);\n</code></pre> <p>521\u200b\u884c\u200b\uff1a</p> <pre><code>ctx-&gt;msg_mask |= (1 &lt;&lt; h.type);\n</code></pre> <p>\u200b\u5c06\u200b\u5176\u4e2d\u200b\u7684\u200b<code>(1 &lt;&lt; h.type)</code>\u200b\u6539\u4e3a\u200b<code>(unsigned long)(1 &lt;&lt; h.type)</code>\u3002</p> <ul> <li><code>nginx-rtmp-module/ngx_rtmp_core_module.c</code>\uff0c\u200b\u5b9a\u4f4d\u200b\u5230\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b\u5757\u200b\uff1a</li> </ul> <p>611\u200b\u884c\u200b\uff1a</p> <pre><code>struct sockaddr  *sa;\n</code></pre> <p>\u200b\u5c06\u200b\u6b64\u5904\u200b\u7684\u200b\u53d8\u91cf\u540d\u200b<code>sa</code>\u200b\u6539\u4e3a\u200b<code>sa2</code>\uff0c\u200b\u5e76\u200b\u5c06\u200b\u6b64\u540e\u200b\u6240\u6709\u200b\u7684\u200b<code>sa</code>\u200b\u53d8\u91cf\u200b\u6539\u4e3a\u200b<code>sa2</code>\u3002</p> <p>\u200b\u542f\u52a8\u200bx64 Native Tools Command Prompt for VS 2019\uff0c\u200b\u5207\u6362\u200b\u5230\u200b\u6e90\u4ee3\u7801\u200b\u76ee\u5f55\u200b\uff0c\u200b\u6267\u884c\u200b<code>nmake</code>\u200b\u542f\u52a8\u200b\u7f16\u8bd1\u200b\u3002\u200b\u8f93\u51fa\u200b\u6587\u4ef6\u200b\u4e3a\u200b<code>objs/nginx.exe</code></p>"},{"location":"coding/configuration/nginx-rtmp/#nginx","title":"nginx\u200b\u914d\u7f6e","text":"<p>\u200b\u5c06\u200b<code>objs/nginx.exe</code>\u200b\u590d\u5236\u200b\u4e00\u4efd\u200b\u3002\u200b\u5728\u200b\u6240\u5728\u200b\u76ee\u5f55\u200b\u4e0b\u200b\u521b\u5efa\u200b\u5982\u4e0b\u200b\u6587\u4ef6\u200b\u7ed3\u6784\u200b\uff1a</p> <pre><code>.\n\u251c\u2500 nginx.exe\n\u251c\u2500 conf/\n\u2502  \u2514\u2500 nginx.conf\n\u251c\u2500 html\n\u2502  \u2514\u2500 index.xsl\n\u251c\u2500 logs\n\u2514\u2500 temp\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff0c<code>html/index.xsl</code>\u200b\u5199\u5165\u200b\u5982\u4e0b\u200b\u5185\u5bb9\u200b</p> html/index.xsl\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b <pre><code>&lt;?xml version=\"1.0\" encoding=\"utf-8\" ?&gt;\n&lt;!--\nCopyright (C) Roman Arutyunyan\n--&gt;\n&lt;xsl:stylesheet version=\"1.0\" xmlns:xsl=\"http://www.w3.org/1999/XSL/Transform\"&gt;\n&lt;xsl:template match=\"/\"&gt;\n    &lt;html&gt;\n        &lt;head&gt;\n            &lt;title&gt;RTMP statistics&lt;/title&gt;\n        &lt;/head&gt;\n        &lt;body&gt;\n            &lt;xsl:apply-templates select=\"rtmp\"/&gt;\n            &lt;hr/&gt;\n            Generated by &lt;a href='https://github.com/arut/nginx-rtmp-module'&gt;\n            nginx-rtmp-module&lt;/a&gt;&amp;#160;&lt;xsl:value-of select=\"/rtmp/nginx_rtmp_version\"/&gt;,\n            &lt;a href=\"http://nginx.org\"&gt;nginx&lt;/a&gt;&amp;#160;&lt;xsl:value-of select=\"/rtmp/nginx_version\"/&gt;,\n            pid &lt;xsl:value-of select=\"/rtmp/pid\"/&gt;,\n            built &lt;xsl:value-of select=\"/rtmp/built\"/&gt;&amp;#160;&lt;xsl:value-of select=\"/rtmp/compiler\"/&gt;\n        &lt;/body&gt;\n    &lt;/html&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template match=\"rtmp\"&gt;\n    &lt;table cellspacing=\"1\" cellpadding=\"5\"&gt;\n        &lt;tr bgcolor=\"#999999\"&gt;\n            &lt;th&gt;RTMP&lt;/th&gt;\n            &lt;th&gt;#clients&lt;/th&gt;\n            &lt;th colspan=\"4\"&gt;Video&lt;/th&gt;\n            &lt;th colspan=\"4\"&gt;Audio&lt;/th&gt;\n            &lt;th&gt;In bytes&lt;/th&gt;\n            &lt;th&gt;Out bytes&lt;/th&gt;\n            &lt;th&gt;In bits/s&lt;/th&gt;\n            &lt;th&gt;Out bits/s&lt;/th&gt;\n            &lt;th&gt;State&lt;/th&gt;\n            &lt;th&gt;Time&lt;/th&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;\n            &lt;td colspan=\"2\"&gt;Accepted: &lt;xsl:value-of select=\"naccepted\"/&gt;&lt;/td&gt;\n            &lt;th bgcolor=\"#999999\"&gt;codec&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;bits/s&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;size&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;fps&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;codec&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;bits/s&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;freq&lt;/th&gt;\n            &lt;th bgcolor=\"#999999\"&gt;chan&lt;/th&gt;\n            &lt;td&gt;\n                &lt;xsl:call-template name=\"showsize\"&gt;\n                    &lt;xsl:with-param name=\"size\" select=\"bytes_in\"/&gt;\n                &lt;/xsl:call-template&gt;\n            &lt;/td&gt;\n            &lt;td&gt;\n                &lt;xsl:call-template name=\"showsize\"&gt;\n                    &lt;xsl:with-param name=\"size\" select=\"bytes_out\"/&gt;\n                &lt;/xsl:call-template&gt;\n            &lt;/td&gt;\n            &lt;td&gt;\n                &lt;xsl:call-template name=\"showsize\"&gt;\n                    &lt;xsl:with-param name=\"size\" select=\"bw_in\"/&gt;\n                    &lt;xsl:with-param name=\"bits\" select=\"1\"/&gt;\n                    &lt;xsl:with-param name=\"persec\" select=\"1\"/&gt;\n                &lt;/xsl:call-template&gt;\n            &lt;/td&gt;\n            &lt;td&gt;\n                &lt;xsl:call-template name=\"showsize\"&gt;\n                    &lt;xsl:with-param name=\"size\" select=\"bw_out\"/&gt;\n                    &lt;xsl:with-param name=\"bits\" select=\"1\"/&gt;\n                    &lt;xsl:with-param name=\"persec\" select=\"1\"/&gt;\n                &lt;/xsl:call-template&gt;\n            &lt;/td&gt;\n            &lt;td/&gt;\n            &lt;td&gt;\n                &lt;xsl:call-template name=\"showtime\"&gt;\n                    &lt;xsl:with-param name=\"time\" select=\"/rtmp/uptime * 1000\"/&gt;\n                &lt;/xsl:call-template&gt;\n            &lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;xsl:apply-templates select=\"server\"/&gt;\n    &lt;/table&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template match=\"server\"&gt;\n    &lt;xsl:apply-templates select=\"application\"/&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template match=\"application\"&gt;\n    &lt;tr bgcolor=\"#999999\"&gt;\n        &lt;td&gt;\n            &lt;b&gt;&lt;xsl:value-of select=\"name\"/&gt;&lt;/b&gt;\n        &lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;xsl:apply-templates select=\"live\"/&gt;\n    &lt;xsl:apply-templates select=\"play\"/&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template match=\"live\"&gt;\n    &lt;tr bgcolor=\"#aaaaaa\"&gt;\n        &lt;td&gt;\n            &lt;i&gt;live streams&lt;/i&gt;\n        &lt;/td&gt;\n        &lt;td align=\"middle\"&gt;\n            &lt;xsl:value-of select=\"nclients\"/&gt;\n        &lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;xsl:apply-templates select=\"stream\"/&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template match=\"play\"&gt;\n    &lt;tr bgcolor=\"#aaaaaa\"&gt;\n        &lt;td&gt;\n            &lt;i&gt;vod streams&lt;/i&gt;\n        &lt;/td&gt;\n        &lt;td align=\"middle\"&gt;\n            &lt;xsl:value-of select=\"nclients\"/&gt;\n        &lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;xsl:apply-templates select=\"stream\"/&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template match=\"stream\"&gt;\n    &lt;tr valign=\"top\"&gt;\n        &lt;xsl:attribute name=\"bgcolor\"&gt;\n            &lt;xsl:choose&gt;\n                &lt;xsl:when test=\"active\"&gt;#cccccc&lt;/xsl:when&gt;\n                &lt;xsl:otherwise&gt;#dddddd&lt;/xsl:otherwise&gt;\n            &lt;/xsl:choose&gt;\n        &lt;/xsl:attribute&gt;\n        &lt;td&gt;\n            &lt;a href=\"\"&gt;\n                &lt;xsl:attribute name=\"onclick\"&gt;\n                    var d=document.getElementById('&lt;xsl:value-of select=\"../../name\"/&gt;-&lt;xsl:value-of select=\"name\"/&gt;');\n                    d.style.display=d.style.display=='none'?'':'none';\n                    return false\n                &lt;/xsl:attribute&gt;\n                &lt;xsl:value-of select=\"name\"/&gt;\n                &lt;xsl:if test=\"string-length(name) = 0\"&gt;\n                    [EMPTY]\n                &lt;/xsl:if&gt;\n            &lt;/a&gt;\n        &lt;/td&gt;\n        &lt;td align=\"middle\"&gt; &lt;xsl:value-of select=\"nclients\"/&gt; &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:value-of select=\"meta/video/codec\"/&gt;&amp;#160;&lt;xsl:value-of select=\"meta/video/profile\"/&gt;&amp;#160;&lt;xsl:value-of select=\"meta/video/level\"/&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showsize\"&gt;\n                &lt;xsl:with-param name=\"size\" select=\"bw_video\"/&gt;\n                &lt;xsl:with-param name=\"bits\" select=\"1\"/&gt;\n                &lt;xsl:with-param name=\"persec\" select=\"1\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:apply-templates select=\"meta/video/width\"/&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:value-of select=\"meta/video/frame_rate\"/&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:value-of select=\"meta/audio/codec\"/&gt;&amp;#160;&lt;xsl:value-of select=\"meta/audio/profile\"/&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showsize\"&gt;\n                &lt;xsl:with-param name=\"size\" select=\"bw_audio\"/&gt;\n                &lt;xsl:with-param name=\"bits\" select=\"1\"/&gt;\n                &lt;xsl:with-param name=\"persec\" select=\"1\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:apply-templates select=\"meta/audio/sample_rate\"/&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:value-of select=\"meta/audio/channels\"/&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showsize\"&gt;\n            &lt;xsl:with-param name=\"size\" select=\"bytes_in\"/&gt;\n        &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showsize\"&gt;\n                &lt;xsl:with-param name=\"size\" select=\"bytes_out\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showsize\"&gt;\n                &lt;xsl:with-param name=\"size\" select=\"bw_in\"/&gt;\n                &lt;xsl:with-param name=\"bits\" select=\"1\"/&gt;\n                &lt;xsl:with-param name=\"persec\" select=\"1\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showsize\"&gt;\n                &lt;xsl:with-param name=\"size\" select=\"bw_out\"/&gt;\n                &lt;xsl:with-param name=\"bits\" select=\"1\"/&gt;\n                &lt;xsl:with-param name=\"persec\" select=\"1\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n        &lt;td&gt;&lt;xsl:call-template name=\"streamstate\"/&gt;&lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showtime\"&gt;\n            &lt;xsl:with-param name=\"time\" select=\"time\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr style=\"display:none\"&gt;\n        &lt;xsl:attribute name=\"id\"&gt;\n            &lt;xsl:value-of select=\"../../name\"/&gt;-&lt;xsl:value-of select=\"name\"/&gt;\n        &lt;/xsl:attribute&gt;\n        &lt;td colspan=\"16\" ngcolor=\"#eeeeee\"&gt;\n            &lt;table cellspacing=\"1\" cellpadding=\"5\"&gt;\n                &lt;tr&gt;\n                    &lt;th&gt;Id&lt;/th&gt;\n                    &lt;th&gt;State&lt;/th&gt;\n                    &lt;th&gt;Address&lt;/th&gt;\n                    &lt;th&gt;Flash version&lt;/th&gt;\n                    &lt;th&gt;Page URL&lt;/th&gt;\n                    &lt;th&gt;SWF URL&lt;/th&gt;\n                    &lt;th&gt;Dropped&lt;/th&gt;\n                    &lt;th&gt;Timestamp&lt;/th&gt;\n                    &lt;th&gt;A-V&lt;/th&gt;\n                    &lt;th&gt;Time&lt;/th&gt;\n                &lt;/tr&gt;\n                &lt;xsl:apply-templates select=\"client\"/&gt;\n            &lt;/table&gt;\n        &lt;/td&gt;\n    &lt;/tr&gt;\n&lt;/xsl:template&gt;\n&lt;xsl:template name=\"showtime\"&gt;\n    &lt;xsl:param name=\"time\"/&gt;\n\n    &lt;xsl:if test=\"$time &amp;gt; 0\"&gt;\n        &lt;xsl:variable name=\"sec\"&gt;\n            &lt;xsl:value-of select=\"floor($time div 1000)\"/&gt;\n        &lt;/xsl:variable&gt;\n\n        &lt;xsl:if test=\"$sec &amp;gt;= 86400\"&gt;\n            &lt;xsl:value-of select=\"floor($sec div 86400)\"/&gt;d\n        &lt;/xsl:if&gt;\n\n        &lt;xsl:if test=\"$sec &amp;gt;= 3600\"&gt;\n            &lt;xsl:value-of select=\"(floor($sec div 3600)) mod 24\"/&gt;h\n        &lt;/xsl:if&gt;\n\n        &lt;xsl:if test=\"$sec &amp;gt;= 60\"&gt;\n            &lt;xsl:value-of select=\"(floor($sec div 60)) mod 60\"/&gt;m\n        &lt;/xsl:if&gt;\n\n        &lt;xsl:value-of select=\"$sec mod 60\"/&gt;s\n    &lt;/xsl:if&gt;\n&lt;/xsl:template&gt;\n\n&lt;xsl:template name=\"showsize\"&gt;\n    &lt;xsl:param name=\"size\"/&gt;\n    &lt;xsl:param name=\"bits\" select=\"0\" /&gt;\n    &lt;xsl:param name=\"persec\" select=\"0\" /&gt;\n    &lt;xsl:variable name=\"sizen\"&gt;\n        &lt;xsl:value-of select=\"floor($size div 1024)\"/&gt;\n    &lt;/xsl:variable&gt;\n    &lt;xsl:choose&gt;\n        &lt;xsl:when test=\"$sizen &amp;gt;= 1073741824\"&gt;\n            &lt;xsl:value-of select=\"format-number($sizen div 1073741824,'#.###')\"/&gt; T&lt;/xsl:when&gt;\n\n        &lt;xsl:when test=\"$sizen &amp;gt;= 1048576\"&gt;\n            &lt;xsl:value-of select=\"format-number($sizen div 1048576,'#.###')\"/&gt; G&lt;/xsl:when&gt;\n\n        &lt;xsl:when test=\"$sizen &amp;gt;= 1024\"&gt;\n            &lt;xsl:value-of select=\"format-number($sizen div 1024,'#.##')\"/&gt; M&lt;/xsl:when&gt;\n        &lt;xsl:when test=\"$sizen &amp;gt;= 0\"&gt;\n            &lt;xsl:value-of select=\"$sizen\"/&gt; K&lt;/xsl:when&gt;\n    &lt;/xsl:choose&gt;\n    &lt;xsl:if test=\"string-length($size) &amp;gt; 0\"&gt;\n        &lt;xsl:choose&gt;\n            &lt;xsl:when test=\"$bits = 1\"&gt;b&lt;/xsl:when&gt;\n            &lt;xsl:otherwise&gt;B&lt;/xsl:otherwise&gt;\n        &lt;/xsl:choose&gt;\n        &lt;xsl:if test=\"$persec = 1\"&gt;/s&lt;/xsl:if&gt;\n    &lt;/xsl:if&gt;\n&lt;/xsl:template&gt;\n\n&lt;xsl:template name=\"streamstate\"&gt;\n    &lt;xsl:choose&gt;\n        &lt;xsl:when test=\"active\"&gt;active&lt;/xsl:when&gt;\n        &lt;xsl:otherwise&gt;idle&lt;/xsl:otherwise&gt;\n    &lt;/xsl:choose&gt;\n&lt;/xsl:template&gt;\n\n\n&lt;xsl:template name=\"clientstate\"&gt;\n    &lt;xsl:choose&gt;\n        &lt;xsl:when test=\"publishing\"&gt;publishing&lt;/xsl:when&gt;\n        &lt;xsl:otherwise&gt;playing&lt;/xsl:otherwise&gt;\n    &lt;/xsl:choose&gt;\n&lt;/xsl:template&gt;\n\n\n&lt;xsl:template match=\"client\"&gt;\n    &lt;tr&gt;\n        &lt;xsl:attribute name=\"bgcolor\"&gt;\n            &lt;xsl:choose&gt;\n                &lt;xsl:when test=\"publishing\"&gt;#cccccc&lt;/xsl:when&gt;\n                &lt;xsl:otherwise&gt;#eeeeee&lt;/xsl:otherwise&gt;\n            &lt;/xsl:choose&gt;\n        &lt;/xsl:attribute&gt;\n        &lt;td&gt;&lt;xsl:value-of select=\"id\"/&gt;&lt;/td&gt;\n        &lt;td&gt;&lt;xsl:call-template name=\"clientstate\"/&gt;&lt;/td&gt;\n        &lt;td&gt;\n            &lt;a target=\"_blank\"&gt;\n                &lt;xsl:attribute name=\"href\"&gt;\n                    http://apps.db.ripe.net/search/query.html&amp;#63;searchtext=&lt;xsl:value-of select=\"address\"/&gt;\n                &lt;/xsl:attribute&gt;\n                &lt;xsl:attribute name=\"title\"&gt;whois&lt;/xsl:attribute&gt;\n                &lt;xsl:value-of select=\"address\"/&gt;\n            &lt;/a&gt;\n        &lt;/td&gt;\n        &lt;td&gt;&lt;xsl:value-of select=\"flashver\"/&gt;&lt;/td&gt;\n        &lt;td&gt;\n            &lt;a target=\"_blank\"&gt;\n                &lt;xsl:attribute name=\"href\"&gt;\n                    &lt;xsl:value-of select=\"pageurl\"/&gt;\n                &lt;/xsl:attribute&gt;\n                &lt;xsl:value-of select=\"pageurl\"/&gt;\n            &lt;/a&gt;\n        &lt;/td&gt;\n        &lt;td&gt;&lt;xsl:value-of select=\"swfurl\"/&gt;&lt;/td&gt;\n        &lt;td&gt;&lt;xsl:value-of select=\"dropped\"/&gt;&lt;/td&gt;\n        &lt;td&gt;&lt;xsl:value-of select=\"timestamp\"/&gt;&lt;/td&gt;\n        &lt;td&gt;&lt;xsl:value-of select=\"avsync\"/&gt;&lt;/td&gt;\n        &lt;td&gt;\n            &lt;xsl:call-template name=\"showtime\"&gt;\n            &lt;xsl:with-param name=\"time\" select=\"time\"/&gt;\n            &lt;/xsl:call-template&gt;\n        &lt;/td&gt;\n    &lt;/tr&gt;\n&lt;/xsl:template&gt;\n\n&lt;xsl:template match=\"publishing\"&gt;\n    publishing\n&lt;/xsl:template&gt;\n\n&lt;xsl:template match=\"active\"&gt;\n    active\n&lt;/xsl:template&gt;\n\n&lt;xsl:template match=\"width\"&gt;\n    &lt;xsl:value-of select=\".\"/&gt;x&lt;xsl:value-of select=\"../height\"/&gt;\n&lt;/xsl:template&gt;\n\n&lt;/xsl:stylesheet&gt;\n</code></pre> <p><code>conf/nginx.conf</code>\u200b\u5199\u5165\u200b\u5982\u4e0b\u200b\u5185\u5bb9\u200b\uff1a</p> <p>conf/nginx.conf\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b</p> <pre><code>worker_processes  8;\nerror_log  logs/error.log info;\nevents {\n    worker_connections  1024;\n}\n\nrtmp {\n    server {\n        listen 1935;\n\n        application live {\n            live on;\n        }\n\n        application hls {\n            live on;\n            hls on;  \n            hls_path temp/hls;  \n            hls_fragment 8s;  \n        }\n    }\n}\n\nhttp {\n    server {\n        listen      8080;\n        root html;\n        location / {\n            rtmp_stat all;\n            rtmp_stat_stylesheet index.xsl;\n        }\n\n        location /index.xsl {\n            root html;\n        }\n\n        location /hls {  \n            types{  \n                application/vnd.apple.mpegurl m3u8;  \n                video/mp2t ts;  \n            }  \n            alias temp/hls;  \n            expires -1;  \n        }  \n    }\n}\n</code></pre> <p>\u200b\u53cc\u51fb\u200b<code>nginx.exe</code>\u200b\u5373\u53ef\u200b\u542f\u52a8\u200b\u670d\u52a1\u5668\u200b\uff0c\u200b\u670d\u52a1\u5668\u200b\u540c\u65f6\u200b\u76d1\u542c\u200b1935\u200b\u7aef\u53e3\u200b\u4e0e\u200b8080\u200b\u7aef\u53e3\u200b</p> <ul> <li>\u200b\u63a8\u6d41\u200b\u4f7f\u7528\u200b\u7684\u200bRTMP\u200b\u670d\u52a1\u5668\u200b\u5730\u5740\u200b\u4e3a\u200b\uff1a<code>rtmp://localhost:1935/live</code></li> <li>\u200b\u8bbf\u95ee\u200b<code>http://localhost:8080</code>\u200b\u53ef\u4ee5\u200b\u67e5\u770b\u200b\u670d\u52a1\u5668\u7aef\u200b\u7684\u200b\u7edf\u8ba1\u200b\u4fe1\u606f\u200b</li> <li>\u200b\u5ba2\u6237\u7aef\u200b\uff08\u200b\u5982\u200bVLC\u200b\u7b49\u200b\uff09\u200b\u53ef\u4ee5\u200b\u8bbf\u95ee\u200b<code>rtmp://localhost:1935/live</code>\u200b\u63a5\u6536\u200b\u6d41\u200b\uff0c<code>localhost</code>\u200b\u53ef\u4ee5\u200b\u66f4\u200b\u6539\u4e3a\u200b\u672c\u673a\u200b\u7684\u200bIP\u200b\u5730\u5740\u200b</li> </ul> <p>\u200b\u4f7f\u7528\u200b\u547d\u4ee4\u200b<code>.\\nginx.exe -s stop</code>\u200b\u5173\u95ed\u200b\u670d\u52a1\u5668\u200b</p>"},{"location":"coding/configuration/ssh/","title":"ssh","text":""},{"location":"coding/configuration/ssh/#_1","title":"\u57fa\u672c\u200b\u4f7f\u7528","text":"<p>SSH\u200b\u53ef\u4ee5\u200b\u8fdc\u7a0b\u200b\u8fde\u63a5\u200b\u670d\u52a1\u5668\u200b\u7684\u200b\u7ec8\u7aef\u200b\u3002\u200b\u547d\u4ee4\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b\uff0c\u200b\u5176\u4e2d\u200b<code>&lt;user&gt;</code>\u200b\u4e3a\u200b\u8fdc\u7a0b\u200b\u670d\u52a1\u5668\u200b\u4e0a\u200b\u7684\u200b\u7528\u6237\u540d\u200b\uff0c<code>&lt;hostname&gt;</code>\u200b\u662f\u200b\u8fdc\u7a0b\u200b\u670d\u52a1\u5668\u200b\u7684\u200b\u7f51\u7edc\u5730\u5740\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u662f\u200bIP\u200b\u5730\u5740\u200b\u6216\u200b\u57df\u540d\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b<code>~/.ssh/config</code>\u200b\u6587\u4ef6\u200b\u4e2d\u200b\u5b9a\u4e49\u200b\u7684\u200b\u4e3b\u673a\u200b\u3002\u200b\u5982\u679c\u200b\u6307\u5b9a\u200b\u4e86\u200b<code>&lt;command&gt;</code>\u200b\u53c2\u6570\u200b\uff0c\u200b\u5219\u200bSSH\u200b\u5728\u200b\u8fde\u63a5\u200b\u6210\u529f\u200b\u540e\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u6267\u884c\u547d\u4ee4\u200b\uff0c\u200b\u5c06\u200b\u7ed3\u679c\u200b\u8f93\u51fa\u200b\u5230\u200b\u7ec8\u7aef\u200b\u5e76\u200b\u5728\u200b\u6267\u884c\u200b\u7ed3\u675f\u200b\u540e\u200b\u65ad\u5f00\u8fde\u63a5\u200b\u3002</p> <pre><code>ssh &lt;user&gt;@&lt;hostname&gt; [&lt;command&gt;]\n</code></pre> <p>\u200b\u82e5\u200b\u5ba2\u6237\u7aef\u200b\u7684\u200b\u7528\u6237\u540d\u200b\u548c\u200b\u670d\u52a1\u5668\u200b\u7684\u200b\u7528\u6237\u540d\u200b\u76f8\u540c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b<code>&lt;user&gt;</code>\u200b\u53c2\u6570\u200b\uff0c\u200b\u5982\u200b</p> <pre><code>ssh &lt;hostname&gt; [&lt;command&gt;]\n</code></pre>"},{"location":"coding/configuration/ssh/#_2","title":"\u7aef\u53e3\u200b\u8f6c\u53d1","text":"<p>SSH\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>-L</code>\u3001<code>-R</code>\u3001<code>-D</code>\u200b\u53c2\u6570\u200b\uff0c\u200b\u5206\u522b\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u672c\u5730\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u3001\u200b\u8fdc\u7a0b\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u4e0e\u200b\u52a8\u6001\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u3002\u200b\u4f7f\u7528\u200b<code>-N</code>\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u5728\u200b\u4e0d\u200b\u8fde\u63a5\u200b\u8fdc\u7a0b\u200b\u7ec8\u7aef\u200b\uff0c\u200b\u53ea\u200b\u5b9e\u73b0\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u3002\u200b\u4e3a\u200b\u6e05\u6670\u200b\u8868\u8ff0\u200b\u5404\u4e2a\u200b\u7aef\u53e3\u200b\uff0c\u200b\u672c\u8282\u200b\u4e2d\u200b\u5047\u8bbe\u200b\u5ba2\u6237\u7aef\u200bA\u3001\u200b\u670d\u52a1\u5668\u200bB\u200b\u548c\u200b\u53e6\u200b\u4e00\u200b\u4e3b\u673a\u200bC\u3002\u200b\u5ba2\u6237\u7aef\u200bA\u200b\u901a\u8fc7\u200bSSH\u200b\u8fde\u63a5\u200b\u81f3\u200b\u670d\u52a1\u5668\u200bB\u3002</p>"},{"location":"coding/configuration/ssh/#_3","title":"\u672c\u5730\u200b\u7aef\u53e3\u200b\u8f6c\u53d1","text":"<p>\u200b\u4f7f\u7528\u200b<code>-L</code>\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u6253\u5f00\u200bSSH\u200b\u7684\u200b\u672c\u5730\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u529f\u80fd\u200b\u3002\u200b\u7531\u4e8e\u200b<code>-L</code>\u200b\u53c2\u6570\u200b\u5c06\u200b\u6307\u5411\u200b\u5ba2\u6237\u7aef\u200b\u672c\u5730\u200b\u7684\u200b\u8bf7\u6c42\u200b\u8f6c\u53d1\u7ed9\u200b\u670d\u52a1\u5668\u200b\uff0c\u200b\u56e0\u6b64\u200b\u79f0\u4e3a\u200b\u672c\u5730\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u3002\u200b\u542f\u52a8\u200b\u672c\u5730\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u7684\u200b\u547d\u4ee4\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>ssh -L [&lt;bind-addr&gt;]:&lt;port-of-A&gt;:&lt;addr-of-C&gt;:&lt;port-of-C&gt; &lt;user&gt;@&lt;hostname&gt;\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u547d\u4ee4\u200b\u8868\u793a\u200b\u5c06\u200b\u5ba2\u6237\u7aef\u200bA\u200b\u4e0a\u200b\u6307\u5411\u200b<code>&lt;port-of-A&gt;</code>\u200b\u7aef\u53e3\u200b\u7684\u200b\u672c\u5730\u200b\u8bf7\u6c42\u200b\u901a\u8fc7\u200bSSH\u200b\u8f6c\u53d1\u7ed9\u200b\u670d\u52a1\u5668\u200bB\uff0c\u200b\u7531\u200b\u670d\u52a1\u5668\u200bB\u200b\u5411\u200b\u8fdc\u7a0b\u200b\u4e3b\u673a\u200b<code>&lt;addr-of-C&gt;</code>\u200b\u7684\u200b<code>&lt;port-of-C&gt;</code>\u200b\u7aef\u53e3\u200b\u53d1\u8d77\u200b\u8bf7\u6c42\u200b\u3002\u200b\u82e5\u200b\u6307\u5b9a\u200b<code>&lt;bind-addr&gt;</code>\uff0c\u200b\u5219\u200bSSH\u200b\u4f1a\u200b\u76d1\u542c\u200b<code>&lt;bind-addr&gt;</code>\u200b\u5730\u5740\u200b\uff0c\u200b\u5e76\u200b\u53ea\u200b\u8f6c\u53d1\u200b\u76ee\u6807\u200b\u4e3a\u200b<code>&lt;bind-addr&gt;</code>\u200b\u7684\u200b\u8bf7\u6c42\u200b\u3002</p>"},{"location":"coding/configuration/ssh/#_4","title":"\u8fdc\u7a0b\u200b\u7aef\u53e3\u200b\u8f6c\u53d1","text":"<p>\u200b\u4f7f\u7528\u200b<code>-R</code>\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u6253\u5f00\u200bSSH\u200b\u7684\u200b\u8fdc\u7a0b\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u529f\u80fd\u200b\u3002<code>-R</code>\u200b\u7aef\u53e3\u200b\u5c06\u200b\u6307\u5411\u200b\u8fdc\u7a0b\u200b\u670d\u52a1\u5668\u200b\u7684\u200b\u8bf7\u6c42\u200b\u8f6c\u53d1\u7ed9\u200b\u5ba2\u6237\u7aef\u200b\uff0c\u200b\u56e0\u6b64\u200b\u79f0\u4e3a\u200b\u8fdc\u7a0b\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u3002\u200b\u542f\u52a8\u200b\u8fdc\u7a0b\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u7684\u200b\u547d\u4ee4\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>ssh -R [&lt;bind-addr&gt;]:&lt;port-of-B&gt;:&lt;addr-of-C&gt;:&lt;port-of-C&gt; &lt;user&gt;@&lt;hostname&gt;\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u547d\u4ee4\u200b\u8868\u793a\u200b\u5c06\u200b\u670d\u52a1\u5668\u200bB\u200b\u4e0a\u200b\u6307\u5411\u200b<code>&lt;port-of-B&gt;</code>\u200b\u7aef\u53e3\u200b\u7684\u200b\u8bf7\u6c42\u200b\u901a\u8fc7\u200bSSH\u200b\u8f6c\u53d1\u7ed9\u200b\u5ba2\u6237\u7aef\u200bA\uff0c\u200b\u7531\u200b\u5ba2\u6237\u7aef\u200bA\u200b\u5411\u200b\u53e6\u4e00\u53f0\u200b\u4e3b\u673a\u200b<code>&lt;addr-of-C&gt;</code>\u200b\u7684\u200b<code>&lt;port-of-C&gt;</code>\u200b\u7aef\u53e3\u200b\u53d1\u8d77\u200b\u8bf7\u6c42\u200b\u3002\u200b\u82e5\u200b\u6307\u5b9a\u200b<code>&lt;bind-addr&gt;</code>\uff0c\u200b\u5219\u200bSSH\u200b\u4f1a\u200b\u76d1\u542c\u200b<code>&lt;bind-addr&gt;</code>\u200b\u5730\u5740\u200b\uff0c\u200b\u5e76\u200b\u53ea\u200b\u8f6c\u53d1\u200b\u76ee\u6807\u200b\u4e3a\u200b<code>&lt;bind-addr&gt;</code>\u200b\u7684\u200b\u8bf7\u6c42\u200b\u3002</p>"},{"location":"coding/configuration/ssh/#_5","title":"\u52a8\u6001\u200b\u7aef\u53e3\u200b\u8f6c\u53d1","text":"<p>\u200b\u52a8\u6001\u200b\u7aef\u53e3\u200b\u8f6c\u53d1\u200b\u7684\u200b\u547d\u4ee4\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>ssh -D &lt;port-of-A&gt; &lt;user&gt;@&lt;hostname&gt;\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u547d\u4ee4\u200b\u8868\u793a\u200b\u5c06\u200b\u5ba2\u6237\u7aef\u200bA\u200b\u4e0a\u200b\u6307\u5411\u200b<code>&lt;port-of-A&gt;</code>\u200b\u7684\u200b\u8bf7\u6c42\u200b\u901a\u8fc7\u200bSSH\u200b\u8f6c\u53d1\u7ed9\u200b\u670d\u52a1\u5668\u200bB\uff0c\u200b\u7531\u200bB\u200b\u4ee3\u4e3a\u200b\u53d1\u8d77\u200b\u8bf7\u6c42\u200b\uff0c\u200b\u5e76\u200b\u5c06\u200b\u8bbf\u95ee\u200b\u7ed3\u679c\u200b\u4f20\u8f93\u200b\u56de\u200b\u672c\u5730\u200b\u3002</p> <p>\u200b\u82e5\u200b\u8981\u200b\u8f6c\u53d1\u200b\u6240\u6709\u200b\u672c\u5730\u7f51\u7edc\u200b\u8bf7\u6c42\u200b\uff0c\u200b\u8bbe\u7f6e\u200bSOCKS5\u200b\u4ee3\u7406\u200b\u5c06\u200b\u672c\u5730\u200b\u8bf7\u6c42\u200b\u91cd\u5b9a\u5411\u200b\u5230\u200b<code>&lt;port-of-A&gt;</code>\u200b\u5373\u53ef\u200b\u3002</p>"},{"location":"coding/configuration/ssh/#_6","title":"\u5bc6\u94a5\u200b\u767b\u5f55","text":"<p>SSH\u200b\u9ed8\u8ba4\u200b\u4f7f\u7528\u200b\u7528\u6237\u540d\u200b-\u200b\u5bc6\u7801\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u9a8c\u8bc1\u200b\uff0c\u200b\u6bcf\u6b21\u200b\u767b\u5f55\u200b\u9700\u8981\u200b\u8f93\u5165\u200b\u5bc6\u7801\u200b\u3002\u200b\u53ef\u4ee5\u200b\u914d\u7f6e\u200bSSH\u200b\u4f7f\u7528\u200b\u5bc6\u94a5\u200b\u5bf9\u200b\u5ba2\u6237\u7aef\u200b\u8fdb\u884c\u200b\u9a8c\u8bc1\u200b\uff0c\u200b\u6b64\u540e\u200b\u4f7f\u7528\u200b\u8be5\u200b\u5ba2\u6237\u7aef\u200b\u767b\u5f55\u200b\u5c31\u200b\u4e0d\u518d\u200b\u9700\u8981\u200b\u8f93\u5165\u200b\u5bc6\u7801\u200b\u3002\u200b\u6b65\u9aa4\u200b\u5982\u4e0b\u200b</p> <ol> <li>\u200b\u5728\u200b\u5ba2\u6237\u7aef\u200b\u6267\u884c\u200b<code>ssh-keygen</code>\u200b\u547d\u4ee4\u200b\u751f\u6210\u200b\u5bc6\u94a5\u200b\u5bf9\u200b\uff0c\u200b\u6267\u884c\u200b\u5b8c\u6210\u200b\u540e\u4f1a\u200b\u5728\u200b<code>~/.ssh</code>\u200b\u76ee\u5f55\u200b\u4e0b\u200b\u751f\u6210\u200b\u79c1\u94a5\u200b\u6587\u4ef6\u200b<code>id_rsa</code>\u200b\u4e0e\u200b\u516c\u94a5\u200b\u6587\u4ef6\u200b<code>id_rsa.pub</code>\uff1b</li> <li>\u200b\u6267\u884c\u200b<code>ssh-copy-id &lt;user&gt;@&lt;hostname&gt;</code>\u200b\u547d\u4ee4\u200b\uff0c\u200b\u5c06\u200b\u516c\u94a5\u200b\u590d\u5236\u5230\u200b\u8fdc\u7a0b\u200b\u4e3b\u673a\u200b\u7684\u200b<code>~/.ssh/authorized_key</code>\u200b\u6587\u4ef6\u200b\u4e2d\u200b\uff08\u200b\u624b\u52a8\u200b\u590d\u5236\u7c98\u8d34\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\uff09\u3002</li> </ol>"},{"location":"coding/configuration/ssh/#_7","title":"\u6587\u4ef6\u4f20\u8f93","text":"<p><code>scp</code>\u200b\u53ef\u4ee5\u200b\u501f\u52a9\u200bSSH tunnel\u200b\u5b9e\u73b0\u200b\u6587\u4ef6\u4f20\u8f93\u200b\u3002\u200b\u547d\u4ee4\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>scp SOURCE DESTINATION\n</code></pre> <p>\u200b\u8868\u793a\u200b\u5c06\u200b\u6587\u4ef6\u200bSOURCE\u200b\u590d\u5236\u5230\u200bDESTINATION\u200b\u4e0b\u200b\u3002SOURCE\u200b\u548c\u200bDESTINATION\u200b\u53ef\u4ee5\u200b\u662f\u200b\uff1a</p> <ol> <li>\u200b\u672c\u5730\u200b\u6709\u6548\u200b\u8def\u5f84\u200b</li> <li>\u200b\u683c\u5f0f\u200b\u4e3a\u200b<code>&lt;user&gt;@&lt;ip&gt;:&lt;path&gt;</code>\u200b\u7684\u200b\u8fdc\u7a0b\u200b\u8def\u5f84\u200b</li> </ol>"},{"location":"coding/configuration/update-alternatives/","title":"update-alternatives","text":"<p><code>update-alternatives</code>\u200b\u662f\u200b\u7528\u4e8e\u200b\u7ba1\u7406\u200b\u8f6f\u200b\u94fe\u63a5\u200b\u7684\u200b\u547d\u4ee4\u200b\u3002</p>"},{"location":"coding/configuration/update-alternatives/#_1","title":"\u57fa\u7840\u200b\u4f7f\u7528","text":""},{"location":"coding/configuration/update-alternatives/#_2","title":"\u521b\u5efa\u200b\u94fe\u63a5","text":"<p>\u200b\u4f7f\u7528\u200b<code>--install</code>\u200b\u53c2\u6570\u200b\u521b\u5efa\u200b\u6307\u5411\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u200b\u7684\u200b\u8f6f\u200b\u94fe\u63a5\u200b\uff0c\u200b\u547d\u4ee4\u200b\u7684\u200b\u57fa\u672c\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>update-alternatives --install &lt;destination&gt; &lt;entry-name&gt; &lt;source&gt; &lt;priority&gt;\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u547d\u4ee4\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b<code>&lt;entry-name&gt;</code>\u200b\u7684\u200b\u8f6f\u200b\u94fe\u63a5\u200b\u6761\u76ee\u200b\uff0c\u200b\u5c06\u200b<code>&lt;source&gt;</code>\u200b\u6307\u5411\u200b<code>&lt;destination&gt;</code>\u3002\u200b\u5f53\u6709\u200b\u591a\u4e2a\u200b<code>&lt;source&gt;</code>\u200b\u6307\u5411\u200b\u540c\u4e00\u4e2a\u200b<code>&lt;destination&gt;</code>\u200b\u65f6\u200b\uff0c\u200b\u9ed8\u8ba4\u200b<code>&lt;priority&gt;</code>\u200b\u66f4\u5927\u200b\u7684\u200b\u751f\u6548\u200b\u3002</p>"},{"location":"coding/configuration/update-alternatives/#_3","title":"\u67e5\u8be2\u200b\u6761\u76ee","text":"<p>\u200b\u4f7f\u7528\u200b<code>--display</code>\u200b\u53c2\u6570\u200b\u67e5\u8be2\u200b<code>&lt;entry-name&gt;</code>\u200b\u4e0b\u200b\u7684\u200b\u547d\u4ee4\u200b\u6307\u5411\u200b\uff1a</p> <pre><code>update-alternatives --display &lt;entry-name&gt;\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>--get-selections</code>\u200b\u53c2\u6570\u200b\u67e5\u8be2\u200b\u6240\u6709\u200b\u7684\u200b\u6761\u76ee\u200b\u548c\u200b\u914d\u7f6e\u200b\u72b6\u6001\u200b\u3002</p> <pre><code>update-alternatives --get-selections\n</code></pre> <p>\u200b\u8f93\u51fa\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>arptables                      auto     /usr/sbin/arptables-nft\nawk                            auto     /usr/bin/gawk\nbuiltins.7.gz                  auto     /usr/share/man/man7/bash-builtins.7.gz\nc++                            auto     /usr/bin/g++\nc89                            auto     /usr/bin/c89-gcc\nc99                            auto     /usr/bin/c99-gcc\ncc                             auto     /usr/bin/gcc\ncpp                            auto     /usr/bin/cpp\ncuda                           auto     /usr/local/cuda-11.7\ncuda-11                        auto     /usr/local/cuda-11.7\n</code></pre>"},{"location":"coding/configuration/update-alternatives/#_4","title":"\u5207\u6362\u200b\u94fe\u63a5","text":"<p>\u200b\u4f7f\u7528\u200b<code>--config</code>\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u5217\u51fa\u200b\u5e76\u200b\u9009\u62e9\u200b\u9700\u8981\u200b\u7684\u200b\u8f6f\u200b\u94fe\u63a5\u200b\uff0c\u200b\u547d\u4ee4\u200b\u7684\u200b\u57fa\u672c\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>update-alternatives --config &lt;entry-name&gt;\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u547d\u4ee4\u200b\u4f1a\u200b\u5217\u51fa\u200b<code>&lt;entry-name&gt;</code>\u200b\u4e0b\u200b\u7684\u200b\u6240\u6709\u200b\u6761\u76ee\u200b\u3002</p> <pre><code>There are 2 choices for the alternative gcc (providing /usr/bin/gcc).\n\n  Selection    Path             Priority   Status\n------------------------------------------------------------\n* 0            /usr/bin/gcc-11   2         auto mode\n  1            /usr/bin/gcc-11   2         manual mode\n  2            /usr/bin/gcc-12   1         manual mode\n\nPress &lt;enter&gt; to keep the current choice[*], or type selection number:\n</code></pre> <p>\u200b\u6216\u8005\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>--set</code>\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\u94fe\u63a5\u200b\u7684\u200b\u76ee\u6807\u200b\uff0c\u200b\u683c\u5f0f\u200b\u4e3a\u200b\uff1a</p> <pre><code>update-alternatives --set &lt;entry-name&gt; &lt;source&gt;\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u547d\u4ee4\u200b\u4f7f\u7528\u200b<code>&lt;source&gt;</code>\u200b\u4f5c\u4e3a\u200b<code>&lt;entry-name&gt;</code>\u200b\u6761\u200b\u76ee\u7684\u200b\u6307\u5411\u200b\u3002</p> <p>\u200b\u4f7f\u7528\u200b<code>--auto</code>\u200b\u53c2\u6570\u200b\u6062\u590d\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u76ee\u6807\u200b\u6307\u5411\u200b\u3002</p> <pre><code>update-alternatives --auto &lt;entry-name&gt;\n</code></pre>"},{"location":"coding/configuration/update-alternatives/#_5","title":"\u5220\u9664\u200b\u94fe\u63a5","text":"<p>\u200b\u4f7f\u7528\u200b<code>--remove</code>\u200b\u53c2\u6570\u200b\u5220\u9664\u200b\u4e00\u4e2a\u200b\u6761\u76ee\u200b\u4e0b\u200b\u7684\u200b\u94fe\u63a5\u200b\u3002</p> <pre><code>update-alternatives --remove &lt;entry-name&gt; &lt;source&gt;\n</code></pre>"},{"location":"coding/configuration/vim/","title":"Vim","text":"<p>Vim\u200b\u662f\u200b\u57fa\u4e8e\u200b\u547d\u4ee4\u884c\u200b\u7684\u200b\u6587\u672c\u7f16\u8f91\u200b\u5668\u200b\u3002</p>"},{"location":"coding/configuration/vim/#_1","title":"\u547d\u4ee4\u884c","text":""},{"location":"coding/configuration/vim/#_2","title":"\u4ea4\u4e92","text":"<p>Vim\u200b\u7684\u200b\u4ea4\u4e92\u200b\u547d\u4ee4\u200b\u5982\u4e0b\u200b\u8868\u6240\u5217\u200b\uff0c\u200b\u4ea4\u4e92\u200b\u547d\u4ee4\u200b\u4ee5\u200b<code>:</code>\u200b\u5f00\u5934\u200b\u3002</p> \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b <code>v</code> \u200b\u9009\u62e9\u200b\u5b57\u7b26\u200b <code>V</code> \u200b\u9009\u62e9\u200b\u884c\u200b <code>Ctrl+V</code> \u200b\u9009\u62e9\u200b\u5757\u200b <code>i</code> \u200b\u5207\u6362\u200b\u5230\u200b\u63d2\u5165\u200b\u6a21\u5f0f\u200b \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b <code>:q</code> \u200b\u9000\u51fa\u200bVim <code>:q!</code> \u200b\u9000\u51fa\u200b\u800c\u200b\u4e0d\u200b\u4fdd\u5b58\u200b <code>:w</code> \u200b\u4fdd\u5b58\u200b <code>&lt;Esc&gt;</code> \u200b\u9000\u51fa\u200b\u63d2\u5165\u200b\u6a21\u5f0f\u200b <code>:h</code> \u200b\u67e5\u8be2\u200b\u5e2e\u52a9\u200b <code>:version</code> \u200b\u67e5\u8be2\u200b\u7248\u672c\u200b\u548c\u200b\u529f\u80fd\u200b <code>:history</code> \u200b\u67e5\u8be2\u200b\u5386\u53f2\u200b\u547d\u4ee4\u200b <code>:s</code> \u200b\u6839\u636e\u200b\u6b63\u5219\u8868\u8fbe\u5f0f\u200b\u66ff\u6362\u200b\u5b57\u7b26\u4e32\u200b <code>:edit</code> \u200b\u7f16\u8f91\u200b\u6587\u4ef6\u200b\uff08\u200b\u652f\u6301\u200bglob pattern\u200b\u5339\u914d\u200b\uff09 <code>:find</code> \u200b\u5728\u200b<code>path</code>\u200b\u4e2d\u200b\u67e5\u627e\u6587\u4ef6\u200b <code>:set</code> \u200b\u67e5\u8be2\u200b/\u200b\u8bbe\u7f6e\u200b\u73af\u5883\u53d8\u91cf\u200b <code>:Rg</code> \u200b\u5728\u200b\u6587\u4ef6\u200b\u4e2d\u200b\u67e5\u627e"},{"location":"coding/configuration/vim/#_3","title":"\u64cd\u4f5c","text":"<p>Vim\u200b\u7684\u200b\u8bed\u53e5\u200b\u7531\u200b\u64cd\u4f5c\u7b26\u200b\u548c\u200b\u52a8\u4f5c\u200b\u7ec4\u6210\u200b\uff0c\u200b\u5728\u200bVim\u200b\u754c\u9762\u200b\u4e2d\u200b\uff0c\u200b\u6309\u200b<code>:</code>\u200b\u8f93\u5165\u200b\u8bed\u53e5\u200b\u3002</p> <p>Vim\u200b\u7684\u200b\u52a8\u4f5c\u200b\u5982\u4e0b\u200b\u8868\u200b\u6240\u793a\u200b</p> \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b <code>h</code> \u200b\u5411\u200b\u5de6\u200b\u79fb\u52a8\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b <code>j</code> \u200b\u5411\u4e0b\u200b\u79fb\u52a8\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b <code>k</code> \u200b\u5411\u4e0a\u200b\u79fb\u52a8\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b <code>l</code> \u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b <code>b</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0a\u200b\u4e00\u8bcd\u200b\u5f00\u5934\u200b <code>w</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0b\u200b\u4e00\u8bcd\u200b\u5f00\u5934\u200b <code>ge</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0a\u200b\u4e00\u8bcd\u200b\u7ed3\u5c3e\u200b <code>e</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0b\u200b\u4e00\u8bcd\u200b\u7ed3\u5c3e\u200b <code>{</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0a\u200b\u4e00\u6bb5\u200b <code>}</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0b\u200b\u4e00\u6bb5\u200b <code>0</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u884c\u9996\u200b <code>$</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u884c\u5c3e\u200b <code>^</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u884c\u9996\u200b\u975e\u200b\u7a7a\u5b57\u7b26\u200b <code>g_</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u884c\u5c3e\u200b\u975e\u200b\u7a7a\u5b57\u7b26\u200b <code>n|</code> \u200b\u79fb\u52a8\u200b\u5230\u200b\u5f53\u524d\u200b\u884c\u200b\u7684\u200b\u7b2c\u200b<code>n</code>\u200b\u5217\u200b <p>Vim\u200b\u7684\u200b\u64cd\u4f5c\u7b26\u200b\u5982\u4e0b\u200b\u8868\u200b\u6240\u793a\u200b</p> \u200b\u8f93\u5165\u200b \u200b\u529f\u80fd\u200b <code>y</code> \u200b\u590d\u5236\u200b <code>d</code> \u200b\u526a\u5207\u200b <code>c</code> \u200b\u526a\u5207\u200b\u5e76\u200b\u7f16\u8f91\u200b <code>p</code> \u200b\u7c98\u8d34\u200b <p>\u200b\u884c\u200b\u64cd\u4f5c\u200b\u6a21\u5f0f\u200b</p> <p>\u200b\u8fde\u7eed\u200b\u8f93\u5165\u200b\u4e24\u4e2a\u200b\u76f8\u540c\u200b\u64cd\u4f5c\u7b26\u200b\u5bf9\u200b\u5f53\u524d\u200b\u884c\u200b\u6267\u884c\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u5982\u200b<code>yy</code>\u200b\u590d\u5236\u200b\u5f53\u524d\u200b\u884c\u200b\u3002</p> <p>\u200b\u5728\u200b\u91cd\u590d\u200b\u7684\u200b\u64cd\u4f5c\u7b26\u200b\u524d\u9762\u200b\u52a0\u200b\u6570\u5b57\u200b<code>n</code>\uff0c\u200b\u8868\u793a\u200b\u4ece\u200b\u5f53\u524d\u200b\u884c\u200b\u5f00\u59cb\u200b\u5230\u200b\u7b2c\u200b<code>n</code>\u200b\u884c\u200b\u6267\u884c\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u5982\u200b<code>3yy</code>\u200b\u4ece\u200b\u5f53\u524d\u200b\u884c\u200b\u5f00\u59cb\u200b\u590d\u5236\u200b3\u200b\u884c\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/","title":"\u6df1\u5ea6\u200b\u5b66\u4e60","text":""},{"location":"coding/dl-from-scratch/#_2","title":"\u795e\u7ecf\u200b\u7f51\u7edc\u5c42","text":"<ul> <li>\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b</li> <li>\u200b\u635f\u5931\u200b\u51fd\u6570\u200b</li> <li>\u200b\u5f52\u4e00\u5316\u200b</li> <li>\u200b\u591a\u5c42\u200b\u611f\u77e5\u673a\u200b</li> <li>\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b</li> </ul>"},{"location":"coding/dl-from-scratch/#llm-from-scratch","title":"LLM from scratch","text":"<ol> <li>\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b</li> <li>\u200b\u7f16\u7801\u5668\u200b\u4e0e\u200b\u89e3\u7801\u5668\u200b<ul> <li>\u200b\u9a8c\u8bc1\u200btransformer\u200b\u5b9e\u73b0\u200b</li> </ul> </li> <li>\u200b\u7edd\u5bf9\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b<ul> <li>\u200b\u9a8c\u8bc1\u200bbert\u200b\u5b9e\u73b0\u200b</li> </ul> </li> <li>RoPE</li> <li>transformer\u200b\u53d8\u79cd\u200b<ul> <li>\u200b\u9a8c\u8bc1\u200bLlama-2\u200b\u5b9e\u73b0\u200b</li> </ul> </li> <li>\u200b\u5927\u200b\u6a21\u578b\u200b\u63a8\u7406\u200b</li> </ol>"},{"location":"coding/dl-from-scratch/activation/","title":"\u6fc0\u6d3b\u200b\u51fd\u6570","text":"<p>\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\u662f\u200b\u795e\u7ecf\u7f51\u7edc\u200b\u4e2d\u200b\u7528\u4e8e\u200b\u5f15\u5165\u200b\u975e\u7ebf\u6027\u200b\u56e0\u7d20\u200b\u7684\u200b\u51fd\u6570\u200b\u3002\u200b\u5e38\u7528\u200b\u7684\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\u6709\u200b\u5982\u4e0b\u200b\u51e0\u79cd\u200b\uff1a</p>"},{"location":"coding/dl-from-scratch/activation/#sigmoid","title":"Sigmoid \u200b\u51fd\u6570","text":"<p>Sigmoid\u200b\u51fd\u6570\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5c06\u200b\u8f93\u5165\u200b\u503c\u200b\u538b\u7f29\u200b\u5230\u200b0\u200b\u5230\u200b1\u200b\u4e4b\u95f4\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5e38\u7528\u200b\\(\\sigma\\)\u200b\u8868\u793a\u200b\u3002Sigmoid\u200b\u51fd\u6570\u200b\u5e38\u7528\u200b\u4e8e\u200b\u4e8c\u200b\u5206\u7c7b\u200b\u95ee\u9898\u200b\u4e2d\u200b\uff0c\u200b\u5c06\u200b\u8f93\u51fa\u200b\u503c\u200b\u6620\u5c04\u200b\u5230\u200b\u6982\u7387\u200b\u7a7a\u95f4\u200b\u3002</p> \\[ \\begin{aligned} \\sigma(x) &amp;= \\frac{1}{1 + e^{-x}} \\\\ \\sigma'(x) &amp;= \\sigma(x)(1 - \\sigma(x)) \\end{aligned} \\] <pre><code>def sigmoid(x: torch.Tensor) -&gt; torch.Tensor:\n    return 1 / (1 + torch.exp(-x))\n\nclass Sigmoid(torch.nn.Module):\n    def __init__(self):\n        super(Sigmoid, self).__init__()\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        return sigmoid(x)\n</code></pre>"},{"location":"coding/dl-from-scratch/activation/#softmax","title":"Softmax \u200b\u51fd\u6570","text":"<p>Softmax\u200b\u51fd\u6570\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5c06\u200b\u8f93\u5165\u200b\u503c\u200b\u538b\u7f29\u200b\u5230\u200b0\u200b\u5230\u200b1\u200b\u4e4b\u95f4\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5e38\u7528\u200b\u4e8e\u200b\u591a\u200b\u5206\u7c7b\u200b\u95ee\u9898\u200b\u4e2d\u200b\uff0c\u200b\u5c06\u200b\u591a\u4e2a\u200b\u5206\u7c7b\u200b\u4e0a\u200b\u8f93\u51fa\u200b\u503c\u200b\u6620\u5c04\u200b\u5230\u200b\u6982\u7387\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u6240\u6709\u200b\u7c7b\u522b\u200b\u7684\u200b\u6982\u7387\u200b\u4e4b\u200b\u548c\u200b\u4e3a\u200b1\u3002</p> \\[ \\begin{aligned} \\text{softmax}(\\bsz)_i &amp;= \\frac{e^{z_i}}{\\sum_{j=1}^n e^{z_j}} \\\\ \\text{softmax}'(\\bsz)_i &amp;= \\text{softmax}(\\bsz)_i(1 - \\text{softmax}(\\bsz)_i) \\end{aligned} \\] <pre><code>def softmax(x: torch.Tensor, dim: int = -1) -&gt; torch.Tensor:\n    return torch.exp(x) / torch.exp(x).sum(dim=dim, keepdim=True)\n\nclass Softmax(torch.nn.Module):\n    def __init__(self, dim: int = -1):\n        super(Softmax, self).__init__()\n        self.dim = dim\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        return softmax(x, dim=self.dim)\n</code></pre> <p>\u200b\u76f8\u5bf9\u200b\u5e94\u200b\u7684\u200b\uff0csoftmin\u200b\u51fd\u6570\u200b\u662f\u200bsoftmax\u200b\u51fd\u6570\u200b\u7684\u200b\u5bf9\u79f0\u200b\u51fd\u6570\u200b\uff0c\u200b\u7528\u4e8e\u200b\u6700\u5c0f\u5316\u200b\u8f93\u51fa\u200b\u503c\u200b\uff0c\\(\\text{softmin}(\\bsz) = \\text{softmax}(-\\bsz)\\)\u3002</p>"},{"location":"coding/dl-from-scratch/attention/","title":"\u6ce8\u610f\u529b\u200b\u673a\u5236","text":""},{"location":"coding/dl-from-scratch/attention/#_2","title":"\u5e8f\u5217\u200b\u5efa\u6a21","text":"<p>\u200b\u8bed\u8a00\u200b\u6a21\u578b\u200b\u662f\u200b\u4e00\u7c7b\u200b\u5e8f\u5217\u200b\u6a21\u578b\u200b\uff0c\u200b\u5176\u200b\u76ee\u6807\u200b\u662f\u200b\u5bf9\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u5e8f\u5217\u200b\u5efa\u6a21\u200b\uff0c\u200b\u5373\u200b\u5bf9\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u4f4d\u7f6e\u200b\u9884\u6d4b\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u6982\u7387\u200b\u3002\u200b\u6587\u672c\u200b\u9996\u5148\u200b\u901a\u8fc7\u200b\u5206\u8bcd\u200b\uff0c\u200b\u83b7\u5f97\u200b\u5bf9\u5e94\u200b\u7684\u200b\u79bb\u6563\u200b\u7b26\u53f7\u200b\uff08token\uff09\uff0c\u200b\u7136\u540e\u200b\u7ecf\u8fc7\u200b\u5d4c\u5165\u200b\u5c42\u200b\uff08embedding layer\uff09\u200b\u5c06\u200b\u7b26\u53f7\u200b\u6620\u5c04\u200b\u5230\u200b\\(D\\)\u200b\u7ef4\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5e8f\u5217\u200b\u5efa\u6a21\u200b\u7684\u200b\u8f93\u5165\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(N\\times L\\times D\\)\u200b\u7684\u200b\u5f20\u91cf\u200b\uff0c\u200b\u5176\u4e2d\u200b\\(N\\)\u200b\u662f\u200bbatch size\uff0c\\(L\\)\u200b\u662f\u200b\u5e8f\u5217\u200b\u957f\u5ea6\u200b\uff0c\\(D\\)\u200b\u662f\u200b\u5d4c\u5165\u200b\u7ef4\u5ea6\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/attention/#transformer","title":"Transformer","text":"<p>Transformer\u200b\u7531\u200bVaswani\u200b\u7b49\u200b\u4eba\u200b\u5728\u200b2017\u200b\u5e74\u200b\u63d0\u51fa\u200b\uff0c\u200b\u5176\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\uff0c\u200b\u5373\u200b\u901a\u8fc7\u200b\u67e5\u8be2\u200b\u3001\u200b\u952e\u200b\u7684\u200b\u5339\u914d\u200b\u5f97\u200b\u5206\u4e3a\u200b\u503c\u200b\u5206\u914d\u200b\u6743\u91cd\u200b\uff0c\u200b\u4ece\u800c\u200b\u5b9e\u73b0\u200b\u5bf9\u200b\u5e8f\u5217\u200b\u7684\u200b\u5efa\u6a21\u200b\u3002\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u7684\u200b\u6570\u5b66\u200b\u8868\u8fbe\u200b\u4e3a\u200b\uff1a</p> \\[ \\text{Attention}(Q, K, V) = \\text{softmax}\\left(\\frac{\\bsQ\\bsK^\\top}{\\sqrt{d_k}}\\right)\\bsV \\] <p>\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u6307\u200b\u7684\u200b\u662f\u200b\u67e5\u8be2\u200b\u3001\u200b\u952e\u548c\u503c\u200b\u90fd\u200b\u6765\u81ea\u200b\u540c\u4e00\u4e2a\u200b\u5e8f\u5217\u200b\u3002\u200b\u5728\u200b\u5b9e\u73b0\u200b\u4e2d\u200b\uff0c\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\\(\\bsx\\)\u200b\u5206\u522b\u200b\u7ecf\u8fc7\u200b\u4e09\u7ec4\u200b\u7ebf\u6027\u53d8\u6362\u200b\\(\\bsW^Q, \\bsW^K, \\bsW^V\\)\u200b\u5f97\u5230\u200b\u67e5\u8be2\u200b\\(\\bsQ\\)\u3001\u200b\u952e\u200b\\(\\bsK\\)\u200b\u548c\u200b\u503c\u200b\\(\\bsV\\)\uff0c\u200b\u7136\u540e\u200b\u8ba1\u7b97\u200b\u6ce8\u610f\u529b\u200b\u5f97\u5206\u200b\uff0c\u200b\u6700\u540e\u200b\u901a\u8fc7\u200b\u503c\u200b\u7684\u200b\u52a0\u6743\u200b\u6c42\u548c\u200b\u5f97\u5230\u200b\u8f93\u51fa\u200b\u3002\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u6307\u200b\u67e5\u8be2\u200b\u548c\u200b</p> <p>\u200b\u4e3a\u4ec0\u4e48\u200b\u9700\u8981\u200b\u7ebf\u6027\u53d8\u6362\u200b</p> <p>\u200b\u7ebf\u6027\u53d8\u6362\u200b\u7684\u200b\u76ee\u7684\u200b\u662f\u200b\u5c06\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u6620\u5c04\u200b\u5230\u200b\u4e0d\u540c\u200b\u7684\u200b\u8bed\u4e49\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u53e6\u5916\u200b\u4e00\u65b9\u9762\u200b\u4e5f\u200b\u4f7f\u5f97\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u4e0d\u518d\u200b\u53d7\u5230\u200b\u77e9\u9635\u200b\u5bf9\u79f0\u200b\u7684\u200b\u7ea6\u675f\u200b\uff08A\u200b\u5bf9\u200bB\u200b\u7684\u200b\u6ce8\u610f\u529b\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u7b49\u4e8e\u200bB\u200b\u5bf9\u200bA\u200b\u7684\u200b\u6ce8\u610f\u529b\u200b\uff09\u3002</p> <p>\u200b\u4e3a\u4ec0\u4e48\u200b\u8981\u200b\u9664\u4ee5\u200b\\(\\sqrt{d_k}\\)</p> <p>\u200b\u5728\u200b\\(\\bsQ\\bsK^\\top\\)\u200b\u7684\u200b\u8ba1\u7b97\u200b\u4e2d\u200b\uff0c\u200b\u7ed3\u679c\u200b\u7684\u200b\u65b9\u5dee\u200b\u4f1a\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u4e3a\u4e86\u200b\u4f7f\u200b\u7ed3\u679c\u200b\u7684\u200b\u65b9\u5dee\u200b\u4fdd\u6301\u200b\u4e0d\u53d8\u200b\uff0c\u200b\u9700\u8981\u200b\u9664\u4ee5\u200b\\(\\sqrt{d_k}\\)\uff0c\u200b\u5176\u4e2d\u200b\\(d_k\\)\u200b\u662f\u200b\u952e\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u3002\u200b\u5426\u5219\u200b\uff0c\u200b\u5f53\u200b\\(d_k\\)\u200b\u8f83\u5927\u200b\u65f6\u200b\uff0c\\(\\bsQ\\bsK^\\top\\)\u200b\u4e2d\u200b\u5143\u7d20\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5dee\u5f02\u200b\u4f1a\u200b\u53d8\u5f97\u200b\u5f88\u5927\u200b\uff0c\u200b\u5bfc\u81f4\u200bsoftmax\u200b\u51fd\u6570\u200b\u7684\u200b\u68af\u5ea6\u200b\u63a5\u8fd1\u200b\u4e8e\u200b0\u3002</p> <p>softmax\u200b\u51fd\u6570\u200b</p> <p>softmax\u200b\u51fd\u6570\u200b\u7684\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\uff1a</p> \\[ \\text{softmax}(\\bsz)_i = \\frac{e^{z_i}}{\\sum_{j=1}^n e^{z_j}} \\] <p>\u200b\u5373\u200b\u5bf9\u200b\u8f93\u5165\u200b\u5411\u91cf\u200b\\(\\bsz\\)\u200b\u4e2d\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u6307\u6570\u200b\u7684\u200b\u5f52\u4e00\u5316\u200b\u3002\u200b\u5176\u4e2d\u200b\\(\\bsz\\)\u200b\u662f\u200b\u8f93\u5165\u200b\u5411\u91cf\u200b\uff0c\\(n\\)\u200b\u662f\u200b\u5411\u91cf\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u3002</p> <p>\u200b\u4e3a\u4e86\u200b\u4ece\u200b\u591a\u4e2a\u200b\u7ef4\u5ea6\u200b\u5b66\u4e60\u200b\u5230\u200b\u5bf9\u200b\u540c\u4e00\u4e2a\u200b\u4f4d\u7f6e\u200b\u4e0d\u540c\u200b\u7279\u5f81\u200b\u7684\u200b\u8868\u5f81\u200b\uff0ctransformer\u200b\u5f15\u5165\u200b\u4e86\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u3002\u200b\u5176\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u5c06\u200b\u8f93\u5165\u200b\u7684\u200b\\(D\\)\u200b\u7ef4\u200b\u5411\u91cf\u200b\u62c6\u5206\u200b\u5230\u200b\\(h\\)\u200b\u4e2a\u5934\u200b\uff0c\u200b\u7136\u540e\u200b\u5206\u522b\u200b\u8ba1\u7b97\u200b\u6ce8\u610f\u529b\u200b\uff0c\u200b\u6700\u540e\u200b\u5c06\u200b\\(h\\)\u200b\u4e2a\u5934\u200b\u7684\u200b\u8f93\u51fa\u200b\u62fc\u63a5\u200b\u5728\u200b\u4e00\u8d77\u200b\uff0c\u200b\u518d\u200b\u7ecf\u8fc7\u200b\u4e00\u4e2a\u200b\u7ebf\u6027\u53d8\u6362\u200b\u5f97\u5230\u200b\u6700\u7ec8\u200b\u8f93\u51fa\u200b\u3002\u200b\u5728\u200b\u5b9e\u73b0\u200b\u4e2d\u200b\uff0c\u200b\u4e00\u822c\u200b\u4f7f\u7528\u200b\u7ef4\u5ea6\u200b\u53d8\u6362\u200b\u6765\u200b\u5b9e\u73b0\u200b\u8f93\u5165\u200b\u5411\u91cf\u200b\u7684\u200b\u62c6\u5206\u200b\u3002</p> <p>\u200b\u5728\u200b\u90e8\u5206\u200b\u4efb\u52a1\u200b\u4e2d\u200b\uff0c\u200b\u9700\u8981\u200b\u63a7\u5236\u200b\u5e8f\u5217\u200b\u67d0\u4e2a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u53ea\u80fd\u200b\u4f9d\u8d56\u4e8e\u200b\u8be5\u200b\u4f4d\u7f6e\u200b\u4e4b\u524d\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u800c\u200b\u4e0d\u80fd\u200b\u4f9d\u8d56\u4e8e\u200b\u4e4b\u540e\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\u200b\u6b64\u65f6\u200b\u9700\u8981\u200b\u4f7f\u7528\u200bmask\u200b\u673a\u5236\u200b\uff0c\u200b\u5373\u200b\u5728\u200b\u8ba1\u7b97\u200b\u6ce8\u610f\u529b\u200b\u5f97\u5206\u200b\u65f6\u200b\uff0c\u200b\u5c06\u200b\u4e0d\u200b\u5408\u6cd5\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5f97\u5206\u200b\u8bbe\u7f6e\u200b\u4e3a\u200b\u8d1f\u200b\u65e0\u7a77\u200b\uff0c\u200b\u4ece\u800c\u200b\u5728\u200bsoftmax\u200b\u51fd\u6570\u200b\u4e2d\u200b\u5f97\u5230\u200b0\u200b\u7684\u200b\u6982\u7387\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/attention/#_3","title":"\u4ee3\u7801\u200b\u5b9e\u73b0","text":"<p>einsum</p> <p><code>einsum</code>\u200b\u662f\u200bEinstein Summation Notation\u200b\u7684\u200b\u7f29\u5199\u200b\uff0c\u200b\u662f\u200b\u4e00\u79cd\u200b\u7528\u4e8e\u200b\u5f20\u91cf\u200b\u8fd0\u7b97\u200b\u7684\u200b\u8bb0\u53f7\u200b\u3002\u200b\u5728\u200bPyTorch\u200b\u4e2d\u200b\uff0c<code>einsum</code>\u200b\u51fd\u6570\u200b\u7684\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\uff1a</p> <pre><code>torch.einsum(equation, *operands)\n</code></pre> <p>\u200b\u5176\u4e2d\u200b<code>equation</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u63cf\u8ff0\u200b\u4e86\u200b\u5f20\u91cf\u200b\u7684\u200b\u8fd0\u7b97\u200b\u65b9\u5f0f\u200b\uff0c<code>operands</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6216\u200b\u591a\u4e2a\u200b\u5f20\u91cf\u200b\u3002\u200b\u4f8b\u5982\u200b\uff0c<code>torch.einsum(\"ij,jk-&gt;ik\", A, B)</code>\u200b\u8868\u793a\u200b\u8ba1\u7b97\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u548c\u200b\\(B\\)\u200b\u7684\u200b\u4e58\u79ef\u200b\u3002\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5176\u200b\u7406\u89e3\u200b\u4e3a\u200b\u4e58\u79ef\u200b-\u200b\u6c42\u548c\u200b\u7684\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u5373\u5c06\u200b<code>equation</code>\u200b\u4e2d\u200b\u7684\u200b\u5b57\u6bcd\u200b\u5bf9\u5e94\u200b\u5230\u200b<code>operands</code>\u200b\u4e2d\u200b\u7684\u200b\u5f20\u91cf\u200b\uff0c\u200b\u7136\u540e\u200b\u5bf9\u200b\u8fd9\u4e9b\u200b\u5f20\u91cf\u200b\u8fdb\u884c\u200b\u4e58\u79ef\u200b\u548c\u200b\u6c42\u548c\u200b\u3002\u200b\u5ffd\u7565\u200bbatch size\uff0c\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u8ba1\u7b97\u200b\\(\\bsQ\\bsK^\\top\\)\uff0c\u200b\u5176\u4e2d\u200b\\(\\bsQ\\)\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u662f\u200b\\((L, D)\\)\uff0c\\(\\bsK\\)\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u662f\u200b\\((L, D)\\)\uff0c\u200b\u8f93\u51fa\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u662f\u200b\\((L, L)\\)\uff0c\u200b\u56e0\u6b64\u200b<code>equation</code>\u200b\u4e3a\u200b<code>\"id,jd-&gt;ij\"</code>\u3002</p> <p>\u200b\u4e0b\u9762\u200b\u6211\u4eec\u200b\u901a\u8fc7\u200bPyTorch\u200b\u5b9e\u73b0\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u3002\u200b\u5728\u200b\u5b9e\u73b0\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u524d\u200b\uff0c\u200b\u9700\u8981\u200b\u5148\u200b\u5b9e\u73b0\u200b\u7ebf\u6027\u53d8\u6362\u200b\u548c\u200bsoftmax\u200b\u51fd\u6570\u200b\u3002\u200b\u4e4b\u540e\u200b\u5b9e\u73b0\u200b\u7528\u4e8e\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\u7684\u200b<code>MultiHeadSelfAttention</code>\u200b\u7c7b\u200b\uff0c\u200b\u5176\u200b\u8f93\u5165\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(N\\times L\\times D\\)\u200b\u7684\u200b\u5f20\u91cf\u200b\uff0c\u200b\u8f93\u51fa\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(N\\times L\\times D\\)\u200b\u7684\u200b\u5f20\u91cf\u200b\u3002</p> <pre><code>class MultiHeadAttention(torch.nn.Module):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadAttention, self).__init__()\n        self.d_model = d_model\n        self.num_heads = num_heads\n        if d_model % num_heads != 0:\n            raise ValueError(\"d_model must be divisible by num_heads\")\n\n        self.d_k = d_model // num_heads\n        self.sqrt_d_k = self.d_k ** 0.5\n\n        self.W_Q = torch.nn.Linear(d_model, d_model)\n        self.W_K = torch.nn.Linear(d_model, d_model)\n        self.W_V = torch.nn.Linear(d_model, d_model)\n        self.W_O = torch.nn.Linear(d_model, d_model)\n\n    def forward(\n        self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,\n        mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        N, L_Q, D = x_q.size()\n        _, L_KV, _ = x_k.size()\n        Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)\n        K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)\n        V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)\n\n        score = torch.einsum(\"nihd,njhd-&gt;nijh\", Q, K) / self.sqrt_d_k\n        score = torch.nn.functional.softmax(score, dim=2)\n        value = torch.einsum(\"nijh,njhd-&gt;nihd\", score, V).reshape(N, L_Q, self.d_model)\n        return self.W_O(value)\n\nclass MultiHeadSelfAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadSelfAttention, self).__init__(d_model, num_heads)\n\n    def forward(self, x: torch.Tensor, mask: torch.Tensor | None = None) -&gt; torch.Tensor:\n        return super().forward(x, x, x, mask)\n\nclass MultiHeadCrossAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadCrossAttention, self).__init__(d_model, num_heads)\n\n    def forward(\n        self, x_q: torch.Tensor, x_kv: torch.Tensor,\n        mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        return super().forward(x_q, x_kv, x_kv, mask)\n</code></pre> <p>\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u4e2d\u200b\u7684\u200bmask</p> <p>\u200b\u5728\u200b\u5b9e\u73b0\u200b\u4e2d\u200b\uff0c\u200b\u6709\u65f6\u200b\u9700\u8981\u200b\u5bf9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u7684\u200b\u5f97\u5206\u200b\u8fdb\u884c\u200bmask\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u4ee5\u200b\u63a7\u5236\u200b\u6a21\u578b\u200b\u7684\u200b\u884c\u4e3a\u200b\u3002mask\u200b\u901a\u5e38\u200b\u5206\u4e3a\u200b\u4e09\u79cd\u200b\uff0c\u200b\u5373\u200b</p> <ol> <li>padding mask\uff1a\u200b\u7ef4\u5ea6\u200b<code>[N, L]</code>\uff0c\u200b\u7528\u4e8e\u200b\u63a9\u76d6\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200bpadding\u200b\u5143\u7d20\u200b\uff0c\u200b\u4f7f\u200b\u5176\u200b\u4e0d\u200b\u53c2\u4e0e\u200b\u6ce8\u610f\u529b\u200b\u8ba1\u7b97\u200b\u3002</li> <li>attention mask\uff1a\u200b\u7ef4\u5ea6\u200b<code>[L, L]</code>\uff0c\u200b\u7528\u4e8e\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u4e2d\u200b\uff0c\u200b\u63a7\u5236\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u4f9d\u8d56\u200b\u5173\u7cfb\u200b\u3002\u200b\u901a\u5e38\u200b\u7528\u200bcausal mask\u200b\u6216\u200blook-ahead mask\u200b\u89c4\u5b9a\u200b\u5e8f\u5217\u200b\u67d0\u4e2a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u53ea\u80fd\u200b\u4f9d\u8d56\u4e8e\u200b\u8be5\u200b\u4f4d\u7f6e\u200b\u4e4b\u524d\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u800c\u200b\u4e0d\u80fd\u200b\u4f9d\u8d56\u4e8e\u200b\u4e4b\u540e\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</li> <li>\u200b\u5728\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u4e2d\u200b\u7684\u200bmemory mask\uff1a\u200b\u7ef4\u5ea6\u200b<code>[L_Q, L_KV]</code>\uff0c\u200b\u7528\u4e8e\u200b\u63a7\u5236\u200b\u67e5\u8be2\u200b\u548c\u952e\u200b\u4e4b\u95f4\u200b\u7684\u200b\u4f9d\u8d56\u200b\u5173\u7cfb\u200b\u3002</li> </ol> <p>\u200b\u5728\u200b\u5b9e\u73b0\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u5c06\u200bmask\u200b\u4e2d\u4e3a\u200b0\u200b\u7684\u200b\u5143\u7d20\u200b\u8bbe\u7f6e\u200b\u4e3a\u200b\u8d1f\u200b\u65e0\u7a77\u200b\uff0c\u200b\u4ece\u800c\u200b\u5728\u200bsoftmax\u200b\u51fd\u6570\u200b\u4e2d\u200b\u5f97\u5230\u200b0\u200b\u7684\u200b\u6982\u7387\u200b\u3002</p> <p>\u200b\u5b8c\u6574\u200b\u7684\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u5b9e\u73b0\u200b\u5982\u4e0b\u200b</p> <pre><code>class MultiHeadAttention(torch.nn.Module):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadAttention, self).__init__()\n        self.d_model = d_model\n        self.num_heads = num_heads\n        if d_model % num_heads != 0:\n            raise ValueError(\"d_model must be divisible by num_heads\")\n\n        self.d_k = d_model // num_heads\n        self.sqrt_d_k = self.d_k ** 0.5\n\n        self.W_Q = torch.nn.Linear(d_model, d_model)\n        self.W_K = torch.nn.Linear(d_model, d_model)\n        self.W_V = torch.nn.Linear(d_model, d_model)\n        self.W_O = torch.nn.Linear(d_model, d_model)\n\n    def forward(\n        self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        # x_q: (N, L_Q, D), x_k: (N, L_KV, D), x_v: (N, L_KV, D)\n        # padding_mask: (N, L_KV), attention_mask: (L_Q, L_KV)\n        N, L_Q, D = x_q.size()\n        _, L_KV, _ = x_k.size()\n\n        # Linear transformation -&gt; Split heads\n        Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)\n        K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)\n        V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)\n\n        # Compute attention score\n        score = torch.einsum('nihd,njhd-&gt;nijh', Q, K) / self.sqrt_d_k\n\n        # Apply attention mask\n        if attention_mask is not None:\n            score = score.masked_fill(\n                attention_mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')\n            )\n\n        # Apply padding mask\n        if padding_mask is not None:\n            score = score.masked_fill(\n                padding_mask.reshape(N, 1, L_KV, 1) == 0, float('-inf')\n            )\n\n        # Softmax -&gt; Weighted sum -&gt; Merge heads -&gt; Output transformation\n        score = torch.nn.functional.softmax(score, dim=2)\n        value = torch.einsum(\n            'nijh,njhd-&gt;nihd', score, V\n        ).reshape(N, L_Q, self.d_model)\n        return self.W_O(value)\n\n\nclass MultiHeadSelfAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadSelfAttention, self).__init__(d_model, num_heads)\n\n    def forward(\n        self, x: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        # Self attention is applied to the same input\n        return super().forward(x, x, x, padding_mask, attention_mask)\n\n\nclass MultiHeadCrossAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadCrossAttention, self).__init__(d_model, num_heads)\n\n    def forward(\n        self, x_q: torch.Tensor, x_kv: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        # Cross attention is applying query on another kv sequence\n        return super().forward(x_q, x_kv, x_kv, padding_mask, attention_mask)\n</code></pre>"},{"location":"coding/dl-from-scratch/bert/","title":"\u5b9e\u73b0\u200bBERT","text":"In\u00a0[1]: Copied! <pre>import transformers\nimport torch\n\nfrom custom_transformer import Encoder\n</pre> import transformers import torch  from custom_transformer import Encoder <p>huggingface\u200b\u4e2d\u200b\u7684\u200b\u6a21\u578b\u200b\u901a\u5e38\u200b\u5bf9\u5e94\u200b\u4e00\u7ec4\u200bconfig\uff0c\u200b\u5b58\u50a8\u200b\u6a21\u578b\u200b\u7684\u200b\u8d85\u200b\u53c2\u6570\u200b\u3002</p> In\u00a0[2]: Copied! <pre>bert_config = transformers.BertConfig.from_pretrained('bert-base-uncased', local_files_only=True)\n# Remove the pooler layer\nbert_config\n</pre> bert_config = transformers.BertConfig.from_pretrained('bert-base-uncased', local_files_only=True) # Remove the pooler layer bert_config <pre>/opt/homebrew/lib/python3.11/site-packages/huggingface_hub/file_download.py:1132: FutureWarning: `resume_download` is deprecated and will be removed in version 1.0.0. Downloads always resume when possible. If you want to force a new download, use `force_download=True`.\n  warnings.warn(\n</pre> Out[2]: <pre>BertConfig {\n  \"architectures\": [\n    \"BertForMaskedLM\"\n  ],\n  \"attention_probs_dropout_prob\": 0.1,\n  \"classifier_dropout\": null,\n  \"gradient_checkpointing\": false,\n  \"hidden_act\": \"gelu\",\n  \"hidden_dropout_prob\": 0.1,\n  \"hidden_size\": 768,\n  \"initializer_range\": 0.02,\n  \"intermediate_size\": 3072,\n  \"layer_norm_eps\": 1e-12,\n  \"max_position_embeddings\": 512,\n  \"model_type\": \"bert\",\n  \"num_attention_heads\": 12,\n  \"num_hidden_layers\": 12,\n  \"pad_token_id\": 0,\n  \"position_embedding_type\": \"absolute\",\n  \"transformers_version\": \"4.41.2\",\n  \"type_vocab_size\": 2,\n  \"use_cache\": true,\n  \"vocab_size\": 30522\n}</pre> <p>\u200b\u9996\u5148\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u52a0\u8f7d\u200b\u9884\u200b\u8bad\u7ec3\u200b\u7684\u200bbert-base-uncased\u200b\u6a21\u578b\u200b\uff0c\u200b\u7528\u4e8e\u200b\u548c\u200b\u6211\u4eec\u200b\u7684\u200b\u6a21\u578b\u200b\u8fdb\u884c\u200b\u5bf9\u6bd4\u200b\u3002</p> In\u00a0[3]: Copied! <pre>bert_tokenizer = transformers.BertTokenizer.from_pretrained(\n    'bert-base-uncased', local_files_only=True\n)\nhf_model = transformers.BertModel.from_pretrained(\n    'bert-base-uncased', config=bert_config, add_pooling_layer=False\n)\nhf_model\n</pre> bert_tokenizer = transformers.BertTokenizer.from_pretrained(     'bert-base-uncased', local_files_only=True ) hf_model = transformers.BertModel.from_pretrained(     'bert-base-uncased', config=bert_config, add_pooling_layer=False ) hf_model Out[3]: <pre>BertModel(\n  (embeddings): BertEmbeddings(\n    (word_embeddings): Embedding(30522, 768, padding_idx=0)\n    (position_embeddings): Embedding(512, 768)\n    (token_type_embeddings): Embedding(2, 768)\n    (LayerNorm): LayerNorm((768,), eps=1e-12, elementwise_affine=True)\n    (dropout): Dropout(p=0.1, inplace=False)\n  )\n  (encoder): BertEncoder(\n    (layer): ModuleList(\n      (0-11): 12 x BertLayer(\n        (attention): BertAttention(\n          (self): BertSdpaSelfAttention(\n            (query): Linear(in_features=768, out_features=768, bias=True)\n            (key): Linear(in_features=768, out_features=768, bias=True)\n            (value): Linear(in_features=768, out_features=768, bias=True)\n            (dropout): Dropout(p=0.1, inplace=False)\n          )\n          (output): BertSelfOutput(\n            (dense): Linear(in_features=768, out_features=768, bias=True)\n            (LayerNorm): LayerNorm((768,), eps=1e-12, elementwise_affine=True)\n            (dropout): Dropout(p=0.1, inplace=False)\n          )\n        )\n        (intermediate): BertIntermediate(\n          (dense): Linear(in_features=768, out_features=3072, bias=True)\n          (intermediate_act_fn): GELUActivation()\n        )\n        (output): BertOutput(\n          (dense): Linear(in_features=3072, out_features=768, bias=True)\n          (LayerNorm): LayerNorm((768,), eps=1e-12, elementwise_affine=True)\n          (dropout): Dropout(p=0.1, inplace=False)\n        )\n      )\n    )\n  )\n)</pre> <p>\u200b\u63a5\u4e0b\u6765\u200b\uff0c\u200b\u6211\u4eec\u200b\u6839\u636e\u200bconfig\u200b\u548c\u200b\u6a21\u578b\u200b\u7684\u200b\u7ed3\u6784\u200b\uff0c\u200b\u5b9e\u73b0\u200b\u4e00\u4e2a\u200bBERT\u200b\u6a21\u578b\u200b\u3002</p> In\u00a0[4]: Copied! <pre>class BertEmbeddings(torch.nn.Module):\n    def __init__(self, config: transformers.BertConfig):\n        super(BertEmbeddings, self).__init__()\n\n        self.word_embeddings = torch.nn.Embedding(\n            config.vocab_size, config.hidden_size, padding_idx=config.pad_token_id\n        )\n        self.positional_embedding = torch.nn.Embedding(\n            config.max_position_embeddings, config.hidden_size\n        )\n        self.token_type_embeddings = torch.nn.Embedding(\n            config.type_vocab_size, config.hidden_size\n        )\n        self.ln = torch.nn.LayerNorm(config.hidden_size, eps=config.layer_norm_eps)\n        self.dropout = torch.nn.Dropout(config.hidden_dropout_prob)\n\n    def forward(self, input_ids):\n        input_embeds = self.word_embeddings(input_ids)\n        pos_embeds = self.positional_embedding(\n            torch.arange(input_ids.size(1), device=input_ids.device)\n        )\n        type_embeds = self.token_type_embeddings(\n            torch.zeros_like(input_ids)\n        )\n        return self.dropout(self.ln(input_embeds + pos_embeds + type_embeds))\n\nclass CustomBertModel(torch.nn.Module):\n    def __init__(self, config: transformers.BertConfig):\n        super(CustomBertModel, self).__init__()\n        self.shared = BertEmbeddings(config)\n        self.encoder = Encoder(\n            num_layers=config.num_hidden_layers,\n            input_dim=config.hidden_size,\n            num_heads=config.num_attention_heads,\n            ffn_dim=config.intermediate_size,\n            dropout=config.hidden_dropout_prob,\n            layer_norm_eps=config.layer_norm_eps,\n            activation=config.hidden_act,\n        )\n\n    def forward(self, input_ids, padding_mask):\n        input_embeds = self.shared(input_ids)\n        return self.encoder(input_embeds, padding_mask=padding_mask)\n\ncustom_model = CustomBertModel(bert_config)\ncustom_model\n</pre> class BertEmbeddings(torch.nn.Module):     def __init__(self, config: transformers.BertConfig):         super(BertEmbeddings, self).__init__()          self.word_embeddings = torch.nn.Embedding(             config.vocab_size, config.hidden_size, padding_idx=config.pad_token_id         )         self.positional_embedding = torch.nn.Embedding(             config.max_position_embeddings, config.hidden_size         )         self.token_type_embeddings = torch.nn.Embedding(             config.type_vocab_size, config.hidden_size         )         self.ln = torch.nn.LayerNorm(config.hidden_size, eps=config.layer_norm_eps)         self.dropout = torch.nn.Dropout(config.hidden_dropout_prob)      def forward(self, input_ids):         input_embeds = self.word_embeddings(input_ids)         pos_embeds = self.positional_embedding(             torch.arange(input_ids.size(1), device=input_ids.device)         )         type_embeds = self.token_type_embeddings(             torch.zeros_like(input_ids)         )         return self.dropout(self.ln(input_embeds + pos_embeds + type_embeds))  class CustomBertModel(torch.nn.Module):     def __init__(self, config: transformers.BertConfig):         super(CustomBertModel, self).__init__()         self.shared = BertEmbeddings(config)         self.encoder = Encoder(             num_layers=config.num_hidden_layers,             input_dim=config.hidden_size,             num_heads=config.num_attention_heads,             ffn_dim=config.intermediate_size,             dropout=config.hidden_dropout_prob,             layer_norm_eps=config.layer_norm_eps,             activation=config.hidden_act,         )      def forward(self, input_ids, padding_mask):         input_embeds = self.shared(input_ids)         return self.encoder(input_embeds, padding_mask=padding_mask)  custom_model = CustomBertModel(bert_config) custom_model Out[4]: <pre>CustomBertModel(\n  (shared): BertEmbeddings(\n    (word_embeddings): Embedding(30522, 768, padding_idx=0)\n    (positional_embedding): Embedding(512, 768)\n    (token_type_embeddings): Embedding(2, 768)\n    (ln): LayerNorm((768,), eps=1e-12, elementwise_affine=True)\n    (dropout): Dropout(p=0.1, inplace=False)\n  )\n  (encoder): Encoder(\n    (layers): ModuleList(\n      (0-11): 12 x EncoderLayer(\n        (attention): MultiHeadSelfAttention(\n          (W_Q): Linear(in_features=768, out_features=768, bias=True)\n          (W_K): Linear(in_features=768, out_features=768, bias=True)\n          (W_V): Linear(in_features=768, out_features=768, bias=True)\n          (W_O): Linear(in_features=768, out_features=768, bias=True)\n        )\n        (norm1): LayerNorm((768,), eps=1e-12, elementwise_affine=True)\n        (dropout1): Dropout(p=0.1, inplace=False)\n        (ffn): FFN(\n          (fc1): Linear(in_features=768, out_features=3072, bias=True)\n          (fc2): Linear(in_features=3072, out_features=768, bias=True)\n          (act): GELU(approximate='none')\n        )\n        (norm2): LayerNorm((768,), eps=1e-12, elementwise_affine=True)\n        (dropout2): Dropout(p=0.1, inplace=False)\n      )\n    )\n  )\n)</pre> <p>\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u5c06\u200b\u9884\u200b\u8bad\u7ec3\u200b\u6a21\u578b\u200b\u7684\u200b\u6743\u91cd\u200b\u52a0\u8f7d\u200b\u5230\u200b\u6211\u4eec\u200b\u7684\u200b\u6a21\u578b\u200b\u4e2d\u200b\u3002</p> In\u00a0[5]: Copied! <pre>def copy_weights(src, dest, bias: bool = False):\n    src.weight.data.copy_(dest.weight.data)\n    if bias:\n        src.bias.data.copy_(dest.bias.data)\n\ndef load_weight_from_hf(model: CustomBertModel, hf_model: transformers.BertModel):\n    # Embeddings\n    unbiased_layer_pairs = [\n        (model.shared.word_embeddings, hf_model.embeddings.word_embeddings),\n        (model.shared.positional_embedding, hf_model.embeddings.position_embeddings),\n        (model.shared.token_type_embeddings, hf_model.embeddings.token_type_embeddings),\n    ]\n    biased_layer_pairs = [\n        (model.shared.ln, hf_model.embeddings.LayerNorm),\n    ]\n    for custom_layer, hf_layer in zip(model.encoder.layers, hf_model.encoder.layer):\n        # attn\n        biased_layer_pairs.append((custom_layer.attention.W_Q, hf_layer.attention.self.query))\n        biased_layer_pairs.append((custom_layer.attention.W_K, hf_layer.attention.self.key))\n        biased_layer_pairs.append((custom_layer.attention.W_V, hf_layer.attention.self.value))\n        biased_layer_pairs.append((custom_layer.attention.W_O, hf_layer.attention.output.dense))\n        biased_layer_pairs.append((custom_layer.norm1, hf_layer.attention.output.LayerNorm))\n        # ffn\n        biased_layer_pairs.append((custom_layer.ffn.fc1, hf_layer.intermediate.dense))\n        biased_layer_pairs.append((custom_layer.ffn.fc2, hf_layer.output.dense))\n        biased_layer_pairs.append((custom_layer.norm2, hf_layer.output.LayerNorm))\n\n    for src, dest in unbiased_layer_pairs:\n        copy_weights(src, dest)\n    for src, dest in biased_layer_pairs:\n        copy_weights(src, dest, bias=True)\n    return model\n</pre> def copy_weights(src, dest, bias: bool = False):     src.weight.data.copy_(dest.weight.data)     if bias:         src.bias.data.copy_(dest.bias.data)  def load_weight_from_hf(model: CustomBertModel, hf_model: transformers.BertModel):     # Embeddings     unbiased_layer_pairs = [         (model.shared.word_embeddings, hf_model.embeddings.word_embeddings),         (model.shared.positional_embedding, hf_model.embeddings.position_embeddings),         (model.shared.token_type_embeddings, hf_model.embeddings.token_type_embeddings),     ]     biased_layer_pairs = [         (model.shared.ln, hf_model.embeddings.LayerNorm),     ]     for custom_layer, hf_layer in zip(model.encoder.layers, hf_model.encoder.layer):         # attn         biased_layer_pairs.append((custom_layer.attention.W_Q, hf_layer.attention.self.query))         biased_layer_pairs.append((custom_layer.attention.W_K, hf_layer.attention.self.key))         biased_layer_pairs.append((custom_layer.attention.W_V, hf_layer.attention.self.value))         biased_layer_pairs.append((custom_layer.attention.W_O, hf_layer.attention.output.dense))         biased_layer_pairs.append((custom_layer.norm1, hf_layer.attention.output.LayerNorm))         # ffn         biased_layer_pairs.append((custom_layer.ffn.fc1, hf_layer.intermediate.dense))         biased_layer_pairs.append((custom_layer.ffn.fc2, hf_layer.output.dense))         biased_layer_pairs.append((custom_layer.norm2, hf_layer.output.LayerNorm))      for src, dest in unbiased_layer_pairs:         copy_weights(src, dest)     for src, dest in biased_layer_pairs:         copy_weights(src, dest, bias=True)     return model In\u00a0[6]: Copied! <pre>custom_model = load_weight_from_hf(custom_model, hf_model)\n</pre> custom_model = load_weight_from_hf(custom_model, hf_model) <p>\u200b\u4f7f\u7528\u200bBERT\u200b\u7f16\u7801\u200b\u4e00\u53e5\u200b\u8bdd\u200b\uff0c\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u5bf9\u200b\u53e5\u5b50\u200b\u5206\u8bcd\u200b\uff0c\u200b\u7136\u540e\u200b\u5c06\u200b\u5176\u200b\u8f93\u5165\u200b\u6a21\u578b\u200b\u5f97\u5230\u200b\u5bf9\u5e94\u200b\u7684\u200b\u8f93\u51fa\u200b\u3002</p> In\u00a0[7]: Copied! <pre>custom_model.eval()\nhf_model.eval()\n\nsentence = \"Hello, World!\"\ntokenized_sentence = bert_tokenizer(sentence, return_tensors='pt')\ninput_ids = tokenized_sentence['input_ids']\npadding_mask = tokenized_sentence['attention_mask']\n\ncustom_output = custom_model(input_ids, padding_mask)\nhf_output = hf_model(input_ids, attention_mask=padding_mask).last_hidden_state\ncustom_output[0, 0, :5], hf_output[0, 0, :5]\n</pre> custom_model.eval() hf_model.eval()  sentence = \"Hello, World!\" tokenized_sentence = bert_tokenizer(sentence, return_tensors='pt') input_ids = tokenized_sentence['input_ids'] padding_mask = tokenized_sentence['attention_mask']  custom_output = custom_model(input_ids, padding_mask) hf_output = hf_model(input_ids, attention_mask=padding_mask).last_hidden_state custom_output[0, 0, :5], hf_output[0, 0, :5] Out[7]: <pre>(tensor([-0.0781,  0.1587,  0.0400, -0.1986, -0.3442], grad_fn=&lt;SliceBackward0&gt;),\n tensor([-0.0781,  0.1587,  0.0400, -0.1986, -0.3442], grad_fn=&lt;SliceBackward0&gt;))</pre> In\u00a0[8]: Copied! <pre>torch.allclose(custom_output, hf_output, atol=1e-5)\n</pre> torch.allclose(custom_output, hf_output, atol=1e-5) Out[8]: <pre>True</pre>"},{"location":"coding/dl-from-scratch/custom_transformer/","title":"Custom transformer","text":"In\u00a0[\u00a0]: Copied! <pre>import torch\n</pre> import torch In\u00a0[\u00a0]: Copied! <pre>class MultiHeadAttention(torch.nn.Module):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadAttention, self).__init__()\n        self.d_model = d_model\n        self.num_heads = num_heads\n        if d_model % num_heads != 0:\n            raise ValueError(\"d_model must be divisible by num_heads\")\n\n        self.d_k = d_model // num_heads\n        self.sqrt_d_k = self.d_k ** 0.5\n\n        self.W_Q = torch.nn.Linear(d_model, d_model)\n        self.W_K = torch.nn.Linear(d_model, d_model)\n        self.W_V = torch.nn.Linear(d_model, d_model)\n        self.W_O = torch.nn.Linear(d_model, d_model)\n\n    def forward(\n        self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        # x_q: (N, L_Q, D), x_k: (N, L_KV, D), x_v: (N, L_KV, D)\n        # padding_mask: (N, L_KV), attention_mask: (L_Q, L_KV)\n        N, L_Q, D = x_q.size()\n        _, L_KV, _ = x_k.size()\n        Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)\n        K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)\n        V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)\n\n        score = torch.einsum('nihd,njhd-&gt;nijh', Q, K) / self.sqrt_d_k\n        # score: (N, L_Q, L_KV, num_heads)\n        # Apply attention mask\n        if attention_mask is not None:\n            score = score.masked_fill(\n                attention_mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')\n            )\n        # Apply padding mask\n        if padding_mask is not None:\n            score = score.masked_fill(\n                padding_mask.reshape(N, 1, L_KV, 1) == 0, float('-inf')\n            )\n        score = torch.nn.functional.softmax(score, dim=2)\n        value = torch.einsum(\n            'nijh,njhd-&gt;nihd', score, V\n        ).reshape(N, L_Q, self.d_model)\n        return self.W_O(value)\n</pre> class MultiHeadAttention(torch.nn.Module):     def __init__(self, d_model: int, num_heads: int):         super(MultiHeadAttention, self).__init__()         self.d_model = d_model         self.num_heads = num_heads         if d_model % num_heads != 0:             raise ValueError(\"d_model must be divisible by num_heads\")          self.d_k = d_model // num_heads         self.sqrt_d_k = self.d_k ** 0.5          self.W_Q = torch.nn.Linear(d_model, d_model)         self.W_K = torch.nn.Linear(d_model, d_model)         self.W_V = torch.nn.Linear(d_model, d_model)         self.W_O = torch.nn.Linear(d_model, d_model)      def forward(         self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,         padding_mask: torch.Tensor | None = None,         attention_mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         # x_q: (N, L_Q, D), x_k: (N, L_KV, D), x_v: (N, L_KV, D)         # padding_mask: (N, L_KV), attention_mask: (L_Q, L_KV)         N, L_Q, D = x_q.size()         _, L_KV, _ = x_k.size()         Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)         K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)         V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)          score = torch.einsum('nihd,njhd-&gt;nijh', Q, K) / self.sqrt_d_k         # score: (N, L_Q, L_KV, num_heads)         # Apply attention mask         if attention_mask is not None:             score = score.masked_fill(                 attention_mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')             )         # Apply padding mask         if padding_mask is not None:             score = score.masked_fill(                 padding_mask.reshape(N, 1, L_KV, 1) == 0, float('-inf')             )         score = torch.nn.functional.softmax(score, dim=2)         value = torch.einsum(             'nijh,njhd-&gt;nihd', score, V         ).reshape(N, L_Q, self.d_model)         return self.W_O(value) In\u00a0[\u00a0]: Copied! <pre>class MultiHeadSelfAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadSelfAttention, self).__init__(d_model, num_heads)\n\n    def forward(\n        self, x: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        return super().forward(x, x, x, padding_mask, attention_mask)\n</pre> class MultiHeadSelfAttention(MultiHeadAttention):     def __init__(self, d_model: int, num_heads: int):         super(MultiHeadSelfAttention, self).__init__(d_model, num_heads)      def forward(         self, x: torch.Tensor,         padding_mask: torch.Tensor | None = None,         attention_mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         return super().forward(x, x, x, padding_mask, attention_mask) In\u00a0[\u00a0]: Copied! <pre>class MultiHeadCrossAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadCrossAttention, self).__init__(d_model, num_heads)\n\n    def forward(\n        self, x_q: torch.Tensor, x_kv: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        return super().forward(x_q, x_kv, x_kv, padding_mask, attention_mask)\n</pre> class MultiHeadCrossAttention(MultiHeadAttention):     def __init__(self, d_model: int, num_heads: int):         super(MultiHeadCrossAttention, self).__init__(d_model, num_heads)      def forward(         self, x_q: torch.Tensor, x_kv: torch.Tensor,         padding_mask: torch.Tensor | None = None,         attention_mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         return super().forward(x_q, x_kv, x_kv, padding_mask, attention_mask) In\u00a0[\u00a0]: Copied! <pre>ACT2FN = {\n    'relu': torch.nn.ReLU,\n    'gelu': torch.nn.GELU,\n}\n</pre> ACT2FN = {     'relu': torch.nn.ReLU,     'gelu': torch.nn.GELU, } In\u00a0[\u00a0]: Copied! <pre>class FFN(torch.nn.Module):\n    def __init__(self, input_dim: int, hidden_dim: int, activation: str = 'relu'):\n        super(FFN, self).__init__()\n        self.fc1 = torch.nn.Linear(input_dim, hidden_dim)\n        self.fc2 = torch.nn.Linear(hidden_dim, input_dim)\n        self.act = ACT2FN[activation]()\n\n    def forward(self, x):\n        return self.fc2(self.act(self.fc1(x)))\n</pre> class FFN(torch.nn.Module):     def __init__(self, input_dim: int, hidden_dim: int, activation: str = 'relu'):         super(FFN, self).__init__()         self.fc1 = torch.nn.Linear(input_dim, hidden_dim)         self.fc2 = torch.nn.Linear(hidden_dim, input_dim)         self.act = ACT2FN[activation]()      def forward(self, x):         return self.fc2(self.act(self.fc1(x))) In\u00a0[\u00a0]: Copied! <pre>class EncoderLayer(torch.nn.Module):\n    def __init__(\n        self, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(EncoderLayer, self).__init__()\n\n        if ffn_dim is None:\n            ffn_dim = input_dim * 4\n\n        self.attention = MultiHeadSelfAttention(input_dim, num_heads)\n        self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout1 = torch.nn.Dropout(dropout)\n\n        self.ffn = FFN(input_dim, ffn_dim, activation)\n        self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout2 = torch.nn.Dropout(dropout)\n\n    def forward(\n        self, x, padding_mask: torch.Tensor | None = None,\n        src_mask: torch.Tensor | None = None\n    ):\n        x = x + self.dropout1(self.attention(x, padding_mask, src_mask))\n        x = self.norm1(x)\n\n        x = x + self.dropout2(self.ffn(x))\n        x = self.norm2(x)\n\n        return x\n</pre> class EncoderLayer(torch.nn.Module):     def __init__(         self, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1,         layer_norm_eps: float = 1e-6, activation: str = 'relu'     ):         super(EncoderLayer, self).__init__()          if ffn_dim is None:             ffn_dim = input_dim * 4          self.attention = MultiHeadSelfAttention(input_dim, num_heads)         self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout1 = torch.nn.Dropout(dropout)          self.ffn = FFN(input_dim, ffn_dim, activation)         self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout2 = torch.nn.Dropout(dropout)      def forward(         self, x, padding_mask: torch.Tensor | None = None,         src_mask: torch.Tensor | None = None     ):         x = x + self.dropout1(self.attention(x, padding_mask, src_mask))         x = self.norm1(x)          x = x + self.dropout2(self.ffn(x))         x = self.norm2(x)          return x In\u00a0[\u00a0]: Copied! <pre>class Encoder(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(Encoder, self).__init__()\n\n        self.layers = torch.nn.ModuleList([\n            EncoderLayer(\n                input_dim, num_heads, ffn_dim,\n                dropout, layer_norm_eps, activation\n            )\n            for _ in range(num_layers)\n        ])\n\n    def forward(\n        self, x, padding_mask: torch.Tensor | None = None,\n        src_mask: torch.Tensor | None = None\n    ):\n        # x: (N, L, D)\n        if src_mask is None:\n            src_mask = torch.ones(x.size(1), x.size(1)).to(x.device)\n        for layer in self.layers:\n            x = layer(x, padding_mask, src_mask)\n        return x\n</pre> class Encoder(torch.nn.Module):     def __init__(         self, num_layers: int, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1,         layer_norm_eps: float = 1e-6, activation: str = 'relu'     ):         super(Encoder, self).__init__()          self.layers = torch.nn.ModuleList([             EncoderLayer(                 input_dim, num_heads, ffn_dim,                 dropout, layer_norm_eps, activation             )             for _ in range(num_layers)         ])      def forward(         self, x, padding_mask: torch.Tensor | None = None,         src_mask: torch.Tensor | None = None     ):         # x: (N, L, D)         if src_mask is None:             src_mask = torch.ones(x.size(1), x.size(1)).to(x.device)         for layer in self.layers:             x = layer(x, padding_mask, src_mask)         return x In\u00a0[\u00a0]: Copied! <pre>class DecoderLayer(torch.nn.Module):\n    def __init__(\n        self, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(DecoderLayer, self).__init__()\n\n        if ffn_dim is None:\n            ffn_dim = input_dim * 4\n\n        self.self_attention = MultiHeadSelfAttention(input_dim, num_heads)\n        self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout1 = torch.nn.Dropout(dropout)\n\n        self.cross_attention = MultiHeadCrossAttention(input_dim, num_heads)\n        self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout2 = torch.nn.Dropout(dropout)\n\n        self.ffn = FFN(input_dim, ffn_dim, activation)\n        self.norm3 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout3 = torch.nn.Dropout(dropout)\n\n    def forward(\n        self, x: torch.Tensor, memory: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        memory_mask: torch.Tensor | None = None,\n        tgt_mask: torch.Tensor | None = None\n    ):\n\n        x = x + self.dropout1(self.self_attention(x, padding_mask, tgt_mask))\n        x = self.norm1(x)\n\n        x = x + self.dropout2(self.cross_attention(x, memory, padding_mask, memory_mask))\n        x = self.norm2(x)\n\n        x = x + self.dropout3(self.ffn(x))\n        x = self.norm3(x)\n\n        return x\n</pre> class DecoderLayer(torch.nn.Module):     def __init__(         self, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1,         layer_norm_eps: float = 1e-6, activation: str = 'relu'     ):         super(DecoderLayer, self).__init__()          if ffn_dim is None:             ffn_dim = input_dim * 4          self.self_attention = MultiHeadSelfAttention(input_dim, num_heads)         self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout1 = torch.nn.Dropout(dropout)          self.cross_attention = MultiHeadCrossAttention(input_dim, num_heads)         self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout2 = torch.nn.Dropout(dropout)          self.ffn = FFN(input_dim, ffn_dim, activation)         self.norm3 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout3 = torch.nn.Dropout(dropout)      def forward(         self, x: torch.Tensor, memory: torch.Tensor,         padding_mask: torch.Tensor | None = None,         memory_mask: torch.Tensor | None = None,         tgt_mask: torch.Tensor | None = None     ):          x = x + self.dropout1(self.self_attention(x, padding_mask, tgt_mask))         x = self.norm1(x)          x = x + self.dropout2(self.cross_attention(x, memory, padding_mask, memory_mask))         x = self.norm2(x)          x = x + self.dropout3(self.ffn(x))         x = self.norm3(x)          return x In\u00a0[\u00a0]: Copied! <pre>class Decoder(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(Decoder, self).__init__()\n\n        self.layers = torch.nn.ModuleList([\n            DecoderLayer(\n                input_dim, num_heads, ffn_dim, dropout,\n                layer_norm_eps, activation\n            )\n            for _ in range(num_layers)\n        ])\n\n    def forward(\n        self, x: torch.Tensor, memory: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        memory_mask: torch.Tensor | None = None,\n        tgt_mask: torch.Tensor | None = None\n    ):\n        # x: (N, L, D)\n        for layer in self.layers:\n            x = layer(x, memory, padding_mask, memory_mask, tgt_mask)\n        return x\n</pre> class Decoder(torch.nn.Module):     def __init__(         self, num_layers: int, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1,         layer_norm_eps: float = 1e-6, activation: str = 'relu'     ):         super(Decoder, self).__init__()          self.layers = torch.nn.ModuleList([             DecoderLayer(                 input_dim, num_heads, ffn_dim, dropout,                 layer_norm_eps, activation             )             for _ in range(num_layers)         ])      def forward(         self, x: torch.Tensor, memory: torch.Tensor,         padding_mask: torch.Tensor | None = None,         memory_mask: torch.Tensor | None = None,         tgt_mask: torch.Tensor | None = None     ):         # x: (N, L, D)         for layer in self.layers:             x = layer(x, memory, padding_mask, memory_mask, tgt_mask)         return x"},{"location":"coding/dl-from-scratch/encoder-decoder/","title":"\u7f16\u7801\u5668\u200b\u4e0e\u200b\u89e3\u7801\u5668","text":"<p>\u200b\u5728\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\uff0cVaswani\u200b\u7b49\u200b\u4eba\u200b\u63d0\u51fa\u200b\u4e86\u200b\u4e24\u79cd\u200btransformer\u200b\u67b6\u6784\u200b\uff0c\u200b\u5373\u200b\u7f16\u7801\u5668\u200b\u548c\u200b\u89e3\u7801\u5668\u200b\u3002\u200b\u7f16\u7801\u5668\u200b\u5229\u7528\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\uff0c\u200b\u5bf9\u200b\u8f93\u5165\u200b\u7684\u200b\u5e8f\u5217\u200b\u8fdb\u884c\u200b\u7f16\u7801\u200b\uff0c\u200b\u89e3\u7801\u5668\u200b\u5219\u200b\u5229\u7528\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u548c\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\uff0c\u200b\u751f\u6210\u200b\u5bf9\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u9884\u6d4b\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/encoder-decoder/#_2","title":"\u7f16\u7801\u5668","text":"<p>\u200b\u7f16\u7801\u5668\u200b\u7531\u200b\u591a\u4e2a\u200b\u76f8\u540c\u200b\u7684\u200b\u5c42\u200b\u7ec4\u6210\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5c42\u200b\u5305\u542b\u200b\u4e00\u4e2a\u200b\u6b8b\u5dee\u200b\u8fde\u63a5\u200b\u7684\u200b\u591a\u5934\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u548c\u200b\u4e00\u4e2a\u200b\u524d\u9988\u200b\u795e\u7ecf\u7f51\u7edc\u200b\u3002\u200b\u5728\u200b\u6b8b\u5dee\u200b\u8fde\u63a5\u200b\u524d\u200b\uff0c\u200b\u4f7f\u7528\u200bdropout\uff1b\u200b\u5728\u200b\u6b8b\u5dee\u200b\u8fde\u63a5\u200b\u540e\u200b\uff0c\u200b\u4f7f\u7528\u200b\u5c42\u200b\u5f52\u4e00\u5316\u200b\u6765\u200b\u9632\u6b62\u200b\u8fc7\u200b\u62df\u5408\u200b\u3002\u200b\u524d\u9988\u200b\u795e\u7ecf\u7f51\u7edc\u200b\u7531\u200b\u4e24\u4e2a\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\u7ec4\u6210\u200b\uff0c\u200b\u4e24\u4e2a\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\u4e4b\u95f4\u200b\u901a\u5e38\u200b\u4f7f\u7528\u200bReLU\u200b\u6216\u200bGeLU\u200b\u7b49\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\u3002</p> <pre><code>ACT2FN = {\n    'relu': torch.nn.ReLU,\n    'gelu': torch.nn.GELU,\n}\n\nclass FFN(torch.nn.Module):\n    def __init__(self, input_dim: int, hidden_dim: int, activation: str = 'relu'):\n        super(FFN, self).__init__()\n        self.fc1 = torch.nn.Linear(input_dim, hidden_dim)\n        self.fc2 = torch.nn.Linear(hidden_dim, input_dim)\n        self.act = ACT2FN[activation]()\n\n    def forward(self, x):\n        return self.fc2(self.act(self.fc1(x)))\n\n\nclass EncoderLayer(torch.nn.Module):\n    def __init__(\n        self, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(EncoderLayer, self).__init__()\n\n        if ffn_dim is None:\n            ffn_dim = input_dim * 4\n\n        self.attention = MultiHeadSelfAttention(input_dim, num_heads)\n        self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout1 = torch.nn.Dropout(dropout)\n\n        self.ffn = FFN(input_dim, ffn_dim, activation)\n        self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout2 = torch.nn.Dropout(dropout)\n\n    def forward(\n        self, x, padding_mask: torch.Tensor | None = None,\n        src_mask: torch.Tensor | None = None\n    ):\n        x = x + self.dropout1(self.attention(x, padding_mask, src_mask))\n        x = self.norm1(x)\n\n        x = x + self.dropout2(self.ffn(x))\n        x = self.norm2(x)\n\n        return x\n\n\nclass Encoder(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(Encoder, self).__init__()\n\n        self.layers = torch.nn.ModuleList([\n            EncoderLayer(\n                input_dim, num_heads, ffn_dim,\n                dropout, layer_norm_eps, activation\n            )\n            for _ in range(num_layers)\n        ])\n\n    def forward(\n        self, x, padding_mask: torch.Tensor | None = None,\n        src_mask: torch.Tensor | None = None\n    ):\n        # x: (N, L, D)\n        if src_mask is None:\n            src_mask = torch.ones(x.size(1), x.size(1)).to(x.device)\n        for layer in self.layers:\n            x = layer(x, padding_mask, src_mask)\n        return x\n</code></pre>"},{"location":"coding/dl-from-scratch/encoder-decoder/#_3","title":"\u89e3\u7801\u5668","text":"<p>\u200b\u89e3\u7801\u5668\u200b\u5728\u200b\u7f16\u7801\u5668\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\uff0c\u200b\u5c06\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u7531\u200b\u53cc\u5411\u200b\u6539\u4e3a\u200b\u5355\u5411\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5728\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\u548c\u200b\u524d\u9988\u200b\u795e\u7ecf\u200b\u7f51\u7edc\u5c42\u200b\u4e4b\u95f4\u200b\u589e\u52a0\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u6b8b\u5dee\u200b\u8fde\u63a5\u200b\u7684\u200b\u591a\u5934\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u3002\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u4f7f\u7528\u200b\u7f16\u7801\u5668\u200b\u7684\u200b\u8f93\u51fa\u200b\u4f5c\u4e3a\u200b\u5168\u5c40\u200b\u952e\u548c\u503c\u200b\uff0c\u200b\u89e3\u7801\u5668\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\u7684\u200b\u8f93\u51fa\u200b\u4f5c\u4e3a\u200b\u67e5\u8be2\u200b\u3002\u200b\u5728\u200b\u6b8b\u5dee\u200b\u8fde\u63a5\u200b\u524d\u540e\u200b\u4f7f\u7528\u200bdropout\u200b\u548c\u200b\u5c42\u200b\u5f52\u4e00\u5316\u200b\u3002\u200b\u6ce8\u610f\u200b\u5bf9\u4e8e\u200bdecoder-only\u200b\u7684\u200btransformer\uff0c\u200b\u6ca1\u6709\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\uff0c\u200b\u800c\u662f\u200b\u4ec5\u200b\u5305\u542b\u200b\u4e00\u4e2a\u200b\u5355\u5411\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\u3002</p> <pre><code>class DecoderLayer(torch.nn.Module):\n    def __init__(\n        self, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(DecoderLayer, self).__init__()\n\n        if ffn_dim is None:\n            ffn_dim = input_dim * 4\n\n        self.self_attention = MultiHeadSelfAttention(input_dim, num_heads)\n        self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout1 = torch.nn.Dropout(dropout)\n\n        self.cross_attention = MultiHeadCrossAttention(input_dim, num_heads)\n        self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout2 = torch.nn.Dropout(dropout)\n\n        self.ffn = FFN(input_dim, ffn_dim, activation)\n        self.norm3 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout3 = torch.nn.Dropout(dropout)\n\n    def forward(\n        self, x: torch.Tensor, memory: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        memory_mask: torch.Tensor | None = None,\n        tgt_mask: torch.Tensor | None = None\n    ):\n        if tgt_mask is None:\n            tgt_mask = torch.triu(\n                torch.ones(x.size(1), x.size(1)), diagonal=1\n            ).to(x.device)\n\n        x = x + self.dropout1(self.self_attention(x, padding_mask, tgt_mask))\n        x = self.norm1(x)\n\n        x = x + self.dropout2(self.cross_attention(x, memory, padding_mask, memory_mask))\n        x = self.norm2(x)\n\n        x = x + self.dropout3(self.ffn(x))\n        x = self.norm3(x)\n\n        return x\n\n\nclass Decoder(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6, activation: str = 'relu'\n    ):\n        super(Decoder, self).__init__()\n\n        self.layers = torch.nn.ModuleList([\n            DecoderLayer(\n                input_dim, num_heads, ffn_dim, dropout,\n                layer_norm_eps, activation\n            )\n            for _ in range(num_layers)\n        ])\n\n    def forward(\n        self, x: torch.Tensor, memory: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        memory_mask: torch.Tensor | None = None,\n        tgt_mask: torch.Tensor | None = None\n    ):\n        # x: (N, L, D)\n        for layer in self.layers:\n            x = layer(x, memory, padding_mask, memory_mask, tgt_mask)\n        return x\n</code></pre>"},{"location":"coding/dl-from-scratch/encoder-decoder/#transformer","title":"transformer\u200b\u7684\u200b\u65f6\u7a7a\u200b\u590d\u6742\u5ea6","text":"<p>\u200b\u7ed9\u5b9a\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u7684\u200b\u957f\u5ea6\u200b\\(L\\)\uff0c\u200b\u8f93\u5165\u200b\u7ef4\u5ea6\u200b\\(D\\)\uff0c\u200b\u7f16\u7801\u5668\u200b\u5c42\u6570\u200b\\(N\\)\uff0c\u200b\u524d\u9988\u200b\u795e\u7ecf\u7f51\u7edc\u200b\u7684\u200b\u9690\u85cf\u200b\u5c42\u200b\u7ef4\u5ea6\u200b\\(D_{ff}\\)\uff0c\u200b\u5219\u200btransformer\u200b\u7684\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b</p> <ul> <li>\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\uff1a\u200b\u6bcf\u5c42\u200b\u5305\u542b\u200b\u56db\u4e2a\u200b\\(D \\rightarrow D\\)\u200b\u7684\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\u5305\u542b\u200b\\(D \\times D + D\\)\u200b\u4e2a\u200b\u53c2\u6570\u200b\uff08\u200b\u6743\u91cd\u200b+\u200b\u504f\u7f6e\u200b\uff09\uff0c\u200b\u5171\u200b\\(4ND(D + 1)\\)\u200b\u4e2a\u200b\u53c2\u6570\u200b\uff1b</li> <li>\u200b\u524d\u9988\u200b\u795e\u7ecf\u7f51\u7edc\u200b\uff1a\u200b\u5171\u200b\\(N\\)\u200b\u5c42\u200b\uff0c\u200b\u6bcf\u5c42\u200b\u5305\u542b\u200b\u4e00\u4e2a\u200b\\(D \\rightarrow D_{ff}\\)\u200b\u7684\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\u548c\u200b\u4e00\u4e2a\u200b\\(D_{ff} \\rightarrow D\\)\u200b\u7684\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\uff0c\u200b\u5171\u200b\\(2NDD_{ff} + ND_{ff} + ND\\)\u200b\u4e2a\u200b\u53c2\u6570\u200b\uff1b</li> <li>\u200b\u5c42\u200b\u5f52\u4e00\u5316\u200b\uff1a\u200b\u6bcf\u5c42\u200b\u7f16\u7801\u5668\u200b\u5305\u542b\u200b\u4e24\u4e2a\u200b\u5f52\u4e00\u5316\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5f52\u4e00\u5316\u200b\u5305\u542b\u200b\u4e24\u4e2a\u200b\\(D\\)\u200b\u7ef4\u200b\u7684\u200b\u7f29\u653e\u200b\u53c2\u6570\u200b\u548c\u200b\u504f\u7f6e\u200b\u53c2\u6570\u200b\uff0c\u200b\u5171\u200b\\(4ND\\)\u200b\u4e2a\u200b\u53c2\u6570\u200b\u3002</li> </ul> <p>\u200b\u56e0\u6b64\u200b\uff0ctransformer\u200b\u7684\u200b\u603b\u53c2\u200b\u6570\u91cf\u200b\u4e3a\u200b\\(4ND(D + 1) + 2NDD_{ff} + ND_{ff} + ND + 2ND\\)\u3002\u200b\u5728\u200bVaswani\u200b\u7b49\u200b\u4eba\u200b\u7684\u200b\u5b9e\u73b0\u200b\u4e2d\u200b\uff0c\\(D_{ff} = 4D\\)\uff0c\u200b\u5219\u200btransformer\u200b\u7684\u200b\u53c2\u200b\u6570\u91cf\u200b\u4e3a\u200b\\(12ND^2 + 13ND\\)\u3002\u200b\u89e3\u7801\u5668\u200b\u6bd4\u200b\u7f16\u7801\u5668\u200b\u591a\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\uff08\\(4ND(D + 1)\\)\uff09\u200b\u548c\u200b1\u200b\u4e2a\u200b\u5f52\u4e00\u5316\u200b\u5c42\u200b\uff08\\(2ND\\)\uff09\uff0c\u200b\u56e0\u6b64\u200b\u89e3\u7801\u5668\u200b\u7684\u200b\u53c2\u200b\u6570\u91cf\u200b\u4e3a\u200b\\(16ND^2 + 19ND\\)\u3002</p> <p>\u200b\u77e9\u9635\u200b\u8fd0\u7b97\u200b\\(\\bbR^{m\\times n}\\times \\bbR^{n\\times p} \\rightarrow \\bbR^{m\\times p}\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(mnp)\\)\uff0c\u200b\u9010\u6b65\u200b\u5206\u6790\u200btransformer\u200b\u7684\u200b\u8ba1\u7b97\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u5ffd\u7565\u200b\u52a0\u6cd5\u200b\u548c\u200b\u5f52\u4e00\u5316\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\uff1a</p> <ol> <li>\u200b\u7ebf\u6027\u53d8\u6362\u200b\uff1a\\(\\bbR^{L\\times D}\\times \\bbR^{D\\times D} \\rightarrow \\bbR^{L\\times D}\\)\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(LD^2)\\)\uff1b</li> <li>\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\uff1a\\(\\bbR^{L\\times D}\\times \\bbR^{D\\times L}\\rightarrow \\bbR^{L\\times L}\\)\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(L^2D)\\)\uff1b</li> <li>\u200b\u524d\u9988\u200b\u795e\u7ecf\u7f51\u7edc\u200b\uff1a\\(\\bbR^{L\\times D}\\times \\bbR^{D\\times D_{ff}}\\rightarrow \\bbR^{L\\times D_{ff}}\\)\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(LDD_{ff})\\)\u3002\u200b\u6309\u7167\u200b\\(D_{ff} = 4D\\)\uff0c\u200b\u5219\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(4LD^2) = O(LD^2)\\)\u3002</li> </ol> <p>\\(D\\)\u200b\u4e3a\u200b\u5e38\u6570\u200b\uff0c\u200b\u6a21\u578b\u200b\u5171\u200b\u53e0\u52a0\u200b\\(N\\)\u200b\u5c42\u200b\u4e32\u884c\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u56e0\u6b64\u200btransformer\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(NL^2D)\\)\u3002</p>"},{"location":"coding/dl-from-scratch/llama-2/","title":"\u5b9e\u73b0\u200bLlama-2","text":"<p>\u200b\u5728\u200b\u672c\u8282\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u5b9e\u73b0\u200b\u4e00\u4e2a\u200bLlama-2\u200b\u6a21\u578b\u200b\u3002</p> In\u00a0[1]: Copied! <pre>import functools\n\nimport torch\nimport transformers\n</pre> import functools  import torch import transformers <p>Llama\u200b\u4f7f\u7528\u200bRoPE\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u3002\u200b\u6ce8\u610f\u200b\u6b64\u5904\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u548c\u200bRoFormer\u200b\u4e2d\u200b\u5b9e\u73b0\u200b\u7684\u200b\u4e0d\u540c\u200b\uff0cRoFormer\u200b\u5c06\u200b\u76f8\u90bb\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5143\u7d20\u200b\u5206\u4e3a\u200b\u4e00\u7ec4\u200b\u8fdb\u884c\u200b\u65cb\u8f6c\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u800c\u200bLlamaRope\u200b\u662f\u200b\u5c06\u200b\u5411\u91cf\u200b\u5206\u4e3a\u200b\u524d\u540e\u200b\u4e24\u6bb5\u200b\uff0c\u200b\u524d\u540e\u200b\u4e24\u6bb5\u200b\u76f8\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u5206\u4e3a\u200b\u4e00\u7ec4\u200b\u8fdb\u884c\u200b\u65cb\u8f6c\u200b\u64cd\u4f5c\u200b\u3002</p> In\u00a0[2]: Copied! <pre>class RoPE(torch.nn.Module):\n    def __init__(self, d_model: int, theta: int | float = 10000):\n        super(RoPE, self).__init__()\n        self.d_model = d_model\n        self.theta = theta ** -(torch.arange(0, d_model, 2) / d_model)\n\n    @functools.lru_cache(maxsize=None)\n    def _forward_l(self, L: int) -&gt; torch.Tensor:\n        # Use lru_cache to avoid redundant computation for the same L\n\n        D = self.d_model\n        pos = torch.einsum(\n            'l,d-&gt;ld',\n            torch.arange(L), self.theta\n        )  # [L, D / 2]\n\n        # 0 paired with d // 2, 1 paired with d // 2 + 1, ...\n        cos = torch.cos(pos).repeat([1, 2])\n        sin = torch.sin(pos)\n\n        # Here we do not use efficient method, but construct a rotary matrix\n        result = torch.zeros(L, D, D)\n        result = torch.diagonal_scatter(result, cos, dim1=1, dim2=2)\n        result = torch.diagonal_scatter(result, sin, dim1=1, dim2=2, offset=-D // 2)\n        result = torch.diagonal_scatter(result, -sin, dim1=1, dim2=2, offset=D // 2)\n        return result\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        # x: [N, L, H, D]\n        _, L, _, D = x.size()\n        rot_matrix = self._forward_l(L).to(x.device)  # [L, D, D]\n        return torch.einsum('lde,nlhe-&gt;nlhd', rot_matrix, x)\n</pre> class RoPE(torch.nn.Module):     def __init__(self, d_model: int, theta: int | float = 10000):         super(RoPE, self).__init__()         self.d_model = d_model         self.theta = theta ** -(torch.arange(0, d_model, 2) / d_model)      @functools.lru_cache(maxsize=None)     def _forward_l(self, L: int) -&gt; torch.Tensor:         # Use lru_cache to avoid redundant computation for the same L          D = self.d_model         pos = torch.einsum(             'l,d-&gt;ld',             torch.arange(L), self.theta         )  # [L, D / 2]          # 0 paired with d // 2, 1 paired with d // 2 + 1, ...         cos = torch.cos(pos).repeat([1, 2])         sin = torch.sin(pos)          # Here we do not use efficient method, but construct a rotary matrix         result = torch.zeros(L, D, D)         result = torch.diagonal_scatter(result, cos, dim1=1, dim2=2)         result = torch.diagonal_scatter(result, sin, dim1=1, dim2=2, offset=-D // 2)         result = torch.diagonal_scatter(result, -sin, dim1=1, dim2=2, offset=D // 2)         return result      def forward(self, x: torch.Tensor) -&gt; torch.Tensor:         # x: [N, L, H, D]         _, L, _, D = x.size()         rot_matrix = self._forward_l(L).to(x.device)  # [L, D, D]         return torch.einsum('lde,nlhe-&gt;nlhd', rot_matrix, x) <p>RoPE\u200b\u662f\u200b\u5728\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u5206\u5934\u200b\u540e\u200b\u624d\u200b\u8fdb\u884c\u200b\u8ba1\u7b97\u200b\u3002\u200b\u5e76\u4e14\u200b\uff0cLlama\u200b\u7684\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u4e2d\u200b\u4e0d\u200b\u5305\u542b\u200b\u504f\u7f6e\u200b\u3002</p> In\u00a0[3]: Copied! <pre>class MultiHeadAttention(torch.nn.Module):\n    def __init__(self, d_model: int, num_heads: int, rope: RoPE):\n        super(MultiHeadAttention, self).__init__()\n        self.d_model = d_model\n        self.num_heads = num_heads\n        if d_model % num_heads != 0:\n            raise ValueError(\"d_model must be divisible by num_heads\")\n\n        self.d_k = d_model // num_heads\n        self.sqrt_d_k = self.d_k ** 0.5\n        self.rope = rope\n\n        self.W_Q = torch.nn.Linear(d_model, d_model, bias=False)\n        self.W_K = torch.nn.Linear(d_model, d_model, bias=False)\n        self.W_V = torch.nn.Linear(d_model, d_model, bias=False)\n        self.W_O = torch.nn.Linear(d_model, d_model, bias=False)\n\n    def forward(\n        self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        # x_q: (N, L_Q, D), x_k: (N, L_KV, D), x_v: (N, L_KV, D)\n        # padding_mask: (N, L_KV), attention_mask: (L_Q, L_KV)\n        N, L_Q, D = x_q.size()\n        _, L_KV, _ = x_k.size()\n        Q = self.rope(self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k))\n        K = self.rope(self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k))\n        V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)\n\n        score = torch.einsum('nihd,njhd-&gt;nijh', Q, K) / self.sqrt_d_k\n        # score: (N, L_Q, L_KV, num_heads)\n        # Apply attention mask\n        if attention_mask is not None:\n            score = score.masked_fill(\n                attention_mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')\n            )\n        # Apply padding mask\n        if padding_mask is not None:\n            score = score.masked_fill(\n                padding_mask.reshape(N, 1, L_KV, 1) == 0, float('-inf')\n            )\n        score = torch.nn.functional.softmax(score, dim=2)\n        value = torch.einsum(\n            'nijh,njhd-&gt;nihd', score, V\n        ).reshape(N, L_Q, self.d_model)\n        return self.W_O(value)\n\n\nclass MultiHeadSelfAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int, rope=RoPE):\n        super(MultiHeadSelfAttention, self).__init__(d_model, num_heads, rope)\n\n    def forward(\n        self, x: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        return super().forward(x, x, x, padding_mask, attention_mask)\n</pre> class MultiHeadAttention(torch.nn.Module):     def __init__(self, d_model: int, num_heads: int, rope: RoPE):         super(MultiHeadAttention, self).__init__()         self.d_model = d_model         self.num_heads = num_heads         if d_model % num_heads != 0:             raise ValueError(\"d_model must be divisible by num_heads\")          self.d_k = d_model // num_heads         self.sqrt_d_k = self.d_k ** 0.5         self.rope = rope          self.W_Q = torch.nn.Linear(d_model, d_model, bias=False)         self.W_K = torch.nn.Linear(d_model, d_model, bias=False)         self.W_V = torch.nn.Linear(d_model, d_model, bias=False)         self.W_O = torch.nn.Linear(d_model, d_model, bias=False)      def forward(         self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,         padding_mask: torch.Tensor | None = None,         attention_mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         # x_q: (N, L_Q, D), x_k: (N, L_KV, D), x_v: (N, L_KV, D)         # padding_mask: (N, L_KV), attention_mask: (L_Q, L_KV)         N, L_Q, D = x_q.size()         _, L_KV, _ = x_k.size()         Q = self.rope(self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k))         K = self.rope(self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k))         V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)          score = torch.einsum('nihd,njhd-&gt;nijh', Q, K) / self.sqrt_d_k         # score: (N, L_Q, L_KV, num_heads)         # Apply attention mask         if attention_mask is not None:             score = score.masked_fill(                 attention_mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')             )         # Apply padding mask         if padding_mask is not None:             score = score.masked_fill(                 padding_mask.reshape(N, 1, L_KV, 1) == 0, float('-inf')             )         score = torch.nn.functional.softmax(score, dim=2)         value = torch.einsum(             'nijh,njhd-&gt;nihd', score, V         ).reshape(N, L_Q, self.d_model)         return self.W_O(value)   class MultiHeadSelfAttention(MultiHeadAttention):     def __init__(self, d_model: int, num_heads: int, rope=RoPE):         super(MultiHeadSelfAttention, self).__init__(d_model, num_heads, rope)      def forward(         self, x: torch.Tensor,         padding_mask: torch.Tensor | None = None,         attention_mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         return super().forward(x, x, x, padding_mask, attention_mask) <p>\u200b\u5728\u200b\u524d\u9988\u200b\u7f51\u7edc\u200b\u90e8\u5206\u200b\uff0cLlama\u200b\u4f7f\u7528\u200bSwiglu\u200b\u4f5c\u4e3a\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\u3002\u200b\u6b64\u200b\u90e8\u5206\u200b\u540c\u6837\u200b\u4e0d\u200b\u4f7f\u7528\u200b\u504f\u7f6e\u200b</p> In\u00a0[4]: Copied! <pre>class FFN(torch.nn.Module):\n    def __init__(self, input_dim: int, hidden_dim: int):\n        super(FFN, self).__init__()\n        self.fc1 = torch.nn.Linear(input_dim, hidden_dim, bias=False)\n        self.fc2 = torch.nn.Linear(hidden_dim, input_dim, bias=False)\n        self.gate = torch.nn.Linear(input_dim, hidden_dim, bias=False)\n        self.act = torch.nn.SiLU()\n\n    def forward(self, x):\n        return self.fc2(self.act(self.gate(x)) * self.fc1(x))\n</pre> class FFN(torch.nn.Module):     def __init__(self, input_dim: int, hidden_dim: int):         super(FFN, self).__init__()         self.fc1 = torch.nn.Linear(input_dim, hidden_dim, bias=False)         self.fc2 = torch.nn.Linear(hidden_dim, input_dim, bias=False)         self.gate = torch.nn.Linear(input_dim, hidden_dim, bias=False)         self.act = torch.nn.SiLU()      def forward(self, x):         return self.fc2(self.act(self.gate(x)) * self.fc1(x)) <p>Llama\u200b\u4f7f\u7528\u200bRMSNorm\u200b\u4f5c\u4e3a\u200bLayerNorm\u3002</p> In\u00a0[5]: Copied! <pre>class RMSNorm(torch.nn.Module):\n    def __init__(self, dim: int, eps: float = 1e-8):\n        super(RMSNorm, self).__init__()\n        self.weight = torch.nn.Parameter(torch.ones(dim))\n        self.eps = eps\n\n    def forward(self, x: torch.Tensor):\n        return x / torch.sqrt(torch.mean(x ** 2, dim=-1, keepdim=True) + self.eps) * self.weight\n</pre> class RMSNorm(torch.nn.Module):     def __init__(self, dim: int, eps: float = 1e-8):         super(RMSNorm, self).__init__()         self.weight = torch.nn.Parameter(torch.ones(dim))         self.eps = eps      def forward(self, x: torch.Tensor):         return x / torch.sqrt(torch.mean(x ** 2, dim=-1, keepdim=True) + self.eps) * self.weight <p>Llama\u200b\u4f7f\u7528\u200bPre-norm\uff0c\u200b\u5373\u5148\u200b\u8fdb\u884c\u200bnorm\uff0c\u200b\u518d\u200b\u8fdb\u884c\u200battention/FFN\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u6700\u540e\u200b\u76f8\u52a0\u200b\u3002\u200b\u636e\u6b64\u200b\u53ef\u4ee5\u200b\u6784\u5efa\u200b\u51fa\u200bLlamaDecoder\u200b\u7684\u200b\u57fa\u672c\u200b\u7ed3\u6784\u200b\u3002</p> In\u00a0[6]: Copied! <pre>class LlamaDecoder(torch.nn.Module):\n    def __init__(\n        self, d_model: int, num_heads: int, rope: RoPE, hidden_dim: int, layernorm_eps: float\n    ):\n        super().__init__()\n        self.norm1 = RMSNorm(d_model, layernorm_eps)\n        self.attention = MultiHeadSelfAttention(d_model, num_heads, rope)\n        self.norm2 = RMSNorm(d_model, layernorm_eps)\n        self.ffn = FFN(d_model, hidden_dim)\n\n    def forward(self, x: torch.Tensor, padding_mask: torch.Tensor | None = None, attention_mask: torch.Tensor | None = None):\n        hidden = x\n        hidden += self.attention(self.norm1(hidden), padding_mask, attention_mask)\n        hidden += self.ffn(self.norm2(hidden))\n        return hidden\n</pre> class LlamaDecoder(torch.nn.Module):     def __init__(         self, d_model: int, num_heads: int, rope: RoPE, hidden_dim: int, layernorm_eps: float     ):         super().__init__()         self.norm1 = RMSNorm(d_model, layernorm_eps)         self.attention = MultiHeadSelfAttention(d_model, num_heads, rope)         self.norm2 = RMSNorm(d_model, layernorm_eps)         self.ffn = FFN(d_model, hidden_dim)      def forward(self, x: torch.Tensor, padding_mask: torch.Tensor | None = None, attention_mask: torch.Tensor | None = None):         hidden = x         hidden += self.attention(self.norm1(hidden), padding_mask, attention_mask)         hidden += self.ffn(self.norm2(hidden))         return hidden In\u00a0[7]: Copied! <pre>class LlamaModel(torch.nn.Module):\n    def __init__(\n        self, config: transformers.LlamaConfig\n    ):\n        super().__init__()\n        self.embedding = torch.nn.Embedding(config.vocab_size, config.hidden_size)\n        self.rope = RoPE(\n            config.hidden_size // config.num_attention_heads,\n            config.rope_theta\n        )\n        self.layers = torch.nn.ModuleList([\n            LlamaDecoder(\n                config.hidden_size,\n                config.num_attention_heads,\n                self.rope,\n                config.intermediate_size,\n                config.rms_norm_eps\n            )\n            for _ in range(config.num_hidden_layers)\n        ])\n        self.norm = RMSNorm(config.hidden_size)\n        self.lm_head = torch.nn.Linear(config.hidden_size, config.vocab_size, bias=False)\n\n    def forward(\n        self, input_ids: torch.Tensor,\n        padding_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        hidden = self.embedding(input_ids)\n        attention_mask = 1 - torch.triu(\n            torch.ones(hidden.size(1), hidden.size(1)), diagonal=1\n        ).to(hidden.device)\n        for layer in self.layers:\n            hidden = layer(hidden, padding_mask, attention_mask)\n        hidden = self.norm(hidden)\n        hidden = self.lm_head(hidden)\n        return hidden\n</pre> class LlamaModel(torch.nn.Module):     def __init__(         self, config: transformers.LlamaConfig     ):         super().__init__()         self.embedding = torch.nn.Embedding(config.vocab_size, config.hidden_size)         self.rope = RoPE(             config.hidden_size // config.num_attention_heads,             config.rope_theta         )         self.layers = torch.nn.ModuleList([             LlamaDecoder(                 config.hidden_size,                 config.num_attention_heads,                 self.rope,                 config.intermediate_size,                 config.rms_norm_eps             )             for _ in range(config.num_hidden_layers)         ])         self.norm = RMSNorm(config.hidden_size)         self.lm_head = torch.nn.Linear(config.hidden_size, config.vocab_size, bias=False)      def forward(         self, input_ids: torch.Tensor,         padding_mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         hidden = self.embedding(input_ids)         attention_mask = 1 - torch.triu(             torch.ones(hidden.size(1), hidden.size(1)), diagonal=1         ).to(hidden.device)         for layer in self.layers:             hidden = layer(hidden, padding_mask, attention_mask)         hidden = self.norm(hidden)         hidden = self.lm_head(hidden)         return hidden In\u00a0[8]: Copied! <pre>llama_config = transformers.AutoConfig.from_pretrained('meta-llama/Llama-2-7b-hf')\nllama = LlamaModel(llama_config)\n</pre> llama_config = transformers.AutoConfig.from_pretrained('meta-llama/Llama-2-7b-hf') llama = LlamaModel(llama_config) <pre>/usr/local/lib/python3.10/dist-packages/huggingface_hub/file_download.py:1132: FutureWarning: `resume_download` is deprecated and will be removed in version 1.0.0. Downloads always resume when possible. If you want to force a new download, use `force_download=True`.\n  warnings.warn(\n</pre> <p>\u200b\u52a0\u8f7d\u200bHuggingface\u200b\u63d0\u4f9b\u200b\u7684\u200bLlama\u200b\u6a21\u578b\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5c06\u200b\u9884\u200b\u8bad\u7ec3\u200b\u7684\u200bLlama\u200b\u53c2\u6570\u200b\u590d\u5236\u5230\u200b\u5b9e\u73b0\u200b\u7684\u200b\u6a21\u578b\u200b\u4e2d\u200b\u3002</p> In\u00a0[9]: Copied! <pre>hf_llama = transformers.LlamaForCausalLM.from_pretrained('meta-llama/Llama-2-7b-hf')\n\ndef load_params(custom_model: LlamaModel, hf_model: transformers.LlamaForCausalLM):\n    layer_pairs = [\n        (custom_model.embedding, hf_model.model.embed_tokens),\n        (custom_model.lm_head, hf_model.lm_head),\n        (custom_model.norm, hf_model.model.norm)\n    ]\n    for custom_layer, hf_layer in zip(custom_model.layers, hf_model.model.layers):\n        layer_pairs.extend([\n            (custom_layer.norm1, hf_layer.input_layernorm),\n            (custom_layer.norm2, hf_layer.post_attention_layernorm),\n            (custom_layer.attention.W_Q, hf_layer.self_attn.q_proj),\n            (custom_layer.attention.W_K, hf_layer.self_attn.k_proj),\n            (custom_layer.attention.W_V, hf_layer.self_attn.v_proj),\n            (custom_layer.attention.W_O, hf_layer.self_attn.o_proj),\n            (custom_layer.ffn.gate, hf_layer.mlp.gate_proj),\n            (custom_layer.ffn.fc1, hf_layer.mlp.up_proj),\n            (custom_layer.ffn.fc2, hf_layer.mlp.down_proj)\n        ])\n    for custom_layer, hf_layer in layer_pairs:\n        custom_layer.weight.data = hf_layer.weight.data.clone().detach()\n    return custom_model\n\nllama = load_params(llama, hf_llama)\n</pre> hf_llama = transformers.LlamaForCausalLM.from_pretrained('meta-llama/Llama-2-7b-hf')  def load_params(custom_model: LlamaModel, hf_model: transformers.LlamaForCausalLM):     layer_pairs = [         (custom_model.embedding, hf_model.model.embed_tokens),         (custom_model.lm_head, hf_model.lm_head),         (custom_model.norm, hf_model.model.norm)     ]     for custom_layer, hf_layer in zip(custom_model.layers, hf_model.model.layers):         layer_pairs.extend([             (custom_layer.norm1, hf_layer.input_layernorm),             (custom_layer.norm2, hf_layer.post_attention_layernorm),             (custom_layer.attention.W_Q, hf_layer.self_attn.q_proj),             (custom_layer.attention.W_K, hf_layer.self_attn.k_proj),             (custom_layer.attention.W_V, hf_layer.self_attn.v_proj),             (custom_layer.attention.W_O, hf_layer.self_attn.o_proj),             (custom_layer.ffn.gate, hf_layer.mlp.gate_proj),             (custom_layer.ffn.fc1, hf_layer.mlp.up_proj),             (custom_layer.ffn.fc2, hf_layer.mlp.down_proj)         ])     for custom_layer, hf_layer in layer_pairs:         custom_layer.weight.data = hf_layer.weight.data.clone().detach()     return custom_model  llama = load_params(llama, hf_llama) <pre>Loading checkpoint shards:   0%|          | 0/2 [00:00&lt;?, ?it/s]</pre> <p>\u200b\u52a0\u8f7d\u200btokenizer\u200b\u5e76\u200b\u751f\u6210\u200b\u8f93\u5165\u200b\u6570\u636e\u200b\uff0c\u200b\u6ce8\u610f\u200bLlama-2\u200b\u6ca1\u6709\u200bpadding token\uff0c\u200b\u9700\u8981\u200b\u624b\u52a8\u200b\u8bbe\u7f6e\u200b\u3002</p> In\u00a0[10]: Copied! <pre>llama_tokenizer = transformers.LlamaTokenizer.from_pretrained('meta-llama/Llama-2-7b-hf')\nllama_tokenizer.pad_token = llama_tokenizer.eos_token\ntokenized_sentence = llama_tokenizer([\n    'User: Hello Llama! How are you?\\nAssistant:',\n    'User: Hello World!\\nAssistant:'\n], return_tensors='pt', padding=True)\ninput_ids = tokenized_sentence['input_ids']\npadding_mask = tokenized_sentence['attention_mask']\n</pre> llama_tokenizer = transformers.LlamaTokenizer.from_pretrained('meta-llama/Llama-2-7b-hf') llama_tokenizer.pad_token = llama_tokenizer.eos_token tokenized_sentence = llama_tokenizer([     'User: Hello Llama! How are you?\\nAssistant:',     'User: Hello World!\\nAssistant:' ], return_tensors='pt', padding=True) input_ids = tokenized_sentence['input_ids'] padding_mask = tokenized_sentence['attention_mask'] <p>\u200b\u6839\u636e\u200b\u8f93\u5165\u200b\u6570\u636e\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u6bcf\u5c42\u200b\u6a21\u578b\u200b\u7684\u200b\u8f93\u51fa\u200b\u3002</p> In\u00a0[11]: Copied! <pre>llama_layers_output = hf_llama(input_ids, attention_mask=padding_mask, output_hidden_states=True)\n</pre> llama_layers_output = hf_llama(input_ids, attention_mask=padding_mask, output_hidden_states=True) <p>\u200b\u4ee5\u4e0b\u200b\u5bf9\u200b\u8f93\u51fa\u200b\u7ed3\u679c\u200b\u8fdb\u884c\u200b\u9a8c\u8bc1\u200b\u3002\u200b\u9996\u5148\u200b\u9a8c\u8bc1\u200bembedding\u200b\u5c42\u200b\u7684\u200b\u8f93\u51fa\u200b\u3002</p> In\u00a0[12]: Copied! <pre>llama_layers_output.hidden_states[0]\nhidden = llama.embedding(input_ids)\nattention_mask = 1 - torch.triu(\n    torch.ones(hidden.size(1), hidden.size(1)), diagonal=1\n).to(device=hidden.device)\n\n# Hidden state 1 equivalent to embedding output\ntorch.allclose(\n    llama_layers_output.hidden_states[0],\n    hidden\n)\n</pre> llama_layers_output.hidden_states[0] hidden = llama.embedding(input_ids) attention_mask = 1 - torch.triu(     torch.ones(hidden.size(1), hidden.size(1)), diagonal=1 ).to(device=hidden.device)  # Hidden state 1 equivalent to embedding output torch.allclose(     llama_layers_output.hidden_states[0],     hidden ) Out[12]: <pre>True</pre> <p>\u200b\u9a8c\u8bc1\u200b\u6bcf\u200b\u4e00\u5c42\u200bdecoder\u200b\u7684\u200b\u8f93\u51fa\u200b\uff0c\u200b\u6700\u540e\u200b\u4e00\u5c42\u200bdecoder\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u51fa\u73b0\u200b\u4e0d\u540c\u200b\u3002</p> In\u00a0[13]: Copied! <pre># Verify layer outputs\nlayer_results = []\nfor i in range(llama_config.num_hidden_layers - 1):\n    hidden = llama.layers[i](hidden, padding_mask, attention_mask)\n    layer_results.append(torch.allclose(\n        llama_layers_output.hidden_states[i + 1], hidden, atol=1e-4\n    ))\nall(layer_results), all(layer_results[:-1])\n</pre> # Verify layer outputs layer_results = [] for i in range(llama_config.num_hidden_layers - 1):     hidden = llama.layers[i](hidden, padding_mask, attention_mask)     layer_results.append(torch.allclose(         llama_layers_output.hidden_states[i + 1], hidden, atol=1e-4     )) all(layer_results), all(layer_results[:-1]) Out[13]: <pre>(False, True)</pre> <p>\u200b\u9a8c\u8bc1\u200bLM-head\u200b\u7684\u200b\u8f93\u51fa\u200b\u3002</p> In\u00a0[14]: Copied! <pre># Verify output logits\nlogits = llama(input_ids, padding_mask)\nhf_logits = llama_layers_output.logits\ntorch.allclose(logits, hf_logits, atol=1e-4)\n</pre> # Verify output logits logits = llama(input_ids, padding_mask) hf_logits = llama_layers_output.logits torch.allclose(logits, hf_logits, atol=1e-4) Out[14]: <pre>True</pre> <p>\u200b\u5c1d\u8bd5\u200b\u5bf9\u200b\u8f93\u51fa\u200b\u7684\u200b\u5355\u8bcd\u200b\u8fdb\u884c\u200b\u89e3\u7801\u200b\u3002</p> In\u00a0[15]: Copied! <pre># Decode the output tokens\nmax_prob_tokens = logits[0, -1].topk(5).indices.tolist()\nfor _ in max_prob_tokens:\n    print(llama_tokenizer.decode(_))\n</pre> # Decode the output tokens max_prob_tokens = logits[0, -1].topk(5).indices.tolist() for _ in max_prob_tokens:     print(llama_tokenizer.decode(_)) <pre>I\nHello\nFine\nHi\nOh\n</pre> In\u00a0[16]: Copied! <pre>del hf_llama\ndel llama_layers_output\ndel hf_logits\nimport gc\ngc.collect()\n</pre> del hf_llama del llama_layers_output del hf_logits import gc gc.collect() Out[16]: <pre>21</pre> In\u00a0[17]: Copied! <pre>def complete_ids(model: LlamaModel, input_ids: torch.Tensor):\n    assert input_ids.size(0) == 1\n\n    next_token = model(input_ids, torch.ones_like(input_ids))[0, -1, :].topk(1).indices[0].item()\n    return next_token\n\ndef complete_string(model: LlamaModel, tokenizer: transformers.LlamaTokenizer, prompt: str, max_length: int):\n    model.eval()\n    gc.collect()\n    tokenized_sentence = tokenizer([prompt], return_tensors='pt')['input_ids'].tolist()\n    next_token = None\n    print(tokenizer.decode(tokenized_sentence[0]))\n    num_tokens = 0\n\n    while next_token != tokenizer.eos_token_id and num_tokens &lt;= max_length:\n        num_tokens += 1\n        next_token = complete_ids(model, torch.tensor(tokenized_sentence))\n        tokenized_sentence[0].append(next_token)\n        print(tokenizer.decode(tokenized_sentence[0]))\n\n        gc.collect()\n\n    return num_tokens\n</pre> def complete_ids(model: LlamaModel, input_ids: torch.Tensor):     assert input_ids.size(0) == 1      next_token = model(input_ids, torch.ones_like(input_ids))[0, -1, :].topk(1).indices[0].item()     return next_token  def complete_string(model: LlamaModel, tokenizer: transformers.LlamaTokenizer, prompt: str, max_length: int):     model.eval()     gc.collect()     tokenized_sentence = tokenizer([prompt], return_tensors='pt')['input_ids'].tolist()     next_token = None     print(tokenizer.decode(tokenized_sentence[0]))     num_tokens = 0      while next_token != tokenizer.eos_token_id and num_tokens &lt;= max_length:         num_tokens += 1         next_token = complete_ids(model, torch.tensor(tokenized_sentence))         tokenized_sentence[0].append(next_token)         print(tokenizer.decode(tokenized_sentence[0]))          gc.collect()      return num_tokens In\u00a0[18]: Copied! <pre>import time\nquery = 'User: Given two numbers, 13.11 and 13.8, which is larger?\\nAssistant:'\n\nstart_time = time.time()\nnum_tokens = complete_string(llama, llama_tokenizer, query, 10)\nend_time = time.time()\n\ntime_span = end_time - start_time\nprint(f'Predicted {num_tokens} in {time_span:.2f} seconds, {num_tokens / time_span:.2f} tokens/s')\n</pre> import time query = 'User: Given two numbers, 13.11 and 13.8, which is larger?\\nAssistant:'  start_time = time.time() num_tokens = complete_string(llama, llama_tokenizer, query, 10) end_time = time.time()  time_span = end_time - start_time print(f'Predicted {num_tokens} in {time_span:.2f} seconds, {num_tokens / time_span:.2f} tokens/s') <pre>&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant:\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is \n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is 1\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is 13\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is 13.\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is 13.8\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is 13.8.\n&lt;s&gt; User: Given two numbers, 13.11 and 13.8, which is larger?\nAssistant: The larger number is 13.8.\n\nPredicted 11 in 37.42 seconds, 0.29 tokens/s\n</pre>"},{"location":"coding/dl-from-scratch/llama-2/","title":"\u6a21\u578b\u200b\u5b9e\u73b0\u200b\u00b6","text":""},{"location":"coding/dl-from-scratch/llama-2/","title":"\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u00b6","text":""},{"location":"coding/dl-from-scratch/llama-2/","title":"\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u00b6","text":""},{"location":"coding/dl-from-scratch/llama-2/#ffn","title":"FFN\u00b6","text":""},{"location":"coding/dl-from-scratch/llama-2/#layernorm","title":"LayerNorm\u00b6","text":""},{"location":"coding/dl-from-scratch/llama-2/#decoder-layer","title":"Decoder 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of sentence\uff09\u200b\u548c\u200beos\uff08end of sentence\uff09\u200b\u7684\u200b\u9884\u6d4b\u200b\u3002\u200b\u6240\u4ee5\u200b\uff0c\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b\u57fa\u5ea7\u200b\u6a21\u578b\u200b\uff0c\u200b\u6a21\u578b\u200b\u4f1a\u200b\u4e0d\u65ad\u200b\u8f93\u51fa\u200btoken\u3002</p>"},{"location":"coding/dl-from-scratch/llm-inference/#kv-cache","title":"KV-cache","text":"<p>\u200b\u628a\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u91cd\u65b0\u200b\u8f93\u51fa\u200bLLM\uff0c\u200b\u9884\u6d4b\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5355\u8bcd\u200b\u63a8\u7406\u200b\u6d41\u7a0b\u200b\u7684\u200b\u8ba1\u7b97\u200b\u6548\u7387\u200b\u975e\u5e38\u4f4e\u200b\uff0c\u200b\u539f\u56e0\u200b\u5728\u4e8e\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u9884\u6d4b\u200b\u90fd\u200b\u9700\u8981\u200b\u5bf9\u200b\u6574\u4e2a\u200b\u5e8f\u5217\u200b\u8ba1\u7b97\u200b\u6ce8\u610f\u529b\u200b\u5f97\u5206\u200b\u3002\u200b\u7136\u800c\u200b\uff0c\u200b\u7531\u4e8e\u200b\u6bcf\u200b\u4e00\u4e2a\u200btoken\u200b\u7684\u200b\u8868\u793a\u200b\u53ea\u200b\u4f9d\u8d56\u4e8e\u200b\u524d\u9762\u200b\u7684\u200btoken\uff0c\u200b\u5728\u200b\u63a8\u7406\u200b\u9636\u6bb5\u200b\uff0c\u200b\u6211\u4eec\u200b\u4e0d\u5fc5\u200b\u91cd\u7b97\u200b\u524d\u8fb9\u200b\u7684\u200btoken\uff0c\u200b\u53ea\u200b\u9700\u8981\u200b\u5b58\u50a8\u200b\u524d\u8fb9\u200b\u7684\u200btoken\u200b\u7684\u200bKV\u200b\u8868\u793a\u200b\u5373\u53ef\u200b\u3002\u200b\u53ea\u200b\u9700\u7528\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200btoken\u200b\u7684\u200bQ\u200b\u8ba1\u7b97\u200b\u6ce8\u610f\u529b\u200b\u5373\u53ef\u200b\u5f97\u5230\u200b\u5bf9\u4e0b\u200b\u4e00\u4e2a\u200btoken\u200b\u7684\u200b\u9884\u6d4b\u200b\u3002\u200b\u8fd9\u79cd\u200b\u6280\u672f\u200b\u5373\u200b\u79f0\u4e3a\u200bKV\u200b\u7f13\u5b58\u200b\u3002\u200b\u5305\u542b\u200bKV\u200b\u7f13\u5b58\u200b\u7684\u200b\u63a8\u7406\u200b\u8fc7\u7a0b\u200b\u5206\u4e3a\u200b\u4e24\u4e2a\u200b\u9636\u6bb5\u200b\uff0c\u200b\u5373\u200b\u7f13\u5b58\u200b\u586b\u5145\u200b\uff08prefill\uff09\u200b\u9636\u6bb5\u200b\u548c\u200b\u89e3\u7801\u200b\uff08decode\uff09\u200b\u9636\u6bb5\u200b\u3002</p> <p>\u200b\u4e3a\u4f55\u200b\u53ea\u200b\u9700\u8981\u200bKV-cache\u200b\u800c\u200b\u4e0d\u200b\u9700\u8981\u200bQ-cache</p> <p>\u200b\u5047\u8bbe\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u7ecf\u8fc7\u200b\\(QKV\\)\u200b\u8ba1\u7b97\u200b\u540e\u200b\u7684\u200b\u5e8f\u5217\u200b\u4e3a\u200b\\(S^Q, S^K, S^V\\)\uff0c\u200b\u5e76\u4e14\u200b\u53ef\u4ee5\u200b\u62c6\u200b\u5206\u4e3a\u200b\u5217\u200b\u5411\u91cf\u200b\u8868\u793a\u200b\uff1a</p> \\[ \\begin{aligned} S^Q &amp;= \\begin{bmatrix} Q_1 &amp; Q_2 &amp; \\cdots &amp; Q_n \\end{bmatrix} \\\\ S^K &amp;= \\begin{bmatrix} K_1 &amp; K_2 &amp; \\cdots &amp; K_n \\end{bmatrix} \\\\ S^V &amp;= \\begin{bmatrix} V_1 &amp; V_2 &amp; \\cdots &amp; V_n \\end{bmatrix} \\] <p>\u200b\u6ce8\u610f\u529b\u200b\u5f97\u5206\u200b\u77e9\u9635\u200b\u4e3a\u200b</p> \\[ \\begin{bmatrix} Q_1K_1 &amp; Q_1K_2 &amp; \\cdots &amp; Q_1K_n \\\\ Q_2K_1 &amp; Q_2K_2 &amp; \\cdots &amp; Q_2K_n \\\\ \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\ Q_nK_1 &amp; Q_nK_2 &amp; \\cdots &amp; Q_nK_n \\] <p>\u200b\u7531\u4e8e\u200bdecoder\u200b\u7684\u200bcausal mask\u200b\u9650\u5236\u200b\uff0c\u200b\u6700\u7ec8\u200b\u8ba1\u7b97\u200b\u7684\u200b\u8868\u793a\u200b\u4e3a\u200b</p> \\[ \\begin{aligned} O_1 &amp;= \\sum_{i=1}^1 \\text{softmax}(Q_1K_i)V_i \\\\ O_2 &amp;= \\sum_{i=1}^2 \\text{softmax}(Q_2K_i)V_i \\\\ \\end{aligned} \\] <p>\u200b\u4e5f\u200b\u5373\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u8868\u793a\u200b\uff0c\u200b\u53ea\u200b\u4f9d\u8d56\u200b\u8be5\u200b\u4f4d\u7f6e\u200b\u7684\u200bQ\uff0c\u200b\u548c\u200b\u8be5\u200b\u4f4d\u7f6e\u200b\u53ca\u200b\u4e4b\u524d\u200b\u7684\u200bK\u200b\u548c\u200bV\u3002\u200b\u56e0\u6b64\u200b\u9884\u6d4b\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5355\u8bcd\u200b\u65f6\u200b\uff0c\u200b\u4e0d\u200b\u9700\u8981\u200b\u91cd\u65b0\u200b\u8ba1\u7b97\u200bKV\uff0c\u200b\u5c06\u200bKV\u200b\u5b58\u50a8\u200b\u4e0b\u6765\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u5c06\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u8f93\u5165\u200bLLM\u200b\u65f6\u200b\uff0c\u200b\u9996\u5148\u200b\u8fdb\u884c\u200b\u7f13\u5b58\u200b\u586b\u5145\u200b\uff0c\u200b\u4fdd\u5b58\u200b\u4e0b\u200b\u6240\u6709\u200b\u5c42\u200b\u7684\u200bKV-cache\uff0c\u200b\u7136\u540e\u200b\u8fdb\u884c\u200b\u89e3\u7801\u200b\uff0c\u200b\u4e0d\u65ad\u200b\u9884\u6d4b\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5355\u8bcd\u200b\uff0c\u200b\u5e76\u200b\u8bb0\u5f55\u200b\u9884\u6d4b\u200b\u5355\u8bcd\u200b\u7684\u200bKV\u200b\u8868\u793a\u200b\u3002\u200b\u5bf9\u4e8e\u200bdecoder\u200b\u7684\u200b\u6bcf\u200b\u4e00\u5c42\u200b\uff0c\u200b\u90fd\u200b\u9700\u8981\u200b\u4fdd\u5b58\u200b\u4e00\u4e2a\u200bKV-cache\uff0c\u200b\u6bcf\u200b\u4e00\u5c42\u200bcache\u200b\u7684\u200b\u5927\u5c0f\u200b\u4e3a\u200b\\(2\\times N\\times L\\times D\\)\u3002\u200b\u5bf9\u4e8e\u200b\u957f\u200b\u5e8f\u5217\u200b\uff0cKV-cache\u200b\u7684\u200b\u5927\u5c0f\u200b\u4f1a\u200b\u975e\u5e38\u200b\u5927\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/loss/","title":"\u635f\u5931\u200b\u51fd\u6570","text":"<p>\u200b\u635f\u5931\u200b\u51fd\u6570\u200b\u662f\u200b\u7528\u6765\u200b\u8bc4\u4ef7\u200b\u6a21\u578b\u200b\u9884\u6d4b\u503c\u200b\u4e0e\u200b\u771f\u5b9e\u200b\u503c\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5dee\u5f02\u200b\u7684\u200b\u51fd\u6570\u200b\u3002\u200b\u7136\u540e\u200b\u901a\u8fc7\u200b\u4f18\u5316\u200b\u7b97\u6cd5\u200b\u6765\u200b\u8c03\u6574\u200b\u6a21\u578b\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u635f\u5931\u200b\u51fd\u6570\u200b\u7684\u200b\u503c\u200b\u6700\u5c0f\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/loss/#_2","title":"\u5747\u65b9\u200b\u8bef\u5dee","text":"<p>\u200b\u5747\u65b9\u200b\u8bef\u5dee\u200b\uff08Mean Squared Error, MSE\uff09\u200b\u662f\u200b\u56de\u5f52\u200b\u4efb\u52a1\u200b\u4e2d\u200b\u6700\u200b\u5e38\u7528\u200b\u7684\u200b\u635f\u5931\u200b\u51fd\u6570\u200b\u3002\u200b\u5b83\u200b\u8ba1\u7b97\u200b\u7684\u200b\u662f\u200b\u9884\u6d4b\u503c\u200b\u4e0e\u200b\u771f\u5b9e\u200b\u503c\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5e73\u65b9\u5dee\u200b\u7684\u200b\u5747\u503c\u200b\u3002</p> \\[ \\begin{aligned} L(y, t) &amp;= \\frac{1}{2} \\sum_{i=1}^{n} (y_i - t_i)^2 \\\\ \\frac{\\partial L}{\\partial y_i} &amp;= y_i - t_i \\end{aligned} \\] <p>\u200b\u5176\u4e2d\u200b\uff0c\\(y\\) \u200b\u662f\u200b\u6a21\u578b\u200b\u7684\u200b\u9884\u6d4b\u503c\u200b\uff0c\\(t\\) \u200b\u662f\u200b\u771f\u5b9e\u200b\u503c\u200b\uff0c\\(n\\) \u200b\u662f\u200b\u6837\u672c\u200b\u6570\u91cf\u200b\u3002</p> <pre><code>def mse(y: torch.Tensor, t: torch.Tensor) -&gt; torch.Tensor:\n    return 0.5 * torch.sum((y - t) ** 2)\n\nclass MSELoss(torch.nn.Module):\n    def __init__(self):\n        super(MSELoss, self).__init__()\n\n    def forward(self, y: torch.Tensor, t: torch.Tensor) -&gt; torch.Tensor:\n        return mse(y, t)\n</code></pre>"},{"location":"coding/dl-from-scratch/loss/#_3","title":"\u4ea4\u53c9\u200b\u71b5","text":"<p>\u200b\u4ea4\u53c9\u200b\u71b5\u200b\uff08Cross Entropy\uff09\u200b\u662f\u200b\u591a\u200b\u5206\u7c7b\u200b\u4efb\u52a1\u200b\u4e2d\u200b\u6700\u200b\u5e38\u7528\u200b\u7684\u200b\u635f\u5931\u200b\u51fd\u6570\u200b\u3002\u200b\u5b83\u200b\u8ba1\u7b97\u200b\u7684\u200b\u662f\u200b\u6a21\u578b\u200b\u8f93\u51fa\u200b\u503c\u200b\u4e0e\u200b\u771f\u5b9e\u200b\u503c\u200b\u4e4b\u95f4\u200b\u7684\u200b\u4ea4\u53c9\u200b\u71b5\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200bKL\u200b\u6563\u5ea6\u200b\u63a8\u5bfc\u200b\u5f97\u5230\u200b\u3002</p> \\[ \\begin{aligned} KL(p||q) &amp;= -\\sum_{i=1}^{n} p_i (\\log q_i - \\log p_i) \\\\ &amp;= -\\sum_{i=1}^{n} p_i \\log q_i + \\sum_{i=1}^{n} p_i \\log p_i \\\\ &amp;= H(p, q) - H(p) \\end{aligned} \\] <p>\u200b\u5176\u4e2d\u200b\uff0c\\(p\\) \u200b\u662f\u200b\u771f\u5b9e\u200b\u6982\u7387\u5206\u5e03\u200b\uff0c\\(q\\) \u200b\u662f\u200b\u6a21\u578b\u200b\u8f93\u51fa\u200b\u6982\u7387\u5206\u5e03\u200b\uff0c\\(H(p)\\)\u200b\u8868\u793a\u200b\u771f\u5b9e\u200b\u5206\u5e03\u200b\\(p\\)\u200b\u7684\u200b\u71b5\u200b\uff0c\u200b\u4e3a\u200b\u5e38\u6570\u200b\u3002\u200b\u5f53\u200b\u8f93\u51fa\u200b\u5206\u5e03\u200b\\(q\\)\u200b\u4e0e\u200b\u771f\u5b9e\u200b\u5206\u5e03\u200b\\(p\\)\u200b\u76f8\u8fd1\u200b\u65f6\u200b\uff0cKL\u200b\u6563\u5ea6\u503c\u200b\u8d8a\u200b\u5c0f\u200b\uff0c\u200b\u4ea4\u53c9\u200b\u71b5\u200b\\(-\\sum_{i=1}^{n} p_i \\log q_i\\)\u200b\u503c\u200b\u4e5f\u200b\u8d8a\u200b\u5c0f\u200b\u3002</p> <p>\u200b\u9700\u8981\u200b\u6ce8\u610f\u200b\u7684\u200b\u662f\u200b\uff0c\u200b\u5728\u200bPyTorch\u200b\u4e2d\u200b\uff0c<code>torch.nn.CrossEntropyLoss</code>\u200b\u51fd\u6570\u200b\u7684\u200b\u8f93\u5165\u200b\u662f\u200b\u6a21\u578b\u200b\u7684\u200b\u8f93\u51fa\u200b\u503c\u200b\u548c\u200b\u771f\u5b9e\u200b\u6807\u7b7e\u200b\uff0c\u200b\u4e0d\u200b\u9700\u8981\u200b\u5bf9\u200b\u8f93\u51fa\u200b\u503c\u200b\u8fdb\u884c\u200bsoftmax\u200b\u64cd\u4f5c\u200b\u3002</p> <pre><code>def cross_entropy(score: torch.Tensor, label: torch.Tensor) -&gt; torch.Tensor:\n    # label: [N], score: [N, C]\n    score = torch.softmax(score, dim=-1)\n    label_logits = score[torch.arange(score.size(0)), label]\n    return -torch.sum(torch.log(label_logits))\n\nclass CrossEntropyLoss(torch.nn.Module):\n    def __init__(self):\n        super(CrossEntropyLoss, self).__init__()\n\n    def forward(self, score: torch.Tensor, label: torch.Tensor) -&gt; torch.Tensor:\n        return cross_entropy(score, label)\n</code></pre>"},{"location":"coding/dl-from-scratch/mlp/","title":"\u591a\u5c42\u200b\u611f\u77e5\u673a","text":"<p>\u200b\u591a\u5c42\u200b\u611f\u77e5\u673a\u200b\uff08Multilayer Perceptron, MLP\uff09\u200b\u662f\u200b\u6df1\u5ea6\u200b\u5b66\u4e60\u200b\u7684\u200b\u57fa\u7840\u200b\uff0c\u200b\u4e5f\u200b\u662f\u200b\u6700\u200b\u7b80\u5355\u200b\u7684\u200b\u6df1\u5ea6\u200b\u5b66\u4e60\u200b\u6a21\u578b\u200b\u4e4b\u4e00\u200b\uff0c\u200b\u7531\u200b\u7ebf\u6027\u53d8\u6362\u200b\u548c\u200b\u975e\u7ebf\u6027\u200b\u7684\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\u7ec4\u6210\u200b\u3002\u200b\u591a\u5c42\u200b\u611f\u77e5\u673a\u200b\u7684\u200b\u7f51\u7edc\u7ed3\u6784\u200b\u662f\u200b\u7531\u200b\u8f93\u5165\u200b\u5c42\u200b\u3001\u200b\u591a\u4e2a\u200b\u9690\u85cf\u200b\u5c42\u200b\u548c\u200b\u8f93\u51fa\u200b\u5c42\u200b\u7ec4\u6210\u200b\uff0c\u200b\u5176\u4e2d\u200b\u9690\u85cf\u200b\u5c42\u200b\u7684\u200b\u795e\u7ecf\u5143\u200b\u6570\u91cf\u200b\u548c\u200b\u5c42\u6570\u200b\u53ef\u4ee5\u200b\u81ea\u7531\u200b\u8bbe\u5b9a\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/mlp/#_2","title":"\u7ebf\u6027\u53d8\u6362","text":"<p>\u200b\u7ebf\u6027\u53d8\u6362\u200b\u662f\u200b\u591a\u5c42\u200b\u611f\u77e5\u673a\u200b\u7684\u200b\u57fa\u7840\u200b\uff0c\u200b\u5b83\u200b\u662f\u200b\u7531\u200b\u8f93\u5165\u200b\u5c42\u5230\u200b\u9690\u85cf\u200b\u5c42\u200b\u548c\u200b\u8f93\u51fa\u200b\u5c42\u200b\u7684\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\u7ec4\u6210\u200b\u3002\u200b\u7ebf\u6027\u53d8\u6362\u200b\u7684\u200b\u6570\u5b66\u200b\u8868\u8fbe\u5f0f\u200b\u4e3a\u200b\uff1a</p> \\[ y = Wx + b \\] <p>\u200b\u5176\u4e2d\u200b \\(W\\) \u200b\u662f\u200b\u6743\u91cd\u200b\u77e9\u9635\u200b\uff0c\\(x\\) \u200b\u662f\u200b\u8f93\u5165\u200b\u5411\u91cf\u200b\uff0c\\(b\\) \u200b\u662f\u200b\u504f\u7f6e\u200b\u5411\u91cf\u200b\uff0c\\(y\\) \u200b\u662f\u200b\u8f93\u51fa\u200b\u5411\u91cf\u200b\u3002</p> <pre><code>class Linear(torch.nn.Module):\n    def __init__(self, in_dim: int, out_dim: int):\n        super(Linear, self).__init__()\n        self.in_dim = in_dim\n        self.out_dim = out_dim\n        self.weight = torch.nn.Parameter(torch.randn(out_dim, in_dim))\n        self.bias = torch.nn.Parameter(torch.randn(out_dim))\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        return torch.einsum(\"ij,j-&gt;i\", self.weight, x) + self.bias\n</code></pre> <p>\u200b\u53cd\u5411\u200b\u4f20\u64ad\u200b\u4e2d\u200b\uff0c\u200b\u68af\u5ea6\u200b\u8ba1\u7b97\u516c\u5f0f\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned} \\frac{\\partial y}{\\partial W} &amp;= x \\\\ \\frac{\\partial y}{\\partial b} &amp;= 1 \\end{aligned} \\]"},{"location":"coding/dl-from-scratch/normalization/","title":"\u5f52\u4e00\u5316","text":"<p>\u200b\u5f52\u4e00\u5316\u200b\u662f\u200b\u6307\u200b\u5c06\u200b\u6570\u636e\u200b\u6309\u7167\u200b\u6bd4\u4f8b\u200b\u7f29\u653e\u200b\uff0c\u200b\u6539\u53d8\u200b\u5176\u200b\u5206\u5e03\u200b\u3002\u200b\u6309\u7167\u200b\u5f52\u4e00\u5316\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u4e0d\u540c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u4ee5\u4e0b\u200b\u51e0\u79cd\u200b\uff1a</p> <ul> <li>\u200b\u6837\u672c\u200b\u5f52\u4e00\u5316\u200b\uff08Batch Normalization\uff09\uff1a\u200b\u5bf9\u200b\u6bcf\u4e2a\u200b\u7279\u5f81\u200b\u8fdb\u884c\u200b\u5f52\u4e00\u5316\u200b\uff0c\u200b\u901a\u5e38\u200b\u7528\u4e8e\u200b\u56fe\u50cf\u200b\u6570\u636e\u200b\\(B \\times C \\times H \\times W\\)\uff0c\u200b\u5728\u200b\u540c\u4e00\u200b\\(C\\)\u200b\u7ef4\u5ea6\u200b\u4e0a\u200b\u8ba1\u7b97\u200b\u5747\u503c\u200b\u548c\u200b\u65b9\u5dee\u200b\u3002</li> <li>\u200b\u5c42\u200b\u5f52\u4e00\u5316\u200b\uff08Layer Normalization\uff09\uff1a\u200b\u5bf9\u200b\u6bcf\u4e2a\u200b\u6837\u672c\u200b\u8fdb\u884c\u200b\u5f52\u4e00\u5316\u200b\uff0c\u200b\u901a\u5e38\u200b\u7528\u4e8e\u200b\u5e8f\u5217\u200b\u6570\u636e\u200b\\(B\\times L \\times D\\)\uff0c\u200b\u5728\u200b\u540c\u4e00\u200b\\(B\\)\u200b\u7ef4\u5ea6\u200b\u4e0a\u200b\u8ba1\u7b97\u200b\u5747\u503c\u200b\u548c\u200b\u65b9\u5dee\u200b\u3002</li> <li>\u200b\u5b9e\u4f8b\u200b\u5f52\u4e00\u5316\u200b\uff08Instance Normalization\uff09\uff1a\u200b\u5bf9\u200b\u6bcf\u4e2a\u200b\u6837\u672c\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u7279\u5f81\u200b\u7ef4\u5ea6\u200b\u8fdb\u884c\u200b\u5f52\u4e00\u5316\u200b\uff0c\u200b\u5728\u200b\u540c\u4e00\u200b\\(B\\times C\\)\u200b\u7ef4\u5ea6\u200b\u4e0a\u200b\u8ba1\u7b97\u200b\u5747\u503c\u200b\u548c\u200b\u65b9\u5dee\u200b\u3002</li> <li>\u200b\u5206\u7ec4\u200b\u5f52\u4e00\u5316\u200b\uff08Group Normalization\uff09\uff1a\u200b\u5bf9\u200b\u6bcf\u4e2a\u200b\u6837\u672c\u200b\u4e2d\u200b\u7684\u200b\u7279\u5f81\u200b\u5206\u6210\u200b\u82e5\u5e72\u7ec4\u200b\u540e\u200b\uff0c\u200b\u5728\u200b\u540c\u200b\u4e00\u7ec4\u200b\u5185\u200b\u8fdb\u884c\u200b\u5f52\u4e00\u5316\u200b\u3002</li> </ul> <p>\u200b\u5f52\u4e00\u5316\u200b\u7684\u200b\u65b9\u5f0f\u200b\u4e3a\u200b</p> \\[ x' = \\frac{x - \\mu}{\\sigma} \\times \\gamma + \\beta \\] <p>\u200b\u5176\u4e2d\u200b\\(\\mu, \\sigma\\)\u200b\u5206\u522b\u200b\u4e3a\u200b\u6837\u672c\u5747\u503c\u200b\u548c\u200b\u6807\u51c6\u5dee\u200b\uff0c\\(\\gamma, \\beta\\)\u200b\u4e3a\u200b\u53ef\u200b\u5b66\u4e60\u200b\u7684\u200b\u91cd\u200b\u7f29\u653e\u200b\u53c2\u6570\u200b\u3002\u200b\u7ed9\u5b9a\u200b\u7279\u5f81\u200b\u7ef4\u5ea6\u200b\\(d\\)\uff0c\u200b\u5f52\u4e00\u5316\u200b\u5c42\u200b\u7684\u200b\u53c2\u200b\u6570\u91cf\u200b\u4e3a\u200b\\(2d\\)\u3002</p>"},{"location":"coding/dl-from-scratch/normalization/#rmsnorm","title":"RMSNorm","text":"<p>Zhang\u200b\u7b49\u200b\u4eba\u200b\u5728\u200bLN\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\u63d0\u51fa\u200b\u4e86\u200bRMSNorm\uff0c\u200b\u76f8\u6bd4\u200b\u4e8e\u200bLN\uff0cRMSNorm\u200b\u4e0d\u200b\u9700\u8981\u200b\u8ba1\u7b97\u200b\u5747\u503c\u200b\u548c\u200b\u65b9\u5dee\u200b\uff0c\u200b\u800c\u662f\u200b\u76f4\u63a5\u200b\u4f7f\u7528\u200b\u5747\u200b\u65b9\u6839\u200b\u4f5c\u4e3a\u200b\u5f52\u4e00\u5316\u200b\u7684\u200b\u6807\u51c6\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5728\u200b\u91cd\u200b\u7f29\u653e\u200b\u9636\u6bb5\u200b\u5220\u53bb\u200b\u4e86\u200b\u504f\u7f6e\u200b\\(\\beta\\)\u3002</p> \\[ x' = \\frac{x}{\\sqrt{\\frac{1}{N} \\sum_{i=1}^{N} x_i^2}} \\times \\gamma \\] <pre><code>class RMSNorm(torch.nn.Module):\n    def __init__(self, dim: int, eps=1e-8):\n        super(RMSNorm, self).__init__()\n        self.gamma = torch.nn.Parameter(torch.ones(dim))\n        self.eps = eps\n\n    def forward(self, x: torch.Tensor):\n        return x / torch.sqrt(torch.mean(x ** 2, dim=-1, keepdim=True) + self.eps) * self.gamma\n</code></pre>"},{"location":"coding/dl-from-scratch/positional-embedding/","title":"\u4f4d\u7f6e\u200b\u7f16\u7801","text":"<p>Attention\u200b\u673a\u5236\u200b\u867d\u7136\u200b\u80fd\u200b\u6355\u6349\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u4e0d\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u4f9d\u8d56\u200b\u5173\u7cfb\u200b\uff0c\u200b\u4f46\u662f\u200b\u65e0\u6cd5\u200b\u533a\u5206\u200b\u4e0d\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\u200b\u4e3a\u4e86\u200b\u89e3\u51b3\u200b\u8fd9\u4e2a\u200b\u95ee\u9898\u200b\uff0cTransformer\u200b\u6a21\u578b\u200b\u5f15\u5165\u200b\u4e86\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff08Positional Encoding\uff09\u3002</p>"},{"location":"coding/dl-from-scratch/positional-embedding/#_2","title":"\u7edd\u5bf9\u200b\u4f4d\u7f6e\u200b\u7f16\u7801","text":"<p>\u200b\u7edd\u5bf9\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u662f\u200bTransformer\u200b\u6a21\u578b\u200b\u4e2d\u200b\u6700\u65e9\u200b\u5f15\u5165\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u65b9\u5f0f\u200b\uff0c\u200b\u5176\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u4e3a\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u4f4d\u7f6e\u200b\u5206\u914d\u200b\u4e00\u4e2a\u200b\u552f\u4e00\u200b\u7684\u200b\u5411\u91cf\u200b\uff0c\u200b\u5c06\u200b\u8fd9\u4e2a\u200b\u5411\u91cf\u200b\u548c\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u76f8\u52a0\u200b\u5f97\u5230\u200b\u6700\u7ec8\u200b\u5e8f\u5217\u200b\u7684\u200b\u8868\u793a\u200b\u3002\u200b\u5982\u200bBERT\u200b\u6a21\u578b\u200b\uff0c\u200b\u9650\u5236\u200b\u4e86\u200b\u5e8f\u5217\u200b\u7684\u200b\u6700\u5927\u200b\u957f\u5ea6\u200b\u4e3a\u200b512\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u56fa\u5b9a\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff0c\u200b\u5c06\u200b\u6bcf\u4e2a\u200b\u4f4d\u7f6e\u200b\u6620\u5c04\u200b\u5230\u200b\u4e00\u4e2a\u200b\u53ef\u200b\u5b66\u4e60\u200b\u7684\u200b\u5411\u91cf\u200b\u3002</p> <pre><code>class AbsolutePE(torch.nn.Module):\n    def __init__(self, max_len: int, d_model: int):\n        super(AbsolutePE, self).__init__()\n        self.pe = torch.nn.Embedding(max_len, d_model)\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        # x: [N, L]\n        pos = torch.arange(x.size(1), device=x.device).unsqueeze(0)\n        return self.pe(pos) # [1, L, D], can be broadcasted to [N, L, D]\n</code></pre> <p>\u200b\u6b64\u7c7b\u200b\u65b9\u6cd5\u200b\u7684\u200b\u7f3a\u70b9\u200b\u662f\u200b\u65e0\u6cd5\u200b\u5904\u7406\u200b\u8d85\u8fc7\u200b\u6700\u5927\u200b\u957f\u5ea6\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5ffd\u89c6\u200b\u4e86\u200b\u5143\u7d20\u200b\u4e4b\u95f4\u200b\u7684\u200b\u76f8\u5bf9\u200b\u4f4d\u7f6e\u200b\u5173\u7cfb\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/positional-embedding/#_3","title":"\u6b63\u5f26\u200b\u4f4d\u7f6e\u200b\u7f16\u7801","text":"<p>Vaswani\u200b\u7b49\u200b\u4eba\u200b\u63d0\u51fa\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u5305\u542b\u200b\u4e00\u4e2a\u200b\u6b63\u5f26\u200b\u51fd\u6570\u200b\u548c\u200b\u4e00\u4e2a\u200b\u4f59\u5f26\u200b\u51fd\u6570\u200b\uff0c\u200b\u5176\u200b\u6570\u5b66\u200b\u8868\u8fbe\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned} PE_{(pos, 2i)} &amp;= \\sin(pos / 10000^{2i / d_{\\text{model}}}) \\\\ PE_{(pos, 2i + 1)} &amp;= \\cos(pos / 10000^{2i / d_{\\text{model}}}) \\\\ \\end{aligned} \\] <p>\u200b\u5176\u4e2d\u200b\uff0c\\(pos\\)\u200b\u8868\u793a\u200b\u4f4d\u7f6e\u200b\uff0c\\(i\\)\u200b\u8868\u793a\u200b\u7ef4\u5ea6\u200b\uff0c\\(d_{\\text{model}}\\)\u200b\u8868\u793a\u200b\u6a21\u578b\u200b\u7684\u200b\u7ef4\u5ea6\u200b\uff0c\u200b\u8f93\u51fa\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u76f4\u63a5\u200b\u4e0e\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u76f8\u52a0\u200b\u5f97\u5230\u200b\u6700\u7ec8\u200b\u8868\u793a\u200b\u3002\u200b\u6b63\u5f26\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u4e0d\u200b\u9700\u8981\u200b\u989d\u5916\u200b\u8fdb\u884c\u200b\u5b66\u4e60\u200b\u3002Vaswani\u200b\u7684\u200b\u5b9e\u9a8c\u200b\u8868\u660e\u200b\uff0c\u200b\u56fa\u5b9a\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u548c\u200b\u53ef\u200b\u5b66\u4e60\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\u5728\u200b\u6027\u80fd\u200b\u4e0a\u200b\u6ca1\u6709\u200b\u663e\u8457\u200b\u5dee\u5f02\u200b\u3002</p> <pre><code>class SinusoidPE(torch.nn.Module):\n    def __init__(self, d_model: int):\n        super(SinusoidPE, self).__init__()\n        self.d_model = d_model\n\n    def _denominator(self, device: torch.device) -&gt; torch.Tensor:\n        return 10000 ** (torch.arange(0, self.d_model, device=device) / self.d_model)\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        # x: [N, L]\n        pos = torch.arange(x.size(1), device=x.device).unsqueeze(0) # [1, L]\n        pos = pos.unsqueeze(-1) / self._denominator(x.device) # [1, L, D]\n        pos[:, :, 0::2] = torch.sin(pos[:, :, 0::2])\n        pos[:, :, 1::2] = torch.cos(pos[:, :, 1::2])\n        return pos # [1, L, D], can be broadcasted to [N, L, D]\n</code></pre>"},{"location":"coding/dl-from-scratch/positional-embedding/#_4","title":"\u65cb\u8f6c\u200b\u4f4d\u7f6e\u200b\u7f16\u7801","text":"<p>Su\u200b\u7b49\u200b\u4eba\u200b\u63d0\u51fa\u200b\u7684\u200b\u65cb\u8f6c\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff08Rotary Positional Embedding\uff0cRoPE\uff09\u3002\u200b\u7f16\u7801\u200b\u7684\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u901a\u8fc7\u200b\u65cb\u8f6c\u200b\u77e9\u9635\u200b\u5c06\u200b\u4f4d\u7f6e\u200b\u4fe1\u606f\u200b\u5d4c\u5165\u200b\u5230\u200b\u7279\u5f81\u200b\u7a7a\u95f4\u200b\u4e2d\u200b\uff0c\u200b\u4ece\u800c\u200b\u4f7f\u200b\u6a21\u578b\u200b\u80fd\u591f\u200b\u5b66\u4e60\u200b\u5230\u200b\u4f4d\u7f6e\u200b\u4fe1\u606f\u200b\u3002</p> \\[ f_{q, k} \\bsx_m = \\bsR_{\\Theta, m}^d W_{q, k} \\bsx_m \\] <p>\u200b\u5176\u4e2d\u200b\uff0c\\(\\bsR_{\\Theta, m}\\)\u200b\u4e3a\u200b\u65cb\u8f6c\u200b\u77e9\u9635\u200b\uff0c\u200b\u5c06\u200b\u5411\u91cf\u200b\u6bcf\u200b\u4e24\u4e2a\u200b\u5206\u91cf\u200b\u8fdb\u884c\u200b\u65cb\u8f6c\u200b\u3002</p> \\[ \\begin{aligned} \\bsR^k_{\\Theta, m} &amp;= \\begin{bmatrix} \\cos(m\\theta_k) &amp; -\\sin(m\\theta_k) \\\\ \\sin(m\\theta_k) &amp; \\cos(m\\theta_k) \\end{bmatrix} \\\\ \\bsR_{\\Theta, m} &amp;= \\begin{bmatrix} \\bsR^1_{\\Theta, m} &amp; 0 &amp; \\cdots &amp; 0 \\\\ 0 &amp; \\bsR^2_{\\Theta, m} &amp; \\cdots &amp; 0 \\\\ \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\ 0 &amp; 0 &amp; \\cdots &amp; \\bsR^{d / 2}_{\\Theta, m} \\end{bmatrix}_{d\\times d} \\end{aligned} \\] <p>\u200b\u6ce8\u610f\u200b\uff0c\\(m\\)\u200b\u4e3a\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\(1, \\ldots, L\\)\uff0c\\(k\\)\u200b\u4e3a\u200b\u7ef4\u5ea6\u200b\u4e0b\u6807\u200b\uff0c\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\(1, \\ldots, d / 2\\)\u3002\u200b\u6700\u7ec8\u200b\u7684\u200b\u65cb\u8f6c\u200b\u64cd\u4f5c\u200b\u5c06\u200b\u6bcf\u4e2a\u200b\u4f4d\u7f6e\u200b\\(m\\)\u200b\u7684\u200b\u5411\u91cf\u200b\\(\\bsx_m\\)\u200b\u5e94\u7528\u200b\u65cb\u8f6c\u200b\u77e9\u9635\u200b\\(\\bsR_{\\Theta, m}\\)\uff0c\u200b\u5f97\u5230\u200b\u65b0\u200b\u7684\u200b\u5411\u91cf\u200b\u3002</p> <pre><code>import functools\n\nclass RoPE(torch.nn.Module):\n    def __init__(self, d_model: int, theta: int | float = 10000):\n        super(RoPE, self).__init__()\n        self.d_model = d_model\n        self.theta = theta ** -(torch.arange(0, d_model, 2) / d_model)\n\n    @functools.lru_cache(maxsize=None)\n    def _forward_l(self, L: int) -&gt; torch.Tensor:\n        # Use lru_cache to avoid redundant computation for the same L\n\n        D = self.d_model\n        pos = torch.einsum(\n            'l,d-&gt;ld',\n            torch.arange(L), self.theta\n        )  # [L, D / 2]\n\n        # Major diagonal is cos, cos, ..., cos; D elements\n        cos = torch.cos(pos).repeat_interleave(2)\n        # Minor diagonal is sin, 0, sin, 0, ..., sin; D - 1 elements\n        sin = torch.stack([\n            torch.sin(pos), torch.zeros_like(pos), dim=-1\n        ]).reshape(L, D)[:, :-1]\n\n        result = torch.zeros(L, D, D)\n        result = torch.diagonal_scatter(result, cos, dim1=1, dim2=2)\n        result = torch.diagonal_scatter(result, sin, dim1=1, dim2=2, offset=-1)\n        result = torch.diagonal_scatter(result, -sin, dim1=1, dim2=2, offset=1)\n        return result\n\n    def forward(self, x: torch.Tensor) -&gt; torch.Tensor:\n        # x: [N, L, H, D]\n        _, L, _, D = x.size()\n        rot_matrix = self._forward_l(L).to(x.device)  # [L, D, D]\n        return torch.einsum('lde,nlhe-&gt;nlhd', rot_matrix, x)\n</code></pre> <p>\u200b\u548c\u200b\u5176\u4ed6\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u65b9\u5f0f\u200b\u4e0d\u540c\u200b\uff0cRoPE\u200b\u5728\u200bQKV\u200b\u53d8\u6362\u200b\u540e\u200b\u8fdb\u884c\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff0c\u200b\u56e0\u6b64\u200b\u9700\u8981\u200b\u5bf9\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u7684\u200b\u5b9e\u73b0\u200b\u8fdb\u884c\u200b\u4fee\u6539\u200b\u3002</p> <pre><code>class MultiHeadAttention(torch.nn.Module):\n    def __init__(self, d_model: int, num_heads: int, rope: RoPE | None = None):\n        ...\n\n        self.rope = rope\n\n        ...\n\n    def forward(\n        self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,\n        padding_mask: torch.Tensor | None = None,\n        attention_mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        ...\n\n        if self.rope is not None:\n            Q = self.rope(self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k))\n            K = self.rope(self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k))\n        else:\n            Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)\n            K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)\n        V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)\n\n        ...\n</code></pre> <p>\u200b\u4e3a\u4e86\u200b\u8ba1\u7b97\u200b\u65b9\u4fbf\u200b\uff0cLlama\u200b\u7b49\u200b\u90e8\u5206\u200b\u6a21\u578b\u200b\u7684\u200bRoPE\u200b\u5c06\u200b\u8f93\u5165\u200b\u5411\u91cf\u200b\u5206\u4e3a\u200b\u524d\u540e\u200b\u4e24\u6bb5\u200b\uff0c\u200b\u524d\u540e\u200b\u4e24\u6bb5\u200b\u76f8\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u5206\u4e3a\u200b\u4e00\u7ec4\u200b\u8fdb\u884c\u200b\u65cb\u8f6c\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u800c\u200b\u4e0d\u662f\u200b\u5c06\u200b\u6574\u4e2a\u200b\u5411\u91cf\u200b\u6309\u7167\u200b\u76f8\u90bb\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5143\u7d20\u200b\u5206\u4e3a\u200b\u4e00\u7ec4\u200b\u8fdb\u884c\u200b\u65cb\u8f6c\u200b\u3002</p>"},{"location":"coding/dl-from-scratch/transformer-variants/","title":"Transformer\u200b\u53d8\u79cd","text":"<p>Transformer\u200b\u53d8\u79cd\u200b\u4e3b\u8981\u200b\u5305\u542b\u200b\u5982\u4e0b\u200b\u51e0\u79cd\u200b\uff1a</p> <ul> <li>\u200b\u6539\u53d8\u200btransformer\u200b\u7684\u200b\u6ce8\u610f\u529b\u200b\u8ba1\u7b97\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u5982\u200bLinformer\u3001Reformer\u200b\u7b49\u200b\uff1b</li> <li>\u200b\u6539\u53d8\u200btransformer\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u65b9\u5f0f\u200b\uff1b</li> <li>\u200b\u6539\u53d8\u200btransformer\u200b\u7684\u200b\u5f52\u4e00\u5316\u200b\u5c42\u200b\uff1b</li> <li>\u200b\u6539\u53d8\u200btransformer\u200b\u7684\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\uff1b</li> <li>\u200b\u90e8\u5206\u200b\u8bed\u8a00\u200b\u6a21\u578b\u200b\uff08\u200b\u5982\u200bLlama\uff09\u200b\u4f7f\u7528\u200b\u7684\u200b\u7ebf\u6027\u200b\u5c42\u4e0d\u5e26\u200b\u504f\u7f6e\u200b\uff1b</li> <li>\u200b\u6539\u53d8\u200btransformer\u200b\u5f52\u4e00\u5316\u200b\u5c42\u200b\u7684\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u662f\u200b\u5148\u200b\u5f52\u4e00\u5316\u200b\u518d\u200b\u52a0\u200b\u6b8b\u5dee\u200b\uff08Pre-Norm\uff09\uff0c\u200b\u8fd8\u662f\u200b\u5148\u52a0\u200b\u6b8b\u5dee\u200b\u518d\u200b\u5f52\u4e00\u5316\u200b\uff08Post-Norm\uff09\u3002</li> </ul>"},{"location":"coding/dl-from-scratch/transformer-variants/#bert","title":"BERT","text":"<p>BERT\u200b\u7684\u200b\u7ed3\u6784\u200b\u7279\u70b9\u200b\u4e3a\u200b\uff1a</p> <ul> <li>encoder-only\uff0c\u200b\u5373\u200b\u4f7f\u7528\u200bBidirectional self-attention\uff1b</li> <li>\u200b\u4f7f\u7528\u200b\u7edd\u5bf9\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff1b</li> </ul>"},{"location":"coding/dl-from-scratch/transformer-variants/#llama","title":"Llama","text":"<p>Llama\u200b\u7684\u200b\u7ed3\u6784\u200b\u7279\u70b9\u200b\u4e3a\u200b\uff1a</p> <ul> <li>decoder-only\uff0c\u200b\u5373\u200b\u4f7f\u7528\u200bCausal self-attention\uff1b</li> <li>\u200b\u5168\u90e8\u200b\u4f7f\u7528\u200b\u4e0d\u5e26\u200b\u504f\u7f6e\u200b\u7684\u200b\u7ebf\u6027\u200b\u5c42\u200b\uff1b</li> <li>\u200b\u5728\u200bFFN\u200b\u4e2d\u200b\u4f7f\u7528\u200bSiLU\u200b\u914d\u5408\u200b\u95e8\u63a7\u200b\uff1b</li> <li>\u200b\u4f7f\u7528\u200bRoPE\u200b\u4f4d\u7f6e\u200b\u7f16\u7801\u200b\uff1b</li> <li>\u200b\u4f7f\u7528\u200bRMSNorm\u200b\u5f52\u4e00\u5316\u200b\u5c42\u200b\u3002\u200b\u5f52\u4e00\u5316\u200b\u65b9\u5f0f\u200b\u4e3a\u200bpre-norm\uff0c\u200b\u5373\u5148\u200b\u8fdb\u884c\u200b\u5f52\u4e00\u5316\u200b\uff0c\u200b\u518d\u200b\u8fdb\u884c\u200b\u6ce8\u610f\u529b\u200b\u6216\u8005\u200b\u5168\u200b\u8fde\u63a5\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u7136\u540e\u200b\u52a0\u200b\u6b8b\u5dee\u200b\u3002</li> </ul>"},{"location":"coding/dl-from-scratch/transformer/","title":"\u5b9e\u73b0\u200btransformer","text":"In\u00a0[1]: Copied! <pre>import torch\n</pre> import torch <p>\u200b\u9996\u5148\u200b\u5b9e\u73b0\u200b\u591a\u5934\u200b\u6ce8\u610f\u529b\u200b\u673a\u5236\u200b\u3002</p> In\u00a0[2]: Copied! <pre>class MultiHeadAttention(torch.nn.Module):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadAttention, self).__init__()\n        self.d_model = d_model\n        self.num_heads = num_heads\n        if d_model % num_heads != 0:\n            raise ValueError(\"d_model must be divisible by num_heads\")\n\n        self.d_k = d_model // num_heads\n        self.sqrt_d_k = self.d_k ** 0.5\n\n        self.W_Q = torch.nn.Linear(d_model, d_model)\n        self.W_K = torch.nn.Linear(d_model, d_model)\n        self.W_V = torch.nn.Linear(d_model, d_model)\n        self.W_O = torch.nn.Linear(d_model, d_model)\n\n    def forward(\n        self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,\n        mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        N, L_Q, D = x_q.size()\n        _, L_KV, _ = x_k.size()\n        Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)\n        K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)\n        V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)\n\n        score = torch.einsum(\"nihd,njhd-&gt;nijh\", Q, K) / self.sqrt_d_k\n        if mask is not None:\n            # mask: (L, L)\n            score = score.masked_fill(\n                mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')\n            )\n        score = torch.nn.functional.softmax(score, dim=2)\n        value = torch.einsum(\"nijh,njhd-&gt;nihd\", score, V).reshape(N, L_Q, self.d_model)\n        return self.W_O(value)\n\n\nclass MultiHeadSelfAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadSelfAttention, self).__init__(d_model, num_heads)\n\n    def forward(self, x: torch.Tensor, mask: torch.Tensor | None = None) -&gt; torch.Tensor:\n        return super().forward(x, x, x, mask)\n\n\nclass MultiHeadCrossAttention(MultiHeadAttention):\n    def __init__(self, d_model: int, num_heads: int):\n        super(MultiHeadCrossAttention, self).__init__(d_model, num_heads)\n\n    def forward(\n        self, x_q: torch.Tensor, x_kv: torch.Tensor,\n        mask: torch.Tensor | None = None\n    ) -&gt; torch.Tensor:\n        return super().forward(x_q, x_kv, x_kv, mask)\n</pre> class MultiHeadAttention(torch.nn.Module):     def __init__(self, d_model: int, num_heads: int):         super(MultiHeadAttention, self).__init__()         self.d_model = d_model         self.num_heads = num_heads         if d_model % num_heads != 0:             raise ValueError(\"d_model must be divisible by num_heads\")          self.d_k = d_model // num_heads         self.sqrt_d_k = self.d_k ** 0.5          self.W_Q = torch.nn.Linear(d_model, d_model)         self.W_K = torch.nn.Linear(d_model, d_model)         self.W_V = torch.nn.Linear(d_model, d_model)         self.W_O = torch.nn.Linear(d_model, d_model)      def forward(         self, x_q: torch.Tensor, x_k: torch.Tensor, x_v: torch.Tensor,         mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         N, L_Q, D = x_q.size()         _, L_KV, _ = x_k.size()         Q = self.W_Q(x_q).reshape(N, L_Q, self.num_heads, self.d_k)         K = self.W_K(x_k).reshape(N, L_KV, self.num_heads, self.d_k)         V = self.W_V(x_v).reshape(N, L_KV, self.num_heads, self.d_k)          score = torch.einsum(\"nihd,njhd-&gt;nijh\", Q, K) / self.sqrt_d_k         if mask is not None:             # mask: (L, L)             score = score.masked_fill(                 mask.reshape(1, L_Q, L_KV, 1) == 0, float('-inf')             )         score = torch.nn.functional.softmax(score, dim=2)         value = torch.einsum(\"nijh,njhd-&gt;nihd\", score, V).reshape(N, L_Q, self.d_model)         return self.W_O(value)   class MultiHeadSelfAttention(MultiHeadAttention):     def __init__(self, d_model: int, num_heads: int):         super(MultiHeadSelfAttention, self).__init__(d_model, num_heads)      def forward(self, x: torch.Tensor, mask: torch.Tensor | None = None) -&gt; torch.Tensor:         return super().forward(x, x, x, mask)   class MultiHeadCrossAttention(MultiHeadAttention):     def __init__(self, d_model: int, num_heads: int):         super(MultiHeadCrossAttention, self).__init__(d_model, num_heads)      def forward(         self, x_q: torch.Tensor, x_kv: torch.Tensor,         mask: torch.Tensor | None = None     ) -&gt; torch.Tensor:         return super().forward(x_q, x_kv, x_kv, mask) <p>\u200b\u524d\u9988\u200b\u795e\u7ecf\u7f51\u7edc\u200b\u7531\u200b\u4e24\u4e2a\u200b\u5168\u200b\u8fde\u63a5\u200b\u5c42\u200b\u548c\u200bReLU\u200b\u6fc0\u6d3b\u200b\u51fd\u6570\u200b\u7ec4\u6210\u200b</p> In\u00a0[3]: Copied! <pre>class FFN(torch.nn.Module):\n    def __init__(self, input_dim: int, hidden_dim: int):\n        super(FFN, self).__init__()\n        self.fc1 = torch.nn.Linear(input_dim, hidden_dim)\n        self.fc2 = torch.nn.Linear(hidden_dim, input_dim)\n        self.relu = torch.nn.ReLU()\n\n    def forward(self, x):\n        return self.fc2(self.relu(self.fc1(x)))\n</pre> class FFN(torch.nn.Module):     def __init__(self, input_dim: int, hidden_dim: int):         super(FFN, self).__init__()         self.fc1 = torch.nn.Linear(input_dim, hidden_dim)         self.fc2 = torch.nn.Linear(hidden_dim, input_dim)         self.relu = torch.nn.ReLU()      def forward(self, x):         return self.fc2(self.relu(self.fc1(x))) <p>\u200b\u7f16\u7801\u5668\u200b\u5c42\u200b\u7531\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200bc\u200b\u5c42\u200b\u548c\u200b\u524d\u9988\u200b\u795e\u7ecf\u200b\u7f51\u7edc\u5c42\u200b\u7ec4\u6210\u200b\uff0c\u200b\u4e00\u4e2a\u200b\u7f16\u7801\u5668\u200b\u7531\u200b\u591a\u4e2a\u200b\u8fd9\u6837\u200b\u7684\u200b\u5c42\u200b\u4e32\u8054\u200b\u7ec4\u6210\u200b</p> In\u00a0[4]: Copied! <pre>class EncoderLayer(torch.nn.Module):\n    def __init__(\n        self, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6\n    ):\n        super(EncoderLayer, self).__init__()\n\n        if ffn_dim is None:\n            ffn_dim = input_dim * 4\n\n        self.attention = MultiHeadSelfAttention(input_dim, num_heads)\n        self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout1 = torch.nn.Dropout(dropout)\n\n        self.ffn = FFN(input_dim, ffn_dim)\n        self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)\n        self.dropout2 = torch.nn.Dropout(dropout)\n\n    def forward(self, x, mask):\n        x = x + self.dropout1(self.attention(x, mask))\n        x = self.norm1(x)\n\n        x = x + self.dropout2(self.ffn(x))\n        x = self.norm2(x)\n\n        return x\n\n\nclass Encoder(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1,\n        layer_norm_eps: float = 1e-6\n    ):\n        super(Encoder, self).__init__()\n\n        self.layers = torch.nn.ModuleList([\n            EncoderLayer(input_dim, num_heads, ffn_dim, dropout, layer_norm_eps)\n            for _ in range(num_layers)\n        ])\n\n    def forward(self, x, mask):\n        # x: (N, L, D)\n        for layer in self.layers:\n            x = layer(x, mask)\n        return x\n</pre> class EncoderLayer(torch.nn.Module):     def __init__(         self, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1,         layer_norm_eps: float = 1e-6     ):         super(EncoderLayer, self).__init__()          if ffn_dim is None:             ffn_dim = input_dim * 4          self.attention = MultiHeadSelfAttention(input_dim, num_heads)         self.norm1 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout1 = torch.nn.Dropout(dropout)          self.ffn = FFN(input_dim, ffn_dim)         self.norm2 = torch.nn.LayerNorm(input_dim, layer_norm_eps)         self.dropout2 = torch.nn.Dropout(dropout)      def forward(self, x, mask):         x = x + self.dropout1(self.attention(x, mask))         x = self.norm1(x)          x = x + self.dropout2(self.ffn(x))         x = self.norm2(x)          return x   class Encoder(torch.nn.Module):     def __init__(         self, num_layers: int, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1,         layer_norm_eps: float = 1e-6     ):         super(Encoder, self).__init__()          self.layers = torch.nn.ModuleList([             EncoderLayer(input_dim, num_heads, ffn_dim, dropout, layer_norm_eps)             for _ in range(num_layers)         ])      def forward(self, x, mask):         # x: (N, L, D)         for layer in self.layers:             x = layer(x, mask)         return x <p>\u200b\u89e3\u7801\u5668\u200b\u5c42\u200b\u7531\u200b\u81ea\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\uff0c\u200b\u4ea4\u53c9\u200b\u6ce8\u610f\u529b\u200b\u5c42\u200b\u548c\u200b\u524d\u9988\u200b\u795e\u7ecf\u200b\u7f51\u7edc\u5c42\u200b\u7ec4\u6210\u200b\u3002</p> In\u00a0[5]: Copied! <pre>class DecoderLayer(torch.nn.Module):\n    def __init__(\n        self, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1\n    ):\n        super(DecoderLayer, self).__init__()\n\n        if ffn_dim is None:\n            ffn_dim = input_dim * 4\n\n        self.self_attention = MultiHeadSelfAttention(input_dim, num_heads)\n        self.norm1 = torch.nn.LayerNorm(input_dim)\n        self.dropout1 = torch.nn.Dropout(dropout)\n\n        self.cross_attention = MultiHeadCrossAttention(input_dim, num_heads)\n        self.norm2 = torch.nn.LayerNorm(input_dim)\n        self.dropout2 = torch.nn.Dropout(dropout)\n\n        self.ffn = FFN(input_dim, ffn_dim)\n        self.norm3 = torch.nn.LayerNorm(input_dim)\n        self.dropout3 = torch.nn.Dropout(dropout)\n\n    def forward(\n        self, x: torch.Tensor, memory: torch.Tensor, tgt_mask: torch.Tensor | None = None\n    ):\n\n        x = x + self.dropout1(self.self_attention(x, tgt_mask))\n        x = self.norm1(x)\n\n        x = x + self.dropout2(self.cross_attention(x, memory))\n        x = self.norm2(x)\n\n        x = x + self.dropout3(self.ffn(x))\n        x = self.norm3(x)\n\n        return x\n\n\nclass Decoder(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1\n    ):\n        super(Decoder, self).__init__()\n\n        self.layers = torch.nn.ModuleList([\n            DecoderLayer(input_dim, num_heads, ffn_dim, dropout)\n            for _ in range(num_layers)\n        ])\n\n    def forward(self, x, memory, tgt_mask):\n        # x: (N, L, D)\n        for layer in self.layers:\n            x = layer(x, memory, tgt_mask)\n        return x\n</pre> class DecoderLayer(torch.nn.Module):     def __init__(         self, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1     ):         super(DecoderLayer, self).__init__()          if ffn_dim is None:             ffn_dim = input_dim * 4          self.self_attention = MultiHeadSelfAttention(input_dim, num_heads)         self.norm1 = torch.nn.LayerNorm(input_dim)         self.dropout1 = torch.nn.Dropout(dropout)          self.cross_attention = MultiHeadCrossAttention(input_dim, num_heads)         self.norm2 = torch.nn.LayerNorm(input_dim)         self.dropout2 = torch.nn.Dropout(dropout)          self.ffn = FFN(input_dim, ffn_dim)         self.norm3 = torch.nn.LayerNorm(input_dim)         self.dropout3 = torch.nn.Dropout(dropout)      def forward(         self, x: torch.Tensor, memory: torch.Tensor, tgt_mask: torch.Tensor | None = None     ):          x = x + self.dropout1(self.self_attention(x, tgt_mask))         x = self.norm1(x)          x = x + self.dropout2(self.cross_attention(x, memory))         x = self.norm2(x)          x = x + self.dropout3(self.ffn(x))         x = self.norm3(x)          return x   class Decoder(torch.nn.Module):     def __init__(         self, num_layers: int, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1     ):         super(Decoder, self).__init__()          self.layers = torch.nn.ModuleList([             DecoderLayer(input_dim, num_heads, ffn_dim, dropout)             for _ in range(num_layers)         ])      def forward(self, x, memory, tgt_mask):         # x: (N, L, D)         for layer in self.layers:             x = layer(x, memory, tgt_mask)         return x <p>Transformer\u200b\u6a21\u578b\u200b\u7531\u200b\u7f16\u7801\u5668\u200b\u548c\u200b\u89e3\u7801\u5668\u200b\u7ec4\u6210\u200b\uff0c\u200b\u7f16\u7801\u5668\u200b\u5904\u7406\u200b\u6e90\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5c06\u200b\u7f16\u7801\u200b\u540e\u200b\u7684\u200b\u5e8f\u5217\u200b\u8f93\u5165\u200b\u5230\u200b\u89e3\u7801\u5668\u200b\u4e2d\u200b\uff0c\u200b\u89e3\u7801\u5668\u200b\u751f\u6210\u200b\u76ee\u6807\u200b\u5e8f\u5217\u200b\u3002</p> In\u00a0[6]: Copied! <pre>class CustomTransformer(torch.nn.Module):\n    def __init__(\n        self, num_layers: int, num_decoder_layers: int, input_dim: int, num_heads: int,\n        ffn_dim: int | None = None, dropout: float = 0.1, layer_norm_eps: float = 1e-6\n    ):\n        super(CustomTransformer, self).__init__()\n        self.encoder = Encoder(num_layers, input_dim, num_heads, ffn_dim, dropout, layer_norm_eps)\n        self.decoder = Decoder(num_decoder_layers, input_dim, num_heads, ffn_dim, dropout)\n\n    def forward(self, src, tgt, src_mask: torch.Tensor | None = None, tgt_mask: torch.Tensor | None = None):\n        memory = self.encoder(src, src_mask)\n        return self.decoder(tgt, memory, tgt_mask)\n</pre> class CustomTransformer(torch.nn.Module):     def __init__(         self, num_layers: int, num_decoder_layers: int, input_dim: int, num_heads: int,         ffn_dim: int | None = None, dropout: float = 0.1, layer_norm_eps: float = 1e-6     ):         super(CustomTransformer, self).__init__()         self.encoder = Encoder(num_layers, input_dim, num_heads, ffn_dim, dropout, layer_norm_eps)         self.decoder = Decoder(num_decoder_layers, input_dim, num_heads, ffn_dim, dropout)      def forward(self, src, tgt, src_mask: torch.Tensor | None = None, tgt_mask: torch.Tensor | None = None):         memory = self.encoder(src, src_mask)         return self.decoder(tgt, memory, tgt_mask) <p>\u200b\u5c06\u200bPyTorch\u200b\u5185\u90e8\u200b\u5b9e\u73b0\u200b\u7684\u200bTransformer\u200b\u6743\u91cd\u200b\u590d\u5236\u5230\u200b\u5b9e\u73b0\u200b\u7684\u200bTransformer\u200b\u4e2d\u200b\u3002</p> In\u00a0[7]: Copied! <pre>def _attn_load_from_torch(\n    custom_attn: MultiHeadAttention, torch_attn: torch.nn.MultiheadAttention\n):\n    embed_dim = custom_attn.d_model\n\n    def split_qkv(weight, embed_dim):\n        return weight[:embed_dim], weight[embed_dim: 2 * embed_dim], weight[2 * embed_dim:3 * embed_dim]\n    custom_attn.W_Q.weight.data, \\\n        custom_attn.W_K.weight.data, \\\n        custom_attn.W_V.weight.data = split_qkv(\n            torch_attn.in_proj_weight.data, embed_dim\n    )\n\n    custom_attn.W_Q.bias.data, \\\n        custom_attn.W_K.bias.data, \\\n        custom_attn.W_V.bias.data = split_qkv(\n            torch_attn.in_proj_bias.data, embed_dim\n    )\n\n    custom_attn.W_O.weight.data = torch_attn.out_proj.weight.data\n    custom_attn.W_O.bias.data = torch_attn.out_proj.bias.data\n\n    return custom_attn\n\n\ndef load_from_torch(\n    custom_transformer: CustomTransformer, torch_transformer: torch.nn.Transformer\n):\n    for custom_layer, torch_layer in zip(\n        [*custom_transformer.encoder.layers, *custom_transformer.decoder.layers],\n        [*torch_transformer.encoder.layers, *torch_transformer.decoder.layers],\n    ):\n\n        if hasattr(custom_layer, 'cross_attention'):\n            # Decoder\n            custom_layer.self_attention = _attn_load_from_torch(\n                custom_layer.self_attention, torch_layer.self_attn\n            )\n            custom_layer.cross_attention = _attn_load_from_torch(\n                custom_layer.cross_attention, torch_layer.multihead_attn\n            )\n        else:\n            # Encoder\n            custom_layer.attention = _attn_load_from_torch(\n                custom_layer.attention, torch_layer.self_attn\n            )\n\n        layer_pairs = [\n            (custom_layer.norm1, torch_layer.norm1),\n            (custom_layer.ffn.fc1, torch_layer.linear1),\n            (custom_layer.ffn.fc2, torch_layer.linear2),\n            (custom_layer.norm2, torch_layer.norm2)\n        ]\n        for custom, torch in layer_pairs:\n            custom.weight.data = torch.weight.data\n            custom.bias.data = torch.bias.data\n    return custom_transformer\n</pre> def _attn_load_from_torch(     custom_attn: MultiHeadAttention, torch_attn: torch.nn.MultiheadAttention ):     embed_dim = custom_attn.d_model      def split_qkv(weight, embed_dim):         return weight[:embed_dim], weight[embed_dim: 2 * embed_dim], weight[2 * embed_dim:3 * embed_dim]     custom_attn.W_Q.weight.data, \\         custom_attn.W_K.weight.data, \\         custom_attn.W_V.weight.data = split_qkv(             torch_attn.in_proj_weight.data, embed_dim     )      custom_attn.W_Q.bias.data, \\         custom_attn.W_K.bias.data, \\         custom_attn.W_V.bias.data = split_qkv(             torch_attn.in_proj_bias.data, embed_dim     )      custom_attn.W_O.weight.data = torch_attn.out_proj.weight.data     custom_attn.W_O.bias.data = torch_attn.out_proj.bias.data      return custom_attn   def load_from_torch(     custom_transformer: CustomTransformer, torch_transformer: torch.nn.Transformer ):     for custom_layer, torch_layer in zip(         [*custom_transformer.encoder.layers, *custom_transformer.decoder.layers],         [*torch_transformer.encoder.layers, *torch_transformer.decoder.layers],     ):          if hasattr(custom_layer, 'cross_attention'):             # Decoder             custom_layer.self_attention = _attn_load_from_torch(                 custom_layer.self_attention, torch_layer.self_attn             )             custom_layer.cross_attention = _attn_load_from_torch(                 custom_layer.cross_attention, torch_layer.multihead_attn             )         else:             # Encoder             custom_layer.attention = _attn_load_from_torch(                 custom_layer.attention, torch_layer.self_attn             )          layer_pairs = [             (custom_layer.norm1, torch_layer.norm1),             (custom_layer.ffn.fc1, torch_layer.linear1),             (custom_layer.ffn.fc2, torch_layer.linear2),             (custom_layer.norm2, torch_layer.norm2)         ]         for custom, torch in layer_pairs:             custom.weight.data = torch.weight.data             custom.bias.data = torch.bias.data     return custom_transformer <p>\u200b\u901a\u8fc7\u200b\u4ee3\u7801\u200b\u9a8c\u8bc1\u200b\u5b9e\u73b0\u200b\u7684\u200bTransformer\u200b\u6a21\u578b\u200b\u7684\u200b\u6b63\u786e\u6027\u200b\u3002</p> In\u00a0[8]: Copied! <pre>transformer_config = {\n    'num_layers': 6,\n    'num_decoder_layers': 6,\n    'input_dim': 512,\n    'num_heads': 8,\n    'ffn_dim': 2048,\n    'dropout': 0.1,\n    'layer_norm_eps': 1e-6\n}\n\ncustom_transformer = CustomTransformer(**transformer_config)\ntorch_transformer = torch.nn.Transformer(\n    d_model=transformer_config['input_dim'],\n    nhead=transformer_config['num_heads'],\n    num_encoder_layers=transformer_config['num_layers'],\n    num_decoder_layers=transformer_config['num_decoder_layers'],\n    dim_feedforward=transformer_config['ffn_dim'],\n    dropout=transformer_config['dropout'],\n    batch_first=True\n)\ncustom_transformer = load_from_torch(custom_transformer, torch_transformer)\ncustom_transformer.eval()\ntorch_transformer.eval()\n\nsrc = torch.randn(32, 10, transformer_config['input_dim'])\ntgt = torch.randn(32, 20, transformer_config['input_dim'])\ntgt_mask = 1 - torch.triu(torch.ones(20, 20), diagonal=1)\n\ncustom_output = custom_transformer(src, tgt, tgt_mask=tgt_mask)\ntorch_output = torch_transformer(src, tgt, tgt_mask=tgt_mask, tgt_is_causal=True)\ncustom_output[0, 0, :5], torch_output[0, 0, :5]\n</pre> transformer_config = {     'num_layers': 6,     'num_decoder_layers': 6,     'input_dim': 512,     'num_heads': 8,     'ffn_dim': 2048,     'dropout': 0.1,     'layer_norm_eps': 1e-6 }  custom_transformer = CustomTransformer(**transformer_config) torch_transformer = torch.nn.Transformer(     d_model=transformer_config['input_dim'],     nhead=transformer_config['num_heads'],     num_encoder_layers=transformer_config['num_layers'],     num_decoder_layers=transformer_config['num_decoder_layers'],     dim_feedforward=transformer_config['ffn_dim'],     dropout=transformer_config['dropout'],     batch_first=True ) custom_transformer = load_from_torch(custom_transformer, torch_transformer) custom_transformer.eval() torch_transformer.eval()  src = torch.randn(32, 10, transformer_config['input_dim']) tgt = torch.randn(32, 20, transformer_config['input_dim']) tgt_mask = 1 - torch.triu(torch.ones(20, 20), diagonal=1)  custom_output = custom_transformer(src, tgt, tgt_mask=tgt_mask) torch_output = torch_transformer(src, tgt, tgt_mask=tgt_mask, tgt_is_causal=True) custom_output[0, 0, :5], torch_output[0, 0, :5] Out[8]: <pre>(tensor([ 0.6672,  0.4102, -0.2145,  1.8782,  0.4108], grad_fn=&lt;SliceBackward0&gt;),\n tensor([ 0.6672,  0.4102, -0.2145,  1.8781,  0.4108], grad_fn=&lt;SliceBackward0&gt;))</pre> In\u00a0[9]: Copied! <pre>torch.allclose(custom_output, torch_output, atol=1e-4)\n</pre> torch.allclose(custom_output, torch_output, atol=1e-4) Out[9]: <pre>True</pre>"},{"location":"coding/dsa/","title":"\u6570\u636e\u7ed3\u6784","text":"<p>\u200b\u4ee5\u4e0b\u200b\u6574\u7406\u200b\u4e86\u200b\u9093\u4fca\u8f89\u200b\u8001\u5e08\u200b\u300a\u200b\u6570\u636e\u7ed3\u6784\u200b\u300b\u200b\u8bfe\u7a0b\u200b\u7684\u200b\u5185\u5bb9\u200b\uff1a</p> <ul> <li>\u200b\u5411\u91cf\u200b\u4e0e\u200b\u5217\u8868\u200b</li> </ul>"},{"location":"coding/dsa/vector-and-list/","title":"\u5411\u91cf\u200b\u4e0e\u200b\u5217\u8868","text":"<p>\u200b\u4e3b\u8981\u200b\u7684\u200b\u57fa\u7840\u200b\u7ebf\u6027\u200b\u6570\u636e\u7ed3\u6784\u200b\u662f\u200b\u5411\u91cf\u200b\u4e0e\u200b\u5217\u8868\u200b\u3002\u200b\u5728\u200b\u672c\u5408\u200b\u96c6\u4e2d\u200b\uff0c\u200b\u5411\u91cf\u200b\u6307\u4ee5\u200b\u6570\u7ec4\u200b\u65b9\u5f0f\u200b\u7ec4\u5408\u200b\u5e76\u200b\u5c01\u88c5\u200b\u901a\u7528\u200b\u62bd\u8c61\u200b\u63a5\u53e3\u200b\u7684\u200b\u6570\u636e\u7c7b\u578b\u200b\uff1b\u200b\u5217\u8868\u200b\u6307\u4ee5\u200b\u94fe\u8868\u200b\u65b9\u5f0f\u200b\u7ec4\u5408\u200b\u5e76\u200b\u5c01\u88c5\u200b\u901a\u7528\u200b\u62bd\u8c61\u200b\u63a5\u53e3\u200b\u7684\u200b\u6570\u636e\u7c7b\u578b\u200b\u3002</p> <p>\u200b\u8f83\u4e3a\u200b\u9ad8\u7ea7\u200b\u7684\u200b\u6570\u636e\u7ed3\u6784\u200b\u2014\u2014\u200b\u6808\u200b\u4e0e\u200b\u961f\u5217\u200b\u5efa\u7acb\u200b\u5728\u200b\u5411\u91cf\u200b\u4e0e\u200b\u5217\u8868\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\u3002\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u4e00\u4e9b\u200b\u53d7\u9650\u5236\u200b\u7684\u200b\u64cd\u4f5c\u200b\u63a5\u53e3\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_2","title":"\u7ebf\u6027\u200b\u6570\u636e\u7ed3\u6784","text":"<p>\u200b\u5bf9\u4e8e\u200b\u57fa\u7840\u200b\u7684\u200b\u7ebf\u6027\u200b\u6570\u636e\u7ed3\u6784\u200b\uff0c\u200b\u4ee5\u4e0b\u200b\u7684\u200b\u63a5\u53e3\u200b\u53ef\u4ee5\u200b\u89c6\u4e3a\u200b\u901a\u7528\u200b\uff1a</p> \u200b\u64cd\u4f5c\u200b \u200b\u63cf\u8ff0\u200b \u200b\u901a\u7528\u6027\u200b <code>size()</code> \u200b\u83b7\u53d6\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u6570\u76ee\u200b <code>get()</code> \u200b\u83b7\u53d6\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u67d0\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u503c\u200b <code>set()</code> \u200b\u4fee\u6539\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u67d0\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u503c\u200b <code>insert()</code> \u200b\u5411\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u67d0\u4e2a\u200b\u4f4d\u7f6e\u200b\u63d2\u5165\u200b\u5143\u7d20\u200b <code>remove()</code> \u200b\u79fb\u9664\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u67d0\u4e2a\u200b\u5143\u7d20\u200b <code>traverse()</code> \u200b\u904d\u5386\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\uff0c\u200b\u5e76\u200b\u6267\u884c\u200b\u67d0\u200b\u64cd\u4f5c\u200b <code>find()</code> \u200b\u5728\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u67e5\u627e\u200b\u67d0\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u4f4d\u7f6e\u200b <code>search()</code> \u200b\u5728\u200b\u6709\u5e8f\u200b\u7684\u200b\u7ed3\u6784\u200b\u4e2d\u200b\u67e5\u627e\u200b\u67d0\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u4f4d\u7f6e\u200b \u200b\u8981\u6c42\u200b\u6709\u5e8f\u200b <code>sort()</code> \u200b\u6392\u5e8f\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b \u200b\u8981\u6c42\u200b\u91cd\u8f7d\u200b<code>&lt;</code>\u200b\u548c\u200b<code>&gt;</code>\u200b\u8fd0\u7b97\u7b26"},{"location":"coding/dsa/vector-and-list/#_3","title":"\u5411\u91cf","text":"<p>\u200b\u5728\u200b\u5411\u91cf\u200b\u4e2d\u200b\uff0c\u200b\u6570\u636e\u200b\u5728\u200b\u5185\u5b58\u200b\u4e2d\u200b\u8fde\u7eed\u200b\u5b58\u50a8\u200b\uff0c\u200b\u5373\u200b\u4e0b\u6807\u200b\u76f8\u90bb\u200b\u7684\u200b\u6570\u636e\u200b\u5728\u200b\u5185\u5b58\u200b\u4e2d\u200b\u76f8\u90bb\u200b\u3002C/C++\u200b\u8bed\u8a00\u200b\u4e2d\u200b\u7684\u200b\u6570\u7ec4\u200b\u6570\u636e\u7ed3\u6784\u200b\u53ef\u4ee5\u200b\u89c6\u4e3a\u200b\u7b80\u5355\u200b\u7684\u200b\u5411\u91cf\u200b\u3002\u200b\u4f46\u200b\u6570\u7ec4\u200b\u4e0d\u200b\u63d0\u4f9b\u200b\u63d2\u5165\u200b\u3001\u200b\u5220\u9664\u200b\u7b49\u200b\u5177\u4f53\u200b\u7684\u200b\u64cd\u4f5c\u200b\u63a5\u53e3\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b\u7a0b\u5e8f\u4ee3\u7801\u200b\u4e2d\u200b\u624b\u52a8\u200b\u5b9e\u73b0\u200b\u3002\u200b\u5c06\u200b\u6570\u7ec4\u200b\u5c01\u88c5\u200b\u6210\u7c7b\u200b\u53ef\u4ee5\u200b\u63d0\u4f9b\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u64cd\u4f5c\u200b\u63a5\u53e3\u200b\uff0c\u200b\u5e76\u4e14\u200b\u4fdd\u8bc1\u6570\u636e\u200b\u88ab\u200b\u5408\u6cd5\u200b\u5730\u200b\u8bbf\u95ee\u200b\u53ca\u200b\u4fee\u6539\u200b\u3002</p> <p>\u200b\u5411\u91cf\u200b\u662f\u200b\u6570\u7ec4\u200b\u7684\u200b\u62bd\u8c61\u200b\u6cdb\u5316\u200b\u3001\u200b\u652f\u6301\u200b\u6240\u6709\u200b\u6570\u7ec4\u200b\u7684\u200b\u529f\u80fd\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u4e0d\u540c\u200b\u7c7b\u578b\u200b\u7684\u200b\u5143\u7d20\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u7edf\u4e00\u200b\u7684\u200b\u64cd\u4f5c\u200b\u63a5\u53e3\u200b\u3002</p> <p>C++\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u6a21\u677f\u200b\u7c7b\u200b\u7684\u200b\u529f\u80fd\u200b\u7528\u6765\u200b\u9488\u5bf9\u200b\u4e0d\u540c\u200b\u7c7b\u578b\u200b\u7684\u200b\u5143\u7d20\u200b\u63d0\u4f9b\u200b\u76f8\u540c\u200b\u7684\u200b\u529f\u80fd\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u67d0\u4e2a\u200bADT\uff08\u200b\u4ee5\u200b\u5411\u91cf\u200b\u4e3a\u4f8b\u200b\uff09\uff0c\u200b\u5176\u200b\u5c5e\u6027\u200b\u548c\u200b\u63a5\u53e3\u200b\u7684\u200b\u7ec4\u7ec7\u200b\u65b9\u5f0f\u200b\u53ef\u200b\u9075\u5faa\u200b\u5982\u4e0b\u200b\u89c4\u5219\u200b\uff1a</p> <pre><code>template &lt;typename T&gt; class Vector { //\u200b\u5411\u91cf\u200b\u6a21\u677f\u200b\u7c7b\u200b\nprivate: Rank _size; int _capacity; T* _elem; //\u200b\u89c4\u6a21\u200b\u3001\u200b\u5bb9\u91cf\u200b\u3001\u200b\u6570\u636e\u200b\u533a\u200b\nprotected:\n/* ... \u200b\u5185\u90e8\u200b\u51fd\u6570\u200b */\npublic:\n/* ... \u200b\u6784\u9020\u51fd\u6570\u200b */\n/* ... \u200b\u6790\u6784\u200b\u51fd\u6570\u200b */\n/* ... \u200b\u53ea\u8bfb\u200b\u63a5\u53e3\u200b */\n/* ... \u200b\u53ef\u200b\u5199\u200b\u63a5\u53e3\u200b */\n/* ... \u200b\u904d\u5386\u200b\u63a5\u53e3\u200b */\n};\n</code></pre> <p>\u200b\u901a\u8fc7\u200b\u5bf9\u200b<code>[]</code>\u200b\u8fd0\u7b97\u7b26\u200b\u8fdb\u884c\u200b\u91cd\u8f7d\u200b\uff0c\u200b\u5411\u91cf\u200b\u548c\u200b\u5217\u8868\u200b\u652f\u6301\u200b\u6570\u7ec4\u200b\u98ce\u683c\u200b\u7684\u200b\u8bbf\u95ee\u200b\u65b9\u5f0f\u200b\u3002</p> <p>\u200b\u5177\u4f53\u200b\u4ee3\u7801\u200b\u53ef\u4ee5\u200b\u53c2\u89c1\u200b\uff1a</p> \u200b\u5411\u91cf\u200b\u6a21\u677f\u200b\u7c7b\u200b\u58f0\u660e\u200b <p><pre><code>/******************************************************************************************\n* Data Structures in C++\n* ISBN: 7-302-33064-6 &amp; 7-302-33065-3 &amp; 7-302-29652-2 &amp; 7-302-26883-3\n* Junhui DENG, deng@tsinghua.edu.cn\n* Computer Science &amp; Technology, Tsinghua University\n* Copyright (c) 2003-2020. All rights reserved.\n******************************************************************************************/\n\ntypedef int Rank; //\u200b\u79e9\u200b\n#define DEFAULT_CAPACITY  3 //\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u521d\u59cb\u200b\u5bb9\u91cf\u200b\uff08\u200b\u5b9e\u9645\u200b\u5e94\u7528\u200b\u4e2d\u200b\u53ef\u200b\u8bbe\u7f6e\u200b\u4e3a\u200b\u66f4\u200b\u5927\u200b\uff09\n\ntemplate &lt;typename T&gt; class Vector { //\u200b\u5411\u91cf\u200b\u6a21\u677f\u200b\u7c7b\u200b\nprotected:\n   Rank _size; int _capacity;  T* _elem; //\u200b\u89c4\u6a21\u200b\u3001\u200b\u5bb9\u91cf\u200b\u3001\u200b\u6570\u636e\u200b\u533a\u200b\n   void copyFrom ( T const* A, Rank lo, Rank hi ); //\u200b\u590d\u5236\u200b\u6570\u7ec4\u200b\u533a\u95f4\u200bA[lo, hi)\n   void expand(); //\u200b\u7a7a\u95f4\u200b\u4e0d\u8db3\u200b\u65f6\u200b\u6269\u5bb9\u200b\n   void shrink(); //\u200b\u88c5\u586b\u200b\u56e0\u5b50\u200b\u8fc7\u200b\u5c0f\u65f6\u200b\u538b\u7f29\u200b\n   bool bubble ( Rank lo, Rank hi ); //\u200b\u626b\u63cf\u200b\u4ea4\u6362\u200b\n   void bubbleSort ( Rank lo, Rank hi ); //\u200b\u8d77\u6ce1\u200b\u6392\u5e8f\u200b\u7b97\u6cd5\u200b\n   Rank max ( Rank lo, Rank hi ); //\u200b\u9009\u53d6\u200b\u6700\u5927\u200b\u5143\u7d20\u200b\n   void selectionSort ( Rank lo, Rank hi ); //\u200b\u9009\u62e9\u200b\u6392\u5e8f\u200b\u7b97\u6cd5\u200b\n   void merge ( Rank lo, Rank mi, Rank hi ); //\u200b\u5f52\u5e76\u200b\u7b97\u6cd5\u200b\n   void mergeSort ( Rank lo, Rank hi ); //\u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b\u7b97\u6cd5\u200b\n   void heapSort ( Rank lo, Rank hi ); //\u200b\u5806\u6392\u5e8f\u200b\uff08\u200b\u7a0d\u540e\u200b\u7ed3\u5408\u200b\u5b8c\u5168\u200b\u5806\u200b\u8bb2\u89e3\u200b\uff09\n   Rank partition ( Rank lo, Rank hi ); //\u200b\u8f74\u70b9\u200b\u6784\u9020\u200b\u7b97\u6cd5\u200b\n   void quickSort ( Rank lo, Rank hi ); //\u200b\u5feb\u901f\u200b\u6392\u5e8f\u200b\u7b97\u6cd5\u200b\n   void shellSort ( Rank lo, Rank hi ); //\u200b\u5e0c\u5c14\u200b\u6392\u5e8f\u200b\u7b97\u6cd5\u200b\npublic:\n// \u200b\u6784\u9020\u51fd\u6570\u200b\n   Vector ( int c = DEFAULT_CAPACITY, int s = 0, T v = 0 ) //\u200b\u5bb9\u91cf\u200b\u4e3a\u200bc\u3001\u200b\u89c4\u6a21\u200b\u4e3a\u200bs\u3001\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u521d\u59cb\u200b\u4e3a\u200bv\n   { _elem = new T[_capacity = c]; for ( _size = 0; _size &lt; s; _elem[_size++] = v ); } //s&lt;=c\n   Vector ( T const* A, Rank n ) { copyFrom ( A, 0, n ); } //\u200b\u6570\u7ec4\u200b\u6574\u4f53\u200b\u590d\u5236\u200b\n   Vector ( T const* A, Rank lo, Rank hi ) { copyFrom ( A, lo, hi ); } //\u200b\u533a\u95f4\u200b\n   Vector ( Vector&lt;T&gt; const&amp; V ) { copyFrom ( V._elem, 0, V._size ); } //\u200b\u5411\u91cf\u200b\u6574\u4f53\u200b\u590d\u5236\u200b\n   Vector ( Vector&lt;T&gt; const&amp; V, Rank lo, Rank hi ) { copyFrom ( V._elem, lo, hi ); } //\u200b\u533a\u95f4\u200b\n// \u200b\u6790\u6784\u200b\u51fd\u6570\u200b\n   ~Vector() { delete [] _elem; } //\u200b\u91ca\u653e\u200b\u5185\u90e8\u7a7a\u95f4\u200b\n// \u200b\u53ea\u8bfb\u200b\u8bbf\u95ee\u200b\u63a5\u53e3\u200b\n   Rank size() const { return _size; } //\u200b\u89c4\u6a21\u200b\n   bool empty() const { return !_size; } //\u200b\u5224\u7a7a\u200b\n   Rank find ( T const&amp; e ) const { return find ( e, 0, _size ); } //\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u6574\u4f53\u200b\u67e5\u627e\u200b\n   Rank find ( T const&amp; e, Rank lo, Rank hi ) const; //\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u533a\u95f4\u200b\u67e5\u627e\u200b\n   Rank search ( T const&amp; e ) const //\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u6574\u4f53\u200b\u67e5\u627e\u200b\n   { return ( 0 &gt;= _size ) ? -1 : search ( e, 0, _size ); }\n   Rank search ( T const&amp; e, Rank lo, Rank hi ) const; //\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u533a\u95f4\u200b\u67e5\u627e\u200b\n// \u200b\u53ef\u200b\u5199\u8bbf\u95ee\u200b\u63a5\u53e3\u200b\n   T&amp; operator[] ( Rank r ); //\u200b\u91cd\u8f7d\u200b\u4e0b\u6807\u200b\u64cd\u4f5c\u7b26\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7c7b\u4f3c\u200b\u4e8e\u200b\u6570\u7ec4\u200b\u5f62\u5f0f\u200b\u5f15\u7528\u200b\u5404\u200b\u5143\u7d20\u200b\n   const T&amp; operator[] ( Rank r ) const; //\u200b\u4ec5\u9650\u4e8e\u200b\u505a\u53f3\u503c\u200b\u7684\u200b\u91cd\u8f7d\u200b\u7248\u672c\u200b\n   Vector&lt;T&gt; &amp; operator= ( Vector&lt;T&gt; const&amp; ); //\u200b\u91cd\u8f7d\u200b\u8d4b\u503c\u200b\u64cd\u4f5c\u7b26\u200b\uff0c\u200b\u4ee5\u4fbf\u200b\u76f4\u63a5\u200b\u514b\u9686\u200b\u5411\u91cf\u200b\n   T remove ( Rank r ); //\u200b\u5220\u9664\u200b\u79e9\u4e3a\u200br\u200b\u7684\u200b\u5143\u7d20\u200b\n   int remove ( Rank lo, Rank hi ); //\u200b\u5220\u9664\u200b\u79e9\u5728\u200b\u533a\u95f4\u200b[lo, hi)\u200b\u4e4b\u5185\u200b\u7684\u200b\u5143\u7d20\u200b\n   Rank insert ( Rank r, T const&amp; e ); //\u200b\u63d2\u5165\u200b\u5143\u7d20\u200b\n   Rank insert ( T const&amp; e ) { return insert ( _size, e ); } //\u200b\u9ed8\u8ba4\u200b\u4f5c\u4e3a\u200b\u672b\u200b\u5143\u7d20\u200b\u63d2\u5165\u200b\n   void sort ( Rank lo, Rank hi ); //\u200b\u5bf9\u200b[lo, hi)\u200b\u6392\u5e8f\u200b\n   void sort() { sort ( 0, _size ); } //\u200b\u6574\u4f53\u200b\u6392\u5e8f\u200b\n   void unsort ( Rank lo, Rank hi ); //\u200b\u5bf9\u200b[lo, hi)\u200b\u7f6e\u4e71\u200b\n   void unsort() { unsort ( 0, _size ); } //\u200b\u6574\u4f53\u200b\u7f6e\u4e71\u200b\n   int deduplicate(); //\u200b\u65e0\u5e8f\u200b\u53bb\u200b\u91cd\u200b\n   int uniquify(); //\u200b\u6709\u5e8f\u200b\u53bb\u200b\u91cd\u200b\n// \u200b\u904d\u5386\u200b\n   void traverse ( void (* ) ( T&amp; ) ); //\u200b\u904d\u5386\u200b\uff08\u200b\u4f7f\u7528\u200b\u51fd\u6570\u6307\u9488\u200b\uff0c\u200b\u53ea\u8bfb\u200b\u6216\u200b\u5c40\u90e8\u6027\u200b\u4fee\u6539\u200b\uff09\n   template &lt;typename VST&gt; void traverse ( VST&amp; ); //\u200b\u904d\u5386\u200b\uff08\u200b\u4f7f\u7528\u200b\u51fd\u6570\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u53ef\u200b\u5168\u5c40\u6027\u200b\u4fee\u6539\u200b\uff09\n}; //Vector\n</code></pre> Reference</p>"},{"location":"coding/dsa/vector-and-list/#_4","title":"\u63d2\u5165","text":"<p>\u200b\u5411\u91cf\u200b\u7684\u200b\u63d2\u5165\u200b\u64cd\u4f5c\u200b\u5206\u4e3a\u200b\u4e09\u6b65\u200b\uff1a</p> <ol> <li>\u200b\u68c0\u67e5\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u4e0e\u200b\u8f93\u5165\u200b\u4e0b\u200b\u6807\u7684\u200b\u5408\u6cd5\u6027\u200b\uff0c\u200b\u5fc5\u8981\u200b\u65f6\u200b\u6539\u53d8\u200b\u5411\u91cf\u200b\u7684\u200b\u957f\u5ea6\u200b\u4ee5\u200b\u5bb9\u7eb3\u200b\u8be5\u200b\u5143\u7d20\u200b</li> <li>\u200b\u5f53\u200b\u8f93\u5165\u200b\u4e0b\u6807\u200b\u4e0d\u662f\u200b\u5411\u91cf\u200b\u7684\u200b\u672b\u5c3e\u200b\u65f6\u200b\uff0c\u200b\u79fb\u52a8\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u5728\u200b\u5bf9\u5e94\u200b\u4f4d\u7f6e\u200b\u817e\u51fa\u200b\u7a7a\u95f4\u200b</li> <li>\u200b\u5c06\u200b\u6570\u636e\u200b\u5199\u5165\u200b\u5411\u91cf\u200b\u4e2d\u200b</li> </ol> <p>\u200b\u5728\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(n\\)\u200b\u7684\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u968f\u673a\u200b\u4f4d\u7f6e\u200b\u63d2\u5165\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u6539\u53d8\u200b\u5411\u91cf\u200b\u957f\u5ea6\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u65f6\u95f4\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)\uff0c\u200b\u79fb\u52a8\u200b\u5143\u7d20\u200b\u4ee5\u200b\u817e\u51fa\u200b\u7a7a\u95f4\u200b\u7684\u200b\u671f\u671b\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)\uff0c\u200b\u5199\u5165\u200b\u6570\u636e\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(1)\\)\uff0c\u200b\u603b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)</p>"},{"location":"coding/dsa/vector-and-list/#_5","title":"\u53ef\u200b\u6269\u5145\u200b\u5411\u91cf","text":"<p>\u200b\u5411\u91cf\u200b\u7684\u200b\u7a7a\u95f4\u200b\u662f\u200b\u6709\u9650\u200b\u7684\u200b\uff08\u200b\u5373\u200b\u4e3a\u200b\u5411\u91cf\u200b\u672c\u8eab\u200b\u7684\u200b\u957f\u5ea6\u200b\uff09\uff0c\u200b\u82e5\u5f53\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u4e0d\u8db3\u200b\u65f6\u4e3a\u200b\u5411\u91cf\u200b\u91cd\u65b0\u5206\u914d\u200b\u4e00\u5757\u200b\u66f4\u957f\u200b\u7684\u200b\u5185\u5b58\u200b\uff0c\u200b\u5c06\u200b\u539f\u200b\u6570\u636e\u200b\u590d\u5236\u5230\u200b\u65b0\u200b\u7684\u200b\u5185\u5b58\u7a7a\u95f4\u200b\u4e2d\u4ee5\u200b\u5b9e\u73b0\u200b\u5bb9\u91cf\u200b\u7684\u200b\u6269\u589e\u200b\uff0c\u200b\u5219\u200b\u5411\u91cf\u200b\u7684\u200b\u7a7a\u95f4\u200b\u53ef\u4ee5\u200b\u8fd1\u4f3c\u200b\u89c6\u4e3a\u200b\u65e0\u9650\u200b\u3002\u200b\u4e00\u822c\u200b\u5b58\u5728\u200b\u4e24\u79cd\u200b\u6269\u5145\u200b\u7b97\u6cd5\u200b\uff1a</p> <ul> <li>\u200b\u5f53\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u4e0d\u8db3\u200b\u65f6\u200b\uff0c\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u589e\u52a0\u200b\u4e00\u4e2a\u200b\u5e38\u91cf\u200b\\(I\\)</li> <li>\u200b\u5f53\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u4e0d\u8db3\u200b\u65f6\u200b\uff0c\u200b\u5411\u91cf\u200b\u7a7a\u95f4\u200b\u4e58\u4ee5\u200b\u4e00\u5b9a\u200b\u500d\u6570\u200b\\(k\\)\uff08\u200b\u901a\u5e38\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\\(k=2\\)\uff09</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u521d\u59cb\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(0\\)\u200b\u7684\u200b\u5411\u91cf\u200b\uff0c\u200b\u8003\u5bdf\u200b\u63d2\u5165\u200b\\(n\\gg 2\\)\u200b\u4e2a\u200b\u5143\u7d20\u200b\u4e0b\u200b\u4e24\u79cd\u200b\u6269\u5145\u200b\u7b97\u6cd5\u200b\u7684\u200b\u8c03\u7528\u200b\u6b21\u6570\u200b\u3002\u200b\u9012\u589e\u200b\u7b56\u7565\u200b\u4e0b\u200b\u63d2\u5165\u200b\u7b2c\u200b\\(mI+1, (m\\in \\mathbb N)\\)\u200b\u4e2a\u200b\u5143\u7d20\u200b\u65f6\u200b\u9700\u8981\u200b\u8fdb\u884c\u200b\u6269\u5145\u200b\uff0c\u200b\u603b\u200b\u7684\u200b\u8c03\u7528\u200b\u6b21\u6570\u200b\u4e3a\u200b\\(\\frac{n}{I}\\)\uff0c\u200b\u5373\u200b\\(\\mathcal O(n)\\)\u3002\u200b\u500d\u589e\u200b\u7b56\u7565\u200b\u4e0b\u200b\u63d2\u5165\u200b\u7b2c\u200b\\(1, 2, 3, 5\\dots 2^m+1, (m\\in \\mathbb N)\\)\u200b\u4e2a\u200b\u5143\u7d20\u200b\u65f6\u200b\u9700\u8981\u200b\u8fdb\u884c\u200b\u6269\u5145\u200b\uff0c\u200b\u603b\u200b\u7684\u200b\u8c03\u7528\u200b\u6b21\u6570\u200b\u4e3a\u200b\\(\\mathcal O(\\log n)\\)\u3002</p> <p>\u200b\u590d\u5236\u200b\u6570\u636e\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)\uff0c\u200b\u56e0\u6b64\u200b\u9012\u589e\u200b\u7b56\u7565\u200b\u6bcf\u6b21\u200b\u6269\u5bb9\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u65f6\u95f4\u200b\u4e3a\u200b\\(0, I, 2I, \\dots\\)\uff0c\u200b\u5171\u200b\u9700\u8981\u200b\u6269\u5bb9\u200b\\(\\frac nI\\)\u200b\u6b21\u200b\uff0c\u200b\u6269\u5bb9\u200b\u64cd\u4f5c\u200b\u7684\u200b\u603b\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n^2)\\)\u3002\u200b\u6bcf\u6b21\u200b\u63d2\u5165\u200b\u64cd\u4f5c\u200b\u5206\u644a\u200b\\(\\mathcal O(n)\\)\u3002\u200b\u500d\u589e\u200b\u7b56\u7565\u200b\u6bcf\u6b21\u200b\u6269\u5bb9\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u65f6\u95f4\u200b\u4e3a\u200b\\(0, 1, 2, \\dots n\\)\uff0c\u200b\u5171\u200b\u9700\u8981\u200b\u6269\u5bb9\u200b\\(\\log n\\)\u200b\u6b21\u200b\u3002\u200b\u6269\u5bb9\u200b\u64cd\u4f5c\u200b\u7684\u200b\u603b\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)\uff0c\u200b\u6bcf\u6b21\u200b\u63d2\u5165\u200b\u64cd\u4f5c\u200b\u5206\u644a\u200b\\(\\mathcal O(1)\\)\u3002</p> <p>\u200b\u4f46\u200b\u500d\u589e\u200b\u7b56\u7565\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u7a7a\u95f4\u200b\u6d6a\u8d39\u200b\uff0c\u200b\u5177\u4f53\u8868\u73b0\u200b\u4e3a\u200b\u5411\u91cf\u200b\u7684\u200b\u88c5\u8f7d\u200b\u56e0\u5b50\u200b\u8f83\u200b\u4f4e\u200b\u3002\u200b\u63d2\u5165\u200b\u5927\u91cf\u200b\u5143\u7d20\u200b\u65f6\u200b\uff0c\u200b\u500d\u589e\u200b\u7b56\u7565\u200b\u7684\u200b\u88c5\u8f7d\u200b\u56e0\u5b50\u200b\u53ea\u80fd\u200b\u4fdd\u8bc1\u200b\\(&gt; 50\\)\uff0c\u200b\u800c\u200b\u9012\u589e\u200b\u7b56\u7565\u200b\u7684\u200b\u88c5\u8f7d\u200b\u56e0\u5b50\u200b\u53ef\u4ee5\u200b\u4fdd\u8bc1\u200b\u5728\u200b\\(1\\)\u200b\u5de6\u53f3\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_6","title":"\u5220\u9664","text":"<p>\u200b\u5411\u91cf\u200b\u7684\u200b\u5220\u9664\u200b\u64cd\u4f5c\u200b\u4e3b\u8981\u200b\u4e3a\u200b\u533a\u95f4\u200b\u5220\u9664\u200b\u4e0e\u200b\u5355\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u5220\u9664\u200b\u3002\u200b\u5176\u4e2d\u200b\u540e\u8005\u200b\u53ef\u4ee5\u200b\u89c6\u4e3a\u200b\u524d\u8005\u200b\u7684\u200b\u7279\u6b8a\u200b\u60c5\u51b5\u200b\u3002\u200b\u5220\u9664\u200b\u64cd\u4f5c\u200b\u4e0d\u200b\u9700\u8981\u200b\u6267\u884c\u200b\u989d\u5916\u200b\u7684\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u53ea\u200b\u9700\u8981\u200b\u5c06\u200b\u5220\u9664\u200b\u533a\u95f4\u200b\u540e\u200b\u7684\u200b\u5143\u7d20\u200b\u5411\u524d\u200b\u79fb\u52a8\u200b\uff0c\u200b\u8986\u76d6\u200b\u5220\u9664\u200b\u533a\u95f4\u200b\u5185\u200b\u7684\u200b\u6570\u636e\u200b\u5e76\u200b\u66f4\u65b0\u200b\u64cd\u4f5c\u200b\u7ed3\u675f\u200b\u540e\u200b\u5411\u91cf\u200b\u7684\u200b\u957f\u5ea6\u200b\u5373\u53ef\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u53ef\u200b\u6269\u5145\u200b\u5411\u91cf\u200b\uff0c\u200b\u5220\u9664\u200b\u540e\u200b\u5e94\u5bf9\u200b\u5411\u91cf\u200b\u7684\u200b\u88c5\u586b\u200b\u56e0\u5b50\u200b\u8fdb\u884c\u200b\u68c0\u67e5\u200b\uff0c\u200b\u5728\u200b\u5fc5\u8981\u200b\u65f6\u200b\u6536\u7f29\u200b\u5411\u91cf\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u591a\u6b21\u200b\u8c03\u7528\u200b\u5220\u9664\u200b\u5355\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u63a5\u53e3\u200b\u5b9e\u73b0\u200b\u533a\u95f4\u200b\u5220\u9664\u200b\uff0c\u200b\u7531\u4e8e\u200b\u6bcf\u6b21\u200b\u5220\u9664\u200b\u90fd\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u4e00\u6b21\u200b\u6570\u636e\u200b\u7684\u200b\u79fb\u52a8\u200b\uff0c\u200b\u603b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n^2)\\)</p> <p>\u200b\u5220\u9664\u200b\u64cd\u4f5c\u200b\u9700\u8981\u200b\u5bf9\u200b\u5220\u9664\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5408\u6cd5\u6027\u200b\u8fdb\u884c\u200b\u68c0\u9a8c\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_7","title":"\u67e5\u627e","text":"<p>\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u4e0e\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u67e5\u627e\u200b\u65b9\u6cd5\u200b\u4e0d\u540c\u200b\uff0c\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e5f\u200b\u4e0d\u540c\u200b\u3002\u200b\u5177\u4f53\u200b\u800c\u8a00\u200b\uff0c\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u987a\u5e8f\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u989d\u5916\u200b\u7684\u200b\u4fe1\u606f\u200b\uff0c\u200b\u4ece\u800c\u200b\u4f7f\u5f97\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u67e5\u627e\u200b\u64cd\u4f5c\u200b\u80fd\u591f\u200b\u5728\u200b\\(\\log n\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u5185\u200b\u5b8c\u6210\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_8","title":"\u987a\u5e8f\u200b\u67e5\u627e","text":"<p>\u200b\u987a\u5e8f\u200b\u67e5\u627e\u200b\u64cd\u4f5c\u200b\u5047\u8bbe\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u7c7b\u578b\u5b9a\u4e49\u200b\u4e86\u200b<code>==</code>\u200b\u8fd0\u7b97\u7b26\u200b\u4e0e\u200b<code>!=</code>\u200b\u8fd0\u7b97\u7b26\u200b\u3002</p> <p>\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u67e5\u627e\u200b\u53ea\u200b\u9700\u8981\u200b\u4ece\u200b\u5411\u91cf\u200b\u7684\u200b\u5f00\u5934\u200b\u4f9d\u6b21\u200b\u5bf9\u6bd4\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u4e0e\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\uff0c\u200b\u5f53\u200b\u5143\u7d20\u200b\u76f8\u540c\u200b\u65f6\u200b\u8fd4\u56de\u200b\u5373\u53ef\u200b\u3002\u200b\u5e73\u5747\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)</p>"},{"location":"coding/dsa/vector-and-list/#_9","title":"\u4e8c\u5206\u200b\u67e5\u627e","text":"<p>\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u67e5\u627e\u200b\u64cd\u4f5c\u200b\u5047\u8bbe\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u7c7b\u578b\u5b9a\u4e49\u200b\u7c7b\u200b<code>==</code>\u200b\u8fd0\u7b97\u7b26\u200b\u3001<code>!=</code>\u200b\u8fd0\u7b97\u7b26\u200b\uff0c\u200b\u56e0\u6b64\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u67e5\u627e\u200b\u65b9\u6cd5\u200b\u7ebf\u6027\u200b\u67e5\u627e\u200b\u3002\u200b\u82e5\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u5143\u7d20\u200b\u7c7b\u578b\u5b9a\u4e49\u200b\u4e86\u200b\u6bd4\u8f83\u200b\u8fd0\u7b97\u7b26\u200b<code>&lt;</code>\u200b\u4e0e\u200b<code>&gt;</code>\uff0c\u200b\u5219\u200b\u57fa\u4e8e\u200b\u6bd4\u8f83\u200b\u7684\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u964d\u200b\u81f3\u200b\\(\\mathcal O(\\log n)\\)\u3002</p> <p>\u200b\u51cf\u800c\u6cbb\u200b\u4e4b\u200b\u662f\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7684\u200b\u6838\u5fc3\u601d\u60f3\u200b\u3002\u200b\u8003\u8651\u200b\u66f4\u200b\u4e00\u822c\u200b\u7684\u200b\u901a\u7528\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(n\\)\u200b\u7684\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5728\u200b\\(\\lambda \\cdot n (0\\leq \\lambda\\leq 1)\\)\u200b\u5904\u8bbe\u200b\u4e3a\u8f74\u70b9\u200b\u3002\u200b\u6bcf\u6b21\u200b\u6bd4\u8f83\u200b\u5f53\u524d\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u7684\u200b\u8f74\u70b9\u200b\\(B\\)\u200b\u4e0e\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\\(A\\)\u3002\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u5bf9\u8f74\u70b9\u200b\u7684\u200b\u6bd4\u8f83\u200b\u6709\u200b\u4e09\u79cd\u200b\u53ef\u80fd\u200b\u7684\u200b\u7ed3\u679c\u200b\uff0c\u200b\u5047\u8bbe\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u6309\u7167\u200b\u5347\u5e8f\u200b\u6392\u5e8f\u200b\uff0c\u200b\u5219\u200b\uff1a</p> <ul> <li>\\(A&lt;B\\)\uff0c\u200b\u8868\u793a\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u53ea\u200b\u53ef\u80fd\u200b\u51fa\u73b0\u200b\u5728\u200b\u8f74\u70b9\u200b\u5de6\u4fa7\u200b\uff0c\u200b\u5c06\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u7f29\u5c0f\u200b\u81f3\u200b\u5de6\u534a\u200b\u90e8\u5206\u200b\uff0c\u200b\u7ee7\u7eed\u200b\u67e5\u627e\u200b\u8fc7\u7a0b\u200b\uff1b</li> <li>\\(A=B\\)\uff0c\u200b\u8868\u793a\u200b\u8f74\u70b9\u200b\u5904\u200b\u51fa\u73b0\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\uff0c\u200b\u76f4\u63a5\u200b\u8fd4\u56de\u200b\u4e2d\u70b9\u200b\u7684\u200b\u4e0b\u6807\u200b\u5373\u53ef\u200b\uff1b</li> <li>\\(B&lt;A\\)\uff0c\u200b\u8868\u793a\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u53ea\u200b\u53ef\u80fd\u200b\u51fa\u73b0\u200b\u5728\u200b\u8f74\u70b9\u200b\u53f3\u4fa7\u200b\uff0c\u200b\u5c06\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u7f29\u5c0f\u200b\u81f3\u200b\u53f3\u534a\u200b\u90e8\u5206\u200b\uff0c\u200b\u7ee7\u7eed\u200b\u67e5\u627e\u200b\u8fc7\u7a0b\u200b\u3002</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\uff0c\u200b\u8f74\u70b9\u200b\u4e3a\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u7684\u200b\u4e2d\u70b9\u200b\uff0c\u200b\u5373\u200b\\(\\lambda = 0.5\\)\u3002</p> <p>1234567891011</p> <p>\u200b\u4f46\u200b\u8be5\u200b\u7248\u672c\u200b\u7684\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u6709\u200b\u591a\u79cd\u200b\u610f\u5916\u200b\u60c5\u5f62\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <ol> <li>\u200b\u5411\u91cf\u200b\u4e2d\u200b\u627e\u200b\u4e0d\u5230\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\uff1b</li> <li>\u200b\u5411\u91cf\u200b\u4e2d\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u5b58\u5728\u200b\u591a\u4e2a\u200b\u3002</li> </ol> <p>\u200b\u6b64\u65f6\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u7684\u200b\u7ed3\u679c\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u552f\u4e00\u200b\u3002\u200b\u7ea6\u5b9a\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7b97\u6cd5\u200b\u8fd4\u56de\u200b\u4e0d\u200b\u5927\u4e8e\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002</p> <p>\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u901a\u8fc7\u200b\u5c3d\u53ef\u80fd\u51cf\u5c11\u200b\u9012\u5f52\u200b\u6df1\u5ea6\u200b\u7684\u200b\u65b9\u5f0f\u200b\u51cf\u5c11\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\uff0c\u200b\u7531\u6b64\u200b\uff0c\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u9012\u5f52\u200b\u7684\u200b\u4e0d\u540c\u200b\u5206\u652f\u200b\u5e94\u6709\u200b\u76f8\u540c\u200b\u7684\u200b\u671f\u671b\u200b\u65f6\u95f4\u200b\u6d88\u8017\u200b\uff08\u200b\u5373\u200b\u8f6c\u5411\u200b\u6210\u672c\u200b\uff09\u3002\u200b\u6b64\u5904\u200b\u7684\u200b\u201c\u200b\u4e8c\u5206\u200b\u201d\u200b\u53ea\u662f\u200b\u5728\u200b\u6570\u7ec4\u200b\u533a\u95f4\u200b\u7684\u200b\u610f\u4e49\u200b\u4e0a\u200b\u4e8c\u7b49\u200b\u5206\u200b\uff0c\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u8bed\u53e5\u200b\uff1a</p> <pre><code>if (A &gt; B)\n    // statement\nelse if (A &lt; B)\n    // statement\nelse\n    // statement\n</code></pre> <p>\u200b\u5de6\u53f3\u4e24\u4e2a\u200b\u8df3\u8f6c\u200b\u4f4d\u4e8e\u200b\u4e0d\u540c\u200b\u7684\u200b\u5206\u652f\u200b\uff0c\u200b\u9700\u8981\u200b\u8fdb\u884c\u200b\u6bd4\u8f83\u200b\u7684\u200b\u6b21\u6570\u200b\u4e0d\u540c\u200b\uff0c\u200b\u56e0\u6b64\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u65f6\u95f4\u200b\u5728\u200b\u4e25\u683c\u200b\u610f\u4e49\u200b\u4e0a\u200b\u662f\u200b\u4e0d\u540c\u200b\u7684\u200b\u3002\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u6539\u53d8\u200b\u8f74\u70b9\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u5c06\u200b\u540c\u4e00\u200b\u9012\u5f52\u200b\u6df1\u5ea6\u200b\u4e0b\u200b\u6bd4\u8f83\u200b\u7684\u200b\u671f\u671b\u200b\u6b21\u6570\u200b\u8c03\u6574\u200b\u4e3a\u200b\u76f8\u540c\u200b\uff0c\u200b\u4ece\u800c\u200b\u964d\u4f4e\u200b\u5e73\u5747\u200b\u67e5\u627e\u200b\u957f\u5ea6\u200b\uff0c\u200b\u5728\u200b\u5e38\u200b\u7cfb\u6570\u200b\u7a0b\u5ea6\u200b\u4e0a\u200b\u5bf9\u200b\u7b97\u6cd5\u200b\u8fdb\u884c\u200b\u4f18\u5316\u200b\u3002</p> <p>+1+2n\u03bbn(1 -\u03bb)\u00b7n</p> <p>\u200b\u8bbe\u200b\u7b97\u6cd5\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\alpha(\\lambda) \\log n\\)\uff0c\u200b\u5219\u200b\uff1a</p> <ul> <li>\u200b\u5de6\u4fa7\u200b\u5b50\u200b\u95ee\u9898\u200b\u6d88\u8017\u200b\u7684\u200b\u65f6\u95f4\u200b\u7b49\u4e8e\u200b\u5224\u65ad\u200b\u6d88\u8017\u200b\u7684\u200b\u65f6\u95f4\u200b\u53ca\u200b\u89e3\u51b3\u200b\u5b50\u200b\u95ee\u9898\u200b\u6d88\u8017\u200b\u7684\u200b\u65f6\u95f4\u200b\uff0c\u200b\u5373\u200b\uff1a\\(\\lambda (1 + \\alpha(\\lambda)  \\log (\\lambda n))\\)</li> <li>\u200b\u540c\u7406\u200b\uff0c\u200b\u53f3\u4fa7\u200b\u5b50\u200b\u95ee\u9898\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\uff1a\\((1 - \\lambda)(2+\\alpha(\\lambda)\\log((1-\\lambda)n))\\)</li> </ul> <p>\u200b\u5f97\u200b\\(\\alpha(\\lambda) = \\frac{\\ln 2(\\lambda - 2)}{\\lambda \\cdot\\ln\\lambda+(1-\\lambda)\\cdot\\ln(1-\\lambda)}\\)</p> <p>\u200b\u5f53\u200b\\(\\lambda=\\frac{\\sqrt 5-1}{2}\\approx 0.618\\)\u200b\u65f6\u200b\uff0c\\(\\alpha(\\lambda)\\)\u200b\u53d6\u200b\u6700\u5c0f\u503c\u200b\u3002</p> <p>\u200b\u6d88\u9664\u200b\u4e0d\u200b\u5bf9\u79f0\u200b\u7684\u200b\u53e6\u200b\u4e00\u79cd\u200b\u65b9\u5f0f\u200b\u662f\u200b\u5c06\u200b\u4e09\u79cd\u200b\u6bd4\u8f83\u200b\u7ed3\u679c\u200b\u53d8\u4e3a\u200b\u4e24\u79cd\u200b\uff0c\u200b\u4ece\u800c\u200b\u4e00\u6b21\u200b\u6bd4\u8f83\u200b\u5373\u53ef\u200b\u8fdb\u884c\u200b\u5212\u5206\u200b\u3002\u200b\u8bb0\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u4e3a\u200b\\(A\\)\uff0c\u200b\u8f74\u70b9\u200b\u4e3a\u200b\\(B\\)\uff0c\u200b\u5219\u200b\uff1a</p> <ul> <li>\\(A&lt;B\\)\uff1a\u200b\u5728\u200b\u8f74\u70b9\u200b\u5de6\u4fa7\u200b\u7684\u200b\u533a\u95f4\u200b\u4e2d\u200b\u67e5\u627e\u200b</li> <li>\\(A\\geq B\\)\uff1a\u200b\u5728\u200b\u8f74\u70b9\u200b\u53f3\u4fa7\u200b \uff08\u200b\u5305\u542b\u200b\u8f74\u70b9\u200b\uff09 \u200b\u7684\u200b\u533a\u95f4\u200b\u4e2d\u200b\u67e5\u627e\u200b</li> <li>\u200b\u5f53\u200b\u533a\u95f4\u200b\u957f\u5ea6\u200b\u7f29\u51cf\u200b\u4e3a\u200b1\u200b\u65f6\u200b\uff0c\u200b\u8868\u793a\u200b\u67e5\u627e\u200b\u8fc7\u7a0b\u200b\u7ed3\u675f\u200b\u3002</li> </ul> <p>\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7684\u200b\u6700\u7ec8\u200b\u5b9e\u73b0\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</p> \u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b <p><pre><code>/******************************************************************************************\n* Data Structures in C++\n* ISBN: 7-302-33064-6 &amp; 7-302-33065-3 &amp; 7-302-29652-2 &amp; 7-302-26883-3\n* Junhui DENG, deng@tsinghua.edu.cn\n* Computer Science &amp; Technology, Tsinghua University\n* Copyright (c) 2003-2020. All rights reserved.\n******************************************************************************************/\n\n// \u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7b97\u6cd5\u200b\uff08\u200b\u7248\u672c\u200bC\uff09\uff1a\u200b\u5728\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u533a\u95f4\u200b[lo, hi)\u200b\u5185\u200b\u67e5\u627e\u200b\u5143\u7d20\u200be\uff0c0 &lt;= lo &lt;= hi &lt;= _size\ntemplate &lt;typename T&gt; static Rank binSearch ( T* S, T const&amp; e, Rank lo, Rank hi ) {\n   while ( lo &lt; hi ) { //\u200b\u6bcf\u6b65\u200b\u8fed\u4ee3\u200b\u4ec5\u200b\u9700\u200b\u505a\u200b\u4e00\u6b21\u200b\u6bd4\u8f83\u200b\u5224\u65ad\u200b\uff0c\u200b\u6709\u200b\u4e24\u4e2a\u200b\u5206\u652f\u200b\n      Rank mi = ( lo + hi ) &gt;&gt; 1; //\u200b\u4ee5\u4e2d\u70b9\u200b\u4e3a\u8f74\u70b9\u200b\uff08\u200b\u533a\u95f4\u200b\u5bbd\u5ea6\u200b\u7684\u200b\u6298\u534a\u200b\uff0c\u200b\u7b49\u6548\u4e8e\u200b\u5bbd\u5ea6\u200b\u4e4b\u200b\u6570\u503c\u200b\u8868\u793a\u200b\u7684\u200b\u53f3\u79fb\u200b\uff09\n      ( e &lt; S[mi] ) ? hi = mi : lo = mi + 1; //\u200b\u7ecf\u200b\u6bd4\u8f83\u200b\u540e\u200b\u786e\u5b9a\u200b\u6df1\u5165\u200b[lo, mi)\u200b\u6216\u200b(mi, hi)\n   } //\u200b\u6210\u529f\u200b\u67e5\u627e\u200b\u4e0d\u80fd\u200b\u63d0\u524d\u200b\u7ec8\u6b62\u200b\n   return lo - 1; //\u200b\u5faa\u73af\u200b\u7ed3\u675f\u200b\u65f6\u200b\uff0clo\u200b\u4e3a\u200b\u5927\u4e8e\u200be\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u6700\u5c0f\u200b\u79e9\u200b\uff0c\u200b\u6545\u200blo - 1\u200b\u5373\u200b\u4e0d\u200b\u5927\u4e8e\u200be\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u6700\u5927\u200b\u79e9\u200b\n} //\u200b\u6709\u200b\u591a\u4e2a\u200b\u547d\u4e2d\u200b\u5143\u7d20\u200b\u65f6\u200b\uff0c\u200b\u603b\u80fd\u200b\u4fdd\u8bc1\u200b\u8fd4\u56de\u200b\u79e9\u200b\u6700\u5927\u8005\u200b\uff1b\u200b\u67e5\u627e\u200b\u5931\u8d25\u200b\u65f6\u200b\uff0c\u200b\u80fd\u591f\u200b\u8fd4\u56de\u200b\u5931\u8d25\u200b\u7684\u200b\u4f4d\u7f6e\u200b\n</code></pre> Reference</p>"},{"location":"coding/dsa/vector-and-list/#_10","title":"\u63d2\u503c\u200b\u67e5\u627e","text":"<p>\u200b\u5047\u8bbe\u200b\u533a\u95f4\u200b\u5185\u200b\u5143\u7d20\u200b\u5206\u5e03\u200b\u7684\u200b\u89c4\u5f8b\u200b\u5df2\u77e5\u200b\uff0c\u200b\u6839\u636e\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u5728\u200b\u8be5\u200b\u5206\u5e03\u200b\u4e2d\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u63a8\u65ad\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u5728\u200b\u533a\u95f4\u200b\u7684\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u4ee5\u6b64\u200b\u786e\u5b9a\u200b\u5212\u5206\u200b\u7684\u200b\u8f74\u70b9\u200b\u4f4d\u7f6e\u200b\u3002</p> <p>\u200b\u5047\u8bbe\u200b\u533a\u95f4\u200b\u5185\u200b\u5143\u7d20\u200b\u670d\u4ece\u200b\u72ec\u7acb\u200b\u7684\u200b\u5e73\u5747\u200b\u5206\u5e03\u200b\uff0c\u200b\u5219\u200b\u6bcf\u6b21\u200b\u67e5\u627e\u200b\u5e73\u5747\u200b\u4f7f\u5f97\u200b\u533a\u95f4\u200b\u7f29\u51cf\u200b\u81f3\u200b\u539f\u6765\u200b\u7684\u200b\\(\\sqrt n\\)\uff0c\u200b\u5e73\u5747\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(\\log\\log n)\\)\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_11","title":"\u53bb\u200b\u91cd","text":"<p>\u200b\u5bf9\u4e8e\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\uff0c\u200b\u53bb\u200b\u91cd\u200b\u64cd\u4f5c\u200b\u53ef\u4ee5\u200b\u5728\u200b\u7ebf\u6027\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u5185\u200b\u5b8c\u6210\u200b\uff0c\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u53bb\u200b\u91cd\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n^2)\\)</p>"},{"location":"coding/dsa/vector-and-list/#_12","title":"\u65e0\u5e8f\u200b\u5411\u91cf","text":"<p>\u200b\u4f7f\u7528\u200b\u6307\u9488\u200b\u6807\u8bb0\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u5df2\u7ecf\u200b\u5b8c\u6210\u200b\u53bb\u200b\u91cd\u200b\u7684\u200b\u90e8\u5206\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u672a\u53bb\u200b\u91cd\u200b\u90e8\u5206\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u5728\u200b\u672a\u200b\u53bb\u200b\u91cd\u200b\u90e8\u5206\u200b\u7684\u200b\u540e\u200b\u534a\u200b\u90e8\u5206\u200b\u4e2d\u200b\u67e5\u627e\u200b\u662f\u5426\u200b\u5b58\u5728\u200b\u76f8\u540c\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u5982\u679c\u200b\u627e\u5230\u200b\u76f8\u540c\u200b\u7684\u200b\u5143\u7d20\u200b\u5219\u200b\u5220\u9664\u200b\u3002\u200b\u5b8c\u6210\u200b\u67e5\u627e\u200b\u540e\u200b\u6307\u9488\u200b\u540e\u79fb\u200b\uff0c\u200b\u5c06\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u6807\u8bb0\u200b\u4e3a\u200b\u5df2\u7ecf\u200b\u5b8c\u6210\u200b\u53bb\u200b\u91cd\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_13","title":"\u6709\u5e8f\u200b\u5411\u91cf","text":"<p>\u200b\u4f7f\u7528\u200b\u53cc\u200b\u6307\u9488\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u4e00\u4e2a\u200b\u6307\u9488\u200b\\(P\\)\u200b\u6807\u8bb0\u200b\u5df2\u53bb\u200b\u91cd\u200b\u90e8\u5206\u200b\u7684\u200b\u7ed3\u5c3e\u200b\uff0c\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u6307\u9488\u200b\\(Q\\)\u200b\u6807\u8bb0\u200b\u672a\u53bb\u200b\u91cd\u200b\u90e8\u5206\u200b\u7684\u200b\u5f00\u5934\u200b\uff0c\u200b\u521d\u59cb\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\\(P\\)\u200b\u4f4d\u4e8e\u200b\u5411\u91cf\u200b\u7b2c\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u4e14\u200b\\(P+1=Q\\)\u3002\u200b\u5f53\u200b\\(P\\)\u200b\u6307\u5411\u200b\u7684\u200b\u5bf9\u8c61\u200b\u4e0e\u200b\\(Q\\)\u200b\u6307\u5411\u200b\u7684\u200b\u5bf9\u8c61\u200b\u76f8\u540c\u200b\u65f6\u200b\uff0c\\(Q\\)\u200b\u9012\u589e\u200b\u800c\u200b\\(P\\)\u200b\u4e0d\u200b\u79fb\u52a8\u200b\uff0c\u200b\u5426\u5219\u200b\u5c06\u200b\\(Q\\)\u200b\u5904\u200b\u7684\u200b\u5143\u7d20\u200b\u590d\u5236\u5230\u200b\\(P+1\\)\u200b\u5904\u200b\uff0c\u200b\u4e24\u200b\u6307\u9488\u200b\u540c\u65f6\u200b\u9012\u589e\u200b\u3002</p> <p> 33355558815333555588153355555881535555588815355558881515 </p>"},{"location":"coding/dsa/vector-and-list/#_14","title":"\u6392\u5e8f","text":"<p>\u200b\u6b64\u5904\u200b\u4ecb\u7ecd\u200b\u5bf9\u200b\u5411\u91cf\u200b\u7684\u200b\u5192\u6ce1\u6392\u5e8f\u200b\u4e0e\u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b\uff0c\u200b\u5176\u4ed6\u200b\u6392\u5e8f\u200b\u65b9\u5f0f\u200b\u5c06\u200b\u5728\u200b\u4ee5\u540e\u200b\u8fdb\u884c\u200b\u4ecb\u7ecd\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_15","title":"\u5192\u6ce1\u6392\u5e8f","text":"<p>\u200b\u5bf9\u4e8e\u200b\u6570\u5217\u200b\\(\\{a_1, \\cdots, a_n\\}\\)\uff0c\u200b\u82e5\u200b\\(a_i&gt;a_{i+1}\\)\uff0c\u200b\u79f0\u200b\\((a_i, a_{i+1})\\)\u200b\u4e3a\u200b\u4e00\u5bf9\u200b\u9006\u5e8f\u200b\u5bf9\u200b\u3002\u200b\u5426\u5219\u200b\u79f0\u4e3a\u200b\u987a\u5e8f\u200b\u5bf9\u200b\u3002\u200b\u5219\u200b\u6709\u5e8f\u200b\u6570\u5217\u200b\u4e2d\u200b\u4e0d\u200b\u5b58\u5728\u200b\u9006\u5e8f\u200b\u5bf9\u200b\uff0c\u200b\u65e0\u5e8f\u200b\u6570\u5217\u200b\u4e2d\u200b\u81f3\u5c11\u200b\u6709\u200b\u4e00\u4e2a\u200b\u9006\u5e8f\u200b\u5bf9\u200b\u3002\u200b\u5982\u679c\u200b\u80fd\u591f\u200b\u901a\u8fc7\u200b\u9006\u5e8f\u200b\u5bf9\u200b\u7684\u200b\u4ea4\u6362\u200b\u4f7f\u5f97\u200b\u6570\u5217\u200b\u4e2d\u200b\u6240\u6709\u200b\u7684\u200b\u9006\u5e8f\u200b\u5bf9\u200b\u8f6c\u4e3a\u200b\u987a\u5e8f\u200b\uff0c\u200b\u5373\u53ef\u200b\u5b8c\u6210\u200b\u5bf9\u200b\u6570\u7ec4\u200b\u7684\u200b\u6392\u5e8f\u200b\u3002\u200b\u8fd9\u662f\u200b\u5192\u6ce1\u6392\u5e8f\u200b\u7b97\u6cd5\u200b\u7684\u200b\u6838\u5fc3\u601d\u60f3\u200b\u3002</p> <ol> <li>\u200b\u8bbe\u200b\u5411\u91cf\u200b\uff08\u200b\u6570\u7ec4\u200b\uff09\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(n\\)\uff0c\u200b\u5f85\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u4e0b\u6807\u200b\u4e3a\u200b\\([0,n)\\)</li> <li>\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u626b\u63cf\u200b\u5f85\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u4e2d\u200b\u7684\u200b\u76f8\u90bb\u200b\u6570\u636e\u200b\uff0c\u200b\u82e5\u200b\u4e3a\u200b\u9006\u5e8f\u200b\u5bf9\u200b\u5219\u200b\u4ea4\u6362\u200b\u4e4b\u200b</li> <li>\u200b\u626b\u63cf\u200b\u7ed3\u675f\u200b\u540e\u200b\uff0c\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u6700\u5927\u200b\u5143\u7d20\u200b\u79fb\u52a8\u200b\u5230\u200b\u5f85\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u672b\u5c3e\u200b\uff0c\u200b\u5373\u200b\u4e3a\u200b\u5df2\u200b\u6392\u5e8f\u200b</li> <li>\u200b\u4e00\u8d9f\u200b\u626b\u63cf\u200b\u540e\u200b\u5c06\u200b\u5f85\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u672b\u5c3e\u200b\u5143\u7d20\u200b\u5212\u5165\u200b\u5df2\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u3002</li> <li>\u200b\u91cd\u590d\u200b\u626b\u63cf\u200b\uff0c\u200b\u76f4\u5230\u200b\u5f85\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u957f\u5ea6\u200b\u7f29\u5c0f\u200b\u4e3a\u200b0\uff0c\u200b\u5373\u200b\u5b8c\u6210\u200b\u6392\u5e8f\u200b\u3002</li> </ol> <p>\u200b\u63d0\u524d\u200b\u7ec8\u6b62\u200b\uff1a\u200b\u82e5\u200b\u67d0\u6b21\u200b\u626b\u63cf\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u6ca1\u6709\u200b\u53d1\u751f\u200b\u4ea4\u6362\u200b\uff0c\u200b\u8bf4\u660e\u200b\u5f85\u200b\u6392\u5e8f\u200b\u533a\u6bb5\u200b\u5df2\u7ecf\u200b\u6709\u5e8f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u63d0\u524d\u7ed3\u675f\u200b\u7b97\u6cd5\u200b</p> <p>\u200b\u7531\u4e8e\u200b\u63d0\u524d\u200b\u7ec8\u6b62\u200b\u7684\u200b\u5b58\u5728\u200b\uff0c\u200b\u5192\u6ce1\u6392\u5e8f\u200b\u5728\u200b\u6700\u597d\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)\uff0c\u200b\u5bf9\u5e94\u200b\u8f93\u5165\u200b\u6570\u636e\u200b\u5df2\u7ecf\u200b\u6709\u5e8f\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002\u200b\u5e73\u5747\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u5192\u6ce1\u6392\u5e8f\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n^2)\\)\u3002</p> <p>\u200b\u5728\u200b\u5192\u6ce1\u6392\u5e8f\u200b\u4e2d\u200b\uff0c\u200b\u76f8\u7b49\u200b\u7684\u200b\u5143\u7d20\u200b\u4e0d\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u9006\u5e8f\u200b\u5bf9\u200b\u3002\u200b\u5982\u679c\u200b\u5c06\u200b\u76f8\u7b49\u200b\u7684\u200b\u5143\u7d20\u200b\u89c6\u4e3a\u200b\u9006\u5e8f\u200b\u5bf9\u200b\uff0c\u200b\u5192\u6ce1\u6392\u5e8f\u200b\u5c06\u200b\u5931\u53bb\u200b\u7a33\u5b9a\u6027\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_16","title":"\u5f52\u5e76\u200b\u6392\u5e8f","text":"<p>\u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b\u91c7\u7528\u200b\u5206\u800c\u6cbb\u4e4b\u200b\u7684\u200b\u65b9\u6cd5\u200b\u5bf9\u200b\u6570\u7ec4\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b\uff0c\u200b\u5176\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u9012\u5f52\u200b\u3002\u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b\u7684\u200b\u7b97\u6cd5\u200b\u8fc7\u7a0b\u200b\u4e3b\u8981\u200b\u5206\u4e3a\u200b\u4e09\u6b65\u200b\uff1a</p> <ul> <li>\u201c\u200b\u5206\u200b\u201d\uff1a\u200b\u5c06\u200b\u5411\u91cf\u200b\u5212\u5206\u200b\u4e3a\u200b\u4e24\u4e2a\u200b\uff08\u200b\u8fd1\u4f3c\u200b\uff09\u200b\u7b49\u957f\u200b\u7684\u200b\u90e8\u5206\u200b\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(1)\\)</li> <li>\u201c\u200b\u6cbb\u200b\u201d\uff1a\u200b\u5bf9\u200b\u4e24\u200b\u90e8\u5206\u200b\u5206\u522b\u200b\u8fdb\u884c\u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b\uff08\u200b\u5df2\u77e5\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(1\\)\u200b\u7684\u200b\u5411\u91cf\u200b\u662f\u200b\u6709\u5e8f\u200b\u5411\u91cf\u200b\uff09\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(2\\times T(n/2)\\)</li> <li>\u201c\u200b\u5408\u200b\u201d\uff1a\u200b\u5c06\u200b\u6392\u5e8f\u200b\u540e\u200b\u7684\u200b\u4e24\u200b\u90e8\u5206\u200b\u5408\u5e76\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u6709\u5e8f\u200b\u7684\u200b\u6574\u4f53\u200b\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)</li> <li>\u200b\u7b97\u6cd5\u200b\u6574\u4f53\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n\\log n)\\)\uff0c\u200b\u5373\u4f7f\u200b\u662f\u200b\u5728\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u6709\u5e8f\u200b\u6216\u200b\u63a5\u8fd1\u200b\u6709\u5e8f\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u3002\u200b\u5f52\u5e76\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)</li> </ul> \u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b <p><pre><code>template &lt;typename T&gt; void Vector&lt;T&gt;::mergeSort( Rank lo, Rank hi ) {\n   if ( hi - lo &lt; 2 ) return; //\u200b\u5355\u200b\u5143\u7d20\u200b\u533a\u95f4\u200b\u81ea\u7136\u200b\u6709\u5e8f\u200b\uff0c\u200b\u5426\u5219\u200b...\n   int mi = (lo + hi) &gt;&gt; 1; //\u200b\u4ee5\u4e2d\u70b9\u200b\u4e3a\u754c\u200b\n   mergeSort( lo, mi ); //\u200b\u5bf9\u200b\u524d\u534a\u6bb5\u200b\u6392\u5e8f\u200b\n   mergeSort( mi, hi ); //\u200b\u5bf9\u200b\u540e\u534a\u6bb5\u200b\u6392\u5e8f\u200b\n   merge( lo, mi, hi ); //\u200b\u5f52\u5e76\u200b\n}\n</code></pre> Reference</p> <p>\u200b\u5f52\u5e76\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u6838\u5fc3\u601d\u60f3\u200b\u5373\u200b\u6309\u200b\u987a\u5e8f\u200b\u904d\u5386\u200b\u4e24\u200b\u90e8\u5206\u200b\uff0c\u200b\u5c06\u200b\u6700\u5c0f\u200b\u7684\u200b\u5199\u5165\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u3002</p> <p> </p> <p>\u200b\u5b9e\u9645\u200b\u6267\u884c\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\uff0c\u200b\u53ea\u200b\u9700\u200b\u5c06\u200b\u6570\u7ec4\u200b\u7684\u200b\u524d\u534a\u90e8\u200b\u5206\u200b\u590d\u5236\u200b\u4e00\u4efd\u200b\uff0c\u200b\u7136\u540e\u200b\u8fdb\u884c\u200b\u5408\u5e76\u200b\u5373\u53ef\u200b\u3002</p> <ul> <li>\u200b\u82e5\u200b\u524d\u534a\u90e8\u200b\u5206\u200b\u63d0\u524d\u7ed3\u675f\u200b\uff0c\u200b\u7531\u4e8e\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\u7684\u200b\u540e\u200b\u534a\u200b\u90e8\u5206\u200b\u4e0e\u200b\u539f\u200b\u6570\u7ec4\u200b\u7684\u200b\u540e\u200b\u534a\u200b\u90e8\u5206\u200b\u91cd\u5408\u200b\uff0c\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\u81ea\u52a8\u200b\u6210\u4e3a\u200b\u6709\u5e8f\u200b</li> <li>\u200b\u82e5\u540e\u200b\u534a\u200b\u90e8\u5206\u200b\u63d0\u524d\u7ed3\u675f\u200b\uff0c\u200b\u5c06\u200b\u524d\u534a\u90e8\u200b\u5206\u200b\u5269\u4f59\u200b\u7684\u200b\u6570\u636e\u200b\u590d\u5236\u5230\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\u540e\u200b\u534a\u200b\u90e8\u5206\u200b\u5bf9\u5e94\u200b\u4f4d\u7f6e\u200b\u5373\u53ef\u200b</li> </ul> <p>\u200b\u5982\u4e0b\u200b\u56fe\u200b\uff1a</p> <p> </p> <p>\u200b\u82e5\u200b\u5728\u200b\u5f52\u5e76\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u76f8\u540c\u200b\u5143\u7d20\u200b\uff0c\u200b\u5de6\u4fa7\u200b\u7684\u200b\u5143\u7d20\u200b\u4f18\u5148\u200b\u5f52\u5165\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u4fdd\u8bc1\u200b\u5f52\u5e76\u200b\u6392\u5e8f\u200b\u7684\u200b\u7a33\u5b9a\u6027\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_17","title":"\u4f4d\u200b\u56fe","text":"<p>\u200b\u4f4d\u56fe\u200b\u662f\u200b\u5176\u4e2d\u200b\u5143\u7d20\u200b\u53ea\u80fd\u200b\u53d6\u200b0/1\u200b\u7684\u200b\u5411\u91cf\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u4e00\u4e2a\u200b\u6709\u9650\u200b\u7684\u200b\u6574\u6570\u200b\u96c6\u5408\u200b\\(U\\)\uff0c\u200b\u4f4d\u200b\u56fe\u200b\u6784\u5efa\u200b\u4e86\u200b\\(U\\)\u200b\u7684\u200b\u5b50\u96c6\u200b\\(S\\)\uff0c\u200b\u5176\u4e2d\u200b\\(k\\in S\\)\u200b\u7b49\u4ef7\u200b\u4e8e\u4f4d\u200b\u56fe\u4e2d\u200b\u4e0b\u6807\u200b\u4e3a\u200b\\(k\\)\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u53d6\u503c\u200b\u4e3a\u200b\\(1\\)\u3002\u200b\u4f4d\u200b\u56fe\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>set()</code>\u3001<code>get()</code>\u3001<code>clear()</code>\u200b\u4e09\u4e2a\u200b\u63a5\u53e3\u200b\uff0c\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u4e8e\u4f4d\u200b\u56fe\u200b\u7684\u200b\u5199\u5165\u200b\u3001\u200b\u67e5\u8be2\u200b\u3001\u200b\u6e05\u9664\u200b\u64cd\u4f5c\u200b\u3002</p> <p>byte11010011</p> <p>\u200b\u57fa\u4e8e\u200b\u4f4d\u200b\u8fd0\u7b97\u200b\u7684\u200b\u5b9e\u73b0\u200b\u5982\u4e0b\u200b\uff1a</p> \u200b\u4f4d\u200b\u56fe\u200b\u65b9\u6cd5\u200b\u5b9e\u73b0\u200b <p><pre><code>bool test( int k ) { return M[ k &gt;&gt; 3 ] &amp; ( 0x80 &gt;&gt; (k &amp; 0x07) ); }\nvoid set( int k ) { expand( k ); M[ k &gt;&gt; 3 ] |= ( 0x80 &gt;&gt; (k &amp; 0x07) ); }\nvoid clear( int k ) { expand( k ); M[ k &gt;&gt; 3 ] &amp;= ~( 0x80 &gt;&gt; (k &amp; 0x07) ); }\n</code></pre> Reference</p> <p>\u200b\u8bfb\u53d6\u200b\u5b57\u8282\u200b\u4e2d\u200b\u7684\u200b\u67d0\u4e2a\u200b\u4f4d\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>&amp;</code>\u200b\u8fd0\u7b97\u7b26\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u5411\u200b\u5b57\u8282\u200b\u4e2d\u200b\u8bbe\u7f6e\u200b\u67d0\u4e2a\u200b\u4f4d\u4e3a\u200b\\(1\\)\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>|=</code>\u200b\u8fd0\u7b97\u7b26\u200b\u5b9e\u73b0\u200b\u3002</p> <p>\u200b\u4f4d\u56fe\u200b\u7684\u200b\u5e94\u7528\u200b\u573a\u666f\u200b\uff1a</p> <ol> <li>\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\u53bb\u200b\u91cd\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u501f\u52a9\u200b\u4f4d\u200b\u56fe\u200b\u53ef\u4ee5\u200b\u5728\u200b\\(O(n)\\)\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u5185\u200b\u5b8c\u6210\u200b\uff0c\u200b\u540c\u65f6\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u5b8c\u6210\u200b\u76f8\u5e94\u200b\u7684\u200b\u6392\u5e8f\u200b\u64cd\u4f5c\u200b\u3002</li> <li>\u200b\u8d28\u6570\u200b\u8ba1\u7b97\u200b\uff1aEratosthenes\u200b\u7b5b\u200b</li> </ol>"},{"location":"coding/dsa/vector-and-list/#_18","title":"\u5feb\u901f\u200b\u521d\u59cb\u5316","text":"<p>\u200b\u5728\u200b\u4e25\u683c\u200b\u610f\u4e49\u200b\u4e0a\u200b\u8bb2\u200b\uff0c\u200b\u5373\u4f7f\u200b\u8c03\u7528\u200b<code>memset()</code>\u200b\u51fd\u6570\u200b\uff0c\u200b\u4f4d\u56fe\u200b\u7684\u200b\u521d\u59cb\u5316\u200b\uff08\u200b\u6e05\u7a7a\u200b\uff09\u200b\u4e5f\u200b\u9700\u8981\u200b\\(\\mathcal O(n)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u3002\u200b\u501f\u52a9\u200b\u6821\u9a8c\u200b\u73af\u200b\u7ed3\u6784\u200b\u53ef\u4ee5\u200b\u5c06\u4f4d\u200b\u56fe\u200b\u7684\u200b\u521d\u59cb\u5316\u200b\u590d\u6742\u5ea6\u200b\u7531\u200b\\(\\mathcal O(n)\\)\u200b\u63d0\u9ad8\u200b\u5230\u200b\\(\\mathcal O(1)\\)\uff08\u200b\u4f46\u200b\u76f8\u5e94\u200b\u5730\u200b\uff0c\u200b\u4f4d\u56fe\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u200b\u4f4d\u200b\u9700\u8981\u200b\u4e24\u4e2a\u200b<code>int</code>\u200b\u7c7b\u578b\u200b\u5373\u200b64\u200b\u5b57\u8282\u200b\u8fdb\u884c\u200b\u5b58\u50a8\u200b\uff0c\u200b\u9020\u6210\u200b\u7a7a\u95f4\u200b\u7684\u200b\u5927\u91cf\u200b\u6d6a\u8d39\u200b\uff09\u3002</p> <p>\u200b\u5c06\u4f4d\u200b\u56fe\u200b\u7684\u200b\u6570\u7ec4\u200b\u62c6\u200b\u5206\u4e3a\u200b<code>A[]</code>\u3001<code>B[]</code>\u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u7b2c\u200b<code>k</code>\u200b\u4f4d\u200b\uff0c\u200b\u6ee1\u8db3\u200b\uff1a</p> <ul> <li><code>A[B[k]] = k</code></li> <li><code>B[A[k]] = k</code></li> </ul> <p>\u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u6784\u6210\u200b\u4e86\u200b\u76f8\u4e92\u200b\u6821\u9a8c\u200b\u7684\u200b\u5173\u7cfb\u200b\uff0c\u200b\u6309\u7167\u200b\u4e0b\u6807\u200b\u67e5\u8be2\u200b\uff0c\u200b\u7b26\u5408\u200b\u6821\u9a8c\u200b\u6761\u4ef6\u200b\u7684\u200b\u4e0b\u6807\u200b\u5b58\u50a8\u200b\u7684\u200b\u4f4d\u4e3a\u200b<code>1</code>\uff0c\u200b\u5426\u5219\u200b\u4e3a\u200b<code>0</code>\u3002</p> <p> 0AB0112512234567</p> <p>\u200b\u53cc\u200b\u6570\u7ec4\u200b\u6a21\u5f0f\u200b\u4e0b\u200b\uff0c\u200b\u6570\u7ec4\u200b<code>B</code>\u200b\u4e3a\u200b\u7c7b\u4f3c\u200b\u4e8e\u6808\u200b\u7684\u200b\u7ed3\u6784\u200b\uff0c\u200b\u4ece\u200b\u5f00\u5934\u200b\u9010\u6e10\u200b\u5411\u200b\u540e\u200b\u5ef6\u4f38\u200b\uff0c\u200b\u7ef4\u62a4\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b<code>top</code>\u200b\u8bb0\u5f55\u200b<code>B</code>\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u6570\u91cf\u200b\u3002\u200b\u6570\u7ec4\u200b<code>A</code>\u200b\u968f\u673a\u200b\u8bbf\u95ee\u200b\u3002</p> <ul> <li><code>set(k)</code>\uff1a\u200b\u5728\u200b\u6570\u7ec4\u200b<code>B</code>\u200b\u4e2d\u200b\u8ffd\u52a0\u200b\u4e00\u4e2a\u200b\u503c\u4e3a\u200b<code>k</code>\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u5728\u200b<code>A[k]</code>\u200b\u4e2d\u200b\u5b58\u50a8\u200b<code>B</code>\u200b\u4e2d\u200b\u5bf9\u5e94\u200b\u5143\u7d20\u200b\u7684\u200b\u4e0b\u6807\u200b\uff1b</li> <li><code>get(k)</code>\uff1a\u200b\u82e5\u200b<code>B[A[k]] == k</code>\u200b\u4e14\u200b<code>A[k]</code>\u200b\u5c0f\u4e8e\u200b\u6808\u9876\u200b\u4f4d\u7f6e\u200b\u5219\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5426\u5219\u200b\u4e3a\u200b\u5047\u200b\uff1b</li> <li><code>clear()</code>\uff1a\u200b\u6e05\u7a7a\u200b\u6570\u7ec4\u200b<code>B</code>\uff0c\u200b\u5373\u5c06\u200b<code>top</code>\u200b\u7f6e\u200b<code>0</code>\u3002</li> <li><code>remove(k)</code>\uff1a\u200b\u5148\u200b\u5c06\u200b<code>A-B</code>\u200b\u6821\u9a8c\u200b\u73af\u4e2d\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u5bf9\u200b\uff08\u200b\u5bf9\u5e94\u200b<code>B[top - 1]</code>\uff09\u200b\u590d\u5236\u5230\u200b<code>B[A[k]]</code>\u200b\u7684\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u518d\u200b<code>top--</code>\u200b\u5220\u9664\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u3002</li> </ul> <p>\u201c\u200b\u6821\u9a8c\u200b\u73af\u200b\u201d\u200b\u4f4d\u56fe\u200b\u7684\u200b\u5b9e\u73b0\u200b\uff1a</p> \u200b\u4f4d\u200b\u56fe\u200b <pre><code>void set(int k)\n{\n   A[k] = top;\n   B[top++] = A[k];\n}\nint get(int k)\n{\n   return B[A[k]] == k &amp;&amp; A[k] &lt; top;\n}\nvoid remove(int k)\n{\n   A[B[--top]] = k;\n   B[k] = B[top];\n}\nvoid clear(int k)\n{\n   top = 0;\n}\n</code></pre>"},{"location":"coding/dsa/vector-and-list/#_19","title":"\u5217\u8868","text":"<p>\u200b\u5411\u91cf\u200b\u662f\u200b\u9759\u6001\u200b\u7684\u200b\u6570\u636e\u7ed3\u6784\u200b\uff0c\u200b\u4e00\u6b21\u200b\u53ea\u80fd\u200b\u5206\u914d\u200b\u56fa\u5b9a\u200b\u957f\u5ea6\u200b\u7684\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u5411\u91cf\u200b\u7684\u200b\u6269\u589e\u200b\u9700\u8981\u200b\u5206\u914d\u200b\u4e00\u5757\u200b\u65b0\u200b\u7684\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u7136\u540e\u200b\u5c06\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u590d\u5236\u5230\u200b\u65b0\u200b\u7684\u200b\u5411\u91cf\u200b\u4e2d\u200b\u3002\u200b\u800c\u200b\u5217\u8868\u200b\u4e0e\u200b\u4e4b\u200b\u4e0d\u540c\u200b\uff0c\u200b\u5217\u8868\u200b\u7684\u200b\u5143\u7d20\u200b\u53ef\u4ee5\u200b\u52a8\u6001\u5206\u914d\u200b\uff0c\u200b\u5217\u8868\u200b\u7684\u200b\u6269\u589e\u200b\u53ea\u200b\u9700\u8981\u200b\u5206\u914d\u200b\u6269\u589e\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u7a7a\u95f4\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u5217\u8868\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u6392\u5217\u200b\u4e0e\u200b\u5217\u8868\u200b\u5143\u7d20\u200b\u5728\u200b\u5185\u5b58\u200b\u4e2d\u200b\u7684\u200b\u6392\u5217\u200b\u987a\u5e8f\u200b\u6ca1\u6709\u200b\u5fc5\u7136\u200b\u7684\u200b\u5173\u7cfb\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u5143\u7d20\u200b\u5728\u200b\u903b\u8f91\u200b\u4e0a\u200b\u5f62\u6210\u200b\u4e00\u4e2a\u200b\u5e8f\u5217\u200b\u3002\u200b\u5217\u8868\u200b\u7684\u200b\u5143\u7d20\u200b\u53c8\u79f0\u200b\u8282\u70b9\u200b\uff0c\u200b\u76f8\u90bb\u200b\u7684\u200b\u8282\u70b9\u200b\u4e92\u79f0\u200b\u524d\u9a71\u200b\u4e0e\u200b\u540e\u7ee7\u200b\u3002\u200b\u6ca1\u6709\u200b\u524d\u9a71\u200b\u7684\u200b\u8282\u70b9\u200b\u79f0\u4e3a\u200b\u9996\u200b\u8282\u70b9\u200b\uff0c\u200b\u6ca1\u6709\u200b\u540e\u7ee7\u200b\u7684\u200b\u8282\u70b9\u200b\u79f0\u4e3a\u200b\u5c3e\u200b\u8282\u70b9\u200b\u3002\u200b\u8282\u70b9\u200b\u9700\u8981\u200b\u989d\u5916\u200b\u7684\u200b\u7a7a\u95f4\u200b\u8bb0\u5f55\u200b\u76f8\u90bb\u200b\u8282\u70b9\u200b\u7684\u200b\u5730\u5740\u200b\u4ee5\u200b\u6784\u5efa\u200b\u903b\u8f91\u200b\u4e0a\u200b\u7684\u200b\u987a\u5e8f\u200b\u5173\u7cfb\u200b\u3002</p> <p>\u200b\u8fd9\u79cd\u200b\u65b9\u6cd5\u200b\u7684\u200b\u7f3a\u70b9\u200b\u5728\u4e8e\u200b\u5217\u8868\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u4e0d\u80fd\u200b\u76f4\u63a5\u200b\u6309\u7167\u200b\u4e0b\u6807\u200b\u8fdb\u884c\u200b\u8bbf\u95ee\u200b\uff0c\u200b\u53ea\u80fd\u200b\u4ece\u200b\u5217\u8868\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u5f00\u59cb\u200b\uff0c\u200b\u4f9d\u5e8f\u200b\u8bbf\u95ee\u200b\u8282\u70b9\u200b\u76f4\u5230\u200b\u8bbf\u95ee\u200b\u5230\u200b\u5bf9\u5e94\u200b\u4e0b\u200b\u6807\u7684\u200b\u8282\u70b9\u200b\uff0c\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(n)\\)</p>"},{"location":"coding/dsa/vector-and-list/#_20","title":"\u8282\u70b9\u200b\u7c7b\u578b","text":"<p>\u200b\u4e3a\u200b\u5b9e\u73b0\u200b\u5217\u8868\u200b\u7684\u200b\u5404\u9879\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u8282\u70b9\u200b\u7c7b\u578b\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b\u4ee5\u4e0b\u200b\u63a5\u53e3\u200b\uff1a</p> <ul> <li>\u200b\u83b7\u53d6\u200b\u4e0e\u200b\u8bbe\u7f6e\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u7684\u200b\u524d\u9a71\u200b</li> <li>\u200b\u83b7\u53d6\u200b\u4e0e\u200b\u8bbe\u7f6e\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u7684\u200b\u540e\u7ee7\u200b</li> <li>\u200b\u83b7\u53d6\u200b\u4e0e\u200b\u8bbe\u7f6e\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u7684\u200b\u53d6\u503c\u200b</li> <li>\u200b\u63d2\u5165\u200b\u524d\u9a71\u200b\u8282\u70b9\u200b</li> <li>\u200b\u63d2\u5165\u200b\u540e\u7ee7\u200b\u8282\u70b9\u200b</li> </ul> <p>\u200b\u6ce8\u610f\u200b\uff1a \u200b\u53cc\u5411\u200b\u94fe\u8868\u200b\u5728\u200b\u5b9e\u73b0\u200b\u65f6\u6709\u200b\u6f5c\u5728\u200b\u7684\u200b\u6570\u636e\u200b\u4e0d\u200b\u4e00\u81f4\u200b\u7684\u200b\u98ce\u9669\u200b\u3002\uff08\u200b\u524d\u9a71\u200b\u8282\u70b9\u200b\u7684\u200b\u540e\u7ee7\u200b\u4e0d\u200b\u7b49\u4e8e\u200b\u540e\u7ee7\u200b\u8282\u70b9\u200b\u7684\u200b\u524d\u9a71\u200b\uff09</p> <p>\u200b\u5982\u4e0b\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u7c7b\u200b\u7684\u200b\u793a\u4f8b\u200b\u63a5\u53e3\u200b\uff0c\u200b\u6ca1\u6709\u200b\u5305\u542b\u200b\u63a5\u53e3\u200b\u7684\u200b\u5177\u4f53\u200b\u5b9e\u73b0\u200b\u3002</p> \u200b\u5217\u8868\u200b\u8282\u70b9\u200b\u7c7b\u200b <p><pre><code>/******************************************************************************************\n * Data Structures in C++\n * ISBN: 7-302-33064-6 &amp; 7-302-33065-3 &amp; 7-302-29652-2 &amp; 7-302-26883-3\n * Junhui DENG, deng@tsinghua.edu.cn\n * Computer Science &amp; Technology, Tsinghua University\n * Copyright (c) 2003-2020. All rights reserved.\n ******************************************************************************************/\n\ntypedef int Rank; //\u200b\u79e9\u200b\n#define ListNodePosi(T) ListNode&lt;T&gt;* //\u200b\u5217\u8868\u200b\u8282\u70b9\u200b\u4f4d\u7f6e\u200b\n\ntemplate &lt;typename T&gt; struct ListNode { //\u200b\u5217\u8868\u200b\u8282\u70b9\u200b\u6a21\u677f\u200b\u7c7b\u200b\uff08\u200b\u4ee5\u200b\u53cc\u5411\u200b\u94fe\u8868\u200b\u5f62\u5f0f\u200b\u5b9e\u73b0\u200b\uff09\n// \u200b\u6210\u5458\u200b\n   T data; ListNodePosi(T) pred; ListNodePosi(T) succ; //\u200b\u6570\u503c\u200b\u3001\u200b\u524d\u9a71\u200b\u3001\u200b\u540e\u7ee7\u200b\n// \u200b\u6784\u9020\u51fd\u6570\u200b\n   ListNode() {} //\u200b\u9488\u5bf9\u200bheader\u200b\u548c\u200btrailer\u200b\u7684\u200b\u6784\u9020\u200b\n   ListNode ( T e, ListNodePosi(T) p = NULL, ListNodePosi(T) s = NULL )\n      : data ( e ), pred ( p ), succ ( s ) {} //\u200b\u9ed8\u8ba4\u200b\u6784\u9020\u200b\u5668\u200b\n// \u200b\u64cd\u4f5c\u200b\u63a5\u53e3\u200b\n   ListNodePosi(T) insertAsPred ( T const&amp; e ); //\u200b\u7d27\u9760\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u4e4b\u524d\u200b\u63d2\u5165\u200b\u65b0\u200b\u8282\u70b9\u200b\n   ListNodePosi(T) insertAsSucc ( T const&amp; e ); //\u200b\u7d27\u968f\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u4e4b\u540e\u200b\u63d2\u5165\u200b\u65b0\u200b\u8282\u70b9\u200b\n};\n</code></pre> Reference</p>"},{"location":"coding/dsa/vector-and-list/#_21","title":"\u63d2\u5165\u200b\u4e0e\u200b\u5220\u9664","text":"<p>\u200b\u5411\u200b\u5217\u8868\u200b\u4e2d\u200b\u63d2\u5165\u200b\u6570\u636e\u200b\u9700\u8981\u200b\u5982\u4e0b\u200b\u8fc7\u7a0b\u200b\uff1a</p> <ul> <li>\u200b\u5b9a\u4f4d\u200b\u5230\u200b\u8282\u70b9\u200b</li> <li>\u200b\u5206\u914d\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u8282\u70b9\u200b</li> <li>\u200b\u66f4\u65b0\u200b\u8282\u70b9\u200b\u4e4b\u95f4\u200b\u7684\u200b\u6307\u9488\u200b\uff0c\u200b\u5c06\u200b\u65b0\u200b\u8282\u70b9\u200b\u4e0e\u200b\u5176\u4ed6\u200b\u8282\u70b9\u200b\u76f8\u8fde\u63a5\u200b</li> </ul> <p>\u200b\u5220\u9664\u200b\u8fc7\u7a0b\u200b\u662f\u200b\u63d2\u5165\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u53cd\u5411\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u66f4\u65b0\u200b\u8282\u70b9\u200b\u4e4b\u95f4\u200b\u6307\u9488\u200b\uff0c\u200b\u5c06\u200b\u5220\u9664\u200b\u8282\u70b9\u200b\u6392\u9664\u200b\u5728\u5916\u200b\u5373\u53ef\u200b\u3002</p>"},{"location":"coding/dsa/vector-and-list/#_22","title":"\u67e5\u627e\u200b\u4e0e\u200b\u53bb\u200b\u91cd","text":"<p>\u200b\u65e0\u5e8f\u200b\u5217\u8868\u200b\u7684\u200b\u67e5\u627e\u200b\u4e0e\u200b\u53bb\u200b\u91cd\u200b\u64cd\u4f5c\u200b\u4e0e\u200b\u65e0\u5e8f\u200b\u5411\u91cf\u200b\u7684\u200b\u67e5\u627e\u200b\u4e0e\u200b\u53bb\u200b\u91cd\u200b\u64cd\u4f5c\u200b\u7684\u200b\u57fa\u672c\u601d\u8def\u200b\u76f8\u540c\u200b\uff0c\u200b\u6b64\u5904\u200b\u4e0d\u518d\u200b\u8d58\u8ff0\u200b\u3002</p> <p>\u200b\u6709\u200b\u5e8f\u5217\u8868\u200b\u7684\u200b\u67e5\u627e\u200b\u4e0d\u80fd\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u800c\u200b\u53ea\u80fd\u200b\u987a\u5e8f\u200b\u67e5\u627e\u200b\u3002</p> <p>\u200b\u6709\u200b\u5e8f\u5217\u8868\u200b\u7684\u200b\u53bb\u200b\u91cd\u200b\uff1a\u200b\u53cd\u590d\u200b\u8003\u5bdf\u200b\u76f8\u90bb\u200b\u7684\u200b\u8282\u70b9\u200b\uff0c\u200b\u82e5\u200b\u8282\u70b9\u200b\u76f8\u540c\u200b\uff0c\u200b\u5219\u200b\u5220\u9664\u200b\u540e\u8fb9\u200b\u7684\u200b\u8282\u70b9\u200b\uff0c\u200b\u7ee7\u7eed\u200b\u8003\u5bdf\u200b\uff0c\u200b\u5426\u5219\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u5411\u200b\u540e\u200b\u79fb\u52a8\u200b\uff0c\u200b\u91cd\u590d\u200b\u8003\u5bdf\u200b\u8fc7\u7a0b\u200b\u3002\u200b\u76f4\u5230\u200b\u5217\u8868\u200b\u5230\u8fbe\u200b\u7ed3\u5c3e\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u7ed3\u675f\u200b\u3002</p>"},{"location":"coding/leetcode/","title":"Leetcode \u200b\u9898\u76ee","text":"<p>\u200b\u6309\u200b\u6807\u7b7e\u200b\u5206\u7c7b\u200b\u7684\u200b\u9898\u76ee\u200b\u5217\u8868\u200b\u53ef\u4ee5\u200b\u53c2\u89c1\u200b\u8fd9\u91cc\u200b</p> \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 1. \u200b\u4e24\u6570\u200b\u4e4b\u200b\u548c\u200b 2. \u200b\u4e24\u200b\u6570\u200b\u76f8\u52a0\u200b 3. \u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b 7. \u200b\u6574\u6570\u200b\u53cd\u8f6c\u200b 8. \u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u200b\u6574\u6570\u200b (atoi) 9. \u200b\u56de\u6587\u200b\u6570\u200b 11. \u200b\u76db\u200b\u6700\u591a\u6c34\u200b\u7684\u200b\u5bb9\u5668\u200b 13. \u200b\u7f57\u9a6c\u6570\u5b57\u200b\u8f6c\u200b\u6574\u6570\u200b 14. \u200b\u6700\u957f\u200b\u516c\u5171\u200b\u524d\u7f00\u200b 17. \u200b\u7535\u8bdd\u53f7\u7801\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408\u200b 19. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u7684\u200b\u5012\u6570\u7b2c\u200b N \u200b\u4e2a\u200b\u7ed3\u70b9\u200b 20. \u200b\u6709\u6548\u200b\u7684\u200b\u62ec\u53f7\u200b 21. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u94fe\u8868\u200b 22. \u200b\u62ec\u53f7\u200b\u751f\u6210\u200b 23. \u200b\u5408\u5e76\u200bK\u200b\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868\u200b 24. \u200b\u4e24\u200b\u4e24\u200b\u4ea4\u6362\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9\u200b 25. K \u200b\u4e2a\u200b\u4e00\u7ec4\u200b\u7ffb\u8f6c\u200b\u94fe\u8868\u200b 26. \u200b\u5220\u9664\u200b\u6709\u5e8f\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u91cd\u590d\u200b\u9879\u200b 27. \u200b\u79fb\u9664\u200b\u5143\u7d20\u200b 28. \u200b\u5b9e\u73b0\u200b strStr() 35. \u200b\u641c\u7d22\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b 36. \u200b\u6709\u6548\u200b\u7684\u200b\u6570\u72ec\u200b 37. \u200b\u89e3\u6570\u200b\u72ec\u200b 38. \u200b\u5916\u89c2\u200b\u6570\u5217\u200b 42. \u200b\u63a5\u200b\u96e8\u6c34\u200b 43. \u200b\u5b57\u7b26\u4e32\u200b\u76f8\u4e58\u200b 46. \u200b\u5168\u200b\u6392\u5217\u200b 48. \u200b\u65cb\u8f6c\u200b\u56fe\u50cf\u200b 50. Pow(x, n) 53. \u200b\u6700\u5927\u200b\u5b50\u5e8f\u200b\u548c\u200b 55. \u200b\u8df3\u8dc3\u200b\u6e38\u620f\u200b 56. \u200b\u5408\u5e76\u200b\u533a\u95f4\u200b 59. \u200b\u87ba\u65cb\u200b\u77e9\u9635\u200b II 61. \u200b\u65cb\u8f6c\u200b\u94fe\u8868\u200b 62. \u200b\u4e0d\u540c\u200b\u8def\u5f84\u200b 64. \u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b 66. \u200b\u52a0\u4e00\u200b 69. x \u200b\u7684\u200b\u5e73\u65b9\u6839\u200b 70. \u200b\u722c\u697c\u68af\u200b 73. \u200b\u77e9\u9635\u200b\u7f6e\u200b\u96f6\u200b 74. \u200b\u641c\u7d22\u200b\u4e8c\u7ef4\u200b\u77e9\u9635\u200b 75. \u200b\u989c\u8272\u200b\u5206\u7c7b\u200b 78. \u200b\u5b50\u96c6\u200b 79. \u200b\u5355\u8bcd\u200b\u641c\u7d22\u200b 87. \u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b 88. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u6570\u7ec4\u200b 90. \u200b\u5b50\u96c6\u200b II 92. \u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b II 98. \u200b\u9a8c\u8bc1\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b 101. \u200b\u5bf9\u79f0\u200b\u4e8c\u53c9\u6811\u200b 102. \u200b\u4e8c\u53c9\u6811\u200b\u7684\u200b\u5c42\u5e8f\u200b\u904d\u5386\u200b 115. \u200b\u4e0d\u540c\u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217\u200b 118. \u200b\u6768\u8f89\u4e09\u89d2\u200b 119. \u200b\u6768\u8f89\u4e09\u89d2\u200b II 120. \u200b\u4e09\u89d2\u5f62\u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b 121. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b 122. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b II 125. \u200b\u9a8c\u8bc1\u200b\u56de\u6587\u200b\u4e32\u200b 136. \u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u7684\u200b\u6570\u5b57\u200b 150. \u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u6c42\u503c\u200b 155. \u200b\u6700\u5c0f\u200b\u6808\u200b 189. \u200b\u65cb\u8f6c\u200b\u6570\u7ec4\u200b 198. \u200b\u6253\u5bb6\u52ab\u820d\u200b 204. \u200b\u8ba1\u6570\u200b\u8d28\u6570\u200b 206. \u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b 217. \u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b 234. \u200b\u56de\u6587\u200b\u94fe\u8868\u200b 237. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9\u200b 242. \u200b\u6709\u6548\u200b\u7684\u200b\u5b57\u6bcd\u200b\u5f02\u4f4d\u200b\u8bcd\u200b 315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570\u200b 341. \u200b\u6241\u5e73\u5316\u200b\u5d4c\u5957\u200b\u5217\u8868\u200b\u8fed\u4ee3\u200b\u5668\u200b 344. \u200b\u53cd\u8f6c\u200b\u5b57\u7b26\u4e32\u200b 350. \u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u7684\u200b\u4ea4\u96c6\u200b II 368. \u200b\u6700\u5927\u200b\u6574\u9664\u200b\u5b50\u96c6\u200b 377. \u200b\u7ec4\u5408\u200b\u603b\u548c\u200b \u2163 384. \u200b\u6253\u4e71\u200b\u6570\u7ec4\u200b 387. \u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u552f\u4e00\u200b\u5b57\u7b26\u200b 403. \u200b\u9752\u86d9\u200b\u8fc7\u6cb3\u200b 433. \u200b\u6700\u5c0f\u200b\u57fa\u56e0\u200b\u53d8\u5316\u200b 488. \u200b\u7956\u739b\u200b\u6e38\u620f\u200b 509. \u200b\u6590\u6ce2\u200b\u90a3\u200b\u5951\u6570\u200b 779. \u200b\u7b2c\u200bK\u200b\u4e2a\u200b\u8bed\u6cd5\u200b\u7b26\u53f7\u200b 781. \u200b\u68ee\u6797\u200b\u4e2d\u200b\u7684\u200b\u5154\u5b50\u200b 917. \u200b\u4ec5\u4ec5\u200b\u53cd\u8f6c\u200b\u5b57\u6bcd\u200b 1006. \u200b\u7b28\u200b\u9636\u4e58\u200b 1011. \u200b\u5728\u200b D \u200b\u5929\u200b\u5185\u200b\u9001\u8fbe\u200b\u5305\u88f9\u200b\u7684\u200b\u80fd\u529b\u200b 1283. \u200b\u4f7f\u200b\u7ed3\u679c\u200b\u4e0d\u200b\u8d85\u8fc7\u200b\u9608\u503c\u200b\u7684\u200b\u6700\u5c0f\u200b\u9664\u6570\u200b 1300. \u200b\u8f6c\u53d8\u200b\u6570\u7ec4\u200b\u540e\u200b\u6700\u200b\u63a5\u8fd1\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u6570\u7ec4\u200b\u548c\u200b 1603. \u200b\u8bbe\u8ba1\u200b\u505c\u8f66\u200b\u7cfb\u7edf\u200b 1835. \u200b\u6240\u6709\u200b\u6570\u5bf9\u200b\u6309\u4f4d\u200b\u4e0e\u200b\u7ed3\u679c\u200b\u7684\u200b\u5f02\u6216\u200b\u548c"},{"location":"coding/leetcode/1/","title":"1. \u200b\u4e24\u6570\u200b\u4e4b\u200b\u548c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/1/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b<code>nums</code>\u200b\u548c\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u503c\u200b<code>target</code>\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u5728\u200b\u8be5\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u627e\u51fa\u200b\u548c\u200b\u4e3a\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u90a3\u200b\u00a0\u200b\u4e24\u4e2a\u200b\u00a0\u200b\u6574\u6570\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b\u4ed6\u4eec\u200b\u7684\u200b\u6570\u7ec4\u200b\u4e0b\u6807\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u6bcf\u79cd\u200b\u8f93\u5165\u200b\u53ea\u4f1a\u200b\u5bf9\u5e94\u200b\u4e00\u4e2a\u200b\u7b54\u6848\u200b\u3002\u200b\u4f46\u662f\u200b\uff0c\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u540c\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u4e0d\u80fd\u200b\u4f7f\u7528\u200b\u4e24\u904d\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b:</p> <pre><code>\u200b\u7ed9\u5b9a\u200b nums = [2, 7, 11, 15], target = 9\n\n\u200b\u56e0\u4e3a\u200b nums[0] + nums[1] = 2 + 7 = 9\n\u200b\u6240\u4ee5\u200b\u8fd4\u56de\u200b [0, 1]\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/1/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a </p> <p>\u200b\u5c06\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u5b58\u50a8\u200b\u5728\u200b\u54c8\u5e0c\u200b\u8868\u4e2d\u200b\uff0c\u200b\u6bcf\u6b21\u200b\u5728\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u67e5\u627e\u200b\u76ee\u6807\u503c\u200b\u4e0e\u200b\u5f53\u524d\u200b\u503c\u4e4b\u5dee\u200b\u662f\u5426\u200b\u5728\u200b\u54c8\u5e0c\u200b\u8868\u4e2d\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\n#define MAX(a, b) (a &gt; b ? a : b)\n#define MIN(a, b) (a &lt; b ? a : b)\nstruct cell\n{\n    int value;\n    int flag;\n};\nint* twoSum(int* nums, int numsSize, int target, int* returnSize){\n    *returnSize = 2;\n    int i = 0, mapSize = numsSize * 2, cur, *ret = malloc(sizeof(int) * 2), targetval, j = 0;\n    struct cell *hashMap = (struct cell *)memset(malloc(sizeof(struct cell) * mapSize), 0, sizeof(struct cell) * mapSize);\n    for (i = 0; i &lt; numsSize; i++)\n    {\n        cur = (nums[i] &gt; 0 ? nums[i] : -nums[i]) % mapSize;\n        while(hashMap[cur].flag != 0 &amp;&amp; hashMap[cur].value != nums[i])\n        {\n            cur++;\n            cur = cur == mapSize ? 0 : cur;\n        }\n        if (hashMap[cur].flag == 0)\n            hashMap[cur].value = nums[i];\n        hashMap[cur].flag++;\n    }\n    for (i = 0; i &lt; numsSize; i++)\n    {\n        targetval = target - nums[i];\n        cur = (targetval &gt; 0 ? targetval : -targetval) % mapSize;\n        while(hashMap[cur].flag != 0 &amp;&amp; hashMap[cur].value != targetval)\n        {\n            cur++;\n            cur = cur == mapSize ? 0 : cur;\n        }\n        if (hashMap[cur].value == targetval)\n        {\n            for (j = 0; j &lt; numsSize; j++)\n                if (nums[j] == targetval &amp;&amp; j != i)\n                    break;\n            if (j != numsSize)\n                break;\n        }\n    }\n    ret[0] = MIN(i, j);\n    ret[1] = MAX(i, j);\n    if (nums[i] + nums[j] == target)\n        return ret;\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/1006/","title":"1006. \u200b\u7b28\u200b\u9636\u4e58","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/1006/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u901a\u5e38\u200b\uff0c\u200b\u6b63\u6574\u6570\u200b <code>n</code> \u200b\u7684\u200b\u9636\u4e58\u200b\u662f\u200b\u6240\u6709\u200b\u5c0f\u4e8e\u200b\u6216\u200b\u7b49\u4e8e\u200b <code>n</code> \u200b\u7684\u200b\u6b63\u6574\u6570\u200b\u7684\u200b\u4e58\u79ef\u200b\u3002\u200b\u4f8b\u5982\u200b\uff0c<code>factorial(10) = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1</code>\u3002</p> <p>\u200b\u76f8\u53cd\u200b\uff0c\u200b\u6211\u4eec\u200b\u8bbe\u8ba1\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u7b28\u200b\u9636\u4e58\u200b <code>clumsy</code>\uff1a\u200b\u5728\u200b\u6574\u6570\u200b\u7684\u200b\u9012\u51cf\u200b\u5e8f\u5217\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u4ee5\u200b\u4e00\u4e2a\u200b\u56fa\u5b9a\u200b\u987a\u5e8f\u200b\u7684\u200b\u64cd\u4f5c\u7b26\u200b\u5e8f\u5217\u200b\u6765\u200b\u4f9d\u6b21\u200b\u66ff\u6362\u200b\u539f\u6709\u200b\u7684\u200b\u4e58\u6cd5\u200b\u64cd\u4f5c\u7b26\u200b\uff1a\u200b\u4e58\u6cd5\u200b(*)\uff0c\u200b\u9664\u6cd5\u200b(/)\uff0c\u200b\u52a0\u6cd5\u200b(+)\u200b\u548c\u200b\u51cf\u6cd5\u200b(-)\u3002</p> <p>\u200b\u4f8b\u5982\u200b\uff0c<code>clumsy(10) = 10 * 9 / 8 + 7 - 6 * 5 / 4 + 3 - 2 * 1</code>\u3002\u200b\u7136\u800c\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u8fd0\u7b97\u200b\u4ecd\u7136\u200b\u4f7f\u7528\u200b\u901a\u5e38\u200b\u7684\u200b\u7b97\u672f\u200b\u8fd0\u7b97\u200b\u987a\u5e8f\u200b\uff1a\u200b\u6211\u4eec\u200b\u5728\u200b\u4efb\u4f55\u200b\u52a0\u200b\u3001\u200b\u51cf\u200b\u6b65\u9aa4\u200b\u4e4b\u524d\u200b\u6267\u884c\u200b\u6240\u6709\u200b\u7684\u200b\u4e58\u6cd5\u200b\u548c\u200b\u9664\u6cd5\u200b\u6b65\u9aa4\u200b\uff0c\u200b\u5e76\u4e14\u200b\u6309\u200b\u4ece\u5de6\u5230\u53f3\u200b\u5904\u7406\u200b\u4e58\u6cd5\u200b\u548c\u200b\u9664\u6cd5\u200b\u6b65\u9aa4\u200b\u3002</p> <p>\u200b\u53e6\u5916\u200b\uff0c\u200b\u6211\u4eec\u200b\u4f7f\u7528\u200b\u7684\u200b\u9664\u6cd5\u200b\u662f\u200b\u5730\u677f\u200b\u9664\u6cd5\u200b\uff08floor division\uff09\uff0c\u200b\u6240\u4ee5\u200b\u00a0<code>10 * 9 / 8</code>\u00a0\u200b\u7b49\u4e8e\u200b\u00a0<code>11</code>\u3002\u200b\u8fd9\u200b\u4fdd\u8bc1\u200b\u7ed3\u679c\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u5b9e\u73b0\u200b\u4e0a\u9762\u200b\u5b9a\u4e49\u200b\u7684\u200b\u7b28\u200b\u51fd\u6570\u200b\uff1a\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b <code>N</code>\uff0c\u200b\u5b83\u200b\u8fd4\u56de\u200b <code>N</code> \u200b\u7684\u200b\u7b28\u200b\u9636\u4e58\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a4\n\u200b\u8f93\u51fa\u200b\uff1a7\n\u200b\u89e3\u91ca\u200b\uff1a7 = 4 * 3 / 2 + 1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a10\n\u200b\u8f93\u51fa\u200b\uff1a12\n\u200b\u89e3\u91ca\u200b\uff1a12 = 10 * 9 / 8 + 7 - 6 * 5 / 4 + 3 - 2 * 1\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= N &lt;= 10000</code></li> <li><code>-2^31 &lt;= answer &lt;= 2^31 - 1</code>  \uff08\u200b\u7b54\u6848\u200b\u4fdd\u8bc1\u200b\u7b26\u5408\u200b <code>32</code> \u200b\u4f4d\u200b\u6574\u6570\u200b\u3002\uff09</li> </ul> <p>Reference</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/1006/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5bf9\u200b\u8ba1\u7b97\u200b\u8fc7\u7a0b\u200b\u8fdb\u884c\u200b\u6a21\u62df\u200b\u5373\u53ef\u200b\uff1a</p> <pre><code>int clumsy(int N){\n    int ret = 0, i = N;\n    bool flag = true, minus = false;\n    for (i = N; i &gt; 0;)\n    {\n        if (flag)\n        {\n            if (i &gt;= 3)\n                ret += (minus ? -1 : 1) * i * (i - 1) / (i - 2);\n            else if (i == 2)\n                ret += (minus ? -1 : 1) * i * (i - 1);\n            else if (i == 1)\n                ret += (minus ? -1 : 1) * i;\n            i -= 3;\n        }\n        else\n            ret += i--;\n        flag = !flag;\n        minus = true;\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/101/","title":"101. \u200b\u5bf9\u79f0\u200b\u4e8c\u53c9\u6811","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6811","\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/101/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u4e8c\u53c9\u6811\u200b\uff0c\u200b\u68c0\u67e5\u200b\u5b83\u200b\u662f\u5426\u662f\u200b\u955c\u50cf\u200b\u5bf9\u79f0\u200b\u7684\u200b\u3002</p> <p>\u200b\u4f8b\u5982\u200b\uff0c\u200b\u4e8c\u53c9\u6811\u200b<code>[1,2,2,3,4,4,3]</code>\u200b\u662f\u200b\u5bf9\u79f0\u200b\u7684\u200b\u3002</p> <pre><code>    1\n   / \\\n  2   2\n / \\ / \\\n3  4 4  3\n</code></pre> <p>\u200b\u4f46\u662f\u200b\u4e0b\u9762\u200b\u8fd9\u4e2a\u200b<code>[1,2,2,null,3,null,3]</code>\u200b\u5219\u200b\u4e0d\u662f\u200b\u955c\u50cf\u200b\u5bf9\u79f0\u200b\u7684\u200b:</p> <pre><code>    1\n   / \\\n  2   2\n   \\   \\\n   3    3\n</code></pre> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u8fd0\u7528\u200b\u9012\u5f52\u200b\u548c\u200b\u8fed\u4ee3\u200b\u4e24\u79cd\u200b\u65b9\u6cd5\u200b\u89e3\u51b3\u200b\u8fd9\u4e2a\u200b\u95ee\u9898\u200b\u5417\u200b\uff1f</p> <p>Reference</p>","tags":["\u6811","\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/101/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u9012\u5f52\u200b\u5224\u65ad\u200b\u4e24\u4e2a\u200b\u6839\u200b\u8282\u70b9\u200b\u662f\u5426\u200b\u6210\u200b\u955c\u50cf\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for a binary tree node.\n * struct TreeNode {\n *     int val;\n *     struct TreeNode *left;\n *     struct TreeNode *right;\n * };\n */\nbool isMirrored(struct TreeNode *root1, struct TreeNode *root2)\n{\n    if (!root1 || !root2)\n        return root1 == root2;\n    if (root1-&gt;val != root2-&gt;val)\n        return false;\n    return isMirrored(root1-&gt;left, root2-&gt;right) &amp;&amp; isMirrored(root2-&gt;left, root1-&gt;right);\n}\n\nbool isSymmetric(struct TreeNode* root){\n    if (root == NULL)\n        return true;\n    return isMirrored(root-&gt;left, root-&gt;right);\n}\n</code></pre>","tags":["\u6811","\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/1011/","title":"1011. \u200b\u5728\u200b D \u200b\u5929\u200b\u5185\u200b\u9001\u8fbe\u200b\u5305\u88f9\u200b\u7684\u200b\u80fd\u529b","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/1011/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u4f20\u9001\u5e26\u200b\u4e0a\u200b\u7684\u200b\u5305\u88f9\u200b\u5fc5\u987b\u200b\u5728\u200b <code>D</code> \u200b\u5929\u5185\u200b\u4ece\u200b\u4e00\u4e2a\u200b\u6e2f\u53e3\u200b\u8fd0\u9001\u200b\u5230\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u6e2f\u53e3\u200b\u3002</p> <p>\u200b\u4f20\u9001\u5e26\u200b\u4e0a\u200b\u7684\u200b\u7b2c\u200b <code>i</code>\u00a0\u200b\u4e2a\u200b\u5305\u88f9\u200b\u7684\u200b\u91cd\u91cf\u200b\u4e3a\u200b\u00a0<code>weights[i]</code>\u3002\u200b\u6bcf\u200b\u4e00\u5929\u200b\uff0c\u200b\u6211\u4eec\u200b\u90fd\u200b\u4f1a\u200b\u6309\u200b\u7ed9\u51fa\u200b\u91cd\u91cf\u200b\u7684\u200b\u987a\u5e8f\u200b\u5f80\u200b\u4f20\u9001\u5e26\u200b\u4e0a\u200b\u88c5\u8f7d\u200b\u5305\u88f9\u200b\u3002\u200b\u6211\u4eec\u200b\u88c5\u8f7d\u200b\u7684\u200b\u91cd\u91cf\u200b\u4e0d\u4f1a\u200b\u8d85\u8fc7\u200b\u8239\u200b\u7684\u200b\u6700\u5927\u200b\u8fd0\u8f7d\u200b\u91cd\u91cf\u200b\u3002</p> <p>\u200b\u8fd4\u56de\u200b\u80fd\u200b\u5728\u200b <code>D</code> \u200b\u5929\u5185\u200b\u5c06\u200b\u4f20\u9001\u5e26\u200b\u4e0a\u200b\u7684\u200b\u6240\u6709\u200b\u5305\u88f9\u200b\u9001\u8fbe\u200b\u7684\u200b\u8239\u200b\u7684\u200b\u6700\u4f4e\u200b\u8fd0\u8f7d\u200b\u80fd\u529b\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aweights = [1,2,3,4,5,6,7,8,9,10], D = 5\n\u200b\u8f93\u51fa\u200b\uff1a15\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u8239\u8236\u200b\u6700\u4f4e\u200b\u8f7d\u91cd\u200b 15 \u200b\u5c31\u200b\u80fd\u591f\u200b\u5728\u200b 5 \u200b\u5929\u200b\u5185\u200b\u9001\u8fbe\u200b\u6240\u6709\u200b\u5305\u88f9\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a\n\u200b\u7b2c\u200b 1 \u200b\u5929\u200b\uff1a1, 2, 3, 4, 5\n\u200b\u7b2c\u200b 2 \u200b\u5929\u200b\uff1a6, 7\n\u200b\u7b2c\u200b 3 \u200b\u5929\u200b\uff1a8\n\u200b\u7b2c\u200b 4 \u200b\u5929\u200b\uff1a9\n\u200b\u7b2c\u200b 5 \u200b\u5929\u200b\uff1a10\n\n\u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u8d27\u7269\u200b\u5fc5\u987b\u200b\u6309\u7167\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u987a\u5e8f\u200b\u88c5\u8fd0\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4f7f\u7528\u200b\u8f7d\u91cd\u200b\u80fd\u529b\u200b\u4e3a\u200b 14 \u200b\u7684\u200b\u8239\u8236\u200b\u5e76\u200b\u5c06\u200b\u5305\u88c5\u200b\u5206\u6210\u200b (2, 3, 4, 5), (1, 6, 7), (8), (9), (10) \u200b\u662f\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u7684\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aweights = [3,2,2,4,1,4], D = 3\n\u200b\u8f93\u51fa\u200b\uff1a6\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u8239\u8236\u200b\u6700\u4f4e\u200b\u8f7d\u91cd\u200b 6 \u200b\u5c31\u200b\u80fd\u591f\u200b\u5728\u200b 3 \u200b\u5929\u200b\u5185\u200b\u9001\u8fbe\u200b\u6240\u6709\u200b\u5305\u88f9\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a\n\u200b\u7b2c\u200b 1 \u200b\u5929\u200b\uff1a3, 2\n\u200b\u7b2c\u200b 2 \u200b\u5929\u200b\uff1a2, 4\n\u200b\u7b2c\u200b 3 \u200b\u5929\u200b\uff1a1, 4\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aweights = [1,2,3,1,1], D = 4\n\u200b\u8f93\u51fa\u200b\uff1a3\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u7b2c\u200b 1 \u200b\u5929\u200b\uff1a1\n\u200b\u7b2c\u200b 2 \u200b\u5929\u200b\uff1a2\n\u200b\u7b2c\u200b 3 \u200b\u5929\u200b\uff1a3\n\u200b\u7b2c\u200b 4 \u200b\u5929\u200b\uff1a1, 1\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= D &lt;= weights.length &lt;= 50000</code></li> <li><code>1 &lt;= weights[i] &lt;= 500</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/1011/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u8bbe\u200b\u5217\u8868\u200b\u4e3a\u200b\\(A=\\{a_i\\}_n\\)\uff0c\u200b\u5219\u200b\u6700\u7ec8\u200b\u7ed3\u679c\u200b\u7684\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\([\\max_{i} \\{a_i\\}, \\sum_{i} a_i]\\)\u3002\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u5728\u200b\u533a\u95f4\u200b\u4e2d\u200b\u67e5\u627e\u200b\u80fd\u591f\u200b\u6ee1\u8db3\u200b\u5728\u200b\\(D\\)\u200b\u5929\u200b\u5185\u200b\u8fd0\u8f93\u200b\u5168\u90e8\u200b\u5305\u88f9\u200b\u7684\u200b\u6700\u5c0f\u200b\u8f7d\u91cd\u200b\u3002</p> <p>\u200b\u6709\u5173\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7684\u200b\u8be6\u7ec6\u200b\u8ba8\u8bba\u200b\uff0c\u200b\u8bf7\u200b\u53c2\u89c1\u200b35. \u200b\u641c\u7d22\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b</p> <pre><code>#define MAX(x, y) ((x) &gt; (y) ? (x) : (y))\n\nbool verify(int *weights, int weightsSize, int D, int limit)\n{\n    int i = 0, ub = limit;\n    for (i = 0; i &lt; weightsSize; i++)\n    {\n        if (weights[i] &gt; ub)\n        {\n            D--;\n            ub = limit;\n        }\n        ub -= weights[i];\n        if (D &lt;= 0)\n            return false;\n    }\n    return true;\n}\n\nint bSearch(int *weights, int weightsSize, int D, int lo, int hi)\n{\n    int mid;\n    while (lo &lt; hi)\n    {\n        mid = (lo + hi) &gt;&gt; 1;\n        if (verify(weights, weightsSize, D, mid))\n            hi = mid;\n        else\n            lo = mid + 1;\n    }\n    return lo;\n}\n\nint shipWithinDays(int* weights, int weightsSize, int D) {\n    int i = 0, sum = 0, max = 0;\n    printf(\"%d\", verify(weights, weightsSize, D, 6));\n    for (i = 0; i &lt; weightsSize; i++)\n    {\n        sum += weights[i];\n        max = MAX(max, weights[i]);\n    }\n    return bSearch(weights, weightsSize, D, max, sum);\n}\n</code></pre>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/102/","title":"102. \u200b\u4e8c\u53c9\u6811\u200b\u7684\u200b\u5c42\u5e8f\u200b\u904d\u5386","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6811","\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/102/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u4e8c\u53c9\u6811\u200b\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u8fd4\u56de\u200b\u5176\u200b\u6309\u200b \u200b\u5c42\u5e8f\u200b\u904d\u5386\u200b \u200b\u5f97\u5230\u200b\u7684\u200b\u8282\u70b9\u200b\u503c\u200b\u3002 \uff08\u200b\u5373\u200b\u9010\u5c42\u200b\u5730\u200b\uff0c\u200b\u4ece\u5de6\u5230\u53f3\u200b\u8bbf\u95ee\u200b\u6240\u6709\u200b\u8282\u70b9\u200b\uff09\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <p>\u200b\u4e8c\u53c9\u6811\u200b\uff1a<code>[3,9,20,null,null,15,7]</code>,</p> <pre><code>    3\n   / \\\n  9  20\n    /  \\\n   15   7\n</code></pre> <p>\u200b\u8fd4\u56de\u200b\u5176\u200b\u5c42\u6b21\u200b\u904d\u5386\u200b\u7ed3\u679c\u200b\uff1a</p> <pre><code>[\n  [3],\n  [9,20],\n  [15,7]\n]\n</code></pre> <p>Reference</p>","tags":["\u6811","\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/102/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u7b2c\u4e00\u5c42\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u5373\u6839\u200b\u8282\u70b9\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u518d\u5c42\u5e8f\u200b\u904d\u5386\u200b\u5de6\u5b50\u200b\u6811\u200b\u548c\u200b\u53f3\u5b50\u200b\u6811\u200b\uff0c\u200b\u5408\u5e76\u200b\u540c\u5c42\u200b\u7684\u200b\u7ed3\u679c\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for a binary tree node.\n * struct TreeNode {\n *     int val;\n *     struct TreeNode *left;\n *     struct TreeNode *right;\n * };\n */\n/**\n * Return an array of arrays of size *returnSize.\n * The sizes of the arrays are returned as *returnColumnSizes array.\n * Note: Both returned array and *columnSizes array must be malloced, assume caller calls free().\n */\n#define MAX(x, y) (x &gt; y ? x : y)\nint *merge(int *leftBranch, int leftSize, int *rightBranch, int rightSize, int *returnSize)\n{\n    *returnSize = leftSize + rightSize;\n    int *ret = (int *)malloc(sizeof(int) * (leftSize + rightSize));\n    if (leftSize != 0)\n    {\n        memcpy(ret, leftBranch, sizeof(int) * leftSize);\n        free(leftBranch);\n    }\n    if (rightSize != 0)\n    {\n        memcpy(ret + leftSize, rightBranch, sizeof(int) * rightSize);\n        free(rightBranch);\n    }\n    return ret;\n}\nint** levelOrder(struct TreeNode* root, int* returnSize, int** returnColumnSizes){\n    if (root == NULL)\n    {\n        *returnSize = 0;\n        *returnColumnSizes = NULL;\n        return NULL;\n    }\n    int leftBranchSize = 0, *leftColumnSizes = NULL, **leftBranch = levelOrder(root-&gt;left, &amp;leftBranchSize, &amp;leftColumnSizes),\n        rightBranchSize = 0, *rightColumnSizes = NULL, **rightBranch = levelOrder(root-&gt;right, &amp;rightBranchSize, &amp;rightColumnSizes);\n    *returnSize = 1 + MAX(leftBranchSize, rightBranchSize);\n    int **ret = (int **)malloc(sizeof(int *) * *returnSize), *returnCols = (int *)malloc(sizeof(int) * *returnSize), i = 0;\n    *returnColumnSizes = returnCols;\n    *returnCols = 1;\n    *ret = malloc(sizeof(int));\n    **ret = root-&gt;val;\n    for (i = 1; i &lt; *returnSize; i++)\n    {\n        if (i &lt;= leftBranchSize &amp;&amp; i &lt;= rightBranchSize)\n            ret[i] = merge(leftBranch[i - 1], leftColumnSizes[i - 1], rightBranch[i - 1], rightColumnSizes[i - 1], returnCols + i);\n        else if (i &lt;= leftBranchSize)\n            ret[i] = merge(leftBranch[i-1], leftColumnSizes[i-1], NULL, 0, returnCols + i);\n        else if (i &lt;= rightBranchSize)\n            ret[i] = merge(NULL, 0, rightBranch[i-1], rightColumnSizes[i-1], returnCols + i);\n    }\n    free(leftBranch);\n    free(rightBranch);\n    free(leftColumnSizes);\n    free(rightColumnSizes);\n    return ret;\n}\n</code></pre>","tags":["\u6811","\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/11/","title":"11. \u200b\u76db\u200b\u6700\u591a\u6c34\u200b\u7684\u200b\u5bb9\u5668","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u5b66","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/11/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b <code>n</code> \u200b\u4e2a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b <code>a_1\uff0ca_2\uff0c...\uff0ca_n</code>\uff0c\u200b\u6bcf\u4e2a\u200b\u6570\u200b\u4ee3\u8868\u200b\u5750\u6807\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u70b9\u200b\u00a0<code>(i,\u00a0a_i)</code> \u3002\u200b\u5728\u200b\u5750\u6807\u200b\u5185\u200b\u753b\u200b <code>n</code> \u200b\u6761\u200b\u5782\u76f4\u7ebf\u200b\uff0c\u200b\u5782\u76f4\u7ebf\u200b <code>i</code>\u00a0\u200b\u7684\u200b\u4e24\u4e2a\u200b\u7aef\u70b9\u200b\u5206\u522b\u200b\u4e3a\u200b\u00a0<code>(i,\u00a0a_i)</code> \u200b\u548c\u200b <code>(i, 0)</code> \u3002\u200b\u627e\u51fa\u200b\u5176\u4e2d\u200b\u7684\u200b\u4e24\u6761\u7ebf\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u5b83\u4eec\u200b\u4e0e\u200b\u00a0<code>x</code>\u00a0\u200b\u8f74\u200b\u5171\u540c\u200b\u6784\u6210\u200b\u7684\u200b\u5bb9\u5668\u200b\u53ef\u4ee5\u200b\u5bb9\u7eb3\u200b\u6700\u591a\u200b\u7684\u200b\u6c34\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u503e\u659c\u200b\u5bb9\u5668\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a[1,8,6,2,5,4,8,3,7]\n\u200b\u8f93\u51fa\u200b\uff1a49 \n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u56fe\u4e2d\u200b\u5782\u76f4\u7ebf\u200b\u4ee3\u8868\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b [1,8,6,2,5,4,8,3,7]\u3002\u200b\u5728\u200b\u6b64\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u5bb9\u5668\u200b\u80fd\u591f\u200b\u5bb9\u7eb3\u200b\u6c34\u200b\uff08\u200b\u8868\u793a\u200b\u4e3a\u200b\u84dd\u8272\u200b\u90e8\u5206\u200b\uff09\u200b\u7684\u200b\u6700\u5927\u503c\u200b\u4e3a\u200b\u00a049\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aheight = [1,1]\n\u200b\u8f93\u51fa\u200b\uff1a1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aheight = [4,3,2,1,4]\n\u200b\u8f93\u51fa\u200b\uff1a16\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aheight = [1,2,1]\n\u200b\u8f93\u51fa\u200b\uff1a2\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>n = height.length</code></li> <li>2 &lt;= n &lt;= 3 * 104</li> <li>0 &lt;= height[i] &lt;= 3 * 104</li> </ul> <p>Reference</p>","tags":["\u6570\u5b66","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/11/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u53cc\u200b\u6307\u9488\u200b\u6cd5\u200b\uff0c\u200b\u4e24\u4e2a\u200b\u6307\u9488\u200b\u5206\u522b\u200b\u6307\u5411\u200b\u76db\u6c34\u200b\u533a\u57df\u200b\u7684\u200b\u8fb9\u754c\u200b\u3002\u200b\u6bcf\u6b21\u200b\u5faa\u73af\u200b\u5c06\u200b\u8f83\u200b\u4f4e\u200b\u7684\u200b\u8fb9\u754c\u200b\u5411\u200b\u4e2d\u95f4\u200b\u6536\u7f29\u200b\uff0c\u200b\u5e76\u200b\u5c1d\u8bd5\u200b\u66f4\u65b0\u200b\u6700\u5927\u200b\u7684\u200b\u76db\u6c34\u200b\u533a\u57df\u200b\u3002</p> <pre><code>#define MIN(x, y) (x &lt; y ? x : y)\n#define MAX(x, y) (x &gt; y ? x : y)\nint maxArea(int* height, int heightSize){\n    int left = 0, right = heightSize - 1,\n        ret = (right - left) * MIN(height[left], height[right]), cur;\n    while (left &lt; right)\n    {\n        if (height[left] &lt; height[right])\n            left++;\n        else\n            right--;\n        cur = (right - left) * MIN(height[left], height[right]);\n        ret = MAX(cur, ret);\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u5b66","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/115/","title":"115. \u200b\u4e0d\u540c\u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u5b57\u7b26\u4e32","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/115/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b <code>s</code> \u200b\u548c\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b <code>t</code> \uff0c\u200b\u8ba1\u7b97\u200b\u5728\u200b <code>s</code> \u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217\u200b\u4e2d\u200b <code>t</code> \u200b\u51fa\u73b0\u200b\u7684\u200b\u4e2a\u6570\u200b\u3002</p> <p>\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u4e00\u4e2a\u200b \u200b\u5b50\u200b\u5e8f\u5217\u200b \u200b\u662f\u200b\u6307\u200b\uff0c\u200b\u901a\u8fc7\u200b\u5220\u9664\u200b\u4e00\u4e9b\u200b\uff08\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u5220\u9664\u200b\uff09\u200b\u5b57\u7b26\u200b\u4e14\u200b\u4e0d\u200b\u5e72\u6270\u200b\u5269\u4f59\u200b\u5b57\u7b26\u200b\u76f8\u5bf9\u200b\u4f4d\u7f6e\u200b\u6240\u200b\u7ec4\u6210\u200b\u7684\u200b\u65b0\u200b\u5b57\u7b26\u4e32\u200b\u3002\uff08\u200b\u4f8b\u5982\u200b\uff0c<code>\"ACE\"</code>\u00a0\u200b\u662f\u200b\u00a0<code>\"ABCDE\"</code>\u00a0\u200b\u7684\u200b\u4e00\u4e2a\u200b\u5b50\u200b\u5e8f\u5217\u200b\uff0c\u200b\u800c\u200b\u00a0<code>\"AEC\"</code>\u00a0\u200b\u4e0d\u662f\u200b\uff09</p> <p>\u200b\u9898\u76ee\u200b\u6570\u636e\u200b\u4fdd\u8bc1\u200b\u7b54\u6848\u200b\u7b26\u5408\u200b 32 \u200b\u4f4d\u200b\u5e26\u7b26\u53f7\u200b\u6574\u6570\u200b\u8303\u56f4\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"rabbbit\", t = \"rabbit\"\n\u200b\u8f93\u51fa\u200b\uff1a3\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u5982\u4e0b\u200b\u56fe\u200b\u6240\u793a\u200b, \u200b\u6709\u200b 3 \u200b\u79cd\u200b\u53ef\u4ee5\u200b\u4ece\u200b s \u200b\u4e2d\u200b\u5f97\u5230\u200b \"rabbit\" \u200b\u7684\u200b\u65b9\u6848\u200b\u3002\n(\u200b\u4e0a\u200b\u7bad\u5934\u200b\u7b26\u53f7\u200b ^ \u200b\u8868\u793a\u200b\u9009\u53d6\u200b\u7684\u200b\u5b57\u6bcd\u200b)\nrabbbit\n^^^^ ^^\nrabbbit\n^^ ^^^^\nrabbbit\n^^^ ^^^\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"babgbag\", t = \"bag\"\n\u200b\u8f93\u51fa\u200b\uff1a5\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u5982\u4e0b\u200b\u56fe\u200b\u6240\u793a\u200b, \u200b\u6709\u200b 5 \u200b\u79cd\u200b\u53ef\u4ee5\u200b\u4ece\u200b s \u200b\u4e2d\u200b\u5f97\u5230\u200b \"bag\" \u200b\u7684\u200b\u65b9\u6848\u200b\u3002 \n(\u200b\u4e0a\u200b\u7bad\u5934\u200b\u7b26\u53f7\u200b ^ \u200b\u8868\u793a\u200b\u9009\u53d6\u200b\u7684\u200b\u5b57\u6bcd\u200b)\nbabgbag\n^^ ^\nbabgbag\n^^    ^\nbabgbag\n^    ^^\nbabgbag\n  ^  ^^\nbabgbag\n    ^^^\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>0 &lt;= s.length, t.length &lt;= 1000</code></li> <li><code>s</code> \u200b\u548c\u200b <code>t</code> \u200b\u7531\u200b\u82f1\u6587\u5b57\u6bcd\u200b\u7ec4\u6210\u200b</li> </ul> <p>Reference</p>","tags":["\u5b57\u7b26\u4e32","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/115/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u89e3\u9898\u200b\u601d\u8def\u200b\uff1a\u200b\u8bbe\u200b\\(f(x, y)\\)\u200b\u4e3a\u200b\u5b57\u7b26\u4e32\u200b\\(x\\)\u200b\u4e2d\u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u5b57\u7b26\u4e32\u200b\\(y\\)\u200b\u51fa\u73b0\u200b\u7684\u200b\u6b21\u6570\u200b\u3002\u200b\u5e76\u4e14\u200b\u4f7f\u7528\u200b\\(x + 1\\)\u200b\u8868\u793a\u200b\u5b57\u7b26\u4e32\u200b\\(x\\)\u200b\u4ece\u200b\u7b2c\u4e8c\u4e2a\u200b\u5b57\u7b26\u200b\u5f00\u59cb\u200b\u7684\u200b\u5b50\u4e32\u200b\u3002</p> <p>\u200b\u5219\u200b\uff1a</p> <ul> <li>\u200b\u5f53\u200b\\(x, y\\)\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u5b57\u7b26\u200b\u4e0d\u200b\u540c\u65f6\u200b\uff0c\u200b\u6709\u200b\\(f(x, y) = f(x + 1, y)\\)</li> <li>\u200b\u5f53\u200b\\(x, y\\)\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u5b57\u7b26\u200b\u5339\u914d\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(f(x, y) = f(x + 1, y + 1) + f(x + 1, y)\\)</li> </ul> <p>\u200b\u7279\u6b8a\u200b\u5730\u200b\uff0c\u200b\u5f53\u200b\\(y\\)\u200b\u4e3a\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u65f6\u200b\uff0c\u200b\u8bf4\u660e\u200b\u5b57\u7b26\u4e32\u200b\u5339\u914d\u200b\u7ed3\u675f\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u4e00\u4e2a\u200b\u5339\u914d\u200b\u6210\u529f\u200b\u7684\u200b\u5b50\u4e32\u200b\uff0c\u200b\u56e0\u6b64\u200b\\(f(x, \\varnothing) = 1\\)\u3002\u200b\u5f53\u200b\\(x\\)\u200b\u4e3a\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u4e14\u200b\\(y\\)\u200b\u4e0d\u4e3a\u200b\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u65f6\u200b\uff0c\u200b\u8bf4\u660e\u200b\u5b57\u7b26\u4e32\u200b\u5339\u914d\u200b\u5931\u8d25\u200b\uff0c\\(f(\\varnothing, y) = \\left\\{\\begin{aligned}&amp;0 &amp;y\\not = \\varnothing \\\\ &amp; 1 &amp;y=\\varnothing\\end{aligned}\\right .\\)\u3002</p> <p>\u200b\u7531\u6b64\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u521d\u6b65\u200b\u5f97\u51fa\u200b\u9012\u5f52\u200b\u4ee3\u7801\u200b\uff1a</p> <pre><code>int numDistinct(char * s, char * t){\n    if (!*t)\n        return 1;\n    if (!*s)\n        return 0;\n    int ret = 0;\n    if (*s == *t)\n        ret += numDistinct(s + 1, t + 1);\n    ret += numDistinct(s + 1, t);\n    return ret;\n}\n</code></pre> <p>\u200b\u5982\u200b\u4e0a\u200b\u4ee3\u7801\u200b\u4f1a\u200b\u9020\u6210\u200b\u8d85\u65f6\u200b\uff0c\u200b\u539f\u56e0\u200b\u662f\u200b\u9012\u5f52\u200b\u65f6\u200b\u51fa\u73b0\u200b\u4e86\u200b\u5206\u652f\u200b\u5bfc\u81f4\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(2^n\\)\u3002\u200b\u4f7f\u7528\u200b\u6570\u7ec4\u200b\u5b58\u50a8\u200b\u5386\u53f2\u200b\u7ed3\u679c\u200b\u53ef\u4ee5\u200b\u907f\u514d\u200b\u9012\u5f52\u200b\u4ea7\u751f\u200b\u5206\u652f\u200b\uff0c\u200b\u4f18\u5316\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\uff1a</p> <pre><code>int cached(char *s, char *t, int x, int y, int **buffer) {\n    int ret = 0;\n    if (!*t)\n        ret = 1;\n    else if (!*s)\n        ret = 0;\n    else\n    {\n        if (buffer[x][y] &gt;= 0)\n            return buffer[x][y];\n        if (*s == *t)\n            ret += cached(s + 1, t + 1, x + 1, y + 1, buffer);\n        ret += cached(s + 1, t, x + 1, y, buffer);\n        buffer[x][y] = ret;\n    }\n    return ret;\n}\nint numDistinct(char * s, char * t){\n    int x = strlen(s), y = strlen(t);\n    int *_buffer = (int *)malloc(sizeof(int) * x * y);\n    int **buffer = (int **)malloc(sizeof(int *) * x);\n    for (int i = 0; i &lt; x; i++)\n        buffer[i] = _buffer + i * y;\n    memset(_buffer, 0xff, sizeof(int) * x * y);\n    return cached(s, t, 0, 0, buffer);\n}\n</code></pre>","tags":["\u5b57\u7b26\u4e32","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/118/","title":"118. \u200b\u6768\u8f89\u4e09\u89d2","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/118/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200bnumRows\uff0c\u200b\u751f\u6210\u200b\u6768\u8f89\u4e09\u89d2\u200b\u7684\u200b\u524d\u200bnumRows\u200b\u884c\u200b\u3002</p> <p></p> <p>\u200b\u5728\u200b\u6768\u8f89\u4e09\u89d2\u200b\u4e2d\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u6570\u662f\u200b\u5b83\u200b\u5de6\u4e0a\u65b9\u200b\u548c\u200b\u53f3\u4e0a\u65b9\u200b\u7684\u200b\u6570\u200b\u7684\u200b\u548c\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a <pre><code>\u200b\u8f93\u5165\u200b: 5\n\u200b\u8f93\u51fa\u200b:\n[\n     [1],\n    [1,1],\n   [1,2,1],\n  [1,3,3,1],\n [1,4,6,4,1]\n]\n</code></pre></p> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/118/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6839\u636e\u200b\u6768\u8f89\u4e09\u89d2\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u901a\u8fc7\u200b\u9012\u5f52\u200b\u9010\u884c\u200b\u6784\u9020\u200b\u51fa\u200b\u6bcf\u200b\u4e00\u884c\u200b\u7684\u200b\u503c\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Return an array of arrays of size *returnSize.\n * The sizes of the arrays are returned as *returnColumnSizes array.\n * Note: Both returned array and *columnSizes array must be malloced, assume caller calls free().\n */\nint **generate(int numRows, int *returnSize, int **returnColumnSizes)\n{\n    *returnSize = numRows;\n    if (numRows == 0)\n        return NULL;\n    if (numRows == 1)\n    {\n        *returnColumnSizes = (int *)malloc(sizeof(int));\n        **returnColumnSizes = 1;\n        int **ret = (int **)malloc(sizeof(int *));\n        *ret = (int *)malloc(sizeof(int));\n        **ret = 1;\n        return ret;\n    }\n    int retSize = 0,\n        *oldRetColSizes = NULL,\n        **rec = generate(numRows - 1, &amp;retSize, &amp;oldRetColSizes), \n        i = 0;\n        **ret = (int **)memcpy(malloc(sizeof(int *) * numRows), rec, (numRows - 1) * sizeof(int *));\n        *newRetColSizes = memcpy((int *)malloc(sizeof(int) * numRows), oldRetColSizes, (numRows - 1) * sizeof(int));\n        *tempArray = (int *)malloc(sizeof(int) * numRows);\n\n    free(oldRetColSizes);\n    free(rec);\n\n    newRetColSizes[numRows - 1] = numRows;\n    for (i = 1; i &lt; numRows - 1; i++)\n        tempArray[i] = *(ret[numRows - 2] + i) + *(ret[numRows - 2] + i - 1);\n    tempArray[0] = 1;\n    tempArray[numRows - 1] = 1;\n    ret[numRows - 1] = tempArray;\n\n    *returnColumnSizes = newRetColSizes;\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/119/","title":"119. \u200b\u6768\u8f89\u4e09\u89d2\u200b II","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/119/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u975e\u8d1f\u200b\u7d22\u5f15\u200b\\(k\\)\uff0c\u200b\u5176\u4e2d\u200b\\(k\\leq33\\)\uff0c\u200b\u8fd4\u56de\u200b\u6768\u8f89\u4e09\u89d2\u200b\u7684\u200b\u7b2c\u200b\\(k\\) \u200b\u884c\u200b\u3002</p> <p></p> <p>\u200b\u5728\u200b\u6768\u8f89\u4e09\u89d2\u200b\u4e2d\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u6570\u662f\u200b\u5b83\u200b\u5de6\u4e0a\u65b9\u200b\u548c\u200b\u53f3\u4e0a\u65b9\u200b\u7684\u200b\u6570\u200b\u7684\u200b\u548c\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a <pre><code>\u200b\u8f93\u5165\u200b: 3\n\u200b\u8f93\u51fa\u200b: [1,3,3,1]\n</code></pre> \u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u4f18\u5316\u200b\u4f60\u200b\u7684\u200b\u7b97\u6cd5\u200b\u5230\u200b\\(O(k)\\)\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u5417\u200b\uff1f</p> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/119/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5f53\u524d\u200b\u884c\u200b\u7684\u200b\u7ed3\u679c\u200b\u4ec5\u200b\u4f9d\u8d56\u4e8e\u200b\u4e0a\u200b\u4e00\u884c\u200b\u7684\u200b\u7ed3\u679c\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\nvoid recur(int rowIndex, int *Filling)\n{\n    if (rowIndex &lt;= 1)\n        return;\n    recur(rowIndex - 1, Filling);\n    for (int i = rowIndex - 1; i &gt; 0; i--)\n        Filling[i] += Filling[i - 1];\n}\n\nint* getRow(int rowIndex, int* returnSize){\n    *returnSize = rowIndex + 1;\n    int *ret = (int *)memset(malloc(sizeof(int) * *returnSize), 0, sizeof(int) * *returnSize);\n    ret[0] = 1;\n    recur(*returnSize, ret);\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/120/","title":"120. \u200b\u4e09\u89d2\u5f62\u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/120/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u4e09\u89d2\u5f62\u200b\uff0c\u200b\u627e\u51fa\u200b\u81ea\u9876\u5411\u4e0b\u200b\u7684\u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b\u3002\u200b\u6bcf\u200b\u4e00\u6b65\u200b\u53ea\u80fd\u200b\u79fb\u52a8\u200b\u5230\u200b\u4e0b\u200b\u4e00\u884c\u200b\u4e2d\u200b\u76f8\u90bb\u200b\u7684\u200b\u7ed3\u70b9\u200b\u4e0a\u200b\u3002</p> <p>\u200b\u76f8\u90bb\u200b\u7684\u200b\u7ed3\u70b9\u200b \u200b\u5728\u200b\u8fd9\u91cc\u200b\u6307\u200b\u7684\u200b\u662f\u200b <code>\u200b\u4e0b\u6807\u200b</code> \u200b\u4e0e\u200b <code>\u200b\u4e0a\u200b\u4e00\u5c42\u200b\u7ed3\u70b9\u200b\u4e0b\u6807\u200b</code> \u200b\u76f8\u540c\u200b\u6216\u8005\u200b\u7b49\u4e8e\u200b <code>\u200b\u4e0a\u200b\u4e00\u5c42\u200b\u7ed3\u70b9\u200b\u4e0b\u6807\u200b + 1</code> \u200b\u7684\u200b\u4e24\u4e2a\u200b\u7ed3\u70b9\u200b\u3002</p> <p>\u200b\u4f8b\u5982\u200b\uff0c\u200b\u7ed9\u5b9a\u200b\u4e09\u89d2\u5f62\u200b\uff1a</p> <pre><code>[\n     [2],\n    [3,4],\n   [6,5,7],\n  [4,1,8,3]\n]\n</code></pre> <p>\u200b\u81ea\u9876\u5411\u4e0b\u200b\u7684\u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b\u4e3a\u200b<code>11</code>\uff08\u200b\u5373\u200b\uff0c2\u00a0+\u00a03\u00a0+\u00a05\u00a0+\u00a01\u00a0= 11\uff09\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u5982\u679c\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u53ea\u200b\u4f7f\u7528\u200b\\(O(n)\\)\u200b\u7684\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\uff08\\(n\\)\u200b\u4e3a\u200b\u4e09\u89d2\u5f62\u200b\u7684\u200b\u603b\u884c\u200b\u6570\u200b\uff09\u200b\u6765\u200b\u89e3\u51b3\u200b\u8fd9\u4e2a\u200b\u95ee\u9898\u200b\uff0c\u200b\u90a3\u4e48\u200b\u4f60\u200b\u7684\u200b\u7b97\u6cd5\u200b\u4f1a\u200b\u5f88\u200b\u52a0\u5206\u200b\u3002</p> <p>Reference</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/120/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\uff0c\u200b\u4f7f\u7528\u200b\u4e09\u89d2\u5f62\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u884c\u200b\u5b58\u653e\u200b\u6700\u5c0f\u200b\u7684\u200b\u548c\u200b\uff0c\u200b\u4ece\u200b\u4e0b\u200b\u5f80\u200b\u4e0a\u200b\u8fdb\u884c\u200b\u8ba1\u7b97\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>#define MIN(x, y) (x &lt; y ? x : y)\nint minimumTotal(int **triangle, int triangleSize, int *triangleColSize)\n{\n    if (!triangleSize)\n        return 0;\n    int *dp1 = triangle[triangleSize - 1], i = 0, j = 0;\n    for (i = triangleSize - 1; i &gt; 0; i--)\n        for (j = 0; j &lt; triangleColSize[i - 1]; j++)\n            dp1[j] = triangle[i - 1][j] + MIN(dp1[j], dp1[j + 1]);\n    return dp1[0];\n}\n</code></pre>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/121/","title":"121. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/121/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5b83\u200b\u7684\u200b\u7b2c\u200b\u00a0i \u200b\u4e2a\u200b\u5143\u7d20\u200b\u662f\u200b\u4e00\u652f\u200b\u7ed9\u5b9a\u200b\u80a1\u7968\u200b\u7b2c\u200b i \u200b\u5929\u200b\u7684\u200b\u4ef7\u683c\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u4f60\u200b\u6700\u200b\u591a\u53ea\u200b\u5141\u8bb8\u200b\u5b8c\u6210\u200b\u4e00\u7b14\u200b\u4ea4\u6613\u200b\uff08\u200b\u5373\u200b\u4e70\u5165\u200b\u548c\u200b\u5356\u51fa\u200b\u4e00\u652f\u200b\u80a1\u7968\u200b\u4e00\u6b21\u200b\uff09\uff0c\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u7b97\u6cd5\u200b\u6765\u200b\u8ba1\u7b97\u200b\u4f60\u200b\u6240\u80fd\u200b\u83b7\u53d6\u200b\u7684\u200b\u6700\u5927\u200b\u5229\u6da6\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b\uff1a\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u5728\u200b\u4e70\u5165\u200b\u80a1\u7968\u200b\u524d\u200b\u5356\u51fa\u200b\u80a1\u7968\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [7,1,5,3,6,4]\n\u200b\u8f93\u51fa\u200b: 5\n\u200b\u89e3\u91ca\u200b: \u200b\u5728\u200b\u7b2c\u200b 2 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 1\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u4e70\u5165\u200b\uff0c\u200b\u5728\u200b\u7b2c\u200b 5 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 6\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u5356\u51fa\u200b\uff0c\u200b\u6700\u5927\u200b\u5229\u6da6\u200b = 6-1 = 5 \u3002\n     \u200b\u6ce8\u610f\u200b\u5229\u6da6\u200b\u4e0d\u80fd\u200b\u662f\u200b 7-1 = 6, \u200b\u56e0\u4e3a\u200b\u5356\u51fa\u200b\u4ef7\u683c\u200b\u9700\u8981\u200b\u5927\u4e8e\u200b\u4e70\u5165\u200b\u4ef7\u683c\u200b\uff1b\u200b\u540c\u65f6\u200b\uff0c\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u5728\u200b\u4e70\u5165\u200b\u524d\u200b\u5356\u51fa\u200b\u80a1\u7968\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [7,6,4,3,1]\n\u200b\u8f93\u51fa\u200b: 0\n\u200b\u89e3\u91ca\u200b: \u200b\u5728\u200b\u8fd9\u79cd\u200b\u60c5\u51b5\u200b\u4e0b\u200b, \u200b\u6ca1\u6709\u200b\u4ea4\u6613\u200b\u5b8c\u6210\u200b, \u200b\u6240\u4ee5\u200b\u6700\u5927\u200b\u5229\u6da6\u200b\u4e3a\u200b 0\u3002\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/121/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u627e\u51fa\u200b\u76f8\u5dee\u200b\u6700\u5927\u200b\u7684\u200b\u6ee1\u8db3\u200b\\(a_i &lt; a_j\\)\u200b\u4e14\u200b\\(i &lt; j\\)\u200b\u7684\u200b\u4e24\u4e2a\u200b\u6570\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int maxProfit(int* prices, int pricesSize){\n    int max = 0, i = pricesSize - 1, value = 0, profit = 0;\n    for (; i&gt;=0; i--)\n    {\n        if (prices[i] &gt; value)\n            value = prices[i];\n        else\n            profit = value - prices[i];\n        max = max &gt; profit ? max : profit;\n    }\n    return max;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/122/","title":"122. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b II","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u8d2a\u5fc3\u7b97\u6cd5","\u6570\u7ec4"]},{"location":"coding/leetcode/122/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5b83\u200b\u7684\u200b\u7b2c\u200b\u00a0i \u200b\u4e2a\u200b\u5143\u7d20\u200b\u662f\u200b\u4e00\u652f\u200b\u7ed9\u5b9a\u200b\u80a1\u7968\u200b\u7b2c\u200b i \u200b\u5929\u200b\u7684\u200b\u4ef7\u683c\u200b\u3002</p> <p>\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u7b97\u6cd5\u200b\u6765\u200b\u8ba1\u7b97\u200b\u4f60\u200b\u6240\u80fd\u200b\u83b7\u53d6\u200b\u7684\u200b\u6700\u5927\u200b\u5229\u6da6\u200b\u3002\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5c3d\u53ef\u80fd\u200b\u5730\u200b\u5b8c\u6210\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u4ea4\u6613\u200b\uff08\u200b\u591a\u6b21\u200b\u4e70\u5356\u200b\u4e00\u652f\u200b\u80a1\u7968\u200b\uff09\u3002</p> <p>\u200b\u6ce8\u610f\u200b\uff1a\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u540c\u65f6\u200b\u53c2\u4e0e\u200b\u591a\u7b14\u200b\u4ea4\u6613\u200b\uff08\u200b\u4f60\u200b\u5fc5\u987b\u200b\u5728\u200b\u518d\u6b21\u200b\u8d2d\u4e70\u200b\u524d\u200b\u51fa\u552e\u200b\u6389\u200b\u4e4b\u524d\u200b\u7684\u200b\u80a1\u7968\u200b\uff09\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [7,1,5,3,6,4]\n\u200b\u8f93\u51fa\u200b: 7\n\u200b\u89e3\u91ca\u200b: \u200b\u5728\u200b\u7b2c\u200b 2 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 1\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u4e70\u5165\u200b\uff0c\u200b\u5728\u200b\u7b2c\u200b 3 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 5\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u5356\u51fa\u200b, \u200b\u8fd9\u7b14\u200b\u4ea4\u6613\u6240\u200b\u80fd\u200b\u83b7\u5f97\u200b\u5229\u6da6\u200b = 5-1 = 4 \u3002\n\u00a0    \u200b\u968f\u540e\u200b\uff0c\u200b\u5728\u200b\u7b2c\u200b 4 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 3\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u4e70\u5165\u200b\uff0c\u200b\u5728\u200b\u7b2c\u200b 5 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 6\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u5356\u51fa\u200b, \u200b\u8fd9\u7b14\u200b\u4ea4\u6613\u6240\u200b\u80fd\u200b\u83b7\u5f97\u200b\u5229\u6da6\u200b = 6-3 = 3 \u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,2,3,4,5]\n\u200b\u8f93\u51fa\u200b: 4\n\u200b\u89e3\u91ca\u200b: \u200b\u5728\u200b\u7b2c\u200b 1 \u200b\u5929\u200b\uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 1\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u4e70\u5165\u200b\uff0c\u200b\u5728\u200b\u7b2c\u200b 5 \u200b\u5929\u200b \uff08\u200b\u80a1\u7968\u4ef7\u683c\u200b = 5\uff09\u200b\u7684\u200b\u65f6\u5019\u200b\u5356\u51fa\u200b, \u200b\u8fd9\u7b14\u200b\u4ea4\u6613\u6240\u200b\u80fd\u200b\u83b7\u5f97\u200b\u5229\u6da6\u200b = 5-1 = 4 \u3002\n\u00a0    \u200b\u6ce8\u610f\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u5728\u200b\u7b2c\u200b 1 \u200b\u5929\u200b\u548c\u200b\u7b2c\u200b 2 \u200b\u5929\u200b\u63a5\u8fde\u200b\u8d2d\u4e70\u200b\u80a1\u7968\u200b\uff0c\u200b\u4e4b\u540e\u200b\u518d\u200b\u5c06\u200b\u5b83\u4eec\u200b\u5356\u51fa\u200b\u3002\n\u00a0    \u200b\u56e0\u4e3a\u200b\u8fd9\u6837\u200b\u5c5e\u4e8e\u200b\u540c\u65f6\u200b\u53c2\u4e0e\u200b\u4e86\u200b\u591a\u7b14\u200b\u4ea4\u6613\u200b\uff0c\u200b\u4f60\u200b\u5fc5\u987b\u200b\u5728\u200b\u518d\u6b21\u200b\u8d2d\u4e70\u200b\u524d\u200b\u51fa\u552e\u200b\u6389\u200b\u4e4b\u524d\u200b\u7684\u200b\u80a1\u7968\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a03\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [7,6,4,3,1]\n\u200b\u8f93\u51fa\u200b: 0\n\u200b\u89e3\u91ca\u200b: \u200b\u5728\u200b\u8fd9\u79cd\u200b\u60c5\u51b5\u200b\u4e0b\u200b, \u200b\u6ca1\u6709\u200b\u4ea4\u6613\u200b\u5b8c\u6210\u200b, \u200b\u6240\u4ee5\u200b\u6700\u5927\u200b\u5229\u6da6\u200b\u4e3a\u200b 0\u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= prices.length &lt;= 3 * 10 ^ 4</code></li> <li><code>0 &lt;= prices[i]\u00a0&lt;= 10 ^ 4</code></li> </ul> <p>Reference</p>","tags":["\u8d2a\u5fc3\u7b97\u6cd5","\u6570\u7ec4"]},{"location":"coding/leetcode/122/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a </p> <ol> <li>\u200b\u5f53\u200b\u6570\u7ec4\u200b\u957f\u5ea6\u200b\u4e3a\u200b1\u200b\u65f6\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u83b7\u53d6\u200b\u4efb\u4f55\u200b\u5229\u6da6\u200b\uff0c\u200b\u8fd4\u56de\u200b<code>0</code>\uff1b</li> <li>\u200b\u4ece\u200b\u6570\u7ec4\u200b\u672b\u5c3e\u200b\u5f00\u59cb\u200b\u641c\u7d22\u200b\u7b2c\u4e00\u4e2a\u200b\u80fd\u200b\u76c8\u5229\u200b\u7684\u200b\u70b9\u200b\uff0c\u200b\u5219\u200b\u6700\u5927\u200b\u5229\u6da6\u200b\u4e3a\u200b\u672c\u6b21\u200b\u76c8\u5229\u200b\u4e0e\u200b\u4e4b\u524d\u200b\u6700\u5927\u200b\u76c8\u5229\u200b\u4e4b\u200b\u548c\u200b\u3002</li> </ol> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int maxProfit(int* prices, int pricesSize){\n    if (pricesSize &lt; 2)\n        return 0;\n    int *cur = prices + pricesSize - 1, currentVal = *cur, len = pricesSize;\n    while(cur &gt; prices)\n    {\n        if (*cur &lt; *(cur - 1))\n            break;\n        len--;\n        cur--;\n    }\n    return (currentVal - *cur) + maxProfit(prices, len - 1);\n}\n</code></pre>","tags":["\u8d2a\u5fc3\u7b97\u6cd5","\u6570\u7ec4"]},{"location":"coding/leetcode/125/","title":"125. \u200b\u9a8c\u8bc1\u200b\u56de\u6587\u200b\u4e32","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/125/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u9a8c\u8bc1\u200b\u5b83\u200b\u662f\u5426\u662f\u200b\u56de\u6587\u200b\u4e32\u200b\uff0c\u200b\u53ea\u200b\u8003\u8651\u200b\u5b57\u6bcd\u200b\u548c\u200b\u6570\u5b57\u200b\u5b57\u7b26\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5ffd\u7565\u200b\u5b57\u6bcd\u200b\u7684\u200b\u5927\u5c0f\u5199\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a \u200b\u672c\u9898\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u5c06\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\u6709\u6548\u200b\u7684\u200b\u56de\u6587\u200b\u4e32\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: \"A man, a plan, a canal: Panama\"\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: \"race a car\"\n\u200b\u8f93\u51fa\u200b: false\n</code></pre> <p>Reference</p>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/125/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4f7f\u7528\u200b\u53cc\u200b\u6307\u9488\u200b\u6cd5\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b\uff0c\u200b\u9700\u8981\u200b\u6ce8\u610f\u200b\u5982\u4e0b\u200b\u7279\u6b8a\u200b\u60c5\u51b5\u200b</p> <ol> <li>\u200b\u9700\u8981\u200b\u5ffd\u7565\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u7b26\u53f7\u200b\u800c\u200b\u53ea\u200b\u8003\u8651\u200b\u5b57\u6bcd\u200b\u548c\u200b\u6570\u5b57\u200b</li> <li>\u200b\u5927\u5c0f\u5199\u200b\u7684\u200bASCII\u200b\u6570\u503c\u200b\u76f8\u5dee\u200b32\uff0c\u200b\u4f46\u200bASCII\u200b\u6570\u503c\u200b\u76f8\u5dee\u200b32\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u5927\u5c0f\u5199\u200b\u5173\u7cfb\u200b\uff0c\u200b\u5982\u200b<code>'0'</code>(48)\u200b\u4e0e\u200b<code>'P'</code>(80)</li> </ol> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>bool isText(char s)\n{\n    return (s &gt;= 'A' &amp;&amp; s &lt;= 'Z') || (s &gt;= 'a' &amp;&amp; s &lt;= 'z') || (s &gt;= '0' &amp;&amp; s &lt;= '9');\n}\nbool isSame(char s1, char s2)\n{\n    return (s1 == s2) || (s2 &gt;= 'A' &amp;&amp; s1 - s2 == 32) || (s1 &gt;= 'A' &amp;&amp; s2 - s1 == 32);\n}\nbool isPalindrome(char * s){\n    int len = strlen(s), head = 0, tail = len - 1;\n    while (head &lt; len &amp;&amp; !isText(s[head]))\n        head++;\n    while (tail &gt;= 0 &amp;&amp; !isText(s[tail]))\n        tail--;\n    while(tail &gt; head &amp;&amp; head &lt; len &amp;&amp; tail &gt;= 0)\n    {\n        if (!isSame(s[head], s[tail]))\n            return false;\n        head++;\n        tail--;\n        while (!isText(s[head]))\n            head++;\n        while (!isText(s[tail]))\n            tail--;\n    }\n    return true;\n}\n</code></pre>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/1283/","title":"1283. \u200b\u4f7f\u200b\u7ed3\u679c\u200b\u4e0d\u200b\u8d85\u8fc7\u200b\u9608\u503c\u200b\u7684\u200b\u6700\u5c0f\u200b\u9664\u6570","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/1283/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\u00a0<code>nums</code> \u200b\u548c\u200b\u4e00\u4e2a\u200b\u6b63\u6574\u6570\u200b\u00a0<code>threshold</code>\u00a0\uff0c\u200b\u4f60\u200b\u9700\u8981\u200b\u9009\u62e9\u200b\u4e00\u4e2a\u200b\u6b63\u6574\u6570\u200b\u4f5c\u4e3a\u200b\u9664\u6570\u200b\uff0c\u200b\u7136\u540e\u200b\u5c06\u200b\u6570\u7ec4\u200b\u91cc\u200b\u6bcf\u4e2a\u200b\u6570\u90fd\u200b\u9664\u4ee5\u200b\u5b83\u200b\uff0c\u200b\u5e76\u200b\u5bf9\u200b\u9664\u6cd5\u200b\u7ed3\u679c\u200b\u6c42\u548c\u200b\u3002</p> <p>\u200b\u8bf7\u200b\u4f60\u200b\u627e\u51fa\u200b\u80fd\u591f\u200b\u4f7f\u200b\u4e0a\u8ff0\u200b\u7ed3\u679c\u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b\u9608\u503c\u200b\u00a0<code>threshold</code>\u00a0\u200b\u7684\u200b\u9664\u6570\u200b\u4e2d\u200b \u200b\u6700\u5c0f\u200b \u200b\u7684\u200b\u90a3\u4e2a\u200b\u3002</p> <p>\u200b\u6bcf\u4e2a\u200b\u6570\u200b\u9664\u4ee5\u200b\u9664\u6570\u200b\u540e\u200b\u90fd\u200b\u5411\u4e0a\u200b\u53d6\u6574\u200b\uff0c\u200b\u6bd4\u65b9\u8bf4\u200b 7/3 = 3 \uff0c 10/2 = 5 \u3002</p> <p>\u200b\u9898\u76ee\u200b\u4fdd\u8bc1\u200b\u4e00\u5b9a\u200b\u6709\u89e3\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1,2,5,9], threshold = 6\n\u200b\u8f93\u51fa\u200b\uff1a5\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5982\u679c\u200b\u9664\u6570\u200b\u4e3a\u200b 1 \uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u548c\u200b\u4e3a\u200b 17 \uff081+2+5+9\uff09\u3002\n\u200b\u5982\u679c\u200b\u9664\u6570\u200b\u4e3a\u200b 4 \uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u548c\u200b\u4e3a\u200b 7 (1+1+2+3) \u3002\u200b\u5982\u679c\u200b\u9664\u6570\u200b\u4e3a\u200b 5 \uff0c\u200b\u548c\u200b\u4e3a\u200b 5 (1+1+1+2)\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [2,3,5,7,11], threshold = 11\n\u200b\u8f93\u51fa\u200b\uff1a3\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [19], threshold = 5\n\u200b\u8f93\u51fa\u200b\uff1a4\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 5 * 10^4</code></li> <li><code>1 &lt;= nums[i] &lt;= 10^6</code></li> <li><code>nums.length &lt;=\u00a0threshold &lt;= 10^6</code></li> </ul> <p>Reference</p>","tags":["\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/1283/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u627e\u5230\u200b\u6700\u5c0f\u200b\u7684\u200b\u9664\u6570\u200b</p> <p>\u200b\u5982\u679c\u200b\u7ed3\u679c\u200b\u5927\u4e8e\u200b<code>threshold</code>\uff0c\u200b\u8bf4\u660e\u200b\u9664\u6570\u200b\u8fc7\u200b\u5c0f\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u53f3\u4fa7\u200b\u67e5\u627e\u200b\uff0c\u200b\u5426\u5219\u200b\u5728\u200b\u5de6\u4fa7\u200b\u67e5\u627e\u200b</p> <pre><code>bool test(int *nums, int numsSize, int threshold, int x)\n{\n    if (!x)\n        return INT_MAX;\n    int ret = 0, i = 0;\n    for (i = 0; i &lt; numsSize; i++)\n        ret += (nums[i] + x - 1) / x;\n    return ret &lt;= threshold;\n}\n\nint bSearch(int *nums, int numsSize, int threshold, int lo, int hi)\n{\n    if (hi - lo == 1)\n        return hi;\n    int mid = (hi + lo) &gt;&gt; 1;\n    if (!test(nums, numsSize, threshold, mid))\n        return bSearch(nums, numsSize, threshold, mid, hi);\n    else\n        return bSearch(nums, numsSize, threshold, lo, mid);\n}\n\nint smallestDivisor(int* nums, int numsSize, int threshold){\n    int max = 0, i = 0;\n    for (i = 0; i &lt; numsSize; i++)\n        if (max &lt; nums[i])\n        max = nums[i];\n    return bSearch(nums, numsSize, threshold, 0, max);\n}\n</code></pre> <pre><code>class Solution:\n    def bSearch(self, nums: List[int], threshold: int, lo: int, hi: int) -&gt; int:\n        if hi - lo == 1:\n            return hi\n        mid = (lo + hi) &gt;&gt; 1\n        return self.bSearch(nums, threshold, lo, mid) \\\n            if sum([math.ceil(_ / mid) for _ in nums]) &lt;= threshold \\\n            else self.bSearch(nums, threshold, mid, hi)\n    def smallestDivisor(self, nums: List[int], threshold: int) -&gt; int:\n        return self.bSearch(nums, threshold, 0, max(nums));\n</code></pre>","tags":["\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/13/","title":"13. \u200b\u7f57\u9a6c\u6570\u5b57\u200b\u8f6c\u200b\u6574\u6570","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/13/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7f57\u9a6c\u6570\u5b57\u200b\u5305\u542b\u200b\u4ee5\u4e0b\u200b\u4e03\u79cd\u200b\u5b57\u7b26\u200b:<code>I</code>\uff0c<code>V</code>\uff0c<code>X</code>\uff0c<code>L</code>\uff0c<code>C</code>\uff0c<code>D</code>\u200b\u548c\u200b<code>M</code>\u3002</p> <p><pre><code>\u200b\u5b57\u7b26\u200b          \u200b\u6570\u503c\u200b\nI             1\nV             5\nX             10\nL             50\nC             100\nD             500\nM             1000\n</code></pre> \u200b\u4f8b\u5982\u200b\uff0c \u200b\u7f57\u9a6c\u6570\u5b57\u200b 2 \u200b\u5199\u200b\u505a\u200b<code>II</code>\u00a0\uff0c\u200b\u5373\u200b\u4e3a\u200b\u4e24\u4e2a\u200b\u5e76\u5217\u200b\u7684\u200b 1\u300212 \u200b\u5199\u200b\u505a\u200b<code>XII</code>\u00a0\uff0c\u200b\u5373\u200b\u4e3a\u200b<code>X</code>\u00a0+\u00a0<code>II</code>\u3002 27 \u200b\u5199\u200b\u505a\u200b<code>XXVII</code>, \u200b\u5373\u200b\u4e3a\u200b\u00a0<code>XX</code>\u00a0+\u00a0<code>V</code>\u00a0+\u00a0<code>II</code>\u3002</p> <p>\u200b\u901a\u5e38\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u7f57\u9a6c\u6570\u5b57\u200b\u4e2d\u5c0f\u200b\u7684\u200b\u6570\u5b57\u200b\u5728\u200b\u5927\u200b\u7684\u200b\u6570\u5b57\u200b\u7684\u200b\u53f3\u8fb9\u200b\u3002\u200b\u4f46\u200b\u4e5f\u200b\u5b58\u5728\u200b\u7279\u4f8b\u200b\uff0c\u200b\u4f8b\u5982\u200b 4 \u200b\u4e0d\u5199\u200b\u505a\u200b<code>IIII</code>\uff0c\u200b\u800c\u662f\u200b<code>IV</code>\u3002\u200b\u6570\u5b57\u200b 1 \u200b\u5728\u200b\u6570\u5b57\u200b 5 \u200b\u7684\u200b\u5de6\u8fb9\u200b\uff0c\u200b\u6240\u200b\u8868\u793a\u200b\u7684\u200b\u6570\u200b\u7b49\u4e8e\u200b\u5927\u6570\u200b 5 \u200b\u51cf\u5c0f\u200b\u6570\u200b 1 \u200b\u5f97\u5230\u200b\u7684\u200b\u6570\u503c\u200b 4 \u3002\u200b\u540c\u6837\u200b\u5730\u200b\uff0c\u200b\u6570\u5b57\u200b 9 \u200b\u8868\u793a\u200b\u4e3a\u200b<code>IX</code>\u3002\u200b\u8fd9\u4e2a\u200b\u7279\u6b8a\u200b\u7684\u200b\u89c4\u5219\u200b\u53ea\u200b\u9002\u7528\u200b\u4e8e\u200b\u4ee5\u4e0b\u200b\u516d\u79cd\u200b\u60c5\u51b5\u200b\uff1a</p> <p><code>I</code>\u200b\u53ef\u4ee5\u200b\u653e\u5728\u200b\u00a0<code>V</code>\u00a0(5) \u200b\u548c\u200b\u00a0<code>X</code>\u00a0(10) \u200b\u7684\u200b\u5de6\u8fb9\u200b\uff0c\u200b\u6765\u200b\u8868\u793a\u200b 4 \u200b\u548c\u200b 9\u3002 <code>X</code>\u200b\u53ef\u4ee5\u200b\u653e\u5728\u200b\u00a0<code>L</code>\u00a0(50) \u200b\u548c\u200b\u00a0<code>C</code>\u00a0(100) \u200b\u7684\u200b\u5de6\u8fb9\u200b\uff0c\u200b\u6765\u200b\u8868\u793a\u200b 40 \u200b\u548c\u200b\u00a090\u3002\u00a0 <code>C</code>\u200b\u53ef\u4ee5\u200b\u653e\u5728\u200b\u00a0<code>D</code>\u00a0(500) \u200b\u548c\u200b\u00a0<code>M</code>\u00a0(1000) \u200b\u7684\u200b\u5de6\u8fb9\u200b\uff0c\u200b\u6765\u200b\u8868\u793a\u200b\u00a0400 \u200b\u548c\u200b\u00a0900\u3002 \u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u7f57\u9a6c\u6570\u5b57\u200b\uff0c\u200b\u5c06\u200b\u5176\u200b\u8f6c\u6362\u6210\u200b\u6574\u6570\u200b\u3002\u200b\u8f93\u5165\u200b\u786e\u4fdd\u200b\u5728\u200b 1\u00a0\u200b\u5230\u200b 3999 \u200b\u7684\u200b\u8303\u56f4\u200b\u5185\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\u00a0\"III\"\n\u200b\u8f93\u51fa\u200b: 3\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\u00a0\"IV\"\n\u200b\u8f93\u51fa\u200b: 4\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a03\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\u00a0\"IX\"\n\u200b\u8f93\u51fa\u200b: 9\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a04\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\u00a0\"LVIII\"\n\u200b\u8f93\u51fa\u200b: 58\n\u200b\u89e3\u91ca\u200b: L = 50, V= 5, III = 3.\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a05\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\u00a0\"MCMXCIV\"\n\u200b\u8f93\u51fa\u200b: 1994\n\u200b\u89e3\u91ca\u200b: M = 1000, CM = 900, XC = 90, IV = 4.\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <p>\u200b\u9898\u76ee\u200b\u6240\u200b\u7ed9\u200b\u6d4b\u8bd5\u7528\u4f8b\u200b\u7686\u200b\u7b26\u5408\u200b\u7f57\u9a6c\u6570\u5b57\u200b\u4e66\u5199\u200b\u89c4\u5219\u200b\uff0c\u200b\u4e0d\u4f1a\u200b\u51fa\u73b0\u200b\u8de8\u4f4d\u200b\u7b49\u200b\u60c5\u51b5\u200b\u3002</p> <p><code>IC</code>\u200b\u548c\u200b<code>IM</code>\u200b\u8fd9\u6837\u200b\u7684\u200b\u4f8b\u5b50\u200b\u5e76\u200b\u4e0d\u200b\u7b26\u5408\u200b\u9898\u76ee\u200b\u8981\u6c42\u200b\uff0c49 \u200b\u5e94\u8be5\u200b\u5199\u4f5c\u200b<code>XLIX</code>\uff0c999 \u200b\u5e94\u8be5\u200b\u5199\u4f5c\u200b <code>CMXCIX</code>\u3002</p> <p>\u200b\u5173\u4e8e\u200b\u7f57\u9a6c\u6570\u5b57\u200b\u7684\u200b\u8be6\u5c3d\u200b\u4e66\u5199\u200b\u89c4\u5219\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u53c2\u8003\u200b\u7f57\u9a6c\u6570\u5b57\u200b - Mathematics\u3002</p> <p>Reference</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/13/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u9010\u4f4d\u200b\u5904\u7406\u200b\u5373\u53ef\u200b\uff0c\u200b\u6ce8\u610f\u200b\u5c0f\u200b\u7684\u200b\u6570\u5b57\u200b\u5728\u200b\u5927\u200b\u7684\u200b\u6570\u5b57\u200b\u5de6\u8fb9\u200b\u7684\u200b\u7279\u6b8a\u200b\u60c5\u51b5\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int romanToInt(char * s){\n    char *cur = s;\n    int ret = 0;\n    while(*cur)\n    {\n        switch(*cur)\n        {\n            case 'I':\n                if (*(cur + 1) == 'V' || *(cur + 1) == 'X')\n                    ret -= 1;\n                else\n                    ret += 1;\n                break;\n            case 'V':\n                ret += 5;\n                break;\n            case 'X':\n                if (*(cur + 1) == 'L' || *(cur + 1) == 'C')\n                    ret -= 10;\n                else\n                    ret += 10;\n                break;\n            case 'L':\n                ret += 50;\n                break;\n            case 'C':\n                if (*(cur + 1) == 'D' || *(cur + 1) == 'M')\n                    ret -= 100;\n                else\n                    ret += 100;\n                break;\n            case 'M':\n                ret += 500;\n            case 'D':\n                ret += 500;\n        }\n        cur++;\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/1300/","title":"1300. \u200b\u8f6c\u53d8\u200b\u6570\u7ec4\u200b\u540e\u200b\u6700\u200b\u63a5\u8fd1\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u6570\u7ec4\u200b\u548c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/1300/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\u00a0<code>arr</code> \u200b\u548c\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u503c\u200b\u00a0<code>target</code> \uff0c\u200b\u8bf7\u200b\u4f60\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u00a0<code>value</code>\u00a0\uff0c\u200b\u4f7f\u5f97\u200b\u5c06\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u6240\u6709\u200b\u5927\u4e8e\u200b\u00a0<code>value</code> \u200b\u7684\u200b\u503c\u200b\u53d8\u6210\u200b\u00a0<code>value</code> \u200b\u540e\u200b\uff0c\u200b\u6570\u7ec4\u200b\u7684\u200b\u548c\u200b\u6700\u200b\u63a5\u8fd1\u200b <code>target</code> \uff08\u200b\u6700\u200b\u63a5\u8fd1\u200b\u8868\u793a\u200b\u4e24\u8005\u200b\u4e4b\u5dee\u200b\u7684\u200b\u7edd\u5bf9\u503c\u200b\u6700\u5c0f\u200b\uff09\u3002</p> <p>\u200b\u5982\u679c\u200b\u6709\u200b\u591a\u79cd\u200b\u4f7f\u5f97\u200b\u548c\u200b\u6700\u200b\u63a5\u8fd1\u200b\u00a0<code>target</code>\u00a0\u200b\u7684\u200b\u65b9\u6848\u200b\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u8fd4\u56de\u200b\u8fd9\u4e9b\u200b\u6574\u6570\u200b\u4e2d\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\u3002</p> <p>\u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u7b54\u6848\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u00a0<code>arr</code> \u200b\u4e2d\u200b\u7684\u200b\u6570\u5b57\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aarr = [4,9,3], target = 10\n\u200b\u8f93\u51fa\u200b\uff1a3\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5f53\u200b\u9009\u62e9\u200b value \u200b\u4e3a\u200b 3 \u200b\u65f6\u200b\uff0c\u200b\u6570\u7ec4\u200b\u4f1a\u200b\u53d8\u6210\u200b [3, 3, 3]\uff0c\u200b\u548c\u200b\u4e3a\u200b 9 \uff0c\u200b\u8fd9\u200b\u662f\u200b\u6700\u200b\u63a5\u8fd1\u200b target \u200b\u7684\u200b\u65b9\u6848\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aarr = [2,3,5], target = 10\n\u200b\u8f93\u51fa\u200b\uff1a5\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aarr = [60864,25176,27249,21296,20204], target = 56803\n\u200b\u8f93\u51fa\u200b\uff1a11361\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= arr.length &lt;= 10^4</code></li> <li><code>1 &lt;= arr[i], target &lt;= 10^5</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/1300/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5728\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\\((lo, hi)\\)\u200b\u4e2d\u200b\u786e\u5b9a\u200b\u4e09\u7b49\u5206\u200b\u70b9\u200b\\(x_1, x_2 (x_1 &lt; x_2)\\)\uff0c\u200b\u5206\u522b\u200b\u8ba1\u7b97\u200b\u4e24\u4e2a\u200b\u4e09\u7b49\u5206\u200b\u70b9\u5904\u200b\u7684\u200b\u5dee\u503c\u200b\u3002\u200b\u8bbe\u200b\\(f(x)\\)\u200b\u4e3a\u200b\u5c06\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5927\u4e8e\u200b\\(x\\)\u200b\u7684\u200b\u6570\u503c\u200b\u8f6c\u53d8\u200b\u4e3a\u200b\\(x\\)\u200b\u540e\u200b\u6570\u7ec4\u200b\u6c42\u548c\u200b\u4e0e\u200b\u76ee\u6807\u503c\u200b\u4e4b\u5dee\u200b\u7684\u200b\u7edd\u5bf9\u503c\u200b\u3002</p> <ul> <li>\u200b\u5f53\u200b\\(f(x_1) &lt; f(x_2)\\)\u200b\u65f6\u200b\uff0c\u200b\u5c06\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u7f29\u51cf\u200b\u4e3a\u200b\\((lo, x_2)\\)</li> <li>\u200b\u5f53\u200b\\(f(x_1) &gt; f(x_2)\\)\u200b\u65f6\u200b\uff0c\u200b\u5c06\u200b\u5f85\u67e5\u200b\u533a\u95f4\u200b\u7f29\u51cf\u200b\u4e3a\u200b\\((x_1, hi)\\)</li> </ul> <p>\\(f(x)\\)\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b+\u200b\u9884\u5904\u7406\u200b\u5728\u200b\\(\\mathcal O(\\log N)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u5185\u200b\u8ba1\u7b97\u200b\u5b8c\u6210\u200b\u3002\u200b\u9884\u5904\u7406\u200b\u9700\u8981\u200b\u82b1\u8d39\u200b\\(\\mathcal O(N)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u3002\u200b\u6392\u5e8f\u200b\u9700\u8981\u200b\u82b1\u8d39\u200b\\(\\mathcal O(N\\log N)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u3002\u200b\u9700\u8981\u200b\\(\\mathcal O(\\log N)\\)\u200b\u6b21\u200b\u4e09\u5206\u200b\u67e5\u627e\u200b\u3002\u200b\u56e0\u6b64\u200b\u603b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(N\\log N)\\)\uff08\u200b\u9884\u5904\u7406\u200b\u9636\u6bb5\u200b\uff09</p> <pre><code>#define ABS(x) ((x) &gt; 0 ? (x) : -(x))\n\nint cmp(const void *a, const void *b)\n{\n    return *((int *)a) - *((int *)b);\n}\n\nint bSearch(int *arr, int target, int lo, int hi)\n{\n    int mid;\n    while (lo &lt; hi)\n    {\n        mid = (lo + hi) &gt;&gt; 1;\n        if (arr[mid] &lt; target)\n            lo = mid + 1;\n        else\n            hi = mid;\n    }\n    return lo - 1;\n}\n\nint test(int *arr, int arrSize, int target, int val, int *presum)\n{\n    int i = bSearch(arr, val, 0, arrSize), ret = val * (arrSize - i - 1) - target;\n    if (i &gt;= 0)\n        ret += presum[i];\n    return ABS(ret);\n}\n\nint tSearch(int *arr, int arrSize, int target, int lo, int hi, int *presum)\n{\n    int left, right;\n    while (hi - lo &gt; 2)\n    {\n        left = (2 * lo + hi) / 3;\n        right = (2 * hi + lo) / 3;\n        if (test(arr, arrSize, target, left, presum) &lt;= test(arr, arrSize, target, right, presum))\n            hi = right;\n        else\n            lo = left;\n    }\n    return (2 * hi + lo) / 3;\n}\n\nint findBestValue(int* arr, int arrSize, int target){\n    int i = 0, presum[arrSize];\n    qsort(arr, arrSize, sizeof(int), cmp);\n    presum[0] = arr[0];\n    for (i = 1; i &lt; arrSize; i++)\n        presum[i] = presum[i - 1] + arr[i];\n\n    return tSearch(arr, arrSize, target, -1, arr[arrSize - 1] + 1, presum);\n}\n</code></pre>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/136/","title":"136. \u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u7684\u200b\u6570\u5b57","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u4f4d\u8fd0\u7b97","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/136/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u975e\u7a7a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\uff0c\u200b\u9664\u4e86\u200b\u67d0\u4e2a\u200b\u5143\u7d20\u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u4ee5\u5916\u200b\uff0c\u200b\u5176\u4f59\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u5747\u200b\u51fa\u73b0\u200b\u4e24\u6b21\u200b\u3002\u200b\u627e\u51fa\u200b\u90a3\u4e2a\u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e86\u200b\u4e00\u6b21\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u7684\u200b\u7b97\u6cd5\u200b\u5e94\u8be5\u200b\u5177\u6709\u200b\u7ebf\u6027\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u3002 \u200b\u4f60\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u4f7f\u7528\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\u6765\u200b\u5b9e\u73b0\u200b\u5417\u200b\uff1f</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [2,2,1]\n\u200b\u8f93\u51fa\u200b: 1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [4,1,2,1,2]\n\u200b\u8f93\u51fa\u200b: 4\n</code></pre> <p>Reference</p>","tags":["\u4f4d\u8fd0\u7b97","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/136/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5f02\u6216\u200b\u8fd0\u7b97\u200b\u5177\u6709\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ol> <li>\u200b\u4ea4\u6362\u5f8b\u200b\uff1a<code>a ^ b = b ^ a</code></li> <li>\u200b\u7ed3\u5408\u5f8b\u200b\uff1a<code>(a ^ b) ^ c = a ^ (b ^ c)</code></li> <li>\u200b\u82e5\u200b<code>a ^ b = c</code>\uff0c\u200b\u5219\u200b<code>a ^ c = b</code>\u3001<code>b ^ c = a</code></li> <li><code>a ^ a = 0</code></li> </ol> <p>\u200b\u6839\u636e\u200b\u6027\u8d28\u200b1\u30012\uff0c\u200b\u5f02\u6216\u200b\u8fd0\u7b97\u200b\u7684\u200b\u7ed3\u679c\u200b\u4e0e\u200b\u53c2\u4e0e\u200b\u8fd0\u7b97\u200b\u7684\u200b\u6240\u6709\u200b\u6570\u503c\u200b\u7684\u200b\u6392\u5217\u200b\u987a\u5e8f\u200b\u65e0\u5173\u200b\u3002\u200b\u6839\u636e\u200b\u6027\u8d28\u200b3\u30014\uff0c\u200b\u76f8\u540c\u200b\u6574\u6570\u200b\u7684\u200b\u5f02\u6216\u200b\u503c\u4e3a\u200b0\uff0c0\u200b\u4e0e\u200b\u4efb\u4f55\u200b\u6570\u200b\u5f02\u6216\u200b\u4ecd\u4e3a\u200b\u8be5\u6570\u200b\u3002\u200b\u56e0\u6b64\u200b\u5c06\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5404\u200b\u5143\u7d20\u200b\u8fdb\u884c\u200b\u6309\u4f4d\u200b\u5f02\u6216\u200b\uff0c\u200b\u5f97\u5230\u200b\u7684\u200b\u7ed3\u679c\u200b\u5373\u200b\u4e3a\u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u7684\u200b\u6570\u503c\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int singleNumber(int* nums, int numsSize){\n    int ret = 0, i = 0;\n    for (i = 0; i &lt; numsSize; i++)\n        ret ^= nums[i];\n    return ret;\n}\n</code></pre>","tags":["\u4f4d\u8fd0\u7b97","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/14/","title":"14. \u200b\u6700\u957f\u200b\u516c\u5171\u200b\u524d\u7f00","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/14/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u6765\u200b\u67e5\u627e\u200b\u5b57\u7b26\u4e32\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u6700\u957f\u200b\u516c\u5171\u200b\u524d\u7f00\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u4e0d\u200b\u5b58\u5728\u200b\u516c\u5171\u200b\u524d\u7f00\u200b\uff0c\u200b\u8fd4\u56de\u200b\u7a7a\u200b\u5b57\u7b26\u4e32\u200b<code>\"\"</code>\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [\"flower\",\"flow\",\"flight\"]\n\u200b\u8f93\u51fa\u200b: \"fl\"\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [\"dog\",\"racecar\",\"car\"]\n\u200b\u8f93\u51fa\u200b: \"\"\n\u200b\u89e3\u91ca\u200b: \u200b\u8f93\u5165\u200b\u4e0d\u200b\u5b58\u5728\u200b\u516c\u5171\u200b\u524d\u7f00\u200b\u3002\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u6240\u6709\u200b\u8f93\u5165\u200b\u53ea\u200b\u5305\u542b\u200b\u5c0f\u5199\u5b57\u6bcd\u200b<code>a-z</code>\u3002</p> <p>Reference</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/14/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5728\u200b\u6bd4\u8f83\u200b\u524d\u7f00\u200b\u65f6\u200b\u53ea\u200b\u9700\u8981\u200b\u5c06\u200b\u524d\u200b n \u200b\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u516c\u5171\u200b\u524d\u7f00\u200b\u4e0e\u200b\u7b2c\u200b n+1 \u200b\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u8fdb\u884c\u200b\u6bd4\u8f83\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>char *compareTwo(char *str1, char *str2)\n{\n    char *cur1 = str1, *cur2 = str2, *ret = NULL;\n    int len = 0;\n    while (*cur1 &amp;&amp; *cur2)\n    {\n        if (*cur1 != *cur2)\n            break;\n        else\n            len++;\n        cur1++;\n        cur2++;\n    }\n    ret = (char *)memset(malloc(sizeof(char) * (len + 1)), 0, sizeof(char) * (len + 1));\n    return memcpy(ret, str1, sizeof(char) * len);\n}\n\nchar * longestCommonPrefix(char ** strs, int strsSize){\n    if (strsSize == 0)\n        return memset(malloc(sizeof(char)), 0, sizeof(char));\n    if (strsSize == 1)\n        return strs[0];\n    if (strsSize == 2)\n        return compareTwo(strs[0], strs[1]);\n    int rec_size = (strsSize + 1) &gt;&gt; 1, i = 0;\n    char **rec = (char **)memset(malloc(sizeof(char *) * rec_size), 0, sizeof(char *) * rec_size);\n    for (i = 0; i &lt; strsSize &gt;&gt; 1; i++)\n        rec[i] = compareTwo(strs[2 * i], strs[2 * i + 1]);\n    if (strsSize &amp; 1)\n        rec[rec_size - 1] = strs[strsSize - 1];\n    return longestCommonPrefix(rec, rec_size);\n}\n</code></pre>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/150/","title":"150. \u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u6c42\u503c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6808"]},{"location":"coding/leetcode/150/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u6839\u636e\u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u793a\u6cd5\u200b\uff0c\u200b\u6c42\u200b\u8868\u8fbe\u5f0f\u200b\u7684\u200b\u503c\u200b\u3002</p> <p>\u200b\u6709\u6548\u200b\u7684\u200b\u7b97\u7b26\u200b\u5305\u62ec\u200b<code>+</code>\u3001<code>-</code>\u3001<code>*</code>\u3001<code>/</code>\u3002\u200b\u6bcf\u4e2a\u200b\u8fd0\u7b97\u200b\u5bf9\u8c61\u200b\u53ef\u4ee5\u200b\u662f\u200b\u6574\u6570\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li>\u200b\u6574\u6570\u200b\u9664\u6cd5\u200b\u53ea\u200b\u4fdd\u7559\u200b\u6574\u6570\u200b\u90e8\u5206\u200b\u3002</li> <li>\u200b\u7ed9\u5b9a\u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u603b\u662f\u200b\u6709\u6548\u200b\u7684\u200b\u3002\u200b\u6362\u53e5\u8bdd\u8bf4\u200b\uff0c\u200b\u8868\u8fbe\u5f0f\u200b\u603b\u4f1a\u200b\u5f97\u51fa\u200b\u6709\u6548\u200b\u6570\u503c\u200b\u4e14\u200b\u4e0d\u200b\u5b58\u5728\u200b\u9664\u6570\u200b\u4e3a\u200b <code>0</code> \u200b\u7684\u200b\u60c5\u51b5\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1atokens = [\"2\",\"1\",\"+\",\"3\",\"*\"]\n\u200b\u8f93\u51fa\u200b\uff1a9\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u8be5\u200b\u7b97\u5f0f\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u5e38\u89c1\u200b\u7684\u200b\u4e2d\u7f00\u200b\u7b97\u672f\u200b\u8868\u8fbe\u5f0f\u200b\u4e3a\u200b\uff1a((2 + 1) * 3) = 9\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1atokens = [\"4\",\"13\",\"5\",\"/\",\"+\"]\n\u200b\u8f93\u51fa\u200b\uff1a6\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u8be5\u200b\u7b97\u5f0f\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u5e38\u89c1\u200b\u7684\u200b\u4e2d\u7f00\u200b\u7b97\u672f\u200b\u8868\u8fbe\u5f0f\u200b\u4e3a\u200b\uff1a(4 + (13 / 5)) = 6\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a03\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1atokens = [\"10\",\"6\",\"9\",\"3\",\"+\",\"-11\",\"*\",\"/\",\"*\",\"17\",\"+\",\"5\",\"+\"]\n\u200b\u8f93\u51fa\u200b\uff1a22\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u8be5\u200b\u7b97\u5f0f\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u5e38\u89c1\u200b\u7684\u200b\u4e2d\u7f00\u200b\u7b97\u672f\u200b\u8868\u8fbe\u5f0f\u200b\u4e3a\u200b\uff1a\n  ((10 * (6 / ((9 + 3) * -11))) + 17) + 5\n= ((10 * (6 / (12 * -11))) + 17) + 5\n= ((10 * (6 / -132)) + 17) + 5\n= ((10 * 0) + 17) + 5\n= (0 + 17) + 5\n= 17 + 5\n= 22\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= tokens.length &lt;= 10^4</code></li> <li><code>tokens[i]</code> \u200b\u8981\u4e48\u200b\u662f\u200b\u4e00\u4e2a\u200b\u7b97\u7b26\u200b\uff08<code>\"+\"</code>\u3001<code>\"-\"</code>\u3001<code>\"*\"</code> \u200b\u6216\u200b <code>\"/\"</code>\uff09\uff0c\u200b\u8981\u4e48\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5728\u200b\u8303\u56f4\u200b <code>[-200, 200]</code> \u200b\u5185\u200b\u7684\u200b\u6574\u6570\u200b</li> </ul> <p>\u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\uff1a</p> <p>\u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u662f\u200b\u4e00\u79cd\u200b\u540e\u7f00\u200b\u8868\u8fbe\u5f0f\u200b\uff0c\u200b\u6240\u8c13\u200b\u540e\u7f00\u200b\u5c31\u662f\u6307\u200b\u7b97\u7b26\u200b\u5199\u200b\u5728\u200b\u540e\u9762\u200b\u3002</p> <p>\u200b\u5e73\u5e38\u200b\u4f7f\u7528\u200b\u7684\u200b\u7b97\u5f0f\u200b\u5219\u200b\u662f\u200b\u4e00\u79cd\u200b\u4e2d\u7f00\u200b\u8868\u8fbe\u5f0f\u200b\uff0c\u200b\u5982\u200b <code>( 1 + 2 ) * ( 3 + 4 )</code> \u3002 \u200b\u8be5\u200b\u7b97\u5f0f\u200b\u7684\u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u5199\u6cd5\u200b\u4e3a\u200b <code>( ( 1 2 + ) ( 3 4 + ) * )</code> \u3002 \u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u4e3b\u8981\u200b\u6709\u200b\u4ee5\u4e0b\u200b\u4e24\u4e2a\u200b\u4f18\u70b9\u200b\uff1a</p> <p>\u200b\u53bb\u6389\u200b\u62ec\u53f7\u200b\u540e\u200b\u8868\u8fbe\u5f0f\u200b\u65e0\u200b\u6b67\u4e49\u200b\uff0c\u200b\u4e0a\u200b\u5f0f\u200b\u5373\u4fbf\u200b\u5199\u6210\u200b <code>1 2 + 3 4 + *</code> \u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4f9d\u636e\u200b\u6b21\u5e8f\u200b\u8ba1\u7b97\u200b\u51fa\u200b\u6b63\u786e\u200b\u7ed3\u679c\u200b\u3002 \u200b\u9002\u5408\u200b\u7528\u6808\u200b\u64cd\u4f5c\u200b\u8fd0\u7b97\u200b\uff1a\u200b\u9047\u5230\u200b\u6570\u5b57\u200b\u5219\u200b\u5165\u6808\u200b\uff1b\u200b\u9047\u5230\u200b\u7b97\u7b26\u200b\u5219\u200b\u53d6\u51fa\u200b\u6808\u9876\u200b\u4e24\u4e2a\u200b\u6570\u5b57\u200b\u8fdb\u884c\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u5e76\u200b\u5c06\u200b\u7ed3\u679c\u200b\u538b\u5165\u200b\u6808\u200b\u4e2d\u200b\u3002</p> <p>Reference</p>","tags":["\u6808"]},{"location":"coding/leetcode/150/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u6839\u636e\u200b\u9898\u76ee\u200b\u4e2d\u200b\u7684\u200b\u63d0\u793a\u200b\u5373\u53ef\u200b\uff1a</p> <p>\u200b\u9002\u5408\u200b\u7528\u6808\u200b\u64cd\u4f5c\u200b\u8fd0\u7b97\u200b\uff1a\u200b\u9047\u5230\u200b\u6570\u5b57\u200b\u5219\u200b\u5165\u6808\u200b\uff1b\u200b\u9047\u5230\u200b\u7b97\u7b26\u200b\u5219\u200b\u53d6\u51fa\u200b\u6808\u9876\u200b\u4e24\u4e2a\u200b\u6570\u5b57\u200b\u8fdb\u884c\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u5e76\u200b\u5c06\u200b\u7ed3\u679c\u200b\u538b\u5165\u200b\u6808\u200b\u4e2d\u200b\u3002</p> <pre><code>struct ListNode *getNode(int val, struct ListNode *next)\n{\n    struct ListNode *ret = (struct ListNode *)malloc(sizeof(struct ListNode));\n    ret-&gt;val = val;\n    ret-&gt;next = next;\n    return ret;\n}\nstruct ListNode *operand(struct ListNode *Stack, char operand)\n{\n    struct ListNode *cur1 = Stack, *cur2 = Stack-&gt;next, *temp = cur2-&gt;next;\n    int ret;\n    switch (operand)\n    {\n        case '+':\n            ret = cur1-&gt;val + cur2-&gt;val;\n            break;\n        case '-':\n            ret = cur2-&gt;val - cur1-&gt;val;\n            break;\n        case '*':\n            ret = cur1-&gt;val * cur2-&gt;val;\n            break;\n        case '/':\n            ret = cur2-&gt;val / cur1-&gt;val;\n            break;\n    }\n    free(cur1);\n    free(cur2);\n    return getNode(ret, temp);\n}\nint evalRPN(char ** tokens, int tokensSize){\n    struct ListNode *Stack = NULL;\n    int i = 0;\n    char *cur;\n    for (i = 0; i &lt; tokensSize; i++)\n    {\n        cur = tokens[i];\n        if (*cur &gt;= '0' &amp;&amp; *cur &lt;= '9' || strlen(cur) &gt; 1)\n            Stack = getNode(atoi(cur), Stack);\n        else\n            Stack = operand(Stack, *cur);\n    }\n    return Stack-&gt;val;\n}\n</code></pre>","tags":["\u6808"]},{"location":"coding/leetcode/155/","title":"155. \u200b\u6700\u5c0f\u200b\u6808","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6808","\u8bbe\u8ba1"]},{"location":"coding/leetcode/155/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u652f\u6301\u200b <code>push</code>\uff0c<code>pop</code>\uff0c<code>top</code>\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u5e76\u200b\u80fd\u200b\u5728\u200b\u5e38\u6570\u200b\u65f6\u95f4\u200b\u5185\u200b\u68c0\u7d22\u200b\u5230\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b\u7684\u200b\u6808\u200b\u3002</p> <ul> <li><code>push(x)</code> \u2014\u2014 \u200b\u5c06\u200b\u5143\u7d20\u200b x \u200b\u63a8\u5165\u200b\u6808\u4e2d\u200b\u3002</li> <li><code>pop()</code>\u00a0\u2014\u2014 \u200b\u5220\u9664\u200b\u6808\u9876\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</li> <li><code>top()</code>\u00a0\u2014\u2014 \u200b\u83b7\u53d6\u200b\u6808\u9876\u200b\u5143\u7d20\u200b\u3002</li> <li><code>getMin()</code> \u2014\u2014 \u200b\u68c0\u7d22\u200b\u6808\u4e2d\u200b\u7684\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a\n[\"MinStack\",\"push\",\"push\",\"push\",\"getMin\",\"pop\",\"top\",\"getMin\"]\n[[],[-2],[0],[-3],[],[],[],[]]\n\n\u200b\u8f93\u51fa\u200b\uff1a\n[null,null,null,null,-3,null,0,-2]\n\n\u200b\u89e3\u91ca\u200b\uff1a\nMinStack minStack = new MinStack();\nminStack.push(-2);\nminStack.push(0);\nminStack.push(-3);\nminStack.getMin();   --&gt; \u200b\u8fd4\u56de\u200b -3.\nminStack.pop();\nminStack.top();      --&gt; \u200b\u8fd4\u56de\u200b 0.\nminStack.getMin();   --&gt; \u200b\u8fd4\u56de\u200b -2.\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <p><code>pop</code>\u3001<code>top</code> \u200b\u548c\u200b <code>getMin</code> \u200b\u64cd\u4f5c\u200b\u603b\u662f\u200b\u5728\u200b \u200b\u975e\u7a7a\u6808\u200b \u200b\u4e0a\u200b\u8c03\u7528\u200b\u3002</p> <p>Reference</p>","tags":["\u6808","\u8bbe\u8ba1"]},{"location":"coding/leetcode/155/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4f7f\u7528\u200b\u4e24\u4e2a\u200b\u57df\u200b\u5206\u522b\u200b\u5b58\u50a8\u200b\u5f53\u524d\u200b\u6570\u503c\u200b\u548c\u200b\u5f53\u524d\u200b\u6808\u4e2d\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>#define MIN(x, y) (x &lt; y ? x : y)\nstruct cell\n{\n    int val;\n    int curMin;\n};\ntypedef struct {\n    struct cell values[10000];\n    int top;\n} MinStack;\n/** initialize your data structure here. */\nMinStack *minStackCreate() \n{\n    return (MinStack *)memset(malloc(sizeof(MinStack)), 0, sizeof(MinStack));\n}\nvoid minStackPush(MinStack* obj, int x)\n{\n    obj-&gt;values[obj-&gt;top].val = x;\n    obj-&gt;values[obj-&gt;top].curMin = obj-&gt;top &gt; 0 ? MIN(x, obj-&gt;values[obj-&gt;top - 1].curMin) : x;\n    obj-&gt;top++;\n}\nvoid minStackPop(MinStack *obj)\n{\n    obj-&gt;top--;\n}\nint minStackTop(MinStack* obj)\n{\n    return obj-&gt;values[obj-&gt;top - 1].val;\n}\nint minStackGetMin(MinStack* obj)\n{\n    return obj-&gt;values[obj-&gt;top - 1].curMin;\n}\nvoid minStackFree(MinStack* obj)\n{\n    free(obj);\n}\n/**\n * Your MinStack struct will be instantiated and called as such:\n * MinStack* obj = minStackCreate();\n * minStackPush(obj, x);\n\n * minStackPop(obj);\n\n * int param_3 = minStackTop(obj);\n\n * int param_4 = minStackGetMin(obj);\n\n * minStackFree(obj);\n*/\n</code></pre>","tags":["\u6808","\u8bbe\u8ba1"]},{"location":"coding/leetcode/1603/","title":"1603. \u200b\u8bbe\u8ba1\u200b\u505c\u8f66\u200b\u7cfb\u7edf","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u8bbe\u8ba1"]},{"location":"coding/leetcode/1603/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u8bf7\u200b\u4f60\u200b\u7ed9\u200b\u4e00\u4e2a\u200b\u505c\u8f66\u573a\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u505c\u8f66\u200b\u7cfb\u7edf\u200b\u3002\u200b\u505c\u8f66\u573a\u200b\u603b\u5171\u200b\u6709\u200b\u4e09\u79cd\u200b\u4e0d\u540c\u200b\u5927\u5c0f\u200b\u7684\u200b\u8f66\u4f4d\u200b\uff1a\u200b\u5927\u200b\uff0c\u200b\u4e2d\u200b\u548c\u200b\u5c0f\u200b\uff0c\u200b\u6bcf\u79cd\u200b\u5c3a\u5bf8\u200b\u5206\u522b\u200b\u6709\u200b\u56fa\u5b9a\u200b\u6570\u76ee\u200b\u7684\u200b\u8f66\u4f4d\u200b\u3002</p> <p>\u200b\u8bf7\u200b\u4f60\u200b\u5b9e\u73b0\u200b\u00a0<code>ParkingSystem</code>\u00a0\u200b\u7c7b\u200b\uff1a</p> <p><code>ParkingSystem(int big, int medium, int small)``\u00a0\u200b\u521d\u59cb\u5316\u200b\u00a0ParkingSystem\u00a0\u200b\u7c7b\u200b\uff0c\u200b\u4e09\u4e2a\u200b\u53c2\u6570\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u6bcf\u79cd\u200b\u505c\u8f66\u4f4d\u200b\u7684\u200b\u6570\u76ee\u200b\u3002</code> <code>bool addCar(int carType)</code>\u00a0\u200b\u68c0\u67e5\u200b\u662f\u5426\u200b\u6709\u200b\u00a0<code>carType</code>\u00a0\u200b\u5bf9\u5e94\u200b\u7684\u200b\u505c\u8f66\u4f4d\u200b\u3002\u00a0<code>carType</code>\u00a0\u200b\u6709\u200b\u4e09\u79cd\u200b\u7c7b\u578b\u200b\uff1a\u200b\u5927\u200b\uff0c\u200b\u4e2d\u200b\uff0c\u200b\u5c0f\u200b\uff0c\u200b\u5206\u522b\u200b\u7528\u200b\u6570\u5b57\u200b\u00a0<code>1</code>\uff0c\u00a0<code>2</code>\u00a0\u200b\u548c\u200b\u00a0<code>3</code>\u00a0\u200b\u8868\u793a\u200b\u3002\u200b\u4e00\u8f86\u8f66\u200b\u53ea\u80fd\u200b\u505c\u200b\u5728\u200b\u00a0<code>carType</code>\u00a0\u200b\u5bf9\u5e94\u200b\u5c3a\u5bf8\u200b\u7684\u200b\u505c\u8f66\u4f4d\u200b\u4e2d\u200b\u3002\u200b\u5982\u679c\u200b\u6ca1\u6709\u200b\u7a7a\u8f66\u200b\u4f4d\u200b\uff0c\u200b\u8bf7\u200b\u8fd4\u56de\u200b\u00a0<code>false</code>\u00a0\uff0c\u200b\u5426\u5219\u200b\u5c06\u200b\u8be5\u8f66\u200b\u505c\u5165\u200b\u8f66\u4f4d\u200b\u5e76\u200b\u8fd4\u56de\u200b\u00a0<code>true</code>\u00a0\u3002\u00a0</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a\n[\"ParkingSystem\", \"addCar\", \"addCar\", \"addCar\", \"addCar\"]\n[[1, 1, 0], [1], [2], [3], [1]]\n\u200b\u8f93\u51fa\u200b\uff1a\n[null, true, true, false, false]\n</code></pre> <p>\u200b\u89e3\u91ca\u200b\uff1a</p> <pre><code>ParkingSystem parkingSystem = new ParkingSystem(1, 1, 0);\nparkingSystem.addCar(1); // \u200b\u8fd4\u56de\u200b true \uff0c\u200b\u56e0\u4e3a\u200b\u6709\u200b 1 \u200b\u4e2a\u7a7a\u200b\u7684\u200b\u5927\u200b\u8f66\u4f4d\u200b\nparkingSystem.addCar(2); // \u200b\u8fd4\u56de\u200b true \uff0c\u200b\u56e0\u4e3a\u200b\u6709\u200b 1 \u200b\u4e2a\u7a7a\u200b\u7684\u200b\u4e2d\u200b\u8f66\u4f4d\u200b\nparkingSystem.addCar(3); // \u200b\u8fd4\u56de\u200b false \uff0c\u200b\u56e0\u4e3a\u200b\u6ca1\u6709\u200b\u7a7a\u200b\u7684\u200b\u5c0f\u8f66\u200b\u4f4d\u200b\nparkingSystem.addCar(1); // \u200b\u8fd4\u56de\u200b false \uff0c\u200b\u56e0\u4e3a\u200b\u6ca1\u6709\u200b\u7a7a\u200b\u7684\u200b\u5927\u200b\u8f66\u4f4d\u200b\uff0c\u200b\u552f\u4e00\u200b\u4e00\u4e2a\u200b\u5927\u200b\u8f66\u4f4d\u200b\u5df2\u7ecf\u200b\u88ab\u200b\u5360\u636e\u200b\u4e86\u200b\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>0 &lt;= big, medium, small &lt;= 1000</code></li> <li><code>carType</code>\u00a0\u200b\u53d6\u503c\u200b\u4e3a\u200b\u00a0<code>1</code>\uff0c\u00a0<code>2</code>\u00a0\u200b\u6216\u200b\u00a0<code>3</code></li> <li>\u200b\u6700\u591a\u4f1a\u200b\u8c03\u7528\u200b\u00a0<code>addCar</code>\u00a0\u200b\u51fd\u6570\u200b\u00a01000\u00a0\u200b\u6b21\u200b</li> </ul> <p>Reference</p>","tags":["\u8bbe\u8ba1"]},{"location":"coding/leetcode/1603/#_2","title":"\u9898\u89e3","text":"<pre><code>typedef struct {\n    int cars[3];\n} ParkingSystem;\n\n\nParkingSystem* parkingSystemCreate(int big, int medium, int small) {\n    ParkingSystem *ret = (ParkingSystem *)malloc(sizeof(ParkingSystem));\n    ret-&gt;cars[0] = big;\n    ret-&gt;cars[1] = medium;\n    ret-&gt;cars[2] = small;\n    return ret;\n}\n\nbool parkingSystemAddCar(ParkingSystem* obj, int carType) {\n    if (obj-&gt;cars[carType - 1] == 0)\n        return false;\n    else\n    {\n        obj-&gt;cars[carType - 1]--;\n        return true;\n    }\n}\n\nvoid parkingSystemFree(ParkingSystem* obj) {\n    free(obj);\n}\n\n/**\n * Your ParkingSystem struct will be instantiated and called as such:\n * ParkingSystem* obj = parkingSystemCreate(big, medium, small);\n * bool param_1 = parkingSystemAddCar(obj, carType);\n\n * parkingSystemFree(obj);\n*/\n</code></pre>","tags":["\u8bbe\u8ba1"]},{"location":"coding/leetcode/17.21/","title":"\u9762\u8bd5\u9898\u200b 17.21. \u200b\u76f4\u65b9\u56fe\u200b\u7684\u200b\u6c34\u91cf","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>"},{"location":"coding/leetcode/17.21/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u76f4\u65b9\u56fe\u200b(\u200b\u4e5f\u200b\u79f0\u200b\u67f1\u72b6\u56fe\u200b)\uff0c\u200b\u5047\u8bbe\u200b\u6709\u4eba\u200b\u4ece\u200b\u4e0a\u9762\u200b\u6e90\u6e90\u4e0d\u65ad\u200b\u5730\u200b\u5012\u6c34\u200b\uff0c\u200b\u6700\u540e\u200b\u76f4\u65b9\u56fe\u200b\u80fd\u5b58\u200b\u591a\u5c11\u200b\u6c34\u91cf\u200b?\u200b\u76f4\u65b9\u56fe\u200b\u7684\u200b\u5bbd\u5ea6\u200b\u4e3a\u200b <code>1</code>\u3002</p> <p></p> <p>\u200b\u4e0a\u9762\u200b\u662f\u200b\u7531\u200b\u6570\u7ec4\u200b <code>[0,1,0,2,1,0,1,3,2,1,2,1]</code> \u200b\u8868\u793a\u200b\u7684\u200b\u76f4\u65b9\u56fe\u200b\uff0c\u200b\u5728\u200b\u8fd9\u79cd\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u63a5\u200b <code>6</code> \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u7684\u200b\u6c34\u200b\uff08\u200b\u84dd\u8272\u200b\u90e8\u5206\u200b\u8868\u793a\u200b\u6c34\u200b\uff09\u3002\u00a0\u200b\u611f\u8c22\u200b Marcos \u200b\u8d21\u732e\u200b\u6b64\u56fe\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b:</p> <pre><code>\u200b\u8f93\u5165\u200b: [0,1,0,2,1,0,1,3,2,1,2,1]\n\u200b\u8f93\u51fa\u200b: 6\n</code></pre> <p>Reference</p> <p>\u200b\u6ce8\u200b\uff1a\u200b\u672c\u9898\u200b\u4e0e\u200b42. \u200b\u63a5\u200b\u96e8\u6c34\u200b\u76f8\u540c\u200b\u3002</p>"},{"location":"coding/leetcode/17.21/#_2","title":"\u9898\u89e3","text":"<pre><code>#define MAX(x, y) ((x) &gt; (y) ? x : y)\n#define MIN(x, y) ((x) &lt; (y) ? x : y)\n#define ABS(x) ((x) &gt; 0 ? (x) : -(x))\nint trap(int* height, int heightSize){\n    int highest = 0, higher = 0, highestIndex = -1, higherIndex = -1, i = 0, ret = 0;\n    for (i = 0; i &lt; heightSize; i++)\n    {\n        if (height[i] &gt; highest)\n        {\n            higher = highest;\n            higherIndex = highestIndex;\n            highest = height[i];\n            highestIndex = i;\n        }\n        else if (height[i] &gt; higher)\n        {\n            higher = height[i];\n            higherIndex = i;\n        }\n    }\n    if (heightSize &lt;= 2 || higherIndex &lt; 0 || highestIndex &lt; 0)\n        return 0;\n    ret = higher * (ABS(higherIndex - highestIndex) - 1);\n    int left = MIN(higherIndex, highestIndex) + 1, right = MAX(higherIndex, highestIndex);\n    for (i = left; i &lt; right; i++)\n        ret -= height[i];\n    return ret + trap(height, left) + trap(height + right, heightSize - right);\n}\n</code></pre>"},{"location":"coding/leetcode/17/","title":"17. \u200b\u7535\u8bdd\u53f7\u7801\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u9012\u5f52","\u5b57\u7b26\u4e32","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/17/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u4ec5\u200b\u5305\u542b\u200b\u6570\u5b57\u200b\u00a0<code>2-9</code>\u00a0\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u8fd4\u56de\u200b\u6240\u6709\u200b\u5b83\u200b\u80fd\u200b\u8868\u793a\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408\u200b\u3002\u200b\u7b54\u6848\u200b\u53ef\u4ee5\u200b\u6309\u200b \u200b\u4efb\u610f\u200b\u987a\u5e8f\u200b \u200b\u8fd4\u56de\u200b\u3002</p> <p>\u200b\u7ed9\u51fa\u200b\u6570\u5b57\u200b\u5230\u200b\u5b57\u6bcd\u200b\u7684\u200b\u6620\u5c04\u200b\u5982\u4e0b\u200b\uff08\u200b\u4e0e\u200b\u7535\u8bdd\u200b\u6309\u952e\u200b\u76f8\u540c\u200b\uff09\u3002\u200b\u6ce8\u610f\u200b 1 \u200b\u4e0d\u200b\u5bf9\u5e94\u200b\u4efb\u4f55\u200b\u5b57\u6bcd\u200b\u3002</p> <p></p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1adigits = \"23\"\n\u200b\u8f93\u51fa\u200b\uff1a[\"ad\",\"ae\",\"af\",\"bd\",\"be\",\"bf\",\"cd\",\"ce\",\"cf\"]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1adigits = \"\"\n\u200b\u8f93\u51fa\u200b\uff1a[]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1adigits = \"2\"\n\u200b\u8f93\u51fa\u200b\uff1a[\"a\",\"b\",\"c\"]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>0 &lt;= digits.length &lt;= 4</li> <li>digits[i] \u200b\u662f\u200b\u8303\u56f4\u200b ['2', '9'] \u200b\u7684\u200b\u4e00\u4e2a\u200b\u6570\u5b57\u200b\u3002</li> </ul> <p>Reference</p>","tags":["\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u9012\u5f52","\u5b57\u7b26\u4e32","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/17/#_2","title":"\u9898\u89e3","text":"<pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\nvoid dfs(char *digits, char **ret, int *subscript, char *current, int end, char **map)\n{\n    if (*digits == 0)\n    {\n        strcpy(ret[(*subscript)++], current);\n        return;\n    }\n    int i = 0, target = *digits - '2';\n    char *cur = map[target];\n    while(*cur)\n    {\n        current[end] = *cur;\n        dfs(digits + 1, ret, subscript, current, end + 1, map);\n        cur++;\n    }\n}\nchar ** letterCombinations(char * digits, int* returnSize){\n    *returnSize = 1;\n    char *cur = digits, len = 0, *charMap[] = {\"abc\", \"def\", \"ghi\", \"jkl\", \"mno\", \"pqrs\", \"tuv\", \"wxyz\"};\n    while(*cur)\n    {\n        *returnSize *= (*cur == '7' || *cur == '9') ? 4 : 3;\n        cur++;\n        len++;\n    }\n    if (len == 0)\n    {\n        *returnSize = 0;\n        return NULL;\n    }\n    char **ret = (char **)malloc(sizeof(char *) * *returnSize),\n         *current = (char *)memset(malloc(sizeof(char) * (len + 1)), 0, sizeof(char) * (len + 1));\n    for (int i = 0; i &lt; *returnSize; i++)\n        ret[i] = (int *)memset(malloc(sizeof(char) * (len + 1)), 0, sizeof(char) * (len + 1));\n    int subscript = 0;\n    dfs(digits, ret, &amp;subscript, current, 0, charMap);\n    return ret;\n}\n</code></pre>","tags":["\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u9012\u5f52","\u5b57\u7b26\u4e32","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/1835/","title":"1835. \u200b\u6240\u6709\u200b\u6570\u5bf9\u200b\u6309\u4f4d\u200b\u4e0e\u200b\u7ed3\u679c\u200b\u7684\u200b\u5f02\u6216\u200b\u548c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/1835/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5217\u8868\u200b\u7684\u200b \u200b\u5f02\u6216\u200b\u548c\u200b\uff08XOR sum\uff09\u200b\u6307\u5bf9\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u8fdb\u884c\u200b\u6309\u4f4d\u200b <code>XOR</code> \u200b\u8fd0\u7b97\u200b\u7684\u200b\u7ed3\u679c\u200b\u3002\u200b\u5982\u679c\u200b\u5217\u8868\u200b\u4e2d\u4ec5\u200b\u6709\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u90a3\u4e48\u200b\u5176\u200b \u200b\u5f02\u6216\u200b\u548c\u200b \u200b\u5c31\u200b\u7b49\u4e8e\u200b\u8be5\u200b\u5143\u7d20\u200b\u3002</p> <ul> <li>\u200b\u4f8b\u5982\u200b\uff0c<code>[1,2,3,4]</code> \u200b\u7684\u200b \u200b\u5f02\u6216\u200b\u548c\u200b \u200b\u7b49\u4e8e\u200b <code>1 XOR 2 XOR 3 XOR 4 = 4</code> \uff0c\u200b\u800c\u200b <code>[3]</code> \u200b\u7684\u200b \u200b\u5f02\u6216\u200b\u548c\u200b \u200b\u7b49\u4e8e\u200b <code>3</code> \u3002</li> </ul> <p>\u200b\u7ed9\u200b\u4f60\u200b\u4e24\u4e2a\u200b\u4e0b\u6807\u200b \u200b\u4ece\u200b 0 \u200b\u5f00\u59cb\u200b \u200b\u8ba1\u6570\u200b\u7684\u200b\u6570\u7ec4\u200b <code>arr1</code> \u200b\u548c\u200b <code>arr2</code> \uff0c\u200b\u4e24\u200b\u6570\u7ec4\u200b\u5747\u200b\u7531\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u7ec4\u6210\u200b\u3002</p> <p>\u200b\u6839\u636e\u200b\u6bcf\u4e2a\u200b\u00a0<code>(i, j)</code> \u200b\u6570\u5bf9\u200b\uff0c\u200b\u6784\u9020\u200b\u4e00\u4e2a\u200b\u7531\u200b <code>arr1[i] AND arr2[j]</code>\uff08\u200b\u6309\u4f4d\u200b <code>AND</code> \u200b\u8fd0\u7b97\u200b\uff09\u200b\u7ed3\u679c\u200b\u7ec4\u6210\u200b\u7684\u200b\u5217\u8868\u200b\u3002\u200b\u5176\u4e2d\u200b <code>0 &lt;= i &lt; arr1.length</code> \u200b\u4e14\u200b <code>0 &lt;= j &lt; arr2.length</code> \u3002</p> <p>\u200b\u8fd4\u56de\u200b\u4e0a\u8ff0\u200b\u5217\u8868\u200b\u7684\u200b \u200b\u5f02\u6216\u200b\u548c\u200b \u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aarr1 = [1,2,3], arr2 = [6,5]\n\u200b\u8f93\u51fa\u200b\uff1a0\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5217\u8868\u200b = [1 AND 6, 1 AND 5, 2 AND 6, 2 AND 5, 3 AND 6, 3 AND 5] = [0,1,2,0,2,1] \uff0c\n\u200b\u5f02\u6216\u200b\u548c\u200b = 0 XOR 1 XOR 2 XOR 0 XOR 2 XOR 1 = 0 \u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aarr1 = [12], arr2 = [4]\n\u200b\u8f93\u51fa\u200b\uff1a4\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5217\u8868\u200b = [12 AND 4] = [4] \uff0c\u200b\u5f02\u6216\u200b\u548c\u200b = 4 \u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= arr1.length, arr2.length &lt;= 105</code></li> <li><code>0 &lt;= arr1[i], arr2[j] &lt;= 109</code></li> </ul> <p>Reference</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/1835/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u8bbe\u200b\u4e24\u4e2a\u200b\u6570\u5217\u200b\\(A=\\{a_i\\}_m\\)\u200b\u4e0e\u200b\\(B=\\{b_j\\}_n\\)\uff0c\u200b\u5e76\u200b\u4f7f\u7528\u200b\u8fd0\u7b97\u7b26\u200b\\(\\otimes\\)\u200b\u8868\u793a\u200b \u200b\u5f02\u6216\u200b \u200b\u64cd\u4f5c\u200b\uff0c\u200b\u4f7f\u7528\u200b\u8fd0\u7b97\u7b26\u200b\\(\\land\\)\u200b\u8868\u793a\u200b \u200b\u4e0e\u200b \u200b\u64cd\u4f5c\u200b\uff0c\u200b\u5219\u200b\u672c\u9898\u200b\u7684\u200b\u7ed3\u679c\u200b\u4e3a\u200b\uff1a</p> \\[ f(A, B) = \\bigotimes_{i=1}^{m} \\bigotimes_{j=1}^n a_i\\land b_j \\] <p>\u200b\u8003\u8651\u200b\u8868\u8fbe\u5f0f\u200b\\((a\\otimes b) \\land (a \\otimes c)\\)\uff1a</p> \\[ \\begin{aligned} &amp; (a\\otimes b) \\land (a \\otimes c) \\\\ \\Leftrightarrow &amp; ((a\\lor b)\\land (\\lnot a\\lor\\lnot b)) \\land ((a\\lor c)\\land (\\lnot a\\lor\\lnot c)) \\\\ \\Leftrightarrow &amp; (a\\lor b)\\land (\\lnot a\\lor\\lnot b) \\land (a\\lor c)\\land (\\lnot a\\lor\\lnot c) \\\\ \\Leftrightarrow &amp; (a\\land (b\\lor c)) \\land (a\\land (\\lnot b\\lor\\lnot c)) \\\\ \\Leftrightarrow &amp; a\\land (b\\otimes c) \\end{aligned} \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\\(\\otimes\\)\u200b\u8fd0\u7b97\u200b\u5bf9\u200b\\(\\land\\)\u200b\u8fd0\u7b97\u200b\u6ee1\u8db3\u200b\u5206\u914d\u5f8b\u200b\uff0c\u200b\u6709\u200b\uff1a</p> \\[ \\begin{aligned} f(A, B) &amp;= \\bigotimes_{i=1}^{m} \\bigotimes_{j=1}^n a_i\\land b_j \\\\ &amp;= \\bigotimes_{i=1}^{m} \\left(a_{i}\\land\\bigotimes_{j=1}^n b_j\\right) \\\\ &amp;= \\left(\\bigotimes_{i=1}^{m} a_i\\right)\\land\\left(\\bigotimes_{j=1}^n b_j\\right) \\end{aligned} \\] <pre><code>int getXORSum(int* arr1, int arr1Size, int* arr2, int arr2Size){\n    int a = 0, b = 0, i = 0;\n    for (i = 0; i &lt; arr1Size; i++)\n        a ^= arr1[i];\n    for (i = 0; i &lt; arr2Size; i++)\n        b ^= arr2[i];\n    return a &amp; b;\n}\n</code></pre>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/189/","title":"189. \u200b\u65cb\u8f6c\u200b\u6570\u7ec4","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/189/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5c06\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u00a0k\u00a0\u200b\u4e2a\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u5176\u4e2d\u200b\u00a0k\u00a0\u200b\u662f\u975e\u200b\u8d1f\u6570\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,2,3,4,5,6,7] \u200b\u548c\u200b k = 3\n\u200b\u8f93\u51fa\u200b: [5,6,7,1,2,3,4]\n\u200b\u89e3\u91ca\u200b:\n\u200b\u5411\u200b\u53f3\u200b\u65cb\u8f6c\u200b 1 \u200b\u6b65\u200b: [7,1,2,3,4,5,6]\n\u200b\u5411\u200b\u53f3\u200b\u65cb\u8f6c\u200b 2 \u200b\u6b65\u200b: [6,7,1,2,3,4,5]\n\u200b\u5411\u200b\u53f3\u200b\u65cb\u8f6c\u200b 3 \u200b\u6b65\u200b: [5,6,7,1,2,3,4]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [-1,-100,3,99] \u200b\u548c\u200b k = 2\n\u200b\u8f93\u51fa\u200b: [3,99,-1,-100]\n\u200b\u89e3\u91ca\u200b: \n\u200b\u5411\u200b\u53f3\u200b\u65cb\u8f6c\u200b 1 \u200b\u6b65\u200b: [99,-1,-100,3]\n\u200b\u5411\u200b\u53f3\u200b\u65cb\u8f6c\u200b 2 \u200b\u6b65\u200b: [3,99,-1,-100]\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li>\u200b\u5c3d\u53ef\u80fd\u200b\u60f3\u51fa\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u89e3\u51b3\u65b9\u6848\u200b\uff0c\u200b\u81f3\u5c11\u200b\u6709\u200b\u4e09\u79cd\u200b\u4e0d\u540c\u200b\u7684\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u89e3\u51b3\u200b\u8fd9\u4e2a\u200b\u95ee\u9898\u200b\u3002</li> <li>\u200b\u8981\u6c42\u200b\u4f7f\u7528\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(O(1)\\)\u200b\u7684\u200b\u539f\u5730\u200b\u7b97\u6cd5\u200b\u3002</li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/189/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5148\u200b\u53cd\u8f6c\u200b\u6574\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u518d\u200b\u4ee5\u200b\u7b2c\u200b\\(k\\)\u200b\u4e2a\u200b\u5143\u7d20\u200b\u4e3a\u754c\u200b\uff0c\u200b\u5206\u522b\u200b\u53cd\u8f6c\u200b\\(k\\)\u200b\u4e4b\u524d\u200b\u7684\u200b\u90e8\u5206\u200b\u4e0e\u200b\\(k\\)\u200b\u53ca\u200b\\(k\\)\u200b\u4e4b\u540e\u200b\u7684\u200b\u90e8\u5206\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>void Reverse(int* nums, int numsSize)\n{\n    int head = 0, tail = numsSize - 1, temp = 0;\n    while(tail &gt; head)\n    {\n        temp = nums[head];\n        nums[head] = nums[tail];\n        nums[tail] = temp;\n        tail--;\n        head++;\n    }\n}\nvoid rotate(int* nums, int numsSize, int k){\n    if (numsSize &lt; 2)\n        return;\n    k %= numsSize;\n    Reverse(nums, numsSize);\n    Reverse(nums, k);\n    Reverse(nums + k, numsSize - k);\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/19/","title":"19. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u7684\u200b\u5012\u6570\u7b2c\u200bN\u200b\u4e2a\u200b\u8282\u70b9","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/19/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b\uff0c\u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u7684\u200b\u5012\u6570\u7b2c\u200b\u00a0n\u00a0\u200b\u4e2a\u200b\u8282\u70b9\u200b\uff0c\u200b\u5e76\u4e14\u200b\u8fd4\u56de\u200b\u94fe\u8868\u200b\u7684\u200b\u5934\u200b\u7ed3\u70b9\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <p>`` \u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b: 1-&gt;2-&gt;3-&gt;4-&gt;5, \u200b\u548c\u200b n = 2.</p> <p>\u200b\u5f53\u200b\u5220\u9664\u200b\u4e86\u200b\u5012\u6570\u200b\u7b2c\u4e8c\u4e2a\u200b\u8282\u70b9\u200b\u540e\u200b\uff0c\u200b\u94fe\u8868\u200b\u53d8\u4e3a\u200b 1-&gt;2-&gt;3-&gt;5. ``</p> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u7ed9\u5b9a\u200b\u7684\u200b n\u00a0\u200b\u4fdd\u8bc1\u200b\u662f\u200b\u6709\u6548\u200b\u7684\u200b\u3002</p> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u80fd\u200b\u5c1d\u8bd5\u200b\u4f7f\u7528\u200b\u4e00\u8d9f\u200b\u626b\u63cf\u200b\u5b9e\u73b0\u200b\u5417\u200b\uff1f</p> <p>Reference</p>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/19/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5148\u200b\u627e\u5230\u200b\u5012\u6570\u7b2c\u200b n \u200b\u4e2a\u200b\u8282\u70b9\u200b\uff0c\u200b\u518d\u200b\u8fdb\u884c\u200b\u5220\u9664\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\nstruct ListNode* removeNthFromEnd(struct ListNode* head, int n){\n    struct ListNode *cur = head, *temp = NULL;\n    int length = 1, i = 0;\n    while(cur-&gt;next != NULL)\n    {\n        cur = cur-&gt;next;\n        length++;\n    }\n    cur = head;\n    for (i = 0; i &lt; length - n - 1; i++)\n        cur = cur-&gt;next;\n    if (n == length)\n        return cur-&gt;next;\n    if (cur-&gt;next)\n    {\n        cur-&gt;next = cur-&gt;next-&gt;next;\n        return head;\n    }\n    return NULL;\n}\n</code></pre>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/198/","title":"198. \u200b\u6253\u5bb6\u52ab\u820d","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/198/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u4f60\u200b\u662f\u200b\u4e00\u4e2a\u200b\u4e13\u4e1a\u200b\u7684\u200b\u5c0f\u5077\u200b\uff0c\u200b\u8ba1\u5212\u200b\u5077\u7a83\u200b\u6cbf\u8857\u200b\u7684\u200b\u623f\u5c4b\u200b\u3002\u200b\u6bcf\u95f4\u623f\u200b\u5185\u200b\u90fd\u200b\u85cf\u6709\u200b\u4e00\u5b9a\u200b\u7684\u200b\u73b0\u91d1\u200b\uff0c\u200b\u5f71\u54cd\u200b\u4f60\u200b\u5077\u7a83\u200b\u7684\u200b\u552f\u4e00\u200b\u5236\u7ea6\u200b\u56e0\u7d20\u200b\u5c31\u662f\u200b\u76f8\u90bb\u200b\u7684\u200b\u623f\u5c4b\u200b\u88c5\u6709\u200b\u76f8\u4e92\u200b\u8fde\u901a\u200b\u7684\u200b\u9632\u76d7\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u5982\u679c\u200b\u4e24\u95f4\u200b\u76f8\u90bb\u200b\u7684\u200b\u623f\u5c4b\u200b\u5728\u200b\u540c\u4e00\u200b\u665a\u4e0a\u200b\u88ab\u200b\u5c0f\u5077\u200b\u95ef\u5165\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u62a5\u8b66\u200b\u3002</p> <p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u4ee3\u8868\u200b\u6bcf\u4e2a\u200b\u623f\u5c4b\u200b\u5b58\u653e\u200b\u91d1\u989d\u200b\u7684\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u4f60\u200b \u200b\u4e0d\u200b\u89e6\u52a8\u200b\u8b66\u62a5\u200b\u88c5\u7f6e\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b \uff0c\u200b\u4e00\u591c\u200b\u4e4b\u5185\u200b\u80fd\u591f\u200b\u5077\u7a83\u200b\u5230\u200b\u7684\u200b\u6700\u9ad8\u200b\u91d1\u989d\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a[1,2,3,1]\n\u200b\u8f93\u51fa\u200b\uff1a4\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5077\u7a83\u200b 1 \u200b\u53f7\u200b\u623f\u5c4b\u200b (\u200b\u91d1\u989d\u200b = 1) \uff0c\u200b\u7136\u540e\u200b\u5077\u7a83\u200b 3 \u200b\u53f7\u200b\u623f\u5c4b\u200b (\u200b\u91d1\u989d\u200b = 3)\u3002\n\u00a0    \u200b\u5077\u7a83\u200b\u5230\u200b\u7684\u200b\u6700\u9ad8\u200b\u91d1\u989d\u200b = 1 + 3 = 4 \u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a[2,7,9,3,1]\n\u200b\u8f93\u51fa\u200b\uff1a12\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5077\u7a83\u200b 1 \u200b\u53f7\u200b\u623f\u5c4b\u200b (\u200b\u91d1\u989d\u200b = 2), \u200b\u5077\u7a83\u200b 3 \u200b\u53f7\u200b\u623f\u5c4b\u200b (\u200b\u91d1\u989d\u200b = 9)\uff0c\u200b\u63a5\u7740\u200b\u5077\u7a83\u200b 5 \u200b\u53f7\u200b\u623f\u5c4b\u200b (\u200b\u91d1\u989d\u200b = 1)\u3002\n\u00a0    \u200b\u5077\u7a83\u200b\u5230\u200b\u7684\u200b\u6700\u9ad8\u200b\u91d1\u989d\u200b = 2 + 9 + 1 = 12 \u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <pre><code>0 &lt;= nums.length &lt;= 100\n0 &lt;= nums[i] &lt;= 400\n</code></pre> <p>Reference</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/198/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\uff0c\u200b\u4ee4\u200b\\(f(i)\\)\u200b\u8868\u793a\u200b\u524d\u200b\\(i\\)\u200b\u4e2a\u200b\u623f\u95f4\u200b\u6240\u200b\u80fd\u200b\u83b7\u53d6\u200b\u7684\u200b\u6700\u5927\u200b\u91d1\u989d\u200b\uff0c\u200b\u5219\u200b\u6709\u200b\uff1a</p> \\[ f(i + 1) = \\max\\left\\{f(i), a_{i + 1} + f(i - 1)\\right\\} \\] <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>#define MAX(x, y) (x &gt; y ? x : y)\nint rob(int* nums, int numsSize){\n    if (numsSize &lt; 2)\n        return numsSize ? *nums : 0;\n    int *dp = (int *)memset(malloc(sizeof(int) * (numsSize)), 0, sizeof(int) * (numsSize)), i = 0;\n    dp[0] = *nums;\n    dp[1] = MAX(nums[0], nums[1]);\n    for (i = 2; i &lt; numsSize; i++)\n        dp[i] = MAX(dp[i-2] + nums[i], dp[i-1]);\n    return dp[numsSize-1];\n}\n</code></pre>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/2/","title":"2. \u200b\u4e24\u200b\u6570\u200b\u76f8\u52a0","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u9012\u5f52","\u94fe\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/2/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e24\u4e2a\u200b\u00a0\u200b\u975e\u7a7a\u200b \u200b\u7684\u200b\u94fe\u8868\u200b\uff0c\u200b\u8868\u793a\u200b\u4e24\u4e2a\u200b\u975e\u8d1f\u200b\u7684\u200b\u6574\u6570\u200b\u3002\u200b\u5b83\u4eec\u200b\u6bcf\u4f4d\u200b\u6570\u5b57\u200b\u90fd\u200b\u662f\u200b\u6309\u7167\u200b\u00a0\u200b\u9006\u5e8f\u200b\u00a0\u200b\u7684\u200b\u65b9\u5f0f\u200b\u5b58\u50a8\u200b\u7684\u200b\uff0c\u200b\u5e76\u4e14\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\u53ea\u80fd\u200b\u5b58\u50a8\u200b\u00a0\u200b\u4e00\u4f4d\u200b\u00a0\u200b\u6570\u5b57\u200b\u3002</p> <p>\u200b\u8bf7\u200b\u4f60\u200b\u5c06\u200b\u4e24\u4e2a\u200b\u6570\u200b\u76f8\u52a0\u200b\uff0c\u200b\u5e76\u200b\u4ee5\u200b\u76f8\u540c\u200b\u5f62\u5f0f\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u8868\u793a\u200b\u548c\u200b\u7684\u200b\u94fe\u8868\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u9664\u4e86\u200b\u6570\u5b57\u200b <code>0</code> \u200b\u4e4b\u5916\u200b\uff0c\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u6570\u90fd\u200b\u4e0d\u4f1a\u200b\u4ee5\u200b <code>0</code>\u00a0\u200b\u5f00\u5934\u200b\u3002</p> <p></p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1al1 = [2,4,3], l2 = [5,6,4]\n\u200b\u8f93\u51fa\u200b\uff1a[7,0,8]\n\u200b\u89e3\u91ca\u200b\uff1a342 + 465 = 807.\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1al1 = [0], l2 = [0]\n\u200b\u8f93\u51fa\u200b\uff1a[0]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1al1 = [9,9,9,9,9,9,9], l2 = [9,9,9,9]\n\u200b\u8f93\u51fa\u200b\uff1a[8,9,9,9,0,0,0,1]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u6bcf\u4e2a\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9\u200b\u6570\u5728\u200b\u8303\u56f4\u200b <code>[1, 100]</code> \u200b\u5185\u200b</li> <li><code>0 &lt;= Node.val &lt;= 9</code></li> <li>\u200b\u9898\u76ee\u200b\u6570\u636e\u200b\u4fdd\u8bc1\u200b\u5217\u8868\u200b\u8868\u793a\u200b\u7684\u200b\u6570\u5b57\u200b\u4e0d\u200b\u542b\u200b\u524d\u5bfc\u200b\u96f6\u200b</li> </ul>","tags":["\u9012\u5f52","\u94fe\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/2/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u9012\u5f52\u200b\u7684\u200b\u601d\u60f3\u200b\uff0c\u200b\u5c06\u200b\u8ba1\u7b97\u200b\u52a0\u6cd5\u200b\u7684\u200b\u95ee\u9898\u200b\u5206\u89e3\u200b\u4e3a\u200b\u8ba1\u7b97\u200b\u6700\u540e\u200b\u4e00\u4f4d\u200b\u76f8\u52a0\u200b\u548c\u200b\u8ba1\u7b97\u200b\u5269\u4f59\u200b\u90e8\u5206\u200b\u76f8\u52a0\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u95ee\u9898\u200b\u3002</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\nstruct ListNode *getNode(int val, struct ListNode *next)\n{\n    struct ListNode *ret = (struct ListNode *)malloc(sizeof(struct ListNode));\n    ret-&gt;val = val;\n    ret-&gt;next = next;\n    return ret;\n}\nstruct ListNode *recur(struct ListNode *l1, struct ListNode *l2, int flag)\n{\n    if (!l1)\n        return flag ? recur(getNode(1, NULL), l2, 0) : l2;\n    if (!l2)\n        return flag ? recur(getNode(1, NULL), l1, 0) : l1;\n    int ret = l1-&gt;val + l2-&gt;val + (flag ? 1 : 0);\n    return getNode(ret - (ret &gt;= 10 ? 10 : 0), recur(l1-&gt;next, l2-&gt;next, ret &gt;= 10));\n}\nstruct ListNode* addTwoNumbers(struct ListNode* l1, struct ListNode* l2){\n    return recur(l1, l2, 0);\n}\n</code></pre> <p>Reference</p>","tags":["\u9012\u5f52","\u94fe\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/20/","title":"20. \u200b\u6709\u6548\u200b\u7684\u200b\u62ec\u53f7","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6808","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/20/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u53ea\u200b\u5305\u62ec\u200b <code>'('</code>\uff0c<code>')'</code>\uff0c<code>'{'</code>\uff0c<code>'}'</code>\uff0c<code>'['</code>\uff0c<code>']'</code>\u00a0\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b <code>s</code> \uff0c\u200b\u5224\u65ad\u200b\u5b57\u7b26\u4e32\u200b\u662f\u5426\u200b\u6709\u6548\u200b\u3002</p> <p>\u200b\u6709\u6548\u200b\u5b57\u7b26\u4e32\u200b\u9700\u200b\u6ee1\u8db3\u200b\uff1a</p> <ol> <li>\u200b\u5de6\u200b\u62ec\u53f7\u200b\u5fc5\u987b\u200b\u7528\u200b\u76f8\u540c\u200b\u7c7b\u578b\u200b\u7684\u200b\u53f3\u200b\u62ec\u53f7\u200b\u95ed\u5408\u200b\u3002</li> <li>\u200b\u5de6\u200b\u62ec\u53f7\u200b\u5fc5\u987b\u200b\u4ee5\u200b\u6b63\u786e\u200b\u7684\u200b\u987a\u5e8f\u200b\u95ed\u5408\u200b\u3002</li> </ol> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"()\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"()[]{}\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a03\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"(]\"\n\u200b\u8f93\u51fa\u200b\uff1afalse\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a04\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"([)]\"\n\u200b\u8f93\u51fa\u200b\uff1afalse\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a05\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as = \"{[]}\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>Reference</p>","tags":["\u6808","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/20/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u6808\u200b\u6a21\u62df\u200b\u5339\u914d\u200b\u8fc7\u7a0b\u200b</p> <ul> <li>\u200b\u9047\u5230\u200b\u5de6\u200b\u62ec\u53f7\u200b\u65f6\u5165\u200b\u6808\u200b\uff1b</li> <li>\u200b\u9047\u5230\u200b\u53f3\u200b\u62ec\u53f7\u200b\u65f6\u200b\uff0c\u200b\u68c0\u67e5\u200b\u6808\u9876\u200b\u62ec\u53f7\u200b\u662f\u5426\u200b\u4e0e\u200b\u53f3\u200b\u62ec\u53f7\u200b\u5339\u914d\u200b\uff0c\u200b\u82e5\u200b\u5339\u914d\u200b\u5219\u200b\u51fa\u6808\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>false</code>\uff1b</li> <li>\u200b\u5b8c\u6210\u200b\u5339\u914d\u200b\u8fc7\u7a0b\u200b\u540e\u200b\u68c0\u67e5\u200b\u6808\u200b\u662f\u5426\u200b\u4e3a\u7a7a\u200b\uff0c\u200b\u82e5\u4e3a\u200b\u7a7a\u200b\u8fd4\u56de\u200b<code>true</code>\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>false</code>\u3002</li> </ul> <pre><code>bool isValid(char * s){\n    int maxLen = strlen(s), top = 0;\n    char *holding = malloc(sizeof(char) * maxLen), *cur = s;\n    while(*cur)\n    {\n        switch(*cur)\n        {\n            case '(':\n            case '[':\n            case '{':\n                holding[top] = *cur;\n                top++;\n                break;\n            case ')':\n                if (top == 0 || holding[top - 1] != '(')\n                    return false;\n                top--;\n                break;\n            case ']':\n            case '}':\n                if (top == 0 || holding[top - 1] != (*cur) - 2)\n                    return false;\n                top--;\n                break;\n        }\n        cur++;\n    }\n    return !top;\n}\n</code></pre>","tags":["\u6808","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/204/","title":"204. \u200b\u8ba1\u6570\u200b\u8d28\u6570","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u54c8\u5e0c\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/204/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7edf\u8ba1\u200b\u6240\u6709\u200b\u5c0f\u4e8e\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b n \u200b\u7684\u200b\u8d28\u6570\u200b\u7684\u200b\u6570\u91cf\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 10\n\u200b\u8f93\u51fa\u200b: 4\n\u200b\u89e3\u91ca\u200b: \u200b\u5c0f\u4e8e\u200b 10 \u200b\u7684\u200b\u8d28\u6570\u200b\u4e00\u5171\u200b\u6709\u200b 4 \u200b\u4e2a\u200b, \u200b\u5b83\u4eec\u200b\u662f\u200b 2, 3, 5, 7 \u3002\n</code></pre> <p>Reference</p>","tags":["\u54c8\u5e0c\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/204/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4ece\u200b2\u200b\u5f00\u59cb\u200b\uff0c\u200b\u5728\u200b\u6570\u5217\u200b\\(2, 3, \\cdots, n\\)\u200b\u4e2d\u5212\u200b\u53bb\u200b\u6240\u6709\u200b2\u200b\u7684\u200b\u500d\u6570\u200b\uff0c\u200b\u5269\u4e0b\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u6ca1\u6709\u200b\u88ab\u5212\u200b\u53bb\u200b\u7684\u200b\u6570\u5b57\u200b3\u200b\u5373\u200b\u4e3a\u200b\u8d28\u6570\u200b\uff0c\u200b\u518d\u200b\u82b1\u65d7\u200b\u6240\u6709\u200b3\u200b\u7684\u200b\u500d\u6570\u200b\uff0c\u200b\u5269\u4e0b\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u6ca1\u6709\u200b\u88ab\u5212\u200b\u53bb\u200b\u7684\u200b\u6570\u5b57\u200b5\u200b\u5373\u200b\u4e3a\u200b\u8d28\u6570\u200b\u3002\u200b\u91cd\u590d\u200b\u5982\u200b\u4e0a\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u76f4\u5230\u200b\u68c0\u67e5\u200b\u5b8c\u200b\u6240\u6709\u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b\\(\\sqrt n\\)\u200b\u7684\u200b\u6574\u6570\u200b\uff0c\u200b\u5373\u200b\u5b8c\u6210\u200b\u3002</p> <p>\u200b\u6570\u5b57\u200b\u662f\u5426\u200b\u5212\u53bb\u200b\u7684\u200b\u72b6\u6001\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\u63a7\u5236\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200b\u4e00\u4e2a\u200b\u7c7b\u578b\u200b\u4e2d\u200b\u7684\u200b1\u200b\u4f4d\u200b\u8868\u793a\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int countPrimes(int n)\n{\n    unsigned char *map = memset(malloc(sizeof(unsigned char) * ((n + 1 &gt;&gt; 3) + 1)), 0x55, sizeof(unsigned char) * ((n + 1 &gt;&gt; 3) + 1));\n    int i = 0, j, ret = 0, sq = (int)sqrt(n), increment = 1, start = 2;\n    *map = 0x51;\n    for (i = 3; i &lt;= sq; i++)\n        if (!(map[i &gt;&gt; 3] &amp; (unsigned char)1 &lt;&lt; (i &amp; 7)))\n            for (j = i * 3; j &lt;= n; j += i &lt;&lt; 1)\n                map[j &gt;&gt; 3] |= (unsigned char)1 &lt;&lt; (j &amp; 7);\n    for (i = 2; i &lt; n; i++)\n        if (!(map[i &gt;&gt; 3] &amp; (unsigned char)1 &lt;&lt; (i &amp; 7)))\n            ret++;\n    return ret;\n}\n</code></pre>","tags":["\u54c8\u5e0c\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/206/","title":"206. \u200b\u53cd\u8f6c\u200b\u94fe\u8868","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/206/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u53cd\u8f6c\u200b\u4e00\u4e2a\u200b\u5355\u94fe\u200b\u8868\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a <pre><code>\u200b\u8f93\u5165\u200b: 1-&gt;2-&gt;3-&gt;4-&gt;5-&gt;NULL\n\u200b\u8f93\u51fa\u200b: 5-&gt;4-&gt;3-&gt;2-&gt;1-&gt;NULL\n</code></pre> \u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u8fed\u4ee3\u200b\u6216\u200b\u9012\u5f52\u200b\u5730\u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b\u3002\u200b\u4f60\u200b\u80fd\u5426\u200b\u7528\u200b\u4e24\u79cd\u200b\u65b9\u6cd5\u200b\u89e3\u51b3\u200b\u8fd9\u9053\u9898\u200b\uff1f</p> <p>Reference</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/206/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u53ea\u200b\u9700\u8981\u200b\u5c06\u200b\u94fe\u8868\u200b\u7ed3\u6784\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u6307\u9488\u200b\u53cd\u8f6c\u200b\uff0c\u200b\u5373\u200b\u6307\u5411\u200b\u524d\u200b\u4e00\u200b\u8282\u70b9\u200b\u7684\u200b\u6307\u9488\u200b\u73b0\u5728\u200b\u6307\u5411\u200b\u540e\u200b\u4e00\u200b\u8282\u70b9\u200b\u3002\u200b\u5b9a\u4e49\u200b\u4e24\u4e2a\u200b\u6307\u9488\u200b\u6307\u5411\u200b\u9700\u8981\u200b\u53cd\u8f6c\u200b\u90e8\u5206\u200b\u7684\u200b\u4e24\u4e2a\u200b\u8282\u70b9\u200b\uff0c\u200b\u5c06\u200b\u540e\u200b\u4e00\u200b\u6307\u9488\u200b\u7684\u200b<code>next</code>\u200b\u57df\u200b\u6307\u5411\u200b\u524d\u200b\u4e00\u200b\u6307\u9488\u200b\u3002\u200b\u6539\u200b\u53cd\u8f6c\u200b\u64cd\u4f5c\u200b\u7834\u574f\u200b\u4e86\u200b\u540e\u200b\u4e00\u200b\u6307\u9488\u200b\u539f\u6709\u200b\u7684\u200b<code>next</code>\u200b\u57df\u200b\uff0c\u200b\u9700\u8981\u200b\u4e00\u4e2a\u200b\u989d\u5916\u200b\u7684\u200b\u6307\u9488\u200b\u7528\u4e8e\u200b\u5411\u524d\u200b\u79fb\u52a8\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\n\nstruct ListNode* reverseList(struct ListNode* head){\n    if (!head || !(head-&gt;next))\n        return head;\n    struct ListNode *cur1 = head, *cur2 = head-&gt;next, *cur3 = cur2-&gt;next;\n    head-&gt;next = NULL;\n    while (cur3)\n    {\n        cur2-&gt;next = cur1;\n        cur1 = cur2;\n        cur2 = cur3;\n        cur3 = cur3-&gt;next;\n    }\n    cur2-&gt;next = cur1;\n    return cur2;\n}\n</code></pre>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/21/","title":"21. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u94fe\u8868","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u9012\u5f52","\u94fe\u8868"]},{"location":"coding/leetcode/21/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5c06\u200b\u4e24\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868\u200b\u5408\u5e76\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b \u200b\u5347\u5e8f\u200b \u200b\u94fe\u8868\u200b\u5e76\u200b\u8fd4\u56de\u200b\u3002\u200b\u65b0\u200b\u94fe\u8868\u200b\u662f\u200b\u901a\u8fc7\u200b\u62fc\u63a5\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u4e24\u4e2a\u200b\u94fe\u8868\u200b\u7684\u200b\u6240\u6709\u200b\u8282\u70b9\u200b\u7ec4\u6210\u200b\u7684\u200b\u3002\u00a0</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a1-&gt;2-&gt;4, 1-&gt;3-&gt;4\n\u200b\u8f93\u51fa\u200b\uff1a1-&gt;1-&gt;2-&gt;3-&gt;4-&gt;4\n</code></pre> <p>Reference</p>","tags":["\u9012\u5f52","\u94fe\u8868"]},{"location":"coding/leetcode/21/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6bd4\u8f83\u200b\u4e24\u4e2a\u200b\u94fe\u8868\u200b\u7684\u200b\u5934\u200b\u8282\u70b9\u200b\uff0c\u200b\u5c06\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u8282\u70b9\u200b\u5e76\u5165\u200b\u7ed3\u679c\u200b\u94fe\u8868\u200b\u4e2d\u200b\uff0c\u200b\u540c\u65f6\u200b\u94fe\u8868\u200b\u5934\u200b\u540e\u79fb\u200b\uff0c\u200b\u9012\u5f52\u200b\u5904\u7406\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\nstruct ListNode* mergeTwoLists(struct ListNode* l1, struct ListNode* l2){\n    if (!l1)\n        return l2;\n    if (!l2)\n        return l1;\n    if (l1-&gt;val &lt; l2-&gt;val)\n    {\n        l1-&gt;next = mergeTwoLists(l1-&gt;next, l2);\n        return l1;\n    }\n    else\n    {\n        l2-&gt;next = mergeTwoLists(l1, l2-&gt;next);\n        return l2;\n    }\n}\n</code></pre>","tags":["\u9012\u5f52","\u94fe\u8868"]},{"location":"coding/leetcode/217/","title":"217. \u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/217/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\uff0c\u200b\u5224\u65ad\u200b\u662f\u5426\u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u4efb\u610f\u200b\u4e00\u503c\u200b\u5728\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u81f3\u5c11\u200b\u4e24\u6b21\u200b\uff0c\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b<code>true</code>\u3002\u200b\u5982\u679c\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u90fd\u200b\u4e0d\u200b\u76f8\u540c\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>false</code>\u3002\u00a0</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,2,3,1]\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,2,3,4]\n\u200b\u8f93\u51fa\u200b: false\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a03\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,1,1,3,3,4,3,2,4,2]\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/217/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5c06\u200b\u6570\u7ec4\u200b\u6392\u5e8f\u200b\u540e\u200b\u68c0\u67e5\u200b\u662f\u5426\u200b\u6709\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int comp(const void *a, const void *b)\n{\n    return *(int*)a - *(int*)b;\n}\nbool containsDuplicate(int* nums, int numsSize){\n    if (numsSize &lt; 2)\n        return false;\n    qsort(nums, numsSize, sizeof(int), comp);\n    int i = 0;\n    for (i = 1; i&lt;numsSize; i++)\n        if (nums[i] == nums[i - 1])\n            return true;\n    return false;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/22/","title":"22. \u200b\u62ec\u53f7\u200b\u751f\u6210","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u5b57\u7b26\u4e32","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/22/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u6570\u5b57\u200b <code>n</code>\u00a0\u200b\u4ee3\u8868\u200b\u751f\u6210\u200b\u62ec\u53f7\u200b\u7684\u200b\u5bf9\u6570\u200b\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u7528\u4e8e\u200b\u80fd\u591f\u200b\u751f\u6210\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u7684\u200b\u5e76\u4e14\u200b \u200b\u6709\u6548\u200b\u7684\u200b \u200b\u62ec\u53f7\u200b\u7ec4\u5408\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1an = 3\n\u200b\u8f93\u51fa\u200b\uff1a[\"((()))\",\"(()())\",\"(())()\",\"()(())\",\"()()()\"]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1an = 1\n\u200b\u8f93\u51fa\u200b\uff1a[\"()\"]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= n &lt;= 8</code></li> </ul> <p>Reference</p>","tags":["\u5b57\u7b26\u4e32","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/22/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u56de\u6eaf\u200b\u7b97\u6cd5\u200b</p> <ul> <li>\u200b\u5f53\u200b\u5de6\u200b\u62ec\u53f7\u200b\u6570\u91cf\u200b\u5c0f\u4e8e\u200b\u62ec\u53f7\u200b\u5bf9\u6570\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6dfb\u52a0\u200b\u5de6\u200b\u62ec\u53f7\u200b</li> <li>\u200b\u5f53\u53f3\u200b\u62ec\u53f7\u200b\u6570\u91cf\u200b\u5c0f\u4e8e\u200b\u5de6\u200b\u62ec\u53f7\u200b\u6570\u91cf\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6dfb\u52a0\u200b\u53f3\u200b\u62ec\u53f7\u200b</li> </ul> <p>\u200b\u6309\u7167\u200b\u5982\u200b\u4e0a\u200b\u65b9\u6cd5\u200b\u751f\u6210\u200b\u7684\u200b\u62ec\u53f7\u200b\u7ec4\u5408\u200b\u53ef\u4ee5\u200b\u4fdd\u8bc1\u200b\u4e00\u5b9a\u200b\u662f\u200b\u6709\u6548\u200b\u7684\u200b\u3002</p> <pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\n\nvoid dfs(char *currentStr, int n, int left, int right, char **ret, int *returnSize)\n{\n    if (left == n &amp;&amp; right == n)\n    {\n        ret[*returnSize] = (char *)malloc(sizeof(char) * (2 * n + 1));\n        strcpy(ret[*returnSize], currentStr);\n        *returnSize += 1;\n        return;\n    }\n    if (left &lt; n)\n    {\n        currentStr[left + right] = '(';\n        dfs(currentStr, n, left + 1, right, ret, returnSize);\n    }\n    if (right &lt; left)\n    {\n        currentStr[left + right] = ')';\n        dfs(currentStr, n, left, right + 1, ret, returnSize);\n    }\n}\nchar ** generateParenthesis(int n, int* returnSize){\n    const int maxLen = 10000;\n    *returnSize = 0;\n    char **ret = (char **)malloc(sizeof(char *) * maxLen),\n         *buffer = (char *)memset(malloc(sizeof(char) * (2 * n + 1)), 0, sizeof(char) * (2 * n + 1));\n    dfs(buffer, n, 0, 0, ret, returnSize);\n    return ret;\n}\n</code></pre>","tags":["\u5b57\u7b26\u4e32","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/23/","title":"23. \u200b\u5408\u5e76\u200bK\u200b\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u5806","\u94fe\u8868","\u5206\u6cbb\u7b97\u6cd5"]},{"location":"coding/leetcode/23/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b\u6570\u7ec4\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u94fe\u8868\u200b\u90fd\u200b\u5df2\u7ecf\u200b\u6309\u200b\u5347\u5e8f\u200b\u6392\u5217\u200b\u3002</p> <p>\u200b\u8bf7\u200b\u4f60\u200b\u5c06\u200b\u6240\u6709\u200b\u94fe\u8868\u200b\u5408\u5e76\u200b\u5230\u200b\u4e00\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868\u200b\u4e2d\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5408\u5e76\u200b\u540e\u200b\u7684\u200b\u94fe\u8868\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1alists = [[1,4,5],[1,3,4],[2,6]]\n\u200b\u8f93\u51fa\u200b\uff1a[1,1,2,3,4,4,5,6]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u94fe\u8868\u200b\u6570\u7ec4\u200b\u5982\u4e0b\u200b\uff1a\n[\n  1-&gt;4-&gt;5,\n  1-&gt;3-&gt;4,\n  2-&gt;6\n]\n\u200b\u5c06\u200b\u5b83\u4eec\u200b\u5408\u5e76\u200b\u5230\u200b\u4e00\u4e2a\u200b\u6709\u5e8f\u200b\u94fe\u8868\u200b\u4e2d\u200b\u5f97\u5230\u200b\u3002\n1-&gt;1-&gt;2-&gt;3-&gt;4-&gt;4-&gt;5-&gt;6\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1alists = []\n\u200b\u8f93\u51fa\u200b\uff1a[]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1alists = [[]]\n\u200b\u8f93\u51fa\u200b\uff1a[]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>k == lists.length</code></li> <li><code>0 &lt;= k &lt;= 10^4</code></li> <li><code>0 &lt;= lists[i].length &lt;= 500</code></li> <li><code>-10^4 &lt;= lists[i][j] &lt;= 10^4</code></li> <li><code>lists[i]</code> \u200b\u6309\u200b \u200b\u5347\u5e8f\u200b \u200b\u6392\u5217\u200b</li> <li><code>lists[i].length</code> \u200b\u7684\u200b\u603b\u548c\u200b\u4e0d\u200b\u8d85\u8fc7\u200b <code>10^4</code></li> </ul> <p>Reference</p>","tags":["\u5806","\u94fe\u8868","\u5206\u6cbb\u7b97\u6cd5"]},{"location":"coding/leetcode/23/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u6700\u200b\u76f4\u63a5\u200b\u7684\u200b\u5408\u5e76\u200b\u601d\u8def\u200b\u662f\u200b\u6309\u7167\u200b\u987a\u5e8f\u200b\u5bf9\u200b\u94fe\u8868\u200b\u8fdb\u884c\u200b\u5408\u5e76\u200b\u3002\u200b\u8003\u8651\u200b\u5230\u200b\u5408\u5e76\u200b\u540e\u200b\u94fe\u8868\u200b\u53d8\u200b\u957f\u200b\uff0c\u200b\u5e73\u5747\u200b\u6bcf\u6b21\u200b\u5408\u5e76\u200b\u9700\u8981\u200b\u82b1\u8d39\u200b\\(\\mathcal O(MN)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\uff0c\u200b\u5171\u9700\u200b\u5408\u5e76\u200b\\(\\mathcal O(M)\\)\u200b\u6b21\u200b\uff0c\u200b\u56e0\u6b64\u200b\u603b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(M^2N)\\)</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\n\nstruct ListNode* mergeLists(struct ListNode* l1, struct ListNode* l2);\nstruct ListNode* mergeKLists(struct ListNode** lists, int listsSize){\n    if (!listsSize)\n        return NULL;\n    struct ListNode *head = lists[0];\n    for (int i = 1; i&lt;listsSize; i++)\n    {\n        if (!head)\n        {\n            head = lists[i];\n            continue;\n        }\n        head = mergeLists(head, lists[i]);\n    }\n    return head;\n}\n\nstruct ListNode* mergeLists(struct ListNode* l1, struct ListNode* l2){\n    if (!l1)\n        return l2;\n    if (!l2)\n        return l1;\n    if (l1-&gt;val &lt; l2-&gt;val)\n    {\n        l1-&gt;next = mergeLists(l1-&gt;next, l2);\n        return l1;\n    }\n    else\n    {\n        l2-&gt;next = mergeLists(l1, l2-&gt;next);\n        return l2;\n    }\n}\n</code></pre> <p>\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b\u4e24\u200b\u4e24\u200b\u5408\u5e76\u200b\u7684\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5408\u5e76\u200b\u64cd\u4f5c\u200b\u7684\u200b\u5e73\u5747\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u964d\u4f4e\u200b\u81f3\u200b\\(\\mathcal O(N\\log M)\\)\uff0c\u200b\u603b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u53d8\u4e3a\u200b\\(\\mathcal O(MN\\log M)\\)\u3002</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\n\nstruct ListNode* mergeLists(struct ListNode* l1, struct ListNode* l2);\nstruct ListNode* mergeKLists(struct ListNode** lists, int listsSize){\n    if (!listsSize)\n        return NULL;\n    else if (listsSize == 1)\n        return lists[0];\n    else if (listsSize == 2)\n        return mergeLists(lists[0], lists[1]);\n    struct ListNode *head = lists[0];\n    int step = (listsSize + 1) &gt;&gt; 1;\n    for (int i = 0; i &lt; step; i++)\n    {\n        if (i + step &lt; listsSize)\n            lists[i] = mergeLists(lists[i], lists[i + step]);\n    }\n    return mergeKLists(lists, step);\n}\n\nstruct ListNode* mergeLists(struct ListNode* l1, struct ListNode* l2){\n    if (!l1)\n        return l2;\n    if (!l2)\n        return l1;\n    if (l1-&gt;val &lt; l2-&gt;val)\n    {\n        l1-&gt;next = mergeLists(l1-&gt;next, l2);\n        return l1;\n    }\n    else\n    {\n        l2-&gt;next = mergeLists(l1, l2-&gt;next);\n        return l2;\n    }\n}\n</code></pre>","tags":["\u5806","\u94fe\u8868","\u5206\u6cbb\u7b97\u6cd5"]},{"location":"coding/leetcode/234/","title":"234. \u200b\u56de\u6587\u200b\u94fe\u8868","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/234/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u8bf7\u200b\u5224\u65ad\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b\u662f\u5426\u200b\u4e3a\u200b\u56de\u6587\u200b\u94fe\u8868\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 1-&gt;2\n\u200b\u8f93\u51fa\u200b: false\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 1-&gt;2-&gt;2-&gt;1\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u80fd\u5426\u200b\u7528\u200b\\(O(n)\\)\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u548c\u200b\\(O(1)\\)\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u89e3\u51b3\u200b\u6b64\u9898\u200b\uff1f</p> <p>Reference</p>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/234/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u53cd\u5e8f\u200b\u6784\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u94fe\u8868\u200b\uff0c\u200b\u6bd4\u8f83\u200b\u4e24\u4e2a\u200b\u94fe\u8868\u200b\u662f\u5426\u200b\u76f8\u540c\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\nstruct ListNode *getNode(int val, struct ListNode *next)\n{\n    struct ListNode *ret = (struct ListNode *)malloc(sizeof(struct ListNode));\n    ret-&gt;val = val;\n    ret-&gt;next = next;\n    return ret;\n}\n\nbool isPalindrome(struct ListNode* head){\n    int len = 0;\n    struct ListNode *reversed = NULL, *cur1 = head, *cur2 = NULL;\n    while(cur1 != NULL)\n    {\n        reversed = getNode(cur1-&gt;val, reversed);\n        cur1 = cur1-&gt;next;\n        len++;\n    }\n    cur1 = head;\n    cur2 = reversed;\n    for (int i = 0; i &lt; len / 2; i++)\n    {\n        if (cur1-&gt;val != cur2-&gt;val)\n            return false;\n        cur1 = cur1-&gt;next;\n        cur2 = cur2-&gt;next;\n    }\n    return true;\n}\n</code></pre>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/237/","title":"237. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/237/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u8bf7\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u4f7f\u200b\u5176\u200b\u53ef\u4ee5\u200b\u5220\u9664\u200b\u67d0\u4e2a\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7ed9\u5b9a\u200b\u7684\u200b\uff08\u200b\u975e\u200b\u672b\u5c3e\u200b\uff09\u200b\u8282\u70b9\u200b\u3002\u200b\u4f20\u5165\u200b\u51fd\u6570\u200b\u7684\u200b\u552f\u4e00\u200b\u53c2\u6570\u200b\u4e3a\u200b \u200b\u8981\u200b\u88ab\u200b\u5220\u9664\u200b\u7684\u200b\u8282\u70b9\u200b \u3002</p> <p>\u200b\u73b0\u6709\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b --\u00a0head =\u00a0[4,5,1,9]\uff0c\u200b\u5b83\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b:</p> <p></p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [4,5,1,9], node = 5\n\u200b\u8f93\u51fa\u200b\uff1a[4,1,9]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u7ed9\u5b9a\u200b\u4f60\u200b\u94fe\u8868\u200b\u4e2d\u503c\u200b\u4e3a\u200b\u00a05\u00a0\u200b\u7684\u200b\u7b2c\u4e8c\u4e2a\u200b\u8282\u70b9\u200b\uff0c\u200b\u90a3\u4e48\u200b\u5728\u200b\u8c03\u7528\u200b\u4e86\u200b\u4f60\u200b\u7684\u200b\u51fd\u6570\u200b\u4e4b\u540e\u200b\uff0c\u200b\u8be5\u200b\u94fe\u8868\u200b\u5e94\u53d8\u200b\u4e3a\u200b 4 -&gt; 1 -&gt; 9.\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [4,5,1,9], node = 1\n\u200b\u8f93\u51fa\u200b\uff1a[4,5,9]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u7ed9\u5b9a\u200b\u4f60\u200b\u94fe\u8868\u200b\u4e2d\u503c\u200b\u4e3a\u200b\u00a01\u00a0\u200b\u7684\u200b\u7b2c\u4e09\u4e2a\u200b\u8282\u70b9\u200b\uff0c\u200b\u90a3\u4e48\u200b\u5728\u200b\u8c03\u7528\u200b\u4e86\u200b\u4f60\u200b\u7684\u200b\u51fd\u6570\u200b\u4e4b\u540e\u200b\uff0c\u200b\u8be5\u200b\u94fe\u8868\u200b\u5e94\u53d8\u200b\u4e3a\u200b 4 -&gt; 5 -&gt; 9.\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u94fe\u8868\u200b\u81f3\u5c11\u200b\u5305\u542b\u200b\u4e24\u4e2a\u200b\u8282\u70b9\u200b\u3002</li> <li>\u200b\u94fe\u8868\u200b\u4e2d\u200b\u6240\u6709\u200b\u8282\u70b9\u200b\u7684\u200b\u503c\u200b\u90fd\u200b\u662f\u200b\u552f\u4e00\u200b\u7684\u200b\u3002</li> <li>\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u8282\u70b9\u200b\u4e3a\u200b\u975e\u200b\u672b\u5c3e\u200b\u8282\u70b9\u200b\u5e76\u4e14\u200b\u4e00\u5b9a\u200b\u662f\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u200b\u6709\u6548\u200b\u8282\u70b9\u200b\u3002</li> <li>\u200b\u4e0d\u8981\u200b\u4ece\u200b\u4f60\u200b\u7684\u200b\u51fd\u6570\u200b\u4e2d\u200b\u8fd4\u56de\u200b\u4efb\u4f55\u200b\u7ed3\u679c\u200b\u3002</li> </ul> <p>Reference</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/237/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5c06\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u7684\u200b\u503c\u200b\u590d\u5236\u5230\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\uff0c\u200b\u7136\u540e\u200b\u5220\u53bb\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u3002\uff08\u200b\u5df2\u77e5\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u4e00\u5b9a\u200b\u4e0d\u662f\u200b\u672b\u5c3e\u200b\u8282\u70b9\u200b\uff09</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>void deleteNode(struct ListNode* node) {\n    node-&gt;val = node-&gt;next-&gt;val;\n    node-&gt;next = node-&gt;next-&gt;next;\n}\n</code></pre>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/24/","title":"24. \u200b\u4e24\u200b\u4e24\u200b\u4ea4\u6362\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u9012\u5f52","\u94fe\u8868"]},{"location":"coding/leetcode/24/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b\uff0c\u200b\u4e24\u200b\u4e24\u200b\u4ea4\u6362\u200b\u5176\u4e2d\u200b\u76f8\u90bb\u200b\u7684\u200b\u8282\u70b9\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b\u4ea4\u6362\u200b\u540e\u200b\u7684\u200b\u94fe\u8868\u200b\u3002</p> <p>\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u53ea\u662f\u200b\u5355\u7eaf\u200b\u7684\u200b\u6539\u53d8\u200b\u8282\u70b9\u200b\u5185\u90e8\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u800c\u662f\u200b\u9700\u8981\u200b\u5b9e\u9645\u200b\u7684\u200b\u8fdb\u884c\u200b\u8282\u70b9\u200b\u4ea4\u6362\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a <pre><code>\u200b\u7ed9\u5b9a\u200b 1-&gt;2-&gt;3-&gt;4, \u200b\u4f60\u200b\u5e94\u8be5\u200b\u8fd4\u56de\u200b 2-&gt;1-&gt;4-&gt;3.\n</code></pre></p> <p>Reference</p>","tags":["\u9012\u5f52","\u94fe\u8868"]},{"location":"coding/leetcode/24/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5148\u200b\u4ea4\u6362\u200b\u524d\u200b\u4e24\u4e2a\u200b\u8282\u70b9\u200b\uff0c\u200b\u518d\u200b\u9012\u5f52\u200b\u5904\u7406\u200b\u540e\u9762\u200b\u7684\u200b\u8282\u70b9\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\n\nstruct ListNode* swapPairs(struct ListNode* head){\n    if (!head || !(head-&gt;next))\n        return head;\n    struct ListNode *temp1 = head-&gt;next, *temp2 = temp1-&gt;next;\n    temp1-&gt;next = head;\n    head-&gt;next = swapPairs(temp2);\n    return temp1;\n}\n</code></pre>","tags":["\u9012\u5f52","\u94fe\u8868"]},{"location":"coding/leetcode/242/","title":"242. \u200b\u6709\u6548\u200b\u7684\u200b\u5b57\u6bcd\u200b\u5f02\u4f4d\u200b\u8bcd","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6392\u5e8f","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/242/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b s \u200b\u548c\u200b t \uff0c\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u6765\u200b\u5224\u65ad\u200b t \u200b\u662f\u5426\u662f\u200b s \u200b\u7684\u200b\u5b57\u6bcd\u200b\u5f02\u4f4d\u200b\u8bcd\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: s = \"anagram\", t = \"nagaram\"\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: s = \"rat\", t = \"car\"\n\u200b\u8f93\u51fa\u200b: false\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u5b57\u7b26\u4e32\u200b\u53ea\u200b\u5305\u542b\u200b\u5c0f\u5199\u5b57\u6bcd\u200b\u3002</p> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u5982\u679c\u200b\u8f93\u5165\u200b\u5b57\u7b26\u4e32\u200b\u5305\u542b\u200b unicode \u200b\u5b57\u7b26\u200b\u600e\u4e48\u529e\u200b\uff1f\u200b\u4f60\u200b\u80fd\u5426\u200b\u8c03\u6574\u200b\u4f60\u200b\u7684\u200b\u89e3\u6cd5\u200b\u6765\u200b\u5e94\u5bf9\u200b\u8fd9\u79cd\u200b\u60c5\u51b5\u200b\uff1f</p> <p>Reference</p>","tags":["\u6392\u5e8f","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/242/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u7edf\u8ba1\u200b\u5e76\u200b\u6bd4\u8f83\u200b\u4e24\u4e2a\u200b\u8f93\u5165\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u5404\u200b\u5b57\u7b26\u200b\u7684\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>bool isAnagram(char * s, char * t){\n    int hashMap1[26] = {0}, hashMap2[26] = {0};\n    char *cur = s;\n    while(*cur)\n    {\n        hashMap1[*cur - 'a']++;\n        cur++;\n    }\n    cur = t;\n    while(*cur)\n    {\n        hashMap2[*cur - 'a']++;\n        cur++;\n    }\n    int i = 0;\n    for (i = 0; i &lt; 26; i++)\n        if (hashMap1[i] != hashMap2[i])\n            return false;\n    return true;\n}\n</code></pre>","tags":["\u6392\u5e8f","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/25/","title":"25. K \u200b\u4e2a\u200b\u4e00\u7ec4\u200b\u7ffb\u8f6c\u200b\u94fe\u8868","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/25/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b\uff0c\u200b\u6bcf\u200b\u00a0k\u00a0\u200b\u4e2a\u200b\u8282\u70b9\u200b\u4e00\u7ec4\u200b\u8fdb\u884c\u200b\u7ffb\u8f6c\u200b\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u8fd4\u56de\u200b\u7ffb\u8f6c\u200b\u540e\u200b\u7684\u200b\u94fe\u8868\u200b\u3002</p> <p>k\u00a0\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6b63\u6574\u6570\u200b\uff0c\u200b\u5b83\u200b\u7684\u200b\u503c\u200b\u5c0f\u4e8e\u200b\u6216\u200b\u7b49\u4e8e\u200b\u94fe\u8868\u200b\u7684\u200b\u957f\u5ea6\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u8282\u70b9\u200b\u603b\u6570\u200b\u4e0d\u662f\u200b\u00a0k\u00a0\u200b\u7684\u200b\u6574\u6570\u500d\u200b\uff0c\u200b\u90a3\u4e48\u200b\u8bf7\u200b\u5c06\u200b\u6700\u540e\u200b\u5269\u4f59\u200b\u7684\u200b\u8282\u70b9\u200b\u4fdd\u6301\u200b\u539f\u6709\u200b\u987a\u5e8f\u200b\u3002</p> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <ul> <li>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u53ea\u200b\u4f7f\u7528\u200b\u5e38\u6570\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\u7684\u200b\u7b97\u6cd5\u200b\u6765\u200b\u89e3\u51b3\u200b\u6b64\u200b\u95ee\u9898\u200b\u5417\u200b\uff1f</li> <li>\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u53ea\u662f\u200b\u5355\u7eaf\u200b\u7684\u200b\u6539\u53d8\u200b\u8282\u70b9\u200b\u5185\u90e8\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u800c\u662f\u200b\u9700\u8981\u200b\u5b9e\u9645\u200b\u8fdb\u884c\u200b\u8282\u70b9\u200b\u4ea4\u6362\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [1,2,3,4,5], k = 2\n\u200b\u8f93\u51fa\u200b\uff1a[2,1,4,3,5]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [1,2,3,4,5], k = 3\n\u200b\u8f93\u51fa\u200b\uff1a[3,2,1,4,5]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [1,2,3,4,5], k = 1\n\u200b\u8f93\u51fa\u200b\uff1a[1,2,3,4,5]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [1], k = 1\n\u200b\u8f93\u51fa\u200b\uff1a[1]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u5217\u8868\u200b\u4e2d\u200b\u8282\u70b9\u200b\u7684\u200b\u6570\u91cf\u200b\u5728\u200b\u8303\u56f4\u200b <code>sz</code> \u200b\u5185\u200b</li> <li><code>1 &lt;= sz &lt;= 5000</code></li> <li><code>0 &lt;= Node.val &lt;= 1000</code></li> <li><code>1 &lt;= k &lt;= sz</code></li> </ul> <p>Reference</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/25/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u904d\u5386\u200b k \u200b\u4e2a\u200b\u5143\u7d20\u200b\u540e\u200b\u5c06\u200b\u94fe\u8868\u200b\u7684\u200b\u7ed3\u5c3e\u200b\u8bbe\u200b\u4e3a\u200b<code>NULL</code>\uff0c\u200b\u6267\u884c\u200b\u53cd\u8f6c\u200b\u64cd\u4f5c\u200b\u540e\u200b\u63a5\u4e0a\u200b\u540e\u9762\u200b\u7684\u200b\u94fe\u8868\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\\(\\mathcal O(1)\\)\u200b\u7684\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b\u3002</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\nstruct ListNode *reverse(struct ListNode *head)\n{\n    struct ListNode *prev = NULL, *cur = head, *next = head-&gt;next;\n    while (next != NULL)\n    {\n        cur-&gt;next = prev;\n        prev = cur;\n        cur = next;\n        next = next-&gt;next;\n    }\n    cur-&gt;next = prev;\n    return cur;\n}\n\nstruct ListNode* reverseKGroup(struct ListNode* head, int k){\n    int i = 0;\n    struct ListNode *ret = NULL, *tempHead = NULL, *cur = head;\n    for (i = 0; i &lt; k - 1 &amp;&amp; cur != NULL; i++)\n        cur = cur-&gt;next;\n    if (cur == NULL)\n        return head;\n    tempHead = cur-&gt;next;\n    cur-&gt;next = NULL;\n    ret = reverse(head);\n    head-&gt;next = reverseKGroup(tempHead, k);\n    return ret;\n}\n</code></pre>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/26/","title":"26. \u200b\u5220\u9664\u200b\u6392\u5e8f\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u91cd\u590d\u200b\u9879","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/26/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6392\u5e8f\u200b\u6570\u7ec4\u200b\uff0c\u200b\u4f60\u200b\u9700\u8981\u200b\u5728\u200b\u539f\u5730\u200b\u5220\u9664\u200b\u91cd\u590d\u200b\u51fa\u73b0\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\uff0c\u200b\u8fd4\u56de\u200b\u79fb\u9664\u200b\u540e\u200b\u6570\u7ec4\u200b\u7684\u200b\u65b0\u200b\u957f\u5ea6\u200b\u3002</p> <p>\u200b\u4e0d\u8981\u200b\u4f7f\u7528\u200b\u989d\u5916\u200b\u7684\u200b\u6570\u7ec4\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u4f60\u200b\u5fc5\u987b\u200b\u5728\u200b\u539f\u5730\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b \u200b\u5e76\u200b\u5728\u200b\u4f7f\u7528\u200b\\(O(1)\\)\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\u7684\u200b\u6761\u4ef6\u200b\u4e0b\u200b\u5b8c\u6210\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u7ed9\u5b9a\u200b\u6570\u7ec4\u200b nums = [1,1,2], \n\n\u200b\u51fd\u6570\u200b\u5e94\u8be5\u200b\u8fd4\u56de\u200b\u65b0\u200b\u7684\u200b\u957f\u5ea6\u200b 2, \u200b\u5e76\u4e14\u200b\u539f\u200b\u6570\u7ec4\u200b nums \u200b\u7684\u200b\u524d\u200b\u4e24\u4e2a\u200b\u5143\u7d20\u200b\u88ab\u200b\u4fee\u6539\u200b\u4e3a\u200b 1, 2\u3002 \n\n\u200b\u4f60\u200b\u4e0d\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u8d85\u51fa\u200b\u65b0\u200b\u957f\u5ea6\u200b\u540e\u9762\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u7ed9\u5b9a\u200b nums = [0,0,1,1,1,2,2,3,3,4],\n\n\u200b\u51fd\u6570\u200b\u5e94\u8be5\u200b\u8fd4\u56de\u200b\u65b0\u200b\u7684\u200b\u957f\u5ea6\u200b 5, \u200b\u5e76\u4e14\u200b\u539f\u200b\u6570\u7ec4\u200b nums \u200b\u7684\u200b\u524d\u200b\u4e94\u4e2a\u200b\u5143\u7d20\u200b\u88ab\u200b\u4fee\u6539\u200b\u4e3a\u200b 0, 1, 2, 3, 4\u3002\n\n\u200b\u4f60\u200b\u4e0d\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u8d85\u51fa\u200b\u65b0\u200b\u957f\u5ea6\u200b\u540e\u9762\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u4e3a\u4ec0\u4e48\u200b\u8fd4\u56de\u200b\u6570\u503c\u200b\u662f\u200b\u6574\u6570\u200b\uff0c\u200b\u4f46\u200b\u8f93\u51fa\u200b\u7684\u200b\u7b54\u6848\u200b\u662f\u200b\u6570\u7ec4\u200b\u5462\u200b?</p> <p>\u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u662f\u200b\u4ee5\u200b \u300c\u200b\u5f15\u7528\u200b\u300d \u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b\u7684\u200b\uff0c\u200b\u8fd9\u200b\u610f\u5473\u7740\u200b\u5728\u200b\u51fd\u6570\u200b\u91cc\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u5bf9\u4e8e\u200b\u8c03\u7528\u8005\u200b\u662f\u200b\u53ef\u89c1\u200b\u7684\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u60f3\u8c61\u200b\u5185\u90e8\u200b\u64cd\u4f5c\u200b\u5982\u4e0b\u200b:</p> <pre><code>// nums \u200b\u662f\u200b\u4ee5\u200b\u201c\u200b\u5f15\u7528\u200b\u201d\u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b\u7684\u200b\u3002\u200b\u4e5f\u5c31\u662f\u8bf4\u200b\uff0c\u200b\u4e0d\u200b\u5bf9\u200b\u5b9e\u53c2\u200b\u505a\u200b\u4efb\u4f55\u200b\u62f7\u8d1d\u200b\nint len = removeDuplicates(nums);\n\n// \u200b\u5728\u200b\u51fd\u6570\u200b\u91cc\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u5bf9\u4e8e\u200b\u8c03\u7528\u8005\u200b\u662f\u200b\u53ef\u89c1\u200b\u7684\u200b\u3002\n// \u200b\u6839\u636e\u200b\u4f60\u200b\u7684\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u7684\u200b\u957f\u5ea6\u200b, \u200b\u5b83\u4f1a\u200b\u6253\u5370\u200b\u51fa\u200b\u6570\u7ec4\u200b\u4e2d\u8be5\u200b\u957f\u5ea6\u200b\u8303\u56f4\u200b\u5185\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u3002\nfor (int i = 0; i &lt; len; i++) {\n\u00a0 \u00a0 print(nums[i]);\n}\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/26/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a </p> <ol> <li>\u200b\u5f53\u200b\u6570\u7ec4\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u65f6\u200b\uff0c\u200b\u4e0d\u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\uff1b</li> <li>\u200b\u4f7f\u7528\u200b\u4e00\u4e2a\u200b\u6307\u9488\u200b\u5206\u522b\u200b\u6307\u5411\u200b\u5220\u9664\u200b\u91cd\u590d\u200b\u503c\u540e\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u672b\u5c3e\u200b\uff0c\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u6570\u503c\u200b\u8868\u793a\u200b\u5220\u53bb\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570\u200b\u3002</li> </ol> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int removeDuplicates(int* nums, int numsSize){\n    if (numsSize &lt; 2)\n        return numsSize;\n    int steps = 0, i = 0;\n    while(i &lt; numsSize - steps - 1)\n    {\n        if (steps)\n            nums[i + 1] = nums[i + steps + 1];\n        if (nums[i] == nums[i + steps + 1])\n            steps++;\n        else\n            i++;\n    }\n    return numsSize - steps;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/27/","title":"27. \u200b\u79fb\u9664\u200b\u5143\u7d20","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/27/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b <code>nums</code>\u00a0\u200b\u548c\u200b\u4e00\u4e2a\u200b\u503c\u200b <code>val</code>\uff0c\u200b\u4f60\u200b\u9700\u8981\u200b \u200b\u539f\u5730\u200b \u200b\u79fb\u9664\u200b\u6240\u6709\u200b\u6570\u503c\u200b\u7b49\u4e8e\u200b\u00a0<code>val</code>\u00a0\u200b\u7684\u200b\u5143\u7d20\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b\u79fb\u9664\u200b\u540e\u200b\u6570\u7ec4\u200b\u7684\u200b\u65b0\u200b\u957f\u5ea6\u200b\u3002</p> <p>\u200b\u4e0d\u8981\u200b\u4f7f\u7528\u200b\u989d\u5916\u200b\u7684\u200b\u6570\u7ec4\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u4f60\u200b\u5fc5\u987b\u200b\u4ec5\u200b\u4f7f\u7528\u200b <code>O(1)</code> \u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\u5e76\u200b \u200b\u539f\u5730\u200b \u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u3002</p> <p>\u200b\u5143\u7d20\u200b\u7684\u200b\u987a\u5e8f\u200b\u53ef\u4ee5\u200b\u6539\u53d8\u200b\u3002\u200b\u4f60\u200b\u4e0d\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u8d85\u51fa\u200b\u65b0\u200b\u957f\u5ea6\u200b\u540e\u9762\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b:</p> <p>\u200b\u4e3a\u4ec0\u4e48\u200b\u8fd4\u56de\u200b\u6570\u503c\u200b\u662f\u200b\u6574\u6570\u200b\uff0c\u200b\u4f46\u200b\u8f93\u51fa\u200b\u7684\u200b\u7b54\u6848\u200b\u662f\u200b\u6570\u7ec4\u200b\u5462\u200b?</p> <p>\u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u662f\u200b\u4ee5\u200b\u300c\u200b\u5f15\u7528\u200b\u300d\u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b\u7684\u200b\uff0c\u200b\u8fd9\u200b\u610f\u5473\u7740\u200b\u5728\u200b\u51fd\u6570\u200b\u91cc\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u5bf9\u4e8e\u200b\u8c03\u7528\u8005\u200b\u662f\u200b\u53ef\u89c1\u200b\u7684\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u60f3\u8c61\u200b\u5185\u90e8\u200b\u64cd\u4f5c\u200b\u5982\u4e0b\u200b:</p> <pre><code>// nums \u200b\u662f\u200b\u4ee5\u200b\u201c\u200b\u5f15\u7528\u200b\u201d\u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b\u7684\u200b\u3002\u200b\u4e5f\u5c31\u662f\u8bf4\u200b\uff0c\u200b\u4e0d\u200b\u5bf9\u200b\u5b9e\u53c2\u200b\u4f5c\u200b\u4efb\u4f55\u200b\u62f7\u8d1d\u200b\nint len = removeElement(nums, val);\n\n// \u200b\u5728\u200b\u51fd\u6570\u200b\u91cc\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u5bf9\u4e8e\u200b\u8c03\u7528\u8005\u200b\u662f\u200b\u53ef\u89c1\u200b\u7684\u200b\u3002\n// \u200b\u6839\u636e\u200b\u4f60\u200b\u7684\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u7684\u200b\u957f\u5ea6\u200b, \u200b\u5b83\u4f1a\u200b\u6253\u5370\u200b\u51fa\u200b\u6570\u7ec4\u200b\u4e2d\u200b \u200b\u8be5\u200b\u957f\u5ea6\u200b\u8303\u56f4\u200b\u5185\u200b \u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u3002\nfor (int i = 0; i &lt; len; i++) {\n\u00a0 \u00a0 print(nums[i]);\n}\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [3,2,2,3], val = 3\n\u200b\u8f93\u51fa\u200b\uff1a2, nums = [2,2]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u51fd\u6570\u200b\u5e94\u8be5\u200b\u8fd4\u56de\u200b\u65b0\u200b\u7684\u200b\u957f\u5ea6\u200b 2, \u200b\u5e76\u4e14\u200b nums \u200b\u4e2d\u200b\u7684\u200b\u524d\u200b\u4e24\u4e2a\u200b\u5143\u7d20\u200b\u5747\u200b\u4e3a\u200b 2\u3002\u200b\u4f60\u200b\u4e0d\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u8d85\u51fa\u200b\u65b0\u200b\u957f\u5ea6\u200b\u540e\u9762\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\u200b\u4f8b\u5982\u200b\uff0c\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u7684\u200b\u65b0\u200b\u957f\u5ea6\u200b\u4e3a\u200b 2 \uff0c\u200b\u800c\u200b nums = [2,2,3,3] \u200b\u6216\u200b nums = [2,2,0,0]\uff0c\u200b\u4e5f\u200b\u4f1a\u200b\u88ab\u200b\u89c6\u4f5c\u200b\u6b63\u786e\u200b\u7b54\u6848\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [0,1,2,2,3,0,4,2], val = 2\n\u200b\u8f93\u51fa\u200b\uff1a5, nums = [0,1,4,0,3]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u51fd\u6570\u200b\u5e94\u8be5\u200b\u8fd4\u56de\u200b\u65b0\u200b\u7684\u200b\u957f\u5ea6\u200b 5, \u200b\u5e76\u4e14\u200b nums \u200b\u4e2d\u200b\u7684\u200b\u524d\u200b\u4e94\u4e2a\u200b\u5143\u7d20\u200b\u4e3a\u200b 0, 1, 3, 0, 4\u3002\u200b\u6ce8\u610f\u200b\u8fd9\u200b\u4e94\u4e2a\u200b\u5143\u7d20\u200b\u53ef\u200b\u4e3a\u200b\u4efb\u610f\u200b\u987a\u5e8f\u200b\u3002\u200b\u4f60\u200b\u4e0d\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u8d85\u51fa\u200b\u65b0\u200b\u957f\u5ea6\u200b\u540e\u9762\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>0 &lt;= nums.length &lt;= 100</code></li> <li><code>0 &lt;= nums[i] &lt;= 50</code></li> <li><code>0 &lt;= val &lt;= 100</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/27/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u53cc\u200b\u6307\u9488\u200b\u6cd5\u200b\uff0c\u200b\u4e24\u4e2a\u200b\u6307\u9488\u200b\u5206\u522b\u200b\u6307\u5411\u200b\uff1a</p> <ul> <li>\u200b\u5c1a\u672a\u200b\u68c0\u67e5\u200b\u8fc7\u200b\u7684\u200b\u90e8\u5206\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u5143\u7d20\u200b</li> <li>\u200b\u5df2\u7ecf\u200b\u68c0\u67e5\u200b\u8fc7\u200b\u7684\u200b\u90e8\u5206\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b</li> </ul> <p>\u200b\u5f53\u200b\u4e24\u4e2a\u200b\u6307\u9488\u200b\u91cd\u5408\u200b\u65f6\u200b\u4e0d\u200b\u9700\u8981\u200b\u8fdb\u884c\u200b\u590d\u5236\u200b\u64cd\u4f5c\u200b\u3002</p> <pre><code>int removeElement(int* nums, int numsSize, int val){\n    int slow = 0, fast = 0, ret = numsSize;\n    if (numsSize == 0)\n        return 0;\n    do\n    {\n        if (fast &gt; slow)\n            nums[slow] = nums[fast];\n        if (nums[fast] == val)\n            ret--;\n        else\n            slow++;\n        fast++;\n    }\n    while (fast &lt; numsSize);\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/28/","title":"28. \u200b\u5b9e\u73b0\u200b strStr()","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/28/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5b9e\u73b0\u200b<code>strStr()</code>\u200b\u51fd\u6570\u200b\u3002</p> <p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b<code>haystack</code>\u200b\u5b57\u7b26\u4e32\u200b\u548c\u200b\u4e00\u4e2a\u200b<code>needle</code>\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u5728\u200b<code>haystack</code>\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u627e\u51fa\u200b needle \u200b\u5b57\u7b26\u4e32\u200b\u51fa\u73b0\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u4f4d\u7f6e\u200b (\u200b\u4ece\u200b0\u200b\u5f00\u59cb\u200b)\u3002\u200b\u5982\u679c\u200b\u4e0d\u200b\u5b58\u5728\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b\u00a0-1\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1:</p> <pre><code>\u200b\u8f93\u5165\u200b: haystack = \"hello\", needle = \"ll\"\n\u200b\u8f93\u51fa\u200b: 2\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2:</p> <pre><code>\u200b\u8f93\u5165\u200b: haystack = \"aaaaa\", needle = \"bba\"\n\u200b\u8f93\u51fa\u200b: -1\n</code></pre> <p>\u200b\u8bf4\u660e\u200b:</p> <p>\u200b\u5f53\u200b<code>needle</code>\u200b\u662f\u200b\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u65f6\u200b\uff0c\u200b\u6211\u4eec\u200b\u5e94\u5f53\u200b\u8fd4\u56de\u200b\u4ec0\u4e48\u200b\u503c\u200b\u5462\u200b\uff1f\u200b\u8fd9\u662f\u200b\u4e00\u4e2a\u200b\u5728\u200b\u9762\u8bd5\u200b\u4e2d\u200b\u5f88\u200b\u597d\u200b\u7684\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u672c\u9898\u200b\u800c\u8a00\u200b\uff0c\u200b\u5f53\u200b<code>needle</code>\u200b\u662f\u200b\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u65f6\u200b\u6211\u4eec\u200b\u5e94\u5f53\u200b\u8fd4\u56de\u200b 0 \u3002\u200b\u8fd9\u200b\u4e0e\u200bC\u200b\u8bed\u8a00\u200b\u7684\u200b<code>strstr()</code>\u200b\u4ee5\u53ca\u200bJava\u200b\u7684\u200b<code>indexOf()</code>\u200b\u5b9a\u4e49\u200b\u76f8\u7b26\u200b\u3002</p> <p>Reference</p>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/28/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u66b4\u529b\u200b\u6309\u4f4d\u200b\u5339\u914d\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int strStr(char * haystack, char * needle){\n    int len1 = strlen(haystack), len2 = strlen(needle), cur1 = 0, cur2 = 0, cur3 = 0, success = 1;\n    if (!len2)\n        return 0;\n    if (len1 &lt; len2)\n        return -1;\n    do\n    {\n        success = 1;\n        if (haystack[cur1] == needle[0])\n        {\n            cur2 = 0;\n            cur3 = 0;\n            while(cur2 &lt; len2 &amp;&amp; cur3 + cur1 &lt; len1 &amp;&amp; success)\n            {\n                if(haystack[cur1 + cur3] != needle[cur2])\n                    success = 0;\n                cur3++;\n                cur2++;\n            }\n            if (success)\n                return cur1;\n        }\n        cur1++;\n    }\n    while (haystack[cur1 + len2 - 1]);\n    return -1;\n}\n</code></pre>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/3/","title":"3. \u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u54c8\u5e0c\u8868","\u53cc\u6307\u9488","\u5b57\u7b26\u4e32","Sliding Window"]},{"location":"coding/leetcode/3/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u627e\u51fa\u200b\u5176\u4e2d\u200b\u4e0d\u200b\u542b\u6709\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u00a0\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b\u00a0\u200b\u7684\u200b\u957f\u5ea6\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01:</p> <pre><code>\u200b\u8f93\u5165\u200b: s = \"abcabcbb\"\n\u200b\u8f93\u51fa\u200b: 3 \n\u200b\u89e3\u91ca\u200b: \u200b\u56e0\u4e3a\u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b\u662f\u200b \"abc\"\uff0c\u200b\u6240\u4ee5\u200b\u5176\u200b\u957f\u5ea6\u200b\u4e3a\u200b 3\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2:</p> <pre><code>\u200b\u8f93\u5165\u200b: s = \"bbbbb\"\n\u200b\u8f93\u51fa\u200b: 1\n\u200b\u89e3\u91ca\u200b: \u200b\u56e0\u4e3a\u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b\u662f\u200b \"b\"\uff0c\u200b\u6240\u4ee5\u200b\u5176\u200b\u957f\u5ea6\u200b\u4e3a\u200b 1\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3:</p> <pre><code>\u200b\u8f93\u5165\u200b: s = \"pwwkew\"\n\u200b\u8f93\u51fa\u200b: 3\n\u200b\u89e3\u91ca\u200b: \u200b\u56e0\u4e3a\u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b\u662f\u200b\u00a0\"wke\"\uff0c\u200b\u6240\u4ee5\u200b\u5176\u200b\u957f\u5ea6\u200b\u4e3a\u200b 3\u3002\n\u00a0    \u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u4f60\u200b\u7684\u200b\u7b54\u6848\u200b\u5fc5\u987b\u200b\u662f\u200b \u200b\u5b50\u4e32\u200b \u200b\u7684\u200b\u957f\u5ea6\u200b\uff0c\"pwke\"\u00a0\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5b50\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4e0d\u662f\u200b\u5b50\u4e32\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4:</p> <pre><code>\u200b\u8f93\u5165\u200b: s = \"\"\n\u200b\u8f93\u51fa\u200b: 0\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>0 &lt;= s.length &lt;= 5 * 104</code></li> <li><code>s</code>\u00a0\u200b\u7531\u200b\u82f1\u6587\u5b57\u6bcd\u200b\u3001\u200b\u6570\u5b57\u200b\u3001\u200b\u7b26\u53f7\u200b\u548c\u200b\u7a7a\u683c\u200b\u7ec4\u6210\u200b</li> </ul>","tags":["\u54c8\u5e0c\u8868","\u53cc\u6307\u9488","\u5b57\u7b26\u4e32","Sliding Window"]},{"location":"coding/leetcode/3/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b \uff1a\u200b\u7528\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\u5b58\u50a8\u200b\u5404\u200b\u5b57\u6bcd\u200b\u51fa\u73b0\u200b\u7684\u200b\u6b21\u6570\u200b\uff08\u200b\u54c8\u5e0c\u200b\u8868\u200b\uff09\u3002\u200b\u5f53\u200b\u4e0d\u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5b57\u6bcd\u200b\u65f6\u200b\u6ed1\u52a8\u200b\u7a97\u53e3\u200b\u4e0d\u65ad\u200b\u5411\u200b\u53f3\u200b\u5ef6\u4f38\u200b\uff0c\u200b\u76f4\u5230\u200b\u51fa\u73b0\u200b\u91cd\u590d\u200b\u5b57\u6bcd\u200b\u6216\u200b\u5230\u8fbe\u200b\u5b57\u7b26\u4e32\u200b\u7ed3\u5c3e\u200b\u3002\u200b\u68c0\u6d4b\u200b\u5230\u200b\u91cd\u590d\u200b\u5b57\u6bcd\u200b\u540e\u200b\uff0c\u200b\u4ece\u200b\u5de6\u5411\u53f3\u200b\u6536\u7f29\u200b\u6ed1\u52a8\u200b\u7a97\u53e3\u200b\uff0c\u200b\u6bcf\u6b21\u200b\u6536\u7f29\u200b\u90fd\u200b\u5bf9\u200b\u54c8\u5e0c\u200b\u8868\u200b\u8fdb\u884c\u200b\u66f4\u65b0\u200b\u3002\u200b\u76f4\u5230\u200b\u4e0d\u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5b57\u6bcd\u200b\u65f6\u200b\uff0c\u200b\u6ed1\u52a8\u200b\u7a97\u53e3\u200b\u624d\u200b\u505c\u6b62\u200b\u6536\u7f29\u200b\uff0c\u200b\u7ee7\u7eed\u200b\u5411\u200b\u53f3\u200b\u5ef6\u4f38\u200b\u3002</p> <p>\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b \uff1a\\(O(N)\\)</p> <p>\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b \uff1a\\(O(1)\\)\uff0c\u200b\u4e0e\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u65e0\u5173\u200b\uff0c\u200b\u4f46\u200b\u4e0e\u200b\u5355\u4e2a\u200b\u5b57\u7b26\u200b\u7684\u200b\u5bbd\u5ea6\u200b\u6709\u5173\u200b\u3002</p> <pre><code>int lengthOfLongestSubstring(char * s){\n    int length = 0, ret = 0;\n    char hashMap[256] = {0}, *cur = s;\n    while(*cur)\n    {\n        while(hashMap[*cur])\n        {\n            hashMap[*(cur - length)] -= 1;\n            length--;\n        }\n        hashMap[*cur] = 1;\n        length++;\n        ret = ret &gt; length ? ret : length;\n        cur++;\n    }\n    return ret;\n}\n</code></pre> <p>Reference</p>","tags":["\u54c8\u5e0c\u8868","\u53cc\u6307\u9488","\u5b57\u7b26\u4e32","Sliding Window"]},{"location":"coding/leetcode/315/","title":"315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u6392\u5e8f","\u6811\u72b6\u6570\u7ec4","\u7ebf\u6bb5\u6811","\u4e8c\u5206\u67e5\u627e","\u5206\u6cbb\u7b97\u6cd5"]},{"location":"coding/leetcode/315/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b nums \uff0c\u200b\u6309\u200b\u8981\u6c42\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u6570\u7ec4\u200b\u00a0counts \u3002\u200b\u6570\u7ec4\u200b counts \u200b\u6709\u200b\u8be5\u200b\u6027\u8d28\u200b\uff1a<code>counts[i]</code>\u200b\u7684\u200b\u503c\u200b\u662f\u200b<code>nums[i]</code>\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b<code>nums[i]</code>\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u6570\u91cf\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [5,2,6,1]\n\u200b\u8f93\u51fa\u200b\uff1a[2,1,1,0] \n\u200b\u89e3\u91ca\u200b\uff1a\n5 \u200b\u7684\u200b\u53f3\u4fa7\u200b\u6709\u200b 2 \u200b\u4e2a\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u5143\u7d20\u200b (2 \u200b\u548c\u200b 1)\n2 \u200b\u7684\u200b\u53f3\u4fa7\u200b\u4ec5\u200b\u6709\u200b 1 \u200b\u4e2a\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u5143\u7d20\u200b (1)\n6 \u200b\u7684\u200b\u53f3\u4fa7\u200b\u6709\u200b 1 \u200b\u4e2a\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u5143\u7d20\u200b (1)\n1 \u200b\u7684\u200b\u53f3\u4fa7\u200b\u6709\u200b 0 \u200b\u4e2a\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u5143\u7d20\u200b\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>0 &lt;= nums.length &lt;= 10^5</code></li> <li><code>-10^4\u00a0&lt;= nums[i] &lt;= 10^4</code></li> </ul> <p>Reference</p>","tags":["\u6392\u5e8f","\u6811\u72b6\u6570\u7ec4","\u7ebf\u6bb5\u6811","\u4e8c\u5206\u67e5\u627e","\u5206\u6cbb\u7b97\u6cd5"]},{"location":"coding/leetcode/315/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4ece\u540e\u200b\u5f80\u524d\u200b\u6784\u9020\u200b\u4e00\u4e2a\u200b\u4e8c\u53c9\u200b\u67e5\u627e\u200b\u6811\u200b\uff0c\u200b\u5728\u200b\u4e8c\u53c9\u200b\u67e5\u627e\u200b\u6811\u4e2d\u200b\u7ef4\u62a4\u200b\u6811\u4e2d\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u7684\u200b\u6570\u5b57\u200b\u7684\u200b\u4e2a\u6570\u200b\u3002</p> <p>\u200b\u4e8c\u53c9\u6811\u200b\u9700\u8981\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u57df\u200b\u7528\u4e8e\u200b\u8ba1\u7b97\u200b\u5de6\u5b50\u200b\u6811\u200b\u7684\u200b\u5143\u7d20\u200b\u603b\u6570\u200b\uff0c\u200b\u5373\u200b\u5f53\u524d\u200b\u6811\u4e2d\u200b\u6570\u503c\u200b\u5c0f\u4e8e\u200b\u8be5\u200b\u8282\u70b9\u200b\u7684\u200b\u5143\u7d20\u200b\u603b\u6570\u200b\u3002\u200b\u5b9e\u73b0\u200b\u65b9\u6cd5\u200b\u662f\u200b\u5728\u200b\u5de6\u5b50\u200b\u6811\u200b\u6dfb\u52a0\u200b\u5143\u7d20\u200b\u540e\u200b\u8be5\u57df\u200b\u7684\u200b\u503c\u200b\u589e\u52a0\u200b1\u3002</p> <p>\u200b\u5728\u200b\u8282\u70b9\u200b\u7684\u200b\u53f3\u5b50\u200b\u6811\u200b\u6dfb\u52a0\u200b\u5143\u7d20\u200b\u65f6\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u548c\u200b\u8282\u70b9\u200b\u5de6\u5b50\u200b\u6811\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u8282\u70b9\u200b\u90fd\u200b\u662f\u200b\u7b26\u5408\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u8282\u70b9\u200b\u3002\u200b\u5728\u200b\u9012\u5f52\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u9700\u8981\u200b\u4f20\u9012\u200b\u4e00\u4e2a\u200b\u7528\u4f5c\u200b\u8ba1\u6570\u5668\u200b\u7684\u200b\u6307\u9488\u200b\u53c2\u6570\u200b\uff08\u200b\u8ba1\u6570\u200b\u7684\u200b\u6570\u503c\u200b\u5373\u200b\u4e3a\u200b\u6240\u6c42\u200b\u7684\u200b\u7ed3\u679c\u200b\uff09\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\nstruct TreeNode2\n{\n    struct TreeNode2 *left;\n    struct TreeNode2 *right;\n    int val;\n    int subs;\n};\n\nstruct TreeNode2 *insert(struct TreeNode2 *dest, int src, int *depth)\n{\n    if (dest == NULL)\n    {\n        struct TreeNode2 *ret = (struct TreeNode2 *)malloc(sizeof(struct TreeNode2));\n        ret-&gt;left = NULL;\n        ret-&gt;right = NULL;\n        ret-&gt;val = src;\n        ret-&gt;subs = 0;\n        return ret;\n    }\n    if (src &gt; dest-&gt;val)\n    {\n        *depth += 1 + dest-&gt;subs;\n        dest-&gt;right = insert(dest-&gt;right, src, depth);\n    }\n    else\n    {\n        dest-&gt;subs++;\n        dest-&gt;left = insert(dest-&gt;left, src, depth);\n    }\n    return dest;\n}\n\nint *countSmaller(int *nums, int numsSize, int *returnSize)\n{\n    *returnSize = numsSize;\n    int *ret = (int *)memset(malloc(sizeof(int) * numsSize), 0, sizeof(int) * numsSize), i = 0;\n    if (numsSize &lt;= 1)\n        return ret;\n    struct TreeNode2 *tempTree = NULL;\n    for (i = numsSize - 1; i &gt;= 0; i--)\n        tempTree = insert(tempTree, nums[i], ret + i);\n    return ret;\n}\n</code></pre>","tags":["\u6392\u5e8f","\u6811\u72b6\u6570\u7ec4","\u7ebf\u6bb5\u6811","\u4e8c\u5206\u67e5\u627e","\u5206\u6cbb\u7b97\u6cd5"]},{"location":"coding/leetcode/341/","title":"341. \u200b\u6241\u5e73\u5316\u200b\u5d4c\u5957\u200b\u5217\u8868\u200b\u8fed\u4ee3\u200b\u5668","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6808","\u8bbe\u8ba1"]},{"location":"coding/leetcode/341/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u5d4c\u5957\u200b\u7684\u200b\u6574\u578b\u200b\u5217\u8868\u200b\u3002\u200b\u8bf7\u200b\u4f60\u200b\u8bbe\u8ba1\u200b\u4e00\u4e2a\u200b\u8fed\u4ee3\u200b\u5668\u200b\uff0c\u200b\u4f7f\u200b\u5176\u200b\u80fd\u591f\u200b\u904d\u5386\u200b\u8fd9\u4e2a\u200b\u6574\u578b\u200b\u5217\u8868\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u5217\u8868\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e00\u9879\u200b\u6216\u8005\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\uff0c\u200b\u6216\u8005\u200b\u662f\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u5217\u8868\u200b\u3002\u200b\u5176\u4e2d\u200b\u5217\u8868\u200b\u7684\u200b\u5143\u7d20\u200b\u4e5f\u200b\u53ef\u80fd\u200b\u662f\u200b\u6574\u6570\u200b\u6216\u662f\u200b\u5176\u4ed6\u200b\u5217\u8868\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [[1,1],2,[1,1]]\n\u200b\u8f93\u51fa\u200b: [1,1,2,1,1]\n\u200b\u89e3\u91ca\u200b: \u200b\u901a\u8fc7\u200b\u91cd\u590d\u200b\u8c03\u7528\u200b\u00a0next \u200b\u76f4\u5230\u200b\u00a0hasNext \u200b\u8fd4\u56de\u200b false\uff0cnext\u00a0\u200b\u8fd4\u56de\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u987a\u5e8f\u200b\u5e94\u8be5\u200b\u662f\u200b: [1,1,2,1,1]\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,[4,[6]]]\n\u200b\u8f93\u51fa\u200b: [1,4,6]\n\u200b\u89e3\u91ca\u200b: \u200b\u901a\u8fc7\u200b\u91cd\u590d\u200b\u8c03\u7528\u200b\u00a0next\u00a0\u200b\u76f4\u5230\u200b\u00a0hasNext \u200b\u8fd4\u56de\u200b false\uff0cnext\u00a0\u200b\u8fd4\u56de\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u987a\u5e8f\u200b\u5e94\u8be5\u200b\u662f\u200b: [1,4,6]\u3002\n</code></pre> <p>Reference</p>","tags":["\u6808","\u8bbe\u8ba1"]},{"location":"coding/leetcode/341/#_2","title":"\u9898\u89e3","text":"<p>C\u200b\u8bed\u8a00\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5d4c\u5957\u200b\u5217\u8868\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u94fe\u8868\u200b\uff0c\u200b\u7136\u540e\u200b\u5bf9\u200b\u94fe\u8868\u200b\u8fdb\u884c\u200b\u8fed\u4ee3\u200b\uff1a</p> <pre><code>/**\n * *********************************************************************\n * // This is the interface that allows for creating nested lists.\n * // You should not implement it, or speculate about its implementation\n * *********************************************************************\n *\n * // Return true if this NestedInteger holds a single integer, rather than a nested list.\n * bool NestedIntegerIsInteger(struct NestedInteger *);\n *\n * // Return the single integer that this NestedInteger holds, if it holds a single integer\n * // The result is undefined if this NestedInteger holds a nested list\n * int NestedIntegerGetInteger(struct NestedInteger *);\n *\n * // Return the nested list that this NestedInteger holds, if it holds a nested list\n * // The result is undefined if this NestedInteger holds a single integer\n * struct NestedInteger **NestedIntegerGetList(struct NestedInteger *);\n *\n * // Return the nested list's size that this NestedInteger holds, if it holds a nested list\n * // The result is undefined if this NestedInteger holds a single integer\n * int NestedIntegerGetListSize(struct NestedInteger *);\n * };\n */\nstruct NestedIterator {\n    struct ListNode *list;\n    struct ListNode *current;\n};\n\nstruct ListNode *getNode(int val, struct ListNode *next)\n{\n    struct ListNode *ret = (struct ListNode *)malloc(sizeof(struct ListNode));\n    ret-&gt;val = val;\n    ret-&gt;next = next;\n    return ret;\n}\n\nstruct ListNode *toList(struct NestedInteger *src, struct ListNode **tail)\n{\n    if (NestedIntegerIsInteger(src))\n    {\n        struct ListNode *ret = getNode(NestedIntegerGetInteger(src), NULL);\n        if (tail)\n            *tail = ret;\n        return ret;\n    }\n    else\n    {\n        struct NestedInteger **content = NestedIntegerGetList(src);\n        struct ListNode *ret = NULL, *temp = NULL;\n        int i = 0, size = NestedIntegerGetListSize(src);\n        for (i = 0; i &lt; size; i++)\n        {\n            if (!ret)\n                ret = toList(content[i], &amp;temp);\n            else\n                (*tail)-&gt;next = toList(content[i], &amp;temp);\n            if (temp)\n                *tail = temp;\n        }\n        return ret;\n    }\n}\n\nstruct NestedIterator *nestedIterCreate(struct NestedInteger** nestedList, int nestedListSize) {\n    struct NestedIterator *ret = (struct NestedIterator *)malloc(sizeof(struct NestedIterator));\n    ret-&gt;list = NULL;\n    ret-&gt;current = NULL;\n    struct ListNode *temp = NULL;\n    for (int i = 0; i &lt; nestedListSize; i++)\n    {\n        if (!ret-&gt;list)\n            ret-&gt;list = toList(nestedList[i], &amp;temp);\n        else\n            ret-&gt;current-&gt;next = toList(nestedList[i], &amp;temp);\n        ret-&gt;current = temp;\n    }\n    ret-&gt;current = ret-&gt;list;\n    return ret;\n}\n\nbool nestedIterHasNext(struct NestedIterator *iter) {\n    return iter-&gt;current;\n}\n\nint nestedIterNext(struct NestedIterator *iter) {\n    int ret = iter-&gt;current-&gt;val;\n    iter-&gt;current = iter-&gt;current-&gt;next;\n    return ret;\n}\n\n/** Deallocates memory previously allocated for the iterator */\nvoid nestedIterFree(struct NestedIterator *iter) {\n    while (iter-&gt;list)\n    {\n        iter-&gt;current = iter-&gt;list-&gt;next;\n        free(iter-&gt;list);\n        iter-&gt;list = iter-&gt;current;\n    }\n    free(iter);\n}\n\n/**\n * Your NestedIterator will be called like this:\n * struct NestedIterator *i = nestedIterCreate(nestedList, nestedListSize);\n * while (nestedIterHasNext(i)) printf(\"%d\\n\", nestedIterNext(i));\n * nestedIterFree(i);\n */\n</code></pre> <p>Python\u200b\u8bed\u8a00\u200b\u652f\u6301\u200b<code>yield</code>\u200b\u8bed\u53e5\u200b\u4e0e\u200b<code>yield from</code>\u200b\u8bed\u53e5\u200b\uff0c\u200b\u56e0\u6b64\u200b\u76f4\u63a5\u200b\u8fed\u4ee3\u200b\u751f\u6210\u5668\u200b\u5373\u53ef\u200b</p> <pre><code># \"\"\"\n# This is the interface that allows for creating nested lists.\n# You should not implement it, or speculate about its implementation\n# \"\"\"\n#class NestedInteger:\n#    def isInteger(self) -&gt; bool:\n#        \"\"\"\n#        @return True if this NestedInteger holds a single integer, rather than a nested list.\n#        \"\"\"\n#\n#    def getInteger(self) -&gt; int:\n#        \"\"\"\n#        @return the single integer that this NestedInteger holds, if it holds a single integer\n#        Return None if this NestedInteger holds a nested list\n#        \"\"\"\n#\n#    def getList(self) -&gt; [NestedInteger]:\n#        \"\"\"\n#        @return the nested list that this NestedInteger holds, if it holds a nested list\n#        Return None if this NestedInteger holds a single integer\n#        \"\"\"\n\ndef listIter(nestNode):\n    for node in nestNode:\n        if node.isInteger():\n            yield node.getInteger()\n        else:\n            yield from listIter(node.getList())\n\nclass NestedIterator:\n    def __init__(self, nestedList):\n        self.nextvalue = 0\n        self.iterator = listIter(nestedList)\n\n    def next(self):\n        return self.nextvalue\n\n    def hasNext(self):\n        try:\n            self.nextvalue = next(self.iterator)\n        except StopIteration:\n            return False\n        return True\n\n# Your NestedIterator object will be instantiated and called as such:\n# i, v = NestedIterator(nestedList), []\n# while i.hasNext(): v.append(i.next())\n</code></pre>","tags":["\u6808","\u8bbe\u8ba1"]},{"location":"coding/leetcode/344/","title":"344. \u200b\u53cd\u8f6c\u200b\u5b57\u7b26\u4e32","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/344/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u5176\u200b\u4f5c\u7528\u200b\u662f\u200b\u5c06\u200b\u8f93\u5165\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\u53cd\u8f6c\u200b\u8fc7\u6765\u200b\u3002\u200b\u8f93\u5165\u200b\u5b57\u7b26\u4e32\u200b\u4ee5\u200b\u5b57\u7b26\u200b\u6570\u7ec4\u200b char[] \u200b\u7684\u200b\u5f62\u5f0f\u200b\u7ed9\u51fa\u200b\u3002</p> <p>\u200b\u4e0d\u8981\u200b\u7ed9\u200b\u53e6\u5916\u200b\u7684\u200b\u6570\u7ec4\u200b\u5206\u914d\u200b\u989d\u5916\u200b\u7684\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u4f60\u200b\u5fc5\u987b\u200b\u539f\u5730\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u3001\u200b\u4f7f\u7528\u200b O(1) \u200b\u7684\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\u89e3\u51b3\u200b\u8fd9\u4e00\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5b57\u7b26\u200b\u90fd\u200b\u662f\u200b ASCII \u200b\u7801\u8868\u200b\u4e2d\u200b\u7684\u200b\u53ef\u200b\u6253\u5370\u200b\u5b57\u7b26\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a <pre><code>\u200b\u8f93\u5165\u200b\uff1a[\"h\",\"e\",\"l\",\"l\",\"o\"]\n\u200b\u8f93\u51fa\u200b\uff1a[\"o\",\"l\",\"l\",\"e\",\"h\"]\n</code></pre> \u200b\u793a\u4f8b\u200b 2\uff1a <pre><code>\u200b\u8f93\u5165\u200b\uff1a[\"H\",\"a\",\"n\",\"n\",\"a\",\"h\"]\n\u200b\u8f93\u51fa\u200b\uff1a[\"h\",\"a\",\"n\",\"n\",\"a\",\"H\"]\n</code></pre></p> <p>Reference</p>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/344/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5206\u522b\u200b\u4f7f\u7528\u200b\u4e24\u4e2a\u200b\u6307\u9488\u200b\u6307\u5411\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5f00\u5934\u200b\u4e0e\u200b\u7ed3\u5c3e\u200b\uff0c\u200b\u4ea4\u6362\u200b\u4e24\u8005\u200b\u7684\u200b\u503c\u200b\u540e\u200b\u6536\u7f29\u200b\u6307\u9488\u200b\u3002\uff08\u200b\u5de6\u200b\u6307\u9488\u200b\u53f3\u79fb\u200b\uff0c\u200b\u53f3\u200b\u6307\u9488\u200b\u5de6\u79fb\u200b\uff09</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>void reverseString(char* s, int sSize)\n{\n    if (sSize &lt;= 1)\n        return;\n    char temp = 0;\n    temp = *s;\n    *s = s[sSize - 1];\n    s[sSize - 1] = temp;\n    reverseString(s + 1, sSize - 2);\n}\n</code></pre>","tags":["\u53cc\u6307\u9488","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/35/","title":"35. \u200b\u641c\u7d22\u200b\u63d2\u5165\u200b\u4f4d\u7f6e","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/35/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6392\u5e8f\u200b\u6570\u7ec4\u200b\u548c\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u503c\u200b\uff0c\u200b\u5728\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u627e\u5230\u200b\u76ee\u6807\u503c\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5176\u200b\u7d22\u5f15\u200b\u3002\u200b\u5982\u679c\u200b\u76ee\u6807\u503c\u200b\u4e0d\u200b\u5b58\u5728\u200b\u4e8e\u200b\u6570\u7ec4\u200b\u4e2d\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5b83\u200b\u5c06\u200b\u4f1a\u200b\u88ab\u200b\u6309\u200b\u987a\u5e8f\u200b\u63d2\u5165\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u65e0\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1:</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,3,5,6], 5\n\u200b\u8f93\u51fa\u200b: 2\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02:</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,3,5,6], 2\n\u200b\u8f93\u51fa\u200b: 1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3:</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,3,5,6], 7\n\u200b\u8f93\u51fa\u200b: 4\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4:</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,3,5,6], 0\n\u200b\u8f93\u51fa\u200b: 0\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/35/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u3002</p> <p>\u200b\u8bbe\u200b\u6570\u5217\u200b\\(\\{a_i\\}\\)\u200b\u4e3a\u200b\u5347\u5e8f\u200b\u6570\u5217\u200b\uff0c\u200b\u76ee\u6807\u503c\u200b\\(b\\)\u200b\u7684\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b\\(j\\)\u200b\u6ee1\u8db3\u200b\uff1a</p> <ul> <li>\\(b \\leq a_j\\)</li> <li>\\(b &gt; a_{j - 1}\\)</li> </ul> <p>\u200b\u6ce8\u610f\u200b\u672c\u9898\u200b\u8981\u6c42\u200b\u7684\u200b\u6570\u636e\u200b\u4e0e\u200b\u666e\u901a\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u610f\u4e49\u200b\u4e0d\u540c\u200b\u3002</p> <ul> <li>\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u8fd4\u56de\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u4e0d\u200b\u5927\u4e8e\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u5143\u7d20\u200b\u4e0b\u6807\u200b\u3002</li> <li>\u200b\u672c\u9898\u200b\u8981\u6c42\u200b\u8fd4\u56de\u200b\u7b2c\u4e00\u4e2a\u200b\u4e0d\u200b\u5c0f\u4e8e\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u5143\u7d20\u200b\u4e0b\u6807\u200b\u3002</li> </ul> <pre><code>int bSearch(int *nums, int lo, int hi, int target)\n{\n    int mid;\n    while (lo &lt; hi)\n    {\n        mid = (lo + hi) &gt;&gt; 1;\n        if (nums[mid] &gt;= target)\n            hi = mid;\n        else\n            lo = mid + 1;\n    }\n    return lo;\n}\n\nint searchInsert(int* nums, int numsSize, int target){\n    return bSearch(nums, 0, numsSize, target);\n}\n</code></pre> <p>\u200b\u9644\u200b\uff1a\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u4f9b\u200b\u6bd4\u8f83\u200b</p> <pre><code>int bSearch(int *nums, int lo, int hi, int target)\n{\n    int mid;\n    while (lo &lt; hi)\n    {\n        mid = (lo + hi) &gt;&gt; 1;\n        if (nums[mid] &gt; target)\n            hi = mid;\n        else\n            lo = mid + 1;\n    }\n    return lo - 1;\n}\n</code></pre> <p>\u200b\u5728\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u4e2d\u200b\uff0c\u200b\u59cb\u7ec8\u200b\u6709\u200b\uff1a</p> <ul> <li><code>nums[lo - 1]</code> \u200b\u662f\u200b\u622a\u81f3\u200b\u5f53\u524d\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u4e0d\u200b\u5927\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5927\u200b\u5143\u7d20\u200b</li> <li><code>nums[hi]</code> \u200b\u662f\u200b\u622a\u81f3\u200b\u5f53\u524d\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u5927\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b</li> </ul> <p>\u200b\u800c\u200b\u5728\u200b\u672c\u9898\u200b\u7b97\u6cd5\u200b\u4e2d\u200b\uff0c\u200b\u59cb\u7ec8\u200b\u6709\u200b\uff1a</p> <ul> <li><code>nums[lo - 1]</code> \u200b\u662f\u200b\u622a\u81f3\u200b\u5f53\u524d\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u5c0f\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5927\u200b\u5143\u7d20\u200b</li> <li><code>nums[hi]</code> \u200b\u662f\u200b\u622a\u81f3\u200b\u5f53\u524d\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u4e0d\u200b\u5c0f\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b</li> </ul> <p>\u200b\u539f\u56e0\u200b\u5728\u4e8e\u200b\uff0c\u200b\u4e24\u200b\u7b97\u6cd5\u200b\u5bf9\u200b\u7b49\u4e8e\u200b\u7684\u200b\u60c5\u51b5\u200b\u7684\u200b\u5904\u7406\u200b\u4e0d\u540c\u200b\uff0c\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u5bf9\u200b<code>nums[mid] == target</code>\u200b\u60c5\u51b5\u200b\u7684\u200b\u5904\u7406\u200b\u4e3a\u200b <code>lo = mid + 1</code>\uff0c\u200b\u4ec5\u5f53\u200b<code>nums[mid] &gt; target</code>\u200b\u65f6\u624d\u200b\u6267\u884c\u200b<code>hi = mid</code>\u3002\u200b\u7531\u4e8e\u200b<code>[lo, hi)</code>\u200b\u662f\u200b\u5f53\u524d\u200b\u5df2\u200b \u200b\u786e\u8ba4\u200b\u7684\u200b\u533a\u95f4\u200b\uff0c\u200b\u5219\u200b<code>nums[hi]</code>\u200b\u4e3a\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u5927\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b\uff0c\u200b\u800c\u200b<code>nums[lo - 1]</code>\u200b\u4e3a\u200b\u4e0d\u200b\u5927\u4e8e\u200b <code>target</code>\u200b\u7684\u200b\u6700\u5927\u200b\u5143\u7d20\u200b\u3002\u200b\u800c\u200b\u672c\u9898\u200b\u7b97\u6cd5\u200b\u5bf9\u200b<code>nums[mid] == target</code>\u200b\u60c5\u51b5\u200b\u7684\u200b\u5904\u7406\u200b\u4e3a\u200b<code>hi = mid</code>\uff0c\u200b\u4ec5\u5f53\u200b <code>nums[mid] &lt; target</code>\u200b\u65f6\u624d\u200b\u6267\u884c\u200b<code>lo = mid + 1</code>\uff0c\u200b\u5219\u200b<code>nums[hi]</code>\u200b\u4e3a\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u4e0d\u200b\u5c0f\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5c0f\u200b \u200b\u5143\u7d20\u200b\uff0c<code>nums[lo - 1]</code>\u200b\u4e3a\u200b\u5df2\u200b\u786e\u8ba4\u200b\u7684\u200b\u5c0f\u4e8e\u200b<code>target</code>\u200b\u7684\u200b\u6700\u5927\u200b\u5143\u7d20\u200b\u3002</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/350/","title":"350. \u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u7684\u200b\u4ea4\u96c6\u200b II","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6392\u5e8f","\u54c8\u5e0c\u8868","\u53cc\u6307\u9488","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/350/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u6765\u200b\u8ba1\u7b97\u200b\u5b83\u4eec\u200b\u7684\u200b\u4ea4\u96c6\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums1 = [1,2,2,1], nums2 = [2,2]\n\u200b\u8f93\u51fa\u200b\uff1a[2,2]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums1 = [4,9,5], nums2 = [9,4,9,8,4]\n\u200b\u8f93\u51fa\u200b\uff1a[4,9]\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li>\u200b\u8f93\u51fa\u200b\u7ed3\u679c\u200b\u4e2d\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u51fa\u73b0\u200b\u7684\u200b\u6b21\u6570\u200b\uff0c\u200b\u5e94\u200b\u4e0e\u200b\u5143\u7d20\u200b\u5728\u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\u4e00\u81f4\u200b\u3002</li> <li>\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u8003\u8651\u200b\u8f93\u51fa\u200b\u7ed3\u679c\u200b\u7684\u200b\u987a\u5e8f\u200b\u3002</li> </ul> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <ul> <li>\u200b\u5982\u679c\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u6570\u7ec4\u200b\u5df2\u7ecf\u200b\u6392\u597d\u5e8f\u200b\u5462\u200b\uff1f\u200b\u4f60\u200b\u5c06\u200b\u5982\u4f55\u200b\u4f18\u5316\u200b\u4f60\u200b\u7684\u200b\u7b97\u6cd5\u200b\uff1f</li> <li>\u200b\u5982\u679c\u200b<code>nums1</code>\u200b\u7684\u200b\u5927\u5c0f\u200b\u6bd4\u200b<code>nums2</code>\u200b\u5c0f\u200b\u5f88\u591a\u200b\uff0c\u200b\u54ea\u200b\u79cd\u200b\u65b9\u6cd5\u200b\u66f4\u4f18\u200b\uff1f</li> <li>\u200b\u5982\u679c\u200b<code>nums2</code>\u200b\u7684\u200b\u5143\u7d20\u200b\u5b58\u50a8\u200b\u5728\u200b\u78c1\u76d8\u200b\u4e0a\u200b\uff0c\u200b\u5185\u5b58\u200b\u662f\u200b\u6709\u9650\u200b\u7684\u200b\uff0c\u200b\u5e76\u4e14\u200b\u4f60\u200b\u4e0d\u80fd\u200b\u4e00\u6b21\u200b\u52a0\u8f7d\u200b\u6240\u6709\u200b\u7684\u200b\u5143\u7d20\u200b\u5230\u200b\u5185\u5b58\u200b\u4e2d\u200b\uff0c\u200b\u4f60\u200b\u8be5\u200b\u600e\u4e48\u529e\u200b\uff1f</li> </ul> <p>Reference</p>","tags":["\u6392\u5e8f","\u54c8\u5e0c\u8868","\u53cc\u6307\u9488","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/350/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4f7f\u7528\u200b\u54c8\u5e0c\u200b\u8868\u200b\u5b58\u50a8\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5404\u200b\u5143\u7d20\u200b\u51fa\u73b0\u200b\u7684\u200b\u6b21\u6570\u200b\uff0c\u200b\u6ce8\u610f\u200b\u6570\u503c\u200b\u5728\u200b\u4ea4\u200b\u96c6\u4e2d\u200b\u53ef\u4ee5\u200b\u91cd\u590d\u200b\u51fa\u73b0\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>#define MIN(x, y) (x &gt; y ? y : x)\nstruct cell\n{\n    int value;\n    int times1;\n    int times2;\n};\nint* intersect(int* nums1, int nums1Size, int* nums2, int nums2Size, int* returnSize){\n    int i = 0, cur, mapSize = nums1Size + nums2Size, *ret = (int *)malloc(sizeof(int) * (nums1Size + nums2Size));\n    *returnSize = 0;\n    struct cell *hashMap = (struct cell *)memset(malloc(sizeof(struct cell) * mapSize), 0, sizeof(struct cell) * mapSize);\n    for (i = 0; i &lt; nums1Size; i++)\n    {\n        cur = (nums1[i] &gt; 0 ? 1 : -1) * (nums1[i] % mapSize);\n        while((hashMap[cur].times1 != 0 || hashMap[cur].times2) &amp;&amp; hashMap[cur].value != nums1[i])\n        {\n            cur++;\n            cur = cur == mapSize ? 0 : cur;\n        }\n        if (hashMap[cur].times1 == 0 &amp;&amp; hashMap[cur].times2 == 0)\n            hashMap[cur].value = nums1[i];\n        hashMap[cur].times1++;\n    }\n    for (i = 0; i &lt; nums2Size; i++)\n    {\n        cur = (nums2[i] &gt; 0 ? 1 : -1) * (nums2[i] % mapSize);\n        while((hashMap[cur].times1 != 0 || hashMap[cur].times2) &amp;&amp; hashMap[cur].value != nums2[i])\n        {\n            cur++;\n            cur = cur == mapSize ? 0 : cur;\n        }\n        if (hashMap[cur].times1 == 0 &amp;&amp; hashMap[cur].times2 == 0)\n            hashMap[cur].value = nums2[i];\n        hashMap[cur].times2++;\n    }\n    for (i = 0; i &lt; mapSize; i++)\n        while ((hashMap[i].times2--) &amp;&amp; (hashMap[i].times1--))\n            ret[(*returnSize)++] = hashMap[i].value;\n    return ret;\n}\n</code></pre>","tags":["\u6392\u5e8f","\u54c8\u5e0c\u8868","\u53cc\u6307\u9488","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/36/","title":"36. \u200b\u6709\u6548\u200b\u7684\u200b\u6570\u72ec","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/36/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5224\u65ad\u200b\u4e00\u4e2a\u200b\u00a09x9 \u200b\u7684\u200b\u6570\u72ec\u200b\u662f\u5426\u200b\u6709\u6548\u200b\u3002\u200b\u53ea\u200b\u9700\u8981\u200b\u6839\u636e\u200b\u4ee5\u4e0b\u200b\u89c4\u5219\u200b\uff0c\u200b\u9a8c\u8bc1\u200b\u5df2\u7ecf\u200b\u586b\u5165\u200b\u7684\u200b\u6570\u5b57\u200b\u662f\u5426\u200b\u6709\u6548\u200b\u5373\u53ef\u200b\u3002</p> <ol> <li>\u200b\u6570\u5b57\u200b<code>1-9</code>\u200b\u5728\u200b\u6bcf\u200b\u4e00\u884c\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002</li> <li>\u200b\u6570\u5b57\u200b<code>1-9</code>\u200b\u5728\u200b\u6bcf\u200b\u4e00\u5217\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002</li> <li>\u200b\u6570\u5b57\u200b<code>1-9</code>\u200b\u5728\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u4ee5\u200b\u7c97\u200b\u5b9e\u7ebf\u200b\u5206\u9694\u200b\u7684\u200b<code>3x3</code>\u200b\u5bab\u5185\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002</li> </ol> <p></p> <p>\u200b\u4e0a\u56fe\u200b\u662f\u200b\u4e00\u4e2a\u200b\u90e8\u5206\u200b\u586b\u5145\u200b\u7684\u200b\u6709\u6548\u200b\u7684\u200b\u6570\u72ec\u200b\u3002</p> <p>\u200b\u6570\u72ec\u200b\u90e8\u5206\u200b\u7a7a\u683c\u200b\u5185\u200b\u5df2\u200b\u586b\u5165\u200b\u4e86\u200b\u6570\u5b57\u200b\uff0c\u200b\u7a7a\u767d\u200b\u683c\u7528\u200b<code>'.'</code>\u200b\u8868\u793a\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\n[\n  [\"5\",\"3\",\".\",\".\",\"7\",\".\",\".\",\".\",\".\"],\n  [\"6\",\".\",\".\",\"1\",\"9\",\"5\",\".\",\".\",\".\"],\n  [\".\",\"9\",\"8\",\".\",\".\",\".\",\".\",\"6\",\".\"],\n  [\"8\",\".\",\".\",\".\",\"6\",\".\",\".\",\".\",\"3\"],\n  [\"4\",\".\",\".\",\"8\",\".\",\"3\",\".\",\".\",\"1\"],\n  [\"7\",\".\",\".\",\".\",\"2\",\".\",\".\",\".\",\"6\"],\n  [\".\",\"6\",\".\",\".\",\".\",\".\",\"2\",\"8\",\".\"],\n  [\".\",\".\",\".\",\"4\",\"1\",\"9\",\".\",\".\",\"5\"],\n  [\".\",\".\",\".\",\".\",\"8\",\".\",\".\",\"7\",\"9\"]\n]\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\n[\n\u00a0 [\"8\",\"3\",\".\",\".\",\"7\",\".\",\".\",\".\",\".\"],\n\u00a0 [\"6\",\".\",\".\",\"1\",\"9\",\"5\",\".\",\".\",\".\"],\n\u00a0 [\".\",\"9\",\"8\",\".\",\".\",\".\",\".\",\"6\",\".\"],\n\u00a0 [\"8\",\".\",\".\",\".\",\"6\",\".\",\".\",\".\",\"3\"],\n\u00a0 [\"4\",\".\",\".\",\"8\",\".\",\"3\",\".\",\".\",\"1\"],\n\u00a0 [\"7\",\".\",\".\",\".\",\"2\",\".\",\".\",\".\",\"6\"],\n\u00a0 [\".\",\"6\",\".\",\".\",\".\",\".\",\"2\",\"8\",\".\"],\n\u00a0 [\".\",\".\",\".\",\"4\",\"1\",\"9\",\".\",\".\",\"5\"],\n\u00a0 [\".\",\".\",\".\",\".\",\"8\",\".\",\".\",\"7\",\"9\"]\n]\n\u200b\u8f93\u51fa\u200b: false\n\u200b\u89e3\u91ca\u200b: \u200b\u9664\u4e86\u200b\u7b2c\u4e00\u884c\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u6570\u5b57\u200b\u4ece\u200b 5 \u200b\u6539\u4e3a\u200b 8 \u200b\u4ee5\u5916\u200b\uff0c\u200b\u7a7a\u683c\u200b\u5185\u200b\u5176\u4ed6\u200b\u6570\u5b57\u200b\u5747\u200b\u4e0e\u200b \u200b\u793a\u4f8b\u200b1 \u200b\u76f8\u540c\u200b\u3002\n     \u200b\u4f46\u200b\u7531\u4e8e\u200b\u4f4d\u4e8e\u200b\u5de6\u4e0a\u89d2\u200b\u7684\u200b 3x3 \u200b\u5bab\u5185\u200b\u6709\u200b\u4e24\u4e2a\u200b 8 \u200b\u5b58\u5728\u200b, \u200b\u56e0\u6b64\u200b\u8fd9\u4e2a\u200b\u6570\u72ec\u200b\u662f\u200b\u65e0\u6548\u200b\u7684\u200b\u3002\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li>\u200b\u4e00\u4e2a\u200b\u6709\u6548\u200b\u7684\u200b\u6570\u72ec\u200b\uff08\u200b\u90e8\u5206\u200b\u5df2\u200b\u88ab\u200b\u586b\u5145\u200b\uff09\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u53ef\u89e3\u200b\u7684\u200b\u3002</li> <li>\u200b\u53ea\u200b\u9700\u8981\u200b\u6839\u636e\u200b\u4ee5\u4e0a\u200b\u89c4\u5219\u200b\uff0c\u200b\u9a8c\u8bc1\u200b\u5df2\u7ecf\u200b\u586b\u5165\u200b\u7684\u200b\u6570\u5b57\u200b\u662f\u5426\u200b\u6709\u6548\u200b\u5373\u53ef\u200b\u3002</li> <li>\u200b\u7ed9\u200b\u5b9a\u6570\u200b\u72ec\u200b\u5e8f\u5217\u200b\u53ea\u200b\u5305\u542b\u200b\u6570\u5b57\u200b\u00a01-9\u00a0\u200b\u548c\u200b\u5b57\u7b26\u200b\u00a0'.'\u00a0\u3002</li> <li>\u200b\u7ed9\u200b\u5b9a\u6570\u200b\u72ec\u200b\u6c38\u8fdc\u200b\u662f\u200b\u00a09x9\u00a0\u200b\u5f62\u5f0f\u200b\u7684\u200b\u3002</li> </ul> <p>Reference</p>","tags":["\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/36/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6839\u636e\u200b\u5b9a\u4e49\u200b\u76f4\u63a5\u5224\u65ad\u200b\u8f93\u5165\u200b\u7684\u200b\u6570\u7ec4\u200b\u662f\u5426\u200b\u6ee1\u8db3\u200b\u6570\u72ec\u200b\u7684\u200b3\u200b\u4e2a\u200b\u6761\u4ef6\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>bool isValidSudoku(char** board, int boardSize, int* boardColSize){\n    int i = 0, j = 0, positions1[boardSize], positions2[boardSize], k = 0;\n    for (i = 0; i &lt; boardSize; i++)\n    {\n        memset(positions1, 0, sizeof(int) * boardSize);\n        memset(positions2, 0, sizeof(int) * boardSize);\n        for (j = 0; j &lt; boardSize; j++)\n        {\n            if (board[i][j] &gt; '0' &amp;&amp; positions1[board[i][j] - '1'])\n                return false;\n            if (board[j][i] &gt; '0' &amp;&amp; positions2[board[j][i] - '1'])\n                return false;\n            if (board[i][j] &gt; '0')\n                positions1[board[i][j] - '1'] = 1;\n            if (board[j][i] &gt; '0')\n                positions2[board[j][i] - '1'] = 1;\n        }\n    }\n    for (i = 0; i &lt; 3; i++)\n    {\n        for (j = 0; j &lt; 3; j++)\n        {\n            memset(positions1, 0, sizeof(int) * boardSize);\n            for (k = 0; k &lt; boardSize; k++)\n            {\n                if (board[i * 3 + k % 3][j * 3 + k / 3] &gt; '0' &amp;&amp; positions1[board[i * 3 + k % 3][j * 3 + k / 3] - '1'])\n                    return false;\n                if (board[i * 3 + k % 3][j * 3 + k / 3] &gt; '0')\n                    positions1[board[i * 3 + k % 3][j * 3 + k / 3] - '1'] = 1;\n            }\n        }\n    }\n    return true;\n}\n</code></pre>","tags":["\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/368/","title":"368. \u200b\u6700\u5927\u200b\u6574\u9664\u200b\u5b50\u96c6","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u5b66","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/368/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u7531\u200b \u200b\u65e0\u200b\u91cd\u590d\u200b \u200b\u6b63\u6574\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b <code>nums</code> \uff0c\u200b\u8bf7\u200b\u4f60\u200b\u627e\u51fa\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5176\u4e2d\u200b\u6700\u5927\u200b\u7684\u200b\u6574\u9664\u200b\u5b50\u96c6\u200b <code>answer</code> \uff0c\u200b\u5b50\u200b\u96c6\u4e2d\u200b\u6bcf\u4e00\u200b\u5143\u7d20\u200b\u5bf9\u200b <code>(answer[i], answer[j])</code> \u200b\u90fd\u200b\u5e94\u5f53\u200b\u6ee1\u8db3\u200b\uff1a</p> <ul> <li><code>answer[i] % answer[j] == 0</code> \uff0c\u200b\u6216\u200b</li> <li><code>answer[j] % answer[i] == 0</code></li> </ul> <p>\u200b\u5982\u679c\u200b\u5b58\u5728\u200b\u591a\u4e2a\u200b\u6709\u6548\u200b\u89e3\u200b\u5b50\u96c6\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5176\u4e2d\u200b\u4efb\u4f55\u200b\u4e00\u4e2a\u200b\u5747\u200b\u53ef\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1,2,3]\n\u200b\u8f93\u51fa\u200b\uff1a[1,2]\n\u200b\u89e3\u91ca\u200b\uff1a[1,3] \u200b\u4e5f\u200b\u4f1a\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u6b63\u786e\u200b\u7b54\u6848\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1,2,4,8]\n\u200b\u8f93\u51fa\u200b\uff1a[1,2,4,8]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 1000</code></li> <li><code>1 &lt;= nums[i] &lt;= 2 * 10^9</code></li> <li><code>nums</code> \u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u6574\u6570\u200b \u200b\u4e92\u4e0d\u200b\u76f8\u540c\u200b</li> </ul> <p>Reference</p>","tags":["\u6570\u5b66","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/368/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u6574\u6570\u200b\u4e4b\u95f4\u200b\u7684\u200b\u6574\u9664\u200b\u5173\u7cfb\u200b\u6ee1\u8db3\u200b\u81ea\u53cd\u6027\u200b\u3001\u200b\u53cd\u5bf9\u79f0\u6027\u200b\u548c\u200b\u4f20\u9012\u6027\u200b\uff0c\u200b\u4f7f\u7528\u200b\\(a|b\\)\u200b\u8868\u793a\u200b\\(a\\)\u200b\u53ef\u4ee5\u200b\u88ab\u200b\\(b\\)\u200b\u6574\u9664\u200b\uff0c\u200b\u5219\u200b\uff1a</p> <ol> <li>\u200b\u81ea\u53cd\u6027\u200b\uff1a\\(a|a = T\\)</li> <li>\u200b\u53cd\u5bf9\u79f0\u6027\u200b\uff1a\\(a|b \\overline \\lor b|a = T\\)</li> <li>\u200b\u4f20\u9012\u6027\u200b\uff1a\\(a|b \\land b|c\\Rightarrow a|c\\)</li> </ol> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u6574\u9664\u200b\u5173\u7cfb\u200b\u662f\u200b\u6b63\u6574\u6570\u200b\u96c6\u200b\\(N_+\\)\u200b\u4e0a\u200b\u7684\u200b\u76f8\u5bb9\u200b\u5173\u7cfb\u200b\u4e0e\u200b\u504f\u5e8f\u200b\u5173\u7cfb\u200b\u3002</p> <p>\u200b\u8bbe\u200b\u6b63\u6574\u6570\u200b\u96c6\u5408\u200b\u4e3a\u200b\\(S\\)\uff0c\u200b\u4ee5\u4e0b\u200b\u57fa\u4e8e\u200b\u504f\u5e8f\u200b\u5173\u7cfb\u200b\\(R\\)\u200b\u4e0e\u200b\u504f\u5e8f\u200b\u96c6\u200b\\(S\\)\u200b\u5b9a\u4e49\u200b\u94fe\u200b\u7684\u200b\u6982\u5ff5\u200b\uff1a</p> <ol> <li>\u200b\u82e5\u200b\\(a\\in S, b\\in S\\)\uff0c\u200b\u4e14\u200b\\(a|b \\lor b|a\\)\uff0c\u200b\u5219\u200b\\(a, b\\)\u200b\u662f\u200b\u53ef\u6bd4\u200b\u7684\u200b\uff1b</li> <li>\u200b\u82e5\u200b\u96c6\u5408\u200b\\(B\\subseteq S\\)\uff0c\u200b\u4e14\u200b\u5bf9\u4e8e\u200b\\(B\\)\u200b\u4e2d\u200b\u4efb\u610f\u200b\u4e00\u5bf9\u200b\u5143\u7d20\u200b\\(x, y\\)\u200b\u90fd\u200b\u662f\u200b\u53ef\u6bd4\u200b\u7684\u200b\uff0c\u200b\u5219\u200b\u79f0\u200b\\(B\\)\u200b\u4e3a\u200b\u504f\u5e8f\u200b\u96c6\u200b\\(S\\)\u200b\u4e0a\u200b\u7684\u200b\u94fe\u200b\u3002</li> <li>\u200b\u504f\u5e8f\u200b\u96c6\u200b\\(S\\)\u200b\u4e2d\u200b\u5305\u542b\u200b\u5143\u7d20\u200b\u6700\u591a\u200b\u7684\u200b\u94fe\u200b\u79f0\u4e3a\u200b\u6700\u957f\u200b\u94fe\u200b\u3002</li> </ol> <p>\u200b\u7531\u6b64\u53ef\u89c1\u200b\uff0c\u200b\u6240\u6c42\u200b\u7684\u200b\u6700\u5927\u200b\u6574\u9664\u200b\u5b50\u96c6\u200b\u5373\u200b\u4e3a\u200b\\(S\\)\u200b\u4e0a\u200b\u7684\u200b\u6700\u957f\u200b\u94fe\u200b\u3002</p> <p>\u200b\u504f\u5e8f\u200b\u96c6\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u54c8\u65af\u200b\u56fe\u200b\u8fdb\u884c\u200b\u8868\u793a\u200b\uff0c\u200b\u5982\u200b\u96c6\u5408\u200b\\(\\{2,3,4,6,8\\}\\)\u200b\u4e0a\u200b\u7684\u200b\u504f\u5e8f\u200b\u5173\u7cfb\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</p> <p></p> <p>\u200b\u54c8\u65af\u200b\u56fe\u662f\u200b\u6709\u200b\u5411\u200b\u65e0\u200b\u73af\u56fe\u200b\uff0c\u200b\u4f46\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u8fde\u901a\u200b\u3002\u200b\u94fe\u200b\u5728\u200b\u54c8\u65af\u200b\u56fe\u4e2d\u200b\u8868\u73b0\u200b\u4e3a\u200b\u4e00\u6bb5\u200b\u8def\u5f84\u200b\uff0c\u200b\u6700\u957f\u200b\u94fe\u5373\u200b\u4e3a\u200b\u54c8\u65af\u200b\u56fe\u4e2d\u200b\u7684\u200b\u6700\u957f\u200b\u8def\u5f84\u200b\uff0c\u200b\u5373\u200b\u5404\u200b\u5206\u91cf\u200b\u7684\u200b\u6700\u957f\u200b\u76f4\u5f84\u200b\uff0c\u200b\u5982\u56fe\u200b\u4e2d\u52a0\u200b\u7c97\u7ebf\u6761\u200b\u6240\u793a\u200b\u3002</p> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u56fe\u200b\u89e3\u51b3\u200b\u672c\u9898\u200b\uff0c\u200b\u7b2c\u4e00\u6b65\u200b\u6784\u9020\u200b\u504f\u5e8f\u200b\u96c6\u200b\u7684\u200b\u54c8\u65af\u200b\u56fe\u200b\uff0c\u200b\u6d41\u7a0b\u200b\u5982\u4e0b\u200b\uff1a</p> <ol> <li>\u200b\u5bf9\u200b\u96c6\u5408\u200b\u6392\u5e8f\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u96c6\u5408\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u5728\u200b\u56fe\u200b\u4e2d\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u4e0e\u200b\u4e4b\u200b\u5bf9\u5e94\u200b\uff1b</li> <li>\u200b\u4ece\u540e\u200b\u5411\u524d\u200b\u904d\u5386\u200b\u96c6\u5408\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\\(a_i\\)\uff1a<ol> <li>\u200b\u4ece\u200b\u7b2c\u200b\\(i + 1\\)\u200b\u4e2a\u200b\u5143\u7d20\u200b\u5f00\u59cb\u200b\uff0c\u200b\u5411\u200b\u540e\u200b\u904d\u5386\u200b\u96c6\u5408\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\\(a_j\\)\uff0c\u200b\u82e5\u200b\\(a_j|a_i\\)\uff0c\u200b\u5e76\u4e14\u200b\u4e0d\u200b\u5b58\u5728\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u8282\u70b9\u200b\u4e0e\u200b\\(a_i\\)\u200b\u76f8\u8fde\u200b\uff0c\u200b\u5219\u200b\u5c06\u200b\u8282\u70b9\u200b\\(i\\)\u200b\u4e0e\u200b\u8282\u70b9\u200b\\(j\\)\u200b\u76f8\u8fde\u63a5\u200b\u3002</li> </ol> </li> </ol> <p>\u200b\u4e3a\u4e86\u200b\u5feb\u901f\u200b\u6c42\u51fa\u200b\u56fe\u200b\u7684\u200b\u76f4\u5f84\u200b\uff0c\u200b\u53ef\u200b\u5728\u200b\u8282\u70b9\u200b\u4e2d\u200b\u5b58\u50a8\u200b\u4e24\u4e2a\u200b\u503c\u200b\uff0c\u200b\u5373\u200b\uff1a</p> <ul> <li>\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u8ddd\u79bb\u200b\u672b\u7aef\u200b\u8282\u70b9\u200b\uff08\u200b\u51fa\u5ea6\u200b\u4e3a\u200b\\(0\\)\u200b\u7684\u200b\u8282\u70b9\u200b\uff09\u200b\u7684\u200b\u6700\u5927\u200b\u8ddd\u79bb\u200b</li> <li>\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u8ddd\u79bb\u200b\u672b\u7aef\u200b\u8282\u70b9\u200b\u6700\u5927\u200b\u8ddd\u79bb\u200b\u6240\u200b\u5bf9\u5e94\u200b\u7684\u200b\u51fa\u8fb9\u200b</li> </ul> <p>\u200b\u7531\u4e8e\u200b\u6574\u9664\u200b\u5173\u7cfb\u200b\u8574\u542b\u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b\u5173\u7cfb\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5728\u200b\u6784\u9020\u200b\u54c8\u65af\u200b\u56fe\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u540c\u65f6\u200b\u6784\u9020\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u503c\u200b\u3002</p> <p>\u200b\u7b2c\u4e8c\u6b65\u200b\u5373\u53ef\u200b\u627e\u51fa\u200b\u6700\u957f\u200b\u76f4\u5f84\u200b\u3002\u200b\u904d\u5386\u200b\u6240\u6709\u200b\u5165\u5ea6\u200b\u4e3a\u200b\\(0\\)\u200b\u7684\u200b\u8282\u70b9\u200b\uff0c\u200b\u627e\u51fa\u200b\u5e76\u200b\u8bb0\u4e0b\u200b\u8ddd\u79bb\u200b\u672b\u7aef\u200b\u8282\u70b9\u200b\u6700\u8fdc\u200b\u7684\u200b\u8282\u70b9\u200b\u3002\u200b\u518d\u200b\u6839\u636e\u200b\u8282\u70b9\u200b\u4e2d\u200b\u5b58\u50a8\u200b\u7684\u200b\u51fa\u8fb9\u200b\u4fe1\u606f\u200b\u5373\u53ef\u200b\u5b8c\u6574\u200b\u6784\u9020\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u6700\u5927\u200b\u6574\u9664\u200b\u5b50\u96c6\u200b\u3002</p> <pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\n#define MAX(x, y) ((x) &gt; (y) ? (x) : (y))\nstruct Edge;\nstruct Vertex;\nstruct Edge\n{\n    struct Vertex *dest;\n    struct Edge *next;\n};\nstruct Vertex\n{\n    int val;\n    int status;\n    struct Edge *edges;\n    int height;\n    struct Edge *maxHeight;\n};\nstruct Edge *getEdge(struct Vertex *dest, struct Edge *next)\n{\n    struct Edge *cur = next;\n    while (cur)\n    {\n        if (dest-&gt;val % cur-&gt;dest-&gt;val == 0)\n            return next;\n        cur = cur-&gt;next;\n    }\n    struct Edge *ret = (struct Edge *)malloc(sizeof(struct Edge));\n    ret-&gt;dest = dest;\n    ret-&gt;next = next;\n    return ret;\n}\nint cmp(const void *a, const void *b)\n{\n    return *((int *)a) - *((int *)b);\n}\nvoid dfs(struct Vertex *graph)\n{\n    if (graph == NULL)\n        return;\n    graph-&gt;status = true;\n    struct Edge *edge = graph-&gt;edges;\n    while (edge)\n    {\n        dfs(edge-&gt;dest);\n        edge = edge-&gt;next;\n    }\n}\nint* largestDivisibleSubset(int* nums, int numsSize, int* returnSize){\n    struct Vertex *nodes = (struct Vertex *)malloc(sizeof(struct Vertex) * numsSize), *maxVertex = NULL;\n    int i = 0, j = 0, max = -1, *ret = NULL;\n    qsort(nums, numsSize, sizeof(int), cmp);\n    for (i = 0; i &lt; numsSize; i++)\n    {\n        nodes[i].val = nums[i];\n        nodes[i].status = 0;\n        nodes[i].edges = NULL;\n        nodes[i].height = 0;\n        nodes[i].maxHeight = NULL;\n    }\n    for (i = numsSize - 1; i &gt;= 0; i--)\n    {\n        for (j = i + 1; j &lt; numsSize; j++)\n        {\n            if (nodes[j].val % nodes[i].val)\n                continue;\n            nodes[i].edges = getEdge(nodes + j, nodes[i].edges);\n            if (nodes[i].height &lt; nodes[j].height + 1)\n            {\n                nodes[i].height = nodes[j].height + 1;\n                nodes[i].maxHeight = nodes[i].edges;\n            }\n        }\n    }\n    for (i = 0; i &lt; numsSize; i++)\n    {\n        if (nodes[i].status)\n            continue;\n        if (nodes[i].height &gt; max)\n        {\n            max = nodes[i].height;\n            maxVertex = nodes + i;\n        }\n        dfs(nodes + i);\n    }\n    *returnSize = max + 1;\n    ret = (int *)malloc(sizeof(int) * (max + 1));\n    for (i = 0; i &lt;= max; i++)\n    {\n        ret[i] = maxVertex-&gt;val;\n        if (maxVertex-&gt;maxHeight == NULL)\n            break;\n        maxVertex = maxVertex-&gt;maxHeight-&gt;dest;\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u5b66","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/37/","title":"37. \u200b\u89e3\u6570\u200b\u72ec","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u54c8\u5e0c\u8868","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/37/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u7a0b\u5e8f\u200b\uff0c\u200b\u901a\u8fc7\u200b\u586b\u5145\u200b\u7a7a\u683c\u200b\u6765\u200b\u89e3\u51b3\u200b\u6570\u72ec\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u6570\u72ec\u200b\u7684\u200b\u89e3\u6cd5\u200b\u9700\u200b \u200b\u9075\u5faa\u200b\u5982\u4e0b\u200b\u89c4\u5219\u200b\uff1a</p> <p>\u200b\u6570\u5b57\u200b\u00a0<code>1-9</code>\u00a0\u200b\u5728\u200b\u6bcf\u200b\u4e00\u884c\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002 \u200b\u6570\u5b57\u200b\u00a0<code>1-9</code>\u00a0\u200b\u5728\u200b\u6bcf\u200b\u4e00\u5217\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002 \u200b\u6570\u5b57\u200b\u00a0<code>1-9</code>\u00a0\u200b\u5728\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u4ee5\u200b\u7c97\u200b\u5b9e\u7ebf\u200b\u5206\u9694\u200b\u7684\u200b\u00a0<code>3x3</code>\u00a0\u200b\u5bab\u5185\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002\uff08\u200b\u8bf7\u200b\u53c2\u8003\u200b\u793a\u4f8b\u200b\u56fe\u200b\uff09 \u200b\u6570\u72ec\u200b\u90e8\u5206\u200b\u7a7a\u683c\u200b\u5185\u200b\u5df2\u200b\u586b\u5165\u200b\u4e86\u200b\u6570\u5b57\u200b\uff0c\u200b\u7a7a\u767d\u200b\u683c\u7528\u200b\u00a0<code>'.'</code>\u00a0\u200b\u8868\u793a\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = [\n    [\"5\",\"3\",\".\",\".\",\"7\",\".\",\".\",\".\",\".\"],\n    [\"6\",\".\",\".\",\"1\",\"9\",\"5\",\".\",\".\",\".\"],\n    [\".\",\"9\",\"8\",\".\",\".\",\".\",\".\",\"6\",\".\"],\n    [\"8\",\".\",\".\",\".\",\"6\",\".\",\".\",\".\",\"3\"],\n    [\"4\",\".\",\".\",\"8\",\".\",\"3\",\".\",\".\",\"1\"],\n    [\"7\",\".\",\".\",\".\",\"2\",\".\",\".\",\".\",\"6\"],\n    [\".\",\"6\",\".\",\".\",\".\",\".\",\"2\",\"8\",\".\"],\n    [\".\",\".\",\".\",\"4\",\"1\",\"9\",\".\",\".\",\"5\"],\n    [\".\",\".\",\".\",\".\",\"8\",\".\",\".\",\"7\",\"9\"]\n]\n\u200b\u8f93\u51fa\u200b\uff1a[\n    [\"5\",\"3\",\"4\",\"6\",\"7\",\"8\",\"9\",\"1\",\"2\"],\n    [\"6\",\"7\",\"2\",\"1\",\"9\",\"5\",\"3\",\"4\",\"8\"],\n    [\"1\",\"9\",\"8\",\"3\",\"4\",\"2\",\"5\",\"6\",\"7\"],\n    [\"8\",\"5\",\"9\",\"7\",\"6\",\"1\",\"4\",\"2\",\"3\"],\n    [\"4\",\"2\",\"6\",\"8\",\"5\",\"3\",\"7\",\"9\",\"1\"],\n    [\"7\",\"1\",\"3\",\"9\",\"2\",\"4\",\"8\",\"5\",\"6\"],\n    [\"9\",\"6\",\"1\",\"5\",\"3\",\"7\",\"2\",\"8\",\"4\"],\n    [\"2\",\"8\",\"7\",\"4\",\"1\",\"9\",\"6\",\"3\",\"5\"],\n    [\"3\",\"4\",\"5\",\"2\",\"8\",\"6\",\"1\",\"7\",\"9\"]\n]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u8f93\u5165\u200b\u7684\u200b\u6570\u72ec\u200b\u5982\u4e0a\u56fe\u200b\u6240\u793a\u200b\uff0c\u200b\u552f\u4e00\u200b\u6709\u6548\u200b\u7684\u200b\u89e3\u51b3\u65b9\u6848\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a\n</code></pre> <p></p> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>board.length == 9</code></li> <li><code>board[i].length == 9</code></li> <li><code>board[i][j]</code> \u200b\u662f\u200b\u4e00\u4f4d\u200b\u6570\u5b57\u200b\u6216\u8005\u200b <code>'.'</code></li> <li>\u200b\u9898\u76ee\u200b\u6570\u636e\u200b \u200b\u4fdd\u8bc1\u200b \u200b\u8f93\u5165\u200b\u6570\u72ec\u4ec5\u200b\u6709\u200b\u4e00\u4e2a\u200b\u89e3\u200b</li> </ul> <p>Reference</p>","tags":["\u54c8\u5e0c\u8868","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/37/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u56de\u6eaf\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u7a7a\u200b\u5355\u5143\u683c\u200b\uff0c\u200b\u68c0\u67e5\u200b\u5176\u200b\u6240\u5728\u200b\u884c\u200b\uff0c\u200b\u6240\u5728\u200b\u5217\u200b\u4e0e\u200b\u6240\u5728\u200b\u65b9\u683c\u200b\uff0c\u200b\u9010\u4e2a\u200b\u68c0\u67e5\u200b\u5176\u200b\u53ef\u4ee5\u200b\u586b\u5165\u200b\u7684\u200b\u6570\u503c\u200b\u3002\u200b\u5982\u679c\u200b \u200b\u6709\u200b\u53ef\u4ee5\u200b\u586b\u5165\u200b\u7684\u200b\u5b57\u7b26\u200b\uff0c\u200b\u5219\u200b\u4f9d\u6b21\u200b\u8fdb\u884c\u200b\u5c1d\u8bd5\u200b\u3002\u200b\u82e5\u200b\u67d0\u4e2a\u200b\u7a7a\u683c\u200b\u6ca1\u6709\u200b\u4efb\u4f55\u200b\u53ef\u4ee5\u200b\u586b\u5165\u200b\u7684\u200b\u6570\u503c\u200b\uff0c\u200b\u6216\u200b\u6240\u6709\u200b\u53ef\u200b\u586b\u5165\u200b\u7684\u200b\u503c\u200b\u90fd\u200b\u662f\u200b\u9519\u8bef\u200b\u7684\u200b \uff0c\u200b\u610f\u5473\u7740\u200b\u5f53\u524d\u200b\u641c\u7d22\u200b\u8def\u5f84\u200b\u5b58\u5728\u200b\u9519\u8bef\u200b\uff0c\u200b\u9700\u8981\u200b\u8fd4\u56de\u200b\u4e0a\u200b\u4e00\u4e2a\u200b\u72b6\u6001\u200b\u8fdb\u884c\u200b\u4e0b\u200b\u4e00\u6b65\u200b\u5c1d\u8bd5\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u65b9\u683c\u200b\u88ab\u200b\u5b8c\u5168\u200b\u586b\u6ee1\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u7acb\u5373\u200b\u8fd4\u56de\u200b\u3002\u200b\u5e76\u200b\u901a\u8fc7\u200b\u8fd4\u56de\u503c\u200b\u544a\u8bc9\u200b\u4e0a\u200b\u4e00\u6b65\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u8fd4\u56de\u200b\u7ed3\u679c\u200b\u3002</p> <pre><code>bool dfs(char **board, int boardSize, int *boardColSize)\n{\n    int x = 0, y = 0, i = 0, j = 0;\n    bool flag = true, candidate[9] = {false};\n    for (x = 0; x &lt; 9 &amp;&amp; flag; x++)\n        for (y = 0; y &lt; 9 &amp;&amp; flag; y++)\n            if (board[x][y] == '.')\n                flag = false;\n    if (flag)\n        return true;\n    x--;\n    y--;\n    for (i = 0; i &lt; 9; i++)\n    {\n        if (board[x][i] != '.')\n            candidate[board[x][i] - '1'] = true;\n        if (board[i][y] != '.')\n            candidate[board[i][y] - '1'] = true;\n        if (board[3 * (x / 3) + i / 3][3 * (y / 3) + i % 3] != '.')\n            candidate[board[3 * (x / 3) + i / 3][3 * (y / 3) + i % 3] - '1'] = true;\n    }\n    for (i = 0; i &lt; 9 &amp;&amp; !flag; i++)\n    {\n        if (!candidate[i])\n        {\n            board[x][y] = '1' + i;\n            flag = flag || dfs(board, boardSize, boardColSize);\n            if (flag)\n                return flag;\n            board[x][y] = '.';\n        }\n    }\n    return flag;\n}\n\nvoid solveSudoku(char** board, int boardSize, int* boardColSize){\n    dfs(board, boardSize, boardColSize);\n}\n</code></pre>","tags":["\u54c8\u5e0c\u8868","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/377/","title":"377. \u200b\u7ec4\u5408\u200b\u603b\u548c\u200b \u2163","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/377/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u7531\u200b \u200b\u4e0d\u540c\u200b \u200b\u6574\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u6570\u7ec4\u200b <code>nums</code> \uff0c\u200b\u548c\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u200b\u6574\u6570\u200b <code>target</code> \u3002\u200b\u8bf7\u200b\u4f60\u200b\u4ece\u200b <code>nums</code> \u200b\u4e2d\u200b\u627e\u51fa\u200b\u5e76\u200b\u8fd4\u56de\u200b\u603b\u548c\u200b\u4e3a\u200b <code>target</code> \u200b\u7684\u200b\u5143\u7d20\u200b\u7ec4\u5408\u200b\u7684\u200b\u4e2a\u6570\u200b\u3002</p> <p>\u200b\u9898\u76ee\u200b\u6570\u636e\u200b\u4fdd\u8bc1\u200b\u7b54\u6848\u200b\u7b26\u5408\u200b <code>32</code> \u200b\u4f4d\u200b\u6574\u6570\u200b\u8303\u56f4\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1,2,3], target = 4\n\u200b\u8f93\u51fa\u200b\uff1a7\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u7684\u200b\u7ec4\u5408\u200b\u4e3a\u200b\uff1a\n(1, 1, 1, 1)\n(1, 1, 2)\n(1, 2, 1)\n(1, 3)\n(2, 1, 1)\n(2, 2)\n(3, 1)\n\u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u987a\u5e8f\u200b\u4e0d\u540c\u200b\u7684\u200b\u5e8f\u5217\u200b\u88ab\u200b\u89c6\u4f5c\u200b\u4e0d\u540c\u200b\u7684\u200b\u7ec4\u5408\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [9], target = 3\n\u200b\u8f93\u51fa\u200b\uff1a0\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 200</code></li> <li><code>1 &lt;= nums[i] &lt;= 1000</code></li> <li><code>nums</code> \u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b \u200b\u4e92\u4e0d\u200b\u76f8\u540c\u200b</li> <li><code>1 &lt;= target &lt;= 1000</code></li> </ul> <p>\u200b\u8fdb\u9636\u200b\uff1a \u200b\u5982\u679c\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u542b\u6709\u200b\u8d1f\u6570\u200b\u4f1a\u200b\u53d1\u751f\u200b\u4ec0\u4e48\u200b\uff1f\u200b\u95ee\u9898\u200b\u4f1a\u200b\u4ea7\u751f\u200b\u4f55\u79cd\u200b\u53d8\u5316\u200b\uff1f\u200b\u5982\u679c\u200b\u5141\u8bb8\u200b\u8d1f\u6570\u200b\u51fa\u73b0\u200b\uff0c\u200b\u9700\u8981\u200b\u5411\u200b\u9898\u76ee\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u54ea\u4e9b\u200b\u9650\u5236\u200b\u6761\u4ef6\u200b\uff1f</p> <p>Reference</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/377/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u672c\u9898\u200b\u7684\u200b\u601d\u8def\u200b\u4e0e\u200b70. \u200b\u722c\u697c\u68af\u200b\u76f8\u540c\u200b\u3002\u200b\u6b64\u5904\u200b\u4f7f\u7528\u200b\u8bb0\u5fc6\u200b\u5316\u200b\u9012\u5f52\u200b\u89e3\u51b3\u200b\u3002</p> <p>\u200b\u8bbe\u200b\\(f(x; S)\\)\u200b\u4e3a\u200b\u7ed9\u5b9a\u200b\u6574\u6570\u200b\u96c6\u5408\u200b\\(S\\)\uff0c\u200b\u7ec4\u6210\u200b\u76ee\u6807\u503c\u200b\\(x\\)\u200b\u7684\u200b\u7ec4\u5408\u200b\u6570\u91cf\u200b\uff0c\u200b\u6211\u4eec\u200b\u9ed8\u8ba4\u200b\\(0\\in S\\)\u3002</p> <ul> <li>\u200b\u5f53\u200b\\(x &lt; 0\\)\uff0c\u200b\u7531\u4e8e\u200b\\(\\forall y\\in S, y\\geq 0\\)\uff0c\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u5f97\u5230\u200b\u5c0f\u4e8e\u200b\\(0\\)\u200b\u7684\u200b\u6570\u503c\u200b\uff0c\u200b\u6709\u200b\\(f(x; S) = 0\\)</li> <li>\\(x = 0\\)\uff0c\u200b\u7531\u4e8e\u200b\u9ed8\u8ba4\u200b\\(0\\in S\\)\uff0c\u200b\u6709\u200b\\(f(x; S) = 1\\)\u3002</li> <li>\u200b\u5f53\u200b\\(x &lt; 0\\)\uff0c\\(S\\)\u200b\u4e2d\u200b\u7684\u200b\u4efb\u610f\u200b\u5143\u7d20\u200b\\(y\\)\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u9009\u62e9\u200b\uff0c\u200b\u9009\u62e9\u200b\\(y\\)\u200b\u5f97\u5230\u200b\u7684\u200b\u7ec4\u5408\u200b\u4e2a\u6570\u200b\u4e3a\u200b\\(f(x - y; S)\\)</li> </ul> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5f97\u5230\u200b\\(f(x; S)\\)\u200b\u7684\u200b\u8868\u8fbe\u5f0f\u200b\uff1a</p> \\[ f(x; S)=\\left\\{ \\begin{aligned}     &amp; 0 &amp; x &lt; 0 \\\\     &amp; 1 &amp; x = 0 \\\\     &amp; \\sum_{y\\in S} f(x - y; S) &amp; x &gt; 0 \\end{aligned} \\right . \\] <p>\u200b\u7531\u4e8e\u200b\u8ba1\u7b97\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u4f1a\u200b\u91cd\u590d\u200b\u8ba1\u7b97\u200b\u67d0\u4e9b\u200b\\(f(x; S)\\)\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\u8bb0\u5f55\u200b\u6240\u6709\u200b\u5df2\u7ecf\u200b\u8ba1\u7b97\u200b\u8fc7\u200b\u7684\u200b\\(f(x; S)\\)\u3002</p> <pre><code>int helper(int *dp, int *nums, int numsSize, int target)\n{\n    if (target == 0)\n        return 1;\n    if (target &lt; 0)\n        return 0;\n    if (dp[target - 1] &lt; 0)\n    {\n        int ret = 0, i = 0;\n        for (i = 0; i &lt; numsSize; i++)\n            ret += helper(dp, nums, numsSize, target - nums[i]);\n        dp[target - 1] = ret;\n    }\n    return dp[target - 1];\n}\n\nint combinationSum4(int* nums, int numsSize, int target) {\n    int *dp = (int *)malloc(sizeof(int) * target);\n    memset(dp, 0xff, sizeof(int) * target);\n    return helper(dp, nums, numsSize, target);\n}\n</code></pre>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/38/","title":"38. \u200b\u5916\u89c2\u200b\u6570\u5217","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/38/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6b63\u6574\u6570\u200b\\(n\\)\uff08\\(1 \\leq\u00a0n\u00a0\\leq 30\\)\uff09\uff0c\u200b\u8f93\u51fa\u200b\u5916\u89c2\u200b\u6570\u5217\u200b\u7684\u200b\u7b2c\u200b\\(n\\)\u200b\u9879\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b\uff1a\u200b\u6574\u6570\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e00\u9879\u200b\u5c06\u200b\u8868\u793a\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u3002</p> <p>\u300c\u200b\u5916\u89c2\u200b\u6570\u5217\u200b\u300d\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4ece\u200b\u6570\u5b57\u200b 1 \u200b\u5f00\u59cb\u200b\uff0c\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e00\u9879\u200b\u90fd\u200b\u662f\u200b\u5bf9\u200b\u524d\u200b\u4e00\u9879\u200b\u7684\u200b\u63cf\u8ff0\u200b\u3002\u200b\u524d\u200b\u4e94\u9879\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>1.     1\n2.     11\n3.     21\n4.     1211\n5.     111221\n</code></pre> <p>\u200b\u7b2c\u4e00\u9879\u200b\u662f\u200b\u6570\u5b57\u200b 1</p> <p>\u200b\u63cf\u8ff0\u200b\u524d\u200b\u4e00\u9879\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u6570\u662f\u200b 1 \u200b\u5373\u200b \u201c\u200b\u4e00\u4e2a\u200b 1 \u201d\uff0c\u200b\u8bb0\u200b\u4f5c\u200b 11</p> <p>\u200b\u63cf\u8ff0\u200b\u524d\u200b\u4e00\u9879\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u6570\u662f\u200b 11 \u200b\u5373\u200b \u201c\u200b\u4e24\u4e2a\u200b 1 \u201d \uff0c\u200b\u8bb0\u200b\u4f5c\u200b 21</p> <p>\u200b\u63cf\u8ff0\u200b\u524d\u200b\u4e00\u9879\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u6570\u662f\u200b 21 \u200b\u5373\u200b \u201c\u200b\u4e00\u4e2a\u200b 2 \u200b\u4e00\u4e2a\u200b 1 \u201d \uff0c\u200b\u8bb0\u200b\u4f5c\u200b 1211</p> <p>\u200b\u63cf\u8ff0\u200b\u524d\u200b\u4e00\u9879\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u6570\u662f\u200b 1211 \u200b\u5373\u200b \u201c\u200b\u4e00\u4e2a\u200b 1 \u200b\u4e00\u4e2a\u200b 2 \u200b\u4e24\u4e2a\u200b 1 \u201d \uff0c\u200b\u8bb0\u200b\u4f5c\u200b 111221</p> <p>\u200b\u793a\u4f8b\u200b\u00a01:</p> <pre><code>\u200b\u8f93\u5165\u200b: 1\n\u200b\u8f93\u51fa\u200b: \"1\"\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u8fd9\u662f\u200b\u4e00\u4e2a\u200b\u57fa\u672c\u200b\u6837\u4f8b\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2:</p> <pre><code>\u200b\u8f93\u5165\u200b: 4\n\u200b\u8f93\u51fa\u200b: \"1211\"\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u5f53\u200b n = 3 \u200b\u65f6\u200b\uff0c\u200b\u5e8f\u5217\u200b\u662f\u200b \"21\"\uff0c\u200b\u5176\u4e2d\u200b\u6211\u4eec\u200b\u6709\u200b \"2\" \u200b\u548c\u200b \"1\" \u200b\u4e24\u7ec4\u200b\uff0c\"2\" \u200b\u53ef\u4ee5\u200b\u8bfb\u4f5c\u200b \"12\"\uff0c\u200b\u4e5f\u200b\u5c31\u662f\u200b\u51fa\u73b0\u200b\u9891\u6b21\u200b = 1 \u200b\u800c\u200b \u200b\u503c\u200b = 2\uff1b\u200b\u7c7b\u4f3c\u200b \"1\" \u200b\u53ef\u4ee5\u200b\u8bfb\u4f5c\u200b \"11\"\u3002\u200b\u6240\u4ee5\u200b\u7b54\u6848\u200b\u662f\u200b \"12\" \u200b\u548c\u200b \"11\" \u200b\u7ec4\u5408\u200b\u5728\u200b\u4e00\u8d77\u200b\uff0c\u200b\u4e5f\u200b\u5c31\u662f\u200b \"1211\"\u3002\n</code></pre> <p>Reference</p>","tags":["\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/38/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6839\u636e\u200b\u5b9a\u4e49\u200b\u8fdb\u884c\u200b\u8ba1\u7b97\u200b\u5373\u53ef\u200b\uff0c\u200b\u6ce8\u610f\u200b\u6b21\u6570\u200b\u90e8\u5206\u200b\u53ef\u80fd\u200b\u8d85\u8fc7\u200b10\uff0c\u200b\u800c\u200b\u6570\u5b57\u200b\u90e8\u5206\u200b\u4e0d\u4f1a\u200b\u8d85\u8fc7\u200b10\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>char * toStr(int n)\n{\n    char *ret = (char *)memset(malloc(sizeof(char) * 10), 0, sizeof(char) * 10), temp;\n    int i = 0, head = 0, tail;\n    while (n)\n    {\n        ret[i] = n % 10 + '0';\n        n /= 10;\n        i++;\n    }\n    tail = i - 1;\n    while(head &lt; tail)\n    {\n        temp = ret[head];\n        ret[head] = ret[tail];\n        ret[tail] = temp;\n    }\n    return ret;\n}\n\nchar * countAndSay(int n){\n    if (n == 1)\n    {\n        char *ret = (char *)malloc(sizeof(char) * 2);\n        ret[0] = '1';\n        ret[1] = 0;\n        return ret;\n    }\n    char *rec = countAndSay(n - 1), *cur = rec, flag = *cur, *temp = NULL,\n         *ret = (char *)memset(malloc(sizeof(char) * (strlen(rec) * 2 + 1)), 0, sizeof(char) * (strlen(rec) * 2 + 1)), *retcur = ret;\n    int counter = 0;\n    while(*cur)\n    {\n        if (*cur == flag)\n            counter++;\n        else\n        {\n            temp = toStr(counter);\n            strcpy(retcur, temp);\n            while(*retcur)\n                retcur++;\n            *retcur = flag;\n            *retcur++;\n            flag = *cur;\n            counter = 1;\n            free(temp);\n        }\n        cur++;\n    }\n    temp = toStr(counter);\n    strcpy(retcur, temp);\n    while (*retcur)\n        retcur++;\n    *retcur = flag;\n    free(temp);\n    free(rec);\n    return ret;\n}\n</code></pre>","tags":["\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/384/","title":"384. \u200b\u6253\u4e71\u200b\u6570\u7ec4","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/384/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u6253\u4e71\u200b\u4e00\u4e2a\u200b\u6ca1\u6709\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\u7684\u200b\u6570\u7ec4\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>// \u200b\u4ee5\u200b\u6570\u5b57\u200b\u96c6\u5408\u200b 1, 2 \u200b\u548c\u200b 3 \u200b\u521d\u59cb\u5316\u200b\u6570\u7ec4\u200b\u3002\nint[] nums = {1,2,3};\nSolution solution = new Solution(nums);\n\n// \u200b\u6253\u4e71\u200b\u6570\u7ec4\u200b [1,2,3] \u200b\u5e76\u200b\u8fd4\u56de\u200b\u7ed3\u679c\u200b\u3002\u200b\u4efb\u4f55\u200b [1,2,3]\u200b\u7684\u200b\u6392\u5217\u200b\u8fd4\u56de\u200b\u7684\u200b\u6982\u7387\u200b\u5e94\u8be5\u200b\u76f8\u540c\u200b\u3002\nsolution.shuffle();\n\n// \u200b\u91cd\u8bbe\u200b\u6570\u7ec4\u200b\u5230\u200b\u5b83\u200b\u7684\u200b\u521d\u59cb\u72b6\u6001\u200b[1,2,3]\u3002\nsolution.reset();\n\n// \u200b\u968f\u673a\u200b\u8fd4\u56de\u200b\u6570\u7ec4\u200b[1,2,3]\u200b\u6253\u4e71\u200b\u540e\u200b\u7684\u200b\u7ed3\u679c\u200b\u3002\nsolution.shuffle();\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/384/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a Fisher-Yates \u200b\u6d17\u724c\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u5173\u952e\u5728\u4e8e\u200b\u5b9e\u73b0\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u65e0\u653e\u56de\u200b\u62bd\u6837\u200b\u3002</p> <p>Fisher-Yates \u200b\u6d17\u724c\u200b\u7b97\u6cd5\u200b\u8ddf\u200b\u66b4\u529b\u200b\u7b97\u6cd5\u200b\u5f88\u200b\u50cf\u200b\u3002\u200b\u5728\u200b\u6bcf\u6b21\u200b\u8fed\u4ee3\u200b\u4e2d\u200b\uff0c\u200b\u751f\u6210\u200b\u4e00\u4e2a\u200b\u8303\u56f4\u200b\u5728\u200b\u5f53\u524d\u200b\u4e0b\u6807\u200b\u5230\u200b\u6570\u7ec4\u200b\u672b\u5c3e\u200b\u5143\u7d20\u200b\u4e0b\u6807\u200b\u4e4b\u95f4\u200b\u7684\u200b\u968f\u673a\u200b\u6574\u6570\u200b\u3002\u200b\u63a5\u4e0b\u6765\u200b\uff0c\u200b\u5c06\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u548c\u200b\u968f\u673a\u200b\u9009\u51fa\u200b\u7684\u200b\u4e0b\u6807\u200b\u6240\u6307\u200b\u7684\u200b\u5143\u7d20\u200b\u4e92\u76f8\u4ea4\u6362\u200b - \u200b\u8fd9\u200b\u4e00\u6b65\u200b\u6a21\u62df\u200b\u4e86\u200b\u6bcf\u6b21\u200b\u4ece\u200b \u201c\u200b\u5e3d\u5b50\u200b\u201d \u200b\u91cc\u9762\u200b\u6478\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u5176\u4e2d\u200b\u9009\u53d6\u200b\u4e0b\u6807\u200b\u8303\u56f4\u200b\u7684\u200b\u4f9d\u636e\u200b\u5728\u4e8e\u200b\u6bcf\u4e2a\u200b\u88ab\u200b\u6478\u200b\u51fa\u200b\u7684\u200b\u5143\u7d20\u200b\u90fd\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u518d\u200b\u88ab\u200b\u6478\u51fa\u6765\u200b\u4e86\u200b\u3002\u200b\u6b64\u5916\u200b\u8fd8\u6709\u200b\u4e00\u4e2a\u200b\u9700\u8981\u200b\u6ce8\u610f\u200b\u7684\u200b\u7ec6\u8282\u200b\uff0c\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u662f\u200b\u53ef\u4ee5\u200b\u548c\u200b\u5b83\u200b\u672c\u8eab\u200b\u4e92\u76f8\u4ea4\u6362\u200b\u7684\u200b - \u200b\u5426\u5219\u200b\u751f\u6210\u200b\u6700\u540e\u200b\u7684\u200b\u6392\u5217\u7ec4\u5408\u200b\u7684\u200b\u6982\u7387\u200b\u5c31\u200b\u4e0d\u200b\u5bf9\u200b\u4e86\u200b\u3002</p> <p>Reference</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>typedef struct {\n    int size;\n    int *nums;\n} Solution;\n\n\nSolution* solutionCreate(int* nums, int numsSize) {\n    Solution *ret = (Solution *)malloc(sizeof(Solution));\n    ret-&gt;size = numsSize;\n    ret-&gt;nums = (int *)memcpy(malloc(sizeof(int) * numsSize), nums, sizeof(int) * numsSize);\n    return ret;\n}\n\n/** Resets the array to its original configuration and return it. */\nint* solutionReset(Solution* obj, int* retSize) {\n    *retSize = obj-&gt;size;\n    return obj-&gt;nums;\n}\n\n/** Returns a random shuffling of the array. */\nint* solutionShuffle(Solution* obj, int* retSize) {\n    *retSize = obj-&gt;size;\n    int *ret = (int *)memcpy(malloc(sizeof(int) * obj-&gt;size), obj-&gt;nums, sizeof(int) * obj-&gt;size), i = 0, temp, x, y;\n    unsigned int *seed = malloc(sizeof(unsigned int));\n    *seed += (unsigned int)seed;\n    srand(*seed);\n    for (i = 0; i &lt; obj-&gt;size; i++)\n    {\n        x = i;\n        y = rand() % (obj-&gt;size - i) + i;\n        temp = ret[x];\n        ret[x] = ret[y];\n        ret[y] = temp;\n    }\n    free(seed);\n    return ret;\n}\n\nvoid solutionFree(Solution* obj) {\n    free(obj-&gt;nums);\n    free(obj);\n}\n\n/**\n * Your Solution struct will be instantiated and called as such:\n * Solution* obj = solutionCreate(nums, numsSize);\n * int* param_1 = solutionReset(obj, retSize);\n\n * int* param_2 = solutionShuffle(obj, retSize);\n\n * solutionFree(obj);\n*/\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/387/","title":"387. \u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u552f\u4e00\u200b\u5b57\u7b26","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u54c8\u5e0c\u8868","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/387/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u627e\u5230\u200b\u5b83\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u4e0d\u200b\u91cd\u590d\u200b\u7684\u200b\u5b57\u7b26\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5b83\u200b\u7684\u200b\u7d22\u5f15\u200b\u3002\u200b\u5982\u679c\u200b\u4e0d\u200b\u5b58\u5728\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>-1</code>\u3002</p> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>s = \"leetcode\"\n\u200b\u8fd4\u56de\u200b 0\n\ns = \"loveleetcode\"\n\u200b\u8fd4\u56de\u200b 2\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a \u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u5b9a\u200b\u8be5\u200b\u5b57\u7b26\u4e32\u200b\u53ea\u200b\u5305\u542b\u200b\u5c0f\u5199\u5b57\u6bcd\u200b\u3002</p> <p>Reference</p>","tags":["\u54c8\u5e0c\u8868","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/387/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4f7f\u7528\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\u5b58\u50a8\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u5404\u200b\u5b57\u7b26\u200b\u7684\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u3002\u200b\u904d\u5386\u200b\u5b57\u7b26\u4e32\u200b\u8fdb\u884c\u200b\u7edf\u8ba1\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int firstUniqChar(char * s){\n    int hashMap[26] = {0};\n    char *cur = s;\n    while(*cur)\n    {\n        hashMap[*cur - 'a']++;\n        cur++;\n    }\n    cur = s;\n    while(*cur)\n    {\n        if (hashMap[*cur - 'a'] == 1)\n            return (int)(cur - s);\n        cur++;\n    }\n    return -1;\n}\n</code></pre>","tags":["\u54c8\u5e0c\u8868","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/403/","title":"403. \u200b\u9752\u86d9\u200b\u8fc7\u6cb3","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p> <p>\u200b\u4e00\u53ea\u200b\u9752\u86d9\u200b\u60f3\u8981\u200b\u8fc7\u6cb3\u200b\u3002 \u200b\u5047\u5b9a\u200b\u6cb3\u6d41\u200b\u88ab\u200b\u7b49\u200b\u5206\u4e3a\u200b\u82e5\u5e72\u4e2a\u200b\u5355\u5143\u683c\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5728\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u5355\u5143\u683c\u200b\u5185\u200b\u90fd\u200b\u6709\u200b\u53ef\u80fd\u200b\u653e\u200b\u6709\u200b\u4e00\u5757\u200b\u77f3\u5b50\u200b\uff08\u200b\u4e5f\u200b\u6709\u200b\u53ef\u80fd\u200b\u6ca1\u6709\u200b\uff09\u3002 \u200b\u9752\u86d9\u200b\u53ef\u4ee5\u200b\u8df3\u4e0a\u200b\u77f3\u5b50\u200b\uff0c\u200b\u4f46\u662f\u200b\u4e0d\u200b\u53ef\u4ee5\u200b\u8df3\u5165\u200b\u6c34\u4e2d\u200b\u3002</p> <p>\u200b\u7ed9\u200b\u4f60\u200b\u77f3\u5b50\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u5217\u8868\u200b <code>stones</code>\uff08\u200b\u7528\u200b\u5355\u5143\u683c\u200b\u5e8f\u53f7\u200b \u200b\u5347\u5e8f\u200b \u200b\u8868\u793a\u200b\uff09\uff0c\u00a0\u200b\u8bf7\u200b\u5224\u5b9a\u200b\u9752\u86d9\u200b\u80fd\u5426\u200b\u6210\u529f\u200b\u8fc7\u6cb3\u200b\uff08\u200b\u5373\u200b\u80fd\u5426\u200b\u5728\u200b\u6700\u540e\u200b\u4e00\u6b65\u200b\u8df3\u81f3\u200b\u6700\u540e\u200b\u4e00\u5757\u200b\u77f3\u5b50\u200b\u4e0a\u200b\uff09\u3002</p> <p>\u200b\u5f00\u59cb\u200b\u65f6\u200b\uff0c\u00a0\u200b\u9752\u86d9\u200b\u9ed8\u8ba4\u200b\u5df2\u7ad9\u200b\u5728\u200b\u7b2c\u4e00\u5757\u200b\u77f3\u5b50\u200b\u4e0a\u200b\uff0c\u200b\u5e76\u200b\u53ef\u4ee5\u200b\u5047\u5b9a\u200b\u5b83\u200b\u7b2c\u4e00\u6b65\u200b\u53ea\u80fd\u200b\u8df3\u8dc3\u200b\u4e00\u4e2a\u200b\u5355\u4f4d\u200b\uff08\u200b\u5373\u200b\u53ea\u80fd\u200b\u4ece\u200b\u5355\u5143\u683c\u200b 1 \u200b\u8df3\u81f3\u200b\u5355\u5143\u683c\u200b 2 \uff09\u3002</p> <p>\u200b\u5982\u679c\u200b\u9752\u86d9\u200b\u4e0a\u200b\u4e00\u6b65\u200b\u8df3\u8dc3\u200b\u4e86\u200b\u00a0<code>k</code>\u00a0\u200b\u4e2a\u200b\u5355\u4f4d\u200b\uff0c\u200b\u90a3\u4e48\u200b\u5b83\u200b\u63a5\u4e0b\u6765\u200b\u7684\u200b\u8df3\u8dc3\u200b\u8ddd\u79bb\u200b\u53ea\u80fd\u200b\u9009\u62e9\u200b\u4e3a\u200b\u00a0<code>k - 1</code>\u3001<code>k</code>\u00a0\u200b\u6216\u200b\u00a0<code>k + 1</code> \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u3002\u00a0\u200b\u53e6\u200b\u8bf7\u200b\u6ce8\u610f\u200b\uff0c\u200b\u9752\u86d9\u200b\u53ea\u80fd\u200b\u5411\u200b\u524d\u65b9\u200b\uff08\u200b\u7ec8\u70b9\u200b\u7684\u200b\u65b9\u5411\u200b\uff09\u200b\u8df3\u8dc3\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1astones = [0,1,3,5,6,8,12,17]\n\u200b\u8f93\u51fa\u200b\uff1atrue\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u9752\u86d9\u200b\u53ef\u4ee5\u200b\u6210\u529f\u200b\u8fc7\u6cb3\u200b\uff0c\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u65b9\u6848\u200b\u8df3\u8dc3\u200b\uff1a\u200b\u8df3\u200b 1 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u5230\u200b\u7b2c\u200b 2 \u200b\u5757\u200b\u77f3\u5b50\u200b, \u200b\u7136\u540e\u200b\u8df3\u200b 2 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u5230\u200b\u7b2c\u200b 3 \u200b\u5757\u200b\u77f3\u5b50\u200b, \u200b\u63a5\u7740\u200b \u200b\u8df3\u200b 2 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u5230\u200b\u7b2c\u200b 4 \u200b\u5757\u200b\u77f3\u5b50\u200b, \u200b\u7136\u540e\u200b\u8df3\u200b 3 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u5230\u200b\u7b2c\u200b 6 \u200b\u5757\u200b\u77f3\u5b50\u200b, \u200b\u8df3\u200b 4 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u5230\u200b\u7b2c\u200b 7 \u200b\u5757\u200b\u77f3\u5b50\u200b, \u200b\u6700\u540e\u200b\uff0c\u200b\u8df3\u200b 5 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u5230\u200b\u7b2c\u200b 8 \u200b\u4e2a\u200b\u77f3\u5b50\u200b\uff08\u200b\u5373\u200b\u6700\u540e\u200b\u4e00\u5757\u200b\u77f3\u5b50\u200b\uff09\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1astones = [0,1,2,3,4,8,9,11]\n\u200b\u8f93\u51fa\u200b\uff1afalse\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u8fd9\u200b\u662f\u56e0\u4e3a\u200b\u7b2c\u200b 5 \u200b\u548c\u200b\u7b2c\u200b 6 \u200b\u4e2a\u200b\u77f3\u5b50\u200b\u4e4b\u95f4\u200b\u7684\u200b\u95f4\u8ddd\u200b\u592a\u200b\u5927\u200b\uff0c\u200b\u6ca1\u6709\u200b\u53ef\u9009\u200b\u7684\u200b\u65b9\u6848\u200b\u4f9b\u200b\u9752\u86d9\u200b\u8df3\u8dc3\u200b\u8fc7\u53bb\u200b\u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>2 &lt;= stones.length &lt;= 2000</code></li> <li><code>0 &lt;= stones[i] &lt;= 2^31 - 1</code></li> <li><code>stones[0] == 0</code></li> </ul> <p>Reference</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/403/#_1","title":"\u9898\u89e3","text":"<p>\u200b\u6839\u636e\u200b\u9898\u610f\u200b\uff0c\u200b\u5728\u200b\u4efb\u4f55\u200b\u72b6\u6001\u200b\uff0c\u200b\u9752\u86d9\u200b\u90fd\u200b\u6700\u200b\u591a\u200b\u53ea\u6709\u200b\u4e09\u79cd\u200b\u72b6\u6001\u200b\u53ef\u4ee5\u200b\u9009\u62e9\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6df1\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b\u67e5\u627e\u200b\u6240\u6709\u200b\u7684\u200b\u72b6\u6001\u200b\u3002\u200b\u5047\u8bbe\u200b\u9752\u86d9\u200b\u8df3\u200b\u4e86\u200b\\(k\\)\u200b\u683c\u200b\u8fbe\u5230\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b</p> <ul> <li>\u200b\u5f53\u200b\u9752\u86d9\u200b\u8fbe\u5230\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u65f6\u200b\u8fd4\u56de\u200b<code>true</code>\uff0c</li> <li>\u200b\u5f53\u200b\u9752\u86d9\u200b\u5230\u8fbe\u200b\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u65f6\u200b\uff0c\u200b\u68c0\u67e5\u200b\u524d\u9762\u200b\u7684\u200b\\(k-1, k, k+1\\)\u200b\u4e2a\u200b\u5355\u5143\u683c\u200b\u662f\u5426\u200b\u6709\u200b\u77f3\u5b50\u200b\uff0c</li> <li>\u200b\u5f53\u200b\u9752\u86d9\u200b\u65e0\u6cd5\u200b\u7ee7\u7eed\u200b\u5411\u524d\u200b\u79fb\u52a8\u200b\u65f6\u200b\u8fd4\u56de\u200b<code>false</code>\u200b\u5e76\u200b\u8fdb\u884c\u200b\u56de\u6eaf\u200b\u3002</li> </ul> <pre><code>bool dfs(int *stones, int stonesSize, int currentPos, int lastStep)\n{\n    if (currentPos == stonesSize - 1)\n        return true;\n    int i = 0, j = 0, cur_id = stones[currentPos];\n    bool ret = false;\n    for (i = -1; i &lt; 2; i++)\n    {\n        if (lastStep + i &lt;= 0)\n            continue;\n        for (j = currentPos + 1; !ret &amp;&amp; j &lt; stonesSize &amp;&amp; stones[j] &lt;= cur_id + lastStep + i; j++)\n            if (stones[j] == cur_id + lastStep + i)\n                ret = ret || dfs(stones, stonesSize, j, lastStep + i);\n    }\n    return ret;\n}\n\nbool canCross(int* stones, int stonesSize){\n    return dfs(stones, stonesSize, 0, 0);\n}\n</code></pre> <p>\u200b\u5982\u200b\u4e0a\u200bdfs\u200b\u7b97\u6cd5\u200b\u4f1a\u200b\u5bf9\u200b\u6240\u6709\u200b\u7684\u200b\u5206\u652f\u200b\u8fdb\u884c\u200b\u904d\u5386\u200b\uff0c\u200b\u76f4\u5230\u200b\u627e\u51fa\u200b\u6700\u7ec8\u200b\u7ed3\u679c\u200b\u3002\u200b\u4f1a\u200b\u9020\u6210\u200b\u5927\u91cf\u200b\u65f6\u95f4\u200b\u7684\u200b\u6d6a\u8d39\u200b\u3002\u200b\u6784\u9020\u200b\u5982\u4e0b\u200b\u8f93\u5165\u200b\u6570\u636e\u200b\u5373\u53ef\u200b\u4f7f\u8be5\u200b\u7b97\u6cd5\u200b\u8d85\u65f6\u200b\uff1a</p> <pre><code>[0,1,2,3,4,5,...,998, +infty]\n</code></pre> <p>\u200b\u5176\u4e2d\u200b<code>+infty</code>\u200b\u6307\u200b\u5145\u5206\u200b\u5927\u200b\u7684\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u52a0\u5165\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8bb0\u5fc6\u200b\u4e0d\u540c\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5230\u200b\u7ec8\u70b9\u200b\u7684\u200b\u53ef\u8fbe\u6027\u200b\uff1a</p> <pre><code>bool dfs(int *stones, int *buf, int stonesSize, int currentPos, int lastStep)\n{\n    if (buf[currentPos] != -1)\n        return buf[currentPos];\n    if (currentPos == stonesSize - 1)\n        return true;\n    int i = 0, j = 0, cur_id = stones[currentPos];\n    bool ret = false;\n    for (i = -1; i &lt; 2; i++)\n    {\n        if (lastStep + i &lt;= 0)\n            continue;\n        for (j = currentPos + 1; !ret &amp;&amp; j &lt; stonesSize &amp;&amp; stones[j] &lt;= cur_id + lastStep + i; j++)\n            if (stones[j] == cur_id + lastStep + i)\n                ret = ret || dfs(stones, buf, stonesSize, j, lastStep + i);\n    }\n    buf[currentPos] = ret;\n    return ret;\n}\n\nbool canCross(int* stones, int stonesSize){\n    int *buf = (int *)malloc(sizeof(int) * stonesSize);\n    memset(buf, 0xff, sizeof(int) * stonesSize);\n    return dfs(stones, buf, stonesSize, 0, 0);\n}\n</code></pre> <p>\u200b\u4f46\u200b\u8fd9\u4e2a\u200b\u7b97\u6cd5\u200b\u53ea\u80fd\u200b\u901a\u8fc7\u200b\u4e00\u90e8\u5206\u200b\u6d4b\u4f8b\u200b\uff0c\u200b\u90e8\u5206\u200b\u6d4b\u4f8b\u200b\u65e0\u6cd5\u200b\u901a\u8fc7\u200b\uff08\u200b\u5982\u200b\u793a\u4f8b\u200b1\uff0c<code>[0,1,3,5,6,8,12,17]</code>\uff09\u3002\u200b\u6211\u4eec\u200b\u67e5\u770b\u200b\u4e00\u4e0b\u200b\u9752\u86d9\u200b\u5728\u200b\u8282\u70b9\u200b\u95f4\u200b\u8df3\u8dc3\u200b\u7684\u200b\u60c5\u51b5\u200b\uff1a</p> <p></p> <p>\u200b\u5f53\u200b\u7b97\u6cd5\u200b\u8d70\u200b\u5230\u200b\u8282\u70b9\u200b<code>5</code>\u200b\u65f6\u200b\uff0c\u200b\u9996\u5148\u200b\u4f1a\u200b\u5c1d\u8bd5\u200b\u8def\u5f84\u200b<code>5-&gt;6-&gt;8</code>\uff0c\u200b\u4f46\u200b\u6b64\u65f6\u200b\u5728\u200b\u8282\u70b9\u200b<code>8</code>\u200b\u9752\u86d9\u200b\u6700\u200b\u591a\u200b\u53ea\u80fd\u200b\u8df3\u200b<code>3</code>\u200b\u6b65\u200b\uff0c\u200b\u4ece\u800c\u200b\u88ab\u200b\u8bb0\u5fc6\u200b\u5316\u200b\u6807\u8bb0\u200b\u4e3a\u200b\u4e0d\u53ef\u200b\u5230\u8fbe\u200b\u7ec8\u70b9\u200b\uff0c\u200b\u4f46\u200b\u5b9e\u9645\u4e0a\u200b\u5982\u679c\u200b\u76f4\u63a5\u200b\u91c7\u53d6\u200b\u8def\u5f84\u200b<code>5-&gt;8</code>\u200b\u5219\u200b\u53ef\u4ee5\u200b\u4ece\u200b\u8282\u70b9\u200b<code>8</code>\u200b\u8df3\u200b\u5230\u200b\u8282\u70b9\u200b<code>12</code>\u3002\u200b\u5373\u200b\u67d0\u4e2a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u7ec8\u70b9\u200b\u53ef\u8fbe\u6027\u200b\u53d7\u5230\u200b\u4e0a\u200b\u4e00\u6b65\u200b\u8df3\u52a8\u200b\u6b65\u6570\u200b\u7684\u200b\u5f71\u54cd\u200b\uff0c\u200b\u6211\u4eec\u200b\u5728\u200b\u8fdb\u884c\u200b\u8bb0\u5fc6\u200b\u5316\u200b\u65f6\u200b\u9700\u8981\u200b\u8bb0\u5fc6\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\u5728\u200b\u6bcf\u4e2a\u200b\u8df3\u52a8\u200b\u6b65\u6570\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u7684\u200b\u53ef\u8fbe\u6027\u200b\u3002</p> <p>\u200b\u5982\u4e0b\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u4ee3\u66ff\u200b\u904d\u5386\u200b\uff0c\u200b\u5e76\u200b\u4f7f\u7528\u200b\u62c9\u94fe\u200b\u6cd5\u200b\u5b58\u50a8\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\u7684\u200b\u53ef\u8fbe\u6027\u200b\uff0c\u200b\u94fe\u8868\u200b\u8282\u70b9\u200b\u4e2d\u200b\u5b58\u50a8\u200b\u76f8\u5e94\u200b\u7684\u200b\u8df3\u52a8\u200b\u6b65\u6570\u200b\u3002</p> <pre><code>struct BufNode\n{\n    int success;\n    int lastStep;\n    struct BufNode *next;\n};\n\nstruct BufNode *getNode(int success, int lastStep, struct BufNode *next)\n{\n    struct BufNode *ret = (struct BufNode *)malloc(sizeof(struct BufNode));\n    ret-&gt;success = success;\n    ret-&gt;lastStep = lastStep;\n    ret-&gt;next = next;\n    return ret;\n}\n\nbool check(struct BufNode *buf, int lastStep, int *success)\n{\n    if (buf == NULL)\n        return NULL;\n    if (buf-&gt;lastStep == lastStep)\n    {\n        if (success)\n            *success = buf-&gt;success;\n        return true;\n    }\n    return check(buf-&gt;next, lastStep, success);\n}\n\nint bSearch(int *stones, int lo, int hi, int target)\n{\n    int mid;\n    while (lo &lt; hi)\n    {\n        mid = (lo + hi) &gt;&gt; 1;\n        if (target &lt; stones[mid])\n            hi = mid;\n        else\n            lo = mid + 1;\n    }\n    return lo - 1;\n}\n\nbool dfs(int *stones, struct BufNode **buf, int stonesSize, int currentPos, int lastStep)\n{\n    int success = false;\n    if (check(buf[currentPos], lastStep, &amp;success))\n        return success;\n    if (currentPos == stonesSize - 1)\n        return true;\n    int i = 0, j = 0, cur_id = stones[currentPos];\n    bool ret = false;\n    for (i = -1; i &lt; 2; i++)\n    {\n        if (lastStep + i &lt;= 0)\n            continue;\n        j = bSearch(stones, currentPos + 1, stonesSize, cur_id + lastStep + i);\n        if (stones[j] == cur_id + lastStep + i)\n            ret = ret || dfs(stones, buf, stonesSize, j, lastStep + i);\n    }\n    buf[currentPos] = getNode(ret, lastStep, buf[currentPos]);\n    return ret;\n}\n\nbool canCross(int* stones, int stonesSize){\n    struct BufNode **buf = (struct BufNode **)malloc(sizeof(struct BufNode *) * stonesSize);\n    memset(buf, 0, sizeof(struct BufNode *) * stonesSize);\n    return dfs(stones, buf, stonesSize, 0, 0);\n}\n</code></pre>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/42/","title":"42. \u200b\u63a5\u200b\u96e8\u6c34","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u6808","\u6570\u7ec4","\u53cc\u6307\u9488","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/42/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u00a0<code>n</code> \u200b\u4e2a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u8868\u793a\u200b\u6bcf\u4e2a\u200b\u5bbd\u5ea6\u200b\u4e3a\u200b <code>1</code> \u200b\u7684\u200b\u67f1\u5b50\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u56fe\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u6309\u6b64\u200b\u6392\u5217\u200b\u7684\u200b\u67f1\u5b50\u200b\uff0c\u200b\u4e0b\u96e8\u200b\u4e4b\u540e\u200b\u80fd\u63a5\u200b\u591a\u5c11\u200b\u96e8\u6c34\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aheight = [0,1,0,2,1,0,1,3,2,1,2,1]\n\u200b\u8f93\u51fa\u200b\uff1a6\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u4e0a\u9762\u200b\u662f\u200b\u7531\u200b\u6570\u7ec4\u200b [0,1,0,2,1,0,1,3,2,1,2,1] \u200b\u8868\u793a\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u56fe\u200b\uff0c\u200b\u5728\u200b\u8fd9\u79cd\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u63a5\u200b 6 \u200b\u4e2a\u200b\u5355\u4f4d\u200b\u7684\u200b\u96e8\u6c34\u200b\uff08\u200b\u84dd\u8272\u200b\u90e8\u5206\u200b\u8868\u793a\u200b\u96e8\u6c34\u200b\uff09\u3002 \n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aheight = [4,2,0,3,2,5]\n\u200b\u8f93\u51fa\u200b\uff1a9\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>n == height.length</code></li> <li><code>0 &lt;= n &lt;= 3 * 10^4</code></li> <li><code>0 &lt;= height[i] &lt;= 10^5</code></li> </ul> <p>Reference</p> <p>\u200b\u6ce8\u200b\uff1a\u200b\u672c\u9898\u200b\u4e0e\u200b\u9762\u8bd5\u9898\u200b 17.21. \u200b\u76f4\u65b9\u56fe\u200b\u7684\u200b\u6c34\u91cf\u200b\u76f8\u540c\u200b\u3002</p>","tags":["\u6808","\u6570\u7ec4","\u53cc\u6307\u9488","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/42/#_2","title":"\u9898\u89e3","text":"<ol> <li> <p>\u200b\u5206\u6cbb\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u6c34\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u4e0d\u4f1a\u200b\u9ad8\u4e8e\u200b\u6700\u9ad8\u200b\u7684\u200b\u67f1\u5b50\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u5728\u200b\u6700\u9ad8\u200b\u7684\u200b\u67f1\u5b50\u200b\\(A\\)\u200b\u4e0e\u200b\u6b21\u200b\u9ad8\u200b\u7684\u200b\u67f1\u5b50\u200b\\(B\\)\u200b\u4e24\u5904\u200b\u8fdb\u884c\u200b\u5206\u5272\u200b\uff0c\u200b\u5206\u522b\u200b\u6c42\u89e3\u200b\u4e09\u6bb5\u200b\u5bb9\u7eb3\u200b\u7684\u200b\u6c34\u91cf\u200b\u3002</p> <p>\\(A, B\\)\u200b\u4e4b\u95f4\u200b\u5fc5\u7136\u200b\u53ef\u4ee5\u200b\u5bb9\u7eb3\u200b\u6c34\u200b\uff0c\u200b\u6c34\u200b\u7684\u200b\u91cf\u200b\u7b49\u4e8e\u200b\u5404\u4e2a\u200b\u67f1\u5b50\u200b\u5230\u6b21\u200b\u9ad8\u200b\u67f1\u5b50\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u4e4b\u200b\u548c\u200b\u3002\u200b\u5176\u4ed6\u200b\u533a\u95f4\u200b\u9700\u8981\u200b\u9012\u5f52\u200b\u8fdb\u884c\u200b\u6c42\u89e3\u200b\u3002\u200b\u9700\u8981\u200b\u6ce8\u610f\u200b\u5206\u5272\u200b\u65f6\u200b\u5206\u5272\u200b\u70b9\u8981\u200b\u540c\u65f6\u200b\u5305\u542b\u200b\u5728\u200b\u5de6\u53f3\u200b\u4e24\u4fa7\u200b\uff0c\u200b\u5426\u5219\u200b\u65e0\u6cd5\u200b\u5bb9\u7eb3\u200b\u6c34\u200b\u3002</p> <p>\u200b\u5f53\u200b\u67f1\u5b50\u200b\u6570\u91cf\u200b\u5c0f\u4e8e\u200b3\u200b\u65f6\u200b\uff0c\u200b\u65e0\u6cd5\u200b\u5bb9\u7eb3\u200b\u4efb\u4f55\u200b\u6c34\u200b\u3002</p> <pre><code>#define MAX(x, y) (x &gt; y ? x : y)\n#define MIN(x, y) (x &lt; y ? x : y)\n\nint trap(int* height, int heightSize){\n    int *maxLeft = (int *)malloc(sizeof(int) * heightSize),\n        *maxRight = (int *)malloc(sizeof(int) * heightSize),\n        i = 0, ret = 0;\n    if (heightSize == 0)\n        return 0;\n    maxLeft[0] = height[0];\n    maxRight[heightSize - 1] = height[heightSize - 1];\n    for (i = 1; i &lt; heightSize; i++)\n    {\n        maxLeft[i] = MAX(maxLeft[i - 1], height[i - 1]);\n        maxRight[heightSize - i - 1] = MAX(maxRight[heightSize - i], height[heightSize - i]);\n    }\n    for (i = 0; i &lt; heightSize; i++)\n        ret += MAX(0, MIN(maxLeft[i], maxRight[i]) - height[i]);\n    return ret;\n}\n</code></pre> <p>\u200b\u603b\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b\\(\\mathcal O(N)\\)\u3002</p> </li> <li> <p>\u200b\u76f4\u63a5\u200b\u601d\u8def\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u67f1\u5b50\u200b\uff0c\u200b\u5176\u200b\u5bb9\u7eb3\u200b\u6c34\u540e\u200b\u7684\u200b\u9ad8\u5ea6\u200b\uff08\u200b\u67f1\u5b50\u200b\u9ad8\u5ea6\u200b\u51cf\u53bb\u200b\u6c34\u200b\u9ad8\u5ea6\u200b\uff09\u200b\u7b49\u4e8e\u200b\u8be5\u200b\u67f1\u5b50\u200b\u5de6\u4fa7\u200b\u6700\u9ad8\u200b\u67f1\u5b50\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u4e0e\u200b\u53f3\u4fa7\u200b\u6700\u9ad8\u200b\u67f1\u5b50\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u67f1\u5b50\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7edf\u8ba1\u200b\u5728\u200b\u5176\u200b\u5de6\u4fa7\u200b\u6700\u9ad8\u200b\u67f1\u5b50\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u4e0e\u200b\u5728\u200b\u5176\u200b\u53f3\u4fa7\u200b\u6700\u9ad8\u200b\u67f1\u5b50\u200b\u7684\u200b\u9ad8\u5ea6\u200b\uff0c\u200b\u5b58\u50a8\u200b\u5728\u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u3002</p> <p>\u200b\u7ed3\u679c\u200b\u4e3a\u200b\u6bcf\u4e2a\u200b\u67f1\u5b50\u200b\u5bb9\u7eb3\u200b\u6c34\u540e\u200b\u7684\u200b\u9ad8\u5ea6\u200b\u4e0e\u200b\u67f1\u5b50\u200b\u9ad8\u5ea6\u200b\u4e4b\u5dee\u200b\u7684\u200b\u6c42\u548c\u200b\u3002</p> <pre><code>#define MAX(x, y) (x &gt; y ? x : y)\n#define MIN(x, y) (x &lt; y ? x : y)\n\nint trap(int* height, int heightSize){\n    int *maxLeft = (int *)malloc(sizeof(int) * heightSize),\n        *maxRight = (int *)malloc(sizeof(int) * heightSize),\n        i = 0, ret = 0;\n    if (heightSize == 0)\n        return 0;\n    maxLeft[0] = height[0];\n    maxRight[heightSize - 1] = height[heightSize - 1];\n    for (i = 1; i &lt; heightSize; i++)\n    {\n        maxLeft[i] = MAX(maxLeft[i - 1], height[i - 1]);\n        maxRight[heightSize - i - 1] = MAX(maxRight[heightSize - i], height[heightSize - i]);\n    }\n    for (i = 0; i &lt; heightSize; i++)\n        ret += MAX(0, MIN(maxLeft[i], maxRight[i]) - height[i]);\n    return ret;\n}\n</code></pre> </li> </ol>","tags":["\u6808","\u6570\u7ec4","\u53cc\u6307\u9488","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/43/","title":"43. \u200b\u5b57\u7b26\u4e32\u200b\u76f8\u4e58","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/43/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e24\u4e2a\u200b\u4ee5\u200b\u5b57\u7b26\u4e32\u200b\u5f62\u5f0f\u200b\u8868\u793a\u200b\u7684\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u00a0<code>num1</code>\u00a0\u200b\u548c\u200b\u00a0<code>num2</code>\uff0c\u200b\u8fd4\u56de\u200b\u00a0<code>num1</code>\u00a0\u200b\u548c\u200b\u00a0<code>num2</code>\u00a0\u200b\u7684\u200b\u4e58\u79ef\u200b\uff0c\u200b\u5b83\u4eec\u200b\u7684\u200b\u4e58\u79ef\u200b\u4e5f\u200b\u8868\u793a\u200b\u4e3a\u200b\u5b57\u7b26\u4e32\u200b\u5f62\u5f0f\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1:</p> <pre><code>\u200b\u8f93\u5165\u200b: num1 = \"2\", num2 = \"3\"\n\u200b\u8f93\u51fa\u200b: \"6\"\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02:</p> <pre><code>\u200b\u8f93\u5165\u200b: num1 = \"123\", num2 = \"456\"\n\u200b\u8f93\u51fa\u200b: \"56088\"\n</code></pre> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li><code>num1</code>\u00a0\u200b\u548c\u200b\u00a0<code>num2</code>\u00a0\u200b\u7684\u200b\u957f\u5ea6\u200b\u5c0f\u4e8e\u200b110\u3002</li> <li><code>num1</code> \u200b\u548c\u200b\u00a0<code>num2</code> \u200b\u53ea\u200b\u5305\u542b\u200b\u6570\u5b57\u200b\u00a0<code>0-9</code>\u3002</li> <li><code>num1</code> \u200b\u548c\u200b\u00a0<code>num2</code>\u00a0\u200b\u5747\u200b\u4e0d\u200b\u4ee5\u200b\u96f6\u200b\u5f00\u5934\u200b\uff0c\u200b\u9664\u975e\u200b\u662f\u200b\u6570\u5b57\u200b <code>0</code> \u200b\u672c\u8eab\u200b\u3002</li> <li>\u200b\u4e0d\u80fd\u200b\u4f7f\u7528\u200b\u4efb\u4f55\u200b\u6807\u51c6\u200b\u5e93\u200b\u7684\u200b\u5927\u6570\u200b\u7c7b\u578b\u200b\uff08\u200b\u6bd4\u5982\u200bBigInteger\uff09\u200b\u6216\u200b\u76f4\u63a5\u200b\u5c06\u200b\u8f93\u5165\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u6574\u6570\u200b\u6765\u200b\u5904\u7406\u200b\u3002</li> </ul> <p>Reference</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/43/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u8ba1\u7b97\u65b9\u6cd5\u200b\u4e0e\u200b\u5217\u7ad6\u5f0f\u200b\u76f8\u540c\u200b\uff0c\u200b\u6ce8\u610f\u200b\u5bf9\u200b\u7279\u6b8a\u200b\u8f93\u5165\u200b<code>0</code>\u200b\u7684\u200b\u5904\u7406\u200b\u3002\u200b\u7ed3\u679c\u200b\u7684\u200b\u957f\u5ea6\u200b\u4e0d\u4f1a\u200b\u8d85\u8fc7\u200b\u8f93\u5165\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u4e4b\u200b\u548c\u200b\u3002</p> <pre><code>char * multiply(char * num1, char * num2){\n    if (*num1 == '0' || *num2 == '0')\n        return *num1 == '0' ? num1 : num2;\n    int len1 = strlen(num1), len2 = strlen(num2), retLen = len1 + len2, i = 0, j = 0, incre, curDigit, pos;\n    char *ret = (char *)memset(malloc(sizeof(char) * (retLen + 1)), '0', sizeof(char) * (retLen + 1)),\n         *cur = ret;\n    ret[retLen] = 0;\n    for (i = len1 - 1; i &gt;= 0; i--)\n    {\n        for (j = len2 - 1; j &gt;= 0; j--)\n        {\n            incre = (num1[i] - '0') * (num2[j] - '0');\n            curDigit = incre % 10;\n            incre /= 10;\n            pos = retLen - ((len1 - i - 1) + (len2 - j - 1)) - 1;\n            while (incre != 0 || curDigit != 0)\n            {\n                ret[pos] += curDigit;\n                incre += ret[pos] &gt; '9';\n                ret[pos] -= ret[pos] &gt; '9' ? 10 : 0;\n                curDigit = incre &gt;= 10 ? incre - 10 : incre;\n                incre = incre &gt;= 10;\n                pos--;\n            }\n        }\n    }\n    while (*cur == '0')\n        cur++;\n    return cur;\n}\n</code></pre>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/433/","title":"433. \u200b\u6700\u5c0f\u200b\u57fa\u56e0\u200b\u53d8\u5316","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/433/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u4e00\u6761\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u7531\u200b\u4e00\u4e2a\u200b\u5e26\u6709\u200b8\u200b\u4e2a\u5b57\u7b26\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\u8868\u793a\u200b\uff0c\u200b\u5176\u4e2d\u200b\u6bcf\u4e2a\u200b\u5b57\u7b26\u200b\u90fd\u200b\u5c5e\u4e8e\u200b <code>\"A\"</code>, <code>\"C\"</code>, <code>\"G\"</code>, <code>\"T\"</code>\u200b\u4e2d\u200b\u7684\u200b\u4efb\u610f\u200b\u4e00\u4e2a\u200b\u3002</p> <p>\u200b\u5047\u8bbe\u200b\u6211\u4eec\u200b\u8981\u200b\u8c03\u67e5\u200b\u4e00\u4e2a\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u7684\u200b\u53d8\u5316\u200b\u3002\u200b\u4e00\u6b21\u200b\u57fa\u56e0\u200b\u53d8\u5316\u200b\u610f\u5473\u7740\u200b\u8fd9\u4e2a\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b\u53d1\u751f\u200b\u4e86\u200b\u53d8\u5316\u200b\u3002</p> <p>\u200b\u4f8b\u5982\u200b\uff0c\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u7531\u200b <code>\"AACCGGTT\"</code>\u00a0\u200b\u53d8\u5316\u200b\u81f3\u200b\u00a0<code>\"AACCGGTA\"</code>\u00a0\u200b\u5373\u200b\u53d1\u751f\u200b\u4e86\u200b\u4e00\u6b21\u200b\u57fa\u56e0\u200b\u53d8\u5316\u200b\u3002</p> <p>\u200b\u4e0e\u6b64\u540c\u65f6\u200b\uff0c\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u57fa\u56e0\u200b\u53d8\u5316\u200b\u7684\u200b\u7ed3\u679c\u200b\uff0c\u200b\u90fd\u200b\u9700\u8981\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5408\u6cd5\u200b\u7684\u200b\u57fa\u56e0\u200b\u4e32\u200b\uff0c\u200b\u5373\u8be5\u200b\u7ed3\u679c\u200b\u5c5e\u4e8e\u200b\u4e00\u4e2a\u200b\u57fa\u56e0\u5e93\u200b\u3002</p> <p>\u200b\u73b0\u5728\u200b\u7ed9\u5b9a\u200b3\u200b\u4e2a\u200b\u53c2\u6570\u200b \u2014 <code>start</code>, <code>end</code>, <code>bank</code>\uff0c\u200b\u5206\u522b\u200b\u4ee3\u8868\u200b\u8d77\u59cb\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\uff0c\u200b\u76ee\u6807\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u53ca\u200b\u57fa\u56e0\u5e93\u200b\uff0c\u200b\u8bf7\u200b\u627e\u51fa\u200b\u80fd\u591f\u200b\u4f7f\u200b\u8d77\u59cb\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u53d8\u5316\u200b\u4e3a\u200b\u76ee\u6807\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u6240\u200b\u9700\u200b\u7684\u200b\u6700\u5c11\u200b\u53d8\u5316\u200b\u6b21\u6570\u200b\u3002\u200b\u5982\u679c\u200b\u65e0\u6cd5\u200b\u5b9e\u73b0\u76ee\u6807\u200b\u53d8\u5316\u200b\uff0c\u200b\u8bf7\u200b\u8fd4\u56de\u200b <code>-1</code>\u3002</p> <p>\u200b\u6ce8\u610f\u200b\uff1a</p> <ul> <li>\u200b\u8d77\u59cb\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u9ed8\u8ba4\u200b\u662f\u200b\u5408\u6cd5\u200b\u7684\u200b\uff0c\u200b\u4f46\u662f\u200b\u5b83\u200b\u5e76\u4e0d\u4e00\u5b9a\u200b\u4f1a\u200b\u51fa\u73b0\u200b\u5728\u200b\u57fa\u56e0\u5e93\u200b\u4e2d\u200b\u3002</li> <li>\u200b\u5982\u679c\u200b\u4e00\u4e2a\u200b\u8d77\u59cb\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u9700\u8981\u200b\u591a\u6b21\u200b\u53d8\u5316\u200b\uff0c\u200b\u90a3\u4e48\u200b\u5b83\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u53d8\u5316\u200b\u4e4b\u540e\u200b\u7684\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u90fd\u200b\u5fc5\u987b\u200b\u662f\u200b\u5408\u6cd5\u200b\u7684\u200b\u3002</li> <li>\u200b\u5047\u5b9a\u200b\u8d77\u59cb\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u4e0e\u200b\u76ee\u6807\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u662f\u200b\u4e0d\u200b\u4e00\u6837\u200b\u7684\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>start: \"AACCGGTT\"\nend:   \"AACCGGTA\"\nbank: [\"AACCGGTA\"]\n\n\u200b\u8fd4\u56de\u503c\u200b: 1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>start: \"AACCGGTT\"\nend:   \"AAACGGTA\"\nbank: [\"AACCGGTA\", \"AACCGCTA\", \"AAACGGTA\"]\n\n\u200b\u8fd4\u56de\u503c\u200b: 2\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>start: \"AAAAACCC\"\nend:   \"AACCCCCC\"\nbank: [\"AAAACCCC\", \"AAACCCCC\", \"AACCCCCC\"]\n\n\u200b\u8fd4\u56de\u503c\u200b: 3\n</code></pre> <p>Reference</p>","tags":["\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/433/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u5e7f\u5ea6\u200b\u4f18\u5148\u200b\u7b97\u6cd5\u200b\u904d\u5386\u200b\u57fa\u56e0\u200b\u7684\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u6539\u52a8\u200b\u60c5\u51b5\u200b\u3002\u200b\u57fa\u56e0\u200b\u5e8f\u5217\u200b\u957f\u5ea6\u200b\u4e3a\u200b8\uff0c\u200b\u6bcf\u6b21\u200b\u53ef\u80fd\u200b\u7684\u200b\u6539\u52a8\u200b\u60c5\u51b5\u200b\u4e3a\u200b\\(8\\times 3 = 24\\)\u200b\u79cd\u200b\uff0c\u200b\u5728\u200b\u8fd9\u200b24\u200b\u79cd\u200b\u6539\u52a8\u200b\u60c5\u51b5\u200b\u4e2d\u200b\u9009\u62e9\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u7ed3\u679c\u200b\uff1a</p> <ul> <li>\u200b\u7ed3\u679c\u200b\u662f\u200b\u5408\u6cd5\u200b\u7684\u200b\u57fa\u56e0\u200b</li> <li>\u200b\u7ed3\u679c\u200b\u672a\u200b\u88ab\u200b\u904d\u5386\u200b\u8fc7\u200b</li> </ul> <p>\u200b\u4f7f\u7528\u200b\u54c8\u5e0c\u200b\u8868\u200b\u5b58\u50a8\u200b\u57fa\u56e0\u5e93\u200b\u548c\u200b\u5df2\u7ecf\u200b\u88ab\u200b\u904d\u5386\u200b\u8fc7\u200b\u7684\u200b\u57fa\u56e0\u200b\u3002</p> <pre><code>char *genes = \"TGCATGCA\";\n\nint getCode(char *s)\n{\n    int ret = 0;\n    switch (*s)\n    {\n    case 'A':\n        ret++;\n    case 'C':\n        ret++;\n    case 'G':\n        ret++;\n    case 'T':\n        break;\n    }\n    return ret;\n}\n\nint hash(char *s)\n{\n    char *cur = s;\n    int ret = 0;\n    while (*cur)\n    {\n        ret &lt;&lt;= 2;\n        ret += getCode(cur);\n        cur++;\n    }\n    return ret;\n}\n\nchar *dehash(int o)\n{\n    char *ret = (char *)malloc(sizeof(char) * 9);\n    ret[8] = 0;\n    for (int i = 7; i &gt;= 0; i--)\n    {\n        ret[i] = genes[o &amp; 3];\n        o &gt;&gt;= 2;\n    }\n    return ret;\n}\n\nstruct Bank {\n    char *bitmap;\n};\n\nvoid deleteBank(struct Bank *dest)\n{\n    free(dest-&gt;bitmap);\n    free(dest);\n}\n\nvoid removeGene(struct Bank *dest, char *seq)\n{\n    int i = hash(seq);\n    dest-&gt;bitmap[i &gt;&gt; 3] &amp;= ~(1 &lt;&lt; (i &amp; 7));\n}\n\nvoid addGene(struct Bank *dest, char *seq)\n{\n    int i = hash(seq);\n    dest-&gt;bitmap[i &gt;&gt; 3] |= 1 &lt;&lt; (i &amp; 7);\n}\n\nbool queryGene(struct Bank *dest, char *seq)\n{\n    int i = hash(seq);\n    return dest-&gt;bitmap[i &gt;&gt; 3] &amp; (1 &lt;&lt; (i &amp; 7));\n}\n\nvoid merge(struct Bank *dest, struct Bank *src)\n{\n    int i = 0;\n    for (i = 0; i &lt; 8192; i++)\n        dest-&gt;bitmap[i] |= src-&gt;bitmap[i];\n    deleteBank(src);\n}\n\nstruct Bank *getBank(char **bank, int bankSize)\n{\n    struct Bank *ret = (struct Bank *)malloc(sizeof(struct Bank));\n    ret-&gt;bitmap = (char *)malloc(sizeof(char) * 8192);\n    memset(ret-&gt;bitmap, 0, sizeof(char) * 8192);\n    for (int i = 0; i &lt; bankSize; i++)\n        addGene(ret, bank[i]);\n    return ret;\n}\n\nstruct Bank *step(char *current, char *end, struct Bank *passed, struct Bank *fullBank, int *count)\n{\n    int startHash = hash(current);\n    struct Bank *ret = getBank(NULL, 0);\n    addGene(passed, current);\n    for (int i = 0; i &lt; 8; i++)\n    {\n        for (int j = 0; j &lt; 4; j++)\n        {\n            current[i] = genes[getCode(current + i) + 1];\n            if (startHash == hash(current))\n                continue;\n            if (!queryGene(fullBank, current))\n                continue;\n            if (queryGene(passed, current))\n                continue;\n            addGene(ret, current);\n            (*count)++;\n        }\n    }\n    return ret;\n}\n\nstruct Bank *stepBank(struct Bank *current, char *end, struct Bank *passed, struct Bank *fullBank, int* count)\n{\n    int index = 0;\n    struct Bank *ret = getBank(NULL, 0);\n    char *temp = NULL;\n    for (int i = 0; i &lt; 8192; i++)\n    {\n        if (current-&gt;bitmap[i])\n        {\n            index = i &lt;&lt; 3;\n            for (int j = 0; j &lt; 8; j++)\n            {\n                if (current-&gt;bitmap[i] &amp; (1 &lt;&lt; j))\n                {\n                    index += j;\n                    temp = dehash(index);\n                    merge(ret, step(temp, end, passed, fullBank, count));\n                    free(temp);\n                }\n            }\n        }\n    }\n    return ret;\n}\n\nint minMutation(char * start, char * end, char ** bank, int bankSize){\n    struct Bank *fullBank = getBank(bank, bankSize), *passed = getBank(NULL, 0);\n    struct Bank *temp = getBank(NULL, 0), *temp2 = NULL;\n    addGene(temp, start);\n    int ret = 0, modified = 0;\n    while (!queryGene(temp, end))\n    {\n        modified = 0;\n        temp2 = stepBank(temp, end, passed, fullBank, &amp;modified);\n        deleteBank(temp);\n        temp = temp2;\n        if (!modified)\n            return -1;\n        ret++;\n    }\n    return ret;\n}\n</code></pre>","tags":["\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22","\u54c8\u5e0c\u8868"]},{"location":"coding/leetcode/46/","title":"46. \u200b\u5168\u200b\u6392\u5217","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/46/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b \u200b\u6ca1\u6709\u200b\u91cd\u590d\u200b \u200b\u6570\u5b57\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5176\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u7684\u200b\u5168\u200b\u6392\u5217\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b:</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,2,3]\n\u200b\u8f93\u51fa\u200b:\n[\n  [1,2,3],\n  [1,3,2],\n  [2,1,3],\n  [2,3,1],\n  [3,1,2],\n  [3,2,1]\n]\n</code></pre> <p>Reference</p>","tags":["\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/46/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u56de\u6eaf\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u5e8f\u5217\u200b\uff1a</p> <ul> <li>\u200b\u82e5\u200b\u957f\u5ea6\u200b\u4e3a\u200b1\uff0c\u200b\u5219\u200b\u53ea\u6709\u200b\u4e00\u79cd\u200b\u6392\u5217\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u76f4\u63a5\u200b\u8fd4\u56de\u200b</li> <li>\u200b\u82e5\u200b\u957f\u5ea6\u200b\u5927\u4e8e\u200b1\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\uff1a<ul> <li>\u200b\u5c06\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7b2c\u4e00\u4e2a\u200b\u6570\u200b\u653e\u5728\u200b\u6392\u5217\u200b\u7684\u200b\u7b2c\u4e00\u4f4d\u200b\uff0c\u200b\u7136\u540e\u200b\u4e0e\u200b\u540e\u9762\u200b\u90e8\u5206\u200b\u7684\u200b\u5168\u200b\u6392\u5217\u200b\u5408\u5e76\u200b</li> <li>\u200b\u5c06\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7b2c\u4e8c\u4e2a\u200b\u6570\u200b\u653e\u5728\u200b\u6392\u5217\u200b\u7684\u200b\u7b2c\u4e00\u4f4d\u200b\uff0c\u200b\u7136\u540e\u200b\u4e0e\u200b\u540e\u9762\u200b\u90e8\u5206\u200b\u7684\u200b\u5168\u200b\u6392\u5217\u200b\u5408\u5e76\u200b</li> <li>\u2026\u2026</li> <li>\u200b\u5c06\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u6570\u200b\u653e\u5728\u200b\u6392\u5217\u200b\u7684\u200b\u7b2c\u4e00\u4f4d\u200b\uff0c\u200b\u7136\u540e\u200b\u4e0e\u200b\u540e\u9762\u200b\u90e8\u5206\u200b\u7684\u200b\u5168\u200b\u6392\u5217\u200b\u5408\u5e76\u200b</li> <li>\u200b\u6700\u7ec8\u200b\u7ed3\u679c\u200b\u5305\u542b\u200b\u5982\u200b\u4e0a\u200b\u6240\u6709\u200b\u7684\u200b\u6392\u5217\u200b\u3002</li> </ul> </li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(n\\)\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5171\u6709\u200b\\(n!\\)\u200b\u79cd\u200b\u6392\u5217\u200b\u3002</p> <pre><code>int fact(int x)\n{\n    if (!x)\n        return 0;\n    int i = 1, ret = 1;\n    for (i = 1; i &lt;= x; i++)\n        ret *= i;\n    return ret;\n}\nvoid swap(int *a, int *b)\n{\n    int temp = *a;\n    *a = *b;\n    *b = temp;\n}\nint** permute(int* nums, int numsSize, int* returnSize, int** returnColumnSizes){\n    if (numsSize &lt; 2)\n    {\n        *returnSize = 1;\n        *returnColumnSizes = (int *)malloc(sizeof(int) * *returnSize);\n        *returnColumnSizes[0] = numsSize;\n        if (numsSize == 0)\n            return NULL;\n        int **ret = (int **)malloc(sizeof(int *) * *returnSize);\n        *ret = (int *)malloc(sizeof(int));\n        **ret = *nums;\n        return ret;\n    }\n    *returnSize = fact(numsSize);\n    *returnColumnSizes = (int *)malloc(sizeof(int *) * *returnSize);\n    int *sub = (int *)malloc(sizeof(int) * numsSize), i = 0, **rec = NULL, recSize, *recColSize,\n        **ret = (int **)malloc(sizeof(int *) * *returnSize), retpos = 0, j = 0;\n    for (i = 0; i &lt; *returnSize; i++)\n        (*returnColumnSizes)[i] = numsSize;\n    for (i = 0; i &lt; numsSize; i++)\n    {\n        sub = memcpy(sub, nums, sizeof(int) * numsSize);\n        swap(sub + i, sub);\n        rec = permute(sub + 1, numsSize - 1, &amp;recSize, &amp;recColSize);\n        for (j = 0; j &lt; recSize; j++)\n        {\n            ret[retpos] = (int *)malloc(sizeof(int) * numsSize);\n            ret[retpos][0] = sub[0];\n            memcpy(ret[retpos] + 1, rec[j], sizeof(int) * (numsSize - 1));\n            retpos++;\n        }\n    }\n    return ret;\n}\n</code></pre>","tags":["\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/48/","title":"48. \u200b\u65cb\u8f6c\u200b\u56fe\u50cf","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/48/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b n\u00a0\u00d7\u00a0n \u200b\u7684\u200b\u4e8c\u7ef4\u200b\u77e9\u9635\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u56fe\u50cf\u200b\u3002</p> <p>\u200b\u5c06\u200b\u56fe\u50cf\u200b\u987a\u65f6\u9488\u200b\u65cb\u8f6c\u200b 90 \u200b\u5ea6\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u5fc5\u987b\u200b\u5728\u200b\u539f\u5730\u200b\u65cb\u8f6c\u200b\u56fe\u50cf\u200b\uff0c\u200b\u8fd9\u200b\u610f\u5473\u7740\u200b\u4f60\u200b\u9700\u8981\u200b\u76f4\u63a5\u200b\u4fee\u6539\u200b\u8f93\u5165\u200b\u7684\u200b\u4e8c\u7ef4\u200b\u77e9\u9635\u200b\u3002\u200b\u8bf7\u200b\u4e0d\u8981\u200b\u4f7f\u7528\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u77e9\u9635\u200b\u6765\u200b\u65cb\u8f6c\u200b\u56fe\u50cf\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u7ed9\u5b9a\u200b matrix = \n[\n  [1,2,3],\n  [4,5,6],\n  [7,8,9]\n],\n\n\u200b\u539f\u5730\u200b\u65cb\u8f6c\u200b\u8f93\u5165\u200b\u77e9\u9635\u200b\uff0c\u200b\u4f7f\u200b\u5176\u200b\u53d8\u4e3a\u200b:\n[\n  [7,4,1],\n  [8,5,2],\n  [9,6,3]\n]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u7ed9\u5b9a\u200b matrix =\n[\n  [ 5, 1, 9,11],\n  [ 2, 4, 8,10],\n  [13, 3, 6, 7],\n  [15,14,12,16]\n], \n\n\u200b\u539f\u5730\u200b\u65cb\u8f6c\u200b\u8f93\u5165\u200b\u77e9\u9635\u200b\uff0c\u200b\u4f7f\u200b\u5176\u200b\u53d8\u4e3a\u200b:\n[\n  [15,13, 2, 5],\n  [14, 3, 4, 1],\n  [12, 6, 8, 9],\n  [16, 7,10,11]\n]\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/48/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5c06\u200b\u77e9\u9635\u200b\u8f6c\u7f6e\u200b\u540e\u200b\u53cd\u8f6c\u200b\u5404\u5217\u200b\u7684\u200b\u987a\u5e8f\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>void rotate(int** matrix, int matrixSize, int* matrixColSize){\n    int i = 0, j = 0, temp;\n    for (i = 0; i &lt; matrixSize; i++)\n    {\n        for (j = i; j &lt; matrixSize; j++)\n        {\n            temp = matrix[i][j];\n            matrix[i][j] = matrix[j][i];\n            matrix[j][i] = temp;\n        }\n    }\n    for (i = 0; i &lt; matrixSize; i++)\n    {\n        for (j = 0; j &lt; matrixSize / 2; j++)\n        {\n            temp = matrix[i][j];\n            matrix[i][j] = matrix[i][matrixSize - j - 1];\n            matrix[i][matrixSize - j - 1] = temp;\n        }\n    }\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/488/","title":"488. \u200b\u7956\u739b\u200b\u6e38\u620f","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u6df1\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/488/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u56de\u5fc6\u200b\u4e00\u4e0b\u200b\u7956\u739b\u200b\u6e38\u620f\u200b\u3002\u200b\u73b0\u5728\u200b\u684c\u4e0a\u200b\u6709\u200b\u4e00\u4e32\u200b\u7403\u200b\uff0c\u200b\u989c\u8272\u200b\u6709\u200b\u7ea2\u8272\u200b(R)\uff0c\u200b\u9ec4\u8272\u200b(Y)\uff0c\u200b\u84dd\u8272\u200b(B)\uff0c\u200b\u7eff\u8272\u200b(G)\uff0c\u200b\u8fd8\u6709\u200b\u767d\u8272\u200b(W)\u3002 \u200b\u73b0\u5728\u200b\u4f60\u200b\u624b\u91cc\u200b\u4e5f\u200b\u6709\u200b\u51e0\u4e2a\u200b\u7403\u200b\u3002</p> <p>\u200b\u6bcf\u200b\u4e00\u6b21\u200b\uff0c\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u4ece\u200b\u624b\u91cc\u200b\u7684\u200b\u7403\u9009\u200b\u4e00\u4e2a\u200b\uff0c\u200b\u7136\u540e\u200b\u628a\u200b\u8fd9\u4e2a\u200b\u7403\u200b\u63d2\u5165\u200b\u5230\u200b\u4e00\u4e32\u200b\u7403\u4e2d\u200b\u7684\u200b\u67d0\u4e2a\u200b\u4f4d\u7f6e\u200b\u4e0a\u200b\uff08\u200b\u5305\u62ec\u200b\u6700\u200b\u5de6\u7aef\u200b\uff0c\u200b\u6700\u200b\u53f3\u7aef\u200b\uff09\u3002\u200b\u63a5\u7740\u200b\uff0c\u200b\u5982\u679c\u200b\u6709\u200b\u51fa\u73b0\u200b\u4e09\u4e2a\u200b\u6216\u8005\u200b\u4e09\u4e2a\u200b\u4ee5\u4e0a\u200b\u989c\u8272\u200b\u76f8\u540c\u200b\u7684\u200b\u7403\u200b\u76f8\u8fde\u200b\u7684\u8bdd\u200b\uff0c\u200b\u5c31\u200b\u628a\u200b\u5b83\u4eec\u200b\u79fb\u200b\u9664\u6389\u200b\u3002\u200b\u91cd\u590d\u200b\u8fd9\u4e00\u200b\u6b65\u9aa4\u200b\u76f4\u5230\u200b\u684c\u4e0a\u200b\u6240\u6709\u200b\u7684\u200b\u7403\u200b\u90fd\u200b\u88ab\u200b\u79fb\u9664\u200b\u3002</p> <p>\u200b\u627e\u5230\u200b\u63d2\u5165\u200b\u5e76\u200b\u53ef\u4ee5\u200b\u79fb\u200b\u9664\u6389\u200b\u684c\u4e0a\u200b\u6240\u6709\u200b\u7403\u6240\u200b\u9700\u200b\u7684\u200b\u6700\u5c11\u200b\u7684\u200b\u7403\u6570\u200b\u3002\u200b\u5982\u679c\u200b\u4e0d\u80fd\u200b\u79fb\u9664\u200b\u684c\u4e0a\u200b\u6240\u6709\u200b\u7684\u200b\u7403\u200b\uff0c\u200b\u8f93\u51fa\u200b <code>-1</code> \u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = \"WRRBBW\", hand = \"RB\"\n\u200b\u8f93\u51fa\u200b\uff1a-1\n\u200b\u89e3\u91ca\u200b\uff1aWRRBBW -&gt; WRR[R]BBW -&gt; WBBW -&gt; WBB[B]W -&gt; WW\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = \"WWRRBBWW\", hand = \"WRBRW\"\n\u200b\u8f93\u51fa\u200b\uff1a2\n\u200b\u89e3\u91ca\u200b\uff1aWWRRBBWW -&gt; WWRR[R]BBWW -&gt; WWBBWW -&gt; WWBB[B]WW -&gt; WWWW -&gt; empty\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = \"G\", hand = \"GGGGG\"\n\u200b\u8f93\u51fa\u200b\uff1a2\n\u200b\u89e3\u91ca\u200b\uff1aG -&gt; G[G] -&gt; GG[G] -&gt; empty \n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = \"RBYYBBRRB\", hand = \"YRBGB\"\n\u200b\u8f93\u51fa\u200b\uff1a3\n\u200b\u89e3\u91ca\u200b\uff1aRBYYBBRRB -&gt; RBYY[Y]BBRRB -&gt; RBBBRRB -&gt; RRRB -&gt; B -&gt; B[B] -&gt; BB[B] -&gt; empty \n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u684c\u4e0a\u200b\u4e00\u200b\u5f00\u59cb\u200b\u7684\u200b\u7403\u200b\u4e2d\u200b\uff0c\u200b\u4e0d\u4f1a\u200b\u6709\u200b\u4e09\u4e2a\u200b\u53ca\u200b\u4e09\u4e2a\u200b\u4ee5\u4e0a\u200b\u989c\u8272\u200b\u76f8\u540c\u200b\u4e14\u200b\u8fde\u200b\u7740\u200b\u7684\u200b\u7403\u200b\u3002</li> <li><code>1 &lt;= board.length &lt;= 16</code></li> <li><code>1 &lt;= hand.length &lt;= 5</code></li> <li>\u200b\u8f93\u5165\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u5747\u200b\u4e3a\u200b\u975e\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u4e14\u200b\u53ea\u200b\u5305\u542b\u200b\u5b57\u7b26\u200b <code>'R','Y','B','G','W'</code>\u3002</li> </ul> <p>Reference</p>","tags":["\u6df1\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/488/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5b9a\u4e49\u200b\u5f53\u524d\u200b\u6e38\u620f\u200b\u7684\u200b\u72b6\u6001\u200b\uff0c\u200b\u4e00\u4e2a\u200b\u72b6\u6001\u200b\u7531\u200b\u684c\u4e0a\u200b\u7684\u200b\u7403\u200b\u5217\u8868\u200b\u4e0e\u200b\u624b\u4e2d\u200b\u7684\u200b\u7403\u200b\u5217\u8868\u200b\u8868\u793a\u200b\u3002\u200b\u6bcf\u6b21\u200b\u63d2\u5165\u200b\u64cd\u4f5c\u200b\u6784\u6210\u200b\u4e00\u4e2a\u200b\u72b6\u6001\u200b\u95f4\u200b\u7684\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\u8fc7\u7a0b\u200b\u3002\u200b\u4f7f\u7528\u200b\u4e24\u4e2a\u200b\u961f\u5217\u200b\u5b58\u50a8\u200b\u5f53\u524d\u200b\u72b6\u6001\u200b\uff0c\u200b\u548c\u200b\u5f53\u524d\u200b\u72b6\u6001\u200b\u7684\u200b \u200b\u6240\u6709\u200b \u200b\u53ef\u80fd\u200b\u8f6c\u79fb\u200b\u5230\u200b\u7684\u200b\u72b6\u6001\u200b\u3002\u200b\u518d\u200b\u5c06\u200b\u8f6c\u79fb\u200b\u5230\u200b\u7684\u200b\u72b6\u6001\u200b\u8bbe\u200b\u4e3a\u200b\u5f53\u524d\u200b\u72b6\u6001\u200b\uff0c\u200b\u7ee7\u7eed\u200b\u8fdb\u884c\u200b\u641c\u7d22\u200b\u8fc7\u7a0b\u200b\u3002\u200b\u672c\u9898\u200b\u4e2d\u200b\uff0c\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\u662f\u200b\u5355\u5411\u200b\u7684\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e0d\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u91cd\u590d\u200b\u7ecf\u8fc7\u200b\u4e00\u79cd\u200b\u72b6\u6001\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002</p> <p>\u200b\u7531\u4e8e\u200b\u4e00\u4e2a\u200b\u72b6\u6001\u200b\u53ef\u4ee5\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u72b6\u6001\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b\u8fd9\u4e9b\u200b\u72b6\u6001\u200b\u95f4\u200b\u8fdb\u884c\u200b\u7b5b\u9009\u200b\u3002\u200b\u7b5b\u9009\u200b\u89c4\u5219\u200b\u5982\u4e0b\u200b\uff1a</p> <ol> <li>\u200b\u5f53\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b\u4e24\u4fa7\u200b\u7684\u200b\u989c\u8272\u200b\u4e0d\u200b\u540c\u65f6\u200b\uff0c\u200b\u65b0\u7403\u200b\u7684\u200b\u989c\u8272\u200b\u9700\u8981\u200b\u4e0e\u200b\u5de6\u53f3\u200b\u4e24\u4fa7\u200b\u5176\u4e2d\u200b\u4e00\u4e2a\u200b\u7403\u200b\u7684\u200b\u989c\u8272\u200b\u76f8\u540c\u200b</li> <li>\u200b\u5f53\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b\u4e24\u4fa7\u200b\u989c\u8272\u200b\u76f8\u540c\u200b\u65f6\u200b\uff0c\u200b\u65b0\u7403\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4efb\u610f\u200b\u53ef\u80fd\u200b\u7684\u200b\u989c\u8272\u200b</li> <li>\u200b\u5f53\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b\u4f4d\u4e8e\u200b\u7aef\u70b9\u200b\u65f6\u200b\uff0c\u200b\u65b0\u7403\u200b\u7684\u200b\u989c\u8272\u200b\u9700\u8981\u200b\u4e0e\u200b\u76f8\u90bb\u200b\u7403\u200b\u7684\u200b\u989c\u8272\u200b\u76f8\u540c\u200b</li> </ol> <p>\u200b\u5982\u200b\u4e0a\u200b\u526a\u679d\u200b\u601d\u8def\u200b\u4e0d\u4f1a\u200b\u5f71\u54cd\u200b\u6d4b\u4f8b\u200b<code>\"RRYGGYYRRYGGYYRR\" \"GGBBB\"</code>\u200b\u4e0e\u200b<code>\"RRWWRRBBRR\" \"WB\"</code>\u200b\u7b49\u200b\u9700\u8981\u200b\u5c06\u200b\u8fde\u7eed\u200b\u5e8f\u5217\u200b\u5206\u5272\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u5982\u200b\u4e0a\u200b\u6761\u4ef6\u200b\u7684\u200b\u72b6\u6001\u200b\u89c6\u4e3a\u200b\u65e0\u6548\u200b\u7684\u200b\u72b6\u6001\u200b\uff0c\u200b\u76f4\u63a5\u200b\u8df3\u8fc7\u200b\u5bf9\u200b\u8fd9\u4e9b\u200b\u72b6\u6001\u200b\u7684\u200b\u641c\u7d22\u200b\u3002</p> <p>\u200b\u4f7f\u7528\u200b\u94fe\u8868\u200b\u6a21\u62df\u200b\u961f\u5217\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u94fe\u8868\u200b\u8282\u70b9\u200b\u8868\u793a\u200b\u95ee\u9898\u200b\u7684\u200b\u4e00\u4e2a\u200b\u72b6\u6001\u200b\uff0c\u200b\u7531\u4e8e\u200b\u94fe\u8868\u200b\u8282\u70b9\u200b\u9700\u8981\u200b\u591a\u6b21\u200b\u5206\u914d\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u63d0\u524d\u200b\u5206\u914d\u5185\u5b58\u200b\u6c60\u200b\u7684\u200b\u7b56\u7565\u200b\u4ee5\u200b\u51cf\u5c11\u200b<code>malloc()</code>\u200b\u7684\u200b\u8c03\u7528\u200b\u6b21\u6570\u200b\u3002</p> <pre><code>struct Queue\n{\n    char *val;\n    int colorsOnHand[5];\n    struct Queue *next;\n};\n\nstruct Queue *nodePool;\nint poolSize;\nint poolAllocated;\n\nstruct Queue *getNode(char *val, struct Queue *next, int *colorsOnHand)\n{\n    int i = 0;\n    struct Queue *ret;\n    if (!nodePool || poolAllocated &gt;= poolSize)\n    {\n        if (poolAllocated)\n            poolSize &lt;&lt;= 1;\n        else\n            poolSize = 1000;\n        poolAllocated = 0;\n        nodePool = (struct Queue *)malloc(sizeof(struct Queue) * poolSize);\n    }\n    ret = nodePool + (poolAllocated++);\n    ret-&gt;val = val;\n    ret-&gt;next = next;\n    for (i = 0; i &lt; 5; i++)\n        ret-&gt;colorsOnHand[i] = colorsOnHand[i];\n    return ret;\n}\n</code></pre> <p>\u200b\u5206\u200b\u4e24\u6b65\u200b\u8fdb\u884c\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\uff0c\u200b\u7b2c\u4e00\u6b65\u200b\u5148\u200b\u63d2\u5165\u200b\u4e00\u4e2a\u200b\u7403\u200b\uff0c\u200b\u7b2c\u4e8c\u6b65\u200b\u6d88\u9664\u200b\u8fde\u7eed\u200b\u7684\u200b\u7403\u200b\uff0c\u200b\u76f4\u5230\u200b\u65e0\u7403\u200b\u53ef\u6d88\u200b\u3002</p> <pre><code>char *strPool;\nint strSize;\nint strAllocated;\n\nchar *getStr(int len)\n{\n    if (!strPool || strAllocated + len &gt;= strSize)\n    {\n        if (strAllocated)\n            strSize &lt;&lt;= 1;\n        else\n            strSize = 32768;\n        strAllocated = 0;\n        strPool = (char *)malloc(sizeof(char) * strSize);\n    }\n    char *ret = strPool + strAllocated;\n    strAllocated += len;\n    return ret;\n}\n\nbool play(char *src)\n{\n    char *ptrfast = src, *ptrslow = src, cur = 0;\n    int i = 0;\n    bool flag = false;\n    while (*ptrfast)\n    {\n        if (cur == *ptrfast)\n            i++;\n        else\n        {\n            if (i &lt; 3)\n            {\n                while (i)\n                {\n                    *(ptrslow++) = cur;\n                    i--;\n                }\n            }\n            else\n                flag = true;\n            i = 1;\n            cur = *ptrfast;\n        }\n        ptrfast++;\n    }\n    if (i &lt; 3)\n    {\n        while (i)\n        {\n            *(ptrslow++) = cur;\n            i--;\n        }\n    }\n    else\n        flag = true;\n    *ptrslow = 0;\n    return flag;\n}\n\nchar *addBall(char *src, int len, char ball, int loc)\n{\n    char *ret = (char *)malloc(sizeof(char) * (len + 2));\n    ret[len + 1] = 0;\n    memcpy(ret, src, sizeof(char) * loc);\n    memcpy(ret + loc + 1, src + loc, sizeof(char) * (len - loc));\n    ret[loc] = ball;\n    return ret;\n}\n</code></pre> <pre><code>int findMinStep(char * board, char * hand) {\n    struct Queue *current = NULL, *temp = NULL;\n    char *cur = hand, *init = getStr(strlen(board) + 1), *newseq = NULL,\n         *colors = \"RYBGW\";\n    int i = 0, j = 0, len = 0, ret = 0, colorsOnHand[5] = {0}, flag;\n\n    nodePool = NULL;\n    poolSize = 0;\n    poolAllocated = 0;\n    strPool = NULL;\n    strSize = 0;\n    strAllocated = 0;\n\n    strcpy(init, board);\n    do\n        for (i = 0; i &lt; 5; i++)\n            if (*cur == colors[i])\n                colorsOnHand[i]++;\n    while (*(++cur));\n\n    current = getNode(init, NULL, colorsOnHand);\n    while (current)\n    {\n        while (current)\n        {\n            len = strlen(current-&gt;val);\n            if (!len)\n                return ret;\n            for (i = 0; i &lt;= len; i++)\n            {\n                flag = false;\n                for (j = 0; j &lt; 5; j++)\n                {\n                    if (current-&gt;colorsOnHand[j] == 0)\n                        continue;\n                    if (i &gt; 0)\n                        flag = flag || current-&gt;val[i - 1] == colors[j];\n                    if (i &lt; len)\n                        flag = flag || current-&gt;val[i] == colors[j];\n                    if (i &gt; 0 &amp;&amp; i &lt; len)\n                        flag = flag || current-&gt;val[i - 1] == current-&gt;val[i];\n                    if (!flag)\n                        continue;\n                    current-&gt;colorsOnHand[j]--;\n                    newseq = addBall(current-&gt;val, len, colors[j], i);\n                    while (play(newseq));\n                    temp = getNode(newseq, temp, current-&gt;colorsOnHand);\n                    current-&gt;colorsOnHand[j]++;\n                }\n            }\n            current = current-&gt;next;\n        }\n        ret++;\n        current = temp;\n        temp = NULL;\n    }\n    return -1;\n}\n</code></pre>","tags":["\u6df1\u5ea6\u4f18\u5148\u641c\u7d22"]},{"location":"coding/leetcode/50/","title":"50. Pow(x, n)","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u5b66","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/50/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5b9e\u73b0\u200b\u00a0pow(x, n)\uff0c\u200b\u5373\u200b\u8ba1\u7b97\u200b x \u200b\u7684\u200b n \u200b\u6b21\u200b\u5e42\u51fd\u6570\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a <pre><code>\u200b\u8f93\u5165\u200b: 2.00000, 10\n\u200b\u8f93\u51fa\u200b: 1024.00000\n</code></pre> \u200b\u793a\u4f8b\u200b\u00a02\uff1a <pre><code>\u200b\u8f93\u5165\u200b: 2.10000, 3\n\u200b\u8f93\u51fa\u200b: 9.26100\n</code></pre> \u200b\u793a\u4f8b\u200b\u00a03\uff1a <pre><code>\u200b\u8f93\u5165\u200b: 2.00000, -2\n\u200b\u8f93\u51fa\u200b: 0.25000\n\u200b\u89e3\u91ca\u200b: 2^(-2) = (1/2)^2 = 1/4 = 0.25\n</code></pre> \u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li>-100.0 &lt;\u00a0x\u00a0&lt; 100.0</li> <li>n\u00a0\u200b\u662f\u200b 32 \u200b\u4f4d\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\uff0c\u200b\u5176\u200b\u6570\u503c\u200b\u8303\u56f4\u200b\u662f\u200b\\([\u22122^{31},\u00a02^{31}\u00a0\u2212 1]\\)\u3002</li> </ul> <p>Reference</p>","tags":["\u6570\u5b66","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/50/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5229\u7528\u200b\u5feb\u901f\u200b\u5e42\u200b\u7684\u200b\u601d\u8def\u200b\uff0c\u200b\u5b58\u50a8\u200b\u6b21\u6570\u200b\u4e3a\u200b2\u200b\u7684\u200b\u6574\u6570\u200b\u5e42\u200b\u7684\u200b\u8ba1\u7b97\u7ed3\u679c\u200b\uff0c\u200b\u7136\u540e\u200b\u7528\u200b\u8fd9\u4e9b\u200b\u7ed3\u679c\u200b\u76f8\u4e58\u200b\u5f97\u5230\u200b\u8f93\u51fa\u200b\u7ed3\u679c\u200b\u3002\u200b\u6ce8\u610f\u200b\u6307\u6570\u200b\u4e3a\u200b\u8d1f\u6570\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002\u200b\u5f53\u200b\u6307\u6570\u200b\u4e3a\u200b<code>-2147483648</code>\u200b\u65f6\u200b\u4e0d\u80fd\u200b\u53d6\u8d1f\u200b\uff0c\u200b\u6b64\u65f6\u200b\u53ef\u4ee5\u200b\u5148\u52a0\u200b\u4e00\u518d\u200b\u53d6\u8d1f\u200b\uff0c\u200b\u6700\u540e\u200b\u9664\u4ee5\u200b\u5e95\u6570\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>double myPow(double x, int n)\n{\n    if (n &lt; 0)\n        return 1 / myPow(x, -(n + 1)) / x;\n    double cache[32] = {0}, ret = 1;\n    int i = 0;\n    cache[0] = x;\n    while (n)\n    {\n        if (n &amp; 1)\n            ret *= cache[i];\n        n &gt;&gt;= 1;\n        i++;\n        cache[i] = cache[i - 1] * cache[i - 1];\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u5b66","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/509/","title":"509. \u200b\u6590\u6ce2\u200b\u90a3\u200b\u5951\u6570","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/509/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u6590\u6ce2\u200b\u90a3\u200b\u5951\u6570\u200b\uff0c\u200b\u901a\u5e38\u200b\u7528\u200b<code>F(n)</code>\u200b\u8868\u793a\u200b\uff0c\u200b\u5f62\u6210\u200b\u7684\u200b\u5e8f\u5217\u200b\u79f0\u4e3a\u200b\u6590\u6ce2\u200b\u90a3\u5951\u200b\u6570\u5217\u200b\u3002\u200b\u8be5\u200b\u6570\u5217\u200b\u7531\u200b<code>0</code>\u200b\u548c\u200b<code>1</code>\u200b\u5f00\u59cb\u200b\uff0c\u200b\u540e\u9762\u200b\u7684\u200b\u6bcf\u4e00\u9879\u200b\u6570\u5b57\u200b\u90fd\u200b\u662f\u200b\u524d\u9762\u200b\u4e24\u9879\u200b\u6570\u5b57\u200b\u7684\u200b\u548c\u200b\u3002\u200b\u4e5f\u200b\u5c31\u662f\u200b\uff1a <pre><code>F(0) = 0,\u00a0 \u00a0F(1)\u00a0= 1\nF(N) = F(N - 1) + F(N - 2), \u200b\u5176\u4e2d\u200b N &gt; 1.\n</code></pre> \u200b\u7ed9\u5b9a\u200b<code>N</code>\uff0c\u200b\u8ba1\u7b97\u200b<code>F(N)</code>\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a <pre><code>\u200b\u8f93\u5165\u200b\uff1a2\n\u200b\u8f93\u51fa\u200b\uff1a1\n\u200b\u89e3\u91ca\u200b\uff1aF(2) = F(1) + F(0) = 1 + 0 = 1.\n</code></pre></p> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a3\n\u200b\u8f93\u51fa\u200b\uff1a2\n\u200b\u89e3\u91ca\u200b\uff1aF(3) = F(2) + F(1) = 1 + 1 = 2.\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a4\n\u200b\u8f93\u51fa\u200b\uff1a3\n\u200b\u89e3\u91ca\u200b\uff1aF(4) = F(3) + F(2) = 2 + 1 = 3.\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <p>\\(0\\leq N\\leq30\\)</p> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/509/#_2","title":"\u9898\u89e3","text":"","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/509/#_3","title":"\u9012\u5f52","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4f7f\u7528\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\u5b58\u653e\u200b\u5386\u53f2\u200b\u7ed3\u679c\u200b\uff0c\u200b\u52a0\u901f\u200b\u9012\u5f52\u8ba1\u7b97\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int fib_cache(int N, int **cache, int *lenCache, int *fullCache){\n    if (!N)\n        return 0;\n    if (!(*cache))\n    {\n        *lenCache = 2;\n        *fullCache = N + 1;\n        *cache = (int *)memset(malloc(sizeof(int) * *fullCache), 0, sizeof(int) * *fullCache);\n        (*cache)[0] = 1;\n        (*cache)[1] = 1;\n    }\n    if (N &gt; *lenCache &amp;&amp; N &lt;= *fullCache)\n        (*cache)[N-1] = fib_cache(N - 1, cache, lenCache, fullCache) + fib_cache(N - 2, cache, lenCache, fullCache);\n        *lenCache = N;\n    return (*cache)[N - 1];\n}\n\nint fib(int N){\n    int *cache = NULL, lenCache = 0, fullCache = 0;\n    return fib_cache(N, &amp;cache, &amp;lenCache, &amp;fullCache);\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/509/#_4","title":"\u8fed\u4ee3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6839\u636e\u200b\u5b9a\u4e49\u200b\u63a8\u7b97\u200b\u6590\u6ce2\u200b\u90a3\u200b\u5951\u6570\u200b</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int fib(int N){\n    int ret = 0, next = 1, tmp;\n    while (N)\n    {\n        tmp = ret;\n        ret = next;\n        next = ret + tmp;\n        N--;\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/53/","title":"53. \u200b\u6700\u5927\u200b\u5b50\u5e8f\u200b\u548c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u5206\u6cbb\u7b97\u6cd5","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/53/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b <code>nums</code>\u00a0\uff0c\u200b\u627e\u5230\u200b\u4e00\u4e2a\u200b\u5177\u6709\u200b\u6700\u5927\u200b\u548c\u200b\u7684\u200b\u8fde\u7eed\u200b\u5b50\u200b\u6570\u7ec4\u200b\uff08\u200b\u5b50\u200b\u6570\u7ec4\u200b\u6700\u5c11\u200b\u5305\u542b\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\uff09\uff0c\u200b\u8fd4\u56de\u200b\u5176\u200b\u6700\u5927\u200b\u548c\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [-2,1,-3,4,-1,2,1,-5,4]\n\u200b\u8f93\u51fa\u200b\uff1a6\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u8fde\u7eed\u200b\u5b50\u200b\u6570\u7ec4\u200b\u00a0[4,-1,2,1] \u200b\u7684\u200b\u548c\u200b\u6700\u5927\u200b\uff0c\u200b\u4e3a\u200b\u00a06 \u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1]\n\u200b\u8f93\u51fa\u200b\uff1a1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [0]\n\u200b\u8f93\u51fa\u200b\uff1a0\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [-1]\n\u200b\u8f93\u51fa\u200b\uff1a-1\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 5\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [-100000]\n\u200b\u8f93\u51fa\u200b\uff1a-100000\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 3 * 10^4</code></li> <li><code>-10^5 &lt;= nums[i] &lt;= 10^5</code></li> </ul> <p>\u200b\u8fdb\u9636\u200b\uff1a\u200b\u5982\u679c\u200b\u4f60\u200b\u5df2\u7ecf\u200b\u5b9e\u73b0\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b <code>O(n)</code> \u200b\u7684\u200b\u89e3\u6cd5\u200b\uff0c\u200b\u5c1d\u8bd5\u200b\u4f7f\u7528\u200b\u66f4\u4e3a\u200b\u7cbe\u5999\u200b\u7684\u200b \u200b\u5206\u6cbb\u200b\u6cd5\u200b \u200b\u6c42\u89e3\u200b\u3002</p> <p>Reference</p>","tags":["\u6570\u7ec4","\u5206\u6cbb\u7b97\u6cd5","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/53/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u4ece\u540e\u200b\u5411\u524d\u200b\u8fdb\u884c\u200b\u8003\u8651\u200b\uff0c\u200b\u8bbe\u200b\u6570\u7ec4\u200b\u4e2d\u5171\u200b\u6709\u200b\\(n\\)\u200b\u4e2a\u200b\u5143\u7d20\u200b\\(\\{a_n\\}\\)\uff0c\u200b\u5f53\u524d\u200b\u4f4d\u7f6e\u200b\\(i\\)\u200b\u4ece\u540e\u200b\u5411\u524d\u200b\u79fb\u52a8\u200b\uff0c\\(i\\)\u200b\u4f4d\u7f6e\u200b\u53ca\u200b\\(i\\)\u200b\u4f4d\u7f6e\u200b\u4e4b\u540e\u200b\u6784\u6210\u200b\u7684\u200b\u6700\u5927\u200b\u5b50\u200b\u5e8f\u5217\u200b\u6570\u503c\u200b\u4e3a\u200b\\(s_i\\)\u3002</p> <ol> <li>\u200b\u521d\u59cb\u200b\u60c5\u51b5\u200b\uff0c\\(s_n = 0\\)</li> <li>\u200b\u8ba8\u8bba\u200b\\(s_{i + 1}\\)\uff1a</li> <li>\u200b\u82e5\u200b\\(s_{i + 1} \\leq 0\\)\uff0c\u200b\u5219\u200b\\(s_{i} = a_{i}\\)</li> <li>\u200b\u82e5\u200b\\(s_{i + 1} &gt; 0\\)\uff0c\u200b\u5219\u200b\\(s_{i} = a_{i} + s_{i + 1}\\)</li> </ol> <p>\u200b\u7531\u6b64\u200b\u4e0d\u65ad\u200b\u8fdb\u884c\u200b\uff0c\u200b\u4e22\u5f03\u200b\u548c\u200b\u4e3a\u200b\u8d1f\u6570\u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217\u200b\uff0c\u200b\u6700\u7ec8\u200b\u5f97\u5230\u200b\u7684\u200b\\(s_0\\)\u200b\u5373\u200b\u4e3a\u200b\u6700\u5927\u200b\u5b50\u5e8f\u200b\u548c\u200b\u3002</p> <pre><code>#define MAX(x, y) (x &gt; y ? x : y)\nint maxSubArray(int* nums, int numsSize){\n    int ret = INT_MIN, value = 0;\n    for (int i = numsSize - 1; i &gt;= 0; i--)\n    {\n        value = MAX(value, 0) + nums[i];\n        ret = MAX(ret, value);\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u5206\u6cbb\u7b97\u6cd5","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/55/","title":"55. \u200b\u8df3\u8dc3\u200b\u6e38\u620f","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p> <p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\u00a0<code>nums</code> \uff0c\u200b\u4f60\u200b\u6700\u521d\u200b\u4f4d\u4e8e\u200b\u6570\u7ec4\u200b\u7684\u200b \u200b\u7b2c\u4e00\u4e2a\u200b\u4e0b\u6807\u200b \u3002</p> <p>\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u4ee3\u8868\u200b\u4f60\u200b\u5728\u200b\u8be5\u200b\u4f4d\u7f6e\u200b\u53ef\u4ee5\u200b\u8df3\u8dc3\u200b\u7684\u200b\u6700\u5927\u200b\u957f\u5ea6\u200b\u3002</p> <p>\u200b\u5224\u65ad\u200b\u4f60\u200b\u662f\u5426\u200b\u80fd\u591f\u200b\u5230\u8fbe\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u4e0b\u6807\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [2,3,1,1,4]\n\u200b\u8f93\u51fa\u200b\uff1atrue\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u53ef\u4ee5\u200b\u5148\u200b\u8df3\u200b 1 \u200b\u6b65\u200b\uff0c\u200b\u4ece\u200b\u4e0b\u6807\u200b 0 \u200b\u5230\u8fbe\u200b\u4e0b\u6807\u200b 1, \u200b\u7136\u540e\u200b\u518d\u200b\u4ece\u200b\u4e0b\u6807\u200b 1 \u200b\u8df3\u200b 3 \u200b\u6b65\u200b\u5230\u8fbe\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u4e0b\u6807\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [3,2,1,0,4]\n\u200b\u8f93\u51fa\u200b\uff1afalse\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u65e0\u8bba\u600e\u6837\u200b\uff0c\u200b\u603b\u4f1a\u200b\u5230\u8fbe\u200b\u4e0b\u6807\u200b\u4e3a\u200b 3 \u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002\u200b\u4f46\u200b\u8be5\u200b\u4e0b\u200b\u6807\u7684\u200b\u6700\u5927\u200b\u8df3\u8dc3\u200b\u957f\u5ea6\u200b\u662f\u200b 0 \uff0c \u200b\u6240\u4ee5\u200b\u6c38\u8fdc\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u5230\u8fbe\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u4e0b\u6807\u200b\u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 3 * 10^4</code></li> <li><code>0 &lt;= nums[i] &lt;= 10^5</code></li> </ul> <p>Reference</p>","tags":["\u8d2a\u5fc3\u7b97\u6cd5","\u6570\u7ec4"]},{"location":"coding/leetcode/55/#_1","title":"\u9898\u89e3","text":"<p>\u200b\u53cd\u5e8f\u200b\u904d\u5386\u200b\u6570\u7ec4\u200b\uff0c\u200b\u4ece\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u4f4d\u7f6e\u200b\u5f00\u59cb\u200b\uff0c\u200b\u68c0\u67e5\u200b\u5404\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u53ef\u200b\u8df3\u200b\u8ddd\u79bb\u200b\uff0c\u200b\u4ece\u800c\u200b\u6784\u9020\u200b\u4e00\u4e2a\u200b\u53ef\u4ee5\u200b\u8df3\u5230\u200b\u4e2d\u70b9\u200b\u7684\u200b\u4e0b\u6807\u200b\u533a\u95f4\u200b\u3002\u200b\u904d\u5386\u200b\u5b8c\u6210\u200b\u540e\u200b\u68c0\u67e5\u200b\u8d77\u70b9\u200b\u662f\u5426\u200b\u5728\u200b\u8be5\u200b\u4e0b\u6807\u200b\u533a\u95f4\u200b\u4e2d\u200b\u3002</p> <pre><code>bool canJump(int* nums, int numsSize) {\n\n    int distance = numsSize - 1, i = 0;\n    for (i = numsSize - 1; i &gt;= 0; i--)\n        if (i + nums[i] &gt;= distance)\n            distance = i;\n    return distance == 0;\n}\n</code></pre>","tags":["\u8d2a\u5fc3\u7b97\u6cd5","\u6570\u7ec4"]},{"location":"coding/leetcode/56/","title":"56. \u200b\u5408\u5e76\u200b\u533a\u95f4","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u6392\u5e8f"]},{"location":"coding/leetcode/56/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u4ee5\u200b\u6570\u7ec4\u200b <code>intervals</code> \u200b\u8868\u793a\u200b\u82e5\u5e72\u4e2a\u200b\u533a\u95f4\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u5176\u4e2d\u200b\u5355\u4e2a\u200b\u533a\u95f4\u200b\u4e3a\u200b <code>intervals[i] = [start_i, end_i]</code> \u3002\u200b\u8bf7\u200b\u4f60\u200b\u5408\u5e76\u200b\u6240\u6709\u200b\u91cd\u53e0\u200b\u7684\u200b\u533a\u95f4\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u4e0d\u200b\u91cd\u53e0\u200b\u7684\u200b\u533a\u95f4\u200b\u6570\u7ec4\u200b\uff0c\u200b\u8be5\u200b\u6570\u7ec4\u200b\u9700\u200b\u6070\u597d\u200b\u8986\u76d6\u200b\u8f93\u5165\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u533a\u95f4\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aintervals = [[1,3],[2,6],[8,10],[15,18]]\n\u200b\u8f93\u51fa\u200b\uff1a[[1,6],[8,10],[15,18]]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u533a\u95f4\u200b [1,3] \u200b\u548c\u200b [2,6] \u200b\u91cd\u53e0\u200b, \u200b\u5c06\u200b\u5b83\u4eec\u200b\u5408\u5e76\u200b\u4e3a\u200b [1,6].\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aintervals = [[1,4],[4,5]]\n\u200b\u8f93\u51fa\u200b\uff1a[[1,5]]\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u533a\u95f4\u200b [1,4] \u200b\u548c\u200b [4,5] \u200b\u53ef\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u91cd\u53e0\u200b\u533a\u95f4\u200b\u3002\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= intervals.length &lt;= 10^4</code></li> <li><code>intervals[i].length == 2</code></li> <li><code>0 &lt;= starti &lt;= endi &lt;= 10^4</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u6392\u5e8f"]},{"location":"coding/leetcode/56/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u9996\u5148\u200b\u5bf9\u200b\u533a\u95f4\u200b\u7684\u200b\u8d77\u70b9\u200b\u6309\u7167\u200b\u5347\u5e8f\u200b\u6392\u5e8f\u200b\uff0c\u200b\u4e4b\u540e\u200b\u904d\u5386\u200b\u533a\u95f4\u200b\u5217\u8868\u200b\uff0c\u200b\u4e0d\u65ad\u200b\u68c0\u67e5\u200b\u5df2\u200b\u5408\u5e76\u200b\u533a\u95f4\u200b\u4e0e\u200b\u5f53\u524d\u200b\u533a\u95f4\u200b\u662f\u5426\u200b\u5b58\u5728\u200b\u91cd\u53e0\u200b\uff0c\u200b\u5982\u679c\u200b\u5b58\u5728\u200b\u91cd\u53e0\u200b\uff0c\u200b\u5219\u200b\u5c06\u200b\u4e24\u200b\u533a\u95f4\u200b\u5408\u5e76\u200b\uff0c\u200b\u5426\u5219\u200b\u5c06\u200b\u5df2\u200b\u5408\u5e76\u200b\u533a\u95f4\u200b\u52a0\u5165\u200b\u7ed3\u679c\u200b\u96c6\u4e2d\u200b\uff0c\u200b\u5f53\u524d\u200b\u533a\u95f4\u200b\u8bbe\u4e3a\u200b\u5df2\u200b\u5408\u5e76\u200b\u533a\u95f4\u200b\u3002</p> <pre><code>/**\n * Return an array of arrays of size *returnSize.\n * The sizes of the arrays are returned as *returnColumnSizes array.\n * Note: Both returned array and *columnSizes array must be malloced, assume caller calls free().\n */\n#define MAX(x, y) (x &gt; y ? x : y)\nint comp(const void *x, const void *y)\n{\n    return **(int **)x - **(int **)y;\n}\nint** merge(int** intervals, int intervalsSize, int* intervalsColSize, int* returnSize, int** returnColumnSizes){\n    if (intervalsSize == 0)\n    {\n        *returnSize = 0;\n        return NULL;\n    }\n    qsort(intervals, intervalsSize, sizeof(int *), comp);\n    int **ret = (int **)malloc(sizeof(int *) * intervalsSize), i = 0;\n    ret[0] = (int *)memcpy(malloc(sizeof(int) * 2), intervals[0], sizeof(int) * 2);\n    *returnSize = 0;\n    for (i = 1; i &lt; intervalsSize; i++)\n    {\n        if (ret[*returnSize][1] &gt;= intervals[i][0])\n            ret[*returnSize][1] = MAX(intervals[i][1], ret[*returnSize][1]);\n        else\n            ret[++(*returnSize)] = (int *)memcpy(malloc(sizeof(int) * 2), intervals[i], sizeof(int) * 2);\n    }\n    (*returnSize)++;\n    *returnColumnSizes = (int *)malloc(sizeof(int) * *returnSize);\n    for (i = 0; i &lt; *returnSize; i++)\n        (*returnColumnSizes)[i] = 2;\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u6392\u5e8f"]},{"location":"coding/leetcode/59/","title":"59. \u200b\u87ba\u65cb\u200b\u77e9\u9635\u200b II","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/59/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u6b63\u6574\u6570\u200b\u00a0<code>n</code> \uff0c\u200b\u751f\u6210\u200b\u4e00\u4e2a\u200b\u5305\u542b\u200b <code>1</code> \u200b\u5230\u200b\u00a0<code>n^2</code>\u00a0\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\uff0c\u200b\u4e14\u200b\u5143\u7d20\u200b\u6309\u200b\u987a\u65f6\u9488\u200b\u987a\u5e8f\u200b\u87ba\u65cb\u200b\u6392\u5217\u200b\u7684\u200b\u00a0<code>n x n</code> \u200b\u6b63\u65b9\u5f62\u200b\u77e9\u9635\u200b <code>matrix</code> \u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1an = 3\n\u200b\u8f93\u51fa\u200b\uff1a[[1,2,3],[8,9,4],[7,6,5]]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1an = 1\n\u200b\u8f93\u51fa\u200b\uff1a[[1]]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= n &lt;= 20</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/59/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u6309\u200b\u987a\u5e8f\u200b\u586b\u5145\u200b\u77e9\u9635\u200b\uff0c\u200b\u5f53\u200b\u8d85\u51fa\u200b\u77e9\u9635\u200b\u6216\u200b\u9047\u5230\u200b\u5df2\u200b\u586b\u5145\u200b\u5143\u7d20\u200b\u65f6\u200b\u8f6c\u5411\u200b\u3002</p> <pre><code>/**\n * Return an array of arrays of size *returnSize.\n * The sizes of the arrays are returned as *returnColumnSizes array.\n * Note: Both returned array and *columnSizes array must be malloced, assume caller calls free().\n */\nvoid change(int direction, int *i, int *j, int incre)\n{\n    switch(direction)\n    {\n        case 0:\n            *j += incre;\n            break;\n        case 1:\n            *i += incre;\n            break;\n        case 2:\n            *j -= incre;\n            break;\n        case 3:\n            *i -= incre;\n            break;\n    }\n}\nint** generateMatrix(int n, int* returnSize, int** returnColumnSizes){\n    *returnSize = n;\n    *returnColumnSizes = (int *)malloc(sizeof(int) * n);\n\n    int i = 0, j = 0, m = 1, direction = 0, **ret = (int **)malloc(sizeof(int *) * n);\n    for (i = 0; i &lt; n; i++)\n    {\n        (*returnColumnSizes)[i] = n;\n        ret[i] = (int *)malloc(sizeof(int) * n);\n        memset(ret[i], 0, sizeof(int) * n);\n    }\n\n    i = 0;\n    while (m &lt; n * n)\n    {\n        ret[i][j] = m;\n        change(direction, &amp;i, &amp;j, 1);\n        m++;\n        if (i &lt; 0 || j &lt; 0 || i &gt;= n || j &gt;= n || ret[i][j])\n        {\n            change(direction, &amp;i, &amp;j, -1);\n            direction = (direction + 1) &amp; 3;\n            m--;\n        }\n    }\n    ret[i][j] = m;\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/61/","title":"61. \u200b\u65cb\u8f6c\u200b\u94fe\u8868","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/61/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u94fe\u8868\u200b\u7684\u200b\u5934\u200b\u8282\u70b9\u200b <code>head</code> \uff0c\u200b\u65cb\u8f6c\u200b\u94fe\u8868\u200b\uff0c\u200b\u5c06\u200b\u94fe\u8868\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u00a0<code>k</code>\u00a0\u200b\u4e2a\u200b\u4f4d\u7f6e\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [1,2,3,4,5], k = 2\n\u200b\u8f93\u51fa\u200b\uff1a[4,5,1,2,3]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [0,1,2], k = 4\n\u200b\u8f93\u51fa\u200b\uff1a[2,0,1]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u94fe\u8868\u200b\u4e2d\u200b\u8282\u70b9\u200b\u7684\u200b\u6570\u76ee\u200b\u5728\u200b\u8303\u56f4\u200b <code>[0, 500]</code> \u200b\u5185\u200b</li> <li><code>-100 &lt;= Node.val &lt;= 100</code></li> <li><code>0 &lt;= k &lt;= 2 * 10^9</code></li> </ul> <p>Reference</p>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/61/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5148\u200b\u5c06\u200b\u94fe\u8868\u200b\u95ed\u73af\u200b\uff0c\u200b\u53f3\u79fb\u200b k \u200b\u4e2a\u200b\u8282\u70b9\u200b\u540e\u200b\u4ee5\u200b\u8be5\u200b\u8282\u70b9\u200b\u4f5c\u4e3a\u200b\u5934\u200b\u8282\u70b9\u200b\uff0c\u200b\u5c06\u200b\u94fe\u8868\u200b\u7684\u200b\u73af\u200b\u6253\u5f00\u200b\u3002</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\n\n\nstruct ListNode* rotateRight(struct ListNode* head, int k) {\n    if (!head)\n        return NULL;\n    struct ListNode *cur = head, *ret = NULL;\n    int len = 1;\n    while (cur-&gt;next != NULL)\n    {\n        cur = cur-&gt;next;\n        len++;\n    }\n    cur-&gt;next = head;\n    cur = head;\n    k %= len;\n    for (int i = k + 1; i &lt; len; i++)\n        cur = cur-&gt;next;\n    ret = cur-&gt;next;\n    cur-&gt;next = NULL;\n    return ret;\n}\n</code></pre>","tags":["\u94fe\u8868","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/62/","title":"62. \u200b\u4e0d\u540c\u200b\u8def\u5f84","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/62/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u4e00\u4e2a\u200b\u673a\u5668\u4eba\u200b\u4f4d\u4e8e\u200b\u4e00\u4e2a\u200b <code>m x n</code>\u00a0\u200b\u7f51\u683c\u200b\u7684\u200b\u5de6\u4e0a\u89d2\u200b \uff08\u200b\u8d77\u59cb\u200b\u70b9\u200b\u5728\u200b\u4e0b\u56fe\u200b\u4e2d\u200b\u6807\u8bb0\u200b\u4e3a\u200b \u201cStart\u201d \uff09\u3002</p> <p>\u200b\u673a\u5668\u4eba\u200b\u6bcf\u6b21\u200b\u53ea\u80fd\u200b\u5411\u4e0b\u200b\u6216\u8005\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u4e00\u6b65\u200b\u3002\u200b\u673a\u5668\u4eba\u200b\u8bd5\u56fe\u200b\u8fbe\u5230\u200b\u7f51\u683c\u200b\u7684\u200b\u53f3\u4e0b\u89d2\u200b\uff08\u200b\u5728\u200b\u4e0b\u56fe\u200b\u4e2d\u200b\u6807\u8bb0\u200b\u4e3a\u200b \u201cFinish\u201d \uff09\u3002</p> <p>\u200b\u95ee\u200b\u603b\u5171\u200b\u6709\u200b\u591a\u5c11\u200b\u6761\u200b\u4e0d\u540c\u200b\u7684\u200b\u8def\u5f84\u200b\uff1f</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1am = 3, n = 7\n\u200b\u8f93\u51fa\u200b\uff1a28\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1am = 3, n = 2\n\u200b\u8f93\u51fa\u200b\uff1a3\n\u200b\u89e3\u91ca\u200b\uff1a\n\u200b\u4ece\u200b\u5de6\u4e0a\u89d2\u200b\u5f00\u59cb\u200b\uff0c\u200b\u603b\u5171\u200b\u6709\u200b 3 \u200b\u6761\u200b\u8def\u5f84\u200b\u53ef\u4ee5\u200b\u5230\u8fbe\u200b\u53f3\u4e0b\u89d2\u200b\u3002\n1. \u200b\u5411\u200b\u53f3\u200b -&gt; \u200b\u5411\u4e0b\u200b -&gt; \u200b\u5411\u4e0b\u200b\n2. \u200b\u5411\u4e0b\u200b -&gt; \u200b\u5411\u4e0b\u200b -&gt; \u200b\u5411\u200b\u53f3\u200b\n3. \u200b\u5411\u4e0b\u200b -&gt; \u200b\u5411\u200b\u53f3\u200b -&gt; \u200b\u5411\u4e0b\u200b\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1am = 7, n = 3\n\u200b\u8f93\u51fa\u200b\uff1a28\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1am = 3, n = 3\n\u200b\u8f93\u51fa\u200b\uff1a6\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= m, n &lt;= 100</code></li> <li>\u200b\u9898\u76ee\u200b\u6570\u636e\u200b\u4fdd\u8bc1\u200b\u7b54\u6848\u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b <code>2 * 10^9</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/62/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4ece\u200b\u6700\u540e\u200b\u4e00\u884c\u200b\u5f00\u59cb\u200b\u8003\u8651\u200b\uff0c\u200b\u8981\u200b\u5230\u8fbe\u200b\u53f3\u4e0b\u89d2\u200b\u7684\u200b\u65b9\u683c\u200b\uff0c\u200b\u6709\u200b\u4e24\u79cd\u200b\u65b9\u6cd5\u200b\uff1a</p> <ol> <li>\u200b\u4ece\u200b\u659c\u200b\u4e0a\u65b9\u200b\u7684\u200b\u65b9\u683c\u200b\u5f00\u59cb\u200b\uff0c\u200b\u5148\u200b\u5411\u4e0b\u200b\u540e\u200b\u5411\u200b\u53f3\u200b</li> <li>\u200b\u4ece\u200b\u659c\u200b\u4e0a\u65b9\u200b\u7684\u200b\u65b9\u683c\u200b\u5f00\u59cb\u200b\uff0c\u200b\u5148\u5411\u200b\u53f3\u540e\u200b\u5411\u4e0b\u200b</li> </ol> <p>\u200b\u7531\u6b64\u200b\u5373\u53ef\u200b\u5f97\u51fa\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\u65b9\u7a0b\u200b\u3002</p> <pre><code>int uniquePaths(int m, int n){\n    if (m &lt; n)\n        return uniquePaths(n, m);\n    int *dp = (int *)malloc(sizeof(int) * m), i = 0, j = 0;\n    for (i = 0; i &lt; m; i++)\n        dp[i] = 1;\n    for (i = 1; i &lt; n; i++)\n        for (j = 1; j &lt; m; j++)\n            dp[j] += dp[j - 1];\n    return dp[m - 1];\n}\n</code></pre> <p>\u200b\u6216\u8005\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6392\u5217\u7ec4\u5408\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u957f\u4e3a\u200b\\(m\\)\uff0c\u200b\u5bbd\u4e3a\u200b\\(n\\)\u200b\u7684\u200b\u65b9\u683c\u200b\uff0c\u200b\u9700\u8981\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\\(m - 1\\)\u200b\u6b21\u200b\uff0c\u200b\u5411\u4e0b\u200b\u79fb\u52a8\u200b\\(n - 1\\)\u200b\u6b21\u200b\uff0c\u200b\u5171\u200b\\(m + n - 2\\)\u200b\u6b21\u200b\u3002</p> <p>\u200b\u56e0\u6b64\u200b\u603b\u200b\u7684\u200b\u8def\u5f84\u200b\u4e2a\u6570\u200b\u4e3a\u200b\\(m + n - 2\\)\u200b\u4e2a\u200b\u9009\u62e9\u200b\u4e2d\u4e2d\u200b\u9009\u62e9\u200b\\(m - 1\\)\u200b\u6b21\u200b\u5411\u4e0b\u200b\u79fb\u52a8\u200b\u7684\u200b\u65b9\u6848\u200b\u603b\u6570\u200b\uff0c\u200b\u8be5\u200b\u65b9\u6848\u200b\u603b\u6570\u200b\u4e3a\u200b\uff1a</p> \\[ \\frac{(m + n - 2)!}{(m - 1)!(n - 1)!} \\]","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/64/","title":"64. \u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p> <p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5305\u542b\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u7684\u200b <code>m\u00a0x\u00a0n</code>\u00a0\u200b\u7f51\u683c\u200b\u00a0<code>grid</code> \uff0c\u200b\u8bf7\u200b\u627e\u51fa\u200b\u4e00\u6761\u200b\u4ece\u200b\u5de6\u4e0a\u89d2\u200b\u5230\u200b\u53f3\u4e0b\u89d2\u200b\u7684\u200b\u8def\u5f84\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u8def\u5f84\u200b\u4e0a\u200b\u7684\u200b\u6570\u5b57\u200b\u603b\u548c\u200b\u4e3a\u200b\u6700\u5c0f\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b\uff1a\u200b\u6bcf\u6b21\u200b\u53ea\u80fd\u200b\u5411\u4e0b\u200b\u6216\u8005\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u4e00\u6b65\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1agrid = [[1,3,1],[1,5,1],[4,2,1]]\n\u200b\u8f93\u51fa\u200b\uff1a7\n\u200b\u89e3\u91ca\u200b\uff1a\u200b\u56e0\u4e3a\u200b\u8def\u5f84\u200b 1\u21923\u21921\u21921\u21921 \u200b\u7684\u200b\u603b\u548c\u200b\u6700\u5c0f\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1agrid = [[1,2,3],[4,5,6]]\n\u200b\u8f93\u51fa\u200b\uff1a12\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>m == grid.length</code></li> <li><code>n == grid[i].length</code></li> <li><code>1 &lt;= m, n &lt;= 200</code></li> <li><code>0 &lt;= grid[i][j] &lt;= 100</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/64/#_1","title":"\u9898\u89e3","text":"<p>\u200b\u8bbe\u200b\u7b2c\u200b\\(i\\)\u200b\u884c\u7b2c\u200b\\(j\\)\u200b\u5217\u200b\u7684\u200b\u5143\u7d20\u200b\\(a_{ij}\\)\u200b\u5230\u200b\u53f3\u4e0b\u89d2\u200b\u8def\u5f84\u200b\u4e0a\u200b\u7684\u200b\u6570\u5b57\u200b\u548c\u200b\u4e3a\u200b\\(f(i, j)\\)\uff0c\u200b\u7279\u522b\u200b\u5730\u200b\uff0c\u200b\u6709\u200b\\(f(m, n) = a_{mn}\\)\uff0c\u200b\u5219\u200b\uff1a</p> <p>\u200b\u5bf9\u4e8e\u200b\u6700\u540e\u200b\u4e00\u884c\u200b\u5143\u7d20\u200b\uff0c\u200b\u53ea\u80fd\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\uff0c\u200b\u6709\u200b\uff1a</p> \\[ f(m, j) = \\sum_{k=j} ^{n} a_{mk} \\] <p>\u200b\u540c\u7406\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6700\u540e\u200b\u4e00\u5217\u200b\u5143\u7d20\u200b\uff0c\u200b\u6709\u200b\uff1a</p> \\[ f(i, n) = \\sum_{k = i} ^ {m} a_{kn} \\] <p>\u200b\u8bbe\u200b\u7b2c\u200b\\(i\\)\u200b\u884c\u200b\u5230\u8fbe\u200b\u53f3\u4e0b\u89d2\u200b\u7684\u200b\u6700\u200b\u77ed\u200b\u8def\u5f84\u200b\u548c\u200b\uff08\u200b\u5373\u200b\\(f(i, j), \\forall j\\)\uff09\u200b\u4ee5\u53ca\u200b\u7b2c\u200b\\(i - 1\\)\u200b\u884c\u200b\u6700\u540e\u200b\u4e00\u5217\u200b\u5230\u8fbe\u200b\u53f3\u4e0b\u89d2\u200b\u7684\u200b\u6700\u200b\u77ed\u200b\u8def\u5f84\u200b\u548c\u200b\uff0c\u200b\u5373\u200b\\(f(i - 1, n)\\)\u200b\u5df2\u77e5\u200b\u3002\u200b\u7531\u4e8e\u200b\u6bcf\u200b\u4e00\u6b65\u200b\u53ea\u80fd\u200b\u5411\u200b\u53f3\u200b\u6216\u200b\u5411\u4e0b\u200b\u79fb\u52a8\u200b\uff0c\\(f(i - 1, j)\\)\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u5982\u4e0b\u200b\u65b9\u7a0b\u200b\u63a8\u5bfc\u200b\uff1a</p> \\[ f(i - 1, j - 1) = \\min\\{f(i - 1, j), f(i, j - 1)\\} + a_{(i - 1)(j - 1)} \\] <p>\u200b\u7531\u6b64\u200b\u5f97\u5230\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\u65b9\u7a0b\u200b\uff0c\u200b\u5373\u53ef\u200b\u4f7f\u7528\u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\u81ea\u200b\u5e95\u5411\u4e0a\u200b\u6c42\u89e3\u200b\u3002\u200b\u6c42\u89e3\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u53ea\u200b\u9700\u8981\u200b\u5b58\u50a8\u200b\u4e00\u884c\u200b\u7684\u200b\u6700\u200b\u77ed\u200b\u8def\u5f84\u200b\u548c\u200b\u5373\u53ef\u200b\u3002</p> <pre><code>#define MIN(x, y) ((x) &lt; (y) ? x : y)\nint minPathSum(int** grid, int gridSize, int* gridColSize){\n    if (gridSize == 0 || *gridColSize == 0)\n        return 0;\n    int *dp = (int *)memset(malloc(sizeof(int) * *gridColSize), 0, sizeof(int) * *gridColSize),\n        i = 0, j = 0;\n    for (i = 0; i &lt; *gridColSize; i++)\n        dp[i] = INT_MAX;\n    dp[0] = 0;\n    for (i = 0; i &lt; gridSize; i++)\n    {\n        dp[0] += grid[i][0];\n        for (j = 1; j &lt; *gridColSize; j++)\n            dp[j] = MIN(dp[j - 1], dp[j]) + grid[i][j];\n    }\n    return dp[*gridColSize - 1];\n}\n</code></pre>","tags":["\u6570\u7ec4","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/66/","title":"66. \u200b\u52a0\u4e00","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/66/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u7531\u200b\u6574\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u975e\u7a7a\u200b\u6570\u7ec4\u200b\u6240\u200b\u8868\u793a\u200b\u7684\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\uff0c\u200b\u5728\u200b\u8be5\u6570\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u52a0\u200b\u4e00\u200b\u3002</p> <p>\u200b\u6700\u9ad8\u200b\u4f4d\u200b\u6570\u5b57\u200b\u5b58\u653e\u200b\u5728\u200b\u6570\u7ec4\u200b\u7684\u200b\u9996\u4f4d\u200b\uff0c \u200b\u6570\u7ec4\u200b\u4e2d\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u53ea\u200b\u5b58\u50a8\u200b\u5355\u4e2a\u200b\u6570\u5b57\u200b\u3002</p> <p>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u9664\u4e86\u200b\u6574\u6570\u200b 0 \u200b\u4e4b\u5916\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u6574\u6570\u200b\u4e0d\u4f1a\u200b\u4ee5\u200b\u96f6\u200b\u5f00\u5934\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [1,2,3]\n\u200b\u8f93\u51fa\u200b: [1,2,4]\n\u200b\u89e3\u91ca\u200b: \u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u8868\u793a\u200b\u6570\u5b57\u200b 123\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: [4,3,2,1]\n\u200b\u8f93\u51fa\u200b: [4,3,2,2]\n\u200b\u89e3\u91ca\u200b: \u200b\u8f93\u5165\u200b\u6570\u7ec4\u200b\u8868\u793a\u200b\u6570\u5b57\u200b 4321\u3002\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/66/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4ece\u200b\u6700\u540e\u200b\u4e00\u4f4d\u200b\u5f00\u59cb\u200b\u52a0\u200b\uff0c\u200b\u5411\u524d\u200b\u5904\u7406\u200b\u8fdb\u4f4d\u200b\u3002\u200b\u6ce8\u610f\u200b9999+1\u200b\u7684\u200b\u7279\u6b8a\u200b\u60c5\u51b5\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Note: The returned array must be malloced, assume caller calls free().\n */\nint* plusOne(int* digits, int digitsSize, int* returnSize){\n    int *ret = (int *)malloc(sizeof(int) * (digitsSize + 1)), i = digitsSize, add = 1;\n    *ret = 0;\n    memcpy(ret + 1, digits, sizeof(int) * digitsSize);\n    for (i = digitsSize; i &gt;= 0 &amp;&amp; add; i--)\n    {\n        if (add = ret[i] == 9)\n            ret[i] = 0;\n        else\n            ret[i]++;\n    }\n    *returnSize = (i + 1) ? digitsSize : digitsSize + 1;\n    return ret + ((i + 1) ? 1 : 0);\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/69/","title":"69. x \u200b\u7684\u200b\u5e73\u65b9\u6839","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u5b66","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/69/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5b9e\u73b0\u200b\u00a0<code>int sqrt(int x)</code>\u00a0\u200b\u51fd\u6570\u200b\u3002</p> <p>\u200b\u8ba1\u7b97\u200b\u5e76\u200b\u8fd4\u56de\u200b\u00a0<code>x</code>\u00a0\u200b\u7684\u200b\u5e73\u65b9\u6839\u200b\uff0c\u200b\u5176\u4e2d\u200b\u00a0<code>x</code> \u200b\u662f\u975e\u200b\u8d1f\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u7531\u4e8e\u200b\u8fd4\u56de\u200b\u7c7b\u578b\u200b\u662f\u200b\u6574\u6570\u200b\uff0c\u200b\u7ed3\u679c\u200b\u53ea\u200b\u4fdd\u7559\u200b\u6574\u6570\u200b\u7684\u200b\u90e8\u5206\u200b\uff0c\u200b\u5c0f\u6570\u200b\u90e8\u5206\u200b\u5c06\u200b\u88ab\u200b\u820d\u53bb\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1:</p> <pre><code>\u200b\u8f93\u5165\u200b: 4\n\u200b\u8f93\u51fa\u200b: 2\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2:</p> <pre><code>\u200b\u8f93\u5165\u200b: 8\n\u200b\u8f93\u51fa\u200b: 2\n\u200b\u8bf4\u660e\u200b: 8 \u200b\u7684\u200b\u5e73\u65b9\u6839\u200b\u662f\u200b 2.82842..., \n\u00a0    \u200b\u7531\u4e8e\u200b\u8fd4\u56de\u200b\u7c7b\u578b\u200b\u662f\u200b\u6574\u6570\u200b\uff0c\u200b\u5c0f\u6570\u200b\u90e8\u5206\u200b\u5c06\u200b\u88ab\u200b\u820d\u53bb\u200b\u3002\n</code></pre> <p>Reference</p>","tags":["\u6570\u5b66","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/69/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u5373\u53ef\u200b\uff0c\u200b\u5173\u4e8e\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7684\u200b\u8ba8\u8bba\u200b\uff0c\u200b\u8bf7\u200b\u53c2\u89c1\u200b35. \u200b\u641c\u7d22\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b</p> <pre><code>unsigned int binsearch(unsigned int target, unsigned int lo, unsigned int hi)\n{\n    int mid;\n    while (lo &lt; hi)\n    {\n        mid = (lo + hi) &gt;&gt; 1;\n        if (mid * mid &gt; target)\n            hi = mid;\n        else\n            lo = mid + 1;\n    }\n    return lo - 1;\n}\nint mySqrt(int x){\n    return binsearch(x, 0, 46341);\n}\n</code></pre>","tags":["\u6570\u5b66","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/7/","title":"7. \u200b\u6574\u6570\u200b\u53cd\u8f6c","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/7/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u51fa\u200b\u4e00\u4e2a\u200b 32 \u200b\u4f4d\u200b\u7684\u200b\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\uff0c\u200b\u4f60\u200b\u9700\u8981\u200b\u5c06\u200b\u8fd9\u4e2a\u200b\u6574\u6570\u200b\u4e2d\u200b\u6bcf\u4f4d\u200b\u4e0a\u200b\u7684\u200b\u6570\u5b57\u200b\u8fdb\u884c\u200b\u53cd\u8f6c\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 123\n\u200b\u8f93\u51fa\u200b: 321\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: -123\n\u200b\u8f93\u51fa\u200b: -321\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 120\n\u200b\u8f93\u51fa\u200b: 21\n</code></pre> <p>\u200b\u6ce8\u610f\u200b\uff1a</p> <p>\u200b\u5047\u8bbe\u200b\u6211\u4eec\u200b\u7684\u200b\u73af\u5883\u200b\u53ea\u80fd\u200b\u5b58\u50a8\u200b\u5f97\u200b\u4e0b\u200b 32 \u200b\u4f4d\u200b\u7684\u200b\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\uff0c\u200b\u5219\u200b\u5176\u200b\u6570\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b \\([\u22122^{31},\u00a0 2^{31}\u00a0\u2212 1]\\) \u3002\u200b\u8bf7\u200b\u6839\u636e\u200b\u8fd9\u4e2a\u200b\u5047\u8bbe\u200b\uff0c\u200b\u5982\u679c\u200b\u53cd\u8f6c\u200b\u540e\u200b\u6574\u6570\u200b\u6ea2\u51fa\u200b\u90a3\u4e48\u200b\u5c31\u200b\u8fd4\u56de\u200b 0\u3002</p> <p>Reference</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/7/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6bcf\u200b\u4e00\u6b21\u200b\u4ece\u200b\u53f3\u4fa7\u200b\u63d0\u53d6\u200b\u51fa\u200b\u6700\u540e\u200b\u4e00\u4f4d\u200b\u52a0\u200b\u5230\u200b\u7ed3\u679c\u200b\u7684\u200b\u53f3\u4fa7\u200b\u3002\u200b\u6ce8\u610f\u200b\u8d1f\u6570\u200b<code>-2147483648</code>\u200b\u5728\u200b 32 \u200b\u4f4d\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\u73af\u5883\u200b\u4e0b\u200b\u4e0d\u80fd\u200b\u53d6\u8d1f\u200b\uff0c\u200b\u5c06\u200b\u4e2d\u95f4\u200b\u53d8\u91cf\u200b\u58f0\u660e\u200b\u4e3a\u200b<code>long</code>\u200b\u5373\u53ef\u200b\u89e3\u51b3\u200b\u3002</p> <pre><code>int reverse(int x){\n    long ret = 0, testcase = x;\n    testcase = testcase &gt; 0 ? testcase : -testcase;\n    while(testcase)\n    {\n        ret *= 10;\n        ret += testcase % 10;\n        testcase /= 10;\n    }\n    return ((ret &gt;&gt; 31) ? 0 : ret) * (x &gt; 0 ? 1 : -1);\n}\n</code></pre>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/70/","title":"70. \u200b\u722c\u697c\u68af","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/70/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5047\u8bbe\u200b\u4f60\u200b\u6b63\u5728\u200b\u722c\u697c\u68af\u200b\u3002\u200b\u9700\u8981\u200b n\u00a0\u200b\u9636\u200b\u4f60\u200b\u624d\u80fd\u200b\u5230\u8fbe\u200b\u697c\u9876\u200b\u3002</p> <p>\u200b\u6bcf\u6b21\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u722c\u200b 1 \u200b\u6216\u200b 2 \u200b\u4e2a\u200b\u53f0\u9636\u200b\u3002\u200b\u4f60\u200b\u6709\u200b\u591a\u5c11\u200b\u79cd\u200b\u4e0d\u540c\u200b\u7684\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u722c\u200b\u5230\u200b\u697c\u9876\u200b\u5462\u200b\uff1f</p> <p>\u200b\u6ce8\u610f\u200b\uff1a\u200b\u7ed9\u5b9a\u200b n \u200b\u662f\u200b\u4e00\u4e2a\u200b\u6b63\u6574\u6570\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a 2\n\u200b\u8f93\u51fa\u200b\uff1a 2\n\u200b\u89e3\u91ca\u200b\uff1a \u200b\u6709\u200b\u4e24\u79cd\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u722c\u200b\u5230\u200b\u697c\u9876\u200b\u3002\n1.  1 \u200b\u9636\u200b + 1 \u200b\u9636\u200b\n2.  2 \u200b\u9636\u200b\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a 3\n\u200b\u8f93\u51fa\u200b\uff1a 3\n\u200b\u89e3\u91ca\u200b\uff1a \u200b\u6709\u200b\u4e09\u79cd\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u722c\u200b\u5230\u200b\u697c\u9876\u200b\u3002\n1.  1 \u200b\u9636\u200b + 1 \u200b\u9636\u200b + 1 \u200b\u9636\u200b\n2.  1 \u200b\u9636\u200b + 2 \u200b\u9636\u200b\n3.  2 \u200b\u9636\u200b + 1 \u200b\u9636\u200b\n</code></pre> <p>Reference</p>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/70/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u4f7f\u7528\u200b\u4e00\u4e2a\u200b\u6570\u7ec4\u200b\u5b58\u50a8\u200b\u4e34\u65f6\u200b\u7ed3\u679c\u200b\uff0c\u200b\u9012\u5f52\u8ba1\u7b97\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int climbStairs_cache(int n, int **cache)\n{\n    if (n &lt;= 2)\n        return n;\n    if (*cache == NULL)\n        *cache = memset(malloc(sizeof(int) * n), 0, sizeof(int) * n);\n    if (!(*cache)[n - 1])\n        (*cache)[n - 1] = climbStairs_cache(n - 1, cache) + climbStairs_cache(n - 2, cache);\n    return (*cache)[n - 1];\n}\n\nint climbStairs(int n){\n    int *cache = NULL;\n    return climbStairs_cache(n, &amp;cache);\n}\n</code></pre>","tags":["\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/73/","title":"73. \u200b\u77e9\u9635\u200b\u7f6e\u200b\u96f6","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/73/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u00a0<code>m x n</code> \u200b\u7684\u200b\u77e9\u9635\u200b\uff0c\u200b\u5982\u679c\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u4e3a\u200b <code>0</code> \uff0c\u200b\u5219\u200b\u5c06\u200b\u5176\u200b\u6240\u5728\u200b\u884c\u548c\u5217\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u90fd\u200b\u8bbe\u4e3a\u200b <code>0</code> \u3002\u200b\u8bf7\u200b\u4f7f\u7528\u200b \u200b\u539f\u5730\u200b \u200b\u7b97\u6cd5\u200b\u3002</p> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <ul> <li>\u200b\u4e00\u4e2a\u200b\u76f4\u89c2\u200b\u7684\u200b\u89e3\u51b3\u65b9\u6848\u200b\u662f\u200b\u4f7f\u7528\u200b \u00a0\\(O(mn)\\)\u00a0\u200b\u7684\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u4f46\u200b\u8fd9\u200b\u5e76\u200b\u4e0d\u662f\u200b\u4e00\u4e2a\u200b\u597d\u200b\u7684\u200b\u89e3\u51b3\u65b9\u6848\u200b\u3002</li> <li>\u200b\u4e00\u4e2a\u200b\u7b80\u5355\u200b\u7684\u200b\u6539\u8fdb\u200b\u65b9\u6848\u200b\u662f\u200b\u4f7f\u7528\u200b \\(O(m\u00a0+\u00a0n)\\) \u200b\u7684\u200b\u989d\u5916\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u4f46\u200b\u8fd9\u200b\u4ecd\u7136\u200b\u4e0d\u662f\u200b\u6700\u597d\u200b\u7684\u200b\u89e3\u51b3\u65b9\u6848\u200b\u3002</li> <li>\u200b\u4f60\u200b\u80fd\u200b\u60f3\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u4ec5\u200b\u4f7f\u7528\u200b\u5e38\u91cf\u200b\u7a7a\u95f4\u200b\u7684\u200b\u89e3\u51b3\u65b9\u6848\u200b\u5417\u200b\uff1f</li> </ul> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1amatrix = [[1,1,1],[1,0,1],[1,1,1]]\n\u200b\u8f93\u51fa\u200b\uff1a[[1,0,1],[0,0,0],[1,0,1]]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1amatrix = [[0,1,2,0],[3,4,5,2],[1,3,1,5]]\n\u200b\u8f93\u51fa\u200b\uff1a[[0,0,0,0],[0,4,5,0],[0,3,1,0]]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>m == matrix.length</code></li> <li><code>n == matrix[0].length</code></li> <li><code>1 &lt;= m, n &lt;= 200</code></li> <li><code>-2^31 &lt;= matrix[i][j] &lt;= 2^31 - 1</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/73/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\\(m\\)\u200b\u4e2a\u200b\u6807\u8bb0\u53d8\u91cf\u200b\u6807\u8bb0\u200b\u884c\u4e2d\u200b\u662f\u5426\u200b\u51fa\u73b0\u200b0\uff0c\u200b\u7528\u200b\\(n\\)\u200b\u4e2a\u200b\u6807\u8bb0\u53d8\u91cf\u200b\u6807\u8bb0\u200b\u5217\u4e2d\u200b\u662f\u5426\u200b\u51fa\u73b0\u200b0\u3002</p> <pre><code>void setZeroes(int** matrix, int matrixSize, int* matrixColSize){\n    int *rows = memset(malloc(sizeof(int) * matrixSize), 0, sizeof(int) * matrixSize),\n        *cols = memset(malloc(sizeof(int) * *matrixColSize), 0, sizeof(int) * *matrixColSize),\n        i = 0, j = 0;\n    for (i = 0; i &lt; matrixSize; i++)\n    {\n        for (j = 0; j &lt; matrixColSize[i]; j++)\n        {\n            if (matrix[i][j] == 0)\n            {\n                rows[i] = 1;\n                cols[j] = 1;\n            }\n        }\n    }\n    for (i = 0; i &lt; matrixSize; i++)\n        for (j = 0; j &lt; matrixColSize[i]; j++)\n            if (rows[i] || cols[j])\n                matrix[i][j] = 0;\n}\n</code></pre>","tags":["\u6570\u7ec4"]},{"location":"coding/leetcode/74/","title":"74. \u200b\u641c\u7d22\u200b\u4e8c\u7ef4\u200b\u77e9\u9635","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/74/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7f16\u5199\u200b\u4e00\u4e2a\u200b\u9ad8\u6548\u200b\u7684\u200b\u7b97\u6cd5\u200b\u6765\u200b\u5224\u65ad\u200b\u00a0<code>m x n</code>\u00a0\u200b\u77e9\u9635\u200b\u4e2d\u200b\uff0c\u200b\u662f\u5426\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u503c\u200b\u3002\u200b\u8be5\u200b\u77e9\u9635\u200b\u5177\u6709\u200b\u5982\u4e0b\u200b\u7279\u6027\u200b\uff1a</p> <ul> <li>\u200b\u6bcf\u884c\u200b\u4e2d\u200b\u7684\u200b\u6574\u6570\u200b\u4ece\u5de6\u5230\u53f3\u200b\u6309\u200b\u5347\u5e8f\u200b\u6392\u5217\u200b\u3002</li> <li>\u200b\u6bcf\u884c\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u6574\u6570\u200b\u5927\u4e8e\u200b\u524d\u200b\u4e00\u884c\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1amatrix = [[1,3,5,7],[10,11,16,20],[23,30,34,60]], target = 3\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1amatrix = [[1,3,5,7],[10,11,16,20],[23,30,34,60]], target = 13\n\u200b\u8f93\u51fa\u200b\uff1afalse\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>m == matrix.length</code></li> <li><code>n == matrix[i].length</code></li> <li><code>1 &lt;= m, n &lt;= 100</code></li> <li><code>-10^4 &lt;= matrix[i][j], target &lt;= 10^4</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/74/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u6839\u636e\u200b\u77e9\u9635\u200b\u7684\u200b\u7279\u6027\u200b\u53ef\u77e5\u200b\uff0c\u200b\u5047\u8bbe\u200b\u77e9\u9635\u200b\u4e3a\u200b\\(\\{a_{ij}\\}_{m\\times n}\\)\uff0c\u200b\u76ee\u6807\u503c\u200b\\(t\\)\u200b\u4f4d\u4e8e\u200b\u5750\u6807\u200b\\((i^*, j^*)\\)\u200b\u5904\u200b\uff0c\u200b\u6709\u200b\uff1a</p> <ul> <li>\u200b\u5bf9\u4e8e\u200b\u4efb\u4f55\u200b\\(i, j\\)\uff0c\u200b\u82e5\u200b\\(i &lt; i^*\\)\uff0c\u200b\u5219\u200b\\(a_{ij} &lt; t\\)\uff0c\u200b\u5f53\u200b\\(i = i^*\\)\u200b\u65f6\u200b\uff0c\u200b\u82e5\u200b\\(j &lt; j^*\\)\uff0c\u200b\u5219\u200b\\(a_{ij} &lt; t\\)</li> <li>\u200b\u5bf9\u4e8e\u200b\u4efb\u4f55\u200b\\(i, j\\)\uff0c\u200b\u82e5\u200b\\(i &gt; i^*\\)\uff0c\u200b\u5219\u200b\\(a_{ij} &gt; t\\)\uff0c\u200b\u5f53\u200b\\(i = i^*\\)\u200b\u65f6\u200b\uff0c\u200b\u82e5\u200b\\(j &gt; j^*\\)\uff0c\u200b\u5219\u200b\\(a_{ij} &lt; t\\)</li> </ul> <p>\u200b\u5373\u200b\u76ee\u6807\u503c\u200b\u5c06\u200b\u77e9\u9635\u200b\u5206\u4e3a\u200b\u4e24\u200b\u90e8\u5206\u200b\uff1a\u200b\u53f3\u4fa7\u200b\u548c\u200b\u4e0b\u4fa7\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u5927\u4e8e\u200b\u7b49\u4e8e\u200b\u76ee\u6807\u503c\u200b\uff0c\u200b\u5de6\u4fa7\u200b\u548c\u200b\u4e0a\u4fa7\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b\u76ee\u6807\u503c\u200b\u3002</p> <p>\u200b\u4ece\u200b\u77e9\u5f62\u200b\u7684\u200b\u5de6\u4e0b\u89d2\u200b\u5f00\u59cb\u200b\uff1a</p> <ul> <li>\u200b\u82e5\u200b\u5f53\u524d\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u6570\u503c\u200b\u5c0f\u4e8e\u200b\u76ee\u6807\u503c\u200b\uff0c\u200b\u8bf4\u660e\u200b\u76ee\u6807\u503c\u200b\u4f4d\u4e8e\u200b\u5f53\u524d\u200b\u884c\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u5373\u53ef\u200b\u627e\u5230\u200b\u76ee\u6807\u503c\u200b</li> <li>\u200b\u82e5\u200b\u5f53\u524d\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u6570\u503c\u200b\u5927\u4e8e\u200b\u76ee\u6807\u503c\u200b\uff0c\u200b\u8bf4\u660e\u200b\u76ee\u6807\u503c\u200b\u4e0d\u200b\u5728\u200b\u5f53\u524d\u200b\u884c\u200b\uff0c\u200b\u9700\u8981\u200b\u5411\u4e0a\u200b\u79fb\u52a8\u200b</li> </ul> <p>\u200b\u91cd\u590d\u200b\u5982\u200b\u4e0a\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u76f4\u81f3\u200b\u627e\u5230\u200b\u76ee\u6807\u200b\u5143\u7d20\u200b\u3002</p> <pre><code>bool searchMatrix(int** matrix, int matrixSize, int* matrixColSize, int target){\n    int x = matrixSize - 1, y = 0;\n    while (x &gt;= 0 &amp;&amp; y &lt; *matrixColSize)\n    {\n        if (matrix[x][y] &lt; target)\n            y++;\n        else if (matrix[x][y] &gt; target)\n            x--;\n        else\n            return true;\n    }\n    return false;\n}\n</code></pre> <p>\u200b\u4ee5\u4e0a\u200b\u7b97\u6cd5\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u8fdb\u884c\u200b\u4f18\u5316\u200b\u3002</p>","tags":["\u6570\u7ec4","\u4e8c\u5206\u67e5\u627e"]},{"location":"coding/leetcode/75/","title":"75. \u200b\u989c\u8272\u200b\u5206\u7c7b","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6392\u5e8f","\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/75/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5305\u542b\u200b\u7ea2\u8272\u200b\u3001\u200b\u767d\u8272\u200b\u548c\u200b\u84dd\u8272\u200b\uff0c\u200b\u4e00\u5171\u200b\u00a0<code>n</code> \u200b\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u6570\u7ec4\u200b\uff0c\u200b\u539f\u5730\u200b\u5bf9\u200b\u5b83\u4eec\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u76f8\u540c\u200b\u989c\u8272\u200b\u7684\u200b\u5143\u7d20\u200b\u76f8\u90bb\u200b\uff0c\u200b\u5e76\u200b\u6309\u7167\u200b\u7ea2\u8272\u200b\u3001\u200b\u767d\u8272\u200b\u3001\u200b\u84dd\u8272\u200b\u987a\u5e8f\u6392\u5217\u200b\u3002</p> <p>\u200b\u6b64\u9898\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u4f7f\u7528\u200b\u6574\u6570\u200b <code>0</code>\u3001\u00a0<code>1</code> \u200b\u548c\u200b <code>2</code> \u200b\u5206\u522b\u200b\u8868\u793a\u200b\u7ea2\u8272\u200b\u3001\u200b\u767d\u8272\u200b\u548c\u200b\u84dd\u8272\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [2,0,2,1,1,0]\n\u200b\u8f93\u51fa\u200b\uff1a[0,0,1,1,2,2]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [2,0,1]\n\u200b\u8f93\u51fa\u200b\uff1a[0,1,2]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [0]\n\u200b\u8f93\u51fa\u200b\uff1a[0]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 4\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1]\n\u200b\u8f93\u51fa\u200b\uff1a[1]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>n == nums.length</code></li> <li><code>1 &lt;= n &lt;= 300</code></li> <li><code>nums[i]</code> \u200b\u4e3a\u200b <code>0</code>\u3001<code>1</code> \u200b\u6216\u200b <code>2</code></li> </ul> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <ul> <li>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u4f7f\u7528\u200b\u4ee3\u7801\u200b\u5e93\u4e2d\u200b\u7684\u200b\u6392\u5e8f\u200b\u51fd\u6570\u200b\u6765\u200b\u89e3\u51b3\u200b\u8fd9\u9053\u9898\u200b\u5417\u200b\uff1f</li> <li>\u200b\u4f60\u200b\u80fd\u200b\u60f3\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u4ec5\u200b\u4f7f\u7528\u200b\u5e38\u6570\u200b\u7a7a\u95f4\u200b\u7684\u200b\u4e00\u8d9f\u200b\u626b\u63cf\u200b\u7b97\u6cd5\u200b\u5417\u200b\uff1f</li> </ul> <p>Reference</p>","tags":["\u6392\u5e8f","\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/75/#_2","title":"\u9898\u89e3","text":"<ol> <li>\u200b\u7edf\u8ba1\u200b\u6570\u7ec4\u200b\u4e2d\u200b<code>0</code>\u3001<code>1</code>\u200b\u548c\u200b<code>2</code>\u200b\u7684\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\uff0c\u200b\u6309\u7167\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u91cd\u65b0\u200b\u751f\u6210\u200b\u6570\u7ec4\u200b\u3002</li> <li>\u200b\u4f7f\u7528\u200b\u5feb\u901f\u200b\u6392\u5e8f\u200b\u7684\u200b\u601d\u8def\u200b\uff0c\u200b\u82e5\u200b\u51fa\u73b0\u200b<code>0</code>\u200b\u5219\u200b\u79fb\u52a8\u200b\u5230\u200b\u6570\u7ec4\u200b\u5f00\u5934\u200b\uff0c\u200b\u82e5\u200b\u51fa\u73b0\u200b<code>2</code>\u200b\u5219\u200b\u79fb\u52a8\u200b\u5230\u200b\u6570\u7ec4\u200b\u7ed3\u5c3e\u200b\u3002</li> </ol> <pre><code>void swap(int *a, int *b)\n{\n    int temp = *a;\n    *a = *b;\n    *b = temp;\n}\nvoid sortColors(int* nums, int numsSize){\n    int head = 0, tail = numsSize - 1, i = 0;\n    for (i = 0; i &lt;= tail &amp;&amp; tail &gt; head; i++)\n    {\n        if (nums[i] == 0)\n            swap(nums + head++, nums + i);\n        if (nums[i] == 2)\n            swap(nums + tail--, nums + i--);\n    }\n}\n</code></pre>","tags":["\u6392\u5e8f","\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/779/","title":"779. \u200b\u7b2c\u200bK\u200b\u4e2a\u200b\u8bed\u6cd5\u200b\u7b26\u53f7","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u9012\u5f52"]},{"location":"coding/leetcode/779/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5728\u200b\u7b2c\u4e00\u884c\u200b\u6211\u4eec\u200b\u5199\u200b\u4e0a\u200b\u4e00\u4e2a\u200b<code>0</code>\u3002\u200b\u63a5\u4e0b\u6765\u200b\u7684\u200b\u6bcf\u200b\u4e00\u884c\u200b\uff0c\u200b\u5c06\u200b\u524d\u200b\u4e00\u884c\u200b\u4e2d\u200b\u7684\u200b<code>0</code>\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>01</code>\uff0c<code>1</code>\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>10</code>\u3002</p> <p>\u200b\u7ed9\u5b9a\u200b\u884c\u200b\u6570\u200b<code>N</code>\u200b\u548c\u200b\u5e8f\u6570\u200b<code>K</code>\uff0c\u200b\u8fd4\u56de\u200b\u7b2c\u200b<code>N</code>\u200b\u884c\u4e2d\u200b\u7b2c\u200b<code>K</code>\u200b\u4e2a\u5b57\u7b26\u200b\u3002\uff08<code>K</code>\u200b\u4ece\u200b 1 \u200b\u5f00\u59cb\u200b\uff09</p> <p>\u200b\u4f8b\u5b50\u200b: <pre><code>\u200b\u8f93\u5165\u200b: N = 1, K = 1\n\u200b\u8f93\u51fa\u200b: 0\n\n\u200b\u8f93\u5165\u200b: N = 2, K = 1\n\u200b\u8f93\u51fa\u200b: 0\n\n\u200b\u8f93\u5165\u200b: N = 2, K = 2\n\u200b\u8f93\u51fa\u200b: 1\n\n\u200b\u8f93\u5165\u200b: N = 4, K = 5\n\u200b\u8f93\u51fa\u200b: 1\n\n\u200b\u89e3\u91ca\u200b:\n\u200b\u7b2c\u4e00\u884c\u200b: 0\n\u200b\u7b2c\u4e8c\u884c\u200b: 01\n\u200b\u7b2c\u4e09\u884c\u200b: 0110\n\u200b\u7b2c\u56db\u884c\u200b: 01101001\n</code></pre> \u200b\u6ce8\u610f\u200b\uff1a</p> <ul> <li><code>N</code>\u200b\u7684\u200b\u8303\u56f4\u200b<code>[1, 30]</code>.</li> <li><code>K</code>\u200b\u7684\u200b\u8303\u56f4\u200b<code>[1, 2^(N-1)]</code>.</li> </ul> <p>Reference</p>","tags":["\u9012\u5f52"]},{"location":"coding/leetcode/779/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u6ce8\u610f\u200b\u5230\u200b\u6bcf\u4e2a\u200b\u7ed3\u679c\u200b\u5b57\u7b26\u4e32\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b4\u200b\u4e2a\u200b\u7b49\u200b\u957f\u200b\u7684\u200b\u90e8\u5206\u200b\uff0c\u200b\u5176\u4e2d\u200b\u7b2c\u200b1\u30014\u200b\u90e8\u5206\u200b\u76f8\u540c\u200b\uff0c2\u30013\u200b\u90e8\u5206\u200b\u76f8\u540c\u200b\uff0c\u200b\u4e24\u7ec4\u200b\u7684\u200b\u4e8c\u8fdb\u5236\u200b\u4e92\u8865\u200b\u3002\u200b\u540c\u65f6\u200b\uff0c\u200b\u4efb\u610f\u200b\u4e00\u884c\u200b\u4ee5\u4e2d\u70b9\u200b\u4e3a\u754c\u200b\uff0c\u200b\u5de6\u53f3\u200b\u4e24\u4fa7\u200b\u4e92\u8865\u200b\u3002\u200b\u56e0\u6b64\u200b\u53ea\u200b\u9700\u8981\u200b\u5728\u200b\u5de6\u4fa7\u200b\u8fdb\u884c\u200b\u67e5\u627e\u200b\uff0c\u200b\u53f3\u4fa7\u200b\u90e8\u5206\u200b\u53d6\u53cd\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int kthGrammar(int N, int K)\n{\n    if (N == 1)\n        return 0;\n    int unit = 1 &lt;&lt; (N - 2);\n    return !(K &lt;= unit) ^ kthGrammar(N - 1, ((K - 1) &amp; (unit - 1)) + 1);\n}\n</code></pre>","tags":["\u9012\u5f52"]},{"location":"coding/leetcode/78/","title":"78. \u200b\u5b50\u96c6","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u4f4d\u8fd0\u7b97","\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/78/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\u00a0<code>nums</code> \uff0c\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b \u200b\u4e92\u4e0d\u200b\u76f8\u540c\u200b \u3002\u200b\u8fd4\u56de\u200b\u8be5\u200b\u6570\u7ec4\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u7684\u200b\u5b50\u96c6\u200b\uff08\u200b\u5e42\u96c6\u200b\uff09\u3002</p> <p>\u200b\u89e3\u96c6\u200b \u200b\u4e0d\u80fd\u200b \u200b\u5305\u542b\u200b\u91cd\u590d\u200b\u7684\u200b\u5b50\u96c6\u200b\u3002\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u6309\u200b \u200b\u4efb\u610f\u200b\u987a\u5e8f\u200b \u200b\u8fd4\u56de\u200b\u89e3\u96c6\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1,2,3]\n\u200b\u8f93\u51fa\u200b\uff1a[[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3]]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [0]\n\u200b\u8f93\u51fa\u200b\uff1a[[],[0]]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 10</code></li> <li><code>-10 &lt;= nums[i] &lt;= 10</code></li> <li><code>nums</code> \u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b \u200b\u4e92\u4e0d\u200b\u76f8\u540c\u200b</li> </ul> <p>Reference</p>","tags":["\u4f4d\u8fd0\u7b97","\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/78/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5df2\u77e5\u200b\u96c6\u5408\u200b\u4e2d\u200b\u4e0d\u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6574\u6570\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u200b\u4e00\u4f4d\u200b\u8868\u793a\u200b\u96c6\u5408\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u9009\u53d6\u200b\u72b6\u6001\u200b\u3002</p> <pre><code>/**\n * Return an array of arrays of size *returnSize.\n * The sizes of the arrays are returned as *returnColumnSizes array.\n * Note: Both returned array and *columnSizes array must be malloced, assume caller calls free().\n */\nint count1(int x)\n{\n    int ret = 0;\n    while (x)\n    {\n        ret += x &amp; 1;\n        x &gt;&gt;= 1;\n    }\n    return ret;\n}\nint** subsets(int* nums, int numsSize, int* returnSize, int** returnColumnSizes){\n    *returnSize = 1 &lt;&lt; numsSize;\n    *returnColumnSizes = (int *)memset(malloc(sizeof(int) * *returnSize), 0, sizeof(int) * *returnSize);\n    int i = 0, j = 0, **ret = (int **)malloc(sizeof(int *) * *returnSize), pos;\n    for (i = 0; i &lt; *returnSize; i++)\n    {\n        (*returnColumnSizes)[i] = count1(i);\n        ret[i] = (int *)malloc(sizeof(int) * (*returnColumnSizes)[i]);\n        pos = 0;\n        for (j = 0; j &lt; numsSize; j++)\n            if (i &amp; (1 &lt;&lt; j))\n                ret[i][pos++] = nums[j];\n    }\n    return ret;\n}\n</code></pre>","tags":["\u4f4d\u8fd0\u7b97","\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/781/","title":"781. \u200b\u68ee\u6797\u200b\u4e2d\u200b\u7684\u200b\u5154\u5b50","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u54c8\u5e0c\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/781/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u68ee\u6797\u200b\u4e2d\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5154\u5b50\u200b\u90fd\u200b\u6709\u200b\u989c\u8272\u200b\u3002\u200b\u5176\u4e2d\u200b\u4e00\u4e9b\u200b\u5154\u5b50\u200b\uff08\u200b\u53ef\u80fd\u200b\u662f\u200b\u5168\u90e8\u200b\uff09\u200b\u544a\u8bc9\u200b\u4f60\u200b\u8fd8\u6709\u200b\u591a\u5c11\u200b\u5176\u4ed6\u200b\u7684\u200b\u5154\u5b50\u200b\u548c\u200b\u81ea\u5df1\u200b\u6709\u200b\u76f8\u540c\u200b\u7684\u200b\u989c\u8272\u200b\u3002\u200b\u6211\u4eec\u200b\u5c06\u200b\u8fd9\u4e9b\u200b\u56de\u7b54\u200b\u653e\u5728\u200b\u00a0<code>answers</code>\u00a0\u200b\u6570\u7ec4\u200b\u91cc\u200b\u3002</p> <p>\u200b\u8fd4\u56de\u200b\u68ee\u6797\u200b\u4e2d\u200b\u5154\u5b50\u200b\u7684\u200b\u6700\u5c11\u200b\u6570\u91cf\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b:</p> <pre><code>\u200b\u8f93\u5165\u200b: answers = [1, 1, 2]\n\u200b\u8f93\u51fa\u200b: 5\n\u200b\u89e3\u91ca\u200b:\n\u200b\u4e24\u53ea\u200b\u56de\u7b54\u200b\u4e86\u200b \"1\" \u200b\u7684\u200b\u5154\u5b50\u200b\u53ef\u80fd\u200b\u6709\u200b\u76f8\u540c\u200b\u7684\u200b\u989c\u8272\u200b\uff0c\u200b\u8bbe\u200b\u4e3a\u200b\u7ea2\u8272\u200b\u3002\n\u200b\u4e4b\u540e\u200b\u56de\u7b54\u200b\u4e86\u200b \"2\" \u200b\u7684\u200b\u5154\u5b50\u200b\u4e0d\u4f1a\u200b\u662f\u200b\u7ea2\u8272\u200b\uff0c\u200b\u5426\u5219\u200b\u4ed6\u4eec\u200b\u7684\u200b\u56de\u7b54\u200b\u4f1a\u200b\u76f8\u4e92\u200b\u77db\u76fe\u200b\u3002\n\u200b\u8bbe\u200b\u56de\u7b54\u200b\u4e86\u200b \"2\" \u200b\u7684\u200b\u5154\u5b50\u200b\u4e3a\u200b\u84dd\u8272\u200b\u3002\n\u200b\u6b64\u5916\u200b\uff0c\u200b\u68ee\u6797\u200b\u4e2d\u200b\u8fd8\u200b\u5e94\u6709\u200b\u53e6\u5916\u200b 2 \u200b\u53ea\u200b\u84dd\u8272\u200b\u5154\u5b50\u200b\u7684\u200b\u56de\u7b54\u200b\u6ca1\u6709\u200b\u5305\u542b\u200b\u5728\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u3002\n\u200b\u56e0\u6b64\u200b\u68ee\u6797\u200b\u4e2d\u200b\u5154\u5b50\u200b\u7684\u200b\u6700\u5c11\u200b\u6570\u91cf\u200b\u662f\u200b 5: 3 \u200b\u53ea\u200b\u56de\u7b54\u200b\u7684\u200b\u548c\u200b 2 \u200b\u53ea\u200b\u6ca1\u6709\u200b\u56de\u7b54\u200b\u7684\u200b\u3002\n\n\u200b\u8f93\u5165\u200b: answers = [10, 10, 10]\n\u200b\u8f93\u51fa\u200b: 11\n\n\u200b\u8f93\u5165\u200b: answers = []\n\u200b\u8f93\u51fa\u200b: 0\n</code></pre> <p>\u200b\u8bf4\u660e\u200b:</p> <ul> <li><code>answers</code>\u00a0\u200b\u7684\u200b\u957f\u5ea6\u200b\u6700\u5927\u200b\u4e3a\u200b<code>1000</code>\u3002</li> <li><code>answers[i]</code>\u00a0\u200b\u662f\u200b\u5728\u200b\u00a0<code>[0, 999]</code>\u00a0\u200b\u8303\u56f4\u200b\u5185\u200b\u7684\u200b\u6574\u6570\u200b\u3002</li> </ul> <p>Reference</p>","tags":["\u54c8\u5e0c\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/781/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5154\u5b50\u200b\u7684\u200b\u5206\u7c7b\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u60c5\u51b5\u200b\uff0c\u200b\u8bbe\u200b\u7b2c\u200b\\(i\\)\u200b\u53ea\u200b\u5154\u5b50\u200b\u7684\u200b\u62a5\u6570\u200b\u4e3a\u200b\\(a_i\\)\uff0c\u200b\u5219\u200b\uff1a</p> <ul> <li>\u200b\u82e5\u200b\\(a_i \\not = a_j\\)\uff0c\u200b\u5219\u200b\u7b2c\u200b\\(i\\)\u200b\u53ea\u200b\u5154\u5b50\u200b\u4e0e\u200b\u7b2c\u200b\\(j\\)\u200b\u53ea\u200b\u5154\u5b50\u200b\u5fc5\u7136\u200b\u4e0d\u200b\u5c5e\u4e8e\u200b\u4e00\u7c7b\u200b</li> <li>\u200b\u82e5\u200b\\(a_i = a_j\\)\uff0c\u200b\u5219\u200b\u7b2c\u200b\\(i\\)\u200b\u53ea\u200b\u5154\u5b50\u200b\u4e0e\u200b\u7b2c\u200b\\(j\\)\u200b\u53ea\u200b\u5154\u5b50\u200b\u53ef\u80fd\u200b\u5c5e\u4e8e\u200b\u4e00\u7c7b\u200b\uff0c</li> <li>\u200b\u7b2c\u200b\\(i\\)\u200b\u53ea\u200b\u5154\u5b50\u200b\u6240\u5c5e\u200b\u7684\u200b\u7c7b\u200b\u4e2d\u200b\u6700\u200b\u591a\u200b\u6709\u200b\\(a_i + 1\\)\u200b\u53ea\u200b\u5154\u5b50\u200b</li> </ul> <p>\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u5f97\u51fa\u200b\u7b97\u6cd5\u200b\uff1a\u200b\u7edf\u8ba1\u200b\u76f8\u540c\u200b\u6570\u5b57\u200b\\(a_i\\)\u200b\u51fa\u73b0\u200b\u7684\u200b\u6b21\u6570\u200b\\(b_j\\)\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u6570\u5b57\u200b\uff0c\u200b\u4e0e\u200b\u4e4b\u200b\u5bf9\u5e94\u200b\u7684\u200b\u6700\u5c11\u200b\u7684\u200b\u5154\u5b50\u200b\u6570\u91cf\u200b\u4e3a\u200b\\(b'_j\\)\uff0c\u200b\u5219\u200b\\(b'_j\\)\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\uff1a</p> <ul> <li>\\(b'_j = k \\times (a_i + 1), k\\in \\N\\)</li> <li>\\(b'_j \\geq b_j\\)</li> <li>\\(b'_j - (a_i + 1) &lt; b_j\\)</li> </ul> <p>\u200b\u5373\u200b\\(b'_j = \\left\\lceil\\frac{b_j}{a_i + 1}\\right\\rceil\\times (a_i + 1)\\)</p> <pre><code>int numRabbits(int* answers, int answersSize){\n    int hash[1000] = {0}, i = 0, ret = 0;\n    for (i = 0; i &lt; answersSize; i++)\n        hash[answers[i]]++;\n    for (i = 0; i &lt; 1000; i++)\n        ret += (hash[i] + i) / (i + 1) * (i + 1);\n    return ret;\n}\n</code></pre> <p>Python\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5185\u7f6e\u200b\u5e93\u200b<code>collections</code>\u200b\u4e2d\u200b\u7684\u200b<code>Counter</code>\u200b\u7c7b\u200b\u7edf\u8ba1\u6570\u5b57\u200b\u7684\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\uff1a</p> <pre><code>import collections\n\nclass Solution:\n    def numRabbits(self, answers: List[int]) -&gt; int:\n        return sum(map(\n            lambda args: sum(args) // (args[0] + 1) * (args[0] + 1)\n            collections.Counter(answers).items()\n        ))\n</code></pre>","tags":["\u54c8\u5e0c\u8868","\u6570\u5b66"]},{"location":"coding/leetcode/79/","title":"79. \u200b\u5355\u8bcd\u200b\u641c\u7d22","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/79/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u00a0<code>m x n</code> \u200b\u4e8c\u7ef4\u200b\u5b57\u7b26\u200b\u7f51\u683c\u200b\u00a0<code>board</code> \u200b\u548c\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u5355\u8bcd\u200b\u00a0<code>word</code> \u3002\u200b\u5982\u679c\u200b\u00a0<code>word</code> \u200b\u5b58\u5728\u200b\u4e8e\u200b\u7f51\u683c\u200b\u4e2d\u200b\uff0c\u200b\u8fd4\u56de\u200b <code>true</code> \uff1b\u200b\u5426\u5219\u200b\uff0c\u200b\u8fd4\u56de\u200b <code>false</code> \u3002</p> <p>\u200b\u5355\u8bcd\u200b\u5fc5\u987b\u200b\u6309\u7167\u200b\u5b57\u6bcd\u200b\u987a\u5e8f\u200b\uff0c\u200b\u901a\u8fc7\u200b\u76f8\u90bb\u200b\u7684\u200b\u5355\u5143\u683c\u200b\u5185\u200b\u7684\u200b\u5b57\u6bcd\u200b\u6784\u6210\u200b\uff0c\u200b\u5176\u4e2d\u200b\u201c\u200b\u76f8\u90bb\u200b\u201d\u200b\u5355\u5143\u683c\u200b\u662f\u200b\u90a3\u4e9b\u200b\u6c34\u5e73\u200b\u76f8\u90bb\u200b\u6216\u200b\u5782\u76f4\u200b\u76f8\u90bb\u200b\u7684\u200b\u5355\u5143\u683c\u200b\u3002\u200b\u540c\u4e00\u4e2a\u200b\u5355\u5143\u683c\u200b\u5185\u200b\u7684\u200b\u5b57\u6bcd\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u88ab\u200b\u91cd\u590d\u4f7f\u7528\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = [[\"A\",\"B\",\"C\",\"E\"],[\"S\",\"F\",\"C\",\"S\"],[\"A\",\"D\",\"E\",\"E\"]], word = \"ABCCED\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = [[\"A\",\"B\",\"C\",\"E\"],[\"S\",\"F\",\"C\",\"S\"],[\"A\",\"D\",\"E\",\"E\"]], word = \"SEE\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <p></p> <pre><code>\u200b\u8f93\u5165\u200b\uff1aboard = [[\"A\",\"B\",\"C\",\"E\"],[\"S\",\"F\",\"C\",\"S\"],[\"A\",\"D\",\"E\",\"E\"]], word = \"ABCB\"\n\u200b\u8f93\u51fa\u200b\uff1afalse\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>m == board.length</code></li> <li><code>n = board[i].length</code></li> <li><code>1 &lt;= m, n &lt;= 6</code></li> <li><code>1 &lt;= word.length &lt;= 15</code></li> <li><code>board</code> \u200b\u548c\u200b <code>word</code> \u200b\u4ec5\u200b\u7531\u200b\u5927\u5c0f\u5199\u200b\u82f1\u6587\u5b57\u6bcd\u200b\u7ec4\u6210\u200b</li> </ul> <p>\u200b\u8fdb\u9636\u200b\uff1a\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u641c\u7d22\u200b\u526a\u679d\u200b\u7684\u200b\u6280\u672f\u200b\u6765\u200b\u4f18\u5316\u200b\u89e3\u51b3\u65b9\u6848\u200b\uff0c\u200b\u4f7f\u200b\u5176\u200b\u5728\u200b <code>board</code> \u200b\u66f4\u5927\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u53ef\u4ee5\u200b\u66f4\u200b\u5feb\u200b\u89e3\u51b3\u95ee\u9898\u200b\uff1f</p> <p>Reference</p>","tags":["\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/79/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u4f7f\u7528\u200b\u6df1\u5ea6\u200b\u4f18\u5148\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u5728\u200b\u4e8c\u7ef4\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5bfb\u627e\u200b\u5355\u8bcd\u200b\u7b2c\u4e00\u4e2a\u200b\u5b57\u6bcd\u200b\u51fa\u73b0\u200b\u7684\u200b\u4f4d\u7f6e\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u51fa\u73b0\u200b\u4f4d\u7f6e\u200b\uff1a</p> <ul> <li>\u200b\u5c06\u200b\u5f53\u524d\u200b\u4f4d\u7f6e\u200b\u6807\u5fd7\u200b\u4e3a\u200b\u7ecf\u5386\u200b\u8fc7\u200b</li> <li>\u200b\u5bfb\u627e\u200b\u5f53\u524d\u200b\u4f4d\u7f6e\u200b\u9644\u8fd1\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u5355\u5143\u683c\u200b<ol> <li>\u200b\u5355\u5143\u683c\u200b\u4e2d\u200b\u7684\u200b\u5b57\u6bcd\u200b\u7b49\u4e8e\u200b\u5355\u8bcd\u200b\u7684\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5b57\u6bcd\u200b</li> <li>\u200b\u5355\u5143\u683c\u200b\u672a\u200b\u88ab\u200b\u904d\u5386\u200b\u8fc7\u200b</li> </ol> </li> <li>\u200b\u5982\u679c\u200b\u627e\u200b\u4e0d\u5230\u200b\u5355\u5143\u683c\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u4e8c\u7ef4\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u65e0\u6cd5\u200b\u627e\u5230\u200b\u76ee\u6807\u200b\u5355\u8bcd\u200b</li> <li>\u200b\u5982\u679c\u200b\u627e\u5230\u200b\u4e00\u4e2a\u200b\u6216\u200b\u591a\u4e2a\u200b\u5355\u5143\u683c\u200b\uff0c\u200b\u5219\u200b\u524d\u8fdb\u200b\u5230\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5355\u5143\u683c\u200b\u7ee7\u7eed\u200b\u8fdb\u884c\u200b\u5339\u914d\u200b</li> <li>\u200b\u5982\u679c\u200b\u5355\u8bcd\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u5b57\u6bcd\u200b\u6210\u529f\u200b\u5339\u914d\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u5339\u914d\u200b\u6210\u529f\u200b</li> </ul> <pre><code>bool dfs(char **board, int boardSize, int* boardColSize, char *word, int posX, int posY, int **map)\n{\n    if (*word == 0)\n        return true;\n    bool ret = false;\n    if (posX &gt; 0 &amp;&amp; board[posX - 1][posY] == *word &amp;&amp; map[posX - 1][posY] == 0)\n    {\n        map[posX - 1][posY] = 1;\n        ret = ret || dfs(board, boardSize, boardColSize, word + 1, posX - 1, posY, map);\n        map[posX - 1][posY] = 0;\n    }\n    if (posX &lt; boardSize - 1 &amp;&amp; board[posX + 1][posY] == *word &amp;&amp; map[posX + 1][posY] == 0)\n    {\n        map[posX + 1][posY] = 1;\n        ret = ret || dfs(board, boardSize, boardColSize, word + 1, posX + 1, posY, map);\n        map[posX + 1][posY] = 0;\n    }\n    if (posY &gt; 0 &amp;&amp; board[posX][posY - 1] == *word &amp;&amp; map[posX][posY - 1] == 0)\n    {\n        map[posX][posY - 1] = 1;\n        ret = ret || dfs(board, boardSize, boardColSize, word + 1, posX, posY - 1, map);\n        map[posX][posY - 1] = 0;\n    }\n    if (posY &lt; boardColSize[0] - 1 &amp;&amp; board[posX][posY + 1] == *word &amp;&amp; map[posX][posY + 1] == 0)\n    {\n        map[posX][posY + 1] = 1;\n        ret = ret || dfs(board, boardSize, boardColSize, word + 1, posX, posY + 1, map);\n        map[posX][posY + 1] = 0;\n    }\n    return ret;\n}\nbool exist(char** board, int boardSize, int* boardColSize, char * word){\n    bool ret = false;\n    int i = 0, j = 0, **map = (int **)malloc(sizeof(int *) * boardSize);\n    for (i = 0; i &lt; boardSize; i++)\n        map[i] = (int *)memset(malloc(sizeof(int) * boardColSize[i]), 0, sizeof(int) * boardColSize[i]);\n    for (i = 0; i &lt; boardSize &amp;&amp; !ret; i++)\n    {\n        for (j = 0; j &lt; boardColSize[0] &amp;&amp; !ret; j++)\n        {\n            if (board[i][j] == *word)\n            {\n                map[i][j] = 1;\n                ret = ret || dfs(board, boardSize, boardColSize, word + 1, i, j, map);\n                map[i][j] = 0;\n            }\n        }\n    }\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/8/","title":"8. \u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u200b\u6574\u6570\u200b (atoi)","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/8/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u8bf7\u200b\u4f60\u200b\u6765\u200b\u5b9e\u73b0\u200b\u4e00\u4e2a\u200b<code>atoi</code>\u200b\u51fd\u6570\u200b\uff0c\u200b\u4f7f\u200b\u5176\u80fd\u200b\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u6210\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u9996\u5148\u200b\uff0c\u200b\u8be5\u200b\u51fd\u6570\u200b\u4f1a\u200b\u6839\u636e\u200b\u9700\u8981\u200b\u4e22\u5f03\u200b\u65e0\u7528\u200b\u7684\u200b\u5f00\u5934\u200b\u7a7a\u683c\u200b\u5b57\u7b26\u200b\uff0c\u200b\u76f4\u5230\u200b\u5bfb\u627e\u200b\u5230\u200b\u7b2c\u4e00\u4e2a\u200b\u975e\u200b\u7a7a\u683c\u200b\u7684\u200b\u5b57\u7b26\u200b\u4e3a\u6b62\u200b\u3002\u200b\u63a5\u4e0b\u6765\u200b\u7684\u200b\u8f6c\u5316\u200b\u89c4\u5219\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u5982\u679c\u200b\u7b2c\u4e00\u4e2a\u200b\u975e\u200b\u7a7a\u5b57\u7b26\u200b\u4e3a\u200b\u6b63\u200b\u6216\u8005\u200b\u8d1f\u53f7\u200b\u65f6\u200b\uff0c\u200b\u5219\u200b\u5c06\u200b\u8be5\u200b\u7b26\u53f7\u200b\u4e0e\u200b\u4e4b\u200b\u540e\u9762\u200b\u5c3d\u53ef\u80fd\u200b\u591a\u200b\u7684\u200b\u8fde\u7eed\u200b\u6570\u5b57\u200b\u5b57\u7b26\u200b\u7ec4\u5408\u200b\u8d77\u6765\u200b\uff0c\u200b\u5f62\u6210\u200b\u4e00\u4e2a\u200b\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\u3002</li> <li>\u200b\u5047\u5982\u200b\u7b2c\u4e00\u4e2a\u200b\u975e\u200b\u7a7a\u5b57\u7b26\u200b\u662f\u200b\u6570\u5b57\u200b\uff0c\u200b\u5219\u200b\u76f4\u63a5\u200b\u5c06\u200b\u5176\u200b\u4e0e\u200b\u4e4b\u540e\u200b\u8fde\u7eed\u200b\u7684\u200b\u6570\u5b57\u200b\u5b57\u7b26\u200b\u7ec4\u5408\u200b\u8d77\u6765\u200b\uff0c\u200b\u5f62\u6210\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u3002</li> <li>\u200b\u8be5\u200b\u5b57\u7b26\u4e32\u200b\u5728\u200b\u6709\u6548\u200b\u7684\u200b\u6574\u6570\u200b\u90e8\u5206\u200b\u4e4b\u540e\u200b\u4e5f\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u5b58\u5728\u200b\u591a\u4f59\u200b\u7684\u200b\u5b57\u7b26\u200b\uff0c\u200b\u90a3\u4e48\u200b\u8fd9\u4e9b\u200b\u5b57\u7b26\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u5ffd\u7565\u200b\uff0c\u200b\u5b83\u4eec\u200b\u5bf9\u200b\u51fd\u6570\u200b\u4e0d\u200b\u5e94\u8be5\u200b\u9020\u6210\u200b\u5f71\u54cd\u200b\u3002</li> </ul> <p>\u200b\u6ce8\u610f\u200b\uff1a\u200b\u5047\u5982\u200b\u8be5\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u975e\u200b\u7a7a\u683c\u200b\u5b57\u7b26\u200b\u4e0d\u662f\u200b\u4e00\u4e2a\u200b\u6709\u6548\u200b\u6574\u6570\u200b\u5b57\u7b26\u200b\u3001\u200b\u5b57\u7b26\u4e32\u200b\u4e3a\u7a7a\u200b\u6216\u200b\u5b57\u7b26\u4e32\u200b\u4ec5\u200b\u5305\u542b\u200b\u7a7a\u767d\u200b\u5b57\u7b26\u200b\u65f6\u200b\uff0c\u200b\u5219\u200b\u4f60\u200b\u7684\u200b\u51fd\u6570\u200b\u4e0d\u200b\u9700\u8981\u200b\u8fdb\u884c\u200b\u8f6c\u6362\u200b\uff0c\u200b\u5373\u200b\u65e0\u6cd5\u200b\u8fdb\u884c\u200b\u6709\u6548\u200b\u8f6c\u6362\u200b\u3002</p> <p>\u200b\u5728\u200b\u4efb\u4f55\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u82e5\u200b\u51fd\u6570\u200b\u4e0d\u80fd\u200b\u8fdb\u884c\u200b\u6709\u6548\u200b\u7684\u200b\u8f6c\u6362\u200b\u65f6\u200b\uff0c\u200b\u8bf7\u200b\u8fd4\u56de\u200b 0 \u3002</p> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u672c\u9898\u200b\u4e2d\u200b\u7684\u200b\u7a7a\u767d\u200b\u5b57\u7b26\u200b\u53ea\u200b\u5305\u62ec\u200b\u7a7a\u683c\u200b\u5b57\u7b26\u200b<code>' '</code>\u3002</li> <li>\u200b\u5047\u8bbe\u200b\u6211\u4eec\u200b\u7684\u200b\u73af\u5883\u200b\u53ea\u80fd\u200b\u5b58\u50a8\u200b 32 \u200b\u4f4d\u200b\u5927\u5c0f\u200b\u7684\u200b\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\uff0c\u200b\u90a3\u4e48\u200b\u5176\u200b\u6570\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\([\u22122^{31},\u00a0 2^{31}\u00a0\u2212 1]\\)\u3002\u200b\u5982\u679c\u200b\u6570\u503c\u200b\u8d85\u8fc7\u200b\u8fd9\u4e2a\u200b\u8303\u56f4\u200b\uff0c\u200b\u8bf7\u200b\u8fd4\u56de\u200b<code>INT_MAX</code>(\\(2^{31}\u00a0\u2212 1\\))\u200b\u6216\u200b<code>INT_MIN</code>(\\(\u22122^{31}\\)) \u3002 \u00a0</li> </ul> <p>\u200b\u793a\u4f8b\u200b\u00a01:</p> <pre><code>\u200b\u8f93\u5165\u200b: \"42\"\n\u200b\u8f93\u51fa\u200b: 42\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02:</p> <pre><code>\u200b\u8f93\u5165\u200b: \"   -42\"\n\u200b\u8f93\u51fa\u200b: -42\n\u200b\u89e3\u91ca\u200b: \u200b\u7b2c\u4e00\u4e2a\u200b\u975e\u200b\u7a7a\u767d\u200b\u5b57\u7b26\u200b\u4e3a\u200b '-', \u200b\u5b83\u200b\u662f\u200b\u4e00\u4e2a\u200b\u8d1f\u53f7\u200b\u3002\n\u00a0    \u200b\u6211\u4eec\u200b\u5c3d\u53ef\u80fd\u200b\u5c06\u200b\u8d1f\u53f7\u200b\u4e0e\u200b\u540e\u9762\u200b\u6240\u6709\u200b\u8fde\u7eed\u200b\u51fa\u73b0\u200b\u7684\u200b\u6570\u5b57\u200b\u7ec4\u5408\u200b\u8d77\u6765\u200b\uff0c\u200b\u6700\u540e\u200b\u5f97\u5230\u200b -42 \u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a03:</p> <pre><code>\u200b\u8f93\u5165\u200b: \"4193 with words\"\n\u200b\u8f93\u51fa\u200b: 4193\n\u200b\u89e3\u91ca\u200b: \u200b\u8f6c\u6362\u200b\u622a\u6b62\u200b\u4e8e\u200b\u6570\u5b57\u200b '3' \uff0c\u200b\u56e0\u4e3a\u200b\u5b83\u200b\u7684\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b\u4e0d\u200b\u4e3a\u200b\u6570\u5b57\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a04:</p> <pre><code>\u200b\u8f93\u5165\u200b: \"words and 987\"\n\u200b\u8f93\u51fa\u200b: 0\n\u200b\u89e3\u91ca\u200b: \u200b\u7b2c\u4e00\u4e2a\u200b\u975e\u200b\u7a7a\u5b57\u7b26\u200b\u662f\u200b 'w', \u200b\u4f46\u200b\u5b83\u200b\u4e0d\u662f\u200b\u6570\u5b57\u200b\u6216\u200b\u6b63\u200b\u3001\u200b\u8d1f\u53f7\u200b\u3002\n     \u200b\u56e0\u6b64\u200b\u65e0\u6cd5\u200b\u6267\u884c\u200b\u6709\u6548\u200b\u7684\u200b\u8f6c\u6362\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a05:</p> <pre><code>\u200b\u8f93\u5165\u200b: \"-91283472332\"\n\u200b\u8f93\u51fa\u200b: -2147483648\n\u200b\u89e3\u91ca\u200b: \u200b\u6570\u5b57\u200b \"-91283472332\" \u200b\u8d85\u8fc7\u200b 32 \u200b\u4f4d\u6709\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\u8303\u56f4\u200b\u3002 \n\u00a0    \u200b\u56e0\u6b64\u200b\u8fd4\u56de\u200b INT_MIN (\u2212231) \u3002\n</code></pre> <p>Reference</p>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/8/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a </p> <ol> <li>\u200b\u6ce8\u610f\u200b\u4e00\u4e2a\u200b\u6709\u6548\u200b\u7684\u200b\u6570\u5b57\u200b\u5b57\u7b26\u4e32\u200b\u53ef\u4ee5\u200b\u4ee5\u200b<code>0-9</code>\u3001<code>'-'</code>\u3001<code>'+'</code>\u200b\u548c\u200b\u7a7a\u683c\u200b<code>' '</code>\u200b\u5f00\u5934\u200b\u3002</li> <li>\u200b\u4e3a\u4e86\u200b\u907f\u514d\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u5e94\u5f53\u200b\u4f7f\u7528\u200b<code>long</code>\u200b\u7c7b\u578b\u200b\u3002</li> </ol> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>int myAtoi(char * str){\n    int flag = 0;\n    char *cur = str;\n    while (*cur &amp;&amp; *cur == ' ')\n        cur++;\n    if (*cur == '-' || *cur == '+')\n        flag = *(cur++) - '+';\n    if (*cur &lt; '0' || *cur &gt; '9')\n        return 0;\n    long ret = 0;\n    while(*cur &gt;= '0' &amp;&amp; *cur &lt;= '9')\n    {\n        ret *= 10;\n        ret += *cur - '0';\n        cur++;\n        if (ret &gt;= -(long)INT_MIN || (ret == (long)INT_MAX &amp;&amp; !flag))\n            return flag ? INT_MIN : INT_MAX;\n    }\n    return flag ? -ret : ret;\n}\n</code></pre>","tags":["\u6570\u5b66","\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/87/","title":"87. \u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u56f0\u96be\u200b</p>","tags":["\u5b57\u7b26\u4e32","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/87/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u4f7f\u7528\u200b\u4e0b\u9762\u200b\u63cf\u8ff0\u200b\u7684\u200b\u7b97\u6cd5\u200b\u53ef\u4ee5\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b <code>s</code> \u200b\u5f97\u5230\u200b\u5b57\u7b26\u4e32\u200b <code>t</code> \uff1a</p> <ol> <li>\u200b\u5982\u679c\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u957f\u5ea6\u200b\u4e3a\u200b 1 \uff0c\u200b\u7b97\u6cd5\u200b\u505c\u6b62\u200b</li> <li>\u200b\u5982\u679c\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u957f\u5ea6\u200b &gt; 1 \uff0c\u200b\u6267\u884c\u200b\u4e0b\u8ff0\u200b\u6b65\u9aa4\u200b\uff1a<ul> <li>\u200b\u5728\u200b\u4e00\u4e2a\u200b\u968f\u673a\u200b\u4e0b\u6807\u200b\u5904\u200b\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u5206\u5272\u200b\u6210\u200b\u4e24\u4e2a\u200b\u975e\u7a7a\u200b\u7684\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u3002\u200b\u5373\u200b\uff0c\u200b\u5982\u679c\u200b\u5df2\u77e5\u200b\u5b57\u7b26\u4e32\u200b <code>s</code> \uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5176\u200b\u5206\u6210\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b <code>x</code> \u200b\u548c\u200b <code>y</code> \uff0c\u200b\u4e14\u200b\u6ee1\u8db3\u200b <code>s = x + y</code> \u3002</li> <li>\u200b\u968f\u673a\u200b \u200b\u51b3\u5b9a\u200b\u662f\u200b\u8981\u200b\u300c\u200b\u4ea4\u6362\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u300d\u200b\u8fd8\u662f\u200b\u8981\u200b\u300c\u200b\u4fdd\u6301\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u987a\u5e8f\u200b\u4e0d\u53d8\u200b\u300d\u3002\u200b\u5373\u200b\uff0c\u200b\u5728\u200b\u6267\u884c\u200b\u8fd9\u4e00\u200b\u6b65\u9aa4\u200b\u4e4b\u540e\u200b\uff0c<code>s</code> \u200b\u53ef\u80fd\u200b\u662f\u200b <code>s = x + y</code> \u200b\u6216\u8005\u200b <code>s = y + x</code> \u3002</li> <li>\u200b\u5728\u200b <code>x</code> \u200b\u548c\u200b <code>y</code> \u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u4e0a\u200b\u7ee7\u7eed\u200b\u4ece\u200b\u6b65\u9aa4\u200b 1 \u200b\u5f00\u59cb\u200b\u9012\u5f52\u200b\u6267\u884c\u200b\u6b64\u200b\u7b97\u6cd5\u200b\u3002</li> </ul> </li> </ol> <p>\u200b\u7ed9\u200b\u4f60\u200b\u4e24\u4e2a\u200b \u200b\u957f\u5ea6\u200b\u76f8\u7b49\u200b \u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b <code>s1</code> \u200b\u548c\u200b\u00a0<code>s2</code>\uff0c\u200b\u5224\u65ad\u200b\u00a0<code>s2</code>\u00a0\u200b\u662f\u5426\u662f\u200b\u00a0<code>s1</code>\u00a0\u200b\u7684\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b\u3002\u200b\u5982\u679c\u200b\u662f\u200b\uff0c\u200b\u8fd4\u56de\u200b <code>true</code> \uff1b\u200b\u5426\u5219\u200b\uff0c\u200b\u8fd4\u56de\u200b <code>false</code> \u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as1 = \"great\", s2 = \"rgeat\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n\u200b\u89e3\u91ca\u200b\uff1as1 \u200b\u4e0a\u200b\u53ef\u80fd\u200b\u53d1\u751f\u200b\u7684\u200b\u4e00\u79cd\u200b\u60c5\u5f62\u200b\u662f\u200b\uff1a\n\"great\" --&gt; \"gr/eat\" // \u200b\u5728\u200b\u4e00\u4e2a\u200b\u968f\u673a\u200b\u4e0b\u6807\u200b\u5904\u200b\u5206\u5272\u200b\u5f97\u5230\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\n\"gr/eat\" --&gt; \"gr/eat\" // \u200b\u968f\u673a\u200b\u51b3\u5b9a\u200b\uff1a\u300c\u200b\u4fdd\u6301\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u987a\u5e8f\u200b\u4e0d\u53d8\u200b\u300d\n\"gr/eat\" --&gt; \"g/r / e/at\" // \u200b\u5728\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u4e0a\u200b\u9012\u5f52\u200b\u6267\u884c\u200b\u6b64\u200b\u7b97\u6cd5\u200b\u3002\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u5206\u522b\u200b\u5728\u200b\u968f\u673a\u200b\u4e0b\u6807\u200b\u5904\u200b\u8fdb\u884c\u200b\u4e00\u8f6e\u200b\u5206\u5272\u200b\n\"g/r / e/at\" --&gt; \"r/g / e/at\" // \u200b\u968f\u673a\u200b\u51b3\u5b9a\u200b\uff1a\u200b\u7b2c\u4e00\u7ec4\u200b\u300c\u200b\u4ea4\u6362\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u300d\uff0c\u200b\u7b2c\u4e8c\u7ec4\u200b\u300c\u200b\u4fdd\u6301\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u987a\u5e8f\u200b\u4e0d\u53d8\u200b\u300d\n\"r/g / e/at\" --&gt; \"r/g / e/ a/t\" // \u200b\u7ee7\u7eed\u200b\u9012\u5f52\u200b\u6267\u884c\u200b\u6b64\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u5c06\u200b \"at\" \u200b\u5206\u5272\u200b\u5f97\u5230\u200b \"a/t\"\n\"r/g / e/ a/t\" --&gt; \"r/g / e/ a/t\" // \u200b\u968f\u673a\u200b\u51b3\u5b9a\u200b\uff1a\u300c\u200b\u4fdd\u6301\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u987a\u5e8f\u200b\u4e0d\u53d8\u200b\u300d\n\u200b\u7b97\u6cd5\u200b\u7ec8\u6b62\u200b\uff0c\u200b\u7ed3\u679c\u200b\u5b57\u7b26\u4e32\u200b\u548c\u200b s2 \u200b\u76f8\u540c\u200b\uff0c\u200b\u90fd\u200b\u662f\u200b \"rgeat\"\n\u200b\u8fd9\u662f\u200b\u4e00\u79cd\u200b\u80fd\u591f\u200b\u6270\u4e71\u200b s1 \u200b\u5f97\u5230\u200b s2 \u200b\u7684\u200b\u60c5\u5f62\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8ba4\u4e3a\u200b s2 \u200b\u662f\u200b s1 \u200b\u7684\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u8fd4\u56de\u200b true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as1 = \"abcde\", s2 = \"caebd\"\n\u200b\u8f93\u51fa\u200b\uff1afalse\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1as1 = \"a\", s2 = \"a\"\n\u200b\u8f93\u51fa\u200b\uff1atrue\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>s1.length == s2.length</code></li> <li><code>1 &lt;= s1.length &lt;= 30</code></li> <li><code>s1</code> \u200b\u548c\u200b <code>s2</code> \u200b\u7531\u200b\u5c0f\u5199\u200b\u82f1\u6587\u5b57\u6bcd\u200b\u7ec4\u6210\u200b</li> </ul> <p>Reference</p>","tags":["\u5b57\u7b26\u4e32","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/87/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u8003\u8651\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\\(s_1, s_2\\)\uff0c\u200b\u51fd\u6570\u200b\\(f(x, y)\\)\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ f(x, y) = \\left\\{ \\begin{aligned}     &amp; T &amp; y\\mathrm{\u200b\u662f\u200b}x\\mathrm{\u200b\u7684\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b} \\\\     &amp; F &amp; y\\mathrm{\u200b\u4e0d\u662f\u200b}x\\mathrm{\u200b\u7684\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b} \\end{aligned} \\right . \\] <p>\u200b\u5219\u200b\u5bf9\u4e8e\u200b\u51fd\u6570\u200b\\(f(x, y)\\)\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ul> <li>\\(f(s_1, s_2) = f(s_2, s_1)\\)</li> <li>\\(f(s_1, s_1) = T\\)</li> <li>\\(f(s_1, s_2) \\land f(s_2, s_3)\\rightarrow f(s_1, s_3)\\)</li> </ul> <p>\u200b\u5373\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b\u5173\u7cfb\u200b\u662f\u200b\u7b49\u4ef7\u5173\u7cfb\u200b\u3002</p> <p>\u200b\u5b9a\u4e49\u200b\u53e6\u200b\u4e00\u79cd\u200b\u7b49\u4ef7\u5173\u7cfb\u200b\\(g(x, y)\\)\uff1a</p> \\[ g(x, y) = \\left\\{ \\begin{aligned}     &amp; T &amp; x, y \\mathrm{\u200b\u5404\u200b\u5b57\u6bcd\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u76f8\u540c\u200b} \\\\     &amp; F &amp; x, y \\mathrm{\u200b\u5404\u200b\u5b57\u6bcd\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u4e0d\u540c\u200b} \\end{aligned} \\right . \\] <p>\u200b\u6839\u636e\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b\u64cd\u4f5c\u200b \u200b\u4e0d\u4f1a\u200b \u200b\u6539\u53d8\u200b</p> <ol> <li>\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u957f\u5ea6\u200b</li> <li>\u200b\u5b57\u7b26\u4e32\u200b\u5185\u200b\u5404\u200b\u5b57\u6bcd\u200b\u7684\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b</li> </ol> <p>\u200b\u5982\u679c\u200b\\(s_1, s_2\\)\u200b\u7684\u200b\u957f\u5ea6\u200b\u4e0d\u200b\u5339\u914d\u200b\u6216\u200b\u5b57\u6bcd\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u4e0d\u200b\u5339\u914d\u200b\uff0c\\(f(s_1, s_2)\\)\u200b\u5373\u53ef\u200b\u63d0\u524d\u200b\u8fd4\u56de\u200b\\(F\\)\u3002</p> <p>\u200b\u8003\u8651\u200b\u5206\u5272\u200b\u540e\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u5c06\u200b\\(s_1\\)\u200b\u5728\u200b\u4e0b\u6807\u200b\\(i\\)\u200b\u4e0e\u200b\u4e0b\u6807\u200b\\(i + 1\\)\u200b\u5904\u200b\u5206\u5272\u200b\u5f97\u5230\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u79f0\u4e3a\u200b \\(s_{1, i}, s'_{1, i}\\)\uff0c\\(s_{1, i}\\)\u200b\u4f4d\u4e8e\u200b\u5de6\u4fa7\u200b\uff0c\\(s'_{1, i}\\)\u200b\u4f4d\u4e8e\u200b\u53f3\u4fa7\u200b\u3002\u200b\u5bf9\u4e8e\u200b\\(s_2\\)\uff0c\u200b\u6709\u200b\u540c\u6837\u200b\u7684\u200b\u5206\u5272\u200b \u200b\u65b9\u6cd5\u200b\u5f97\u5230\u200b\\(s_{2, i}, s'_{2, i}\\)\u3002\u200b\u8bbe\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(s\\)\uff0c\u200b\u4ee4\u200b\\(-i\\)\u200b\u4e3a\u200b\u4ece\u200b\u5b57\u7b26\u4e32\u200b\u53f3\u4fa7\u200b\u5f00\u59cb\u200b\u7684\u200b\u4e0b\u6807\u200b\\(i\\)\u3002</p> <p>\u200b\u6839\u636e\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u82e5\u4efb\u200b\u4e00\u7ec4\u200b\u5206\u5272\u200b\u540e\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u6ee1\u8db3\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5173\u7cfb\u200b\uff0c\u200b\u5219\u200b\u539f\u6765\u200b\u7684\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u6ee1\u8db3\u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5173\u7cfb\u200b \uff0c\u200b\u5e76\u4e14\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5173\u7cfb\u200b\u4e0e\u200b\u539f\u200b\u5b57\u7b26\u4e32\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5173\u7cfb\u200b\u65e0\u5173\u200b\uff0c\u200b\u4e0d\u4f1a\u200b\u76f8\u4e92\u5f71\u54cd\u200b\u3002\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u91c7\u7528\u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\uff0c\u200b\u5c06\u200b\u539f\u200b \u200b\u95ee\u9898\u200b\u5206\u5272\u200b\u4e3a\u200b\u4e00\u7cfb\u5217\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u5b50\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u7531\u4e8e\u200b\\(f(s_1, s_2)\\)\u200b\u5b58\u5728\u200b\u5bf9\u79f0\u6027\u200b\uff0c\u200b\u4e0d\u59a8\u200b\u56fa\u5b9a\u200b\\(s_1\\)\u200b\u7684\u200b\u5206\u5272\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u8003\u8651\u200b\\(s_2\\)\u200b\u7684\u200b\u5206\u5272\u200b\u65b9\u5f0f\u200b\u3002\\(s_2\\)\u200b\u6709\u200b\u4e24\u79cd\u200b\u5206\u5272\u200b \u200b\u65b9\u5f0f\u200b\uff0c\u200b\u5373\u200b\u201c\u200b\u4fdd\u6301\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u987a\u5e8f\u200b\u4e0d\u53d8\u200b\u201d\u200b\u6216\u200b\u201c\u200b\u4ea4\u6362\u200b\u4e24\u4e2a\u200b\u5b50\u200b\u5b57\u7b26\u4e32\u200b\u201d\u3002\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u4e8e\u200b\u5982\u4e0b\u200b\u60c5\u51b5\u200b\uff1a</p> <ul> <li>\\(s_{1, i}\\)\u200b\u5bf9\u5e94\u200b\u4e8e\u200b\\(s_{2, i}\\)\uff0c\\(s'_{1, i}\\)\u200b\u5bf9\u5e94\u200b\u4e8e\u200b\\(s'_{2, i}\\)\u3002</li> <li>\\(s_{1, i}\\)\u200b\u5bf9\u5e94\u200b\u4e8e\u200b\\(s'_{2, -i}\\)\uff0c\\(s'_{1, i}\\)\u200b\u5bf9\u5e94\u200b\u4e8e\u200b\\(s_{2, -i}\\)\u3002</li> </ul> <p>\u200b\u5219\u200b\\(f(s_1, s_2)\\)\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u65b9\u5f0f\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u8bbe\u200b\\(\\mathrm{len}(s_1) = \\mathrm{len}(s_2) = s\\)\u3002</p> \\[ f(s_1, s_2) = \\left\\{ \\begin{aligned}     &amp; F &amp; \\lnot g(s_1, s_2) \\\\     &amp; T &amp; s\\leq 3 \\land g(s_1, s_2) \\\\     &amp; \\left(\\bigvee_{i} f(s_{1, i}, s_{2, i})\\land f(s'_{1, i}, s'_{2, i})\\right)     \\lor     \\left(\\bigvee_{i} f(s_{1, i}, s'_{2, -i})\\land f(s'_{1, i}, s_{2, -i})\\right) &amp; s &gt; 3 \\land g(s_1, s_2)\\\\ \\end{aligned} \\right . \\] <p>\u200b\u7531\u6b64\u200b\u5373\u53ef\u200b\u5f97\u51fa\u200b\u521d\u6b65\u200b\u7684\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u4f46\u200b\u591a\u6b21\u200b\u9012\u5f52\u200b\u4f1a\u200b\u4f7f\u5f97\u200b\u7b97\u6cd5\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e0a\u5347\u200b\uff0c\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u6570\u7ec4\u200b\u4fdd\u5b58\u200b\u5b50\u200b\u95ee\u9898\u200b\u7684\u200b\u89e3\u200b\u3002\u200b\u5171\u6709\u200b \\(\\mathcal O(n^3)\\)\u200b\u89c4\u6a21\u200b\u7684\u200b\u5b50\u200b\u95ee\u9898\u200b\uff0c\u200b\u56e0\u6b64\u200b\u9700\u8981\u200b\u82b1\u8d39\u200b\\(\\mathcal O(n^3)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u590d\u6742\u5ea6\u200b\u4e0e\u200b\u7a7a\u95f4\u200b\u590d\u6742\u5ea6\u200b\u3002</p> <pre><code>inline int hash(char *s, int len)\n{\n    int i, hashMap[26] = {0};\n    unsigned long long ret = 0;\n    for (i = 0; i &lt; len; i++)\n        hashMap[s[i] - 'a']++;\n    for (i = 0; i &lt; 26; i++)\n    {\n        ret *= 31;\n        ret += hashMap[i] + 1;\n        if (ret &gt; INT_MAX)\n            ret = ret &amp; INT_MAX;\n    }\n    return (int)ret;\n}\n\nbool helper(char *s1, int offset1, char *s2, int offset2, int len, int ***dp)\n{\n    if (dp[offset1][offset2][len - 1] &gt;= 0)\n        return dp[offset1][offset2][len - 1];\n    else if (len &lt; 4)\n    {\n        dp[offset1][offset2][len - 1] = hash(s1 + offset1, len) == hash(s2 + offset2, len);\n        return dp[offset1][offset2][len - 1];\n    }\n    int flag = false, a, i;\n    dp[offset1][offset2][len - 1] = 0;\n    for (i = 1; i &lt; len; i++)\n    {\n        a = hash(s1 + offset1, i);\n        if (a == hash(s2 + offset2, i))\n            flag = flag || (\n                helper(s1, offset1, s2, offset2, i, dp) &amp;&amp; \n                helper(s1, offset1 + i, s2, offset2 + i, len - i, dp)\n            );\n        if (!flag &amp;&amp; a == hash(s2 + offset2 + len - i, i))\n            flag = flag || (\n                helper(s1, offset1, s2, offset2 + len - i, i, dp) &amp;&amp; \n                helper(s1, offset1 + i, s2, offset2, len - i, dp)\n            );\n        if (flag)\n            break;\n    }\n    dp[offset1][offset2][len - 1] = flag;\n    return dp[offset1][offset2][len - 1];\n}\n\nbool isScramble(char * s1, char * s2){\n    int len = strlen(s1), i, j,\n        ***dp = (int ***)malloc(sizeof(int **) * len),\n        **_dp = (int **)malloc(sizeof(int *) * len * len),\n        *__dp = (int *)malloc(sizeof(int) * len * len * len);\n    memset(__dp, 0xff, sizeof(int) * len * len * len);\n    for (i = 0; i &lt; len; i++)\n    {\n        dp[i] = _dp + i * len;\n        for (j = 0; j &lt; len; j++)\n            dp[i][j] = __dp + i * len * len + j * len;\n    }\n    return helper(s1, 0, s2, 0, strlen(s1), dp);\n}\n</code></pre>","tags":["\u5b57\u7b26\u4e32","\u52a8\u6001\u89c4\u5212"]},{"location":"coding/leetcode/88/","title":"88. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u6570\u7ec4","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/88/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b\u00a0<code>nums1</code> \u200b\u548c\u200b <code>nums2</code>\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u5c06\u200b <code>nums2</code> \u200b\u5408\u5e76\u200b\u5230\u200b\u00a0<code>nums1</code>\u00a0\u200b\u4e2d\u200b\uff0c\u200b\u4f7f\u200b <code>nums1</code> \u200b\u6210\u4e3a\u200b\u4e00\u4e2a\u200b\u6709\u5e8f\u200b\u6570\u7ec4\u200b\u3002</p> <p>\u200b\u8bf4\u660e\u200b:</p> <ul> <li>\u200b\u521d\u59cb\u5316\u200b\u00a0<code>nums1</code> \u200b\u548c\u200b <code>nums2</code> \u200b\u7684\u200b\u5143\u7d20\u200b\u6570\u91cf\u200b\u5206\u522b\u200b\u4e3a\u200b\u00a0<code>m</code> \u200b\u548c\u200b <code>n</code> \u3002</li> <li>\u200b\u4f60\u200b\u53ef\u4ee5\u200b\u5047\u8bbe\u200b\u00a0<code>nums1</code>\u00a0\u200b\u6709\u200b\u8db3\u591f\u200b\u7684\u200b\u7a7a\u95f4\u200b\uff08\u200b\u7a7a\u95f4\u200b\u5927\u5c0f\u200b\u5927\u4e8e\u200b\u6216\u200b\u7b49\u4e8e\u200b\u00a0<code>m + n</code>\uff09\u200b\u6765\u200b\u4fdd\u5b58\u200b <code>nums2</code> \u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\nnums1 = [1,2,3,0,0,0], m = 3\nnums2 = [2,5,6],       n = 3\n\n\u200b\u8f93\u51fa\u200b:\u00a0[1,2,2,3,5,6]\n</code></pre> <p>Reference</p>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/88/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u76f4\u63a5\u200b\u5728\u200b<code>nums1</code>\u200b\u7684\u200b\u5185\u5b58\u7a7a\u95f4\u200b\u4e0a\u200b\u4ece\u200b\u540e\u200b\u5411\u524d\u200b\u539f\u4f4d\u200b\u5408\u5e76\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>#define MAX(x, y) (x &gt; y ? x : y)\nvoid merge(int* nums1, int nums1Size, int m, int* nums2, int nums2Size, int n){\n    int x = m - 1, y = n - 1;\n    while(y &gt;= 0 &amp;&amp; x &gt;= 0)\n    {\n        nums1[x + y + 1] = MAX(nums1[x], nums2[y]);\n        if (nums1[x] &gt; nums2[y])\n            x--;\n        else\n            y--;\n    }\n    if (y &gt;= 0)\n        memcpy(nums1, nums2, sizeof(int) * (y + 1));\n}\n</code></pre>","tags":["\u6570\u7ec4","\u53cc\u6307\u9488"]},{"location":"coding/leetcode/9/","title":"9. \u200b\u56de\u6587\u200b\u6570","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/9/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u5224\u65ad\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u662f\u5426\u662f\u200b\u56de\u6587\u200b\u6570\u200b\u3002\u200b\u56de\u6587\u200b\u6570\u662f\u200b\u6307\u6b63\u200b\u5e8f\u200b\uff08\u200b\u4ece\u200b\u5de6\u5411\u53f3\u200b\uff09\u200b\u548c\u200b\u5012\u5e8f\u200b\uff08\u200b\u4ece\u53f3\u200b\u5411\u200b\u5de6\u200b\uff09\u200b\u8bfb\u200b\u90fd\u200b\u662f\u200b\u4e00\u6837\u200b\u7684\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 121\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: -121\n\u200b\u8f93\u51fa\u200b: false\n\u200b\u89e3\u91ca\u200b: \u200b\u4ece\u200b\u5de6\u5411\u53f3\u200b\u8bfb\u200b, \u200b\u4e3a\u200b -121 \u3002 \u200b\u4ece\u53f3\u200b\u5411\u200b\u5de6\u8bfb\u200b, \u200b\u4e3a\u200b 121- \u3002\u200b\u56e0\u6b64\u200b\u5b83\u200b\u4e0d\u662f\u200b\u4e00\u4e2a\u200b\u56de\u6587\u200b\u6570\u200b\u3002\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b: 10\n\u200b\u8f93\u51fa\u200b: false\n\u200b\u89e3\u91ca\u200b: \u200b\u4ece\u53f3\u200b\u5411\u200b\u5de6\u8bfb\u200b, \u200b\u4e3a\u200b 01 \u3002\u200b\u56e0\u6b64\u200b\u5b83\u200b\u4e0d\u662f\u200b\u4e00\u4e2a\u200b\u56de\u6587\u200b\u6570\u200b\u3002\n</code></pre> <p>\u200b\u8fdb\u9636\u200b\uff1a</p> <p>\u200b\u4f60\u200b\u80fd\u200b\u4e0d\u200b\u5c06\u200b\u6574\u6570\u200b\u8f6c\u4e3a\u200b\u5b57\u7b26\u4e32\u200b\u6765\u200b\u89e3\u51b3\u200b\u8fd9\u4e2a\u200b\u95ee\u9898\u200b\u5417\u200b\uff1f</p> <p>Reference</p>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/9/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a </p> <ol> <li>\u200b\u5c0f\u4e8e\u200b0\u200b\u7684\u200b\u6570\u200b\u5fc5\u7136\u200b\u4e0d\u662f\u200b\u56de\u6587\u200b\u6570\u200b\uff1b</li> <li>\u200b\u5bf9\u4e8e\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\uff0c\u200b\u53ea\u200b\u9700\u8981\u200b\u5224\u65ad\u200b\u6309\u4f4d\u200b\u53cd\u8f6c\u200b\u540e\u200b\u7684\u200b\u6570\u503c\u200b\u4e0e\u200b\u539f\u200b\u6570\u503c\u200b\u662f\u5426\u200b\u76f8\u7b49\u200b\u5373\u53ef\u200b\u3002</li> </ol> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>bool isPalindrome(int x){\n    if (x &lt; 0)\n        return false;\n    long transformed = 0, original = x;\n    while (x)\n    {\n        transformed *= 10;\n        transformed += x % 10;\n        x /= 10;\n    }\n    return transformed == original;\n}\n</code></pre>","tags":["\u6570\u5b66"]},{"location":"coding/leetcode/90/","title":"90. \u200b\u5b50\u96c6\u200b II","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/90/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\u6570\u7ec4\u200b <code>nums</code> \uff0c\u200b\u5176\u4e2d\u200b\u53ef\u80fd\u200b\u5305\u542b\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b\uff0c\u200b\u8bf7\u200b\u4f60\u200b\u8fd4\u56de\u200b\u8be5\u200b\u6570\u7ec4\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u7684\u200b\u5b50\u96c6\u200b\uff08\u200b\u5e42\u96c6\u200b\uff09\u3002</p> <p>\u200b\u89e3\u96c6\u200b \u200b\u4e0d\u80fd\u200b \u200b\u5305\u542b\u200b\u91cd\u590d\u200b\u7684\u200b\u5b50\u96c6\u200b\u3002\u200b\u8fd4\u56de\u200b\u7684\u200b\u89e3\u200b\u96c6\u4e2d\u200b\uff0c\u200b\u5b50\u96c6\u200b\u53ef\u4ee5\u200b\u6309\u200b \u200b\u4efb\u610f\u200b\u987a\u5e8f\u200b \u200b\u6392\u5217\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [1,2,2]\n\u200b\u8f93\u51fa\u200b\uff1a[[],[1],[1,2],[1,2,2],[2],[2,2]]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1anums = [0]\n\u200b\u8f93\u51fa\u200b\uff1a[[],[0]]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>1 &lt;= nums.length &lt;= 10</code></li> <li><code>-10 &lt;= nums[i] &lt;= 10</code></li> </ul> <p>Reference</p>","tags":["\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/90/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5148\u200b\u7edf\u8ba1\u200b\u96c6\u5408\u200b\u4e2d\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u7684\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\uff0c\u200b\u518d\u200b\u6839\u636e\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u8fdb\u884c\u200b\u56de\u6eaf\u200b\uff0c\u200b\u6700\u540e\u200b\u91cd\u65b0\u200b\u751f\u6210\u200b\u6bcf\u4e2a\u200b\u96c6\u5408\u200b\u3002</p> <pre><code>/**\n * Return an array of arrays of size *returnSize.\n * The sizes of the arrays are returned as *returnColumnSizes array.\n * Note: Both returned array and *columnSizes array must be malloced, assume caller calls free().\n */\nint *convert(int *count, int countLen, int *returnSize)\n{\n    int i = 0, j = 0, k = 0;\n    *returnSize = 0;\n    for (i = 0; i &lt; countLen; i++)\n        (*returnSize) += count[i];\n    int *ret = (int *)malloc(sizeof(int) * *returnSize);\n    for (i = 0; i &lt; countLen; i++)\n        for (j = 0; j &lt; count[i]; j++)\n            ret[k++] = i - 10;\n    return ret;\n}\n\nvoid dfs(int **ret, int *retSize, int *count, int *current, int currentPos, int maxPos, int *number)\n{\n    if (currentPos == maxPos)\n    {\n        ret[*number] = convert(current, maxPos, retSize + *number);\n        (*number)++;\n    }\n    else\n    {\n        if (count[currentPos] == 0)\n            dfs(ret, retSize, count, current, currentPos + 1, maxPos, number);\n        else\n        {\n            int i = 0;\n            for (i = 0; i &lt;= count[currentPos]; i++)\n            {\n                current[currentPos] = i;\n                dfs(ret, retSize, count, current, currentPos + 1, maxPos, number);\n                current[currentPos] = 0;\n            }\n        }\n    }\n}\n\nint** subsetsWithDup(int* nums, int numsSize, int* returnSize, int** returnColumnSizes){\n    int *count = (int *)malloc(sizeof(int) * 21), *buffer = (int *)malloc(sizeof(int) * 21),\n        i = 0, **ret = NULL;\n    memset(count, 0, sizeof(int) * 21);\n    memset(buffer, 0, sizeof(int) * 21);\n    *returnSize = 1;\n    for (i = 0; i &lt; numsSize; i++)\n        count[nums[i] + 10]++;\n    for (i = 0; i &lt; 21; i++)\n        *returnSize *= count[i] + 1;\n    *returnColumnSizes = (int *)malloc(sizeof(int) * *returnSize);\n    ret = (int **)malloc(sizeof(int *) * *returnSize);\n    i = 0;\n    dfs(ret, *returnColumnSizes, count, buffer, 0, 21, &amp;i);\n    return ret;\n}\n</code></pre>","tags":["\u6570\u7ec4","\u56de\u6eaf\u7b97\u6cd5"]},{"location":"coding/leetcode/917/","title":"917. \u200b\u4ec5\u4ec5\u200b\u53cd\u8f6c\u200b\u5b57\u6bcd","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u7b80\u5355\u200b</p>","tags":["\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/917/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u00a0<code>S</code>\uff0c\u200b\u8fd4\u56de\u200b\u00a0\u201c\u200b\u53cd\u8f6c\u200b\u540e\u200b\u7684\u200b\u201d\u00a0\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u5176\u4e2d\u200b\u4e0d\u662f\u200b\u5b57\u6bcd\u200b\u7684\u200b\u5b57\u7b26\u200b\u90fd\u200b\u4fdd\u7559\u200b\u5728\u200b\u539f\u5730\u200b\uff0c\u200b\u800c\u200b\u6240\u6709\u200b\u5b57\u6bcd\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u53d1\u751f\u200b\u53cd\u8f6c\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a\"ab-cd\"\n\u200b\u8f93\u51fa\u200b\uff1a\"dc-ba\"\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a\"a-bC-dEf-ghIj\"\n\u200b\u8f93\u51fa\u200b\uff1a\"j-Ih-gfE-dCba\"\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 3\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1a\"Test1ng-Leet=code-Q!\"\n\u200b\u8f93\u51fa\u200b\uff1a\"Qedo1ct-eeLg=ntse-T!\"\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li><code>S.length &lt;= 100</code></li> <li><code>33 &lt;= S[i].ASCIIcode &lt;= 122</code></li> <li><code>S</code> \u200b\u4e2d\u200b\u4e0d\u200b\u5305\u542b\u200b\u00a0<code>\\</code> or <code>\"</code></li> </ul> <p>Reference</p>","tags":["\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/917/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u548c\u200b\u53cd\u8f6c\u200b\u5b57\u7b26\u4e32\u200b\u4f7f\u7528\u200b\u76f8\u540c\u200b\u7684\u200b\u601d\u8def\u200b\uff0c\u200b\u53ea\u200b\u9700\u200b\u52a0\u4e0a\u200b\u5bf9\u200b\u5f53\u524d\u200b\u5b57\u7b26\u200b\u7684\u200b\u5224\u65ad\u200b\u5373\u53ef\u200b\u3002</p> <pre><code>bool isChar(char *s)\n{\n    return (*s &gt;= 'a' &amp;&amp; *s &lt;= 'z') || (*s &gt;= 'A' &amp;&amp; *s &lt;= 'Z');\n}\nchar * reverseOnlyLetters(char * S) {\n    int len = strlen(S);\n    char *head = S, *tail = S + len - 1, temp;\n    while (tail &gt; head)\n    {\n        while (!isChar(head) &amp;&amp; head &lt; tail)\n            head++;\n        while (!isChar(tail) &amp;&amp; head &lt; tail)\n            tail--;\n        temp = *head;\n        *head = *tail;\n        *tail = temp;\n        head++;\n        tail--;\n    }\n    return S;\n}\n</code></pre>","tags":["\u5b57\u7b26\u4e32"]},{"location":"coding/leetcode/92/","title":"92. \u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b II","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/92/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u200b\u4f60\u200b\u5355\u94fe\u200b\u8868\u200b\u7684\u200b\u5934\u200b\u8282\u70b9\u200b\u00a0<code>head</code> \u200b\u548c\u200b\u4e24\u4e2a\u200b\u6574\u6570\u200b\u00a0<code>left</code> \u200b\u548c\u200b <code>right</code> \uff0c\u200b\u5176\u4e2d\u200b\u00a0<code>left &lt;= right</code> \u3002\u200b\u8bf7\u200b\u4f60\u200b\u53cd\u8f6c\u200b\u4ece\u200b\u4f4d\u7f6e\u200b <code>left</code> \u200b\u5230\u200b\u4f4d\u7f6e\u200b <code>right</code> \u200b\u7684\u200b\u94fe\u8868\u200b\u8282\u70b9\u200b\uff0c\u200b\u8fd4\u56de\u200b \u200b\u53cd\u8f6c\u200b\u540e\u200b\u7684\u200b\u94fe\u8868\u200b \u3002</p> <p></p> <p>\u200b\u793a\u4f8b\u200b 1\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [1,2,3,4,5], left = 2, right = 4\n\u200b\u8f93\u51fa\u200b\uff1a[1,4,3,2,5]\n</code></pre> <p>\u200b\u793a\u4f8b\u200b 2\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b\uff1ahead = [5], left = 1, right = 1\n\u200b\u8f93\u51fa\u200b\uff1a[5]\n</code></pre> <p>\u200b\u63d0\u793a\u200b\uff1a</p> <ul> <li>\u200b\u94fe\u8868\u200b\u4e2d\u200b\u8282\u70b9\u200b\u6570\u76ee\u200b\u4e3a\u200b <code>n</code></li> <li><code>1 &lt;= n &lt;= 500</code></li> <li><code>-500 &lt;= Node.val &lt;= 500</code></li> <li><code>1 &lt;= left &lt;= right &lt;= n</code></li> </ul> <p>Reference</p>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/92/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u5148\u200b\u8003\u8651\u200b\u53cd\u8f6c\u200b\u6574\u4e2a\u200b\u94fe\u8868\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e09\u200b\u6307\u9488\u200b\u6cd5\u5728\u200b\\(\\mathcal O(N)\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u5185\u200b\u4f7f\u7528\u200b\\(\\mathcal O(1)\\)\u200b\u7684\u200b\u7a7a\u95f4\u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b</p> <p>\u200b\u94fe\u8868\u200b\u53cd\u8f6c\u200b\u7684\u200b\u4e09\u200b\u6307\u9488\u200b\u6cd5\u200b\uff1a\u200b\u5b9a\u4e49\u200b<code>cur1, cur2, cur3</code>\u200b\u4e09\u4e2a\u200b\u6307\u9488\u200b\uff0c\u200b\u5206\u522b\u200b\u6307\u5411\u200b\u94fe\u8868\u200b\u7684\u200b\u5934\u200b\u8282\u70b9\u200b\u3001\u200b\u7b2c\u4e8c\u4e2a\u200b\u8282\u70b9\u200b\u4e0e\u200b\u7b2c\u4e09\u4e2a\u200b\u8282\u70b9\u200b\u3002\u200b\u6307\u9488\u200b<code>cur1, cur2</code>\u200b\u7528\u4e8e\u200b\u4ea4\u6362\u200b\u94fe\u8868\u200b\u8282\u70b9\u200b\u7684\u200b\u8fde\u63a5\u200b\u65b9\u5411\u200b\u3002\u200b\u7531\u4e8e\u200b\u4ea4\u6362\u200b\u8282\u70b9\u200b\u64cd\u4f5c\u200b\u7834\u574f\u200b\u4e86\u200b\u539f\u6709\u200b\u7684\u200b\u8fde\u63a5\u7ed3\u6784\u200b\uff0c\u200b\u56e0\u6b64\u200b\u9700\u8981\u200b\u4e00\u4e2a\u200b\u989d\u5916\u200b\u7684\u200b\u6307\u9488\u200b<code>cur3</code>\u200b\u7528\u4e8e\u200b\u79fb\u52a8\u200b\u6307\u9488\u200b\u3002</p> <p>\u200b\u4e09\u200b\u6307\u9488\u200b\u6cd5\u200b\u7684\u200b\u4e3b\u8981\u200b\u6d41\u7a0b\u200b\u5982\u4e0b\u200b\u56fe\u200b\uff1a</p> <ol> <li>\u200b\u521d\u59cb\u5316\u200b<code>cur1, cur2, cur3</code>\uff0c\u200b\u7f6e\u200b<code>cur1-&gt;next</code>\u200b\u4e3a\u200b<code>NULL</code></li> <li>\u200b\u4ee4\u200b<code>cur2-&gt;next = cur1</code>\uff0c\u200b\u5b8c\u6210\u200b\u4ea4\u6362\u200b\u64cd\u4f5c\u200b</li> <li>\u200b\u4ee4\u200b<code>cur1 = cur2</code></li> <li>\u200b\u4ee4\u200b<code>cur2 = cur3</code></li> <li>\u200b\u5982\u679c\u200b<code>cur3</code>\u200b\u4e0d\u200b\u4e3a\u200b<code>NULL</code>\uff0c<code>cur3</code>\u200b\u5728\u200b\u94fe\u8868\u200b\u4e0a\u200b\u5411\u524d\u200b\u79fb\u52a8\u200b\uff0c\u200b\u7136\u540e\u200b\u8fd4\u56de\u200b2</li> <li>\u200b\u5982\u679c\u200b<code>cur3</code>\u200b\u4e3a\u200b<code>NULL</code>\uff0c\u200b\u7ed3\u675f\u200b</li> <li>\u200b\u8fd4\u56de\u200b<code>cur2</code></li> </ol> <p></p> <p>\u200b\u53cd\u8f6c\u200b\u7ed9\u5b9a\u200b\u8303\u56f4\u200b\u5185\u200b\u7684\u200b\u94fe\u8868\u200b\uff0c\u200b\u9700\u8981\u200b\u6ce8\u610f\u200b\u5982\u4e0b\u200b\u60c5\u51b5\u200b\uff1a</p> <ul> <li>\u200b\u5982\u679c\u200b\u53cd\u8f6c\u200b\u533a\u95f4\u200b\u4ece\u200b\u7b2c\u4e00\u4e2a\u200b\u8282\u70b9\u200b\u5f00\u59cb\u200b\uff0c\u200b\u94fe\u8868\u200b\u7684\u200b\u5934\u200b\u8282\u70b9\u200b\u4f1a\u200b\u6539\u53d8\u200b\uff0c\u200b\u5426\u5219\u200b\u4e0d\u200b\u6539\u53d8\u200b</li> <li>\u200b\u5982\u679c\u200b\u533a\u95f4\u200b\u957f\u5ea6\u200b\u4e3a\u200b1\uff0c\u200b\u5219\u200b\u65e0\u9700\u200b\u53cd\u8f6c\u200b</li> <li>\u200b\u53cd\u8f6c\u200b\u540e\u200b\u9700\u8981\u200b\u66f4\u65b0\u200b\u53cd\u8f6c\u200b\u533a\u95f4\u200b\u524d\u540e\u200b\u7684\u200b\u8282\u70b9\u200b\u8fde\u63a5\u200b\u5173\u7cfb\u200b</li> </ul> <p>\u200b\u6574\u4f53\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for singly-linked list.\n * struct ListNode {\n *     int val;\n *     struct ListNode *next;\n * };\n */\nstruct ListNode* reverseBetween(struct ListNode* head, int left, int right){\n    if (left == right)\n        return head;\n    struct ListNode *cur = head, *prev = NULL;\n    left--;\n    right--;\n    while (left)\n    {\n        left--;\n        right--;\n        prev = cur;\n        cur = cur-&gt;next;\n    }\n    struct ListNode * cur2 = cur-&gt;next, *cur3 = cur2-&gt;next, *tail = cur;\n    while (right)\n    {\n        cur2-&gt;next = cur;\n        cur = cur2;\n        cur2 = cur3;\n        if (cur3)\n            cur3 = cur3-&gt;next;\n        else\n            break;\n        right--;\n    }\n    tail-&gt;next = cur2;\n    if (prev)\n        prev-&gt;next = cur;\n    return prev ? head : cur;\n}\n</code></pre>","tags":["\u94fe\u8868"]},{"location":"coding/leetcode/98/","title":"98. \u200b\u9a8c\u8bc1\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811","text":"<p>\u200b\u96be\u5ea6\u200b\uff1a\u200b\u4e2d\u7b49\u200b</p>","tags":["\u6811","\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u9012\u5f52"]},{"location":"coding/leetcode/98/#_1","title":"\u9898\u76ee","text":"<p>\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u4e8c\u53c9\u6811\u200b\uff0c\u200b\u5224\u65ad\u200b\u5176\u200b\u662f\u5426\u662f\u200b\u4e00\u4e2a\u200b\u6709\u6548\u200b\u7684\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b\u3002</p> <p>\u200b\u5047\u8bbe\u200b\u4e00\u4e2a\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b\u5177\u6709\u200b\u5982\u4e0b\u200b\u7279\u5f81\u200b\uff1a</p> <ul> <li>\u200b\u8282\u70b9\u200b\u7684\u200b\u5de6\u5b50\u200b\u6811\u200b\u53ea\u200b\u5305\u542b\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u7684\u200b\u6570\u200b\u3002</li> <li>\u200b\u8282\u70b9\u200b\u7684\u200b\u53f3\u5b50\u200b\u6811\u200b\u53ea\u200b\u5305\u542b\u200b\u5927\u4e8e\u200b\u5f53\u524d\u200b\u8282\u70b9\u200b\u7684\u200b\u6570\u200b\u3002</li> <li>\u200b\u6240\u6709\u200b\u5de6\u5b50\u200b\u6811\u200b\u548c\u200b\u53f3\u5b50\u200b\u6811\u200b\u81ea\u8eab\u200b\u5fc5\u987b\u200b\u4e5f\u200b\u662f\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b\u3002</li> </ul> <p>\u200b\u793a\u4f8b\u200b\u00a01\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\n    2\n   / \\\n  1   3\n\u200b\u8f93\u51fa\u200b: true\n</code></pre> <p>\u200b\u793a\u4f8b\u200b\u00a02\uff1a</p> <pre><code>\u200b\u8f93\u5165\u200b:\n    5\n   / \\\n  1   4\n\u00a0    / \\\n\u00a0   3   6\n\u200b\u8f93\u51fa\u200b: false\n\u200b\u89e3\u91ca\u200b: \u200b\u8f93\u5165\u200b\u4e3a\u200b: [5,1,4,null,null,3,6]\u3002\n\u00a0    \u200b\u6839\u200b\u8282\u70b9\u200b\u7684\u200b\u503c\u200b\u4e3a\u200b 5 \uff0c\u200b\u4f46\u662f\u200b\u5176\u53f3\u5b50\u200b\u8282\u70b9\u200b\u503c\u4e3a\u200b 4 \u3002\n</code></pre> <p>Reference</p>","tags":["\u6811","\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u9012\u5f52"]},{"location":"coding/leetcode/98/#_2","title":"\u9898\u89e3","text":"<p>\u200b\u57fa\u672c\u601d\u8def\u200b\uff1a \u200b\u5c06\u200b\u4e8c\u53c9\u200b\u67e5\u627e\u200b\u6811\u4e2d\u5e8f\u200b\u5c55\u5f00\u200b\u4e3a\u200b\u94fe\u8868\u200b\uff0c\u200b\u5224\u65ad\u200b\u94fe\u8868\u200b\u662f\u5426\u662f\u200b\u6709\u5e8f\u200b\u7684\u200b\u3002</p> <p>\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>/**\n * Definition for a binary tree node.\n * struct TreeNode {\n *     int val;\n *     struct TreeNode *left;\n *     struct TreeNode *right;\n * };\n */\nstruct ListNode *insert(struct ListNode *List, int val)\n{\n    struct ListNode *ret = malloc(sizeof(struct ListNode));\n    ret-&gt;val = val;\n    ret-&gt;next = List;\n    return ret;\n}\nstruct ListNode *toList(struct TreeNode *root, struct ListNode *List)\n{\n    if (root == NULL)\n        return List;\n    List = toList(root-&gt;right, List);\n    List = insert(List, root-&gt;val);\n    List = toList(root-&gt;left, List);\n    return List;\n}\nbool isValidBST(struct TreeNode* root){\n    if (root == NULL)\n        return true;\n    struct ListNode *rec = toList(root, NULL), *cur = rec;\n    do\n    {\n        if (cur-&gt;next == NULL)\n            break;\n        if (cur-&gt;val &gt;= cur-&gt;next-&gt;val)\n            return false;\n        cur = cur-&gt;next;\n    }\n    while (true);\n    return true;\n}\n</code></pre>","tags":["\u6811","\u6df1\u5ea6\u4f18\u5148\u641c\u7d22","\u9012\u5f52"]},{"location":"coding/leetcode/tags/","title":"Leetcode \u200b\u9898\u76ee","text":"<p>\u200b\u70b9\u51fb\u200b\u53f3\u4fa7\u200b\u76ee\u5f55\u200b\u53ef\u4ee5\u200b\u8df3\u8f6c\u200b\u5230\u200b\u5bf9\u5e94\u200b\u7684\u200b\u6807\u7b7e\u200b</p>"},{"location":"coding/leetcode/tags/#_1","title":"\u6570\u7ec4","text":"<p>\u200b\u6570\u7ec4\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b31\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6570\u7ec4\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 1. \u200b\u4e24\u6570\u200b\u4e4b\u200b\u548c\u200b 26. \u200b\u5220\u9664\u200b\u6392\u5e8f\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u91cd\u590d\u200b\u9879\u200b 27. \u200b\u79fb\u9664\u200b\u5143\u7d20\u200b 35. \u200b\u641c\u7d22\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b 42. \u200b\u63a5\u200b\u96e8\u6c34\u200b 48. \u200b\u65cb\u8f6c\u200b\u56fe\u50cf\u200b 53. \u200b\u6700\u5927\u200b\u5b50\u5e8f\u200b\u548c\u200b 55. \u200b\u8df3\u8dc3\u200b\u6e38\u620f\u200b 56. \u200b\u5408\u5e76\u200b\u533a\u95f4\u200b 59. \u200b\u87ba\u65cb\u200b\u77e9\u9635\u200b II 62. \u200b\u4e0d\u540c\u200b\u8def\u5f84\u200b 64. \u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b 66. \u200b\u52a0\u4e00\u200b 73. \u200b\u77e9\u9635\u200b\u7f6e\u200b\u96f6\u200b 74. \u200b\u641c\u7d22\u200b\u4e8c\u7ef4\u200b\u77e9\u9635\u200b 75. \u200b\u989c\u8272\u200b\u5206\u7c7b\u200b 78. \u200b\u5b50\u96c6\u200b 79. \u200b\u5355\u8bcd\u200b\u641c\u7d22\u200b 88. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u6570\u7ec4\u200b 90. \u200b\u5b50\u96c6\u200b II 118. \u200b\u6768\u8f89\u4e09\u89d2\u200b 119. \u200b\u6768\u8f89\u4e09\u89d2\u200b II 120. \u200b\u4e09\u89d2\u5f62\u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b 121. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b 122. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b II 189. \u200b\u65cb\u8f6c\u200b\u6570\u7ec4\u200b 217. \u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b 384. \u200b\u6253\u4e71\u200b\u6570\u7ec4\u200b 509. \u200b\u6590\u6ce2\u200b\u90a3\u200b\u5951\u6570\u200b 1011. \u200b\u5728\u200b D \u200b\u5929\u200b\u5185\u200b\u9001\u8fbe\u200b\u5305\u88f9\u200b\u7684\u200b\u80fd\u529b\u200b 1300. \u200b\u8f6c\u53d8\u200b\u6570\u7ec4\u200b\u540e\u200b\u6700\u200b\u63a5\u8fd1\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u6570\u7ec4\u200b\u548c"},{"location":"coding/leetcode/tags/#_2","title":"\u5b57\u7b26\u4e32","text":"<p>\u200b\u5b57\u7b26\u4e32\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b16\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u5b57\u7b26\u4e32\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 3. \u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b 8. \u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u200b\u6574\u6570\u200b (atoi) 13. \u200b\u7f57\u9a6c\u6570\u5b57\u200b\u8f6c\u200b\u6574\u6570\u200b 14. \u200b\u6700\u957f\u200b\u516c\u5171\u200b\u524d\u7f00\u200b 17. \u200b\u7535\u8bdd\u53f7\u7801\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408\u200b 20. \u200b\u6709\u6548\u200b\u7684\u200b\u62ec\u53f7\u200b 22. \u200b\u62ec\u53f7\u200b\u751f\u6210\u200b 28. \u200b\u5b9e\u73b0\u200b strStr() 38. \u200b\u5916\u89c2\u200b\u6570\u5217\u200b 43. \u200b\u5b57\u7b26\u4e32\u200b\u76f8\u4e58\u200b 87. \u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b 115. \u200b\u4e0d\u540c\u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217\u200b 125. \u200b\u9a8c\u8bc1\u200b\u56de\u6587\u200b\u4e32\u200b 344. \u200b\u53cd\u8f6c\u200b\u5b57\u7b26\u4e32\u200b 387. \u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u552f\u4e00\u200b\u5b57\u7b26\u200b 917. \u200b\u4ec5\u4ec5\u200b\u53cd\u8f6c\u200b\u5b57\u6bcd"},{"location":"coding/leetcode/tags/#_3","title":"\u6570\u5b66","text":"<p>\u200b\u6570\u5b66\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b15\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6570\u5b66\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 2. \u200b\u4e24\u200b\u6570\u200b\u76f8\u52a0\u200b 7. \u200b\u6574\u6570\u200b\u53cd\u8f6c\u200b 8. \u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u200b\u6574\u6570\u200b (atoi) 9. \u200b\u56de\u6587\u200b\u6570\u200b 11. \u200b\u76db\u200b\u6700\u591a\u6c34\u200b\u7684\u200b\u5bb9\u5668\u200b 13. \u200b\u7f57\u9a6c\u6570\u5b57\u200b\u8f6c\u200b\u6574\u6570\u200b 14. \u200b\u6700\u957f\u200b\u516c\u5171\u200b\u524d\u7f00\u200b 43. \u200b\u5b57\u7b26\u4e32\u200b\u76f8\u4e58\u200b 50. Pow(x, n) 69. x \u200b\u7684\u200b\u5e73\u65b9\u6839\u200b 204. \u200b\u8ba1\u6570\u200b\u8d28\u6570\u200b 368. \u200b\u6700\u5927\u200b\u6574\u9664\u200b\u5b50\u96c6\u200b 781. \u200b\u68ee\u6797\u200b\u4e2d\u200b\u7684\u200b\u5154\u5b50\u200b 1006. \u200b\u7b28\u200b\u9636\u4e58\u200b 1835. \u200b\u6240\u6709\u200b\u6570\u5bf9\u200b\u6309\u4f4d\u200b\u4e0e\u200b\u7ed3\u679c\u200b\u7684\u200b\u5f02\u6216\u200b\u548c"},{"location":"coding/leetcode/tags/#_4","title":"\u53cc\u200b\u6307\u9488","text":"<p>\u200b\u53cc\u200b\u6307\u9488\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b14\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u53cc\u200b\u6307\u9488\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 3. \u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b 11. \u200b\u76db\u200b\u6700\u591a\u6c34\u200b\u7684\u200b\u5bb9\u5668\u200b 19. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u7684\u200b\u5012\u6570\u7b2c\u200bN\u200b\u4e2a\u200b\u8282\u70b9\u200b 26. \u200b\u5220\u9664\u200b\u6392\u5e8f\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u91cd\u590d\u200b\u9879\u200b 27. \u200b\u79fb\u9664\u200b\u5143\u7d20\u200b 28. \u200b\u5b9e\u73b0\u200b strStr() 42. \u200b\u63a5\u200b\u96e8\u6c34\u200b 61. \u200b\u65cb\u8f6c\u200b\u94fe\u8868\u200b 75. \u200b\u989c\u8272\u200b\u5206\u7c7b\u200b 88. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u6570\u7ec4\u200b 125. \u200b\u9a8c\u8bc1\u200b\u56de\u6587\u200b\u4e32\u200b 234. \u200b\u56de\u6587\u200b\u94fe\u8868\u200b 344. \u200b\u53cd\u8f6c\u200b\u5b57\u7b26\u4e32\u200b 350. \u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u7684\u200b\u4ea4\u96c6\u200b II"},{"location":"coding/leetcode/tags/#_5","title":"\u52a8\u6001\u200b\u89c4\u5212","text":"<p>\u200b\u52a8\u6001\u200b\u89c4\u5212\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b13\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u52a8\u6001\u200b\u89c4\u5212\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 42. \u200b\u63a5\u200b\u96e8\u6c34\u200b 53. \u200b\u6700\u5927\u200b\u5b50\u5e8f\u200b\u548c\u200b 62. \u200b\u4e0d\u540c\u200b\u8def\u5f84\u200b 64. \u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b 70. \u200b\u722c\u697c\u68af\u200b 87. \u200b\u6270\u4e71\u200b\u5b57\u7b26\u4e32\u200b 115. \u200b\u4e0d\u540c\u200b\u7684\u200b\u5b50\u200b\u5e8f\u5217\u200b 120. \u200b\u4e09\u89d2\u5f62\u200b\u6700\u5c0f\u200b\u8def\u5f84\u200b\u548c\u200b 121. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b 198. \u200b\u6253\u5bb6\u52ab\u820d\u200b 368. \u200b\u6700\u5927\u200b\u6574\u9664\u200b\u5b50\u96c6\u200b 377. \u200b\u7ec4\u5408\u200b\u603b\u548c\u200b \u2163 403. \u200b\u9752\u86d9\u200b\u8fc7\u6cb3"},{"location":"coding/leetcode/tags/#_6","title":"\u54c8\u5e0c\u200b\u8868","text":"<p>\u200b\u54c8\u5e0c\u200b\u8868\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b12\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u54c8\u5e0c\u200b\u8868\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 1. \u200b\u4e24\u6570\u200b\u4e4b\u200b\u548c\u200b 3. \u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32\u200b 36. \u200b\u6709\u6548\u200b\u7684\u200b\u6570\u72ec\u200b 37. \u200b\u89e3\u6570\u200b\u72ec\u200b 136. \u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u7684\u200b\u6570\u5b57\u200b 204. \u200b\u8ba1\u6570\u200b\u8d28\u6570\u200b 217. \u200b\u5b58\u5728\u200b\u91cd\u590d\u200b\u5143\u7d20\u200b 242. \u200b\u6709\u6548\u200b\u7684\u200b\u5b57\u6bcd\u200b\u5f02\u4f4d\u200b\u8bcd\u200b 350. \u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u7684\u200b\u4ea4\u96c6\u200b II 387. \u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u552f\u4e00\u200b\u5b57\u7b26\u200b 433. \u200b\u6700\u5c0f\u200b\u57fa\u56e0\u200b\u53d8\u5316\u200b 781. \u200b\u68ee\u6797\u200b\u4e2d\u200b\u7684\u200b\u5154\u5b50"},{"location":"coding/leetcode/tags/#_7","title":"\u94fe\u8868","text":"<p>\u200b\u94fe\u8868\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b11\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u94fe\u8868\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 2. \u200b\u4e24\u200b\u6570\u200b\u76f8\u52a0\u200b 19. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u7684\u200b\u5012\u6570\u7b2c\u200bN\u200b\u4e2a\u200b\u8282\u70b9\u200b 21. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u94fe\u8868\u200b 23. \u200b\u5408\u5e76\u200bK\u200b\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868\u200b 24. \u200b\u4e24\u200b\u4e24\u200b\u4ea4\u6362\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9\u200b 25. K \u200b\u4e2a\u200b\u4e00\u7ec4\u200b\u7ffb\u8f6c\u200b\u94fe\u8868\u200b 61. \u200b\u65cb\u8f6c\u200b\u94fe\u8868\u200b 92. \u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b II 206. \u200b\u53cd\u8f6c\u200b\u94fe\u8868\u200b 234. \u200b\u56de\u6587\u200b\u94fe\u8868\u200b 237. \u200b\u5220\u9664\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9"},{"location":"coding/leetcode/tags/#_8","title":"\u4e8c\u5206\u200b\u67e5\u627e","text":"<p>\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b9\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 35. \u200b\u641c\u7d22\u200b\u63d2\u5165\u200b\u4f4d\u7f6e\u200b 50. Pow(x, n) 69. x \u200b\u7684\u200b\u5e73\u65b9\u6839\u200b 74. \u200b\u641c\u7d22\u200b\u4e8c\u7ef4\u200b\u77e9\u9635\u200b 315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570\u200b 350. \u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u7684\u200b\u4ea4\u96c6\u200b II 1011. \u200b\u5728\u200b D \u200b\u5929\u200b\u5185\u200b\u9001\u8fbe\u200b\u5305\u88f9\u200b\u7684\u200b\u80fd\u529b\u200b 1283. \u200b\u4f7f\u200b\u7ed3\u679c\u200b\u4e0d\u200b\u8d85\u8fc7\u200b\u9608\u503c\u200b\u7684\u200b\u6700\u5c0f\u200b\u9664\u6570\u200b 1300. \u200b\u8f6c\u53d8\u200b\u6570\u7ec4\u200b\u540e\u200b\u6700\u200b\u63a5\u8fd1\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u6570\u7ec4\u200b\u548c"},{"location":"coding/leetcode/tags/#_9","title":"\u56de\u6eaf\u200b\u7b97\u6cd5","text":"<p>\u200b\u56de\u6eaf\u200b\u7b97\u6cd5\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b7\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u56de\u6eaf\u200b\u7b97\u6cd5\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 17. \u200b\u7535\u8bdd\u53f7\u7801\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408\u200b 22. \u200b\u62ec\u53f7\u200b\u751f\u6210\u200b 37. \u200b\u89e3\u6570\u200b\u72ec\u200b 46. \u200b\u5168\u200b\u6392\u5217\u200b 78. \u200b\u5b50\u96c6\u200b 79. \u200b\u5355\u8bcd\u200b\u641c\u7d22\u200b 90. \u200b\u5b50\u96c6\u200b II"},{"location":"coding/leetcode/tags/#_10","title":"\u9012\u5f52","text":"<p>\u200b\u9012\u5f52\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b6\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u9012\u5f52\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 2. \u200b\u4e24\u200b\u6570\u200b\u76f8\u52a0\u200b 17. \u200b\u7535\u8bdd\u53f7\u7801\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408\u200b 21. \u200b\u5408\u5e76\u200b\u4e24\u4e2a\u200b\u6709\u5e8f\u200b\u94fe\u8868\u200b 24. \u200b\u4e24\u200b\u4e24\u200b\u4ea4\u6362\u200b\u94fe\u8868\u200b\u4e2d\u200b\u7684\u200b\u8282\u70b9\u200b 98. \u200b\u9a8c\u8bc1\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b 779. \u200b\u7b2c\u200bK\u200b\u4e2a\u200b\u8bed\u6cd5\u200b\u7b26\u53f7"},{"location":"coding/leetcode/tags/#_11","title":"\u6808","text":"<p>\u200b\u6808\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b5\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6808\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 20. \u200b\u6709\u6548\u200b\u7684\u200b\u62ec\u53f7\u200b 42. \u200b\u63a5\u200b\u96e8\u6c34\u200b 150. \u200b\u9006\u200b\u6ce2\u5170\u200b\u8868\u8fbe\u5f0f\u200b\u6c42\u503c\u200b 155. \u200b\u6700\u5c0f\u200b\u6808\u200b 341. \u200b\u6241\u5e73\u5316\u200b\u5d4c\u5957\u200b\u5217\u8868\u200b\u8fed\u4ee3\u200b\u5668"},{"location":"coding/leetcode/tags/#_12","title":"\u6392\u5e8f","text":"<p>\u200b\u6392\u5e8f\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b5\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6392\u5e8f\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 56. \u200b\u5408\u5e76\u200b\u533a\u95f4\u200b 75. \u200b\u989c\u8272\u200b\u5206\u7c7b\u200b 242. \u200b\u6709\u6548\u200b\u7684\u200b\u5b57\u6bcd\u200b\u5f02\u4f4d\u200b\u8bcd\u200b 315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570\u200b 350. \u200b\u4e24\u4e2a\u200b\u6570\u7ec4\u200b\u7684\u200b\u4ea4\u96c6\u200b II"},{"location":"coding/leetcode/tags/#_13","title":"\u6df1\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22","text":"<p>\u200b\u6df1\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b4\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6df1\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 17. \u200b\u7535\u8bdd\u53f7\u7801\u200b\u7684\u200b\u5b57\u6bcd\u7ec4\u5408\u200b 98. \u200b\u9a8c\u8bc1\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b 101. \u200b\u5bf9\u79f0\u200b\u4e8c\u53c9\u6811\u200b 488. \u200b\u7956\u739b\u200b\u6e38\u620f"},{"location":"coding/leetcode/tags/#_14","title":"\u6811","text":"<p>\u200b\u6811\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b3\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6811\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 98. \u200b\u9a8c\u8bc1\u200b\u4e8c\u53c9\u200b\u641c\u7d22\u200b\u6811\u200b 101. \u200b\u5bf9\u79f0\u200b\u4e8c\u53c9\u6811\u200b 102. \u200b\u4e8c\u53c9\u6811\u200b\u7684\u200b\u5c42\u5e8f\u200b\u904d\u5386"},{"location":"coding/leetcode/tags/#_15","title":"\u5e7f\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22","text":"<p>\u200b\u5e7f\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b3\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u5e7f\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 101. \u200b\u5bf9\u79f0\u200b\u4e8c\u53c9\u6811\u200b 102. \u200b\u4e8c\u53c9\u6811\u200b\u7684\u200b\u5c42\u5e8f\u200b\u904d\u5386\u200b 433. \u200b\u6700\u5c0f\u200b\u57fa\u56e0\u200b\u53d8\u5316"},{"location":"coding/leetcode/tags/#_16","title":"\u8bbe\u8ba1","text":"<p>\u200b\u8bbe\u8ba1\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b3\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u8bbe\u8ba1\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 155. \u200b\u6700\u5c0f\u200b\u6808\u200b 341. \u200b\u6241\u5e73\u5316\u200b\u5d4c\u5957\u200b\u5217\u8868\u200b\u8fed\u4ee3\u200b\u5668\u200b 1603. \u200b\u8bbe\u8ba1\u200b\u505c\u8f66\u200b\u7cfb\u7edf"},{"location":"coding/leetcode/tags/#_17","title":"\u5206\u6cbb\u200b\u7b97\u6cd5","text":"<p>\u200b\u5206\u6cbb\u200b\u7b97\u6cd5\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b3\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u5206\u6cbb\u200b\u7b97\u6cd5\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 23. \u200b\u5408\u5e76\u200bK\u200b\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868\u200b 53. \u200b\u6700\u5927\u200b\u5b50\u5e8f\u200b\u548c\u200b 315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570"},{"location":"coding/leetcode/tags/#_18","title":"\u8d2a\u5fc3\u200b\u7b97\u6cd5","text":"<p>\u200b\u8d2a\u5fc3\u200b\u7b97\u6cd5\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b2\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u8d2a\u5fc3\u200b\u7b97\u6cd5\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 55. \u200b\u8df3\u8dc3\u200b\u6e38\u620f\u200b 122. \u200b\u4e70\u5356\u200b\u80a1\u7968\u200b\u7684\u200b\u6700\u4f73\u65f6\u673a\u200b II"},{"location":"coding/leetcode/tags/#_19","title":"\u4f4d\u200b\u8fd0\u7b97","text":"<p>\u200b\u4f4d\u200b\u8fd0\u7b97\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b2\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u4f4d\u200b\u8fd0\u7b97\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 78. \u200b\u5b50\u96c6\u200b 136. \u200b\u53ea\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u7684\u200b\u6570\u5b57"},{"location":"coding/leetcode/tags/#_20","title":"\u5806","text":"<p>\u200b\u5806\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b1\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u5806\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 23. \u200b\u5408\u5e76\u200bK\u200b\u4e2a\u200b\u5347\u5e8f\u200b\u94fe\u8868"},{"location":"coding/leetcode/tags/#sliding-window","title":"Sliding Window","text":"<p>Sliding Window\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b1\u200b\u9053\u200b\u9898\u76ee\u200b</p> Sliding Window \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 3. \u200b\u65e0\u200b\u91cd\u590d\u200b\u5b57\u7b26\u200b\u7684\u200b\u6700\u957f\u200b\u5b50\u4e32"},{"location":"coding/leetcode/tags/#_21","title":"\u6811\u72b6\u200b\u6570\u7ec4","text":"<p>\u200b\u6811\u72b6\u200b\u6570\u7ec4\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b1\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u6811\u72b6\u200b\u6570\u7ec4\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570"},{"location":"coding/leetcode/tags/#_22","title":"\u7ebf\u6bb5\u200b\u6811","text":"<p>\u200b\u7ebf\u6bb5\u200b\u6811\u200b\u6807\u7b7e\u200b\u4e0b\u200b\u5171\u6709\u200b1\u200b\u9053\u200b\u9898\u76ee\u200b</p> \u200b\u7ebf\u6bb5\u200b\u6811\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b \u200b\u9898\u76ee\u200b 315. \u200b\u8ba1\u7b97\u200b\u53f3\u4fa7\u200b\u5c0f\u4e8e\u200b\u5f53\u524d\u200b\u5143\u7d20\u200b\u7684\u200b\u4e2a\u6570"},{"location":"coding/machine-learning/","title":"\u673a\u5668\u200b\u5b66\u4e60","text":"<p>\u200b\u90e8\u5206\u200b\u673a\u5668\u200b\u5b66\u4e60\u200b\u7b97\u6cd5\u200b\u7684\u200bPython\u200b\u5b9e\u73b0\u200b</p> <ul> <li>\u200b\u7ebf\u6027\u200b\u6a21\u578b\u200b</li> <li>\u200b\u51b3\u7b56\u6811\u200b</li> <li>\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b</li> <li>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b</li> <li>\u200b\u7b97\u6cd5\u200b\u6d41\u7a0b\u200b</li> <li>\u200b\u4e3b\u9898\u200b\u6a21\u578b\u200b</li> <li>VAE-NTM</li> </ul>"},{"location":"coding/machine-learning/bayesian-optimization/","title":"\u8d1d\u53f6\u65af\u200b\u4f18\u5316","text":"In\u00a0[1]: Copied! <pre>import numpy as np\nfrom scipy.stats import norm\n\ndef objective(x):\n    return norm.pdf(x, 3, 2) * 1.5 + norm.pdf(x, 7, 1) + norm.pdf(x, 11, 2)\n</pre> import numpy as np from scipy.stats import norm  def objective(x):     return norm.pdf(x, 3, 2) * 1.5 + norm.pdf(x, 7, 1) + norm.pdf(x, 11, 2) <p>\u200b\u521d\u6b65\u200b\u5bf9\u200b\u8be5\u200b\u51fd\u6570\u200b\u8fdb\u884c\u200b\u53ef\u89c6\u5316\u200b</p> In\u00a0[2]: Copied! <pre>from matplotlib import pyplot as plt\n%config InlineBackend.figure_format = 'svg'\n\nx = np.linspace(-5, 20, 50)\nplt.plot(x, objective(x))\nplt.xlabel('x')\nplt.ylabel('f(x)')\nplt.title('Objective function')\nplt.show()\n</pre> from matplotlib import pyplot as plt %config InlineBackend.figure_format = 'svg'  x = np.linspace(-5, 20, 50) plt.plot(x, objective(x)) plt.xlabel('x') plt.ylabel('f(x)') plt.title('Objective function') plt.show() <p>\u200b\u91c7\u96c6\u200b\u51fd\u6570\u200b\u7684\u200b\u76ee\u6807\u200b\u662f\u200b\u9009\u51fa\u200b\u540e\u9a8c\u200b\u4e2d\u200b\u6700\u4f18\u200b\u7684\u200b\u70b9\u200b\u3002\u200b\u5728\u200b\u6b64\u5904\u200b\uff0c\u200b\u6211\u4eec\u200b\u4f7f\u7528\u200b\u671f\u671b\u200b\u6539\u8fdb\u200b\uff08Expected Improvement, EI\uff09\u200b\u6765\u200b\u9009\u62e9\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u91c7\u96c6\u200b\u70b9\u200b\u3002EI\u200b\u7684\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff1a</p> <p>$$ \\begin{aligned} \\text{EI}(x) &amp;= \\mathbb{E}[\\max(0, \\tilde f(x) - f(x^*) - \\xi)] \\\\ \\end{aligned} $$</p> <p>\u200b\u5176\u4e2d\u200b$\\tilde f(x)$\u200b\u662f\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u5728\u200b$x$\u200b\u5904\u200b\u7684\u200b\u9884\u6d4b\u503c\u200b\uff0c$f(x^*)$\u200b\u662f\u200b\u5f53\u524d\u200b\u6700\u4f18\u200b\u7684\u200b\u51fd\u6570\u200b\u503c\u200b\uff0c$\\xi$\u200b\u662f\u200b\u4e00\u4e2a\u200b\u63a7\u5236\u200b\u63a2\u7d22\u200b\u548c\u200b\u5f00\u53d1\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5f53\u200b$\\xi$\u200b\u8f83\u5927\u200b\u65f6\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u66f4\u200b\u503e\u5411\u200b\u4e8e\u200b\u63a2\u7d22\u200b\uff0c\u200b\u5f53\u200b$\\xi$\u200b\u8f83\u200b\u5c0f\u65f6\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u66f4\u200b\u503e\u5411\u200b\u4e8e\u200b\u5f00\u53d1\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u9ad8\u65af\u5206\u5e03\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u7ed9\u5b9a\u200b$x$, $\\tilde f(x)$\u200b\u7684\u200b\u5206\u5e03\u200b\u662f\u200b\u4e00\u4e2a\u200b\u9ad8\u65af\u5206\u5e03\u200b\uff0c\u200b\u5176\u200b\u5747\u503c\u200b\u4e3a\u200b$\\mu(x)$\uff0c\u200b\u65b9\u5dee\u200b\u4e3a\u200b$\\sigma^2(x)$\uff0c\u200b\u4ee4\u200b</p> <p>$$ z(x) = \\frac{\\mu(x) - f(x^*) - \\xi}{\\sigma(x)} $$</p> <p>\u200b\u5219\u200b\u671f\u671b\u200b\u6539\u8fdb\u200b\u4e3a\u200b</p> <p>$$ \\text{EI}(x) = \\sigma(x)[z(x)\\Phi(z(x)) + \\phi(z(x))] $$</p> In\u00a0[3]: Copied! <pre>from typing import Callable\nfrom sklearn.gaussian_process import GaussianProcessRegressor\n\nGP = GaussianProcessRegressor\nAcquisition = Callable[[np.ndarray, GP, float], np.ndarray]\n\ndef expected_improvement(x: np.ndarray, gp: GP, y_max: float, xi: float = 0.01):\n    x = np.array(x).reshape(-1, 1)\n    mu, sigma = gp.predict(x, return_std=True) # type: ignore\n    sigma = sigma.reshape(-1, 1)\n    mu = mu.reshape(-1, 1)\n\n    with np.errstate(divide='warn'):\n        Z = (mu - y_max - xi) / sigma # Normalized Z-score\n        ei = sigma * (Z * norm.cdf(Z) + norm.pdf(Z))\n\n    return ei\n</pre> from typing import Callable from sklearn.gaussian_process import GaussianProcessRegressor  GP = GaussianProcessRegressor Acquisition = Callable[[np.ndarray, GP, float], np.ndarray]  def expected_improvement(x: np.ndarray, gp: GP, y_max: float, xi: float = 0.01):     x = np.array(x).reshape(-1, 1)     mu, sigma = gp.predict(x, return_std=True) # type: ignore     sigma = sigma.reshape(-1, 1)     mu = mu.reshape(-1, 1)      with np.errstate(divide='warn'):         Z = (mu - y_max - xi) / sigma # Normalized Z-score         ei = sigma * (Z * norm.cdf(Z) + norm.pdf(Z))      return ei <p>\u200b\u6839\u636e\u200b\u91c7\u96c6\u200b\u51fd\u6570\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u5728\u200b\u8303\u56f4\u200b\u4e2d\u200b\u9009\u62e9\u200b\u4e00\u4e2a\u70b9\u200b\u4f7f\u5f97\u200bEI\u200b\u6700\u5927\u200b\uff0c\u200b\u7136\u540e\u200b\u89c2\u6d4b\u200b\u8be5\u70b9\u200b\u7684\u200b\u51fd\u6570\u200b\u503c\u200b\uff0c\u200b\u66f4\u65b0\u200b\u540e\u9a8c\u200b\uff0c\u200b\u91cd\u590d\u200b\u8fd9\u4e2a\u200b\u8fc7\u7a0b\u200b\u76f4\u5230\u200b\u8fbe\u5230\u200b\u8fed\u4ee3\u200b\u6b21\u6570\u200b\u3002</p> In\u00a0[4]: Copied! <pre>def next_point(\n    x_min: float, x_max: float, n_samples: int,\n    acquisition: Acquisition, gp: GP, y_max: float\n):\n    x = np.random.uniform(x_min, x_max, n_samples).reshape(-1, 1)\n    ei = acquisition(x, gp, y_max)\n    return x[np.argmax(ei)]\n</pre> def next_point(     x_min: float, x_max: float, n_samples: int,     acquisition: Acquisition, gp: GP, y_max: float ):     x = np.random.uniform(x_min, x_max, n_samples).reshape(-1, 1)     ei = acquisition(x, gp, y_max)     return x[np.argmax(ei)] <p>\u200b\u6b64\u540e\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u6765\u200b\u62df\u5408\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u3002<code>sklearn</code>\u200b\u4e2d\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u5b9e\u73b0\u200b\u3002</p> In\u00a0[5]: Copied! <pre>from sklearn.gaussian_process.kernels import RBF\n\nkernel = RBF(length_scale=1.0, length_scale_bounds=(1e-2, 1e2))\ngp = GP(kernel=kernel, alpha=1e-5, n_restarts_optimizer=10)\n\ndef plot_gp(ax, gp: GP, x_min: float, x_max: float, ci: float = 0.95):\n    x = np.linspace(x_min, x_max, 50).reshape(-1, 1)\n    y, sigma = gp.predict(x, return_std=True)\n\n    ci_coef = norm.ppf((1 + ci) / 2)\n\n    ax.plot(x, y, linewidth=1)\n    ax.fill_between(x.flatten(), y - ci_coef * sigma, y + ci_coef * sigma, alpha=0.2)\n    ax.set_xlabel('x')\n    ax.set_ylabel('f(x)')\n\nfig, ax = plt.subplots(1, 1)\nplot_gp(ax, gp, -5, 20)\nfig.show()\n</pre> from sklearn.gaussian_process.kernels import RBF  kernel = RBF(length_scale=1.0, length_scale_bounds=(1e-2, 1e2)) gp = GP(kernel=kernel, alpha=1e-5, n_restarts_optimizer=10)  def plot_gp(ax, gp: GP, x_min: float, x_max: float, ci: float = 0.95):     x = np.linspace(x_min, x_max, 50).reshape(-1, 1)     y, sigma = gp.predict(x, return_std=True)      ci_coef = norm.ppf((1 + ci) / 2)      ax.plot(x, y, linewidth=1)     ax.fill_between(x.flatten(), y - ci_coef * sigma, y + ci_coef * sigma, alpha=0.2)     ax.set_xlabel('x')     ax.set_ylabel('f(x)')  fig, ax = plt.subplots(1, 1) plot_gp(ax, gp, -5, 20) fig.show() <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_54343/3771838051.py:19: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <p>\u200b\u6b64\u540e\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u8fed\u4ee3\u200b\u5730\u200b\u4f18\u5316\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u3002</p> In\u00a0[6]: Copied! <pre>fig, ax = plt.subplots(4, 4, sharex=False, sharey=False, figsize=(9, 9))\nax_flatten = ax.flatten()\n\nx_min = -5\nx_max = 20\nx_samples = np.random.uniform(x_min, x_max, 3).reshape(-1, 1)\ny_samples = objective(x_samples)\ny_max = y_samples.max()\ngp = gp.fit(x_samples, y_samples)\n\nfor i, current_ax in enumerate(ax_flatten):\n    # Setup plot\n    current_ax.set_xlim(x_min, x_max)\n    current_ax.set_ylim(-1, 1)\n    if i // 4 != 3:\n        current_ax.get_xaxis().set_visible(False)\n    if i % 4 != 0:\n        current_ax.get_yaxis().set_visible(False)\n\n    # Plot prediction\n    plot_gp(current_ax, gp, x_min, x_max)\n\n    # Plot function\n    x_grid = np.linspace(x_min, x_max, 50).reshape(-1, 1)\n\n    # Plot samples\n    current_ax.plot(\n        x_grid, objective(x_grid), '--', color='gray', linewidth=0.5\n    )\n    current_ax.scatter(x_samples, y_samples, color='red', s=4, zorder=10)\n\n    # Plot optimal point\n    x_optimal = x_samples[np.argmax(y_samples)].item()\n    current_ax.scatter(x_optimal, y_max, color='green', s=10, zorder=20)\n    current_ax.text(\n        0.5, 0.9,\n        f'Best: x={x_optimal:.2f}, y={y_max:.2f}',\n        ha='center', transform=current_ax.transAxes\n    )\n\n    point = next_point(x_min, x_max, 100, expected_improvement, gp, y_max)\n\n    # Plot next point\n    current_ax.plot(\n        [point, point], [-0.8, 0.8], '--', color='red', linewidth=0.5\n    )\n    current_ax.text(\n        0.5, 0.05,\n        f'Next x: {point[0]:.2f}',\n        ha='center', transform=current_ax.transAxes\n    )\n\n    # Update samples\n    y = objective(point).item()\n    x_samples = np.vstack([x_samples, point])\n    y_samples = np.append(y_samples, y)\n\n    y_max = max(y_max, y)\n\n    # Re-fit model\n    gp = gp.fit(x_samples, y_samples)\n\nfig.show()\n</pre> fig, ax = plt.subplots(4, 4, sharex=False, sharey=False, figsize=(9, 9)) ax_flatten = ax.flatten()  x_min = -5 x_max = 20 x_samples = np.random.uniform(x_min, x_max, 3).reshape(-1, 1) y_samples = objective(x_samples) y_max = y_samples.max() gp = gp.fit(x_samples, y_samples)  for i, current_ax in enumerate(ax_flatten):     # Setup plot     current_ax.set_xlim(x_min, x_max)     current_ax.set_ylim(-1, 1)     if i // 4 != 3:         current_ax.get_xaxis().set_visible(False)     if i % 4 != 0:         current_ax.get_yaxis().set_visible(False)      # Plot prediction     plot_gp(current_ax, gp, x_min, x_max)      # Plot function     x_grid = np.linspace(x_min, x_max, 50).reshape(-1, 1)      # Plot samples     current_ax.plot(         x_grid, objective(x_grid), '--', color='gray', linewidth=0.5     )     current_ax.scatter(x_samples, y_samples, color='red', s=4, zorder=10)      # Plot optimal point     x_optimal = x_samples[np.argmax(y_samples)].item()     current_ax.scatter(x_optimal, y_max, color='green', s=10, zorder=20)     current_ax.text(         0.5, 0.9,         f'Best: x={x_optimal:.2f}, y={y_max:.2f}',         ha='center', transform=current_ax.transAxes     )      point = next_point(x_min, x_max, 100, expected_improvement, gp, y_max)      # Plot next point     current_ax.plot(         [point, point], [-0.8, 0.8], '--', color='red', linewidth=0.5     )     current_ax.text(         0.5, 0.05,         f'Next x: {point[0]:.2f}',         ha='center', transform=current_ax.transAxes     )      # Update samples     y = objective(point).item()     x_samples = np.vstack([x_samples, point])     y_samples = np.append(y_samples, y)      y_max = max(y_max, y)      # Re-fit model     gp = gp.fit(x_samples, y_samples)  fig.show() <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_54343/3753732084.py:63: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre>"},{"location":"coding/machine-learning/bayesian-optimization/","title":"\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u00b6","text":"<p>\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u662f\u200b\u4e00\u7c7b\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u4f18\u5316\u200b\u9ed1\u76d2\u200b\u51fd\u6570\u200b\u7684\u200b\u4f18\u5316\u200b\u65b9\u6cd5\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u9ed1\u76d2\u200b\u51fd\u6570\u200b\u6765\u8bf4\u200b\uff0c\u200b\u6211\u4eec\u200b\u65e0\u6cd5\u200b\u5f97\u77e5\u200b\u5176\u200b\u663e\u5f0f\u200b\u7684\u200b\u8868\u8fbe\u5f0f\u200b\uff0c\u200b\u53ea\u80fd\u200b\u901a\u8fc7\u200b\u8f93\u5165\u200b\u4e00\u4e2a\u200b\u6570\u636e\u200b\uff0c\u200b\u89c2\u6d4b\u200b\u5176\u200b\u8f93\u51fa\u200b\u7684\u200b\u65b9\u6cd5\u200b\u5bfb\u627e\u200b\u5b83\u200b\u7684\u200b\u6700\u5927\u503c\u200b\u3002\u200b\u501f\u52a9\u4e8e\u200b\u8d1d\u53f6\u65af\u200b\u7edf\u8ba1\u200b\u7684\u200b\u601d\u60f3\u200b\uff0c\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u6700\u521d\u200b\u8ba4\u4e3a\u200b\u51fd\u6570\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5b8c\u5168\u200b\u968f\u673a\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4e0d\u65ad\u200b\u5730\u200b\u89c2\u6d4b\u200b\u6765\u200b\u66f4\u65b0\u200b\u5bf9\u200b\u51fd\u6570\u200b\u7684\u200b\u4f30\u8ba1\u200b\uff08\u200b\u540e\u9a8c\u200b\uff09\u3002\u200b\u5728\u200b\u6bcf\u200b\u4e00\u6b65\u200b\u8fed\u4ee3\u200b\u4e2d\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u4f1a\u200b\u6839\u636e\u200b\u5f53\u524d\u200b\u7684\u200b\u540e\u9a8c\u200b\uff0c\u200b\u901a\u8fc7\u200b\u91c7\u96c6\u200b\u51fd\u6570\u200b\u6765\u200b\u9009\u62e9\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u89c2\u6d4b\u200b\u7684\u200b\u70b9\u200b\u3002\u200b\u8d1d\u53f6\u65af\u200b\u4f18\u5316\u200b\u7684\u200b\u4f18\u70b9\u200b\u662f\u200b\u53ef\u4ee5\u200b\u5728\u200b\u8f83\u200b\u5c11\u200b\u7684\u200b\u8fed\u4ee3\u200b\u6b21\u6570\u200b\u5185\u200b\u627e\u5230\u200b\u5168\u5c40\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5728\u200b\u5b9e\u9645\u200b\u5e94\u7528\u200b\u4e2d\u200b\u88ab\u200b\u5e7f\u6cdb\u200b\u4f7f\u7528\u200b\u3002</p> <p>\u200b\u5047\u8bbe\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7684\u200b\u5f62\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p>"},{"location":"coding/machine-learning/decision-tree/","title":"\u51b3\u7b56\u6811","text":"In\u00a0[30]: Copied! <pre>import numpy as np\nimport pandas as pd\nimport typing as T\n</pre> import numpy as np import pandas as pd import typing as T <p>\u200b\u4ee5\u4e0b\u200b\u8003\u8651\u200b\u897f\u74dc\u200b\u6570\u636e\u200b\u96c6\u200b3.0\u200b\u4e2d\u200b\u7b2c\u4e00\u6b65\u200b\u5212\u5206\u200b\u7684\u200b\u5c5e\u6027\u200b\u9009\u62e9\u200b\u3002</p> In\u00a0[31]: Copied! <pre>df = pd.read_csv('data/melon3.0a.csv')\nattributes = ['\u200b\u8272\u6cfd\u200b', '\u200b\u6839\u8482\u200b', '\u200b\u6572\u58f0\u200b', '\u200b\u7eb9\u7406\u200b', '\u200b\u8110\u90e8\u200b', '\u200b\u89e6\u611f\u200b']\ncategory = '\u200b\u597d\u74dc\u200b'\n</pre> df = pd.read_csv('data/melon3.0a.csv') attributes = ['\u200b\u8272\u6cfd\u200b', '\u200b\u6839\u8482\u200b', '\u200b\u6572\u58f0\u200b', '\u200b\u7eb9\u7406\u200b', '\u200b\u8110\u90e8\u200b', '\u200b\u89e6\u611f\u200b'] category = '\u200b\u597d\u74dc\u200b' <p>\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b\u57fa\u4e8e\u200b\u4fe1\u606f\u71b5\u200b\uff0c\u200b\u5728\u200b\u6bcf\u200b\u4e00\u6b21\u200b\u5212\u5206\u200b\u540e\u200b\uff0c\u200b\u96c6\u5408\u200b\u7684\u200b\u4fe1\u606f\u71b5\u200b\u4e4b\u200b\u548c\u200b\uff08\u200b\u6309\u200b\u6bd4\u4f8b\u200b\u52a0\u6743\u200b\uff09\u200b\u4f1a\u200b\u51cf\u5c0f\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b\u5373\u200b\u4e3a\u200b\u539f\u200b\u4fe1\u606f\u71b5\u200b\u4e0e\u200b\u5212\u5206\u200b\u540e\u200b\u7684\u200b\u4fe1\u606f\u71b5\u200b\u4e4b\u5dee\u200b\u3002</p> In\u00a0[32]: Copied! <pre>def entropy(df, target_col):\n    p = df[target_col].value_counts() / len(df)\n    return -sum(p * np.log2(p))\n\ndef conditional_entropy(df, discrete_col, target_col):\n    ret = 0\n    for val in df[discrete_col].unique():\n        p = len(df[df[discrete_col] == val]) / len(df) # \u200b\u6bd4\u4f8b\u200b\n        ret += p * entropy(df[df[discrete_col] == val], target_col)\n    return ret\n\ndef calc_gain(df, attributes, category) -&gt; T.Dict[str, float]:\n    entropy_original = entropy(df, category)\n    return {\n        k: entropy_original - conditional_entropy(df, k, category)\n        for k in attributes\n    }\n\ngain = pd.DataFrame([calc_gain(df, attributes, category)], index=['\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b'])\ngain\n</pre> def entropy(df, target_col):     p = df[target_col].value_counts() / len(df)     return -sum(p * np.log2(p))  def conditional_entropy(df, discrete_col, target_col):     ret = 0     for val in df[discrete_col].unique():         p = len(df[df[discrete_col] == val]) / len(df) # \u200b\u6bd4\u4f8b\u200b         ret += p * entropy(df[df[discrete_col] == val], target_col)     return ret  def calc_gain(df, attributes, category) -&gt; T.Dict[str, float]:     entropy_original = entropy(df, category)     return {         k: entropy_original - conditional_entropy(df, k, category)         for k in attributes     }  gain = pd.DataFrame([calc_gain(df, attributes, category)], index=['\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b']) gain Out[32]: \u200b\u8272\u6cfd\u200b \u200b\u6839\u8482\u200b \u200b\u6572\u58f0\u200b \u200b\u7eb9\u7406\u200b \u200b\u8110\u90e8\u200b \u200b\u89e6\u611f\u200b \u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b 0.108125 0.142675 0.140781 0.380592 0.289159 0.006046 <p>\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b\u4f1a\u200b\u66f4\u52a0\u200b\u503e\u5411\u200b\u4e8e\u200b\u9009\u62e9\u200b\u5305\u542b\u200b\u7c7b\u522b\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u5c5e\u6027\u200b\uff0c\u200b\u4f7f\u7528\u200b\u589e\u76ca\u200b\u7387\u200b\u6307\u6807\u200b\u53ef\u4ee5\u200b\u5e73\u8861\u200b\u4e0d\u540c\u200b\u7c7b\u522b\u200b\u4e4b\u95f4\u200b\u7684\u200b\u6743\u91cd\u200b\u3002\u200b\u589e\u76ca\u200b\u7387\u200b\u6307\u6807\u200b\u4e3a\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b\u4e0e\u200b\u5c5e\u6027\u200b\u56fa\u6709\u200b\u503c\u200b $\\mathrm{IV}$ \u200b\u4e4b\u200b\u6bd4\u200b\u3002</p> In\u00a0[33]: Copied! <pre>def iv(df, discrete_col):\n    p = df[discrete_col].value_counts() / len(df)\n    return -sum(p * np.log2(p))\n\ndef calc_gain_ratio(df, attributes, category) -&gt; T.Dict[str, float]:\n    entropy_original = entropy(df, category)\n    return {k: (entropy_original - conditional_entropy(df, k, category)) / iv(df, k)\n        for k in attributes\n    }\n\ngain_ratio = pd.DataFrame(\n    [calc_gain_ratio(df, attributes, category)], index=['\u200b\u589e\u76ca\u200b\u7387\u200b']\n)\ngain_ratio\n</pre> def iv(df, discrete_col):     p = df[discrete_col].value_counts() / len(df)     return -sum(p * np.log2(p))  def calc_gain_ratio(df, attributes, category) -&gt; T.Dict[str, float]:     entropy_original = entropy(df, category)     return {k: (entropy_original - conditional_entropy(df, k, category)) / iv(df, k)         for k in attributes     }  gain_ratio = pd.DataFrame(     [calc_gain_ratio(df, attributes, category)], index=['\u200b\u589e\u76ca\u200b\u7387\u200b'] ) gain_ratio Out[33]: \u200b\u8272\u6cfd\u200b \u200b\u6839\u8482\u200b \u200b\u6572\u58f0\u200b \u200b\u7eb9\u7406\u200b \u200b\u8110\u90e8\u200b \u200b\u89e6\u611f\u200b \u200b\u589e\u76ca\u200b\u7387\u200b 0.06844 0.101759 0.105627 0.263085 0.186727 0.006918 <p>\u200b\u57fa\u5c3c\u200b\u6307\u6570\u200b\u8861\u91cf\u200b\u4ece\u200b\u6570\u636e\u200b\u96c6\u4e2d\u200b\u968f\u673a\u200b\u62bd\u53d6\u200b\u4e24\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u6807\u7b7e\u200b\u53d6\u503c\u200b\u4e0d\u540c\u200b\u7684\u200b\u6982\u7387\u200b\u3002</p> In\u00a0[34]: Copied! <pre>def gini(df, target_col):\n    p = df[target_col].value_counts() / len(df)\n    return 1 - sum(p ** 2)\n\ndef gini_conditional(df, discrete_col, target_col):\n    ret = 0\n    for col in df[discrete_col].unique():\n        p = len(df[df[discrete_col] == col]) / len(df)\n        ret += p * gini(df[df[discrete_col] == col], target_col)\n    return ret\n\ndef calc_gini_index(df, attributes, category) -&gt; T.Dict[str, float]:\n    return {\n        k: sum((df[k].value_counts() / len(df)) * gini_conditional(df, k, category))\n        for k in attributes\n    }\n\ngini_index = pd.DataFrame([calc_gini_index(df, attributes, category)], index=['\u200b\u57fa\u5c3c\u200b\u6307\u6570\u200b'])\ngini_index\n</pre> def gini(df, target_col):     p = df[target_col].value_counts() / len(df)     return 1 - sum(p ** 2)  def gini_conditional(df, discrete_col, target_col):     ret = 0     for col in df[discrete_col].unique():         p = len(df[df[discrete_col] == col]) / len(df)         ret += p * gini(df[df[discrete_col] == col], target_col)     return ret  def calc_gini_index(df, attributes, category) -&gt; T.Dict[str, float]:     return {         k: sum((df[k].value_counts() / len(df)) * gini_conditional(df, k, category))         for k in attributes     }  gini_index = pd.DataFrame([calc_gini_index(df, attributes, category)], index=['\u200b\u57fa\u5c3c\u200b\u6307\u6570\u200b']) gini_index Out[34]: \u200b\u8272\u6cfd\u200b \u200b\u6839\u8482\u200b \u200b\u6572\u58f0\u200b \u200b\u7eb9\u7406\u200b \u200b\u8110\u90e8\u200b \u200b\u89e6\u611f\u200b \u200b\u57fa\u5c3c\u200b\u6307\u6570\u200b 0.427451 0.422269 0.423529 0.277124 0.344538 0.494118 In\u00a0[35]: Copied! <pre>class TreeNode():\n\n    def __init__(self, key):\n        self.key = key\n        self.mapping = {}\n\n    def add_child(self, value, child):\n        self.mapping[value] = child\n\n    def predict(self, x):\n        if self.key not in x or x[self.key] not in self.mapping:\n            return None\n        if isinstance(self.mapping[x[self.key]], TreeNode):\n            return self.mapping[x[self.key]].predict(x)\n        return self.mapping[x[self.key]]\n\ndef build_tree(df, attributes, category):\n    if len(df[category].unique()) == 1:\n        return df[category].unique()[0]\n    if len(attributes) == 0:\n        return df[category].value_counts().index[0]\n    gain = calc_gain(df, attributes, category)\n    best_attr = max(gain, key=lambda _: gain[_])\n    for attr in attributes:\n        if entropy(df, category) - conditional_entropy(df, attr, category) &gt; entropy(df, category) - conditional_entropy(df, best_attr, category):\n            best_attr = attr\n    tree = TreeNode(best_attr)\n    for value in df[best_attr].unique():\n        tree.add_child(value, build_tree(df[df[best_attr] == value], [attr for attr in attributes if attr != best_attr], category))\n    return tree\n\ntree = build_tree(df, attributes, category)\n</pre> class TreeNode():      def __init__(self, key):         self.key = key         self.mapping = {}      def add_child(self, value, child):         self.mapping[value] = child      def predict(self, x):         if self.key not in x or x[self.key] not in self.mapping:             return None         if isinstance(self.mapping[x[self.key]], TreeNode):             return self.mapping[x[self.key]].predict(x)         return self.mapping[x[self.key]]  def build_tree(df, attributes, category):     if len(df[category].unique()) == 1:         return df[category].unique()[0]     if len(attributes) == 0:         return df[category].value_counts().index[0]     gain = calc_gain(df, attributes, category)     best_attr = max(gain, key=lambda _: gain[_])     for attr in attributes:         if entropy(df, category) - conditional_entropy(df, attr, category) &gt; entropy(df, category) - conditional_entropy(df, best_attr, category):             best_attr = attr     tree = TreeNode(best_attr)     for value in df[best_attr].unique():         tree.add_child(value, build_tree(df[df[best_attr] == value], [attr for attr in attributes if attr != best_attr], category))     return tree  tree = build_tree(df, attributes, category) <p>\u200b\u51b3\u7b56\u6811\u200b\u91c7\u53d6\u200b\u81ea\u9876\u5411\u4e0b\u200b\u7684\u200bpredict\u200b\u65b9\u5f0f\u200b</p> In\u00a0[36]: Copied! <pre>assert isinstance(tree, TreeNode)\ntree.predict({\n    '\u200b\u8272\u6cfd\u200b': '\u200b\u9752\u7eff\u200b',\n    '\u200b\u6839\u8482\u200b': '\u200b\u8737\u7f29\u200b',\n    '\u200b\u6572\u58f0\u200b': '\u200b\u6d4a\u54cd\u200b',\n    '\u200b\u7eb9\u7406\u200b': '\u200b\u6e05\u6670\u200b',\n    '\u200b\u8110\u90e8\u200b': '\u200b\u51f9\u9677\u200b',\n    '\u200b\u89e6\u611f\u200b': '\u200b\u786c\u6ed1\u200b'\n})\n</pre> assert isinstance(tree, TreeNode) tree.predict({     '\u200b\u8272\u6cfd\u200b': '\u200b\u9752\u7eff\u200b',     '\u200b\u6839\u8482\u200b': '\u200b\u8737\u7f29\u200b',     '\u200b\u6572\u58f0\u200b': '\u200b\u6d4a\u54cd\u200b',     '\u200b\u7eb9\u7406\u200b': '\u200b\u6e05\u6670\u200b',     '\u200b\u8110\u90e8\u200b': '\u200b\u51f9\u9677\u200b',     '\u200b\u89e6\u611f\u200b': '\u200b\u786c\u6ed1\u200b' }) Out[36]: <pre>'\u200b\u662f\u200b'</pre> In\u00a0[119]: Copied! <pre>from sklearn.model_selection import train_test_split\n\ndef no_split(train, test, attributes, category):\n    return train[category].value_counts().index[0]\n\ndef build_pre_prune(train, test, attributes, category, tree_root=None):\n    if len(train[category].unique()) == 1:\n        return train[category].unique()[0]\n    no_split_result = no_split(train, test, attributes, category)\n    no_split_metric = np.sum(test[category] == no_split_result)\n    if len(attributes) == 0:\n        return no_split_result\n\n    # \u200b\u9009\u62e9\u200b\u5212\u5206\u200b\u5c5e\u6027\u200b\n    gain = calc_gain(train, attributes, category)\n    best_attr = max(gain, key=lambda _: gain[_])\n\n    # \u200b\u521b\u5efa\u200b\u8282\u70b9\u200b\n    tree = TreeNode(best_attr)\n    if tree_root is None:\n        tree_root = tree\n    for value in train[best_attr].unique():\n        tree.add_child(value, no_split(\n            train[train[best_attr] == value], test,\n            [attr for attr in attributes if attr != best_attr], category\n        ))\n\n    # \u200b\u8ba1\u7b97\u200b\u526a\u679d\u200b\u6548\u679c\u200b\n    result = []\n    for row in test[attributes].itertuples():\n        row_dict = row._asdict()\n        result.append(tree_root.predict(row_dict))\n    split_metric = np.sum([a == b for a, b in zip(result, test[category])])\n\n    # \u200b\u526a\u679d\u200b\n    print(f'\u200b\u526a\u679d\u200b\u524d\u200b: {no_split_metric}, \u200b\u526a\u679d\u200b\u540e\u200b: {split_metric}')\n    if split_metric &lt; no_split_metric:\n        print('\u200b\u4e0d\u200b\u526a\u679d\u200b')\n        return no_split_result\n    print('\u200b\u526a\u679d\u200b')\n    for value in train[best_attr].unique():\n        child = build_pre_prune(\n            train[train[best_attr] == value], test,\n            [attr for attr in attributes if attr != best_attr], category,\n            tree_root\n        )\n        tree.add_child(value, child)\n    return tree\n\ntrain, test = train_test_split(df, test_size=0.5)\ntree = build_pre_prune(train, test, attributes, category)\n\nresult = [\n    tree.predict(row._asdict()) if isinstance(tree, TreeNode) else tree\n    for row in test[attributes].itertuples()\n]\nsplit_metric = np.mean([a == b for a, b in zip(result, test[category])])\nsplit_metric\n</pre> from sklearn.model_selection import train_test_split  def no_split(train, test, attributes, category):     return train[category].value_counts().index[0]  def build_pre_prune(train, test, attributes, category, tree_root=None):     if len(train[category].unique()) == 1:         return train[category].unique()[0]     no_split_result = no_split(train, test, attributes, category)     no_split_metric = np.sum(test[category] == no_split_result)     if len(attributes) == 0:         return no_split_result      # \u200b\u9009\u62e9\u200b\u5212\u5206\u200b\u5c5e\u6027\u200b     gain = calc_gain(train, attributes, category)     best_attr = max(gain, key=lambda _: gain[_])      # \u200b\u521b\u5efa\u200b\u8282\u70b9\u200b     tree = TreeNode(best_attr)     if tree_root is None:         tree_root = tree     for value in train[best_attr].unique():         tree.add_child(value, no_split(             train[train[best_attr] == value], test,             [attr for attr in attributes if attr != best_attr], category         ))      # \u200b\u8ba1\u7b97\u200b\u526a\u679d\u200b\u6548\u679c\u200b     result = []     for row in test[attributes].itertuples():         row_dict = row._asdict()         result.append(tree_root.predict(row_dict))     split_metric = np.sum([a == b for a, b in zip(result, test[category])])      # \u200b\u526a\u679d\u200b     print(f'\u200b\u526a\u679d\u200b\u524d\u200b: {no_split_metric}, \u200b\u526a\u679d\u200b\u540e\u200b: {split_metric}')     if split_metric &lt; no_split_metric:         print('\u200b\u4e0d\u200b\u526a\u679d\u200b')         return no_split_result     print('\u200b\u526a\u679d\u200b')     for value in train[best_attr].unique():         child = build_pre_prune(             train[train[best_attr] == value], test,             [attr for attr in attributes if attr != best_attr], category,             tree_root         )         tree.add_child(value, child)     return tree  train, test = train_test_split(df, test_size=0.5) tree = build_pre_prune(train, test, attributes, category)  result = [     tree.predict(row._asdict()) if isinstance(tree, TreeNode) else tree     for row in test[attributes].itertuples() ] split_metric = np.mean([a == b for a, b in zip(result, test[category])]) split_metric <pre>\u200b\u526a\u679d\u200b\u524d\u200b: 4, \u200b\u526a\u679d\u200b\u540e\u200b: 4\n\u200b\u526a\u679d\u200b\n\u200b\u526a\u679d\u200b\u524d\u200b: 5, \u200b\u526a\u679d\u200b\u540e\u200b: 0\n\u200b\u4e0d\u200b\u526a\u679d\u200b\n</pre> Out[119]: <pre>0.4444444444444444</pre> In\u00a0[120]: Copied! <pre>def build_post_prune(train, test, attributes, category):\n    tree = build_tree(train, attributes, category)\n    # TODO: apply post pruning\n</pre> def build_post_prune(train, test, attributes, category):     tree = build_tree(train, attributes, category)     # TODO: apply post pruning"},{"location":"coding/machine-learning/decision-tree/","title":"\u51b3\u7b56\u6811\u200b\u00b6","text":"<p>\u200b\u51b3\u7b56\u6811\u200b\u662f\u200b\u4e00\u7c7b\u200b\u57fa\u4e8e\u200b\u89c4\u5219\u200b\u7684\u200b\u673a\u5668\u200b\u5b66\u4e60\u200b\u7b97\u6cd5\u200b</p>"},{"location":"coding/machine-learning/decision-tree/","title":"\u5212\u5206\u200b\u6307\u6807\u200b\u00b6","text":"<p>\u200b\u51b3\u7b56\u6811\u200b\u7b97\u6cd5\u200b\u4e2d\u200b\u7684\u200b\u6838\u5fc3\u200b\u662f\u200b\u6bcf\u200b\u4e00\u6b65\u200b\u5212\u5206\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u5c5e\u6027\u200b\u7684\u200b\u9009\u62e9\u200b\uff0c\u200b\u6bcf\u200b\u4e00\u6b65\u200b\u5212\u5206\u200b\u9700\u8981\u200b\u53d6\u5f97\u200b\u6700\u5927\u200b\u7684\u200b\u5212\u5206\u200b\u6548\u679c\u200b\uff0c\u200b\u5373\u200b\u4ea7\u751f\u200b\u7684\u200b\u5b50\u96c6\u200b\u201c\u200b\u7eaf\u5ea6\u200b\u201d\u200b\u6700\u9ad8\u200b\u3002\u200b\u5e38\u89c1\u200b\u7684\u200b\u4e09\u79cd\u200b\u8861\u91cf\u200b\u96c6\u5408\u200b\u201c\u200b\u7eaf\u5ea6\u200b\u201d\u200b\u7684\u200b\u6307\u6807\u200b\u4e3a\u200b\u4fe1\u606f\u200b\u589e\u76ca\u200b\u3001\u200b\u589e\u76ca\u200b\u7387\u200b\u4e0e\u200b\u57fa\u5c3c\u200b\u6307\u6570\u200b\u3002</p>"},{"location":"coding/machine-learning/decision-tree/","title":"\u51b3\u7b56\u6811\u200b\u6784\u5efa\u200b\u00b6","text":"<p>\u200b\u672c\u200b\u8282\u200b\u5b9e\u73b0\u200b\u539f\u4e66\u56fe\u200b4.2\u200b\u7684\u200b\u51b3\u7b56\u6811\u200b\u5b66\u4e60\u200b\u7b97\u6cd5\u200b\u3002</p>"},{"location":"coding/machine-learning/decision-tree/","title":"\u51b3\u7b56\u6811\u200b\u526a\u679d\u200b\u5904\u7406\u200b\u00b6","text":"<p>\u200b\u51b3\u7b56\u6811\u200b\u6709\u9884\u200b\u526a\u679d\u200b\u548c\u200b\u540e\u200b\u526a\u679d\u200b\u4e24\u4e2a\u200b\u526a\u679d\u200b\u6b65\u9aa4\u200b\uff0c\u200b\u526a\u679d\u200b\u53ef\u4ee5\u200b\u7f13\u89e3\u200b\u6a21\u578b\u200b\u51fa\u73b0\u200b\u7684\u200b\u8fc7\u200b\u62df\u5408\u200b\u73b0\u8c61\u200b\u3002</p>"},{"location":"coding/machine-learning/decision-tree/","title":"\u9884\u200b\u526a\u679d\u200b\u00b6","text":"<p>\u200b\u9884\u200b\u526a\u679d\u200b\u5728\u200b\u6a21\u578b\u200b\u7684\u200b\u8bad\u7ec3\u200b\u9636\u6bb5\u200b\u5c31\u200b\u901a\u8fc7\u200b\u9a8c\u8bc1\u200b\u96c6\u200b\u7684\u200b\u51c6\u786e\u6027\u200b\u6765\u200b\u51b3\u5b9a\u200b\u662f\u5426\u200b\u8fdb\u884c\u200b\u5212\u5206\u200b\u3002</p>"},{"location":"coding/machine-learning/decision-tree/","title":"\u540e\u200b\u526a\u679d\u200b\u00b6","text":"<p>\u200b\u540e\u200b\u526a\u679d\u200b\u5728\u200b\u51b3\u7b56\u6811\u200b\u751f\u6210\u200b\u540e\u200b\u518d\u200b\u904d\u5386\u200b\u51b3\u7b56\u6811\u200b\uff0c\u200b\u5220\u53bb\u200b\u8fc7\u591a\u200b\u7684\u200b\u8282\u70b9\u200b</p>"},{"location":"coding/machine-learning/gaussian-process/","title":"\u9ad8\u65af\u200b\u8fc7\u7a0b","text":"In\u00a0[1]: Copied! <pre>from scipy.stats import norm\nfrom matplotlib import pyplot as plt\nimport numpy as np\n\n%config InlineBackend.figure_format = 'svg'\n</pre> from scipy.stats import norm from matplotlib import pyplot as plt import numpy as np  %config InlineBackend.figure_format = 'svg' <p>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\uff08Gaussian Process\uff0cGP\uff09\u200b\u662f\u200b\u5c06\u200b\u591a\u200b\u53d8\u91cf\u200b\u9ad8\u65af\u5206\u5e03\u200b\u63a8\u5e7f\u200b\u5230\u200b\u65e0\u9650\u200b\u7ef4\u5ea6\u200b$\\mathcal X\\subseteq \\mathbb R^n$\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b\u3002\u200b\u5177\u4f53\u200b\u5730\u200b\uff0c\u200b\u5bf9\u4e8e\u200b$X = \\{x_1, \\ldots, x_n\\}\\subseteq \\mathcal X$\uff0c\u200b\u968f\u673a\u53d8\u91cf\u200b$f(x_1), \\ldots, f(x_n)$\u200b\u670d\u4ece\u200b\u591a\u5143\u200b\u9ad8\u65af\u5206\u5e03\u200b$\\mathcal N(\\mu(X), \\Sigma(X))$\u3002\u200b\u5176\u4e2d\u200b\uff0c$\\mu(X)$\u200b\u4e3a\u200b\u5747\u503c\u200b\u51fd\u6570\u200b\uff0c$\\Sigma(X)$\u200b\u4e3a\u200b\u534f\u65b9\u5dee\u200b\u51fd\u6570\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u9ad8\u65af\u5206\u5e03\u200b\u7814\u7a76\u200b\u7684\u200b\u662f\u200b\u51fd\u6570\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b\u3002</p> <p>\u200b\u591a\u5143\u200b\u9ad8\u65af\u5206\u5e03\u200b\u7684\u200b\u6761\u4ef6\u200b\u5206\u5e03\u200b\u4f9d\u7136\u200b\u662f\u200b\u9ad8\u65af\u5206\u5e03\u200b\u3002\u200b\u8bbe\u200b\u968f\u673a\u53d8\u91cf\u200b$X = (X_1, X_2)$\u200b\u670d\u4ece\u200b\u591a\u5143\u200b\u9ad8\u65af\u5206\u5e03\u200b$\\mathcal N(\\mu, \\Sigma)$\uff0c\u200b\u5176\u4e2d\u200b$\\mu = (\\mu_1, \\mu_2)$\uff0c$\\Sigma = \\begin{bmatrix} \\Sigma_{11} &amp; \\Sigma_{12} \\\\ \\Sigma_{21} &amp; \\Sigma_{22} \\end{bmatrix}$\uff0c\u200b\u5219\u200b\u7ed9\u5b9a\u200b$X_1 = x_1$\u200b\u540e\u200b\uff0c$X_2$\u200b\u7684\u200b\u6761\u4ef6\u200b\u5206\u5e03\u200b\u4e3a\u200b\uff1a</p> <p>$$ (X_2\\mid X_1 = x_1) \\sim \\mathcal N(\\mu_2 + \\Sigma_{21}\\Sigma_{11}^{-1}(x_1 - \\mu_1), \\Sigma_{22} - \\Sigma_{21}\\Sigma_{11}^{-1}\\Sigma_{12}) $$</p> <p>\u200b\u8bbe\u200b\u4e00\u7ec4\u200b\u89c2\u6d4b\u200b\u6570\u636e\u200b\u6570\u636e\u200b$\\boldsymbol X = x_1, \\ldots, x_n$\u200b\u53ca\u5176\u200b\u5bf9\u5e94\u200b\u7684\u200b\u51fd\u6570\u200b\u503c\u200b$\\boldsymbol Y = y_1, \\ldots, y_n$\u3002\u200b\u5bf9\u4e8e\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u6570\u636e\u200b\u70b9\u200b$x$\uff0c\u200b\u8981\u200b\u9884\u6d4b\u200b\u5176\u200b\u5bf9\u5e94\u200b\u7684\u200b\u51fd\u6570\u200b\u503c\u200b$y = f(x)$\u200b\u7684\u200b\u5206\u5e03\u200b\u3002\u200b\u5982\u679c\u200b\u6211\u4eec\u200b\u80fd\u200b\u8ba1\u7b97\u200b\u51fa\u200b\u89c2\u6d4b\u200b\u6570\u636e\u200b$\\boldsymbol Y$\u200b\u548c\u200b\u9884\u6d4b\u200b\u53d8\u91cf\u200b$y$\u200b\u4e4b\u95f4\u200b\u7684\u200b\u534f\u65b9\u5dee\u200b\uff0c\u200b\u4fbf\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u9ad8\u65af\u5206\u5e03\u200b\u7684\u200b\u6761\u4ef6\u200b\u5206\u5e03\u200b\u8ba1\u7b97\u200b\u5f97\u5230\u200b$f(x)$\u200b\u7684\u200b\u5747\u503c\u200b\u548c\u200b\u65b9\u5dee\u200b\u3002</p> <p>\u200b\u6b64\u5904\u200b\u4ee4\u200b$n = 1$\uff0c\u200b\u5373\u200b$X$\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\u6574\u4e2a\u200b\u5b9e\u6570\u200b\u57df\u200b\u3002</p> In\u00a0[2]: Copied! <pre>def y_posterior(x, x_obs, y_obs, kernel_func):\n    # x: (num_features)\n    # x_obs: (num_observations, num_features)\n    # y_obs: (num_observations)\n\n    sigma_21 = kernel_func(x, x_obs)  # dim: (num_observations)\n    sigma_22 = kernel_func(x, x)  # dim: scalar\n    # dim: (num_observations, num_observations)\n    sigma_11 = kernel_func(x_obs, x_obs)\n\n    inv_11 = np.linalg.inv(sigma_11 + 1e-8 * np.eye(sigma_11.shape[0]))\n    mu = np.einsum('i,ij,j-&gt;', sigma_21, inv_11, y_obs)\n    sigma = sigma_22 - np.einsum('j,jk,k-&gt;', sigma_21, inv_11, sigma_21)\n    return mu, sigma\n\ndef generate_samples(num_samples=5, target_function=None, sigma=1):\n    X = np.random.uniform(-5, 5, num_samples).reshape(-1, 1)\n    Y = np.random.normal(0, sigma, num_samples)\n    if target_function is not None:\n        Y += target_function(X)\n    return X, Y\n\ndef plot_posterior(kernel, num_samples=5, X_obs=None, Y_obs=None, ax=None, grid_size=200):\n    if X_obs is None != Y_obs is None:\n        raise ValueError('X_obs must be provided if Y_obs is provided.')\n    if (X_obs is None or Y_obs is None) and num_samples is None:\n        raise ValueError('num_samples must be provided if X_obs is provided.')\n    if X_obs is None:\n        assert Y_obs is None\n        X_obs, Y_obs = generate_samples(num_samples)\n    assert X_obs is not None and Y_obs is not None\n    if X_obs.shape[0] != Y_obs.shape[0]:\n        raise ValueError('The number of observations must be the same.')\n\n    X_min, X_max = np.min(X_obs), np.max(X_obs)\n    alpha_95 = norm.ppf(0.975)\n\n    if ax is None:\n        fig, ax = plt.subplots()\n    else:\n        fig = None\n\n    X = np.linspace(X_min, X_max, grid_size)\n    y = [y_posterior(x, X_obs, Y_obs, kernel) for x in X.reshape(-1, 1)]\n    ax.plot(X, [y[0] for y in y], linewidth=1, label='Function')\n    ax.scatter(X_obs, Y_obs, s=20, label='Observations')\n    ax.fill_between(X,\n        [y[0] - np.sqrt(y[1]) * alpha_95 for y in y],\n        [y[0] + np.sqrt(y[1]) * alpha_95 for y in y],\n        alpha=0.5, label='95% CI'\n    )\n\n    ax.set_xlabel('x')\n    ax.set_xlabel('y')\n    ax.legend()\n    if fig is not None:\n        fig.show()\n</pre> def y_posterior(x, x_obs, y_obs, kernel_func):     # x: (num_features)     # x_obs: (num_observations, num_features)     # y_obs: (num_observations)      sigma_21 = kernel_func(x, x_obs)  # dim: (num_observations)     sigma_22 = kernel_func(x, x)  # dim: scalar     # dim: (num_observations, num_observations)     sigma_11 = kernel_func(x_obs, x_obs)      inv_11 = np.linalg.inv(sigma_11 + 1e-8 * np.eye(sigma_11.shape[0]))     mu = np.einsum('i,ij,j-&gt;', sigma_21, inv_11, y_obs)     sigma = sigma_22 - np.einsum('j,jk,k-&gt;', sigma_21, inv_11, sigma_21)     return mu, sigma  def generate_samples(num_samples=5, target_function=None, sigma=1):     X = np.random.uniform(-5, 5, num_samples).reshape(-1, 1)     Y = np.random.normal(0, sigma, num_samples)     if target_function is not None:         Y += target_function(X)     return X, Y  def plot_posterior(kernel, num_samples=5, X_obs=None, Y_obs=None, ax=None, grid_size=200):     if X_obs is None != Y_obs is None:         raise ValueError('X_obs must be provided if Y_obs is provided.')     if (X_obs is None or Y_obs is None) and num_samples is None:         raise ValueError('num_samples must be provided if X_obs is provided.')     if X_obs is None:         assert Y_obs is None         X_obs, Y_obs = generate_samples(num_samples)     assert X_obs is not None and Y_obs is not None     if X_obs.shape[0] != Y_obs.shape[0]:         raise ValueError('The number of observations must be the same.')      X_min, X_max = np.min(X_obs), np.max(X_obs)     alpha_95 = norm.ppf(0.975)      if ax is None:         fig, ax = plt.subplots()     else:         fig = None      X = np.linspace(X_min, X_max, grid_size)     y = [y_posterior(x, X_obs, Y_obs, kernel) for x in X.reshape(-1, 1)]     ax.plot(X, [y[0] for y in y], linewidth=1, label='Function')     ax.scatter(X_obs, Y_obs, s=20, label='Observations')     ax.fill_between(X,         [y[0] - np.sqrt(y[1]) * alpha_95 for y in y],         [y[0] + np.sqrt(y[1]) * alpha_95 for y in y],         alpha=0.5, label='95% CI'     )      ax.set_xlabel('x')     ax.set_xlabel('y')     ax.legend()     if fig is not None:         fig.show() <p>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u5047\u8bbe\u200b\u66f4\u200b\u76f8\u4f3c\u200b\u7684\u200b$x$\u200b\u6709\u200b\u66f4\u200b\u76f8\u4f3c\u200b\u7684\u200b$y$\uff0c\u200b\u5373\u200b$\\text{Cov}(f(x), f(x')) = d_{x, x'}$\u3002\u200b\u7528\u4e8e\u200b\u63cf\u8ff0\u200b$x$\u200b\u4e4b\u95f4\u200b\u76f8\u4f3c\u200b\u5ea6\u200b\u7684\u200b\u51fd\u6570\u200b\u79f0\u4e3a\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a$d_{x, x'} = k(x, x')$\u3002\u200b\u6838\u200b\u51fd\u6570\u200b\u7684\u200b\u6570\u503c\u200b\u8d8a\u5927\u200b\uff0c\u200b\u8bf4\u660e\u200b$x$\u200b\u548c\u200b$x'$\u200b\u8d8a\u200b\u76f8\u4f3c\u200b\uff0c\u200b\u5426\u5219\u200b\u8d8a\u200b\u4e0d\u200b\u76f8\u4f3c\u200b\u3002</p> In\u00a0[3]: Copied! <pre>import functools\n\ndef kernel_wrapper(*args, **kwargs):\n    if args:\n        _kernel = args[0]\n        if kwargs:\n            _kernel = functools.partial(_kernel, **kwargs)\n\n        @functools.wraps(_kernel)\n        def new_kernel(a, b):\n            if a.shape[-1] != b.shape[-1]:\n                raise ValueError('The last dimension of a and b must be the same.')\n\n            num_a = 1 if a.ndim == 1 else a.shape[0]\n            num_b = 1 if b.ndim == 1 else b.shape[0]\n            num_hidden = a.shape[-1]\n            target_shape = (num_a, num_b, num_hidden)\n\n            x_a = np.broadcast_to(a.reshape((num_a, 1, num_hidden)), target_shape)\n            x_b = np.broadcast_to(b.reshape((1, num_b, num_hidden)), target_shape)\n\n            result = _kernel(x_a, x_b)\n            result_shape = [\n                *([] if a.ndim == 1 else [num_a]),\n                *([] if b.ndim == 1 else [num_b])\n            ]\n            result = result.reshape(result_shape)\n            return result\n\n        return new_kernel\n    else:\n        return functools.partial(kernel_wrapper, **kwargs)\n\n# Use the following method to define a kernel function\n# @kernel_wrapper - for kernels with no hyperparameters\n# @kernel_wrapper(hyperparameter=value) - for kernels with hyperparameters\n</pre> import functools  def kernel_wrapper(*args, **kwargs):     if args:         _kernel = args[0]         if kwargs:             _kernel = functools.partial(_kernel, **kwargs)          @functools.wraps(_kernel)         def new_kernel(a, b):             if a.shape[-1] != b.shape[-1]:                 raise ValueError('The last dimension of a and b must be the same.')              num_a = 1 if a.ndim == 1 else a.shape[0]             num_b = 1 if b.ndim == 1 else b.shape[0]             num_hidden = a.shape[-1]             target_shape = (num_a, num_b, num_hidden)              x_a = np.broadcast_to(a.reshape((num_a, 1, num_hidden)), target_shape)             x_b = np.broadcast_to(b.reshape((1, num_b, num_hidden)), target_shape)              result = _kernel(x_a, x_b)             result_shape = [                 *([] if a.ndim == 1 else [num_a]),                 *([] if b.ndim == 1 else [num_b])             ]             result = result.reshape(result_shape)             return result          return new_kernel     else:         return functools.partial(kernel_wrapper, **kwargs)  # Use the following method to define a kernel function # @kernel_wrapper - for kernels with no hyperparameters # @kernel_wrapper(hyperparameter=value) - for kernels with hyperparameters <p>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u5e38\u7528\u200b\u7684\u200b\u6838\u200b\u51fd\u6570\u200b\u6709\u200b\uff1a</p> <ul> <li>\u200b\u7ebf\u6027\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a$k(x, x'; v) = vx^Tx'$\uff0c\u200b\u5176\u4e2d\u200b$v$\u200b\u4e3a\u200b\u8d85\u200b\u53c2\u6570\u200b\u3002</li> </ul> In\u00a0[4]: Copied! <pre>def linear_kernel(a, b, nu=1):\n    # Input: a, b: (num_a, num_b, num_hidden)\n    return np.einsum('ijk,ijk-&gt;ij', a, b) * nu\n\nplot_posterior(kernel_wrapper(nu=1)(linear_kernel), grid_size=10)\n</pre> def linear_kernel(a, b, nu=1):     # Input: a, b: (num_a, num_b, num_hidden)     return np.einsum('ijk,ijk-&gt;ij', a, b) * nu  plot_posterior(kernel_wrapper(nu=1)(linear_kernel), grid_size=10) <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:48: RuntimeWarning: invalid value encountered in sqrt\n  [y[0] - np.sqrt(y[1]) * alpha_95 for y in y],\n/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:49: RuntimeWarning: invalid value encountered in sqrt\n  [y[0] + np.sqrt(y[1]) * alpha_95 for y in y],\n/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <ul> <li>\u200b\u5e73\u65b9\u200b\u6307\u6570\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a$k(x, x'; \\sigma, l) = \\sigma^2\\exp\\left(-\\frac{\\|x - x'\\|^2}{2l^2}\\right)$\uff0c$\\sigma$\u200b\u4e3a\u200b\u5e45\u5ea6\u200b\u53c2\u6570\u200b\uff0c\u200b\u63a7\u5236\u200b\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u968f\u673a\u200b\u6ce2\u52a8\u200b\u8303\u56f4\u200b\uff1b$l$\u200b\u4e3a\u200b\u957f\u5ea6\u200b\u53c2\u6570\u200b\uff0c\u200b\u63a7\u5236\u200b\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u968f\u673a\u200b\u6ce2\u52a8\u200b\u9891\u7387\u200b\u3002</li> </ul> In\u00a0[5]: Copied! <pre>def rbf_kernel(a, b, sigma, l):\n    # Input: a, b: (num_a, num_b, num_hidden)\n    return sigma ** 2 * np.exp(\n        -0.5 * np.linalg.norm(a - b, axis=-1) ** 2 / l ** 2\n    )\n\nplot_posterior(kernel_wrapper(sigma=1, l=1)(rbf_kernel), grid_size=80)\n</pre> def rbf_kernel(a, b, sigma, l):     # Input: a, b: (num_a, num_b, num_hidden)     return sigma ** 2 * np.exp(         -0.5 * np.linalg.norm(a - b, axis=-1) ** 2 / l ** 2     )  plot_posterior(kernel_wrapper(sigma=1, l=1)(rbf_kernel), grid_size=80) <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <ul> <li>Matern\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a$k(x, x'; \\sigma, l, \\nu) = \\frac{2^{1-\\nu}}{\\Gamma(\\nu)}\\left(\\frac{\\sqrt{2\\nu}\\|x - x'\\|}{l}\\right)^\\nu K_\\nu\\left(\\frac{\\sqrt{2\\nu}\\|x - x'\\|}{l}\\right)$\uff0c\u200b\u5176\u4e2d\u200b$\\nu$\u200b\u4e3a\u200b\u8d85\u200b\u53c2\u6570\u200b\uff0c\u200b\u7528\u4e8e\u200b\u63a7\u5236\u200b\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u5149\u6ed1\u5ea6\u200b\uff1b$K_\\nu$\u200b\u4e3a\u200b\u4fee\u6b63\u200bBessel\u200b\u51fd\u6570\u200b\u3002</li> </ul> In\u00a0[6]: Copied! <pre>from scipy.special import kn, gamma\n\ndef matern_kernel(a, b, l, nu):\n    # Input: a, b: (num_a, num_b, num_hidden)\n    x = (np.sqrt(2 * nu) * np.linalg.norm(a - b, axis=-1) / l)\n    y = 2 ** (1 - nu) / gamma(nu) * x ** nu * kn(nu, x)\n    # Replace inf with 1\n    y = np.where(x &lt; 1e-6, 1, y)\n    return y\n\nplot_posterior(kernel_wrapper(l=1, nu=2)(matern_kernel), 4, grid_size=80)\n</pre> from scipy.special import kn, gamma  def matern_kernel(a, b, l, nu):     # Input: a, b: (num_a, num_b, num_hidden)     x = (np.sqrt(2 * nu) * np.linalg.norm(a - b, axis=-1) / l)     y = 2 ** (1 - nu) / gamma(nu) * x ** nu * kn(nu, x)     # Replace inf with 1     y = np.where(x &lt; 1e-6, 1, y)     return y  plot_posterior(kernel_wrapper(l=1, nu=2)(matern_kernel), 4, grid_size=80) <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/497016935.py:6: RuntimeWarning: invalid value encountered in multiply\n  y = 2 ** (1 - nu) / gamma(nu) * x ** nu * kn(nu, x)\n/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <ul> <li>\u200b\u5468\u671f\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a$k(x, x'; \\sigma, l, p) = \\sigma^2\\exp\\left(-\\frac{2\\sin^2(\\pi\\|x - x'\\|/p)}{l^2}\\right)$\uff0c\u200b\u5176\u4e2d\u200b$p$\u200b\u4e3a\u200b\u5468\u671f\u200b\u53c2\u6570\u200b\uff0c$l$\u200b\u4e3a\u200b\u957f\u5ea6\u200b\u53c2\u6570\u200b\uff0c$\\sigma$\u200b\u4e3a\u200b\u5e45\u5ea6\u200b\u53c2\u6570\u200b\u3002</li> </ul> In\u00a0[7]: Copied! <pre>def periodic_kernel(a, b, sigma, l, p):\n    # Input: a, b: (num_a, num_b, num_hidden)\n    return sigma ** 2 * np.exp(\n        -2 * np.sin(np.pi * np.linalg.norm(a - b, axis=-1) / p) ** 2 / l ** 2\n    )\n\nplot_posterior(kernel_wrapper(sigma=1, l=1, p=2)(periodic_kernel), 4, grid_size=150)\n</pre> def periodic_kernel(a, b, sigma, l, p):     # Input: a, b: (num_a, num_b, num_hidden)     return sigma ** 2 * np.exp(         -2 * np.sin(np.pi * np.linalg.norm(a - b, axis=-1) / p) ** 2 / l ** 2     )  plot_posterior(kernel_wrapper(sigma=1, l=1, p=2)(periodic_kernel), 4, grid_size=150) <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <ul> <li><p>\u200b\u566a\u58f0\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a$k(x, x'; \\sigma, l) = \\sigma^2\\delta_{x, x'}$\uff0c\u200b\u5176\u4e2d\u200b$\\delta_{x, x'}$\u200b\u4e3a\u200bKronecker delta\u200b\u51fd\u6570\u200b\uff0c\u200b\u7528\u4e8e\u200b\u6355\u83b7\u200b\u89c2\u6d4b\u200b\u6570\u636e\u200b\u7684\u200b\u566a\u58f0\u200b\u3002</p> <p>$$   \\delta(x, x') = \\begin{cases}   1, &amp; x = x' \\\\   0, &amp; x \\neq x'   \\end{cases}   $$</p> </li> </ul> In\u00a0[8]: Copied! <pre>def noise_kernel(a, b, sigma):\n    return sigma ** 2 * np.all((a == b), axis=-1)\n\nplot_posterior(kernel_wrapper(sigma=1)(noise_kernel), 5)\n</pre> def noise_kernel(a, b, sigma):     return sigma ** 2 * np.all((a == b), axis=-1)  plot_posterior(kernel_wrapper(sigma=1)(noise_kernel), 5) <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <p>\u200b\u5728\u200b\u5b9e\u9645\u200b\u5e94\u7528\u200b\u4e2d\u200b\uff0c\u200b\u9700\u8981\u200b\u6839\u636e\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u5df2\u77e5\u200b\u7684\u200b\u7279\u6027\u200b\uff0c\u200b\u8bbe\u8ba1\u200b\u5408\u9002\u200b\u7684\u200b\u6838\u200b\u51fd\u6570\u200b\uff1a\u200b\u5982\u679c\u200b\u51fd\u6570\u200b\u4e2d\u200b\u5305\u542b\u200b\u5468\u671f\u6027\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5468\u671f\u200b\u6838\u200b\u51fd\u6570\u200b\uff1b\u200b\u5982\u679c\u200b\u51fd\u6570\u200b\u503c\u200b\u5728\u200b\u76f8\u4f3c\u200b\u7684\u200b$x$\u200b\u4e4b\u95f4\u200b\u53d8\u5316\u200b\u8f83\u200b\u5c0f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5e73\u65b9\u200b\u6307\u6570\u200b\u6838\u200b\u51fd\u6570\u200b\u3002</p> <p>\u200b\u5982\u200b\u5bf9\u4e8e\u200b\u51fd\u6570\u200b</p> <p>$$ f(x) = \\sin(\\pi x) + 0.3 x^2 + \\varepsilon $$</p> <p>\u200b\u9700\u8981\u200b\u540c\u65f6\u200b\u5f15\u5165\u200b\u5468\u671f\u200b\u6838\u200b\u51fd\u6570\u200b\u548c\u200b\u5e73\u65b9\u200b\u6307\u6570\u200b\u6838\u200b\u51fd\u6570\u200b\uff0c\u200b\u624d\u80fd\u200b\u8f83\u200b\u597d\u200b\u5730\u200b\u9884\u6d4b\u200b\u51fd\u6570\u200b\u503c\u200b\u3002</p> In\u00a0[9]: Copied! <pre>def target_function(X):\n    x = X.reshape(-1)\n    return np.sin(x * 2 * 3.14 / 2) + x ** 2 * 0.3\n\n\ndef plot_function(func, ax, X_min=-5, X_max=5, grid_size=200, **kwargs):\n    X = np.linspace(X_min, X_max, grid_size)\n    Y = func(X.reshape(-1, 1))\n    ax.plot(X, Y, **kwargs)\n\nfig, ax = plt.subplots(figsize=(4, 4))\nplot_function(target_function, ax, grid_size=100)\nax.set_xlabel('x')\nax.set_ylabel('y')\nax.set_title('Target Function')\nfig.show()\n</pre> def target_function(X):     x = X.reshape(-1)     return np.sin(x * 2 * 3.14 / 2) + x ** 2 * 0.3   def plot_function(func, ax, X_min=-5, X_max=5, grid_size=200, **kwargs):     X = np.linspace(X_min, X_max, grid_size)     Y = func(X.reshape(-1, 1))     ax.plot(X, Y, **kwargs)  fig, ax = plt.subplots(figsize=(4, 4)) plot_function(target_function, ax, grid_size=100) ax.set_xlabel('x') ax.set_ylabel('y') ax.set_title('Target Function') fig.show() <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/4237481582.py:16: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> In\u00a0[10]: Copied! <pre>periodic = kernel_wrapper(sigma=1, l=1, p=2)(periodic_kernel)\nrbf = kernel_wrapper(sigma=1, l=1)(rbf_kernel)\nnoise = kernel_wrapper(sigma=0.1)(noise_kernel)\n\ndef added(a, b):\n    return periodic(a, b) + rbf(a, b) + noise(a, b)\n\nkernels = {\n    'Periodic': periodic,\n    'RBF': rbf,\n    'Hybrid': added\n}\n\nX_obs, Y_obs = generate_samples(10, target_function, sigma=0.5)\nX_min, X_max = np.min(X_obs), np.max(X_obs)\n\nfig, axes = plt.subplots(1, 3, figsize=(10, 3))\nfor ax, (name, kernel) in zip(axes, kernels.items()):\n    plot_posterior(kernel, X_obs=X_obs, Y_obs=Y_obs, ax=ax, grid_size=150)\n    plot_function(target_function, ax, X_min, X_max, color='red', label='Target Function', grid_size=100)\n    ax.set_title(f'{name} Kernel')\n    ax.get_legend().remove()\nfig.show()\n</pre> periodic = kernel_wrapper(sigma=1, l=1, p=2)(periodic_kernel) rbf = kernel_wrapper(sigma=1, l=1)(rbf_kernel) noise = kernel_wrapper(sigma=0.1)(noise_kernel)  def added(a, b):     return periodic(a, b) + rbf(a, b) + noise(a, b)  kernels = {     'Periodic': periodic,     'RBF': rbf,     'Hybrid': added }  X_obs, Y_obs = generate_samples(10, target_function, sigma=0.5) X_min, X_max = np.min(X_obs), np.max(X_obs)  fig, axes = plt.subplots(1, 3, figsize=(10, 3)) for ax, (name, kernel) in zip(axes, kernels.items()):     plot_posterior(kernel, X_obs=X_obs, Y_obs=Y_obs, ax=ax, grid_size=150)     plot_function(target_function, ax, X_min, X_max, color='red', label='Target Function', grid_size=100)     ax.set_title(f'{name} Kernel')     ax.get_legend().remove() fig.show() <pre>/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:48: RuntimeWarning: invalid value encountered in sqrt\n  [y[0] - np.sqrt(y[1]) * alpha_95 for y in y],\n/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:49: RuntimeWarning: invalid value encountered in sqrt\n  [y[0] + np.sqrt(y[1]) * alpha_95 for y in y],\n/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/1232705018.py:23: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n  fig.show()\n</pre> <p>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u4e0d\u9002\u200b\u7528\u4e8e\u200b\u4ee5\u4e0b\u200b\u573a\u666f\u200b\uff1a</p> <ul> <li>\u200b\u6570\u636e\u91cf\u200b\u8f83\u5927\u200b\u65f6\u200b\uff0c\u200b\u7531\u4e8e\u200b\u8ba1\u7b97\u200b$m$\u200b\u7ef4\u200b\u77e9\u9635\u200b\u9006\u200b\u7684\u200b\u590d\u6742\u5ea6\u200b\u4e3a\u200b$O(m^3)$\uff0c\u200b\u8ba1\u7b97\u200b\u590d\u6742\u5ea6\u200b\u8f83\u200b\u9ad8\u200b\u3002</li> <li>\u200b\u6570\u636e\u200b\u7ef4\u5ea6\u200b\u8f83\u200b\u9ad8\u65f6\u200b\uff0c\u200b\u7ef4\u5ea6\u200b\u707e\u96be\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u6838\u200b\u51fd\u6570\u200b\u5bb9\u6613\u200b\u9000\u5316\u200b\uff0c\u200b\u65e0\u6cd5\u200b\u6355\u83b7\u200b\u6570\u636e\u200b\u4e4b\u95f4\u200b\u7684\u200b\u76f8\u4f3c\u6027\u200b\u3002</li> <li>\u200b\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u9884\u6d4b\u200b\u7ed3\u679c\u200b\u662f\u200b\u8fde\u7eed\u200b\u7684\u200b\uff0c\u200b\u65e0\u6cd5\u200b\u76f4\u63a5\u200b\u5904\u7406\u200b\u8fde\u7eed\u200b\u6570\u636e\u200b\u3002</li> </ul>"},{"location":"coding/machine-learning/gaussian-process/","title":"\u9ad8\u65af\u200b\u8fc7\u7a0b\u200b\u00b6","text":""},{"location":"coding/machine-learning/linear-models/","title":"\u7ebf\u6027\u200b\u6a21\u578b","text":"In\u00a0[\u00a0]: Copied! <pre>import numpy as np\nimport pandas as pd\n\ndf = pd.read_csv('data/melon3.0a.csv')\nX_rcol, y_rcol = df.columns[1:-2], df.columns[-2] # columns used for regression\nX_ccol, y_ccol = df.columns[1:-1], df.columns[-1] # columns used for classification\nvalue_map = {\n    '\u200b\u8272\u6cfd\u200b': {'\u200b\u6d45\u767d\u200b': 0, '\u200b\u9752\u7eff\u200b': 1, '\u200b\u4e4c\u9ed1\u200b': 2},\n    '\u200b\u6839\u8482\u200b': {'\u200b\u8737\u7f29\u200b': 0, '\u200b\u7a0d\u200b\u8737\u200b': 1, '\u200b\u786c\u633a\u200b': 2},\n    '\u200b\u6572\u58f0\u200b': {'\u200b\u6c89\u95f7\u200b': 0, '\u200b\u6d4a\u54cd\u200b': 1, '\u200b\u6e05\u8106\u200b': 2},\n    '\u200b\u7eb9\u7406\u200b': {'\u200b\u6a21\u7cca\u200b': 0, '\u200b\u7a0d\u7cca\u200b': 1, '\u200b\u6e05\u6670\u200b': 2},\n    '\u200b\u8110\u90e8\u200b': {'\u200b\u51f9\u9677\u200b': 0, '\u200b\u7a0d\u200b\u51f9\u200b': 1, '\u200b\u5e73\u5766\u200b': 2},\n    '\u200b\u89e6\u611f\u200b': {'\u200b\u786c\u6ed1\u200b': 0, '\u200b\u8f6f\u200b\u7c98\u200b': 1},\n    '\u200b\u597d\u74dc\u200b': {'\u200b\u662f\u200b': 1, '\u200b\u5426\u200b': 0},\n}\nfor col in value_map:\n    df[col] = df[col].map(value_map[col])\n\nX = np.concatenate([df[X_rcol].values, np.ones((df.shape[0], 1))], axis=1)\ny = df[y_rcol].values.reshape(-1, 1)\n\ndef linear_regression(X, y):\n    return np.linalg.inv(X.T @ X) @ X.T @ y\n\nlinear_regression(X, y).T\n</pre> import numpy as np import pandas as pd  df = pd.read_csv('data/melon3.0a.csv') X_rcol, y_rcol = df.columns[1:-2], df.columns[-2] # columns used for regression X_ccol, y_ccol = df.columns[1:-1], df.columns[-1] # columns used for classification value_map = {     '\u200b\u8272\u6cfd\u200b': {'\u200b\u6d45\u767d\u200b': 0, '\u200b\u9752\u7eff\u200b': 1, '\u200b\u4e4c\u9ed1\u200b': 2},     '\u200b\u6839\u8482\u200b': {'\u200b\u8737\u7f29\u200b': 0, '\u200b\u7a0d\u200b\u8737\u200b': 1, '\u200b\u786c\u633a\u200b': 2},     '\u200b\u6572\u58f0\u200b': {'\u200b\u6c89\u95f7\u200b': 0, '\u200b\u6d4a\u54cd\u200b': 1, '\u200b\u6e05\u8106\u200b': 2},     '\u200b\u7eb9\u7406\u200b': {'\u200b\u6a21\u7cca\u200b': 0, '\u200b\u7a0d\u7cca\u200b': 1, '\u200b\u6e05\u6670\u200b': 2},     '\u200b\u8110\u90e8\u200b': {'\u200b\u51f9\u9677\u200b': 0, '\u200b\u7a0d\u200b\u51f9\u200b': 1, '\u200b\u5e73\u5766\u200b': 2},     '\u200b\u89e6\u611f\u200b': {'\u200b\u786c\u6ed1\u200b': 0, '\u200b\u8f6f\u200b\u7c98\u200b': 1},     '\u200b\u597d\u74dc\u200b': {'\u200b\u662f\u200b': 1, '\u200b\u5426\u200b': 0}, } for col in value_map:     df[col] = df[col].map(value_map[col])  X = np.concatenate([df[X_rcol].values, np.ones((df.shape[0], 1))], axis=1) y = df[y_rcol].values.reshape(-1, 1)  def linear_regression(X, y):     return np.linalg.inv(X.T @ X) @ X.T @ y  linear_regression(X, y).T <p>\u200b\u635f\u5931\u200b\u51fd\u6570\u200b\u4e3a\u5747\u65b9\u200b\u8bef\u5dee\u200b</p> In\u00a0[\u00a0]: Copied! <pre>def mse_loss(X, y, w):\n    return np.mean((X @ w - y) ** 2)\n\nmse_loss(X, y, linear_regression(X, y))\n</pre> def mse_loss(X, y, w):     return np.mean((X @ w - y) ** 2)  mse_loss(X, y, linear_regression(X, y)) <p>\u200b\u5f53\u200b\u6837\u672c\u200b\u77e9\u9635\u200b\u975e\u6ee1\u200b\u79e9\u65f6\u200b\uff0c\u200b\u5b58\u5728\u200b\u591a\u4e2a\u200b\u6ee1\u8db3\u200b\u8bad\u7ec3\u200b\u96c6\u200b\u7684\u200b\u6a21\u578b\u200b\uff0c\u200b\u6b64\u65f6\u200b\u53ef\u4ee5\u200b\u5728\u200b\u4f18\u5316\u200b\u76ee\u6807\u200b\u4e2d\u200b\u52a0\u5165\u200b\u6b63\u5219\u200b\u5316\u9879\u200b\uff0c\u200b\u5982\u200bL2-norm\u200b\u5373\u200b\u52a0\u5165\u200b\u6743\u91cd\u200b\u7684\u200b\u5e73\u65b9\u200b\u4e4b\u200b\u548c\u200b\u3002\u200b\u7136\u540e\u200b\u4f7f\u7528\u200b\u68af\u5ea6\u200b\u4e0b\u964d\u200b\u7b49\u200b\u6570\u503c\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u8ba1\u7b97\u200b\u3002</p> In\u00a0[\u00a0]: Copied! <pre>class SGD():\n    def __init__(self, d, lr=0.01, epochs=1000):\n        self.d = d\n        self.lr = lr\n        self.epochs = epochs\n\n    def __call__(self, X, y):\n        w = np.random.normal(0, 1, size=(X.shape[1], 1))\n        for _ in range(self.epochs):\n            w -= self.lr * self.d(X, y, w)\n        return w\n\noptim_l2 = SGD(lambda X, y, w: X.T @ (X @ w - y) + 0.1 * w)\n\nw = optim_l2(X, y)\nprint(w.T, mse_loss(X, y, w))\n</pre> class SGD():     def __init__(self, d, lr=0.01, epochs=1000):         self.d = d         self.lr = lr         self.epochs = epochs      def __call__(self, X, y):         w = np.random.normal(0, 1, size=(X.shape[1], 1))         for _ in range(self.epochs):             w -= self.lr * self.d(X, y, w)         return w  optim_l2 = SGD(lambda X, y, w: X.T @ (X @ w - y) + 0.1 * w)  w = optim_l2(X, y) print(w.T, mse_loss(X, y, w)) <p>\u200b\u5bf9\u7387\u200b\u56de\u5f52\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u7ebf\u6027\u200b\u6a21\u578b\u200b\u5e94\u7528\u200b\u5230\u200b\u4e8c\u200b\u5206\u7c7b\u200b\u95ee\u9898\u200b\u4e0a\u200b\u3002\u200b\u5bf9\u7387\u200b\u56de\u5f52\u200b\u9700\u8981\u200b\u7528\u5230\u200blogit\u200b\u51fd\u6570\u200b\u5c06\u200b\u8fde\u7eed\u200b\u7684\u200b\u56de\u5f52\u200b\u503c\u200b\u6620\u5c04\u200b\u5230\u200b $(0, 1)$ \u200b\u4e0a\u200b</p> <p>$$ f(x) = \\frac{1}{1 + e^{-x}} $$</p> In\u00a0[\u00a0]: Copied! <pre>X = df[X_ccol].values\ny = df[y_ccol].values.reshape(-1, 1)\n\ndef sigmoid(x):\n    return 1 / (1 + np.exp(-x))\n\ndef precision(X, y, w):\n    return np.mean((sigmoid(X @ w) &gt; 0.5) == y)\n</pre> X = df[X_ccol].values y = df[y_ccol].values.reshape(-1, 1)  def sigmoid(x):     return 1 / (1 + np.exp(-x))  def precision(X, y, w):     return np.mean((sigmoid(X @ w) &gt; 0.5) == y) <p>\u200b\u4f7f\u7528\u200b\u6781\u5927\u200b\u4f3c\u7136\u6cd5\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u5bf9\u7387\u200b\u56de\u5f52\u200b\u7684\u200b\u4f18\u5316\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b</p> <p>$$ l(\\boldsymbol w) = \\sum_{i=1}^m \\left(-\\boldsymbol y_i\\boldsymbol \\beta^\\top \\boldsymbol x_i + \\ln \\left(1 + e^{\\boldsymbol \\beta^\\top \\boldsymbol x_i}\\right) \\right) $$</p> In\u00a0[\u00a0]: Copied! <pre>optim_logit = SGD(lambda X, y, w: X.T @ (sigmoid(X @ w) - y))\n\nw = optim_logit(X, y)\nprint(w.T, precision(X, y, w))\n</pre> optim_logit = SGD(lambda X, y, w: X.T @ (sigmoid(X @ w) - y))  w = optim_logit(X, y) print(w.T, precision(X, y, w)) In\u00a0[\u00a0]: Copied! <pre>def cov(X, a, b):\n    return np.mean((X[:, a] - np.mean(X[:, a])) * (X[:, b] - np.mean(X[:, b])))\n\ndef cov_matrix(X):\n    return np.array([\n        [\n            cov(X, i, j)\n            for i in range(X.shape[1])\n        ]\n        for j in range(X.shape[1])\n    ])\n\nsw = cov_matrix(X[y[:, 0] == 0]) + cov_matrix(X[y[:, 0] == 1])\nmu0, mu1 = np.mean(X[y[:, 0] == 0], axis=0), np.mean(X[y[:, 0] == 1], axis=0)\nw = np.linalg.inv(sw) @ (mu0 - mu1).reshape(-1, 1)\nc0, c1 = w.T @ mu0, w.T @ mu1\n\ndef precision(X, y, w):\n    return np.mean((X @ w &lt; (c0 + c1) / 2) == y)\n\nprecision(X, y, w)\n</pre> def cov(X, a, b):     return np.mean((X[:, a] - np.mean(X[:, a])) * (X[:, b] - np.mean(X[:, b])))  def cov_matrix(X):     return np.array([         [             cov(X, i, j)             for i in range(X.shape[1])         ]         for j in range(X.shape[1])     ])  sw = cov_matrix(X[y[:, 0] == 0]) + cov_matrix(X[y[:, 0] == 1]) mu0, mu1 = np.mean(X[y[:, 0] == 0], axis=0), np.mean(X[y[:, 0] == 1], axis=0) w = np.linalg.inv(sw) @ (mu0 - mu1).reshape(-1, 1) c0, c1 = w.T @ mu0, w.T @ mu1  def precision(X, y, w):     return np.mean((X @ w &lt; (c0 + c1) / 2) == y)  precision(X, y, w)"},{"location":"coding/machine-learning/linear-models/","title":"\u7ebf\u6027\u200b\u6a21\u578b\u200b\u00b6","text":"<p>\u200b\u4ee5\u4e0b\u200b\u4f7f\u7528\u200bPython\u200b\u5b9e\u73b0\u200b\u57fa\u4e8e\u200b\u7ebf\u6027\u5173\u7cfb\u200b\u7684\u200b\u5404\u79cd\u200b\u673a\u5668\u200b\u5b66\u4e60\u200b\u6a21\u578b\u200b</p>"},{"location":"coding/machine-learning/linear-models/","title":"\u591a\u5143\u200b\u7ebf\u6027\u200b\u56de\u5f52\u200b\u00b6","text":"<p>\u200b\u591a\u5143\u200b\u7ebf\u6027\u200b\u56de\u5f52\u200b\u7684\u200b\u4e00\u822c\u200b\u5f62\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> <p>$$ \\hat{\\boldsymbol y} = f(\\boldsymbol x) = \\boldsymbol w^\\top \\boldsymbol x + b $$</p> <p>\u200b\u5f53\u200b\u6837\u672c\u200b\u77e9\u9635\u200b$\\boldsymbol X$\u200b\u6ee1\u200b\u79e9\u65f6\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u4e3a\u200b</p> <p>$$ \\newcommand{\\bmX}{\\boldsymbol X} \\hat{\\boldsymbol w}^* = (\\bmX^\\top\\bmX)^{-1}\\bmX^\\top \\boldsymbol y $$</p>"},{"location":"coding/machine-learning/linear-models/#lda","title":"LDA\u00b6","text":"<p>LDA\u200b\u662f\u200b\u7ebf\u6027\u200b\u5224\u522b\u5206\u6790\u200b\u7684\u200b\u7b80\u79f0\u200b\uff0c\u200b\u5c5e\u4e8e\u200b\u5206\u7c7b\u200b\u7b97\u6cd5\u200b\u3002\u200b\u8be5\u200b\u7b97\u6cd5\u200b\u7684\u200b\u6838\u5fc3\u200b\u601d\u8def\u200b\u4e3a\u200b\u5c06\u200b\u6837\u672c\u200b\u70b9\u200b\u6295\u5f71\u200b\u5230\u200b $n$ \u200b\u7ef4\u7a7a\u95f4\u200b\u7684\u200b\u5e73\u9762\u200b\u4e0a\u200b\uff0c\u200b\u901a\u8fc7\u200b\u9009\u62e9\u200b\u5e73\u9762\u200b\uff0c\u200b\u6700\u5c0f\u5316\u200b\u540c\u4e00\u200b\u7c7b\u522b\u200b\u5185\u200b\u6837\u672c\u200b\u70b9\u200b\u6295\u5f71\u200b\u7684\u200b\u8ddd\u79bb\u200b\uff0c\u200b\u540c\u65f6\u200b\u6700\u5927\u5316\u200b\u4e0d\u540c\u200b\u7c7b\u522b\u200b\u6837\u672c\u200b\u70b9\u200b\u6295\u5f71\u200b\u7684\u200b\u8ddd\u79bb\u200b\u3002</p>"},{"location":"coding/python/","title":"Python","text":"<ul> <li>\u200b\u5173\u95ed\u200bPython\u200b\u7ebf\u7a0b\u200b</li> <li>Python\u200b\u4e2d\u200b\u7684\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b</li> <li>\u200b\u5e8f\u5217\u200b\u7c7b\u578b\u200b</li> <li>\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b</li> <li>\u200b\u5b57\u8282\u200b\u7c7b\u578b\u200b</li> <li>\u200b\u51fd\u6570\u200b</li> </ul>"},{"location":"coding/python/bytes/","title":"Python\u200b\u4e2d\u200b\u7684\u200b\u5b57\u8282\u200b\u7c7b\u578b","text":"<p>Python\u200b\u4e2d\u200b\u7684\u200b\u5b57\u8282\u200b\u7c7b\u578b\u200b\u5305\u62ec\u200b\u5b57\u8282\u200b\u6d41\u200b\u4e0e\u200b\u5b57\u7b26\u4e32\u200b\u3002</p> <p>\u200b\u5b57\u7b26\u4e32\u200b\u662f\u200b\u7531\u200b\u5b57\u7b26\u200b\u7ec4\u6210\u200b\u7684\u200b\u5e8f\u5217\u200b\u3002\u200b\u5b57\u7b26\u200b\u662f\u200b\u7ec4\u6210\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u57fa\u672c\u200b\u5355\u4f4d\u200b\uff0c\u200b\u5bf9\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5207\u7247\u200b\u7b49\u200b\u64cd\u4f5c\u200b\u4ee5\u200b\u5b57\u7b26\u200b\u4e3a\u200b\u5355\u4f4d\u200b\u8fdb\u884c\u200b\u3002\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u200b\u7531\u200b\u4e24\u4e2a\u200b\u90e8\u5206\u200b\u8fdb\u884c\u200b\u5b9a\u4e49\u200b\uff1a</p> <ul> <li>\u200b\u7801\u4f4d\u200b\uff0c\u200b\u5373\u200b\u5b57\u7b26\u200b\u7684\u200b\u5b57\u8282\u200b\u6570\u503c\u200b\u3002</li> <li>\u200b\u7f16\u7801\u65b9\u5f0f\u200b\uff0c\u200b\u5373\u200b\u5b57\u8282\u200b\u6570\u503c\u200b\u4e0e\u200b\u5b57\u7b26\u200b\u7684\u200b\u5bf9\u5e94\u200b\u5173\u7cfb\u200b\u3002</li> </ul> <p>\u200b\u901a\u8fc7\u200b\u7f16\u7801\u200b\u4e0e\u200b\u89e3\u7801\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u5b57\u7b26\u4e32\u200b\u4e0e\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u4e4b\u95f4\u200b\u7684\u200b\u8f6c\u6362\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; \"abc\".encode() \nb'abc'\n&gt;&gt;&gt; b\"abc\".decode()\n'abc'\n&gt;&gt;&gt;\n</code></pre> <p>Python\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u4e24\u79cd\u200b\u5b57\u8282\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5373\u200b<code>bytes</code>\u200b\u4e0e\u200b<code>bytearray</code>\uff0c\u200b\u4e24\u8005\u200b\u90fd\u200b\u662f\u200b\u7531\u200b\u65e0\u200b\u7b26\u53f7\u200b\u5b57\u8282\u200b\uff08\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b0~255\uff09\u200b\u4e3a\u200b\u5355\u4f4d\u200b\u7ec4\u6210\u200b\u7684\u200b\u5e8f\u5217\u200b\u3002<code>bytes</code>\u200b\u662f\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\u3002</p> <pre><code>&gt;&gt;&gt; b\"\\xff\\xff\"[0]\n255\n&gt;&gt;&gt; bytes(3)[0]=1   \nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: 'bytes' object does not support item assignment\n&gt;&gt;&gt;\n</code></pre> <p>New in version 3.5</p> <p><code>bytes</code>\u200b\u5bf9\u8c61\u200b\u4e0e\u200b<code>bytearray</code>\u200b\u5bf9\u8c61\u200b\u6dfb\u52a0\u200b\u4e86\u200b<code>hex</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5b57\u8282\u200b\u7684\u200b\u5341\u516d\u8fdb\u5236\u200b\u8868\u793a\u200b\u5f62\u5f0f\u200b\u3002\u200b\u8be5\u200b\u51fd\u6570\u200b\u4e0e\u200b<code>bytes</code>\u200b\u5bf9\u8c61\u200b\u7684\u200b<code>fromhex</code>\u200b\u65b9\u6cd5\u200b\u76f8\u53cd\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import random\n&gt;&gt;&gt; seq = random.randbytes(8) # New in Python 3.9\n&gt;&gt;&gt; seq.hex() # doctest: +SKIP\n'f04be4376519e9ce'\n&gt;&gt;&gt;\n</code></pre> <p>Changed in version 3.8</p> <p><code>hex</code>\u200b\u65b9\u6cd5\u200b\u65b0\u589e\u200b\u4e86\u200b\u53ef\u9009\u200b\u7684\u200b<code>sep</code>\u200b\u53c2\u6570\u200b\u4e0e\u200b<code>bytes_per_sep</code>\u200b\u53c2\u6570\u200b\u3002</p> <ul> <li><code>sep</code>\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\u533a\u6bb5\u200b\u95f4\u200b\u7684\u200b\u8fde\u63a5\u200b\u5b57\u7b26\u4e32\u200b\uff1b</li> <li><code>bytes_per_sep</code>\u200b\u53c2\u6570\u200b\u7528\u4e8e\u200b\u5212\u5206\u200b\u8fde\u7eed\u200b\u7684\u200b\u5b57\u8282\u200b\u6bb5\u200b\uff0c\u200b\u5b57\u8282\u200b\u6bb5\u200b\u4ece\u200b\u53f3\u200b\u5f80\u200b\u5de6\u200b\u8fdb\u884c\u200b\u5212\u5206\u200b\u3002</li> </ul> <pre><code># Following the previous example\n&gt;&gt;&gt; seq.hex('-') # doctest: +SKIP\n'f0-4b-e4-37-65-19-e9-ce'\n&gt;&gt;&gt; seq.hex('-', 2) # doctest: +SKIP\n'f04b-e437-6519-e9ce'\n&gt;&gt;&gt; seq.hex('-', 3) # doctest: +SKIP\n'f04b-e43765-19e9ce'\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u5b57\u8282\u200b\u6709\u200b\u4e09\u79cd\u200b\u8868\u793a\u200b\u65b9\u5f0f\u200b\uff1a</p> <ul> <li>ASCII\u200b\u4e2d\u200b\u89c4\u5b9a\u200b\u7684\u200b\u53ef\u200b\u6253\u5370\u200b\u5b57\u7b26\u200b\uff0c\u200b\u4f7f\u7528\u200b\u8be5\u200b\u5b57\u7b26\u200b\u672c\u8eab\u200b</li> <li>\u200b\u5236\u8868\u7b26\u200b\u3001\u200b\u6362\u884c\u7b26\u200b\u3001\u200b\u56de\u8f66\u7b26\u200b\u4e0e\u200b\u53cd\u200b\u659c\u6760\u200b\u4f7f\u7528\u200b\u5bf9\u5e94\u200b\u7684\u200b\u8f6c\u4e49\u5e8f\u5217\u200b\u8868\u793a\u200b\uff0c\u200b\u5373\u200b<code>\\t, \\n, \\r, \\\\</code></li> <li>\u200b\u6240\u6709\u200b\u5b57\u8282\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5341\u516d\u8fdb\u5236\u200b\u8f6c\u4e49\u5e8f\u5217\u200b\u8868\u793a\u200b\uff0c\u200b\u5982\u200b<code>\\x00</code></li> </ul> <pre><code>&gt;&gt;&gt; b\"\\t\" == b\"\\x09\"\nTrue\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5b57\u7b26\u4e32\u200b\u4e0e\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u7684\u200b\u533a\u522b\u200b\u5728\u4e8e\u200b\uff1a\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u7d22\u5f15\u200b\u4e0e\u200b\u5207\u7247\u200b\u64cd\u4f5c\u200b\u8fd4\u56de\u200b\u7684\u200b\u5bf9\u8c61\u200b\u90fd\u200b\u662f\u200b\u5b57\u7b26\u4e32\u200b\u7c7b\u578b\u200b\uff1b\u200b\u800c\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u7684\u200b\u7d22\u5f15\u200b\u64cd\u4f5c\u200b\u8fd4\u56de\u200b<code>int</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5207\u7247\u200b\u64cd\u4f5c\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u3002</p>"},{"location":"coding/python/bytes/#_1","title":"\u6784\u9020","text":"<p>\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u6784\u9020\u200b\u975e\u5e38\u7b80\u5355\u200b\uff0c\u200b\u6b64\u5904\u200b\u4e0d\u200b\u505a\u200b\u8ba8\u8bba\u200b\u3002\u200b\u5728\u200b\u5b57\u7b26\u4e32\u200b\u524d\u52a0\u200b<code>r</code>\u200b\u53ef\u4ee5\u200b\u53d6\u6d88\u200b\u5b57\u7b26\u4e32\u200b\u5185\u90e8\u200b\u7684\u200b\u8f6c\u4e49\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; print(r\"ab\\n\") \nab\\n\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5728\u200b\u5b57\u7b26\u4e32\u200b\u524d\u52a0\u200b<code>b</code>\u200b\u53ef\u4ee5\u200b\u6784\u9020\u200b\u4e00\u4e2a\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u3002\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u53ea\u80fd\u200b\u5305\u542b\u200bASCII\u200b\u53ef\u200b\u6253\u5370\u200b\u5b57\u7b26\u200b\u3002</p> <p>\u200b\u9664\u6b64\u4e4b\u5916\u200b\uff0c\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u8fd8\u6709\u200b\u5982\u4e0b\u200b\u6784\u9020\u200b\u65b9\u5f0f\u200b\uff1a</p> <ul> <li>\u200b\u6307\u5b9a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u548c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u7f16\u7801\u65b9\u5f0f\u200b\uff0c\u200b\u5c06\u200b\u8be5\u200b\u5b57\u7b26\u4e32\u200b\u7f16\u7801\u200b\u4e3a\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b</li> <li>\u200b\u4e00\u4e2a\u200b\u4ec5\u200b\u5305\u542b\u200b<code>0~255</code>\u200b\u5185\u200b\u6570\u503c\u200b\u7684\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b</li> <li>\u200b\u4e00\u4e2a\u200b\u5b9e\u73b0\u200b\u7f13\u51b2\u200b\u534f\u8bae\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5c06\u200b\u8be5\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u590d\u5236\u5230\u200b\u65b0\u200b\u7684\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u4e2d\u200b\uff08\u200b\u53ef\u80fd\u200b\u6d89\u53ca\u200b\u7c7b\u578b\u8f6c\u6362\u200b\uff09</li> <li>\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\uff0c\u200b\u521b\u5efa\u200b\u5bf9\u5e94\u200b\u957f\u5ea6\u200b\u7684\u200b\u7a7a\u200b\u5b57\u8282\u200b\u5bf9\u8c61\u200b</li> </ul> <p>\u200b\u5982\u200b\uff0c\u200b\u4ece\u200b<code>array.array</code>\u200b\u5bf9\u8c61\u200b\u521b\u5efa\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; import array  \n&gt;&gt;&gt; import random\n&gt;&gt;&gt; a = array.array(\"H\", [51417, 45016, 65120, 9976])\n&gt;&gt;&gt; b = bytes(a)\n&gt;&gt;&gt; b\nb'\\xd9\\xc8\\xd8\\xaf`\\xfe\\xf8&amp;'\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/bytes/#_2","title":"\u7ed3\u6784\u200b\u4f53","text":"<p><code>struct</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5c06\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u4e0d\u540c\u200b\u7c7b\u578b\u200b\u5b57\u6bb5\u200b\u7684\u200b\u5143\u7ec4\u200b\uff0c\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bC\u200b\u8bed\u8a00\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7684\u200b\u529f\u80fd\u200b\u3002</p>"},{"location":"coding/python/bytes/#_3","title":"\u7ed3\u6784\u200b\u5b9a\u4e49","text":"<p>\u200b\u7ed3\u6784\u200b\u5b9a\u4e49\u200b\u5305\u542b\u200b\u4e24\u4e2a\u200b\u90e8\u5206\u200b\uff0c\u200b\u5373\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\u4e0e\u200b\u5b57\u200b\u6bb5\u200b\u3002<code>struct</code>\u200b\u6a21\u5757\u200b\u5141\u8bb8\u200b\u591a\u79cd\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b</p> \u200b\u5b57\u7b26\u200b \u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b \u200b\u5927\u5c0f\u200b \u200b\u5bf9\u9f50\u200b\u65b9\u5f0f\u200b <code>@</code> native native native <code>=</code> native standard none <code>&lt;</code> little-endian standard none <code>&gt;</code> big-endian standard none <code>!</code> network (= big-endian) standard none <p>\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\u4e3a\u200b<code>@</code>\u3002</p> <p><code>struct</code>\u200b\u6a21\u5757\u200b\u7684\u200b\u5b57\u200b\u6bb5\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff0c\u200b\u6240\u6709\u200b\u7684\u200b\u5b57\u200b\u6bb5\u200b\u5728\u200bC\u200b\u8bed\u8a00\u200b\u4e2d\u200b\u90fd\u200b\u6709\u200b\u5bf9\u5e94\u200b\u7684\u200b\u7c7b\u578b\u200b\uff1a</p> \u200b\u5b57\u7b26\u200b C \u200b\u7c7b\u578b\u200b Python \u200b\u7c7b\u578b\u200b \u200b\u5b57\u200b\u5bbd\u200b <code>x</code> \uff08\u200b\u586b\u5145\u200b\u5b57\u8282\u200b\uff09 N/A <code>c</code> <code>char</code> \u200b\u957f\u5ea6\u200b\u4e3a\u200b1\u200b\u7684\u200b\u5b57\u8282\u200b 1 <code>b</code> <code>signed char</code> <code>int</code> 1 <code>B</code> <code>unsigned char</code> <code>int</code> 1 <code>?</code> <code>_Bool</code> <code>bool</code> 1 <code>h</code> <code>short</code> <code>int</code> 2 <code>H</code> <code>unsigned short</code> <code>int</code> 2 <code>i</code> <code>int</code> <code>int</code> 4 <code>I</code> <code>unsigned int</code> <code>int</code> 4 <code>l</code> <code>long</code> <code>int</code> 4 <code>L</code> <code>unsigned long</code> <code>int</code> 4 <code>q</code> <code>long long</code> <code>int</code> 8 <code>Q</code> <code>unsigned long long</code> <code>int</code> 8 <code>n</code> <code>ssize_t</code> <code>int</code> \uff08\u200b\u4ec5\u200b\u9002\u7528\u200b\u4e8e\u200b\u9ed8\u8ba4\u200b\u6216\u200b<code>@</code>\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\uff09 <code>N</code> <code>size_t</code> <code>int</code> \uff08\u200b\u4ec5\u200b\u9002\u7528\u200b\u4e8e\u200b\u9ed8\u8ba4\u200b\u6216\u200b<code>@</code>\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\uff09 <code>e</code> \uff08\u200b\u534a\u200b\u7cbe\u5ea6\u200b\uff09 <code>float</code> 2 <code>f</code> <code>float</code> <code>float</code> 4 <code>d</code> <code>double</code> <code>float</code> 8 <code>s</code> <code>char[]</code> <code>bytes</code> \u200b\u4e0e\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u6709\u5173\u200b <code>p</code> <code>char[]</code> <code>bytes</code> \u200b\u4e0e\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u6709\u5173\u200b <code>P</code> <code>void *</code> <code>int</code> <p>\u200b\u5f53\u200b\u8bd5\u56fe\u200b\u5c06\u200b\u975e\u200b\u6574\u6570\u200b\u5bf9\u8c61\u200b\u6253\u5305\u200b\u4e3a\u200b\u6574\u6570\u200b\u7c7b\u578b\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u7684\u200b<code>__index__</code>\u200b\u65b9\u6cd5\u200b\u3002</p> <p>\u200b\u4e00\u4e2a\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7684\u200b\u5b9a\u4e49\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u7ed3\u6784\u200b\u7ec4\u7ec7\u200b\uff1a</p> <ol> <li>\u200b\u7b2c\u4e00\u4e2a\u200b\u5b57\u7b26\u200b\u8868\u793a\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b</li> <li>\u200b\u6b64\u540e\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\u8868\u793a\u200b\u7ed3\u6784\u200b\u4f53\u4e2d\u200b\u7684\u200b\u5b57\u200b\u6bb5\u200b\u7c7b\u578b\u200b</li> <li>\u200b\u9664\u200b<code>s</code>\u200b\u548c\u200b<code>p</code>\u200b\u4ee5\u5916\u200b\uff0c\u200b\u5b57\u6bcd\u200b\u524d\u200b\u7684\u200b\u6570\u5b57\u200b\u8868\u660e\u200b\u8be5\u5b57\u6bb5\u200b\u91cd\u590d\u200b\u51fa\u73b0\u200b\u7684\u200b\u6b21\u6570\u200b</li> <li><code>s</code>\u3001<code>p</code>\u200b\u524d\u200b\u7684\u200b\u6570\u5b57\u200b\u8868\u660e\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u957f\u5ea6\u200b</li> </ol> <p>\u200b\u7ed3\u6784\u200b\u4f53\u200b\u5b57\u7b26\u4e32\u200b\u53ef\u4ee5\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b<code>struct.Struct</code>\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u4e24\u8005\u200b\u5b9e\u73b0\u200b\u76f8\u540c\u200b\u7684\u200b\u529f\u80fd\u200b\u3002</p>"},{"location":"coding/python/bytes/#_4","title":"\u7ed3\u6784\u200b\u64cd\u4f5c","text":"<p>\u200b\u5bf9\u4e8e\u200b\u4e00\u4e2a\u200b\u5b9a\u4e49\u200b\u7684\u200b\u7ed3\u6784\u200b\u4f53\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u6570\u636e\u200b\u6309\u7167\u200b\u7ed3\u6784\u200b\u4f53\u200b\u8fdb\u884c\u200b\u6253\u5305\u200b\uff0c\u200b\u6216\u200b\u5c06\u200b\u7ed3\u6784\u200b\u4f53\u4e2d\u200b\u7684\u200b\u6570\u636e\u200b\u89e3\u5305\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u663e\u793a\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7684\u200b\u5927\u5c0f\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import struct   \n&gt;&gt;&gt; struct.pack('&lt;hhf', 1, 2, 3)\nb'\\x01\\x00\\x02\\x00\\x00\\x00@@'\n&gt;&gt;&gt; struct.unpack('&lt;hhf', b'\\x01\\x00\\x02\\x00\\x00\\x00@@')\n(1, 2, 3.0)\n&gt;&gt;&gt;\n</code></pre> <p>New in version 3.4</p> <p><code>struct</code>\u200b\u5bf9\u8c61\u200b\u65b0\u589e\u200b\u4e86\u200b<code>iter_unpack</code>\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u4e0d\u540c\u4e8e\u200b<code>unpack</code>\u200b\u51fd\u6570\u200b\uff0c<code>iter_unpack</code>\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002</p> <p><code>struct</code>\u200b\u6a21\u5757\u200b\u4e0d\u4f1a\u200b\u5bf9\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\u8fdb\u884c\u200b\u68c0\u6d4b\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5bf9\u4e8e\u200b\u540c\u4e00\u4e2a\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4e0d\u540c\u200b\u7684\u200b\u7ed3\u6784\u200b\u4f53\u200b\u5b9a\u4e49\u200b\u5728\u200b\u89e3\u5305\u200b\u540e\u4f1a\u200b\u6709\u200b\u4e0d\u540c\u200b\u7684\u200b\u7ed3\u679c\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; struct.unpack('&gt;hhf', b'\\x01\\x00\\x02\\x00\\x00\\x00@@') \n(256, 512, 2.304855714121459e-41)\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6bcf\u4e2a\u200b\u5b57\u6bb5\u200b\u90fd\u200b\u6709\u200b\u8303\u56f4\u200b\u9650\u5236\u200b\uff0c\u200b\u5f53\u200b\u4f20\u5165\u200b\u7684\u200b\u53c2\u6570\u200b\u8d85\u8fc7\u200b\u5b57\u6bb5\u200b\u6240\u200b\u5141\u8bb8\u200b\u7684\u200b\u8303\u56f4\u200b\uff0c\u200b\u5219\u200b\u4f1a\u200b\u629b\u51fa\u200b\u5f02\u5e38\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; struct.pack('&lt;hhf', 32768, -32769, 3)\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nstruct.error: short format requires (-32768) &lt;= number &lt;= 32767\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5f53\u200b\u89e3\u5305\u200b\u7684\u200b\u5b57\u8282\u200b\u957f\u5ea6\u200b\u4e0e\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7684\u200b\u957f\u5ea6\u200b\u4e0d\u200b\u5bf9\u5e94\u200b\u65f6\u200b\uff0c\u200b\u4e5f\u200b\u4f1a\u200b\u629b\u51fa\u200b\u5f02\u5e38\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; struct.unpack('&gt;hhf', b'\\x01\\x00\\x02\\x00\\x00\\x00@')  \nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nstruct.error: unpack requires a buffer of 8 bytes\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u4f55\u200b\u7ed3\u6784\u200b\u4f53\u200b\uff0c<code>struct</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>calcsize</code>\u200b\u65b9\u6cd5\u200b\u7528\u4e8e\u200b\u68c0\u67e5\u200b\u7ed3\u6784\u200b\u4f53\u200b\u957f\u5ea6\u200b\u3002</p>"},{"location":"coding/python/bytes/#_5","title":"\u5185\u5b58\u200b\u89c6\u56fe","text":"<p>\u200b\u5185\u5b58\u200b\u89c6\u56fe\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5728\u200b\u4e0d\u540c\u200b\u5bf9\u8c61\u200b\u95f4\u200b\u5171\u4eab\u5185\u5b58\u200b\u7684\u200b\u65b9\u5f0f\u200b\u3002</p> <p>\u200b\u6709\u5173\u200b\u5185\u5b58\u200b\u89c6\u56fe\u200b\u7684\u200b\u5185\u5bb9\u200b\uff0c\u200b\u8bf7\u200b\u53c2\u89c1\u200b\u5e8f\u5217\u200b\u7c7b\u578b\u200b\u7684\u200b\u5185\u5b58\u200b\u89c6\u56fe\u200b\u90e8\u5206\u200b\u3002</p>"},{"location":"coding/python/bytes/#_6","title":"\u5b57\u7b26\u4e32","text":"<p>\u200b\u6b64\u5904\u200b\u7740\u91cd\u200b\u8ba8\u8bba\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u76f8\u5173\u200b\u95ee\u9898\u200b</p>"},{"location":"coding/python/bytes/#_7","title":"\u7f16\u7801\u200b\u4e0e\u200b\u89e3\u7801","text":"<p>\u200b\u5982\u524d\u6240\u8ff0\u200b\uff0c<code>str.encode</code>\u200b\u65b9\u6cd5\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u4ece\u200b\u5b57\u7b26\u4e32\u200b\u5230\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u7684\u200b\u8f6c\u6362\u200b\u65b9\u5f0f\u200b\uff0c<code>encoding</code>\u200b\u53c2\u6570\u200b\u6307\u660e\u200b\u4e86\u200b\u6240\u200b\u4f7f\u7528\u200b\u7684\u200b\u7f16\u7801\u5668\u200b\u3002</p> <pre><code>&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"utf-8\") \nb'\\xe6\\xb5\\x8b\\xe8\\xaf\\x95'\n&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"utf-16\")\nb'\\xff\\xfeKm\\xd5\\x8b'\n&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"gb2312\")  \nb'\\xb2\\xe2\\xca\\xd4'\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5f53\u200b\u7f16\u7801\u200b\u8fc7\u7a0b\u200b\u51fa\u73b0\u200b\u9519\u8bef\u200b\uff0c\u200b\u5982\u200b\u7f16\u7801\u5668\u200b\u65e0\u6cd5\u200b\u8bc6\u522b\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u5b57\u7b26\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u629b\u51fa\u200b<code>UnicodeEncodeError</code>\u200b\u5f02\u5e38\u200b\u3002</p> <pre><code>&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"latin-1\")\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nUnicodeEncodeError: 'latin-1' codec can't encode characters in position 0-1: ordinal not in range(256)\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u51fa\u73b0\u200b\u9519\u8bef\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\u4ee5\u4e0b\u200b\u89e3\u51b3\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>errors</code>\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\uff1a</p> <ul> <li><code>ignore</code>\uff1a\u200b\u8df3\u8fc7\u200b\u65e0\u6cd5\u200b\u7f16\u7801\u200b\u7684\u200b\u5b57\u7b26\u200b</li> <li><code>replace</code>\uff1a\u200b\u5c06\u200b\u65e0\u6cd5\u200b\u7f16\u7801\u200b\u7684\u200b\u5b57\u7b26\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>?</code></li> <li><code>xmlcharrefreplace</code>\uff1a\u200b\u5c06\u200b\u65e0\u6cd5\u200b\u7f16\u7801\u200b\u7684\u200b\u5b57\u7b26\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>xml</code>\u200b\u5b9e\u4f53\u200b\uff08\u200b\u5373\u200bXML\u200b\u4e2d\u200b\u6240\u200b\u4f7f\u7528\u200b\u7684\u200b\u5b57\u7b26\u200b\u8f6c\u6362\u200b\u65b9\u5f0f\u200b\uff09</li> </ul> <pre><code>&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"latin-1\", errors=\"ignore\")\nb''\n&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"latin-1\", errors=\"replace\")\nb'??'\n&gt;&gt;&gt; \"\u200b\u6d4b\u8bd5\u200b\".encode(\"latin-1\", errors=\"xmlcharrefreplace\")\nb'&amp;#27979;&amp;#35797;'\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5bf9\u5e94\u200b\u5730\u200b\uff0c\u200b\u89e3\u7801\u5668\u200b\u65e0\u6cd5\u200b\u8bc6\u522b\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u7684\u200b\u5b57\u8282\u200b\u65f6\u4f1a\u200b\u4ea7\u751f\u200b<code>UnicodeDecodeError</code>\u200b\u5f02\u5e38\u200b\u3002\u200b\u4f46\u200b\u4e0d\u200b\u629b\u51fa\u200b\u5f02\u5e38\u200b\u4e0d\u200b\u4ee3\u8868\u200b\u89e3\u7801\u200b\u6210\u529f\u200b\uff0c\u200b\u89e3\u7801\u200b\u5f97\u5230\u200b\u7684\u200b\u6570\u636e\u200b\u53ef\u80fd\u200b\u662f\u200b\u65e0\u7528\u200b\u6570\u636e\u200b\u3002<code>errors</code>\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\u4e86\u200b\u89e3\u7801\u5668\u200b\u5728\u200b\u51fa\u9519\u200b\u65f6\u200b\u7684\u200b\u884c\u4e3a\u200b\uff0c<code>replace</code>\u200b\u5c06\u200b\u65e0\u6cd5\u200b\u7f16\u7801\u200b\u7684\u200b\u5b57\u7b26\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>\ufffd</code>\u3002</p> <p><code>chardet</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u57fa\u4e8e\u200bPython\u200b\u7684\u200b\u5b57\u7b26\u200b\u7f16\u7801\u200b\u68c0\u6d4b\u5de5\u5177\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4e8c\u8fdb\u5236\u200b\u5e8f\u5217\u200b\u5bf9\u200b\u539f\u59cb\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u7f16\u7801\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u63a8\u65ad\u200b\u3002\u200b\u4e0d\u8fc7\u200b\u63a8\u65ad\u200b\u4ec5\u200b\u9002\u7528\u200b\u4e8e\u200b\u8f83\u200b\u957f\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u56e0\u4e3a\u200b\u4efb\u4f55\u200b\u5b57\u7b26\u4e32\u200b\u4f1a\u200b\u6709\u200b\u591a\u4e2a\u200b\u7f16\u7801\u65b9\u5f0f\u200b\u9002\u7528\u200b\u4e8e\u200b\u540c\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u6240\u4ee5\u200b\u65e0\u6cd5\u200b\u5b8c\u5168\u200b\u786e\u5b9a\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u7f16\u7801\u65b9\u5f0f\u200b\u3002</p>"},{"location":"coding/python/bytes/#bom","title":"BOM","text":"<p>BOM\u200b\u662f\u200b\u5b57\u8282\u200b\u5e8f\u200b\u6807\u8bb0\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200bUnicode\u200b\u5b57\u7b26\u200b\u4e3a\u200b<code>U+FEFF</code>\uff08\u200b\u4e0d\u200b\u5b58\u5728\u200b<code>U+FFFE</code>\u200b\u5b57\u7b26\u200b\uff0c\u200b\u56e0\u6b64\u200b\u8be5\u200b\u5b57\u7b26\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u63a8\u65ad\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\uff09</p> <p>\u200b\u5728\u200bUTF-16\u200b\u7f16\u7801\u200b\u7684\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u5f00\u5934\u200b\u4f1a\u200b\u5199\u5165\u200bBOM\uff0c\u200b\u5982\u679c\u200b\u5f00\u5934\u200b\u662f\u200b<code>b'\\xff\\xfe'</code>\u200b\u4e24\u4e2a\u200b\u5b57\u8282\u200b\uff0c\u200b\u6307\u660e\u200b\u7f16\u7801\u200b\u65f6\u200b\u4f7f\u7528\u200blittle endian\u200b\u5b57\u8282\u200b\u7f16\u7801\u200b\u987a\u5e8f\u200b\u3002\u200b\u5982\u679c\u200b\u662f\u200b<code>b'\\xfe\\xff'</code>\u200b\u4e24\u4e2a\u200b\u5b57\u8282\u200b\uff0c\u200b\u8bf4\u660e\u200b\u7f16\u7801\u200b\u65f6\u200b\u4f7f\u7528\u200b\u7684\u200b\u662f\u200bbig endian\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\u3002\u200b\u5982\u679c\u200b\u6307\u660e\u200bUTF-16\u200b\u6240\u200b\u4f7f\u7528\u200b\u7684\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\uff0c\u200b\u5982\u200bUTF-16LE\u200b\u6216\u200bUTF-16BE\uff0c\u200b\u5219\u200b\u4e0d\u4f1a\u200b\u751f\u6210\u200bBOM\u3002</p> <p>BOM\u200b\u4ec5\u200b\u7528\u4e8e\u200b\u63a8\u65ad\u200b\u5b57\u8282\u200b\u987a\u5e8f\u200b\u800c\u200b\u4e0d\u4f1a\u200b\u51fa\u73b0\u200b\u5728\u200b\u6700\u7ec8\u200b\u89e3\u7801\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\u3002</p>"},{"location":"coding/python/bytes/#_8","title":"\u6587\u672c\u6587\u4ef6","text":"<p>\u200b\u4f7f\u7528\u200b<code>open</code>\u200b\u51fd\u6570\u200b\u4ee5\u200b\u6587\u672c\u200b\u6a21\u5f0f\u200b\u6253\u5f00\u200b\u4e00\u4e2a\u200b\u6587\u4ef6\u200b\u65f6\u200b\uff0c\u200b\u6700\u597d\u200b\u6307\u5b9a\u200b\u6587\u4ef6\u200b\u7684\u200b\u7f16\u7801\u65b9\u5f0f\u200b\u3002</p> <p>\u200b\u4e0d\u8981\u200b\u4f7f\u7528\u200b\u4e8c\u8fdb\u5236\u200b\u65b9\u5f0f\u200b\u6253\u5f00\u200b\u6587\u672c\u6587\u4ef6\u200b\u3002</p>"},{"location":"coding/python/bytes/#unicode","title":"Unicode\u200b\u89c4\u8303\u5316","text":"<p>\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a = 'caf\u00e9'\n&gt;&gt;&gt; b = 'cafe\\u0301'             \n&gt;&gt;&gt; print(a, b)\ncaf\u00e9 cafe\u0301\n&gt;&gt;&gt; a == b\nFalse\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u76f8\u540c\u200b\u7684\u200b\u6253\u5370\u200b\u7ed3\u679c\u200b\uff0c\u200b\u5374\u200b\u5bf9\u5e94\u200b\u4e0d\u540c\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u539f\u56e0\u200b\u5728\u4e8e\u200b\u5b57\u7b26\u4e32\u200b<code>b</code>\u200b\u4f7f\u7528\u200b\u4e86\u200b<code>U+0301</code>\u200b\u5b57\u7b26\u200b\u4f5c\u4e3a\u200b\u91cd\u97f3\u200b\u6807\u8bb0\u200b\uff08\u200b\u7ec4\u5408\u200b\u5b57\u7b26\u200b\uff09\uff0c\u200b\u591a\u200b\u4f7f\u7528\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u5b57\u8282\u200b\u3002\u200b\u5bf9\u4e8e\u200bPython\u200b\u800c\u8a00\u200b\uff0c\u200b\u8fd9\u200b\u4e00\u6bb5\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u5e76\u4e0d\u76f8\u540c\u200b\uff0c\u200b\u56e0\u6b64\u200b\u8ba4\u4e3a\u200b<code>a != b</code>\u3002</p> <pre><code>&gt;&gt;&gt; a.encode(\"utf-8\")     \nb'caf\\xc3\\xa9'\n&gt;&gt;&gt; b.encode(\"utf-8\") \nb'cafe\\xcc\\x81'\n&gt;&gt;&gt;\n</code></pre> <p><code>unicodedata</code>\u200b\u6a21\u5757\u200b\u4e2d\u200b\u7684\u200b<code>normalize</code>\u200b\u51fd\u6570\u200b\u63d0\u4f9b\u200b\u4e86\u200bUnicode\u200b\u89c4\u8303\u5316\u200b\u7684\u200b\u529f\u80fd\u200b\uff0c\u200b\u8be5\u200b\u51fd\u6570\u200b\u63a5\u6536\u200b\u5982\u4e0b\u200b\u53c2\u6570\u200b\u7528\u4e8e\u200b\u786e\u5b9a\u200b\u8f6c\u6362\u200b\u6807\u51c6\u200b\uff1a</p> <ul> <li><code>'NFC'</code>\uff1a\u200b\u4f7f\u7528\u200b\u6700\u5c11\u200b\u7801\u4f4d\u200b\u6784\u6210\u200b\u7b49\u4ef7\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b</li> <li><code>'NFD'</code>\uff1a\u200b\u5c06\u200b\u7ec4\u5408\u200b\u5b57\u7b26\u200b\u5206\u89e3\u200b\u4e3a\u57fa\u200b\u5b57\u7b26\u200b\u4e0e\u200b\u5355\u72ec\u200b\u7684\u200b\u7ec4\u5408\u200b\u5b57\u7b26\u200b\uff08\u200b\u5982\u200b<code>U+0301</code>\uff09\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u53bb\u9664\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u53d8\u97f3\u200b\u7b26\u53f7\u200b</li> <li><code>NFKC</code>\u3001<code>NFKD</code>\u200b\u4f1a\u200b\u989d\u5916\u200b\u5c06\u200b\u517c\u5bb9\u200b\u5b57\u7b26\u200b\u5206\u89e3\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u6216\u200b\u591a\u4e2a\u200b\u517c\u5bb9\u200b\u5206\u89e3\u200b\uff0c\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u6570\u636e\u200b\u635f\u5931\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</li> </ul> <pre><code>&gt;&gt;&gt; from unicodedata import normalize\n&gt;&gt;&gt; normalize('NFKC', '\u337f') \n'\u200b\u682a\u5f0f\u4f1a\u793e\u200b'\n&gt;&gt;&gt;\n</code></pre> <p><code>NKFC</code>\u3001<code>NFKD</code>\u200b\u89c4\u8303\u5316\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u539f\u610f\u200b\u53d8\u5316\u200b\uff0c\u200b\u4f46\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u641c\u7d22\u5f15\u64ce\u200b\u3002</p> <p><code>str.casefold</code>\u200b\u51fd\u6570\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u53e6\u200b\u4e00\u79cd\u200b\u89c4\u8303\u5316\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u5373\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5927\u5199\u5b57\u6bcd\u200b\u8f6c\u4e3a\u200b\u5c0f\u5199\u200b\u3002\u200b\u4e0e\u200b<code>str.lower</code>\u200b\u4e0d\u540c\u200b\uff0c\u200b\u90e8\u5206\u200b\u5b57\u7b26\u200b\u4f1a\u200b\u88ab\u200b\u66ff\u6362\u6210\u200b\u65b0\u200b\u7684\u200b\u5b57\u7b26\u200b\u3002</p>"},{"location":"coding/python/bytes/#_9","title":"\u5b57\u7b26\u4e32\u200b\u6392\u5e8f","text":"<p>\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u6392\u5e8f\u200b\u4e0e\u200b\u5176\u4ed6\u200b\u6570\u636e\u200b\u7684\u200b\u6392\u5e8f\u200b\u65b9\u5f0f\u200b\u76f8\u540c\u200b\uff0c\u200b\u90fd\u200b\u662f\u200b\u6309\u7167\u200b\u7801\u4f4d\u200b\u5347\u5e8f\u200b\u6392\u5e8f\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u975e\u200b<code>ASCII</code>\u200b\u5b57\u7b26\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u4e00\u4e9b\u200b\u95ee\u9898\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; sorted(['caf\u00e9', 'cafu'])       \n['cafu', 'caf\u00e9']\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6a21\u5757\u200b<code>locale</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>strxfrm</code>\u200b\u51fd\u6570\u200b\uff0c\u200b\u7528\u4e8e\u200b\u6309\u7167\u200b\u533a\u57df\u200b\u8bbe\u7f6e\u200b\u5bf9\u200b\u5b57\u7b26\u4e32\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b\u3002\u200b\u5982\u679c\u200b\u64cd\u4f5c\u7cfb\u7edf\u200b\u652f\u6301\u200b\uff0c\u200b\u533a\u57df\u200b\u8bbe\u7f6e\u200b\u53ef\u4ee5\u200b\u5728\u200b<code>setlocale</code>\u200b\u51fd\u6570\u200b\u4e2d\u200b\u5168\u5c40\u200b\u6307\u5b9a\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import locale\n&gt;&gt;&gt; locale.setlocale(locale.LC_COLLATE, \"en_US.UTF-8\")\n'en_US.UTF-8'\n&gt;&gt;&gt; sorted(['caf\u00e9', 'cafu'], key=locale.strxfrm)\n['caf\u00e9', 'cafu']\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/bytes/#api","title":"API","text":"<p>\u200b\u90e8\u5206\u200b\u6d89\u53ca\u200b\u5230\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u4e00\u4e9b\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u8f93\u5165\u200b\u5b57\u7b26\u4e32\u200b\u6216\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u3002</p>"},{"location":"coding/python/bytes/#re","title":"re\u200b\u6a21\u5757","text":"<p>\u200b\u4e0d\u540c\u4e8e\u200b\u4f7f\u7528\u200b\u5b57\u7b26\u4e32\u200b\u6784\u9020\u200b\u7684\u200b\u6b63\u5219\u8868\u8fbe\u5f0f\u200b\uff0c\u200b\u4f7f\u7528\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u6784\u9020\u200b\u7684\u200b\u6b63\u5219\u8868\u8fbe\u5f0f\u200b\uff0c<code>\\d</code>\u200b\u4e0e\u200b<code>\\w</code>\u200b\u53ea\u80fd\u200b\u5339\u914d\u200bASCII\u200b\u5b57\u7b26\u200b\u3002</p>"},{"location":"coding/python/bytes/#os","title":"os\u200b\u6a21\u5757","text":"<p><code>os</code>\u200b\u6a21\u5757\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5b57\u7b26\u4e32\u200b\u53c2\u6570\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u66ff\u6362\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u51fd\u6570\u200b\u4f1a\u200b\u4f7f\u7528\u200b<code>sys.getfilesystemencoding()</code>\u200b\u51fd\u6570\u200b\u53d6\u5f97\u200b\u5408\u9002\u200b\u7684\u200b\u7f16\u7801\u5668\u200b\u3002\u200b\u6b64\u5916\u200b\uff0c<code>os</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b<code>fsencode</code>\u200b\u4e0e\u200b<code>fsdecode</code>\u200b\u51fd\u6570\u200b\u7528\u4e8e\u200b\u624b\u52a8\u200b\u8fdb\u884c\u200b\u7f16\u7801\u200b\u4e0e\u200b\u89e3\u7801\u200b\u64cd\u4f5c\u200b\u3002</p>"},{"location":"coding/python/decorator/","title":"\u88c5\u9970\u200b\u5668","text":"<p>\u200b\u88c5\u9970\u200b\u5668\u662f\u200b\u4e00\u4e2a\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u5e76\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u5982\u4e0b\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u88c5\u9970\u200b\u5668\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; def decorator(x):\n...     print(x)\n...     return x\n...\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u88c5\u9970\u200b\u5668\u200b\u5728\u200b\u51fd\u6570\u200b\u5b9a\u4e49\u200b\u524d\u200b\u4f7f\u7528\u200b\uff0c<code>@</code>\u200b\u6807\u8bb0\u200b\u4e00\u4e2a\u200b\u88c5\u9970\u200b\u5668\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; @decorator    \n... def target(x):\n...     return x + 1\n... \n&lt;function target at ...&gt;\n&gt;&gt;&gt;\n</code></pre> <p><code>@decorator</code>\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b\uff0c\u200b\u51fd\u6570\u200b\u88ab\u200b\u88c5\u9970\u200b\u5668\u200b\u8fd4\u56de\u200b\u7684\u200b\u51fd\u6570\u200b\u66ff\u6362\u200b</p> <pre><code>&gt;&gt;&gt; def target(x):\n...     return x + 1\n... \n&gt;&gt;&gt; target = decorator(target)\n&lt;function target at ...&gt;\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u88c5\u9970\u200b\u5668\u200b\u5728\u200b\u88ab\u200b\u4fee\u9970\u200b\u7684\u200b\u51fd\u6570\u200b\u5b9a\u4e49\u200b\u540e\u200b\u7acb\u5373\u200b\u8fd0\u884c\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5f53\u200b\u5bfc\u5165\u200b\u6a21\u5757\u200b\u65f6\u200b\uff0c\u200b\u88c5\u9970\u200b\u5668\u200b\u51fd\u6570\u200b\u4f1a\u200b\u7acb\u5373\u200b\u6267\u884c\u200b\uff0c\u200b\u800c\u200b\u88ab\u200b\u88c5\u9970\u200b\u7684\u200b\u51fd\u6570\u200b\u53ea\u4f1a\u200b\u8c03\u7528\u200b\u6267\u884c\u200b\u3002</p> <pre><code>&gt;&gt;&gt; src = \"\"\"def decorator(_):\n...     print(_)\n...     return _\n... print(\"Before A\")\n... @decorator\n... def a():\n...     return 1\n... print(\"Before B\")\n... @decorator  \n... def b():\n...     return 1\n... print(\"After B\")\n... \"\"\"\n&gt;&gt;&gt; exec(src)\nBefore A\n&lt;function a at ...&gt;\nBefore B\n&lt;function b at ...&gt;\nAfter B\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/decorator/#_2","title":"\u53d8\u91cf\u200b\u7684\u200b\u4f5c\u7528\u57df","text":"<p>Python\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u5224\u65ad\u200b\u51fd\u6570\u200b\u4ee3\u7801\u200b\u4e2d\u200b\u7684\u200b\u53d8\u91cf\u200b\u662f\u5426\u200b\u4e3a\u200b\u5c40\u90e8\u53d8\u91cf\u200b\u3002\u200b\u6240\u6709\u200b\u5728\u200b\u51fd\u6570\u200b\u4ee3\u7801\u200b\u5185\u200b\u51fa\u73b0\u200b\u7684\u200b\u53d8\u91cf\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u5c40\u90e8\u53d8\u91cf\u200b\u3002</p> <pre><code>&gt;&gt;&gt; b = 2\n&gt;&gt;&gt; def A():\n...     print(b)\n... \n&gt;&gt;&gt; def B():\n...     print(b)\n...     b = 2\n... \n&gt;&gt;&gt; A()\n2\n&gt;&gt;&gt; B()\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\n  File \"&lt;stdin&gt;\", line 2, in B\nUnboundLocalError: local variable 'b' referenced before assignment\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5982\u679c\u200b\u9700\u8981\u200b\u5728\u200b\u51fd\u6570\u200b\u5185\u90e8\u200b\u8bbf\u95ee\u200b\u5168\u5c40\u53d8\u91cf\u200b\uff0c\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>global</code>\u200b\u5173\u952e\u5b57\u200b\u58f0\u660e\u200b\u3002</p> <pre><code>&gt;&gt;&gt; def C():\n...     global b\n...     print(b)\n...     b = 3\n... \n&gt;&gt;&gt; b\n2\n&gt;&gt;&gt; C()\n2\n&gt;&gt;&gt; b\n3\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u68c0\u67e5\u200b\u7f16\u8bd1\u200b\u540e\u200b\u7684\u200b\u5b57\u8282\u200b\u7801\u200b\uff0c\u200b\u53d1\u73b0\u200b<code>global</code>\u200b\u5173\u952e\u5b57\u200b\u58f0\u660e\u200b\u53d8\u91cf\u200b\u7531\u200b<code>LOAD_FAST</code>\u200b\u4e0e\u200b<code>STORE_FAST</code>\u200b\u6539\u4e3a\u200b\u4e86\u200b<code>LOAD_GLOBAL</code>\u200b\u4e0e\u200b<code>STORE_GLOBAL</code>\uff1a</p>"},{"location":"coding/python/decorator/#_3","title":"\u95ed\u5305","text":"<p>\u200b\u95ed\u5305\u200b\u662f\u200b\u6269\u5c55\u200b\u4e86\u200b\u51fd\u6570\u200b\u4f5c\u7528\u57df\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8bbf\u95ee\u200b\u51fd\u6570\u200b\u5b9a\u4e49\u200b\u4e4b\u5916\u200b\u7684\u200b\u5c40\u90e8\u53d8\u91cf\u200b\u3002\u200b\u4ee5\u4e0b\u200b\u901a\u8fc7\u200b\u4e0d\u540c\u200b\u65b9\u5f0f\u200b\u5b9e\u73b0\u200b\u6ed1\u52a8\u200b\u5e73\u5747\u503c\u200b\u7684\u200b\u8ba1\u7b97\u200b\uff1a</p> <ul> <li>\u200b\u4f7f\u7528\u200b\u7c7b\u200b</li> </ul> <pre><code>&gt;&gt;&gt; class avg1():\n...     def __init__(self):\n...             self.values = []\n...     def __call__(self, value):\n...             self.values.append(value)\n...             return sum(self.values) / len(self.values)\n... \n&gt;&gt;&gt; avg = avg1()\n&gt;&gt;&gt; avg(10)\n10.0\n&gt;&gt;&gt; avg(11)\n10.5\n&gt;&gt;&gt;\n</code></pre> <ul> <li>\u200b\u4f7f\u7528\u200b\u9ad8\u9636\u200b\u51fd\u6570\u200b</li> </ul> <pre><code>&gt;&gt;&gt; def avg2():\n...     values = []\n...     def avg(value): \n...             values.append(value)\n...             return sum(values) / len(values)\n...     return avg\n&gt;&gt;&gt; \n&gt;&gt;&gt; avg = avg2()\n&gt;&gt;&gt; avg(10)\n10.0\n&gt;&gt;&gt; avg(11)\n10.5\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5728\u200b<code>avg2</code>\u200b\u51fd\u6570\u200b\u7ed3\u675f\u200b\u540e\u200b\uff0c\u200b\u5c40\u90e8\u53d8\u91cf\u200b<code>values</code>\u200b\u539f\u6709\u200b\u7684\u200b\u4f5c\u7528\u57df\u200b\u6d88\u5931\u200b\uff0c\u200b\u6210\u4e3a\u200b\u81ea\u7531\u200b\u53d8\u91cf\u200b\u3002\u200b\u81ea\u7531\u200b\u53d8\u91cf\u200b\u6307\u200b\u6ca1\u6709\u200b\u5728\u200b\u672c\u5730\u200b\u4f5c\u7528\u57df\u200b\u4e2d\u200b\u7ed1\u5b9a\u200b\u7684\u200b\u53d8\u91cf\u200b\u3002<code>values</code>\u200b\u53d8\u91cf\u200b\u7684\u200b\u5f15\u7528\u200b\u5b58\u50a8\u200b\u5728\u200b<code>avg</code>\u200b\u51fd\u6570\u200b\u7684\u200b<code>__closure__</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\u3002\u200b\u5373\u4f7f\u200b<code>avg2</code>\u200b\u51fd\u6570\u200b\u5df2\u7ecf\u200b\u7ed3\u675f\u200b\u6267\u884c\u200b\uff0c<code>avg</code>\u200b\u51fd\u6570\u200b\u4ecd\u7136\u200b\u80fd\u200b\u8bbf\u95ee\u200b<code>values</code>\u200b\u53d8\u91cf\u200b\u3002</p>"},{"location":"coding/python/functions/","title":"Python\u200b\u4e2d\u200b\u7684\u200b\u51fd\u6570","text":"<p>\u201c\u200b\u4e00\u7b49\u200b\u5bf9\u8c61\u200b\u201d\u200b\u6307\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff1a</p> <ul> <li>\u200b\u5728\u200b\u7a0b\u5e8f\u6267\u884c\u200b\u65f6\u200b\u521b\u5efa\u200b</li> <li>\u200b\u80fd\u591f\u200b\u8d4b\u503c\u200b\u7ed9\u200b\u53d8\u91cf\u200b\u6216\u200b\u6570\u636e\u7ed3\u6784\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b</li> <li>\u200b\u80fd\u200b\u4f5c\u4e3a\u200b\u51fd\u6570\u53c2\u6570\u200b\u4e0e\u200b\u8fd4\u56de\u503c\u200b</li> </ul> <p>Python\u200b\u4e2d\u200b\u7684\u200b\u51fd\u6570\u200b\u662f\u200b\u4e00\u7b49\u200b\u5bf9\u8c61\u200b</p>"},{"location":"coding/python/functions/#_1","title":"\u51fd\u6570\u200b\u4e0e\u200b\u9ad8\u9636\u200b\u51fd\u6570","text":"<p>\u200b\u51fd\u6570\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u901a\u5e38\u200b\u4f7f\u7528\u200b<code>def</code>\u200b\u5173\u952e\u5b57\u200b\u5b9a\u4e49\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; def constant():  \n...     return 1\n... \n&gt;&gt;&gt; type(constant)\n&lt;class 'function'&gt;\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u5728\u200b\u5176\u4ed6\u200b\u51fd\u6570\u200b\u5185\u90e8\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; def outer():\n...     def inner():\n...             return \"ok\"\n...     return inner\n... \n&gt;&gt;&gt; func = outer()\n&gt;&gt;&gt; func()\n'ok'\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u9ad8\u9636\u200b\u51fd\u6570\u200b\u662f\u200b\u63a5\u53d7\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u6216\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5982\u200b<code>map</code>\u3001<code>filter</code>\u200b\u7b49\u200b\u3002</p> <p>\u200b\u9664\u200b\u4f7f\u7528\u200b<code>def</code>\u200b\u5173\u952e\u5b57\u200b\u5b9a\u4e49\u200b\u51fd\u6570\u200b\u4ee5\u5916\u200b\uff0c<code>lambda</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u53e6\u200b\u4e00\u79cd\u200b\u521b\u5efa\u200b\u7b80\u5355\u200b\u51fd\u6570\u200b\u7684\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u8fd9\u79cd\u200b\u65b9\u6cd5\u200b\u5e38\u7528\u200b\u4e8e\u200b\u5411\u200b\u9ad8\u9636\u200b\u51fd\u6570\u200b\u4f20\u9012\u200b\u53c2\u6570\u200b\u3002<code>lambda</code>\u200b\u8868\u8fbe\u5f0f\u200b\u53ea\u80fd\u200b\u52a0\u5165\u200b\u8868\u8fbe\u5f0f\u200b\u800c\u200b\u65e0\u6cd5\u200b\u6dfb\u52a0\u200b\u63a7\u5236\u200b\u903b\u8f91\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a = lambda : \"ok\"  \n&gt;&gt;&gt; a()\n'ok'\n&gt;&gt;&gt;\n</code></pre> <p><code>lambda</code>\u200b\u8868\u8fbe\u5f0f\u200b\u53ef\u4ee5\u200b\u521b\u5efa\u200b\u4e0d\u540c\u200b\u53c2\u6570\u200b\u6570\u91cf\u200b\u7684\u200b\u51fd\u6570\u200b\uff1a</p> <ul> <li>\u200b\u6ca1\u6709\u200b\u53c2\u6570\u200b\uff0c\u200b\u5982\u200b<code>lambda : \"ok\"</code></li> <li>\u200b\u7ed9\u5b9a\u200b\u6570\u91cf\u200b\u53c2\u6570\u200b\uff0c\u200b\u5982\u200b<code>lambda x, y : x + y</code></li> <li>\u200b\u4efb\u610f\u200b\u53c2\u6570\u200b\uff0c\u200b\u5982\u200b<code>lambda *_ : sum(_)</code>\uff08\u200b\u65e0\u540d\u200b\u53c2\u6570\u200b\uff09\u200b\u6216\u200b<code>lambda **_ : sum(_)</code>\uff08\u200b\u5177\u540d\u200b\u53c2\u6570\u200b\uff09</li> </ul> <p>\u200b\u82e5\u200b<code>lambda</code>\u200b\u8868\u8fbe\u5f0f\u200b\u7684\u200b\u53f3\u4fa7\u200b\u662f\u200b\u9017\u53f7\u200b\u5206\u9694\u200b\u7684\u200b\u8868\u8fbe\u5f0f\u200b\uff0c\u200b\u751f\u6210\u200b\u7684\u200b\u662f\u200b<code>lambda</code>\u200b\u5bf9\u8c61\u200b\u7ec4\u6210\u200b\u7684\u200b\u5143\u7ec4\u200b\u3002</p> <p>\u200b\u4f7f\u7528\u200b<code>dir()</code>\u200b\u67e5\u770b\u200b\u51fd\u6570\u200b\u7684\u200b\u5185\u7f6e\u200b\u5c5e\u6027\u200b\u3002\u200b\u76f8\u6bd4\u200b\u4e8e\u200b\u4e00\u822c\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u51fd\u6570\u200b\u6709\u200b\u4ee5\u4e0b\u200b\u7684\u200b\u989d\u5916\u200b\u5c5e\u6027\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; class Obj():\n...     pass\n... \n&gt;&gt;&gt; def func():\n...     pass\n... \n&gt;&gt;&gt; sorted(list(set(dir(func)) - set(dir(Obj())))) # doctest: +SKIP\n['__annotations__', '__call__', '__closure__', '__code__', '__defaults__', '__get__', '__globals__', '__kwdefaults__', '__name__', '__qualname__']\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/functions/#_2","title":"\u5e7f\u4e49\u200b\u51fd\u6570","text":"<p>\u200b\u4efb\u4f55\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>__call__</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u5bf9\u8c61\u200b\u90fd\u200b\u662f\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u7c7b\u4f3c\u200b\u4e8e\u200b\u51fd\u6570\u8c03\u7528\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u8c03\u7528\u200b\u3002\u200b\u4ee5\u4e0b\u200b\u7c7b\u578b\u200b\u4e3a\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\uff1a</p> <ul> <li>\u200b\u51fd\u6570\u200b\u6216\u200b<code>lambda</code>\u200b\u8868\u8fbe\u5f0f\u200b</li> <li>\u200b\u5185\u7f6e\u200b\u51fd\u6570\u200b\u6216\u200b\u7c7b\u578b\u200b\u65b9\u6cd5\u200b</li> <li>\u200b\u7c7b\u578b\u200b\u65b9\u6cd5\u200b</li> <li>\u200b\u7c7b\u200b\uff08\u200b\u76f8\u5f53\u4e8e\u200b\u4e00\u4e2a\u200b\u8fd4\u56de\u200b\u7c7b\u200b\u5bf9\u8c61\u200b\u7684\u200b\u51fd\u6570\u200b\uff09</li> <li>\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>__call__</code>\u200b\u7684\u200b\u5bf9\u8c61\u200b</li> <li>\u200b\u751f\u6210\u5668\u200b\u51fd\u6570\u200b</li> </ul> <p>\u200b\u5185\u7f6e\u200b\u51fd\u6570\u200b<code>callable()</code>\u200b\u7528\u4e8e\u200b\u68c0\u6d4b\u200b\u4e00\u4e2a\u200b\u5bf9\u8c61\u200b\u662f\u5426\u662f\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u3002</p>"},{"location":"coding/python/functions/#_3","title":"\u51fd\u6570\u200b\u7684\u200b\u53c2\u6570","text":"<p>Python\u200b\u4e2d\u200b\uff0c\u200b\u51fd\u6570\u200b\u7684\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b\uff1a</p> <ul> <li>\u200b\u9017\u53f7\u200b\u5206\u9694\u200b\uff0c\u200b\u6309\u200b\u51fd\u6570\u200b\u58f0\u660e\u200b\u987a\u5e8f\u200b\u4f20\u5165\u200b\u53c2\u6570\u200b</li> <li>\u200b\u5173\u952e\u5b57\u200b\u53c2\u6570\u200b\uff0c\u200b\u5373\u200b\u53c2\u6570\u200b\u540d\u200b=\u200b\u53c2\u6570\u503c\u200b</li> <li>\u200b\u6709\u200b\u9ed8\u8ba4\u503c\u200b\u7684\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b</li> <li><code>*</code>\u200b\u89e3\u5305\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff08\u200b\u6309\u200b\u53c2\u6570\u200b\u4f4d\u7f6e\u200b\u4f20\u9012\u200b\uff09\u200b\u6216\u200b<code>**</code>\u200b\u89e3\u5305\u200b\u6620\u5c04\u200b\u5bf9\u8c61\u200b\uff08\u200b\u6309\u200b\u5173\u952e\u5b57\u200b\u4f20\u9012\u200b\uff09</li> </ul> <p>\u200b\u5bf9\u5e94\u200b\u5730\u200b\uff0c\u200b\u51fd\u6570\u200b\u5bf9\u200b\u53c2\u6570\u200b\u6709\u200b\u5982\u4e0b\u200b\u5904\u7406\u200b\u65b9\u5f0f\u200b\uff1a</p> <ul> <li>\u200b\u6309\u7167\u200b\u53c2\u6570\u200b\u4f4d\u7f6e\u200b\u4e0e\u200b\u53c2\u6570\u200b\u540d\u200b\u83b7\u53d6\u200b\u53c2\u6570\u200b</li> <li>\u200b\u5176\u4ed6\u200b\u7684\u200b\u65e0\u8bb0\u540d\u200b\u53c2\u6570\u200b\u88ab\u200b<code>*</code>\u200b\u53c2\u6570\u200b\u83b7\u53d6\u200b\uff0c\u200b\u5355\u4e2a\u200b<code>*</code>\u200b\u6216\u200b<code>*</code>\u200b\u53c2\u6570\u200b\u540e\u200b\u7684\u200b\u666e\u901a\u200b\u53c2\u6570\u200b\u53ea\u80fd\u200b\u6309\u7167\u200b\u5173\u952e\u5b57\u200b\u53c2\u6570\u200b\u7684\u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b</li> <li>\u200b\u5176\u4ed6\u200b\u7684\u200b\u8bb0\u540d\u200b\u53c2\u6570\u200b\u88ab\u200b<code>**</code>\u200b\u53c2\u6570\u200b\u83b7\u53d6\u200b</li> </ul> <p>\u200b\u51fd\u6570\u200b\u58f0\u660e\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u7684\u200b\u5355\u4e2a\u200b<code>*</code>\u200b\u4e0d\u4f1a\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u53c2\u6570\u200b\uff0c\u200b\u53ea\u200b\u7528\u4e8e\u200b\u9650\u5b9a\u200b\u6b64\u540e\u200b\u6240\u6709\u200b\u53c2\u6570\u200b\u7684\u200b\u4f20\u5165\u200b\u65b9\u5f0f\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; def a(*, a, b):\n...     return a + b\n... \n&gt;&gt;&gt; a(1, 2) \nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: a() takes 0 positional arguments but 2 were given\n&gt;&gt;&gt; a(a=1, b=2)\n3\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u51fd\u6570\u200b\u4f20\u53c2\u6709\u200b\u4e24\u79cd\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u5373\u503c\u200b\u4f20\u9012\u200b\u4e0e\u200b\u5f15\u7528\u200b\u4f20\u9012\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u53ef\u53d8\u200b\u5bf9\u8c61\u200b\u6309\u200b\u5f15\u7528\u200b\u4f20\u9012\u200b\uff0c\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5bf9\u8c61\u200b\u6309\u503c\u200b\u4f20\u9012\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; def argPass(x):\n...     x += x      \n...           \n&gt;&gt;&gt; a = \"test\"      \n&gt;&gt;&gt; b = [\"test\"]\n&gt;&gt;&gt; argPass(a)\n&gt;&gt;&gt; argPass(b)\n&gt;&gt;&gt; a\n'test'\n&gt;&gt;&gt; b\n['test', 'test']\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/functions/#_4","title":"\u51fd\u6570\u200b\u5185\u7701","text":"<p>\u200b\u51fd\u6570\u200b\u5bf9\u8c61\u200b\u7279\u6709\u200b\u7684\u200b\u5c5e\u6027\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u6709\u5173\u200b\u51fd\u6570\u200b\u7684\u200b\u4fe1\u606f\u200b\uff1a</p> \u200b\u5c5e\u6027\u200b\u540d\u200b \u200b\u5c5e\u6027\u200b\u7c7b\u578b\u200b \u200b\u5c5e\u6027\u200b\u8bf4\u660e\u200b <code>__annotations__</code> <code>dict</code> \u200b\u51fd\u6570\u53c2\u6570\u200b\u548c\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u7c7b\u578b\u200b\u6ce8\u89e3\u200b\uff0c\u200b\u952e\u4e3a\u200b\u53c2\u6570\u200b\u540d\u200b\uff0c\u200b\u503c\u4e3a\u200b\u53c2\u6570\u200b\u7c7b\u578b\u200b <code>__call__</code> <code>method-wrapper</code> \u200b\u5b9e\u73b0\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u7684\u200b\u8c03\u7528\u200b\u65b9\u6cd5\u200b <code>__closure__</code> <code>tuple</code> \u200b\u51fd\u6570\u200b\u95ed\u5305\u200b\uff0c\u200b\u53c2\u89c1\u200b\u76f8\u5173\u200b\u7ae0\u8282\u200b <code>__code__</code> <code>code</code> \u200b\u7f16\u8bd1\u6210\u200b\u4e8c\u8fdb\u5236\u200b\u683c\u5f0f\u200b\u7684\u200b\u51fd\u6570\u200b\u5143\u200b\u6570\u636e\u200b\u548c\u200b\u5b9a\u4e49\u200b\u4f53\u200b <code>__defaults__</code> <code>tuple</code> \u200b\u5f62\u53c2\u200b\u7684\u200b\u9ed8\u8ba4\u503c\u200b\u5217\u8868\u200b <code>__get__</code> <code>method-wrapper</code> \u200b\u5b9e\u73b0\u200b\u53ea\u8bfb\u200b\u63cf\u8ff0\u7b26\u200b\u534f\u8bae\u200b <code>__globals__</code> <code>dict</code> \u200b\u51fd\u6570\u200b\u6240\u5728\u200b\u6a21\u5757\u200b\u4e2d\u200b\u7684\u200b\u5168\u5c40\u53d8\u91cf\u200b <code>__kwdefaults__</code> <code>dict</code> \u200b\u9650\u5b9a\u200b\u5173\u952e\u5b57\u200b\u5f62\u5f0f\u200b\u8bbf\u95ee\u200b\u7684\u200b\u5f62\u53c2\u200b\u9ed8\u8ba4\u503c\u200b <code>__name__</code> <code>str</code> \u200b\u51fd\u6570\u200b\u540d\u79f0\u200b <code>__qualname__</code> <code>str</code> \u200b\u51fd\u6570\u200b\u7684\u200b\u9650\u5b9a\u200b\u540d\u79f0\u200b <pre><code>&gt;&gt;&gt; def dummy(a, b: int, c = 10, *args, d, e=50, **kwargs):\n...     return (a, b, c, args, d, e, kwargs) \n... \n&gt;&gt;&gt; dummy.__annotations__\n{'b': &lt;class 'int'&gt;}\n&gt;&gt;&gt; dummy.__defaults__\n(10,)\n&gt;&gt;&gt; dummy.__kwdefaults__\n{'e': 50}\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u51fd\u6570\u200b\u7684\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\u4e0e\u200b\u6570\u91cf\u200b\u5b58\u50a8\u200b\u5728\u200b<code>__code__</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; dummy.__code__.co_varnames\n('a', 'b', 'c', 'd', 'e', 'args', 'kwargs')\n&gt;&gt;&gt; dummy.__code__.co_argcount\n3\n&gt;&gt;&gt;\n</code></pre> <p><code>inspect</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u68c0\u67e5\u200b\u51fd\u6570\u53c2\u6570\u200b\u4fe1\u606f\u200b\u7684\u200b\u5de5\u5177\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import inspect\n&gt;&gt;&gt; inspect.signature(dummy)\n&lt;Signature (a, b: int, c=10, *args, d, e=50, **kwargs)&gt;\n&gt;&gt;&gt; sig = inspect.signature(dummy) \n&gt;&gt;&gt; sig.parameters\nmappingproxy(OrderedDict([('a', &lt;Parameter \"a\"&gt;), ('b', &lt;Parameter \"b: int\"&gt;), ('c', &lt;Parameter \"c=10\"&gt;), ('args', &lt;Parameter \"*args\"&gt;), ('d', &lt;Parameter \"d\"&gt;), ('e', &lt;Parameter \"e=50\"&gt;), ('kwargs', &lt;Parameter \"**kwargs\"&gt;)]))\n&gt;&gt;&gt;\n</code></pre> <p><code>parameters</code>\u200b\u5b58\u50a8\u200b\u4e86\u200b\u53c2\u6570\u200b\u540d\u200b\u4e0e\u200b\u53c2\u6570\u4fe1\u606f\u200b\u7684\u200b\u6620\u5c04\u200b\u5173\u7cfb\u200b\u3002\u200b\u6bcf\u4e2a\u200b<code>Parameter</code>\u200b\u5bf9\u8c61\u200b\u6709\u200b<code>name</code>\u3001<code>default</code>\u200b\u548c\u200b<code>kind</code>\u200b\u5c5e\u6027\u200b\u3002<code>dummy</code>\u200b\u51fd\u6570\u200b\u5404\u4e2a\u200b\u53c2\u6570\u200b\u7684\u200b\u5c5e\u6027\u200b\u4fe1\u606f\u200b\u5982\u4e0b\u200b\uff0c<code>inspect._empty</code>\u200b\u8868\u793a\u200b\u53c2\u6570\u200b\u6ca1\u6709\u200b\u9ed8\u8ba4\u503c\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; par = sig.parameters \n&gt;&gt;&gt; for parName in par:\n...     print(par[parName].name, par[parName].default, par[parName].kind)\n... \na &lt;class 'inspect._empty'&gt; POSITIONAL_OR_KEYWORD\nb &lt;class 'inspect._empty'&gt; POSITIONAL_OR_KEYWORD\nc 10 POSITIONAL_OR_KEYWORD\nargs &lt;class 'inspect._empty'&gt; VAR_POSITIONAL\nd &lt;class 'inspect._empty'&gt; KEYWORD_ONLY\ne 50 KEYWORD_ONLY\nkwargs &lt;class 'inspect._empty'&gt; VAR_KEYWORD\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u4ece\u200b\u8f93\u51fa\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u770b\u51fa\u200b\uff0c<code>dummy</code>\u200b\u5171\u6709\u200b\u56db\u79cd\u200b\u7c7b\u578b\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5373\u200b\uff1a</p> <ul> <li><code>POSITIONAL_OR_KEYWORD</code>\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u5b9a\u4f4d\u200b\u53c2\u6570\u200b\u6216\u200b\u5173\u952e\u5b57\u200b\u53c2\u6570\u200b\u4f20\u5165\u200b</li> <li><code>VAR_POSITIONAL</code>\uff0c\u200b\u591a\u4f59\u200b\u7684\u200b\u5b9a\u4f4d\u200b\u53c2\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u5143\u7ec4\u200b</li> <li><code>KEYWORD_ONLY</code>\uff0c\u200b\u53ea\u80fd\u200b\u901a\u8fc7\u200b\u5173\u952e\u5b57\u200b\u53c2\u6570\u200b\u4f20\u5165\u200b</li> <li><code>VAR_KEYWORD</code>\uff0c\u200b\u591a\u4f59\u200b\u7684\u200b\u5173\u952e\u5b57\u200b\u53c2\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u5b57\u5178\u200b</li> </ul> <p>\u200b\u53e6\u200b\u4e00\u79cd\u200b\u53c2\u6570\u200b\u7c7b\u578b\u200b\u662f\u200b\u53ea\u80fd\u200b\u901a\u8fc7\u200b\u5b9a\u4f4d\u200b\u53c2\u6570\u200b\u4f20\u5165\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5373\u200b<code>POSITIONAL_ONLY</code>\uff0cPython\u200b\u8bed\u6cd5\u200b\u4e0d\u200b\u652f\u6301\u200b\u58f0\u660e\u200b\u8fd9\u79cd\u200b\u7c7b\u578b\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5728\u200bC\u200b\u8bed\u8a00\u200bAPI\u200b\u4e2d\u200b\u53ef\u80fd\u200b\u51fa\u73b0\u200b\u3002</p> <p>Changed in version 3.8</p> <p>\u200b\u5728\u200b\u51fd\u6570\u200b\u5b9a\u4e49\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>/</code>\u200b\u5206\u9694\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\uff0c\u200b\u51fa\u73b0\u200b\u5728\u200b<code>/</code>\u200b\u4e4b\u524d\u200b\u7684\u200b\u53c2\u6570\u200b\u5fc5\u987b\u200b\u901a\u8fc7\u200b\u5b9a\u4f4d\u200b\u53c2\u6570\u200b\u4f20\u5165\u200b\u3002</p> <pre><code>&gt;&gt;&gt; def test(a, /, b, *args, c, **kwargs):\n...     pass\n...\n&gt;&gt;&gt; import inspect\n&gt;&gt;&gt; sig = inspect.signature(test)\n&gt;&gt;&gt; for val in sig.parameters.values():\n...     print(val.name, val.kind)\n...\na POSITIONAL_ONLY\nb POSITIONAL_OR_KEYWORD\nargs VAR_POSITIONAL\nc KEYWORD_ONLY\nkwargs VAR_KEYWORD\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/functions/#_5","title":"\u53c2\u6570\u200b\u7684\u200b\u7c7b\u578b\u200b\u9650\u5236","text":"<p>Python\u200b\u4e2d\u200b\u5bf9\u200b\u51fd\u6570\u53c2\u6570\u200b\u7684\u200b\u7c7b\u578b\u200b\u68c0\u67e5\u200b\u4ee5\u200b\u51fd\u6570\u200b\u6ce8\u89e3\u200b\u7684\u200b\u5f62\u5f0f\u200b\u5b9e\u73b0\u200b\u3002\u200b\u51fd\u6570\u200b\u6ce8\u89e3\u200b\u89c4\u5b9a\u200b\u4e86\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u63a5\u53d7\u200b\u7684\u200b\u53c2\u6570\u200b\u7c7b\u578b\u200b\u3002</p> <p>\u200b\u51fd\u6570\u200b\u6ce8\u89e3\u200b\u53ef\u4ee5\u200b\u4e3a\u200b\u7c7b\u578b\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u3002<code>:</code>\u200b\u7528\u4e8e\u200b\u5bf9\u200b\u53c2\u6570\u200b\u7684\u200b\u6ce8\u89e3\u200b\uff0c<code>-&gt;</code>\u200b\u8868\u793a\u200b\u5bf9\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u6ce8\u89e3\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; def another_dummy(x: int) -&gt; int:       \n...     return -x\n... \n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6ce8\u610f\u200b\uff0c\u200b\u6ce8\u89e3\u200b\u5e76\u200b\u4e0d\u4f1a\u200b\u5bfc\u81f4\u200bPython\u200b\u5bf9\u200b\u4f20\u5165\u200b\u53c2\u6570\u200b\u7684\u200b\u7c7b\u578b\u200b\u8fdb\u884c\u200b\u68c0\u67e5\u200b\u3002</p> <pre><code>&gt;&gt;&gt; another_dummy(0.5)         \n-0.5\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6240\u6709\u200b\u7684\u200b\u6ce8\u89e3\u200b\u5b58\u50a8\u200b\u5728\u200b\u51fd\u6570\u200b\u7684\u200b<code>__annotations__</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\u3002</p> <pre><code>&gt;&gt;&gt; another_dummy.__annotations__\n{'x': &lt;class 'int'&gt;, 'return': &lt;class 'int'&gt;}\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/functions/#_6","title":"\u51fd\u6570\u200b\u5f0f\u200b\u7f16\u7a0b","text":""},{"location":"coding/python/functions/#operator","title":"operator\u200b\u6a21\u5757","text":"<p><code>operator</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\u4e86\u200bPython\u200b\u4e2d\u200b\u90e8\u5206\u200b\u8fd0\u7b97\u7b26\u200b\u6216\u200b\u64cd\u4f5c\u200b\u7684\u200b\u51fd\u6570\u200b\u5f0f\u200b\u5b9e\u73b0\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import operator\n&gt;&gt;&gt; operator.add(5, 3)\n8\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6b64\u5916\u200b\uff0c<code>__getitem__</code>\u3001<code>__getattribute__</code>\u200b\u65b9\u6cd5\u200b\u5728\u200b<code>operator</code>\u200b\u6a21\u5757\u200b\u4e2d\u200b\u4e5f\u200b\u6709\u200b\u5bf9\u5e94\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u4e3a\u200b<code>itemgetter</code>\u3001<code>attrgetter</code>\u3002<code>methodcaller</code>\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u5185\u200b\u7684\u200b\u67d0\u4e2a\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u4e0e\u200b<code>operator</code>\u200b\u6a21\u5757\u200b\u4e2d\u200b\u7684\u200b\u5176\u4ed6\u200b\u51fd\u6570\u200b\u4e0d\u540c\u200b\uff0c\u200b\u8c03\u7528\u200b\u8be5\u200b\u51fd\u6570\u200b\u4f1a\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u3002</p> <pre><code>&gt;&gt;&gt; sort = operator.methodcaller('sort')\n&gt;&gt;&gt; sort_rev = operator.methodcaller('sort', reverse=True)\n&gt;&gt;&gt; a = [1, 3, 2]\n&gt;&gt;&gt; sort(a)\n&gt;&gt;&gt; a\n[1, 2, 3]\n&gt;&gt;&gt; sort_rev(a)\n&gt;&gt;&gt; a\n[3, 2, 1]\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/functions/#functools","title":"functools\u200b\u6a21\u5757","text":"<p>functools\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u4e0e\u200b\u51fd\u6570\u200b\u5f0f\u200b\u7f16\u7a0b\u200b\u76f8\u5173\u200b\u7684\u200b\u4e00\u7cfb\u5217\u200b\u51fd\u6570\u200b\u5de5\u5177\u200b\u3002</p> <p><code>partial</code>\u200b\u7528\u4e8e\u200b\u9501\u5b9a\u200b\u51fd\u6570\u200b\u7684\u200b\u4e00\u90e8\u5206\u200b\u53c2\u6570\u200b\u3002\u200b\u5176\u4f59\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u5728\u200b\u8c03\u7528\u200b\u65f6\u200b\u6307\u5b9a\u200b\uff08\u200b\u5b9a\u4f4d\u200b\u53c2\u6570\u200b\u53ea\u80fd\u200b\u6309\u200b\u987a\u5e8f\u200b\u6307\u5b9a\u200b\uff0c\u200b\u5173\u952e\u5b57\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u4efb\u610f\u200b\u6307\u5b9a\u200b\uff09\uff1a</p> <pre><code>&gt;&gt;&gt; from functools import partial\n&gt;&gt;&gt; part = partial(divmod, 3) \n&gt;&gt;&gt; part(2)\n(1, 1)\n&gt;&gt;&gt;\n</code></pre> <p><code>reduce</code>\u200b\u5c06\u200b\u63a5\u53d7\u200b\u4e24\u4e2a\u200b\u53c2\u6570\u200b\u7684\u200b\u51fd\u6570\u200b\u5e94\u7528\u200b\u5230\u200b\u4e00\u4e2a\u200b\u5e8f\u5217\u200b\u4e0a\u200b\uff0c\u200b\u5f97\u5230\u200b\u4e00\u4e2a\u200b\u7ed3\u679c\u200b\u3002\u200b\u501f\u52a9\u200b<code>operator</code>\u200b\u6a21\u5757\u200b\u4e2d\u200b\u63d0\u4f9b\u200b\u7684\u200b\u8fd0\u7b97\u7b26\u200b\uff0c<code>reduce</code>\u200b\u6a21\u5757\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u9636\u4e58\u200b\u7b49\u200b\u9700\u8981\u200b\u5faa\u73af\u200b\u624d\u80fd\u200b\u5b9e\u73b0\u200b\u7684\u200b\u8fd0\u7b97\u200b\u64cd\u4f5c\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; from functools import reduce\n&gt;&gt;&gt; def fact(x: int):\n...     return reduce(operator.mul, range(1, x + 1))\n... \n&gt;&gt;&gt; fact(5)\n120\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/killable-thread/","title":"Python\u200b\u5173\u95ed\u200b\u7ebf\u7a0b","text":"<p>\u200b\u4f7f\u7528\u200b<code>PyThreadState_SetAsyncExc</code>\u200b\u51fd\u6570\u200b\u5728\u5b50\u200b\u7ebf\u7a0b\u200b\u4e2d\u200b\u5f15\u53d1\u200b\u5f02\u5e38\u200b\uff0c\u200b\u4ece\u800c\u200b\u4f7f\u200b\u7ebf\u7a0b\u200b\u63d0\u524d\u7ed3\u675f\u200b\u3002</p> <p>\u200b\u793a\u4f8b\u200b\u4ee3\u7801\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>import threading \nimport ctypes \nimport time \n\nclass thread_with_exception(threading.Thread): \n    def __init__(self, name): \n        threading.Thread.__init__(self) \n        self.name = name \n\n    def run(self): \n\n        # target function of the thread class \n        try: \n            while True: \n                print('running ' + self.name) \n        finally: \n            print('ended') \n\n    def get_id(self): \n\n        # returns id of the respective thread \n        if hasattr(self, '_thread_id'): \n            return self._thread_id \n        for id, thread in threading._active.items(): \n            if thread is self: \n                return id\n\n    def raise_exception(self): \n        thread_id = self.get_id() \n        res = ctypes.pythonapi.PyThreadState_SetAsyncExc(thread_id, \n              ctypes.py_object(SystemExit)) \n        if res &gt; 1: \n            ctypes.pythonapi.PyThreadState_SetAsyncExc(thread_id, 0) \n            print('Exception raise failure') \n\nt1 = thread_with_exception('Thread 1') \nt1.start() \ntime.sleep(2) \nt1.raise_exception() \nt1.join() \n</code></pre> <p>\u200b\u628a\u200b<code>run()</code>\u200b\u4e2d\u200b\u6267\u884c\u200b\u7684\u200b\u64cd\u4f5c\u200b\u4ee5\u200b\u53c2\u6570\u200b\u7684\u200b\u5f62\u5f0f\u200b\u4f20\u9012\u200b\uff1a</p> <pre><code>class KillableThread(threading.Thread):\n    \"\"\"\n    A thread class extending threading.Thread, provides a kill() method to stop the thread and a getResult() method to get the return value of the thread.\n    \"\"\"\n\n    def __init__(self, func: Callable, *args, **kwargs):\n        super().__init__()\n        self.func: Callable = func\n        self.funcArg: dict = kwargs\n        self.funcTup: Tuple = args\n        self.result: Any = None\n        self.exception: Any = None\n\n    def run(self):\n        \"\"\"\n        Executes the function here\n        \"\"\"\n        try:\n            self.result = self.func(*self.funcTup, **self.funcArg)\n        except BaseException as e:\n            self.exception = e\n\n    def get_id(self):\n        \"\"\"\n        Get the id of the thread\n        \"\"\"\n        if hasattr(self, '_thread_id'):\n            return self._thread_id\n        for id, thread in threading._active.items():\n            if thread is self:\n                return id\n\n    def kill(self):\n        \"\"\"\n        Stops the thread\n        \"\"\"\n        thread_id = self.get_id()\n        res = ctypes.pythonapi.PyThreadState_SetAsyncExc(thread_id,\n                                                         ctypes.py_object(SystemExit))\n        if res &gt; 1:\n            ctypes.pythonapi.PyThreadState_SetAsyncExc(thread_id, 0)\n            print('Exception raise failure')\n\n    def getResult(self):\n        if self.exception is None:\n            return self.result\n        else:\n            raise self.exception\n</code></pre> <p>\u200b\u6765\u6e90\u200b\uff1ahttps://www.geeksforgeeks.org/python-different-ways-to-kill-a-thread</p>"},{"location":"coding/python/magic-functions/","title":"\u9b54\u672f\u200b\u65b9\u6cd5","text":"<p>\u200b\u672c\u6587\u200b\u4e3b\u8981\u200b\u6db5\u76d6\u200bPython\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\u3002\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\u662f\u200b\u7c7b\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u4e00\u79cd\u200b\u7279\u6b8a\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u65b9\u6cd5\u200b\u62e5\u6709\u200b\u56fa\u5b9a\u200b\u7684\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\u548c\u200b\u7279\u6b8a\u200b\u7684\u200b\u8c03\u7528\u200b\u65f6\u673a\u200b\u3002</p> <p>\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\u4ee5\u53cc\u200b\u4e0b\u5212\u7ebf\u200b<code>__</code>\u200b\u5f00\u5934\u200b\uff0c\u200b\u4ee5\u53cc\u200b\u4e0b\u5212\u7ebf\u200b\u7ed3\u5c3e\u200b\u3002</p> <p>Python\u200b\u4e2d\u200b\u7684\u200b\u8fd0\u7b97\u7b26\u200b\u91cd\u8f7d\u200b\u901a\u8fc7\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\u5206\u4e3a\u200b\u4e0e\u200b\u8fd0\u7b97\u7b26\u200b\u65e0\u5173\u200b\u7684\u200b\u7279\u6b8a\u200b\u65b9\u6cd5\u200b\u53ca\u200b\u4e0e\u200b\u8fd0\u7b97\u7b26\u200b\u6709\u5173\u200b\u7684\u200b\u7279\u6b8a\u200b\u65b9\u6cd5\u200b</p>"},{"location":"coding/python/magic-functions/#_2","title":"\u4e0e\u200b\u8fd0\u7b97\u7b26\u200b\u65e0\u5173\u200b\u7684\u200b\u9b54\u672f\u200b\u65b9\u6cd5","text":"<p>\u200b\u6d89\u53ca\u200b\u5230\u200b\u5b57\u7b26\u4e32\u200b\u4e0e\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u7684\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\uff1a</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__repr__</code> <code>self</code> \u200b\u5f53\u200b\u8c03\u7528\u200b<code>repr()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>str</code> \u200b\u8fd4\u56de\u503c\u200b\u5e94\u4e3a\u200b\u91cd\u5efa\u200b\u8be5\u200b\u5bf9\u8c61\u200b\u7684\u200bPython\u200b\u8868\u8fbe\u5f0f\u200b\uff08\u200b\u5982\u679c\u200b\u53ef\u80fd\u200b\uff09\uff0c\u200b\u6216\u200b\u5305\u542b\u200b\u5176\u4ed6\u200b\u4fe1\u606f\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b<code>&lt;...&gt;</code> <code>__str__</code> <code>self</code> \u200b\u5f53\u200b\u8c03\u7528\u200b<code>str()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>str</code> \u200b\u8c03\u7528\u200b<code>print()</code>\u200b\u7b49\u200b\u5176\u4ed6\u200b\u51fd\u6570\u200b\u65f6\u4f1a\u200b\u9690\u5f0f\u200b\u8c03\u7528\u200b<code>__str__()</code>\uff0c\u200b\u5f53\u200b\u4e0d\u200b\u5b58\u5728\u200b<code>__str__()</code>\u200b\u65f6\u4f1a\u200b\u81ea\u52a8\u200b\u8c03\u7528\u200b<code>__repr__()</code> <code>__format__</code> <code>self, format_spec</code> \u200b\u5f53\u200b\u8c03\u7528\u200b<code>format()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>str</code> <code>object.__format__(x, '')</code>\u200b\u7b49\u4ef7\u200b\u4e8e\u200b<code>str(x)</code> <code>__bytes__</code> <code>self</code> \u200b\u5f53\u200b\u8c03\u7528\u200b<code>bytes()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>bytes</code> \u200b\u8fd4\u56de\u503c\u200b\u662f\u200b\u5bf9\u8c61\u200b\u7684\u200b\u5b57\u8282\u200b\u8868\u793a\u200b <p>\u200b\u6d89\u53ca\u200b\u5230\u200b\u6570\u503c\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u7684\u200b\u9b54\u672f\u200b\u65b9\u6cd5\u200b\uff1a</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__bool__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>bool()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>bool</code> \u200b\u5f53\u200b<code>__bool__</code>\u200b\u51fd\u6570\u200b\u6ca1\u6709\u200b\u5b9a\u4e49\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u9690\u5f0f\u200b\u8c03\u7528\u200b<code>__len__</code>\uff0c\u200b\u5982\u679c\u200b<code>__len__</code>\u200b\u4e5f\u200b\u6ca1\u6709\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>True</code> <code>__complex__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>commplex()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>complex</code> <code>__int__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>int()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>int</code> <code>__float__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>float()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>float</code> <code>__hash__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>hash()</code> \u200b\u51fd\u6570\u200b\u65f6\u200b <code>hash()</code>\u200b\u51fd\u6570\u200b\u5c06\u200b<code>__hash__()</code>\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u622a\u65ad\u200b\u5230\u200b<code>Py_ssize_t</code>\u200b\u7684\u200b\u8303\u56f4\u200b\u5185\u200b\uff0c64\u200b\u4f4d\u200b\u7cfb\u7edf\u200b\u4e0b\u200b<code>Py_ssize_t</code>\u200b\u4e3a\u200b8\u200b\u5b57\u8282\u200b\uff1b<code>set</code>\u200b\u7b49\u200b\u7c7b\u578b\u200b\u7684\u200b\u64cd\u4f5c\u200b\u4e5f\u200b\u4f1a\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u7684\u200b<code>__hash__</code>\u200b\u51fd\u6570\u200b <code>__index__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>operator.index()</code>\u200b\u6216\u200bPython\u200b\u9700\u8981\u200b\u6574\u6570\u200b\u503c\u65f6\u200b <code>int</code> \u200b\u5f53\u200b<code>__int__</code>\u3001<code>__float__</code>\u3001<code>__complex__</code>\u200b\u6ca1\u6709\u200b\u5b9e\u73b0\u200b\u65f6\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u51fd\u6570\u200b\u9690\u5f0f\u200b\u5730\u200b\u8c03\u7528\u200b<code>__index__</code> <p>\u200b\u96c6\u5408\u200b\u76f8\u5173\u200b\u64cd\u4f5c\u200b</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__len__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>len()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>int</code> \u200b\u8fd4\u56de\u503c\u200b\u5fc5\u987b\u200b\u4e3a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\uff0c\u200b\u5728\u200bCPython\u200b\u4e2d\u200b\u8fd4\u56de\u503c\u200b\u4e0d\u80fd\u200b\u8d85\u8fc7\u200b<code>sys.maxsize</code>\u200b\u4e2d\u200b\u5b9a\u4e49\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u5426\u5219\u200b<code>len()</code>\u200b\u7b49\u200b\u51fd\u6570\u200b\u4f1a\u200b\u629b\u51fa\u200b<code>OverflowError</code>\u200b\u7b49\u200b\u5f02\u5e38\u200b <code>__getitemm__</code> <code>self, key</code> \u200b\u4f7f\u7528\u200b<code>[]</code>\u200b\u8fd0\u7b97\u7b26\u200b\u65f6\u200b\uff08\u200b\u53f3\u503c\u200b\uff09 <code>key</code>\u200b\u53ef\u4ee5\u200b\u4e3a\u200b\u5408\u6cd5\u200b\u7684\u200b\u6570\u7ec4\u200b\u4e0b\u6807\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u53cd\u5411\u200b\u7684\u200b\u4e0b\u6807\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u5e8f\u5217\u200b\u7684\u200b\u5207\u7247\u200b\uff0c\u200b\u8be5\u200b\u51fd\u6570\u200b\u5e94\u200b\u5728\u200b<code>key</code>\u200b\u7c7b\u578b\u200b\u4e0d\u200b\u5bf9\u5e94\u200b\u65f6\u200b\u629b\u51fa\u200b<code>TypeError</code>\uff0c<code>key</code>\u200b\u8d85\u51fa\u8303\u56f4\u200b\u65f6\u200b\u629b\u51fa\u200b<code>IndexError</code>\u200b\u6216\u200b<code>KeyError</code>\uff08\u200b\u5bf9\u4e8e\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\uff09 <code>__setitem__</code> <code>self, key, value</code> \u200b\u4f7f\u7528\u200b<code>[]</code>\u200b\u8fd0\u7b97\u7b26\u200b\u65f6\u200b\uff08\u200b\u5de6\u503c\u200b\uff09 <code>None</code> <code>key</code>\u200b\u53c2\u6570\u200b\u7684\u200b\u53d6\u503c\u200b\u53ca\u200b\u5f02\u5e38\u200b\u5904\u7406\u200b\u4e0e\u200b<code>__getitem__</code>\u200b\u51fd\u6570\u200b\u76f8\u540c\u200b <code>__delitem__</code> <code>self, key</code> \u200b\u4f7f\u7528\u200b<code>del</code>\u200b\u5220\u9664\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u65f6\u200b <code>None</code> \u200b\u5bf9\u4e8e\u200b\u4e0d\u53ef\u200b\u5220\u9664\u200b\u7684\u200b\u5e8f\u5217\u200b\u6216\u200b\u6620\u5c04\u200b\uff0c\u200b\u65e0\u9700\u200b\u5b9e\u73b0\u200b\u8be5\u200b\u51fd\u6570\u200b\uff0c\u200b\u5176\u4ed6\u200b\u540c\u200b<code>__getitem__</code> <code>__contains__</code> <code>self, item</code> \u200b\u4f7f\u7528\u200b<code>in</code>\u200b\u4f5c\u4e3a\u200b\u8fd0\u7b97\u7b26\u200b\u65f6\u200b <code>bool</code> \u200b\u5bf9\u4e8e\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\uff0c<code>__contains__</code>\u200b\u63a5\u6536\u200b\u7684\u200b<code>item</code>\u200b\u53c2\u6570\u200b\u5bf9\u5e94\u200b\u6620\u5c04\u200b\u7684\u200b<code>key</code>\u200b\u90e8\u5206\u200b\u800c\u200b\u4e0d\u662f\u200b<code>value</code>\u200b\u90e8\u5206\u200b\u3002\u200b\u82e5\u200b\u6ca1\u6709\u200b\u5b9a\u4e49\u200b\u8be5\u200b\u51fd\u6570\u200b\uff0cPython\u200b\u9996\u5148\u200b\u901a\u8fc7\u200b<code>__iter__</code>\u200b\u8fdb\u884c\u200b\u67e5\u627e\u200b\uff0c\u200b\u5176\u6b21\u200b\u4f7f\u7528\u200b<code>__getitem__</code>\u200b\u8fdb\u884c\u200b\u67e5\u627e\u200b <p>\u200b\u8fed\u4ee3\u200b\u5668\u200b\u4e0e\u200b\u679a\u4e3e\u200b\u7c7b\u578b\u200b</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__iter__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>iter()</code>\u200b\u51fd\u6570\u200b\u6216\u200b\u4f7f\u7528\u200b\u5faa\u73af\u200b\u65f6\u200b \u200b\u8fd4\u56de\u503c\u200b\u5e94\u4e3a\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff0c<code>__iter__</code>\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u81ea\u8eab\u200b <code>__reversed__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>reversed()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b \u200b\u8fd4\u56de\u503c\u200b\u5e94\u4e3a\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u4e14\u200b\u662f\u200b\u539f\u200b\u5e8f\u5217\u200b\u7684\u200b\u9006\u5e8f\u200b <code>__next__</code> <code>self</code> \u200b\u8c03\u7528\u200b<code>next()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b \u200b\u8fd4\u56de\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u5f53\u200b\u5230\u8fbe\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u65f6\u200b\u629b\u51fa\u200b<code>StopIteration</code> <p>\u200b\u51fd\u6570\u8c03\u7528\u200b</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__call__</code> <code>self, *args, **kwargs</code> \u200b\u5f53\u200b\u5bf9\u8c61\u200b\u4ee5\u200b\u51fd\u6570\u200b\u7684\u200b\u65b9\u5f0f\u200b\u88ab\u200b\u8c03\u7528\u200b\u65f6\u200b \u200b\u5b9a\u4e49\u200b\u4e86\u200b<code>__call__</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u5bf9\u8c61\u200b\u5c5e\u4e8e\u200b<code>typing.Callable</code>\u200b\u7c7b\u578b\u200b <p>\u200b\u4e0a\u4e0b\u6587\u200b\u7ba1\u7406\u200b\uff08<code>with</code>\uff09\u200b\u5173\u952e\u8bcd\u200b\uff1a</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__enter__</code> <code>self</code> \u200b\u4f7f\u7528\u200b<code>with</code>\u200b\u5173\u952e\u8bcd\u200b\u58f0\u660e\u200b\u4e0a\u4e0b\u6587\u200b\u65f6\u200b <code>Any</code> \u200b\u8fd4\u56de\u503c\u200b\u5206\u914d\u200b\u7ed9\u200b<code>as</code>\u200b\u5173\u952e\u8bcd\u200b\u6307\u660e\u200b\u7684\u200b\u5bf9\u8c61\u200b <code>__exit__</code> <code>self, exc_type, exc_value, traceback</code> \u200b\u79bb\u5f00\u200b<code>with</code>\u200b\u5757\u200b\u65f6\u200b <code>Any</code> \u200b\u82e5\u200b<code>with</code>\u200b\u5757\u200b\u6267\u884c\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u6ca1\u6709\u200b\u51fa\u73b0\u5f02\u5e38\u200b\uff0c\u200b\u5219\u200b<code>exc_type, exc_value, traceback</code>\u200b\u53c2\u6570\u5747\u200b\u4e3a\u200b<code>None</code>\uff0c\u200b\u5426\u5219\u200b<code>exit</code>\u200b\u5757\u200b\u53ef\u4ee5\u200b\u5bf9\u200b\u5f02\u5e38\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b <p>\u200b\u5bf9\u8c61\u200b\u7ba1\u7406\u200b</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__new__</code> <code>cls[, ...]</code> \u200b\u521b\u5efa\u200b\u7c7b\u200b\u5b9e\u4f8b\u200b\u65f6\u200b <code>Any</code> <code>__new__</code>\u200b\u4e3a\u200b\u9759\u6001\u65b9\u6cd5\u200b\u4e14\u200b\u5148\u4e8e\u200b<code>__init__</code>\u200b\u6267\u884c\u200b\uff0c<code>__new__</code>\u200b\u7684\u200b\u5176\u4f59\u200b\u53c2\u6570\u200b\u4f1a\u200b\u4f20\u9012\u200b\u7ed9\u200b<code>__init__</code>\u3002\u200b\u82e5\u200b<code>__new__</code>\u200b\u8fd4\u56de\u200b\u7684\u200b\u7c7b\u578b\u200b\u4e0e\u200b\u521b\u5efa\u200b\u7684\u200b\u7c7b\u578b\u200b\u4e0d\u200b\u5bf9\u5e94\u200b\uff0c\u200b\u5219\u200b\u4e0d\u4f1a\u200b\u6267\u884c\u200b<code>__init__</code> <code>__init__</code> <code>self, ...</code> \u200b\u521d\u59cb\u5316\u200b\u7c7b\u200b\u5b9e\u4f8b\u200b\u65f6\u200b <code>None</code> \u200b\u82e5\u200b\u57fa\u7c7b\u200b\u5b9a\u4e49\u200b\u4e86\u200b<code>__init__</code>\uff0c\u200b\u5219\u200b\u5728\u200b\u5b50\u7c7b\u200b\u7684\u200b<code>__init__</code>\u200b\u4e2d\u200b\u5fc5\u987b\u200b\u6709\u200b<code>super().__init()</code>\u200b\u7684\u200b\u663e\u5f0f\u200b\u8c03\u7528\u200b <code>__del__</code> <code>self</code> \u200b\u5bf9\u8c61\u200b\u9500\u6bc1\u200b\u65f6\u200b <code>None</code> \u200b\u5f53\u200b\u5bf9\u8c61\u200b\u7684\u200b\u5f15\u7528\u200b\u8ba1\u6570\u200b\u51cf\u200b\u4e3a\u200b0\u200b\u65f6\u200b\uff0c\u200b\u5bf9\u8c61\u200b\u624d\u200b\u53ef\u80fd\u200b\u88ab\u200b\u5220\u9664\u200b <p>\u200b\u5c5e\u6027\u200b\u7ba1\u7406\u200b</p> \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u53c2\u6570\u200b\u5217\u8868\u200b \u200b\u8c03\u7528\u200b\u65f6\u673a\u200b \u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b \u200b\u5907\u6ce8\u200b <code>__getattribute__</code> <code>self, name</code> \u200b\u4f7f\u7528\u200b<code>self.name</code>\u200b\u8bbf\u95ee\u200b\u5bf9\u8c61\u200b\u7684\u200b\u5c5e\u6027\u200b\u65f6\u200b <code>Any</code> \u200b\u4e3a\u200b\u907f\u514d\u200b\u65e0\u9650\u200b\u9012\u5f52\u200b\uff0c\u200b\u5b50\u7c7b\u200b\u7684\u200b<code>__getattribute__</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u5b9e\u73b0\u200b\u4e2d\u200b\u5fc5\u987b\u200b\u5305\u542b\u200b\u7236\u7c7b\u200b\u7684\u200b<code>__getattribute__</code>\u200b\u65b9\u6cd5\u200b\u8c03\u7528\u200b <code>__getattr__</code> <code>self, name</code> \u200b\u4f7f\u7528\u200b<code>self.name</code>\u200b\u8bbf\u95ee\u200b\u5bf9\u8c61\u200b\u7684\u200b\u4e0d\u200b\u5b58\u5728\u200b\u5c5e\u6027\u200b\u6216\u200b<code>__getattribute__</code>\u200b\u51fd\u6570\u200b\u629b\u51fa\u200b<code>AttributeError</code>\u200b\u5f02\u5e38\u200b <code>Any</code> \u200b\u5f53\u200b<code>__getattribute__</code>\u200b\u6b63\u5e38\u200b\u6267\u884c\u200b\u65f6\u200b<code>__getattr__</code>\u200b\u4e0d\u4f1a\u200b\u6267\u884c\u200b <code>__setattr__</code> <code>self, name, value</code> \u200b\u4f7f\u7528\u200b<code>self.name</code>\u200b\u8bbe\u7f6e\u200b\u5c5e\u6027\u200b\u7684\u200b\u503c\u65f6\u200b <code>None</code> \u200b\u5f53\u200b\u8bd5\u56fe\u200b\u901a\u8fc7\u200b<code>__setattr__</code>\u200b\u5411\u200b\u5bf9\u8c61\u200b\u8bbe\u7f6e\u200b\u5c5e\u6027\u200b\u503c\u65f6\u200b\u5fc5\u987b\u200b\u901a\u8fc7\u200b\u7c7b\u578b\u200b\u7684\u200b<code>__setattr__</code>\u200b\u65b9\u6cd5\u200b\u8fdb\u884c\u200b\u8bbe\u7f6e\u200b <code>__delattr__</code> <code>self, name</code> \u200b\u6267\u884c\u200b<code>del self.name</code>\u200b\u65f6\u200b <code>None</code> \u200b\u4ec5\u5f53\u200b<code>del self.name</code>\u200b\u8bed\u53e5\u200b\u6709\u200b\u610f\u4e49\u200b\u65f6\u200b\uff0c\u200b\u624d\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b\u8be5\u200b\u65b9\u6cd5\u200b <code>__dir__</code> <code>self</code> \u200b\u6267\u884c\u200b<code>dir()</code>\u200b\u51fd\u6570\u200b\u65f6\u200b <code>Sequence</code> <code>dir()</code>\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b\u5904\u7406"},{"location":"coding/python/magic-functions/#_3","title":"\u4e0e\u200b\u8fd0\u7b97\u7b26\u200b\u6709\u5173\u200b\u7684\u200b\u9b54\u672f\u200b\u65b9\u6cd5","text":"<p>\u200b\u4e00\u5143\u200b\u8fd0\u7b97\u7b26\u200b</p> \u200b\u63cf\u8ff0\u200b \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u8fd0\u7b97\u7b26\u200b \u200b\u53d6\u8d1f\u200b <code>__neg__</code> <code>-</code> \u200b\u53d6\u6b63\u200b <code>__pos__</code> <code>+</code> \u200b\u7edd\u5bf9\u503c\u200b <code>__abs__</code> <code>abs()</code> <p>\u200b\u6bd4\u8f83\u200b\u8fd0\u7b97\u7b26\u200b</p> \u200b\u63cf\u8ff0\u200b \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u8fd0\u7b97\u7b26\u200b \u200b\u5c0f\u4e8e\u200b <code>__lt__</code> <code>&lt;</code> \u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b <code>__le__</code> <code>&lt;=</code> \u200b\u7b49\u4e8e\u200b <code>__eq__</code> <code>==</code> \u200b\u4e0d\u200b\u7b49\u4e8e\u200b <code>__ne__</code> <code>!=</code> \u200b\u5927\u4e8e\u200b <code>__gt__</code> <code>&gt;</code> \u200b\u5927\u4e8e\u200b\u7b49\u4e8e\u200b <code>__ge__</code> <code>&gt;=</code> <p>\u200b\u7b97\u672f\u200b\u8fd0\u7b97\u7b26\u200b</p> \u200b\u63cf\u8ff0\u200b \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u8fd0\u7b97\u7b26\u200b \u200b\u76f8\u52a0\u200b <code>__add__</code> <code>+</code> \u200b\u76f8\u51cf\u200b <code>__sub__</code> <code>-</code> \u200b\u76f8\u4e58\u200b <code>__mul__</code> <code>*</code> \u200b\u5b9e\u6570\u200b\u9664\u200b <code>__truediv__</code> <code>/</code> \u200b\u6574\u6570\u200b\u9664\u200b <code>__floordiv__</code> <code>//</code> \u200b\u53d6\u4f59\u200b <code>__mod__</code> <code>%</code> \u200b\u6574\u6570\u200b\u9664\u200b+\u200b\u53d6\u4f59\u200b <code>__divmod__</code> <code>divmod()</code> \u200b\u4e58\u65b9\u200b <code>__pow__</code> <code>**</code> \u200b\u53d6\u6574\u200b <code>__round__</code> <code>round()</code> <p>\u200b\u53cd\u5411\u200b\u7b97\u672f\u200b\u8fd0\u7b97\u7b26\u200b\u5728\u200b\u65b9\u6cd5\u200b\u540d\u200b\u524d\u9762\u200b\u52a0\u200b<code>r</code>\uff0c\u200b\u5982\u200b<code>__radd__</code>\u3002\u200b\u589e\u91cf\u200b\u8d4b\u503c\u200b\u7b97\u672f\u200b\u8fd0\u7b97\u7b26\u200b\u5728\u200b\u65b9\u6cd5\u200b\u540d\u200b\u524d\u9762\u200b\u52a0\u200b<code>i</code>\uff0c\u200b\u5982\u200b<code>__iadd__</code>\u3002</p> <p>\u200b\u4f4d\u200b\u8fd0\u7b97\u7b26\u200b</p> \u200b\u63cf\u8ff0\u200b \u200b\u65b9\u6cd5\u200b\u540d\u200b \u200b\u8fd0\u7b97\u7b26\u200b \u200b\u6309\u4f4d\u200b\u53d6\u53cd\u200b <code>__invert__</code> <code>~</code> \u200b\u5de6\u79fb\u200b <code>__lshift__</code> <code>&lt;&lt;</code> \u200b\u53f3\u79fb\u200b <code>__rshift__</code> <code>&gt;&gt;</code> \u200b\u6309\u4f4d\u200b\u4e0e\u200b <code>__and__</code> <code>&amp;</code> \u200b\u6309\u4f4d\u200b\u6216\u200b <code>__or__</code> <code>|</code> \u200b\u6309\u4f4d\u200b\u5f02\u6216\u200b <code>__xor__</code> <code>^</code> <p>\u200b\u53cd\u5411\u200b\u7b97\u672f\u200b\u8fd0\u7b97\u7b26\u200b\u5728\u200b\u65b9\u6cd5\u200b\u540d\u200b\u524d\u9762\u200b\u52a0\u200b<code>r</code>\uff0c\u200b\u589e\u91cf\u200b\u8d4b\u503c\u200b\u7b97\u672f\u200b\u8fd0\u7b97\u7b26\u200b\u5728\u200b\u65b9\u6cd5\u200b\u540d\u200b\u524d\u9762\u200b\u52a0\u200b<code>i</code>\u3002</p>"},{"location":"coding/python/mapping/","title":"Python\u200b\u4e2d\u200b\u7684\u200b\u6620\u5c04\u200b\u7c7b\u578b","text":"<p>Python\u200b\u4e2d\u200b\u6700\u200b\u5e38\u89c1\u200b\u7684\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\u662f\u200b\u5b57\u5178\u200b\u3002\u200b\u672c\u6587\u200b\u5c06\u200b\u4ece\u200b\u5b57\u5178\u200b\u5165\u624b\u200b\u5bf9\u200bPython\u200b\u4e2d\u200b\u7684\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\u8fdb\u884c\u200b\u63a2\u8ba8\u200b\u3002</p>"},{"location":"coding/python/mapping/#_1","title":"\u5b57\u5178","text":"<p>\u200b\u5b57\u5178\u200b\u7c7b\u578b\u200b<code>dict</code>\u200b\u5bf9\u5e94\u200b\u7684\u200b\u62bd\u8c61\u200b\u57fa\u7c7b\u200b\u4e3a\u200b<code>MutableMapping</code>\u3002\u200b\u800c\u200b<code>MutableMapping</code>\u200b\u662f\u200b<code>Mapping</code>\u200b\u7684\u200b\u5b50\u7c7b\u200b\u3002</p> <p><code>dict</code>\u200b\u4e2d\u200b\u7684\u200b\u6570\u636e\u7ec4\u7ec7\u200b\u4e3a\u952e\u200b-\u200b\u503c\u200b\u5bf9\u200b\u7684\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u5b57\u5178\u200b\u5bf9\u503c\u200b\u7684\u200b\u7c7b\u578b\u200b\u6ca1\u6709\u200b\u8981\u6c42\u200b\uff0c\u200b\u4f46\u200b\u8981\u6c42\u200b\u952e\u200b\u7684\u200b\u7c7b\u578b\u200b\u5fc5\u987b\u200b\u53ef\u6563\u5217\u200b\uff08hashable\uff09\u3002\u200b\u7531\u6b64\u53ef\u89c1\u200b\uff0c\u200b\u5b57\u5178\u200b\u7ed3\u6784\u200b\u7684\u200b\u5e95\u5c42\u200b\u5b9e\u73b0\u200b\u7c7b\u4f3c\u200b\u6563\u200b\u5217\u8868\u200b\uff0c\u200b\u4ee5\u200b\u52a0\u5feb\u200b\u5b57\u5178\u200b\u5185\u200b\u5143\u7d20\u200b\u7684\u200b\u8bbf\u95ee\u901f\u5ea6\u200b\u3002\u200b\u5b57\u5178\u200b\u7684\u200b\u6267\u884c\u200b\u6548\u7387\u200b\u8fdc\u9ad8\u4e8e\u200b\u5217\u8868\u200b\u3002</p> <p>Python 3.6\u200b\u53ca\u200b\u4ee5\u540e\u200b\uff0c<code>dict</code>\u200b\u7c7b\u578b\u200b\u5728\u200b\u5b9e\u73b0\u200b\u4e0a\u200b\u662f\u200b\u6709\u5e8f\u200b\u7684\u200b\uff0c\u200b\u952e\u200b\u7684\u200b\u987a\u5e8f\u200b\u53d6\u51b3\u4e8e\u200b\u6dfb\u52a0\u200b\u987a\u5e8f\u200b\uff0c\u200b\u4f46\u200b\u6807\u51c6\u200b\u5e76\u200b\u6ca1\u6709\u200b\u89c4\u5b9a\u200b<code>dict</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u6709\u5e8f\u6027\u200b\u3002\u200b\u56e0\u6b64\u200b\u4e24\u4e2a\u200b\u952e\u200b\u987a\u5e8f\u200b\u4e0d\u540c\u200b\u7684\u200b\u5b57\u5178\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u76f8\u540c\u200b\u3002</p> <pre><code>&gt;&gt;&gt; a = {1: 2, 2: 1}\n&gt;&gt;&gt; b = {2: 1, 1: 2}\n&gt;&gt;&gt; a == b\nTrue\n&gt;&gt;&gt;\n</code></pre> <p>New in version 3.9</p> <p>Python 3.9\u200b\u4e2d\u200b\uff0c\u200b\u5b57\u5178\u200b\u5bf9\u8c61\u200b\u652f\u6301\u200b<code>|</code>\u200b\u8fd0\u7b97\u7b26\u200b\uff0c\u200b\u7528\u4e8e\u200b\u5c06\u200b\u4e24\u4e2a\u200b\u5b57\u5178\u200b\u5408\u5e76\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u3002\u200b\u5f53\u200b\u4e24\u4e2a\u200b\u5b57\u5178\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u91cd\u590d\u200b\u503c\u65f6\u200b\u53f3\u4fa7\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u4f18\u5148\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import sys \n&gt;&gt;&gt; ver = sys.version_info\n&gt;&gt;&gt; if ver.major == 3 and ver.minor &gt; 8:\n...     a = {1: 2}\n...     b = {3: 4}\n...     print(a | b == {1: 2, 3: 4})\n... else:\n...     print(True)\nTrue\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/mapping/#_2","title":"\u8fed\u4ee3","text":"<p><code>dict</code>\u200b\u5bf9\u8c61\u200b\u662f\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5f53\u5bf9\u200b<code>dict</code>\u200b\u5bf9\u8c61\u200b\u8fdb\u884c\u200b\u8fed\u4ee3\u200b\u65f6\u200b\uff0c\u200b\u5b9e\u9645\u4e0a\u200b\u662f\u200b\u8fed\u4ee3\u200b<code>dict</code>\u200b\u7684\u200b\u952e\u200b\u3002</p> <pre><code>&gt;&gt;&gt; for key in {1: 2, 3: 4}:\n...     print(key)\n...\n1\n3\n&gt;&gt;&gt;\n</code></pre> <ul> <li><code>dict.keys()</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b\u7531\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7684\u200b\u952e\u200b\u7ec4\u6210\u200b\u7684\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff1b</li> <li><code>dict.values()</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b\u7531\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u7ec4\u6210\u200b\u7684\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff1b</li> <li><code>dict.items()</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b\u7531\u952e\u200b-\u200b\u503c\u200b\u5143\u7ec4\u200b\u7ec4\u6210\u200b\u7684\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u3002</li> </ul> <p>\u200b\u6ce8\u610f\u200b\uff0c\u200b\u6bcf\u6b21\u200b\u8c03\u7528\u200b<code>values</code>\u200b\u65b9\u6cd5\u200b\u65f6\u200b\uff0c\u200b\u90fd\u200b\u4f1a\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5373\u4f7f\u200b\u662f\u200b\u5bf9\u200b\u540c\u4e00\u4e2a\u200b\u5b57\u5178\u200b\u8c03\u7528\u200b\u591a\u6b21\u200b<code>values</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5176\u200b\u8fd4\u56de\u503c\u200b\u4e5f\u200b\u4e0d\u60f3\u200b\u7b49\u200b\u3002</p> <p>Changed in version 3.8</p> <p>\u200b\u5185\u7f6e\u200b\u51fd\u6570\u200b<code>reversed</code>\u200b\u652f\u6301\u200b\u4f20\u9012\u200b\u5b57\u5178\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7684\u200b\u952e\u200b\u9006\u5e8f\u200b\u7ec4\u6210\u200b\u7684\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u3002</p> <p>\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\u901a\u5e38\u200b\u662f\u200b\u53ef\u200b\u4fee\u6539\u200b\u7684\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>types.MappingProxyType</code>\u200b\u4f1a\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u539f\u200b\u6620\u5c04\u200b\u7684\u200b\u89c6\u56fe\u200b\u3002\u200b\u5bf9\u539f\u200b\u6620\u5c04\u200b\u7684\u200b\u4efb\u4f55\u200b\u4fee\u6539\u200b\u90fd\u200b\u4f1a\u200b\u53cd\u6620\u200b\u5728\u200b\u89c6\u56fe\u200b\u4e2d\u200b\uff0c\u200b\u4f46\u200b\u4e0d\u80fd\u200b\u5bf9\u200b\u89c6\u56fe\u200b\u8fdb\u884c\u200b\u4efb\u4f55\u200b\u76f4\u63a5\u200b\u7684\u200b\u4fee\u6539\u200b\u3002\u200b\u6b64\u200b\u65b9\u6cd5\u200b\u95f4\u63a5\u200b\u5730\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u4e0d\u53ef\u200b\u4fee\u6539\u200b\u7684\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>&gt;&gt;&gt; from types import MappingProxyType\n&gt;&gt;&gt; a = {1: 2, 3: 4}\n&gt;&gt;&gt; b = MappingProxyType(a)\n&gt;&gt;&gt; b\nmappingproxy({1: 2, 3: 4})\n&gt;&gt;&gt; a.update({1: 5})\n&gt;&gt;&gt; b\nmappingproxy({1: 5, 3: 4})\n&gt;&gt;&gt; b[1] = 2\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: 'mappingproxy' object does not support item assignment\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/mapping/#_3","title":"\u53ef\u6563\u5217\u200b\u7c7b\u578b","text":"<p>\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u5bf9\u8c61\u200b\u79f0\u4e3a\u200b\u53ef\u6563\u5217\u200b\u5bf9\u8c61\u200b\uff1a</p> <ol> <li>\u200b\u5982\u679c\u200b\u8be5\u200b\u5bf9\u8c61\u200b\u53ef\u6563\u5217\u200b\uff0c\u200b\u5219\u200b\u5728\u200b\u5176\u200b\u751f\u547d\u5468\u671f\u200b\u4e2d\u200b\uff0c\u200b\u8be5\u6563\u200b\u5217\u503c\u200b\u4e0d\u53d8\u200b</li> <li>\u200b\u5bf9\u8c61\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>__hash__()</code>\u200b\u4e0e\u200b<code>__eq__()</code>\u200b\u65b9\u6cd5\u200b\uff08\u200b\u6ca1\u6709\u200b\u5b9e\u73b0\u200b<code>__eq__()</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u5bf9\u8c61\u200b\u4e0d\u200b\u5e94\u200b\u5b9e\u73b0\u200b<code>__hash__()</code>\u200b\u65b9\u6cd5\u200b\uff09</li> <li>\u200b\u5982\u679c\u200b\u5b83\u200b\u548c\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u5bf9\u8c61\u200b\u76f8\u7b49\u200b\uff0c\u200b\u5219\u200b\u4e24\u8005\u200b\u6709\u200b\u76f8\u540c\u200b\u7684\u200b\u6563\u5217\u503c\u200b</li> </ol> <p>\u200b\u5982\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a = \"abc\"\n&gt;&gt;&gt; b = \"\".join([\"a\", \"b\", \"c\"])\n&gt;&gt;&gt; a is b\nFalse\n&gt;&gt;&gt; hash(a) == hash(b)\nTrue\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6ce8\u200b\uff1a\u200b\u82e5\u200b\u4e00\u4e2a\u200b\u81ea\u5b9a\u4e49\u200b\u7c7b\u578b\u5b9a\u4e49\u200b\u4e86\u200b<code>__eq__</code>\uff0c\u200b\u4e14\u200b\u6ca1\u6709\u200b\u5b9a\u4e49\u200b<code>__hash__</code>\uff0c<code>__hash__</code>\u200b\u4f1a\u200b\u88ab\u200b\u8bbe\u4e3a\u200b<code>None</code>\u3002\u200b\u53ef\u6563\u5217\u200b\u7684\u200b\u81ea\u5b9a\u4e49\u200b\u5bf9\u8c61\u200b\u5fc5\u987b\u200b\u6709\u200b<code>__hash__</code>\u200b\u65b9\u6cd5\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u5185\u5bb9\u200b\u6765\u81ea\u200b\u4e8e\u200bPython\u200b\u6587\u6863\u200b </p> <p>\u200b\u6839\u636e\u200b\u5982\u200b\u4e0a\u200b\u89c4\u5219\u200b\u53ef\u4ee5\u200b\u5224\u65ad\u200b\u4e00\u4e9b\u200b\u5185\u7f6e\u200b\u6570\u636e\u7ed3\u6784\u200b\u662f\u5426\u200b\u4e3a\u200b\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b</p> <ul> <li>\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\u5728\u200b\u751f\u547d\u5468\u671f\u200b\u4e2d\u200b\u5e8f\u5217\u200b\u7684\u200b\u5185\u5bb9\u200b\u53ef\u80fd\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\uff08\u200b\u5982\u200b<code>list</code>\u3001<code>set</code>\u200b\u7b49\u200b\uff09\u200b\u4e0d\u662f\u200b\u53ef\u6563\u5217\u200b\u5bf9\u8c61\u200b\u3002</li> <li>\u200b\u539f\u5b50\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u6570\u636e\u7c7b\u578b\u200b\u6ee1\u8db3\u200b\u5982\u200b\u4e0a\u200b\u6240\u6709\u200b\u6761\u4ef6\u200b\uff0c\u200b\u662f\u200b\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b</li> <li>\u200b\u5bb9\u5668\u200b<code>frozenset</code>\u200b\u8981\u6c42\u200b\u5176\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u5747\u200b\u4e3a\u200b\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5176\u4e2d\u200b\u5185\u5bb9\u200b\u5728\u200b\u6574\u4e2a\u200b\u751f\u547d\u5468\u671f\u200b\u4e2d\u200b\u4e0d\u4f1a\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u56e0\u6b64\u200b\u662f\u200b\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b</li> <li>\u200b\u5bb9\u5668\u200b<code>tuple</code>\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u5bb9\u7eb3\u200b\u4e0d\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u56e0\u6b64\u200b<code>tuple</code>\u200b\u4e0d\u662f\u200b\u53ef\u6563\u5217\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u4f46\u200b\u4e0d\u200b\u542b\u200b\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b\u7684\u200b\u5143\u7ec4\u200b\u662f\u200b\u53ef\u200b\u6563\u5217\u200b\u5bf9\u8c61\u200b\u3002</li> </ul> <pre><code>&gt;&gt;&gt; t1 = hash((1, 2))\n&gt;&gt;&gt; t2 = hash((1, [2]))\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: unhashable type: 'list'\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>hash</code>\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u53d6\u5f97\u200b\u53ef\u6563\u5217\u200b\u5bf9\u8c61\u200b\u7684\u200b\u6563\u5217\u503c\u200b\u3002\u200b\u4e3a\u4e86\u200b\u907f\u514d\u200bDOS\u200b\u653b\u51fb\u200b\uff0c\u200b\u5728\u200b\u8ba1\u7b97\u200b\u6563\u5217\u503c\u200b\u65f6\u4f1a\u200b\u52a0\u4e0a\u200b\u4e00\u4e2a\u200b\u968f\u673a\u200b\u7684\u200b\u566a\u58f0\u200b\u3002\u200b\u566a\u58f0\u200b\u7684\u200b\u503c\u200b\u4ec5\u200b\u5728\u200b\u540c\u4e00\u200bPython\u200b\u5b9e\u4f8b\u200b\u4e2d\u200b\u76f8\u540c\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u4e00\u4e2a\u200b\u6574\u6570\u200b\uff0c\u200b\u82e5\u200b\u5b83\u200b\u7684\u200b\u503c\u200b\u5c0f\u4e8e\u200b\u6563\u5217\u503c\u200b\u6240\u200b\u5141\u8bb8\u200b\u7684\u200b\u6700\u5927\u503c\u200b\uff08\u200b\u5373\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u653e\u8fdb\u200b\u6563\u5217\u503c\u200b\u6240\u200b\u9650\u5b9a\u200b\u7684\u200b\u7a7a\u95f4\u200b\u5185\u200b\uff09\uff0c\u200b\u5219\u200b\u6563\u5217\u200b\u7684\u200b\u503c\u200b\u5c31\u662f\u200b\u8be5\u200b\u6574\u6570\u200b\u7684\u200b\u503c\u200b\u3002</p> <p>\u200b\u6563\u200b\u5217\u8868\u200b\u7684\u200b\u4f7f\u7528\u200b\u662f\u200b\u201c\u200b\u7a7a\u95f4\u200b\u6362\u200b\u65f6\u95f4\u200b\u201d\u200b\u7b56\u7565\u200b\u7684\u200b\u5178\u578b\u200b\u4f53\u73b0\u200b\u3002\u200b\u4e3a\u4e86\u200b\u5c3d\u53ef\u80fd\u51cf\u5c11\u200b\u51b2\u7a81\u200b\uff0c\u200b\u6563\u200b\u5217\u8868\u200b\u5fc5\u987b\u200b\u6709\u200b\u8f83\u200b\u4f4e\u200b\u7684\u200b\u88c5\u586b\u200b\u56e0\u5b50\u200b\uff0c\u200b\u8fd9\u200b\u610f\u5473\u7740\u200b\u6563\u200b\u5217\u8868\u200b\u4f1a\u200b\u5360\u7528\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u5185\u5b58\u7a7a\u95f4\u200b\u3002</p>"},{"location":"coding/python/mapping/#_4","title":"\u6784\u9020","text":"<p>\u200b\u5b57\u5178\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u591a\u79cd\u200b\u6784\u9020\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6784\u9020\u200b\u5b57\u5178\u200b<code>{'a': 1, 'b': 2}</code></p> <ol> <li>\u200b\u82b1\u200b\u62ec\u53f7\u200b\u6784\u9020\u200b\uff0c\u200b\u5982\u200b<code>{'a': 1, 'b': 2}</code></li> <li>\u200b\u901a\u8fc7\u200b\u7c7b\u200b\u6784\u9020\u51fd\u6570\u200b\uff0c\u200b\u5982\u200b<code>dict(a=1, b=2)</code></li> <li>\u200b\u901a\u8fc7\u200b\u5143\u7ec4\u200b/\u200b\u5217\u8868\u200b\uff08\u200b\u53ea\u200b\u9700\u200b\u662f\u200b\u957f\u5ea6\u200b\u4e3a\u200b2\u200b\u7684\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff09\u200b\u6784\u9020\u200b\uff0c\u200b\u5982\u200b<code>dict([('a', 1), ['b', 2]])</code></li> <li>\u200b\u901a\u8fc7\u200b<code>zip</code>\u200b\u6253\u5305\u200b\u6784\u9020\u200b\uff0c\u200b\u5982\u200b<code>dict(zip(['a', 'b'], [1, 2]))</code></li> <li>\u200b\u901a\u8fc7\u200b\u5b57\u5178\u200b\u63a8\u5bfc\u200b\u6784\u9020\u200b\uff0c\u200b\u5982\u200b<code>{chr(97 + _): _ + 1 for _ in range(2)}</code></li> </ol>"},{"location":"coding/python/mapping/#_5","title":"\u53d8\u79cd","text":"<p><code>collections</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5b57\u5178\u200b\u7684\u200b\u4e24\u4e2a\u200b\u53d8\u79cd\u200b\uff0c\u200b\u5373\u200b<code>defaultdict</code>\u200b\u4e0e\u200b<code>OrderDict</code></p>"},{"location":"coding/python/mapping/#defaultdict","title":"defaultdict","text":"<p>\u200b\u5f53\u200b\u5728\u200b\u4e00\u4e2a\u200b\u5b57\u5178\u200b<code>d</code>\u200b\u4e2d\u200b\u627e\u200b\u4e0d\u5230\u200b\u952e\u200b<code>k</code>\u200b\u65f6\u200b\uff0cPython\u200b\u4f1a\u200b\u4ea7\u751f\u200b<code>KeyError</code>\u200b\u5f02\u5e38\u200b\u3002\u200b\u4f7f\u7528\u200b<code>d.get(k, default)</code>\u200b\u53ef\u4ee5\u200b\u4e3a\u200b\u627e\u200b\u4e0d\u5230\u200b\u7684\u200b\u952e\u200b\u8bbe\u7f6e\u200b\u9ed8\u8ba4\u200b\u8fd4\u56de\u503c\u200b\u3002\u200b\u4f46\u200b<code>dict.get</code>\u200b\u51fd\u6570\u200b\u4e0d\u4f1a\u200b\u5bf9\u200b\u5b57\u5178\u200b\u8fdb\u884c\u200b\u66f4\u65b0\u200b\uff0c\u200b\u63d2\u5165\u200b\u627e\u200b\u4e0d\u5230\u200b\u7684\u200b\u952e\u200b\u3002</p> <p><code>dict.setdefault</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u53e6\u200b\u4e00\u79cd\u200b\u4ece\u200b\u5b57\u5178\u200b\u4e2d\u200b\u83b7\u53d6\u200b\u503c\u200b\u7684\u200b\u65b9\u6cd5\u200b\u3002<code>setdefault</code>\u200b\u63a5\u6536\u200b\u4e24\u4e2a\u200b\u53c2\u6570\u200b\uff0c\u200b\u7b2c\u4e00\u4e2a\u200b\u53c2\u6570\u200b\u4e3a\u200b<code>key</code>\uff0c\u200b\u7b2c\u4e8c\u4e2a\u200b\u53c2\u6570\u200b\u4e3a\u200b\u53ef\u9009\u200b\u7684\u200b\u9ed8\u8ba4\u503c\u200b<code>default</code>\u3002\u200b\u5f53\u200b\u5b57\u5178\u200b\u4e2d\u200b\u627e\u5230\u200b<code>key</code>\u200b\u6240\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5143\u7d20\u200b\u65f6\u200b\uff0c<code>setdefault</code>\u200b\u76f4\u63a5\u200b\u8fd4\u56de\u200b\u8be5\u503c\u200b\uff0c\u200b\u5426\u5219\u200b<code>setdefault</code>\u200b\u4f1a\u200b\u5728\u200b\u5b57\u5178\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u952e\u503c\u200b\u5bf9\u200b<code>key: default</code>\uff0c\u200b\u7136\u540e\u200b\u8fd4\u56de\u200b<code>default</code>\u3002</p> <pre><code>&gt;&gt;&gt; a = {}\n&gt;&gt;&gt; a.setdefault(0, True)\nTrue\n&gt;&gt;&gt; a[0] = False\n&gt;&gt;&gt; a.setdefault(0, False) \nFalse\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>setdefault</code>\u200b\u4ece\u200b\u5b57\u5178\u200b\u4e2d\u200b\u83b7\u53d6\u200b\u503c\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u51cf\u5c11\u200b\u5b57\u5178\u200b\u7684\u200b\u67e5\u8be2\u200b\u6b21\u6570\u200b\uff0c\u200b\u63d0\u9ad8\u200b\u5b57\u5178\u200b\u6267\u884c\u200b\u7684\u200b\u6548\u7387\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a = {}\n&gt;&gt;&gt; try:\n...     a[0].append(True)\n... except KeyError:\n...     a[0] = []\n...     a[0].append(True)\n...        \n&gt;&gt;&gt; a\n{0: [True]}\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5982\u200b\u4e0a\u200b\u4f7f\u7528\u200b<code>try-except</code>\u200b\u7ed3\u6784\u200b\u5904\u7406\u200b\u5b57\u5178\u200b\u4e2d\u200b\u627e\u200b\u4e0d\u5230\u200b\u503c\u200b\u7684\u200b\u7ed3\u679c\u200b\uff0c\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b<code>setdefault</code>\u200b\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u6709\u6548\u200b\u51cf\u5c11\u200b\u4ee3\u7801\u200b\u957f\u5ea6\u200b\uff0c\u200b\u589e\u52a0\u200b\u4ee3\u7801\u200b\u53ef\u8bfb\u6027\u200b\u3002</p> <pre><code>&gt;&gt;&gt; a = {}\n&gt;&gt;&gt; a.setdefault(0, []).append(True)\n&gt;&gt;&gt; a\n{0: [True]}\n&gt;&gt;&gt;\n</code></pre> <p><code>defaultdict</code>\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u5904\u7406\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7f3a\u5931\u200b\u7684\u200b\u952e\u200b\u3002\u200b\u51c6\u786e\u200b\u5730\u8bf4\u200b\uff0c<code>defaultdict</code>\u200b\u5728\u200b\u521b\u5efa\u200b\u65f6\u200b\u9700\u8981\u200b\u63d0\u4f9b\u200b\u4e00\u4e2a\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\uff0c\u200b\u5b57\u5178\u200b\u4e2d\u200b\u627e\u200b\u4e0d\u5230\u200b\u952e\u65f6\u200b\uff0c<code>defaultdict</code>\u200b\u4f1a\u200b\u901a\u8fc7\u200b\u8c03\u7528\u200b\u8fd9\u4e2a\u200b\u53ef\u200b\u8c03\u7528\u200b\u5bf9\u8c61\u200b\u6765\u200b\u5f97\u5230\u200b\u8be5\u952e\u200b\u5bf9\u5e94\u200b\u7684\u200b\u9ed8\u8ba4\u503c\u200b\u3002</p> <pre><code>&gt;&gt;&gt; from collections import defaultdict\n&gt;&gt;&gt; a = defaultdict(list)\n&gt;&gt;&gt; a\ndefaultdict(&lt;class 'list'&gt;, {})\n&gt;&gt;&gt; a[False] = {}\n&gt;&gt;&gt; a[False]\n{}\n&gt;&gt;&gt; a[True]\n[]\n&gt;&gt;&gt; a\ndefaultdict(&lt;class 'list'&gt;, {False: {}, True: []})\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5982\u679c\u200b\u521b\u5efa\u200b<code>defaultdict</code>\u200b\u65f6\u200b\u6ca1\u6709\u200b\u6307\u5b9a\u200b\u53c2\u6570\u200b\uff0c\u200b\u8bbf\u95ee\u200b\u4e0d\u200b\u5b58\u5728\u200b\u7684\u200b\u952e\u200b\u65f6\u200b\u4ecd\u4f1a\u200b\u629b\u51fa\u200b<code>KeyError</code>\u3002</p> <p>\u200b\u4e8b\u5b9e\u4e0a\u200b\uff0c<code>__getitem__</code>\u200b\u901a\u8fc7\u200b\u7279\u6b8a\u200b\u65b9\u6cd5\u200b<code>__missing__</code>\u200b\u5904\u7406\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7f3a\u5931\u200b\u7684\u200b\u952e\u200b\u3002\u200b\u624b\u52a8\u200b\u8c03\u7528\u200b<code>__missing__</code>\u200b\u65b9\u6cd5\u200b\u80fd\u200b\u5b9e\u73b0\u200b\u76f8\u540c\u200b\u7684\u200b\u6548\u679c\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; from collections import defaultdict \n&gt;&gt;&gt; a = defaultdict(list)\n&gt;&gt;&gt; a \ndefaultdict(&lt;class 'list'&gt;, {})\n&gt;&gt;&gt; a.__missing__(0)  \n[]\n&gt;&gt;&gt; a\ndefaultdict(&lt;class 'list'&gt;, {0: []})\n&gt;&gt;&gt;\n</code></pre> <p><code>__missing__</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u53c2\u6570\u200b\u4e3a\u200b<code>self, key</code>\uff0c\u200b\u5176\u4e2d\u200b<code>key</code>\u200b\u4e3a\u200b\u7f3a\u5931\u200b\u7684\u200b\u952e\u200b\u3002\u200b\u5373\u4f7f\u200b<code>__missing__</code>\u200b\u6b63\u5e38\u200b\u6267\u884c\u200b\uff0c\u200b\u8fd4\u56de\u503c\u200b\u4e5f\u200b\u4e0d\u4f1a\u200b\u88ab\u200b\u6dfb\u52a0\u200b\u5230\u200b\u5b57\u5178\u200b\u4e2d\u200b\u3002</p> <pre><code>&gt;&gt;&gt; class NumDict(dict):\n...     def __init__(self, *args, **kwargs):\n...             super().__init__(*args, **kwargs)\n...     def __missing__(self, key):\n...             if isinstance(key, int):\n...                     return 0\n...             else:\n...                     raise KeyError(key)\n...\n&gt;&gt;&gt; a = NumDict([(1, 2), (\"a\", \"b\")])\n&gt;&gt;&gt; a\n{1: 2, 'a': 'b'}\n&gt;&gt;&gt; a[0]\n0\n&gt;&gt;&gt; a['b']\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\n  File \"&lt;stdin&gt;\", line 8, in __missing__\nKeyError: 'b'\n&gt;&gt;&gt; a\n{1: 2, 'a': 'b'}\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/mapping/#ordereddict","title":"OrderedDict","text":"<p><code>OrderedDict</code>\u200b\u662f\u200b\u4e25\u683c\u200b\uff08\u200b\u5728\u200b\u4efb\u4f55\u200b\u7248\u672c\u200b\u4e2d\u200b\uff09\u200b\u4fdd\u6301\u200b\u987a\u5e8f\u200b\u7684\u200b\u5b57\u5178\u200b\u3002\u200b\u5728\u200b\u8fed\u4ee3\u200b\u8fc7\u7a0b\u200b\u4e2d\u952e\u200b\u7684\u200b\u987a\u5e8f\u200b\u4fdd\u6301\u4e00\u81f4\u200b\u3002</p> <p><code>OrderedDict</code>\u200b\u65b0\u589e\u200b\u4e86\u200b<code>popitem</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u9ed8\u8ba4\u200b\u79fb\u9664\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5b57\u5178\u200b\u4e2d\u200b\u6700\u540e\u200b\u6dfb\u52a0\u200b\u7684\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u3002\u200b\u5982\u679c\u200b\u624b\u52a8\u200b\u6307\u5b9a\u200b<code>last</code>\u200b\u53c2\u6570\u200b\u4e3a\u200b<code>False</code>\uff0c\u200b\u5219\u200b\u79fb\u9664\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7b2c\u4e00\u4e2a\u200b\u6dfb\u52a0\u200b\u7684\u200b\u5143\u7d20\u200b\u3002\u200b\u540c\u65f6\u200b\uff0c<code>move_to_end</code>\u200b\u662f\u200b\u4e0e\u200b\u4e4b\u200b\u914d\u5408\u200b\u7684\u200b\u79fb\u52a8\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5b57\u5178\u200b\u4e2d\u200b\u7684\u200b\u952e\u503c\u200b\u5bf9\u200b\uff08\u200b\u7531\u200b<code>key</code>\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\uff09\u200b\u79fb\u52a8\u200b\u5230\u200b\u5b57\u5178\u200b\u4e00\u7aef\u200b\uff08\u200b\u7531\u200b<code>last</code>\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\uff09\u3002</p>"},{"location":"coding/python/mapping/#chainmap","title":"ChainMap","text":"<p><code>ChainMap</code>\u200b\u53ef\u4ee5\u200b\u89c6\u4e3a\u200b\u4e00\u79cd\u200b\u5bb9\u7eb3\u200b\u6620\u5c04\u200b\u7684\u200b\u5e8f\u5217\u200b\u3002\u200b\u76f8\u6bd4\u200b\u4e8e\u200b\u4f7f\u7528\u200b<code>dict.update()</code>\u200b\u6216\u200b<code>|</code>\u200b\u8fd0\u7b97\u7b26\u200b\u9010\u4e2a\u200b\u5408\u5e76\u200b\u6620\u5c04\u200b\uff0c<code>ChainMap</code>\u200b\u6709\u200b\u66f4\u200b\u9ad8\u200b\u7684\u200b\u6548\u7387\u200b\u3002</p> <p><code>ChainMap</code>\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u6620\u5c04\u200b\uff0c\u200b\u5f53\u200b\u5728\u200b<code>ChainMap</code>\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u67e5\u8be2\u200b\u65f6\u200b\uff0c<code>ChainMap</code>\u200b\u4f1a\u200b\u9010\u4e2a\u200b\u67e5\u8be2\u200b\u5176\u4e2d\u200b\u7684\u200b\u6620\u5c04\u200b\u3002\u200b\u5f53\u200b\u540c\u4e00\u4e2a\u200b\u952e\u200b\u51fa\u73b0\u200b\u5728\u200b\u4e0d\u540c\u200b\u7684\u200b\u6620\u5c04\u200b\u4e2d\u200b\u65f6\u200b\uff0c<code>ChainMap</code>\u200b\u8fd4\u56de\u200b\u67e5\u8be2\u200b\u5230\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u7ed3\u679c\u200b\u3002</p> <pre><code>&gt;&gt;&gt; from collections import ChainMap\n&gt;&gt;&gt; a = ChainMap({'a': 1, 'b': 2}, {'a': 3, 'b': 4})\n&gt;&gt;&gt; a['b']\n2\n&gt;&gt;&gt; a['a']\n1\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5b58\u50a8\u200b\u7684\u200b\u6620\u5c04\u200b\u5728\u200b<code>ChainMap.maps</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\uff0c\u200b\u5bf9\u200b\u8be5\u200b\u5c5e\u6027\u200b\u7684\u200b\u4fee\u6539\u200b\u4f1a\u200b\u540c\u65f6\u200b\u53cd\u6620\u200b\u5728\u200b<code>ChainMap</code>\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u3002<code>parents</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u901a\u8fc7\u200b\u5c5e\u6027\u200b\u65b9\u5f0f\u200b\u8bbf\u95ee\u200b\u7684\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u8fd4\u56de\u200b\u53bb\u9664\u200b\u539f\u200b<code>ChainMap</code>\u200b\u4e2d\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u6620\u5c04\u200b\u540e\u200b\u7684\u200b\u65b0\u200b<code>ChainMap</code>\uff0c\u200b\u539f\u200b<code>ChainMap</code>\u200b\u4fdd\u6301\u200b\u4e0d\u53d8\u200b\u3002</p> <p><code>new_child</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5411\u200b<code>ChainMap</code>\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u6620\u5c04\u200b\u7684\u200b\u65b9\u6cd5\u200b\u3002\u200b\u65b0\u200b\u7684\u200b\u6620\u5c04\u200b\u88ab\u200b\u6dfb\u52a0\u200b\u5230\u200b<code>ChainMap</code>\u200b\u7684\u200b\u5f00\u5934\u200b\uff0c\u200b\u5982\u679c\u200b\u6ca1\u6709\u200b\u63d0\u4f9b\u200b\u6620\u5c04\u200b\uff0c\u200b\u5219\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u7a7a\u200b\u5b57\u5178\u200b\u3002</p> <p><code>ChainMap</code>\u200b\u5e76\u200b\u6ca1\u6709\u200b\u5408\u5e76\u200b\u539f\u6709\u200b\u7684\u200b\u6620\u5c04\u200b\uff0c\u200b\u56e0\u6b64\u200b\u8fed\u4ee3\u200b<code>ChainMap</code>\u200b\u5bf9\u8c61\u200b\u6216\u200b\u5bf9\u200b<code>ChainMap</code>\u200b\u5bf9\u8c61\u200b\u8c03\u7528\u200b<code>dict()</code>\u200b\u65f6\u200b\u53ea\u4f1a\u200b\u5e94\u7528\u200b\u5230\u200b\u5176\u4e2d\u200b\u5305\u542b\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u6620\u5c04\u200b\u3002</p> <pre><code>&gt;&gt;&gt; dict(a)\n{'a': 1, 'b': 2}\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/mapping/#counter","title":"Counter","text":"<p><code>Counter</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5bf9\u200b\u4e00\u7cfb\u5217\u200b\u503c\u200b\u7684\u200b\u7edf\u8ba1\u200b\u65b9\u6cd5\u200b\u3002\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u521d\u59cb\u5316\u200b<code>Counter</code>\u200b\u7c7b\u578b\u200b\uff0c<code>Counter</code>\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u7edf\u8ba1\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u6bcf\u4e2a\u200b\u503c\u200b\u7684\u200b\u51fa\u73b0\u200b\u987a\u5e8f\u200b\u3002</p> <p><code>Counter</code>\u200b\u4e0d\u4f1a\u200b\u5bf9\u200b\u8f93\u5165\u200b\u7684\u200b\u503c\u200b\u8fdb\u884c\u200b\u68c0\u67e5\u200b\uff0c\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b\u5b57\u5178\u200b\u521d\u59cb\u5316\u200b\u5e76\u200b\u63d0\u4f9b\u200b\u975e\u200b\u6574\u6570\u200b\u7c7b\u578b\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u5728\u200b\u8c03\u7528\u200b<code>Counter.most_common()</code>\u200b\u65b9\u6cd5\u200b\u6216\u200b\u8fed\u4ee3\u200b<code>Counter.elements()</code>\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u65f6\u4f1a\u200b\u629b\u51fa\u200b<code>TypeError</code>\u200b\u5f02\u5e38\u200b\u3002 <pre><code>&gt;&gt;&gt; from collections import Counter\n&gt;&gt;&gt; a = Counter({'0': '0'})\n&gt;&gt;&gt; for i in a.elements():\n...     pass\n...\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: 'str' object cannot be interpreted as an integer\n</code></pre></p> <p>\u200b\u8fd9\u200b\u4e00\u70b9\u200b\u5728\u200bPython\u200b\u6587\u6863\u200b\u4e2d\u6709\u200b\u76f8\u5e94\u200b\u8bf4\u660e\u200b\uff1a</p> <p></p> <pre><code>&gt;&gt;&gt; from collections import Counter \n&gt;&gt;&gt; a = Counter([1, 1, 1, 2, 2, 3])          \n&gt;&gt;&gt; a\nCounter({1: 3, 2: 2, 3: 1})\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u66f4\u65b0\u200b<code>Counter</code>\u200b\u5bf9\u8c61\u200b\u65f6\u200b\uff0c\u200b\u8ba1\u6570\u5668\u200b\u4f1a\u200b\u76f8\u5e94\u200b\u5730\u200b\u66f4\u65b0\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a.update([3, 3, 3, 2, 2, 1])\n&gt;&gt;&gt; a\nCounter({1: 4, 2: 4, 3: 4})\n&gt;&gt;&gt;\n</code></pre> <p><code>Counter.elements()</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u8fed\u4ee3\u200b\u5668\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u6309\u7167\u200b\u5176\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u91cd\u590d\u200b\u3002<code>Counter.most_common</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b\u8ba1\u6570\u5668\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u6b21\u6570\u200b\u6700\u591a\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</p> <p><code>Counter</code>\u200b\u7c7b\u578b\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>+</code>\u3001<code>-</code>\u3001<code>&amp;</code>\u200b\u548c\u200b<code>|</code>\u200b\u8fd0\u7b97\u7b26\u200b\uff0c\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u8ba1\u6570\u5668\u200b\u7684\u200b\u76f8\u52a0\u200b\u3001\u200b\u76f8\u51cf\u200b\u3001\u200b\u4ea4\u200b\u548c\u200b\u5e76\u200b\u64cd\u4f5c\u200b\u3002\u200b\u5f53\u200b<code>+</code>\u3001<code>-</code>\u200b\u88ab\u200b\u7528\u4f5c\u200b\u4e00\u5143\u200b\u8fd0\u7b97\u7b26\u200b\u65f6\u200b\uff0c\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u64cd\u4f5c\u6570\u200b\u9ed8\u8ba4\u200b\u4e3a\u7a7a\u200b\u8ba1\u6570\u5668\u200b\u3002</p>"},{"location":"coding/python/mapping/#_6","title":"\u6269\u5c55","text":"<p><code>collections.UserDict</code>\u200b\u7c7b\u200b\u662f\u200b\u5bf9\u200b<code>dict</code>\u200b\u7684\u200bPython\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u81ea\u5b9a\u4e49\u200b\u6620\u5c04\u200b\u7c7b\u578b\u200b\u7684\u200b\u5b9e\u73b0\u200b\u3002</p> <p><code>UserDict</code>\u200b\u4e0d\u662f\u200b<code>dict</code>\u200b\u7684\u200b\u5b50\u7c7b\u200b\uff0c\u200b\u4f46\u200b<code>UserDict</code>\u200b\u4e2d\u6709\u200b\u4e00\u4e2a\u200b<code>dict</code>\u200b\u7c7b\u578b\u200b\u7684\u200b<code>data</code>\u200b\u5c5e\u6027\u200b\uff0c\u200b\u662f\u200b<code>UserDict</code>\u200b\u5b58\u50a8\u200b\u6570\u636e\u200b\u7684\u200b\u6700\u7ec8\u200b\u4f4d\u7f6e\u200b\u3002\u200b\u76f8\u6bd4\u200b\u4e8e\u200b\u76f4\u63a5\u200b\u5bf9\u200b<code>dict</code>\u200b\u7c7b\u578b\u200b\u8fdb\u884c\u200b\u6269\u5c55\u200b\uff0c<code>data</code>\u200b\u5c5e\u6027\u200b\u4fdd\u8bc1\u200b\u4e86\u200b<code>__getitem__</code>\u200b\u65b9\u6cd5\u200b\u4e0d\u4f1a\u200b\u4ea7\u751f\u200b\u5faa\u73af\u200b\u9012\u5f52\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002</p> <p><code>UserDict</code>\u200b\u7c7b\u200b\u7ee7\u627f\u200b\u81ea\u200b<code>_collections_abc.MutableMapping</code>\u200b\u4e0e\u200b<code>_collections_abc.Mapping</code>\u200b\u7c7b\u200b\u3002<code>MutableMapping</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>update()</code>\u200b\u65b9\u6cd5\u200b\u3002\u200b\u65b9\u6cd5\u200b\u7684\u200b\u5185\u90e8\u200b\u5b9e\u73b0\u200b\u8868\u660e\u200b\uff0c<code>update()</code>\u200b\u5728\u200b\u5185\u90e8\u200b\u901a\u8fc7\u200b<code>__setitem__</code>\u200b\u5b9e\u73b0\u200b\u3002<code>Mapping</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>get()</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u8be5\u200b\u65b9\u6cd5\u200b\u5728\u200b\u5185\u90e8\u200b\u901a\u8fc7\u200b<code>__getitem__</code>\u200b\u5b9e\u73b0\u200b\u3002</p> <pre><code>&gt;&gt;&gt; from collections import UserDict\n&gt;&gt;&gt; class myDict(UserDict):\n...     def __init__(self, *args, **kwargs):\n...             super().__init__(*args, **kwargs)  \n...     def __setitem__(self, index, value):\n...             print(\"Modification: \", index, \":\", value)\n...             super().__setitem__(index, value)\n... \n&gt;&gt;&gt; a = myDict()\n&gt;&gt;&gt; a.update(a=5, b=4)\nModification:  a : 5\nModification:  b : 4\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/mapping/#_7","title":"\u96c6\u5408","text":"<p>\u200b\u96c6\u5408\u200b<code>set</code>\u200b\u662f\u200b\u4e00\u79cd\u200b\u5e8f\u5217\u200b\u7c7b\u578b\u200b\uff0c\u200b\u96c6\u5408\u200b\u4e2d\u200b\u7684\u200b\u5404\u4e2a\u200b\u5143\u7d20\u200b\u6700\u200b\u591a\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u6b21\u200b\u3002\u200b\u4efb\u4f55\u200b\u975e\u200b\u7a7a\u96c6\u5408\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5b57\u9762\u200b\u8868\u793a\u6cd5\u200b<code>{}</code>\u200b\u8fdb\u884c\u200b\u8868\u793a\u200b\uff0c\u200b\u7a7a\u96c6\u200b\u53ea\u80fd\u200b\u7528\u200b<code>set()</code>\u200b\u8868\u793a\u200b\u3002\u200b\u96c6\u5408\u200b\u540c\u6837\u200b\u9002\u7528\u200b\u4e8e\u200b\u4e0e\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u76f8\u4f3c\u200b\u7684\u200b\u63a8\u5bfc\u200b\u65b9\u5f0f\u200b\u3002</p> <p>\u200b\u96c6\u5408\u200b\u6709\u200b\u5bf9\u5e94\u200b\u7684\u200b\u53ea\u8bfb\u200b\u53d8\u79cd\u200b<code>frozenset</code>\u3002<code>set</code>\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u5b57\u9762\u200b\u8868\u793a\u6cd5\u200b\u8fdb\u884c\u200b\u6784\u9020\u200b\uff0c\u200b\u6216\u200b\u4f7f\u7528\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u8fdb\u884c\u200b\u6784\u9020\u200b\u3002<code>frozenset</code>\u200b\u53ea\u80fd\u200b\u4f7f\u7528\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u6784\u9020\u200b\u3002</p> <p>\u200b\u53ef\u4ee5\u200b\u5229\u7528\u200b<code>set</code>\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u552f\u4e00\u6027\u200b\u53bb\u9664\u200b\u5217\u8868\u200b\u4e2d\u200b\u7684\u200b\u91cd\u590d\u200b\u503c\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; src_list = [1, 1, 2, 3, 4, 2, 3]\n&gt;&gt;&gt; list(set(src_list))\n[1, 2, 3, 4]\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u53e6\u200b\u4e00\u79cd\u200b\u65b9\u6cd5\u200b\u662f\u200b\u901a\u8fc7\u200b<code>collections.Counter</code>\u200b\u53bb\u200b\u91cd\u200b\u3002</p> <pre><code>&gt;&gt;&gt; from collections import Counter\n&gt;&gt;&gt; src_list = [1, 1, 2, 3, 4, 2, 3]\n&gt;&gt;&gt; list(Counter(src_list).keys())\n[1, 2, 3, 4]\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u548c\u200b\u5b57\u5178\u200b\u7684\u200b\u952e\u200b\u4e00\u6837\u200b\uff0c\u200b\u96c6\u5408\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u5fc5\u987b\u200b\u662f\u200b\u53ef\u200b\u6563\u5217\u200b\u7684\u200b\u3002<code>set</code>\u200b\u7c7b\u578b\u200b\u672c\u8eab\u200b\u4e0d\u53ef\u200b\u6563\u5217\u200b\uff0c<code>frozenset</code>\u200b\u7c7b\u578b\u200b\u53ef\u6563\u5217\u200b\u3002</p>"},{"location":"coding/python/mapping/#_8","title":"\u8fd0\u7b97","text":"<p>\u200b\u96c6\u5408\u200b\u6709\u200b\u5982\u4e0b\u200b\u8fd0\u7b97\u200b\uff0c\u200b\u7531\u4e8e\u200b\u96c6\u5408\u200b\u7684\u200b\u5e95\u5c42\u200b\u6570\u636e\u7ed3\u6784\u200b\u662f\u200b\u6563\u200b\u5217\u8868\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u8fd0\u7b97\u200b\u65b9\u6cd5\u200b\u6709\u200b\u8f83\u200b\u9ad8\u200b\u7684\u200b\u8ba1\u7b97\u200b\u6548\u7387\u200b\u3002</p> <ul> <li>\u200b\u4ea4\u96c6\u200b\u3001\u200b\u5e76\u96c6\u200b\u3001\u200b\u5dee\u96c6\u200b\u3001\u200b\u5bf9\u79f0\u200b\u5dee\u200b\uff0c\u200b\u5373\u200b<code>&amp;</code>\u3001<code>|</code>\u3001<code>-</code>\u200b\u4e0e\u200b<code>^</code>\uff1b</li> <li>\u200b\u5b50\u96c6\u200b\u3001\u200b\u771f\u200b\u5b50\u96c6\u200b\u3001\u200b\u8d85\u96c6\u200b\u3001\u200b\u771f\u8d85\u96c6\u200b\uff0c\u200b\u5373\u200b<code>&lt;=</code>\u3001<code>&lt;</code>\u3001<code>&gt;=</code>\u200b\u4e0e\u200b<code>&gt;</code>\uff1b</li> <li>\u200b\u5c5e\u4e8e\u200b\uff0c\u200b\u5373\u200b<code>in</code></li> </ul> <p>\u200b\u4ee5\u200b<code>in</code>\u200b\u64cd\u4f5c\u200b\u4e3a\u4f8b\u200b\uff0c<code>set</code>\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u67e5\u8be2\u200b\u6563\u200b\u5217\u8868\u200b\u83b7\u53d6\u200b\u7ed3\u679c\u200b\uff0c\u200b\u4f46\u200b<code>list</code>\u200b\u5fc5\u987b\u200b\u626b\u63cf\u200b\u4e00\u904d\u200b\u5217\u8868\u200b\u624d\u80fd\u200b\u83b7\u53d6\u200b\u7ed3\u679c\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u542b\u6709\u200b\u8bb8\u591a\u200b\u5143\u7d20\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4e24\u8005\u200b\u5728\u200b\u6267\u884c\u200b\u901f\u5ea6\u200b\u4e0a\u200b\u6709\u200b\u660e\u663e\u200b\u7684\u200b\u5dee\u522b\u200b\u3002</p> <p><code>set</code>\u200b\u8fd8\u200b\u652f\u6301\u200b<code>add</code>\u3001<code>clear</code>\u3001<code>copy</code>\u3001<code>pop</code>\u3001<code>remove</code>\u200b\u7b49\u200b\u65b9\u6cd5\u200b\u3002</p>"},{"location":"coding/python/sequence/","title":"Python\u200b\u4e2d\u200b\u7684\u200b\u5e8f\u5217\u200b\u7c7b\u578b","text":"<p>\u200b\u5e8f\u5217\u200b\u6307\u200b\u53ef\u4ee5\u200b\u5b58\u50a8\u200b\u591a\u4e2a\u200b\u6570\u636e\u200b\u7684\u200b\u6570\u636e\u7ed3\u6784\u200b\u3002</p> <p>\u200b\u6309\u7167\u200b\u5b58\u50a8\u200b\u7684\u200b\u6570\u636e\u7c7b\u578b\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\uff1a</p> <ul> <li>\u200b\u5bb9\u5668\u200b\u53ef\u4ee5\u200b\u5b58\u50a8\u200b\u4e0d\u540c\u200b\u7c7b\u578b\u200b\u7684\u200b\u6570\u636e\u200b\uff0c\u200b\u8fd9\u90e8\u5206\u200b\u7c7b\u578b\u200b\u6709\u200b<code>list</code>\u3001<code>tuple</code>\u3001<code>collections.deque</code></li> <li>\u200b\u6241\u5e73\u200b\u5e8f\u5217\u200b\u53ea\u80fd\u200b\u5b58\u50a8\u200b\u67d0\u79cd\u200b\u7c7b\u578b\u200b\u7684\u200b\u6570\u636e\u200b\uff0c\u200b\u8fd9\u90e8\u5206\u200b\u7c7b\u578b\u200b\u6709\u200b<code>str</code>\u3001<code>bytes</code>\u3001<code>bytearray</code>\u3001<code>memoryview</code>\u3001<code>array.array</code></li> </ul> <p>\u200b\u6309\u7167\u200b\u5e8f\u5217\u200b\u662f\u5426\u200b\u53ef\u4ee5\u200b\u4fee\u6539\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b</p> <ul> <li>\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\uff1a<code>list</code>\u3001<code>bytearray</code>\u3001<code>array.array</code>\u3001<code>collections.deque</code>\u3001<code>memoryview</code></li> <li>\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\uff1a<code>tuple</code>\u3001<code>str</code>\u3001<code>bytes</code></li> </ul>"},{"location":"coding/python/sequence/#_1","title":"\u5217\u8868","text":"<p>\u200b\u5217\u8868\u200b\u662f\u200b\u53ef\u4ee5\u200b\u4fee\u6539\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5b58\u50a8\u200b\u591a\u79cd\u4e0d\u540c\u200b\u7c7b\u578b\u200b\u7684\u200b\u6570\u636e\u7ed3\u6784\u200b\u3002\u200b\u5982\u4e0b\u200b\u7740\u91cd\u200b\u8ba8\u8bba\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u7684\u200b\u4e66\u5199\u200b\u5f62\u5f0f\u200b\u3002</p> <p>\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u53ef\u200b\u7528\u4e8e\u200b\u4ece\u200b\u4e00\u4e2a\u200b\u5217\u8868\u200b\u751f\u6210\u200b\u65b0\u200b\u7684\u200b\u5217\u8868\u200b\uff0c\u200b\u76f8\u6bd4\u200b\u4e8e\u200bfor\u200b\u5faa\u73af\u200b\u52a0\u200b<code>append()</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u7684\u200b\u5199\u6cd5\u200b\u6709\u200b\u66f4\u597d\u200b\u7684\u200b\u53ef\u8bfb\u6027\u200b\u3002</p> <p>\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u793a\u4f8b\u200b\uff1a</p> <pre><code>a = [1, 2, 3]\nb = [_ ** 2 for _ in a]\n</code></pre> <p>Python 3.x\u200b\u4e2d\u200b\u7684\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u4e0d\u4f1a\u200b\u4ea7\u751f\u200b\u53d8\u91cf\u200b\u6cc4\u9732\u200b\u7684\u200b\u95ee\u9898\u200b\uff0c\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>x = 'ABC'\ny = [x for x in x]\nprint(x)\n</code></pre> <p>\u200b\u8f93\u51fa\u200b\u4e3a\u200b<code>ABC</code>\uff0c\u200b\u8bf4\u660e\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u53d8\u91cf\u200b<code>x</code>\u200b\u7684\u200b\u503c\u200b\u6ca1\u6709\u200b\u4f20\u9012\u200b\u5230\u200b\u5916\u90e8\u200b\u3002</p> <p>\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u7b49\u4ef7\u200b\u4e8e\u200b<code>map</code>\u200b\u4e0e\u200b<code>filter</code>\u200b\u7684\u200b\u7ed3\u5408\u200b\u3002<code>map</code>\u200b\u51fd\u6570\u200b\u63a5\u6536\u200b\u4e24\u4e2a\u200b\u53c2\u6570\u200b\u3002\u200b\u7b2c\u4e00\u4e2a\u200b\u53c2\u6570\u200b\u4e3a\u200b\u51fd\u6570\u200b\uff0c\u200b\u7b2c\u4e8c\u4e2a\u200b\u53c2\u6570\u200b\u4e3a\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u3002<code>map</code>\u200b\u51fd\u6570\u200b\u8fed\u4ee3\u200b\u5730\u200b\u5c06\u200b\u51fd\u6570\u200b\u4f5c\u7528\u200b\u4e8e\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u4ece\u800c\u200b\u5f97\u5230\u200b\u7ed3\u679c\u200b\u3002</p> <p>\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u8ba1\u7b97\u200b\u5217\u8868\u200b\u7684\u200b\u7b1b\u5361\u5c14\u200b\u79ef\u200b\uff0c\u200b\u5982\u200b\u4e0b\u5217\u200b\u4ee3\u7801\u200b\uff1a</p> <pre><code>[(x, y) for x in range(10) for y in range(11, 20)]\n</code></pre> <p>\u200b\u5faa\u73af\u200b\u7684\u200b\u987a\u5e8f\u200b\u4e0e\u200b<code>for</code>\u200b\u5faa\u73af\u200b\u4e00\u81f4\u200b\uff0c\u200b\u5148\u200b\u6267\u884c\u200b\u5916\u5c42\u200b\u7684\u200b\u5faa\u73af\u200b\uff0c\u200b\u518d\u200b\u6267\u884c\u200b\u5185\u5c42\u200b\u7684\u200b\u5faa\u73af\u200b\u3002</p> <p>\u200b\u5c06\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u7684\u200b\u65b9\u62ec\u53f7\u200b\u5706\u62ec\u53f7\u200b\uff0c\u200b\u5373\u200b\u4e3a\u200b\u751f\u6210\u5668\u200b\u8868\u8fbe\u5f0f\u200b\u3002\u200b\u5982\u679c\u200b\u751f\u6210\u5668\u200b\u8868\u8fbe\u5f0f\u200b\u662f\u200b\u51fd\u6570\u8c03\u7528\u200b\u7684\u200b\u552f\u4e00\u200b\u53c2\u6570\u200b\uff0c\u200b\u5706\u62ec\u53f7\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u3002</p>"},{"location":"coding/python/sequence/#_2","title":"\u5143\u7ec4","text":"<p>\u200b\u5143\u7ec4\u200b\u662f\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7406\u89e3\u200b\u4e3a\u200b\u5b57\u200b\u6bb5\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u6570\u636e\u8868\u200b\u4e2d\u200b\u7684\u200b\u67d0\u200b\u4e00\u884c\u200b\u3002\u200b\u5982\u4e0b\u200b\u7740\u91cd\u200b\u8ba8\u8bba\u200b\u5143\u7ec4\u200b\u89e3\u5305\u200b\u7684\u200b\u5185\u5bb9\u200b\u3002</p> <p>\u200b\u5e73\u884c\u200b\u8d4b\u503c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u4e00\u4e2a\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u590d\u5236\u5230\u200b\u4e00\u7ec4\u200b\u53d8\u91cf\u200b\u4e2d\u200b\u3002\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u5bf9\u8c61\u200b\u4e0d\u200b\u8981\u6c42\u200b\u662f\u200b\u5143\u7ec4\u200b\u3002\u200b\u5982\u200b\uff1a</p> <pre><code>a, b = (1, 2)\nc, d = [3, 4]\ne, f = {1: 2, 3: 4} # \u200b\u6309\u7167\u200bkey\u200b\u7684\u200b\u53d6\u503c\u200b\u8fdb\u884c\u200b\u5206\u914d\u200b\n</code></pre> <p><code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u89e3\u5305\u200b\u5143\u7ec4\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u4f5c\u4e3a\u200b\u51fd\u6570\u53c2\u6570\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <pre><code>print(*(1, 2))\n</code></pre> <p><code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u4ece\u200b\u51fd\u6570\u200b\u4e2d\u200b\u83b7\u53d6\u200b\u53ef\u53d8\u200b\u6570\u91cf\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5e38\u89c1\u200b\u7528\u6cd5\u200b\u4e3a\u200b<code>*args</code>\u200b\u4e0e\u200b<code>**kwargs</code>\u3002</p> <p><code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u5e73\u884c\u200b\u8d4b\u503c\u200b\u4e2d\u200b\uff0c\u200b\u4f46\u200b\u5e73\u884c\u200b\u8d4b\u503c\u200b\u4e2d\u200b\u7684\u200b\u4e00\u7ec4\u200b\u53d8\u91cf\u200b\u53ea\u80fd\u200b\u51fa\u73b0\u200b\u4e00\u4e2a\u200b<code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u3002\u200b\u5e26\u200b<code>*</code>\u200b\u7684\u200b\u53d8\u91cf\u200b\u89e3\u5305\u200b\u540e\u200b\u6210\u4e3a\u200b\u5217\u8868\u200b\u3002</p> <pre><code>a, b, c, *d = range(5) # d = [3, 4]\na, b, *c, d = range(5) # c = [2, 3]\n</code></pre> <p>\u200b\u5f53\u5f85\u200b\u89e3\u5305\u200b\u7684\u200b\u5143\u7ec4\u200b\u672c\u8eab\u200b\u662f\u200b\u5143\u7ec4\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u62ec\u53f7\u200b\u89e3\u5305\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <pre><code>a, (b, c), d = (1, (2, 3), 4)\n</code></pre> <p>\u200b\u5143\u7ec4\u200b\u4e0d\u200b\u652f\u6301\u200b\u5217\u8868\u200b\u7684\u200b<code>append</code>\u3001<code>clear</code>\u3001<code>copy</code>\u3001<code>extend</code>\u3001<code>insert</code>\u3001<code>pop</code>\u3001<code>remove</code>\u3001<code>reverse</code>\u200b\u7b49\u200b\u65b9\u6cd5\u200b\u3002</p>"},{"location":"coding/python/sequence/#_3","title":"\u5177\u540d\u200b\u5143\u7ec4","text":"<p><code>collections.namedtuple</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u7528\u4e8e\u200b\u521b\u5efa\u200b\u5177\u540d\u200b\u5143\u7ec4\u200b\u7c7b\u578b\u200b\u7684\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5176\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>collections.namedtuple(typename, field_names, *, rename=False, defaults=None, module=None)\n</code></pre> <ul> <li><code>typename</code>\u200b\u6307\u200b\u7c7b\u578b\u200b\u7684\u200b\u540d\u79f0\u200b\uff0c\u200b\u8be5\u200b\u53d8\u91cf\u200b\u88ab\u200b\u5199\u5165\u200b<code>__repr__</code>\u200b\u65b9\u6cd5\u200b\u4e2d\u200b\u3002\u200b\u8be5\u200b\u53c2\u6570\u200b\u7684\u200b\u53d6\u503c\u200b\u4e0e\u200b\u4ee3\u7801\u200b\u4e2d\u200b\u7c7b\u578b\u200b\u88ab\u200b\u5b9e\u9645\u200b\u5206\u914d\u200b\u7684\u200b\u540d\u79f0\u200b\u65e0\u5173\u200b\u3002</li> <li><code>field_names</code>\u200b\u662f\u200b\u5143\u7ec4\u200b\u4e2d\u5b57\u200b\u6bb5\u200b\u7684\u200b\u540d\u79f0\u200b</li> <li>\u200b\u5f53\u200b<code>rename</code>\u200b\u4e3a\u200b<code>True</code>\u200b\u91cd\u590d\u200b\u6216\u200b\u4e0e\u200b\u5173\u952e\u5b57\u200b\u51b2\u7a81\u200b\u7684\u200b\u5b57\u6bb5\u540d\u200b\u79f0\u4f1a\u200b\u88ab\u200b\u81ea\u52a8\u200b\u5904\u7406\u200b\uff0c\u200b\u5426\u5219\u200b\u4f1a\u200b\u629b\u51fa\u200b<code>ValueError</code></li> <li><code>defaults</code>\u200b\u4e3a\u200b\u5b57\u200b\u6bb5\u200b\u7684\u200b\u9ed8\u8ba4\u503c\u200b\uff0c\u200b\u8be5\u200b\u5217\u8868\u200b\u4e0e\u200b\u5b57\u6bb5\u540d\u200b\u79f0\u200b\u5217\u8868\u200b\u6309\u53f3\u200b\u5bf9\u9f50\u200b\u8fdb\u884c\u200b\u5339\u914d\u200b\u3002</li> <li>\u200b\u82e5\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>module</code>\u200b\u53c2\u6570\u200b\uff0c\u200b\u8be5\u503c\u200b\u4f1a\u200b\u88ab\u200b\u5199\u5165\u200b<code>__module__</code>\u200b\u5c5e\u6027\u200b\u4e2d\u200b\u3002</li> </ul> <p><code>namedtuple</code>\u200b\u8fd4\u56de\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u521b\u5efa\u5bf9\u8c61\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>&gt;&gt;&gt; from collections import namedtuple\n&gt;&gt;&gt; Point = namedtuple('Point', [\"x\", \"y\"], defaults=[1, 2]) \n&gt;&gt;&gt; Point()\nPoint(x=1, y=2)\n&gt;&gt;&gt; Point(2)\nPoint(x=2, y=2)\n&gt;&gt;&gt; Point(2, 3)\nPoint(x=2, y=3)\n&gt;&gt;&gt;\n</code></pre>"},{"location":"coding/python/sequence/#_4","title":"\u5207\u7247","text":"<p>\u200b\u5207\u7247\u200b\u662f\u200b\u5728\u200b\u5e8f\u5217\u200b\u4e0a\u200b\u6267\u884c\u200b\u7684\u200b\u64cd\u4f5c\u200b\uff0c\u200b\u5207\u7247\u200b\u64cd\u4f5c\u200b\u5f97\u5230\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u5e8f\u5217\u200b\u3002\u200b\u5207\u7247\u200b\u4ee5\u200b<code>s[a:b:c]</code>\u200b\u7684\u200b\u5f62\u5f0f\u200b\u6307\u5b9a\u200b\uff0c<code>a</code>\u200b\u4e3a\u200b\u8d77\u59cb\u200b\u4e0b\u6807\u200b\u3001<code>b</code>\u200b\u4e3a\u200b\u7ec8\u6b62\u200b\u4e0b\u6807\u200b\uff08\u200b\u4e0d\u200b\u5305\u542b\u200b\uff09\u3001<code>c</code>\u200b\u4e3a\u200b\u6b65\u957f\u200b\uff08\u200b\u53ef\u4ee5\u200b\u4e3a\u200b\u8d1f\u200b\uff09\u3002</p> <ul> <li><code>s[::-1]</code>\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u53cd\u8f6c\u200b<code>s</code>\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b</li> <li>\u200b\u5207\u7247\u200b\u5e76\u200b\u6ca1\u6709\u200b\u521b\u5efa\u200b\u65b0\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5bf9\u200b\u5207\u7247\u200b\u7684\u200b\u4fee\u6539\u200b\u4f1a\u200b\u53cd\u6620\u200b\u5230\u200b\u539f\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u3002\u200b\u4f46\u200b\u5bf9\u200b\u5207\u7247\u200b\u7684\u200b\u4fee\u6539\u200b\u5fc5\u987b\u200b\u6ee1\u8db3\u200b\u5143\u7d20\u200b\u6570\u91cf\u200b\u76f8\u540c\u200b\u7684\u200b\u6761\u4ef6\u200b\uff0c\u200b\u5426\u5219\u200b\u4f1a\u200b\u629b\u51fa\u200b<code>ValueError</code>\u3002</li> </ul> <p>\u200b\u53e6\u200b\u4e00\u79cd\u200b\u5207\u7247\u200b\u7684\u200b\u65b9\u5f0f\u200b\u4e3a\u200b<code>s[slice(a, b, c)]</code>\uff0c\u200b\u7701\u7565\u200b\u7684\u200b\u53c2\u6570\u200b\u4f7f\u7528\u200b<code>None</code>\u200b\u8868\u793a\u200b\u3002\u200b\u56e0\u6b64\u200b<code>s[::-1]</code>\u200b\u7b49\u4ef7\u200b\u4e8e\u200b<code>s[slice(None, None, -1)]</code>\u3002\u200b\u4e8b\u5b9e\u4e0a\u200b\uff0cPython\u200b\u5728\u200b\u5904\u7406\u200b\u5f62\u200b\u5982\u200b<code>s[a:b:c]</code>\u200b\u7684\u200b\u5207\u7247\u200b\u65f6\u4f1a\u200b\u8c03\u7528\u200b<code>s.__getitem__(slice(start, stop, step))</code></p> <p>\u200b\u5728\u200bNumPy\u200b\u6570\u7ec4\u200b\u7684\u200b\u5207\u7247\u200b\u4e2d\u200b\u6709\u65f6\u200b\u4f1a\u200b\u51fa\u73b0\u200b\u7701\u7565\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u6b64\u65f6\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>...</code>\u3002<code>...</code>\u200b\u7b26\u53f7\u200b\u4f1a\u200b\u88ab\u200bPython\u200b\u89e3\u91ca\u200b\u4e3a\u200b<code>EllipsisType</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u5e38\u91cf\u200b\u3002</p>"},{"location":"coding/python/sequence/#_5","title":"\u5e8f\u5217\u200b\u8fd0\u7b97","text":"<p>\u200b\u5e8f\u5217\u200b\u901a\u5e38\u200b\u652f\u6301\u200b<code>+</code>\u200b\u8fd0\u7b97\u200b\u4e0e\u200b<code>*</code>\u200b\u8fd0\u7b97\u200b\u3002\u200b\u4e0e\u200b\u5207\u7247\u200b\u4e0d\u540c\u200b\uff0c\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u8fd0\u7b97\u200b\u76f4\u63a5\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u5e8f\u5217\u200b\u3002</p> <pre><code>&gt;&gt;&gt; a = [1]\n&gt;&gt;&gt; b = [2] \n&gt;&gt;&gt; c = a + b\n&gt;&gt;&gt; c is a\nFalse\n&gt;&gt;&gt;\n</code></pre> <p><code>+</code>\u200b\u8fd0\u7b97\u200b\u5c40\u9650\u4e8e\u200b\u76f8\u540c\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5bf9\u4e8e\u200b<code>list</code>\u200b\u7b49\u200b\u53ef\u53d8\u200b\u7c7b\u578b\u200b\uff0c<code>a + b</code>\u200b\u7b49\u4ef7\u200b\u4e8e\u200b<code>a.extend(b)</code>\u3002\u200b\u4f46\u200b<code>a.extend(b)</code>\u200b\u65b9\u6cd5\u200b\u5bf9\u200b<code>b</code>\u200b\u7684\u200b\u7c7b\u578b\u200b\u65e0\u200b\u8981\u6c42\u200b\uff0c\u200b\u4ec5\u200b\u9700\u8981\u200b<code>b</code>\u200b\u53ef\u200b\u8fed\u4ee3\u200b\u3002</p> <p><code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u7684\u200b\u7b2c\u4e8c\u4e2a\u200b\u64cd\u4f5c\u6570\u200b<code>y</code>\u200b\u5fc5\u987b\u200b\u4e3a\u200b\u6574\u6570\u200b\uff0c\u200b\u8868\u793a\u200b\u5c06\u200b\u5e8f\u5217\u200b\u91cd\u590d\u200b<code>y</code>\u200b\u6b21\u200b\u3002\u200b\u82e5\u200b<code>y&lt;=0</code>\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b\u7a7a\u200b\u5217\u8868\u200b</p> <p>\u200b\u6ce8\u610f\u200b\uff0c<code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u5f97\u5230\u200b\u7684\u200b\u5217\u8868\u200b\u4e2d\u200b\uff0c\u200b\u5217\u8868\u200b\u591a\u6b21\u91cd\u590d\u200b\u7684\u200b\u90e8\u5206\u200b\u4ee5\u200b\u5f15\u7528\u200b\u7684\u200b\u5f62\u5f0f\u200b\u5b58\u50a8\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; b = [[]] * 3\n&gt;&gt;&gt; b[0].append(1)\n&gt;&gt;&gt; b\n[[1], [1], [1]]\n&gt;&gt;&gt;\n</code></pre> <p><code>append</code>\u200b\u5904\u7406\u200b\u53ef\u53d8\u200b\u5bf9\u8c61\u200b\u7684\u200b\u65b9\u5f0f\u200b\u540c\u6837\u200b\u662f\u200b\u590d\u5236\u200b\u5f15\u7528\u200b\u800c\u200b\u4e0d\u662f\u200b\u590d\u5236\u200b\u503c\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e5f\u200b\u4f1a\u200b\u9020\u6210\u200b\u76f8\u540c\u200b\u7684\u200b\u95ee\u9898\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a = []\n&gt;&gt;&gt; b = []\n&gt;&gt;&gt; for i in range(3):\n...     a.append(b)\n... \n&gt;&gt;&gt; a[0].append(1)\n&gt;&gt;&gt; a\n[[1], [1], [1]]\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u56e0\u6b64\u200b\u8981\u200b\u907f\u514d\u200b\u4f7f\u7528\u200b<code>*</code>\u200b\u8fd0\u7b97\u7b26\u200b\u5904\u7406\u200b\u5305\u542b\u200b\u53ef\u53d8\u200b\u5bf9\u8c61\u200b\u7684\u200b\u5217\u8868\u200b\u3002\u200b\u5efa\u8bae\u200b\u4f7f\u7528\u200b\u5217\u8868\u200b\u63a8\u5bfc\u200b\u521b\u5efa\u200b\u5d4c\u5957\u200b\u5217\u8868\u200b\u3002</p> <p>\u200b\u4e0e\u200b\u5e8f\u5217\u200b\u7684\u200b\u8fd0\u7b97\u200b\u4e0d\u540c\u200b\uff0c\uff08\u200b\u53ef\u53d8\u200b\uff09\u200b\u5e8f\u5217\u200b\u7684\u200b\u589e\u91cf\u200b\u8fd0\u7b97\u200b<code>+=</code>\u3001<code>*=</code>\u200b\u662f\u200b\u5728\u200b\u539f\u200b\u5e8f\u5217\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\u8fdb\u884c\u200b\u7684\u200b\uff0c\u200b\u4e0d\u4f1a\u200b\u521b\u5efa\u200b\u65b0\u200b\u7684\u200b\u5e8f\u5217\u200b\u5bf9\u8c61\u200b\u3002\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\u4e0d\u200b\u652f\u6301\u200b\u589e\u91cf\u200b\u8fd0\u7b97\u200b\u3002</p> <p>\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b\uff1a</p> <pre><code>&gt;&gt;&gt; a = [1, 2, 3]\n&gt;&gt;&gt; a[2] += 5\n&gt;&gt;&gt; a\n[1, 2, 8]\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\u901a\u8fc7\u200b<code>__setitem__</code>\u200b\u65b9\u6cd5\u200b\u5b9e\u73b0\u200b\u4e86\u200b\u5207\u7247\u200b\u7684\u200b\u4fee\u6539\u200b\u3002\u200b\u82e5\u200b<code>a</code>\u200b\u4e3a\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\uff0c\u200b\u6267\u884c\u200b\u7ed3\u679c\u200b\u5e94\u200b\u4e3a\u4f55\u200b\uff1f</p> <p>\u200b\u5bf9\u4e8e\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5176\u200b\u672c\u8eab\u200b\u4e0d\u200b\u652f\u6301\u200b\u5bf9\u200b\u5143\u7d20\u200b\u8fdb\u884c\u200b\u7684\u200b\u4fee\u6539\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5e8f\u5217\u200b\u672c\u8eab\u200b\u7684\u200b\u503c\u200b\u4e0d\u4f1a\u200b\u6539\u53d8\u200b\u3002</p> <pre><code>&gt;&gt;&gt; a = (1, 2, 3)\n&gt;&gt;&gt; a[2] += 5\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: 'tuple' object does not support item assignment\n&gt;&gt;&gt; a\n(1, 2, 3)\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u4f46\u200b\u5982\u679c\u200b\u8981\u200b\u4fee\u6539\u200b\u7684\u200b\u5143\u7d20\u200b\u672c\u8eab\u200b\u662f\u200b\u5f15\u7528\u200b\uff0c\u200b\u5982\u200b\u4e00\u4e2a\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\uff0c\u200b\u6b64\u65f6\u200b\u5bf9\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\u7684\u200b\u4fee\u6539\u200b\u4e0d\u4f1a\u200b\u5f71\u54cd\u200b\u5230\u200b\u5f15\u7528\u200b\u672c\u8eab\u200b\uff0c\u200b\u56e0\u6b64\u200b<code>+=</code>\u200b\u8fd0\u7b97\u200b\u4ecd\u4f1a\u200b\u6b63\u5e38\u200b\u4f5c\u7528\u200b\u3002\u200b\u589e\u91cf\u200b\u64cd\u4f5c\u200b\u4e0d\u662f\u200b\u539f\u5b50\u200b\u64cd\u4f5c\u200b\u3002</p> <pre><code>&gt;&gt;&gt; t = (1, 2, [30, 40])\n&gt;&gt;&gt; t[2] += [50, 60]\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: 'tuple' object does not support item assignment\n&gt;&gt;&gt; t\n(1, 2, [30, 40, 50, 60])\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b<code>append</code>\u200b\u6216\u200b<code>extend</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u907f\u514d\u200b\u5f02\u5e38\u200b\u7684\u200b\u4ea7\u751f\u200b\u3002\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\u88ab\u200b \u200b\u6b63\u5e38\u200b\u4fee\u6539\u200b \u3002\u200b\u4e3a\u200b\u4fdd\u8bc1\u6570\u636e\u200b\u4e0d\u200b\u88ab\u200b\u9519\u8bef\u200b\u5730\u200b\u4fee\u6539\u200b\uff0c\u200b\u5e94\u200b\u907f\u514d\u200b\u5728\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u5bb9\u7eb3\u200b\u53ef\u53d8\u200b\u5bf9\u8c61\u200b\u3002</p>"},{"location":"coding/python/sequence/#_6","title":"\u6392\u5e8f","text":"<p><code>list.sort</code>\u200b\u65b9\u6cd5\u200b\u4f5c\u7528\u200b\u4e0e\u200b\u5217\u8868\u200b\u672c\u8eab\u200b\u800c\u200b\u4e0d\u200b\u8fd4\u56de\u200b\uff08\u200b\u8fd4\u56de\u503c\u200b\u4e3a\u200b<code>None</code>\uff09\u3002\u200b\u8981\u200b\u4ea7\u751f\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u5df2\u200b\u6392\u5e8f\u200b\u5217\u8868\u200b\u800c\u200b\u4e0d\u200b\u5f71\u54cd\u200b\u539f\u200b\u5217\u8868\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>sorted()</code>\u200b\u5185\u7f6e\u200b\u51fd\u6570\u200b\u3002</p> <pre><code>&gt;&gt;&gt; a = [6, 3, 2, 1]\n&gt;&gt;&gt; b = sorted(a)\n&gt;&gt;&gt; b is a\nFalse\n&gt;&gt;&gt;\n</code></pre> <p><code>sorted</code>\u200b\u4e0e\u200b<code>list.sort</code>\u200b\u652f\u6301\u200b\u6309\u7167\u200b\u5176\u4ed6\u200b\u89c4\u5219\u200b\u5bf9\u200b\u5e8f\u5217\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b\uff0c\u200b\u5373\u200b<code>key</code>\u200b\u53c2\u6570\u200b\u3002<code>key</code>\u200b\u53c2\u6570\u200b\u4e3a\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u53c2\u6570\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5982\u200b<code>key=len</code>\u200b\u5373\u200b\u4e3a\u200b\u5bf9\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u5347\u5e8f\u200b\u6392\u5e8f\u200b\uff0c<code>key=lower</code>\u200b\u5373\u200b\u4e3a\u200b\u5ffd\u7565\u200b\u5b57\u7b26\u4e32\u200b\u5927\u5c0f\u5199\u200b\u8fdb\u884c\u200b\u5347\u5e8f\u200b\u6392\u5e8f\u200b\u3002</p> <p>Python\u200b\u7684\u200b<code>bisect</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u5bf9\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u7684\u200b\u652f\u6301\u200b\u3002<code>bisect.bisect</code>\u200b\u4e3a\u200b\u4e8c\u5206\u200b\u67e5\u627e\u200b\u51fd\u6570\u200b\uff0c\u200b\u8fd4\u56de\u200b\u5728\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5927\u4e8e\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u503c\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002<code>bisect.insort</code>\u200b\u4e3a\u200b\u63d2\u5165\u200b\u51fd\u6570\u200b\uff0c\u200b\u5728\u200b<code>bisect</code>\u200b\u51fd\u6570\u200b\u67e5\u8be2\u200b\u4f4d\u7f6e\u200b\u63d2\u5165\u200b\u5143\u7d20\u200b\u3002</p> <p><code>bisect</code>\u200b\u4e0e\u200b<code>insort</code>\u200b\u51fd\u6570\u200b\u6709\u200b\u53d8\u79cd\u200b<code>bisect_left</code>\u200b\u4e0e\u200b<code>insort_left</code>\u3002<code>bisect_left</code>\u200b\u8fd4\u56de\u200b\u5728\u200b\u76ee\u6807\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u4e0d\u200b\u5927\u4e8e\u200b\u5f85\u67e5\u200b\u5143\u7d20\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u503c\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002</p>"},{"location":"coding/python/sequence/#_7","title":"\u5176\u4ed6\u200b\u5e8f\u5217\u200b\u7c7b\u578b","text":""},{"location":"coding/python/sequence/#_8","title":"\u6570\u7ec4","text":"<p><code>array</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u6570\u7ec4\u200b\u7ed3\u6784\u200b\u7684\u200b\u652f\u6301\u200b\u3002\u200b\u6570\u7ec4\u200b\u662f\u200b\u53ea\u80fd\u200b\u5b58\u50a8\u200b\u4e00\u79cd\u200b\u7c7b\u578b\u200b\u5143\u7d20\u200b\u7684\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\uff0c\u200b\u652f\u6301\u200b<code>list</code>\u200b\u7684\u200b\u591a\u6570\u200b\u65b9\u6cd5\u200b\u3002\u200b\u540c\u65f6\u200b\u6570\u7ec4\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>fromfile, tofile</code>\u200b\u4e0e\u200b<code>frombytes, tobytes</code>\uff0c\u200b\u4ece\u800c\u200b\u652f\u6301\u200b\u76f4\u63a5\u200b\u8bfb\u5199\u200b\u6587\u4ef6\u200b\u6216\u200b\u6bd4\u7279\u6d41\u200b\u3002</p> <p>\u200b\u5728\u200b\u521b\u5efa\u200b\u6570\u7ec4\u200b\u65f6\u200b\u9700\u8981\u200b\u6307\u5b9a\u200b\u7c7b\u578b\u200b\u7801\u200b\uff0c\u200b\u5373\u200b\u6570\u7ec4\u200b\u4e2d\u200b\u5b58\u50a8\u200b\u7684\u200b\u6570\u636e\u7c7b\u578b\u200b\u3002\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u7c7b\u578b\u200b\u7801\u200b\u5bf9\u5e94\u200b\u7684\u200b\u7c7b\u578b\u200b\u53ea\u200b\u5360\u7528\u200b\u6709\u9650\u200b\u7684\u200b\u5185\u5b58\u7a7a\u95f4\u200b\u3002\u200b\u5728\u200b\u5b58\u50a8\u200b\u5927\u91cf\u200b\u6570\u636e\u200b\u65f6\u200b\u4f18\u4e8e\u200b\u5217\u8868\u200b\u7b49\u200b\u5176\u4ed6\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\u3002</p> \u200b\u7c7b\u578b\u200b\u7801\u200b C \u200b\u7c7b\u578b\u200b Python \u200b\u7c7b\u578b\u200b \u200b\u5b57\u200b\u5bbd\u200b <code>'b'</code> <code>signed char</code> <code>int</code> 1 <code>'B'</code> <code>unsigned char</code> <code>int</code> 1 <code>'u'</code> <code>wchar_t</code> Unicode character 2 <code>'h'</code> <code>signed short</code> <code>int</code> 2 <code>'H'</code> <code>unsigned short</code> <code>int</code> 2 <code>'i'</code> <code>signed int</code> <code>int</code> 2 <code>'I'</code> <code>unsigned int</code> <code>int</code> 2 <code>'l'</code> <code>signed long</code> <code>int</code> 4 <code>'L'</code> <code>unsigned long</code> <code>int</code> 4 <code>'q'</code> <code>signed long long</code> <code>int</code> 8 <code>'Q'</code> <code>unsigned long long</code> <code>int</code> 8 <code>'f'</code> <code>float</code> <code>float</code> 4 <code>'d'</code> <code>double</code> <code>float</code> 8 <p>\u200b\u6570\u7ec4\u200b\u4e0d\u200b\u652f\u6301\u200b\u5217\u8868\u200b\u7684\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li><code>clear</code></li> <li><code>copy</code></li> <li><code>sort</code></li> </ul> <p>\u200b\u4f46\u200b\u6570\u7ec4\u200b\u652f\u6301\u200b\u4ece\u200b\u5217\u8868\u200b\u6dfb\u52a0\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5373\u200b<code>fromlist</code>\u200b\u65b9\u6cd5\u200b\u3002\u200b\u5f53\u200b\u5217\u8868\u200b\u4e2d\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u5728\u200b\u6dfb\u52a0\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u51fa\u9519\u200b\uff0c\u200b\u5219\u200b\u6240\u6709\u200b\u7684\u200b\u6dfb\u52a0\u200b\u90fd\u200b\u4f1a\u200b\u88ab\u200b\u56de\u6eda\u200b\u3002</p> <pre><code>&gt;&gt;&gt; import array\n&gt;&gt;&gt; a = array.array('f')\n&gt;&gt;&gt; b = [1.0, 2.0, 'a']\n&gt;&gt;&gt; a\narray('f')\n&gt;&gt;&gt; a.fromlist(b)\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: must be real number, not str\n&gt;&gt;&gt; a\narray('f')\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u6570\u7ec4\u200b\u662f\u200bbytes-like object\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u5185\u5b58\u200b\u89c6\u56fe\u200b\u4e2d\u200b\u3002</p>"},{"location":"coding/python/sequence/#_9","title":"\u5185\u5b58\u200b\u89c6\u56fe","text":"<p><code>memoryview</code>\u200b\u5141\u8bb8\u200b\u4f7f\u7528\u200b\u4e0d\u540c\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8bfb\u5199\u200b\u540c\u4e00\u200b\u5757\u5185\u5b58\u200b\u7684\u200b\u6570\u636e\u200b\u3002</p> <p>\u200b\u5982\u4e0b\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u6620\u5c04\u200b\u5230\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\u7684\u200b<code>memoryview</code>\uff0c\u200b\u5728\u200b\u4e0d\u200b\u6267\u884c\u200b\u64cd\u4f5c\u200b\u65f6\u200b\uff0c<code>memory</code>\u200b\u7684\u200b\u6570\u636e\u200b\u5185\u5bb9\u200b\u4e0e\u200b\u539f\u200b\u5e8f\u5217\u200b\u7684\u200b\u6570\u636e\u200b\u5185\u5bb9\u200b\u76f8\u540c\u200b\u3002</p> <pre><code>&gt;&gt;&gt; a = [1, 2, 3, 4]\n&gt;&gt;&gt; b = memoryview(array.array('i', a))\n&gt;&gt;&gt; b[0]\n1\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>memoryview.cast</code>\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u5c06\u200b<code>memoryview</code>\u200b\u6620\u5c04\u200b\u7684\u200b\u5185\u5b58\u200b\u533a\u57df\u200b\u4ee5\u200b\u4e0d\u540c\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u89e3\u91ca\u200b\u3002</p> <pre><code>&gt;&gt;&gt; c = b.cast('B')\n&gt;&gt;&gt; list(c)\n[1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0]\n&gt;&gt;&gt;\n</code></pre> <p><code>memoryview</code>\u200b\u7684\u200b\u53ef\u53d8\u6027\u200b\u53d6\u51b3\u4e8e\u200b\u5176\u6240\u200b\u6620\u5c04\u200b\u5185\u5b58\u200b\u533a\u57df\u200b\u7684\u200b\u6027\u8d28\u200b\uff0c\u200b\u82e5\u200b\u4f7f\u7528\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\u521b\u5efa\u200b<code>memoryview</code>\uff0c\u200b\u5219\u200b<code>memoryview</code>\u200b\u4e3a\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5e8f\u5217\u200b\u3002</p> <pre><code>&gt;&gt;&gt; d = memoryview(b\"abc\") \n&gt;&gt;&gt; d[0] = \"d\"\nTraceback (most recent call last):\n  File \"&lt;stdin&gt;\", line 1, in &lt;module&gt;\nTypeError: cannot modify read-only memory\n&gt;&gt;&gt;\n</code></pre> <p>\u200b\u4f5c\u4e3a\u200b\u53ef\u53d8\u200b\u5e8f\u5217\u200b\u7684\u200b<code>memoryview</code>\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u4fee\u6539\u200b\uff0c\u200b\u518d\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u539f\u6765\u200b\u7684\u200b\u89e3\u91ca\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u5373\u53ef\u200b\u5b9e\u73b0\u200b\u5bf9\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u67d0\u4e2a\u200b\u7279\u5b9a\u200b\u5b57\u8282\u200b\u7684\u200b\u4fee\u6539\u200b\u3002</p> <pre><code>&gt;&gt;&gt; c[1] = 1 \n&gt;&gt;&gt; b = c.cast(\"i\")\n&gt;&gt;&gt; list(b)\n[257, 2, 3, 4]\n&gt;&gt;&gt;\n</code></pre> <p><code>memoryview</code>\u200b\u7684\u200b\u5207\u7247\u200b\u662f\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b<code>memoryview</code>\u200b\u5bf9\u8c61\u200b\uff08\u200b\u57fa\u4e8e\u200b\u539f\u200b\u5b57\u8282\u200b\u5e8f\u5217\u200b\uff09</p>"},{"location":"coding/python/sequence/#_10","title":"\u961f\u5217","text":"<p>\u200b\u5217\u8868\u200b\u7684\u200b<code>append</code>\u200b\u4e0e\u200b<code>pop</code>\u200b\u65b9\u6cd5\u200b\u7ec4\u5408\u200b\u4f7f\u7528\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u5bf9\u6808\u200b\u6216\u200b\u961f\u5217\u200b\u7684\u200b\u6a21\u62df\u200b\u3002<code>collection.deque</code>\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u961f\u5217\u200b\u7684\u200b\u66f4\u200b\u9ad8\u6548\u200b\u7684\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u5e76\u4e14\u200b\u6ee1\u8db3\u200b\u7ebf\u7a0b\u200b\u5b89\u5168\u200b\u7684\u200b\u6761\u4ef6\u200b\u3002</p> <p>\u200b\u5728\u200b\u521b\u5efa\u200b\u961f\u5217\u200b\u65f6\u200b\uff0c\u200b\u53ef\u200b\u9009\u200b\u63d0\u4f9b\u200b\u961f\u5217\u200b\u7684\u200b\u6700\u5927\u200b\u957f\u5ea6\u200b<code>maxlen</code>\u3002\u200b\u8be5\u200b\u53c2\u6570\u200b\u53ea\u80fd\u200b\u5728\u200b\u6784\u9020\u51fd\u6570\u200b\u4e2d\u200b\u6307\u5b9a\u200b\uff0c\u200b\u5728\u200b\u5bf9\u8c61\u200b\u4e2d\u200b\u53ea\u8bfb\u200b\u3002</p> <p>\u200b\u961f\u5217\u200b\u63d0\u4f9b\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li><code>rotate</code>\uff0c\u200b\u5f53\u200b\u53c2\u6570\u200b<code>n &gt; 0</code>\u200b\u65f6\u200b\u5c06\u200b\u961f\u5217\u200b\u53f3\u4fa7\u200b\u7684\u200b<code>n</code>\u200b\u4e2a\u200b\u5143\u7d20\u200b\u79fb\u52a8\u200b\u5230\u200b\u961f\u5217\u200b\u5de6\u4fa7\u200b\uff0c\u200b\u53cd\u4e4b\u200b\u5c06\u200b\u961f\u5217\u200b\u5de6\u4fa7\u200b\u7684\u200b<code>n</code>\u200b\u4e2a\u200b\u5143\u7d20\u200b\u79fb\u52a8\u200b\u5230\u200b\u961f\u5217\u200b\u53f3\u4fa7\u200b</li> <li><code>appendleft</code>\uff0c\u200b\u5728\u200b\u961f\u5217\u200b\u7684\u200b\u5de6\u4fa7\u200b\u6dfb\u52a0\u200b\u5143\u7d20\u200b\uff08\u200b\u5f53\u200b\u8d85\u51fa\u200b\u961f\u5217\u200b\u957f\u5ea6\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u4ece\u200b\u961f\u5217\u200b\u53e6\u4e00\u4fa7\u200b\u5220\u9664\u200b\uff09</li> <li><code>extendleft</code>\uff0c\u200b\u5728\u200b\u961f\u5217\u200b\u7684\u200b\u5de6\u4fa7\u200b\u6dfb\u52a0\u200b\u4e00\u7ec4\u200b\u5143\u7d20\u200b\uff08\u200b\u5f53\u200b\u8d85\u51fa\u200b\u961f\u5217\u200b\u957f\u5ea6\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u4ece\u200b\u961f\u5217\u200b\u53e6\u4e00\u4fa7\u200b\u5220\u9664\u200b\uff09</li> <li>\u200b\u961f\u5217\u200b\u652f\u6301\u200b\u5217\u8868\u200b\u7684\u200b\u5927\u591a\u6570\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u9664\u200b<code>copy</code>\u3001<code>sort</code>\u3001<code>insert</code>\u3001<code>index</code>\u200b\u7b49\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u961f\u5217\u200b\u4ec5\u200b\u652f\u6301\u200b<code>+=</code>\u200b\u8fd0\u7b97\u7b26\u200b\u7684\u200b\u4f5c\u7528\u200b\u3002</li> </ul>"},{"location":"coding/reinforcement-learning/","title":"\u5f3a\u5316\u200b\u5b66\u4e60","text":"<p>\u200b\u8bb0\u5f55\u200b\u5f3a\u5316\u200b\u5b66\u4e60\u200b\u7684\u200b\u5b66\u4e60\u200b\u8fc7\u7a0b\u200b\u3002</p> <ul> <li>\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b</li> <li>\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u7b97\u6cd5\u200b</li> </ul>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/","title":"\u591a\u81c2\u200b\u8001\u864e\u673a","text":"In\u00a0[1]: Copied! <pre>import itertools\nimport random\nfrom typing import List, Optional\n\nfrom scipy.stats import norm\n\n\nclass MultiArmedBandit():\n    def __init__(self, means, stds: Optional[List[float]] = None):\n        self.means = means\n        if stds is not None:\n            if len(stds) != len(means):\n                raise ValueError(\n                    'The length of means and stds should be the same.')\n            self.stds = stds\n        else:\n            self.stds = [1.0] * len(means)\n        self.k = len(means)\n        self.dists = [\n            norm(loc=mean, scale=std) for mean, std in zip(means, self.stds)\n        ]\n\n    def __len__(self) -&gt; int:\n        return self.k\n\n    @property\n    def optimal_reward(self) -&gt; float:\n        return max(self.means)\n\n    def pull(self, action: int) -&gt; float:\n        return self.dists[action].rvs()\n\nrewards = [*range(0, 10, 2)]\nstds = [*range(1, 6)]\nargs = itertools.product(rewards, stds)\nbandit = MultiArmedBandit(*zip(*args))\nprint(bandit.pull(0))\n</pre> import itertools import random from typing import List, Optional  from scipy.stats import norm   class MultiArmedBandit():     def __init__(self, means, stds: Optional[List[float]] = None):         self.means = means         if stds is not None:             if len(stds) != len(means):                 raise ValueError(                     'The length of means and stds should be the same.')             self.stds = stds         else:             self.stds = [1.0] * len(means)         self.k = len(means)         self.dists = [             norm(loc=mean, scale=std) for mean, std in zip(means, self.stds)         ]      def __len__(self) -&gt; int:         return self.k      @property     def optimal_reward(self) -&gt; float:         return max(self.means)      def pull(self, action: int) -&gt; float:         return self.dists[action].rvs()  rewards = [*range(0, 10, 2)] stds = [*range(1, 6)] args = itertools.product(rewards, stds) bandit = MultiArmedBandit(*zip(*args)) print(bandit.pull(0)) <pre>-1.435877863923347\n</pre> In\u00a0[2]: Copied! <pre>import abc\nfrom typing import Tuple\n\nclass AbstractAgent(abc.ABC):\n    def __init__(self, T: int, k: int):\n        self.T = T\n        self.k = k\n\n        self.history: List[Tuple[int, float]] = []\n\n    @abc.abstractmethod\n    def strategy(self) -&gt; int:\n        pass\n\n    def run(self, bandit: MultiArmedBandit) -&gt; float:\n        rewards = []\n        for t in range(self.T):\n            action = self.strategy()\n            if action &gt;= self.k:\n                raise ValueError('Invalid action.')\n\n            reward = bandit.pull(action)\n            self.history.append((action, reward))\n            rewards.append(reward)\n        return self.T * bandit.optimal_reward - sum(rewards)\n</pre> import abc from typing import Tuple  class AbstractAgent(abc.ABC):     def __init__(self, T: int, k: int):         self.T = T         self.k = k          self.history: List[Tuple[int, float]] = []      @abc.abstractmethod     def strategy(self) -&gt; int:         pass      def run(self, bandit: MultiArmedBandit) -&gt; float:         rewards = []         for t in range(self.T):             action = self.strategy()             if action &gt;= self.k:                 raise ValueError('Invalid action.')              reward = bandit.pull(action)             self.history.append((action, reward))             rewards.append(reward)         return self.T * bandit.optimal_reward - sum(rewards) In\u00a0[3]: Copied! <pre>import numpy as np\n\ndef show_result(x: List[float], name: str) -&gt; str:\n    x_array = np.array(x)\n    return f'{name} - Mean: {x_array.mean():.2f} Std: {x_array.std():.2f}'\n</pre> import numpy as np  def show_result(x: List[float], name: str) -&gt; str:     x_array = np.array(x)     return f'{name} - Mean: {x_array.mean():.2f} Std: {x_array.std():.2f}' In\u00a0[4]: Copied! <pre>class RandomAgent(AbstractAgent):\n    def strategy(self) -&gt; int:\n        return random.randint(0, self.k - 1)\n\nrandom_agents = [RandomAgent(T=100, k=len(bandit)) for _ in range(1000)]\nregrets = [agent.run(bandit) for agent in random_agents]\nshow_result(regrets, 'Random Agent')\n</pre> class RandomAgent(AbstractAgent):     def strategy(self) -&gt; int:         return random.randint(0, self.k - 1)  random_agents = [RandomAgent(T=100, k=len(bandit)) for _ in range(1000)] regrets = [agent.run(bandit) for agent in random_agents] show_result(regrets, 'Random Agent') Out[4]: <pre>'Random Agent - Mean: 399.66 Std: 44.19'</pre> In\u00a0[5]: Copied! <pre>from typing import Dict\n\nclass EpsilonGreedyAgent(AbstractAgent):\n    def __init__(self, T: int, k: int, epsilon: float):\n        super().__init__(T, k)\n        self.epsilon = epsilon\n\n    def _random(self) -&gt; int:\n        return random.randint(0, self.k - 1)\n\n    def _greedy(self) -&gt; int:\n        history_dict: Dict[int, List[float]] = {}\n        for action, reward in self.history:\n            if action not in history_dict:\n                history_dict[action] = []\n            history_dict[action].append(reward)\n        return max(\n            history_dict,\n            key=lambda x: sum(history_dict[x]) / len(history_dict[x])\n        )\n\n    def strategy(self) -&gt; int:\n        if random.random() &lt; self.epsilon or not self.history:\n            return self._random()\n        return self._greedy()\n\n\nepsilon_greedy = [\n    EpsilonGreedyAgent(T=100, k=len(bandit), epsilon=0.2) for _ in range(1000)\n]\nregrets = [agent.run(bandit) for agent in epsilon_greedy]\nshow_result(regrets, 'Epsilon-Greedy Agent')\n</pre> from typing import Dict  class EpsilonGreedyAgent(AbstractAgent):     def __init__(self, T: int, k: int, epsilon: float):         super().__init__(T, k)         self.epsilon = epsilon      def _random(self) -&gt; int:         return random.randint(0, self.k - 1)      def _greedy(self) -&gt; int:         history_dict: Dict[int, List[float]] = {}         for action, reward in self.history:             if action not in history_dict:                 history_dict[action] = []             history_dict[action].append(reward)         return max(             history_dict,             key=lambda x: sum(history_dict[x]) / len(history_dict[x])         )      def strategy(self) -&gt; int:         if random.random() &lt; self.epsilon or not self.history:             return self._random()         return self._greedy()   epsilon_greedy = [     EpsilonGreedyAgent(T=100, k=len(bandit), epsilon=0.2) for _ in range(1000) ] regrets = [agent.run(bandit) for agent in epsilon_greedy] show_result(regrets, 'Epsilon-Greedy Agent') Out[5]: <pre>'Epsilon-Greedy Agent - Mean: 155.49 Std: 79.32'</pre> In\u00a0[6]: Copied! <pre>import math\n\nclass UCBAgent(AbstractAgent):\n    def __init__(self, T: int, k: int, c: float):\n        super().__init__(T, k)\n        self.c = c\n\n    def ucb(self, action: int) -&gt; float:\n        n = len([a for a, _ in self.history if a == action])\n        if n == 0:\n            return float('inf')\n        mean = sum(r for a, r in self.history if a == action) / n\n        std = math.sqrt(math.log(len(self.history)) / n)\n        return mean + self.c * std\n\n    def strategy(self) -&gt; int:\n        # If there are still unselected actions, select one of them.\n        if len(self.history) &lt; self.k:\n            return len(self.history)\n\n        return max(range(self.k), key=self.ucb)\n\nucb_agents = [UCBAgent(T=100, k=len(bandit), c=0.3) for _ in range(1000)]\nregrets = [agent.run(bandit) for agent in ucb_agents]\nshow_result(regrets, 'UCB Agent')\n</pre> import math  class UCBAgent(AbstractAgent):     def __init__(self, T: int, k: int, c: float):         super().__init__(T, k)         self.c = c      def ucb(self, action: int) -&gt; float:         n = len([a for a, _ in self.history if a == action])         if n == 0:             return float('inf')         mean = sum(r for a, r in self.history if a == action) / n         std = math.sqrt(math.log(len(self.history)) / n)         return mean + self.c * std      def strategy(self) -&gt; int:         # If there are still unselected actions, select one of them.         if len(self.history) &lt; self.k:             return len(self.history)          return max(range(self.k), key=self.ucb)  ucb_agents = [UCBAgent(T=100, k=len(bandit), c=0.3) for _ in range(1000)] regrets = [agent.run(bandit) for agent in ucb_agents] show_result(regrets, 'UCB Agent') Out[6]: <pre>'UCB Agent - Mean: 116.76 Std: 29.87'</pre> In\u00a0[7]: Copied! <pre>from scipy.stats import invgamma\n\nclass ThompsonSamplingAgent(AbstractAgent):\n    def __init__(self, T: int, k: int):\n        super().__init__(T, k)\n\n        # We know that the reward is gaussian, so we can use normal-inverse-gamma\n        self.mu = [0.0] * k\n        self.lambd = [1.0] * k\n        self.alpha = [1.0] * k\n        self.beta = [1.0] * k\n\n    def update(self, action: int, reward: float) -&gt; None:\n        self.mu[action] = (\n            self.lambd[action] * self.mu[action] + reward\n        ) / (self.lambd[action] + 1)\n        self.lambd[action] += 1\n        self.alpha[action] += 0.5\n        self.beta[action] += (\n            self.lambd[action] * (reward - self.mu[action]) ** 2\n        ) / (self.lambd[action] + 1) / 2\n\n    def strategy(self) -&gt; int:\n        if self.history:\n            self.update(*self.history[-1])\n\n        samples = [\n            norm.rvs(\n                loc=mu, scale=invgamma.rvs(a, scale=b) / math.sqrt(lambd)\n            )\n            for mu, lambd, a, b in zip(self.mu, self.lambd, self.alpha, self.beta)\n        ]\n        return max(range(self.k), key=lambda x: samples[x])\n\nthompson_sampling_agents = [\n    ThompsonSamplingAgent(T=100, k=len(bandit)) for _ in range(1000)\n]\nregrets = [agent.run(bandit) for agent in thompson_sampling_agents]\nshow_result(regrets, 'Thompson Sampling Agent')\n</pre> from scipy.stats import invgamma  class ThompsonSamplingAgent(AbstractAgent):     def __init__(self, T: int, k: int):         super().__init__(T, k)          # We know that the reward is gaussian, so we can use normal-inverse-gamma         self.mu = [0.0] * k         self.lambd = [1.0] * k         self.alpha = [1.0] * k         self.beta = [1.0] * k      def update(self, action: int, reward: float) -&gt; None:         self.mu[action] = (             self.lambd[action] * self.mu[action] + reward         ) / (self.lambd[action] + 1)         self.lambd[action] += 1         self.alpha[action] += 0.5         self.beta[action] += (             self.lambd[action] * (reward - self.mu[action]) ** 2         ) / (self.lambd[action] + 1) / 2      def strategy(self) -&gt; int:         if self.history:             self.update(*self.history[-1])          samples = [             norm.rvs(                 loc=mu, scale=invgamma.rvs(a, scale=b) / math.sqrt(lambd)             )             for mu, lambd, a, b in zip(self.mu, self.lambd, self.alpha, self.beta)         ]         return max(range(self.k), key=lambda x: samples[x])  thompson_sampling_agents = [     ThompsonSamplingAgent(T=100, k=len(bandit)) for _ in range(1000) ] regrets = [agent.run(bandit) for agent in thompson_sampling_agents] show_result(regrets, 'Thompson Sampling Agent') Out[7]: <pre>'Thompson Sampling Agent - Mean: 231.88 Std: 49.94'</pre>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/","title":"\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u00b6","text":"<p>\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\uff08Multi-armed bandit\uff09\u200b\u95ee\u9898\u200b\u6765\u81ea\u200b\u4e8e\u200b\u73b0\u5b9e\u200b\u4e2d\u200b\u7684\u200b\u8001\u864e\u673a\u200b\u95ee\u9898\u200b\u3002\u200b\u6bcf\u53f0\u200b\u8001\u864e\u673a\u200b\u90fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u62c9\u6746\u200b\uff0c\u200b\u62c9\u52a8\u200b\u62c9\u6746\u200b\u540e\u200b\uff0c\u200b\u8001\u864e\u200b\u673a\u4f1a\u200b\u968f\u673a\u200b\u7ed9\u51fa\u200b\u4e00\u4e2a\u200b\u5956\u52b1\u200b\u3002\u200b\u73a9\u5bb6\u200b\u7684\u200b\u4efb\u52a1\u200b\u662f\u200b\u5728\u200b\u6709\u9650\u200b\u7684\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u5185\u200b\u5c3d\u53ef\u80fd\u200b\u83b7\u5f97\u200b\u66f4\u200b\u591a\u200b\u7684\u200b\u5956\u52b1\u200b\u3002</p> <p>\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u95ee\u9898\u200b\u7684\u200b\u6570\u5b66\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\uff1a\u200b\u5df2\u77e5\u200b\u4e00\u7ec4\u200b$K\\in \\mathbb N^{+}$\u200b\u4e2a\u200b\u5b9a\u4e49\u200b\u5728\u200b$\\mathbb R$\u200b\u4e0a\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b$\\mathcal B = \\{D_1, \\ldots, D_K\\}$\uff0c\u200b\u5176\u4e2d\u200b$D_k$\u200b\u8868\u793a\u200b\u7b2c\u200b$k$\u200b\u4e2a\u200b\u8001\u864e\u673a\u200b\u7ed9\u51fa\u200b\u5956\u52b1\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b\uff0c$D_k$\u200b\u7684\u200b\u5747\u503c\u200b\u4e3a\u200b$\\mu_k$\u3002\u200b\u5728\u200b\u7b2c\u200b$t$\u200b\u8f6e\u4e2d\u200b\uff0c\u200b\u667a\u80fd\u200b\u4f53\u4ece\u200b$\\mathcal B$\u200b\u4e2d\u200b\u968f\u673a\u200b\u9009\u51fa\u200b\u4e00\u4e2a\u200b\u5206\u5e03\u200b$D^{(t)}\\in\\mathcal B$\uff0c\u200b\u5e76\u200b\u4ece\u4e2d\u200b\u91c7\u6837\u200b\u4e00\u4e2a\u200b\u5956\u52b1\u200b$X^{(t)}\\sim D^{(t)}$\u3002\u200b\u667a\u80fd\u200b\u4f53\u200b\u7684\u200b\u76ee\u6807\u200b\u662f\u200b\u5728\u200b\u6709\u9650\u200b\u7684\u200b\u8f6e\u6b21\u200b\u5185\u200b\u6700\u5927\u5316\u200b\u7d2f\u79ef\u200b\u5956\u52b1\u200b$\\sum_{t=1}^T X^{(t)}$\u3002\u200b\u540e\u6094\u200b\u503c\u200b$\\rho$\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\u6700\u4f18\u200b\u7b56\u7565\u200b\u7684\u200b$T$\u200b\u8f6e\u200b\u7d2f\u8ba1\u200b\u5956\u52b1\u200b\u7684\u200b\u671f\u671b\u503c\u200b\u4e0e\u200b\u667a\u80fd\u200b\u4f53\u200b\u7684\u200b$T$\u200b\u8f6e\u200b\u7d2f\u8ba1\u200b\u5956\u52b1\u200b\u4e4b\u5dee\u200b\uff0c\u200b\u5373\u200b</p> <p>$$ \\rho = T\\max\\{\\mu_k\\} - \\mathbb E\\left[\\sum_{t=1}^T X^{(t)}\\right] $$</p> <p>\u200b\u7528\u200bPython\u200b\u53ef\u4ee5\u200b\u7b80\u5355\u200b\u5b9e\u73b0\u200b\u4e00\u4e2a\u200b\u4e0e\u200b\u667a\u80fd\u200b\u4f53\u200b\u4ea4\u4e92\u200b\u7684\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u73af\u5883\u200b\uff1a</p>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/","title":"\u7b97\u6cd5\u200b\u00b6","text":"<p>\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u7b97\u6cd5\u200b\u7684\u200b\u6838\u5fc3\u200b\u662f\u200b\u5728\u200b\u63a2\u7d22\u200b\uff08Exploration\uff09\u200b\u548c\u200b\u5229\u7528\u200b\uff08Exploitation\uff09\u200b\u4e4b\u95f4\u200b\u5bfb\u627e\u200b\u5e73\u8861\u200b\u3002\u200b\u63a2\u7d22\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u627e\u5230\u200b\uff08\u200b\u76f8\u8f83\u200b\u4e8e\u200b\u5f53\u524d\u200b\u5df2\u77e5\u200b\u7684\u200b\uff09\u200b\u66f4\u597d\u200b\u7684\u200b\u5956\u52b1\u200b\uff0c\u200b\u800c\u200b\u5229\u7528\u200b\u5219\u200b\u662f\u200b\u76f4\u63a5\u200b\u9009\u62e9\u200b\u5f53\u524d\u200b\u5df2\u77e5\u200b\u7684\u200b\u6700\u597d\u200b\u7684\u200b\u5956\u52b1\u200b\u3002</p>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/","title":"\u5b8c\u5168\u200b\u968f\u673a\u200b\u7b56\u7565\u200b\u00b6","text":"<p>\u200b\u5728\u200b\u5b8c\u5168\u200b\u968f\u673a\u200b\u7b56\u7565\u200b\u4e0b\u200b\uff0c\u200b\u667a\u80fd\u200b\u4f53\u200b\u968f\u673a\u200b\u9009\u62e9\u200b\u4e00\u4e2a\u200b\u81c2\u200b\uff0c\u200b\u83b7\u53d6\u200b\u5956\u52b1\u200b\uff0c\u200b\u540e\u6094\u200b\u503c\u200b\u7684\u200b\u671f\u671b\u200b\u4e3a\u200b$\\mathbb E\\rho = \\sum_{k=1}^K (\\mu^* - \\mu_k)$\u3002</p>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/#epsilon-","title":"$\\epsilon$-\u200b\u8d2a\u5fc3\u200b\u7b97\u6cd5\u200b\u00b6","text":"<p>$\\epsilon$-\u200b\u8d2a\u5fc3\u200b\u7b97\u6cd5\u200b\u662f\u200b\u6700\u200b\u7b80\u5355\u200b\u7684\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u7b97\u6cd5\u200b\u4e4b\u4e00\u200b\u3002\u200b\u5728\u200b\u6bcf\u200b\u4e00\u8f6e\u200b\u4e2d\u200b\uff0c\u200b\u4ee5\u200b$\\epsilon$\u200b\u7684\u200b\u6982\u7387\u200b\u968f\u673a\u200b\u9009\u62e9\u200b\u4e00\u4e2a\u200b\u8001\u864e\u673a\u200b\uff0c\u200b\u4ee5\u200b$1-\\epsilon$\u200b\u7684\u200b\u6982\u7387\u200b\u9009\u62e9\u200b\u5f53\u524d\u200b\u5df2\u77e5\u200b\u7684\u200b\u6700\u597d\u200b\u7684\u200b\u8001\u864e\u673a\u200b\u3002</p>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/","title":"\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u7b97\u6cd5\u200b\u00b6","text":"<p>\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u7b97\u6cd5\u200b\uff08Upper Confidence Bound, UCB\uff09\u200b\u662f\u200b\u4e00\u79cd\u200b\u57fa\u4e8e\u200b\u7f6e\u4fe1\u533a\u95f4\u200b\u7684\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u7b97\u6cd5\u200b\u3002\u200b\u5728\u200b\u6bcf\u200b\u4e00\u8f6e\u200b\u4e2d\u200b\uff0c\u200b\u667a\u80fd\u200b\u4f53\u200b\u9009\u62e9\u200b\u4e00\u4e2a\u200b\u8001\u864e\u673a\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u8be5\u200b\u8001\u864e\u673a\u200b\u7684\u200b\u7f6e\u4fe1\u533a\u95f4\u200b\u4e0a\u200b\u754c\u200b\u6700\u5927\u200b\u3002\u200b\u7f6e\u4fe1\u533a\u95f4\u200b\u4e0a\u754c\u200b\u7684\u200b\u8ba1\u7b97\u516c\u5f0f\u200b\u4e3a\u200b</p> <p>$$ \\bar X_k + c\\cdot \\sqrt{\\frac{\\log t}{N_k}} $$</p> <p>\u200b\u5176\u4e2d\u200b$\\bar X_k$\u200b\u4e3a\u200b\u7b2c\u200b$k$\u200b\u4e2a\u200b\u8001\u864e\u673a\u200b\u7684\u200b\u5e73\u5747\u200b\u5956\u52b1\u200b\uff0c$N_k$\u200b\u4e3a\u200b\u7b2c\u200b$k$\u200b\u4e2a\u200b\u8001\u864e\u673a\u200b\u88ab\u200b\u9009\u62e9\u200b\u7684\u200b\u6b21\u6570\u200b\uff0c$c$\u200b\u4e3a\u200b\u4e00\u4e2a\u200b\u5e38\u6570\u200b\uff0c$t$\u200b\u4e3a\u200b\u5f53\u524d\u200b\u8f6e\u6b21\u200b\u3002\u200b\u5176\u200b\u601d\u60f3\u200b\u5728\u4e8e\u200b\uff0c\u200b\u667a\u80fd\u200b\u4f53\u4f1a\u200b\u9996\u5148\u200b\u904d\u5386\u200b\u6240\u6709\u200b\u81c2\u200b\uff0c\u200b\u4e4b\u540e\u200b\u4f18\u5148\u9009\u62e9\u200b\u90a3\u4e9b\u200b\u88ab\u200b\u9009\u62e9\u200b\u6b21\u6570\u200b\u8f83\u5c11\u200b\u7684\u200b\u8001\u864e\u673a\u200b\uff0c\u200b\u4ee5\u4fbf\u200b\u66f4\u597d\u200b\u5730\u200b\u4f30\u8ba1\u200b\u5176\u200b\u5956\u52b1\u200b\u3002</p>"},{"location":"coding/reinforcement-learning/multi-armed-bandit/#thompson","title":"Thompson\u200b\u91c7\u6837\u200b\u00b6","text":"<p>Thompson\u200b\u91c7\u6837\u200b\u662f\u200b\u4e00\u79cd\u200b\u57fa\u4e8e\u200b\u8d1d\u53f6\u65af\u200b\u63a8\u65ad\u200b\u7684\u200b\u591a\u81c2\u200b\u8001\u864e\u673a\u200b\u7b97\u6cd5\u200b\uff0c\u200b\u9700\u8981\u200b\u5df2\u77e5\u200b\u6bcf\u4e2a\u200b\u8001\u864e\u673a\u200b\u7684\u200b\u5956\u52b1\u200b\u670d\u4ece\u200b\u67d0\u79cd\u200b\u5206\u5e03\u200b\u3002\u200b\u5728\u200b\u6bcf\u200b\u4e00\u8f6e\u200b\u4e2d\u200b\uff0c\u200b\u667a\u80fd\u200b\u4f53\u4ece\u200b\u6bcf\u4e2a\u200b\u8001\u864e\u673a\u200b\u7684\u200b\u5956\u52b1\u200b\u5206\u5e03\u200b\u4e2d\u200b\u91c7\u6837\u200b\u4e00\u4e2a\u200b\u5956\u52b1\u200b\uff0c\u200b\u9009\u62e9\u200b\u5956\u52b1\u200b\u6700\u5927\u200b\u7684\u200b\u8001\u864e\u673a\u200b\u3002\u200b\u6839\u636e\u200b\u73af\u5883\u200b\u7684\u200b\u53cd\u9988\u200b\uff0c\u200b\u66f4\u65b0\u200b\u6bcf\u4e2a\u200b\u8001\u864e\u673a\u200b\u7684\u200b\u540e\u9a8c\u200b\u5956\u52b1\u200b\u5206\u5e03\u200b\u3002</p>"},{"location":"coding/reinforcement-learning/upper-confidence-bound/","title":"\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u7b97\u6cd5","text":"<p>\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\uff08Upper Confidence Bound, UCB\uff09\u200b\u7b97\u6cd5\u200b\u7684\u200b\u6838\u5fc3\u601d\u60f3\u200b\u662f\u200b\u5c06\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u7eb3\u5165\u200b\u8003\u8651\u200b\uff0c\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u8f83\u5c11\u200b\u7684\u200b\u52a8\u4f5c\u200b\u6709\u200b\u66f4\u200b\u5927\u200b\u7684\u200b\u4e0d\u786e\u5b9a\u6027\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u4f1a\u200b\u5c3d\u53ef\u80fd\u200b\u4e50\u89c2\u200b\u5730\u200b\u8003\u8651\u200b\u8fd9\u4e9b\u200b\u6ca1\u6709\u200b\u5c1d\u8bd5\u200b\u8fc7\u200b\u7684\u200b\u52a8\u4f5c\u200b\uff0c\u200b\u5e76\u200b\u4f18\u5148\u200b\u63a2\u7d22\u200b\u8fd9\u4e9b\u200b\u52a8\u4f5c\u200b\u3002\u200b\u5728\u200b\u6bcf\u4e2a\u200b\u52a8\u4f5c\u200b\u63a2\u7d22\u200b\u6bd4\u8f83\u200b\u6df1\u5165\u200b\u540e\u200b\uff0c\u200b\u518d\u200b\u8f6c\u5411\u200b\u5229\u7528\u200b\u5956\u52b1\u200b\u8f83\u200b\u9ad8\u200b\u7684\u200b\u52a8\u4f5c\u200b\u3002</p> <p>\u200b\u5f15\u7406\u200b</p> <p>\u200b\u8bbe\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X_1, \\ldots, X_n\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u5747\u503c\u200b\u4e3a\u200b\\(0\\)\uff0c\u200b\u5bf9\u200b\u5747\u503c\u200b\u7684\u200b\u4f30\u8ba1\u200b\u4e3a\u200b</p> \\[ \\hat \\mu = \\frac{1}{n}\\sum_{i=1}^n X_i \\] <p>\u200b\u5219\u200b\u5bf9\u200b\u4efb\u610f\u200b\\(\\delta &gt; 0\\)\uff0c\u200b\u6709\u200b</p> \\[ P(\\hat\\mu &gt; \\delta) \\leq \\exp(-n\\delta^2 / 2) \\] <p>\u200b\u8be5\u200b\u5f15\u7406\u200b\u79f0\u4e3a\u200bHoeffding\u200b\u4e0d\u7b49\u5f0f\u200b\u3002</p> <p>\u200b\u4ee4\u200b\\(\\varepsilon = \\exp(-2n\\delta^2)\\)\uff0c\u200b\u5373\u200b\\(\\delta = \\sqrt{\\frac{-2\\log\\varepsilon}{n}}\\)</p> \\[ P\\left(\\hat\\mu &gt; \\sqrt{\\frac{-2\\log\\varepsilon}{n}}\\right) \\leq \\varepsilon \\] <p>\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u52a8\u4f5c\u200b\\(k\\)\uff0c\u200b\u6211\u4eec\u200b\u5c1d\u8bd5\u200b\u4e86\u200b\\(n_k\\)\u200b\u6b21\u200b\uff0c\u200b\u6bcf\u6b21\u200b\u5c1d\u8bd5\u200b\u7684\u200b\u5956\u52b1\u200b\u4e3a\u200b\\(X_1, \\ldots, X_{n_k}\\)\uff0c\u200b\u89c2\u6d4b\u200b\u5230\u200b\u7684\u200b\u5e73\u5747\u200b\u5956\u52b1\u200b\u4e3a\u200b\\(\\hat\\mu_k\\)\uff0c\u200b\u5b9a\u4e49\u200b\\(\\tilde X_i = X_i - \\hat\\mu_k\\)\uff0c\u200b\u5219\u200b\\(\\tilde X_1, \\ldots, \\tilde X_{n_k}\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u5747\u503c\u200b\u4e3a\u200b\\(0\\)\u3002</p> <p>\u200b\u4ee4\u200b\u5bf9\u200b\\(\\tilde X_1, \\ldots, \\tilde X_{n_k}\\)\u200b\u4f30\u8ba1\u200b\u7684\u200b\u5747\u503c\u200b\u4e3a\u200b\\(\\hat\\mu_0\\)\uff0c\u200b\u6211\u4eec\u200b\u6709\u200b\\(1 - \\varepsilon\\)\u200b\u7684\u200b\u6982\u7387\u200b\u4fdd\u8bc1\u200b\\(\\hat\\mu_0\\)\u200b\u4e0d\u200b\u8d85\u8fc7\u200b\\(\\sqrt{\\frac{-2\\log\\varepsilon}{n_k}}\\)\u3002</p> \\[ P\\left(\\hat\\mu_0 &gt; \\sqrt{\\frac{-2\\log\\varepsilon}{n_k}}\\right) \\leq \\varepsilon \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u6211\u4eec\u200b\u53d6\u200b\\(\\hat\\mu_k + \\sqrt{\\frac{-2\\log\\varepsilon}{n_k}}\\)\u200b\u4f5c\u4e3a\u200b\u52a8\u4f5c\u200b\\(k\\)\u200b\u7684\u200b\u7f6e\u4fe1\u200b\u4e0a\u200b\u754c\u200b\u3002\u200b\u901a\u8fc7\u200b\u8c03\u6574\u200b\\(\\varepsilon\\)\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u63a7\u5236\u200b\u7f6e\u4fe1\u200b\u4e0a\u754c\u200b\u7684\u200b\u5bbd\u5ea6\u200b\u4ee5\u200b\u5728\u200b\u63a2\u7d22\u200b\u548c\u200b\u5229\u7528\u200b\u4e4b\u524d\u200b\u53d6\u5f97\u200b\u5e73\u8861\u200b\u3002</p> <p>\u200b\u901a\u5e38\u200b\u91c7\u53d6\u200b\u7684\u200b\u7b56\u7565\u200b\u662f\u200b\u53d6\u200b\\(\\varepsilon = 1/t\\)\uff0c\\(t\\)\u200b\u4e3a\u200b\u603b\u200b\u8f6e\u6b21\u200b\uff0c\u200b\u4ee5\u200b\u4fdd\u8bc1\u200b\u968f\u7740\u200b\u5c1d\u8bd5\u200b\u6b21\u6570\u200b\u7684\u200b\u589e\u52a0\u200b\uff0c\u200b\u7f6e\u4fe1\u200b\u4e0a\u754c\u200b\u7684\u200b\u5bbd\u5ea6\u200b\u9010\u6e10\u200b\u51cf\u5c0f\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u7b97\u6cd5\u200b\u9010\u6b65\u200b\u8f6c\u5411\u200b\u5229\u7528\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b</p> \\[ \\hat\\mu_k + \\sqrt{\\frac{2\\log t}{n_k}} \\] <p>\u200b\u6b64\u5916\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5176\u4ed6\u200b\u7684\u200b\u4e0a\u754c\u200b\u5bbd\u5ea6\u200b\uff0c\u200b\u5982\u200b</p> \\[ \\varepsilon = \\frac{1}{1 + t\\log^2 t} \\] <p>\u200b\u5728\u5b9e\u8df5\u4e2d\u200b\uff0c\u200b\u901a\u5e38\u200b\u8fd8\u4f1a\u200b\u5f15\u5165\u200b\u4e00\u4e2a\u200b\u53c2\u6570\u200b\\(c\\)\u200b\u63a7\u5236\u200b\u5bbd\u5ea6\u200b\uff0c\u200b\u5373\u200b</p> \\[ \\hat \\mu_k + c\\cdot\\sqrt{\\frac{\\log t}{n_k}} \\]"},{"location":"coding/rust/","title":"Rust","text":"<p>\u200b\u8bb0\u5f55\u200bRust\u200b\u7684\u200b\u5b66\u4e60\u200b\u3002</p> <ul> <li>Introduction</li> <li>Variables</li> <li>Control Flows</li> <li>Complex Objects</li> <li>Lifetimes</li> <li>Error Handling</li> <li>Supplements</li> </ul>"},{"location":"coding/rust/0/","title":"Introduction","text":"<p>Rust\u200b\u7b80\u4ecb\u200b</p> <p>Rust\u200b\u6709\u200b\u5982\u4e0b\u200b\u4e8c\u8fdb\u5236\u200b\u6587\u4ef6\u200b</p> <ul> <li><code>rustc</code>\uff1aRust\u200b\u7f16\u8bd1\u5668\u200b</li> <li><code>cargo</code>\uff1aRust\u200b\u5305\u200b\u7ba1\u7406\u5668\u200b\u548c\u200b\u6784\u5efa\u200b\u5de5\u5177\u200b</li> <li><code>rustup</code>\uff1aRust\u200b\u7248\u672c\u200b\u7ba1\u7406\u5668\u200b</li> <li><code>rustdoc</code>\uff1aRust\u200b\u6587\u6863\u200b</li> <li><code>rls</code>\uff1aRust language server</li> <li><code>rustfmt</code>\uff1aRust\u200b\u4ee3\u7801\u200b\u683c\u5f0f\u5316\u200b\u5de5\u5177\u200b</li> </ul>"},{"location":"coding/rust/0/#hello-world","title":"Hello, world!","text":"<p>  Download source code</p> <pre><code>fn main() {\n    println!(\"Hello, world!\");\n}\n</code></pre> <p><code>println!</code>\u200b\u4e2d\u200b\u7684\u200b<code>!</code>\u200b\u8868\u793a\u200b\u8fd9\u662f\u200b\u4e00\u4e2a\u200b\u5b8f\u200b\uff0c\u200b\u800c\u200b\u4e0d\u662f\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u3002\u200b\u5b8f\u662f\u200b\u4e00\u79cd\u200b\u5728\u200b\u7f16\u8bd1\u200b\u65f6\u200b\u6267\u884c\u200b\u7684\u200b\u4ee3\u7801\u200b\u3002<code>println</code>\u200b\u6709\u200bPython\u200b\u4e2d\u200b<code>str.format</code>\u200b\u7c7b\u4f3c\u200b\u7684\u200b\u529f\u80fd\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6267\u884c\u200b\u5b57\u7b26\u4e32\u200b\u683c\u5f0f\u5316\u200b\u3002<code>print!</code>\u200b\u4e0e\u200b<code>println!</code>\u200b\u7c7b\u4f3c\u200b\uff0c\u200b\u4f46\u662f\u200b\u4e0d\u4f1a\u200b\u5728\u200b\u7ed3\u5c3e\u200b\u6dfb\u52a0\u200b\u6362\u884c\u7b26\u200b\u3002</p> <pre><code>println!(\"{} days\", 31);\nprintln!(\"{0}, this is {1}. {1}, this is {0}\", \"Alice\", \"Bob\");\nprintln!(\"{greeting}, {name}!\", greeting=\"Hello\", name=\"world\");\n</code></pre>"},{"location":"coding/rust/0/#cargo","title":"Cargo","text":"<p>Cargo\u200b\u662f\u200bRust\u200b\u7684\u200b\u5305\u200b\u7ba1\u7406\u5668\u200b\u548c\u200b\u6784\u5efa\u200b\u5de5\u5177\u200b\u3002Cargo\u200b\u7684\u200b\u5de5\u4f5c\u200b\u65b9\u5f0f\u200b\u4e0e\u200bPython\u200b\u7684\u200b<code>pip</code>\u200b\u548c\u200b<code>setuptools</code>\u200b\u7c7b\u4f3c\u200b\u3002</p> <pre><code>cargo new # \u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200bRust\u200b\u9879\u76ee\u200b\ncargo init # \uff08\u200b\u5728\u200b\u5df2\u7ecf\u200b\u5b58\u5728\u200b\u7684\u200b\u76ee\u5f55\u200b\u4e2d\u200b\uff09\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200bRust\u200b\u9879\u76ee\u200b\ncargo build # \u200b\u7f16\u8bd1\u200b\u9879\u76ee\u200b\ncargo run # \u200b\u7f16\u8bd1\u200b\u5e76\u200b\u8fd0\u884c\u200b\u9879\u76ee\u200b\ncargo update # \u200b\u66f4\u65b0\u200bCargo.lock\u200b\u6587\u4ef6\u200b\u4e2d\u200b\u7684\u200b\u4f9d\u8d56\u200b\ncargo doc # \u200b\u751f\u6210\u200b\u9879\u76ee\u200b\u6587\u6863\u200b\ncargo test # \u200b\u8fd0\u884c\u200b\u9879\u76ee\u200b\u6d4b\u8bd5\u200b\n\ncargo install ...\ncargo uninstall ...\n</code></pre> <p>Rust\u200b\u7684\u200b\u4ed3\u5e93\u200b\u662f\u200b<code>crates.io</code>\uff0c\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bPython\u200b\u7684\u200b<code>PyPI</code>\u3002</p>"},{"location":"coding/rust/0/#_1","title":"\u5305\u4e0e\u200b\u6a21\u5757","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u5305\u200b\u79f0\u4e3a\u200bcrate\u3002crate\u200b\u7684\u200b\u6839\u76ee\u5f55\u200b\u4e0b\u200b\u6709\u200b\u4e00\u4e2a\u200b<code>Cargo.toml</code>\u200b\u6587\u4ef6\u200b\uff0c\u200b\u5176\u4e2d\u200b\u5305\u542b\u200bcrate\u200b\u7684\u200b\u5143\u200b\u6570\u636e\u200b\u548c\u200b\u4f9d\u8d56\u200b\u3002Rust\u200b\u9879\u76ee\u200b\u7684\u200b\u6587\u4ef6\u200b\u7ed3\u6784\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>project\n\u251c\u2500\u2500 Cargo.toml\n\u251c\u2500\u2500 Cargo.lock\n\u251c\u2500\u2500 tests\n\u2502   \u2514\u2500\u2500 ...\n\u251c\u2500\u2500 benches\n\u2502   \u2514\u2500\u2500 ...\n\u251c\u2500\u2500 examples\n\u2502   \u2514\u2500\u2500 ...\n\u2514\u2500\u2500 src\n    \u251c\u2500\u2500 bin     // \u200b\u5176\u4ed6\u200b\u7684\u200b\u4e8c\u8fdb\u5236\u200b\u7a0b\u5e8f\u200b\n    \u251c\u2500\u2500 main.rs // \u200b\u9ed8\u8ba4\u200b\u7684\u200b\u4e8c\u8fdb\u5236\u200b\u7a0b\u5e8f\u200b\n    \u2514\u2500\u2500 lib.rs  // \u200b\u5e93\u5305\u200b\uff08\u200b\u53ea\u80fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\uff09\n</code></pre> <p>Rust\u200b\u4e2d\u200b\u7684\u200b\u6a21\u5757\u200b\u7528\u200b<code>mod</code>\u200b\u5173\u952e\u5b57\u200b\u58f0\u660e\u200b\uff0c\u200b\u6a21\u5757\u200b\u53ef\u4ee5\u200b\u5d4c\u5957\u200b\u3002\u200b\u7528\u200b<code>use</code>\u200b\u5f15\u7528\u200b\u6a21\u5757\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>crate</code>\u200b\u5f00\u59cb\u200b\u7684\u200b\u7edd\u5bf9\u200b\u5f15\u7528\u200b\u8def\u5f84\u200b\u548c\u200b<code>self</code>\uff08\u200b\u672c\u200b\u6a21\u5757\u200b\uff09\u200b\u6216\u200b<code>super</code>\uff08\u200b\u7236\u200b\u6a21\u5757\u200b\uff09\u200b\u5f00\u59cb\u200b\u7684\u200b\u76f8\u5bf9\u200b\u5f15\u7528\u200b\u8def\u5f84\u200b\uff0c\u200b\u6216\u8005\u200b\u7528\u200b<code>*</code>\u200b\u5f15\u5165\u200b\u6a21\u5757\u200b\u7684\u200b\u6240\u6709\u200b\u5185\u5bb9\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u6a21\u5757\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u9879\u200b\u90fd\u200b\u662f\u200b\u79c1\u6709\u200b\u7684\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u5916\u754c\u200b\u8bbf\u95ee\u200b\uff0c\u200b\u9664\u975e\u200b\u7528\u200b<code>pub</code>\u200b\u5173\u952e\u5b57\u200b\u58f0\u660e\u200b\u4e3a\u200b\u516c\u6709\u200b\u3002\u200b\u516c\u6709\u200b\u6a21\u5757\u200b\u5185\u200b\u6240\u6709\u200b\u7684\u200b\u9879\u200b\u5fc5\u987b\u200b\u90fd\u200b\u7528\u200b<code>pub</code>\u200b\u58f0\u660e\u200b\u4e3a\u200b\u516c\u6709\u200b\u3002<code>pub</code>\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>(in &lt;module path&gt;)</code>\u200b\u8bed\u6cd5\u200b\u58f0\u660e\u200b\u5bf9\u8c61\u200b\u4ec5\u200b\u5728\u200b<code>&lt;module path&gt;</code>\u200b\u4e2d\u200b\u53ef\u89c1\u200b\u3002</p> <p>\u200b\u6bcf\u4e2a\u200b<code>.rs</code>\u200b\u6587\u4ef6\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u89c6\u4e3a\u200b\u4e00\u4e2a\u200b\u4ee5\u200b\u6587\u4ef6\u540d\u200b\u547d\u540d\u200b\u7684\u200b\u6a21\u5757\u200b\u3002\u200b\u5982\u679c\u200b\u8981\u200b\u628a\u200b\u6587\u4ef6\u5939\u200b\u4f5c\u4e3a\u200b\u6a21\u5757\u200b\uff0c\u200b\u5219\u200b\u5fc5\u987b\u200b\u5728\u200b\u6587\u4ef6\u5939\u200b\u4e2d\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b<code>mod.rs</code>\u200b\u6587\u4ef6\u200b\u6216\u200b\u4e0e\u200b\u6587\u4ef6\u5939\u200b\u540c\u540d\u200b\u7684\u200b<code>.rs</code>\u200b\u6587\u4ef6\u200b\uff0c\u200b\u5176\u4e2d\u200b\u58f0\u660e\u200b\u5bf9\u5916\u5f00\u653e\u200b\u7684\u200b\u9879\u200b\u3002</p> <p>\u200b\u5f53\u200b\u591a\u4e2a\u200b\u6a21\u5757\u200b\u4e2d\u200b\u5b58\u5728\u200b\u76f8\u540c\u200b\u540d\u79f0\u200b\u7684\u200b\u9879\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b<code>\u200b\u6a21\u5757\u200b\u540d\u200b::\u200b\u9879\u540d\u200b</code>\u200b\u6765\u200b\u6307\u5b9a\u200b\u5177\u4f53\u200b\u7684\u200b\u9879\u200b\uff0c\u200b\u6216\u8005\u200b\u7528\u200b<code>as</code>\u200b\u58f0\u660e\u200b\u522b\u540d\u200b\u3002</p>"},{"location":"coding/rust/0/#_2","title":"\u6ce8\u91ca\u200b\u4e0e\u200b\u6587\u6863","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u6ce8\u91ca\u200b\u4e0e\u200bC/C++\u200b\u7c7b\u4f3c\u200b\uff0c\u200b\u7528\u200b<code>//</code>\u200b\u8868\u793a\u200b\u5355\u884c\u200b\u6ce8\u91ca\u200b\uff0c\u200b\u7528\u200b<code>/* */</code>\u200b\u8868\u793a\u200b\u591a\u884c\u200b\u6ce8\u91ca\u200b\u3002\u200b\u5176\u4e2d\u200b\uff0c\u200b\u5e38\u7528\u200b\u7684\u200b\u6ce8\u91ca\u200b\u65b9\u6cd5\u200b\u4e3a\u200b\uff1a</p> <ol> <li>\u200b\u5305\u200b\u6ce8\u91ca\u200b<code>//! ...</code>\u200b\u4e0e\u200b<code>/*! ... */</code>\uff1a()\u200b\u7528\u4e8e\u200b\u6574\u4e2a\u200b\u5305\u200b\u7684\u200b\u6ce8\u91ca\u200b\uff08Inner comments\uff09</li> <li>\u200b\u6587\u6863\u200b\u6ce8\u91ca\u200b<code>/// ...</code>\u200b\u4e0e\u200b<code>/** ... */</code>\uff1a\u200b\u7528\u4e8e\u200b\u6a21\u5757\u200b\u3001\u200b\u51fd\u6570\u200b\u3001\u200b\u7ed3\u6784\u200b\u4f53\u200b\u3001\u200b\u679a\u4e3e\u200b\u3001\u200b\u65b9\u6cd5\u200b\u548c\u200btrait\u200b\u7684\u200b\u6ce8\u91ca\u200b\uff08Outer comments\uff09</li> <li>\u200b\u901a\u5e38\u200b\u4e0d\u7528\u200b<code>/* */</code>\u200b\u6ce8\u91ca\u200b</li> </ol> <p>\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200bdoc attribute\u200b\u6765\u200b\u6ce8\u91ca\u200b\uff0c\u200b\u5176\u200b\u8bed\u6cd5\u200b\u4e3a\u200b<code>#![doc=...]</code>\u200b\u548c\u200b<code>#[doc=...]</code>\uff0c\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200bOuter comments\u200b\u548c\u200bInner comments\u3002</p> <p>Rust\u200b\u652f\u6301\u200b\u5728\u200b\u6587\u6863\u200b\u4e2d\u200b\u8fdb\u884c\u200b\u6d4b\u8bd5\u200b\uff0c\u200b\u88ab\u200b\u6d4b\u8bd5\u200b\u7684\u200b\u51fd\u6570\u200b\u5fc5\u987b\u200b\u662f\u200b\u516c\u6709\u200b\u7684\u200b\u3002\u200b\u4ee5\u200b<code>#</code>\u200b\u5f00\u59cb\u200b\u7684\u200b\u884c\u200b\u8868\u793a\u200b\u8be5\u884c\u200b\u4e0d\u4f1a\u200b\u5728\u200b\u6587\u6863\u200b\u4e2d\u200b\u663e\u793a\u200b\uff0c\u200b\u53ea\u4f1a\u200b\u7528\u4e8e\u200b\u6d4b\u8bd5\u200b\u3002</p> <pre><code>// Assume the name of crate is `playground`\n\n/// ```rust\n/// assert_eq!(playground::add(1, 2), 3);\n/// ```\npub fn add(a: i32, b: i32) -&gt; i32 {\n    a + b\n}\n/// ```rust, should_panic\n/// // This function calls [`panic!`] macro.\n/// playground::panic_fn();\n/// ```\npub fn panic_fn() {\n    panic!();\n}\n</code></pre>"},{"location":"coding/rust/1/","title":"Variables","text":""},{"location":"coding/rust/1/#_1","title":"\u58f0\u660e","text":"<p>Rust\u200b\u4e2d\u200b\u7528\u4e8e\u200b\u58f0\u660e\u200b\u53d8\u91cf\u200b\u7684\u200b\u5173\u952e\u5b57\u200b\u662f\u200b<code>let</code>\uff0c\u200b\u5b83\u200b\u7684\u200b\u8bed\u6cd5\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>let [mut] &lt;variable&gt;[: &lt;type&gt;] [= &lt;value&gt;];\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff0c\u200b\u7c7b\u578b\u200b\u548c\u200b\u521d\u59cb\u5316\u200b\u7684\u200b\u503c\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u3002\u200b\u5982\u679c\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\u4ece\u200b\u58f0\u660e\u200b\u5f00\u59cb\u200b\u6ca1\u6709\u200b\u88ab\u200b\u8d4b\u503c\u200b\uff0c\u200b\u5c31\u200b\u4e0d\u80fd\u200b\u4f7f\u7528\u200b\u8fd9\u4e2a\u200b\u53d8\u91cf\u200b\u3002\u200b\u5982\u679c\u200b\u6ca1\u6709\u200b\u6307\u5b9a\u200b\u7c7b\u578b\u200b\u4f46\u200b\u6307\u5b9a\u200b\u4e86\u200b\u521d\u59cb\u5316\u200b\u503c\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u5219\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u63a8\u65ad\u200b\u53d8\u91cf\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002Rust\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>{}</code>\u200b\u6765\u200b\u7528\u200b\u4e00\u7cfb\u5217\u200b\u8bed\u53e5\u200b\u6765\u200b\u521d\u59cb\u5316\u200b\u53d8\u91cf\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>()</code>\u200b\u6765\u200b\u4e00\u6b21\u6027\u200b\u58f0\u660e\u200b\u591a\u4e2a\u200b\u53d8\u91cf\u200b\u3002\u200b\u6ca1\u6709\u200b\u7528\u200b<code>mut</code>\u200b\u58f0\u660e\u200b\u7684\u200b\u53d8\u91cf\u200b\u662f\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u7684\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u4fee\u6539\u200b\u3002</p> <p>Rust\u200b\u4e2d\u200b\u7528\u4e8e\u200b\u58f0\u660e\u200b\u5e38\u91cf\u200b\u7684\u200b\u5173\u952e\u5b57\u200b\u662f\u200b<code>const</code>\uff0c\u200b\u7528\u4e8e\u200b\u58f0\u660e\u200b\u9759\u6001\u200b\u53d8\u91cf\u200b\u7684\u200b\u5173\u952e\u5b57\u200b\u662f\u200b<code>static</code>\u3002\u200b\u8bed\u6cd5\u200b\u4e0e\u200b<code>let</code>\u200b\u76f8\u4f3c\u200b\uff0c\u200b\u4f46\u200b\u9700\u8981\u200b\u540c\u65f6\u200b\u6307\u5b9a\u200b\u7c7b\u578b\u200b\u548c\u200b\u521d\u59cb\u5316\u200b\u503c\u200b\u3002\u200b\u901a\u5e38\u200b<code>const</code>\u200b\u548c\u200b<code>static</code>\u200b\u8bed\u53e5\u200b\u90fd\u200b\u662f\u200b\u5728\u200b\u5168\u5c40\u200b\u4f5c\u7528\u57df\u200b\uff08\u200b\u5373\u200b\u51fd\u6570\u200b\u5916\u200b\uff09\u200b\u4e2d\u200b\u58f0\u660e\u200b\u7684\u200b\u3002</p> <p><code>let</code>\u200b\u53ef\u4ee5\u200b\u7528\u6765\u200b\u91cd\u65b0\u200b\u58f0\u660e\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\uff0c\u200b\u9690\u85cf\u200b\u4e4b\u524d\u200b\u7684\u200b\u53d8\u91cf\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>mut</code>\u200b\u6765\u200b\u6539\u53d8\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\u7684\u200b\u53ef\u53d8\u6027\u200b\u3002</p> <p>  Download source code</p> <pre><code>fn main() {\n    let x: f64 = -20.48; // float\n    let x: i64 = x.floor() as i64; // int\n    println!(\"{}\", x); // -21\n\n    let s: &amp;str = \"hello\"; // &amp;str\n    let s: String = s.to_uppercase(); // String\n    println!(\"{}\", s) // HELLO\n}\n</code></pre>"},{"location":"coding/rust/1/#_2","title":"\u53d8\u91cf\u200b\u7c7b\u578b","text":"<p>Rust\u200b\u7684\u200b\u53d8\u91cf\u200b\u7c7b\u578b\u200b\u6709\u200b</p> <ul> <li>\u200b\u4e00\u822c\u200b\u7c7b\u578b\u200b<ul> <li>\u200b\u5e03\u5c14\u200b\uff1a<code>bool</code>\uff0c\u200b\u53d6\u503c\u200b\u4e3a\u200b<code>true</code>\u200b\u6216\u200b<code>false</code>\uff0c\u200b\u5360\u7528\u200b1\u200b\u4e2a\u200b\u5b57\u8282\u200b\u3002</li> <li>\u200b\u5b57\u7b26\u200b\uff1a<code>char</code>\uff0c\u200b\u4f7f\u7528\u200b\u5355\u5f15\u53f7\u200b\uff0c\u200b\u5360\u7528\u200b4\u200b\u4e2a\u200b\u5b57\u8282\u200b\u3002</li> <li>\u200b\u6574\u6570\u200b\uff1a<code>i8</code>\uff5e<code>i128</code>\u200b\u548c\u200b<code>isize</code>\uff0c\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\([-2^{n-1}, 2^{n-1} - 1]\\)\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>min_value()</code>\u200b\u548c\u200b<code>max_value()</code>\u200b\u51fd\u6570\u200b\u6765\u200b\u83b7\u53d6\u200b\u7c7b\u578b\u200b\u7684\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u3002<code>isize</code>\u200b\u7684\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u548c\u200b\u7cfb\u7edf\u200b\u6307\u9488\u200b\u957f\u5ea6\u200b\u76f8\u540c\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u6574\u6570\u200b\u7c7b\u578b\u200b\u662f\u200b<code>i32</code>\u3002</li> <li>\u200b\u65e0\u200b\u7b26\u53f7\u200b\u6574\u6570\u200b\uff1a<code>u8</code>\uff5e<code>u128</code>\u200b\u548c\u200b<code>usize</code>\uff0c\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\([0, 2^n - 1]\\)\u3002</li> <li>\u200b\u6d6e\u70b9\u6570\u200b\uff1a<code>f32</code>\u200b\u548c\u200b<code>f64</code>\uff0c\u200b\u5206\u522b\u200b\u4e3a\u200b\u5355\u7cbe\u5ea6\u200b\u548c\u200b\u53cc\u200b\u7cbe\u5ea6\u200b\u6d6e\u70b9\u6570\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u6d6e\u70b9\u6570\u200b\u7c7b\u578b\u200b\u662f\u200b<code>f64</code>\u3002</li> <li>\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\uff1a<code>str</code>\uff0c\u200b\u4f7f\u7528\u200b\u53cc\u5f15\u53f7\u200b\uff0c\u200b\u662f\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u7c7b\u578b\u200b\u3002\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\uff08\u200b\u53d8\u91cf\u200b\uff09\u200b\u7684\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>&amp;str</code>\uff0c\u200b\u662f\u200b\u5f15\u7528\u200b\u7c7b\u578b\u200b\u3002</li> <li>\u200b\u5b57\u7b26\u4e32\u200b\uff1a<code>String</code>\uff0c\u200b\u662f\u200b\u53ef\u53d8\u200b\u7c7b\u578b\u200b\uff0c\u200b\u4f46\u200b\u4e0d\u200b\u652f\u6301\u200b\u7d22\u5f15\u200b\u3002\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5f15\u7528\u200b\u6216\u200b\u5207\u7247\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>&amp;str</code>\u3002</li> <li>\u200b\u5355\u5143\u200b\u7c7b\u578b\u200b\uff1a<code>()</code>\uff0c\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u503c\u200b<code>()</code>\uff0c\u200b\u6ca1\u6709\u200b<code>return</code>\u200b\u8bed\u53e5\u200b\u7684\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>()</code>\uff0c\u200b\u4e0d\u200b\u5360\u7528\u200b\u5185\u5b58\u200b\u3002</li> </ul> </li> <li>\u200b\u5bb9\u5668\u200b\u53ca\u200b\u76f8\u5173\u200b\u7c7b\u578b\u200b<ul> <li>\u200b\u4f7f\u7528\u200b<code>println!</code>\u200b\u6253\u5370\u200b\u5bb9\u5668\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b<code>{:?}</code>\u200b\u6216\u200b<code>{:#?}</code>\uff0c\u200b\u524d\u8005\u200b\u4e3a\u200b\u4e00\u884c\u200b\u8f93\u51fa\u200b\uff0c\u200b\u540e\u8005\u200b\u5728\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u540e\u9762\u200b\u6dfb\u52a0\u200b\u6362\u884c\u7b26\u200b\u3002</li> <li>\u200b\u5b9a\u200b\u957f\u200b\u6570\u7ec4\u200b\uff1a<code>[T; N]</code>\uff0c\u200b\u5176\u4e2d\u200b<code>T</code>\u200b\u4e3a\u200b\u6570\u7ec4\u200b\u5143\u7d20\u200b\u7c7b\u578b\u200b\u6216\u200b\u91cd\u590d\u200b\u7684\u200b\u6570\u7ec4\u200b\uff0c<code>N</code>\u200b\u4e3a\u200b\u6570\u7ec4\u200b\u957f\u5ea6\u200b\uff0c\u200b\u5fc5\u987b\u200b\u662f\u200b\u5e38\u91cf\u200b\u3002\u200b\u53ea\u6709\u200b\u7528\u200b<code>mut</code>\u200b\u58f0\u660e\u200b\u7684\u200b\u6570\u7ec4\u200b\u624d\u80fd\u200b\u4fee\u6539\u200b\u5176\u4e2d\u200b\u5143\u7d20\u200b\u3002<ul> <li>\u200b\u6ce8\u610f\u200b\u4e0d\u200b\u652f\u6301\u200b<code>Copy</code> trait\u200b\u7684\u200b\u7c7b\u578b\u200b\u4e0d\u80fd\u200b\u7528\u200b<code>[T; N]</code>\u200b\u8bed\u6cd5\u200b\uff0c\u200b\u5fc5\u987b\u200b\u9010\u4e2a\u200b\u8d4b\u503c\u200b\u3002</li> <li>\u200b\u4e0d\u540c\u200b\u957f\u5ea6\u200b\u7684\u200b\u6570\u7ec4\u200b\u662f\u200b\u4e0d\u540c\u200b\u7684\u200b\u7c7b\u578b\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u76f8\u4e92\u200b\u8d4b\u503c\u200b\u3002</li> </ul> </li> <li>\u200b\u5b9a\u200b\u957f\u200b\u5143\u7ec4\u200b\uff1a<code>(T1, T2, ..., Tn)</code>\uff0c\u200b\u5176\u4e2d\u200b<code>T1</code>\uff5e<code>Tn</code>\u200b\u4e3a\u200b\u5143\u7ec4\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u53ea\u6709\u200b\u7528\u200b<code>mut</code>\u200b\u58f0\u660e\u200b\u7684\u200b\u5143\u7ec4\u200b\u624d\u80fd\u200b\u4fee\u6539\u200b\u5176\u4e2d\u200b\u5143\u7d20\u200b\uff08\u200b\u7c7b\u578b\u200b\u5fc5\u987b\u200b\u5339\u914d\u200b\uff09\u3002\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bPython\uff0c\u200b\u5728\u200b\u53d8\u91cf\u200b\u58f0\u660e\u200b\u9636\u6bb5\u200b\u6216\u200b\u8d4b\u503c\u200b\u9636\u6bb5\u200b\uff0c\u200b\u4ec5\u200b\u5305\u542b\u200b\uff08\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u8d4b\u503c\u200b\u7684\u200b\uff09\u200b\u53d8\u91cf\u200b\u5143\u7ec4\u200b\u53ef\u4ee5\u200b\u4f5c\u4e3a\u200b\u8d4b\u503c\u200b\u8fd0\u7b97\u200b\u7684\u200b\u5de6\u7aef\u200b\u9879\u200b\uff0c\u200b\u6b64\u65f6\u200b\u4f1a\u200b\u5206\u522b\u200b\u4e3a\u200b\u6bcf\u4e2a\u200b\u53d8\u91cf\u200b\u5206\u914d\u200b\u5bf9\u5e94\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u7701\u7565\u200b\u7684\u200b\u53d8\u91cf\u200b\u7528\u200b<code>_</code>\u200b\u8868\u793a\u200b\u3002\u200b\u5143\u7ec4\u200b\u652f\u6301\u200b\u5d4c\u5957\u200b\u3002\u200b\u5355\u200b\u5143\u7d20\u200b\u5143\u7ec4\u200b\u7684\u200b\u8bed\u6cd5\u200b\u4e3a\u200b<code>(&lt;value&gt;,)</code>\u3002</li> <li>\u200b\u6570\u7ec4\u200b\u5207\u7247\u200b\uff1a<code>&amp;&lt;parent&gt;[&lt;start&gt;..&lt;end&gt;]</code>\uff0c\u200b\u6ce8\u610f\u200b\u5207\u7247\u200b\u662f\u200b\u5bf9\u200b<code>&lt;parent&gt;</code>\u200b\u7684\u200b\u5f15\u7528\u200b\uff0c\u200b\u662f\u200b\u5f15\u7528\u200b\u7c7b\u578b\u200b\u3002\u200b\u5143\u7ec4\u200b\u4e0d\u200b\u652f\u6301\u200b\u5207\u7247\u200b\uff0c\u200b\u53ea\u80fd\u200b\u7528\u200b<code>.0</code>\u3001<code>.1</code>\u200b\u7b49\u200b\u6765\u200b\u8bbf\u95ee\u200b\u5143\u7d20\u200b\uff0c\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>[T]</code>\u3002</li> </ul> </li> <li>\u200b\u8fed\u4ee3\u200b\u5668\u200b\uff0c\u200b\u901a\u5e38\u200b\u7531\u200b\u5bb9\u5668\u200b\u5bf9\u8c61\u200b\u7684\u200b<code>iter()</code>\u200b\u65b9\u6cd5\u200b\u751f\u6210\u200b\u3002</li> <li>\u200b\u51fd\u6570\u200b\uff1a\u200b\u9700\u8981\u200b\u7528\u200b<code>fn(arg1: type1, ...) -&gt; return_type</code>\u200b\u8868\u793a\u200b\uff0c\u200b\u51fd\u6570\u200b\u4ee5\u200b\u5f15\u7528\u200b\u7684\u200b\u65b9\u5f0f\u200b\u4f20\u9012\u200b\u3002</li> </ul>"},{"location":"coding/rust/1/#_3","title":"\u6574\u6570\u200b\u7c7b\u578b","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u6574\u6570\u200b\u5b57\u9762\u200b\u91cf\u200b\u6709\u200b\u5982\u4e0b\u200b\u5199\u6cd5\u200b\uff1a</p> <ul> <li>\u200b\u5341\u8fdb\u5236\u200b\uff1a<code>123_456</code></li> <li>\u200b\u5341\u516d\u8fdb\u5236\u200b\uff1a<code>0x123</code>\uff0c<code>0x</code>\u200b\u5f00\u5934\u200b\uff0c\u200b\u6570\u5b57\u200b\u4e3a\u200b<code>0-9</code>\u200b\u548c\u200b<code>a-f</code></li> <li>\u200b\u516b\u8fdb\u5236\u200b\uff1a<code>0o123</code>\uff0c<code>0o</code>\u200b\u5f00\u5934\u200b\uff0c\u200b\u6570\u5b57\u200b\u4e3a\u200b<code>0-7</code></li> <li>\u200b\u4e8c\u8fdb\u5236\u200b\uff1a<code>0b101</code>\uff0c<code>0b</code>\u200b\u5f00\u5934\u200b\uff0c\u200b\u6570\u5b57\u200b\u4e3a\u200b<code>0</code>\u200b\u548c\u200b<code>1</code></li> <li>\u200b\u5b57\u8282\u200b\uff1a<code>b'A'</code></li> <li>\u200b\u4e0b\u5212\u7ebf\u200b<code>_</code>\u200b\u7c7b\u4f3c\u200b\u4e8e\u200b\u5343\u4f4d\u200b\u5206\u9694\u7b26\u200b\uff0c\u200b\u5b58\u5728\u200b\u4e0e\u5426\u200b\u4e0d\u200b\u5f71\u54cd\u200b\u6570\u5b57\u200b\u7684\u200b\u503c\u200b</li> </ul> <p>\u200b\u6574\u6570\u200b\u6ea2\u51fa\u200b</p> <p>\u200b\u5728\u200bDebug\u200b\u6a21\u5f0f\u200b\u4e0b\u200b\uff0cRust\u200b\u4f1a\u200b\u68c0\u67e5\u200b\u6574\u6570\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u5982\u679c\u200b\u53d1\u751f\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u7a0b\u5e8f\u200b\u4f1a\u200bpanic\u3002\u200b\u5728\u200bRelease\u200b\u6a21\u5f0f\u200b\u4e0b\u200b\uff0cRust\u200b\u4e0d\u4f1a\u200b\u68c0\u67e5\u200b\u6574\u6570\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u6ea2\u51fa\u200b\u540e\u200b\u4f1a\u200b\u6309\u7167\u200b\u4e8c\u8fdb\u5236\u200b\u8865\u7801\u200b\u8fdb\u884c\u200b\u5faa\u73af\u200b\u3002</p> <p>\u200b\u6574\u6570\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li>\u200b\u4e0b\u6587\u200b\u4e2d\u200b\u7684\u200b<code>*</code>\u200b\u6307\u200b<code>add</code>\u3001<code>sub</code>\u3001<code>mul</code>\u3001<code>div</code>\u200b\u7b49\u200b\u8fd0\u7b97\u200b</li> <li><code>wrapping_*</code>\uff1a\u200b\u5982\u679c\u200b\u8fd0\u7b97\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u5219\u200b\u53d6\u200b\u6ea2\u51fa\u200b\u540e\u200b\u7684\u200b\u503c\u200b\u3002</li> <li><code>checked_*</code>\uff1a\u200b\u5982\u679c\u200b\u8fd0\u7b97\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>None</code>\u3002</li> <li><code>saturating_*</code>\uff1a\u200b\u5982\u679c\u200b\u8fd0\u7b97\u200b\u6ea2\u51fa\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b\u76ee\u6807\u200b\u7c7b\u578b\u200b\u7684\u200b\u6700\u5927\u503c\u200b\u6216\u200b\u6700\u5c0f\u503c\u200b\u3002</li> <li> <p><code>overflowing_*</code>\uff1a\u200b\u8fd4\u56de\u200b\u8fd0\u7b97\u200b\u7ed3\u679c\u200b\u548c\u200b\u662f\u5426\u200b\u6ea2\u51fa\u200b\u7684\u200b\u6807\u8bb0\u200b</p> <pre><code>let a: i32 = 1;\nlet b: i32 = 2;\nlet (c, overflow) = a.overflowing_add(b);\nprintln!(\"{} {}\", c, overflow); // 3 false\n</code></pre> </li> </ul>"},{"location":"coding/rust/1/#_4","title":"\u6d6e\u70b9\u6570\u200b\u7c7b\u578b","text":"<ol> <li>\u200b\u6d6e\u70b9\u6570\u200b\u6ca1\u6709\u200b\u5b9e\u73b0\u200b<code>Eq</code> trait\uff0c\u200b\u56e0\u6b64\u200b\u4e0d\u80fd\u200b\u76f4\u63a5\u200b\u7528\u200b<code>==</code>\u200b\u548c\u200b<code>!=</code>\u200b\u6765\u200b\u6bd4\u8f83\u200b\uff0c\u200b\u4e5f\u200b\u4e0d\u80fd\u200b\u4f5c\u4e3a\u200b\u5b57\u5178\u200b\u7684\u200b\u952e\u200b\u3002</li> <li>\u200b\u6570\u5b66\u200b\u4e0a\u200b\u65e0\u200b\u5b9a\u4e49\u200b\u7684\u200b\u8fd0\u7b97\u200b\uff0c\u200b\u5982\u200b<code>0.0 / 0.0</code>\uff0cRust\u200b\u4f1a\u200b\u8fd4\u56de\u200b<code>NaN</code>\uff08Not a Number\uff09\u3002\u200b\u4e0e\u200b<code>NaN</code>\u200b\u7684\u200b\u8fd0\u7b97\u200b\u7ed3\u679c\u200b\u603b\u662f\u200b<code>NaN</code>\uff0c\u200b\u5e76\u4e14\u200b<code>NaN</code>\u200b\u4e0d\u80fd\u200b\u53c2\u4e0e\u200b\u6bd4\u8f83\u200b\u8fd0\u7b97\u200b\u3002</li> <li>\u200b\u7531\u4e8e\u200b<code>NaN</code>\u200b\u7684\u200b\u5b58\u5728\u200b\uff0c\u200b\u6d6e\u70b9\u6570\u200b\u6ca1\u6709\u200b\u5b9e\u73b0\u200b<code>Ord</code> trait\uff0c\u200b\u56e0\u6b64\u200b\u4e0d\u80fd\u200b\u76f4\u63a5\u200b\u7528\u4e8e\u200b<code>sort()</code>\u200b\u7b49\u200b\u9700\u8981\u200b<code>Ord</code> trait\u200b\u7684\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b<code>partial_cmp().unwrap()</code>\u200b\u6765\u200b\u8fdb\u884c\u200b\u6bd4\u8f83\u200b\u3002</li> <li>\u200b\u7531\u4e8e\u200b\u6d6e\u70b9\u6570\u200b\u7684\u200b\u7cbe\u5ea6\u200b\u95ee\u9898\u200b\uff0c\u200b\u6d6e\u70b9\u6570\u200b\u7684\u200b\u6bd4\u8f83\u200b\u5e94\u8be5\u200b\u7528\u200b<code>abs()</code>\u200b\u548c\u200b<code>&lt;</code>\u3001<code>&gt;</code>\u200b\u6765\u200b\u8fdb\u884c\u200b\u3002</li> </ol> <p>\u200b\u6d6e\u70b9\u6570\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li><code>is_nan()</code>\uff1a\u200b\u5224\u65ad\u200b\u662f\u5426\u200b\u4e3a\u200b<code>NaN</code>\u3002</li> </ul>"},{"location":"coding/rust/1/#_5","title":"\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\u7c7b\u578b","text":"<p>\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li> <p><code>len()</code>\uff1a\u200b\u8fd4\u56de\u200b\u5b57\u7b26\u4e32\u200b\u957f\u5ea6\u200b\u3002</p> <pre><code>let s = \"hello\";\nprintln!(\"{}\", s.len()); // 5\n</code></pre> </li> <li> <p><code>lines()</code>\uff1a\u200b\u6309\u200b\u6362\u884c\u7b26\u200b<code>\\n</code>\u200b\u5206\u5272\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u8fd4\u56de\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002</p> <pre><code>let s = \"hello\\nworld\";\nfor line in s.lines() {\n    println!(\"{}\", line);\n}\n</code></pre> </li> <li> <p><code>split()</code>\uff1a\u200b\u6309\u200b\u7ed9\u5b9a\u200b\u5b57\u7b26\u200b\u5206\u5272\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u8fd4\u56de\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002</p> <pre><code>let s = \"hello world\";\nfor word in s.split(\" \") {\n    println!(\"{}\", word);\n}\n</code></pre> </li> <li> <p><code>trim()</code>\uff1a\u200b\u53bb\u9664\u200b\u5b57\u7b26\u4e32\u200b\u9996\u5c3e\u200b\u7684\u200b\u7a7a\u767d\u200b\u5b57\u7b26\u200b\u3002</p> <pre><code>let s = \" hello world \";\nprintln!(\"{}\", s.trim()); // \"hello world\"\n</code></pre> </li> <li> <p><code>parse()</code>\uff1a\u200b\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u5176\u4ed6\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5982\u200b<code>parse::&lt;i32&gt;()</code>\u200b\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u8f6c\u6362\u200b\u4e3a\u200b<code>i32</code>\u200b\u7c7b\u578b\u200b\u3002\u200b\u5f53\u200b\u8f6c\u4e3a\u200b\u6570\u5b57\u200b\u65f6\u200b\uff0c\u200b\u6570\u5b57\u200b\u4e0d\u80fd\u200b\u8d85\u8fc7\u200b\u7c7b\u578b\u200b\u7684\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\uff0c\u200b\u5426\u5219\u200b\u4f1a\u200b\u62a5\u9519\u200b\u3002</p> <pre><code>let s = \"123\";\nlet n: i32 = s.parse().unwrap(); // \u200b\u8fd4\u56de\u503c\u200b\u662f\u200b`Result`\u200b\u7c7b\u578b\u200b\n</code></pre> <p>\u200b\u81ea\u5b9a\u4e49\u200b\u7c7b\u578b\u200b\u5982\u679c\u200b\u9700\u8981\u200b\u901a\u8fc7\u200b<code>parse()</code>\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b<code>std::str::FromStr</code> trait\uff0c\u200b\u53c2\u89c1\u200bImpl &amp; Trait</p> </li> <li> <p><code>to_string()</code>\uff1a\u200b\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\u8f6c\u6362\u200b\u4e3a\u200b<code>String</code>\u200b\u7c7b\u578b\u200b\u3002</p> </li> <li><code>chars()</code>\uff1a\u200b\u8fd4\u56de\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5b57\u7b26\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002</li> <li><code>bytes()</code>\uff1a\u200b\u8fd4\u56de\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u5b57\u8282\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002</li> </ul>"},{"location":"coding/rust/1/#_6","title":"\u5b57\u7b26\u4e32\u200b\u7c7b\u578b","text":"<p>\u200b\u5b57\u7b26\u4e32\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li><code>push()</code>\u3001<code>push_str()</code>\uff1a\u200b\u5206\u522b\u200b\u7528\u4e8e\u200b\u5728\u200b\u5b57\u7b26\u4e32\u200b\u672b\u5c3e\u200b\u6dfb\u52a0\u200b\u5b57\u7b26\u200b\u548c\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\u3002</li> <li><code>insert()</code>\u3001<code>insert_str()</code>\uff1a\u200b\u5206\u522b\u200b\u7528\u4e8e\u200b\u5728\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b\u6307\u5b9a\u200b\u4f4d\u7f6e\u200b\u63d2\u5165\u200b\u5b57\u7b26\u200b\u548c\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\u3002</li> <li><code>replace(A, B)</code>\u3001<code>replacen(A, B, n)</code>\uff1a\u200b\u5206\u522b\u200b\u7528\u4e8e\u200b\u5c06\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u7684\u200b<code>A</code>\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>B</code>\uff0c<code>replacen</code>\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u6307\u5b9a\u200b\u66ff\u6362\u200b\u6b21\u6570\u200b<code>n</code>\uff0c\u200b\u8fd4\u56de\u200b\u66ff\u6362\u200b\u540e\u200b\u7684\u200b\u65b0\u200b\u5b57\u7b26\u4e32\u200b\u3002</li> <li><code>replace_range(range, A)</code>\uff1a\u200b\u7528\u4e8e\u200b\u5c06\u200b\u539f\u200b\u5b57\u7b26\u4e32\u200b\u7684\u200b\u6307\u5b9a\u200b\u8303\u56f4\u200b\u66ff\u6362\u200b\u4e3a\u200b\u65b0\u200b\u7684\u200b\u5b57\u7b26\u4e32\u200b<code>A</code>\uff0c\u200b\u8fd4\u56de\u200b\u66ff\u6362\u200b\u540e\u200b\u7684\u200b\u539f\u200b\u5b57\u7b26\u4e32\u200b\u3002</li> <li><code>pop()</code>\uff1a\u200b\u5220\u9664\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5b57\u7b26\u4e32\u200b\u672b\u5c3e\u200b\u7684\u200b\u5b57\u7b26\u200b\uff0c\u200b\u5982\u679c\u200b\u5b57\u7b26\u4e32\u200b\u4e3a\u200b\u7a7a\u5219\u200b\u8fd4\u56de\u200b<code>None</code>\u3002</li> <li><code>remove()</code>\uff1a\u200b\u5220\u9664\u200b\u5e76\u200b\u8fd4\u56de\u200b\u5b57\u7b26\u4e32\u200b\u4e2d\u200b\u6307\u5b9a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5b57\u7b26\u200b\u3002\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u7d22\u5f15\u200b\u5fc5\u987b\u200b\u662f\u200b\u6709\u6548\u200b\u7684\u200b\u5b57\u7b26\u200b\u8fb9\u754c\u200b\u3002</li> <li><code>truncate()</code>\uff1a\u200b\u5220\u9664\u200b\u4ece\u200b\u6307\u5b9a\u200b\u4f4d\u7f6e\u200b\u5f00\u59cb\u200b\u5230\u200b\u5b57\u7b26\u4e32\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5168\u90e8\u200b\u5b57\u7b26\u200b\u3002</li> <li><code>clear()</code>\uff1a\u200b\u6e05\u7a7a\u200b\u5b57\u7b26\u4e32\u200b\u3002</li> <li><code>+</code>\u200b\u8fd0\u7b97\u7b26\u200b\uff1a\u200b\u7528\u4e8e\u200b\u8fde\u63a5\u200b\u4e24\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\uff0c\u200b\u53f3\u7aef\u200b\u9879\u4e3a\u200b<code>&amp;str</code>\u200b\u7c7b\u578b\u200b\u3002</li> </ul>"},{"location":"coding/rust/1/#_7","title":"\u6570\u7ec4\u200b\u7c7b\u578b","text":"<p>\u200b\u6570\u7ec4\u200b\u8d8a\u754c\u200b</p> <p>Rust\u200b\u4f1a\u200b\u68c0\u67e5\u200b\u6570\u7ec4\u200b\u8bbf\u95ee\u200b\u8d8a\u754c\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u6570\u7ec4\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li> <p><code>std::array::from_fn()</code>\uff1a\u200b\u6839\u636e\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\u751f\u6210\u200b\u6570\u7ec4\u200b\u3002</p> <pre><code>let a: [String; 8] = std::array::from_fn(|_i| String::from(\"This is Rust\"));\n</code></pre> </li> </ul>"},{"location":"coding/rust/1/#_8","title":"\u8fed\u4ee3\u200b\u5668\u200b\u7c7b\u578b","text":"<p>\u200b\u8fed\u4ee3\u200b\u5668\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li> <p><code>map(fn)</code>\uff1a\u200b\u5bf9\u200b\u8fed\u4ee3\u200b\u5668\u4e2d\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u5e94\u7528\u200b\u51fd\u6570\u200b<code>fn</code>\uff0c\u200b\u8fd4\u56de\u200b\u65b0\u200b\u7684\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002</p> <pre><code>let v = vec![1, 2, 3];\nfor i in v.iter().map(|x| x * 2) {\n    print!(\"{}\", i); // 246\n}\n</code></pre> </li> <li> <p><code>collect()</code>\uff1a\u200b\u5c06\u200b\u8fed\u4ee3\u200b\u5668\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u5bb9\u5668\u200b\uff08\u200b\u5b9a\u200b\u957f\u200b\u6570\u7ec4\u200b\uff09\u3002</p> <pre><code>let v = vec![1, 2, 3];\nlet v2: Vec&lt;i32&gt; = v.iter().map(|x| x * 2).collect();\n</code></pre> </li> </ul>"},{"location":"coding/rust/1/#_9","title":"\u8fd0\u7b97\u7b26","text":"<p>Rust\u200b\u7684\u200b\u8fd0\u7b97\u7b26\u200b\u4e0e\u200bC/C++\u200b\u7c7b\u4f3c\u200b</p> <ul> <li>\u200b\u56db\u5219\u8fd0\u7b97\u200b\u7b26\u200b\uff1a<code>+ - * / %</code></li> <li>\u200b\u6bd4\u8f83\u200b\u8fd0\u7b97\u7b26\u200b\uff1a<code>== != &gt; &lt; &gt;= &lt;=</code></li> <li>\u200b\u903b\u8f91\u200b\u8fd0\u7b97\u7b26\u200b\uff1a<code>&amp;&amp; || !</code></li> <li>\u200b\u4f4d\u200b\u8fd0\u7b97\u7b26\u200b\uff1a<code>&amp; | ^ &lt;&lt; &gt;&gt; !</code>\uff08<code>!</code>\u200b\u4e3a\u200b\u6309\u4f4d\u200b\u53d6\u53cd\u200b\uff09</li> <li>\u200b\u539f\u4f4d\u200b\u8d4b\u503c\u200b\u8fd0\u7b97\u7b26\u200b\uff1a<code>+= -= *= /= %= &amp;= |= ^= &lt;&lt;= &gt;&gt;=</code>\u200b\u7b49\u200b</li> <li>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u8fd0\u7b97\u7b26\u200b\uff1a<code>as</code>\uff0c\u200b\u5982\u200b<code>let b = (a as f64) / 2.0;</code></li> <li>\u200b\u5f15\u7528\u200b\u4e0e\u200b\u89e3\u200b\u5f15\u7528\u200b\u8fd0\u7b97\u7b26\u200b\uff1a<code>&amp; &amp;mut *</code>\uff0c\u200b\u5176\u4e2d\u200b<code>&amp;</code>\u200b\u7528\u4e8e\u200b\u83b7\u53d6\u200b\u53d8\u91cf\u200b\u7684\u200b\u5f15\u7528\u200b\uff08borrow\uff09\uff0c<code>&amp;mut</code>\u200b\u7528\u4e8e\u200b\u83b7\u53d6\u200b\u53d8\u91cf\u200b\u7684\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\uff0c<code>*</code>\u200b\u7528\u4e8e\u200b\u89e3\u200b\u5f15\u7528\u200b\u3002</li> </ul>"},{"location":"coding/rust/2/","title":"Control Flows","text":"<p>\u200b\u672c\u200b\u8282\u200b\u4ecb\u7ecd\u200bRust\u200b\u4e2d\u200b\u7684\u200b\u51fd\u6570\u200b\u548c\u200b\u63a7\u5236\u6d41\u200b\u8bed\u53e5\u200b\u3002</p>"},{"location":"coding/rust/2/#_1","title":"\u51fd\u6570\u200b\u4e0e\u200b\u51fd\u6570\u6307\u9488","text":"<p>Rust\u200b\u4e2d\u200b\uff0c\u200b\u7528\u200b\u5173\u952e\u5b57\u200b<code>fn</code>\u200b\u58f0\u660e\u200b\u51fd\u6570\u200b\uff0c\u200b\u51fd\u6570\u200b\u7684\u200b\u683c\u5f0f\u200b\u4e3a\u200b\uff1a</p> <pre><code>fn &lt;function name&gt;([&lt;parameter name&gt;: &lt;parameter type&gt;, ...]) -&gt; [&lt;return type&gt;] {\n    &lt;function body&gt;\n}\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff0c\u200b\u51fd\u6570\u200b\u5fc5\u987b\u200b\u6307\u660e\u200b\u6240\u6709\u200b\u53c2\u6570\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u5982\u679c\u200b\u6ca1\u6709\u200b\u6307\u660e\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5219\u200b\u9ed8\u8ba4\u200b\u4e3a\u200b<code>()</code>\uff0c\u200b\u5373\u7a7a\u200b\u5143\u7ec4\u200b\u3002\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>return</code>\u200b\u5173\u952e\u5b57\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u6700\u540e\u200b\u4e00\u884c\u200b\u7684\u200b\u8868\u8fbe\u5f0f\u200b\u8fd4\u56de\u503c\u200b\u3002\u200b\u4efb\u4f55\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u5982\u679c\u200b\u901a\u8fc7\u200b\u8868\u8fbe\u5f0f\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5219\u200b\u4e0d\u80fd\u200b\u52a0\u5206\u200b\u53f7\u200b<code>;</code>\u3002</p> <p>  Download source code</p> <pre><code>fn add(a: i32, b: i32) -&gt; i32 {\n    a + b // Return by expression\n}\nfn sub(a: i32, b: i32) -&gt; i32 {\n    return a - b; // Return by return keyword\n}\n\nfn main() {\n    let (a, b) = (2, 1);\n    println!(\"{} + {} = {}\", a, b, add(a, b));\n    println!(\"{} - {} = {}\", a, b, sub(a, b));\n}\n</code></pre> <p>\u200b\u51fd\u6570\u6307\u9488\u200b\u7684\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>fn(&lt;parameter type&gt;, ...) -&gt; &lt;return type&gt;</code>\uff0c\u200b\u5c06\u200b\u51fd\u6570\u200b\u8d4b\u503c\u200b\u7ed9\u200b\u51fd\u6570\u6307\u9488\u200b\u65f6\u200b\uff0c\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\u548c\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u5fc5\u987b\u200b\u5168\u90e8\u200b\u5339\u914d\u200b\u3002</p> <p>\u200b\u5728\u200b\u51fd\u6570\u200b\u5185\u90e8\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u58f0\u660e\u200b\u51fd\u6570\u200b\uff0c\u200b\u79f0\u4e3a\u200blambda\u200b\u51fd\u6570\u200b\uff0c\u200b\u58f0\u660e\u200b\u65b9\u5f0f\u200b\u4e3a\u200b</p> <pre><code>let &lt;function name&gt; = |[&lt;parameter name&gt;: &lt;parameter type&gt;, ...]| -&gt; [&lt;return type&gt;] {\n    &lt;function body&gt;\n}\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff1a</p> <ol> <li> <p>\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>let add = |a, b| { return a + b; };\n</code></pre> </li> <li> <p>\u200b\u5982\u679c\u200b\u51fd\u6570\u200b\u4f53\u200b\u53ea\u6709\u200b\u4e00\u884c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b<code>{}</code>\uff0c\u200b\u6b64\u65f6\u200b\u4e0d\u80fd\u200b\u52a0\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>let add = |a, b| a + b;\n</code></pre> </li> <li> <p>\u200b\u53ef\u4ee5\u200b\u5728\u200b\u58f0\u660e\u200b\u7684\u200b\u65f6\u5019\u200b\u76f4\u63a5\u200b\u8c03\u7528\u200b\uff0c\u200b\u6b64\u65f6\u200b\u4e0d\u80fd\u200b\u7701\u7565\u200b<code>{}</code>\u200b\u548c\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>let result = |a, b| -&gt; i32 { a + b }(1, 2); // result = 3\n</code></pre> </li> </ol> <p>  Download source code</p> <pre><code>fn main() {\n    let (a, b) = (2, 1);\n    let add = |a, b| a + b;\n    println!(\"{} + {} = {}\", a, b, add(a, b));\n    println!(\"{} - {} = {}\", a, b, |a, b| -&gt; i32 { return a - b; }(a, b));\n}\n</code></pre>"},{"location":"coding/rust/2/#if","title":"<code>if</code>\u200b\u8bed\u53e5","text":"<p>Rust\u200b\u4e2d\u200b<code>if</code>\u200b\u8bed\u53e5\u200b\u7684\u200b\u683c\u5f0f\u200b\u4e0e\u200bC/C++\u200b\u76f8\u4f3c\u200b\uff0c\u200b\u4f46\u662f\u200b\u6761\u4ef6\u200b\u8868\u8fbe\u5f0f\u200b\u4e0d\u200b\u9700\u8981\u200b\u52a0\u200b\u62ec\u53f7\u200b\u3002</p> <pre><code>if &lt;condition&gt; {\n    &lt;expression&gt;\n} [ else if &lt;condition&gt; {\n    &lt;expression&gt;\n} ] [ else {\n    &lt;expression&gt;\n} ]\n</code></pre> <p><code>if</code>\u200b\u8bed\u53e5\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200b\u5728\u200b\u8868\u8fbe\u5f0f\u200b\u4e2d\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6bcf\u4e2a\u200b\u5206\u652f\u200b\u5757\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u884c\u200b\u4f5c\u4e3a\u200b\u8be5\u200b\u5206\u652f\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002\u200b\u6b64\u65f6\u200b\u6bcf\u4e2a\u200b\u5206\u652f\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u5fc5\u987b\u200b\u76f8\u540c\u200b\u3002</p> <p>  Download source code</p> <pre><code>fn main() {\n    let (a, b) = (5, 3);\n    println!(\"max({}, {}) = {}\", a, b, if a &gt; b { a } else { b });\n    if a &lt; b {\n        println!(\"min({}, {}) = {}\", a, b, a);\n    } else {\n        println!(\"min({}, {}) = {}\", a, b, b);\n    }\n}\n</code></pre>"},{"location":"coding/rust/2/#match","title":"<code>match</code>\u200b\u8bed\u53e5","text":"<p><code>match</code>\u200b\u8bed\u53e5\u200b\u7528\u4e8e\u200b\u5bf9\u200b\u53d8\u91cf\u200b\u7684\u200b\u503c\u200b\u8fdb\u884c\u200b\u6a21\u5f0f\u5339\u914d\u200b\uff0c\u200b\u683c\u5f0f\u200b\u4e3a\u200b\uff1a</p> <pre><code>match &lt;variable&gt; {\n    &lt;pattern&gt; =&gt; &lt;expression&gt;,\n    ...\n    _ =&gt; &lt;expression&gt;\n}\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff0c<code>&lt;pattern&gt;</code>\u200b\u53ef\u4ee5\u200b\u662f\u200b</p> <ol> <li>\u200b\u5355\u4e2a\u200b\u6570\u503c\u200b\uff0c\u200b\u5982\u200b<code>5 =&gt; \"five\"</code></li> <li>\u200b\u8303\u56f4\u200b\uff0c\u200b\u5982\u200b<code>1 ..= 5 =&gt; \"one to five\"</code></li> <li><code>|</code>\u200b\u5206\u9694\u200b\u7684\u200b\u591a\u4e2a\u200b\u6a21\u5f0f\u200b\uff0c\u200b\u5982\u200b<code>1 | 2 | 3 =&gt; \"one to three\"</code></li> <li>\u200b\u6a21\u677f\u200b\u5b9e\u4f8b\u200b\uff0c\u200b\u5982\u200b<code>Some(x) =&gt; Some(x + 1)</code></li> <li><code>_</code>\uff0c\u200b\u8868\u793a\u200b\u5339\u914d\u200b\u6240\u6709\u200b\u60c5\u51b5\u200b\u3002</li> <li>\u200b\u5982\u679c\u200b\u88ab\u200b\u5339\u914d\u200b\u7684\u200b\u53d8\u91cf\u200b\u662f\u200b\u5143\u7ec4\u200b\uff0c\u200b\u5219\u200b\u5143\u7ec4\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u5143\u7d20\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u662f\u200b\u4e0a\u8ff0\u200b\u6a21\u5f0f\u200b\u3002</li> <li>\u200b\u6761\u4ef6\u200b\u8bed\u53e5\u200b\uff0c\u200b\u5982\u200b<code>x if x &gt; 5 =&gt; \"greater than five\"</code>\uff0c\u200b\u6b64\u65f6\u200b\u5143\u7ec4\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6574\u4f53\u200b\u53c2\u4e0e\u200b\u6761\u4ef6\u200b\u8bed\u53e5\u200b\uff0c\u200b\u5982\u200b<code>(x, y) if x &gt; 5 &amp;&amp; y &gt; 5 =&gt; \"both greater than five\"</code></li> </ol> <p><code>&lt;expression&gt;</code>\u200b\u53ef\u4ee5\u200b\u662f\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u4ee3\u7801\u200b\u5757\u200b\uff0c\u200b\u6b64\u65f6\u200b\u4ee3\u7801\u200b\u5757\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u884c\u200b\u4f5c\u4e3a\u200b\u8fd4\u56de\u503c\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b<code>match</code>\u200b\u7684\u200b\u5206\u652f\u200b\u5fc5\u987b\u200b\u5305\u62ec\u200b\u6240\u6709\u200b\u53ef\u80fd\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u5426\u5219\u200b\u4f1a\u200b\u62a5\u9519\u200b\u3002</p> <p>  Download source code</p> <pre><code>fn main() {\n    let marks_paper_a: u8 = 25;\n    let marks_paper_b: u8 = 30;\n\n    let output = match (marks_paper_a, marks_paper_b) {\n        (50, 50) =&gt; \"Full marks for both papers\",\n        (50, _) =&gt; \"Full marks for paper A\",\n        (_, 50) =&gt; \"Full marks for paper B\",\n        (x, y) if x &gt; 25 &amp;&amp; y &gt; 25 =&gt; \"Good\",\n        (_, _) =&gt; \"Work hard\"\n    };\n\n    println!(\"{}\", output); // Work hard\n}\n</code></pre>"},{"location":"coding/rust/2/#if-let","title":"<code>if let</code>\u200b\u8bed\u53e5","text":"<p>\u200b\u548c\u200b<code>match</code>\u200b\u7c7b\u4f3c\u200b\uff0c<code>if let</code>\u200b\u8bed\u53e5\u200b\u7528\u4e8e\u200b\u5339\u914d\u200b\u5355\u4e2a\u200b\u6a21\u5f0f\u200b\uff0c\u200b\u683c\u5f0f\u200b\u4e3a\u200b\uff1a</p> <pre><code>if let &lt;pattern&gt; = &lt;variable&gt; {\n    &lt;expression&gt;\n}\n</code></pre> <p><code>let &lt;pattern&gt; = &lt;variable&gt;</code>\u200b\u53ef\u4ee5\u200b\u770b\u6210\u200b\u4e00\u4e2a\u200b\u6761\u4ef6\u200b\uff08\u200b\u4f46\u200b\u5b9e\u9645\u200b\u5e76\u200b\u4e0d\u662f\u200b\uff09\uff0c\u200b\u5982\u679c\u200b\u80fd\u200b\u5339\u914d\u200b\u5219\u200b\u5c06\u200b<code>&lt;variable&gt;</code>\u200b\u7684\u200b\u503c\u200b\u6309\u7167\u200b<code>&lt;pattern&gt;</code>\u200b\u8fdb\u884c\u200b\u7ed1\u5b9a\u200b\uff0c\u200b\u5e76\u200b\u8fd4\u56de\u200b<code>true</code>\uff0c\u200b\u56e0\u6b64\u200b\u4e5f\u200b\u4f1a\u200b\u6709\u200b<code>while let</code>\u200b\u7b49\u200b\u5176\u4ed6\u200b\u8bed\u53e5\u200b\u3002</p>"},{"location":"coding/rust/2/#loop","title":"<code>loop</code>\u200b\u8bed\u53e5","text":"<p><code>loop</code>\u200b\u8bed\u53e5\u200b\u7528\u4e8e\u200b\u65e0\u9650\u200b\u5faa\u73af\u200b\u3002</p> <ol> <li>\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>continue</code>\u200b\u8df3\u8f6c\u200b\u5230\u200b\u4e0b\u200b\u4e00\u6b21\u200b\u5faa\u73af\u200b\uff0c\u200b\u6216\u200b<code>break</code>\u200b\u8df3\u51fa\u200b\u5faa\u73af\u200b\u3002</li> <li><code>loop</code>\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>break &lt;expression&gt;</code>\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6240\u6709\u200b<code>break</code>\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u5fc5\u987b\u200b\u76f8\u540c\u200b\u3002</li> <li>\u200b\u6bcf\u4e2a\u200b\u5faa\u73af\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>'&lt;label&gt;: loop</code>\u200b\u6807\u8bb0\u200b\uff0c\u200b\u5728\u200b<code>break</code>\u200b\u548c\u200b<code>continue</code>\u200b\u65f6\u200b\u53ef\u4ee5\u200b\u6307\u5b9a\u200b\u8df3\u51fa\u200b/\u200b\u8df3\u8fc7\u200b\u7684\u200b\u5faa\u73af\u200b\u3002</li> </ol> <p>  Download source code</p> <pre><code>fn main() {\n    let mut b1 = 1;\n\n    let (c1, c2) = 'outer_loop: loop {\n      let mut b2 = 1;\n\n      'inner_loop: loop {\n        println!(\"Current Value : [{}][{}]\", b1, b2);\n\n        if b1 == 2 &amp;&amp; b2 == 2 {\n            break 'outer_loop (b1, b2); // Leave outer_loop with return value\n        } else if b2 == 5 {\n            break; // Leave inner_loop by default\n        }\n\n        b2 += 1;\n      }\n\n      b1 += 1;\n    };\n    println!(\"b1 = {}, b2 = {}\", c1, c2);\n}\n</code></pre>"},{"location":"coding/rust/2/#while","title":"<code>while</code>\u200b\u8bed\u53e5","text":"<p><code>while</code>\u200b\u8bed\u53e5\u200b\u7528\u4e8e\u200b\u662f\u200b\u6709\u200b\u6761\u4ef6\u200b\u7684\u200b\u5faa\u73af\u200b\uff0c<code>while</code>\u200b\u8bed\u53e5\u200b\u4e0d\u80fd\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5176\u4ed6\u200b\u4e0e\u200b<code>loop</code>\u200b\u8bed\u53e5\u200b\u76f8\u540c\u200b\u3002</p> <pre><code>while &lt;condition&gt; {\n    &lt;expression&gt;\n}\n</code></pre>"},{"location":"coding/rust/2/#for","title":"<code>for</code>\u200b\u8bed\u53e5","text":"<p><code>for</code>\u200b\u8bed\u53e5\u200b\u7528\u4e8e\u200b\u904d\u5386\u200b\u8fed\u4ee3\u200b\u5668\u200b\uff0c\u200b\u683c\u5f0f\u200b\u4e3a\u200b\uff1a</p> <pre><code>for &lt;variable&gt; in &lt;iterator&gt; {\n    &lt;expression&gt;\n}\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff0c<code>&lt;iterator&gt;</code>\u200b\u53ef\u4ee5\u200b\u662f\u200b</p> <ol> <li><code>start .. end</code>\uff0c\u200b\u8868\u793a\u200b\u4ece\u200b<code>start</code>\u200b\u5230\u200b<code>end - 1</code>\u200b\u7684\u200b\u6574\u6570\u200b\u8303\u56f4\u200b\u3002</li> <li><code>start ..= end</code>\uff0c\u200b\u8868\u793a\u200b\u4ece\u200b<code>start</code>\u200b\u5230\u200b<code>end</code>\u200b\u7684\u200b\u6574\u6570\u200b\u8303\u56f4\u200b\u3002</li> <li>\u200b\u7531\u200b\u65b9\u6cd5\u200b<code>&lt;iterable&gt;.iter()</code>\u200b\u8fd4\u56de\u200b\u7684\u200b\u8fed\u4ee3\u200b\u5668\u200b\uff0c\u200b\u5982\u200b<code>[1, 2, 3].iter()</code>\u3002</li> </ol> <p><code>for</code>\u200b\u5faa\u73af\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u6dfb\u52a0\u200b\u6807\u7b7e\u200b\u3002</p> <p>  Download source code</p> <pre><code>fn main() {\n    let group : [&amp;str; 4] = [\"Mark\", \"Larry\", \"Bill\", \"Steve\"];\n\n    for n in 0..group.len() { // group.len() = 4 -&gt; 0..4 \ud83d\udc4e check group.len()on each iteration\n    println!(\"Current Person : {}\", group[n]);\n    }\n\n    for person in group.iter() { // \ud83d\udc4d group.iter() turn the array into a simple iterator\n    println!(\"Current Person : {}\", person);\n    }\n}\n</code></pre> <p>\u200b\u4e25\u683c\u200b\u610f\u4e49\u200b\u4e0a\u200b<code>for</code>\u200b\u8bed\u53e5\u200b\u4e0d\u80fd\u200b\u76f4\u63a5\u200b\u904d\u5386\u200b\u5bb9\u5668\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b<code>iter()</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b\u8fed\u4ee3\u200b\u5668\u200b\u3002\u200b\u4f46\u200b<code>for</code>\u200b\u4f1a\u200b\u5c1d\u8bd5\u200b\u9690\u5f0f\u200b\u8c03\u7528\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e0b\u9762\u200b\u7684\u200b\u4ee3\u7801\u200b\u4e5f\u200b\u80fd\u200b\u6b63\u5e38\u200b\u6267\u884c\u200b\u3002</p> <pre><code>fn main() {\n    let a = [1, 2, 3];\n    for i in a {\n        println!(\"{}\", i);\n    }\n}\n</code></pre>"},{"location":"coding/rust/3/","title":"Complex objects","text":"<p>\u200b\u672c\u200b\u8282\u200b\u4ecb\u7ecd\u200bRust\u200b\u4e2d\u200b\u5e38\u7528\u200b\u7684\u200b\u590d\u6742\u200b\u7c7b\u578b\u200b</p>"},{"location":"coding/rust/3/#_1","title":"\u5bb9\u5668\u200b\u7c7b\u578b","text":""},{"location":"coding/rust/3/#vec","title":"<code>Vec</code>","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u5411\u91cf\u200b\u7c7b\u578b\u200b\u662f\u200b\u957f\u5ea6\u200b\u53ef\u53d8\u200b\u7684\u200b\u6570\u7ec4\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>Vec::new()</code>\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u7a7a\u200b\u7684\u200b\u5411\u91cf\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>vec![]</code>\u200b\u5b8f\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u5e26\u6709\u200b\u521d\u59cb\u503c\u200b\u7684\u200b\u5411\u91cf\u200b\u3002<code>Vec</code>\u200b\u662f\u200b\u6cdb\u578b\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>Vec&lt;type&gt;</code>\u200b\u53ef\u4ee5\u200b\u6307\u5b9a\u200b\u5411\u91cf\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u4e00\u4e2a\u200b\u5411\u91cf\u200b\u4e2d\u200b\u5b58\u50a8\u200b\u5143\u7d20\u200b\u7684\u200b\u7c7b\u578b\u200b\u5168\u90e8\u200b\u76f8\u540c\u200b\u3002\u200b\u5411\u91cf\u200b\u53ca\u5176\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6574\u4f53\u200b\uff0c\u200b\u5bf9\u200b\u5176\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u4e5f\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u5411\u91cf\u200b\u672c\u8eab\u200b\u65e0\u6cd5\u200b\u518d\u200b\u8fdb\u884c\u200b\u5f15\u7528\u200b\u3002</p> <pre><code>let v1 = Vec::new(); // Empty vector\nlet v2: Vec&lt;i32&gt; = Vec::new(); // Empty vector with type annotation\nlet v3 = vec![1, 2, 3]; // Vector with initial values\nlet v4 = vec![0; 10]; // Repeat values using colon\nlet v5 = vec![1i32, 2, 3]; // Vector with type specified to the first element\nlet v6 = Vec::with_capacity(10); // Vector with pre-allocated capacity 10\n</code></pre> <p>\u200b\u7528\u200b<code>let mut</code>\u200b\u58f0\u660e\u200b\u7684\u200b\u5411\u91cf\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u4fee\u6539\u200b\u3002<code>Vec</code>\u200b\u7c7b\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li><code>push()</code>\uff1a\u200b\u5411\u200b\u5411\u91cf\u200b\u7684\u200b\u5c3e\u90e8\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b</li> <li><code>pop()</code>\uff1a\u200b\u4ece\u200b\u5411\u91cf\u200b\u7684\u200b\u5c3e\u90e8\u200b\u5f39\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b</li> <li><code>extend()</code>\uff1a\u200b\u5411\u200b\u5411\u91cf\u200b\u7684\u200b\u5c3e\u90e8\u200b\u6dfb\u52a0\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u5411\u91cf\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b</li> <li><code>reserve()</code>\uff1a\u200b\u8c03\u6574\u200b\u5411\u91cf\u200b\u7684\u200b\u5bb9\u91cf\u200b</li> <li><code>len()</code>\uff1a\u200b\u8fd4\u56de\u200b\u5411\u91cf\u200b\u7684\u200b\u957f\u5ea6\u200b</li> <li><code>capacity()</code>\uff1a\u200b\u8fd4\u56de\u200b\u5411\u91cf\u200b\u7684\u200b\u5bb9\u91cf\u200b\uff08\u200b\u5f53\u200b\u8d85\u51fa\u200b\u5bb9\u91cf\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u91cd\u65b0\u5206\u914d\u200b\u5185\u5b58\u7a7a\u95f4\u200b\uff09</li> <li><code>get()</code>\uff1a\u200b\u8fd4\u56de\u200b\u5411\u91cf\u200b\u4e2d\u200b\u6307\u5b9a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\u7684\u200b\u5f15\u7528\u200b\uff0c\u200b\u5982\u679c\u200b\u7d22\u5f15\u200b\u8d8a\u754c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>None</code></li> <li><code>shrink_to_fit()</code>\uff1a\u200b\u5c06\u200b\u5411\u91cf\u200b\u7684\u200b\u5bb9\u91cf\u200b\u8c03\u6574\u200b\u4e3a\u200b\u5411\u91cf\u200b\u7684\u200b\u957f\u5ea6\u200b\uff0c\u200b\u91ca\u653e\u200b\u591a\u4f59\u200b\u7684\u200b\u5185\u5b58\u7a7a\u95f4\u200b</li> <li><code>sort()</code>\u3001<code>sort_unstable()</code>\uff1a\u200b\u5bf9\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b</li> <li><code>sort_by()</code>\u3001<code>sort_unstable_by()</code>\uff1a\u200b\u5bf9\u200b\u5411\u91cf\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u8fdb\u884c\u200b\u6392\u5e8f\u200b\uff0c\u200b\u4f7f\u7528\u200b\u81ea\u5b9a\u4e49\u200b\u7684\u200b\u6bd4\u8f83\u200b\u51fd\u6570\u200b</li> <li><code>[]</code>\u200b\u8fd0\u7b97\u7b26\u200b\uff1a\u200b\u8fd4\u56de\u200b\u5411\u91cf\u200b\u4e2d\u200b\u6307\u5b9a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\uff08\u200b\u53f3\u7aef\u200b\u9879\u200b\uff09\uff0c\u200b\u6216\u8005\u200b\u4fee\u6539\u200b\u5411\u91cf\u200b\u4e2d\u200b\u6307\u5b9a\u200b\u4f4d\u7f6e\u200b\u7684\u200b\u5143\u7d20\u200b\uff08\u200b\u5de6\u7aef\u200b\u9879\u200b\uff09\u3002</li> </ul> <p>\u200b\u5411\u91cf\u200b\u53ca\u5176\u200b\u5f15\u7528\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u7528\u200b\u5728\u200b<code>for</code>\u200b\u5faa\u73af\u200b\u4e2d\u200b\u3002</p>"},{"location":"coding/rust/3/#_2","title":"\u7ed3\u6784\u200b\u4f53","text":"<p>\u200b\u6709\u200b\u4e09\u79cd\u200b\u65b9\u6cd5\u200b\u5728\u200bRust\u200b\u4e2d\u200b\u58f0\u660e\u200b\u7ed3\u6784\u200b\u4f53\u200b\uff1a</p> <ol> <li>C\u200b\u8bed\u8a00\u200b\u98ce\u683c\u200b</li> <li>\u200b\u547d\u540d\u200b\u5143\u7ec4\u200b</li> <li>unit struct</li> </ol> <pre><code>struct CStyle {\n    x: i32,\n    y: i32,\n} // C-style struct with two fields\nstruct NamedTuple(i32, i32); // Named tuple struct with two fields\nstruct TypePattern(i32); // Type pattern struct with one field, can be used in `match` statement\nstruct UnitStruct; // Unit struct without fields\n</code></pre> <p>\u200b\u7528\u200b<code>pub</code>\u200b\u5173\u952e\u5b57\u200b\u6807\u8bb0\u200b\u7ed3\u6784\u200b\u4f53\u540e\u200b\uff0c\u200b\u5176\u4e2d\u200b\u7684\u200b\u5b57\u200b\u6bb5\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u5916\u754c\u200b\u8bbf\u95ee\u200b\u3002</p>"},{"location":"coding/rust/3/#_3","title":"\u679a\u4e3e\u200b\u7c7b\u578b","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u679a\u4e3e\u200b\u7c7b\u578b\u200b\u5173\u952e\u5b57\u200b\u4e3a\u200b<code>enum</code>\uff0c\u200b\u53ef\u4ee5\u200b\u5305\u542b\u200b\u4e0d\u540c\u200b\u7c7b\u578b\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u503c\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u6709\u200b\u4e0d\u540c\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u679a\u4e3e\u200b\u7c7b\u578b\u200b\u7684\u200b\u503c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>match</code>\u200b\u8bed\u53e5\u200b\u8fdb\u884c\u200b\u5339\u914d\u200b\u3002</p> <pre><code>enum Week {\n    Monday,\n    Tuesday,\n    Wednesday,\n    Thursday,\n    Friday,\n    Saturday,\n    Sunday,\n} // Enum type with 7 variants\n</code></pre> <p>Rust\u200b\u4e2d\u200b\u7684\u200b\u679a\u4e3e\u200b\u5143\u7d20\u200b\u53ef\u4ee5\u200b\u662f\u200b\u4e0d\u540c\u200b\u7684\u200bpattern\uff0c\u200b\u53ef\u4ee5\u200b\u662f\u200b\u7ed3\u6784\u200b\u4f53\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u5143\u7ec4\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200bunit struct\u3002</p> <pre><code>enum IpAddr {\n    V4(u8, u8, u8, u8), // Tuple\n    V6{ addr: String }, // Struct\n    LocalHost, // Unit\n}\n</code></pre> <p>\u200b\u8fd9\u4e9b\u200bpattern\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b<code>match</code>\u200b\u8bed\u53e5\u200b\u4e2d\u200b\u8fdb\u884c\u200b\u6a21\u5f0f\u5339\u914d\u200b\uff1a</p> <p>  Download source code</p> <pre><code>enum IpAddr {\n    V4(u8, u8, u8, u8), // Tuple\n    V6{ addr: String }, // Struct\n    LocalHost, // Unit\n}\n\nfn main() {\n    let v4 = IpAddr::V4(4, 4, 4, 4);\n    let v4_invalid = IpAddr::V4(0, 0, 0, 0);\n    let v6 = IpAddr::V6{ addr: String::from(\"::1\") };\n    let local = IpAddr::LocalHost;\n    print_ip(v4);\n    print_ip(v4_invalid);\n    print_ip(v6);\n    print_ip(local);\n}\n\nfn print_ip(ip: IpAddr) {\n    match ip {\n        IpAddr::V4(a, b, c, d) if a != 0 || b != 0 || c != 0 || d != 0\n            =&gt; println!(\"{}.{}.{}.{}\", a, b, c, d),\n        IpAddr::V4(a, b, c, d)\n            =&gt; println!(\"{}.{}.{}.{} is not an valid IPv4 address\", a, b, c, d),\n        IpAddr::V6{ addr } =&gt; println!(\"{}\", addr),\n        IpAddr::LocalHost =&gt; println!(\"localhost\"),\n    }\n}\n</code></pre> <p>\u200b\u7528\u200b<code>pub</code>\u200b\u5173\u952e\u5b57\u200b\u6807\u8bb0\u200b\u679a\u4e3e\u200b\u7c7b\u578b\u200b\u540e\u200b\uff0c\u200b\u5176\u4e2d\u200b\u7684\u200b\u5404\u4e2a\u200b\u5b57\u6bb5\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u5916\u754c\u200b\u8bbf\u95ee\u200b\u3002</p>"},{"location":"coding/rust/3/#_4","title":"\u6cdb\u578b","text":"<p>\u200b\u6cdb\u578b\u200b\u662f\u200b\u4e00\u79cd\u200b\u7c7b\u578b\u200b\u6a21\u677f\u200b\uff0c\u200b\u5f53\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b/\u200b\u679a\u4e3e\u200b/\u200b\u7ed3\u6784\u200b\u4f53\u200b\u9700\u8981\u200b\u5904\u7406\u200b\u591a\u79cd\u7c7b\u578b\u200b\u7684\u200b\u53c2\u6570\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6cdb\u578b\u200b\u3002\u200b\u6cdb\u578b\u200b\u7684\u200b\u7c7b\u578b\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u5728\u200b\u51fd\u6570\u200b\u540d\u200b\u540e\u9762\u200b\u7528\u200b\u5c16\u62ec\u53f7\u200b\u6307\u5b9a\u200b\uff0c\u200b\u4e4b\u540e\u200b\u53c2\u6570\u200b\u5217\u8868\u200b\u4e2d\u200b\u7528\u200b\u5192\u53f7\u200b\u6307\u5b9a\u200b\u6cdb\u578b\u200b\u7c7b\u578b\u200b\u3002</p> <p>  Download source code</p> <pre><code>fn max&lt;T&gt; (a: T, b: T) -&gt; T\n    where T: std::cmp::PartialOrd\n{\n    if a &gt; b { a } else { b }\n}\n\nfn main() {\n    println!(\"{}\", max(1, 2)); // i32\n    println!(\"{}\", max(1.0, 2.0)); // f64\n    println!(\"{}\", max(\"a\", \"b\")); // &amp;str\n}\n</code></pre> <p>Rust\u200b\u6807\u51c6\u200b\u5e93\u4e2d\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u4e9b\u200b\u5e38\u7528\u200b\u7684\u200b\u6cdb\u578b\u200b\u7c7b\u578b\u200b\uff1a</p> <pre><code>enum Option&lt;T&gt; {\n    Some(T),\n    None,\n}\n\nenum Result&lt;T, E&gt; {\n    Ok(T),\n    Err(E),\n}\n</code></pre> <ul> <li><code>Option&lt;T&gt;</code>\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u53ef\u80fd\u200b\u5b58\u5728\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u662f\u200b<code>Some(T)</code>\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b<code>None</code>\u3002\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bPython\u200b\u4e2d\u200b\u7684\u200b<code>Optional</code>\u3002</li> <li><code>Result&lt;T, E&gt;</code>\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u53ef\u80fd\u200b\u51fa\u9519\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5982\u679c\u200b\u6210\u529f\u200b\u5219\u200b\u662f\u200b<code>Ok(T)</code>\uff0c\u200b\u5982\u679c\u200b\u5931\u8d25\u200b\u5219\u200b\u662f\u200b<code>Err(E)</code>\u3002</li> <li>\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b<code>match</code>\u200b\u8bed\u53e5\u200b\u4e2d\u200b\u8fdb\u884c\u200b\u6a21\u5f0f\u5339\u914d\u200b\u3002</li> </ul>"},{"location":"coding/rust/3/#impl-trait","title":"Impl &amp; Trait","text":"<p>\u200b\u5728\u200bRust\u200b\u4e2d\u200b\uff0c<code>impl</code>\u200b\u5173\u952e\u5b57\u200b\u7528\u4e8e\u200b\u4e3a\u200b\u7ed3\u6784\u200b\u4f53\u200b\u5b9e\u73b0\u200b\u65b9\u6cd5\u200b\uff0c<code>trait</code>\u200b\u5173\u952e\u5b57\u200b\u7528\u4e8e\u200b\u58f0\u660e\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u4e0e\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7ec4\u5408\u200b\u540e\u200b\u76f8\u5f53\u4e8e\u200bC++\u200b\u4e2d\u200b\u7684\u200b\u7c7b\u200b\u3002<code>impl</code>\u200b\u53ef\u4ee5\u200b\u8131\u79bb\u200b<code>trait</code>\u200b\u5355\u72ec\u200b\u4f7f\u7528\u200b\u3002</p> <ol> <li>\u200b\u5f53\u200b\u6ca1\u6709\u200b<code>trait</code>\u200b\u65f6\u200b\uff0c<code>impl</code>\u200b\u7684\u200b\u8bed\u6cd5\u200b\u4e3a\u200b<code>impl &lt;struct name&gt; { &lt;methods&gt;}</code>\uff0c\u200b\u5176\u4e2d\u200b<code>&lt;struct name&gt;</code>\u200b\u53ef\u4ee5\u200b\u662f\u200b\u7ed3\u6784\u200b\u4f53\u540d\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u6cdb\u578b\u200b\u53c2\u6570\u200b\u3002</li> <li>\u200b\u5f53\u6709\u200b<code>trait</code>\u200b\u65f6\u200b\uff0c<code>impl</code>\u200b\u7684\u200b\u8bed\u6cd5\u200b\u4e3a\u200b<code>impl &lt;trait name&gt; for &lt;struct name&gt; { &lt;methods&gt;}</code>\uff0c\u200b\u5176\u4e2d\u200b<code>&lt;trait name&gt;</code>\u200b\u662f\u200b\u5bf9\u5e94\u200b\u7684\u200b<code>trait</code>\u200b\u540d\u200b\uff0c<code>&lt;struct name&gt;</code>\u200b\u662f\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b\u8be5\u200b<code>trait</code>\u200b\u7684\u200b\u7ed3\u6784\u200b\u4f53\u540d\u200b\u3002</li> <li>\u200b\u7279\u6b8a\u200b\u5bf9\u8c61\u200b<code>self</code>\u200b\u8868\u793a\u200b\u5f53\u524d\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7684\u200b\u5b9e\u4f8b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4ee5\u200b\u5f15\u7528\u200b\uff08<code>&amp;self</code>\u3001<code>&amp;mut self</code>\uff09\u200b\u6216\u200b\u81ea\u8eab\u200b\u5f62\u6001\u200b\uff08<code>self</code>\uff09\u200b\u51fa\u73b0\u200b\u3002\u200b\u5982\u679c\u200b\u4e0d\u200b\u4f7f\u7528\u200b<code>self</code>\uff0c\u200b\u5219\u200b\u4e3a\u200b\u9759\u6001\u65b9\u6cd5\u200b/\u200b\u7c7b\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bPython\u200b\u4e2d\u200b\u7684\u200b<code>@staticmethod</code>\u200b\u6216\u200b<code>@classmethod</code>\u3002\u200b\u6b64\u65f6\u200b\u9700\u8981\u200b\u7528\u200b<code>&lt;struct&gt;::&lt;method&gt;</code>\u200b\u8c03\u7528\u200b\uff0c\u200b\u800c\u200b\u4e0d\u662f\u200b<code>&lt;object&gt;.&lt;method&gt;</code>\u3002</li> <li>\u200b\u7279\u6b8a\u200b\u7c7b\u578b\u200b<code>Self</code>\u200b\u8868\u793a\u200b\u5f53\u524d\u200b\u7ed3\u6784\u200b\u4f53\u200b\u7684\u200b\u7c7b\u578b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u7684\u200b\u58f0\u660e\u200b\u3002</li> </ol> <p>\u200b\u5b64\u513f\u200b\u89c4\u5219\u200b</p> <p>\u200b\u5982\u679c\u200b\u9700\u8981\u200b\u5bf9\u200b\u7c7b\u578b\u200b<code>A</code>\u200b\u5b9e\u73b0\u200b\u7279\u5f81\u200b<code>T</code>\uff0c\u200b\u5219\u200b<code>T</code>\u200b\u6216\u200b<code>A</code>\u200b\u5176\u4e2d\u200b\u4e4b\u4e00\u200b\u5fc5\u987b\u200b\u5728\u200b\u5f53\u524d\u200b\u4f5c\u7528\u57df\u200b\u5185\u200b\u5b9a\u4e49\u200b\u3002</p> <pre><code>struct Player {\n    first_name: String,\n    last_name: String,\n}\n\ntrait GetFullName {\n    fn full_name(&amp;self) -&gt; String;\n}\n\nimpl GetFullName for Player {\n    fn full_name(&amp;self) -&gt; String {\n        format!(\"{} {}\", self.first_name, self.last_name)\n    }\n}\n\nfn main() {\n    let player_1 = Player {\n        first_name: \"Rafael\".to_string(),\n        last_name: \"Nadal\".to_string(),\n    };\n\n    println!(\"Player 01: {}\", player_1.full_name());\n}\n</code></pre> <p><code>trait</code>\u200b\u53ef\u4ee5\u200b\u548c\u200b\u6cdb\u578b\u200b\u7ed3\u5408\u200b\u4f7f\u7528\u200b\uff0c<code>impl</code>\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u5bf9\u200b\u5185\u7f6e\u200b\u7684\u200b\u7c7b\u578b\u200b\u4f7f\u7528\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <p>  Download source code</p> <pre><code>trait From&lt;T&gt; {\n    fn from_u8(x: T) -&gt; Self;\n}\nimpl From&lt;u8&gt; for u8 {\n    fn from_u8(x: u8) -&gt; Self {\n        x\n    }\n}\nimpl From&lt;u8&gt; for u16 {\n    fn from_u8(x: u8) -&gt; Self {\n        x as u16\n    }\n}\nimpl From&lt;u8&gt; for i8 {\n    fn from_u8(x: u8) -&gt; Self {\n        x as i8\n    }\n}\nimpl From&lt;u8&gt; for i16 {\n    fn from_u8(x: u8) -&gt; Self {\n        x as i16\n    }\n}\n\nfn main() {\n    println!(\"{}\", i16::from_u8(1 as u8));\n}\n</code></pre> <p><code>trait</code>\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u7ee7\u627f\u200b\u548c\u200b\u591a\u200b\u7ee7\u627f\u200b\u3002</p> <pre><code>trait Person {\n    fn full_name(&amp;self) -&gt; String;\n}\n\ntrait Employee : Person { // Employee inherits from Person\n    fn job_title(&amp;self) -&gt; String;\n}\n\ntrait ExpatEmployee : Employee + Expat { // ExpatEmployee inherits from Employee and Expat\n    fn additional_tax(&amp;self) -&gt; f64;\n}\n</code></pre> <p>\u200b\u5728\u200b<code>trait</code>\u200b\u7684\u200b\u58f0\u660e\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u7528\u200b\u51fd\u6570\u200b\u4f53\u200b\uff0c\u200b\u6b64\u65f6\u200b\u8868\u793a\u200b\u65b9\u6cd5\u200b\u7684\u200b\u9ed8\u8ba4\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u5728\u200b<code>impl</code>\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u91cd\u8f7d\u200b\u8fd9\u4e2a\u200b\u65b9\u6cd5\u200b\u3002</p> <pre><code>struct Blog {};\nstruct Tweet {};\n\npub trait Summary {\n    fn summarize(&amp;self) -&gt; String {\n        String::from(\"(Read more...)\")\n    }\n}\n\nimpl Summary for Blog {} // Use default implementation\nimpl Summary for Tweet {\n    fn summarize(&amp;self) -&gt; String { // Override default implementation\n        String::from(\"Tweet\")\n    }\n}\n</code></pre>"},{"location":"coding/rust/3/#_5","title":"\u7279\u5f81\u200b\u7ea6\u675f","text":"<p>\u200b\u7279\u5f81\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u7ea6\u675f\u200b\u51fd\u6570\u200b\u6216\u6cdb\u200b\u578b\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u5982\u200b</p> <ol> <li>\u200b\u51fd\u6570\u53c2\u6570\u200b\u7684\u200b\u9650\u5b9a\u200b\uff0c\u200b\u5982\u200b<code>fn foo(bar: impl Copy)</code>\u200b\u8868\u793a\u200b<code>bar</code>\u200b\u53c2\u6570\u200b\u5fc5\u987b\u200b\u5b9e\u73b0\u200b\u540d\u4e3a\u200b<code>Copy</code>\u200b\u7684\u200b<code>trait</code>\u3002</li> <li>\u200b\u6cdb\u578b\u200b\u53c2\u6570\u200b\u7684\u200b\u9650\u5b9a\u200b\uff0c\u200b\u5982\u200b<code>&lt;T: Copy&gt;</code>\u200b\u8868\u793a\u200b<code>T</code>\u200b\u5fc5\u987b\u200b\u5b9e\u73b0\u200b\u540d\u4e3a\u200b<code>Copy</code>\u200b\u7684\u200b<code>trait</code>\uff0c\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u66f4\u200b\u590d\u6742\u200b\u7684\u200b\u7c7b\u578b\u200b\u63a7\u5236\u200b\u3002</li> </ol> <p>\u200b\u4f7f\u7528\u200b<code>+</code>\u200b\u8fde\u63a5\u200b\u591a\u4e2a\u200b\u7279\u5f81\u200b\uff0c\u200b\u8868\u793a\u200b\u53c2\u6570\u200b\u5fc5\u987b\u200b\u540c\u65f6\u200b\u5b9e\u73b0\u200b\u591a\u4e2a\u200b\u7279\u5f81\u200b\u3002</p> <pre><code>fn foo&lt;T: Copy + Display&gt;(bar: T) {\n    println!(\"{}\", bar); // `bar` must implement `Display` and `Copy`\n}\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>where</code>\u200b\u5173\u952e\u5b57\u200b\u53ef\u4ee5\u200b\u4f7f\u200b\u7279\u5f81\u200b\u7ea6\u675f\u200b\u66f4\u52a0\u200b\u6e05\u6670\u200b\u3002</p> <pre><code>fn foo&lt;T, U&gt;(bar: T, baz: U) -&gt; U\n    where   T: Copy + Display,\n            U: Add&lt;Output = U&gt;\n{\n    println!(\"{}\", bar); // `bar` must implement `Display` and `Copy`\n    baz + 1 // `baz` must implement `Add`\n}\n</code></pre> <p>\u200b\u7279\u5f81\u200b\u7ea6\u675f\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u7528\u200b\u5728\u200b<code>impl</code>\u200b\u8bed\u53e5\u200b\u4e2d\u200b\u7528\u4e8e\u200b\u9009\u62e9\u6027\u200b\u5730\u200b\u5b9e\u73b0\u200b\u7279\u5f81\u200b</p> <pre><code>trait Increment {\n    fn increment(&amp;self) -&gt; Self;\n}\n\nimpl&lt;T&gt; Increment for T\n    where T: Add&lt;Output = T&gt; + Copy\n{\n    fn increment(&amp;self) -&gt; Self {\n        *self + 1\n    }\n}\n</code></pre> <p>\u200b\u8868\u793a\u200b<code>Increment</code>\u200b\u7279\u5f81\u200b\u53ea\u80fd\u200b\u7528\u4e8e\u200b\u540c\u65f6\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>Add&lt;Output = T&gt;</code>\u200b\u548c\u200b<code>Copy</code>\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002</p> <p>\u200b\u7279\u5f81\u200b\u7ea6\u675f\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u9650\u5b9a\u200b\uff0c\u200b\u53ea\u80fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u7279\u5f81\u200b\uff0c\u200b\u8868\u793a\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u5b9e\u73b0\u200b\u4e86\u200b\u7279\u5b9a\u200b\u7279\u5f81\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u4f46\u200b\u51fd\u6570\u200b\u4f53\u200b\u5185\u90e8\u200b\u4e0d\u80fd\u200b\u8fd4\u56de\u200b\u591a\u4e2a\u200b\u4e0d\u540c\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>fn foo(x: bool) -&gt; impl Display {\n    if x {\n        1\n    } else {\n        \"hello\" // Error, since 1 and \"hello\" have different types\n    }\n}\n</code></pre>"},{"location":"coding/rust/4/","title":"Lifetimes","text":"<p>\u200b\u5728\u200bRust\u200b\u4e2d\u200b\uff0c\u200b\u5bf9\u8c61\u200b\u751f\u547d\u5468\u671f\u200b\u548c\u200b\u6240\u6709\u6743\u200b\u662f\u200b\u7d27\u5bc6\u200b\u76f8\u5173\u200b\u7684\u200b\u3002\u200b\u6240\u6709\u6743\u200b\u89c4\u5219\u200b\u662f\u200bRust\u200b\u7684\u200b\u6838\u5fc3\u200b\u7279\u6027\u200b\uff0c\u200b\u4e5f\u200b\u662f\u200bRust\u200b\u4e0e\u200b\u5176\u4ed6\u200b\u8bed\u8a00\u200b\u6700\u5927\u200b\u7684\u200b\u4e0d\u540c\u4e4b\u5904\u200b\u3002</p>"},{"location":"coding/rust/4/#ownership","title":"\u6240\u6709\u6743\u200b\uff08Ownership\uff09","text":"<p>\u200b\u6bcf\u200b\u4e00\u5757\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u90fd\u200b\u548c\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u540d\u200b\u7ed1\u5b9a\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u53d8\u91cf\u540d\u200b\u5c31\u662f\u200b\u8fd9\u5757\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u7684\u200b\u6240\u6709\u8005\u200b\u3002\u200b\u5f53\u200b\u6240\u6709\u8005\u200b\u8d85\u51fa\u200b\u4f5c\u7528\u57df\u200b\u65f6\u200b\uff0c\u200b\u8fd9\u5757\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u5c31\u200b\u4f1a\u200b\u88ab\u200b\u91ca\u653e\u200b\u3002\u200b\u5728\u200b\u672c\u8282\u200b\u4e2d\u200b\uff0c\u201c\u200b\u53d8\u91cf\u200b\u201d\u200b\u548c\u200b\u201c\u200b\u7ed1\u5b9a\u200b\u201d\u200b\u90fd\u200b\u662f\u200b\u6307\u200b\u901a\u8fc7\u200b<code>let</code>\u200b\u5173\u952e\u5b57\u200b\u58f0\u660e\u200b\u7684\u200b\u53d8\u91cf\u200b\u3002</p> <p>\u200b\u5f53\u200b\u4f7f\u7528\u200b<code>=</code>\u200b\u5c06\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\u8d4b\u503c\u200b\u7ed9\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\uff0c\u200b\u6216\u662f\u200b\u5c06\u200b\u53d8\u91cf\u200b\u4f5c\u4e3a\u200b\u51fd\u6570\u200b\u53c2\u6570\u4f20\u9012\u200b\u65f6\u200b\uff0c\u200b\u6240\u6709\u6743\u200b\u4f1a\u200b\u53d1\u751f\u200b\u8f6c\u79fb\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b\u4e24\u79cd\u200b\u53ef\u80fd\u200b\uff1a</p> <ul> <li>\u200b\u503c\u200b\u62f7\u8d1d\u200b\uff1a\u200b\u539f\u6765\u200b\u7684\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u4f1a\u200b\u88ab\u200b\u590d\u5236\u200b\u4e00\u4efd\u200b\u7ed9\u200b\u65b0\u200b\u7684\u200b\u53d8\u91cf\u200b\uff0c\u200b\u4e24\u4e2a\u200b\u53d8\u91cf\u540d\u200b\u7ed1\u5b9a\u200b\u7684\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u4e0d\u540c\u200b\u3002</li> <li>\u200b\u503c\u200b\u79fb\u52a8\u200b\uff1a\u200b\u539f\u6765\u200b\u7684\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u4f1a\u200b\u88ab\u200b\u91cd\u65b0\u200b\u7ed1\u5b9a\u200b\u7ed9\u200b\u65b0\u200b\u7684\u200b\u53d8\u91cf\u200b\uff0c\u200b\u6b64\u65f6\u200b\u4e0d\u80fd\u200b\u518d\u200b\u901a\u8fc7\u200b\u539f\u6765\u200b\u7684\u200b\u53d8\u91cf\u540d\u200b\u8bbf\u95ee\u200b\u5185\u5b58\u200b\u6570\u636e\u200b\u3002</li> </ul> <p>\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0cRust\u200b\u4f1a\u200b\u91c7\u53d6\u200b\u503c\u200b\u79fb\u52a8\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u8d4b\u503c\u200b\uff0c\u200b\u9664\u975e\u200b\u5bf9\u8c61\u200b\u7684\u200b\u7c7b\u578b\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>Copy</code> trait\u3002\u200b\u5927\u591a\u6570\u200b\u57fa\u672c\u200b\u7c7b\u578b\u200b\u90fd\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>Copy</code> trait\u3002</p> <pre><code>fn main() {\n    let a = vec![1, 2, 3]; // vec is not a primitive type, and it does not implement Copy trait\n    let b = a;\n    println!(\"{:?} {:?}\", a, b); // Error; use of moved value: `a`\n}\n</code></pre>"},{"location":"coding/rust/4/#borrowing","title":"\u5f15\u7528\u200b\uff08Borrowing\uff09","text":"<p>\u200b\u5728\u200bRust\u200b\u4e2d\u200b\uff0c\u200b\u53d8\u91cf\u200b\u7684\u200b\u5f15\u7528\u200b\u6709\u200b\u4e24\u79cd\u200b\uff0c\u200b\u5373\u200b\u53ea\u8bfb\u200b\u5f15\u7528\u200b\uff08<code>&amp;T</code>\uff09\u200b\u548c\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\uff08<code>&amp;mut T</code>\uff09\u3002\u200b\u53ea\u8bfb\u200b\u5f15\u7528\u200b\u548c\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u90fd\u200b\u662f\u200b\u6307\u5411\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u7684\u200b\u6307\u9488\u200b\uff0c\u200b\u4f46\u200b\u53ea\u8bfb\u200b\u5f15\u7528\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u4fee\u6539\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u800c\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u5141\u8bb8\u200b\u4fee\u6539\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u7684\u200b\u503c\u200b\u3002\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u53ea\u80fd\u200b\u6307\u5411\u200b\u53ef\u53d8\u200b\u53d8\u91cf\u200b\u3002\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u89e3\u200b\u5f15\u7528\u200b\u8fd0\u7b97\u7b26\u200b<code>*</code>\u200b\u6765\u200b\u8bbf\u95ee\u200b\u5f15\u7528\u200b\u6307\u5411\u200b\u7684\u200b\u503c\u200b\u3002</p> <pre><code>fn main() {\n    let const_var = 5;\n    let mutable_ref = &amp;mut const_var;\n    // error[E0596]: cannot borrow `const_var` as mutable, as it is not declared as mutable\n}\n</code></pre> <p>\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u662f\u4ece\u200b\u5f15\u7528\u200b\u58f0\u660e\u200b\u5f00\u59cb\u200b\u5230\u200b\u5f15\u7528\u200b\u6700\u540e\u200b\u4e00\u6b21\u200b\u4f7f\u7528\u200b\u7ed3\u675f\u200b\u3002\u200b\u5173\u4e8e\u200b\u5f15\u7528\u200b\u7684\u200b\u4f7f\u7528\u200b\u6709\u200b\u5982\u4e0b\u200b\u9650\u5236\u200b\uff1a</p> <ol> <li> <p>\u200b\u5728\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u5185\u200b\uff0c\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u548c\u200b\u53ea\u8bfb\u200b\u5f15\u7528\u200b\u4e0d\u80fd\u200b\u540c\u65f6\u200b\u5b58\u5728\u200b\u3002</p> \u200b\u9519\u8bef\u200b\u6b63\u786e\u200b <pre><code>fn main() {\n    let mut owner = 5;\n    let shared_borrow = &amp;owner; // share_borrow starts here\n    let mutable_borrow = &amp;mut owner; // mutable_borrow starts here\n    println!(\"{}\", mutable_borrow); // Error\n    // mutable_borrow ends here\n    println!(\"{}\", shared_borrow);\n    // share_borrow ends here\n}\n</code></pre> <pre><code>fn main() {\n    let mut owner = 5;\n    let shared_borrow = &amp;owner; // shared_borrow starts here\n    println!(\"{}\", shared_borrow);\n    // shared_borrow ends here\n    // mutable_borrow starts here\n    let mutable_borrow = &amp;mut owner; // mutable_borrow starts here\n    println!(\"{}\", mutable_borrow);\n    // mutable_borrow ends here\n}\n</code></pre> </li> <li> <p>\u200b\u5728\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u5185\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u5bf9\u200b\u540c\u4e00\u200b\u53d8\u91cf\u200b\u58f0\u660e\u200b\u65b0\u200b\u7684\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u3002</p> \u200b\u9519\u8bef\u200b\u6b63\u786e\u200b <pre><code>fn main() {\n    let mut owner = 5;\n    let mutable_borrow = &amp;mut owner;\n    let another_mutable_borrow = &amp;mut owner;\n    println!(\"{} {}\", mutable_borrow, another_mutable_borrow);\n    // error[E0499]: cannot borrow `owner` as mutable more than once at a time\n}\n</code></pre> <pre><code>fn main() {\n    let mut owner = 5;\n    let mutable_borrow = &amp;mut owner;\n    println!(\"{}\", mutable_borrow);\n    let another_mutable_borrow = &amp;mut owner;\n    println!(\"{}\", another_mutable_borrow);\n}\n</code></pre> </li> <li> <p>\u200b\u5728\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u5185\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u8bbf\u95ee\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u6216\u200b\u5bf9\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u8fdb\u884c\u200b\u8d4b\u503c\u200b\u3002</p> \u200b\u9519\u8bef\u200b\u6b63\u786e\u200b <pre><code>fn main() {\n    let mut owner = 5;\n    let mutable_borrow = &amp;mut owner;\n    println!(\"{}\", owner + *mutable_borrow);\n    // error[E0503]: cannot use `owner` because it was mutably borrowed\n}\n</code></pre> <pre><code>fn main() {\n    let mut owner = 5;\n    let mutable_borrow = &amp;mut owner;\n    println!(\"{}\", *mutable_borrow + owner); // 10\n}\n</code></pre> </li> <li> <p>\u200b\u53ef\u4ee5\u200b\u7528\u200b\u6570\u636e\u5e93\u200b\u4e2d\u200b\u201c\u200b\u5171\u4eab\u200b\u9501\u200b\u201d\u200b\u548c\u200b\u201c\u200b\u6392\u200b\u4ed6\u200b\u9501\u200b\u201d\u200b\u7684\u200b\u6982\u5ff5\u200b\u6765\u200b\u7406\u89e3\u200b\u53ea\u8bfb\u200b\u5f15\u7528\u200b\u548c\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u3002\u200b\u53ea\u8bfb\u200b\u5f15\u7528\u200b\u76f8\u5f53\u4e8e\u200b\u5bf9\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u52a0\u200b\u5171\u4eab\u200b\u9501\u200b\uff0c\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\u76f8\u5f53\u4e8e\u200b\u5bf9\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u52a0\u6392\u200b\u4ed6\u9501\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u540c\u200b\u4e00\u5757\u200b\u6570\u636e\u200b\uff0c\u200b\u5171\u4eab\u200b\u9501\u200b\u53ef\u4ee5\u200b\u540c\u65f6\u200b\u5b58\u5728\u200b\uff0c\u200b\u4f46\u6392\u200b\u4ed6\u200b\u9501\u200b\u53ea\u80fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u3002</p> </li> <li> <p>\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u4e0d\u80fd\u200b\u8d85\u8fc7\u200b\u539f\u59cb\u200b\u53d8\u91cf\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u3002</p> E0597: ... does not live long enoughE0515: cannot return value referencing local variable ... <pre><code>struct Foo&lt;'a&gt; {\n    x: Option&lt;&amp;'a u32&gt;,\n}\n\nfn main() {\n    let mut x = Foo { x: None };\n    {\n        let y = 0;\n        x.x = Some(&amp;y);\n    } // Leaving the scope of y, y is dropped\n    println!(\"{:?}\", x.x) // error[E0597]: `y` does not live long enough\n}\n</code></pre> <pre><code>fn one(x: &amp;i32) -&gt; &amp;i32 {\n    let ret = *x + 1;\n    &amp;ret // error[E0515]: cannot return value referencing local variable `ret`\n} // Leaving the scope of ret, ret is dropped\n\nfn main() {\n    println!(\"{}\", *one(&amp;5));\n}\n</code></pre> </li> </ol>"},{"location":"coding/rust/4/#_1","title":"\u751f\u547d\u5468\u671f","text":"<p>\u200b\u5f15\u7528\u200b\u76f8\u5f53\u4e8e\u200b\u6307\u5411\u200b\u4e00\u6bb5\u200b\u5185\u5b58\u7a7a\u95f4\u200b\u7684\u200b\u6307\u9488\u200b\uff0c\u200b\u5728\u200b\u6700\u540e\u200b\u4e00\u6b21\u200b\u8bbf\u95ee\u200b\u5f15\u7528\u200b\u540e\u200b\uff0c\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u5c31\u200b\u7ed3\u675f\u200b\u4e86\u200b\uff0c\u200b\u4f46\u200b\u5728\u200b\u4e00\u4e9b\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u9700\u8981\u200b\u7ee7\u7eed\u200b\u4fdd\u6301\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\uff0c\u200b\u6bd4\u5982\u200b\u5728\u200b\u51fd\u6570\u200b\u4e2d\u200b\u8fd4\u56de\u200b\u5f15\u7528\u200b\u3002\u200b\u5728\u200b\u8fd9\u79cd\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u6765\u200b\u6307\u5b9a\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u3002</p> <pre><code>fn main() {\n    let reference_to_nothing = dangle();\n    // error[E0106]: missing lifetime specifier\n}\n\nfn dangle() -&gt; &amp;String {\n    let s = String::from(\"hello\");\n\n    &amp;s // The lifetime of s is over, so this reference is dangling\n}\n</code></pre> <p>\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u6cdb\u578b\u200b\u53c2\u6570\u200b\u3001\u200b\u51fd\u6570\u53c2\u6570\u200b\u3001\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u800c\u200b\u4e0d\u80fd\u200b\u7528\u4e8e\u200b\u5f15\u7528\u200b\u58f0\u660e\u200b\u3002\u200b\u6709\u200b\u76f8\u540c\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u7684\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u76f8\u540c\u200b\u3002<code>'static</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u7279\u6b8a\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\uff0c\u200b\u8868\u793a\u200b\u5f15\u7528\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u662f\u200b\u6574\u4e2a\u200b\u7a0b\u5e8f\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\uff0c\u200b\u5373\u200b\u5f15\u7528\u200b\u6307\u5411\u200b\u7684\u200b\u5185\u5b58\u7a7a\u95f4\u200b\u5728\u200b\u7a0b\u5e8f\u200b\u7ed3\u675f\u200b\u65f6\u624d\u200b\u4f1a\u200b\u88ab\u200b\u91ca\u653e\u200b\u3002\u200b\u6240\u6709\u200b\u6307\u5411\u200b\u5b57\u7b26\u4e32\u200b\u5b57\u9762\u200b\u91cf\u200b\u7684\u200b\u5f15\u7528\u200b\u90fd\u200b\u662f\u200b<code>'static</code>\u200b\u7c7b\u578b\u200b\u3002</p> \u200b\u51fd\u6570\u200b\u590d\u6742\u200b\u5bf9\u8c61\u200bImpl &amp; Trait\u200b\u6cdb\u578b\u200b <pre><code>fn function&lt;'a&gt;() -&gt; &amp;'a str {}\nfn function&lt;'a&gt;(x: &amp;'a str) {}\nfn function&lt;'a&gt;(x: &amp;'a str) -&gt; &amp;'a str {}\nfn function&lt;'a&gt;(x: &amp;'a str, y: &amp;'a str) -&gt; &amp;'a str {}\nfn function&lt;'a, 'b&gt;(x: &amp;'a str, y: &amp;'b str) -&gt; &amp;'a str {}\n</code></pre> <pre><code>enum Enum&lt;'a&gt; {\n    Variant1(&amp;'a str),\n    Variant2(&amp;'a str) // exist as long as parent object exists\n}\nstruct Struct&lt;'a&gt; {\n    field: &amp;'a str // exist as long as parent object exists\n}\n</code></pre> <pre><code>struct Struct&lt;'a&gt; {\n    field: &amp;'a str\n}\ntrait New&lt;'a&gt; {\n    fn new() -&gt; Self;\n}\nimpl&lt;'a&gt; New&lt;'a&gt; for Struct&lt;'a&gt; {\n    fn new() -&gt; Self {\n        Struct { field: \"hello\" }\n    }\n}\n</code></pre> <pre><code>fn function&lt;T&gt;(x: T) where for&lt;'a&gt; T: Fn(&amp;'a str) -&gt; &amp;'a str {}\nenum Enum&lt;T&gt; where for&lt;'a&gt; T: Fn(&amp;'a str) -&gt; &amp;'a str { V(T) }\nstruct Struct&lt;T&gt; where for&lt;'a&gt; T: Fn(&amp;'a str) -&gt; &amp;'a str { field: T }\nimpl&lt;T&gt; Struct&lt;T&gt; where for&lt;'a&gt; T: Fn(&amp;'a str) -&gt; &amp;'a str {\n    fn function(&amp;self) -&gt; &amp;F { &amp;self.field }\n}\n</code></pre> <p>\u200b\u5176\u4e2d\u200b\uff0c<code>where for</code>\u200b\u6709\u200b\u53e6\u200b\u4e00\u79cd\u200b\u5199\u6cd5\u200b</p> <pre><code>fn function&lt;T&gt;(x: T) where T: for&lt;'a&gt; Fn(&amp;'a str) -&gt; &amp;'a str {}\n</code></pre> <p><code>for</code>\u200b\u5bf9\u200b<code>T</code>\u200b\u8fdb\u884c\u200b\u4e86\u200b\u9650\u5236\u200b\uff0c\u200b\u8868\u793a\u200b<code>T</code>\u200b\u5fc5\u987b\u200b\u662f\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u4e14\u200b\u8fd9\u4e2a\u200b\u51fd\u6570\u200b\u7684\u200b\u53c2\u6570\u200b\u548c\u200b\u751f\u547d\u5468\u671f\u200b\u5fc5\u987b\u200b\u548c\u200b\u6a21\u677f\u200b\u5339\u914d\u200b\u3002</p> <pre><code>struct Closure&lt;F&gt; {\n    data: (u8, u16),\n    func: F,\n}\n\nimpl&lt;F&gt; Closure&lt;F&gt;\n    where for&lt;'a&gt; F: Fn(&amp;'a (u8, u16)) -&gt; &amp;'a u8,\n{\n    fn call(&amp;self) -&gt; &amp;u8 {\n        (self.func)(&amp;self.data)\n    }\n}\n\nfn do_it(data: &amp;(u8, u16)) -&gt; &amp;u8 { &amp;data.0 }\n\nfn main() {\n    let clo = Closure { data: (0, 1), func: do_it };\n    println!(\"{}\", clo.call());\n}\n</code></pre> <p>\u200b\u4e0d\u540c\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u7684\u200b\u5f15\u7528\u200b\u4e0d\u80fd\u200b\u76f8\u4e92\u200b\u8d4b\u503c\u200b\uff0c\u200b\u5728\u200b\u4f5c\u4e3a\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\u65f6\u200b\uff0cRust\u200b\u4e5f\u200b\u4f1a\u200b\u68c0\u67e5\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u662f\u5426\u200b\u5339\u914d\u200b\u3002\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>'a: 'b</code>\u200b\u6765\u200b\u7ea6\u675f\u200b<code>'a</code>\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u4e0d\u77ed\u4e8e\u200b<code>'b</code>\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u3002</p> \u200b\u9519\u8bef\u200b\u6b63\u786e\u200b\u6b63\u786e\u200b <pre><code>fn two_args&lt;'a, 'b&gt;(a: &amp;'a mut i32, b: &amp;'b i32) -&gt; &amp;'b mut i32 {\n    *a += b;\n    a // error: lifetime may not live long enough\n}\nfn main() {\n    let mut a = 3;\n    let b: &amp;mut i32 = two_args(&amp;mut a, &amp;4);\n    println!(\"{}\", b);\n}\n</code></pre> <pre><code>fn two_args&lt;'a, 'b&gt;(a: &amp;'a mut i32, b: &amp;'b i32) -&gt; &amp;'a mut i32 {\n    *a += b;\n    a\n}\nfn main() {\n    let mut a = 3;\n    let b: &amp;mut i32 = two_args(&amp;mut a, &amp;4);\n    println!(\"{}\", b);\n}\n</code></pre> <pre><code>fn two_args&lt;'a: 'b, 'b&gt;(a: &amp;'a mut i32, b: &amp;'b i32) -&gt; &amp;'b mut i32 {\n    *a += b;\n    a\n}\nfn main() {\n    let mut a = 3;\n    let b: &amp;mut i32 = two_args(&amp;mut a, &amp;4);\n    println!(\"{}\", b);\n}\n</code></pre> <p>\u200b\u901a\u5e38\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u51fd\u6570\u200b\u58f0\u660e\u200b\u4e2d\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\uff0cRust\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u63a8\u65ad\u200b\u6bcf\u4e2a\u200b\u53c2\u6570\u200b\u548c\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b</p> <ol> <li> <p>\u200b\u6bcf\u4e2a\u200b\u8f93\u51fa\u200b\u53c2\u6570\u200b\u90fd\u200b\u4f1a\u200b\u5206\u914d\u200b\u4e00\u4e2a\u200b\u72ec\u7acb\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u3002</p> \u200b\u7701\u7565\u200b\u6807\u8bb0\u200b\u663e\u793a\u200b\u6807\u8bb0\u200b <pre><code>fn four_args(a: &amp;i32, b: &amp;i32, c: &amp;i32, d: &amp;i32) -&gt; i32 {\n    a + b + c + d\n}\n</code></pre> <pre><code>fn four_args&lt;'a, 'b, 'c, 'd&gt;(a: &amp;'a i32, b: &amp;'b i32, c: &amp;'c i32, d: &amp;'d i32) -&gt; i32 {\n    a + b + c + d\n}\n</code></pre> </li> <li> <p>\u200b\u5982\u679c\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u8f93\u5165\u200b\u53c2\u6570\u200b\uff0c\u200b\u90a3\u4e48\u200b\u8fd9\u4e2a\u200b\u53c2\u6570\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u5c31\u662f\u200b\uff08\u200b\u6240\u6709\u200b\uff09\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u3002</p> \u200b\u7701\u7565\u200b\u6807\u8bb0\u200b\u663e\u793a\u200b\u6807\u8bb0\u200b <pre><code>fn one_arg(a: &amp;i32) -&gt; &amp;i32 {\n    a\n}\n</code></pre> <pre><code>fn one_arg&lt;'a&gt;(a: &amp;'a i32) -&gt; &amp;'a i32 {\n    a\n}\n</code></pre> </li> <li> <p>\u200b\u5f53\u200b\u51fd\u6570\u200b\u7684\u200b\u67d0\u4e2a\u200b\u53c2\u6570\u200b\u4e3a\u200b<code>&amp;self</code>\u200b\u6216\u200b<code>&amp;mut self</code>\u200b\u65f6\u200b\uff0c\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u548c\u200b<code>&amp;self</code>\u200b\u7684\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u76f8\u540c\u200b\u3002</p> \u200b\u7701\u7565\u200b\u6807\u8bb0\u200b\u663e\u793a\u200b\u6807\u8bb0\u200b <pre><code>struct Player {\n    first_name: String,\n    last_name: String,\n}\n\nimpl Player {\n    fn full_name(&amp;self) -&gt; String {\n        format!(\"{} {}\", self.first_name, self.last_name)\n    }\n}\n</code></pre> <pre><code>struct Player {\n    first_name: String,\n    last_name: String,\n}\n\nimpl Player {\n    fn full_name&lt;'a&gt;(&amp;'a self) -&gt; String {\n        format!(\"{} {}\", self.first_name, self.last_name)\n    }\n}\n</code></pre> </li> <li> <p>\u200b\u5982\u679c\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u4e0a\u8ff0\u200b\u6761\u4ef6\u200b\uff0c\u200b\u5219\u200b\u9700\u8981\u200b\u663e\u5f0f\u200b\u6307\u5b9a\u200b\u5f15\u7528\u200b\u6807\u8bb0\u200b\u3002</p> \u200b\u9519\u8bef\u200b\u6b63\u786e\u200b <pre><code>fn two_args(a: &amp;mut i32, b: &amp;i32) -&gt; &amp;mut i32 {\n    *a += b;\n    a // error[E0106]: missing lifetime specifier\n}\nfn main() {\n    let mut a = 3;\n    let b: &amp;mut i32 = two_args(&amp;mut a, &amp;4);\n    println!(\"{}\", b);\n}\n</code></pre> <pre><code>fn two_args&lt;'a, 'b&gt;(a: &amp;'a mut i32, b: &amp;'b i32) -&gt; &amp;'a mut i32 {\n    *a += b;\n    a\n}\nfn main() {\n    let mut a = 3;\n    let b: &amp;mut i32 = two_args(&amp;mut a, &amp;4);\n    println!(\"{}\", b);\n}\n</code></pre> </li> </ol>"},{"location":"coding/rust/5/","title":"Error Handling","text":"<p>\u200b\u672c\u200b\u8282\u200b\u4ecb\u7ecd\u200bRust\u200b\u7684\u200b\u5f02\u5e38\u200b\u5904\u7406\u200b\u673a\u5236\u200b\u3002</p>"},{"location":"coding/rust/5/#_1","title":"\u7f16\u8bd1\u5668\u200b\u9519\u8bef","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u90e8\u5206\u200b\u9519\u8bef\u200b\u53ef\u4ee5\u200b\u5728\u200b\u7f16\u8bd1\u200b\u65f6\u200b\u88ab\u200b\u68c0\u6d4b\u200b\u51fa\u6765\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u9519\u8bef\u200b\u88ab\u200b\u79f0\u4e3a\u200b\u7f16\u8bd1\u200b\u65f6\u200b\u9519\u8bef\u200b\uff08compile-time error\uff09\u3002\u200b\u4f7f\u7528\u200b<code>rustc --explain</code>\u200b\u53ef\u4ee5\u200b\u67e5\u770b\u200b\u7f16\u8bd1\u200b\u65f6\u200b\u9519\u8bef\u200b\u7684\u200b\u8be6\u7ec6\u4fe1\u606f\u200b\u3002</p> <pre><code>rustc --explain E0382\n# A variable was used after its contents were moved elsewhere.\n#\n# Erroneous code example:\n#\n# struct MyStruct { s: u32 }\n#\n# fn main() {\n#    let mut x = MyStruct { s: 5u32 };\n#    let y = x;\n#    x.s = 6;\n#    println!(\"{}\", x.s);\n# }\n# ...\n</code></pre>"},{"location":"coding/rust/5/#panic","title":"\u5f02\u5e38\u200b\uff08Panic\uff09","text":"<p>Rust\u200b\u4e2d\u200b\u7684\u200b\u5f02\u5e38\u200b\u88ab\u200b\u79f0\u4e3a\u200bpanic\u3002\u200b\u5f53\u200b\u7a0b\u5e8f\u200b\u53d1\u751f\u200bpanic\u200b\u65f6\u200b\uff0cRust\u200b\u4f1a\u200b\u6253\u5370\u200b\u51fa\u200b\u5f02\u5e38\u200b\u4fe1\u606f\u200b\u5e76\u200b\u9000\u51fa\u200b\u7a0b\u5e8f\u200b\u3002\u200b\u6709\u200b\u591a\u4e2a\u200b\u5b8f\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\u3002</p> <pre><code>// \u200b\u4ee5\u4e0b\u200b\u7684\u200b\u5b8f\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u63a5\u53d7\u200b\u7c7b\u4f3c\u200bprintln!\u200b\u7684\u200b\u683c\u5f0f\u5316\u200b\u5b57\u7b26\u4e32\u200b\npanic!(\"This is a panic!\");\nunimplemented!(); // \u200b\u7528\u4e8e\u200b\u6807\u8bb0\u200b\u672a\u200b\u5b9e\u73b0\u200b\u7684\u200b\u4ee3\u7801\u200b\nunreachable!(); // \u200b\u7528\u4e8e\u200b\u6807\u8bb0\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u5230\u8fbe\u200b\u7684\u200b\u5206\u652f\u200b\n\n// \u200b\u4ee5\u4e0b\u200b\u5b8f\u200b\u7528\u4e8e\u200b\u6d4b\u8bd5\u200b\nassert!(false); // \u200b\u7528\u4e8e\u200b\u6d4b\u8bd5\u200b\u8868\u8fbe\u5f0f\u200b\u662f\u5426\u200b\u4e3a\u200btrue\nassert_eq!(1, 2); // \u200b\u7528\u4e8e\u200b\u6d4b\u8bd5\u200b\u4e24\u4e2a\u200b\u8868\u8fbe\u5f0f\u200b\u7684\u200b\u503c\u200b\u662f\u5426\u200b\u76f8\u7b49\u200b\nassert_ne!(1, 1); // \u200b\u7528\u4e8e\u200b\u6d4b\u8bd5\u200b\u4e24\u4e2a\u200b\u8868\u8fbe\u5f0f\u200b\u7684\u200b\u503c\u200b\u662f\u5426\u200b\u4e0d\u200b\u76f8\u7b49\u200b\n\n// \u200b\u5728\u200bDebug\u200b\u6a21\u5f0f\u200b\u6216\u200b\u4f7f\u7528\u200b-C debug-assertions\u200b\u53c2\u6570\u200b\u65f6\u200b\uff0c\u200b\u4ee5\u4e0b\u200b\u5b8f\u624d\u200b\u4f1a\u200b\u89e6\u53d1\u200b\ndebug_assert!(false);\ndebug_assert_eq!(1, 2);\ndebug_assert_ne!(1, 1);\n</code></pre>"},{"location":"coding/rust/5/#_2","title":"\u5f02\u5e38\u200b\u5904\u7406","text":"<p>Rust\u200b\u4e2d\u200b\u5185\u7f6e\u200b\u4e86\u200b\u4e24\u4e2a\u200b\u7c7b\u578b\u200b<code>Option&lt;T&gt;</code>\u200b\u548c\u200b<code>Result&lt;T, E&gt;</code>\u200b\u7528\u4e8e\u200b\u5f02\u5e38\u200b\u5904\u7406\u200b\uff0c\u200b\u5176\u200b\u5b9a\u4e49\u200b\u4e3a\u200b</p> <pre><code>enum Option&lt;T&gt; {\n    None,\n    Some(T),\n}\nenum Result&lt;T, E&gt; {\n    Ok(T),\n    Err(E),\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\uff0c\u200b\u5747\u200b\u4ee5\u200b<code>T</code>\u200b\u4ee3\u8868\u200b<code>Option::Some</code>\u200b\u6216\u200b<code>Result::Ok</code>\u200b\u7684\u200b\u7c7b\u578b\u200b\uff0c<code>E</code>\u200b\u4ee3\u8868\u200b<code>Result::Err</code>\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002</p> <ul> <li> <p><code>Option</code>\u200b\u7c7b\u578b\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u53ef\u80fd\u200b\u4e0d\u200b\u5b58\u5728\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u5e38\u7528\u200b\u4e8e\u200b\u51fd\u6570\u53c2\u6570\u200b\u3001\u200b\u7ed3\u6784\u200b\u4f53\u5b57\u200b\u6bb5\u200b\u6216\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>match</code>\u200b\u8bed\u53e5\u200b\u53ef\u4ee5\u200b\u5339\u914d\u200b<code>Option</code>\u200b\u7c7b\u578b\u200b\u53d6\u503c\u200b\u7684\u200b\u4e0d\u540c\u200b\u60c5\u51b5\u200b\u3002</p> <pre><code>fn main() {\n    let x: Option&lt;i32&gt; = Some(5);\n    match x {\n        Some(i) =&gt; println!(\"{}\", i),\n        None =&gt; println!(\"None\"),\n    }\n}\n</code></pre> <ol> <li>\u200b\u4f7f\u7528\u200b<code>is_some</code>\u200b\u548c\u200b<code>is_none</code>\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u5feb\u901f\u200b\u5224\u65ad\u200b<code>Option</code>\u200b\u7c7b\u578b\u200b\u662f\u5426\u200b\u4e3a\u200b<code>Some</code>\u200b\u6216\u200b<code>None</code>\u3002</li> <li>\u200b\u4f7f\u7528\u200b<code>unwrap</code>\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u5c06\u200b<code>Option</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b<code>T</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5982\u679c\u200b<code>Option</code>\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>None</code>\uff0c\u200b\u5219\u200b\u4f1a\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\u3002</li> </ol> </li> <li> <p><code>Result</code>\u200b\u7c7b\u578b\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u53ef\u80fd\u200b\u51fa\u9519\u200b\u7684\u200b\u503c\u200b\uff0c\u200b\u5e38\u7528\u200b\u4e8e\u200b\u51fd\u6570\u200b\u8fd4\u56de\u503c\u200b\u3002<code>Ok</code>\u200b\u8868\u793a\u200b\u51fd\u6570\u200b\u6267\u884c\u200b\u6210\u529f\u200b\uff0c<code>Err</code>\u200b\u8868\u793a\u200b\u51fd\u6570\u200b\u6267\u884c\u200b\u5931\u8d25\u200b\u3002</p> <pre><code>fn main() {\n    let x: Result&lt;i32, &amp;str&gt; = Ok(5);\n    match x {\n        Ok(i) =&gt; println!(\"{}\", i),\n        Err(e) =&gt; println!(\"{}\", e),\n    }\n}\n</code></pre> <ol> <li>\u200b\u4f7f\u7528\u200b<code>is_ok</code>\u200b\u548c\u200b<code>is_err</code>\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u5feb\u901f\u200b\u5224\u65ad\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u662f\u5426\u200b\u4e3a\u200b<code>Ok</code>\u200b\u6216\u200b<code>Err</code>\u3002</li> <li> <p>\u200b\u4f7f\u7528\u200b<code>ok</code>\u200b\u548c\u200b<code>err</code>\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u5c06\u200b<code>Result&lt;T, E&gt;</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b<code>Option&lt;T&gt;</code>\u200b\u7c7b\u578b\u200b\u3002\u200b\u5982\u679c\u200b\u8c03\u7528\u200b\u7684\u200b\u65b9\u6cd5\u200b\u548c\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u53d6\u503c\u200b\u76f8\u540c\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>Some(T)</code>\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>None</code>\u3002</p> <pre><code>fn main() {\n    let x: Result&lt;i32, &amp;str&gt; = Ok(5);\n    let y: Option&lt;i32&gt; = x.ok();\n    println!(\"{:?}\", y); // Some(5)\n    let x: Result&lt;i32, &amp;str&gt; = Err(\"error\");\n    let y: Option&lt;i32&gt; = x.ok();\n    println!(\"{:?}\", y); // None\n}\n</code></pre> </li> <li> <p>\u200b\u4f7f\u7528\u200b<code>unwrap</code>\u200b\u65b9\u6cd5\u200b\u6216\u200b<code>expect</code>\u200b\u65b9\u6cd5\u200b\u53ef\u4ee5\u200b\u5c06\u200b<code>Result</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b<code>T</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5982\u679c\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>Err</code>\uff0c\u200b\u5219\u200b\u4f1a\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\uff0c\u200b\u8f93\u51fa\u200b\u7684\u200b\u5f02\u5e38\u200b\u4fe1\u606f\u200b\u5bf9\u5e94\u200b\u4e8e\u200b<code>Err</code>\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u6216\u200b<code>expect</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u53c2\u6570\u200b\u3002</p> <pre><code>fn main() {\n    let x: Result&lt;i32, &amp;str&gt; = Ok(5);\n    let y: i32 = x.unwrap();\n    println!(\"{}\", y); // 5\n    let x: Result&lt;i32, &amp;str&gt; = Err(\"error\");\n    let y: i32 = x.unwrap(); // thread 'main' panicked at 'called `Result::unwrap()` on an `Err` value: \"error\"', src/main.rs:2:26\n    println!(\"{}\", y);\n}\n</code></pre> </li> </ol> </li> </ul> <p><code>unwrap</code>\u200b\u65b9\u6cd5\u200b\u6709\u200b\u5982\u4e0b\u200b\u53d8\u79cd\u200b\uff1a</p> <ul> <li><code>expect</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u5b57\u7b26\u4e32\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\uff0c\u200b\u7528\u4e8e\u200b\u6307\u5b9a\u200b\u8df3\u51fa\u200b\u5f02\u5e38\u200b\u65f6\u200b\u7684\u200b\u63d0\u793a\u4fe1\u606f\u200b\u3002</li> <li>\uff08\u200b\u4ec5\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\uff09<code>unwrap_err</code>\u200b\u65b9\u6cd5\u200b\u548c\u200b<code>expect_err</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u4e0e\u200b<code>unwrap</code>\u200b\u76f8\u53cd\u200b\uff0c\u200b\u5f53\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>Ok</code>\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>Err</code>\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u3002</li> <li><code>unwrap_or</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b<code>T</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5728\u200b<code>unwrap</code>\u200b\u65b9\u6cd5\u200b\u4f1a\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u8fd4\u56de\u200b\u8be5\u200b\u53c2\u6570\u200b\u3002</li> <li> <p><code>unwrap_or_default</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5728\u200b<code>unwrap</code>\u200b\u65b9\u6cd5\u200b\u4f1a\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u8fd4\u56de\u200b\u8be5\u200b\u9ed8\u8ba4\u503c\u200b\u3002\u200b\u8be5\u200b\u65b9\u6cd5\u200b\u8981\u6c42\u200b<code>T</code>\u200b\u7c7b\u578b\u200b\u5b9e\u73b0\u200b\u4e86\u200b<code>Default</code>\u200b\u7279\u6027\u200b\u3002</p> <pre><code>struct MyStruct {\n    s: i32\n}\n\nfn main() {\n    let a: Option&lt;MyStruct&gt; = None;\n    println!(\"{}\", a.unwrap_or_default().s);\n    // error[E0277]: the trait bound `MyStruct: Default` is not satisfied\n}\n</code></pre> </li> <li> <p><code>unwrap_or_else</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\uff0c\u200b\u5728\u200b<code>unwrap</code>\u200b\u65b9\u6cd5\u200b\u4f1a\u200b\u89e6\u53d1\u200b\u5f02\u5e38\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u8fd4\u56de\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002\u200b\u5176\u4e2d\u200b\uff0c<code>Result</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u95ed\u5305\u200b\u53c2\u6570\u200b\u4e3a\u200b<code>E</code>\u200b\u7c7b\u578b\u200b\uff0c<code>Option</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u95ed\u5305\u200b\u53c2\u6570\u200b\u4e3a\u200b<code>()</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u9700\u8981\u200b\u548c\u200b<code>Some</code>\u200b\u6216\u200b<code>Ok</code>\u200b\u7684\u200b\u7c7b\u578b\u200b\u76f8\u540c\u200b\u3002</p> <pre><code>fn increment(x: i32) -&gt; i32 {\n    x + 1\n}\n\nfn main() {\n    let a: Result&lt;i32, i32&gt; = Err(2);\n    println!(\"{}\", a.unwrap_or_else(increment));\n    // 3\n}\n</code></pre> </li> </ul>"},{"location":"coding/rust/5/#_3","title":"\u8fd0\u7b97","text":"<p>Rust\u200b\u53ef\u4ee5\u200b\u5728\u200b\u4e0d\u200b\u53d6\u51fa\u200b<code>Option</code>\u200b\u6216\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u503c\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u5bf9\u200b\u5176\u200b\u8fdb\u884c\u200b\u8fd0\u7b97\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u4ee5\u4e0b\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u8bb0\u200b<code>A</code>\u200b\u4e3a\u200b\u8c03\u7528\u200b\u65b9\u6cd5\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff0c<code>B</code>\u200b\u4e3a\u200b\u65b9\u6cd5\u200b\u7684\u200b\u53c2\u6570\u200b\uff0c\u200b\u5373\u200b<code>A.method(B)</code>\u3002</p> <p><code>Option</code>\u200b\u548c\u200b<code>Result</code>\u200b\u7684\u200b\u771f\u503c\u200b</p> <p>\u200b\u4e0d\u59a8\u200b\u5c06\u200b<code>Option</code>\u200b\u548c\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u7684\u200b<code>Some</code>\u200b\u548c\u200b<code>Ok</code>\u200b\u89c6\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5176\u4e2d\u200b\u7684\u200b\u503c\u200b<code>T</code>\u200b\u79f0\u4e3a\u200b\u771f\u503c\u200b\uff0c<code>None</code>\u200b\u548c\u200b<code>Err</code>\u200b\u89c6\u4e3a\u200b\u5047\u200b\uff0c<code>Err</code>\u200b\u7684\u200b\u503c\u200b<code>E</code>\u200b\u89c6\u4e3a\u200b\u5047\u503c\u200b\u3002</p> <ol> <li>\u200b\u903b\u8f91\u8fd0\u7b97\u200b<code>A.or(B)</code>\u200b\u4e0e\u200b<code>A.and(B)</code>\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u548c\u200b\u81ea\u8eab\u200b\u7c7b\u578b\u200b\u76f8\u540c\u200b\u7684\u200b\u53c2\u6570\u200b\u3002<ul> <li>\u200b\u5bf9\u4e8e\u200b<code>or()</code>\uff0c\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>B</code>\u200b\u503c\u200b\u3002</li> <li>\u200b\u5bf9\u4e8e\u200b<code>and()</code>\uff0c\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u5047\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u5047\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>B</code>\u3002</li> </ul> </li> <li><code>A.or_else()</code>\u200b\u4e3a\u200b<code>A.or(B)</code>\u200b\u7684\u200b\u53d8\u79cd\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002</li> <li><code>A.and_then()</code>\u200b\u4e3a\u200b<code>A.and(B)</code>\u200b\u7684\u200b\u53d8\u79cd\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b\u4e3a\u200b\u5047\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u5047\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002</li> <li>\uff08\u200b\u4ec5\u200b<code>Option</code>\u200b\u7c7b\u578b\u200b\uff09<code>A.filter()</code>\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b<code>fn (T) -&gt; bool</code>\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u4e14\u200b\u4ee5\u200b\u5176\u200b\u771f\u503c\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u540e\u200b\u8c03\u7528\u51fd\u6570\u200b\u8fd4\u56de\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>None</code>\u3002</li> <li><code>A.map()</code>\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b<code>fn (T) -&gt; U</code>\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u540e\u200b\u8c03\u7528\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u5047\u503c\u200b\u3002\u200b\u6ce8\u610f\u200b\u8fd4\u56de\u503c\u200b\u7684\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>Option&lt;U&gt;</code>\u200b\u6216\u200b<code>Result&lt;U, E&gt;</code>\u3002<ul> <li>\uff08\u200b\u4ec5\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\uff09<code>A.map_err()</code>\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b<code>fn (E) -&gt; F</code>\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u5047\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u5047\u503c\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u540e\u200b\u8c03\u7528\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\u3002</li> <li>\uff08\u200b\u4ec5\u200b<code>Option</code>\u200b\u7c7b\u578b\u200b\uff09<code>A.map_or()</code>\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b<code>U</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u9ed8\u8ba4\u503c\u200b\u548c\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b<code>fn (T) -&gt; U</code>\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u540e\u200b\u8c03\u7528\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b\u9ed8\u8ba4\u503c\u200b\u3002</li> <li><code>A.map_or_else()</code>\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b<code>fn () -&gt; U</code>\u200b\u548c\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b<code>fn (T) -&gt; U</code>\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u540e\u200b\u8c03\u7528\u200b\u7b2c\u4e8c\u4e2a\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b\u7b2c\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002</li> </ul> </li> <li>\uff08\u200b\u4ec5\u200b<code>Option</code>\u200b\u7c7b\u578b\u200b\uff09<code>A.ok_or()</code>\u200b\u7528\u4e8e\u200b\u5c06\u200b<code>Option</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b<code>E</code>\u200b\u7c7b\u578b\u200b\u7684\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b\u53c2\u6570\u200b\u4f5c\u4e3a\u200b<code>Err</code>\u200b\u7684\u200b\u503c\u200b\u3002<ul> <li><code>A.ok_or_else()</code>\u200b\u4e3a\u200b<code>A.ok_or()</code>\u200b\u7684\u200b\u53d8\u79cd\u200b\uff0c\u200b\u63a5\u53d7\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\u4f5c\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u5982\u679c\u200b<code>A</code>\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u5219\u200b\u8fd4\u56de\u200b<code>A</code>\u200b\u7684\u200b\u771f\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u8fd4\u56de\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002</li> </ul> </li> <li><code>A.as_ref()</code>\u200b\u548c\u200b<code>A.as_mut()</code>\uff0c\u200b\u7528\u4e8e\u200b\u5c06\u200b<code>Option&lt;T&gt;</code>\u200b\u6216\u200b<code>Result&lt;T, E&gt;</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u5176\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5f15\u7528\u200b\u3002\u200b\u5176\u4e2d\u200b<code>A.as_ref()</code>\u200b\u8fd4\u56de\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5f15\u7528\u200b\uff0c\u200b\u5373\u200b<code>Option&lt;&amp;T&gt;</code>\u200b\u6216\u200b<code>Result&lt;&amp;T, &amp;E&gt;</code>\uff1b<code>A.as_mut()</code>\u200b\u8fd4\u56de\u200b\u53ef\u53d8\u200b\u5f15\u7528\u200b\uff0c\u200b\u5373\u200b<code>Option&lt;&amp;mut T&gt;</code>\u200b\u6216\u200b<code>Result&lt;&amp;mut T, &amp;mut E&gt;</code>\u3002</li> </ol>"},{"location":"coding/rust/5/#_4","title":"\u5f02\u5e38\u200b\u4f20\u9012","text":"<p>Rust\u200b\u4e2d\u200b\uff0c\u200b\u6709\u200b\u4e24\u79cd\u200b\u65b9\u5f0f\u200b\u53ef\u4ee5\u200b\u5411\u4e0a\u200b\u8fd4\u56de\u200b\u5f02\u5e38\u200b\u3002</p> <ul> <li> <p><code>?</code>\u200b\u8fd0\u7b97\u7b26\u200b\uff0c\u200b\u7528\u4e8e\u200b\u5c06\u200b<code>Option</code>\u200b\u6216\u200b<code>Result</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b<code>T</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5982\u679c\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\u4e3a\u200b<code>Err</code>\u200b\u6216\u200b\uff0c\u200b\u5219\u200b\u4f1a\u200b\u76f4\u63a5\u200b\u8fd4\u56de\u200b<code>Err</code>\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u3002</p> <pre><code>fn fn_with_error&lt;'a&gt;() -&gt; Result&lt;i32, &amp;'a str&gt; {\n    Err(\"Deterministic error\")\n}\nfn use_fn&lt;'a&gt;() -&gt; Result&lt;i32, &amp;'a str&gt; {\n    let x = fn_with_error()?; // Directly return Err(\"Deterministic error\")\n    Ok(x) // unreachable\n}\nfn main() {\n    let x: &amp;str = use_fn().unwrap_err();\n    println!(\"{}\", x); // Deterministic error\n}\n</code></pre> </li> <li> <p><code>try!</code>\u200b\u5b8f\u200b\uff0c\u200b\u4e0e\u200b<code>?</code>\u200b\u7684\u200b\u6548\u679c\u200b\u7b49\u540c\u200b\uff0c\u200b\u4f7f\u7528\u200b\u65b9\u5f0f\u200b\u4e3a\u200b<code>try!(expression)</code>\u3002\u200b\u76ee\u524d\u200b\u5df2\u7ecf\u200b\u88ab\u200b\u5f03\u7528\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>?</code>\u200b\u4ee3\u66ff\u200b\u3002</p> <pre><code>fn fn_with_error&lt;'a&gt;() -&gt; Result&lt;i32, &amp;'a str&gt; {\n    Err(\"Deterministic error\")\n}\nfn use_fn&lt;'a&gt;() -&gt; Result&lt;i32, &amp;'a str&gt; {\n    let x = try!(fn_with_error()); // error: use of deprecated `try` macro\n    Ok(x) // unreachable\n}\nfn main() {\n    let x: &amp;str = use_fn().unwrap_err();\n    println!(\"{}\", x); // Deterministic error\n}\n</code></pre> </li> </ul>"},{"location":"coding/rust/5/#_5","title":"\u4e3b\u200b\u51fd\u6570","text":"<p>Rust\u200b\u7684\u200b<code>main</code>\u200b\u51fd\u6570\u200b\u53ea\u80fd\u200b\u53d6\u200b<code>()</code>\u200b\u6216\u200b<code>Result&lt;(), E&gt;</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5176\u4e2d\u200b<code>E</code>\u200b\u4e3a\u200b<code>std::error::Error</code>\u200b\u7684\u200b\u5b9e\u73b0\u200b\u7c7b\u578b\u200b\u3002\u200b\u5982\u679c\u200b<code>main</code>\u200b\u51fd\u6570\u200b\u8fd4\u56de\u200b<code>Result</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5219\u200b\u4f1a\u200b\u5c06\u200b<code>Err</code>\u200b\u4e2d\u200b\u7684\u200b\u503c\u200b\u6253\u5370\u200b\u5230\u200b<code>stderr</code>\u200b\u5e76\u200b\u9000\u51fa\u200b\u7a0b\u5e8f\u200b\u3002</p> <pre><code>use std::fs::File;\n\nfn main() -&gt; std::io::Result&lt;()&gt; {\n    let _ = File::open(\"not-existing-file.txt\")?; // Result::Err\n\n    Ok(()) // Default return value\n    // Must with this line, otherwise return () which is incompatible\n}\n</code></pre> <p>\u200b\u5982\u679c\u200b<code>main</code>\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u7c7b\u578b\u200b\u4e0d\u662f\u200b<code>()</code>\u200b\u6216\u200b<code>Result</code>\uff0c\u200b\u5219\u200b\u65e0\u6cd5\u200b\u901a\u8fc7\u200b\u7f16\u8bd1\u200b\u3002</p> <pre><code>fn main() -&gt; i32 {\n    0\n}\n// error[E0277]: `main` has invalid return type `i32`\n</code></pre>"},{"location":"coding/rust/5/#_6","title":"\u81ea\u5b9a\u4e49\u200b\u5f02\u5e38","text":"<p>\u200b\u5728\u200bRust\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u81ea\u5b9a\u4e49\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u200b\u3002\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b<code>std::error::Error</code>\u200b\u7279\u6027\u200b\uff0c\u200b\u8be5\u200b\u7279\u6027\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4ee5\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <ul> <li><code>fn source(&amp;self) -&gt; Option&lt;&amp;(dyn Error + 'static)&gt;</code>\uff0c\u200b\u7528\u4e8e\u200b\u8fd4\u56de\u200b\u5f15\u8d77\u200b\u5f02\u5e38\u200b\u7684\u200b\u539f\u56e0\u200b\uff08\u200b\u53ef\u200b\u9009\u200b\uff09\u3002</li> <li><code>fn Debug::fmt(&amp;self, f: &amp;mut Formatter&lt;'_&gt;) -&gt; Result</code>\uff0c\u200b\u7528\u4e8e\u200b\u683c\u5f0f\u5316\u200b\u5f02\u5e38\u200b\u4fe1\u606f\u200b\uff08\u200b\u7528\u4e8e\u200b\u8c03\u8bd5\u200b\uff09\u3002</li> <li><code>fn Display::fmt(&amp;self, f: &amp;mut Formatter&lt;'_&gt;) -&gt; Result</code>\uff0c\u200b\u7528\u4e8e\u200b\u683c\u5f0f\u5316\u200b\u5f02\u5e38\u200b\u4fe1\u606f\u200b\uff08\u200b\u7528\u4e8e\u200b\u8f93\u51fa\u200b\uff09\u3002</li> </ul> <pre><code>struct MyError {\n    code: i32,\n    message: String\n}\n\nimpl std::error::Error for MyError {\n    fn source(&amp;self) -&gt; Option&lt;&amp;(dyn std::error::Error + 'static)&gt; {\n        None\n    }\n}\n\nimpl std::fmt::Debug for MyError {\n    fn fmt(&amp;self, f: &amp;mut std::fmt::Formatter&lt;'_&gt;) -&gt; std::fmt::Result {\n        write!(\n            f, \"MyError {{ code: {}, message: {} }} at file {} line {}\",\n            self.code, self.message, file!(), line!()\n            // file!() and line!() are macros to get the current file and line\n        )\n    }\n}\n\nimpl std::fmt::Display for MyError {\n    fn fmt(&amp;self, f: &amp;mut std::fmt::Formatter&lt;'_&gt;) -&gt; std::fmt::Result {\n        write!(f, \"MyError {{ code: {}, message: {} }}\", self.code, self.message)\n    }\n}\n\n// Use the error\n\nfn raise_error() -&gt; Result&lt;(), MyError&gt; {\n    Err(MyError { code: 1, message: \"Error message\".to_string() })\n}\n\nfn main() -&gt; Result&lt;(), MyError&gt; {\n    let _ = raise_error()?;\n    // MyError { code: 1, message: Error message } at file src/main.rs line 16\n    Ok(())\n}\n</code></pre>"},{"location":"coding/rust/5/#_7","title":"\u5f02\u5e38\u200b\u8f6c\u6362","text":"<p>\u200b\u7531\u4e8e\u200bRust\u200b\u4e2d\u200b\u7684\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u200b\u662f\u200b\u9759\u6001\u200b\u7684\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5728\u200b\u5f02\u5e38\u200b\u4f20\u9012\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\uff0c\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u9047\u5230\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u200b\u4e0d\u200b\u5339\u914d\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002\u200b\u6b64\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>From</code>\u200b\u7279\u6027\u200b\u5c06\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b\u53e6\u200b\u4e00\u79cd\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4ee5\u200b<code>std::io::Error</code>\u200b\u4e3a\u4f8b\u200b\u3002</p> <pre><code>impl std::convert::From&lt;std::io::Error&gt; for MyError {\n    fn from(error: std::io::Error) -&gt; Self {\n        MyError { code: 2, message: error.to_string() }\n    }\n}\n</code></pre> <p>\u200b\u6b64\u540e\u200b\uff0c\u200b\u65e0\u9700\u200b\u663e\u5f0f\u200b\u8c03\u7528\u200b<code>From</code>\u200b\u7279\u6027\u200b\u7684\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5373\u53ef\u200b\u5c06\u200b<code>std::io::Error</code>\u200b\u7c7b\u578b\u8f6c\u6362\u200b\u4e3a\u200b<code>MyError</code>\u200b\u7c7b\u578b\u200b\u3002</p>"},{"location":"coding/rust/6/","title":"Supplements for Basics","text":""},{"location":"coding/rust/6/#_1","title":"\u5ffd\u7565\u200b\u53d8\u91cf","text":"<p>\u200b\u5728\u200b\u58f0\u660e\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\u5374\u200b\u6ca1\u6709\u200b\u4f7f\u7528\u200b\u65f6\u200b\uff0cRust\u200b\u4f1a\u200b\u7ed9\u51fa\u200b\u8b66\u544a\u200b\u3002\u200b\u5982\u679c\u200b\u4e0d\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u8fd9\u4e2a\u200b\u53d8\u91cf\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5728\u200b\u53d8\u91cf\u540d\u200b\u524d\u200b\u52a0\u4e0a\u200b\u4e0b\u5212\u7ebf\u200b\u524d\u7f00\u200b\u4f5c\u4e3a\u200b\u6807\u8bb0\u200b\u3002\u200b\u540d\u4e3a\u200b<code>_</code>\u200b\u7684\u200b\u53d8\u91cf\u200b\u662f\u200b\u4e00\u4e2a\u200b\u7279\u6b8a\u200b\u7684\u200b\u53d8\u91cf\u200b\uff0c\u200b\u5b83\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u58f0\u660e\u200b\u591a\u6b21\u200b\uff0c\u200b\u4f46\u662f\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u4f7f\u7528\u200b\u3002</p> <p>  Download source code</p> <pre><code>fn main() {\n    let _: i32 = 5;\n    println!(\"{}\", _);\n    // error: in expressions, `_` can only be used on the left-hand side of an assignment\n}\n</code></pre>"},{"location":"coding/rust/6/#_2","title":"\u6a21\u5f0f\u5339\u914d","text":""},{"location":"coding/rust-gui/","title":"Rust GUI library","text":"<p>The following pages are notes from Building a simple text editor with iced, a cross-platform GUI library for Rust using iced lib.</p> <ol> <li>Hello World</li> <li>Multi-line Input</li> <li>Theme and Cursor Indicator</li> <li>Async File Loading</li> <li>File Picker</li> <li>File Path Indicator</li> <li>New and Save</li> <li>Button prettify</li> <li>Syntax Highlighting</li> <li>Misc</li> </ol>"},{"location":"coding/rust-gui/1-hello-world/","title":"Hello World App","text":"<p>Keywords: <code>iced::Sandbox</code>, <code>iced::Settings</code>, <code>iced::widget::text</code></p>"},{"location":"coding/rust-gui/1-hello-world/#_1","title":"\u521b\u5efa\u200b\u9879\u76ee","text":"<p>\u200b\u9996\u5148\u200b\uff0c\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b Rust \u200b\u9879\u76ee\u200b\uff0c\u200b\u5e76\u200b\u6dfb\u52a0\u200b\u5bf9\u200b<code>iced</code>\u200b\u5e93\u200b\u7684\u200b\u4f9d\u8d56\u200b\u3002</p> <pre><code>cargo new text-editor\ncd text-editor\ncargo add iced\n</code></pre> <p>\u200b\u6b64\u65f6\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>fn main() {\n    println!(\"Hello, world!\");\n}\n</code></pre>"},{"location":"coding/rust-gui/1-hello-world/#gui","title":"\u521b\u5efa\u200bGUI\u200b\u5e94\u7528","text":"<p><code>iced</code>\u200b\u7684\u200bGUI\u200b\u7c7b\u6709\u200b<code>Application</code>\u200b\u548c\u200b<code>Sandbox</code>\u200b\u4e24\u79cd\u200b\u3002\u200b\u5176\u4e2d\u200b<code>Sandbox</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u7b80\u5316\u7248\u200b\u7684\u200b<code>Application</code>\uff0c\u200b\u63d0\u4f9b\u200b\u4e86\u200b\u4e00\u4e9b\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u884c\u4e3a\u200b\u3002<code>Sandbox</code>\u200b\u7279\u6027\u200b\u5305\u542b\u200b\u5982\u4e0b\u200b\u65b9\u6cd5\u200b\uff1a</p> <pre><code>pub trait Sandbox {\n    type Message: std::fmt::Debug + Send;\n    fn new() -&gt; Self;\n    fn title(&amp;self) -&gt; String;\n    fn update(&amp;mut self, message: Self::Message);\n    fn view(&amp;self) -&gt; Element&lt;'_, Self::Message&gt;;\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::default()\n    }\n    fn style(&amp;self) -&gt; theme::Application {\n        theme::Application::default()\n    }\n    fn scale_factor(&amp;self) -&gt; f64 {\n        1.0\n    }\n    fn run(settings: Settings&lt;()&gt;) -&gt; Result&lt;(), Error&gt;\n    where\n        Self: 'static + Sized,\n    {\n        &lt;Self as Application&gt;::run(settings)\n    }\n}\n</code></pre> <p>\u200b\u4e3a\u4e86\u200b\u4f7f\u7528\u200b<code>Sandbox</code>\uff0c\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b\u5176\u4e2d\u200b\u672a\u200b\u5b9e\u73b0\u200b\u7684\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5373\u200b<code>Message</code>\u3001<code>new</code>\u3001<code>title</code>\u3001<code>update</code>\u200b\u548c\u200b<code>view</code>\u3002</p> <ul> <li> <p><code>Message</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u679a\u4e3e\u200b\u7c7b\u578b\u200b\uff0c\u200b\u7528\u4e8e\u200b\u5b9a\u4e49\u200b\u5e94\u7528\u200b\u4f1a\u200b\u4ea7\u751f\u200b\u7684\u200b\u6d88\u606f\u200b\u7c7b\u578b\u200b\uff0c\u200b\u9700\u8981\u200b\u5b9e\u73b0\u200b<code>std::fmt::Debug</code>\u200b\u548c\u200b<code>Send</code>\u200b\u7279\u6027\u200b\u3002</p> <pre><code>#[derive(Debug)] // Inherit the Debug trait\nenum Message {} // No message required\n</code></pre> </li> <li> <p><code>new</code>\u200b\u65b9\u6cd5\u200b\u7528\u4e8e\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b<code>Sandbox</code>\u200b\u5b9e\u4f8b\u200b\uff0c\u200b\u521d\u59cb\u5316\u200b\u5b9e\u4f8b\u200b\u72b6\u6001\u200b\uff0c\u200b\u4e00\u822c\u200b\u60c5\u51b5\u200b\u4e0b\u200b\u76f4\u63a5\u200b\u8fd4\u56de\u200b<code>Self</code>\u200b\u5373\u53ef\u200b\u3002</p> <pre><code>fn new() -&gt; Self {\n    Self\n}\n</code></pre> </li> <li> <p><code>title</code>\u200b\u65b9\u6cd5\u200b\u7528\u4e8e\u200b\u8fd4\u56de\u200bGUI\u200b\u7a97\u53e3\u200b\u7684\u200b\u6807\u9898\u200b\u3002</p> <pre><code>fn title(&amp;self) -&gt; String {\n    String::from(\"A text editor\")\n}\n</code></pre> </li> <li> <p><code>update</code>\u200b\u65b9\u6cd5\u200b\u548c\u200b<code>view</code>\u200b\u65b9\u6cd5\u200b\u5171\u540c\u200b\u7ec4\u6210\u200b\u5e94\u7528\u200b\u7684\u200b\u6d88\u606f\u200b\u5faa\u73af\u200b\uff1a<code>update</code>\u200b\u65b9\u6cd5\u200b\u7528\u4e8e\u200b\u5904\u7406\u200b\u6d88\u606f\u200b\uff0c\u200b\u66f4\u65b0\u200b\u5e94\u7528\u200b\u72b6\u6001\u200b\uff0c<code>view</code>\u200b\u65b9\u6cd5\u200b\u7528\u4e8e\u200b\u5728\u200b\u72b6\u6001\u200b\u66f4\u65b0\u200b\u540e\u200b\u66f4\u65b0\u200b\u5e94\u7528\u200b\u754c\u9762\u200b\u3002\u200b\u6b64\u5904\u200b\u6211\u4eec\u200b\u5728\u200b<code>view</code>\u200b\u65b9\u6cd5\u200b\u4e2d\u200b\u653e\u7f6e\u200b\u4e00\u4e2a\u200b\u6587\u672c\u200b\u63a7\u4ef6\u200b\u3002</p> <pre><code>fn update(&amp;mut self, message: Self::Message) {\n    match message {\n        // No message to handle\n    }\n}\n\nfn view(&amp;self) -&gt; Element&lt;'_, Self::Message&gt; {\n    text(\"Hello, world!\").into()\n}\n</code></pre> </li> </ul> <p>\u200b\u901a\u8fc7\u200b<code>Sandbox::run</code>\u200b\u65b9\u6cd5\u200b\u542f\u52a8\u200b\u5e94\u7528\u200b\uff0c\u200b\u8be5\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b<code>Result&lt;(), Error&gt;</code>\u200b\u7c7b\u578b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u4f5c\u4e3a\u200b<code>main</code>\u200b\u51fd\u6570\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use iced::{Element, Sandbox, Settings};\nuse iced::widget::text;\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor;\n\n#[derive(Debug)]\nenum Message {}\n\nimpl Sandbox for Editor {\n    type Message = Message; // Define the type of messages\n    fn new() -&gt; Self {\n        Self\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) {\n        // Handle messages here\n        match message {}\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        text(\"Hello, world!\").into()\n    }\n}\n</code></pre>"},{"location":"coding/rust-gui/10-misc/","title":"Miscellaneous","text":"<p>\u200b\u672c\u200b\u8282\u200b\u6dfb\u52a0\u200b\u4e00\u4e9b\u200b\u989d\u5916\u200b\u529f\u80fd\u200b\u3002</p> <p>\u200b\u9996\u5148\u200b\uff0c\u200b\u5f53\u200b\u6587\u4ef6\u200b\u6ca1\u6709\u200b\u88ab\u200b\u4fee\u6539\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8bbe\u7f6e\u200b\u7981\u7528\u200b\u4fdd\u5b58\u200b\u6309\u94ae\u200b\u3002\u200b\u8fd9\u6837\u200b\u53ef\u4ee5\u200b\u907f\u514d\u200b\u7528\u6237\u200b\u8bef\u64cd\u4f5c\u200b\u3002\u200b\u5728\u200b<code>on_press_maybe</code>\u200b\u4e2d\u200b\u4f20\u5165\u200b<code>None</code>\u200b\u5373\u53ef\u200b\u3002\u200b\u540c\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6839\u636e\u200b\u6587\u4ef6\u200b\u7684\u200b\u4fee\u6539\u200b\u60c5\u51b5\u200b\u8bbe\u7f6e\u200b\u6309\u94ae\u200b\u7684\u200b\u6837\u5f0f\u200b\u3002</p> <pre><code>fn toolbar_button&lt;'a&gt;(description: &amp;str, callback: Option&lt;Message&gt;) -&gt; Element&lt;'a, Message&gt; {\n    let font = Font::with_name(\"editor-icon\");\n    let lower = description.to_lowercase();\n    let icon = text(match lower.as_str() {\n        \"new\" =&gt; '\\u{E800}',\n        \"open\" =&gt; '\\u{F115}',\n        \"save\" =&gt; '\\u{E801}',\n        _ =&gt; ' '\n    }).font(font);\n    let is_disabled = callback.is_none();\n    tooltip(\n        button(container(icon)\n            .width(30)  // Set the width of the button\n            .center_x() // Center the icon\n        ).on_press_maybe(callback).style(\n            if is_disabled {\n                theme::Button::Secondary\n            } else {\n                theme::Button::Primary\n            }\n        ),\n        description, tooltip::Position::FollowCursor\n    ).style(theme::Container::Box).into()\n}\n</code></pre> <p>\u200b\u540c\u65f6\u200b\u4fee\u6539\u200b<code>toolbar_button</code>\u200b\u7684\u200b\u8c03\u7528\u200b\u3002</p> <pre><code>// ... In `view` function\nlet controls = row![\n    toolbar_button(\"New\", Some(Message::NewButtonPressed)),\n    toolbar_button(\"Open\", Some(Message::OpenButtonPressed)),\n    toolbar_button(\"Save\", if self.modified { Some(Message::SaveButtonPressed) } else { None }),\n    horizontal_space(Length::Fill),\n    pick_list(highlighter::Theme::ALL, Some(self.theme), Message::ThemeChanged)\n].spacing(10);\n</code></pre> <p>\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u6dfb\u52a0\u200b\u4e0d\u540c\u200b\u7684\u200b\u5feb\u6377\u952e\u200b\uff0c\u200b\u4ee5\u200b\u65b9\u4fbf\u200b\u7528\u6237\u200b\u64cd\u4f5c\u200b\u3002</p> <pre><code>// In `impl Application for Editor`\nfn subscription(&amp;self) -&gt; Subscription&lt;Message&gt; {\n    keyboard::on_key_press(|keycode, modifier| {\n        match (keycode, modifier) {\n            (keyboard::KeyCode::S, keyboard::Modifiers::COMMAND) =&gt; {\n                Some(Message::SaveButtonPressed)\n            },\n            (keyboard::KeyCode::O, keyboard::Modifiers::COMMAND) =&gt; {\n                Some(Message::OpenButtonPressed)\n            },\n            (keyboard::KeyCode::N, keyboard::Modifiers::COMMAND) =&gt; {\n                Some(Message::NewButtonPressed)\n            },\n            _ =&gt; None\n        }\n    })\n}\n</code></pre> <p>\u200b\u8fd9\u6837\u200b\uff0c\u200b\u7528\u6237\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>Command + S</code>\u200b\u6765\u200b\u4fdd\u5b58\u200b\u6587\u4ef6\u200b\uff0c<code>Command + O</code>\u200b\u6765\u200b\u6253\u5f00\u200b\u6587\u4ef6\u200b\uff0c<code>Command + N</code>\u200b\u6765\u200b\u65b0\u5efa\u200b\u6587\u4ef6\u200b\u3002</p> <p>\u200b\u6587\u4ef6\u200b\u7684\u200b\u6807\u9898\u680f\u200b\u901a\u5e38\u200b\u663e\u793a\u6587\u4ef6\u200b\u8def\u5f84\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u548c\u200b\u5de6\u4e0b\u89d2\u200b\u7684\u200b\u72b6\u6001\u680f\u200b\u4fdd\u6301\u200b\u540c\u6b65\u200b\u3002</p> <pre><code>// ... In `impl Application for Editor`\nfn title(&amp;self) -&gt; String {\n    let path_text = match &amp;self.path {\n        None =&gt; String::from(\"New file\"),\n        Some(path) =&gt; path.to_string_lossy().to_string()\n    };\n    let suffix = if self.modified { \"*\" } else { \"\" };\n    format!(\"{}{}\", path_text, suffix)\n}\n\n// ... In `view` function\nlet path_indicator = if let Some(error) = &amp;self.error {\n    match error {\n        Error::DialogClosed =&gt; text(\"Dialog closed\"),\n        Error::IO(kind) =&gt; text(format!(\"I/O error: {:?}\", kind))\n    }\n} else {\n    text(self.title())\n};\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use std::io;\nuse std::path::{Path, PathBuf};\nuse std::sync::Arc;\n\nuse iced::{executor, keyboard, theme, Application, Command, Element, Font, Length, Settings, Subscription, Theme};\nuse iced::highlighter::{self, Highlighter};\nuse iced::widget::{button, column, container, horizontal_space, pick_list, row, text, text_editor, tooltip};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings {\n        fonts: vec![include_bytes!(\"../fonts/editor-icon.ttf\").as_slice().into()],\n        ..Default::default()\n    })\n}\n\nstruct Editor {\n    path: Option&lt;PathBuf&gt;,\n    content: text_editor::Content,\n    modified: bool,\n    error: Option&lt;Error&gt;,\n    theme: highlighter::Theme\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    FileSaved(Result&lt;PathBuf, Error&gt;),\n    ThemeChanged(highlighter::Theme),\n    NewButtonPressed,\n    OpenButtonPressed,\n    SaveButtonPressed\n}\n\nimpl Application for Editor {\n    type Message = Message; // Define the type of messages\n    type Theme = Theme;\n    type Executor = executor::Default; // Engine for running async tasks\n    type Flags = ();\n\n    fn new(_flags: Self::Flags) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                content: text_editor::Content::new(),\n                error: None,\n                modified: false,\n                path: None,\n                theme: highlighter::Theme::SolarizedDark\n            },\n            Command::perform(\n            load_file(format!(\"{}/src/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n            Message::FileOpened\n            )\n        )\n    }\n\n    fn title(&amp;self) -&gt; String {\n        let path_text = match &amp;self.path {\n            None =&gt; String::from(\"New file\"),\n            Some(path) =&gt; path.to_string_lossy().to_string()\n        };\n        let suffix = if self.modified { \"*\" } else { \"\" };\n        format!(\"{}{}\", path_text, suffix)\n    }\n\n    fn subscription(&amp;self) -&gt; Subscription&lt;Message&gt; {\n        keyboard::on_key_press(|keycode, modifier| {\n            match (keycode, modifier) {\n                (keyboard::KeyCode::S, keyboard::Modifiers::COMMAND) =&gt; {\n                    Some(Message::SaveButtonPressed)\n                },\n                (keyboard::KeyCode::O, keyboard::Modifiers::COMMAND) =&gt; {\n                    Some(Message::OpenButtonPressed)\n                },\n                (keyboard::KeyCode::N, keyboard::Modifiers::COMMAND) =&gt; {\n                    Some(Message::NewButtonPressed)\n                },\n                _ =&gt; None\n            }\n        })\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                match &amp;action {\n                    text_editor::Action::Edit(_) =&gt; self.modified = true,\n                    _ =&gt; {}\n                }\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            Message::FileSaved(Ok(path)) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                Command::none()\n            },\n            Message::FileOpened(Err(error)) | Message::FileSaved(Err(error)) =&gt; {\n                self.error = Some(error);\n                Command::none()\n            },\n            Message::ThemeChanged(theme) =&gt; {\n                self.theme = theme;\n                Command::none()\n            },\n            Message::NewButtonPressed =&gt; {\n                self.content = text_editor::Content::new();\n                self.error = None;\n                self.path = None;\n                self.modified = false;\n                Command::none()\n            },\n            Message::OpenButtonPressed =&gt; {\n                self.modified = false;\n                Command::perform(pick_file(), Message::FileOpened)\n            },\n            Message::SaveButtonPressed =&gt; {\n                let content = self.content.text();\n                match self.modified {\n                    false =&gt; Command::none(),\n                    true =&gt; Command::perform(\n                        save_file(self.path.clone(), content),\n                        Message::FileSaved\n                    )\n                }\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content)\n            .on_edit(Message::EditorEdit)\n            .highlight::&lt;Highlighter&gt;(\n                highlighter::Settings {\n                    theme: self.theme,\n                    extension: self.path\n                        .as_ref()\n                        .and_then(|path| path.extension()?.to_str())\n                        .unwrap_or(\"rs\")\n                        .to_string()\n                }, |highlighter, _theme| {\n                    highlighter.to_format()\n                }\n            )\n            .font(Font::MONOSPACE);\n        let controls = row![\n            toolbar_button(\"New\", Some(Message::NewButtonPressed)),\n            toolbar_button(\"Open\", Some(Message::OpenButtonPressed)),\n            toolbar_button(\"Save\", if self.modified { Some(Message::SaveButtonPressed) } else { None }),\n            horizontal_space(Length::Fill),\n            pick_list(highlighter::Theme::ALL, Some(self.theme), Message::ThemeChanged)\n        ].spacing(10);\n\n        // Query cursor position\n        let path_indicator = if let Some(error) = &amp;self.error {\n            match error {\n                Error::DialogClosed =&gt; text(\"Dialog closed\"),\n                Error::IO(kind) =&gt; text(format!(\"I/O error: {:?}\", kind))\n            }\n        } else {\n            text(self.title())\n        };\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![\n            path_indicator,\n            horizontal_space(Length::Fill),\n            cursor_indicator\n        ];\n\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        if self.theme.is_dark() {\n            Theme::Dark\n        } else {\n            Theme::Light\n        }\n    }\n}\n\nasync fn pick_file() -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let file_handle = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?;\n    load_file(file_handle.path()).await\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let content = tokio::fs::read_to_string(path.as_ref())\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO);\n    content.and_then(|content| Ok((path.as_ref().to_path_buf(), content)))\n}\n\nasync fn save_file(path: Option&lt;PathBuf&gt;, content: String) -&gt; Result&lt;PathBuf, Error&gt; {\n    let path = if let Some(path) = path {\n        path\n    } else {\n        rfd::AsyncFileDialog::new()\n            .set_title(\"Save the file to...\")\n            .save_file()\n            .await\n            .ok_or(Error::DialogClosed)?\n            .path()\n            .to_path_buf()\n    };\n    tokio::fs::write(&amp;path, content)\n        .await\n        .map_err(|err| err.kind())\n        .map_err(Error::IO)\n        .map(|_| path)\n}\n\nfn toolbar_button&lt;'a&gt;(description: &amp;str, callback: Option&lt;Message&gt;) -&gt; Element&lt;'a, Message&gt; {\n    let font = Font::with_name(\"editor-icon\");\n    let lower = description.to_lowercase();\n    let icon = text(match lower.as_str() {\n        \"new\" =&gt; '\\u{E800}',\n        \"open\" =&gt; '\\u{F115}',\n        \"save\" =&gt; '\\u{E801}',\n        _ =&gt; ' '\n    }).font(font);\n    let is_disabled = callback.is_none();\n    tooltip(\n        button(container(icon)\n            .width(30)  // Set the width of the button\n            .center_x() // Center the icon\n        ).on_press_maybe(callback).style(\n            if is_disabled {\n                theme::Button::Secondary\n            } else {\n                theme::Button::Primary\n            }\n        ),\n        description, tooltip::Position::FollowCursor\n    ).style(theme::Container::Box).into()\n}\n\n#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n</code></pre>"},{"location":"coding/rust-gui/2-multi-line-input/","title":"Add Multi-line Input","text":"<p>Keywords: <code>iced::widget::text_editor</code>, <code>iced::widget::container</code>.</p> <p>\u200b\u672c\u200b\u8282\u200b\u6211\u4eec\u200b\u5c06\u200b<code>text</code>\u200b\u63a7\u4ef6\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>text_editor</code>\u200b\u63a7\u4ef6\u200b\uff0c\u200b\u4ee5\u200b\u5b9e\u73b0\u200b\u591a\u884c\u200b\u6587\u672c\u200b\u7684\u200b\u8f93\u5165\u200b\u3002</p> <p>iced\u200b\u5e93\u200b\u7248\u672c\u200b</p> <p><code>text_editor</code>\u200b\u63a7\u4ef6\u200b\u5c1a\u672a\u200b\u5728\u200b\u6b63\u5f0f\u200b\u7248\u672c\u200b\u4e2d\u200b\u53d1\u5e03\u200b\uff0c\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>git</code>\u200b\u4ed3\u5e93\u200b\u4e2d\u200b\u7684\u200b\u4ee3\u7801\u200b\u3002</p> <pre><code>[dependencies]\niced = { git = \"https://github.com/iced-rs/iced.git\", rev = \"refs/tags/text-editor\" }\n</code></pre> <p>\u200b\u9996\u5148\u200b\uff0c\u200b\u5c06\u200b\u5f15\u5165\u200b\u7684\u200b<code>iced::widget::text</code>\u200b\u66ff\u6362\u200b\u4e3a\u200b<code>iced::widget::text_editor</code>\u3002<code>text_editor</code>\u200b\u662f\u200b\u6709\u200b\u72b6\u6001\u200b\u7684\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b<code>new</code>\u200b\u4e2d\u200b\u521d\u59cb\u5316\u200b\u7a7a\u95f4\u200b\u72b6\u6001\u200b\uff0c\u200b\u5e76\u4e14\u200b\u5728\u200b<code>view</code>\u200b\u4e2d\u200b\u6839\u636e\u200b\u72b6\u6001\u200b\u66f4\u65b0\u200b\u63a7\u4ef6\u200b\u5185\u5bb9\u200b\u3002</p> <pre><code>struct Editor {\n    content: text_editor::Content\n}\n\n// ...\n\nimpl Sandbox for Editor {\n    type Message = Message;\n\n    fn new() -&gt; Self {\n        Editor {\n            content: text_editor::Content::new() // Initialize the content\n        }\n    }\n\n    // ...\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        text_editor(&amp;self.content).into() // Link the content state\n    }\n}\n</code></pre> <p>\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c<code>into</code>\u200b\u65b9\u6cd5\u200b\u4f1a\u200b\u4f7f\u200b\u63a7\u4ef6\u200b\u5145\u6ee1\u200b\u6574\u4e2a\u200b\u7a97\u53e3\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>container</code>\u200b\u5305\u56f4\u200b<code>text_editor</code>\u200b\u63a7\u4ef6\u200b\u8f85\u52a9\u200b\u5e03\u5c40\u200b\u3002</p> <pre><code>use iced::widget::container;\n\n// ...\n\nimpl Sandbox for Editor {\n    // ...\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        let editor = text_editor(&amp;self.content);\n        container(editor).padding(10).into() // Add padding and wrap the editor\n    }\n}\n</code></pre> <p>\u200b\u6b64\u65f6\u200b\uff0c\u200b\u591a\u884c\u200b\u6587\u672c\u200b\u8f93\u5165\u6846\u200b\u7684\u200b\u5e03\u5c40\u200b\u5df2\u7ecf\u200b\u5b8c\u6210\u200b\u3002\u200b\u4f46\u200b\u7531\u4e8e\u200b\u8f93\u5165\u6846\u200b\u6ca1\u6709\u200b\u7ed1\u5b9a\u200b\u4e8b\u4ef6\u5904\u7406\u200b\uff0c\u200b\u56e0\u6b64\u200b\u76ee\u524d\u200b\u65e0\u6cd5\u200b\u8f93\u5165\u200b\u6587\u672c\u200b\u3002<code>text_editor</code>\u200b\u63a7\u4ef6\u200b\u652f\u6301\u200b\u7684\u200b\u4e8b\u4ef6\u200b\u6709\u200b</p> <pre><code>pub enum Action {\n    Move(Motion),\n    Select(Motion),\n    SelectWord,\n    SelectLine,\n    Edit(Edit),\n    Click(Point),\n    Drag(Point),\n    Scroll { lines: i32 },\n}\n</code></pre> <p>\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u5c06\u200b<code>Edit</code>\u200b\u4e8b\u4ef6\u200b\u5728\u200b<code>update</code>\u200b\u548c\u200b<code>view</code>\u200b\u4e2d\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b\uff0c\u200b\u540c\u65f6\u200b\u66f4\u65b0\u200b<code>Message</code>\u200b\u679a\u4e3e\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action)\n}\n\n// ...\n\nimpl Sandbox for Editor {\n    // ...\n    fn update(&amp;mut self, message: Message) {\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                self.content = self.content.edit(action);\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit); // Bind the edit event\n        container(editor).padding(10).into() // Add padding and wrap the editor\n    }\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use iced::{Element, Sandbox, Settings};\nuse iced::widget::{text_editor, container};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor {\n    content: text_editor::Content\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action)\n}\n\nimpl Sandbox for Editor {\n    type Message = Message; // Define the type of messages\n\n    fn new() -&gt; Self {\n        Self {\n            content: text_editor::Content::new()\n        }\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                self.content.edit(action);\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n        container(editor).padding(10).into()\n    }\n}\n</code></pre>"},{"location":"coding/rust-gui/3-theme-and-cursor-indicator/","title":"Theme and Cursor Indicator","text":"<p>Keywords: <code>iced::widget::column</code>, <code>iced::widget::row</code>, <code>iced::widget::horizontal_space</code>, <code>iced::Theme</code>, <code>iced::Length</code></p> <p>\u200b\u672c\u8282\u200b\u5c06\u200b\u7a97\u53e3\u200b\u8bbe\u7f6e\u200b\u4e3a\u200b\u6697\u8272\u200b\u8c03\u200b\uff0c\u200b\u5e76\u4e14\u200b\u901a\u8fc7\u200b<code>text</code>\u200b\u63a7\u4ef6\u200b\u663e\u793a\u200b\u5149\u6807\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002</p>"},{"location":"coding/rust-gui/3-theme-and-cursor-indicator/#_1","title":"\u8bbe\u7f6e\u200b\u7a97\u53e3\u200b\u4e3b\u9898","text":"<p>\u200b\u7a97\u53e3\u200b\u4e3b\u9898\u200b\u901a\u8fc7\u200b<code>theme</code>\u200b\u51fd\u6570\u200b\u8bbe\u7f6e\u200b\uff0c\u200b\u5b9a\u4e49\u200b\u4e3b\u9898\u200b\u98ce\u683c\u200b\u7684\u200b\u7c7b\u200b\u4e3a\u200b<code>iced::Theme</code>\uff0c<code>Light</code>\u200b\u548c\u200b<code>Dark</code>\u200b\u5206\u522b\u200b\u8868\u793a\u200b\u4eae\u8272\u200b\u548c\u200b\u6697\u8272\u200b\u8c03\u200b\u3002</p> <pre><code>use iced::Theme\n\n// ...\n\nimpl Sandbox for Editor {\n    // ...\n\n    fn theme(&amp;self) -&gt; iced::Theme {\n        iced::Theme::Dark // Set the window theme to dark\n    }\n}\n</code></pre>"},{"location":"coding/rust-gui/3-theme-and-cursor-indicator/#_2","title":"\u8bfb\u53d6\u200b\u6587\u4ef6\u200b\u5185\u5bb9","text":"<p>\u200b\u5728\u200b\u7a0b\u5e8f\u200b\u542f\u52a8\u200b\u65f6\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u8bbe\u7f6e\u200b\u4e00\u4e2a\u200b\u9ed8\u8ba4\u200b\u7684\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff0c\u200b\u6b64\u5904\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>include_str!</code>\u200b\u5b8f\u5728\u200b\u7f16\u8bd1\u200b\u65f6\u200b\u8bfb\u53d6\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\u5e76\u200b\u7528\u4e8e\u200b\u521d\u59cb\u5316\u200b\u6587\u672c\u6846\u200b\u7684\u200b\u72b6\u6001\u200b\u3002</p> <pre><code>impl Sandbox for Editor {\n    // ...\n\n    fn new() -&gt; Self {\n        Self {\n            content: text_editor::Content::with(include_str!(\"main.rs\"))\n        }\n    }\n}\n</code></pre>"},{"location":"coding/rust-gui/3-theme-and-cursor-indicator/#_3","title":"\u663e\u793a\u200b\u5149\u6807\u200b\u4f4d\u7f6e","text":"<p>\u200b\u6587\u672c\u6846\u200b\u7684\u200b\u5149\u6807\u200b\u4f4d\u7f6e\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>self.content</code>\u200b\u72b6\u6001\u200b\u7684\u200b<code>cursor_position</code>\u200b\u65b9\u6cd5\u200b\u83b7\u53d6\u200b\uff0c\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u5305\u542b\u200b\u5149\u6807\u200b\u4f4d\u7f6e\u200b\uff08\u200b\u4ece\u200b0\u200b\u5f00\u59cb\u200b\uff09\u200b\u7684\u200b\u5143\u7ec4\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6839\u636e\u200b\u5149\u6807\u200b\u4f4d\u7f6e\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b<code>text</code>\u200b\u63a7\u4ef6\u200b\u7528\u4e8e\u200b\u663e\u793a\u200b\u3002</p> <ul> <li>\u200b\u901a\u8fc7\u200b<code>column!</code>\u200b\u5b8f\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u5217\u200b\u5e03\u5c40\u200b\uff0c\u200b\u4e0a\u8fb9\u200b\u5305\u542b\u200b\u8f93\u5165\u200b\u6587\u672c\u6846\u200b\uff0c\u200b\u4e0b\u8fb9\u200b\u5305\u542b\u200b\u5149\u6807\u200b\u4f4d\u7f6e\u200b\u663e\u793a\u200b\u3002</li> <li>\u200b\u901a\u8fc7\u200b<code>row!</code>\u200b\u5b8f\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u884c\u200b\u5e03\u5c40\u200b\uff0c\u200b\u7ed3\u5408\u200b<code>Length::Fill</code>\u200b\u628a\u200b\u5149\u6807\u200b\u4f4d\u7f6e\u200b\u538b\u7f29\u200b\u5230\u200b\u53f3\u4fa7\u200b\u5bf9\u9f50\u200b\u3002</li> </ul> <pre><code>use iced::widget::{column, row, horizontal_space, text};\nuse iced::Length;\n\n// ...\n\nimpl Sandbox for Editor {\n    // ...\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        let editor = text_editor(&amp;self.content);\n        // Query cursor position\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![horizontal_space(Length::Fill), cursor_indicator];\n\n        container(column![editor, status_bar].spacing(10)).padding(10).into()\n    }\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use iced::{Theme, Element, Sandbox, Settings, Length};\nuse iced::widget::{column, container, horizontal_space, row, text, text_editor};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor {\n    content: text_editor::Content\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action)\n}\n\nimpl Sandbox for Editor {\n    type Message = Message; // Define the type of messages\n\n    fn new() -&gt; Self {\n        Self {\n            content: text_editor::Content::with(include_str!(\"main.rs\"))\n        }\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                self.content.edit(action);\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n\n        // Query cursor position\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![horizontal_space(Length::Fill), cursor_indicator];\n\n        container(column![editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::Dark\n    }\n}\n</code></pre>"},{"location":"coding/rust-gui/4-async-file-loading/","title":"Asynchronous File Loading","text":"<p>Keywords: <code>iced::Application</code>, <code>iced::Command</code>, <code>iced::executor</code>, <code>std::io</code>, <code>std::path::Path</code>, <code>std::sync::Arc</code>, <code>tokio::fs::read_to_string</code></p> <p>\u200b\u4e3a\u4e86\u200b\u5b9e\u73b0\u200b\u5f02\u6b65\u200b\u6587\u4ef6\u200b\u52a0\u8f7d\u200b\uff0c\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>tokio</code>\u200b\u5e93\u200b</p> <pre><code>[dependencies]\ntokio = { version = \"1.32\", features = [\"fs\"] }\n</code></pre> <p><code>Sandbox</code>\u200b\u4e0d\u80fd\u200b\u76f4\u63a5\u200b\u6267\u884c\u200b\u5f02\u6b65\u200b\u51fd\u6570\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b<code>Application</code>\u3002\u200b\u76f8\u6bd4\u200b\u4e8e\u200b<code>Sandbox</code>\uff0c<code>Application</code>\u200b\u9700\u8981\u200b\u989d\u5916\u200b\u5b9e\u73b0\u200b\u5982\u4e0b\u200b\u7c7b\u578b\u200b</p> <pre><code>impl Application for Editor {\n    // ...\n    type Theme; // Color theme\n    type Executor; // Engine for running async tasks\n    type Flags; // Initial state\n}\n</code></pre> <p>\u200b\u6b64\u5904\u200b\uff0c<code>Theme</code>\u200b\u548c\u200b<code>Flags</code>\u200b\u7c7b\u578b\u200b\u4e0d\u200b\u9700\u8981\u200b\u989d\u5916\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5206\u522b\u200b\u7528\u200b<code>iced::Theme</code>\u200b\u548c\u200b<code>()</code>\u200b\u4ee3\u66ff\u200b\u3002\u200b\u9ed8\u8ba4\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c<code>Executor</code>\u200b\u7c7b\u578b\u200b\u9700\u8981\u200b\u7528\u200b<code>iced::executor::Default</code>\u200b\u4ee3\u66ff\u200b\u3002</p> <p>\u200b\u5728\u200b\u4fee\u6539\u200b\u4e3a\u200b<code>Application</code>\u200b\u540e\u200b\uff0c<code>new</code>\u200b\u548c\u200b<code>update</code>\u200b\u65b9\u6cd5\u200b\u7684\u200b\u8fd4\u56de\u503c\u200b\u6807\u7b7e\u200b\u4e5f\u200b\u9700\u8981\u200b\u4fee\u6539\u200b\uff0c<code>new</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b<code>(Self, Command&lt;Message&gt;)</code>\uff0c<code>update</code>\u200b\u65b9\u6cd5\u200b\u8fd4\u56de\u200b<code>Command&lt;Message&gt;</code>\u3002\u200b\u5176\u4e2d\u200b<code>Command&lt;Message&gt;</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u5f02\u6b65\u200b\u4efb\u52a1\u200b\uff0c\u200b\u5728\u200b\u6267\u884c\u200b\u5b8c\u6bd5\u200b\u540e\u200b\u4f1a\u200b\u53d1\u9001\u200b\u4e00\u4e2a\u200b\u5bf9\u5e94\u200b\u7c7b\u578b\u200b\u7684\u200b<code>Message</code>\u3002</p> <pre><code>impl Application for Editor {\n    // ...\n\n    fn new(_flags: ()) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                // ...\n            },\n            Command::none(),\n        )\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        match message {\n            // ...\n        }\n\n        Command::none()\n    }\n}\n</code></pre> <p>\u200b\u6b64\u540e\u200b\uff0c\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u7f16\u5199\u200b\u7528\u4e8e\u200b\u8bfb\u53d6\u200b\u6587\u4ef6\u200b\u7684\u200b\u51fd\u6570\u200b\u3002</p> <pre><code>use std::io;\nuse std::path::Path;\nuse std::sync::Arc;\n\nasync fn load_file&lt;T&gt;(path: T) -&gt; Result&lt;Arc&lt;String&gt;, io::ErrorKind&gt;\n    where T: AsRef&lt;Path&gt;\n{\n    tokio::fs::read_to_string(path)\n        .await\n        .map(Arc::new)\n        .map_err(|e| e.kind())\n}\n</code></pre> <p>\u200b\u5728\u200b<code>new</code>\u200b\u65b9\u6cd5\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>Command</code>\u200b\u6765\u200b\u8c03\u7528\u200b<code>load_file</code>\u200b\u51fd\u6570\u200b\u3002</p> <pre><code>enum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;Arc&lt;String&gt;, io::ErrorKind&gt;)\n}\n\nimpl Application for Editor {\n    // ...\n\n    fn new(_flags: ()) -&gt; (Self, Command&lt;Message&gt;) {\n        let path = \"path/to/file.txt\";\n        let file = load_file(path);\n\n        (\n            Self {\n                // ...\n            },\n            Command::perform(\n                load_file(format!(\"{}/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n                Message::FileLoaded,\n            ),\n        )\n    }\n}\n</code></pre> <p>\u200b\u8bfb\u53d6\u200b\u6587\u4ef6\u200b\u65f6\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u53d1\u751f\u200b\u9519\u8bef\u200b\uff0c\u200b\u9700\u8981\u200b\u5bf9\u200b\u5f02\u5e38\u200b\u6d88\u606f\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b\u3002\u200b\u5728\u200b<code>load_file</code>\u200b\u51fd\u6570\u200b\u4e2d\u200b\u5df2\u7ecf\u200b\u901a\u8fc7\u200b<code>Result</code>\u200b\u8fd4\u56de\u200b\u4e86\u200b\u5bf9\u5e94\u200b\u7684\u200b\u9519\u8bef\u200b\u7c7b\u578b\u200b\uff0c\u200b\u53ea\u200b\u9700\u8981\u200b\u5728\u200b<code>update</code>\u200b\u65b9\u6cd5\u200b\u4e2d\u200b\u5904\u7406\u200b<code>Message::FileOpened</code>\u200b\u5373\u53ef\u200b\u3002</p> <pre><code>struct Editor {\n    // ...\n    error: Option&lt;io::ErrorKind&gt;, // Use Option to store error\n}\n\n// ...\n\nimpl Application for Editor {\n    // ...\n\n    fn new(_flags: ()) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                // ...\n                error: None, // Initialize error as None\n            },\n            // ...\n        )\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // ...\n        match message {\n            // ...\n            Message::FileOpened(Error(e)) =&gt; {\n                self.error = Some(e); // Store error\n            }\n        }\n        // ...\n    }\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use iced::{Theme, Element, Sandbox, Settings, Length};\nuse iced::widget::{column, container, horizontal_space, row, text, text_editor};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor {\n    content: text_editor::Content\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action)\n}\n\nimpl Sandbox for Editor {\n    type Message = Message; // Define the type of messages\n\n    fn new() -&gt; Self {\n        Self {\n            content: text_editor::Content::with(include_str!(\"main.rs\"))\n        }\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                self.content.edit(action);\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n\n        // Query cursor position\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![horizontal_space(Length::Fill), cursor_indicator];\n\n        container(column![editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::Dark\n    }\n}\n</code></pre>"},{"location":"coding/rust-gui/5-file-picker/","title":"File Picker","text":"<p>Keywords: <code>iced::widget::button</code>, <code>rfd::AsyncFileDialog</code></p> <p>\u200b\u672c\u8282\u200b\u5728\u200b\u7a97\u53e3\u200b\u4e0a\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u6309\u94ae\u200b\uff0c\u200b\u70b9\u51fb\u200b\u540e\u5f39\u200b\u51fa\u200b\u6587\u4ef6\u200b\u9009\u62e9\u200b\u5bf9\u8bdd\u6846\u200b\uff0c\u200b\u9009\u62e9\u200b\u6587\u4ef6\u200b\u540e\u200b\u5c06\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\u663e\u793a\u200b\u5728\u200b\u7a97\u53e3\u200b\u4e0a\u200b\u3002\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u6dfb\u52a0\u200b<code>rfd</code>\u200b\u5e93\u200b\u7684\u200b\u4f9d\u8d56\u200b\uff0c\u200b\u7528\u4e8e\u200b\u6587\u4ef6\u200b\u9009\u62e9\u200b\u5bf9\u8bdd\u6846\u200b\u3002</p> <pre><code>cargo add rfd\n</code></pre> <p>\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u7f16\u5199\u200b\u7528\u4e8e\u200b\u663e\u793a\u6587\u4ef6\u200b\u9009\u62e9\u200b\u7a97\u53e3\u200b\u7684\u200b\u51fd\u6570\u200b\u3002\u200b\u6ce8\u610f\u200b<code>rfd::AsyncFileDialog</code>\u200b\u4f1a\u200b\u8fd4\u56de\u200b\u4e00\u4e2a\u200b\u5f02\u5e38\u200b\uff0c\u200b\u56e0\u6b64\u200b\u9700\u8981\u200b\u81ea\u5b9a\u4e49\u200b\u5f02\u5e38\u200b\u7c7b\u578b\u200b\u3002</p> <pre><code>#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n\nasync fn pick_file() -&gt; Result&lt;Arc&lt;String&gt;, Error&gt; {\n    let handler = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?; // If error, return DialogClosed error\n    load_file(handler.path()).await\n}\n</code></pre> <p>\u200b\u8c03\u6574\u200b\u7a97\u53e3\u200b\u5e03\u5c40\u200b\uff0c\u200b\u5728\u200b\u7a97\u53e3\u200b\u4e0a\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u6309\u94ae\u200b\u3002</p> <pre><code>impl Application for Editor {\n    // ...\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        let controls = row![button(\"Open\")];\n        // ...\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n}\n</code></pre> <p>\u200b\u4e4b\u540e\u200b\u9700\u8981\u200b\u6dfb\u52a0\u200b\u6309\u94ae\u200b\u7684\u200b\u70b9\u51fb\u200b\u4e8b\u4ef6\u5904\u7406\u200b\uff0c\u200b\u8c03\u7528\u200b<code>pick_file</code>\u200b\u51fd\u6570\u200b\u3002</p> <pre><code>enum Message {\n    // ...\n    OpenButtonPressed\n}\n\nimpl Application for Editor {\n    // ...\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            // ...\n            Message::OpenButtonPressed =&gt; {\n                Command::perform(pick_file(), Message::FileOpened)\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        let controls = row![button(\"Open\").on_press(Message::OpenButtonPressed)];\n        // ...\n    }\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use std::io;\nuse std::path::Path;\nuse std::sync::Arc;\n\nuse iced::{executor, Application, Command, Element, Length, Settings, Theme};\nuse iced::widget::{button, column, container, horizontal_space, row, text, text_editor};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor {\n    content: text_editor::Content,\n    error: Option&lt;Error&gt;\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;Arc&lt;String&gt;, Error&gt;),\n    OpenButtonPressed\n}\n\nimpl Application for Editor {\n    type Message = Message; // Define the type of messages\n    type Theme = Theme;\n    type Executor = executor::Default; // Engine for running async tasks\n    type Flags = ();\n\n    fn new(_flags: Self::Flags) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                content: text_editor::Content::new(),\n                error: None\n            },\n            Command::perform(\n            load_file(format!(\"{}/src/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n            Message::FileOpened\n            )\n        )\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok(result)) =&gt; {\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            Message::FileOpened(Err(error)) =&gt; {\n                self.error = Some(error);\n                Command::none()\n            },\n            Message::OpenButtonPressed =&gt; {\n                Command::perform(pick_file(), Message::FileOpened)\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n        let controls = row![button(\"Open\").on_press(Message::OpenButtonPressed)];\n\n        // Query cursor position\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![horizontal_space(Length::Fill), cursor_indicator];\n\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::Dark\n    }\n}\n\nasync fn pick_file() -&gt; Result&lt;Arc&lt;String&gt;, Error&gt; {\n    let file_handle = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?;\n    load_file(file_handle.path()).await\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;Arc&lt;String&gt;, Error&gt; {\n    tokio::fs::read_to_string(path)\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO)\n}\n\n#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n</code></pre>"},{"location":"coding/rust-gui/6-file-path-indicator/","title":"File Path Indicator","text":"<p>\u200b\u672c\u8282\u200b\u5728\u200b\u72b6\u6001\u680f\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u663e\u793a\u6587\u4ef6\u200b\u8def\u5f84\u200b\u7684\u200b\u6587\u672c\u200b\u63a7\u4ef6\u200b\u3002\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u5b58\u50a8\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u72b6\u6001\u200b\uff0c\u200b\u5e76\u200b\u5728\u200b\u521d\u59cb\u5316\u200b\u9636\u6bb5\u200b\u8bbe\u7f6e\u200b\u4e3a\u200b<code>None</code>\u3002</p> <pre><code>struct Editor {\n    path: Option&lt;PathBuf&gt;,\n    // ...\n}\n\nimpl Application for Editor {\n    // ...\n    fn new() -&gt; (Editor, Command&lt;Message&gt;) {\n        (\n            Editor {\n                path: None,\n                // ...\n            },\n            // ...\n        )\n    }\n    // ...\n}\n</code></pre> <p>Path\u200b\u4e0e\u200bPathBuf</p> <p><code>std::path::Path</code>\u200b\u548c\u200b<code>std::path::PathBuf</code>\u200b\u7684\u200b\u5173\u7cfb\u200b\u7c7b\u4f3c\u200b\u4e8e\u200b<code>&amp;str</code>\u200b\u548c\u200b<code>String</code>\u200b\u7684\u200b\u5173\u7cfb\u200b\u3002<code>Path</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u4e0d\u53ef\u200b\u53d8\u200b\u5f15\u7528\u200b\uff0c<code>PathBuf</code>\u200b\u662f\u200b\u4e00\u4e2a\u200b\u53ef\u53d8\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5b58\u50a8\u200b\u8def\u5f84\u200b\u72b6\u6001\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>PathBuf</code>\uff0c\u200b\u5f15\u7528\u200b\u8def\u5f84\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>Path</code>\u3002</p> <p>\u200b\u4e4b\u524d\u200b\u7684\u200b<code>load_file</code>\u200b\u51fd\u6570\u200b\u53ea\u200b\u8fd4\u56de\u200b\u4e86\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff0c\u200b\u6b64\u5904\u200b\u9700\u8981\u200b\u5c06\u200b\u6587\u4ef6\u200b\u7684\u200b\u8def\u5f84\u200b\u4e00\u540c\u200b\u8fd4\u56de\u200b\u3002\u200b\u6ce8\u610f\u200b\u51fd\u6570\u200b\u4e0d\u80fd\u200b\u8fd4\u56de\u200b\u5f15\u7528\u200b\uff0c\u200b\u56e0\u6b64\u200b\u9700\u8981\u200b\u5c06\u200b\u8def\u5f84\u200b\u8f6c\u6362\u200b\u4e3a\u200b<code>PathBuf</code>\u200b\u7c7b\u578b\u200b\u3002<code>and_then</code>\u200b\u65b9\u6cd5\u200b\u5904\u7406\u200b<code>Result</code>\u200b\u7684\u200b<code>Ok</code>\u200b\u503c\u200b\uff0c\u200b\u5728\u200b\u6210\u529f\u200b\u8bfb\u53d6\u200b\u6587\u4ef6\u200b\u540e\u200b\u5c06\u200b<code>Path</code>\u200b\u8f6c\u6362\u200b\u4e3a\u200b<code>PathBuf</code>\u200b\u5e76\u200b\u8fd4\u56de\u200b\u3002\u200b\u8c03\u7528\u200b<code>load_file</code>\u200b\u7684\u200b<code>pick_file</code>\u200b\u51fd\u6570\u200b\u4e5f\u200b\u9700\u8981\u200b\u4e00\u5e76\u200b\u4fee\u6539\u200b\uff0c\u200b\u540c\u65f6\u200b\u8fd4\u56de\u200b\u8def\u5f84\u200b\u548c\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\u3002</p> <pre><code>async fn pick_file() -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    // ...\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let content = tokio::fs::read_to_string(path.as_ref())\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO);\n    content.and_then(|content| Ok((path.as_ref().to_path_buf(), content)))\n}\n</code></pre> <p>\u200b\u5728\u200b\u4fee\u6539\u200b\u8bfb\u53d6\u200b\u6587\u4ef6\u200b\u7684\u200b\u51fd\u6570\u200b\u540e\u200b\uff0c\u200b\u9700\u8981\u200b\u4fee\u6539\u200b\u51fd\u6570\u200b\u56de\u8c03\u200b\u4e8b\u4ef6\u200b\u7684\u200b\u7c7b\u578b\u200b\u548c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5904\u7406\u51fd\u6570\u200b</p> \u200b\u4fee\u6539\u200b\u540e\u200b\u4fee\u6539\u200b\u524d\u200b <pre><code>enum Message {\n    // ...\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n}\n\nimpl Application for Editor {\n    // ...\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        match message {\n            // ...\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n        }\n    }\n}\n</code></pre> <pre><code>enum Message {\n    // ...\n    FileOpened(Result&lt;Arc&lt;String&gt;, Error&gt;),\n}\n\nimpl Application for Editor {\n    // ...\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        match message {\n            // ...\n            Message::FileOpened(Ok(result)) =&gt; {\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n        }\n    }\n}\n</code></pre> <p>\u200b\u6700\u540e\u200b\u5728\u200b\u72b6\u6001\u680f\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u6587\u672c\u200b\u63a7\u4ef6\u200b\u663e\u793a\u6587\u4ef6\u200b\u8def\u5f84\u200b\u5373\u53ef\u200b\u3002</p> <pre><code>impl Application for Editor {\n    // ...\n    fn view(&amp;mut self) -&gt; Element&lt;Message&gt; {\n        // ...\n        let path_indicator = match &amp;self.path {\n            None =&gt; text(\"\"),\n            Some(path) =&gt; text(path.to_string_lossy())\n        };\n        let status_bar = row![\n            path_indicator, // Add path indicator here\n            horizontal_space(Length::Fill),\n            cursor_indicator\n        ];\n        // ...\n    }\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use std::io;\nuse std::path::{Path, PathBuf};\nuse std::sync::Arc;\n\nuse iced::{executor, Application, Command, Element, Length, Settings, Theme};\nuse iced::widget::{button, column, container, horizontal_space, row, text, text_editor};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor {\n    path: Option&lt;PathBuf&gt;,\n    content: text_editor::Content,\n    error: Option&lt;Error&gt;\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    OpenButtonPressed\n}\n\nimpl Application for Editor {\n    type Message = Message; // Define the type of messages\n    type Theme = Theme;\n    type Executor = executor::Default; // Engine for running async tasks\n    type Flags = ();\n\n    fn new(_flags: Self::Flags) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                content: text_editor::Content::new(),\n                error: None,\n                path: None\n            },\n            Command::perform(\n            load_file(format!(\"{}/src/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n            Message::FileOpened\n            )\n        )\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            Message::FileOpened(Err(error)) =&gt; {\n                self.error = Some(error);\n                Command::none()\n            },\n            Message::OpenButtonPressed =&gt; {\n                Command::perform(pick_file(), Message::FileOpened)\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n        let controls = row![button(\"Open\").on_press(Message::OpenButtonPressed)];\n\n        // Query cursor position\n        let path_indicator = match &amp;self.path {\n            None =&gt; text(\"\"),\n            Some(path) =&gt; text(path.to_string_lossy())\n        };\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![\n            path_indicator,\n            horizontal_space(Length::Fill),\n            cursor_indicator\n        ];\n\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::Dark\n    }\n}\n\nasync fn pick_file() -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let file_handle = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?;\n    load_file(file_handle.path()).await\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let content = tokio::fs::read_to_string(path.as_ref())\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO);\n    content.and_then(|content| Ok((path.as_ref().to_path_buf(), content)))\n}\n\n#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n</code></pre>"},{"location":"coding/rust-gui/7-new-and-save/","title":"New and Save","text":"<p>\u200b\u672c\u200b\u8282\u200b\u6dfb\u52a0\u200b\u65b0\u5efa\u200b\u548c\u200b\u4fdd\u5b58\u200b\u6587\u4ef6\u200b\u7684\u200b\u529f\u80fd\u200b\u3002</p> <p>\u200b\u9996\u5148\u200b\uff0c\u200b\u5728\u200b<code>EditorEdit</code>\u200b\u6d88\u606f\u200b\u5904\u7406\u200b\u4e2d\u200b\u68c0\u67e5\u200b\u662f\u5426\u200b\u5bf9\u200b\u6587\u4ef6\u200b\u8fdb\u884c\u200b\u4e86\u200b\u4fee\u6539\u200b\uff0c\u200b\u5e76\u4e14\u200b\u8bb0\u5f55\u200b\u6587\u4ef6\u200b\u7684\u200b\u4fee\u6539\u200b\u72b6\u6001\u200b\u3002</p> <pre><code>struct Editor {\n    // ...\n    modified: bool\n}\n// ... In `update` function\n            Message::EditorEdit(action) =&gt; {\n                match &amp;action {\n                    text_editor::Action::Edit(_) =&gt; self.modified = true,\n                    _ =&gt; {}\n                }\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            // ...\n            Message::OpenButtonPressed =&gt; {\n                self.modified = false;\n                Command::perform(pick_file(), Message::FileOpened)\n            },\n</code></pre> <p>\u200b\u5176\u6b21\u200b\uff0c\u200b\u8c03\u6574\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u63a7\u4ef6\u200b\u7684\u200b\u4e0d\u540c\u200b\u663e\u793a\u200b\u72b6\u6001\u200b\uff1a</p> <ol> <li>\u200b\u5f53\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u4e3a\u200b\u7a7a\u65f6\u200b\uff0c\u200b\u4e3a\u200b\u65b0\u200b\u6587\u4ef6\u200b\uff0c\u200b\u663e\u793a\u200b\u201cNew File\u201d\u3002</li> <li>\u200b\u5f53\u200b\u6253\u5f00\u200b\u4e00\u4e2a\u200b\u6587\u4ef6\u200b\u65f6\u200b\uff0c\u200b\u663e\u793a\u6587\u4ef6\u200b\u8def\u5f84\u200b\u3002</li> <li>\u200b\u5f53\u200b\u6587\u4ef6\u200b\u88ab\u200b\u4fee\u6539\u200b\u540e\u200b\uff0c\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u540e\u200b\u52a0\u4e0a\u200b\u201c*\u201d\u3002</li> <li>\u200b\u5f53\u200b\u6253\u5f00\u200b\u6587\u4ef6\u200b\u51fa\u9519\u200b\u65f6\u200b\uff0c\u200b\u663e\u793a\u200b\u9519\u8bef\u4fe1\u606f\u200b\u3002</li> </ol> <pre><code>let path_indicator = if let Some(error) = &amp;self.error {\n    match error {\n        Error::DialogClosed =&gt; text(\"Dialog closed\"),\n        Error::IO(kind) =&gt; text(format!(\"I/O error: {:?}\", kind))\n    }\n} else {\n    let path_text = match &amp;self.path {\n        None =&gt; String::from(\"New file\"),\n        Some(path) =&gt; path.to_string_lossy().to_string()\n    };\n    let suffix = if self.modified { \"*\" } else { \"\" };\n    text(format!(\"{}{}\", path_text, suffix))\n};\n</code></pre> <p>\u200b\u5728\u200b\u521b\u5efa\u200b\u6587\u4ef6\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u6e05\u7a7a\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u548c\u200b\u5185\u5bb9\u200b\uff0c\u200b\u4ee5\u53ca\u200b\u6e05\u7a7a\u200b\u9519\u8bef\u4fe1\u606f\u200b\u3002\u200b\u52a0\u5165\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u6d88\u606f\u200b\u7c7b\u578b\u200b<code>NewButtonPressed</code>\uff0c\u200b\u7531\u200b\u4e00\u4e2a\u200b\u6309\u94ae\u200b\u89e6\u53d1\u200b\uff0c\u200b\u5728\u200b<code>update</code>\u200b\u51fd\u6570\u200b\u4e2d\u200b\u6267\u884c\u200b\u8fd9\u4e2a\u200b\u903b\u8f91\u200b\u3002</p> \u200b\u6d88\u606f\u200b\u7c7b\u578b\u200b\u58f0\u660e\u200b\u6d88\u606f\u200b\u89e6\u53d1\u200b\u6267\u884c\u200b\u903b\u8f91\u200b <pre><code>enum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    NewButtonPressed,\n    OpenButtonPressed\n}\n</code></pre> <pre><code>// ... In `view` function\nlet controls = row![\n    button(\"New\").on_press(Message::NewButtonPressed),\n    button(\"Open\").on_press(Message::OpenButtonPressed)\n];\n</code></pre> <pre><code>// ... In matching logic in `update` function\nMessage::NewButtonPressed =&gt; {\n    self.content = text_editor::Content::new();\n    self.error = None;\n    self.path = None;\n    self.modified = false;\n    Command::none()\n},\n// ...\n</code></pre> <p>\u200b\u63a5\u4e0b\u6765\u200b\u5904\u7406\u200b\u4fdd\u5b58\u200b\u6587\u4ef6\u200b\u7684\u200b\u903b\u8f91\u200b\uff0c\u200b\u5f53\u200b\u5b58\u5728\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u65f6\u200b\uff0c\u200b\u4fdd\u5b58\u200b\u6587\u4ef6\u200b\uff0c\u200b\u5426\u5219\u200b\u6253\u5f00\u200b\u6587\u4ef6\u200b\u9009\u62e9\u200b\u5bf9\u8bdd\u6846\u200b\u3002</p> <pre><code>async fn save_file(path: Option&lt;PathBuf&gt;, content: String) -&gt; Result&lt;PathBuf, Error&gt; {\n    let path = if let Some(path) = path {\n        path\n    } else {\n        rfd::AsyncFileDialog::new()\n            .set_title(\"Save the file to...\")\n            .save_file()\n            .await\n            .ok_or(Error::DialogClosed)?\n            .path()\n            .to_path_buf()\n    };\n    tokio::fs::write(&amp;path, content)\n        .await\n        .map_err(|err| err.kind())\n        .map_err(Error::IO)\n        .map(|_| path)\n}\n</code></pre> <p>\u200b\u5728\u200b\u4fdd\u5b58\u200b\u6587\u4ef6\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b</p> <ol> <li>\u200b\u68c0\u67e5\u200b\u6587\u4ef6\u200b\u7684\u200b\u4fee\u6539\u200b\u72b6\u6001\u200b\uff0c\u200b\u5982\u679c\u200b\u6587\u4ef6\u200b\u6ca1\u6709\u200b\u4fee\u6539\u200b\uff0c\u200b\u4e0d\u200b\u6267\u884c\u200b\u4fdd\u5b58\u200b\u64cd\u4f5c\u200b\u3002</li> <li>\u200b\u68c0\u67e5\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u662f\u5426\u200b\u4e3a\u7a7a\u200b\uff0c\u200b\u5982\u679c\u200b\u4e3a\u7a7a\u200b\uff0c\u200b\u6253\u5f00\u200b\u6587\u4ef6\u200b\u9009\u62e9\u200b\u5bf9\u8bdd\u6846\u200b\uff0c\u200b\u5426\u5219\u200b\u76f4\u63a5\u200b\u4fdd\u5b58\u200b\u6587\u4ef6\u200b\u3002</li> <li>\u200b\u52a0\u5165\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u6d88\u606f\u200b\u7c7b\u578b\u200b<code>SaveButtonPressed</code>\uff0c\u200b\u7531\u200b\u4e00\u4e2a\u200b\u6309\u94ae\u200b\u89e6\u53d1\u200b\uff0c\u200b\u5728\u200b<code>update</code>\u200b\u51fd\u6570\u200b\u4e2d\u200b\u6267\u884c\u200b\u8fd9\u4e2a\u200b\u903b\u8f91\u200b\u3002</li> </ol> \u200b\u6d88\u606f\u200b\u7c7b\u578b\u200b\u58f0\u660e\u200b\u6d88\u606f\u200b\u89e6\u53d1\u200b\u6267\u884c\u200b\u903b\u8f91\u200b <pre><code>enum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    FileSaved(Result&lt;PathBuf, Error&gt;),\n    NewButtonPressed,\n    OpenButtonPressed,\n    SaveButtonPressed\n}\n</code></pre> <pre><code>// ... In `view` function\nlet controls = row![\n    button(\"New\").on_press(Message::NewButtonPressed),\n    button(\"Open\").on_press(Message::OpenButtonPressed),\n    button(\"Save\").on_press(Message::SaveButtonPressed)\n];\n</code></pre> <pre><code>// ... In matching logic in `update` function\nMessage::FileSaved(Ok(path)) =&gt; {\n    self.path = Some(path);\n    self.modified = false;\n    Command::none()\n},\nMessage::FileOpened(Err(error)) | Message::FileSaved(Err(error)) =&gt; {\n    self.error = Some(error);\n    Command::none()\n},\n// ...\nMessage::SaveButtonPressed =&gt; {\n    let content = self.content.text();\n    match self.modified {\n        false =&gt; Command::none(),\n        true =&gt; Command::perform(\n            save_file(self.path.clone(), content),\n            Message::FileSaved\n        )\n    }\n}\n</code></pre> <p>\u200b\u6700\u540e\u200b\uff0c\u200b\u8c03\u6574\u200b\u4e09\u4e2a\u200b\u6309\u94ae\u200b\u4e4b\u95f4\u200b\u7684\u200b\u95f4\u8ddd\u200b</p> <pre><code>let controls = row![\n    button(\"New\").on_press(Message::NewButtonPressed),\n    button(\"Open\").on_press(Message::OpenButtonPressed),\n    button(\"Save\").on_press(Message::SaveButtonPressed)\n].spacing(10);\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use std::io;\nuse std::path::{Path, PathBuf};\nuse std::sync::Arc;\n\nuse iced::{executor, Application, Command, Element, Length, Settings, Theme};\nuse iced::widget::{button, column, container, horizontal_space, row, text, text_editor};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings::default())\n}\n\nstruct Editor {\n    path: Option&lt;PathBuf&gt;,\n    content: text_editor::Content,\n    modified: bool,\n    error: Option&lt;Error&gt;\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    FileSaved(Result&lt;PathBuf, Error&gt;),\n    NewButtonPressed,\n    OpenButtonPressed,\n    SaveButtonPressed\n}\n\nimpl Application for Editor {\n    type Message = Message; // Define the type of messages\n    type Theme = Theme;\n    type Executor = executor::Default; // Engine for running async tasks\n    type Flags = ();\n\n    fn new(_flags: Self::Flags) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                content: text_editor::Content::new(),\n                error: None,\n                modified: false,\n                path: None\n            },\n            Command::perform(\n            load_file(format!(\"{}/src/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n            Message::FileOpened\n            )\n        )\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                match &amp;action {\n                    text_editor::Action::Edit(_) =&gt; self.modified = true,\n                    _ =&gt; {}\n                }\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            Message::FileSaved(Ok(path)) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                Command::none()\n            },\n            Message::FileOpened(Err(error)) | Message::FileSaved(Err(error)) =&gt; {\n                self.error = Some(error);\n                Command::none()\n            },\n            Message::NewButtonPressed =&gt; {\n                self.content = text_editor::Content::new();\n                self.error = None;\n                self.path = None;\n                self.modified = false;\n                Command::none()\n            },\n            Message::OpenButtonPressed =&gt; {\n                self.modified = false;\n                Command::perform(pick_file(), Message::FileOpened)\n            },\n            Message::SaveButtonPressed =&gt; {\n                let content = self.content.text();\n                match self.modified {\n                    false =&gt; Command::none(),\n                    true =&gt; Command::perform(\n                        save_file(self.path.clone(), content),\n                        Message::FileSaved\n                    )\n                }\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n        let controls = row![\n            button(\"New\").on_press(Message::NewButtonPressed),\n            button(\"Open\").on_press(Message::OpenButtonPressed),\n            button(\"Save\").on_press(Message::SaveButtonPressed)\n        ].spacing(10);\n\n        // Query cursor position\n        let path_indicator = if let Some(error) = &amp;self.error {\n            match error {\n                Error::DialogClosed =&gt; text(\"Dialog closed\"),\n                Error::IO(kind) =&gt; text(format!(\"I/O error: {:?}\", kind))\n            }\n        } else {\n            let path_text = match &amp;self.path {\n                None =&gt; String::from(\"New file\"),\n                Some(path) =&gt; path.to_string_lossy().to_string()\n            };\n            let suffix = if self.modified { \"*\" } else { \"\" };\n            text(format!(\"{}{}\", path_text, suffix))\n        };\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![\n            path_indicator,\n            horizontal_space(Length::Fill),\n            cursor_indicator\n        ];\n\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::Dark\n    }\n}\n\nasync fn pick_file() -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let file_handle = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?;\n    load_file(file_handle.path()).await\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let content = tokio::fs::read_to_string(path.as_ref())\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO);\n    content.and_then(|content| Ok((path.as_ref().to_path_buf(), content)))\n}\n\nasync fn save_file(path: Option&lt;PathBuf&gt;, content: String) -&gt; Result&lt;PathBuf, Error&gt; {\n    let path = if let Some(path) = path {\n        path\n    } else {\n        rfd::AsyncFileDialog::new()\n            .set_title(\"Save the file to...\")\n            .save_file()\n            .await\n            .ok_or(Error::DialogClosed)?\n            .path()\n            .to_path_buf()\n    };\n    tokio::fs::write(&amp;path, content)\n        .await\n        .map_err(|err| err.kind())\n        .map_err(Error::IO)\n        .map(|_| path)\n}\n\n#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n</code></pre>"},{"location":"coding/rust-gui/8-button-prettify/","title":"Button Prettify","text":"<p>\u200b\u5728\u200b\u672c\u8282\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u628a\u200b\u6309\u94ae\u200b\u7684\u200b\u6587\u672c\u200b\u66ff\u6362\u200b\u4e3a\u200b\u56fe\u6807\u200b\uff0c\u200b\u5e76\u4e14\u200b\u6dfb\u52a0\u200b\u6587\u672c\u200b\u60ac\u6d6e\u200b\u63d0\u793a\u200b\u3002</p> <p>\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u521b\u5efa\u200b\u5305\u542b\u200b\u56fe\u6807\u200b\u7684\u200b\u5b57\u4f53\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5728\u200bFontello\u200b\u4e0a\u200b\u9009\u62e9\u200b\u56fe\u6807\u200b\uff0c\u200b\u7136\u540e\u200b\u4e0b\u8f7d\u200b\u5b57\u4f53\u200b\u6587\u4ef6\u200b\u3002\u200b\u5c06\u200b<code>ttf</code>\u200b\u7248\u672c\u200b\u7684\u200b\u5b57\u4f53\u200b\u5b58\u653e\u200b\u5728\u200b\u9879\u76ee\u200b\u4e0b\u200b\u7684\u200b<code>fonts/editor-icon.ttf</code>\u200b\u4e2d\u200b\u3002</p> <p>\u200b\u7136\u540e\u200b\u5728\u200b\u4ee3\u7801\u200b\u4e2d\u200b\u52a0\u8f7d\u200b\u5b57\u4f53\u200b\u6587\u4ef6\u200b\uff0c\u200b\u5728\u200b<code>iced::Settings</code>\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u5b57\u4f53\u200b\uff1a</p> <pre><code>fn main() -&gt; iced::Result {\n    Editor::run(Settings {\n        fonts: vec![include_bytes!(\"../fonts/editor-icon.ttf\").as_slice().into()],\n        ..Default::default() // Expand the default settings\n    })\n}\n</code></pre> <p>\u200b\u52a0\u8f7d\u200b\u5b57\u4f53\u200b\u540e\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u6309\u94ae\u200b\u7684\u200b\u8f93\u5165\u200b\u6587\u672c\u200b\u66ff\u6362\u200b\u4e3a\u200b\u56fe\u6807\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>text</code>\u200b\u63a7\u4ef6\u200b\u521b\u5efa\u200b\u56fe\u6807\u200b\u3002\u200b\u5728\u200b\u7f51\u9875\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u8bfb\u53d6\u200b\u5230\u200b\u5bf9\u5e94\u200b\u65b0\u5efa\u200b\u3001\u200b\u6253\u5f00\u200b\u3001\u200b\u4fdd\u5b58\u200b\u7684\u200bUnicode\u200b\u7f16\u7801\u200b\u5206\u522b\u200b\u4e3a\u200b<code>\\u{E800}</code>\u3001<code>\\u{F115}</code>\u3001<code>\\u{E801}</code>\u3002</p> <pre><code>fn toolbar_button&lt;'a&gt;(description: &amp;str, callback: Message) -&gt; Element&lt;'a, Message&gt; {\n    let font = Font::with_name(\"editor-icon\");\n\n    let icon = text(match description {\n        \"new\" =&gt; '\\u{E800}',\n        \"open\" =&gt; '\\u{F115}',\n        \"save\" =&gt; '\\u{E801}',\n        _ =&gt; ' '\n    }).font(font);\n\n    button(container(icon)\n        .width(30)  // Set the width of the button\n        .center_x() // Center the icon\n    ).on_press(callback).into()\n}\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>button_icon</code>\u200b\u51fd\u6570\u200b\u66ff\u6362\u200b\u6309\u94ae\u200b\u539f\u672c\u200b\u7684\u200b\u8f93\u5165\u200b</p> <pre><code>// ... In `view` function\nlet controls = row![\n    toolbar_button(\"new\", Message::NewButtonPressed),\n    toolbar_button(\"open\", Message::OpenButtonPressed),\n    toolbar_button(\"save\", Message::SaveButtonPressed)\n].spacing(10);\n</code></pre> <p>\u200b\u6700\u540e\u200b\uff0c\u200b\u5b9e\u73b0\u200b\u60ac\u6d6e\u200b\u63d0\u793a\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>Tooltip</code>\u200b\u63a7\u4ef6\u200b\u5305\u88f9\u200b\u6309\u94ae\u200b\u5373\u53ef\u200b\u3002\u200b\u4e3a\u4e86\u200b\u7f8e\u89c2\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>style</code>\u200b\u65b9\u6cd5\u200b\u8bbe\u7f6e\u200b\u63d0\u793a\u6846\u200b\u7684\u200b\u6837\u5f0f\u200b\u3002</p> <pre><code>// ... In `view` function\nlet controls = row![\n    toolbar_button(\"New\", Message::NewButtonPressed),\n    toolbar_button(\"Open\", Message::OpenButtonPressed),\n    toolbar_button(\"Save\", Message::SaveButtonPressed)\n].spacing(10);\n\n// ... In the outer scope\nfn toolbar_button&lt;'a&gt;(description: &amp;str, callback: Message) -&gt; Element&lt;'a, Message&gt; {\n    let font = Font::with_name(\"editor-icon\");\n    let lower = description.to_lowercase();\n    let icon = text(match lower.as_str() {\n        \"new\" =&gt; '\\u{E800}',\n        \"open\" =&gt; '\\u{F115}',\n        \"save\" =&gt; '\\u{E801}',\n        _ =&gt; ' '\n    }).font(font);\n\n    tooltip(\n        button(container(icon)\n            .width(30)  // Set the width of the button\n            .center_x() // Center the icon\n        ).on_press(callback),\n        description, tooltip::Position::FollowCursor\n    ).style(theme::Container::Box).into() // Set the style of the tooltip\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use std::io;\nuse std::path::{Path, PathBuf};\nuse std::sync::Arc;\n\nuse iced::{executor, theme, Application, Command, Element, Font, Length, Settings, Theme};\nuse iced::widget::{button, column, container, horizontal_space, row, text, text_editor, tooltip};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings {\n        fonts: vec![include_bytes!(\"../fonts/editor-icon.ttf\").as_slice().into()],\n        ..Default::default()\n    })\n}\n\nstruct Editor {\n    path: Option&lt;PathBuf&gt;,\n    content: text_editor::Content,\n    modified: bool,\n    error: Option&lt;Error&gt;\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    FileSaved(Result&lt;PathBuf, Error&gt;),\n    NewButtonPressed,\n    OpenButtonPressed,\n    SaveButtonPressed\n}\n\nimpl Application for Editor {\n    type Message = Message; // Define the type of messages\n    type Theme = Theme;\n    type Executor = executor::Default; // Engine for running async tasks\n    type Flags = ();\n\n    fn new(_flags: Self::Flags) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                content: text_editor::Content::new(),\n                error: None,\n                modified: false,\n                path: None\n            },\n            Command::perform(\n            load_file(format!(\"{}/src/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n            Message::FileOpened\n            )\n        )\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                match &amp;action {\n                    text_editor::Action::Edit(_) =&gt; self.modified = true,\n                    _ =&gt; {}\n                }\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            Message::FileSaved(Ok(path)) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                Command::none()\n            },\n            Message::FileOpened(Err(error)) | Message::FileSaved(Err(error)) =&gt; {\n                self.error = Some(error);\n                Command::none()\n            },\n            Message::NewButtonPressed =&gt; {\n                self.content = text_editor::Content::new();\n                self.error = None;\n                self.path = None;\n                self.modified = false;\n                Command::none()\n            },\n            Message::OpenButtonPressed =&gt; {\n                self.modified = false;\n                Command::perform(pick_file(), Message::FileOpened)\n            },\n            Message::SaveButtonPressed =&gt; {\n                let content = self.content.text();\n                match self.modified {\n                    false =&gt; Command::none(),\n                    true =&gt; Command::perform(\n                        save_file(self.path.clone(), content),\n                        Message::FileSaved\n                    )\n                }\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content).on_edit(Message::EditorEdit);\n        let controls = row![\n            toolbar_button(\"New\", Message::NewButtonPressed),\n            toolbar_button(\"Open\", Message::OpenButtonPressed),\n            toolbar_button(\"Save\", Message::SaveButtonPressed)\n        ].spacing(10);\n\n        // Query cursor position\n        let path_indicator = if let Some(error) = &amp;self.error {\n            match error {\n                Error::DialogClosed =&gt; text(\"Dialog closed\"),\n                Error::IO(kind) =&gt; text(format!(\"I/O error: {:?}\", kind))\n            }\n        } else {\n            let path_text = match &amp;self.path {\n                None =&gt; String::from(\"New file\"),\n                Some(path) =&gt; path.to_string_lossy().to_string()\n            };\n            let suffix = if self.modified { \"*\" } else { \"\" };\n            text(format!(\"{}{}\", path_text, suffix))\n        };\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![\n            path_indicator,\n            horizontal_space(Length::Fill),\n            cursor_indicator\n        ];\n\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        Theme::Dark\n    }\n}\n\nasync fn pick_file() -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let file_handle = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?;\n    load_file(file_handle.path()).await\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let content = tokio::fs::read_to_string(path.as_ref())\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO);\n    content.and_then(|content| Ok((path.as_ref().to_path_buf(), content)))\n}\n\nasync fn save_file(path: Option&lt;PathBuf&gt;, content: String) -&gt; Result&lt;PathBuf, Error&gt; {\n    let path = if let Some(path) = path {\n        path\n    } else {\n        rfd::AsyncFileDialog::new()\n            .set_title(\"Save the file to...\")\n            .save_file()\n            .await\n            .ok_or(Error::DialogClosed)?\n            .path()\n            .to_path_buf()\n    };\n    tokio::fs::write(&amp;path, content)\n        .await\n        .map_err(|err| err.kind())\n        .map_err(Error::IO)\n        .map(|_| path)\n}\n\nfn toolbar_button&lt;'a&gt;(description: &amp;str, callback: Message) -&gt; Element&lt;'a, Message&gt; {\n    let font = Font::with_name(\"editor-icon\");\n    let lower = description.to_lowercase();\n    let icon = text(match lower.as_str() {\n        \"new\" =&gt; '\\u{E800}',\n        \"open\" =&gt; '\\u{F115}',\n        \"save\" =&gt; '\\u{E801}',\n        _ =&gt; ' '\n    }).font(font);\n\n    tooltip(\n        button(container(icon)\n            .width(30)  // Set the width of the button\n            .center_x() // Center the icon\n        ).on_press(callback),\n        description, tooltip::Position::FollowCursor\n    ).style(theme::Container::Box).into()\n}\n\n#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n</code></pre>"},{"location":"coding/rust-gui/9-syntax-highlighting/","title":"Syntax Highlighting","text":"<p>Keywords: <code>iced::widget::pick_list</code></p> <p>\u200b\u672c\u200b\u8282\u200b\u6dfb\u52a0\u200b\u4ee3\u7801\u200b\u7f16\u8f91\u5668\u200b\u7684\u200b\u8bed\u6cd5\u200b\u9ad8\u4eae\u200b\u529f\u80fd\u200b\u3002</p> <p>\u200b\u9996\u5148\u200b\uff0c\u200b\u4fee\u6539\u200b\u7f16\u8f91\u5668\u200b\u7684\u200b\u9ed8\u8ba4\u200b\u5b57\u4f53\u200b\u4e3a\u200b\u7b49\u200b\u5bbd\u200b\u5b57\u4f53\u200b\uff0c\u200b\u4ee5\u4fbf\u200b\u66f4\u597d\u200b\u5730\u200b\u663e\u793a\u200b\u4ee3\u7801\u200b\u3002</p> <pre><code>// ... In `view` function\nlet editor = text_editor(&amp;self.content)\n    .on_edit(Message::EditorEdit)\n    .font(Font::MONOSPACE);\n</code></pre> <p>\u200b\u63a5\u4e0b\u6765\u200b\uff0c\u200b\u6dfb\u52a0\u200b\u9ad8\u4eae\u200b\u5904\u7406\u200b\u7684\u200b\u903b\u8f91\u200b\u3002\u200b\u9ad8\u4eae\u200b\u529f\u80fd\u200b\u7531\u200b<code>iced</code>\u200b\u7684\u200b<code>highlighter</code>\u200b\u6a21\u5757\u200b\u63d0\u4f9b\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b<code>Cargo.toml</code>\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u4f9d\u8d56\u200b\u3002</p> <pre><code>[dependencies]\niced = { git = \"https://github.com/iced-rs/iced.git\", rev = \"refs/tags/text-editor\", features = [ \"highlighter\" ] }\n</code></pre> <p>\u200b\u5728\u200b\u6587\u672c\u6846\u200b\u7684\u200b<code>highlight</code>\u200b\u65b9\u6cd5\u200b\u4e2d\u200b\u8bbe\u7f6e\u200b\u9ad8\u4eae\u200b\u3002<code>highlight</code>\u200b\u65b9\u6cd5\u200b\u9700\u8981\u200b\u6307\u5b9a\u200b\u914d\u8272\u200b\u65b9\u6848\u200b<code>theme</code>\u200b\u548c\u200b\u8bed\u6cd5\u200b<code>extension</code>\u3002<code>extension</code>\u200b\u53ef\u4ee5\u200b\u6839\u636e\u200b\u6587\u4ef6\u200b\u8def\u5f84\u200b\u7684\u200b\u540e\u7f00\u540d\u200b\u81ea\u52a8\u8bc6\u522b\u200b\uff0c\u200b\u5982\u679c\u200b\u6ca1\u6709\u200b\u540e\u7f00\u540d\u200b\uff0c\u200b\u5219\u200b\u9ed8\u8ba4\u200b\u4e3a\u200b<code>rs</code>\u3002</p> <pre><code>let editor = text_editor(&amp;self.content)\n    .on_edit(Message::EditorEdit)\n    .highlight::&lt;Highlighter&gt;( // use iced::highlighter::Highlighter\n        highlighter::Settings {\n            theme: highlighter::Theme::SolarizedDark, // Set the theme\n            extension: self.path\n                .as_ref()\n                .and_then(|path| path.extension()?.to_str())\n                .unwrap_or(\"rs\") // If extension is not found, use `rs`\n                .to_string()\n        }, |highlighter, _theme| {\n            highlighter.to_format()\n        }\n    )\n    .font(Font::MONOSPACE);\n</code></pre> <p>\u200b\u63a5\u4e0b\u6765\u200b\u6dfb\u52a0\u200b\u9ad8\u4eae\u200b\u98ce\u683c\u200b\u7684\u200b\u9009\u62e9\u200b\u529f\u80fd\u200b\u3002\u200b\u9996\u5148\u200b\u5728\u200b<code>Editor</code>\u200b\u4e2d\u200b\u6dfb\u52a0\u200b<code>theme</code>\u200b\u72b6\u6001\u200b\u4f5c\u4e3a\u200b\u5f53\u524d\u200b\u7684\u200b\u4e3b\u9898\u200b\uff0c\u200b\u5e76\u4e14\u200b\u8bbe\u7f6e\u200b\u521d\u59cb\u5316\u200b\u72b6\u6001\u200b\u3002</p> <pre><code>struct Editor {\n    // ...\n    theme: highlighter::Theme\n}\n// ... In `new` function\nSelf {\n    // ...\n    theme: highlighter::Theme::SolarizedDark\n},\n// ... In `view` function\nhighlighter::Settings {\n    theme: self.theme,\n    // ...\n}\n</code></pre> <p>\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b<code>pick_list</code>\u200b\u63a7\u4ef6\u200b\u7528\u4e8e\u200b\u9009\u62e9\u200b\u9ad8\u4eae\u200b\u98ce\u683c\u200b\uff0c<code>pick_list</code>\u200b\u9700\u8981\u200b\u8bbe\u7f6e\u200b\u9009\u9879\u200b\u3001\u200b\u5f53\u524d\u200b\u9009\u4e2d\u200b\u9879\u200b\u548c\u200b\u89e6\u53d1\u200b\u4e8b\u4ef6\u200b\u3002\u200b\u6bcf\u6b21\u200b\u66f4\u65b0\u200b\u65f6\u200b\u90fd\u200b\u4f1a\u200b\u4ee5\u200b\u9009\u4e2d\u200b\u9879\u4e3a\u200b\u53c2\u6570\u200b\u89e6\u53d1\u200b\u4e8b\u4ef6\u200b\u3002</p> <pre><code>let controls = row![\n    toolbar_button(\"New\", Message::NewButtonPressed),\n    toolbar_button(\"Open\", Message::OpenButtonPressed),\n    toolbar_button(\"Save\", Message::SaveButtonPressed),\n    horizontal_space(Length::Fill),\n    pick_list(highlighter::Theme::ALL, Some(self.theme), Message::ThemeChanged)\n].spacing(10);\n</code></pre> <p>\u200b\u540c\u65f6\u200b\u66f4\u65b0\u200b<code>ThemeChanged</code>\u200b\u4e8b\u4ef6\u200b\u7684\u200b\u5904\u7406\u200b\u903b\u8f91\u200b</p> <pre><code>// ... In `update` function\nMessage::ThemeChanged(theme) =&gt; {\n    self.theme = theme;\n    Command::none()\n}\n</code></pre> <p>\u200b\u5185\u7f6e\u200b\u7684\u200b\u90e8\u5206\u200b\u4e3b\u9898\u200b\u5e76\u200b\u4e0d\u662f\u200b\u6697\u8272\u200b\u4e3b\u9898\u200b\uff0c\u200b\u9700\u8981\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u6839\u636e\u200b\u4e3b\u9898\u200b\u7684\u200b\u4eae\u200b\u6697\u200b\u5207\u6362\u200b\u7a97\u53e3\u200b\u914d\u8272\u200b\u7684\u200b\u529f\u80fd\u200b\uff1a</p> <pre><code>// ... In `impl Application for Editor`\nfn theme(&amp;self) -&gt; Theme {\n    if self.theme.is_dark() {\n        Theme::Dark\n    } else {\n        Theme::Light\n    }\n}\n</code></pre> <p>\u200b\u4ee5\u4e0b\u200b\u4e3a\u200b\u5b8c\u6574\u200b\u7684\u200b<code>main.rs</code>\u200b\u6587\u4ef6\u200b\u5185\u5bb9\u200b\uff1a</p> <p>  Download source code</p> <pre><code>use std::io;\nuse std::path::{Path, PathBuf};\nuse std::sync::Arc;\n\nuse iced::{executor, theme, Application, Command, Element, Font, Length, Settings, Theme};\nuse iced::highlighter::{self, Highlighter};\nuse iced::widget::{button, column, container, horizontal_space, pick_list, row, text, text_editor, tooltip};\n\nfn main() -&gt; iced::Result {\n    Editor::run(Settings {\n        fonts: vec![include_bytes!(\"../fonts/editor-icon.ttf\").as_slice().into()],\n        ..Default::default()\n    })\n}\n\nstruct Editor {\n    path: Option&lt;PathBuf&gt;,\n    content: text_editor::Content,\n    modified: bool,\n    error: Option&lt;Error&gt;,\n    theme: highlighter::Theme\n}\n\n#[derive(Debug, Clone)]\nenum Message {\n    EditorEdit(text_editor::Action),\n    FileOpened(Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt;),\n    FileSaved(Result&lt;PathBuf, Error&gt;),\n    ThemeChanged(highlighter::Theme),\n    NewButtonPressed,\n    OpenButtonPressed,\n    SaveButtonPressed\n}\n\nimpl Application for Editor {\n    type Message = Message; // Define the type of messages\n    type Theme = Theme;\n    type Executor = executor::Default; // Engine for running async tasks\n    type Flags = ();\n\n    fn new(_flags: Self::Flags) -&gt; (Self, Command&lt;Message&gt;) {\n        (\n            Self {\n                content: text_editor::Content::new(),\n                error: None,\n                modified: false,\n                path: None,\n                theme: highlighter::Theme::SolarizedDark\n            },\n            Command::perform(\n            load_file(format!(\"{}/src/main.rs\", env!(\"CARGO_MANIFEST_DIR\"))),\n            Message::FileOpened\n            )\n        )\n    }\n\n    fn title(&amp;self) -&gt; String {\n        String::from(\"A text editor\")\n    }\n\n    fn update(&amp;mut self, message: Message) -&gt; Command&lt;Message&gt; {\n        // Handle messages here\n        match message {\n            Message::EditorEdit(action) =&gt; {\n                match &amp;action {\n                    text_editor::Action::Edit(_) =&gt; self.modified = true,\n                    _ =&gt; {}\n                }\n                self.content.edit(action);\n                Command::none()\n            },\n            Message::FileOpened(Ok((path, result))) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                self.content = text_editor::Content::with(&amp;result);\n                Command::none()\n            },\n            Message::FileSaved(Ok(path)) =&gt; {\n                self.path = Some(path);\n                self.modified = false;\n                Command::none()\n            },\n            Message::FileOpened(Err(error)) | Message::FileSaved(Err(error)) =&gt; {\n                self.error = Some(error);\n                Command::none()\n            },\n            Message::ThemeChanged(theme) =&gt; {\n                self.theme = theme;\n                Command::none()\n            },\n            Message::NewButtonPressed =&gt; {\n                self.content = text_editor::Content::new();\n                self.error = None;\n                self.path = None;\n                self.modified = false;\n                Command::none()\n            },\n            Message::OpenButtonPressed =&gt; {\n                self.modified = false;\n                Command::perform(pick_file(), Message::FileOpened)\n            },\n            Message::SaveButtonPressed =&gt; {\n                let content = self.content.text();\n                match self.modified {\n                    false =&gt; Command::none(),\n                    true =&gt; Command::perform(\n                        save_file(self.path.clone(), content),\n                        Message::FileSaved\n                    )\n                }\n            }\n        }\n    }\n\n    fn view(&amp;self) -&gt; Element&lt;'_, Message&gt; {\n        // Create the user interface here\n        let editor = text_editor(&amp;self.content)\n            .on_edit(Message::EditorEdit)\n            .highlight::&lt;Highlighter&gt;(\n                highlighter::Settings {\n                    theme: self.theme,\n                    extension: self.path\n                        .as_ref()\n                        .and_then(|path| path.extension()?.to_str())\n                        .unwrap_or(\"rs\")\n                        .to_string()\n                }, |highlighter, _theme| {\n                    highlighter.to_format()\n                }\n            )\n            .font(Font::MONOSPACE);\n        let controls = row![\n            toolbar_button(\"New\", Message::NewButtonPressed),\n            toolbar_button(\"Open\", Message::OpenButtonPressed),\n            toolbar_button(\"Save\", Message::SaveButtonPressed),\n            horizontal_space(Length::Fill),\n            pick_list(highlighter::Theme::ALL, Some(self.theme), Message::ThemeChanged)\n        ].spacing(10);\n\n        // Query cursor position\n        let path_indicator = if let Some(error) = &amp;self.error {\n            match error {\n                Error::DialogClosed =&gt; text(\"Dialog closed\"),\n                Error::IO(kind) =&gt; text(format!(\"I/O error: {:?}\", kind))\n            }\n        } else {\n            let path_text = match &amp;self.path {\n                None =&gt; String::from(\"New file\"),\n                Some(path) =&gt; path.to_string_lossy().to_string()\n            };\n            let suffix = if self.modified { \"*\" } else { \"\" };\n            text(format!(\"{}{}\", path_text, suffix))\n        };\n        let cursor_indicator = {\n            let (line, column) = self.content.cursor_position();\n\n            text(format!(\"Line: {}, Column: {}\", line + 1, column + 1))\n        };\n        let status_bar = row![\n            path_indicator,\n            horizontal_space(Length::Fill),\n            cursor_indicator\n        ];\n\n        container(column![controls, editor, status_bar].spacing(10)).padding(10).into()\n    }\n\n    fn theme(&amp;self) -&gt; Theme {\n        if self.theme.is_dark() {\n            Theme::Dark\n        } else {\n            Theme::Light\n        }\n    }\n}\n\nasync fn pick_file() -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let file_handle = rfd::AsyncFileDialog::new()\n        .set_title(\"Choose a text file...\")\n        .pick_file()\n        .await\n        .ok_or(Error::DialogClosed)?;\n    load_file(file_handle.path()).await\n}\n\nasync fn load_file(path: impl AsRef&lt;Path&gt;) -&gt; Result&lt;(PathBuf, Arc&lt;String&gt;), Error&gt; {\n    let content = tokio::fs::read_to_string(path.as_ref())\n        .await\n        .map(Arc::new)\n        .map_err(|err| err.kind())\n        .map_err(Error::IO);\n    content.and_then(|content| Ok((path.as_ref().to_path_buf(), content)))\n}\n\nasync fn save_file(path: Option&lt;PathBuf&gt;, content: String) -&gt; Result&lt;PathBuf, Error&gt; {\n    let path = if let Some(path) = path {\n        path\n    } else {\n        rfd::AsyncFileDialog::new()\n            .set_title(\"Save the file to...\")\n            .save_file()\n            .await\n            .ok_or(Error::DialogClosed)?\n            .path()\n            .to_path_buf()\n    };\n    tokio::fs::write(&amp;path, content)\n        .await\n        .map_err(|err| err.kind())\n        .map_err(Error::IO)\n        .map(|_| path)\n}\n\nfn toolbar_button&lt;'a&gt;(description: &amp;str, callback: Message) -&gt; Element&lt;'a, Message&gt; {\n    let font = Font::with_name(\"editor-icon\");\n    let lower = description.to_lowercase();\n    let icon = text(match lower.as_str() {\n        \"new\" =&gt; '\\u{E800}',\n        \"open\" =&gt; '\\u{F115}',\n        \"save\" =&gt; '\\u{E801}',\n        _ =&gt; ' '\n    }).font(font);\n\n    tooltip(\n        button(container(icon)\n            .width(30)  // Set the width of the button\n            .center_x() // Center the icon\n        ).on_press(callback),\n        description, tooltip::Position::FollowCursor\n    ).style(theme::Container::Box).into()\n}\n\n#[derive(Debug, Clone)]\nenum Error {\n    DialogClosed,\n    IO(io::ErrorKind)\n}\n</code></pre>"},{"location":"japanese/","title":"\u65e5\u8bed","text":"<p>\u200b\u65e5\u8bed\u200b\u5b66\u4e60\u200b\u7b14\u8bb0\u200b</p> <ol> <li>\u200b\u65e5\u8bed\u200b(1) - 2023-2024\u200b\u5b66\u5e74\u200b\u79cb\u5b63\u200b\u5b66\u671f\u200b<ul> <li>\u200b\u77ed\u8bed\u200b\u4e0e\u200b\u8868\u8fbe\u200b</li> <li>\u200b\u8bed\u6cd5\u200b</li> <li>\u200b\u5355\u8bcd\u8868\u200b</li> </ul> </li> </ol>"},{"location":"japanese/japanese-1-grammar/","title":"\u65e5\u8bed\u200b(1)\u200b\u8bed\u6cd5","text":""},{"location":"japanese/japanese-1-grammar/#_1","title":"\u8bcd\u6cd5","text":"<p>\u200b\u65e5\u8bed\u200b\u4e2d\u200b\u7684\u200b\u5355\u8bcd\u200b\u8bcd\u6027\u200b\u4e3b\u8981\u200b\u6709\u200b\u540d\u8bcd\u200b\u3001\u200b\u52a8\u8bcd\u200b\u3001\u200b\u5f62\u5bb9\u8bcd\u200b\u3001\u200b\u526f\u8bcd\u200b\u3001\u200b\u52a9\u8bcd\u200b\u3001\u200b\u63a5\u7eed\u8bcd\u200b\u7b49\u200b\u3002\u200b\u5176\u4e2d\u200b\uff0c\u200b\u52a8\u8bcd\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3001III\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3001III\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3002\u200b\u5f62\u5bb9\u8bcd\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u3001II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u3002</p> <ul> <li>\u200b\u540d\u8bcd\u200b\u3001\u200b\u4ee3\u8bcd\u200b\u3001\u200b\u6570\u8bcd\u200b\u79f0\u4e3a\u200b\u300c\u200b\u4f53\u8a00\u200b\u300d\u3002</li> <li>\u200b\u52a8\u8bcd\u200b\u3001\u200b\u5f62\u5bb9\u8bcd\u200b\u3001\u200b\u5f62\u5bb9\u200b\u52a8\u8bcd\u200b\u79f0\u4e3a\u200b\u300c\u200b\u7528\u8a00\u200b\u300d\u3002</li> </ul>"},{"location":"japanese/japanese-1-grammar/#_2","title":"\u300c\u30a2\u30bd\u30b3\u30c9\u300d\u200b\u7cfb\u7edf","text":"<p>\u200b\u5728\u200b\u65e5\u8bed\u200b\u4e2d\u200b\uff0c\u300c\u30a2\u30bd\u30b3\u30c9\u300d\u200b\u548c\u200b\u300c\u308c\u300d\uff08\u200b\u4e8b\u7269\u200b\uff09\u3001\u300c\u3053\u300d\uff08\u200b\u5730\u70b9\u200b\uff09\u3001\u300c\u3061\u3089\u300d\uff08\u200b\u65b9\u4f4d\u200b\uff09\u200b\u7b49\u200b\u6307\u793a\u200b\u4ee3\u8bcd\u200b\u8fde\u7528\u200b\u65f6\u200b\uff0c\u200b\u8868\u793a\u200b\u8be5\u8bcd\u200b\u7684\u200b\u8fdc\u8fd1\u200b\u3002</p> \u200b\u524d\u7f00\u200b \u200b\u4e8b\u7269\u200b \u200b\u5730\u70b9\u200b \u200b\u65b9\u4f4d\u200b \u200b\u72b6\u6001\u200b \u200b\u8fd1\u79f0\u200b \u3053 \u3053\u308c \u3053\u3053 \u3053\u3061\u3089 \u3053\u3093\u306a \u200b\u4e2d\u79f0\u200b \u305d \u305d\u308c \u305d\u3053 \u305d\u3061\u3089 \u305d\u3093\u306a \u200b\u8fdc\u79f0\u200b \u3042 \u3042\u308c \u3042\u305d\u3053 \u3042\u3061\u3089 \u3042\u3093\u306a \u200b\u4e0d\u5b9a\u200b\u79f0\u200b \u3069 \u3069\u308c \u3069\u3053 \u3069\u3061\u3089 \u3069\u3093\u306a <p>\u200b\u4e0d\u5b9a\u200b\u79f0\u200b\u662f\u200b\u7591\u95ee\u8bcd\u200b\uff0c\u200b\u7528\u4e8e\u200b\u63d0\u95ee\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_3","title":"\u5f62\u5bb9\u8bcd","text":"<p>\u200b\u5f62\u5bb9\u8bcd\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u548c\u200bII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u3002I\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u8bcd\u5c3e\u200b\u662f\u200b\u300c\u3044\u300d\uff0c\u200b\u5176\u4ed6\u200b\u7684\u200b\u90fd\u200b\u662f\u200bII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0cII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u53c8\u200b\u79f0\u4e3a\u200b\u5f62\u5bb9\u200b\u52a8\u8bcd\u200b\uff0c\u200b\u5176\u200b\u539f\u578b\u200b\u7684\u200b\u8bcd\u5c3e\u200b\u4e3a\u200b\u300c\u3060\u300d\u3002</p> I\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b \u200b\u542b\u4e49\u200b II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b \u200b\u542b\u4e49\u200b \u3042\u305f\u3089\u3057\u3044 \u200b\u65b0\u200b\u7684\u200b \u3060\u3044\u3058\u3087\u3046\u3076 \u200b\u6ca1\u5173\u7cfb\u200b \u305f\u304b\u3044 \u200b\u9ad8\u200b\u7684\u200b \u3079\u3093\u308a \u200b\u65b9\u4fbf\u200b \u3072\u308d\u3044 \u200b\u5bbd\u200b\u7684\u200b \u3059\u304d \u200b\u559c\u6b22\u200b <p>\u200b\u4f8b\u5916\u200b\uff1a\u200b\u7dba\u9e97\u200b\uff08\u304d\u308c\u3044\uff09\u3001\u200b\u5acc\u200b\u3044\uff08\u304d\u3089\u3044\uff09\u200b\u662f\u200bII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0c\u200b\u4f46\u662f\u200b\u8bcd\u5c3e\u200b\u662f\u200b\u300c\u3044\u300d\u3002</p> <p>\u200b\u5f62\u5bb9\u8bcd\u200b\u6709\u200b\u3066\u200b\u578b\u200b\u53d8\u5f62\u200b\u3002</p> \u3066\u200b\u578b\u200b\u306a\u3044\u200b\u578b\u200b\u3063\u305f\u200b\u578b\u200b <p>\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u8fde\u7528\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8fde\u63a5\u200b\u4e24\u4e2a\u200b\u5206\u53e5\u200b\u3002</p> <ol> <li>I\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u8bcd\u5c3e\u200b\u7684\u200b\u300c\u3044\u300d\u200b\u53d8\u4e3a\u200b\u300c\u304f\u3066\u300d\u3002</li> <li>II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5728\u200b\u8bcd\u5c3e\u200b\u76f4\u63a5\u200b\u52a0\u4e0a\u200b\u300c\u3067\u300d\u3002</li> </ol> \u200b\u5f62\u5bb9\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u3066\u200b\u578b\u200b \u3042\u305f\u3089\u3057\u3044 I\u200b\u7c7b\u200b \u3042\u305f\u3089\u3057\u304f\u3066 \u305f\u304b\u3044 I\u200b\u7c7b\u200b \u305f\u304b\u304f\u3066 \u3072\u308d\u3044 I\u200b\u7c7b\u200b \u3072\u308d\u304f\u3066 \u3079\u3093\u308a II\u200b\u7c7b\u200b \u3079\u3093\u308a\u3067 \u3059\u304d II\u200b\u7c7b\u200b \u3059\u304d\u3067 <p>\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u7b80\u4f53\u200b\u5426\u5b9a\u53e5\u200b\u4e2d\u200b\u3002</p> <ol> <li>I\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u8bcd\u5c3e\u200b\u7684\u200b\u300c\u3044\u300d\u200b\u53d8\u4e3a\u200b\u300c\u304f\u306a\u3044\u300d\u3002</li> <li>II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u5728\u200b\u8bcd\u5c3e\u200b\u76f4\u63a5\u200b\u52a0\u4e0a\u200b\u300c\u3067\u306f\u306a\u3044\u300d\u3002</li> </ol> \u200b\u5f62\u5bb9\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u306a\u3044\u200b\u578b\u200b \u3042\u305f\u3089\u3057\u3044 I\u200b\u7c7b\u200b \u3042\u305f\u3089\u3057\u304f\u306a\u3044 \u305f\u304b\u3044 I\u200b\u7c7b\u200b \u305f\u304b\u304f\u306a\u3044 \u3072\u308d\u3044 I\u200b\u7c7b\u200b \u3072\u308d\u304f\u306a\u3044 \u3079\u3093\u308a II\u200b\u7c7b\u200b \u3079\u3093\u308a\u3067\u306f\u306a\u3044 \u3059\u304d II\u200b\u7c7b\u200b \u3059\u304d\u3067\u306f\u306a\u3044 <p>\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u3063\u305f\u200b\u578b\u200b\u662f\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u8fc7\u53bb\u200b\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u7b80\u4f53\u200b\u8fc7\u53bb\u200b\u53e5\u200b\u4e2d\u200b\u3002</p> <ol> <li>I\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u3063\u305f\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u8bcd\u5c3e\u200b\u7684\u200b\u300c\u3044\u300d\u200b\u53d8\u4e3a\u200b\u300c\u304b\u3063\u305f\u300d\u3002</li> <li>II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u3063\u305f\u200b\u578b\u200b\u662f\u200b\u5728\u200b\u8bcd\u5c3e\u200b\u76f4\u63a5\u200b\u52a0\u4e0a\u200b\u300c\u3060\u3063\u305f\u300d\uff08\u200b\u76f8\u5f53\u4e8e\u200b\u539f\u578b\u200b\u76f4\u63a5\u200b\u52a0\u200b\u300c\u3063\u305f\u300d\uff09\u3002</li> </ol> \u200b\u5f62\u5bb9\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u3063\u305f\u200b\u578b\u200b \u3042\u305f\u3089\u3057\u3044 I\u200b\u7c7b\u200b \u3042\u305f\u3089\u3057\u304b\u3063\u305f \u305f\u304b\u3044 I\u200b\u7c7b\u200b \u305f\u304b\u304b\u3063\u305f \u3072\u308d\u3044 I\u200b\u7c7b\u200b \u3072\u308d\u304b\u3063\u305f \u3079\u3093\u308a II\u200b\u7c7b\u200b \u3079\u3093\u308a\u3060\u3063\u305f \u3059\u304d II\u200b\u7c7b\u200b \u3059\u304d\u3060\u3063\u305f <p>\u200b\u6ce8\u610f\u200b\u300c\u306a\u3044\u300d\u200b\u578b\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u770b\u4f5c\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0c\u200b\u5176\u200b\u8fc7\u53bb\u200b\u5f62\u5f0f\u200b\u4e3a\u200b\u300c\u306a\u304b\u3063\u305f\u300d\u3002</p> <ul> <li>I\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u4fee\u9970\u200b\u4f53\u8a00\u65f6\u200b\uff0c\u200b\u76f4\u63a5\u200b\u52a0\u4f53\u8a00\u200b\u5373\u53ef\u200b\uff0c\u200b\u4fee\u9970\u200b\u7528\u8a00\u65f6\u200b\uff08\u200b\u5373\u200b\u526f\u8bcd\u200b\u5f62\u5f0f\u200b\uff09\u200b\u9700\u8981\u200b\u5c06\u200b\u300c\u3044\u300d\u200b\u53d8\u4e3a\u200b\u300c\u304f\u300d\u3002</li> <li>II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u4fee\u9970\u200b\u4f53\u8a00\u65f6\u200b\u9700\u8981\u200b\u52a0\u200b\u300c\u306a\u300d\uff0c\u200b\u4fee\u9970\u200b\u7528\u8a00\u65f6\u200b\u9700\u8981\u200b\u52a0\u200b\u300c\u306b\u300d\u3002</li> </ul>"},{"location":"japanese/japanese-1-grammar/#_4","title":"\u52a8\u8bcd","text":"<p>\u200b\u52a8\u8bcd\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3001II\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3001III\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3002</p> <ol> <li>II\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u8bcd\u5c3e\u200b\u662f\u200b\u300c\u308b\u300d\uff0c\u300c\u308b\u300d\u200b\u524d\u200b\u7684\u200b\u5047\u540d\u200b\u662f\u200b\u300c\u3044\u300d\u200b\u6bb5\u200b\u6216\u200b\u300c\u3048\u300d\u200b\u6bb5\u200b\u3002</li> <li>III\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u8bcd\u5c3e\u200b\u662f\u200b\u300c\u3059\u308b\u300d\u200b\u6216\u200b\u300c\u304f\u308b\u300d\u3002\u300c\u3059\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u52a8\u8bcd\u200b\u53c8\u200b\u79f0\u4e3a\u200b\u30b5\u200b\u53d8\u52a8\u200b\u8bcd\u200b\uff0c\u300c\u304f\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u52a8\u8bcd\u200b\u53c8\u200b\u79f0\u4e3a\u200b\u30ab\u200b\u53d8\u52a8\u200b\u8bcd\u200b\u3002</li> <li>\u200b\u5269\u4e0b\u200b\u6240\u6709\u200b\u7684\u200b\u52a8\u8bcd\u200b\u5747\u200b\u4e3a\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u3002I\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u7ed3\u5c3e\u200b\u662f\u200b\u300c\u3046\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\u3002</li> </ol> <p>\u200b\u52a8\u8bcd\u200b\u6709\u200b\u307e\u3059\u200b\u578b\u200b\u3001\u306a\u3044\u200b\u578b\u200b\u3001\u305f\u200b\u578b\u200b\u3001\u3066\u200b\u578b\u200b\u7b49\u200b\u4e0d\u540c\u200b\u7684\u200b\u53d8\u5f62\u200b\u3002</p> \u307e\u3059\u200b\u578b\u200b\u306a\u3044\u200b\u578b\u200b\u305f\u200b\u578b\u200b\u3066\u200b\u578b\u200b <p>\u200b\u52a8\u8bcd\u200b\u7684\u200b\u307e\u3059\u200b\u578b\u200b\u662f\u200b\u52a8\u8bcd\u200b\u7684\u200b\u656c\u4f53\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u656c\u4f53\u200b\u53e5\u200b\u4e2d\u200b\u3002</p> <ol> <li>II\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u307e\u3059\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u307e\u3059\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3059\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u307e\u3059\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u3059\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u3057\u307e\u3059\u300d\uff0c\u200b\u4ee5\u200b\u300c\u304f\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u307e\u3059\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u304f\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u304d\u307e\u3059\u300d\u3002</li> <li>I\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u307e\u3059\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u300c\u3046\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u5bf9\u5e94\u200b\u7684\u200b\u300c\u3044\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\uff0c\u200b\u518d\u200b\u52a0\u4e0a\u200b\u300c\u307e\u3059\u300d\u3002</li> </ol> \u200b\u52a8\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u307e\u3059\u200b\u578b\u200b \u3042\u308b I\u200b\u7c7b\u200b \u3042\u308a\u307e\u3059 \u3067\u304d\u308b II\u200b\u7c7b\u200b \u3067\u304d\u307e\u3059 \u3068\u308b I\u200b\u7c7b\u200b \u3068\u308a\u307e\u3059 \u308c\u3093\u3057\u3085\u3046\u3059\u308b III\u200b\u7c7b\u200b \u308c\u3093\u3057\u3085\u3046\u3057\u307e\u3059 \u3082\u3064 I\u200b\u7c7b\u200b \u3082\u3061\u307e\u3059 <p>\u200b\u52a8\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u52a8\u8bcd\u200b\u7684\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u7b80\u4f53\u200b\u5426\u5b9a\u53e5\u200b\u4e2d\u200b\u3002</p> <ol> <li>II\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u306a\u3044\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3059\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u3059\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u3057\u306a\u3044\u300d\uff0c\u200b\u4ee5\u200b\u300c\u304f\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u304f\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u3053\u306a\u3044\u300d\u3002</li> <li>I\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u306a\u3044\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u300c\u3046\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u5bf9\u5e94\u200b\u7684\u200b\u300c\u3042\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\uff0c\u200b\u518d\u200b\u52a0\u4e0a\u200b\u300c\u306a\u3044\u300d\u3002</li> </ol> \u200b\u52a8\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u306a\u3044\u200b\u578b\u200b \u3042\u308b I\u200b\u7c7b\u200b \u3042\u304b\u306a\u3044 \u3067\u304d\u308b II\u200b\u7c7b\u200b \u3067\u304d\u306a\u3044 \u3068\u308b I\u200b\u7c7b\u200b \u3068\u3089\u306a\u3044 \u308c\u3093\u3057\u3085\u3046\u3059\u308b III\u200b\u7c7b\u200b \u308c\u3093\u3057\u3085\u3046\u3057\u306a\u3044 \u3082\u3064 I\u200b\u7c7b\u200b \u3082\u305f\u306a\u3044 <p>\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u52a8\u8bcd\u200b\u7684\u200b\u8fc7\u53bb\u200b\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u7b80\u4f53\u200b\u8fc7\u53bb\u200b\u53e5\u200b\u4e2d\u200b\u3002</p> <ol> <li>II\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u305f\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3059\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u3059\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u3057\u305f\u300d\uff0c\u200b\u4ee5\u200b\u300c\u304f\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u304f\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u304d\u305f\u300d\u3002</li> <li>I\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u8bcd\u5c3e\u200b\u7684\u200b\u300c\u3046\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\u3002<ul> <li>\u200b\u4ee5\u200b\u300c\u3046\u300d\u3001\u300c\u3064\u300d\u3001\u300c\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3063\u305f\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3080\u300d\u3001\u300c\u3076\u300d\u3001\u300c\u306c\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3093\u3060\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u304f\u300d\u3001\u300c\u3050\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3044\u305f\u300d\u200b\u548c\u200b\u300c\u3044\u3060\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3059\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3057\u305f\u300d\u3002</li> </ul> </li> <li>\u200b\u7279\u4f8b\u200b\uff1a\u300c\u200b\u884c\u200b\u304f\u300d\u200b\u7684\u200b\u305f\u200b\u578b\u200b\u662f\u200b\u300c\u3044\u3063\u305f\u300d\u3002</li> </ol> \u200b\u52a8\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u305f\u200b\u578b\u200b \u3042\u308b I\u200b\u7c7b\u200b \u3042\u3063\u305f \u3067\u304d\u308b II\u200b\u7c7b\u200b \u3067\u304d\u305f \u3068\u308b I\u200b\u7c7b\u200b \u3068\u3063\u305f \u308c\u3093\u3057\u3085\u3046\u3059\u308b III\u200b\u7c7b\u200b \u308c\u3093\u3057\u3085\u3046\u3057\u305f \u3082\u3064 I\u200b\u7c7b\u200b \u3082\u3063\u305f <p>\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u52a8\u8bcd\u200b\u7684\u200b\u8fde\u7528\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8fde\u63a5\u200b\u4e24\u4e2a\u200b\u5206\u53e5\u200b\u3002</p> <ol> <li>II\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u3066\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3059\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u3059\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u3057\u3066\u300d\uff0c\u200b\u4ee5\u200b\u300c\u304f\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u53bb\u6389\u200b\u8bcd\u5c3e\u200b\u300c\u304f\u308b\u300d\uff0c\u200b\u52a0\u4e0a\u200b\u300c\u304d\u3066\u300d\u3002</li> <li>I\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u9700\u8981\u200b\u8003\u8651\u200b\u8bcd\u5c3e\u200b\u7684\u200b\u300c\u3046\u300d\u200b\u6bb5\u200b\u5047\u540d\u200b\u3002<ul> <li>\u200b\u4ee5\u200b\u300c\u3046\u300d\u3001\u300c\u3064\u300d\u3001\u300c\u308b\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3063\u3066\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3080\u300d\u3001\u300c\u3076\u300d\u3001\u300c\u306c\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3093\u3067\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u304f\u300d\u3001\u300c\u3050\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3044\u3066\u300d\u200b\u548c\u200b\u300c\u3044\u3067\u300d\u3002</li> <li>\u200b\u4ee5\u200b\u300c\u3059\u300d\u200b\u7ed3\u5c3e\u200b\u7684\u200bI\u200b\u7c7b\u200b\u52a8\u8bcd\u200b\u7684\u200b\u3066\u200b\u578b\u200b\u662f\u200b\u5c06\u200b\u7ed3\u5c3e\u200b\u7684\u200b\u5047\u540d\u200b\u53d8\u4e3a\u200b\u300c\u3057\u3066\u300d\u3002</li> </ul> </li> </ol> \u200b\u52a8\u8bcd\u200b \u200b\u5206\u7c7b\u200b \u3066\u200b\u578b\u200b \u3042\u308b I\u200b\u7c7b\u200b \u3042\u3063\u3066 \u3067\u304d\u308b II\u200b\u7c7b\u200b \u3067\u304d\u3066 \u3068\u308b I\u200b\u7c7b\u200b \u3068\u3063\u3066 \u308c\u3093\u3057\u3085\u3046\u3059\u308b III\u200b\u7c7b\u200b \u308c\u3093\u3057\u3085\u3046\u3057\u3066 \u3082\u3064 I\u200b\u7c7b\u200b \u3082\u3063\u3066"},{"location":"japanese/japanese-1-grammar/#_5","title":"\u53e5\u6cd5","text":""},{"location":"japanese/japanese-1-grammar/#_6","title":"\u8bed\u53e5\u200b\u5206\u7c7b","text":"<p>\u200b\u65e5\u8bed\u200b\u4e2d\u200b\u7684\u200b\u8bed\u53e5\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u5b58\u5728\u200b\u53e5\u200b\uff08\u3042\u308a\u307e\u3059\u3001\u3044\u307e\u3059\uff09\u3001\u200b\u5224\u65ad\u53e5\u200b\uff08\u200b\u540d\u8bcd\u200b\uff09\u3001\u200b\u53d9\u8ff0\u200b\u53e5\u200b\uff08\u200b\u4e00\u822c\u200b\u52a8\u8bcd\u200b\uff09\u3001\u200b\u63cf\u5199\u200b\u53e5\u200b\uff08\u200b\u5f62\u5bb9\u8bcd\u200b\u3001\u200b\u5f62\u5bb9\u200b\u52a8\u8bcd\u200b\uff09\u3002\u200b\u5224\u65ad\u53e5\u200b\u548c\u200b\u63cf\u5199\u200b\u53e5\u200b\u7684\u200b\u656c\u4f53\u200b\u5f62\u5f0f\u200b\u662f\u200b\u300c\u301c\u3067\u3059\u300d\uff0c\u200b\u7b80\u4f53\u200b\u5f62\u5f0f\u200b\u662f\u200b\u300c\u301c\u3060\u300d\u3002\u200b\u53d9\u8ff0\u200b\u53e5\u200b\u548c\u200b\u5b58\u5728\u200b\u53e5\u200b\u7684\u200b\u656c\u4f53\u200b\u5f62\u5f0f\u200b\u662f\u200b\u300c\u301c\u307e\u3059\u300d\uff0c\u200b\u53c8\u79f0\u200b\u307e\u3059\u200b\u578b\u200b\u3002</p> <p>\u200b\u53e5\u5b50\u200b\u7531\u200b\u4e0d\u540c\u200b\u7684\u200b\u300c\u200b\u6587\u8282\u200b\u300d\u200b\u7ec4\u6210\u200b\uff0c\u200b\u6587\u8282\u200b\u662f\u200b\u65e5\u8bed\u200b\u8bed\u6cd5\u7ed3\u6784\u200b\u4e2d\u200b\u7684\u200b\u5355\u5143\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u6587\u8282\u200b\u90fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u300c\u200b\u6838\u5fc3\u200b\u8bcd\u200b\u300d\uff0c\u200b\u6838\u5fc3\u200b\u8bcd\u200b\u7684\u200b\u8bcd\u6027\u200b\u51b3\u5b9a\u200b\u4e86\u200b\u6587\u8282\u200b\u7684\u200b\u7c7b\u578b\u200b\u3002\u200b\u6587\u8282\u200b\u4e2d\u200b\u7684\u200b\u9644\u5c5e\u200b\u8bcd\u200b\uff08\u200b\u52a9\u8bcd\u200b\uff09\u200b\u7528\u4e8e\u200b\u4fee\u9970\u200b\u6838\u5fc3\u200b\u8bcd\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b\uff1a\u200b\u52a9\u8bcd\u200b\u53ea\u80fd\u200b\u7528\u4e8e\u200b\u4fee\u9970\u200b\u540d\u8bcd\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_7","title":"\u683c\u200b\u52a9\u8bcd","text":"<p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u7528\u4e8e\u200b\u540d\u8bcd\u200b\u540e\u200b\uff0c\u200b\u8868\u793a\u200b\u540d\u8bcd\u200b\u5728\u200b\u53e5\u5b50\u200b\u4e2d\u200b\u7684\u200b\u8bed\u6cd5\u200b\u6210\u5206\u200b\u3002</p> \u304c\u306e\u306b\u3067\u3092\u304b\u3089\u3078\u3088\u308a\u307b\u3069 <p>\u200b\u52a9\u8bcd\u200b\u300c\u304c\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e3b\u8bed\u200b\uff0c\u200b\u5176\u524d\u200b\u7684\u200b\u540d\u8bcd\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u6267\u884c\u8005\u200b\uff0c\u200b\u6216\u8005\u200b\u63cf\u5199\u200b\u3001\u200b\u5224\u65ad\u200b\u6216\u200b\u5b58\u5728\u200b\u7684\u200b\u5bf9\u8c61\u200b\u3002</p> <p>\u300c\u200b\u79c1\u200b\u304c\u200b\u884c\u200b\u304d\u307e\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u53bb\u200b\u3002\uff08\u200b\u53d9\u8ff0\u200b\u53e5\u200b\uff09 \u300c\u200b\u79c1\u200b\u304c\u200b\u5b66\u751f\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u662f\u200b\u5b66\u751f\u200b\u3002\uff08\u200b\u5224\u65ad\u53e5\u200b\uff09 \u300c\u200b\u79c1\u200b\u304c\u3044\u307e\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u5728\u200b\u3002\uff08\u200b\u5b58\u5728\u200b\u53e5\u200b\uff09</p> <p>\u200b\u52a9\u8bcd\u200b\u300c\u306e\u300d\u200b\u4f5c\u4e3a\u200b\u683c\u200b\u52a9\u8bcd\u200b\u65f6\u200b\u8868\u793a\u200b\u6240\u6709\u200b\u5173\u7cfb\u200b\uff0c\u200b\u5176\u524d\u200b\u7684\u200b\u540d\u8bcd\u200b\u8868\u793a\u200b\u88ab\u200b\u4fee\u9970\u200b\u7684\u200b\u5bf9\u8c61\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b\uff0c\u300c\u306e\u300d\u200b\u8fd8\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u540c\u4f4d\u8bed\u200b\uff0c\u200b\u56e0\u6b64\u200b\u300c\u306e\u300d\u200b\u7684\u200b\u5b9e\u9645\u200b\u542b\u4e49\u200b\u9700\u8981\u200b\u6839\u636e\u200b\u4e0a\u4e0b\u6587\u200b\u6765\u200b\u786e\u5b9a\u200b\u3002</p> <p>\u300c\u30de\u30a4\u30b1\u30eb\u3055\u3093\u306e\u200b\u5f8c\u200b\u8f29\u200b\u306e\u200b\u571f\u5c4b\u200b\u3068\u200b\u7533\u200b\u3057\u307e\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u662f\u200b\u8fc8\u514b\u5c14\u200b\u7684\u200b\u540e\u8f88\u200b\u571f\u5c4b\u200b\u3002</p> <p>\u200b\u53e5\u200b\u4e2d\u200b\u7684\u200b\u300c\u30de\u30a4\u30b1\u30eb\u3055\u3093\u306e\u200b\u5f8c\u200b\u8f29\u200b\u300d\u200b\u8868\u793a\u200b\u8fc8\u514b\u5c14\u200b\u7684\u200b\u540e\u8f88\u200b\uff0c\u200b\u662f\u200b\u6240\u6709\u200b\u5173\u7cfb\u200b\uff0c\u300c\u200b\u5f8c\u200b\u8f29\u200b\u306e\u200b\u571f\u5c4b\u200b\u300d\u200b\u8868\u793a\u200b\u571f\u5c4b\u200b\u662f\u200b\u8fc8\u514b\u5c14\u200b\u7684\u200b\u540e\u8f88\u200b\uff0c\u200b\u662f\u200b\u540c\u4f4d\u8bed\u200b\u5173\u7cfb\u200b\u3002</p> <p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u300c\u306b\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b</p> <ol> <li> <p>\u200b\u52a8\u4f5c\u200b\u53e5\u200b\u4e2d\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u6267\u884c\u200b\u65f6\u95f4\u200b\u3001\u200b\u5730\u70b9\u200b\u3001\u200b\u5bf9\u8c61\u200b\u7b49\u200b\u3002\u200b\u52a8\u8bcd\u200b\u9700\u8981\u200b\u662f\u200b\u79fb\u52a8\u200b\u52a8\u8bcd\u200b\u3002</p> <p>\u300c7\u200b\u6642\u200b\u306b\u200b\u56f3\u200b\u66f8\u9928\u200b\u306b\u200b\u884c\u200b\u304d\u307e\u3059\u3002\u300d\uff1a7\u200b\u70b9\u53bb\u200b\u56fe\u4e66\u9986\u200b\u3002</p> </li> <li> <p>\u200b\u5b58\u5728\u200b\u53e5\u200b\u4e2d\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u5b58\u5728\u200b\u7684\u200b\u5730\u70b9\u200b\u3002</p> <p>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u306b\u200b\u672c\u200b\u304c\u3042\u308a\u307e\u3059\u3002\u300d\uff1a\u200b\u56fe\u4e66\u9986\u200b\u91cc\u200b\u6709\u200b\u4e66\u200b\u3002 \u300c\u200b\u79c1\u200b\u306b\u306f\u200b\u59c9\u200b\u304c\u200b\u4e00\u4eba\u200b\u3044\u307e\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u6709\u200b\u4e00\u4e2a\u200b\u59d0\u59d0\u200b\u3002</p> </li> </ol> <p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u300c\u3067\u300d\u200b\u7528\u4e8e\u200b</p> <ol> <li> <p>\u200b\u63a5\u200b\u540d\u8bcd\u200b\u65f6\u200b\uff0c\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u6267\u884c\u200b\u624b\u6bb5\u200b\u6216\u200b\u65b9\u6cd5\u200b\u3002</p> <p>\u300c\u200b\u96fb\u8eca\u200b\u3067\u200b\u884c\u200b\u304d\u307e\u3059\u3002\u300d\uff1a\u200b\u5750\u200b\u7535\u8f66\u200b\u53bb\u200b\u3002</p> </li> <li> <p>\u200b\u63a5\u200b\u5730\u70b9\u200b\u65f6\u200b\uff0c\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u53d1\u751f\u200b\u5730\u70b9\u200b\u3002</p> <p>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u3067\u200b\u52c9\u5f37\u200b\u3057\u307e\u3059\u3002\u300d\uff1a\u200b\u5728\u200b\u56fe\u4e66\u9986\u200b\u5b66\u4e60\u200b\u3002</p> </li> </ol> <p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u300c\u3092\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff08\u200b\u4ed6\u200b\u52a8\u8bcd\u200b\uff09\u200b\u6216\u200b\u4f7f\u5f79\u200b\u6001\u200b\uff08\u200b\u81ea\u52a8\u8bcd\u200b\uff09\u3002</p> <p>\u300c\u200b\u672c\u200b\u3092\u200b\u8aad\u200b\u307f\u307e\u3059\u3002\u300d\uff1a\u200b\u770b\u4e66\u200b\u3002</p> <p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u300c\u304b\u3089\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u8d77\u70b9\u200b\u3002</p> <p>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u304b\u3089\u200b\u51fa\u200b\u767a\u200b\u3057\u307e\u3059\u3002\u300d\uff1a\u200b\u4ece\u200b\u56fe\u4e66\u9986\u200b\u51fa\u53d1\u200b\u3002</p> <p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u300c\u3078\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\uff08\u200b\u79fb\u52a8\u200b\u52a8\u8bcd\u200b\uff09\u200b\u7684\u200b\u7ec8\u70b9\u200b\u3002\uff08\u200b\u53ef\u4ee5\u200b\u4e0e\u200b\u300c\u306b\u300d\u200b\u4e92\u6362\u200b\uff09</p> <p>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u3078\u200b\u884c\u200b\u304d\u307e\u3059\u3002\u300d\uff1a\u200b\u53bb\u200b\u56fe\u4e66\u9986\u200b\u3002</p> <p>\u200b\u683c\u200b\u52a9\u8bcd\u200b\u300c\u3088\u308a\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u6bd4\u8f83\u200b\u7684\u200b\u5bf9\u8c61\u200b\uff0c\u200b\u5176\u524d\u200b\u7684\u200b\u540d\u8bcd\u200b\u7684\u200b\u7a0b\u5ea6\u200b\u6bd4\u200b\u4e3b\u8bed\u200b\u7684\u200b\u7a0b\u5ea6\u200b\u5f31\u200b\u3002</p> <p>\u300c\u200b\u6e05\u83ef\u200b\u5927\u5b66\u200b\u306f\u200b\u5317\u4eac\u5927\u5b66\u200b\u3088\u308a\u200b\u6709\u540d\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u6e05\u534e\u5927\u5b66\u200b\u6bd4\u200b\u5317\u4eac\u5927\u5b66\u200b\u6709\u540d\u200b\u3002</p> <p>\u200b\u5728\u200b\u63cf\u8ff0\u200b\u53e5\u200b\u4e2d\u683c\u200b\u52a9\u8bcd\u200b\u300c\u307b\u3069\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u7a0b\u5ea6\u200b\uff0c\u200b\u7528\u4e8e\u200b\u5426\u5b9a\u53e5\u200b\u4e2d\u200b\u8868\u793a\u200b\u5176\u524d\u200b\u7684\u200b\u540d\u8bcd\u200b\u7684\u200b\u7a0b\u5ea6\u200b\u6bd4\u200b\u4e3b\u8bed\u200b\u7684\u200b\u7a0b\u5ea6\u200b\u5f3a\u200b\u3002</p> <p>\u300c\u200b\u5317\u4eac\u5927\u5b66\u200b\u306f\u200b\u6e05\u83ef\u200b\u5927\u5b66\u200b\u307b\u3069\u200b\u6709\u540d\u200b\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u300d\uff1a\u200b\u5317\u4eac\u5927\u5b66\u200b\u6ca1\u6709\u200b\u6e05\u534e\u5927\u5b66\u200b\u6709\u540d\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_8","title":"\u526f\u200b\u52a9\u8bcd","text":"<p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u7528\u4e8e\u200b\u4fee\u9970\u8bed\u200b\u6cd5\u200b\u6210\u5206\u200b\uff0c\u200b\u4efb\u4f55\u200b\u4e0d\u662f\u200b\u683c\u200b\u52a9\u8bcd\u200b\u3001\u200b\u63a5\u7eed\u200b\u52a9\u8bcd\u200b\u3001\u200b\u7ec8\u200b\u52a9\u8bcd\u200b\u7684\u200b\u52a9\u8bcd\u200b\u90fd\u200b\u662f\u200b\u526f\u200b\u52a9\u8bcd\u200b\u3002</p> \u306f\u307e\u3067\u3082\u306e\u307f\u3060\u3051\u3068\u304b\u304f\u3089\u3044\u304b <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u306f\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u53e5\u5b50\u200b\u7684\u200b\u4e3b\u9898\u200b\uff0c\u200b\u8054\u7ed3\u200b\u524d\u200b\u4e00\u4e2a\u200b\u53e5\u5b50\u200b\u3002\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u5c06\u200b\u5176\u4ed6\u200b\u7684\u200b\u683c\u200b\u52a9\u8bcd\u200b\u4fee\u9970\u200b\u7684\u200b\u8bed\u6cd5\u200b\u6210\u5206\u200b\u4f5c\u4e3a\u200b\u4e3b\u9898\u200b\u3002\u300c\u306f\u300d\u200b\u6240\u200b\u4fee\u9970\u200b\u7684\u200b\u4e3b\u9898\u200b\u901a\u5e38\u200b\u5728\u200b\u53e5\u5b50\u200b\u7684\u200b\u5f00\u5934\u200b\u3002</p> <ul> <li> <p>\u200b\u5c06\u200b\u4e3b\u8bed\u200b\u4f5c\u4e3a\u200b\u4e3b\u9898\u200b\uff1a\u300c\u304c\u300d\u200b\u53d8\u4e3a\u200b\u300c\u306f\u300d</p> <p>\u300c\u200b\u79c1\u200b\u306f\u200b\u5b66\u751f\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u662f\u200b\u5b66\u751f\u200b\u3002</p> </li> <li> <p>\u200b\u5c06\u200b\u5730\u70b9\u200b\u4f5c\u4e3a\u200b\u4e3b\u9898\u200b\uff1a\u300c\u306b\u300d\u200b\u540e\u52a0\u200b\u300c\u306f\u300d</p> <p>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u306e\u200b\u8fd1\u200b\u304f\u306b\u306f\u30b3\u30f3\u30d3\u30cb\u304c\u3042\u308a\u307e\u3059\u3002\u300d\uff1a\u200b\u56fe\u4e66\u9986\u200b\u9644\u8fd1\u200b\u6709\u200b\u4fbf\u5229\u5e97\u200b\u3002</p> </li> </ul> <p>\u200b\u6ce8\u610f\u200b\uff0c\u200b\u7591\u95ee\u8bcd\u200b\u4e0d\u80fd\u200b\u4f5c\u4e3a\u200b\u4e3b\u9898\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e0d\u80fd\u200b\u7528\u200b\u300c\u306f\u300d\u200b\u4fee\u9970\u200b\u7591\u95ee\u8bcd\u200b\uff0c\u200b\u53ea\u80fd\u200b\u7528\u200b\u300c\u304c\u300d\u3002</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u307e\u3067\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u65f6\u95f4\u200b\u3001\u200b\u5730\u70b9\u200b\u3001\u200b\u8303\u56f4\u200b\u7b49\u200b\u7684\u200b\u7ec8\u70b9\u200b\uff0c\u200b\u8868\u793a\u200b\u201c\u200b\u76f4\u5230\u200b\u201d\u3001\u201c\u200b\u5230\u200b\u2026\u2026\u200b\u4e3a\u6b62\u200b\u201d\u3002\u200b\u53ef\u4ee5\u200b\u548c\u200b\u300c\u304b\u3089\u300d\u200b\u4e00\u8d77\u200b\u7528\u200b\uff0c\u200b\u8868\u793a\u200b\u201c\u200b\u4ece\u200b\u2026\u2026\u200b\u5230\u200b\u2026\u2026\u201d\u3002</p> <p>\u300c\u3055\u3063\u304d\u307e\u3067\u200b\u56f3\u200b\u66f8\u9928\u200b\u306b\u3044\u307e\u3057\u305f\u3002\u300d\uff1a\u200b\u521a\u624d\u200b\u4e00\u76f4\u200b\u5728\u200b\u56fe\u4e66\u9986\u200b\u3002\uff08\u200b\u65f6\u95f4\u200b\uff09 \u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u307e\u3067\u200b\u884c\u200b\u304d\u307e\u3057\u3087\u3046\u3002\u300d\uff1a\u200b\u53bb\u200b\u56fe\u4e66\u9986\u200b\u5427\u200b\u3002\uff08\u200b\u5730\u70b9\u200b\uff09</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u3082\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u201c\u200b\u4e5f\u200b\u201d\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e0e\u200b\u524d\u9762\u200b\u7684\u200b\u5185\u5bb9\u200b\u76f8\u540c\u200b\u7684\u200b\u4e8b\u7269\u200b\u3002</p> <p>\u300c\u200b\u79c1\u200b\u3082\u200b\u5b66\u751f\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u4e5f\u200b\u662f\u200b\u5b66\u751f\u200b\u3002 \u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u8fd1\u200b\u304f\u306b\u3082\u200b\u7dcf\u200b\u5408\u200b\u68df\u200b\u304c\u3042\u308a\u307e\u3059\u3002\u300d\uff1a\u200b\u56fe\u4e66\u9986\u200b\u9644\u8fd1\u200b\u4e5f\u200b\u6709\u200b\u7efc\u5408\u697c\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b\u300c\u304c\u300d\u3001\u300c\u306f\u300d\u3001\u300c\u3082\u300d\u200b\u4e0d\u80fd\u200b\u540c\u65f6\u200b\u7528\u4e8e\u200b\u540c\u4e00\u4e2a\u200b\u4e3b\u8bed\u200b\u3002</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u306e\u307f\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u201c\u200b\u53ea\u6709\u200b\u201d\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e0e\u200b\u524d\u9762\u200b\u7684\u200b\u5185\u5bb9\u200b\u4e0d\u540c\u200b\u7684\u200b\u4e8b\u7269\u200b\u3002</p> <p>\u300c\u304a\u200b\u4f1a\u8a08\u200b\u306f\u200b\u73fe\u91d1\u200b\u306e\u307f\u3067\u3059\u3002\u300d\uff1a\u200b\u53ea\u6536\u200b\u73b0\u91d1\u200b\u3002</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u3060\u3051\u300d\u200b\u7684\u200b\u542b\u4e49\u200b\u540c\u200b\u300c\u306e\u307f\u300d\uff0c\u200b\u4f46\u662f\u200b\u300c\u3060\u3051\u300d\u200b\u7684\u200b\u8bed\u6c14\u200b\u66f4\u52a0\u200b\u53e3\u8bed\u200b\u3002</p> <p>\u300c\u304a\u200b\u4f1a\u8a08\u200b\u306f\u200b\u73fe\u91d1\u200b\u3060\u3051\u3067\u3059\u3002\u300d\uff1a\u200b\u53ea\u6536\u200b\u73b0\u91d1\u200b\u3002</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u3068\u304b\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u201c\u200b\u4f8b\u5982\u200b\u201d\uff0c\u200b\u7528\u4e8e\u200b\u5217\u4e3e\u200b\u4e8b\u7269\u200b\u3002</p> <p>\u300c\u200b\u79c1\u200b\u306f\u200b\u65e5\u672c\u200b\u8a9e\u200b\u3068\u304b\u200b\u82f1\u8a9e\u200b\u3068\u304b\u200b\u4e2d\u56fd\u200b\u8a9e\u200b\u3068\u304b\u200b\u8a71\u200b\u305b\u307e\u3059\u3002\u300d\uff1a\u200b\u6211\u4f1a\u200b\u8bf4\u200b\u65e5\u8bed\u200b\u3001\u200b\u82f1\u8bed\u200b\u3001\u200b\u6c49\u8bed\u200b\u7b49\u200b\u3002</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u304f\u3089\u3044\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u201c\u200b\u5927\u7ea6\u200b\u201d\u3001\u201c\u200b\u5de6\u53f3\u200b\u201d\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u6570\u91cf\u200b\u3002\u200b\u524d\u9762\u200b\u7684\u200b\u540d\u8bcd\u200b\u4ee5\u200b\u62e8\u97f3\u200b\u7ed3\u5c3e\u200b\u65f6\u200b\uff0c\u300c\u304f\u3089\u3044\u300d\u200b\u53d8\u4e3a\u200b\u300c\u3050\u3089\u3044\u300d\u3002</p> <p>\u300c\u200b\u5168\u90e8\u200b\u3067100\u200b\u5186\u200b\u304f\u3089\u3044\u3067\u3059\u3002\u300d\uff1a\u200b\u4e00\u5171\u200b\u5927\u7ea6\u200b100\u200b\u65e5\u5143\u200b\u3002</p> <p>\u200b\u526f\u200b\u52a9\u8bcd\u200b\u300c\u304b\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e0d\u200b\u786e\u5b9a\u200b\u7684\u200b\u4e8b\u7269\u200b\u3002</p> <p>\u300c\u3044\u3064\u304b\u200b\u56f3\u200b\u66f8\u9928\u200b\u306b\u200b\u884c\u200b\u304d\u307e\u3057\u3087\u3046\u3002\u300d\uff1a\u200b\u4ec0\u4e48\u200b\u65f6\u5019\u200b\u53bb\u200b\u56fe\u4e66\u9986\u200b\u5427\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_9","title":"\u63a5\u7eed\u200b\u52a9\u8bcd","text":"<p>\u200b\u63a5\u7eed\u200b\u52a9\u8bcd\u200b\u7528\u4e8e\u200b\u8fde\u63a5\u200b\u4e24\u4e2a\u200b\u5206\u53e5\u200b\uff0c\u200b\u63a5\u200b\u5728\u200b\u7b2c\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u7684\u200b\u7ed3\u5c3e\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e24\u4e2a\u200b\u5206\u53e5\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5173\u7cfb\u200b\u3002</p> \u304c\u3051\u3069\u304b\u3089\u3068\u3082\u306e\u3067 <p>\u300c\u304c\u300d\u200b\u5728\u200b\u5206\u53e5\u200b\u7684\u200b\u7ed3\u5c3e\u200b\u65f6\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u8f7b\u5fae\u200b\u7684\u200b\u8f6c\u6298\u200b\u8bed\u6c14\u200b\u3002\u200b\u524d\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u867d\u7136\u200b\u2026\u2026\u201d\uff0c\u200b\u540e\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u4f46\u662f\u200b\u2026\u2026\u201d\u3002</p> <p>\u300c\u200b\u79c1\u200b\u3067\u3059\u304c\u3001\u200b\u4f55\u200b\u3067\u3059\u304b\u3002\u300d\uff1a\u200b\u6211\u200b\u662f\u200b\uff0c\u200b\u6709\u200b\u4ec0\u4e48\u200b\u4e8b\u200b\u5417\u200b\uff1f</p> <p>\u300c\u3051\u3069\u300d\u200b\u5728\u200b\u5206\u53e5\u200b\u7684\u200b\u7ed3\u5c3e\u200b\u65f6\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u8f6c\u6298\u200b\u8bed\u6c14\u200b\u3002\u200b\u524d\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u867d\u7136\u200b\u2026\u2026\u201d\uff0c\u200b\u540e\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u4f46\u662f\u200b\u2026\u2026\u201d\u3002\u300c\u3051\u3069\u300d\u200b\u7684\u200b\u8bed\u6c14\u200b\u6bd4\u200b\u300c\u304c\u300d\u200b\u66f4\u52a0\u200b\u53e3\u8bed\u200b\uff0c\u200b\u662f\u200b\u300c\u3051\u308c\u3069\u3082\u300d\u200b\u7684\u200b\u7f29\u5199\u200b\u3002\u200b\u9664\u4e86\u200b\u300c\u3051\u3069\u300d\uff0c\u300c\u3051\u308c\u3069\u3082\u300d\u200b\u8fd8\u6709\u200b\u300c\u3051\u308c\u3069\u300d\u3001\u300c\u3051\u3069\u3082\u300d\u200b\u7b49\u200b\u7f29\u5199\u200b\u5f62\u5f0f\u200b\u3002</p> <p>\u300c\u305d\u3046\u306a\u3093\u3067\u3059\u3051\u3069\u3001\u3084\u306f\u308a\u200b\u5927\u200b\u5909\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u867d\u7136\u200b\u662f\u200b\u8fd9\u6837\u200b\uff0c\u200b\u4f46\u662f\u200b\u8fd8\u662f\u200b\u5f88\u200b\u8f9b\u82e6\u200b\u3002</p> <p>\u300c\u304b\u3089\u300d\u200b\u5728\u200b\u5206\u53e5\u200b\u7684\u200b\u7ed3\u5c3e\u200b\u65f6\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u56e0\u679c\u5173\u7cfb\u200b\u3002\u200b\u524d\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u56e0\u4e3a\u200b\u2026\u2026\u201d\uff0c\u200b\u540e\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u6240\u4ee5\u200b\u2026\u2026\u201d\u3002\u200b\u5728\u200b\u8868\u793a\u200b\u56e0\u679c\u5173\u7cfb\u200b\u65f6\u200b\uff0c\u300c\u304b\u3089\u300d\u200b\u524d\u200b\u5fc5\u987b\u200b\u662f\u200b\u4e00\u4e2a\u200b\u53e5\u5b50\u200b\u3002\u300c\u304b\u3089\u300d\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u52a0\u200b\u540e\u8fb9\u200b\u7684\u200b\u5206\u53e5\u200b\uff0c\u200b\u7528\u4e8e\u200b\u9648\u8ff0\u200b\u539f\u56e0\u200b\u3002</p> <p>\u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u304b\u3089\u3001\u200b\u6dbc\u200b\u3057\u3044\u200b\u6240\u200b\u306b\u200b\u884c\u200b\u304d\u307e\u3057\u3087\u3046\u3002\u300d\uff1a\u200b\u56e0\u4e3a\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\uff0c\u200b\u6240\u4ee5\u200b\u53bb\u200b\u51c9\u5feb\u200b\u7684\u200b\u5730\u65b9\u200b\u5427\u200b\u3002</p> <p>\u300c\u3068\u3082\u300d\u200b\u8868\u793a\u200b\u603b\u6570\u200b\uff0c\u200b\u76f8\u5f53\u4e8e\u200b\u6c49\u8bed\u200b\u4e2d\u200b\u7684\u200b\u201c\u200b\u5171\u200b\u201d\u3001\u201c\u200b\u90fd\u200b\u201d\u3002</p> <p>\u300c\u304a\u200b\u4e8c\u4eba\u200b\u3068\u3082\u30db\u30c3\u30c8\u30b3\u30fc\u30d2\u30fc\u3067\u3088\u308d\u3057\u3044\u3067\u3059\u304b\u3002\u300d\uff1a\u200b\u4e24\u4f4d\u200b\u90fd\u200b\u8981\u200b\u559d\u200b\u70ed\u5496\u5561\u200b\u5417\u200b\uff1f</p> <p>\u300c\u306e\u3067\u300d\u200b\u5728\u200b\u5206\u53e5\u200b\u7684\u200b\u7ed3\u5c3e\u200b\u65f6\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u56e0\u679c\u5173\u7cfb\u200b\u3002\u200b\u524d\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u56e0\u4e3a\u200b\u2026\u2026\u201d\uff0c\u200b\u540e\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u8868\u793a\u200b\u201c\u200b\u6240\u4ee5\u200b\u2026\u2026\u201d\u3002\u200b\u5728\u200b\u8868\u793a\u200b\u56e0\u679c\u5173\u7cfb\u200b\u65f6\u200b\uff0c\u300c\u306e\u3067\u300d\u200b\u524d\u200b\u662f\u200b\u4e00\u4e2a\u200b\u7b80\u4f53\u200b\u53e5\u200b\u3002\u200b\u5982\u679c\u200b\u300c\u306e\u3067\u300d\u200b\u524d\u200b\u662f\u200b\u4e00\u4e2a\u200b\u540d\u8bcd\u200b\u6216\u200bII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0c\u200b\u9700\u8981\u200b\u5c06\u200b\u7b80\u4f53\u200b\u53e5\u200b\u7684\u200b\u300c\u3060\u300d\u200b\u53d8\u4e3a\u200b\u300c\u306a\u300d\u3002</p> <p>\u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u306e\u3067\u3001\u200b\u6dbc\u200b\u3057\u3044\u200b\u6240\u200b\u306b\u200b\u884c\u200b\u304d\u307e\u3057\u3087\u3046\u3002\u300d\uff1a\u200b\u56e0\u4e3a\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\uff0c\u200b\u6240\u4ee5\u200b\u53bb\u200b\u51c9\u5feb\u200b\u7684\u200b\u5730\u65b9\u200b\u5427\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_10","title":"\u7ec8\u200b\u52a9\u8bcd","text":"<p>\u200b\u65e5\u8bed\u200b\u4e2d\u200b\u7684\u200b\u7ec8\u200b\u52a9\u8bcd\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u53e5\u5b50\u200b\u7684\u200b\u8bed\u6c14\u200b\u53d8\u5316\u200b\u3002</p> \u304b\u306d\u3088\u3088\u306d <p>\u300c\u304b\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u7591\u95ee\u200b\u7684\u200b\u8bed\u6c14\u200b\u3002\u200b\u5728\u200b\u53e5\u5b50\u200b\u7684\u200b\u7ed3\u5c3e\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u7591\u95ee\u53e5\u200b\u3002</p> <p>\u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u3067\u3059\u304b\u3002\u300d\uff1a\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\u5417\u200b\uff1f</p> <p>\u300c\u306d\u300d\u200b\u53ef\u4ee5\u200b\u8868\u8fbe\u200b\u786e\u8ba4\u200b\u6216\u200b\u611f\u53f9\u200b\u7684\u200b\u8bed\u6c14\u200b\u3002\u200b\u5f53\u200b\u300c\u306d\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u786e\u8ba4\u200b\u7684\u200b\u8bed\u6c14\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u8bfb\u200b\u5347\u8c03\u200b\uff1b\u200b\u5f53\u200b\u300c\u306d\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u611f\u53f9\u200b\u7684\u200b\u8bed\u6c14\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u8bfb\u200b\u964d\u8c03\u200b\u3002</p> <p>\u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u571f\u66dc\u65e5\u200b\u3067\u3059\u306d\u3002\u300d\u200b\u4eca\u5929\u200b\u662f\u200b\u661f\u671f\u516d\u200b\u5427\u200b\uff1f\uff08\u200b\u786e\u8ba4\u200b\uff09 \u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u3067\u3059\u306d\u3002\u300d\uff1a\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\u554a\u200b\u3002\uff08\u200b\u611f\u53f9\u200b\uff09</p> <p>\u300c\u3088\u300d\u200b\u7528\u4e8e\u200b\u5f3a\u8c03\u200b\u81ea\u5df1\u200b\u7684\u200b\u89c2\u70b9\u200b\u3002</p> <p>\u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u3067\u3059\u3088\u3002\u300d\uff1a\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\u554a\u200b\u3002</p> <p>\u300c\u3088\u306d\u300d\u200b\u540c\u65f6\u200b\u5305\u542b\u200b\u4e86\u200b\u300c\u3088\u300d\u200b\u548c\u200b\u300c\u306d\u300d\u200b\u7684\u200b\u8bed\u6c14\u200b\u3002</p> <p>\u300c\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u3067\u3059\u3088\u306d\u3002\u300d\uff1a\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\uff0c\u200b\u5bf9\u200b\u5427\u200b\uff1f</p>"},{"location":"japanese/japanese-1-grammar/#_11","title":"\u300c\u306f\u300d\u200b\u4e0e\u200b\u300c\u304c\u300d","text":"<p>\u200b\u52a9\u8bcd\u200b\u300c\u304c\u300d\u200b\u662f\u200b\u683c\u200b\u52a9\u8bcd\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e3b\u8bed\u200b\uff0c\u300c\u306f\u300d\u200b\u662f\u200b\u526f\u200b\u52a9\u8bcd\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u4e3b\u9898\u200b\u3002\u200b\u4e3b\u9898\u200b\u662f\u200b\u6307\u200b\u53e5\u5b50\u200b\u4e3b\u8981\u200b\u63cf\u8ff0\u200b\u7684\u200b\u8bdd\u9898\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7531\u200b\u4e3b\u8bed\u200b\u6216\u200b\u5176\u4ed6\u200b\u6210\u5206\u200b\u5145\u5f53\u200b\u3002</p> <ul> <li>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u306e\u200b\u8fd1\u200b\u304f\u306b\u306f\u30b3\u30f3\u30d3\u30cb\u304c\u3042\u308a\u307e\u3059\u3002\u300d</li> <li>\u300c\u30b3\u30f3\u30d3\u30cb\u306f\u200b\u56f3\u200b\u66f8\u9928\u200b\u306e\u200b\u8fd1\u200b\u304f\u306b\u3042\u308a\u307e\u3059\u3002\u300d</li> <li>\u300c\u200b\u56f3\u200b\u66f8\u9928\u200b\u306e\u3061\u304b\u304f\u306b\u30b3\u30f3\u30d3\u30cb\u304c\u3042\u308a\u307e\u3059\u3002\u300d</li> <li>\u200b\u7b2c\u200b1\u200b\u53e5\u200b\u4e2d\u200b\u4e3b\u8981\u200b\u5f3a\u8c03\u200b\u56fe\u4e66\u9986\u200b\u7684\u200b\u9644\u8fd1\u200b\u6709\u200b\u4fbf\u5229\u5e97\u200b\uff0c\u200b\u53e5\u5b50\u200b\u7684\u200b\u4e3b\u9898\u200b\u662f\u200b\u56fe\u4e66\u9986\u200b\u9644\u8fd1\u200b\u7684\u200b\u4e1c\u897f\u200b\uff1b\u200b\u7b2c\u200b2\u200b\u53e5\u200b\u4e2d\u200b\u4e3b\u8981\u200b\u5f3a\u8c03\u200b\u4fbf\u5229\u5e97\u200b\u5728\u200b\u56fe\u4e66\u9986\u200b\u9644\u8fd1\u200b\uff0c\u200b\u53e5\u5b50\u200b\u7684\u200b\u4e3b\u9898\u200b\u662f\u200b\u4fbf\u5229\u5e97\u200b\u3002</li> <li>\u200b\u7b2c\u200b3\u200b\u53e5\u200b\u4e0d\u200b\u5305\u542b\u200b\u4e3b\u9898\u200b\uff0c\u200b\u4ec5\u4ec5\u200b\u9648\u8ff0\u200b\u4e86\u200b\u56fe\u4e66\u9986\u200b\u9644\u8fd1\u200b\u6709\u200b\u4fbf\u5229\u5e97\u200b\u3002</li> </ul> <p>\u200b\u5f53\u200b\u56de\u7b54\u200b\u7591\u95ee\u53e5\u200b\u65f6\u200b\uff0c\u200b\u5982\u679c\u200b\u7591\u95ee\u53e5\u200b\u662f\u200b\u7528\u200b\u300c\u304c\u300d\u200b\u63d0\u95ee\u200b\u7684\u200b\uff0c\u200b\u5728\u200b\u56de\u7b54\u200b\u7684\u200b\u65f6\u5019\u200b\u4e5f\u200b\u9700\u8981\u200b\u7528\u200b\u300c\u304c\u300d\u3002</p> <p>\u200b\u901a\u5e38\u200b\u6765\u8bf4\u200b\uff1a</p> <ol> <li>\u300c\u306f\u300d\u200b\u901a\u5e38\u200b\u7528\u4e8e\u200b\u63d0\u793a\u200b\u5df2\u6709\u200b\u7684\u200b\u4fe1\u606f\u200b\uff0c\u300c\u304c\u300d\u200b\u7528\u4e8e\u200b\u63d0\u793a\u200b\u65b0\u200b\u7684\u200b\u4fe1\u606f\u200b\u3002</li> <li>\u300c\u306f\u300d\u200b\u5f3a\u8c03\u200b\uff08\u200b\u4f20\u8fbe\u200b\uff09\u200b\u7684\u200b\u901a\u5e38\u200b\u662f\u200b\u300c\u306f\u300d\u200b\u540e\u9762\u200b\u7684\u200b\u5185\u5bb9\u200b\uff0c\u300c\u304c\u300d\u200b\u5f3a\u8c03\u200b\uff08\u200b\u4f20\u8fbe\u200b\uff09\u200b\u7684\u200b\u901a\u5e38\u200b\u662f\u200b\u300c\u304c\u300d\u200b\u524d\u9762\u200b\u7684\u200b\u5185\u5bb9\u200b\u3002</li> <li>\u200b\u7528\u4e8e\u200b\u5f62\u5bb9\u200b\u80fd\u529b\u200b\u7684\u200bII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff08\u300c\u3058\u3087\u3046\u305a\u300d\u3001\u300c\u3078\u305f\u300d\u3001\u300c\u200b\u597d\u200b\u304d\u300d\u3001\u300c\u3068\u304f\u3044\u300d\uff09\u200b\u6216\u200b\u52a8\u8bcd\u200b\uff08\u300c\u3067\u304d\u308b\u300d\u3001\u300c\u308f\u304b\u308b\u300d\uff09\u200b\u7684\u200b\u4e3b\u8bed\u200b\u53ea\u80fd\u200b\u7528\u200b\u300c\u304c\u300d\uff0c\u200b\u4e0d\u80fd\u200b\u7528\u200b\u300c\u306f\u300d\u3002</li> </ol>"},{"location":"japanese/japanese-1-grammar/#_12","title":"\u63a5\u7eed\u8bcd","text":"<p>\u200b\u63a5\u7eed\u8bcd\u200b\u653e\u5728\u200b\u53e5\u5b50\u200b\u7684\u200b\u5f00\u5934\u200b\uff0c\u200b\u540e\u9762\u200b\u53ef\u4ee5\u200b\u63a5\u200b\u53e5\u5b50\u200b\u6216\u200b\u77ed\u8bed\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u8be5\u53e5\u200b\u548c\u200b\u524d\u200b\u4e00\u53e5\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5173\u7cfb\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_13","title":"\u3057\u304b\u3057","text":"<p>\u300c\u3057\u304b\u3057\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u8f6c\u6298\u200b\u5173\u7cfb\u200b\uff0c\u200b\u76f8\u5f53\u4e8e\u200b\u6c49\u8bed\u200b\u4e2d\u200b\u7684\u200b\u201c\u200b\u4f46\u662f\u200b\u201d\uff0c\u200b\u7528\u4e8e\u200b\u53e5\u5b50\u200b\u7684\u200b\u5f00\u5934\u200b\u3002</p> <p>\u300c\u3057\u304b\u3057\u3001\u200b\u4eca\u65e5\u200b\u306f\u200b\u6691\u200b\u3044\u3067\u3059\u306d\u3002\u300d\uff1a\u200b\u4f46\u662f\u200b\uff0c\u200b\u4eca\u5929\u200b\u5f88\u200b\u70ed\u200b\u5427\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_14","title":"\u3067\u3082","text":"<p>\u300c\u3067\u3082\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u8f6c\u6298\u200b\u5173\u7cfb\u200b\uff0c\u200b\u76f8\u5f53\u4e8e\u200b\u6c49\u8bed\u200b\u4e2d\u200b\u7684\u200b\u201c\u200b\u4f46\u662f\u200b\u201d\uff0c\u200b\u8bed\u6c14\u200b\u6bd4\u200b\u300c\u3057\u304b\u3057\u300d\u200b\u66f4\u52a0\u200b\u53e3\u8bed\u200b\u3002</p> <p>\u300c\u3067\u3082\u3001\u200b\u5348\u200b\u5f8c\u200b\u304b\u3089\u306f\u200b\u6dbc\u200b\u3057\u304f\u306a\u308a\u307e\u3059\u3002\u300d\uff1a\u200b\u4f46\u662f\u200b\uff0c\u200b\u4ece\u200b\u4e0b\u5348\u200b\u5f00\u59cb\u200b\u4f1a\u200b\u51c9\u5feb\u200b\u4e00\u4e9b\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_15","title":"\u305d\u308c\u304b\u3089","text":"<p>\u300c\u305d\u308c\u304b\u3089\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u987a\u63a5\u200b\u5173\u7cfb\u200b\uff0c\u200b\u76f8\u5f53\u4e8e\u200b\u6c49\u8bed\u200b\u4e2d\u200b\u7684\u200b\u201c\u200b\u7136\u540e\u200b\u201d\uff0c\u200b\u7528\u4e8e\u200b\u53e5\u5b50\u200b\u7684\u200b\u5f00\u5934\u200b\u3002</p> <p>\u300c\u305d\u308c\u304b\u3089\u3001\u200b\u56f3\u200b\u66f8\u9928\u200b\u306b\u200b\u884c\u200b\u304d\u307e\u3057\u3087\u3046\u3002\u300d\uff1a\u200b\u7136\u540e\u200b\uff0c\u200b\u53bb\u200b\u56fe\u4e66\u9986\u200b\u5427\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_16","title":"\u8bed\u6cd5","text":""},{"location":"japanese/japanese-1-grammar/#_17","title":"\u656c\u4f53\u200b\u53e5\u200b\u53ca\u200b\u65f6\u6001\u53d8\u5316","text":""},{"location":"japanese/japanese-1-grammar/#_18","title":"\u540d\u8bcd\u200b\u53e5\u200b\u4e0e\u200b\u52a8\u200b\u8bcd\u53e5","text":"<p>\u200b\u65e5\u8bed\u200b\u4e2d\u200b\u7684\u200b\u65f6\u6001\u53d8\u5316\u200b\u4e3b\u8981\u200b\u6709\u200b\u73b0\u5728\u200b\u548c\u200b\u8fc7\u53bb\u200b\u65f6\u200b\u3002\u200b\u540d\u8bcd\u200b\u656c\u4f53\u200b\u53e5\u200b\u4ee5\u200b\u300c\u3067\u3059\u300d\u200b\u7ed3\u53e5\u200b\uff0c\u200b\u5176\u200b\u5728\u200b\u4e0d\u540c\u200b\u65f6\u6001\u200b\u4e0b\u200b\u7684\u200b\u53d8\u5f62\u200b\u5982\u4e0b\u200b\uff1a</p> \u200b\u80af\u5b9a\u200b \u200b\u5426\u5b9a\u200b \u200b\u73b0\u5728\u200b \u301c\u3067\u3059\u3002 \u301c\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002 \u200b\u8fc7\u53bb\u200b \u301c\u3067\u3057\u305f\u3002 \u301c\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3067\u3057\u305f\u3002 <p>\u300c\u3067\u306f\u300d\u200b\u53ef\u4ee5\u200b\u66ff\u6362\u6210\u200b\u300c\u3058\u3083\u300d\u3002</p> <p>\u200b\u52a8\u8bcd\u200b\u656c\u4f53\u200b\u53e5\u200b\u4ee5\u200b\u300c\u307e\u3059\u300d\u200b\u7ed3\u53e5\u200b\uff0c\u200b\u5176\u200b\u5728\u200b\u4e0d\u540c\u200b\u65f6\u6001\u200b\u4e0b\u200b\u7684\u200b\u53d8\u5f62\u200b\u5982\u4e0b\u200b\uff1a</p> \u200b\u80af\u5b9a\u200b \u200b\u5426\u5b9a\u200b \u200b\u73b0\u5728\u200b \u301c\u307e\u3059\u3002 \u301c\u307e\u305b\u3093\u3002 \u200b\u8fc7\u53bb\u200b \u301c\u307e\u3057\u305f\u3002 \u301c\u307e\u305b\u3093\u3067\u3057\u305f\u3002 <p>\u200b\u5b58\u5728\u200b\u53e5\u200b\u7684\u200b\u52a8\u8bcd\u200b\u6709\u200b\u4e24\u79cd\u200b\uff0c\u200b\u5373\u200b\u300c\u3042\u308b\u300d\uff08\u3042\u308a\u307e\u3059\uff09\u200b\u548c\u200b\u300c\u3044\u308b\u300d\uff08\u3044\u307e\u3059\uff09\uff0c\u200b\u5176\u4e2d\u200b\u300c\u3042\u308b\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u65e0\u200b\u751f\u547d\u200b\u7684\u200b\u4e8b\u7269\u200b\uff0c\u300c\u3044\u308b\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u6709\u200b\u751f\u547d\u200b\u7684\u200b\u4e8b\u7269\u200b\u3002</p> <p>\u200b\u8868\u793a\u200b\u63d0\u8bae\u200b\u65f6\u200b\uff0c\u200b\u7528\u200b\u300c\u307e\u3057\u3087\u3046\u300d\u200b\u6216\u200b\u300c\u307e\u3057\u3087\u3046\u304b\u300d\u200b\u4ee3\u66ff\u200b\u300c\u307e\u3059\u300d\u3002\u200b\u8868\u793a\u200b\u63a8\u6d4b\u200b\u65f6\u200b\uff0c\u200b\u7528\u200b\u300c\u3067\u3057\u3087\u3046\u300d\u200b\u4ee3\u66ff\u200b\u300c\u3067\u3059\u300d\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_19","title":"\u63cf\u8ff0\u200b\u53e5","text":"<p>\u200b\u5bf9\u4e8e\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0c\u200b\u5176\u200b\u5728\u200b\u4e0d\u540c\u200b\u65f6\u6001\u200b\u4e0b\u200b\u7684\u200b\u53d8\u5f62\u200b\u5982\u4e0b\u200b\uff08\u201c\uff5e\u201d\u200b\u6307\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u8bcd\u5e72\u200b\uff0c\u200b\u5373\u200b\u53bb\u6389\u200b\u300c\u3044\u300d\u200b\u7684\u200b\u90e8\u5206\u200b\uff09\uff1a</p> \u200b\u80af\u5b9a\u200b \u200b\u5426\u5b9a\u200b \u200b\u73b0\u5728\u200b \u301c\u3044\u3067\u3059\u3002 \u301c\u304f\u3042\u308a\u307e\u305b\u3093\u3002 \u200b\u8fc7\u53bb\u200b \u301c\u304b\u3063\u305f\u3067\u3059\u3002 \u301c\u304f\u3042\u308a\u307e\u305b\u3093\u3067\u3057\u305f\u3002 <p>\u200b\u6ce8\u610f\u200b\u300c\u306a\u3044\u300d\u200b\u4e5f\u200b\u662f\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0c\u300c\u3042\u308a\u307e\u305b\u3093\u300d\u200b\u548c\u200b\u300c\u306a\u3044\u3067\u3059\u300d\u200b\u7b49\u4ef7\u200b\u3002\u200b\u56e0\u6b64\u200b\u8fd8\u6709\u200b\u5982\u4e0b\u200b\u53d8\u5f62\u200b\uff1a</p> \u200b\u80af\u5b9a\u200b \u200b\u5426\u5b9a\u200b \u200b\u73b0\u5728\u200b \u301c\u3044\u3067\u3059\u3002 \uff5e\u304f\u306a\u3044\u3067\u3059\u3002 \u200b\u8fc7\u53bb\u200b \u301c\u304b\u3063\u305f\u3067\u3057\u305f\u3002 \u301c\u304f\u306a\u304b\u3063\u305f\u3067\u3059 <p>\u200b\u4ee5\u4e0b\u200b\u4e24\u4e2a\u200b\u53e5\u5b50\u200b\u7b49\u4ef7\u200b\uff1a</p> <p>\u300c\u200b\u79c1\u200b\u306f\u200b\u9ad8\u200b\u304f\u306a\u3044\u3067\u3059\u3002\u300d \u300c\u200b\u79c1\u200b\u306f\u200b\u9ad8\u200b\u304f\u3042\u308a\u307e\u305b\u3093\u3002\u300d</p> <p>II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff08\u200b\u5f62\u5bb9\u200b\u52a8\u8bcd\u200b\uff09\u200b\u7684\u200b\u53d8\u5f62\u200b\u548c\u200b\u540d\u8bcd\u200b\u76f8\u540c\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_20","title":"\u52a8\u200b\u8bcd\u53e5\u200b\u7684\u200b\u8fdb\u884c\u200b\u65f6\u200b\u4e0e\u200b\u5b8c\u6210\u200b\u65f6","text":"<p>\u300c\u3066\u3044\u307e\u3059\u300d\u200b\u4e0e\u200b\u6301\u7eed\u6027\u200b\u52a8\u8bcd\u200b\u8fde\u7528\u200b\u65f6\u200b\uff0c\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u8fdb\u884c\u200b\u65f6\u200b\uff1b\u200b\u4e0e\u200b\u77ac\u95f4\u200b\u52a8\u8bcd\u200b\u8fde\u7528\u200b\u65f6\u200b\uff0c\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u5b8c\u6210\u200b\u65f6\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_21","title":"\u987a\u63a5\u200b\u5173\u7cfb","text":"<p>\u200b\u4e24\u4e2a\u200b\u5206\u53e5\u200b\u5982\u679c\u200b\u4e0d\u662f\u200b\u8868\u793a\u200b\u8f6c\u6298\u200b\u3001\u200b\u6761\u4ef6\u200b\u7b49\u200b\u5173\u7cfb\u200b\uff0c\u200b\u8868\u793a\u200b\u987a\u63a5\u200b\u5173\u7cfb\u200b\u65f6\u200b\uff0c\u200b\u6784\u6210\u200b\u987a\u63a5\u200b\u5173\u7cfb\u200b\u3002\u200b\u7b2c\u4e00\u4e2a\u200b\u4ece\u53e5\u200b\u7684\u200b\u53e5\u200b\u5c3e\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u300c\u3066\u300d\u200b\u578b\u200b\uff08\u200b\u8fde\u7528\u200b\u578b\u200b\uff09\u200b\u7ed3\u5c3e\u200b\u3002</p> <p>\u300c\u200b\u79c1\u200b\u306f\u200b\u65e5\u672c\u200b\u8a9e\u200b\u3092\u200b\u52c9\u5f37\u200b\u3057\u3066\u3001\u200b\u4e2d\u56fd\u200b\u8a9e\u200b\u3092\u200b\u52c9\u5f37\u200b\u3057\u307e\u3059\u3002\u300d\uff1a\u200b\u6211\u200b\u5b66\u4e60\u200b\u65e5\u8bed\u200b\uff0c\u200b\u4e5f\u200b\u5b66\u4e60\u200b\u6c49\u8bed\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_22","title":"\u5f62\u5f0f\u200b\u4f53\u8a00","text":"<p>\u200b\u5f62\u5f0f\u200b\u4f53\u8a00\u200b\u662f\u200b\u6307\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u62bd\u8c61\u6982\u5ff5\u200b\u7684\u200b\u540d\u8bcd\u200b\uff0c\u200b\u5982\u200b\u300c\u3053\u3068\u300d\u3001\u300c\u306e\u300d\u3001\u300c\u3082\u306e\u300d\u200b\u7b49\u200b\u3002\u200b\u5f62\u5f0f\u200b\u4f53\u8a00\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u52a8\u4f5c\u200b\u7684\u200b\u4e3b\u8bed\u200b\u6216\u200b\u5bbe\u8bed\u200b\uff0c\u200b\u4e0e\u200b\u52a8\u8bcd\u200b\u539f\u578b\u200b\u8fde\u7528\u200b\u65f6\u200b\u76f8\u5f53\u4e8e\u200b\u52a8\u540d\u8bcd\u200b\u3002</p> \u200b\u5f62\u5f0f\u200b\u4f53\u8a00\u200b \u200b\u542b\u4e49\u200b \u3053\u3068 \u200b\u4e8b\u60c5\u200b\uff08\u200b\u4e00\u822c\u200b\u8868\u793a\u200b\u4e8b\u60c5\u200b\u3001\u200b\u4e8b\u5b9e\u200b\u3001\u200b\u5185\u5bb9\u200b\uff09 \u306e \u200b\u4e8b\u60c5\u200b\uff08\u200b\u4e00\u822c\u200b\u8868\u793a\u200b\u4eba\u200b\u3001\u200b\u4e8b\u200b\u3001\u200b\u7269\u200b\uff09 \u3068\u3053\u308d \u200b\u5730\u70b9\u200b\uff08\u200b\u7528\u6765\u200b\u505a\u200b\u2026\u2026\uff09 \u3082\u306e \u200b\u7269\u54c1\u200b\uff08\u200b\u7528\u6765\u200b\u505a\u200b\u2026\u2026\uff09 <p>\u200b\u56f3\u200b\u66f8\u9928\u200b\u306f\u200b\u52c9\u5f37\u200b\u3059\u308b\u3068\u3053\u308d\u3067\u3059\u3002\uff1a\u200b\u56fe\u4e66\u9986\u200b\u662f\u200b\u5b66\u4e60\u200b\u7684\u200b\u5730\u65b9\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_23","title":"\u6bd4\u8f83\u7ea7\u200b\u4e0e\u200b\u6700\u9ad8\u7ea7","text":"<p>\u200b\u65e5\u8bed\u200b\u4e2d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u6bd4\u8f83\u200b\u7684\u200b\u52a9\u8bcd\u200b\u4e3a\u200b\u300c\u3088\u308a\u300d\u200b\u548c\u200b\u300c\u307b\u3069\u300d\u3002\u200b\u8981\u200b\u8868\u793a\u200b\u201cA\u200b\u6bd4\u200bB\u200b\u66f4\u200b\u2026\u2026\u201d\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u7684\u200b\u8868\u8fbe\u65b9\u5f0f\u200b\uff1a</p> <ol> <li>\u300cA\u306fB\u3088\u308a\u301c\u300d\uff0c\u200b\u8868\u793a\u200b\u201cA\u200b\u6bd4\u200bB\u200b\u66f4\u200b\u2026\u2026\u201d\uff0c\u200b\u5728\u200b\u53e3\u8bed\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u300cA\u306f\u300d\u200b\u6765\u200b\u66f4\u200b\u591a\u200b\u5f3a\u8c03\u200b\u7528\u4e8e\u200b\u6bd4\u8f83\u200b\u7684\u200b\u57fa\u51c6\u200b\u3002</li> <li>\u300cB\u3088\u308aA\u306e\u200b\u65b9\u200b\u304c\u300d\u200b\u8868\u793a\u200b\u201cA\u200b\u6bd4\u200bB\u200b\u66f4\u200b\u2026\u2026\u201d\uff0c\u200b\u5728\u200b\u53e3\u8bed\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u300cB\u3088\u308a\u300d\uff0c\u200b\u66f4\u200b\u591a\u200b\u5f3a\u8c03\u200b\u4e3b\u8bed\u200bA\u3002</li> <li>\u300cB\u306fA\u307b\u3069\u301c\u3042\u308a\u307e\u305b\u3093\u300d\uff0c\u200b\u8868\u793a\u200b\u201cB\u200b\u4e0d\u5982\u200bA\u2026\u2026\u201d\u3002</li> </ol> <p>\u200b\u5728\u200b\u7591\u95ee\u53e5\u200b\u4e2d\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b\u300cA\u3068B\u3068\u3069\u3061\u3089\u306e\u200b\u65b9\u200b\u304c\u301c\u3067\u3059\u304b\u300d\u200b\u6765\u200b\u8be2\u95ee\u200bA\u200b\u548c\u200bB\u200b\u54ea\u4e2a\u200b\u66f4\u200b\u2026\u2026\u3002\u200b\u5176\u4e2d\u200b\u300c\u306e\u200b\u65b9\u200b\u300d\u200b\u8868\u793a\u200b\u201c\u2026\u2026\u200b\u7684\u200b\u7a0b\u5ea6\u200b\u201d\uff0c\u200b\u5728\u200b\u95ee\u53e5\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u3002\u200b\u6ce8\u610f\u200b\u5728\u200b\u56de\u7b54\u200b\u7591\u95ee\u53e5\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b\u300c\u304c\u300d\u3002\u200b\u5982\u679c\u200b\u4e24\u8005\u200b\u7a0b\u5ea6\u200b\u5dee\u4e0d\u591a\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b\u300c\u3069\u3061\u3089\u3082\u301c\u300d\u200b\u56de\u7b54\u200b\u3002</p> <p>\u300c\u200b\u5317\u4eac\u5927\u5b66\u200b\u3068\u200b\u6e05\u83ef\u200b\u5927\u5b66\u200b\u3068\u3069\u3061\u3089\u306e\u200b\u65b9\u200b\u304c\u200b\u6709\u540d\u200b\u3067\u3059\u304b\u3002\u300d\uff1a\u200b\u5317\u4eac\u5927\u5b66\u200b\u548c\u200b\u6e05\u534e\u5927\u5b66\u200b\u54ea\u4e2a\u200b\u66f4\u200b\u6709\u540d\u200b\uff1f \u300c\u200b\u6e05\u83ef\u200b\u5927\u5b66\u200b\u306e\u200b\u65b9\u200b\u304c\u200b\u6709\u540d\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u5317\u4eac\u5927\u5b66\u200b\u66f4\u200b\u6709\u540d\u200b\u3002 \u200b\u6216\u200b \u300c\u3069\u3061\u3089\u3082\u200b\u6709\u540d\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u4e24\u4e2a\u200b\u90fd\u200b\u6709\u540d\u200b\u3002</p> <p>\u200b\u65e5\u8bed\u200b\u4e2d\u200b\u8868\u8fbe\u200b\u6700\u9ad8\u7ea7\u200b\u7684\u200b\u65b9\u5f0f\u200b\u6709\u200b\u4e24\u79cd\u200b\uff1a</p> <ol> <li> <p>\u200b\u526f\u8bcd\u200b\u300c\u200b\u4e00\u756a\u200b\u300d\uff0c\u200b\u8868\u793a\u200b\u201c\u200b\u6700\u200b\u2026\u2026\u201d\uff0c\u200b\u5728\u200b\u53e5\u5b50\u200b\u4e2d\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u300c\u301c\u306e\u200b\u4e2d\u200b\u3067\u300d\u200b\u6765\u200b\u9650\u5236\u200b\u6bd4\u8f83\u200b\u7684\u200b\u8303\u56f4\u200b\u3002</p> <p>\u300c\u200b\u5317\u4eac\u200b\u306e\u200b\u5927\u5b66\u200b\u306e\u200b\u4e2d\u200b\u3067\u200b\u6e05\u83ef\u200b\u5927\u5b66\u200b\u304c\u200b\u4e00\u756a\u200b\u6709\u540d\u200b\u3067\u3059\u3002\u300d\uff1a\u200b\u5728\u200b\u5317\u4eac\u200b\u7684\u200b\u5927\u5b66\u200b\u4e2d\u200b\uff0c\u200b\u6e05\u534e\u5927\u5b66\u200b\u6700\u200b\u6709\u540d\u200b\u3002</p> </li> <li> <p>\u200b\u4f7f\u7528\u200b\u201c\u200b\u6ca1\u6709\u200b\u6bd4\u200b\u2026\u2026\u200b\u66f4\u200b\u2026\u2026\u201d\u200b\u7684\u200b\u53e5\u5f0f\u200b\uff0c\u200b\u5373\u200b\u300cA\u307b\u3069\u301cB\u306f\u3042\u308a\u307e\u305b\u3093\u300d\u3002\u200b\u5176\u4e2d\u200b\uff0cB\u200b\u4e3a\u200b\u5f62\u5f0f\u200b\u4f53\u8a00\u200b\uff0c\u200b\u8868\u793a\u200b\u201c\u2026\u2026\u200b\u7684\u200b\u4e1c\u897f\u200b\u201d\u3002\u200b\u6ce8\u610f\u200b\u7528\u4e8e\u200b\u6709\u200b\u751f\u547d\u200b\u7684\u200b\u4e8b\u7269\u200b\u65f6\u200b\uff0c\u200b\u9700\u8981\u200b\u7528\u200b\u300c\u3044\u307e\u305b\u3093\u300d\u3002</p> <p>\u300c\u200b\u6e05\u83ef\u200b\u5927\u5b66\u200b\u307b\u3069\u200b\u6709\u540d\u200b\u306a\u200b\u5927\u5b66\u200b\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u300d\uff1a\u200b\u6ca1\u6709\u200b\u6bd4\u200b\u6e05\u534e\u5927\u5b66\u200b\u66f4\u200b\u6709\u540d\u200b\u7684\u200b\u5927\u5b66\u200b\u4e86\u200b\u3002</p> </li> </ol> <p>\u200b\u63d0\u95ee\u200b\u6700\u9ad8\u7ea7\u200b\u65f6\u200b\u7684\u200b\u4e00\u822c\u200b\u53e5\u5f0f\u200b\u4e3a\u200b\u300cA\u3067\u3069\u3053\u304c\u200b\u4e00\u756a\u200b\u301c\u3067\u3059\u304b\u300d\u200b\u6216\u200b\u300cA\u3067\u3069\u306e\u301c\u304c\u200b\u4e00\u756a\u200b\u301c\u3067\u3059\u304b\u300d\uff0c\u200b\u610f\u200b\u4e3a\u200bA\u200b\u4e2d\u200b\u6700\u200b\u2026\u2026\u200b\u7684\u200b\u662f\u200b\u54ea\u4e2a\u200b\u2026\u2026\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_24","title":"\u7b80\u4f53\u200b\u53e5","text":"<p>\u200b\u5f53\u200b\u4f7f\u7528\u200b\u63a5\u7eed\u200b\u52a9\u8bcd\u200b\u300c\u306e\u3067\u300d\u200b\u65f6\u200b\uff0c\u200b\u524d\u200b\u4e00\u4e2a\u200b\u5206\u53e5\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u7b80\u4f53\u200b\u53e5\u200b\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_25","title":"\u540d\u8bcd\u200b\u53e5","text":"<p>\u200b\u5bf9\u4e8e\u200b\u53e5\u5b50\u200b\u7ed3\u5c3e\u200b\u662f\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u7b80\u4f53\u200b\u53e5\u200b\u65e0\u9700\u200b\u505a\u51fa\u200b\u53d8\u5316\u200b\u3002</p> \u200b\u80af\u5b9a\u200b \u200b\u5426\u5b9a\u200b \u200b\u73b0\u5728\u200b \uff5e\u3044 \uff5e\u304f\u306a\u3044 \u200b\u8fc7\u53bb\u200b \uff5e\u304b\u3063\u305f \uff5e\u304f\u306a\u304b\u3063\u305f <p>\u200b\u5bf9\u4e8e\u200b\u53e5\u5b50\u200b\u7ed3\u5c3e\u200b\u662f\u200b\u540d\u8bcd\u200b\u548c\u200bII\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u7b80\u4f53\u200b\u53e5\u200b\u662f\u200b\u5728\u200b\u53e5\u200b\u5c3e\u200b\u52a0\u4e0a\u200b\u300c\u3060\u300d\uff08II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u7684\u200b\u539f\u578b\u200b\uff09\u3002\u200b\u8fc7\u53bb\u200b\u65f6\u200b\u7b80\u4f53\u200b\u53e5\u200b\u662f\u200b\u5728\u200b\u53e5\u200b\u5c3e\u200b\u52a0\u4e0a\u200b\u300c\u3060\u3063\u305f\u300d\u3002\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b\u52a0\u4e0a\u200b\u300c\u3067\u306f\u306a\u3044\u300d\u200b\u6216\u200b\u300c\u3067\u306f\u306a\u3063\u304b\u305f\u300d\u3002</p> \u200b\u80af\u5b9a\u200b \u200b\u5426\u5b9a\u200b \u200b\u73b0\u5728\u200b \u301c\u3060 \u301c\u3067\u306f\u306a\u3044 \u200b\u8fc7\u53bb\u200b \u301c\u3060\u3063\u305f \u301c\u3067\u306f\u306a\u304b\u3063\u305f"},{"location":"japanese/japanese-1-grammar/#_26","title":"\u52a8\u200b\u8bcd\u53e5","text":"<p>\u200b\u5728\u200b\u7b80\u4f53\u200b\u53e5\u200b\u4e2d\u200b\uff0c\u200b\u52a8\u8bcd\u200b\u4f7f\u7528\u200b\u5176\u200b\u539f\u578b\u200b\u3002\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b\u4e2d\u200b\u4f7f\u7528\u200b\u5176\u200b\u306a\u3044\u200b\u578b\u200b\uff0c\u200b\u8fc7\u53bb\u200b\u65f6\u200b\u4f7f\u7528\u200b\u5176\u200b\u305f\u200b\u578b\u200b\u3002\u200b\u6ce8\u610f\u200b\u300c\u306a\u3044\u300d\u200b\u662f\u200bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5176\u200b\u8fc7\u53bb\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b\u4e3a\u200b\u300c\u306a\u304b\u3063\u305f\u300d\u3002</p>"},{"location":"japanese/japanese-1-grammar/#_27","title":"\u306e\u3067\u3059/\u3093\u3067\u3059\u200b\u7ed3\u6784","text":"<p>\u300c\u306e\u3067\u3059\u300d\u200b\u548c\u200b\u300c\u3093\u3067\u3059\u300d\u200b\u591a\u200b\u7528\u4e8e\u200b\u53e3\u8bed\u200b\uff0c\u200b\u8868\u793a\u200b\u89e3\u91ca\u200b\u3001\u200b\u8bf4\u660e\u200b\u3001\u200b\u8bf7\u6c42\u200b\u3001\u200b\u5efa\u8bae\u200b\u7b49\u200b\u8bed\u6c14\u200b\u3002II\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u6216\u200b\u540d\u8bcd\u200b\u9700\u8981\u200b\u5728\u200b\u7b80\u4f53\u200b\u53e5\u200b\u5f62\u5f0f\u200b\u540e\u200b\u52a0\u4e0a\u200b\u300c\u306a\u300d\uff0c\u200b\u518d\u200b\u52a0\u200b\u300c\u306e\u3067\u3059\u300d\u200b\u6216\u200b\u300c\u3093\u3067\u3059\u300d\uff1bI\u200b\u7c7b\u200b\u5f62\u5bb9\u8bcd\u200b\u76f4\u63a5\u200b\u52a0\u200b\u300c\u306e\u3067\u3059\u300d\u200b\u6216\u200b\u300c\u3093\u3067\u3059\u300d\u3002</p>"},{"location":"japanese/japanese-1-pattern/","title":"\u65e5\u8bed\u200b(1)\u200b\u77ed\u8bed\u200b\u4e0e\u200b\u8868\u8fbe","text":""},{"location":"japanese/japanese-1-pattern/#1-4","title":"1-4\u200b\u8bfe","text":"<ol> <li>\u300c\u304a\u3084\u3059\u307f\u306a\u3055\u3044\u300d\uff1a\u200b\u665a\u5b89\u200b</li> <li>\u300c\u304a\u200b\u75b2\u200b\u308c\u200b\u69d8\u200b\u300d\uff1a\u200b\u8f9b\u82e6\u200b\u4e86\u200b</li> <li>\u300c\u308f\u304b\u308a\u307e\u3057\u305f\u300d\uff1a\u200b\u6211\u200b\u77e5\u9053\u200b\u4e86\u200b</li> <li>\u300c\u200b\u521d\u200b\u3081\u307e\u3057\u3066\u300d\uff1a\u200b\u521d\u6b21\u89c1\u9762\u200b</li> <li>\u300c\u304a\u200b\u5143\u200b\u6c17\u200b\u3067\u3059\u304b\u300d\uff1a\u200b\u60a8\u597d\u200b\u5417\u200b\uff1f</li> <li>\u300c\u304a\u304b\u3052\u3055\u307e\u3067\u3001\u200b\u5143\u200b\u6c17\u200b\u3067\u3059\u300d\uff1a\u200b\u8c22\u8c22\u60a8\u200b\uff0c\u200b\u6211\u200b\u5f88\u200b\u597d\u200b\u3002</li> <li>\u300c\u200b\u884c\u200b\u3063\u3066\u304d\u307e\u3059\u300d\uff1a\u200b\u6211\u200b\u8d70\u200b\u4e86\u200b</li> <li>\u300c\u200b\u884c\u200b\u3063\u3066\u3089\u3063\u3057\u3083\u3044\u300d\uff1a\u200b\u4e00\u8def\u987a\u98ce\u200b</li> <li>\u300c\u305f\u3060\u3044\u307e\u300d\uff1a\u200b\u6211\u200b\u56de\u6765\u200b\u4e86\u200b</li> <li>\u300c\u304a\u200b\u5e30\u200b\u308a\u306a\u3055\u3044\u300d\uff1a\u200b\u6b22\u8fce\u200b\u56de\u6765\u200b</li> <li>\u300c\u3055\u3088\u3046\u306a\u3089\u300d\uff1a\u200b\u518d\u89c1\u200b</li> <li>\u300c\u3042\u308a\u304c\u3068\u3046\u3054\u3056\u3044\u307e\u3059\u300d\uff1a\u200b\u975e\u5e38\u200b\u611f\u8c22\u60a8\u200b</li> <li>\u300c\u3042\u308a\u304c\u3068\u3046\u3054\u3056\u3044\u307e\u3057\u305f\u300d\uff1a\u200b\u975e\u5e38\u200b\u611f\u8c22\u60a8\u200b</li> </ol>"},{"location":"japanese/japanese-1-pattern/#5","title":"\u7b2c\u200b5\u200b\u8bfe","text":"<ol> <li>\u300c\u304a\u306f\u3088\u3046\u3054\u3056\u3044\u307e\u3059\u300d\uff1a\u200b\u65e9\u4e0a\u597d\u200b</li> <li>\u300c\u3053\u3093\u306b\u3061\u306f\u300d\uff1a\u200b\u4f60\u597d\u200b</li> <li>\u300c\u3053\u3093\u3070\u3093\u306f\u300d\uff1a\u200b\u665a\u4e0a\u200b\u597d\u200b</li> <li>\u300c\u3088\u308d\u3057\u304f\u304a\u200b\u9858\u200b\u3044\u3057\u307e\u3059\u300d\u3001\u300c\u3088\u308d\u3057\u304f\u304a\u200b\u9858\u200b\u3044\u3044\u305f\u3057\u307e\u3059\u300d\uff1a\u200b\u8bf7\u591a\u5173\u7167\u200b</li> <li>\u200b\u5173\u4e8e\u200b\u95ee\u53e5\u200b<ul> <li>\u200b\u80af\u5b9a\u200b\u56de\u590d\u200b\uff1a\u300c\u306f\u3044\u3001\u305d\u3046\u3067\u3059\u300d\uff0c\u200b\u6ce8\u610f\u200b\u300c\u306f\u3044\u300d\u200b\u4e0d\u80fd\u200b\u591a\u6b21\u91cd\u590d\u200b\u3002</li> <li>\u200b\u5426\u5b9a\u200b\u56de\u590d\u200b\uff1a\u300c\u3044\u3044\u3048\u3001\u200b\u9055\u200b\u3044\u307e\u3059\u300d</li> </ul> </li> </ol>"},{"location":"japanese/japanese-1-pattern/#6","title":"\u7b2c\u200b6\u200b\u8bfe","text":"<ol> <li>\u300c\u3048\u3048\u3068\u300d\uff1a\u200b\u55ef\u200b\u2026\u2026\uff0c\u200b\u8868\u793a\u200b\u77ed\u6682\u200b\u7684\u200b\u505c\u987f\u200b\u6216\u200b\u601d\u8003\u200b</li> <li>\u200b\u5207\u5165\u200b\u8bdd\u9898\u200b<ul> <li>\u300c\u3042\u306e\u3046\u300d\uff1a\u200b\u90a3\u4e2a\u200b\u2026\u2026</li> <li>\u300c\u3059\u307f\u307e\u305b\u3093\uff08\u304c\uff09\u300d\uff1a\u200b\u4e0d\u597d\u610f\u601d\u200b</li> </ul> </li> <li>\u200b\u7ed3\u675f\u200b\u8bdd\u9898\u200b<ul> <li>\u300c\u200b\u5931\u793c\u200b\u3057\u307e\u3059\u300d\uff1a\u200b\u5931\u966a\u200b\u4e86\u200b</li> <li>\u300c\u3042\u308a\u304c\u3068\u3046\u3054\u3056\u3044\u307e\u3057\u305f\u300d\uff1a\u200b\u4eca\u5929\u200b\u975e\u5e38\u200b\u611f\u8c22\u60a8\u200b</li> <li>\u300c\u3067\u306f\u3001\u307e\u305f\u300d\uff1a\u200b\u90a3\u4e48\u200b\uff0c\u200b\u518d\u89c1\u200b</li> </ul> </li> </ol>"},{"location":"japanese/japanese-1-pattern/#7","title":"\u7b2c\u200b7\u200b\u8bfe","text":"<ol> <li>\u300c\u200b\u5931\u793c\u200b\u3057\u307e\u3057\u305f\u300d\uff1a\u200b\u4e0d\u597d\u610f\u601d\u200b\uff0c\u200b\u7528\u4e8e\u200b\u5931\u793c\u200b\u3001\u200b\u5931\u656c\u200b\u7684\u200b\u884c\u4e3a\u200b\u3002</li> <li>\u300c\u304a\u200b\u4e16\u8a71\u200b\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u300d\uff1a\u200b\u611f\u8c22\u60a8\u200b\u4e00\u76f4\u200b\u4ee5\u6765\u200b\u7684\u200b\u7167\u987e\u200b\u3002</li> <li>\u200b\u4ecb\u7ecd\u200b\u5bb6\u5ead\u6210\u5458\u200b<ul> <li>\u300c\u200b\u5fa1\u5b85\u200b\u300d\u200b\u7528\u4e8e\u200b\u5c0a\u79f0\u200b\u5bf9\u65b9\u200b\u7684\u200b\u5bb6\u5ead\u200b\uff0c\u300c\u3046\u3061\u300d\u200b\u7528\u4e8e\u200b\u79f0\u547c\u200b\u81ea\u5df1\u200b\u7684\u200b\u5bb6\u5ead\u200b\u3002</li> <li>\u300cA\u306b\u306f\uff5e\u304c\u3044\u307e\u3059\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200bA\u200b\u6709\u200b\uff5e\u200b\u5bb6\u5ead\u6210\u5458\u200b\uff0c\u200b\u5982\u679c\u200b\u6709\u200b\u591a\u4f4d\u200b\uff0c\u200b\u5219\u200b\u5728\u200b\u300c\u304c\u300d\u200b\u540e\u52a0\u200b\u6570\u91cf\u8bcd\u200b\uff0c\u300c\u3068\u300d\u200b\u7528\u4e8e\u200b\u8868\u793a\u200b\u5e76\u5217\u200b\u3002</li> </ul> </li> <li>\u300c\u305d\u3046\u306a\u3093\u3067\u3059\u300d\uff1a\u200b\u8868\u793a\u200b\u53cd\u590d\u200b\u786e\u8ba4\u200b\u540e\u200b\u7684\u200b\u5f3a\u8c03\u200b\u3002</li> </ol>"},{"location":"japanese/japanese-1-pattern/#8","title":"\u7b2c\u200b8\u200b\u8bfe","text":"<ol> <li>\u300c\u301c\u3067\u200b\u5b9c\u200b\u3057\u3044\u3067\u3059\u304b\u300d\u200b\u7528\u4e8e\u200b\u5411\u200b\u5bf9\u65b9\u200b\u786e\u8ba4\u200b\uff5e\u200b\u4e2d\u200b\u7684\u200b\u5185\u5bb9\u200b\u662f\u5426\u200b\u5408\u9002\u200b\u3002\u200b\u53ef\u4ee5\u200b\u5982\u4e0b\u200b\u56de\u7b54\u200b<ul> <li>\u300c\u306f\u3044\u3001\u200b\u5927\u4e08\u592b\u200b\u3067\u3059\u3002\u300d</li> <li>\u300c\u306f\u3044\u3001\u304a\u200b\u9858\u200b\u3044\u3057\u307e\u3059\u3002\u300d</li> </ul> </li> <li>\u200b\u8d2d\u7269\u200b<ul> <li>\u200b\u63d0\u51fa\u200b\u9700\u6c42\u200b\uff1a\u300c\u301c\u306f\u3042\u308a\u307e\u3059\u304b\u300d\u3001\u300c\u301c\u3092\u304f\u3060\u3055\u3044\u300d\u3001\u300c\u301c\u3092\u304a\u200b\u9858\u200b\u3044\u3057\u307e\u3059\u300d\u3002</li> <li>\u200b\u8be2\u95ee\u200b\u4ef7\u683c\u200b\uff1a\u300c\u301c\uff08\u304a\uff09\u3044\u304f\u3089\u3067\u3059\u304b\u300d</li> <li>\u200b\u9009\u62e9\u200b\uff1a\u300c\u301c\u306b\u3057\u307e\u3059\u300d\u3001\u300c\u301c\u3067\u300d</li> </ul> </li> <li>\u200b\u5c31\u9910\u200b<ul> <li>\u200b\u9910\u524d\u200b\uff1a\u300c\u3044\u305f\u3060\u304d\u307e\u3059\u300d\uff1a\u200b\u6211\u8981\u200b\u5f00\u59cb\u200b\u5403\u200b\u4e86\u200b\u3002</li> <li>\u200b\u9910\u540e\u200b\uff1a\u300c\u3054\u200b\u99b3\u200b\u8d70\u200b\u69d8\u200b\u3067\u3057\u305f\u300d\uff1a\u200b\u60a8\u200b\u7684\u200b\u6b3e\u5f85\u200b\u771f\u662f\u200b\u592a\u597d\u4e86\u200b\u3002</li> </ul> </li> <li>\u300c\u3054\u3061\u3085\u3046\u3082\u3093\u306f\uff1f\u300d\uff1a\u200b\u60a8\u200b\u70b9\u200b\u4ec0\u4e48\u200b\uff1f</li> </ol>"},{"location":"japanese/japanese-1-pattern/#9","title":"\u7b2c\u200b9\u200b\u8bfe","text":"<ol> <li>\u300c\u304a\u200b\u4e45\u200b\u3057\u3076\u308a\u3067\u3059\u300d\uff1a\u200b\u597d\u4e45\u4e0d\u89c1\u200b</li> <li>\u300c\uff08\u305d\u308c\u306f\uff09\u3088\u304b\u3063\u305f\u3067\u3059\u306d\u300d\uff1a\u200b\u592a\u597d\u4e86\u200b<ul> <li>\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u81ea\u5df1\u200b\uff0c\u200b\u5bf9\u200b\u81ea\u5df1\u200b\u65f6\u4e0d\u52a0\u200b\u300c\u306d\u300d</li> </ul> </li> <li>\u300c\u200b\u52c9\u5f37\u200b\u306b\u306a\u308a\u307e\u3059\u300d\uff1a\u200b\u53d7\u76ca\u532a\u6d45\u200b</li> <li>\u300c\u200b\u9811\u5f35\u200b\u3063\u3066\uff08\u304f\u3060\u3055\u3044\uff09\u300d\uff1a\u200b\u52a0\u6cb9\u200b</li> <li>\u300c\u304a\u305f\u3093\u3058\u3087\u3046\u3073\u306f\u3044\u3064\u3067\u3059\u304b\u300d\uff1a\u200b\u60a8\u200b\u7684\u200b\u751f\u65e5\u200b\u662f\u200b\u4ec0\u4e48\u200b\u65f6\u5019\u200b\uff1f<ul> <li>\u300c\u200b\u79c1\u200b\u306e\u200b\u8a95\u200b\u751f\u65e5\u200b\u306f\u306a\u306a\u304c\u3064\u306b\u3058\u3085\u3046\u308d\u304f\u306b\u3061\u3067\u3059\u300d\uff1a\u200b\u6211\u200b\u7684\u200b\u751f\u65e5\u200b\u662f\u200b7\u200b\u6708\u200b26\u200b\u65e5\u200b\u3002</li> </ul> </li> <li>\u300c\u304a\u3081\u3067\u3068\u3046\u3054\u3056\u3044\u307e\u3059\u300d\uff1a\u200b\u606d\u559c\u200b\u60a8\u200b</li> </ol>"},{"location":"japanese/japanese-1-pattern/#10","title":"\u7b2c\u200b10\u200b\u8bfe","text":"<ol> <li>\u300c\u301c\u306f\u200b\u7a7a\u200b\u3044\u3066\u3044\u307e\u3059\u300d\uff1a\uff5e\u200b\u6709\u7a7a\u200b</li> <li>\u300c\u301c\u305f\u3044\u3067\u3059\u300d\uff1a\u200b\u60f3\u8981\u200b\uff5e</li> <li>\u300c\u301c\u304c\u200b\u697d\u200b\u3057\u307f\u3067\u3059\u300d\uff1a\uff5e\u200b\u4ee4\u4eba\u200b\u671f\u5f85\u200b</li> <li>\u200b\u63d0\u51fa\u200b\u9080\u8bf7\u200b\uff1a<ul> <li>\u300c\uff08\u200b\u4e00\u7dd2\u200b\u306b\uff09\u301c\u307e\u3057\u3087\u3046\u304b\u300d\uff1a\u200b\u4e00\u8d77\u200b\uff5e\u200b\u5427\u200b\uff1f</li> <li>\u300c\u301c\u306f\u3069\u3046\u3067\u3059\u304b\u300d\uff1a\uff5e\u200b\u600e\u4e48\u6837\u200b\uff1f</li> <li>\u300c\u301c\u304c\u3044\u3044\u3067\u3059\u304b\u300d\uff1a\uff5e\u200b\u597d\u200b\u5417\u200b\uff1f\uff08\uff5e\u200b\u5fc5\u987b\u200b\u662f\u200b\u7591\u95ee\u8bcd\u200b\uff09</li> </ul> </li> <li>\u200b\u56de\u7b54\u200b\u9080\u8bf7\u200b<ul> <li>\u200b\u63a5\u53d7\u200b\u9080\u8bf7\u200b\uff1a\u300c\u306f\u3044\u3001\u3044\u3044\u3067\u3059\u306d\u300d\u3001\u300c\u305d\u3057\u307e\u3057\u3087\u3046\u300d</li> <li>\u200b\u62d2\u7edd\u200b\u9080\u8bf7\u200b\uff1a\u300c\u3059\u307f\u307e\u305b\u3093\u3001\u3061\u3087\u3063\u3068\u2026\u2026\u300d\u3001\u300c\u3044\u3044\u3048\u3001\u3051\u3063\u3053\u3046\u3067\u3059\u3002\u300d\u3001\u300c\u3044\u3044\u3067\u3059\u300d</li> </ul> </li> </ol>"},{"location":"japanese/japanese-1-pattern/#11","title":"\u7b2c\u200b11\u200b\u8bfe","text":"<ol> <li>\u300c\u200b\u5168\u7136\u200b\u301c\u300d\uff1a\u200b\u5b8c\u5168\u200b\u4e0d\u200b\uff5e\uff0c\u200b\u7528\u4e8e\u200b\u5426\u5b9a\u53e5\u200b\u3002</li> <li>\u300c\u305d\u3046\u306a\u3093\u3067\u3059\u304b\u300d\uff1a\u200b\u662f\u200b\u5417\u200b\uff1f\u200b\u8868\u793a\u200b\u66f4\u200b\u60ca\u8bb6\u200b\u7684\u200b\u8bed\u6c14\u200b\u3002</li> </ol>"},{"location":"japanese/japanese-1-pattern/#12","title":"\u7b2c\u200b12\u200b\u8bfe","text":"<ol> <li>\u300c\uff08\u3059\u307f\u307e\u305b\u3093\u3001\uff09\u301c\u3066\u304f\u3060\u3055\u3044\u300d\uff1a\u200b\u8bf7\u200b\uff5e</li> <li>\u300c\u301c\u3066\u304f\u3060\u3055\u3044\u307e\u305b\u3093\u304b\u300d\uff1a\u200b\u8bf7\u200b\uff5e\u200b\u5417\u200b\uff1f\uff08\u200b\u7528\u4e8e\u200b\u4e0b\u7ea7\u200b\u5bf9\u200b\u4e0a\u7ea7\u200b\uff09</li> </ol>"},{"location":"japanese/japanese-1-pattern/#1311","title":"\u7b2c\u200b13\u200b\u8bfe\u200b\uff08\u200b\u65b0\u4e16\u7eaa\u200b\u521d\u7ea7\u200b\u7b2c\u200b11\u200b\u8bfe\u200b\uff09","text":"<ol> <li>\u300c\u200b\u6559\u200b\u3048\u3066\u304f\u3060\u3055\u3044\u300d\uff1a\u200b\u8bf7\u200b\u544a\u8bc9\u200b\u6211\u200b</li> <li>\u300c\u200b\u5b89\u5fc3\u200b\u3057\u3066\u304f\u3060\u3055\u3044\u300d\uff1a\u200b\u8bf7\u200b\u653e\u5fc3\u200b</li> <li>\u300c\u3067\u306f\u3001\u304a\u200b\u5143\u200b\u6c17\u200b\u3067\u300d\uff1a\u200b\u90a3\u4e48\u200b\uff0c\u200b\u4fdd\u91cd\u200b</li> <li>\u300c\u3054\u200b\u5bb6\u65cf\u200b\u306e\u200b\u7686\u200b\u69d8\u200b\u306b\u3082\u3088\u308d\u3057\u304f\u300d\uff1a\u200b\u8bf7\u4ee3\u200b\u6211\u200b\u5411\u200b\u60a8\u200b\u7684\u200b\u5bb6\u4eba\u200b\u95ee\u597d\u200b</li> <li>\u300c\u3069\u3046\u3082\u300d\uff1a\u200b\u975e\u5e38\u611f\u8c22\u200b</li> </ol>"},{"location":"japanese/japanese-1-vocab/","title":"\u65e5\u8bed\u200b(1) \u200b\u5355\u8bcd\u8868","text":""},{"location":"japanese/japanese-1-vocab/#_1","title":"\u52a8\u8bcd","text":"I\u200b\u7c7b\u200b\u52a8\u8bcd\u200bII\u200b\u7c7b\u200b\u52a8\u8bcd\u200bIII\u200b\u7c7b\u200b\u52a8\u8bcd\u200b \u200b\u52a8\u8bcd\u200b \u200b\u5047\u540d\u200b \u200b\u5206\u7c7b\u200b \u200b\u542b\u4e49\u200b \u200b\u9055\u200b\u3046 \u3061\u304c\u3046 \u200b\u81ea\u200bI \u200b\u4e0d\u662f\u200b \u200b\u5408\u200b\u3046 \u3042\u3046 \u200b\u81ea\u200bI \u200b\u5408\u9002\u200b \u200b\u4ed8\u200b\u304d\u200b\u5408\u200b\u3046 \u3064\u304d\u3042\u3046 \u200b\u81ea\u200bI \u200b\u4ea4\u5f80\u200b \u3053\u305f\u308f\u308b \u200b\u81ea\u200bI \u200b\u8bb2\u7a76\u200b \u200b\u9811\u5f35\u200b\u308b \u304c\u3093\u3070\u308b \u200b\u81ea\u200bI \u200b\u52a0\u6cb9\u200b \u200b\u884c\u200b\u304f \u3044\u304f \u200b\u81ea\u200bI \u200b\u53bb\u200b \u200b\u7a7a\u200b\u304f \u3042\u304f \u200b\u81ea\u200bI \u200b\u7a7a\u95f2\u200b \u200b\u5206\u200b\u304b\u308b \u308f\u304b\u308b \u200b\u81ea\u200bI \u200b\u660e\u767d\u200b \u304b\u304b\u308b \u200b\u81ea\u200bI \u200b\u82b1\u8d39\u200b \u200b\u5e30\u200b\u308b \u304b\u3048\u308b \u200b\u81ea\u200bI \u200b\u56de\u5bb6\u200b \u200b\u8981\u200b\u308b \u3044\u308b \u200b\u81ea\u200bI \u200b\u9700\u8981\u200b \u200b\u6e07\u200b\u304f \u304b\u308f\u304f \u200b\u81ea\u200bI \u200b\u53e3\u6e34\u200b \u200b\u7b11\u200b\u3046 \u308f\u3089\u3046 \u200b\u81ea\u200bI \u200b\u7b11\u200b \u200b\u6b8b\u200b\u308b \u306e\u3053\u308b \u200b\u81ea\u200bI \u200b\u5269\u4e0b\u200b \u200b\u558b\u200b\u308b \u3057\u3083\u3079\u308b 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\u3068\u304f\u3044 \u200b\u64c5\u957f\u200b \u200b\u5fc3\u914d\u200b \u3057\u3093\u3071\u3044 \u200b\u62c5\u5fc3\u200b \u200b\u5b89\u5fc3\u200b \u3042\u3093\u3057\u3093 \u200b\u5b89\u5fc3\u200b \u200b\u5143\u200b\u6c17\u200b \u3052\u3093\u304d \u200b\u7cbe\u795e\u200b \u200b\u4e0d\u8981\u200b \u3075\u3088\u3046 \u200b\u4e0d\u8981\u200b \u200b\u7c21\u200b\u5358\u200b \u304b\u3093\u305f\u3093 \u200b\u7b80\u5355\u200b \u200b\u4e00\u751f\u200b\u61f8\u547d\u200b \u3044\u3063\u3057\u3087\u3046\u3051\u3093\u3081\u3044 \u200b\u62fc\u547d\u200b \u200b\u5927\u5207\u200b \u305f\u3044\u305b\u3064 \u200b\u91cd\u8981"},{"location":"japanese/japanese-1-vocab/#_3","title":"\u6570\u5b57\u200b\u3001\u200b\u65f6\u95f4\u200b\u3001\u200b\u65e5\u671f","text":""},{"location":"japanese/japanese-1-vocab/#_4","title":"\u6570\u5b57","text":"<p>\u200b\u6ce8\u610f\u200b\uff1a</p> <ol> <li>1\u200b\u4eba\u200b\u548c\u200b2\u200b\u4eba\u200b\u7684\u200b\u7279\u6b8a\u200b\u5199\u6cd5\u200b\u3002</li> <li>4\u200b\u4eba\u200b\u7684\u200b\u5199\u6cd5\u200b\u662f\u200b\u300c\u3088\u306b\u3093\u300d\uff0c\u200b\u800c\u200b\u4e0d\u662f\u200b\u300c\u3088\u3093\u306b\u3093\u300d\u3002</li> <li>7\u30019\u200b\u4eba\u200b\u7684\u200b\u5199\u6cd5\u200b\u6709\u200b\u4e24\u79cd\u200b\u3002</li> <li>\u300c\u3072\u3083\u304f\u300d\u200b\u548c\u200b\u300c\u305b\u3093\u300d\u200b\u53ef\u4ee5\u200b\u7528\u6765\u200b\u76f4\u63a5\u200b\u8868\u793a\u200b100\u200b\u548c\u200b1000\uff0c\u200b\u800c\u200b\u300c\u307e\u3093\u300d\u200b\u548c\u200b\u300c\u304a\u304f\u300d\u200b\u9700\u8981\u200b\u52a0\u4e0a\u200b\u300c\u3044\u3061\u300d\u200b\u6765\u200b\u8868\u793a\u200b1\u200b\u4e07\u200b\u548c\u200b1\u200b\u4ebf\u200b\u3002</li> </ol> \u200b\u6570\u5b57\u200b \u200b\u5047\u540d\u200b \u200b\u6570\u5b57\u200b \u200b\u5047\u540d\u200b 1 \u3044\u3061 1\u200b\u4eba\u200b \u3072\u3068\u308a 2 \u306b 2\u200b\u4eba\u200b \u3075\u305f\u308a 3 \u3055\u3093 3\u200b\u4eba\u200b \u3055\u3093\u306b\u3093 4 \u3088\u3093 4\u200b\u4eba\u200b \u3088\u306b\u3093 5 \u3054 5\u200b\u4eba\u200b \u3054\u306b\u3093 6 \u308d\u304f 6\u200b\u4eba\u200b \u308d\u304f\u306b\u3093 7 \u306a\u306a 7\u200b\u4eba\u200b \u3057\u3061\u306b\u3093/\u306a\u306a\u306b\u3093 8 \u306f\u3061 8\u200b\u4eba\u200b \u306f\u3061\u306b\u3093 9 \u304d\u3085\u3046 9\u200b\u4eba\u200b \u304d\u3085\u3046\u306b\u3093/\u304f\u306b\u3093 10 \u3058\u3085\u3046 10\u200b\u4eba\u200b \u3058\u3085\u3046\u306b\u3093 \u200b\u767e\u200b \u3072\u3083\u304f \u200b\u5343\u200b \u305b\u3093 \u200b\u4e07\u200b \u3044\u3061\u307e\u3093 \u200b\u5104\u200b \u304a\u304f"},{"location":"japanese/japanese-1-vocab/#_5","title":"\u91cf\u8bcd","text":"<p>\u200b\u6ce8\u610f\u200b\uff1a</p> <ol> <li>\u200b\u6570\u5b57\u200b+\u300c\u3064\u300d\u200b\u7684\u200b\u5199\u6cd5\u200b\uff0c\u200b\u9664\u4e86\u200b7\u200b\u662f\u200b\u76f4\u63a5\u200b\u52a0\u200b\u300c\u3064\u300d\u200b\u4ee5\u5916\u200b\uff0c\u200b\u5176\u4ed6\u200b\u90fd\u200b\u6709\u200b\u660e\u663e\u200b\u7684\u200b\u53d8\u5316\u200b\u3002</li> <li>\u200b\u6570\u5b57\u200b+\u300c\u200b\u500b\u200b\u3001\u200b\u672c\u200b\u3001\u200b\u518a\u200b\u300d\u200b\u7684\u200b\u5199\u6cd5\u200b\uff0c\u200b\u9664\u4e86\u200b1\u30016\u30018\u300110\u200b\u662f\u200b\u76f4\u63a5\u200b\u52a0\u200b\u300c\u200b\u500b\u200b\u300d\u200b\u4ee5\u5916\u200b\uff0c\u200b\u5176\u4ed6\u200b\u90fd\u200b\u5c06\u200b\u6570\u5b57\u200b\u8bfb\u97f3\u200b\u7684\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u5047\u540d\u200b\u8f6c\u4e3a\u200b\u501a\u97f3\u200b\u300c\u3063\u300d\uff0c\u200b\u518d\u200b\u52a0\u200b\u300c\u200b\u500b\u200b\u300d\uff0c\u200b\u5982\u200b\u300c\u3044\u3063\u3053\u300d\u3001\u300c\u308d\u3063\u3053\u300d\u3001\u300c\u306f\u3063\u3053\u300d\u3001\u300c\u3058\u3085\u3063\u3053\u300d\u3002</li> <li>\u200b\u6570\u5b57\u200b+\u300c\u200b\u672c\u200b\u300d\u200b\u7684\u200b\u5199\u6cd5\u200b\uff0c\u200b\u9700\u8981\u200b\u6ce8\u610f\u200b1\u30013\u30016\u30018\u300110\u200b\u9700\u8981\u200b\u5c06\u200b\u300c\u307b\u300d\u200b\u7684\u200b\u8bfb\u97f3\u200b\u53d8\u4e3a\u200b\u300c\u307d\u300d\u3002</li> <li>\u200b\u6570\u5b57\u200b+\u300c\u200b\u679a\u200b\u3001\u200b\u53f0\u200b\u300d\uff0c\u200b\u76f4\u63a5\u200b\u5728\u200b\u6570\u5b57\u200b\u540e\u9762\u200b\u52a0\u200b\u300c\u200b\u679a\u200b\u300d\u200b\u6216\u200b\u300c\u200b\u53f0\u200b\u300d\u200b\u5373\u53ef\u200b\u3002</li> </ol> \u200b\u6570\u91cf\u200b \u3064 \u200b\u500b\u200b \u200b\u672c\u200b \u200b\u518a\u200b \u200b\u679a\u200b \u200b\u53f0\u200b 1 \u3072\u3068\u3064 \u3044\u3063\u3053 \u3044\u3063\u307d\u3093 \u3044\u3063\u3055\u3064 \u3044\u3061\u307e\u3044 \u3044\u3061\u3060\u3044 2 \u3075\u305f\u3064 \u306b\u3053 \u306b\u307b\u3093 \u306b\u3055\u3064 \u306b\u307e\u3044 \u306b\u3060\u3044 3 \u307f\u3063\u3064 \u3055\u3093\u3053 \u3055\u3093\u307d\u3093 \u3055\u3093\u3055\u3064 \u3055\u3093\u307e\u3044 \u3055\u3093\u3060\u3044 4 \u3088\u3063\u3064 \u3088\u3093\u3053 \u3088\u3093\u307b\u3093 \u3088\u3093\u3055\u3064 \u3088\u3093\u307e\u3044 \u3088\u3093\u3060\u3044 5 \u3044\u3064\u3064 \u3054\u3053 \u3054\u307b\u3093 \u3054\u3055\u3064 \u3054\u307e\u3044 \u3054\u3060\u3044 6 \u3080\u3063\u3064 \u308d\u3063\u3053 \u308d\u3063\u307d\u3093 \u308d\u304f\u3055\u3064 \u308d\u304f\u307e\u3044 \u308d\u304f\u3060\u3044 7 \u306a\u306a\u3064 \u306a\u306a\u3053 \u306a\u306a\u307b\u3093 \u306a\u306a\u3055\u3064 \u306a\u306a\u307e\u3044 \u306a\u306a\u3060\u3044 8 \u3084\u3063\u3064 \u306f\u3063\u3053 \u306f\u3063\u307d\u3093 \u306f\u3063\u3055\u3064 \u306f\u3061\u307e\u3044 \u306f\u3061\u3060\u3044 9 \u3053\u3053\u306e\u3064 \u304d\u3085\u3046\u3053 \u304d\u3085\u3046\u307b\u3093 \u304d\u3085\u3046\u3055\u3064 \u304d\u3085\u3046\u307e\u3044 \u304d\u3085\u3046\u3060\u3044 10 \u3068\u304a \u3058\u3085\u3063\u3053 \u3058\u3085\u3063\u307d\u3093 \u3058\u3085\u3063\u3055\u3064 \u3058\u3085\u3046\u307e\u3044 \u3058\u3085\u3046\u3060\u3044"},{"location":"japanese/japanese-1-vocab/#_6","title":"\u5c0f\u65f6","text":"<p>\u200b\u534a\u5c0f\u65f6\u200b\u5728\u200b\u300c\u200b\u6642\u9593\u200b\u300d\u200b\u540e\u52a0\u200b\u300c\u200b\u534a\u200b\uff08\u306f\u3093\uff09\u300d\u200b\u5373\u53ef\u200b\u3002</p> \u200b\u65f6\u95f4\u200b \u200b\u5047\u540d\u200b \u200b\u65f6\u95f4\u200b \u200b\u5047\u540d\u200b \u200b\u65f6\u95f4\u200b \u200b\u5047\u540d\u200b 1\u200b\u6642\u9593\u200b \u3044\u3061\u3058\u304b\u3093 9\u200b\u6642\u9593\u200b \u304f\u3058\u304b\u3093 17\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u306a\u306a\u3058\u304b\u3093 2\u200b\u6642\u9593\u200b \u306b\u3058\u304b\u3093 10\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u3058\u304b\u3093 18\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u306f\u3061\u3058\u304b\u3093 3\u200b\u6642\u9593\u200b \u3055\u3093\u3058\u304b\u3093 11\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u3044\u3061\u3058\u304b\u3093 19\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u304f\u3058\u304b\u3093 4\u200b\u6642\u9593\u200b \u3088\u3058\u304b\u3093 12\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u306b\u3058\u304b\u3093 20\u200b\u6642\u9593\u200b \u306b\u3058\u3085\u3046\u3058\u304b\u3093 5\u200b\u6642\u9593\u200b \u3054\u3058\u304b\u3093 13\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u3055\u3093\u3058\u304b\u3093 21\u200b\u6642\u9593\u200b \u306b\u3058\u3085\u3046\u3044\u3061\u3058\u304b\u3093 6\u200b\u6642\u9593\u200b \u308d\u304f\u3058\u304b\u3093 14\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u3088\u3058\u304b\u3093 22\u200b\u6642\u9593\u200b \u306b\u3058\u3085\u3046\u306b\u3058\u304b\u3093 7\u200b\u6642\u9593\u200b \u3057\u3061\u3058\u304b\u3093 15\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u3054\u3058\u304b\u3093 23\u200b\u6642\u9593\u200b \u306b\u3058\u3085\u3046\u3055\u3093\u3058\u304b\u3093 8\u200b\u6642\u9593\u200b \u306f\u3061\u3058\u304b\u3093 16\u200b\u6642\u9593\u200b \u3058\u3085\u3046\u308d\u304f\u3058\u304b\u3093 24\u200b\u6642\u9593\u200b \u306b\u3058\u3085\u3046\u3088\u3058\u304b\u3093"},{"location":"japanese/japanese-1-vocab/#_7","title":"\u5206\u949f","text":"\u65f6\u95f4\u200b \u200b\u5047\u540d\u200b \u200b\u65f6\u95f4\u200b \u200b\u5047\u540d\u200b 1\u200b\u5206\u200b \u3044\u3063\u3077\u3093 1\u200b\u5206\u534a\u200b \u3044\u3063\u3077\u3093\u306f\u3093 2\u200b\u5206\u200b \u306b\u3075\u3093 2\u200b\u5206\u534a\u200b \u306b\u3075\u3093\u306f\u3093 3\u200b\u5206\u200b \u3055\u3093\u3077\u3093 3\u200b\u5206\u534a\u200b \u3055\u3093\u3077\u3093\u306f\u3093 4\u200b\u5206\u200b \u3088\u3093\u3077\u3093 4\u200b\u5206\u534a\u200b \u3088\u3093\u3077\u3093\u306f\u3093 5\u200b\u5206\u200b \u3054\u3075\u3093 5\u200b\u5206\u534a\u200b \u3054\u3075\u3093\u306f\u3093 6\u200b\u5206\u200b \u308d\u3063\u3077\u3093 6\u200b\u5206\u534a\u200b \u308d\u3063\u3077\u3093\u306f\u3093 7\u200b\u5206\u200b \u306a\u306a\u3075\u3093 7\u200b\u5206\u534a\u200b \u306a\u306a\u3075\u3093\u306f\u3093 8\u200b\u5206\u200b \u306f\u3063\u3077\u3093 8\u200b\u5206\u534a\u200b \u306f\u3063\u3077\u3093\u306f\u3093 9\u200b\u5206\u200b \u304d\u3085\u3046\u3075\u3093 9\u200b\u5206\u534a\u200b \u304d\u3085\u3046\u3075\u3093\u306f\u3093 10\u200b\u5206\u200b \u3058\u3085\u3063\u3077\u3093 10\u200b\u5206\u534a\u200b \u3058\u3085\u3063\u3077\u3093\u306f\u3093"},{"location":"japanese/japanese-1-vocab/#_8","title":"\u65e5\u671f","text":"<p>\u200b\u6ce8\u610f\u200b\uff1a1-10\u200b\u65e5\u200b\u7684\u200b\u53d8\u5316\u200b\u6bd4\u8f83\u200b\u7279\u6b8a\u200b\uff0c\u200b\u5176\u4f59\u200b\u7684\u200b\u90fd\u200b\u662f\u200b\u6570\u5b57\u200b+\u300c\u200b\u65e5\u200b\u300d\u3002</p> \u200b\u65e5\u671f\u200b \u200b\u5047\u540d\u200b \u200b\u65e5\u671f\u200b \u200b\u5047\u540d\u200b \u200b\u65e5\u671f\u200b \u200b\u5047\u540d\u200b 1\u200b\u65e5\u200b 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7\u200b\u65e5\u200b \u306a\u306e\u304b 17\u200b\u65e5\u200b \u3058\u3085\u3046\u3057\u3061\u306b\u3061 27\u200b\u65e5\u200b \u306b\u3058\u3085\u3046\u3057\u3061\u306b\u3061 8\u200b\u65e5\u200b \u3088\u3046\u304b 18\u200b\u65e5\u200b \u3058\u3085\u3046\u306f\u3061\u306b\u3061 28\u200b\u65e5\u200b \u306b\u3058\u3085\u3046\u306f\u3061\u306b\u3061 9\u200b\u65e5\u200b \u3053\u3053\u306e\u304b 19\u200b\u65e5\u200b \u3058\u3085\u3046\u304f\u306b\u3061 29\u200b\u65e5\u200b \u306b\u3058\u3085\u3046\u304f\u306b\u3061 10\u200b\u65e5\u200b \u3068\u304a\u304b 20\u200b\u65e5\u200b \u306f\u3064\u304b 30\u200b\u65e5\u200b \u3055\u3093\u3058\u3085\u3046\u306b\u3061 31\u200b\u65e5\u200b \u3055\u3093\u3058\u3085\u3046\u3044\u3061\u306b\u3061"},{"location":"japanese/japanese-1-vocab/#_9","title":"\u661f\u671f","text":"\u661f\u671f\u200b \u200b\u661f\u671f\u200b \u200b\u661f\u671f\u4e00\u200b \u200b\u6708\u66dc\u65e5\u200b \u3052\u3064\u3088\u3046\u3073 \u200b\u661f\u671f\u4e8c\u200b 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\u200b\u53f8\u673a\u200b \u200b\u64ae\u200b\u308b \u3068\u308b \u200b\u62cd\u7167\u200b \u200b\u904b\u200b\u8ee2\u200b \u3046\u3093\u3066\u3093 \u200b\u5f00\u8f66\u200b \u200b\u672c\u5c4b\u200b \u307b\u3093\u3084 \u200b\u4e66\u5e97\u200b \u301c\u200b\u5c4b\u200b \u3084 ~\u200b\u5e97\u200b \u200b\u5929\u5a66\u200b\u7f85\u200b \u3066\u3093\u3077\u3089 \u200b\u5929\u5987\u200b\u7f57\u200b \u200b\u96e8\u200b \u3042\u3081 \u200b\u96e8\u200b \u200b\u6dbc\u200b\u3057\u3044 \u3059\u305a\u3057\u3044 \u200b\u51c9\u723d\u200b \u200b\u6674\u200b\u308c \u306f\u308c \u200b\u6674\u5929\u200b \u200b\u5546\u54c1\u200b \u3057\u3087\u3046\u3072\u3093 \u200b\u5546\u54c1\u200b \u200b\u685c\u200b \u3055\u304f\u3089 \u200b\u6a31\u82b1\u200b \u200b\u5473\u200b \u3042\u3058 \u200b\u5473\u9053\u200b \u200b\u8272\u200b \u3044\u308d \u200b\u989c\u8272\u200b \u200b\u521d\u200b\u3081\u3066 \u306f\u3058\u3081\u3066 \u200b\u7b2c\u4e00\u6b21\u200b \u200b\u7389\u5b50\u200b\u713c\u200b\u304d \u305f\u307e\u3054\u3084\u304d \u200b\u714e\u86cb\u200b \u3064\u307e\u3089\u306a\u3044 \u200b\u65e0\u804a\u200b \u200b\u697d\u200b\u5668\u200b \u304c\u3063\u304d \u200b\u4e50\u5668\u200b \u200b\u9eba\u200b \u3081\u3093 \u200b\u9762\u6761\u200b \u200b\u9055\u200b\u3046 \u3061\u304c\u3046 \u200b\u4e0d\u200b\u4e00\u6837"},{"location":"japanese/japanese-1-vocab/#1311","title":"\u7b2c\u200b13\u200b\u8bfe\u200b\uff08\u200b\u65b0\u4e16\u7eaa\u200b\u521d\u7ea7\u200b\u7b2c\u200b11\u200b\u8bfe\u200b\uff09","text":"\u5355\u8bcd\u200b \u200b\u5047\u540d\u200b \u200b\u542b\u4e49\u200b \u200b\u5355\u8bcd\u200b \u200b\u5047\u540d\u200b \u200b\u542b\u4e49\u200b \u200b\u7a7a\u6e2f\u200b \u304f\u3046\u3053\u3046 \u200b\u673a\u573a\u200b \u200b\u6c7a\u5b9a\u200b \u3051\u3063\u3066\u3044 \u200b\u51b3\u5b9a\u200b \u200b\u6765\u200b\u65e5\u200b \u3089\u3044\u306b\u3061 \u200b\u6765\u200b\u65e5\u672c\u200b \u200b\u5230\u200b\u7740\u200b \u3068\u3046\u3061\u3083\u304f \u200b\u5230\u8fbe\u200b \u200b\u4f55\u6642\u200b \u306a\u3093\u3058 \u200b\u51e0\u70b9\u200b \u200b\u4e88\u5b9a\u200b \u3088\u3066\u3044 \u200b\u9884\u5b9a\u200b \u200b\u8fce\u200b\u3048\u308b \u3080\u304b\u3048\u308b \u200b\u8fce\u63a5\u200b \u200b\u5e30\u200b\u308b \u304b\u3048\u308b \u200b\u56de\u5bb6\u200b \u200b\u5c31\u200b\u8077\u200b \u3057\u3085\u3046\u3057\u3087\u304f \u200b\u5c31\u804c\u200b \u200b\u8cbf\u6613\u200b \u307c\u3046\u3048\u304d \u200b\u8d38\u6613\u200b \u200b\u4f7f\u200b\u3046 \u3064\u304b\u3046 \u200b\u4f7f\u7528\u200b \u200b\u6b8b\u696d\u200b \u3056\u3093\u304e\u3087\u3046 \u200b\u52a0\u73ed\u200b \u200b\u65e5\u200b\u3005\u200b \u3072\u3073 \u200b\u65e5\u5b50\u200b \u200b\u61d0\u200b\u304b\u3057\u3044 \u306a\u3064\u304b\u3057\u3044 \u200b\u6000\u5ff5\u200b \u200b\u4f4f\u5c45\u200b \u3058\u3085\u3046\u304d\u3087 \u200b\u4f4f\u6240\u200b \u200b\u6c7a\u200b\u3057\u3066 \u3051\u3063\u3057\u3066 \u200b\u7edd\u4e0d\u200b \u200b\u5fc3\u914d\u200b \u3057\u3093\u3071\u3044 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\u304a\u3066\u3042\u3089\u3044 \u200b\u6d17\u624b\u95f4\u200b \u307e\u3064 \u200b\u7b49"},{"location":"japanese/japanese-1-vocab/#1412","title":"\u7b2c\u200b14\u200b\u8bfe\u200b\uff08\u200b\u65b0\u4e16\u7eaa\u200b\u521d\u7ea7\u200b\u7b2c\u200b12\u200b\u8bfe\u200b\uff09","text":"\u5355\u8bcd\u200b \u200b\u5047\u540d\u200b \u200b\u542b\u4e49\u200b \u200b\u5355\u8bcd\u200b \u200b\u5047\u540d\u200b \u200b\u542b\u4e49\u200b \u200b\u6d77\u5916\u200b \u304b\u3044\u304c\u3044 \u200b\u6d77\u5916\u200b \u200b\u65c5\u884c\u200b \u308a\u3087\u3053\u3046 \u200b\u65c5\u884c\u200b \u200b\u73cd\u200b\u3057\u3044 \u3081\u305a\u3089\u3057\u3044 \u200b\u73cd\u8d35\u200b \u3053\u3068 \u200b\u4e8b\u60c5\u200b \u200b\u4f8b\u200b\u3048\u3070 \u305f\u3068\u3048\u3070 \u200b\u4f8b\u5982\u200b \u200b\u65b0\u5a5a\u200b \u3057\u3093\u3053\u3093 \u200b\u65b0\u5a5a\u200b \u200b\u56fd\u5185\u200b \u3053\u304f\u306a\u3044 \u200b\u56fd\u5185\u200b \u200b\u9078\u200b\u3076 \u3048\u3089\u3076 \u200b\u9009\u62e9\u200b 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\u304f\u308b \u200b\u6765\u200b \u200b\u4e07\u91cc\u200b\u9577\u200b\u57ce\u200b \u3070\u3093\u308a\u3061\u3087\u3046\u3058\u3087\u3046 \u200b\u4e07\u91cc\u957f\u57ce\u200b \u200b\u6848\u5185\u200b \u3042\u3093\u306a\u3044 \u200b\u5bfc\u6e38\u200b \u305d\u3046 \u200b\u8fd9\u6837\u200b \u200b\u826f\u200b\u3044 \u3088\u3044 \u200b\u597d"},{"location":"latex/","title":"LaTeX","text":"<p>\u200b\u8bb0\u5f55\u200bLaTeX\u200b\u7684\u200b\u4f7f\u7528\u200b\u3002</p>"},{"location":"latex/#_1","title":"\u57fa\u672c\u200b\u6392\u7248","text":"<ul> <li>\u200b\u5b57\u4f53\u200b\u4e0e\u200b\u5b57\u53f7\u200b</li> <li>\u200b\u5217\u8868\u200b</li> </ul>"},{"location":"latex/#_2","title":"\u6570\u5b66\u516c\u5f0f\u200b\u6392\u7248","text":"<ul> <li>\u200b\u6570\u5b66\u200b\u7b26\u53f7\u200b</li> <li>\u200b\u7b26\u53f7\u200b\u5806\u53e0\u200b</li> <li>\u200b\u7279\u6b8a\u200b\u6280\u5de7\u200b</li> </ul>"},{"location":"latex/#_3","title":"\u56fe\u8868\u200b\u6392\u7248","text":""},{"location":"latex/#tikz","title":"tikz\u200b\u7ed8\u56fe","text":"<ul> <li>pgfplots\u200b\u56fe\u8868\u200b</li> </ul>"},{"location":"latex/#_4","title":"\u8868\u683c\u200b\u6392\u7248","text":"<ul> <li>tabular\u200b\u73af\u5883\u200b</li> </ul>"},{"location":"latex/#_5","title":"\u5176\u4ed6","text":"<ul> <li>musixtex</li> <li>\u200b\u4f7f\u7528\u200bLaTeX\u200b\u521b\u5efa\u200b\u5e7b\u706f\u7247\u200b</li> <li>\u200b\u9012\u5f52\u200b</li> </ul>"},{"location":"latex/advanced/recursion/","title":"\u9012\u5f52","text":"<p>LaTeX\u200b\u4e2d\u200b\u7684\u200b\u9012\u5f52\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b<code>\\def</code>\u200b\u547d\u4ee4\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u5728\u200b\u5b9a\u4e49\u200b\u7684\u200b\u5b8f\u4e2d\u200b\u5305\u542b\u200b\u81ea\u8eab\u200b\u5373\u53ef\u200b\u3002</p> <pre><code>\\def\\recursion#1{\n    \\ifx#1\\relax\n        \\relax\n    \\else\n        do something\n        \\expandafter\\recursion\n    \\fi\n}\n</code></pre> <p>\u200b\u5728\u200b\u5b8f\u200b\u7b2c\u4e00\u6b21\u200b\u5c55\u5f00\u200b\u540e\u200b\uff0c\u200b\u5982\u679c\u200b<code>#1</code>\u200b\u4e0d\u662f\u200b<code>\\relax</code>\uff0c\u200b\u5219\u200b\u4f1a\u200b\u8f6c\u5165\u200b<code>\\else</code>\u200b\u8bed\u53e5\u200b\u8fdb\u884c\u200b\u7b2c\u4e8c\u6b21\u200b\u5c55\u5f00\u200b\u3002</p> <ul> <li>\u200b\u5b8f\u200b\u5b9a\u4e49\u200b\u4e2d\u200b\u5f15\u7528\u200b\u7684\u200b\u5b8f\u200b\u540d\u79f0\u200b\u524d\u200b\u5fc5\u987b\u200b\u52a0\u200b<code>\\expandafter</code>\uff0c\u200b\u8868\u793a\u200b\u5728\u200b\u5c55\u5f00\u200b\u5f53\u524d\u200b\u5b8f\u540e\u200b\u624d\u200b\u5c55\u5f00\u200b<code>\\recursion</code>\uff1b</li> <li><code>\\relax</code>\u200b\u662f\u200b\u5b8f\u200b\u7684\u200b\u7ec8\u6b62\u200b\u6761\u4ef6\u200b\uff0c\u200b\u8f93\u5165\u200b\u5e8f\u5217\u200b\u9700\u8981\u200b\u4ee5\u200b<code>\\relax</code>\u200b\u7ed3\u5c3e\u200b\u3002</li> </ul>"},{"location":"latex/basics/fonts/","title":"LaTeX\u200b\u4e2d\u200b\u7684\u200b\u5b57\u4f53\u200b\u4e0e\u200b\u5b57\u53f7","text":""},{"location":"latex/basics/fonts/#_1","title":"\u5b57\u53f7","text":"<p>\\(\\LaTeX\\)\u200b\u4e2d\u5171\u200b\u6709\u200b<code>\\tiny</code>\u3001<code>\\scriptsize</code>\u3001<code>\\footnotesize</code>\u3001<code>\\small</code>\u3001<code>\\normalsize</code>\u3001<code>\\large</code>\u3001<code>\\Large</code>\u3001<code>\\LARGE</code>\u3001<code>\\huge</code>\u3001<code>\\Huge</code>\u200b\u7b49\u200b10\u200b\u79cd\u200b\u76f8\u5bf9\u200b\u5c3a\u5bf8\u200b\u7684\u200b\u5b57\u53f7\u200b\u8bbe\u7f6e\u200b\u3002\u200b\u6bcf\u6b21\u200b\u4fee\u6539\u200b\u5b57\u53f7\u200b\u540e\u200b\uff0c\u200b\u4f1a\u200b\u5bf9\u200b\u547d\u4ee4\u200b\u540e\u200b\u6240\u6709\u200b\u7684\u200b\u6587\u672c\u200b\u751f\u6548\u200b\u3002\u200b\u5982\u679c\u200b\u4ec5\u200b\u9700\u8981\u200b\u4fee\u6539\u200b\u4e00\u90e8\u5206\u200b\u6587\u5b57\u200b\u7684\u200b\u5b57\u53f7\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u9700\u8981\u200b\u8c03\u6574\u200b\u5b57\u53f7\u200b\u7684\u200b\u90e8\u5206\u200b\u4f7f\u7528\u200b\u82b1\u200b\u62ec\u53f7\u200b<code>{}</code>\u200b\u5305\u56f4\u200b\u3002</p> <p>LaTeX\u200b\u4e2d\u200b\u7684\u200b\u5b57\u53f7\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\centering\n\\tiny tiny \\\\\n\\scriptsize scriptsize \\\\\n\\footnotesize footnotesize \\\\\n\\small small \\\\\n\\normalsize normalsize \\\\\n\\large large \\\\\n\\Large Large \\\\\n\\LARGE LARGE \\\\\n\\huge huge \\\\\n\\Huge Huge \\\\\n</code></pre> <p></p>"},{"location":"latex/basics/lists/","title":"\u5217\u8868","text":"<p>\u200b\u5728\u200bLaTeX\u200b\u4e2d\u200b\uff0c<code>enumerate</code>\u200b\u73af\u5883\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u7f16\u53f7\u200b\u5217\u8868\u200b\uff0c<code>itemize</code>\u200b\u73af\u5883\u200b\u4e0e\u200b<code>description</code>\u200b\u73af\u5883\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u7b26\u53f7\u200b\u5217\u8868\u200b\u3002\u200b\u6240\u6709\u200b\u7684\u200b\u5217\u8868\u200b\u4f7f\u7528\u200b<code>\\item</code>\u200b\u547d\u4ee4\u200b\u5b9a\u4e49\u200b\u5217\u8868\u200b\u9879\u200b\u3002\u200b\u5217\u8868\u200b\u53ef\u4ee5\u200b\u81ea\u5b9a\u4e49\u200b\u4f7f\u7528\u200b\u7684\u200b\u7f16\u53f7\u200b\u4e0e\u200b\u7b26\u53f7\u200b\u3002</p> <p>\u200b\u5f53\u200b\u5217\u8868\u200b\u88ab\u200b\u5d4c\u5957\u200b\u65f6\u200b\uff0c\u200b\u4e0d\u540c\u200b\u5c42\u6b21\u200b\u7684\u200b\u5217\u8868\u200b\u4f1a\u200b\u4f7f\u7528\u200b\u4e0d\u540c\u200b\u7684\u200b\u7f16\u53f7\u200b\u4ee5\u200b\u907f\u514d\u200b\u6df7\u6dc6\u200b\u3002</p>"},{"location":"latex/basics/lists/#_2","title":"\u57fa\u672c\u200b\u4f7f\u7528","text":""},{"location":"latex/basics/lists/#enumerate","title":"<code>enumerate</code>\u200b\u73af\u5883","text":"<p><code>enumerate</code>\u200b\u73af\u5883\u200b\u7684\u200b\u4f7f\u7528\u200b\u65b9\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>\\begin{enumerate}\n    \\item This is an item.\n    \\item This is an item.\n    \\item This is an item.\n\\end{enumerate}\n</code></pre> <p>\u200b\u6e32\u67d3\u200b\u540e\u200b\u7684\u200b\u6548\u679c\u200b\uff1a</p> <ol> <li>This is an item.</li> <li>This is an item.</li> <li>This is an item.</li> </ol> <p><code>enumerate</code>\u200b\u73af\u5883\u200b\u9ed8\u8ba4\u200b\u4f7f\u7528\u200b<code>1. 2. 3. ...</code>\u200b\u8fdb\u884c\u200b\u7f16\u53f7\u200b\uff0c\u200b\u81ea\u5b9a\u4e49\u200b\u7f16\u53f7\u200b\u901a\u8fc7\u200b\u5728\u200b<code>\\begin{enumerate}</code>\u200b\u540e\u200b\u6dfb\u52a0\u200b\u53c2\u6570\u200b\u5b9e\u73b0\u200b\u3002</p> <pre><code>\\begin{enumerate}[A.]\n    \\item This is an item.\n    \\item This is an item.\n    \\item This is an item.\n\\end{enumerate}\n</code></pre> <p>\u200b\u6e32\u67d3\u200b\u540e\u200b\u7684\u200b\u6548\u679c\u200b\uff1a</p> <p>A. This is an item. B. This is an item. C. This is an item.</p> <p>\u200b\u5728\u200b<code>enumerate</code>\u200b\u4e0e\u200b<code>item</code>\u200b\u73af\u5883\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>\\setlength</code>\u200b\u8c03\u8282\u200b\u5217\u8868\u200b\u9879\u200b\u4e4b\u95f4\u200b\u7684\u200b\u8ddd\u79bb\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</p> <pre><code>\\setlength{\\itemsep}{0pt}\n\\setlength{\\parskip}{0pt}\n\\setlength{\\itemindent}{1em}\n</code></pre> <p>\u200b\u81ea\u5b9a\u4e49\u200b\u7f16\u53f7\u200b\u5217\u8868\u200b\u9700\u8981\u200b\u5f15\u5165\u200b<code>enumerate</code>\u200b\u5305\u200b\uff1a<code>\\usepackage{enumerate}</code></p>"},{"location":"latex/basics/lists/#itemize","title":"<code>itemize</code>\u200b\u73af\u5883","text":"<p><code>itemize</code>\u200b\u73af\u5883\u200b\u7528\u4e8e\u200b\u751f\u6210\u200b\u7b26\u53f7\u200b\u5217\u8868\u200b\uff0c\u200b\u4f7f\u7528\u200b\u65b9\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> <pre><code>\\begin{itemize}\n    \\item This is an item.\n    \\item This is an item.\n    \\item This is an item.\n\\end{itemize}\n</code></pre> <p>\u200b\u6e32\u67d3\u200b\u540e\u200b\u7684\u200b\u6548\u679c\u200b\uff1a</p> <ul> <li>This is an item.</li> <li>This is an item.</li> <li>This is an item.</li> </ul> <p><code>\\item</code>\u200b\u547d\u4ee4\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e2d\u62ec\u53f7\u200b\u6307\u5b9a\u200b\u5f53\u524d\u200b\u9879\u200b\u4f7f\u7528\u200b\u7684\u200b\u7b26\u53f7\u200b\uff0c\u200b\u5982\u200b<code>\\item[-]</code>\u200b\u4f7f\u7528\u200b<code>-</code>\u200b\u4f5c\u4e3a\u200b\u7b26\u53f7\u200b</p>"},{"location":"latex/floats/table/","title":"tabular\u200b\u73af\u5883","text":"<p><code>tabular</code>\u200b\u73af\u5883\u200b\u662f\u200b\\(\\LaTeX{}\\)\u200b\u7528\u4e8e\u200b\u521b\u5efa\u8868\u683c\u200b\u7684\u200b\u73af\u5883\u200b\u3002</p>"},{"location":"latex/floats/table/#_1","title":"\u57fa\u672c\u200b\u4f7f\u7528","text":"<p>\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b\u6f14\u793a\u200b\u4e86\u200b<code>tabular</code>\u200b\u8bed\u53e5\u200b\u7684\u200b\u4f7f\u7528\u200b\u65b9\u6cd5\u200b,\u200b\u751f\u6210\u200b\u4e00\u4e2a\u200b3\u200b\u884c\u200b3\u200b\u5217\u200b\u7684\u200b\u8868\u683c\u200b\u3002</p> <p>tabular\u200b\u57fa\u672c\u200b\u4f7f\u7528\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{tabular}{|c|c|c|}\n    \\hline 2    &amp; 9     &amp; 4     \\\\\n    \\hline 7    &amp; 5     &amp; 3     \\\\\n    \\hline 6    &amp; 1     &amp; 8     \\\\\n    \\hline\n\\end{tabular}\n</code></pre> <p></p>"},{"location":"latex/floats/table/#_2","title":"\u5bfc\u8a00\u200b\u533a","text":"<p><code>\\begin{tabular}</code>\u200b\u5176\u540e\u200b\u7d27\u8ddf\u200b\u7684\u200b\u4e00\u5bf9\u200b\u62ec\u53f7\u200b\u5185\u200b\u662f\u200b\u5bfc\u8a00\u200b\u533a\u200b\uff0c\u200b\u5bfc\u8a00\u200b\u533a\u200b\u89c4\u5b9a\u200b\u4e86\u200b\u5355\u5143\u683c\u200b\u7684\u200b\u7eb5\u5411\u200b\u8fb9\u6846\u200b\u548c\u200b\u5355\u5143\u683c\u200b\u5185\u200b\u5143\u7d20\u200b\u7684\u200b\u5bf9\u9f50\u200b\u65b9\u5f0f\u200b\u3002</p> <ul> <li><code>l</code>\u3001<code>c</code>\u3001<code>r</code>\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u5de6\u200b\u5bf9\u9f50\u200b\uff0c\u200b\u5c45\u4e2d\u200b\uff0c\u200b\u53f3\u200b\u5bf9\u9f50\u200b</li> <li><code>|</code>\u3001<code>||</code>\u200b\u5bf9\u5e94\u200b\u8868\u683c\u200b\u7eb5\u5411\u200b\u6846\u7ebf\u200b\u7c7b\u578b\u200b\uff08\u200b\u5355\u7ebf\u200b\u6216\u200b\u53cc\u7ebf\u200b\uff09</li> <li><code>@{exp}</code>\u200b\u6307\u5b9a\u200b\u63d2\u5165\u200b\u5230\u200b\u5217\u4e2d\u200b\u7684\u200b\u6587\u672c\u200b</li> <li><code>*{n}{pre}</code>\u200b\u6307\u200b\u91cd\u590d\u200b<code>pre</code>\u200b\u5185\u5bb9\u200b<code>n</code>\u200b\u6b21\u200b</li> <li><code>p{len}</code>\u200b\u5c06\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5217\u200b\u653e\u5165\u200b\u4e00\u4e2a\u200bparbox\u200b\u4e2d\u200b</li> </ul> <p>\u200b\u6839\u636e\u200b\u5982\u200b\u4e0a\u200b\u8868\u8ff0\u200b\uff0c\u200b\u5bfc\u8a00\u200b\u533a\u200b<code>{|c|c|c|}</code>\u200b\u7b49\u4ef7\u200b\u4e8e\u200b<code>{*{3}{|c}|}</code></p> <p>\u200b\u5bfc\u8a00\u200b\u533a\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{tabular}{||l|c|r||}\n    \\hline 2    &amp; 91    &amp; 4     \\\\\n    \\hline 71   &amp; 5     &amp; 32    \\\\\n    \\hline 652  &amp; 251   &amp; 89    \\\\\n    \\hline\n\\end{tabular}\n</code></pre> <p></p>"},{"location":"latex/floats/table/#_3","title":"\u5408\u5e76\u200b\u5355\u5143\u683c","text":"<p>\u200b\u6a2a\u5411\u200b\u5408\u5e76\u200b\u5355\u5143\u683c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>\\multicolumn</code>\u200b\u547d\u4ee4\u200b\uff0c\u200b\u7eb5\u5411\u200b\u5408\u5e76\u200b\u5355\u5143\u683c\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>multirow</code>\u200b\u5b8f\u5305\u200b\u4e2d\u200b\u7684\u200b<code>\\multirow</code>\u200b\u547d\u4ee4\u200b\uff1a</p> <ul> <li><code>\\multicolumn</code>\u200b\u547d\u4ee4\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\u5408\u5e76\u200b\u5217\u200b\u7684\u200b\u6570\u91cf\u200b\uff0c\u200b\u7b2c\u4e8c\u4e2a\u200b\u53c2\u6570\u200b\u662f\u200b\u5408\u5e76\u200b\u540e\u200b\u5355\u5143\u683c\u200b\u7684\u200b\u5bfc\u8a00\u200b\u5217\u200b</li> <li><code>\\multirow</code>\u200b\u547d\u4ee4\u200b\u7684\u200b\u7b2c\u4e00\u4e2a\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200b\u5408\u5e76\u200b\u884c\u200b\u7684\u200b\u6570\u91cf\u200b\uff0c\u200b\u7b2c\u4e8c\u4e2a\u200b\u53c2\u6570\u200b\u6307\u5b9a\u200bLaTeX\u200b\u81ea\u884c\u200b\u8bbe\u7f6e\u200b\u5bbd\u5ea6\u200b\uff0c\u200b\u7b2c\u4e09\u4e2a\u200b\u53c2\u6570\u200b\u4e3a\u200b\u5355\u5143\u683c\u200b\u5185\u5bb9\u200b</li> </ul> <p>\u200b\u5408\u5e76\u200b\u5355\u5143\u683c\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{tabular}{|ccc|}\n    \\hline\n    2   &amp; 9     &amp; 4\\\\\n    7   &amp; \\multicolumn{2}{c|} {\\multirow{2}*{{?}}} \\\\\n    6   &amp;       &amp;\\\\\n    \\hline\n\\end{tabular}\n</code></pre> <p></p>"},{"location":"latex/floats/table/#_4","title":"\u4e09\u7ebf\u200b\u8868","text":"<p>LaTeX\u200b\u4e2d\u200b\u4f7f\u7528\u200b\u4e09\u7ebf\u200b\u8868\u200b\u9700\u8981\u200b\u7528\u5230\u200b<code>booktabs</code>\u200b\u5b8f\u5305\u200b\uff0c\u200b\u52a0\u5165\u200b\u5b8f\u5305\u200b\u540e\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>\\toprule</code>\u3001<code>\\midrule</code>\u200b\u4e0e\u200b<code>\\bottomrule</code>\u200b\u753b\u7ebf\u200b\uff08\u200b\u4e0d\u5e94\u200b\u518d\u200b\u4f7f\u7528\u200b<code>\\hline</code>\u200b\u547d\u4ee4\u200b\uff09\u3002</p> <p>\u200b\u4e09\u7ebf\u200b\u8868\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{tabular}{ccc}\n    \\toprule\n    2   &amp; 91    &amp; 4     \\\\\n    \\midrule\n    71  &amp; 5     &amp; 32    \\\\\n    652 &amp; 251   &amp; 89    \\\\\n    \\bottomrule\n\\end{tabular}\n</code></pre> <p></p>"},{"location":"latex/floats/table/#_5","title":"\u989c\u8272\u200b\u586b\u5145","text":"<p>\u200b\u989c\u8272\u200b\u586b\u5145\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b<code>colortbl</code>\u200b\u5b8f\u5305\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>\\rowcolor</code>\u200b\u547d\u4ee4\u200b\u6307\u5b9a\u200b\u884c\u200b\u7684\u200b\u586b\u5145\u200b\u989c\u8272\u200b\uff0c\u200b\u6216\u200b<code>\\cellcolor</code>\u200b\u547d\u4ee4\u200b\u6307\u5b9a\u200b\u5355\u5143\u683c\u200b\u7684\u200b\u586b\u5145\u200b\u989c\u8272\u200b\u3002\u200b\u586b\u5145\u200b\u989c\u8272\u200b\u53ef\u4ee5\u200b\u9009\u62e9\u200b\u7070\u5ea6\u200b<code>[gray]</code>\u200b\u6216\u200b\u5f69\u8272\u200b<code>[rgb]</code>\u3002</p> <p>\u200b\u989c\u8272\u200b\u586b\u5145\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{tabular}{ccc}\n    \\rowcolor[gray]{0.6} 2    &amp; 9     &amp; 4     \\\\\n    \\rowcolor[gray]{0.7} 7    &amp; 5     &amp; 3     \\\\\n    \\rowcolor[gray]{0.8} 6    &amp; 1     &amp; 8     \\\\\n\\end{tabular}\n</code></pre> <p></p>"},{"location":"latex/floats/table/#_6","title":"\u659c\u7ebf\u200b\u8868\u5934","text":"<p>\u200b\u659c\u7ebf\u200b\u8868\u5934\u200b\u7531\u200b<code>diagbox</code>\u200b\u5b8f\u5305\u200b\u63d0\u4f9b\u200b\uff0c\u200b\u4f7f\u7528\u200b\u65b9\u6cd5\u200b\u5982\u4e0b\u200b\uff1a</p> <p>\u200b\u659c\u7ebf\u200b\u8868\u5934\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{tabular}{|l|ccc|}\n    \\hline\n    \\diagbox{Time}{Room}{Day} &amp; Mon &amp; Tue &amp; Wed \\\\\n    \\hline\n    Morning &amp; used &amp; used &amp; \\\\\n    Afternoon &amp; &amp; used &amp; used \\\\\n    \\hline\n\\end{tabular}\n</code></pre> <p></p>"},{"location":"latex/math/stack/","title":"LaTeX \u200b\u7b26\u53f7\u200b\u5806\u53e0","text":""},{"location":"latex/math/stack/#_1","title":"\u7b26\u53f7\u200b\u4e0a\u4e0b\u200b\u52a0\u200b\u6587\u672c","text":"<p>\u200b\u90e8\u5206\u200b\u5927\u578b\u200b\u8fd0\u7b97\u7b26\u200b\uff08\u200b\u53c2\u89c1\u200b\u6570\u5b66\u200b\u7b26\u53f7\u200b\uff09\u200b\u53ef\u4ee5\u200b\u52a0\u5165\u200b\u4e0a\u4e0b\u200b\u6807\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u884c\u95f4\u200b\u516c\u5f0f\u200b\uff0c\u200b\u8fd0\u7b97\u7b26\u200b\u7684\u200b\u4e0a\u4e0b\u200b\u6807\u6709\u200b\u4e24\u79cd\u200b\u4f4d\u7f6e\u200b\u9009\u62e9\u200b\uff1a</p> <ul> <li>\u200b\u4e0a\u4e0b\u200b\u6807\u5728\u200b\u7b26\u53f7\u200b\u4e0a\u4e0b\u200b\u4e24\u4fa7\u200b\uff0c\u200b\u5982\u200b   $$   \\sum_{i=0}^\\infty   $$</li> <li>\u200b\u4e0a\u4e0b\u200b\u6807\u5728\u200b\u7b26\u53f7\u200b\u53f3\u4fa7\u200b\uff0c\u200b\u5982\u200b   $$   \\int_0^\\infty   $$</li> </ul> <p>\u200b\u884c\u5185\u200b\u516c\u5f0f\u200b\u9ed8\u8ba4\u200b\u6240\u6709\u200b\u7684\u200b\u4e0a\u4e0b\u200b\u6807\u90fd\u200b\u5728\u200b\u7b26\u53f7\u200b\u53f3\u4fa7\u200b\uff0c\u200b\u5982\u200b\\(\\sum_{i=0}^\\infty\\)\u3002\u200b\u4f7f\u7528\u200b<code>\\limits</code>\u200b\u53ef\u4ee5\u200b\u5f3a\u5236\u200b\u4e0a\u4e0b\u200b\u6807\u51fa\u200b\u73b0\u5728\u200b\u8fd0\u7b97\u7b26\u200b\u4e0a\u4e0b\u200b\u4e24\u4fa7\u200b\uff0c\u200b\u5982\u200b</p> <p><code>\\limits</code>\u200b\u547d\u4ee4\u200b</p> LaTeX\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>$\\sum\\limits_{i=0}^\\infty$\n</code></pre> <p>\\(\\sum\\limits_{i=0}^\\infty\\)</p> <p><code>\\overset</code>\u3001<code>\\underset</code>\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u5728\u200b\u7b26\u53f7\u200b\u4e0a\u4e0b\u200b\u63d2\u5165\u200b\u7b26\u53f7\u200b\u3002<code>\\stackrel</code>\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u4e0e\u200b<code>\\overset</code>\u200b\u76f8\u540c\u200b\u7684\u200b\u529f\u80fd\u200b\u3002</p> <p><code>\\overset</code>\u200b\u547d\u4ee4\u200b\u4e0e\u200b<code>\\underset</code>\u200b\u547d\u4ee4\u200b</p> LaTeX\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{equation*}\n    \\begin{aligned}\n        &amp;\\overset{above}{\\rightarrow} \\\\\n        &amp;\\underset{below}{\\rightarrow} \\\\\n    \\end{aligned}\n\\end{\\equation*}\n</code></pre> \\[ \\begin{aligned}     &amp;\\overset{above}{\\rightarrow} \\\\     &amp;\\underset{below}{\\rightarrow} \\\\ \\end{aligned} \\] <p>\u200b\u5982\u679c\u200b\u9700\u8981\u200b\u6362\u884c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>\\substack</code>\u200b\u547d\u4ee4\u200b\uff1a</p> <p><code>\\substack</code>\u200b\u547d\u4ee4\u200b</p> LaTeX\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{equation*}\n    \\begin{aligned}\n        &amp;\\overset{\\substack{above1 \\\\ above2}}{\\rightarrow} \\\\\n        &amp;\\underset{\\substack{below1 \\\\ below2}}{\\rightarrow} \\\\\n    \\end{aligned}\n\\end{\\equation*}\n</code></pre> \\[ \\begin{aligned}     &amp;\\overset{\\substack{above1 \\\\ above2}}{\\rightarrow} \\\\     &amp;\\underset{\\substack{below1 \\\\ below2}}{\\rightarrow} \\\\ \\end{aligned} \\]"},{"location":"latex/math/stack/#_2","title":"\u7bad\u5934\u200b\u4e0a\u4e0b\u200b\u52a0\u200b\u6587\u672c","text":"<p><code>\\overset</code>\u3001<code>\\stackrel</code>\u200b\u4e0e\u200b<code>\\underset</code>\u200b\u53ea\u80fd\u200b\u5b9e\u73b0\u200b\u7bad\u5934\u200b\u4e0a\u4e0b\u200b\u52a0\u200b\u6587\u672c\u200b\uff0c\u200b\u7bad\u5934\u200b\u7684\u200b\u957f\u5ea6\u200b\u4e0d\u80fd\u200b\u968f\u200b\u6587\u672c\u200b\u957f\u5ea6\u200b\u81ea\u52a8\u200b\u8c03\u6574\u200b\u3002<code>amsmath</code>\u200b\u5b8f\u5305\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>\\xleftarrow</code>\u200b\u4e0e\u200b<code>\\xrightarrow</code>\uff0c\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b\u7bad\u5934\u200b\u4e0a\u4e0b\u200b\u52a0\u200b\u6587\u672c\u200b\uff0c\u200b\u540c\u65f6\u200b\u7bad\u5934\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u9002\u5e94\u200b\u6587\u672c\u200b\u7684\u200b\u957f\u5ea6\u200b\u3002<code>extarrows</code>\u200b\u5b8f\u5305\u200b\u63d0\u4f9b\u200b\u4e86\u200b<code>\\xlongequal</code>\u200b\u547d\u4ee4\u200b\uff0c\u200b\u5b9e\u73b0\u200b\u7b49\u200b\u53f7\u200b\u4e0a\u4e0b\u200b\u52a0\u200b\u6587\u672c\u200b\uff0c\u200b\u793a\u4f8b\u200b\u5982\u4e0b\u200b\uff1a</p> <p><code>\\xrightarrow</code>\u200b\u547d\u4ee4\u200b</p> LaTeX\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{itemize}\n    \\item $\\xleftarrow[\\text{below}]{\\text{above}}$\n    \\item $\\xrightarrow[\\text{very very long below}]{\\text{very very long above}}$\n    \\item $\\xlongequal[\\text{very very long below}]{\\text{very very long above}}$\n\\end{itemize}\n</code></pre> <p></p> <p>\u200b\u5b9a\u4e49\u200b\u7b26\u53f7\u200b\\(\\triangleq\\)\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u4f7f\u7528\u200b<code>\\triangleq</code>\u200b\u8f93\u5165\u200b\u3002</p>"},{"location":"latex/math/symbols/","title":"LaTeX \u200b\u6570\u5b66\u200b\u7b26\u53f7","text":"<p>\u200b\u6536\u96c6\u200b\\(\\LaTeX\\)\u200b\u5e38\u7528\u200b\u7684\u200b\u6570\u5b66\u200b\u7b26\u53f7\u200b\u547d\u4ee4\u200b</p>"},{"location":"latex/math/symbols/#_1","title":"\u5b57\u6bcd\u200b\u53d8\u4f53","text":"<p>\u200b\u5e38\u7528\u547d\u4ee4\u200b\u6709\u200b\uff1a<code>\\mathcal</code>\u3001<code>\\mathbb</code>\u3001<code>\\mathfrak</code>\u3001<code>\\mathsf</code>\u3001<code>\\mathbf</code>\u3001<code>\\boldsymbol</code>\u3001<code>\\mathit</code>\u3001<code>\\mathrm</code>\u200b\u7b49\u200b\u3002\u200b\u5176\u4e2d\u200b<code>\\mathcal</code>\u3001<code>\\mathbb</code>\u200b\u4ec5\u200b\u9002\u7528\u200b\u4e8e\u200b\u5927\u5199\u5b57\u6bcd\u200b\uff0c\u200b\u5176\u4f59\u200b\u547d\u4ee4\u200b\u5927\u5199\u200b\u3001\u200b\u5c0f\u5199\u5b57\u6bcd\u200b\u901a\u7528\u200b</p> <ul> <li><code>\\mathcal{A}</code>\uff1a\\(\\mathcal A\\)</li> <li><code>\\mathbb{A}</code>\uff1a\\(\\mathbb A\\)</li> <li><code>\\mathfrak{A}</code>\uff1a\\(\\mathfrak A\\)</li> <li><code>\\mathsf{A}</code>\uff1a\\(\\mathsf A\\)</li> <li><code>\\mathbf{A}</code>\uff1a\\(\\mathbf A\\)</li> <li><code>\\boldsymbol{A}</code>\uff1a\\(\\boldsymbol A\\)</li> <li><code>\\mathit{A}</code>\uff1a\\(\\mathit A\\)\uff08\u200b\u65e0\u200b\u6548\u679c\u200b\uff09</li> <li><code>\\mathrm{A}</code>\uff1a\\(\\mathrm A\\)</li> <li><code>\\mathscr{A}</code>\uff1a\\(\\mathscr A\\)\uff08\u200b\u9700\u8981\u200b\u5f15\u5165\u200b\u5b8f\u5305\u200b<code>mathrsfs</code>\uff09</li> </ul>"},{"location":"latex/math/symbols/#_2","title":"\u5e0c\u814a\u5b57\u6bcd","text":"\u5c0f\u5199\u200b \u200b\u547d\u4ee4\u200b \u200b\u5927\u5199\u200b \u200b\u547d\u4ee4\u200b \u200b\u53d8\u4f53\u200b \u200b\u547d\u4ee4\u200b \\(\\alpha\\) <code>\\alpha</code> \\(A\\) <code>A</code> \\(\\beta\\) <code>\\beta</code> \\(B\\) <code>B</code> \\(\\gamma\\) <code>\\gamma</code> \\(\\Gamma\\) <code>\\Gamma</code> \\(\\varGamma\\) <code>\\varGamma</code> \\(\\delta\\) <code>\\delta</code> \\(\\Delta\\) <code>\\Delta</code> \\(\\varDelta\\) <code>\\varDelta</code> \\(\\epsilon\\) <code>\\epsilonn</code> \\(E\\) <code>E</code> \\(\\varepsilon\\) <code>\\varepsilon</code> \\(\\zeta\\) <code>\\zeta</code> \\(Z\\) <code>Z</code> \\(\\eta\\) <code>\\eta</code> \\(H\\) <code>H</code> \\(\\theta\\) <code>\\theta</code> \\(\\Theta\\) <code>\\Theta</code> \\(\\vartheta,\\varTheta\\) <code>\\vartheta,\\varTheta</code> \\(\\iota\\) <code>\\iota</code> \\(I\\) <code>I</code> \\(\\kappa\\) <code>\\kappa</code> \\(K\\) <code>K</code> \\(\\varkappa\\) <code>\\varkappa</code> \\(\\lambda\\) <code>\\lambda</code> \\(\\Lambda\\) <code>\\Lambda</code> \\(\\varLambda\\) <code>\\varLambda</code> \\(\\mu\\) <code>\\mu</code> \\(M\\) <code>M</code> \\(\\nu\\) <code>\\nu</code> \\(N\\) <code>N</code> \\(\\xi\\) <code>\\xi</code> \\(\\Xi\\) <code>\\Xi</code> \\(\\varXi\\) <code>\\varXi</code> \\(o\\) <code>o</code> \\(O\\) <code>O</code> \\(\\pi\\) <code>\\pi</code> \\(\\Pi\\) <code>\\Pi</code> \\(\\varpi, \\varPi\\) <code>\\varpi,\\varPi</code> \\(\\rho\\) <code>\\rho</code> \\(P\\) <code>P</code> \\(\\varrho\\) <code>\\varrho</code> \\(\\sigma\\) <code>\\sigma</code> \\(\\Sigma\\) <code>\\Sigma</code> \\(\\varsigma, \\varSigma\\) <code>\\varsigma,\\varSigma</code> \\(\\tau\\) <code>\\tau</code> \\(T\\) <code>T</code> \\(\\upsilon\\) <code>\\upsilon</code> \\(\\Upsilon\\) <code>\\Upsilon</code> \\(\\varUpsilon\\) <code>\\varUpsilon</code> \\(\\phi\\) <code>\\phi</code> \\(\\Phi\\) <code>\\Phi</code> \\(\\varphi, \\varPhi\\) <code>\\varphi,\\varPhi</code> \\(\\chi\\) <code>\\chi</code> \\(X\\) <code>X</code> \\(\\psi\\) <code>\\psi</code> \\(\\Psi\\) <code>\\Psi</code> \\(\\varPsi\\) <code>\\varPsi</code> \\(\\omega\\) <code>\\omega</code> \\(\\Omega\\) <code>\\Omega</code> \\(\\varOmega\\) <code>\\varOmega</code>"},{"location":"latex/math/symbols/#_3","title":"\u8fd0\u7b97\u7b26","text":"<p>\u200b\u5e38\u89c1\u200b\u8fd0\u7b97\u7b26\u200b\u5217\u4e8e\u200b\u4e0b\u8868\u200b\uff1a</p>"},{"location":"latex/math/symbols/#_4","title":"\u6570\u5b66\u200b\u8fd0\u7b97","text":"\u8fd0\u7b97\u7b26\u200b \u200b\u547d\u4ee4\u200b \\(\\times\\) <code>\\times</code> \\(\\div\\) <code>\\div</code> \\(\\odot\\) <code>\\odot</code> \\(\\oplus\\) <code>\\oplus</code> \\(\\otimes\\) <code>\\otimes</code>"},{"location":"latex/math/symbols/#_5","title":"\u96c6\u5408\u200b\u8fd0\u7b97","text":"\u8fd0\u7b97\u7b26\u200b \u200b\u547d\u4ee4\u200b \\(\\in\\) <code>\\in</code> \\(\\ni\\) <code>\\ni</code> \\(\\subset\\) <code>\\subset</code> \\(\\subseteq\\) <code>\\subseteq</code> \\(\\subseteqq\\) <code>\\oplsubseteqqu</code> \\(\\subsetneq\\) <code>\\subsetneq</code> \\(\\subsetneqq\\) <code>\\subsetneqq</code> \\(\\supset\\) <code>\\subset</code> \\(\\supseteq\\) <code>\\subseteq</code> \\(\\supseteqq\\) <code>\\oplsubseteqqu</code> \\(\\supsetneq\\) <code>\\subsetneq</code> \\(\\supsetneqq\\) <code>\\subsetneqq</code> \\(\\cap\\) <code>\\cap</code> \\(\\cup\\) <code>\\cup</code>"},{"location":"latex/math/symbols/#_6","title":"\u5173\u7cfb\u200b\u8fd0\u7b97\u7b26","text":"\u8fd0\u7b97\u7b26\u200b \u200b\u547d\u4ee4\u200b \\(\\geq\\) <code>\\geq</code> \\(\\geqq\\) <code>\\geqq</code> \\(\\geqslant\\) <code>\\geqslant</code> \\(\\gg\\) <code>\\gg</code> \\(\\ggg\\) <code>\\ggg</code> \\(\\leq\\) <code>\\leq</code> \\(\\leqq\\) <code>\\leqq</code> \\(\\leqslant\\) <code>\\leqslant</code> \\(\\ll\\) <code>\\ll</code> \\(\\lll\\) <code>\\lll</code> \\(\\sim\\) <code>\\sim</code> \\(\\approx\\) <code>\\approx</code>"},{"location":"latex/math/symbols/#_7","title":"\u903b\u8f91\u200b\u8fd0\u7b97\u7b26","text":"\u8fd0\u7b97\u7b26\u200b \u200b\u547d\u4ee4\u200b \\(\\lnot\\) <code>\\lnot</code> \\(\\land\\) <code>\\land</code> \\(\\lor\\) <code>\\lor</code> \\(\\forall\\) <code>\\forall</code> \\(\\exists\\) <code>\\exists</code>"},{"location":"latex/math/symbols/#_8","title":"\u51fd\u6570\u200b\u4e0e\u200b\u590d\u6742\u200b\u8fd0\u7b97","text":"\u8fd0\u7b97\u7b26\u200b \u200b\u547d\u4ee4\u200b \\(\\frac ab\\) <code>\\frac{a}{b}</code> \\(\\sum_i^n f\\) <code>\\sum_{i}^{n} f</code> \\(\\prod_i^n f\\) <code>\\prod_{i}^{n} f</code> \\(\\sqrt[x]{y}\\) <code>\\sqrt[x]{y}</code> \\(\\int_a^bf(x)\\mathrm dx\\) <code>\\int_a^bf(x)\\mathrm dx</code> \\(\\oint_Df(x)\\mathrm dx\\) <code>\\oint_Df(x)\\mathrm dx</code> \\(\\iint_Df(x)\\mathrm dx\\) <code>\\iint_Df(x)\\mathrm dx</code> \\(\\iiint_Df(x)\\mathrm dx\\) <code>\\iiint_Df(x)\\mathrm dx</code> \\(\\partial\\) <code>\\partial</code> \\(\\log\\) <code>\\log</code> \\(\\ln\\) <code>\\ln</code> \\(\\lg\\) <code>\\lg</code> \\(\\exp\\) <code>\\exp</code> \\(a\\mod b\\) <code>a\\mod b</code> \\(\\max\\) <code>\\max</code> \\(\\min\\) <code>\\min</code> \\(\\arg\\) <code>\\arg</code> \\(\\sin\\) <code>\\sin</code> \\(\\cos\\) <code>\\cos</code> \\(\\tan\\) <code>\\tan</code> \\(\\cot\\) <code>\\cot</code> \\(\\sinh\\) <code>\\sinh</code> \\(\\cosh\\) <code>\\cosh</code> \\(\\tanh\\) <code>\\tanh</code> \\(\\coth\\) <code>\\coth</code> \\(\\arcsin\\) <code>\\arcsin</code> \\(\\arccos\\) <code>\\arccos</code> \\(\\arctan\\) <code>\\arctan</code> \\(\\ker\\) <code>\\ker</code> \\(\\dim\\) <code>\\dim</code> \\(\\det\\) <code>\\det</code> \\(\\lim\\) <code>\\lim</code> \\(\\inf\\) <code>\\inf</code>"},{"location":"latex/math/symbols/#_9","title":"\u62ec\u53f7","text":"\u8fd0\u7b97\u7b26\u200b \u200b\u547d\u4ee4\u200b \\(\\{\\) <code>\\{</code> \\(\\}\\) <code>\\}</code> \\(\\langle\\) <code>\\langle</code> \\(\\rangle\\) <code>\\rangle</code> \\(\\lceil\\) <code>\\lceil</code> \\(\\rceil\\) <code>\\rceil</code> \\(\\lfloor\\) <code>\\lfloor</code> \\(\\rfloor\\) <code>\\rfloor</code> \\(\\|\\) <code>\\|</code>"},{"location":"latex/math/tricks/","title":"\u7279\u6b8a\u200b\u6280\u5de7","text":""},{"location":"latex/math/tricks/#_2","title":"\u516c\u5f0f\u200b\u5185\u200b\u4e2d\u6587","text":"<p>\u200b\u4f7f\u7528\u200b\u547d\u4ee4\u200b<code>\\text</code>\u200b\u6216\u200b<code>\\mbox</code>\u200b\u53ef\u4ee5\u200b\u5728\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u52a0\u5165\u200b\u4e2d\u6587\u200b\u6587\u672c\u200b\u3002\u200b\u5982\u679c\u200b\u4f7f\u7528\u200b<code>\\mathrm</code>\uff0c\u200b\u5219\u200b\u4e0d\u4f1a\u200b\u663e\u793a\u200b\u4e2d\u6587\u200b\u3002</p>"},{"location":"latex/math/tricks/#_3","title":"\u9690\u5f62\u200b\u5b57\u7b26","text":"<p><code>\\phantom</code>\u200b\u547d\u4ee4\u200b\u53ef\u4ee5\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u5360\u636e\u200b\u4e00\u5b9a\u200b\u7a7a\u95f4\u200b\u7684\u200b\u7a7a\u767d\u200b\uff0c\u200b\u7a7a\u767d\u200b\u7684\u200b\u957f\u5bbd\u200b\u7531\u200b\u53c2\u6570\u200b\u7684\u200b\u5c3a\u5bf8\u200b\u51b3\u5b9a\u200b\uff1a</p> \\[ \\begin{aligned}     \\bbox[border: 2px solid magenta]{\\phantom{\\int_0^1 x\\mathrm dx}} \\; &amp; \\bbox[border: 2px solid magenta]{\\int_0^1 x\\mathrm dx} \\\\     \\bbox[border: 2px solid magenta]{\\int_0^1 x\\mathrm dx} \\; &amp; \\end{aligned} \\] <p>\u200b\u6b64\u5916\u200b\uff0c<code>\\phantom</code>\u200b\u6709\u200b\u4e24\u4e2a\u200b\u53d8\u79cd\u200b<code>\\vphantom</code>\u200b\u4e0e\u200b<code>\\hphantom</code>\uff0c\u200b\u524d\u8005\u200b\u521b\u9020\u200b\u4e00\u4e2a\u200b\u5bbd\u5ea6\u200b\u4e3a\u200b0\uff0c\u200b\u9ad8\u5ea6\u200b\u4e3a\u200b\u6307\u5b9a\u200b\u9ad8\u5ea6\u200b\u7684\u200b\u7a7a\u767d\u200b\uff1b\u200b\u540e\u8005\u200b\u521b\u9020\u200b\u4e00\u4e2a\u200b\u9ad8\u5ea6\u200b\u4e3a\u200b0\uff0c\u200b\u5bbd\u5ea6\u200b\u4e3a\u200b\u6307\u5b9a\u200b\u5bbd\u5ea6\u200b\u7684\u200b\u7a7a\u767d\u200b\u3002\u200b\u5728\u200b\u5c3a\u5bf8\u200b\u6839\u636e\u200b\u5185\u5bb9\u200b\u800c\u53d8\u200b\u7684\u200b\u5bb9\u5668\u200b\u4e2d\u200b\uff0c<code>\\phantom</code>\u200b\u7b49\u200b\u4e00\u7cfb\u5217\u200b\u547d\u4ee4\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u63a7\u5236\u200b\u5bb9\u5668\u200b\u7684\u200b\u5927\u5c0f\u200b\u3002</p> <p><code>\\vphantom</code>\u200b\u4f7f\u7528\u200b\u573a\u666f\u200b</p> <p>\u200b\u5728\u200b<code>\\left</code>\u3001<code>\\right</code>\u200b\u7b49\u200b\u547d\u4ee4\u200b\u53ea\u80fd\u200b\u5728\u200b\u4e00\u884c\u200b\u4e0a\u200b\u63a7\u5236\u200b\u5de6\u53f3\u200b\u4e24\u4fa7\u200b\u62ec\u53f7\u200b\u7684\u200b\u5927\u5c0f\u200b\uff0c\u200b\u800c\u200b\u5f53\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u8de8\u884c\u200b\uff0c<code>\\left</code>\u3001<code>\\right</code>\u200b\u5219\u200b\u65e0\u6cd5\u200b\u6b63\u786e\u200b\u5339\u914d\u200b\u3002\u200b\u6b64\u65f6\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b<code>\\vphantom</code>\u200b\u547d\u4ee4\u200b\u5c06\u200b\u62ec\u53f7\u200b\u6491\u5f00\u200b\u3002</p> \u200b\u4e0d\u200b\u4f7f\u7528\u200b<code>\\vphantom</code>\u200b\u4f7f\u7528\u200b<code>\\vphantom</code> <pre><code>\\begin{equation}\n    \\begin{aligned}\n        \\frac{\\partial ^2\\Pi(x, t)}{\\partial x\\partial t} =\n        &amp; -(p-s)\\mathbb E_{Q, B, L}\\left[D(t)|Q(t)\\leq x\\right]f_{Q(t)}(x) \\\\\n        &amp; + h\\mathbb P(t &lt; B + L) - h\\mathbb E_{B, L} \\\\\n        &amp; \\times \\left[\\int_{t}^\\mathbb{B+L}\\mathbb E_Q\\left[D(t)|Q(t) - Q(\\tau)\\leq x, B\\leq t, L\\right]\\right. \\\\\n        &amp; \\color{magenta} \\times \\left. f_{Q(t) - Q(\\tau) | B\\leq t}(x)\\mathbb P(B\\leq t)\\mathrm d \\tau\\right]\n    \\end{aligned}\n\\end{equation}\n</code></pre> \\[ \\begin{aligned}     \\frac{\\partial ^2\\Pi(x, t)}{\\partial x\\partial t} =     &amp; -(p-s)\\mathbb E_{Q, B, L}\\left[D(t)|Q(t)\\leq x\\right]f_{Q(t)}(x) \\\\     &amp; + h\\mathbb P(t &lt; B + L) - h\\mathbb E_{B, L} \\\\     &amp; \\times \\left[\\int_{t}^\\mathbb{B+L}\\mathbb E_Q\\left[D(t)|Q(t) - Q(\\tau)\\leq x, B\\leq t, L\\right]\\right. \\\\     &amp; \\color{magenta} \\times \\left. f_{Q(t) - Q(\\tau) | B\\leq t}(x)\\mathbb P(B\\leq t)\\mathrm d \\tau\\right] \\end{aligned} \\] <pre><code>\\begin{equation}\n    \\begin{aligned}\n        \\frac{\\partial ^2\\Pi(x, t)}{\\partial x\\partial t} =\n        &amp; -(p-s)\\mathbb E_{Q, B, L}\\left[D(t)|Q(t)\\leq x\\right]f_{Q(t)}(x) \\\\\n        &amp; + h\\mathbb P(t &lt; B + L) - h\\mathbb E_{B, L} \\\\\n        &amp; \\times \\left[\\int_{t}^\\mathbb{B+L}\\mathbb E_Q\\left[D(t)|Q(t) - Q(\\tau)\\leq x, B\\leq t, L\\right]\\right. \\\\\n        &amp; \\color{magenta} \\times \\left.\\vphantom{\\int_t^{B+L}} f_{Q(t) - Q(\\tau) | B\\leq t}(x)\\mathbb P(B\\leq t)\\mathrm d \\tau\\right]\n    \\end{aligned}\n\\end{equation}\n</code></pre> \\[ \\begin{aligned}     \\frac{\\partial ^2\\Pi(x, t)}{\\partial x\\partial t} =     &amp; -(p-s)\\mathbb E_{Q, B, L}\\left[D(t)|Q(t)\\leq x\\right]f_{Q(t)}(x) \\\\     &amp; + h\\mathbb P(t &lt; B + L) - h\\mathbb E_{B, L} \\\\     &amp; \\times \\left[\\int_{t}^\\mathbb{B+L}\\mathbb E_Q\\left[D(t)|Q(t) - Q(\\tau)\\leq x, B\\leq t, L\\right]\\right. \\\\     &amp; \\color{magenta} \\times \\left.\\vphantom{\\int_t^{B+L}} f_{Q(t) - Q(\\tau) | B\\leq t}(x)\\mathbb P(B\\leq t)\\mathrm d \\tau\\right] \\end{aligned} \\]"},{"location":"latex/misc/beamer/","title":"\u4f7f\u7528\u200bLaTeX\u200b\u521b\u5efa\u200b\u5e7b\u706f\u7247","text":"<p>\u200b\u9996\u5148\u200b\uff0c\u200b\u9700\u8981\u200b\u6307\u5b9a\u200b<code>documentclass</code>\u200b\u4e3a\u200b<code>beamer</code>\u200b\u7c7b\u578b\u200b\uff1a</p> <pre><code>\\documentclass{beamer}\n</code></pre> <p>\u200b\u6bcf\u200b\u4e00\u5f20\u200b\u5e7b\u706f\u7247\u200b\u4f7f\u7528\u200b<code>frame</code>\u200b\u73af\u5883\u200b\uff0c\u200b\u5982\u4e0b\u200b\u4ee3\u7801\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u65b0\u200b\u7684\u200b\u7a7a\u767d\u200b\u5e7b\u706f\u7247\u200b\uff1a</p> <pre><code>\\begin{frame}\n\\end{frame}\n</code></pre> <p>\u200b\u7b2c\u4e00\u9875\u200bPPT\u200b\u901a\u5e38\u200b\u663e\u793a\u200bPPT\u200b\u7684\u200b\u6807\u9898\u200b\u3001\u200b\u4f5c\u8005\u200b\u7684\u200b\u4e2a\u4eba\u4fe1\u606f\u200b\u7b49\u200b\u5185\u5bb9\u200b\uff1a</p> <pre><code>\\title{\u200b\u6807\u9898\u200b}\n\\subtitle{\u200b\u526f\u6807\u9898\u200b}\n\\author{\u200b\u4f5c\u8005\u200b}\n\\institute{\u200b\u7ec4\u7ec7\u200b}\n\\date{\\today}  % \u200b\u663e\u793a\u200b\u65e5\u671f\u200b\n\\titlepage\n</code></pre>"},{"location":"latex/misc/musixtex/","title":"\u4e50\u8c31\u200b\u6392\u7248","text":"<p><code>musixtex</code>\u200b\u662f\u200bLaTeX\u200b\u4e2d\u200b\u7528\u6765\u200b\u6392\u7248\u200b\u4e50\u8c31\u200b\u7684\u200b\u5b8f\u5305\u200b\u3002\u200b\u4ee5\u4e0b\u5185\u5bb9\u200b\u9488\u5bf9\u200b\u9f13\u8c31\u200b\u6392\u7248\u200b\u3002</p> <pre><code>\\usepackage{musixtex}\n\\input{musixper} % \u200b\u52a0\u5165\u200b\u4e86\u200b\u9f13\u8c31\u200b\u6392\u7248\u200b\u7684\u200b\u4e00\u4e9b\u200b\u5185\u5bb9\u200b\n</code></pre>"},{"location":"latex/misc/musixtex/#_2","title":"\u97f3\u9ad8\u200b\u5bf9\u7167\u8868","text":"<p>\u200b\u4ece\u200b\u5de6\u5411\u53f3\u200b\u4f9d\u6b21\u200b\u4e3a\u200bA-Q</p> <p></p>"},{"location":"latex/misc/musixtex/#_3","title":"\u57fa\u672c\u200b\u4f7f\u7528","text":"<p><code>musixtex</code>\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u4e86\u200b<code>music</code>\u200b\u73af\u5883\u200b\uff0c\u200b\u5728\u200b<code>music</code>\u200b\u73af\u5883\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u5bf9\u200b\u4e50\u8c31\u200b\u8fdb\u884c\u200b\u8bbe\u7f6e\u200b\u3002\u200b\u7136\u540e\u200b\u5728\u200b\u547d\u4ee4\u200b<code>\\startextract</code>\u200b\u548c\u200b<code>\\zendextract</code>\u200b\u4e4b\u95f4\u200b\u8f93\u5165\u200b\u97f3\u7b26\u200b\u3002</p> <pre><code>\\begin{music}\n    \\instrumentnumber{1}        % \u200b\u4e50\u5668\u200b\n    \\setclefsymbol1\\empty       % \u200b\u8c31\u53f7\u200b\n    \\nobarnumbers               % \u200b\u53d6\u6d88\u200b\u5c0f\u8282\u200b\u7f16\u53f7\u200b\n    \\generalmeter{\\meterfrac44} % \u200b\u8282\u62cd\u200b\n    \\startextract\n        % Enter notes here\n    \\zendextract\n\\end{music}\n</code></pre> <p>\u200b\u5728\u200b\u672c\u200b\u5c0f\u8282\u200b\u4e2d\u200b\uff0c<code>p</code>\u200b\u8868\u793a\u200b\u97f3\u7b26\u200b\u5bf9\u5e94\u200b\u7684\u200b\u97f3\u9ad8\u200b\u3002</p>"},{"location":"latex/misc/musixtex/#_4","title":"\u786e\u5b9a\u200b\u97f3\u7b26\u200b\u5206\u5e03\u200b\u95f4\u8ddd","text":"<p>\u200b\u4f7f\u7528\u200b<code>\\notes</code>\u200b\u7684\u200b\u4e0d\u540c\u200b\u5927\u5c0f\u5199\u200b\u5f62\u5f0f\u200b\u6765\u200b\u786e\u5b9a\u200b\u97f3\u7b26\u200b\u7684\u200b\u5206\u5e03\u200b\u95f4\u8ddd\u200b\u3002</p> <pre><code>\\notes ... \\en % Suitable for sixteenth notes\n\\Notes ... \\en % Suitable for eighth notes\n\\NOtes ... \\en % Suitable for quarter notes\n</code></pre> <code>\\notes</code><code>\\Notes</code><code>\\NOtes</code> <p></p> <p></p> <p></p>"},{"location":"latex/misc/musixtex/#_5","title":"\u97f3\u7b26","text":"<p>\u200b\u97f3\u7b26\u200b\u547d\u4ee4\u200b\u7531\u200b\u4e24\u200b\u90e8\u5206\u200b\u7ec4\u6210\u200b\uff1a\u200b\u65f6\u503c\u200b+\u200b\u65b9\u5411\u200b\u3002\u200b\u4e0d\u540c\u200b\u97f3\u7b26\u200b\u7684\u200b\u65f6\u503c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u547d\u4ee4\u200b\u5982\u4e0b\u200b\u8868\u200b\u6240\u793a\u200b</p> \u200b\u65f6\u503c\u200b \u200b\u5168\u97f3\u7b26\u200b \u200b\u4e8c\u5206\u200b \u200b\u56db\u5206\u200b \u200b\u516b\u5206\u200b \u200b\u5341\u516d\u5206\u200b \u200b\u547d\u4ee4\u200b <code>w</code> <code>h</code> <code>q</code> <code>c</code> <code>cc</code> <p><code>u</code>\u200b\u8868\u793a\u200b\u5411\u4e0a\u200b\u7684\u200b\u97f3\u7b26\u200b\uff0c<code>l</code>\u200b\u8868\u793a\u200b\u5411\u4e0b\u200b\u7684\u200b\u97f3\u7b26\u200b\uff0c<code>a</code>\u200b\u8868\u793a\u200b\u81ea\u52a8\u200b\u8bbe\u7f6e\u200b\u97f3\u7b26\u200b\u65b9\u5411\u200b\u3002\u200b\u5982\u200b<code>\\qu{p}</code>\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u5411\u4e0a\u200b\u7684\u200b\u56db\u5206\u97f3\u7b26\u200b\u3002\u200b\u5982\u679c\u200b\u4e0d\u200b\u52a0\u200b\u65b9\u5411\u200b\uff0c\u200b\u5219\u200b\u6ca1\u6709\u200b\u97f3\u7b26\u200b\u7684\u200b\u201c\u200b\u6746\u200b\u201d\u3002</p> <p>\u200b\u5728\u200b\u65b9\u5411\u200b\u540e\u9762\u200b\u52a0\u4e0a\u200b<code>p</code>\u200b\u8868\u793a\u200b\u9644\u70b9\u200b\uff0c<code>pp</code>\u200b\u8868\u793a\u200b\u53cc\u200b\u9644\u70b9\u200b\u3002</p> <p>\u200b\u6bcf\u200b\u8f93\u5165\u200b\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\uff0c\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u547d\u4ee4\u200b\u8f93\u5165\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u4fbf\u4f1a\u200b\u5411\u200b\u540e\u200b\u79fb\u52a8\u200b\u3002\u200b\u5982\u679c\u200b\u8981\u200b\u8f93\u5165\u200b\u548c\u5f26\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\u4e4b\u524d\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>z</code>+\u200b\u65f6\u503c\u200b\u8868\u793a\u200b\u8fd9\u4e2a\u200b\u97f3\u7b26\u200b\u4e0d\u5411\u200b\u540e\u200b\u79fb\u52a8\u200b\u8f93\u5165\u200b\u4f4d\u7f6e\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u4e24\u4e2a\u200b\u97f3\u7b26\u200b\u79bb\u5f97\u200b\u8fc7\u8fd1\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>l</code>\u200b\u5c06\u200b\u97f3\u7b26\u200b\u5411\u200b\u5de6\u200b\u79fb\u52a8\u200b\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\u7684\u200b\u4f4d\u7f6e\u200b\uff0c<code>r</code>\u200b\u5c06\u200b\u97f3\u7b26\u200b\u5411\u200b\u53f3\u200b\u79fb\u52a8\u200b\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u3002</p> <pre><code>\\NOtes \\zq d\\qu g\\rq g\\qu f\\zq e\\qu h\\zq f\\qu k \\en\n</code></pre> <p></p>"},{"location":"latex/misc/musixtex/#_6","title":"\u7b26\u6760","text":"<p>\u200b\u5982\u679c\u200b\u516b\u5206\u200b\u6216\u200b\u66f4\u200b\u77ed\u200b\u7684\u200b\u97f3\u7b26\u200b\u8fde\u7eed\u200b\u51fa\u73b0\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u97f3\u7b26\u200b\u4e4b\u95f4\u200b\u4f7f\u7528\u200b\u7b26\u6760\u200b\u76f8\u8fde\u200b\uff0c\u200b\u901a\u5e38\u200b\u6bcf\u200b\u56db\u4e2a\u200b\u97f3\u7b26\u200b\u8fde\u6210\u200b\u4e00\u7ec4\u200b\u3002\u200b\u7b26\u6760\u200b\u7684\u200b\u5b9a\u4e49\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b</p> <pre><code>% \u200b\u4e0a\u65b9\u200b\n\\ibu{n}{p}{s} &lt;notes-under-beam&gt; \\tbu{n} &lt;the-last-note&gt;\n% \u200b\u4e0b\u65b9\u200b\n\\ibl{n}{p}{s} &lt;notes-under-beam&gt; \\tbl{n} &lt;the-last-note&gt;\n</code></pre> <p>\u200b\u5176\u4e2d\u200b<code>n</code>\u200b\u4e3a\u200b\u7f16\u53f7\u200b\uff0c<code>p</code>\u200b\u4e3a\u200b\u5f00\u59cb\u200b\u7684\u200b\u97f3\u7b26\u200b\u9ad8\u5ea6\u200b\uff0c<code>s</code>\u200b\u4e3a\u200b\u659c\u5ea6\u200b\u3002\u200b\u547d\u4ee4\u200b\u91cc\u200b<code>b</code>\u200b\u7684\u200b\u4e2a\u6570\u200b\u7b49\u4e8e\u200b\u663e\u793a\u200b\u7684\u200b\u7b26\u200b\u6760\u6570\u200b\u3002\u200b\u4e0e\u200b\u7b26\u6760\u200b\u76f8\u8fde\u200b\u7684\u200b\u97f3\u7b26\u200b\u6807\u8bb0\u200b\u4e3a\u200b\u65f6\u503c\u200b+<code>b</code>+\u200b\u7f16\u53f7\u200b\uff0c\u200b\u5982\u679c\u200b\u540c\u4e00\u200b\u4f4d\u7f6e\u200b\u6709\u200b\u591a\u4e2a\u200b\u97f3\u7b26\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b\u65f6\u503c\u200b\u524d\u9762\u200b\u52a0\u200b<code>z</code>\u3002\u200b\u6ce8\u610f\u200b\u97f3\u7b26\u200b\u4e4b\u95f4\u200b\u4e0d\u8981\u200b\u52a0\u200b\u7a7a\u683c\u200b\u3002</p> <pre><code>\\Notes\\ibu0j0\\qb0j\\qb0j\\qb0j\\tbu0\\qb0j\\en\n\\notes\\ibbu0j0\\qb0j\\qb0j\\qb0j\\tbu0\\qb0j\\en\n\\NOtes\\qu j\\en\n</code></pre> <p></p> <p><code>\\tbu{n}\\qb{n}{p}</code>\u200b\u53ef\u4ee5\u200b\u7f29\u5199\u200b\u4e3a\u200b<code>\\tqu{n}{p}</code>\uff0c<code>\\tbl{n}{p}\\qb{n}{p}</code>\u200b\u53ef\u4ee5\u200b\u7f29\u5199\u200b\u4e3a\u200b<code>\\tql{n}{p}</code>\uff0c<code>\\tbu{n}\\zqb{n}{p}</code>\u200b\u53ef\u4ee5\u200b\u7f29\u5199\u200b\u4e3a\u200b<code>\\zqu{n}{p}</code>\uff0c<code>\\tbl{n}{p}\\zqb{n}{p}</code>\u200b\u53ef\u4ee5\u200b\u7f29\u5199\u200b\u4e3a\u200b<code>\\zql{n}{p}</code></p> <p><code>\\ibu</code>\u200b\u4e2d\u200b\u9700\u8981\u200b\u624b\u52a8\u200b\u786e\u5b9a\u200b\u659c\u7387\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b\u5bf9\u5e94\u200b\u7684\u200b<code>\\Ibu</code>\u200b\u547d\u4ee4\u200b\u81ea\u52a8\u200b\u786e\u5b9a\u200b\uff1a</p> <pre><code>\\Ibu{n}{p1}{p2}{np}\n</code></pre> <p>\u200b\u5176\u4e2d\u200b<code>{n}</code>\u200b\u4e3a\u200b\u7f16\u53f7\u200b\uff0c<code>p1</code>\u200b\u4e3a\u200b\u7b2c\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\u9ad8\u5ea6\u200b\uff0c<code>p2</code>\u200b\u4e3a\u200b\u6700\u540e\u200b\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\u9ad8\u5ea6\u200b\uff0c<code>np</code>\u200b\u4e3a\u200b\u97f3\u7b26\u200b\u4e2a\u6570\u200b\u3002</p> <pre><code>\\Notes\\Ibu0dk4\\qb0d\\qb0h\\qb0f\\tbu0\\qb0k\\en\n\\Notes\\Ibu0kd4\\qb0k\\qb0f\\qb0h\\tbu0\\qb0d\\en\n</code></pre> <p></p> <p>\u200b\u5f53\u200b\u4e0d\u200b\u5305\u542b\u200b\u548c\u5f26\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>D</code>\u3001<code>T</code>\u3001<code>Q</code>\u200b\u5feb\u901f\u200b\u8f93\u5165\u200b\u5305\u542b\u200b2\u30013\u30014\u200b\u4e2a\u200b\u97f3\u7b26\u200b\u7684\u200b\u7b26\u6760\u200b\u3002</p> <pre><code>\\Notes\\Qqbu dhfk\\en\n\\notes\\Qqbbu kfhd\\Qqbbl dhfk\\en\n</code></pre> <p></p> <p>\u200b\u5f53\u200b\u5728\u200b\u7b26\u6760\u200b\u4e2d\u200b\u9700\u8981\u200b\u6539\u53d8\u200b\u7b49\u7ea7\u200b\u65f6\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>\\nbu</code>\u200b\u5c06\u200b\u7b49\u7ea7\u200b\u589e\u52a0\u200b\u5230\u200b<code>b</code>\u200b\u7684\u200b\u4e2a\u6570\u200b\u5bf9\u5e94\u200b\u7684\u200b\u7ea7\u6570\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>\\ibu</code>\u200b\u5c06\u200b\u7b49\u7ea7\u200b\u51cf\u5c11\u200b\u5230\u200b<code>b</code>\u200b\u7684\u200b\u4e2a\u6570\u200b\u51cf\u4e00\u200b\u5bf9\u5e94\u200b\u7684\u200b\u7ea7\u6570\u200b\u3002</p> <pre><code>\\Notes\\ibu0k0\\qb0j\\en\\notes\\nbbu0\\qb0j\\tbu0\\qb0j\\en\n\\notes\\ibbu0j0\\qb0j\\tbbu0\\qb0j\\en\\Notes\\tbu0\\qb0j\\en\n</code></pre> <p>\u200b\u4f7f\u7528\u200b<code>\\roff{}</code>\u200b\u5728\u200b\u5f53\u524d\u200b\u97f3\u7b26\u200b\u53f3\u4fa7\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u5bf9\u5e94\u200b\u7b49\u7ea7\u200b\u7684\u200b\u77ed\u6760\u200b\uff0c\u200b\u5f53\u200b<code>\\tbbu</code>\u200b\u914d\u5408\u200b\u66f4\u200b\u9ad8\u7b49\u7ea7\u200b\u8f93\u5165\u200b\u65f6\u200b\uff0c\u200b\u5728\u200b\u5f53\u524d\u200b\u97f3\u7b26\u200b\u5de6\u4fa7\u200b\u4f4d\u7f6e\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u5bf9\u5e94\u200b\u7b49\u7ea7\u200b\u77ed\u6760\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u6765\u200b\u8f93\u5165\u200b\u5207\u5206\u97f3\u200b\u3002</p> <pre><code>\\notes\\ibbu0j0\\qb0j\\tbbu0\\qb0j\\en\n\\Notes\\tbu0\\qb0j\\en\n\n\\notes\\ibbu0j0\\roff{\\tbbu0}\\qb0j\\en\n\\Notes\\qb0j\\tbbu0\\en\n\\notes\\tbu0\\qb0j\\en\n</code></pre> <p></p>"},{"location":"latex/misc/musixtex/#_7","title":"\u4f11\u6b62\u7b26","text":"<p>\u200b\u4f11\u6b62\u7b26\u200b\u7684\u200b\u547d\u4ee4\u200b\u4e3a\u200b</p> \u200b\u65f6\u503c\u200b \u200b\u5168\u97f3\u7b26\u200b \u200b\u4e8c\u5206\u200b \u200b\u56db\u5206\u200b \u200b\u516b\u5206\u200b \u200b\u5341\u516d\u5206\u200b \u200b\u547d\u4ee4\u200b <code>\\pause</code> <code>\\hp</code> <code>\\qp</code> <code>\\ds</code> <code>\\qs</code> <p>\u200b\u5728\u200b\u547d\u4ee4\u200b\u524d\u200b\u4f7f\u7528\u200b<code>\\raise</code>\u200b\u8c03\u6574\u200b\u4f11\u6b62\u7b26\u200b\u9ad8\u5ea6\u200b\uff0c<code>\\Interligne</code>\u200b\u8868\u793a\u200b\u7ebf\u200b\u95f4\u8ddd\u200b\uff1b\u200b\u4f7f\u7528\u200b<code>\\rlap</code>\u200b\u4f7f\u4e0b\u200b\u4e00\u4e2a\u200b\u97f3\u7b26\u200b\u505c\u7559\u200b\u5728\u200b\u539f\u4f4d\u200b\u3002</p> <pre><code>\\NOtes\\qp\\en\n\\Notes\\ds\\ds\\en\n\\NOtes\\raise2\\Interligne\\rlap\\qp\\raise-2\\Interligne\\qp\\qp \\en \\doublebar\n</code></pre> <p></p>"},{"location":"latex/misc/musixtex/#_8","title":"\u6587\u672c","text":"<p>\u200b\u4f7f\u7528\u200b<code>charnote{p}{text}</code>\u200b\u524d\u9762\u200b\u52a0\u200b<code>\\z</code>\u3001<code>\\c</code>\u3001<code>\\l</code>\u200b\u5728\u200b\u8c31\u200b\u4e0a\u200b\u6dfb\u52a0\u200b\u6587\u5b57\u200b\uff0c<code>\\z</code>\u3001<code>\\c</code>\u3001<code>\\l</code>\u200b\u5206\u522b\u200b\u8868\u793a\u200b\u5de6\u200b\u3001\u200b\u4e2d\u200b\u3001\u200b\u53f3\u200b\u5bf9\u9f50\u200b\u5f53\u524d\u200b\u97f3\u7b26\u200b\u3002<code>char{num}{text}</code>\u200b\u548c\u200b<code>charnote</code>\u200b\u7528\u6cd5\u200b\u76f8\u540c\u200b\uff0c\u200b\u4f46\u200b\u4f7f\u7528\u200b\u6570\u5b57\u200b\u6765\u200b\u8868\u793a\u200b\u4f4d\u7f6e\u200b\uff1a\u200b\u4ece\u200b\u6700\u200b\u4e0b\u65b9\u200b\u7684\u200b\u7ebf\u200b\u5f00\u59cb\u200b\uff0c\u200b\u6587\u672c\u200b\u57fa\u7ebf\u200b\u5bf9\u9f50\u200b\u6570\u5b57\u200b\u8868\u793a\u200b\u7684\u200b\u7ebf\u200b\u3002</p> <p></p>"},{"location":"latex/misc/musixtex/#_9","title":"\u8282\u62cd","text":"<p>\u200b\u4f7f\u7528\u200b<code>\\metron{\\qu}{60}</code>\u200b\u6765\u200b\u8bbe\u7f6e\u200b\u8282\u62cd\u200b\uff0c\u200b\u8282\u62cd\u200b\u4e5f\u200b\u662f\u200b\u6587\u672c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b\u76f8\u5173\u200b\u547d\u4ee4\u200b\u653e\u5728\u200b\u8c31\u200b\u4e0a\u200b\u3002\u200b\u53ef\u4ee5\u200b\u7528\u200b<code>\\smallnotesize</code>\u3001<code>\\tinynotesize</code>\u200b\u8c03\u8282\u200b\u97f3\u7b26\u200b\u5927\u5c0f\u200b\u3002</p> <p></p>"},{"location":"latex/misc/musixtex/#_10","title":"\u9f13\u8c31","text":"<p>\u200b\u9f13\u8c31\u200b\u4e2d\u200b\u4ee3\u8868\u200b\u51fb\u6253\u200b\u9572\u200b\u7247\u200b\u7684\u200b\u97f3\u7b26\u200b\u7531\u200b<code>musixper</code>\u200b\u6269\u5c55\u200b\u5305\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u4f7f\u7528\u200b<code>\\input</code>\u200b\u547d\u4ee4\u200b\u5f15\u5165\u200b\u3002\u200b\u5982\u4e0b\u200b\u7b26\u53f7\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b<code>\\dc, \\dh, \\do, \\y, \\x, \\ox, \\ro, k</code>\uff0c\u200b\u52a0\u200b\u5728\u200b\u5bf9\u5e94\u200b\u7684\u200b\u97f3\u7b26\u200b\u65f6\u503c\u200b\u524d\u200b\u3002</p> <pre><code>\\Notes\\ibu0o0\\dcqb0o\\dhqb0o\\doqb0o\\tbu0\\yqb0o\\en\n\\Notes\\ibu0o0\\xqb0o\\oxqb0o\\roqb0o\\tbu0\\kqb0o\\en\n</code></pre> <p></p> <p>\u200b\u6253\u51fb\u4e50\u200b\u4f7f\u7528\u200b\u7684\u200b\u8c31\u53f7\u200b\u7528\u200b<code>\\setclefsymbol1{\\drumclef}</code>\u200b\u547d\u4ee4\u200b\u5f15\u5165\u200b\u3002\u200b\u97f3\u9ad8\u200b\u8bb0\u53f7\u200b\u548c\u200b\u9ad8\u97f3\u200b\u8c31\u53f7\u200b\u4e00\u81f4\u200b\uff0c\u200b\u7235\u58eb\u9f13\u200b\u4e2d\u200b\u7684\u200b\u4e0d\u540c\u200b\u58f0\u90e8\u200b\u5bf9\u5e94\u200b\u4e94\u7ebf\u8c31\u200b\u7684\u200b\u4e0d\u540c\u200b\u97f3\u9ad8\u200b\u5982\u4e0b\u200b\u56fe\u200b\u6240\u793a\u200b</p> <p></p>"},{"location":"latex/tikz/pgfplots/","title":"pgfplots","text":"<p>\u200b\u501f\u52a9\u200b\u5b8f\u5305\u200b<code>pgfplots</code>\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200bTikZ\u200b\u7ed8\u5236\u200b\u6298\u7ebf\u56fe\u200b\uff0c\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\u3002</p> <p>pgfplots\u200b\u7ed8\u5236\u200b\u6298\u7ebf\u56fe\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{figure}[htbp]\n    \\begin{tikzpicture}\n        \\begin{axis}[sharp plot]\n            \\addplot coordinates{(0, 5)(1, 9)(2, 7)(3, 4)(4, 10)};\n        \\end{axis}\n    \\end{tikzpicture}\n\\end{figure}\n</code></pre> <p></p> <p><code>axis</code>\u200b\u63a7\u5236\u200b\u4e00\u4e2a\u200b\u56fe\u50cf\u200b\uff0c<code>\\begin{axis}</code>\u200b\u540e\u200b\u53ef\u4ee5\u200b\u52a0\u200b\u4e00\u5bf9\u200b\u65b9\u62ec\u53f7\u200b\uff0c\u200b\u7528\u4e8e\u200b\u8bbe\u5b9a\u200b\u56fe\u7247\u200b\u7684\u200b\u53c2\u6570\u200b\u3002<code>addplot</code>\u200b\u63a7\u5236\u200b\u56fe\u50cf\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5176\u540e\u200b\u540c\u6837\u200b\u53ef\u4ee5\u200b\u52a0\u200b\u4e00\u5bf9\u200b\u65b9\u62ec\u53f7\u200b\u7528\u4e8e\u200b\u63a7\u5236\u53c2\u6570\u200b\u3002</p>"},{"location":"latex/tikz/pgfplots/#axis","title":"<code>axis</code>\u200b\u73af\u5883","text":"<p><code>axis</code>\u200b\u73af\u5883\u200b\u53ef\u7528\u200b\u7684\u200b\u53c2\u6570\u200b\u5982\u4e0b\u200b\u6240\u200b\u5217\u200b\uff1a</p> <ul> <li><code>[sharp plot, smooth]</code>\u200b\u7b49\u200b\uff0c\u200b\u8868\u660e\u200b\u6298\u7ebf\u56fe\u200b\u7684\u200b\u66f2\u7ebf\u200b\u662f\u5426\u200b\u7ecf\u8fc7\u200b\u5e73\u6ed1\u200b</li> <li><code>xlabel</code>\u200b\u4e0e\u200b<code>ylabel</code>\uff0c\u200b\u63a7\u5236\u200b\u5750\u6807\u8f74\u200b\u7684\u200b\u6807\u6ce8\u200b\u3002</li> <li><code>title</code>\u200b\u63a7\u5236\u200b\u56fe\u50cf\u200b\u7684\u200b\u6807\u9898\u200b</li> <li><code>xmode</code>\u200b\u4e0e\u200b<code>ymode</code>\u200b\u63a7\u5236\u200b\u5750\u6807\u8f74\u200b\u662f\u200b\u7ebf\u6027\u200b\u5750\u6807\u8f74\u200b\uff08<code>normal</code>\uff09\u200b\u8fd8\u662f\u200b\u5bf9\u6570\u200b\u5750\u6807\u8f74\u200b\uff08<code>log</code>\uff09</li> <li><code>xtick distance</code>\u200b\u4e0e\u200b<code>ytick distance</code>\u200b\u63a7\u5236\u200b\u5750\u6807\u8f74\u200b\u523b\u5ea6\u200b\u5927\u5c0f\u200b</li> <li><code>xmin, xmax</code>\u200b\u4e0e\u200b<code>ymin, ymax</code>\u200b\u63a7\u5236\u200b\u5750\u6807\u8f74\u200b\u7684\u200b\u8303\u56f4\u200b<ul> <li>\u200b\u5f53\u200b\u6307\u5b9a\u200b\u5750\u6807\u8f74\u200b\u8303\u56f4\u200b\u65f6\u200b\uff0c\u200b\u5982\u679c\u200b<code>enlargelimits</code>\u200b\u6307\u5b9a\u200b\u4e3a\u200b<code>true</code>\uff0c\u200b\u5219\u200b\u5750\u6807\u8f74\u200b\u4f1a\u200b\u81ea\u52a8\u200b\u653e\u5927\u200b\u4ee5\u200b\u7b26\u5408\u200b\u8f93\u5165\u200b\u6570\u636e\u200b</li> </ul> </li> </ul> <p>\u200b\u5982\u4e0b\u200b\u521b\u5efa\u200b\u4e00\u4e2a\u200b\u7a7a\u200b\u7684\u200b\u56fe\u50cf\u200b\uff0cx\u200b\u5750\u6807\u8f74\u200b\u4e3a\u200b\u7ebf\u6027\u200b\uff0cy\u200b\u5750\u6807\u8f74\u200b\u4e3a\u200b\u5bf9\u6570\u200b\uff1a</p> <p>axis\u200b\u53c2\u6570\u200b\u793a\u4f8b\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{figure}[htbp]\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xlabel=Year,\n            ylabel=Inventory,\n            xmin=2019, xmax=2021,\n            ymin=100, ymax=1000,\n            xtick distance=1,\n            title=Inventory Graph,\n            xmode=normal, ymode=log\n        ]\n        \\end{axis}\n    \\end{tikzpicture}\n\\end{figure}\n</code></pre> <p></p> <p>\u200b\u6709\u5173\u200b\u66f4\u200b\u591a\u200b\u53c2\u6570\u200b\uff0c\u200b\u8bf7\u200b\u53c2\u89c1\u200bpgfplots \u200b\u624b\u518c\u200b</p>"},{"location":"latex/tikz/pgfplots/#addplot","title":"<code>\\addplot</code>","text":"<p><code>\\addplot</code>\u200b\u547d\u4ee4\u200b\u53ef\u4ee5\u200b\u5411\u200b\u56fe\u50cf\u200b\u4e2d\u200b\u6dfb\u52a0\u200b\u6298\u7ebf\u56fe\u200b\u6216\u200b\u51fd\u6570\u200b\u3002</p> <ol> <li><code>\\addplot coordinates{}</code> \u200b\u7528\u4e8e\u200b\u7ed8\u5236\u200b\u6298\u7ebf\u56fe\u200b\uff0c\u200b\u5176\u4e2d\u200b<code>coordinates</code>\u200b\u4e2d\u200b\u7684\u200b\u5750\u6807\u200b\u7528\u200b\u5706\u62ec\u53f7\u200b\u62ec\u8d77\u200b</li> <li><code>\\addplot [smooth] {f(x)}</code> \u200b\u7528\u4e8e\u200b\u7ed8\u5236\u200b\u5e73\u6ed1\u200b\u7684\u200b\u6298\u7ebf\u56fe\u200b\uff0c\u200b\u5176\u4e2d\u200b<code>f(x)</code>\u200b\u4e3a\u200b\u51fd\u6570\u200b\u8868\u8fbe\u5f0f\u200b</li> </ol> <p>axis\u200b\u53c2\u6570\u200b\u793a\u4f8b\u200b</p> tex\u200b\u4ee3\u7801\u200b\u6e32\u67d3\u200b\u7ed3\u679c\u200b <pre><code>\\begin{figure}[htbp]\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xlabel=$x$ variable,\n            ylabel=$y$ variable,\n            xmin=0, xmax=7,\n            title=Regression Example,\n        ]\n            \\addplot coordinates{(0, 5)(1, 9)(2, 7)(3, 4)(4, 10)};\n            \\addlegendentry{\\small Data}\n            \\addplot [smooth, red]{0.5*x + 6};\n            \\addlegendentry{\\small Regressor}\n        \\end{axis}\n    \\end{tikzpicture}\n\\end{figure}\n</code></pre> <p></p> <p><code>\\addplot</code>\u200b\u6709\u200b\u5982\u4e0b\u200b\u53c2\u6570\u200b\uff1a</p> <ul> <li><code>smooth</code>\u200b\u5982\u679c\u200b\u5b58\u5728\u200b\uff0c\u200b\u7ed8\u5236\u200b\u4e0d\u200b\u5305\u542b\u200b\u70b9\u200b\u5750\u6807\u200b\u7684\u200b\u5e73\u6ed1\u200b\u56fe\u50cf\u200b</li> <li><code>[color]</code>\u200b\u8bbe\u7f6e\u200b\u989c\u8272\u200b\u3002</li> </ul>"},{"location":"math/","title":"\u8bfe\u7a0b\u200b\u7b14\u8bb0","text":"<p>\u200b\u4ee5\u4e0b\u200b\u6574\u7406\u200b\u4e86\u200b\u8bfe\u4e0a\u200b\u7684\u200b\u5185\u5bb9\u200b</p> <ol> <li>\u200b\u79bb\u6563\u6570\u5b66\u200b</li> <li>\u200b\u8fd0\u7b79\u5b66\u200b\u4e0e\u200b\u7ebf\u6027\u89c4\u5212\u200b</li> <li>\u200b\u5bf9\u7b56\u200b\u3001\u200b\u51b3\u7b56\u200b\u4e0e\u200b\u6392\u961f\u200b\u8bba\u200b</li> <li>\u200b\u8fd0\u4f5c\u200b\u7ba1\u7406\u200b</li> <li>\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b</li> <li>\u200b\u51f8\u200b\u4f18\u5316\u200b</li> <li>\u200b\u56e0\u679c\u200b\u63a8\u65ad\u200b</li> </ol>"},{"location":"math/#_2","title":"\u6269\u5c55\u200b\u9605\u8bfb","text":""},{"location":"math/#_3","title":"\u6570\u5b66\u5206\u6790","text":"<ul> <li>\u200b\u521d\u4e2d\u200b\u7ade\u8d5b\u200b:\\(x+y+z=1\\)\uff0c\u200b\u6c42\u200b\\(xy+2yz+3zx\\)\u200b\u7684\u200b\u6700\u5927\u503c\u200b? - \u200b\u4e88\u4e00\u4eba\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u628a\u200b\\([-0.5,0.5]\\)\u200b\u5185\u200b\u6240\u6709\u200b\u5b9e\u6570\u200b\u76f8\u52a0\u200b\uff0c\u200b\u5f97\u51fa\u200b\u7684\u200b\u7ed3\u679c\u200b\u662f\u200b\\(0\\)\u200b\u5417\u200b\uff1f - Be2SiO4\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u5982\u4f55\u200b\u6c42\u200b\u5fae\u5206\u65b9\u7a0b\u200b \\(y'' + y = 0\\) \u200b\u7684\u200b\u901a\u89e3\u200b\uff1f - \u200b\u4e88\u4e00\u4eba\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u4e3a\u4ec0\u4e48\u200b \\(\\ln x\\) \u200b\u6c42\u5bfc\u200b\u662f\u200b \\(1/x\\)\uff1f - \u200b\u8881\u8d85\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\\(\\ln x (x&gt;1, x\\in \\mathbb Z)\\) \u200b\u53ef\u80fd\u200b\u662f\u200b\u6709\u7406\u6570\u200b\u5417\u200b\uff1f - \u200b\u4e88\u4e00\u4eba\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u79ef\u5206\u200b\u7b26\u53f7\u200b\u5185\u53d6\u200b\u5fae\u5206\u200b\u662f\u200b\u4e00\u79cd\u200b\u4ec0\u4e48\u200b\u65b9\u6cd5\u200b\uff1f - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u4e00\u6761\u200b\u4ece\u200b\\((0,0)\\)\u200b\u51fa\u53d1\u200b\u7684\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(1\\)\u200b\u7684\u200b\u66f2\u7ebf\u200b\uff0c\u200b\u56f4\u7ed5\u200b\\(x\\)\u200b\u8f74\u200b\u65cb\u8f6c\u200b\u4ea7\u751f\u200b\u7684\u200b\u65cb\u8f6c\u4f53\u200b\u7684\u200b\u6700\u5927\u200b\u4f53\u79ef\u200b\u662f\u200b\u591a\u5c11\u200b\uff1f - \u200b\u865a\u200b\u8c03\u5b50\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> </ul>"},{"location":"math/#_4","title":"\u6982\u7387\u8bba\u200b\u4e0e\u200b\u968f\u673a\u200b\u8fc7\u7a0b","text":"<ul> <li>\u200b\u6570\u5b66\u200b\u671f\u671b\u200b\u5927\u4e8e\u200b\u96f6\u200b\u7684\u200b\u6e38\u620f\u200b\u4e00\u5b9a\u200b\u80fd\u200b\u8d5a\u94b1\u200b\u5417\u200b - \u200b\u9648\u58a8\u77b3\u200bnono\u200b\u7684\u200b\u6587\u7ae0\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u7532\u4e59\u200b\u4e24\u4eba\u200b\u4e0b\u56f4\u68cb\u200b\uff0c\u200b\u7532\u80dc\u200b\u7684\u200b\u6982\u7387\u200b\u4e3a\u200b\\(a\\)\uff0c\u200b\u4e59\u80dc\u200b\u7684\u200b\u6982\u7387\u200b\u4e3a\u200b\\(b\\)\uff0c\\(a+b=1\\)\uff0c\u200b\u6bd4\u200b\u5bf9\u65b9\u200b\u591a\u200b\u8d62\u200b\u4e24\u5c40\u200b\u8005\u200b\u83b7\u80dc\u200b\uff0c\u200b\u6c42\u7532\u200b\u8d62\u200b\u7684\u200b\u6982\u7387\u200b? - Dylaaan\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u5f3a\u5927\u200b\u6570\u5f8b\u200b\uff08SLLN) - \u200b\u4f55\u5bb6\u5fd7\u200b\u7684\u200b\u6587\u7ae0\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u5982\u679c\u200b\u6211\u200b\u6709\u200b \\(2^{-n}\\) \u200b\u6982\u7387\u200b\u83b7\u5f97\u200b \\(2^n\\) \u200b\u5143\u200b\uff0c\u200b\u90a3\u200b\u83b7\u5f97\u200b\u94b1\u200b\u7684\u200b\u6570\u5b66\u200b\u671f\u671b\u503c\u200b\u5e94\u8be5\u200b\u662f\u200b\u65e0\u7a77\u200b\uff0c\u200b\u4f46\u200b\u5b9e\u9645\u200b\u4e0d\u662f\u200b\u8fd9\u6837\u200b\uff0c\u200b\u4e3a\u4ec0\u4e48\u200b\uff1f - ClefRybak\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> </ul>"},{"location":"math/#_5","title":"\u7ebf\u6027\u4ee3\u6570","text":"<ul> <li>\u200b\u5982\u4f55\u200b\u5224\u5b9a\u200b\u4e0d\u53ef\u200b\u7ea6\u200b\u591a\u9879\u5f0f\u200b\uff1f - \u200b\u5982\u8564\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u8ba1\u7b97\u673a\u200b\u56fe\u5f62\u5b66\u200b\u4e00\u200b\uff1a\u200b\u57fa\u7840\u200b\u53d8\u6362\u200b\u77e9\u9635\u200b\u603b\u7ed3\u200b(\u200b\u7f29\u653e\u200b\uff0c\u200b\u65cb\u8f6c\u200b\uff0c\u200b\u4f4d\u79fb\u200b)</li> </ul>"},{"location":"math/#_6","title":"\u79bb\u6563\u6570\u5b66\u200b\u4e0e\u200b\u6570\u8bba","text":"<ul> <li>\u200b\u4e00\u822c\u200b\u4e00\u5143\u200b\u4e94\u6b21\u200b\u65b9\u7a0b\u200b\u6ca1\u6709\u200b\u6839\u5f0f\u200b\u89e3\u200b\u7684\u200b\u79d8\u5bc6\u200b\u7a76\u7adf\u200b\u5728\u200b\u54ea\u91cc\u200b\uff1f\u200b\u82b1\u8d39\u200b\u4eba\u7c7b\u200b\u4e09\u767e\u5e74\u200b.? - Geometrie\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u7ec4\u5408\u200b\u6570\u5b66\u200b\u5b66\u4e60\u200b\u7b14\u8bb0\u200b\uff08\u200b\u5341\u200b\uff09 - sola\u200b\u7684\u200b\u6587\u7ae0\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>ACM\u2014\u2014\u200b\u5e38\u89c1\u200b\u7684\u200b\u51e0\u79cd\u200b\u5206\u89e3\u200b\u8d28\u200b\u56e0\u5b50\u200b\u7684\u200b\u65b9\u6cd5\u200b - \u200b\u4e25\u683c\u200b\u9e3d\u200b\u7684\u200b\u6587\u7ae0\u200b - \u200b\u77e5\u4e4e\u200b</li> </ul>"},{"location":"math/#_7","title":"\u5176\u4ed6","text":"<ul> <li>\u200b\u4e94\u4e2a\u200b\u56da\u72af\u200b\u5148\u540e\u200b\u4ece\u200b100\u200b\u9897\u200b\u7eff\u8c46\u200b\u4e2d\u200b\u6293\u200b\u7eff\u8c46\u200b\u3002\u200b\u6293\u5f97\u200b\u6700\u200b\u591a\u200b\u548c\u200b\u6700\u5c11\u200b\u7684\u200b\u4eba\u200b\u5c06\u200b\u88ab\u200b\u5904\u6b7b\u200b\uff0c\u200b\u4e0d\u80fd\u200b\u4ea4\u6d41\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6478\u200b\u51fa\u200b\u5269\u4e0b\u200b\u7eff\u8c46\u200b\u7684\u200b\u6570\u91cf\u200b\uff0c\u200b\u8c01\u200b\u7684\u200b\u5b58\u6d3b\u200b\u51e0\u7387\u200b\u6700\u5927\u200b\uff1f - \u200b\u66fe\u52a0\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>1000\u200b\u6876\u200b\u6c34\u200b\uff0c\u200b\u5176\u4e2d\u200b\u4e00\u6876\u200b\u6709\u6bd2\u200b\uff0c\u200b\u732a\u200b\u559d\u200b\u6bd2\u6c34\u200b\u540e\u4f1a\u200b\u5728\u200b15\u200b\u5206\u949f\u200b\u5185\u200b\u6b7b\u53bb\u200b\uff0c\u200b\u60f3\u200b\u7528\u200b\u4e00\u4e2a\u200b\u5c0f\u65f6\u200b\u627e\u5230\u200b\u8fd9\u6876\u200b\u6bd2\u6c34\u200b\uff0c\u200b\u81f3\u5c11\u200b\u9700\u8981\u200b\u51e0\u5934\u200b\u732a\u200b\uff1f - \u200b\u82d7\u534e\u680b\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u9664\u4e86\u200b 3\uff0c4\uff0c5 \u200b\u4ee5\u5916\u200b\u662f\u5426\u200b\u8fd8\u6709\u200b\u522b\u7684\u200b\u4e09\u89d2\u5f62\u200b\uff0c\u200b\u5b83\u200b\u7684\u200b\u4e09\u6761\u200b\u8fb9\u200b\u662f\u200b\u8fde\u7eed\u200b\u81ea\u7136\u6570\u200b\uff0c\u200b\u5b83\u200b\u7684\u200b\u9762\u79ef\u200b\u4e5f\u200b\u662f\u200b\u81ea\u7136\u6570\u200b\uff1f - Pandora Eartha\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u72b6\u6001\u200b\u7a7a\u95f4\u200b\u8868\u8fbe\u5f0f\u200b\u7684\u200b\u89e3\u200b - Anonymity67\u200b\u7684\u200b\u6587\u7ae0\u200b - \u200b\u77e5\u4e4e\u200b</li> <li>\u200b\u5982\u4f55\u200b\u6c42\u200b\u6570\u5217\u200b\\(a_{n+1} = a_n^2 - 2a_n\\)? - Reuental\u200b\u7684\u200b\u56de\u7b54\u200b - \u200b\u77e5\u4e4e\u200b</li> </ul>"},{"location":"math/causal-inference/","title":"\u56e0\u679c\u200b\u63a8\u65ad","text":"<p>According to Causal Inference: What If.</p> <ol> <li>A definition of causal effect</li> <li>Randomized experiments</li> <li>Observational studies</li> <li>Effect modification</li> <li>Interaction</li> <li>Graphical representation of causal effects</li> <li>Confounding</li> <li>Selection bias</li> <li>Measurement bias and \"noncausal\" diagrams</li> </ol>"},{"location":"math/causal-inference/causal-inference-stale/","title":"\u56e0\u679c\u200b\u63a8\u65ad","text":"<p>Casual inference is about studying the consequence of doing one thing vs another.</p> <p>Simpson\u2019s Paradox</p> <p>\u200b\u67d0\u4e2a\u200b\u7edf\u8ba1\u200b\u5173\u8054\u200b\uff0c\u200b\u53ef\u80fd\u200b\u5728\u200b\u6240\u6709\u200b\u5b50\u200b\u7fa4\u4f53\u200b\u4e0a\u200b\u7684\u200b\u6548\u679c\u200b\u90fd\u200b\u662f\u200b\u76f8\u53cd\u200b\u7684\u200b\u3002</p> <p>A statistical association that holds for the entire population can be reversed in every subpopulation.</p> <p></p> <p>Berkson\u2019s Paradox</p> <p>Variables that have no relationships can appear to have strong associations in a selected subpopulation.</p> <p>\u200b\u56e0\u679c\u200b\u63a8\u65ad\u200b\u6709\u200b\u4e24\u4e2a\u200b\u4e3b\u8981\u200b\u7684\u200b\u7814\u7a76\u200b\u6846\u67b6\u200b\uff0c\u200b\u5373\u200b Potential outcome framework \u200b\u548c\u200b Causal graph \u3002</p>"},{"location":"math/causal-inference/causal-inference-stale/#potential-outcome-framework","title":"Potential Outcome Framework","text":""},{"location":"math/causal-inference/causal-inference-stale/#_2","title":"\u6846\u67b6\u200b\u8bbe\u5b9a","text":"<ul> <li>Unit of analysis\uff1a\u200b\u4e0d\u540c\u200b\u65f6\u95f4\u200b\u4e0b\u200b\u7684\u200b\u540c\u4e00\u4e2a\u200b\u4e2a\u4f53\u200b\u5e94\u5f53\u200b\u770b\u4f5c\u200b\u4e0d\u540c\u200b\u7684\u200b\u4e2a\u4f53\u200b</li> <li>Treatment \\(D\\)\uff1a\u200b\u901a\u5e38\u200b\u5212\u5206\u200b\u4e3a\u200b\u5b9e\u9a8c\u7ec4\u200b\uff08\\(D=1\\)\uff09\u200b\u4e0e\u200b\u5bf9\u7167\u7ec4\u200b\uff08\\(D=0\\)\uff09\uff0c\u200b\u4f46\u200b\\(D\\)\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u53d6\u591a\u503c\u200b\u6216\u200b\u8fde\u7eed\u200b</li> <li>Potential outcome \\(Y(d)\\)\uff1a\u200b\u5f53\u200b\u5e72\u9884\u200b\u6c34\u5e73\u200b\u4e3a\u200b\\(d\\)\u200b\u65f6\u200b\u5e94\u5f53\u200b\u51fa\u73b0\u200b\u7684\u200b\u7ed3\u679c\u200b\u3002</li> <li>Counterfactual outcome\uff1a\u200b\u5f53\u200b\u5e72\u9884\u200b\u6c34\u5e73\u200b\u4e3a\u200b\\(D\\)\u200b\u65f6\u200b\u6240\u6709\u200b\\(d\\not = D\\)\u200b\u7684\u200b\u7ed3\u679c\u200b\u2014\u2014\u200b\u5b58\u5728\u200b\u4f46\u200b\u65e0\u6cd5\u200b\u89c2\u6d4b\u200b\u5230\u200b</li> <li>Casual effect\uff1a\u200b\u6bd4\u8f83\u200b\u4e24\u4e2a\u200b\u5e73\u884c\u200b\u7ed3\u679c\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5dee\u5f02\u200b\uff0c\\(Y(1) - Y(0)\\)</li> </ul> <p>\u200b\u5f53\u200b\u4e00\u4e2a\u200b\u4e2a\u4f53\u200b\u63a5\u6536\u200b\u5230\u200b\\(d=D\\)\u200b\u4e4b\u540e\u200b\uff0c\u200b\u5176\u200b\u5e72\u9884\u200b\u72b6\u6001\u200b\u5c31\u200b\u4e0d\u4f1a\u200b\u518d\u200b\u6539\u53d8\u200b\u3002\u200b\u56e0\u6b64\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u89c2\u6d4b\u200b\u5230\u200b\u4e00\u4e2a\u200b\u4e2a\u4f53\u6240\u6709\u200b\u53ef\u80fd\u200b\u51fa\u73b0\u200b\u7684\u200b\u5e72\u9884\u200b\u7ed3\u679c\u200b\u3002</p> <p>SUTVA assumption</p> <p>\u200b\u5047\u8bbe\u200b\u4e2a\u4f53\u200b\\(i\\)\u200b\u7684\u200b\u5e72\u9884\u200b\u72b6\u6001\u200b\\(D_i\\)\u3001\u200b\u5e72\u9884\u200b\u548c\u200b\u4e0d\u200b\u5e72\u9884\u200b\u7684\u200b\u6548\u679c\u200b\\(Y_i(0), Y_i(1)\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u5373\u200b</p> \\[ \\{D_i, Y_i(0), Y_i(1)\\}\\sim\\{D, Y(0), Y(1)\\} \\] <p>\u200b\u540c\u65f6\u200b\u5047\u8bbe\u200b</p> <ol> <li>\u200b\u6bcf\u4e2a\u200b\u4e2a\u4f53\u200b\u7684\u200b\u7ed3\u679c\u200b\u4e0d\u200b\u53d7\u5230\u200b\u5176\u4ed6\u200b\u4e2a\u4f53\u200b\u5e72\u9884\u200b\u6c34\u5e73\u200b\u7684\u200b\u5f71\u54cd\u200b\uff08\u200b\u4e0d\u200b\u5b58\u5728\u200b\u4e2a\u4f53\u200b\u4e4b\u95f4\u200b\u7684\u200b\u76f8\u4e92\u5f71\u54cd\u200b\uff0c\u200b\u4e2a\u4f53\u200b\u7684\u200b\u7ed3\u679c\u200b\u53ea\u53d7\u200b\u5e72\u9884\u200b\u6c34\u5e73\u200b\u7684\u200b\u5f71\u54cd\u200b\uff09</li> <li>\u200b\u4e0d\u200b\u5b58\u5728\u200b\u5e72\u9884\u200b\u6c34\u5e73\u200b\u7684\u200b\u9690\u85cf\u200b\u72b6\u6001\u200b\uff08\u200b\u5373\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5e72\u9884\u200b\u72b6\u6001\u200b\u90fd\u200b\u662f\u200b\u539f\u5b50\u200b\u7684\u200b\uff0c\u200b\u4e0d\u200b\u53ef\u4ee5\u200b\u8fdb\u4e00\u6b65\u200b\u5212\u5206\u200b\uff09</li> </ol> <p>\u200b\u8003\u8651\u200b\u603b\u4f53\u200b\u7684\u200b\u671f\u671b\u200b\u5e72\u9884\u200b\u6548\u679c\u200b\uff08 Average Treatment Effect, ATE\uff09</p> \\[ \\tau = \\mathbb E[Y(1) - Y(0)] \\] <p>\u200b\u7531\u4e8e\u200b\u5bf9\u4e8e\u200b\u4e2a\u4f53\u200b\\(i\\)\uff0c\u200b\u65e0\u6cd5\u200b\u540c\u65f6\u200b\u89c2\u6d4b\u200b\\(Y_i(1), Y_i(0)\\)\uff0c\u200b\u56e0\u6b64\u200b\\(\\tau\\)\u200b\u7684\u200b\u786e\u5207\u200b\u6570\u503c\u200b\u540c\u6837\u200b\u65e0\u6cd5\u200b\u89c2\u6d4b\u200b\u3002</p> \\[ \\begin{aligned}     \\hat\\tau &amp;= \\frac{1}{n_1} \\sum_{i;D_i = 1}Y_i - \\frac{1}{n_0} \\sum_{i;D_i = 0}Y_i \\\\     &amp;= \\mathbb E[Y(1) | D = 1] - \\mathbb E[Y(0) | D = 0] \\end{aligned} \\] <p>\u200b\u9700\u8981\u200b\u8003\u8651\u200b\\(D\\)\u200b\u7684\u200b\u5206\u5e03\u200b\uff0c\u200b\u5373\u200b\u5b9e\u9a8c\u7ec4\u200b\u4e0e\u200b\u5bf9\u7167\u7ec4\u200b\u5982\u4f55\u200b\u4ea7\u751f\u200b\uff0c\u200b\u6216\u200b\\(Y, D\\)\u200b\u7684\u200b\u72ec\u7acb\u6027\u200b\u3002</p> <p>\u200b\u5f53\u200b\\(Y, D\\)\u200b\u6ee1\u8db3\u200b</p> \\[ (Y(0), Y(1))\\perp D \\label{1} \\] <p>\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(\\hat\\tau = \\tau\\)</p>"},{"location":"math/causal-inference/causal-inference-stale/#_3","title":"\u6df7\u6dc6\u200b\u56e0\u5b50","text":"<p>\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u573a\u666f\u200b\uff1a</p> <ol> <li>\u200b\u4e00\u7fa4\u200b\u88ab\u8bd5\u200b\u968f\u673a\u200b\u63a5\u53d7\u200b\u836f\u7269\u200b\u8bd5\u9a8c\u200b</li> <li>\u200b\u88ab\u200b\u8bd5\u4e2d\u200b\u75c7\u72b6\u200b\u6bd4\u8f83\u4e25\u91cd\u200b\u7684\u200b\u4e2a\u4f53\u200b\u63a5\u53d7\u200b\u836f\u7269\u200b\u5b9e\u9a8c\u200b</li> </ol> <ol> <li>\u200b\u968f\u673a\u200b\u8bd5\u9a8c\u200b\u6307\u200b\u5206\u914d\u673a\u5236\u200b\u7531\u200b\u7814\u7a76\u8005\u200b\u53ef\u63a7\u200b\u3002\u200b\u5982\u200b\u6309\u7167\u200b\\(p = 0.5\\)\u200b\u7684\u200b\u6982\u7387\u200b\u8fdb\u884c\u200b\u4f2f\u52aa\u5229\u200b\u5b9e\u9a8c\u200b\u968f\u673a\u200b\u5206\u914d\u200b\uff0c\u200b\u5219\u200b\\((Y(1)\uff0c Y(0))\\perp D\\)\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b\\(\\hat\\tau = \\tau\\)</li> <li>\u200b\u89c2\u6d4b\u200b\u7814\u7a76\u200b\u6307\u200b\u5206\u914d\u673a\u5236\u200b\u4e0d\u7531\u200b\u7814\u7a76\u8005\u200b\u63a7\u5236\u200b\u3002\u200b\u5982\u200b\u88ab\u200b\u8bd5\u200b\u6839\u636e\u200b\u4e2a\u4f53\u200b\u611f\u53d7\u200b\u81ea\u884c\u51b3\u5b9a\u200b\u5403\u836f\u200b\u3002\u200b\u6b64\u65f6\u200b\u5b58\u5728\u200b\uff08\u200b\u53ef\u4ee5\u200b\u89c2\u6d4b\u200b\u5230\u200b\u7684\u200b\uff09\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\\(X\\)</li> </ol> <p>\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b</p> <p>\u200b\u53d8\u91cf\u200b\\(X\\)\u200b\u662f\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\uff0c\u200b\u5982\u679c\u200b\uff1a</p> <ol> <li>\\(X\\)\u200b\u4f1a\u200b\u5f71\u54cd\u200b\u5206\u914d\u200b\uff0c\u200b\u5373\u200b\\(X\\not \\perp D\\)\uff0c\u200b\u6216\u200b</li> <li>\\(X\\)\u200b\u4f1a\u200b\u5f71\u54cd\u200b\u7ed3\u679c\u200b\uff0c\u200b\u5373\u200b\\(X\\not \\perp (Y(1)\uff0c Y(0))\\)</li> </ol> <p>\u200b\u901a\u5e38\u200b\u5c06\u200b\u53ef\u200b\u89c2\u6d4b\u200b\u5230\u200b\u7684\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u8bb0\u200b\u4e3a\u200b\\(X\\)\uff0c\u200b\u4e0d\u80fd\u200b\u89c2\u6d4b\u200b\u5230\u200b\u7684\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u8bb0\u200b\u4e3a\u200b\\(U\\)\u3002</p> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u5f53\u200b\u5b58\u5728\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u65f6\u200b\uff0c\\(\\eqref{1}\\)\u200b\u4e0d\u518d\u200b\u6210\u7acb\u200b\uff0c\u200b\u5176\u4e2d\u200b\u7684\u200b\\(D\\)\u200b\u53d8\u4e3a\u200b\\(D|X\\)</p> <p>\u200b\u5f53\u200b\u6240\u6709\u200b\u7684\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u90fd\u200b\u88ab\u200b\u89c2\u6d4b\u200b\u5230\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u9002\u7528\u200b unconfoundedness assumption</p> <p>Unconfoundedness assumption</p> <p>\u200b\u5047\u8bbe\u200b\u6210\u7acb\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b</p> \\[ (Y(1), Y(0)) \\perp D|X \\] <p>\u200b\u63a7\u5236\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b</p> \\[\\begin{aligned}     &amp; \\mathbb E [\\mathbb E[Y | D = 1, X] - \\mathbb E[Y | D = 0, X]] \\\\     =&amp; \\mathbb E [\\mathbb E[Y(1) | D = 1, X] - \\mathbb E[Y(0) | D = 0, X]] \\\\     =&amp; \\mathbb E [\\mathbb E[Y(1) | X] - \\mathbb E[Y(0) | X]] \\\\     =&amp; \\mathbb E [\\mathbb E[Y(1) - Y(0) | X]] \\\\     =&amp; \\mathbb E[Y(1) - Y(0)] \\end{aligned} \\] <p>\u200b\u4f46\u200b\u5b9e\u9645\u4e0a\u200b\uff0c\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u89c2\u6d4b\u200b\u5230\u200b\u6240\u6709\u200b\u7684\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u3002\u200b\u6b64\u65f6\u200b</p> \\[ \\begin{aligned}     &amp; \\bbE [Y | D = 1] - \\bbE [Y| D = 0] \\\\     =&amp; \\bbE[\\bbE[Y | D = 1, X] | D = 1] - \\bbE[\\bbE[Y | D = 0, X] | D = 0] \\end{aligned} \\] <p>\u200b\u4f46\u200b\u5b9e\u9645\u4e0a\u200b\uff0c\\(\\bbP(X|D = 1) \\not = \\bbP(X|D = 0)\\)\uff0c\u200b\u56e0\u6b64\u200b\u9020\u6210\u200b\u504f\u5dee\u200b\u3002</p> <p>\u200b\u5e72\u9884\u200b\u4e0e\u200b\u6761\u4ef6\u200b</p> <ul> <li>\\(\\bbE(Y|D = 0)\\)\u200b\u53cd\u6620\u200b\u6761\u4ef6\u200b\uff0c\u200b\u4f5c\u7528\u200b\u5728\u200b\u7fa4\u4f53\u200b\\(D = 0\\)\u200b\u4e0a\u200b\u3002</li> <li>\\(\\bbE(Y(0))\\)\u200b\u53cd\u6620\u200b\u5e72\u9884\u200b\uff0c\u200b\u4f5c\u7528\u200b\u5728\u200b\u5168\u4f53\u200b\u4e0a\u200b\u3002</li> </ul> <p>\u200b\u5f53\u200b\\((Y(1), Y(0))\\perp D | X\\)\u200b\u6210\u7acb\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\bbE(Y(1) | X) = \\bbE(Y | D = 1, X) \\] <p>\u200b\u524d\u63d0\u200b\u662f\u200b\uff0c overlap assumption \u200b\u6210\u7acb\u200b\u3002</p> <p>Overlap assumption</p> <p>\u200b\u5bf9\u4e8e\u200b\u6240\u6709\u200b\u7684\u200b\\(X\\)\uff0c\\(Y(0), Y(1)\\)\u200b\u5fc5\u987b\u200b\u90fd\u200b\u5b58\u5728\u200b\uff0c\u200b\u5373\u200b</p> \\[ \\bbP(D=1|X)\\in (0, 1) \\] <p>\u200b\u5b9a\u4e49\u200b\\(\\bbP(D=1|X)\\in (0, 1)\\)\u200b\u4e3a\u200bpropensity score\u3002</p>"},{"location":"math/causal-inference/causal-inference-stale/#_4","title":"\u5b9e\u8bc1\u200b\u7814\u7a76","text":"<p>\u200b\u5b9e\u9645\u4e0a\u200b\uff0c unconfoundedness \u200b\u548c\u200b overlap \u200b\u5047\u8bbe\u200b\u5747\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u6210\u7acb\u200b\uff0c\u200b\u56e0\u4e3a\u200b\u6709\u200b\u8bb8\u591a\u200b\u9690\u85cf\u200b\u7684\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u524d\u200b\u6587\u4e2d\u200b\u7684\u200b Simpson\u2019s Paradox \uff0c\u200b\u51fa\u73b0\u200b\u53cd\u200b\u76f4\u89c9\u200b\u73b0\u8c61\u200b\u7684\u200b\u539f\u56e0\u200b\u5728\u4e8e\u200b\uff0c\u200b\u6027\u522b\u200b\u662f\u200b\u4e00\u4e2a\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\uff1a</p> <ol> <li>\u200b\u5973\u6027\u200b\u7684\u200b\u88ab\u200b\u8bd5\u200b\u6bd4\u4f8b\u200b\u66f4\u9ad8\u200b\uff08\\(D\\not\\perp X\\)\uff09</li> <li>\u200b\u5973\u6027\u200b\u6574\u4f53\u200b\u7684\u200b\u5eb7\u590d\u200b\u7387\u200b\u66f4\u200b\u4f4e\u200b\uff08\\(Y\\not\\perp X\\)\uff09</li> </ol> <p>\u200b\u5728\u200b\u5b9e\u8bc1\u200b\u7814\u7a76\u200b\u4e2d\u200b\uff0c\u200b\u5f80\u5f80\u200b\u8fd8\u6709\u200b\u7814\u7a76\u8005\u200b\u672a\u80fd\u200b\u89c2\u6d4b\u200b\u5230\u200b\u7684\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u3002</p> <p>\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u65e2\u200b\u662f\u200b\u969c\u788d\u200b\uff0c\u200b\u4e5f\u200b\u662f\u200b\u5de5\u5177\u200b\u3002\u200b\u5728\u200b\u7406\u60f3\u5316\u200b\u7684\u200b\u7edf\u8ba1\u200b\u63a8\u65ad\u200b\u4e2d\u200b\uff0c\u200b\u76f8\u5173\u6027\u200b\u4e0d\u80fd\u200b\u63a8\u51fa\u200b\u56e0\u679c\u6027\u200b\u3002\u200b\u4f46\u200b\u5728\u200b\u5b9e\u8bc1\u200b\u7814\u7a76\u200b\u4e2d\u200b\uff0c\u200b\u7531\u4e8e\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u7684\u200b\u5b58\u5728\u200b\uff0c\u200b\u4f7f\u5f97\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u533a\u5206\u200b\u76f8\u5173\u6027\u200b\u4e2d\u200b\u7684\u200b\u56e0\u679c\u5173\u7cfb\u200b\u3002</p> <p>\u200b\u4e00\u4e9b\u200b\u8861\u91cf\u200b\u5e72\u9884\u200b\u6548\u679c\u200b\u7684\u200b\u6307\u6807\u200b\uff08\u200b\u53c2\u6570\u200b\uff09</p> <p>ATE</p> \\[ \\tau = \\bbE[Y(1) - Y(0)] \\] <p>\u200b\u88ab\u8bd5\u200b\u7684\u200b ATE</p> \\[ \\tau_1 = \\bbE[Y(1) - Y(0) | D = 1] \\] <p>\u200b\u6761\u4ef6\u200b ATE\uff08 CATE \uff09</p> \\[ \\tau_C = \\bbE[Y(1) - Y(0) | (D, X)\\in \\text{subgroup}] \\] <p>QTE\uff08 Quantile treatment effect \uff09</p> \\[ \\begin{gathered}     \\text{QTE} = F_1^{-1}(\\gamma) - F_0^{-1}(\\gamma) \\\\     F_d^{-1}(\\gamma) = \\inf\\{y|\\bbP(Y(d)\\leq y)\\geq \\gamma\\} \\end{gathered} \\] <p>Causal risk ratio\uff08\u200b\u53ea\u200b\u9002\u7528\u200b\u4e8e\u200b\u7ed3\u679c\u200b\u4e8c\u5206\u200b\u7684\u200b\u60c5\u51b5\u200b\uff09</p> \\[ \\text{CRR} = \\bbP(Y(1) = 1) / \\bbP(Y(0) = 1) \\] <p>\u200b\u4ee5\u4e0a\u200b\u8fd9\u4e9b\u200b\u53c2\u6570\u200b\u90fd\u200b\u53d7\u5230\u200b\u6df7\u6dc6\u200b\u53d8\u91cf\u200b\uff08\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u80fd\u200b\u88ab\u200b\u89c2\u6d4b\u200b\u5230\u200b\uff09\u200b\u7684\u200b\u5f71\u54cd\u200b\u3002\u200b\u76f8\u5bf9\u200b\u5e94\u200b\u7684\u200b\uff0c\u200b\u4ece\u200b\u5b9e\u9a8c\u200b\u7ed3\u679c\u200b\u7684\u200b\u7edf\u8ba1\u200b\u6307\u6807\u200b\u4e2d\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u8fd9\u4e9b\u200b\u6307\u6807\u200b\u7684\u200b\u4e00\u4e2a\u200b\u4f30\u8ba1\u200b\u3002</p> <p>\u200b\u5982\u200b\u5bf9\u4e8e\u200b ATE \u200b\u6307\u6807\u200b\\(\\tau = \\bbE[Y(1) - Y(0)]\\)\uff0c\\(\\tilde \\tau = \\bbE[Y | D = 1] - \\bbE[Y | D = 0]\\)\u200b\u5c31\u662f\u200b\\(\\tau\\)\u200b\u7684\u200b\u4e00\u4e2a\u200b\u4f30\u8ba1\u200b\u3002</p> <p>\\(\\tau\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\u56e0\u679c\u200b\u53c2\u6570\u200b\uff0c\u200b\u53d7\u5230\u200b\\(X, D, Y(0), Y(1)\\)\u200b\u7684\u200b\u5171\u540c\u200b\u5f71\u54cd\u200b\u3002\\(\\tilde\\tau\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\u53ef\u4ee5\u200b\u4ece\u200b\u7edf\u8ba1\u6570\u636e\u200b\u4e2d\u200b\u89c2\u6d4b\u200b\u5230\u200b\u7684\u200b\u6307\u6807\u200b\uff0c\u200b\u53d7\u5230\u200b\\(X, D, Y\\)\u200b\u7684\u200b\u5171\u540c\u200b\u5f71\u54cd\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u67d0\u4e2a\u200b\u53c2\u6570\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4ece\u200b\u7edf\u8ba1\u6570\u636e\u200b\u4e2d\u200b\u89c2\u6d4b\u200b\u5230\u200b\u4e00\u4e2a\u200b\u552f\u4e00\u200b\u7684\u200b\u65e0\u504f\u200b\u4f30\u8ba1\u200b\uff0c\u200b\u5219\u200b\u79f0\u4e3a\u200b identifiable \u3002\u200b\u7531\u4e8e\u200b\u53cd\u200b\u4e8b\u5b9e\u200b\u7684\u200b\u5b58\u5728\u200b\uff0c\u200b\u6211\u4eec\u200b\u53ea\u80fd\u200b\u89c2\u6d4b\u200b\u5230\u200b\\(Y = Y(D)\\)\uff0c\u200b\u56e0\u6b64\u200b\u5373\u4f7f\u200b\u6709\u200b\u65e0\u9650\u200b\u591a\u200b\u7684\u200b\u6570\u636e\u200b\u96c6\u200b\uff0c\u200b\u4e5f\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u80fd\u200b\u5b9e\u73b0\u200b identification \u3002\u200b\u5728\u200b unconfoundedness \u200b\u5047\u8bbe\u200b\u6210\u7acb\u200b\u7684\u200b\u6761\u4ef6\u200b\u4e0b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5b9e\u73b0\u200b identification \u3002</p> <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u4e3a\u4e86\u200b\u9a8c\u8bc1\u200b\u4e00\u4e2a\u200b\u56e0\u679c\u5173\u7cfb\u200b\u662f\u5426\u200b\u6210\u7acb\u200b\uff0c\u200b\u6211\u4eec\u200b\u9700\u8981\u200b\u4e24\u200b\u65b9\u9762\u200b\u7684\u200b\u4fe1\u606f\u200b</p> <ol> <li>\u200b\u5145\u8db3\u200b\u7684\u200b\u6570\u636e\u200b\u96c6\u4ee5\u200b\u53cd\u6620\u200b\\(X, D, Y\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u3002</li> <li>\u200b\u5173\u4e8e\u200b\\(X, D, Y(0), Y(1)\\)\u200b\u7684\u200b\u4e00\u4e9b\u200b\u5047\u8bbe\u200b\uff0c\u200b\u5982\u200b\u968f\u673a\u200b\u8bd5\u9a8c\u200b\u5047\u8bbe\u200b\u6216\u200b unconfoundedness \u200b\u5047\u8bbe\u200b\u3002</li> </ol>"},{"location":"math/causal-inference/causal-inference-stale/#causal-graph","title":"Causal Graph","text":"<p>\u200b\u4e00\u7cfb\u5217\u200b\u53d8\u91cf\u200b\u7684\u200b\u56e0\u679c\u5173\u7cfb\u200b\u53ef\u4ee5\u200b\u6784\u6210\u200b\u56e0\u679c\u200b\u56fe\u200b\u3002\u200b\u56e0\u679c\u200b\u56fe\u662f\u200b\u4e00\u4e2a\u200b\u6709\u200b\u5411\u200b\u65e0\u200b\u73af\u56fe\u200b\uff08 DAG \uff09\uff0c\u200b\u5176\u4e2d\u200b\u7684\u200b\u8282\u70b9\u200b\u8868\u793a\u200b\u53d8\u91cf\u200b\uff0c\u200b\u6bcf\u6761\u200b\u8fb9\u200b\u4ece\u7236\u200b\u53d8\u91cf\u200b\uff08\u200b\u8282\u70b9\u200b\uff09\u200b\u51fa\u53d1\u200b\uff0c\u200b\u6307\u5411\u200b\u5b50\u200b\u53d8\u91cf\u200b\uff08\u200b\u8282\u70b9\u200b\uff09\u3002</p> <ul> <li>\u200b\u76f8\u90bb\u200b\uff1a\u200b\u5982\u679c\u200b\u4e24\u4e2a\u200b\u8282\u70b9\u200b\u4e4b\u95f4\u200b\u6709\u200b\u4e00\u6761\u200b\u8fb9\u200b\uff0c\u200b\u79f0\u200b\u4e24\u4e2a\u200b\u8282\u70b9\u200b\u76f8\u90bb\u200b</li> <li>\uff08\u200b\u56e0\u679c\u200b\uff09\u200b\u8def\u5f84\u200b\uff1a\u200b\u4e00\u7cfb\u5217\u200b\u76f8\u90bb\u200b\u7684\u200b\u8282\u70b9\u200b\u7ec4\u6210\u200b\u4e00\u6761\u200b\u8def\u5f84\u200b\uff08\u200b\u65e0\u89c6\u200b\u8fb9\u200b\u7684\u200b\u65b9\u5411\u200b\uff09</li> <li>\u200b\u6709\u200b\u5411\u200b\u8def\u5f84\u200b\uff1a\u200b\u65b9\u5411\u200b\u76f8\u540c\u200b\u7684\u200b\u4e00\u7cfb\u5217\u200b\u8def\u5f84\u200b\u6240\u200b\u8fde\u63a5\u200b\u7684\u200b\u8282\u70b9\u200b\u5f62\u6210\u200b\u7684\u200b\u8def\u5f84\u200b</li> </ul> <p>\u200b\u8def\u5f84\u200b\u6709\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\uff1a</p> <ul> <li>\u200b\u94fe\u200b\uff1a\\(A\\ra B\\ra C\\)</li> <li>\u200b\u6c47\u805a\u200b\uff1a\\(A\\ra B\\la C\\)</li> <li>\u200b\u5206\u652f\u200b\uff1a\\(A\\la B\\ra C\\)</li> </ul> <p>\u200b\u5728\u200b\u56e0\u679c\u200b\u56fe\u4e2d\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u8282\u70b9\u200b\u90fd\u200b\u662f\u200b\u5176\u5b50\u200b\u8282\u70b9\u200b\uff08\u200b\u540e\u4ee3\u200b\uff09\u200b\u7684\u200b\u76f4\u63a5\u200b\u539f\u56e0\u200b\uff08\u200b\u7956\u5148\u200b\uff09\uff08\u200b\u5373\u200b\uff0c\u200b\u4e0d\u200b\u5b58\u5728\u200b\u4e2d\u95f4\u200b\u7684\u200b\u53d8\u91cf\u200b\u5f71\u54cd\u200b\u5b50\u200b\u8282\u70b9\u200b\uff09\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u5982\u4e0b\u200b\u56fe\u200b\u6240\u793a\u200b\u7684\u200b\u56e0\u679c\u200b\u56fe\u200b</p> <p></p> <ol> <li>\\(X_1\\)\u200b\u5f71\u54cd\u200b\u53d8\u91cf\u200b\\(X_4\\)\uff0c\u200b\u4f46\u200b\u4e0d\u662f\u200b\u76f4\u63a5\u200b\u5f71\u54cd\u200b\u3002</li> <li> <p>\u200b\u5982\u679c\u200b\u56fa\u5b9a\u200b\\(X_3\\)\uff0c\u200b\u5219\u200b\\(X_1\\)\u200b\u7684\u200b\u53d8\u5316\u200b\u4e0d\u4f1a\u200b\u5f71\u54cd\u200b\\(X_4\\)\uff0c\u200b\u5373\u200b</p> \\[ X_4\\perp (X_1, X_2) | X_3 \\] </li> <li> <p>\u200b\u4f46\u200b\\(X_4\\)\u200b\u4ecd\u7136\u200b\u4e0e\u200b\\(X_5\\)\u200b\u6709\u5173\u200b\u3002</p> </li> </ol> <p>\u200b\u8fdb\u4e00\u6b65\u200b\u5730\u200b\uff0c\u200b\u6211\u4eec\u200b\u4e0d\u59a8\u200b\u5047\u8bbe\u200b\uff0c\u200b\u5982\u679c\u200b\u63a7\u5236\u200b\u4e86\u200b\u67d0\u4e2a\u200b\u53d8\u91cf\u200b\\(X_i\\)\u200b\u5168\u90e8\u200b\u7684\u200b\u76f4\u63a5\u200b\u7956\u5148\u200b\uff0c\u200b\u5219\u200b\\(X_i\\)\u200b\u53ea\u200b\u4e0e\u5176\u200b\u540e\u4ee3\u200b\u6709\u5173\u200b\uff08\u200b\u56e0\u679c\u200bMarkov\u200b\u5047\u8bbe\u200b\uff09\u3002</p> <p>Markov\u200b\u56e0\u5b50\u200b\u5206\u89e3\u200b</p> <p>\u200b\u5f53\u200bMarkov\u200b\u5047\u8bbe\u200b\u6210\u7acb\u200b\u65f6\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\\(n\\)\u200b\u4e2a\u200b\u53d8\u91cf\u200b\\(X_1, \\ldots, X_n\\)\uff0c\u200b\u6709\u200b</p> \\[ \\bbP(X_1, \\ldots, X_n) = \\prod_{i=1}^n \\bbP(X_i | A_i) \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\uff0c\\(A_i\\)\u200b\u4e3a\u200b\\(X_i\\)\u200b\u5168\u90e8\u200b\u7684\u200b\u7956\u5148\u200b\u3002</p>"},{"location":"math/causal-inference/causal-inference-stale/#d-","title":"d-\u200b\u5206\u5272","text":"<p>\u200b\u5bf9\u4e8e\u200b\u6709\u200b\u4e24\u4e2a\u200b\u8282\u70b9\u200b\u7684\u200b\u56e0\u679c\u200b\u56fe\u200b\uff0c\\(X_1\\la X_2\\)\u200b\u548c\u200b\\(X_1\\ra X_2\\)\u200b\u662f\u200b\u4e24\u4e2a\u200b\u4e0d\u540c\u200b\u7684\u200b\u56e0\u679c\u5173\u7cfb\u200b\uff0c\u200b\u4f46\u200b\u5728\u200b\u76f8\u5173\u6027\u200b\u4e0a\u200b\u6709\u200b\u76f8\u540c\u200b\u7684\u200b\u8868\u73b0\u200b\uff0c\u200b\u65e0\u6cd5\u200b\u533a\u5206\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u6709\u200b\u4e09\u4e2a\u200b\u8282\u70b9\u200b\u7684\u200b\u56e0\u679c\u200b\u56fe\u200b\uff1a</p> <ul> <li>\\(X_1\\ra X_2\\ra X_3\\)\u200b\u548c\u200b\\(X_1\\la X_2\\ra X_3\\)\u200b\u662f\u200b\u4e24\u4e2a\u200b\u4e0d\u540c\u200b\u7684\u200b\u56e0\u679c\u5173\u7cfb\u200b\uff0c\u200b\u4f46\u200b\u5b83\u4eec\u200b\u90fd\u200b\u6ee1\u8db3\u200b\\(X_1\\perp X_3 | X_2, X_1\\not\\perp X_3\\)\uff0c\u200b\u4ecd\u7136\u200b\u65e0\u6cd5\u200b\u533a\u5206\u200b\u3002\u200b\u5e76\u4e14\u200b\u7531\u4e8e\u200b\u53d8\u91cf\u200b\\(X_2\\)\u200b\u7684\u200b\u5b58\u5728\u200b\uff0c\u200b\u5373\u4f7f\u200b\\(X_1, X_3\\)\u200b\u5b58\u5728\u200b\u76f8\u5173\u200b\u5173\u7cfb\u200b\uff0c\u200b\u4e5f\u200b\u4e0d\u80fd\u200b\u8bf4\u660e\u200b\u5176\u200b\u5b58\u5728\u200b\u76f4\u63a5\u200b\u6216\u200b\u95f4\u63a5\u200b\u7684\u200b\u56e0\u679c\u5173\u7cfb\u200b\u3002\u200b\u63a7\u5236\u200b\\(X_2\\)\u200b\u53ef\u4ee5\u200b\u963b\u65ad\u200b\\(X_1, X_3\\)\u200b\u7684\u200b\u56e0\u679c\u200b\u94fe\u200b\u3002</li> <li>\u200b\u5bf9\u4e8e\u200b\\(\\X_1\\ra X_2\\la X_3\\)\uff0c\u200b\u5982\u679c\u200b\u63a7\u5236\u200b\\(X_2\\)\uff0c\u200b\u5219\u200b\u6709\u200b\\(X_1\\not \\perp X_3 | X_2\\)\uff0c\u200b\u4f46\u200b\\(X_1\\perp X_3\\)\u3002\\(X_2\\)\u200b\u963b\u65ad\u200b\u4e86\u200b\\(X_1, X_3\\)\u200b\u7684\u200b\u56e0\u679c\u200b\u94fe\u200b\u3002<ul> <li>\u200b\u5982\u679c\u200b\u6709\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\\(X_4\\)\u200b\u6ee1\u8db3\u200b\\(X_2\\ra X_4\\)\uff0c\u200b\u5219\u200b\u63a7\u5236\u200b\\(X_4\\)\u200b\u4e5f\u200b\u4f1a\u200b\u4f7f\u200b\\(X_1, X_3\\)\u200b\u8868\u73b0\u200b\u51fa\u200b\u76f8\u5173\u200b\uff0c\u200b\u5373\u200b\\(X_1\\not \\perp X_3 | X_4\\)\u3002</li> </ul> </li> </ul> <p>\u200b\u963b\u65ad\u200b\u56e0\u679c\u200b\u94fe\u200b</p> <p>\u200b\u4e24\u4e2a\u200b\u53d8\u91cf\u200b\u4e4b\u95f4\u200b\u7684\u200b\u8def\u5f84\u200b\u5982\u679c\u200b\u6ee1\u8db3\u200b\u4e0b\u5217\u200b\u4e4b\u4e00\u200b\uff0c\u200b\u5219\u200b\u56e0\u679c\u5173\u7cfb\u200b\u4e0d\u80fd\u200b\u901a\u8fc7\u200b\u8fd9\u200b\u6761\u200b\u8def\u5f84\u200b\uff08\u200b\u88ab\u200b\u963b\u65ad\u200b\uff09\u3002</p> <ol> <li>\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u6ca1\u6709\u200b\u88ab\u200b\u63a7\u5236\u200b\u7684\u200b\u6c47\u805a\u200b\u8282\u70b9\u200b\uff1b</li> <li>\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u88ab\u200b\u63a7\u5236\u200b\u7684\u200b\u975e\u200b\u6c47\u805a\u200b\u8282\u70b9\u200b\u3002</li> </ol> <p>\u200b\u5982\u679c\u200b\u4e24\u4e2a\u200b\u53d8\u91cf\u200b\u4e4b\u95f4\u200b\u7684\u200b\u6240\u6709\u200b\u8def\u5f84\u200b\u90fd\u200b\u88ab\u200b\u963b\u65ad\u200b\uff0c\u200b\u5219\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u53d8\u91cf\u200b\u88ab\u200b\u5206\u5272\u200b\u3002</p> <p>\u200b\u4e00\u822c\u200b\u5b9a\u4e49\u200b\u88ab\u200b\u63a7\u5236\u200b\u7684\u200b\u53d8\u91cf\u200b\u96c6\u5408\u200b\u4e3a\u200b\\(Z\\)\u3002\u200b\u5982\u679c\u200b\u56e0\u679c\u200b\u56fe\u200b\\(\\calG\\)\u200b\u4e2d\u200b\uff0c\u200b\u63a7\u5236\u53d8\u91cf\u200b\\(Z\\)\u200b\u5c06\u200b\u56e0\u679c\u200b\u56fe\u200b\u5206\u4e3a\u200b\u4e86\u200b\u4e24\u4e2a\u200b\u90e8\u5206\u200b\uff0c\u200b\u5219\u200b\u8fd9\u200b\u4e24\u4e2a\u200b\u90e8\u5206\u200b\\(X, Y\\)\u200b\u6784\u6210\u200b\u4e86\u200b\u56e0\u679c\u200b\u56fe\u200b\u7684\u200b\u4e00\u4e2a\u200bd-\u200b\u5206\u5272\u200b\uff0c\u200b\u5373\u200b\\(X\\perp_\\calG Y | Z\\)\u3002\u200b\u8fdb\u4e00\u6b65\u200b\u5730\u200b\uff0c\u200b\u5982\u679c\u200b\u6ee1\u8db3\u200b Markov \u200b\u5047\u8bbe\u200b\uff0c\u200b\u5219\u200b\\(X\\)\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u53d8\u91cf\u200b\u548c\u200b\\(Y\\)\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u53d8\u91cf\u200b\u72ec\u7acb\u200b\u3002</p> <p>\u200b\u4f46\u662f\u200b\u53d8\u91cf\u200b\u72ec\u7acb\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u80fd\u200b\u5bfc\u51fa\u200bd-\u200b\u5206\u5272\u200b\uff0c\u200b\u7531\u6b64\u200b\u5f97\u5230\u200b faithfulness assumption</p> <p>faithfulness assumption</p> <p>\u200b\u5bf9\u4e8e\u200b\u56e0\u679c\u200b\u56fe\u200b\\(\\calG\\)\uff0c\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\u4e92\u4e0d\u200b\u76f8\u4ea4\u200b\u7684\u200b\\(X, Y, Z\\)\u200b\u4e14\u200b\\(X\\perp Y | Z\\)\uff0c\u200b\u5219\u200b\\(X, Y\\)\u200b\u6784\u6210\u200b\u4e00\u4e2a\u200bd-\u200b\u5206\u5272\u200b\u3002</p> <p>\u200b\u5728\u200b Markov \u200b\u5047\u8bbe\u200b\u548c\u200b faithfulness \u200b\u5047\u8bbe\u200b\u6210\u7acb\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0cd-\u200b\u5206\u5272\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u53d8\u91cf\u200b\u72ec\u7acb\u200b\u3002</p>"},{"location":"math/causal-inference/causal-inference-stale/#_5","title":"\u76f8\u5173\u6027\u200b\u4e0e\u200b\u56e0\u679c\u6027\u200b\u7684\u200b\u8def\u5f84","text":"<p>\u200b\u5728\u200b\u56e0\u679c\u200b\u56fe\u4e0a\u200b\uff1a</p> <ul> <li>\u200b\u76f8\u5173\u6027\u200b\u6cbf\u200b\u8def\u5f84\u200b\u6d41\u52a8\u200b</li> <li>\u200b\u56e0\u679c\u6027\u200b\u6cbf\u6709\u200b\u5411\u200b\u8def\u5f84\u200b\u6d41\u52a8\u200b</li> </ul> <p>\u200b\u5982\u679c\u200b\u963b\u65ad\u200b\u4e86\u200b\u6240\u6709\u200b\u975e\u200b\u56e0\u679c\u6027\u200b\u7684\u200b\u8def\u5f84\u200b\uff0c\u200b\u5219\u200b\u4e24\u4e2a\u200b\u53d8\u91cf\u200b\u4e4b\u95f4\u200b\u5b58\u5728\u200b\u56e0\u679c\u5173\u7cfb\u200b\u3002</p> <p>do\u200b\u7b97\u5b50\u200b</p> <p>\u200b\u5728\u200b\u56e0\u679c\u200b\u56fe\u4e2d\u200b\uff0c\u200b\u5bf9\u200b\u4e00\u4e2a\u200b\u53d8\u91cf\u200b\u7684\u200b\u5e72\u9884\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u963b\u65ad\u200b\u6240\u6709\u200b\u6307\u5411\u200b\u8be5\u200b\u53d8\u91cf\u200b\u7684\u200b\u8fb9\u200b\u5b9e\u73b0\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(\\mathrm {do}(\\cdot)\\)</p> <p>\u200b\u975e\u200b\u56e0\u679c\u200b\u7684\u200b\u8def\u5f84\u200b\u6709\u200b\u4e24\u4e2a\u200b\u6765\u6e90\u200b\uff1a</p> <ul> <li>\u200b\u6ca1\u6709\u200b\u63a7\u5236\u200b\u5206\u652f\u200b\u53d8\u91cf\u200b\uff0c\u200b\u5373\u200b confounding bias</li> <li>\u200b\u63a7\u5236\u200b\u4e86\u200b\u6c47\u805a\u200b\u53d8\u91cf\u200b\uff0c\u200b\u5373\u200b selection bias</li> </ul>"},{"location":"math/causal-inference/causal-inference-stale/#confounding","title":"Confounding \u200b\u4e0e\u200b\u540e\u95e8\u200b\u8c03\u6574","text":"<p>Confounding \u200b\u6307\u200b\u56e0\u679c\u200b\u8def\u5f84\u200b\u7684\u200b\u8d77\u59cb\u200b\u548c\u200b\u7ec8\u7ed3\u200b\u53d8\u91cf\u200b\u90fd\u200b\u53d7\u5230\u200b\u540c\u4e00\u4e2a\u200b\u6df7\u6dc6\u200b\u56e0\u5b50\u200b\u7684\u200b\u5f71\u54cd\u200b\u3002\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u540e\u95e8\u200b\u8c03\u6574\u200b\u6d88\u9664\u200b\u5f71\u54cd\u200b\u3002</p>"},{"location":"math/causal-inference/chapter-1/","title":"A Definition of Causal Effect","text":"<p>Key concepts: treatment, outcome, average causal effect, potential outcome, consistency, causation-association difference, identifiability</p>"},{"location":"math/causal-inference/chapter-1/#treatment-and-outcome","title":"Treatment and Outcome","text":"<p>We want to compare the result when a specified action is or is not applied to an individual. The action is called a treatment, and the result is called an outcome, which are denoted by \\(A\\) and \\(Y\\) respectively. Usually we take \\(A\\) and \\(Y\\) to be both binary.</p> <p>The outcome under a specified treatment \\(A = a\\) is denoted by \\(Y^{A = a}\\). In reality, we can only observe one value of \\(Y^{A = 1}\\) or \\(Y^{A = 0}\\) for each individual. Each \\(Y^{A = a}\\) is called a potential outcome. If \\(A = a\\) is observed, \\(Y^{A = a}\\) is factual, and \\(Y^{A = 1 - a}\\) is counterfactual. The probability of potential outcome \\(Y = 1\\) is called risk.</p> <p>Consistency assumption</p> <p>If \\(A = a\\) is observed, then \\(Y = Y^{A = a}\\).</p>"},{"location":"math/causal-inference/chapter-1/#causal-effects","title":"Causal Effects","text":"<p>Identifiability</p> <p>A quantity is identifiable if it can be expressed (unbiasedly) as a function of the distribution of the observed data.</p> <p>Before we clarify the definition of causal effect, we need to make some assumptions on individual outcome.</p> <p>No interference assumption</p> <p>The potential outcome under \\(A = a\\) does not depend on the treatment of other individuals (i.e. no interference).</p>"},{"location":"math/causal-inference/chapter-1/#individual-causal-effect","title":"Individual Causal Effect","text":"<p>If for individual \\(i\\), \\(Y_i^{A = 1} \\not = Y_i^{A = 0}\\), then we say that the treatment has a causal effect for individual \\(i\\).</p>"},{"location":"math/causal-inference/chapter-1/#average-causal-effect","title":"Average Causal Effect","text":"<p>The average causal effect is defined as</p> \\[ P(Y^{A = 1} = 1) - P(Y^{A = 0} = 1) \\] <p>or</p> \\[ E(Y^{A = 1}) - E(Y^{A = 0}) \\] <p>The two forms of average causal effect are equivalent. We can test average causal effect by testing the causal null hypothesis \\(H_0: P(Y^{A = 1} = 1) = P(Y^{A = 0} = 1)\\). However, the absence of average causal effect does not imply the absence of individual causal effect. We call the null hypothesis \\(H_0: \\forall i, Y_i^{a = 1} = Y_i^{a = 0}\\) as sharp causal null hypothesis.</p>"},{"location":"math/causal-inference/chapter-1/#functional-causal-effect","title":"Functional Causal Effect","text":"<p>We can measure the outcome \\(Y\\) over a function \\(f(y)\\). For example \\(V(Y^{a = 1}) - V(Y^{a = 0})\\) as the causal effect on sample variance. Notice that \\(V(Y^{a = 1}) - V(Y^{a = 0})\\) usually does not equal to \\(V(Y^{a = 1} - Y^{a = 0})\\).</p>"},{"location":"math/causal-inference/chapter-1/#measuring-causal-effects","title":"Measuring Causal Effects","text":"<p>The causal null hypothesis can be written as</p> <ol> <li>Difference: \\(H_0: P(Y^{a = 1} = 1) - P(Y^{a = 0} = 1) = 0\\)</li> <li>Ratio: \\(H_0: \\frac{P(Y^{a = 1} = 1)}{P(Y^{a = 0} = 1)} = 1\\)</li> <li>Odds ratio: \\(H_0: \\frac{P(Y^{a = 1} = 1) / P(Y^{a = 1} = 0)}{P(Y^{a = 0} = 1) / P(Y^{a = 0} = 0)} = 1\\)</li> </ol> <p>When \\(H_0\\) is rejected, we can use number needed to treat (NNT) to measure the strength of causal effect. NNT is defined as the number of individuals needed to be treated to prevent one additional bad outcome. NNT is given by:</p> \\[ \\text{NNT} = \\frac{-1}{P(Y^{a = 1} = 1) - P(Y^{a = 0} = 1)} \\]"},{"location":"math/causal-inference/chapter-1/#random-variability","title":"Random Variability","text":"<p>In reality, we can only collect a sample of individuals in the population, and our estimate on causal effect suffers from random variability.</p> <ol> <li>Sampling variablility: We use \\(\\hat P(Y^{a = 1} = 1) = N(Y^{a = 1}) / N(Y)\\) as an estimator of \\(P(Y^{a = 1} = 1)\\). The estimator suffers from random variability.</li> <li>Potential outcomes are usually not deterministic, which makes another source of random variability.</li> </ol>"},{"location":"math/causal-inference/chapter-1/#causation-and-association","title":"Causation and Association","text":"<p>In the real world, we can only observe one potential outcome for each individual. Therefore, we cannot directly measure the outcome \\(P(Y^{a = 1} = 1)\\) and \\(P(Y^{a = 0} = 1)\\). Instead, we can only use \\(P(Y = 1 | A = 1)\\) and \\(P(Y = 1 | A = 0)\\) as estimators.</p> <p>Like the causal null hypothesis, we can also define the independent (\\(A\\perp Y\\)) null hypothesis in three forms:</p> <ol> <li>Risk difference: \\(H_0: P(Y = 1 | A = 1) - P(Y = 1 | A = 0) = 0\\)</li> <li>Risk ratio: \\(H_0: \\frac{P(Y = 1 | A = 1)}{P(Y = 1 | A = 0)} = 1\\)</li> <li>Odds ratio: \\(H_0: \\frac{P(Y = 1 | A = 1) / P(Y = 0 | A = 1)}{P(Y = 1 | A = 0) / P(Y = 0 | A = 0)} = 1\\)</li> </ol> <p>Notice that any terms in the form of conditional probability or expectation are associations. Only marginal probability or expectation can be causal effects, since the marginal probability takes all other factors into consideration.</p> <p></p>"},{"location":"math/causal-inference/chapter-2/","title":"Randomized Experiments","text":"<p>Key concepts: random experiment, exchangeability, effect modification</p>"},{"location":"math/causal-inference/chapter-2/#randomized-experiments_1","title":"Randomized Experiments","text":"<p>Consider the following treatment assignment:</p> <p></p> <p>Assume that the treatment assignment is determined by a random procedure, the experiment is called a random experiment. Then exchanging the individual in treated group with those in control group will not affect the association measure.</p> <p>If a random experiment satisfies the following conditions, it is called an ideal randomized experiment:</p> <ol> <li>Full adherence: All individuals follow the treatment assignment.</li> <li>No loss to follow-up: All individuals are followed up and their outcomes are observed.</li> <li>Single treatment: Each individual is assigned to one treatment only.</li> <li>Double-blind assignment: The treatment assignment is unknown to both the individuals and the researchers.</li> </ol> <p>If all the possible values of treatment variable \\(A = a\\) are sequentially assigned to individuals, the experiment is called crossover experiment. Individual causal effects can be estimated using crossover experiments if the following conditions are met:</p> <ol> <li>No carryover effect: The treatment effect of \\(a\\) does not depend on the previous treatments.</li> <li>The individual causal effect \\(\\alpha_i = Y^{a_t = 1}_{it} - Y^{a_t = 0}_{it}\\) does not depend on time.</li> <li>The potential outcome under no treatment does not depend on time.</li> </ol> <p>By assigning the treatment value sequence randomly, we can remove the required conditiona 3. In such case:</p> \\[ (Y_{i1} - Y_{i0})A_{i1} - (Y_{i0} - Y_{i1})A_{i0} \\] <p>estimates the individual causal effect \\(E(\\alpha_i)\\) under condition 2. If condition 2 is not met, the estimator estimates \\(E(\\alpha_{i0} + \\alpha_{i1}) / 2\\).</p>"},{"location":"math/causal-inference/chapter-2/#exchangeability","title":"Exchangeability","text":"<p>Exchangeability</p> <ul> <li>(Full) exchangeability indicates that swapping the treatment assignment of two individuals does not affect the association measure, or \\(Y^a \\perp A\\).</li> <li>Mean exchangeability indicates that \\(\\forall a', E(Y^a | A = a') = E(Y^a)\\), or the expectation of switched potential outcome is the same as the original one.</li> </ul> <p>When \\(Y\\) and \\(A\\) are binary, full exchangeability is equivalent to mean exchangeability. Mean exchangeability is sufficient for deriving \\(E(Y^a) = E(Y | A = a)\\).</p> <p>Exchangeability indicates that all the potential outcome and the treatment assignment are independent, i.e. \\(\\forall a, Y^{A = a} \\perp A\\), or the treatment assignment is exogenous. Under exchangeability, \\(P(Y = 1 | A = a)\\) is an unbiased estimator of \\(P(Y^{a} = 1)\\).</p> <p>\\(Y^a\\perp A\\) and \\(Y\\perp A\\)</p> <p>Notice that \\(Y^a \\perp A\\) is not equivalent to \\(Y \\perp A\\). The latter indicates that treatment \\(A\\) does not have any causal effect on the outcome \\(Y\\). If \\(\\forall a, E(Y^a) = E(Y)\\), then \\(Y\\perp A\\).</p> <p>Notice that a randomized experiment could be unexchangeable even if infinite samples are collected.</p>"},{"location":"math/causal-inference/chapter-2/#conditional-randomization","title":"Conditional Randomization","text":"<p>Consider an additional factor \\(L\\) affects the assignment of treatments, i.e., when \\(L\\) takes different values, the distribution of \\(A\\) is different. Such a randomized experiment is called conditional randomized experiment. In a conditional randomized experiment, the corresponding randomized experiment using the overall distribution of \\(A\\) is called marginally randomized experiment.</p> <p>Although the marginally randomized experiment is exchangeable, the conditional randomized experiment is not exchangeable in general. Only conditional exchangeability holds in such randomized experiments, i.e. \\(\\forall a, Y^a \\perp A | L\\).</p> <p>We can measure the causal effects on sub-groups \\(P(Y^{a = 1} = 1 | L = l) - P(Y^{a = 0} = 1 | L = l)\\) using conditional exchangeability (stratification) or directly measure the overall causal effect \\(P(Y^{a = 1} = 1) - P(Y^{a = 0} = 1)\\) on the population using marginal exchangeability. For different sub-groups, the causal effects may be different, which is called effect modification or treatment effect heterogeneity.</p> <p>Stratification</p> <p>Stratification is to split the population into mutually exclusive sub-groups in which the treatment assignment is exchangeable. The causal effect can be estimated in each sub-group.</p>"},{"location":"math/causal-inference/chapter-2/#standardization","title":"Standardization","text":"<p>We can view the marginal risk as a weighted average of the conditional risks:</p> \\[ P(Y^a = 1) = \\sum_{l\\in \\calL} P(Y^a = 1 | L = l) P(L = l) \\] <p>With conditional exchangeability holds, we can derive that \\(P(Y^a = 1) = \\sum_{l\\in \\calL} P(Y = 1 | A = a, L = l) P(L = l)\\), which is called standardization, making the unobserved potential outcome \\(P(Y^a = 1)\\) to be identifiable.</p>"},{"location":"math/causal-inference/chapter-2/#inverse-probability-weighting","title":"Inverse Probability Weighting","text":"<p>Notice that</p> \\[ P(A = a | L = l) = \\frac{P(A = a, L = l)}{P(L = l)} \\] <p>We can assign the weight \\(w_l = \\frac{1}{P(A = a | L = l)}\\) to each individual with \\(L = l\\) to construct a pseudo-population. The pseudo-population satisfies exchangeability, and we can use the pseudo-population to estimate the causal effect. The IP weighted mean of outcome \\(Y\\) is \\(E\\left( I(A = a) Y / P(A = a | L)\\right)\\), where \\(I(A = a)\\) is the indicator function.</p> <p>Assuming that \\(\\forall l, P(A = a | L = l) &gt; 0\\), we can derive that</p> \\[ \\begin{aligned}     &amp; E\\left( \\frac{I(A = a) Y}{P(A = a | L)}\\right) \\\\     =&amp; \\sum_{l} \\frac{E\\left( I(A = a) Y | L = l\\right) P(L = l)}{P(A = a | L = l)} \\\\     =&amp; \\sum_{l} \\frac{E\\left( Y | A = a, L = l\\right) P(A = a | L = l)P(L = l)}{P(A = a | L = l)} \\\\     =&amp; \\sum_{l} E\\left( Y | A = a, L = l\\right) P(L = l) \\\\     =&amp; \\sum_{l} E(Y^a | L = l) P(L = l) \\\\     =&amp; E(Y^a) \\end{aligned} \\] <p>Therefore, under positivity assumption, IP weighting is equivalent to standardization.</p> <p>Positivity</p> <p>For any individual in any sub-group \\(l\\) has a non-zero probability of receiving treatment \\(a\\), i.e. \\(\\forall l, \\forall a, P(A = a | L = l) &gt; 0\\).</p>"},{"location":"math/causal-inference/chapter-3/","title":"Observational Studies","text":"<p>In an observational study, the variables are neither assigned nor controlled by the researcher. The researcher only observes the values of the variables. Compared with randomized experiments, observational studies are more limited to the researchers.</p>"},{"location":"math/causal-inference/chapter-3/#identifiability-conditions","title":"Identifiability Conditions","text":"<p>If the following identifiability conditions are met, the observational study can be regarded as a conditionally randomized experiment.</p> <p>Identifiability condition</p> <ol> <li>Consistency: The values of treatment under comparison correspond to well-defined interventions that, in turn, correspond to the versions of treatment in the data.</li> <li>Exchangeability: The conditional probability of receiving any value of treatment depends only on observed covariates \\(L\\).</li> <li>Positivity: The probability of receiving any value of treatment is strictly positive for all combinations of covariates \\(L\\).</li> </ol> <p>If the identifiability conditions are not met, we can use other methods including instrumental variable to establish another group of identifiability condition.</p>"},{"location":"math/causal-inference/chapter-3/#exchangeability","title":"Exchangeability","text":"<p>In observational studies, the exchangeablitiy condition may or may not hold, as the observed \\(L\\) may not be the only predictor of the treatment. The true predictor may even be unobserved or unobservable.</p> <p>Exchangeability is given by</p> \\[ Y^a \\perp A \\mid L \\Longleftrightarrow \\forall a, P(Y^a  = 1 \\mid A = a, L = l) = P(Y^a = 1\\mid A \\not = a, L = l) \\] <p>However, in observational studies, \\(P(Y^a = 1\\mid A\\not = a, L = l)\\) is unidentifiable. Thus, the exchangeablitiy cannot be empirically verified.</p>"},{"location":"math/causal-inference/chapter-3/#positivity","title":"Positivity","text":"<p>To measure causal effects, we need to measure the distribution of outcome under treatment and under no treatment. That is, in the whole sample or any observed subgroup, the probability of receiving any value of treatment should be strictly positive, which is called positivity. Notice that positivity is only required for these observed covariates that requires exchangeability.</p> <p>Standardization and IP weighting are meaningful only if positivity holds.</p> <p>IP weighting without positivity</p> <p>If positivity does not hold, IP weighting is undefined since the denominator in</p> \\[ E\\left[\\frac{I(A = a)Y}{P(A = a | L)}\\right] \\] <p>is zero. However, the IP weighted mean is still defined.</p> \\[ E\\left[\\frac{I(A = a)Y}{N(A = a, L) / N(L)}\\right] \\] <p>Notice that \\(N(A = a, L) / N(L)\\) is defined on observed data. Let \\(Q(a) = \\{l | N(A = a, L) / N(L) &gt; 0\\}\\). Now, \\(E\\left[\\frac{I(A = a)Y}{N(a, L) / N(L)}\\right] = E[Y^a | L \\in Q(a)]P(L\\in Q(a))\\) is a biased estimator of \\(E[Y^a]\\).</p> <p>When positivity does not hold, then \\(Q(0) \\not = Q(1)\\), under this condition, the difference between IP weighted means (\\(E\\left[\\frac{I(A = 1)Y}{N(A = 1, L) / N(L)}\\right] - E\\left[\\frac{I(A = 0)Y}{N(A = 0, L) / N(L)}\\right]\\)) cannot be interpreted causally.</p>"},{"location":"math/causal-inference/chapter-3/#consistency","title":"Consistency","text":"<p>Consider the following examples in heart transplant studies:</p> <ol> <li>Two patients received heart transplants performed by different surgeons.</li> <li>Two patients received heart transplants performed under different procedures.</li> </ol> <p>Due to the complexity of real-world situations, we always need to aggregate the different treatments into a single treatment value. The map between discrete values of the treatment variable and real-world intervention is called treatment protocol. The first step for investigating the causal effect is to precisely specify the treatment protocol, so that any variation regarded as treatment will lead to the same outcome, or treatment variation irrelevance. Under this condition can we define the potential outcome under the specified treatment value. In addition, the treatment protocol of interest and the treatment protocol of the observed data should be the same.</p> <p>Under the treatment variation irrelevance, we can define the potential outcome \\(Y^a\\).</p> <p>Treatment variation irrelevance</p> <p>Suppose that receiving heart transplant is denoted as \\(a\\), the heart transplant is performed by Alex and Bob. But the treatment protocol is observed as the heart transplant is performed by Alex. Then, the treatment variation irrelevance is violated and consistency assumption \\(Y = Y^{a = 1}\\) does not hold.</p>"},{"location":"math/causal-inference/chapter-3/#target-trial","title":"Target Trial","text":"<p>Under the identifiability conditions, we can regard the observational study as a conditionally randomized experiment. The equivalent hypothetical, randomized experiment is called target trial.</p>"},{"location":"math/causal-inference/chapter-4/","title":"Effect Modification","text":"<p>A null average causal effect does not imply a null average causal effect for a subset of the population. A variable \\(V\\) is an effect modifier if the average causal effect of \\(A\\) on \\(Y\\) differs across levels of \\(V\\).</p> <p>An effect modifier does need to directly have causal relationship with the different causal effect. If the difference in causal effects can be interpreted causally, the variable is a causal effect modifier, otherwise it is a surrogate effect modifier.</p> <p>Effects in the treated</p> <p>Notice the treatment variable itself can be regarded as a subgroup. The average causal effect in the treated is measured on subgroup \\(A = 1\\).</p> \\[ P(Y^{a = 1} = 1 | A = 1) - P(Y^{a = 0} = 1 | A = 1) \\] <ol> <li>If the distribution of individual causal effects varies between the treated and untreated, the average causal effect in the treated is not equal to the average causal effect in the population.</li> <li>\\(P(Y^{a = 0} = 1 | A = 1)\\) is unidentifiable. Estimating \\(P(Y^{a = 0} = 1 | A = 1)\\) requires partial exchangeability \\(Y^{a = 0} \\perp A \\mid L\\).</li> <li>Following the definition of effects in the treated, we can define effects in the untreated, which is defined on subgroup \\(a = 0\\). Effects in the untreated requires partial exchangeability \\(Y^{a = 1} \\perp A \\mid L\\).</li> </ol> <p>The counterfactual \\(P(Y^{a} = 1 | A = a')\\) can be estimated using standardization or IP weighting.</p> <ol> <li>Standardization: \\(P(Y^{a} = 1 | A = a') = \\sum_{l} P(Y = 1 | A = a, L = l) P(L = l | A = a')\\)</li> <li>IP weighting: \\(P(Y^{a} = 1 | A = a') = E\\left[\\frac{I(A = a)Y}{f(A | L)P(A = a' | L)}\\right]/E\\left[\\frac{I(A = a)}{f(A | L)P(A = a' | L)}\\right]\\)</li> </ol> <p>The extrapolation of computed causal effects across populations are called transportability. Transportability is an unverifiable assumption. If the effect modification measure, version of treatment and the in-group interference are similar, we can assume transportability.</p> <p>From effect modification, we can identify the subgroup that benefits most from the treatment or intervention. Moreover, effect modification helps us to learn underlying mechanisms of the outcome.</p>"},{"location":"math/causal-inference/chapter-4/#effect-modification-vs-prognostic-factor","title":"Effect Modification v.s. Prognostic Factor","text":"<p>When marginal randomization does not hold, there exists a prognostic factor \\(L\\) is a variable that affect the treatment assignment, while an effect modifier \\(V\\) affects the effect. To estimate causal effects in such conditions, extra adjustments are required</p>"},{"location":"math/causal-inference/chapter-4/#stratification","title":"Stratification","text":"<p>Stratification can be used to adjust the marginal causal effect of \\(A\\) on \\(V\\). However, it requires to take all the combinations of \\(V\\) and \\(L\\) into account. And stratification requires positivity condition to be met on \\(L\\).</p> <p>Using stratification, we can yield the conditional causal effect of \\(A\\) on \\(Y\\) for each stratum. The per-stratum causal effect can be further weighted to get a pooled causal effect.</p> <p>Notice that stratification will introduce sampling variability when the stratification is getting finer.</p>"},{"location":"math/causal-inference/chapter-4/#matching","title":"Matching","text":"<p>Matching strategy is similar but inversed approach to IP weighting.</p> <p>The goal of matching is to construct a subset of the population in which the distribution of \\(L\\) is similar between the treated and untreated by sampling in the treated part or untreated part. Stratas with only treated or untreated individuals are excluded.</p> <p>Usually, the group with more individuals between the treated and untreated is sampled to match the other group. The subgroup being matched is used to calculate the average effect in the treated or untreated.</p> <p>However, as matching changes the distribution of \\(L\\), the distribution of causal modifier \\(V\\) will also be changed.</p>"},{"location":"math/causal-inference/chapter-4/#overview-of-adjustment-methods","title":"Overview of Adjustment Methods","text":"<p>There are 4 methods to adjust the causal effect of \\(A\\) on \\(Y\\), including standardization, IP weighting, matching and stratification.</p> <ol> <li>Standardization and IP weighting can be used to compute marginal or conditional causal effects.</li> <li>Matching and stratification can be used to compute conditional causal effects in subgroups.</li> <li>With effect modification, the causal effect of different adjustment methods may be different.</li> </ol>"},{"location":"math/causal-inference/chapter-5/","title":"Interaction","text":"<p>This chapter considers a system that consists multiple treatment variables.</p>"},{"location":"math/causal-inference/chapter-5/#from-joint-intervention-to-interaction","title":"From Joint Intervention to Interaction","text":"<p>Intervention is to manually set the value of treatment variable ignoring the original distribution of the variable. Intervention on multiple variables are called joint intervention.</p> <p>Let the treatment variable be \\(A\\) and \\(B\\). If the causal effect (measured in risk difference) of treatment \\(A\\) is different with different values of \\(B\\), then \\(B\\) has an additive interaction on \\(A\\). Notice that \\(B\\) has an interaction on \\(A\\) is equivalent to \\(A\\) has an interaction on \\(B\\).</p> \\[ P(Y^{a = 1, b = 1} = 1) - P(Y^{a = 0, b = 1} = 1) \\not = P(Y^{a = 1, b = 0} = 1) - P(Y^{a = 0, b = 0} = 1) \\] <p>Superadditive and subadditive</p> <p>Rewrite the above inequality as</p> \\[ \\begin{aligned}     P(Y^{a = 1, b = 1} = 1) - P(Y^{a = 0, b = 0} = 1) &amp;\\not = [P(Y^{a = 1, b = 1} = 1) - P(Y^{a = 0, b = 1} = 1)] \\\\     &amp;+ [P(Y^{a = 1, b = 0} = 1) - P(Y^{a = 0, b = 0} = 1)] \\end{aligned} \\] <p>If the \\(\\not =\\) can be replaced by \\(&gt;\\) or \\(&lt;\\), then the interaction is called superadditive or subadditive, respectively.</p> <p>Similarly, if the causal effect (measured in risk ratio) of treatment \\(A\\) is different with different values of \\(B\\), or vice versa, then there is a multiplicative interaction between \\(A\\) and \\(B\\). If the \\(\\not =\\) can be replaced by \\(&gt;\\) or \\(&lt;\\), then the interaction is called supermultiplicative or submultiplicative, respectively.</p> \\[ \\frac{P(Y^{a = 1, b = 1} = 1)}{P(Y^{a = 0, b = 0} = 1)} \\not = \\frac{P(Y^{a = 1, b = 0} = 1)}{P(Y^{a = 0, b = 0} = 1)} \\times \\frac{P(Y^{a = 1, b = 1} = 1)}{P(Y^{a = 0, b = 1} = 1)} \\]"},{"location":"math/causal-inference/chapter-5/#idenfication-of-interactions","title":"Idenfication of Interactions","text":"<p>To identify the interaction between \\(A\\) and \\(B\\), we need make sure that consistency, exchangeability, positivity for \\(A\\) is satisfied for all values of both \\(A\\) and \\(B\\).</p> <p>Notice the difference between effect modification and interaction. If \\(B\\) is randomly assigned, then \\(B\\) interacts with \\(A\\) is equivalent to that \\(B\\) serves as an effect modifier for \\(A\\).</p> <p>Effect modification v.s. interaction</p> <p>In an interaction, the two variables are of equal status, we can intervene on either or both of them. In effect modification, one variable is the treatment and the other is the modifier, we can only intervene on the treatment variable.</p> <p>For treatment variables, we need consistency, exchangeability, positivity. But such conditions are not required for modifier variables.</p>"},{"location":"math/causal-inference/chapter-5/#response-type","title":"Response Type","text":"<p>For each individual, the potential outcome pattern for receiving different levels of treatment is called response type.</p> <p></p> <p>For binary outcome and one binary treatment, there are \\(2^{2^1} = 4\\) response types. For binary outcome and two binary treatments, there are \\(2^{2 ^ 2} = 16\\) response types.</p> <p>Monotonicity</p> <p>For a response type, when treatment variables except \\(A\\) are fixed, if \\(Y^{a = 1} \\geq Y^{a = 0}\\), then the causal effect of \\(A\\) on \\(Y\\) is monotonic.</p>"},{"location":"math/causal-inference/chapter-5/#sufficient-cause-interaction","title":"Sufficient Cause Interaction","text":"<p>Consider the mechanism of yielding certain outcome by controlling one binary treatment variable \\(A\\). There are three types of mechanisms to inevitably yield the outcome.</p> <ol> <li>With treatment \\(A\\).</li> <li>Without treatment \\(A\\).</li> <li>\\(A\\) is irrelevant to the outcome.</li> </ol> <p>Since the outcome is stochastic, there exists some unknown factors \\(U_1, U_2, U_0\\) that leads to the outcome, respectively. For example, we can say \\(\\{A = 1, U_1 = 1\\}, \\{A = 0, U_2 = 1\\}\\) or \\(\\{U_3 = 1\\}\\) are sufficient-component causes for the outcome.</p> <p>For models with \\(k\\) binary treatment variables, there are \\(3^k\\) possible mechanisms. If any individual \\(U_i = 1\\) corresponds to multiple variables exists, then there exists sufficient cause interaction among these variables.</p> <p>Let \\(k = 2\\) and the treatment variables be \\(A, B\\), respectively. The sufficient cause interaction is synergistic if \\(A = 1, B = 1\\) is a sufficient-component cause for the outcome. The sufficient cause interaction is antagonistic if \\(A = 1, B = 0\\) or \\(A = 0, B = 1\\) are sufficient-component causes for the outcome.</p> <p>When monotonicity holds, some sufficient causes are not possible.</p>"},{"location":"math/causal-inference/chapter-5/#summary","title":"Summary","text":"<p>Sufficient-component-cause framework and the counterfactual framework focus on different interaction questions:</p> <ol> <li>Sufficient-component-cause framework focuses on the causal mechanism (reason) of the outcome.</li> <li>Counterfactual framework focuses on the causal effect of the treatment.</li> </ol>"},{"location":"math/causal-inference/chapter-6/","title":"Graphical Representation of Causal Effects","text":"<p>A causal graph consists the following elements:</p> <ul> <li>Nodes in causal graphs represent random variables.</li> <li>Edges in causal graphs represent direct causal effects.<ul> <li>A trail is a sequence of edges that does not meet the same node twice.</li> <li>A path is a trail that all the edges are pointing in the same direction.</li> </ul> </li> <li>Conventionally, time flows from left to right in causal graphs.</li> <li>Causal graphs follow the properties of Bayesian networks (acyclic, markov factorization).</li> </ul> <p>Markov Factorization</p> <p>The joint distribution of a set of random variables can be factorized into a product of conditional distributions of each variable given its parents in the graph.</p> \\[ P(X_1, \\dots, X_n) = \\prod_{i=1}^n P(X_i \\mid \\text{Pa}(X_i)) \\] <p>where \\(\\text{Pa}(X_i)\\) is the set of parents of \\(X_i\\).</p> <p>Markov factorization is equivalent to the following local independence property:</p> \\[ X_i \\perp \\text{ND}(X_i) \\mid \\text{Pa}(X_i) \\] <p>where \\(\\text{ND}(X_i)\\) is the set of non-descendants of \\(X_i\\).</p> <p>A causal graph contains the following building blocks: fork, chain, collider.</p> <p></p>"},{"location":"math/causal-inference/chapter-6/#d-seperation-and-independence","title":"D-seperation and Independence","text":"<p>Conditional independence relations are encoded in the graph structure.</p>"},{"location":"math/causal-inference/chapter-6/#blocked-trails-and-d-seperation","title":"Blocked Trails and D-seperation","text":"<p>Trails in causal graphs are regarded as blocked or open according to the following rules:</p> <ol> <li>If there are no nodes being observed, a path is blocked if and only if there exists one or more colliders on the path.</li> <li>If the trail contains a non-collider that is observed, the trail is blocked.</li> <li>A collider which is observed does not block the trail.</li> <li>A collider which any of its descendants is observed does not block the trail.</li> </ol> <p>A trail is blocked if it satisfies any of the following conditions:</p> <ol> <li>There exists a collider on the trail that itself or all of its descendants are not observed.</li> <li>There exists a non-collider on the trail that is observed.</li> </ol> <p>If all possible trails between two nodes \\(X, Y\\) are blocked \\(Z\\), the two nodes are d-separated. D-separated nodes are conditionally independent given the observed nodes, i.e., \\(X \\perp Y \\mid Z\\).</p> <p>Faithfulness assumption</p> <p>Faithfulness assumptions states that the observed conditional independence relations are consistent with the graph structure. That is: observed conditional independence \\(X \\perp Y \\mid Z\\) implies that \\(X\\) and \\(Y\\) are d-separated given \\(Z\\) in the graph.</p>"},{"location":"math/causal-inference/chapter-6/#flows-of-association-and-causation","title":"Flows of Association and Causation","text":"<p>If two nodes are not d-separated, there exists at least one open trail between the nodes.</p> <ul> <li>Association flows along open trails.</li> <li>Causation flows along open paths.</li> </ul> <p>There may be multiple trails and paths between two nodes, each trail provides a flow of association or causation. The overall observed association is affected by all the flows of association and causation. Existence of undesired flows of association may lead to lack of exchangeability and biased estimation of causal effects. If there are no flow of association between two nodes, the association is causal.</p> <p>There two types of flows of association:</p> <ol> <li> <p>Common cause: two nodes are associated because they are both affected by a common cause.</p> <p></p> </li> <li> <p>Condition on common effect: two nodes are associated because they are both affecting an observed common effect.</p> <p></p> </li> </ol> <p>Blocking a trail can be regarded as blocking the flow of association or causation on the graph.</p>"},{"location":"math/causal-inference/chapter-6/#hidden-properties","title":"Hidden Properties","text":"<p>Causal graphs only encode conditional independence relations and flows of causation or association. Some properties or assumptions cannot be represent in causal graphs.</p> <ul> <li>Positivity: Positivity in causal graphs can be represented that all of the edges in the graph are not deterministic.</li> <li>Consistency: Consistency is implicitly assumed in causal graphs.</li> </ul> <p>In a causal graph, the treatment node should lead to well-defined treatment.\\</p>"},{"location":"math/causal-inference/chapter-7/","title":"Confounding","text":"<p>Confounding is a situation in which the effect of an exposure on an outcome is distorted by the presence of another variable that is associated with both the exposure and the outcome. The confounding effect is a non-causal association that leads to biased estimation of causal effect.</p> <p>In a causal graph, the confounding effect is caused by an open backdoor path. A backdoor path is a non-path trail from treatment variable \\(A\\) to outcome variable \\(Y\\) that points to \\(A\\). The simplest structure of confounding contains a variable \\(L\\) pointing to treatment \\(A\\) and outcome \\(Y\\).</p> <p></p> <p>Suppose the causal graph is the true causal graph and satisfies faithfulness.</p>"},{"location":"math/causal-inference/chapter-7/#exchangeability","title":"Exchangeability","text":"<p>Following the definition of backdoor path, the definition of confounding can be alternatively expressed as</p> <p>Alternative definition of confounding</p> <p>Confounding is any systematic bias that can be removed by randomized assignment of treatment \\(A\\) (eliminating the backdoor path).</p> <p>When backdoor criterion is satisfied, exchangeability stands. The backdoor criterion requires on of the following:</p> <ol> <li>No confounding: no common cause of treatment \\(A\\) and outcome \\(Y\\).</li> <li>No unmeasured confounding: common cause of \\(A\\) and \\(Y\\) exists but a subset \\(L\\) of measured (observed) non-descendants of \\(A\\) blocks all the backdoor paths. The corresponding set \\(L\\) is called a sufficient set for confounding adjustment. Variable \\(L\\) is also called a confounder.</li> </ol> <p>Unconditional causal effect can be measured when unconditional exchangeability is satisfied. Conditional causal effect can be measured when conditional exchangeability is satisfied. The following table shows causal graph structures holding different exchangeability. The conditioned variable is \\(L\\).</p> Causal identifiability table Unconditional exchangeable Not unconditional exchangeable Conditional exchangeable Not conditional exchangeable <p>To identify conditional or unconditional causal effect, we can apply standardization on measured variables when conditional or unconditional exchangeability is satisfied. Take the following graph as an example.</p> <p></p> <p>When either \\(L_2\\) or \\(L\\) is observed, neither unconditional nor conditional exchangeability is satisfied. However, when both \\(L_2\\) and \\(L\\) are observed, both unconditional and conditional exchangeability is satisfied. We can identify the following causal effect based on observation.</p> <ol> <li>Conditional causal effect within strata of \\(L\\) and \\(L_2\\): \\(E[Y | A = a, L = l, L_2 = l_2]\\)</li> <li>Unconditional causal effect: \\(E[Y | A = a, L = l, L_2 = l_2]P(L_2 = l_2, L = l)\\)</li> <li>Conditional causal effect within strata of \\(L_2\\): \\(E[Y | A = a, L_2 = l_2, L = l]P(L = l | L_2 = l_2)\\)</li> <li>Conditional causal effect within strata of \\(L\\): \\(E[Y | A = a, L = l, L_2 = l_2]P(L_2 = l_2 | L = l)\\)</li> </ol>"},{"location":"math/causal-inference/chapter-7/#confounders","title":"Confounders","text":"<p>The sufficient set for identifying confounder depends on variables observed or measured. For a causal graph, there may also exist multiple sufficient sets for confounding adjustment.</p>"},{"location":"math/causal-inference/chapter-7/#traditional-confounder","title":"Traditional Confounder","text":"<p>A confounder satisfies the following conditions, which is the definition of a traditional confounder.</p> <ol> <li>The variable associates with the treatment.</li> <li>Condition on the treatment, the variable associates with the outcome.</li> <li>The variable is not on the causal path from the treatment to the outcome.</li> </ol> <p>However, a variable that satisfies the above conditions is not necessarily a confounder. Variable \\(L\\) in the following graph is a traditional confounder. When \\(L\\) is not adjusted, unconditional causal effect can be directly identified, adjusting for \\(L\\) will introduce extra selection bias.</p> <p></p>"},{"location":"math/causal-inference/chapter-7/#surrogate-confounder","title":"Surrogate Confounder","text":"<p>A surrogate confounder is a variable that is the result of an unobserved common cause of the treatment and the outcome. The surrogate confounder itself does not lie on a backdoor path. The following graph shows a surrogate confounder \\(L\\).</p> <p></p> <p>Although \\(L\\) does not lie on the backdoor path, observing distribution of \\(L\\) may provide information about the unobserved common cause \\(U\\). In this case, adjusting for \\(L\\) can partially reduce the confounding bias.</p>"},{"location":"math/causal-inference/chapter-7/#single-world-intervention-graph","title":"Single-world Intervention Graph","text":"<p>A single-world intervention graph (SWIG) is a causal graph that explicitly include the conterfactual variables on the graph. Unlike the causal graph that represents the real world, the SWIG represents a counterfactual world created by an intervention.</p> Causal graph Single-world intervention graph <p>In a SWIG, the original treatment variable \\(A\\) is now regarded as two distinct nodes \\(A\\) and \\(a\\). \\(A\\) encodes the possible value of \\(A\\) under no intervention, \\(a\\) is the value of intervened treatment variable. Any descendant \\(X\\) of \\(A\\) is replaced by \\(X^a\\), which is the value of \\(X\\) under intervention \\(A = a\\). From the SWIG, we can see that after conditioning on \\(L\\), the trail \\(A\\leftarrow L\\leftarrow U\\rightarrow Y^a\\) is blocked, indicating that exchangeability \\(A \\perp Y^a | L\\) holds.</p> <p>Confounder cannot be descendants of treatment</p> <p>A confounder cannot be a descendant of the treatment variable. Suppose \\(L\\) is a descendant of \\(A\\), and \\(L\\) connects to \\(Y\\) through some trail. If the trail is a path from \\(L\\) to \\(Y\\), then the whole path through \\(L\\) is a causal path, adjustment on \\(L\\) will block such path. Otherwise, there should be a collider on the path. If the collider is \\(L\\), adjustment on \\(L\\) will open the non-causal trail, introducing extra selection bias. If the collider is not \\(L\\), adjustment on \\(L\\) have no effect.</p>"},{"location":"math/causal-inference/chapter-7/#confounding-adjustment","title":"Confounding Adjustment","text":"<p>Methods adjust for confounder \\(L\\) can be classified into two categories:</p> <ol> <li>G-methods: methods that adjust for \\(L\\) by conditioning on \\(L\\) - IP weighting, G-estimation, standardization.</li> <li>Stratification methods: methods that adjust for \\(L\\) by stratifying on \\(L\\) - stratification, matching.</li> <li>Other methods: difference-in-difference, instrumental variable, front-door adjustment. These methods does not rely on conditional exchangeability.</li> </ol>"},{"location":"math/causal-inference/chapter-7/#difference-in-difference","title":"Difference-in-Difference","text":"<p>Consider the following causal graph. We want to measure causal effect on treated \\(E(Y^1 | A = 1) - E(Y^0 | A = 1)\\). But there exists some unknown factors \\(U\\) that affects both treatment \\(A\\) and outcome \\(Y\\). For example, \\(U\\) can be the severity of disease, which affects both the treatment decision and the outcome.</p> <p></p> <p>Notice that</p> \\[ \\begin{aligned} &amp;&amp;&amp; \\underbrace{E(Y | A = 1) - E(Y | A = 0)}_{\\text{Measured association}} \\\\ &amp;=&amp;&amp; E(Y^1 | A = 1) - E(Y^0 | A = 0) \\\\ &amp;=&amp;&amp; \\underbrace{(E(Y^1 | A = 1) - E(Y^0 | A = 1))}_{\\text{Causal effect on treated}} + \\underbrace{(E(Y^0 | A = 1) - E(Y^0 | A = 0))}_{\\text{Confounding effect}} \\end{aligned} \\] <p>To estimate confounding effect \\(E(Y^0 | A = 1) - E(Y^0 | A = 0)\\) we can measure a pre-treatment outcome \\(C\\) before the treatment is applied. There is no direct causal relationship between pre-treatment outcome \\(C\\) and treatment \\(A\\), but \\(C\\) and \\(A\\) show some association since backdoor \\(C\\leftarrow U\\rightarrow A\\) presents, that is, \\(E(C | A = 1) - E(C | A = 0) \\neq 0\\). The metric \\(E(C | A = 1) - E(C | A = 0) \\neq 0\\) measures the strength of confounding effect of \\(A\\) on \\(C\\). If the confounding effect of \\(A\\) on \\(Y\\) caused by \\(Y\\leftarrow U\\rightarrow A\\) is the same as the confounding effect of \\(A\\) on \\(C\\) (additive equi-confounding assumption), then the confounding effect of \\(A\\) on \\(Y\\) can be removed by difference-in-difference.</p> \\[ E(Y^0 | A = 1) - E(Y^0 | A = 0) = E(C | A = 1) - E(C | A = 0) \\] <p>The causal effect can be measured in</p> \\[ \\begin{aligned} &amp;&amp;&amp; E(Y^1 | A = 1) - E(Y^0 | A = 1) \\\\ &amp;=&amp;&amp; (E(Y | A = 1) - E(Y | A = 0)) - (E(C | A = 1) - E(C | A = 0)) \\end{aligned} \\]"},{"location":"math/causal-inference/chapter-7/#front-door-criterion","title":"Front-door Criterion","text":"<p>Consider a causal graph with a treatment \\(A\\), a mediator \\(M\\) and an outcome \\(Y\\). The causal effect of \\(A\\) on \\(Y\\) is confounded by an unobserved variable \\(U\\). The mediator fully mediates the effect of \\(A\\) on \\(Y\\), i.e. all the causal path from \\(A\\) to \\(Y\\) passes through \\(M\\).</p> <p></p> <p>First we have</p> \\[ P(Y^a = 1) = \\sum_{m} P(Y^a = 1 | M^a = m) P(M^a = m) \\] <p>\\(M\\) is not confounded, so</p> \\[ P(M^a = m) = P(M = m | A = a) \\] <p>Notice that</p> \\[ P(Y^a = 1 | M^a = m) = P(Y^m = 1) = P(Y = 1 | M = m) \\] <p>and</p> \\[ P(Y = 1 | M = m) = \\sum_{a'} P(Y^m = 1 | M = m, A = a')P(A = a') \\] <p>by conditional exchangeability \\(Y^m \\perp M \\mid A\\). Under this case, \\(P(Y^a)\\) can be identified by</p> \\[ P(Y^a = 1) = \\sum_{m} P(M = m | A = a)\\sum_{a'} P(Y = 1 | M = m, A = a')P(A = a') \\]"},{"location":"math/causal-inference/chapter-8/","title":"Selection bias","text":"<p>Selection bias is caused by extra association caused by only part of the population is selected for analysis. Selection bias is caused by conditioning on a common effect of treatment and outcome, even if the treatment actually has no individual causal effect on the outcome.</p> <p>On the causal graph, selection bias is caused by either condition on a collider or descendant of a collider. In reality, selection bias can both appear in observational studies or randomized experiments since participants may be removed from the study before the outcome is observed. If participants are removed not in random, selection bias is introduced.</p> <p></p> \\[ \\frac{P(Y = 1 | A = 1)}{P(Y = 1 | A = 0)} = \\frac{P(Y^{a = 1} = 1)}{P(Y^{a = 0} = 1)} \\not = \\frac{P(Y = 1 | A = 1, L = 0)}{P(Y = 0 | A = 1, L = 0)} \\] <p>Selection bias and hazard ratio</p> <p>Hazards is defined as the probability of a participant to die at a certain time. Following the definition of risk, the hazard ratio is the same as risk ratio. Consider the following causal graph.</p> <p></p> <p>In the graph, treatment \\(A\\) denote the heart transplant. The outcome \\(Y_1\\) and \\(Y_2\\) denote the death of the patient. Unmeasured variable \\(U\\) affect the overall death rate of the patient. For each time, we can define the associational hazard ratio as</p> \\[ \\begin{aligned}     aRR_{AY_1} &amp;= \\frac{P(Y_1 = 1 | A = 1)}{P(Y_1 = 1 | A = 0)} \\\\     aRR_{AY_2} &amp;= \\frac{P(Y_2 = 1 | A = 1)}{P(Y_2 = 1 | A = 0)} \\end{aligned} \\] <p>However, we can only measure the hazard ratio among the patients who are still alive at that time, that is:</p> \\[ aRR_{AY_2 \\mid Y_1 = 0} = \\frac{P(Y_1 = 1 | A = 1, Y_1 = 0)}{P(Y_1 = 1 | A = 0, Y_1 = 0)} \\] <p>However, condition on \\(Y_1\\) opens a trail \\(A \\ra Y_1 \\la U\\ra Y_2\\). Therefore, unless \\(U\\) is measured, from the data collected we cannot distinguish the existence of the path \\(A \\ra Y_2\\).</p>"},{"location":"math/causal-inference/chapter-8/#selection-without-bias","title":"Selection without Bias","text":"<p>Selection will cause bias within the study, but in some cases such bias can be restricted to some strata of the study. Consider the following causal graph. \\(Y = 0\\) if and only if \\(Y_A = Y_E = Y_O = 0\\).</p> <p></p> <ol> <li>\\(Y = 0\\) is equivalent to \\(Y_A = 0\\) and \\(Y_E = 0\\). In such case, \\(A\\) is independent of \\(E\\).</li> <li>Consider the case when \\(Y = 1\\) and \\(Y_O = 0\\), then \\(Y_A = 0\\) indicates \\(Y_E = 1\\) and vice versa. In such case, \\(A\\) is dependent of \\(E\\).</li> </ol>"},{"location":"math/causal-inference/chapter-8/#adjustment-for-selection-bias","title":"Adjustment for Selection Bias","text":"<p>Assume that positivity holds for \\(C = 0\\) and consistency holds for the analysis. Selection bias arises when the participants are not randomly removed from the study, causing the distribution of remaining participants to be different from the original population, i.e. the joint distribution \\(P'(A, L) = P(A, L | C = 0)\\) is no longer identical to \\(P(A, L)\\).</p> <p>Selection bias is often unavoidable. IP weighting and standardization can be used to adjust for selection bias. The inverse probability weight \\(W^C\\) is defined as</p> \\[ W^C = \\frac{1}{P(C = 0 | \\cdot)} \\] <p>where \\(\\cdot\\) denote all the variable that directly affects \\(C\\). Since we can only observed variables for uncensored (\\(C = 0\\)) individuals, the IP weight only uses \\(C = 0\\). IP weighting assigns different weight to the probability distribution of each pair of \\((A, L)\\), so that the distribution of the weighted sample is identical to the original population.</p> \\[ \\begin{aligned} &amp;&amp;&amp; \\frac{P(A = a, L = l, C = 0)}{P(C = 0 | A = a, L = l)} \\\\ &amp;=&amp;&amp; \\frac{P(A = a, L = l, C = 0)}{P(A = a, L = l, C = 0) / P(A = a, L = l)} \\\\ &amp;=&amp;&amp; P(A = a, L = l) \\end{aligned} \\] <p>Difference in confounding bias and selection bias</p> <p>In confounding bias, IP weighting is applied on the treatment variable \\(A\\), while in selection bias, IP weighting is applied on the censoring variable \\(C\\).</p> <p>When there are measured variable \\(L\\) on the trail through \\(C\\) that is able to block the trail causing selection bias, we can use stratification to adjust for selection bias by conditioning on \\(L\\).</p>"},{"location":"math/causal-inference/chapter-9/","title":"Measurement bias and \"noncausal\" diagrams","text":""},{"location":"math/causal-inference/chapter-9/#measurement-bias","title":"Measurement Bias","text":"<p>Measurement bias is caused by errors in measuring values of variables, i.e. \\(A^* \\not = A\\). Measurement error is defined as the difference between the measured value and the true value of a variable, \\(e_A = A^* - A\\). Taken measurement error into consideration, the causal diagram is modified as follows:</p> <p></p> <p>Measurement error follow two properties:</p> <ol> <li>Indepedence: \\(e_A \\perp e_Y\\).     </li> <li>Nondifferentiality: \\(e_A \\perp Y\\) and \\(e_Y \\perp A\\).     </li> </ol> <p>Lack of either property will bring extra association and lead to bias.</p> <ul> <li>Edge \\(Y\\ra U_A\\) will introduce recall bias.</li> <li>Edge \\(A\\ra U_Y\\) will introduce reverse causation bias.</li> <li>Edge \\(U_A\\la U_{AY}\\ra U_Y\\) will introduce independent measurement error.</li> </ul> <p>Correcting for measurement error usually requires additional validated non-biased samples.</p>"},{"location":"math/causal-inference/chapter-9/#noncausal-diagrams","title":"\"Noncausal\" Diagrams","text":"<p>A causal graph requires that all of the edges in the graph can be interpreted causally, together with well-defined intervention. For graphs with non-causal edges, adjustments might fail to remove bias, as the adjusted variable is not on the true causal path.</p> <p></p>"},{"location":"math/convex-optimization/","title":"\u51f8\u200b\u4f18\u5316","text":""},{"location":"math/convex-optimization/#_2","title":"\u76ee\u5f55","text":"<ul> <li>\u200b\u51f8\u96c6\u200b</li> <li>\u200b\u51f8\u51fd\u6570\u200b</li> </ul>"},{"location":"math/convex-optimization/#notations-and-definitions","title":"Notations and Definitions","text":"<p>\u200b\u672c\u200b\u8282\u200b\u5217\u4e3e\u200b\u51f8\u200b\u4f18\u5316\u200b\u4e2d\u200b\u6240\u7528\u200b\u7b26\u53f7\u200b\u7684\u200b\u8868\u8ff0\u200b\uff0c\u200b\u5e76\u200b\u501f\u6b64\u200b\u56de\u987e\u200b\u76f8\u5173\u200b\u7684\u200b\u6570\u5b66\u200b\u6982\u5ff5\u200b</p>"},{"location":"math/convex-optimization/#_3","title":"\u6570\u57df","text":"<ul> <li>\\(\\mathbb R, \\mathbb N, \\mathbb Z, \\mathbb C\\)\uff1a\u200b\u5b9e\u6570\u200b\u57df\u200b\u3001\u200b\u81ea\u7136\u6570\u200b\u57df\u200b\u3001\u200b\u6574\u6570\u200b\u57df\u200b\u3001\u200b\u590d\u6570\u200b\u57df\u200b</li> <li>\u200b\u4e0b\u6807\u200b \\({}_\\plus, {}_{\\plus\\plus}\\) \u200b\u5728\u200b\u5b9e\u6570\u200b\u57df\u548c\u5b9e\u200b\u5411\u91cf\u200b\u57df\u200b\u4e0a\u200b\u5206\u522b\u200b\u8868\u793a\u200b\u975e\u8d1f\u200b\u3001\u200b\u4e25\u683c\u200b\u6b63\u200b\uff0c\u200b\u5728\u200b\u5b9e\u77e9\u9635\u200b\u57df\u200b\u4e0a\u200b\u8868\u793a\u200b\u534a\u200b\u6b63\u5b9a\u200b\u3001\u200b\u6b63\u5b9a\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/#_4","title":"\u77e9\u9635\u200b\u4e0e\u200b\u5411\u91cf","text":"<p>\\(A\\in \\mathbb R^{m\\times n}\\)\u200b\u8868\u793a\u200b\\(A\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(m\\)\u200b\u884c\u200b\\(n\\)\u200b\u5217\u200b\u7684\u200b\u5b9e\u77e9\u9635\u200b\uff0c\u200b\u5176\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u4e3a\u200b\\(a_{ij}\\)\uff0c\u200b\u67d0\u5217\u200b\uff08\u200b\u6216\u200b\u67d0\u884c\u200b\uff09\u200b\u7684\u200b\u5207\u7247\u200b\u4e3a\u200b\\(A_{\\cdot j}\\)</p> <p>\\(\\boldsymbol x\\in \\mathbb R^m\\)\u200b\u8868\u793a\u200b\\(\\boldsymbol x\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(m\\)\u200b\u7ef4\u7684\u5b9e\u200b\u5411\u91cf\u200b\uff0c\\(\\boldsymbol 1\\)\u200b\u8868\u793a\u200b\u6240\u6709\u200b\u5206\u91cf\u200b\u5747\u200b\u4e3a\u200b\\(1\\)\u200b\u7684\u200b\u5411\u91cf\u200b\u3002</p> <p>\u200b\u65b9\u9635\u200b\u662f\u200b\u884c\u6570\u200b\u4e0e\u200b\u5217\u6570\u200b\u76f8\u7b49\u200b\u7684\u200b\u77e9\u9635\u200b\uff0c\\(A\\in \\mathbb R^{n\\times n}\\)\u3002\u200b\u5bf9\u79f0\u200b\u77e9\u9635\u200b\u6ee1\u8db3\u200b\\(A^\\top = A\\)\uff0c\u200b\u53cd\u5bf9\u200b\u79f0\u200b\u77e9\u9635\u200b\u6ee1\u8db3\u200b\\(A^\\top = - A\\)\u3002\u200b\u6b63\u5b9a\u200b\u77e9\u9635\u200b\u6ee1\u8db3\u200b\\(\\forall x\\in \\mathbb R^n, x^\\top Ax &gt; 0\\)\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(A\\succ 0\\)\uff1b\u200b\u534a\u200b\u6b63\u5b9a\u200b\u77e9\u9635\u200b\u6ee1\u8db3\u200b\\(\\forall x\\in \\mathbb R^n, x^\\top Ax \\geq 0\\)\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(A\\succeq 0\\)\u3002\u200b\u5bf9\u4e8e\u200b\u590d\u200b\u65b9\u9635\u200b\\(A\\in \\mathbb C^{n\\times n}\\)\uff0c\u200b\u5b9a\u4e49\u200b\u5171\u8f6d\u200b\u8f6c\u7f6e\u200b\uff08Hermitian\uff09\u200b\u77e9\u9635\u200b\u4e3a\u200b\\(A^* = \\overline {A^\\top}\\)\u3002\u200b\u5168\u4f53\u200b\\(k\\)\u200b\u9636\u200b\u5bf9\u79f0\u200b\u77e9\u9635\u200b\u7684\u200b\u96c6\u5408\u200b\u4e3a\u200b\\(\\mathbb S^k\\)\u3002</p> <p>\u200b\u77e9\u9635\u7684\u8ff9\u200b\u662f\u200b\u5bf9\u89d2\u7ebf\u200b\u4e0a\u200b\u5404\u4e2a\u200b\u5143\u7d20\u200b\u4e4b\u200b\u548c\u200b</p> \\[ \\mathrm {tr}(A) = \\sum_{i=1}^n a_{ii} \\] \u200b\u8ff9\u200b\u7684\u200b\u6027\u8d28\u200b <ol> <li>\\(\\mathrm {tr}(A) = \\mathrm {tr}(A^\\top)\\)</li> <li>\\(\\mathrm {tr}(A+B) = \\mathrm {tr}(A) + \\mathrm {tr}(B)\\)</li> <li>\\(\\mathrm {tr}(t\\cdot A) = t\\cdot \\mathrm {tr}(A)\\)</li> <li>\\(\\mathrm {tr}(AB) = \\mathrm {tr}(BA)\\)\uff08\u200b\u4e0d\u200b\u8981\u6c42\u200b\\(A, B, \\cdots\\)\u200b\u4e3a\u200b\u65b9\u9635\u200b\uff0c\u200b\u53ea\u200b\u9700\u8981\u200b\u6ee1\u8db3\u200b\u8fd0\u7b97\u200b\u7ed3\u679c\u200b\u4e3a\u200b\u65b9\u9635\u200b\u5373\u53ef\u200b\uff1b\u200b\u5bf9\u4e8e\u200b\u591a\u4e2a\u200b\u5143\u7d20\u200b\uff0c\u200b\u4e58\u6cd5\u200b\u987a\u5e8f\u200b\u9700\u8981\u200b\u6ee1\u8db3\u200b\u8f6e\u6362\u200b\u5bf9\u79f0\u6027\u200b\u624d\u80fd\u200b\u4fdd\u8bc1\u200b\u7b49\u5f0f\u200b\u6210\u7acb\u200b\uff09</li> </ol> <p>\u200b\u5b9a\u4e49\u200b\u4e24\u4e2a\u200b\u77e9\u9635\u200b\\(X, Y\\)\u200b\u7684\u200b\u5185\u79ef\u200b\u4e3a\u200b\u5176\u200b\u4e58\u79ef\u200b\u7684\u200b\u8ff9\u200b\\(\\mathrm{tr}(XY)\\)</p>"},{"location":"math/convex-optimization/#_5","title":"\u8303\u6570","text":"<p>\u200b\u8303\u6570\u200b\u662f\u200b\u6ee1\u8db3\u200b\u4ee5\u4e0b\u200b\u6027\u8d28\u200b\u7684\u200b\u51fd\u6570\u200b\\(f\\)</p> <ol> <li>\u200b\u975e\u200b\u8d1f\u6027\u200b\uff1a\\(f(\\boldsymbol{x}) \\geq 0\\)</li> <li>\u200b\u89c4\u8303\u6027\u200b\uff1a\\(f(\\boldsymbol{x}) = 0 \\Leftrightarrow \\boldsymbol x = 0\\)</li> <li>\u200b\u9f50\u6b21\u200b\u6027\u200b\uff1a\\(f(t\\cdot \\boldsymbol{x}) = t\\cdot f(\\boldsymbol{x})\\)</li> <li>\u200b\u4e09\u89d2\u200b\u4e0d\u7b49\u5f0f\u200b\uff1a\\(f(\\boldsymbol x + \\boldsymbol y)\\leq f(\\boldsymbol x) + f(\\boldsymbol y)\\)</li> </ol> \u200b\u5e38\u89c1\u200b\u7684\u200b\u8303\u6570\u200b <ol> <li>1-\u200b\u8303\u6570\u200b\uff1a\\(f(\\boldsymbol{x}) = \\sum_{i=1}^n \\left|x_i\\right|\\)</li> <li>2-\u200b\u8303\u6570\u200b\uff1a\\(f(\\boldsymbol{x}) = \\sqrt{\\sum_{i=1}^n \\left|x_i^2\\right|}\\)</li> <li>\\(\\infty\\)-\u200b\u8303\u6570\u200b\uff1a\\(f(\\boldsymbol{x}) = \\sqrt[\\infty]{\\sum_{i=1}^n \\left|x_i^\\infty\\right|}\\)</li> <li>0-\u200b\u8303\u6570\u200b\uff08\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u5b9a\u4e49\u200b3\uff0c\u200b\u56e0\u6b64\u200b\u4e0d\u662f\u200b\u8303\u6570\u200b\uff09\uff1a\\(f(\\boldsymbol{x}) = \\sum_{i=1}^n \\boldsymbol{1}_{\\{x_i \\not = 0\\}}\\)</li> <li>\\(l_p\\)-\u200b\u8303\u6570\u200b\uff1a\\(f(\\boldsymbol{x}) = \\sqrt[p]{\\sum_{i=1}^n \\left|x_i^p\\right|}\\)</li> <li>Frobenius\u200b\u8303\u6570\u200b\uff1a\\(f(A) = \\sqrt{\\sum_i\\sum_j a_{ij}^2} = \\sqrt{\\mathrm{tr}(A^\\top A)}\\)</li> </ol> <p>\u200b\u8303\u6570\u200b\u7b49\u4ef7\u200b\u5b9a\u7406\u200b\uff1a\u200b\u5bf9\u4e8e\u200b\u4e24\u4e2a\u200b\u8303\u6570\u200b\\(f, g\\)\uff0c\u200b\u5b58\u5728\u200b\u6b63\u5e38\u200b\u6570\u200b\\(c\\)\u200b\u4f7f\u5f97\u200b\\(\\forall \\boldsymbol{x}, f(\\boldsymbol{x}) \\leq cg(\\boldsymbol{x})\\)</p> <p>\u200b\u8bbe\u200b\\(f\\)\u200b\u4e3a\u200b\\(\\mathbb R^n\\)\u200b\u4e0a\u200b\u7684\u200b\u8303\u6570\u200b\uff0c\u200b\u5bf9\u5076\u200b\u8303\u6570\u200b\\(f_*\\)\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b</p> \\[ f_*(\\boldsymbol{z}) = \\sup_{\\boldsymbol{x}} \\{\\boldsymbol{z^\\top x} | f(\\boldsymbol{x}) \\leq 1\\} \\] \u200b\u5bf9\u5076\u200b\u8303\u6570\u200b <ul> <li>\u200b\u5bf9\u5076\u200b\u8303\u6570\u200b\u7684\u200b\u5bf9\u5076\u200b\u8303\u6570\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u539f\u200b\u8303\u6570\u200b\u3002</li> <li> <p>\\(p\\)-\u200b\u8303\u6570\u200b\u7684\u200b\u5bf9\u5076\u200b\u8303\u6570\u200b\u662f\u200b\\(q\\)-\u200b\u8303\u6570\u200b\uff0c\u200b\u6ee1\u8db3\u200b</p> \\[ \\frac{1}{p} + \\frac{1}{q} = 1\\qquad p, q\\geq 1 \\] </li> </ul> <p>\u200b\u6839\u636e\u200b\u8303\u6570\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u7403\u200b\u7684\u200b\u6982\u5ff5\u200b</p> \u200b\u7403\u200b <p>\u200b\u7ed9\u5b9a\u200b\\(\\mathbb R^n\\)\u200b\u7a7a\u95f4\u200b\u4e2d\u200b\u7684\u200b\u8303\u6570\u200b\\(\\Vert\\cdot\\Vert\\)\uff0c\\(B(x, r)\\)\u200b\u8868\u793a\u200b\u8303\u6570\u200b\\(\\Vert \\cdot\\Vert\\)\u200b\u4e2d\u200b\u7684\u200b\uff08\u200b\u5f00\u200b\uff09\u200b\u7403\u200b\uff0c\u200b\u5373\u200b</p> \\[ B(x, r) = \\{y | \\Vert y - x\\Vert &lt; r\\} \\] <p>\\(\\bar B(x, r)\\)\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u95ed\u7403\u200b</p> \\[ \\bar B(x, r) = \\{y | \\Vert y - x\\Vert \\leq r\\} \\]"},{"location":"math/convex-optimization/#_6","title":"\u96c6\u5408","text":"<p>\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(C\\subseteq \\mathbb R^n, x\\in C\\)\uff0c\u200b\u4f7f\u7528\u200b2-\u200b\u8303\u6570\u200b\u5ea6\u91cf\u200b\u8ddd\u79bb\u200b\u3002</p> <ul> <li>\u200b\u82e5\u200b\\(\\exists r \\in \\mathbb R_{\\plus\\plus}, B(x, r)\\subseteq C\\)\uff0c\u200b\u79f0\u200b\\(x\\)\u200b\u4e3a\u200b\u5185\u70b9\u200b\u3002\u200b\u5168\u4f53\u200b\u5185\u70b9\u200b\u6784\u6210\u200b\u96c6\u5408\u200b\u7684\u200b\u5185\u90e8\u200b\\(C^\\circ\\)\uff0c\u200b\u663e\u7136\u200b\\(C^\\circ \\subseteq C\\)\u3002</li> <li>\u200b\u82e5\u200b\\(\\forall r\\in \\mathbb R_{+}, B(x, r)\\cap C \\not = \\varnothing, B(x, r) \\cap C^C \\not = \\varnothing\\)\uff0c\u200b\u79f0\u200b\\(x\\)\u200b\u4e3a\u200b\u8fb9\u754c\u70b9\u200b\u3002\u200b\u5168\u4f53\u200b\u8fb9\u754c\u70b9\u200b\u6784\u6210\u200b\u96c6\u5408\u200b\u7684\u200b\u8fb9\u754c\u200b\\(\\partial C\\)\u3002<ul> <li>\u200b\u6839\u636e\u200b\u8fb9\u754c\u70b9\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u6709\u200b\\(\\partial C = \\partial C^C\\)\u3002</li> </ul> </li> <li>\u200b\u82e5\u200b\\(x\\)\u200b\u4e0d\u662f\u200b\u5185\u70b9\u200b\uff0c\u200b\u5219\u200b\\(x\\)\u200b\u4e3a\u200b\u8fb9\u754c\u70b9\u200b\uff0c\u200b\u5373\u200b\\(\\partial C \\cup C^\\circ = C, \\partial C\\cap C^\\circ = \\varnothing\\)\u3002</li> </ul> <p>\u200b\u82e5\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u70b9\u200b\u90fd\u200b\u662f\u200b\u5185\u70b9\u200b\uff0c\u200b\u5373\u200b\\(C^\\circ = C\\)\uff0c\u200b\u79f0\u200b\\(C\\)\u200b\u4e3a\u200b\u5f00\u96c6\u200b\u3002\u200b\u82e5\u200b\\(C\\)\u200b\u4e0d\u662f\u200b\u5f00\u96c6\u200b\uff0c\u200b\u5219\u200b\\(C\\)\u200b\u4e3a\u200b\u95ed\u96c6\u200b\u3002\u200b\u5f00\u96c6\u200b\u7684\u200b\u8865\u96c6\u200b\u662f\u200b\u95ed\u96c6\u200b\u3002</p> <p>\u200b\u5b9a\u4e49\u200b\u95ed\u5305\u200b\u4e3a\u200b</p> \\[ \\mathbf{cl}C = \\{x \\in \\mathbb R^n| \\forall r\\in \\mathbb R_{++}, B(x, r)\\cap C\\not = \\varnothing\\} \\] <p>\u200b\u76f4\u89c2\u200b\u4e0a\u200b\uff0c\u200b\u95ed\u5305\u200b\u53ef\u4ee5\u200b\u7406\u89e3\u200b\u4e3a\u200b\\(C\\cup \\partial C^C\\)\uff0c\u200b\u5373\u200b\u96c6\u5408\u200b\u4e0e\u5176\u200b\u8865\u96c6\u200b\u7684\u200b\u8fb9\u754c\u200b\u7684\u200b\u5e76\u200b\u3002</p> <p>\\(\\varnothing, \\mathbb R^n\\)\u200b\u65e2\u200b\u662f\u200b\u5f00\u96c6\u200b\u4e5f\u200b\u662f\u200b\u95ed\u96c6\u200b\uff1a</p> <ul> <li>\\(\\varnothing\\)\u200b\u7684\u200b\u95ed\u5305\u200b\u4ecd\u200b\u4e3a\u200b\\(\\varnothing\\)\uff0c\u200b\u6240\u4ee5\u200b\u662f\u200b\u95ed\u96c6\u200b\uff1b\u200b\u7136\u800c\u200b\\(\\varnothing^\\circ = \\varnothing\\)\uff0c\u200b\u6240\u4ee5\u200b\u662f\u200b\u5f00\u96c6\u200b\u3002</li> <li>\\(\\varnothing^C = \\mathbb R^n\\)\uff0c\u200b\u6240\u4ee5\u200b\\(\\mathbb R^n\\)\u200b\u65e2\u200b\u662f\u200b\u5f00\u96c6\u200b\u4e5f\u200b\u662f\u200b\u95ed\u96c6\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-function/","title":"\u51f8\u51fd\u6570\u200b\u53ca\u5176\u200b\u6027\u8d28","text":"<p>\u200b\u51f8\u51fd\u6570\u200b\u662f\u200b\u4e00\u7c7b\u200b\u51fd\u6570\u200b\u4e0a\u65b9\u200b\uff08\u200b\u4e0a\u5883\u200b\u56fe\u200b\uff0cepigraph\uff09\u200b\u662f\u200b\u51f8\u96c6\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5373\u200b\u51fd\u6570\u200b\u66f2\u7ebf\u200b\u4e0a\u200b\u4efb\u610f\u200b\u4e24\u70b9\u200b\u8fde\u6210\u200b\u7684\u200b\u7ebf\u6bb5\u200b\u90fd\u200b\u5728\u200b\u51fd\u6570\u200b\u66f2\u7ebf\u200b\u7684\u200b\u4e0a\u65b9\u200b\u3002</p> <p>\u200b\u82e5\u200b\u51fd\u6570\u200b\\(f: \\mathbb R^n\\rightarrow \\mathbb R\\)\u200b\u6ee1\u8db3\u200b\u7434\u751f\u200b\u4e0d\u7b49\u5f0f\u200b\uff1a</p> \\[ f(\\theta x + (1 - \\theta)y)\\leq \\theta f(x) + (1 - \\theta)f(y), 0\\leq \\theta\\leq 1 \\] <p>\uff0c\u200b\u5219\u200b\u79f0\u200b\\(f\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\u3002</p> \u200b\u7434\u751f\u200b\u4e0d\u7b49\u5f0f\u200b\u7684\u200b\u6269\u5c55\u200b <p>\u200b\u7434\u751f\u200b\u4e0d\u7b49\u5f0f\u200b\u53ef\u4ee5\u200b\u6269\u5c55\u200b\u5230\u200b\u66f4\u200b\u591a\u70b9\u200b\u7684\u200b\u51f8\u200b\u7ec4\u5408\u200b\u4e0a\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u51f8\u51fd\u6570\u200b\\(f\\)\uff0c\u200b\u8bbe\u200b\\(\\theta_1, \\ldots, \\theta_k\\geq 0, \\theta_1 + \\cdots + \\theta_k = 1\\)\uff0c\u200b\u5219\u200b</p> \\[ f(\\theta_1 x_1 + \\cdots + \\theta_k x_k)\\leq \\theta_1 f(x_1) + \\cdots + \\theta_k f(x_k) \\] <p>\u200b\u8be5\u5f0f\u200b\u53ef\u4ee5\u200b\u8fdb\u4e00\u6b65\u200b\u62d3\u5c55\u200b\u81f3\u200b\u7ea7\u6570\u200b\u3001\u200b\u79ef\u5206\u200b\u3001\u200b\u671f\u671b\u200b\u7b49\u200b\u5f62\u5f0f\u200b\u3002</p> <p>\u200b\u82e5\u200b\\(f(\\theta x + (1 - \\theta)y)&lt; \\theta f(x) + (1 - \\theta)f(y), 0\\leq \\theta\\leq 1\\)\uff0c\u200b\u5219\u200b\u4e3a\u200b\u4e25\u683c\u200b\u51f8\u51fd\u6570\u200b\u3002\u200b\u51f8\u51fd\u6570\u200b\u7684\u200b\u4e0a\u200b\u5883\u56fe\u200b\uff08\\(\\mathbf{epi}f\\)\uff09\u200b\u662f\u200b\u51f8\u96c6\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\\(-f\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\u51f9\u200b\u51fd\u6570\u200b\uff0c\u200b\u82e5\u200b\\(-f\\)\u200b\u662f\u200b\u4e25\u683c\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\u4e25\u683c\u200b\u51f9\u200b\u51fd\u6570\u200b\u3002\u200b\u51f9\u200b\u51fd\u6570\u200b\u7684\u200b\u4e9a\u56fe\u200b\uff08\\(\\mathbf{hypo}f\\)\uff09\u200b\u662f\u200b\u51f8\u96c6\u200b\u3002</p> <p>\u200b\u6240\u6709\u200b\u7684\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\u90fd\u200b\u65e2\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u4e5f\u200b\u662f\u200b\u51f9\u200b\u51fd\u6570\u200b\u3002</p> <p>\\(f\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u5728\u200b\u5b9a\u4e49\u57df\u200b\u5185\u200b\u4efb\u4f55\u200b\u76f4\u7ebf\u200b\u4e0a\u200b\u90fd\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5373\u200b</p> \\[ \\forall v,\\forall x, g(t; v, x) = f(x + tv) \\] <p>\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002</p> <p>\u200b\u8bb0\u200b\\(f\\)\u200b\u7684\u200b\u5ef6\u4f38\u200b\\(\\tilde f\\)\u200b\u4e3a\u200b\u5206\u6bb5\u200b\u51fd\u6570\u200b</p> \\[ \\tilde f(x) = \\left\\{\\begin{aligned}     &amp; f(x) &amp; x\\in \\mathbf{dom}f \\\\     &amp; \\infty &amp; x\\not\\in \\mathbf{dom}f \\end{aligned}\\right. \\] <p>\u200b\u7c7b\u4f3c\u200b\u5730\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u51f9\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5176\u200b\u5b9a\u4e49\u57df\u200b\u5916\u200b\u7684\u200b\u90e8\u5206\u200b\u5ef6\u4f38\u200b\u81f3\u200b\\(-\\infty\\)\u3002 \u200b\u5728\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e2d\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u793a\u6027\u200b\u51fd\u6570\u200b\\(\\tilde I_C(x) = 0, x\\in C\\)\u200b\u6765\u200b\u5c06\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u89e3\u200b\u9650\u5b9a\u200b\u5230\u200b\u51f8\u96c6\u200b\\(C\\)\u200b\u4e0a\u200b\u3002\\(\\tilde I_C(x)\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002</p> <p>\u200b\u5176\u4ed6\u200b\u7684\u200b\u51f8\u51fd\u6570\u200b\u8fd8\u6709\u200b</p> <ul> <li>\u200b\u6307\u6570\u51fd\u6570\u200b\\(f: \\mathbb R\\rightarrow\\mathbb R = e^{ax}, a\\in \\mathbb R\\)</li> <li>\u200b\u5e42\u51fd\u6570\u200b\\(f: \\mathbb R_{ + + }\\rightarrow\\mathbb R = x^a, a\\geq 1\\)\u200b\u6216\u200b\\(a\\leq 0\\)</li> <li>\u200b\u7edd\u5bf9\u503c\u200b\u5e42\u51fd\u6570\u200b\\(f: \\mathbb R\\rightarrow\\mathbb R = |x|^p, p\\geq 1\\)</li> <li>\u200b\u8d1f\u200b\u5bf9\u6570\u51fd\u6570\u200b\\(f: \\mathbb R_{\\plus\\plus}\\rightarrow \\mathbb R = -\\log x\\)</li> <li>\u200b\u8d1f\u71b5\u200b\\(f: \\mathbb R_{\\plus\\plus}\\rightarrow \\mathbb R = x\\log x\\)</li> <li>\u200b\u8303\u6570\u200b\uff08\u200b\u4e09\u89d2\u200b\u4e0d\u7b49\u5f0f\u200b\uff09<ul> <li>\u200b\u6700\u5927\u503c\u200b\u51fd\u6570\u200b\u76f8\u5f53\u4e8e\u200b\u65e0\u7a77\u200b\u9636\u200b\u8303\u6570\u200b\\(\\Vert\\cdot\\Vert_\\infty\\)\uff0c\u200b\u56e0\u6b64\u200b\u4e5f\u200b\u662f\u200b\u51f8\u200b\u7684\u200b\u3002</li> </ul> </li> <li>\u200b\u6307\u6570\u200b\u548c\u200b\u7684\u200b\u5bf9\u6570\u200b\\(f: \\mathbb R^n\\rightarrow \\mathbb R = \\log \\sum_{i=1}^n e^{x_i}\\)</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(\\alpha\\)\uff0c\u200b\u51f8\u51fd\u6570\u200b\u7684\u200b\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u200b\\(\\{x\\in \\mathbf{dom} f | f(x) \\leq \\alpha\\}\\)\u200b\u662f\u200b\u51f8\u96c6\u200b\u3002\u200b\u6216\u8005\u8bf4\u200b\uff0c\u200b\u51f8\u51fd\u6570\u200b\u7684\u200b\u7b49\u9ad8\u7ebf\u200b\u662f\u200b\u51f8\u96c6\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-function/#_2","title":"\u4e00\u9636\u200b\u6761\u4ef6\u200b\u4e0e\u200b\u4e8c\u9636\u200b\u6761\u4ef6","text":""},{"location":"math/convex-optimization/convex-function/#_3","title":"\u4e00\u9636\u200b\u6761\u4ef6","text":"<p>\u200b\u82e5\u200b\\(f\\)\u200b\u7684\u200b\u5b9a\u4e49\u57df\u200b\u4e3a\u200b\u51f8\u200b\u96c6\u4e14\u200b\u53ef\u5fae\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u7b49\u4ef7\u200b\u4e8e\u200b</p> \\[ f(y) \\geq f(x) + (\\nabla f(x))^\\top (y - x) \\] <p>\u200b\u51f8\u51fd\u6570\u200b\u7684\u200b\u4e00\u9636\u200b\u8fd1\u4f3c\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u5168\u5c40\u200b\u4e0b\u200b\u4f30\u8ba1\u200b\u3002</p> <p></p> <p>\u200b\u4e00\u9636\u200b\u6761\u4ef6\u200b\u540c\u6837\u200b\u4e5f\u200b\u80fd\u200b\u7528\u6765\u200b\u53cd\u6620\u200b\u4e25\u683c\u200b\u51f8\u51fd\u6570\u200b\u3001\u200b\u51f9\u200b\u51fd\u6570\u200b\u548c\u200b\u4e25\u683c\u200b\u51f9\u200b\u51fd\u6570\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-function/#_4","title":"\u4e8c\u9636\u200b\u6761\u4ef6","text":"<p>\u200b\u82e5\u200b\\(f\\)\u200b\u7684\u200b\u5b9a\u4e49\u57df\u200b\u4e3a\u200b\u51f8\u200b\u96c6\u4e14\u200b\u4e8c\u9636\u200b\u53ef\u5fae\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u7b49\u4ef7\u200b\u4e8e\u200b</p> \\[ \\nabla ^2f(x)\\succeq 0 \\] <p>\u200b\u4e8c\u9636\u200b\u6761\u4ef6\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u6765\u200b\u53cd\u6620\u200b\u4e25\u683c\u200b\u51f8\u51fd\u6570\u200b\u3001\u200b\u51f9\u200b\u51fd\u6570\u200b\u548c\u200b\u4e25\u683c\u200b\u51f9\u200b\u51fd\u6570\u200b\u3002\u200b\u6ce8\u610f\u200b\u4e25\u683c\u200b\u51f8\u51fd\u6570\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u6ee1\u8db3\u200b\\(\\nabla^2 f(x)\\succ 0\\)\uff0c\u200b\u4e25\u683c\u200b\u51f9\u200b\u51fd\u6570\u200b\u540c\u7406\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-function/#_5","title":"\u4fdd\u51f8\u200b\u8fd0\u7b97","text":"<p>\u200b\u5bf9\u4e8e\u200b\u51fd\u6570\u200b\\(f, g, f_1, \\ldots f_k: \\bbR^n\\rightarrow \\bbR\\)\uff08\u200b\u5982\u672a\u200b\u6307\u660e\u200b\uff0c\u200b\u5219\u200b\u5047\u5b9a\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff09\uff0c\u200b\u4ee5\u4e0b\u200b\u51fd\u6570\u200b\u4ecd\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff1a</p> <ul> <li> <p>\u200b\u51f8\u51fd\u6570\u200b\u7684\u200b\u7ebf\u6027\u7ec4\u5408\u200b\uff1a\\(f'(x) = \\sum_{i=1}^k \\theta_i f_i(x)\\)</p> \u200b\u6269\u5c55\u200b\u5230\u200b\u79ef\u5206\u200b\u5f62\u5f0f\u200b <p>\u200b\u8bbe\u200b\\(f(x, y)\\)\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(y\\)\uff0c\u200b\u5173\u4e8e\u200b\\(x\\)\u200b\u90fd\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u51fd\u6570\u200b\\(w(y)\\)\u200b\u6ee1\u8db3\u200b\\(w(y)\\geq 0\\)\uff0c\u200b\u5219\u200b</p> \\[ g(x) = \\int_{\\mathcal A} w(y)f(x, y)\\dd y \\] <p>\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\u3002</p> </li> <li> <p>\u200b\u4eff\u5c04\u53d8\u6362\u200b\uff1a\\(f'(x): \\bbR^m\\rightarrow \\bbR = f(Ax + b)\\)\uff0c\u200b\u5176\u4e2d\u200b\\(A\\in \\bbR^{m\\times n}, b\\in \\bbR^n\\)</p> </li> <li> <p>\u200b\u9010\u70b9\u200b\u6700\u5927\u200b\uff1a\\(f'(x) = \\max\\{f_1(x), \\ldots, f_k(x)\\}\\)</p> <p>\u200b\u9010\u70b9\u200b\u6700\u5927\u200b\u53ef\u4ee5\u200b\u7406\u89e3\u200b\u4e3a\u200b\u4e00\u7cfb\u5217\u200b\u51fd\u6570\u200b\u4e0a\u5883\u200b\u56fe\u200b\u7684\u200b\u4ea4\u96c6\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\\(f'\\)\uff0c\u200b\u6709\u200b</p> \\[ \\mathbf{epi} f' = \\bigcap_{i=1}^k \\mathbf{epi} f_k \\] <p>\u200b\u6839\u636e\u200b\u51f8\u96c6\u200b\u7684\u200b\u4fdd\u51f8\u200b\u6027\u8d28\u200b\u53ef\u77e5\u200b\uff0c\\(\\mathbf{epi} f'\\)\u200b\u662f\u200b\u51f8\u96c6\u200b\u3002</p> \u200b\u6269\u5c55\u200b\u5230\u200b\u65e0\u9650\u200b\u4e2a\u200b\u51fd\u6570\u200b\u7684\u200b\u5f62\u5f0f\u200b <p>\u200b\u8bbe\u200b\\(f(x, y)\\)\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(y\\)\uff0c\u200b\u5173\u4e8e\u200b\\(x\\)\u200b\u90fd\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b</p> \\[ g(x) = \\sup_{y\\in \\calA} f(x, y) \\] <p>\u200b\u4e5f\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\u3002</p> </li> <li> <p>\u200b\u590d\u5408\u200b\u51fd\u6570\u200b\uff1a\\(g\\circ f\\)</p> <ul> <li>\u200b\u5f53\u200b\\(g: \\bbR\\rightarrow \\bbR\\)\u200b\u65f6\u200b\uff0c\\(f' = g\\circ f = g(f(x))\\)\u200b\u6ee1\u8db3\u200b<ol> <li>\u200b\u82e5\u200b\\(g\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u4e14\u200b\u975e\u51cf\u200b\uff0c\\(f\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f'\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b</li> <li>\u200b\u82e5\u200b\\(g\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u4e14\u200b\u975e\u589e\u200b\uff0c\\(f\\)\u200b\u4e3a\u200b\u51f9\u200b\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f'\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b</li> </ol> </li> <li>\u200b\u5f53\u200b\\(g: \\bbR^k\\rightarrow \\bbR\\)\u200b\u65f6\u200b\uff0c\\(f' = g\\circ f = g(f_1(x), \\ldots, f_k(x))\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002<ol> <li>\u200b\u82e5\u200b\\(g\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u4e14\u200b\u5728\u200b\u5404\u4e2a\u200b\u5206\u91cf\u200b\u4e0a\u200b\u975e\u51cf\u200b\uff0c\\(f_i\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f'\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b</li> <li>\u200b\u82e5\u200b\\(g\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u4e14\u200b\u5728\u200b\u5404\u4e2a\u200b\u5206\u91cf\u200b\u4e0a\u200b\u975e\u589e\u200b\uff0c\\(f_i\\)\u200b\u4e3a\u200b\u51f9\u200b\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f'\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b</li> </ol> </li> </ul> </li> <li> <p>\u200b\u6700\u5c0f\u5316\u200b\uff1a\u200b\u82e5\u200b\\(f(x, y): \\bbR^m\\times \\bbR^n\\rightarrow \\bbR\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\u5bf9\u4e8e\u200b\u975e\u7a7a\u200b\u51f8\u96c6\u200b\\(C\\)\uff0c\\(\\inf_{y\\in C}f(x, y)\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\u3002</p> </li> <li>\u200b\u900f\u89c6\u200b\u51fd\u6570\u200b\uff1a\\(g(x, t) = tf(x / t)\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-function/#_6","title":"\u5171\u8f6d\u200b\u51fd\u6570","text":"<p>\u200b\u51fd\u6570\u200b\\(f: \\bbR^n\\rightarrow \\bbR\\)\u200b\u7684\u200b\u5171\u8f6d\u200b\u51fd\u6570\u200b\\(f^*: \\bbR^n\\rightarrow\\bbR\\)\u200b\u4e3a\u200b</p> \\[ f^*(y)\\triangleq \\sup_{x\\in \\mathbf{dom}f} (y^\\top x - f(x)) \\] <p>\u200b\u5171\u8f6d\u200b\u51fd\u6570\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002\\(f^{**}\\)\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u548c\u200b\\(f\\)\u200b\u76f8\u7b49\u200b\uff0c\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\u65f6\u200b\uff0c\\(f^{**} = f\\)\u3002</p> \u200b\u5171\u8f6d\u200b\u51fd\u6570\u200b\u7684\u200b\u51f8\u6027\u200b <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\u51fd\u6570\u200b\\(f: \\bbR^n\\rightarrow \\bbR\\)\u200b\u548c\u200b\\(x\\in \\bbR^n\\)\uff0c\\(g(y, x) = y^\\top x - f(x)\\)\u200b\u4e3a\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\uff08\u200b\u4e5f\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff09</p> <p>\\(f^*(y)\\)\u200b\u4e3a\u200b\\(g(y, x)\\)\u200b\u5173\u4e8e\u200b\\(x\\)\u200b\u7684\u200b\u9010\u70b9\u200b\u4e0a\u200b\u786e\u754c\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\u3002</p> <p>\u200b\u6839\u636e\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u51fd\u6570\u200b\\(f\\)\u200b\u53ca\u5176\u200b\u5171\u8f6d\u200b\u6ee1\u8db3\u200b</p> \\[ f(x) + f^*(y)\\geq x^\\top y \\] \u200b\u4e8c\u6b21\u200b\u51fd\u6570\u200b\u7684\u200b\u5171\u8f6d\u200b <p>\u200b\u5bf9\u4e8e\u200b\u4e8c\u6b21\u200b\u51fd\u6570\u200b\\(f(x) = \\frac{1}{2} x^\\top Qx, Q\\in \\bbS_{\\plus\\plus}^n\\)\uff0c\u200b\u5176\u200b\u5171\u8f6d\u200b\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ f^*(y) = \\frac{1}{2} y^\\top Q^{-1}y \\] \u200b\u8303\u6570\u200b\u7684\u200b\u5171\u8f6d\u200b <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\u8303\u6570\u200b\\(\\Vert\\cdot\\Vert\\)\uff0c\u200b\u5176\u200b\u5bf9\u5076\u200b\u8303\u6570\u200b\u4e3a\u200b\\(\\Vert\\cdot\\Vert_*\\)\uff0c\u200b\u5219\u200b\\(f(x) = \\Vert x\\Vert\\)\u200b\u7684\u200b\u5171\u8f6d\u200b\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ f^*(y) = \\left\\{ \\begin{aligned}     &amp; 0 &amp; \\Vert y\\Vert_* \\leq 1 \\\\     &amp; \\infty &amp; \\text{otherwise} \\end{aligned} \\right. \\] <p>\u200b\u51fd\u6570\u200b\u548c\u200b\u7684\u200b\u5171\u8f6d\u200b\u7b49\u4e8e\u200b\u5176\u200b\u5171\u8f6d\u200b\u7684\u200b\u548c\u200b</p> \\[ (f + g)^* = f^* + g^* \\] <p>\u200b\u5bf9\u4e8e\u200b\u53ef\u5fae\u200b\u51fd\u6570\u200b\\(f\\)\uff0c\u200b\u6709\u200b</p> \\[ \\begin{gather*}     f(y) = z^\\top \\nabla f(z) - f(z) \\\\     \\nabla f(z) = y \\end{gather*} \\] \u200b\u7ebf\u6027\u53d8\u6362\u200b\u7684\u200b\u5171\u8f6d\u200b <p>\u200b\u8bbe\u200b\\(g(x) = f(Ax + b)\\)\uff0c\u200b\u5176\u4e2d\u200b\\(A\\in \\bbR^{n\\times n}\\)\u200b\u4e14\u200b\u6ee1\u200b\u79e9\u200b\uff0c\u200b\u5219\u200b</p> \\[ \\begin{aligned} g^*(y) &amp;= \\sup_{x} (x^\\top y - f(Ax + b)) \\\\ &amp;= \\sup_{x} [A^{-1}[(Ax + b) - b]]^\\top y - f(Ax + b) \\\\ &amp;= \\sup_x [(A^{-1}(Ax + b))^\\top y - f(Ax + b)] - b^\\top A^{-\\top}y \\\\ &amp;= \\sup_x [(Ax+b)^\\top (A^{-\\top}y) - f(Ax + b)] - b^\\top A^{-\\top}y \\\\ &amp;= f^*(A^{-\\top}y) - b^\\top A^{-\\top}y \\end{aligned} \\] <p>\u200b\u8bbe\u200b\\(h(x) = af(x) + b\\)\uff0c\u200b\u5219\u200b\uff1a</p> \\[ \\begin{aligned} h^*(y) &amp;= \\sup_x y^\\top x - af(x) - b \\\\ &amp;= \\sup_{x} a(a^{-1}y^\\top x - f(x)) - b \\\\ &amp;= af^*(a^{-1}y) - b \\end{aligned} \\]"},{"location":"math/convex-optimization/convex-function/#_7","title":"\u62df\u51f8\u51fd\u6570","text":"<p>\u200b\u82e5\u200b\u51fd\u6570\u200b\\(f\\)\u200b\u7684\u200b\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u200b\\(\\{x|f(x)\\leq \\alpha\\}\\)\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(\\alpha\\)\u200b\u90fd\u200b\u4e3a\u200b\u51f8\u96c6\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\u62df\u51f8\u51fd\u6570\u200b\u3002\u200b\u82e5\u200b\\(-f\\)\u200b\u4e3a\u200b\u62df\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\u62df\u51f9\u51fd\u6570\u200b\u3002\u200b\u82e5\u200b\\(f\\)\u200b\u65e2\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\uff0c\u200b\u53c8\u200b\u662f\u200b\u62df\u51f9\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\u62df\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\u3002</p> <p>\u200b\u6240\u6709\u200b\u7684\u200b\u51f8\u51fd\u6570\u200b\u90fd\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\u3002\u200b\u4f46\u200b\u62df\u51f8\u51fd\u6570\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff0c\\(\\nabla f(x_0) = 0\\not \\Rightarrow f(x)\\geq f(x_0)\\)</p>"},{"location":"math/convex-optimization/convex-function/#_8","title":"\u6027\u8d28","text":"<p>\u200b\u62df\u51f8\u51fd\u6570\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ul> <li> <p>\u200b\u51fd\u6570\u200b\\(f\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\mathbf{dom} f\\)\u200b\u662f\u200b\u51f8\u96c6\u200b\uff0c\u200b\u4e14\u200b\\(f\\)\u200b\u6ee1\u8db3\u200b</p> \\[ f(\\theta x + (1 - \\theta)y)\\leq \\max\\{f(x), f(y)\\} \\] </li> <li> <p>\u200b\u5bf9\u4e8e\u200b\u5b9a\u4e49\u200b\u5728\u200b\\(\\bbR\\)\u200b\u4e0a\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\\(f\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u5176\u4e2d\u200b\u4e4b\u4e00\u200b</p> <ul> <li>\\(f\\)\u200b\u662f\u200b\u5355\u8c03\u200b\u51fd\u6570\u200b</li> <li>\\(\\exists x_0\\)\uff0c\u200b\u5bf9\u4e8e\u200b\\(x\\geq x_0, f\\)\u200b\u975e\u51cf\u200b\uff0c\\(x\\leq x_0, f\\)\u200b\u975e\u589e\u200b</li> </ul> </li> <li>\u200b\u5bf9\u4e8e\u200b\\(\\bbR\\rightarrow \\bbR\\)\u200b\u4e0a\u200b\u7684\u200b\u4efb\u610f\u200b\u5355\u8c03\u200b\u51fd\u6570\u200b\\(f\\)\uff0c\\(-f\\)\u200b\u4e5f\u200b\u662f\u200b\u5355\u8c03\u200b\u51fd\u6570\u200b\uff0c\u200b\u56e0\u6b64\u200b\\(f\\)\u200b\u662f\u200b\u62df\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-function/#_9","title":"\u4e00\u9636\u200b\u6761\u4ef6\u200b\u4e0e\u200b\u4e8c\u9636\u200b\u6761\u4ef6","text":"<p>\uff08\u200b\u4e00\u9636\u200b\u6761\u4ef6\u200b\uff09\u200b\u5bf9\u4e8e\u200b\u5b9a\u4e49\u200b\u5728\u200b\u51f8\u96c6\u200b\u4e0a\u200b\u7684\u200b\u53ef\u5fae\u200b\u51fd\u6570\u200b\\(f: \\bbR^n\\rightarrow \\bbR, f\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u6ee1\u8db3\u200b</p> \\[ \\forall x, y\\in \\mathbf{dom}f, f(y) \\leq f(x)\\Longrightarrow \\nabla f(x)^\\top (y - x)\\leq 0 \\] \u200b\u4e00\u9636\u200b\u6761\u4ef6\u200b\u7684\u200b\u51e0\u4f55\u200b\u89e3\u91ca\u200b <p>\u200b\u6839\u636e\u200b\u62df\u51f8\u51fd\u6570\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(x\\)\uff0c\u200b\u5176\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u200b\\(S_x = \\{y | f(y)\\leq f(x)\\}\\)\u200b\u5747\u200b\u4e3a\u200b\u51f8\u96c6\u200b\u3002</p> <p>\u200b\u6839\u636e\u200b\u51f8\u96c6\u200b\u7684\u200b\u6027\u8d28\u200b\uff0c\u200b\u4efb\u4f55\u200b\u51f8\u96c6\u200b\u90fd\u200b\u80fd\u200b\u8868\u793a\u200b\u6210\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u5bf9\u5e94\u200b\u534a\u200b\u7a7a\u95f4\u200b\u7684\u200b\u4ea4\u96c6\u200b\u3002\u200b\u96c6\u5408\u200b\\(\\{y | \\nabla f(x)^\\top (y - x)\\leq 0\\}\\)\u200b\u662f\u200b\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u200b\\(S_x\\)\u200b\u5728\u200b\\(x\\)\u200b\u5904\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u5bf9\u5e94\u200b\u7684\u200b\u534a\u200b\u7a7a\u95f4\u200b\u3002</p> <p>\uff08\u200b\u4e8c\u9636\u200b\u6761\u4ef6\u200b\uff09\u200b\u5bf9\u4e8e\u200b\u5b9a\u4e49\u200b\u5728\u200b\u51f8\u96c6\u200b\u4e0a\u200b\u7684\u200b\u4e8c\u6b21\u200b\u53ef\u5fae\u200b\u51fd\u6570\u200b\\(f: \\bbR^n\\rightarrow \\bbR, f\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u6ee1\u8db3\u200b</p> \\[ \\forall x\\in \\mathbf{dom} f, \\nabla f(x) = 0\\Longrightarrow \\nabla^2 f(x)\\succeq 0 \\] <p>\u200b\u5373\u200b\\(f\\)\u200b\u5728\u200b\u68af\u5ea6\u200b\u4e3a\u200b\\(0\\)\u200b\u7684\u200b\u5730\u65b9\u200b\uff0c\u200b\u4e8c\u9636\u200b\u68af\u5ea6\u200b\u6b63\u5b9a\u200b\uff08\u200b\u975e\u8d1f\u200b\uff09\u3002</p> <p>\u200b\u62df\u51f8\u51fd\u6570\u200b\u7684\u200b\u4e8c\u9636\u200b\u68af\u5ea6\u200b\u6700\u200b\u591a\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u8d1f\u200b\u7279\u5f81\u503c\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-function/#_10","title":"\u4fdd\u62df\u200b\u51f8\u200b\u8fd0\u7b97","text":"<p>\u200b\u8bbe\u200b\\(f, g, f_1,\\ldots, f_k\\)\u200b\u4e3a\u200b\u62df\u51f8\u51fd\u6570\u200b\u3002\u200b\u5982\u672a\u200b\u6307\u660e\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u51fd\u6570\u200b\u5b9a\u4e49\u200b\u5728\u200b\\(\\bbR^n \\rightarrow \\bbR\\)\u200b\u4e0a\u200b\u3002\u200b\u4ee5\u4e0b\u200b\u51fd\u6570\u200b\u4e3a\u200b\u62df\u51f8\u51fd\u6570\u200b\uff1a</p> <ul> <li>\u200b\u975e\u8d1f\u200b\u52a0\u6743\u200b\u6700\u5927\u503c\u200b\uff1a\u200b\u8bbe\u200b\\(w_1, \\ldots, w_k\\geq 0, f' = \\max\\{w_1f_1(x), \\ldots, w_kf_k(x)\\}\\)\u200b\u4e3a\u200b\u62df\u51f8\u51fd\u6570\u200b\u3002</li> <li>\u200b\u590d\u5408\u200b\u51fd\u6570\u200b\uff1a\\(f' = g\\circ f\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5176\u4e2d\u200b\\(g: \\bbR\\rightarrow \\bbR\\)\u200b\u5355\u8c03\u200b\u975e\u51cf\u200b\u3002</li> <li>\u200b\u6700\u5c0f\u503c\u200b\uff1a\\(f(x, y): \\bbR^m\\times \\bbR^n\\rightarrow \\bbR\\)\u200b\u5173\u4e8e\u200b\\(x, y\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\u51f8\u96c6\u200b\\(C, g(x) = \\inf_{y\\in C}f(x, y)\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-function/#-","title":"\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570","text":"<p>\u200b\u5b9a\u4e49\u200b\\(\\log 0 = -\\infty\\)\u3002\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u51fd\u6570\u200b\\(f: \\bbR^n\\rightarrow \\bbR_{\\plus}\\)\uff0c\u200b\u6709\u200b\\(\\log f\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\u79f0\u200b\\(f\\)\u200b\u4e3a\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\u3002\u200b\u82e5\u200b\\(1 / f\\)\u200b\u4e3a\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\u3002</p> <p>\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff08\u200b\u6307\u6570\u51fd\u6570\u200b\\(e^x\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\uff09\uff0c\u200b\u975e\u8d1f\u200b\u7684\u200b\u51f9\u200b\u51fd\u6570\u200b\u662f\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\uff0c\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\u662f\u200b\u62df\u51f9\u51fd\u6570\u200b\u3002</p> \u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\u7684\u200b\u5145\u8981\u6761\u4ef6\u200b <p>\\(f: \\bbR^n \\rightarrow \\bbR\\)\u200b\u662f\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b</p> \\[ f(\\theta x + (1 - \\theta)y)\\geq f^\\theta(x)f^{1 - \\theta}(y) \\] <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(x, y\\in \\mathbf{dom} f, 0\\leq\\theta\\leq 1\\)\u200b\u6210\u7acb\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u4e8c\u6b21\u200b\u53ef\u5fae\u200b\u7684\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\\(f\\)\uff0c\u200b\u6709\u200b</p> \\[ f(x)\\nabla^2 f(x)\\preceq \\nabla f(x)^\\top \\nabla f(x) \\] <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(x\\in \\mathbf{dom} f\\)\u200b\u6210\u7acb\u200b\uff0c\u200b\u53cd\u4e4b\u4ea6\u7136\u200b\u3002</p> <p>\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\u4e4b\u200b\u548c\u200b\u4ecd\u200b\u662f\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\uff08\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\u4e4b\u200b\u548c\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u5bf9\u6570\u200b-\u200b\u51f9\u200b\u51fd\u6570\u200b\uff09\uff0c\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u200b\uff08\u200b\u51f9\u200b\uff09\u200b\u51fd\u6570\u200b\u7684\u200b\u4e58\u79ef\u200b\u662f\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u200b\uff08\u200b\u51f9\u200b\uff09\u200b\u51fd\u6570\u200b\u3002</p> \u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\u4e4b\u200b\u548c\u200b\u6027\u8d28\u200b\u7684\u200b\u62d3\u5c55\u200b <p>\u200b\u8bbe\u200b\\(f(x, y): \\bbR^n\\times \\bbR^m\\rightarrow \\bbR\\)\u200b\u6ee1\u8db3\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(y\\)\uff0c\\(f(x, y)\\)\u200b\u5173\u4e8e\u200b\\(x\\)\u200b\u662f\u200b\u5bf9\u6570\u200b-\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b</p> \\[ \\int_{y} f(x, y)\\dd y \\] <p>\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-function/#_11","title":"\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u5b9a\u4e49\u200b\u7684\u200b\u51f8\u6027","text":"<p>\u200b\u8bbe\u200b\\(K\\)\u200b\u4e3a\u200b\u6b63\u5e38\u200b\u9525\u200b\uff0c\u200b\u5b9a\u4e49\u200b\\(f(x)\\)\u200b\u7684\u200b\u589e\u51cf\u200b\u6027\u200b\u5982\u4e0b\u200b\u3002</p> <ul> <li>\\(f\\)\u200b\u4e3a\u200b\\(K\\)-\u200b\u589e\u200b\uff0c\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u6ee1\u8db3\u200b\\(x\\succeq_K y, x\\not = y\\Longrightarrow f(x)\\succ_K f(y)\\)\uff0c\u200b\u82e5\u200b\\(-f\\)\u200b\u4e3a\u200b\\(K\\)-\u200b\u589e\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\\(f\\)\u200b\u4e3a\u200b\\(K\\)-\u200b\u975e\u200b\u589e\u51fd\u6570\u200b\u3002</li> <li>\\(f\\)\u200b\u4e3a\u200b\\(K\\)-\u200b\u975e\u51cf\u200b\uff0c\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(f\\)\u200b\u6ee1\u8db3\u200b\\(x\\succeq_K y\\Longrightarrow f(x)\\succeq_K f(y)\\)</li> </ul> <p>\\(f\\)\u200b\u662f\u200b\\(K\\)-\u200b\u975e\u51cf\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\nabla f\\succeq_{K^*} 0\\)\uff0c\\(f\\)\u200b\u662f\u200b\\(K\\)-\u200b\u589e\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\nabla f\\succ_{K^*} 0\\)\u3002</p> <p>\\(f\\)\u200b\u662f\u200b\\(K\\)-\u200b\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\forall x, y, \\theta f(x) + (1 - \\theta) f(y)\\succeq_K f(\\theta x + (1 - \\theta) y)\\)</p> <p>\\(f\\)\u200b\u662f\u200b\u4e25\u683c\u200b\\(K\\)-\u200b\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\forall x\\not = y, \\theta f(x) + (1 - \\theta) f(y)\\succ_K f(\\theta x + (1 - \\theta) y)\\)</p> <p>\\(f\\)\u200b\u662f\u200b\\(K\\)-\u200b\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\forall w\\succeq_{K^*} 0, w^\\top f(x)\\)\u200b\u4e3a\u200b\u51f8\u51fd\u6570\u200b\uff1b\\(f\\)\u200b\u662f\u200b\u4e25\u683c\u200b\\(K\\)-\u200b\u51f8\u51fd\u6570\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\forall w\\succeq_{K^*} 0, w\\not = 0, w^\\top f\\)\u200b\u662f\u200b\u4e25\u683c\u200b\u51f8\u51fd\u6570\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/","title":"\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898","text":"<p>\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u662f\u200b\u6307\u200b\u5728\u200b\u4e00\u5b9a\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff08\u200b\u7b49\u5f0f\u200b\u6216\u200b\u4e0d\u7b49\u5f0f\u200b\uff09\u200b\u4e0b\u200b\u4f7f\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u53d6\u5f97\u200b\u6700\u5927\u200b\uff08\u200b\u6700\u5c0f\u200b\uff09\u200b\u503c\u200b\u7684\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_2","title":"\u4e00\u822c\u200b\u4f18\u5316\u200b\u95ee\u9898","text":"<p>\\(\\eqref{1}\\)\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p> \\[ \\label{1} \\optim{\\min}{f_0(x)}{\\cases{     \\begin{aligned}     &amp; f_i(x)\\leq 0 &amp; i = \\oneto m\\\\     &amp; h_i(x) = 0 &amp; i = \\oneto p     \\end{aligned} }} \\] <p>\u200b\u5f0f\u200b\\(\\eqref{1}\\)\u200b\u4e2d\u200b\uff0c\\(x\\in \\bbR^n\\)\u200b\u4e3a\u200b\u4f18\u5316\u200b\u53d8\u91cf\u200b\uff0c\\(f_0(x): \\bbR^n \\ra \\bbR\\)\u200b\u4e3a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\uff0c\u200b\u4e0d\u7b49\u5f0f\u200b\\(f_i(x)\\leq 0\\)\u200b\u79f0\u4e3a\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\uff0c\\(h_0(x) = 0\\)\u200b\u79f0\u4e3a\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u3002\\(m = p = 0\\)\u200b\u7684\u200b\u95ee\u9898\u200b\u79f0\u4e3a\u200b\u65e0\u7ea6\u675f\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u548c\u200b\u5168\u4f53\u200b\u7ea6\u675f\u200b\u51fd\u6570\u200b\u7684\u200b\u5b9a\u4e49\u57df\u200b\u4ea4\u96c6\u200b\u662f\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u5b9a\u4e49\u57df\u200b\\(\\calD\\)\uff0c\u200b\u5f53\u200b\u5b58\u5728\u200b\\(x\\)\u200b\u6ee1\u8db3\u200b\u6240\u6709\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u65f6\u200b\uff0c\u200b\u79f0\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u662f\u200b\u53ef\u884c\u200b\u7684\u200b\uff0c\u200b\u5426\u5219\u200b\u662f\u200b\u4e0d\u200b\u53ef\u884c\u200b\u7684\u200b\u3002\u200b\u5b9a\u4e49\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u53ef\u884c\u200b\u57df\u200b\u4e3a\u200b\u5168\u4f53\u200b\u53ef\u884c\u200b\u70b9\u200b\u7684\u200b\u96c6\u5408\u200b\u3002</p> \\[ X = \\{x | f_i(x)\\leq 0, h_j(x) = 0, i = \\oneto m, j = \\oneto p\\} \\] <p>\u200b\u5bf9\u4e8e\u200b\u6700\u5c0f\u5316\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u5176\u200b\u6700\u4f18\u200b\u503c\u4e3a\u200b</p> \\[ p^\\ast = \\inf\\{f_0(x) | x\\in X\\} \\] <p>\u200b\u53ef\u884c\u200b\u57df\u200b\u4e3a\u200b\u7a7a\u96c6\u200b\u7684\u200b\u60c5\u51b5\u200b</p> <p>\u200b\u6211\u4eec\u200b\u5141\u8bb8\u200b\\(p^*\\)\u200b\u53d6\u200b\\(\\pm\\infty\\)\uff0c\u200b\u5e76\u4e14\u200b\u5b9a\u4e49\u200b</p> \\[ \\begin{aligned} \\inf\\varnothing &amp;= \\infty \\\\ \\sup\\varnothing &amp;= -\\infty \\end{aligned} \\] <p>\u200b\u5982\u679c\u200b\u5b58\u5728\u200b\u4e00\u7ec4\u200b\u53ef\u884c\u200b\u89e3\u200b\\(x_1, \\ldots, x_k\\)\u200b\u6ee1\u8db3\u200b\\(k\\ra\\infty\\)\u200b\u65f6\u200b\\(f_0(x_k)\\ra -\\infty\\)\uff0c\u200b\u79f0\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u65e0\u200b\u4e0b\u754c\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u5b58\u5728\u200b\\(x^*\\in X, f_0(x^*) = p^*\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\\(x^*\\)\u200b\u4e3a\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u5168\u4f53\u200b\u6700\u4f18\u200b\u89e3\u200b\u7684\u200b\u96c6\u5408\u200b\u4e3a\u200b\u6700\u4f18\u200b\u96c6\u200b\u3002\u200b\u82e5\u200b\u6700\u4f18\u200b\u96c6\u975e\u7a7a\u200b\uff0c\u200b\u79f0\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u53ef\u89e3\u200b\u3002\u200b\u5f53\u200b\u95ee\u9898\u200b\u4e0d\u53ef\u200b\u89e3\u65f6\u200b\uff0c\u200b\u96c6\u5408\u200b\\(\\{x | x\\in X, f_0(x) \\leq p^* + \\varepsilon\\}\\)\u200b\u79f0\u4e3a\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\\(\\varepsilon\\)-\u200b\u6b21\u4f18\u200b\u96c6\u200b\u3002</p> <p>\u200b\u6700\u4f18\u200b\u503c\u200b\u4e0d\u53ef\u200b\u8fbe\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b</p> <p>\u200b\u8bbe\u200b\u7ea6\u675f\u200b\u95ee\u9898\u200b\u4e3a\u200b\\(\\eqref{2}\\)\uff0c\u200b\u4ee5\u4e0b\u200b\u95ee\u9898\u200b\u90fd\u200b\u662f\u200b\u6700\u4f18\u200b\u503c\u200b\u4e0d\u53ef\u200b\u8fbe\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\optim{\\min}{f_0(x)}{x &gt; 0} \\label{2} \\] <ul> <li>\u200b\u5f53\u200b\\(f_0(x) = 1/x\\)\u200b\u65f6\u200b\uff0c\\(p^* = 0\\)\uff0c\u200b\u4f46\u200b\u6700\u4f18\u200b\u96c6\u4e3a\u200b\u7a7a\u200b\u3002</li> <li>\u200b\u5f53\u200b\\(f_0(x) = -\\log x\\)\u200b\u65f6\u200b\uff0c\\(p^* = -\\infty\\)\uff0c\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u65e0\u200b\u4e0b\u754c\u200b\u3002</li> </ul> <p>\u200b\u5982\u679c\u200b\\(x_0\\)\u200b\u6ee1\u8db3\u200b\\(\\eqref{3}\\)\uff0c\u200b\u79f0\u4e3a\u200b\u5c40\u90e8\u200b\u6700\u4f18\u200b\u3002</p> \\[ \\begin{gathered}     &amp; x_0\\in X \\\\     &amp; \\forall z\\in X, \\Vert z - x_0\\Vert_2\\leq R, f_0(z)\\geq f_0(x_0) \\end{gathered} \\label{3} \\] <p>\u200b\u82e5\u200b\\(f_i(x^*) = 0\\)\uff0c\u200b\u79f0\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\\(f_i\\)\u200b\u5728\u200b\u6700\u4f18\u200b\u503c\u5904\u200b\u8d77\u200b\u4f5c\u7528\u200b\uff0c\u200b\u5426\u5219\u200b\u4e0d\u8d77\u4f5c\u7528\u200b\u3002\u200b\u5982\u679c\u200b\u4ece\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e2d\u200b\u5220\u9664\u200b\u67d0\u4e2a\u200b\u7ea6\u675f\u200b\u4e0d\u4f1a\u200b\u5f71\u54cd\u200b\u53ef\u884c\u200b\u57df\u200b\uff0c\u200b\u79f0\u200b\u7ea6\u675f\u200b\u5197\u4f59\u200b\u3002</p> <p>\u200b\u53ef\u884c\u6027\u200b\u95ee\u9898\u200b</p> <p>\u200b\u5f53\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\\(f_0(x) = 0\\)\u200b\u65f6\u200b\uff0c\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u662f\u200b\u4e00\u4e2a\u200b\u53ef\u884c\u6027\u200b\u95ee\u9898\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b</p> \\[ \\begin{gathered} p^* = -\\infty \\Lolra X = \\varnothing \\\\ p^* = 0 \\Lolra X \\not = \\varnothing \\end{gathered} \\] <p>\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\uff1a</p> <ul> <li>\u200b\u7ea6\u675f\u200b\u7684\u200b\u53f3\u7aef\u200b\u9879\u4e3a\u200b\u96f6\u200b</li> <li>\u200b\u4f18\u5316\u200b\u76ee\u6807\u200b\u4e3a\u200b\u6781\u5c0f\u200b\u5316\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b</li> <li>\u200b\u9664\u4e86\u200b\u975e\u8d1f\u200b\u7ea6\u675f\u200b\u4ee5\u5916\u200b\uff0c\u200b\u4e0d\u200b\u5305\u542b\u200b\\(\\geq\\)\u200b\u7ea6\u675f\u200b</li> </ul>"},{"location":"math/convex-optimization/convex-optimization/#_3","title":"\u95ee\u9898\u200b\u7684\u200b\u7b49\u4ef7\u200b\u53d8\u6362","text":"<p>\u200b\u4e00\u4e9b\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u6027\u8d28\u200b\u53ef\u80fd\u200b\u4e0d\u591f\u200b\u5f3a\u200b\u5bfc\u81f4\u200b\u96be\u4ee5\u200b\u8fdb\u884c\u200b\u4f18\u5316\u200b\uff0c\u200b\u6b64\u65f6\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4e00\u4e9b\u200b\u7b49\u4ef7\u200b\u53d8\u6362\u200b\uff0c\u200b\u5c06\u200b\u539f\u672c\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u8f6c\u53d8\u200b\u4e3a\u200b\u6027\u8d28\u200b\u8f83\u5f3a\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_4","title":"\u53d8\u91cf\u200b\u4ee3\u6362","text":"<p>\u200b\u8bbe\u200b\u51fd\u6570\u200b\\(\\phi: \\bbR^n\\ra\\bbR^n\\)\u200b\u662f\u200b\u53cc\u5c04\u200b\uff0c\u200b\u4e14\u200b\u51fd\u6570\u200b\u7684\u200b\u503c\u57df\u200b\u5305\u542b\u200b\u95ee\u9898\u200b\u7684\u200b\u5b9a\u4e49\u57df\u200b\uff0c\u200b\u5219\u200b\u5982\u4e0b\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e0e\u200b\u539f\u200b\u95ee\u9898\u200b\\(\\eqref{1}\\)\u200b\u7b49\u4ef7\u200b\uff1a</p> \\[ \\optim{\\min}{f_0(\\phi(z))}{\\cases{     \\begin{aligned}         &amp; f_i(\\phi(z))\\leq 0 &amp; i = \\oneto m \\\\         &amp; h_i(\\phi(z)) = 0 &amp; i = \\oneto p     \\end{aligned} }} \\]"},{"location":"math/convex-optimization/convex-optimization/#_5","title":"\u51fd\u6570\u200b\u4ee3\u6362","text":"<p>\u200b\u8bbe\u200b\u51fd\u6570\u200b\\(\\psi_0: \\bbR\\ra\\bbR\\)\u200b\u5355\u8c03\u200b\u9012\u589e\u200b\uff0c\\(\\psi_{1}, \\ldots, \\psi_{m}\\)\u200b\u6ee1\u8db3\u200b\\(u\\leq 0\\Lora \\psi_{i}(u) = 0\\)\uff0c\\(\\phi_{1}, \\ldots, \\psi_{p}\\)\u200b\u6ee1\u8db3\u200b\\(u = 0\\Lora \\phi_i(u) = 0\\)\uff0c\u200b\u5219\u200b\u5982\u4e0b\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e0e\u200b\u539f\u200b\u95ee\u9898\u200b\\(\\eqref{1}\\)\u200b\u7b49\u4ef7\u200b\uff1a</p> \\[ \\optim{\\min}{\\psi_0(f_0(x))}{\\cases{     \\begin{aligned}         &amp; \\psi_i(f_i(z))\\leq 0 &amp; i = \\oneto m \\\\         &amp; \\phi_i(h_i(z)) = 0 &amp; i = \\oneto p     \\end{aligned} }} \\]"},{"location":"math/convex-optimization/convex-optimization/#_6","title":"\u6d88\u9664\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f","text":"<p>\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u52a0\u5165\u200b\u975e\u8d1f\u200b\u53d8\u91cf\u200b\u7684\u200b\u65b9\u5f0f\u200b\u5c06\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u548c\u200b\u975e\u8d1f\u200b\u7ea6\u675f\u200b\u3002</p> \\[ \\optim{\\min}{f_0(x)}{\\cases{     \\begin{aligned}         &amp; s_i\\geq 0 &amp; i = \\oneto m \\\\         &amp; f_i(x) + s_i 0 &amp; i = \\oneto m \\\\         &amp; h_i(x) = 0 &amp; i = \\oneto p     \\end{aligned} }} \\] <p>\u200b\u79f0\u200b\\(s_i\\)\u200b\u4e3a\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b\uff0c\u200b\u6b64\u65f6\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u5171\u6709\u200b\\(n + m\\)\u200b\u4e2a\u200b\u4f18\u5316\u200b\u53d8\u91cf\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_7","title":"\u6d88\u9664\u200b\u7b49\u5f0f\u200b\u7ea6\u675f","text":"<p>\u200b\u8bbe\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u7684\u200b\u89e3\u96c6\u200b\u53ef\u4ee5\u200b\u7528\u200b\u53cc\u5c04\u200b\\(\\phi: \\bbR^k\\ra\\bbR^n\\)\u200b\u8868\u793a\u200b\uff0c\u200b\u5219\u200b\u539f\u200b\u95ee\u9898\u200b\\(\\eqref{1}\\)\u200b\u4e0e\u200b\u5982\u4e0b\u200b\u95ee\u9898\u200b\u7b49\u4ef7\u200b\uff1a</p> \\[ \\optim{\\min}{f_0(\\phi(z))}{f_i(\\phi(z))\\leq 0} \\] <p>\u200b\u5bf9\u4e8e\u200b\u7ebf\u6027\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\\(Ax - b = 0\\)\uff0c\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\\(\\phi(z) = Fz + x_0\\)\uff0c\u200b\u5176\u4e2d\u200b\\(x_0\\)\u200b\u4e3a\u200b\\(Ax - b = 0\\)\u200b\u7684\u200b\u4e00\u4e2a\u200b\u7279\u89e3\u200b\uff0c\\(F\\)\u200b\u4e3a\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u7684\u200b\u96f6\u7a7a\u95f4\u200b\u7684\u200b\u4e00\u7ec4\u200b\u57fa\u200b\u6784\u6210\u200b\u7684\u200b\u77e9\u9635\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_8","title":"\u4f18\u5316\u200b\u90e8\u5206\u200b\u53d8\u91cf","text":"<p>\u200b\u6ce8\u610f\u200b\u5230\u200b\\(\\inf_{x}\\inf{y} f(x, y) = \\inf_{(x, y)}f(x, y) = \\inf_{y}\\inf_x f(x, y)\\)\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u5148\u200b\u4f18\u5316\u200b\u4e00\u90e8\u5206\u200b\u53d8\u91cf\u200b\uff0c\u200b\u518d\u200b\u4f18\u5316\u200b\u53e6\u200b\u4e00\u90e8\u5206\u200b\u53d8\u91cf\u200b\uff0c\u200b\u7531\u6b64\u200b\u5f97\u5230\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e0e\u200b\u539f\u200b\u95ee\u9898\u200b\\(\\eqref{1}\\)\u200b\u7b49\u4ef7\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_9","title":"\u4e0a\u5883\u200b\u56fe\u200b\u95ee\u9898","text":"<p>\u200b\u5982\u4e0b\u200b\u95ee\u9898\u200b\u4e0e\u200b\u539f\u200b\u95ee\u9898\u200b\\(\\eqref{1}\\)\u200b\u7b49\u4ef7\u200b\uff1a</p> \\[ \\optim{\\min}{t}{\\cases{     \\begin{aligned}         &amp; f_0(x) - t\\leq 0 \\\\         &amp; f_i(x)\\leq 0 &amp; i = \\oneto m \\\\         &amp; h_i(x) = 0 &amp; i= \\oneto p     \\end{aligned} }} \\]"},{"location":"math/convex-optimization/convex-optimization/#_10","title":"\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u53ca\u5176\u200b\u6027\u8d28","text":"<p>\\(\\eqref{4}\\)\u200b\u662f\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\optim{\\min}{f_0(x)}{\\cases{     \\begin{aligned}         &amp; f_i(x)\\leq 0 &amp; i = \\oneto m \\\\         &amp; A^\\top x = b     \\end{aligned} }} \\label{4} \\] <p>\u200b\u5e76\u4e14\u200b\uff0c\u200b\u6709\u200b</p> <ul> <li>\\(A\\in \\bbR^{p\\times n}, b\\in \\bbR^p\\)</li> <li>\\(f_i, i\\in 0, \\ldots, m\\)\u200b\u662f\u200b\u51f8\u51fd\u6570\u200b\u3002</li> </ul> <p>\u200b\u5982\u679c\u200b\\(f_0(x)\\)\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5219\u200b\u4e3a\u200b\u62df\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_11","title":"\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u6027\u8d28","text":"<p>\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ul> <li>\u200b\u53ef\u884c\u200b\u57df\u200b\u662f\u200b\u51f8\u96c6\u200b</li> <li>\u200b\u4efb\u610f\u200b\u5c40\u90e8\u200b\u6700\u4f18\u200b\u89e3\u90fd\u200b\u662f\u200b\u5168\u5c40\u200b\u6700\u4f18\u200b\u89e3\u200b</li> <li>\u200b\u5982\u679c\u200b\\(f_0\\)\u200b\u53ef\u5fae\u200b\uff0c\u200b\u5219\u200b\\(\\forall x\\in X, \\nabla f_0(x)^\\top (x - x^*)\\geq 0\\)<ul> <li>\u200b\u5bf9\u4e8e\u200b\u65e0\u7ea6\u675f\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u65b9\u7a0b\u200b\\(\\nabla f_0(x) = 0\\)\u200b\u7684\u200b\u89e3\u5373\u200b\u4e3a\u200b\u6700\u4f18\u200b</li> <li>\u200b\u5bf9\u4e8e\u200b\u4ec5\u200b\u5305\u542b\u200b\u7ebf\u6027\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\\(Ax - b = 0\\)\u200b\u7684\u200b\u95ee\u9898\u200b\uff0c\u200b\u65b9\u7a0b\u200b\\(\\nabla f_0(x)^\\top v = 0, v\\in \\ker A\\)\u200b\u7684\u200b\u89e3\u4e3a\u200b\u6700\u4f18\u200b</li> <li>\u200b\u5bf9\u4e8e\u200b\u4ec5\u200b\u5305\u542b\u200b\u7b26\u53f7\u200b\u7ea6\u675f\u200b\\(x\\succeq 0\\)\u200b\u7684\u200b\u95ee\u9898\u200b\uff0c\\(\\nabla f_0(x)\\succeq 0, \\nabla f_0(x)^\\top x = 0\\)\u200b\u7684\u200b\u89e3\u5373\u200b\u4e3a\u200b\u6700\u4f18\u200b\uff08\u200b\u89e3\u200b\u548c\u200b\u6b64\u5904\u200b\u7684\u200b\u68af\u5ea6\u200b\u81f3\u5c11\u200b\u6709\u200b\u4e00\u4e2a\u200b\u5206\u91cf\u200b\u4e3a\u200b\u96f6\u200b\uff09\u3002</li> </ul> </li> </ul>"},{"location":"math/convex-optimization/convex-optimization/#_12","title":"\u4fdd\u51f8\u200b\u8fd0\u7b97","text":"<p>\u200b\u5bf9\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u8fdb\u884c\u200b\u5982\u4e0b\u200b\u53d8\u6362\u200b\uff0c\u200b\u5f97\u5230\u200b\u7684\u200b\u95ee\u9898\u200b\u4ecd\u7136\u200b\u662f\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b</p> <ul> <li>\u200b\u6d88\u9664\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b</li> <li>\u200b\u5f15\u5165\u200b\uff08\u200b\u7ebf\u6027\u200b\uff09\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b</li> <li>\u200b\u5f15\u5165\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b</li> <li>\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u4e0a\u200b\u5883\u56fe\u200b\u95ee\u9898\u200b</li> <li>\u200b\u5206\u6b65\u200b\u4f18\u5316\u200b\u53d8\u91cf\u200b</li> </ul>"},{"location":"math/convex-optimization/convex-optimization/#_13","title":"\u62df\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u6027\u8d28","text":"<p>\u200b\u62df\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u5b58\u5728\u200b\uff08\u200b\u4e0d\u662f\u200b\u5168\u5c40\u200b\u6700\u4f18\u200b\u7684\u200b\uff09\u200b\u5c40\u90e8\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u53ef\u5fae\u200b\u7684\u200b\u62df\u51f8\u51fd\u6570\u200b\\(f_0\\)\uff0c\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\u4ecd\u7136\u200b\u6210\u7acb\u200b\uff1a</p> \\[ \\forall x\\in X, \\nabla f_0(x)^\\top (x - x^*)\\geq 0 \\] <p>\u200b\u4f46\u200b\u4f7f\u5f97\u200b\\(\\forall x\\in X, \\nabla f_0(x)^\\top (x - y)\\geq 0\\)\u200b\u6210\u7acb\u200b\u7684\u200b\\(y\\)\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u5168\u5c40\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002\u200b\u62df\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u89e3\u200b\u7684\u200b\u5145\u8981\u6761\u4ef6\u200b\u4e3a\u200b</p> \\[ \\nabla f_0(x)^\\top (y - x) &gt; 0, y\\in X - \\{x\\} \\] <p>\u200b\u62df\u51f8\u51fd\u6570\u200b\u7684\u200b\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u662f\u200b\u51f8\u96c6\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e00\u7ec4\u200b\u51f8\u51fd\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u4e0d\u7b49\u5f0f\u200b\u6761\u4ef6\u200b\u8868\u793a\u200b\u5176\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u200b\u3002\u200b\u7531\u6b64\u200b\uff0c\u200b\u62df\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u53ef\u4ee5\u200b\u5bf9\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u8fdb\u884c\u200b\u4e8c\u5206\u200b\u4ee5\u200b\u6784\u9020\u200b\u76f8\u5e94\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u6c42\u89e3\u200b\u6700\u4f18\u200b\u503c\u200b\\(p^*\\)\u3002\u200b\u5bf9\u4e8e\u200b\u5f62\u200b\u5982\u200b\\(\\eqref{4}\\)\u200b\u7684\u200b\u62df\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u6784\u9020\u200b\u53ef\u884c\u6027\u200b\u95ee\u9898\u200b\\(\\eqref{5}\\)\uff1a</p> \\[ \\optim{\\min}{0}{\\cases{     \\begin{aligned}         &amp; \\phi(x; t)\\leq 0 \\\\         &amp; f_i(x)\\leq 0 &amp; i = \\oneto m \\\\         &amp; Ax=b     \\end{aligned} }} \\label{5} \\] <p>\u200b\u5176\u4e2d\u200b\\(\\phi(x, t): \\bbR^n\\times \\bbR\\ra\\bbR\\)\u200b\u662f\u200b\u6ee1\u8db3\u200b\\(f_0(x) \\leq t\\Lolra \\phi(x, t)\\leq 0\\)\u200b\u7684\u200b\u51f8\u51fd\u6570\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u5bf9\u4e8e\u200b\u67d0\u4e2a\u200b\\(t\\)\uff0c\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\\(\\eqref{5}\\)\u200b\u53ef\u884c\u200b\uff0c\u200b\u5219\u200b\u5728\u200b\\(f_0(x) \\geq t\\)\u200b\u7684\u200b\u533a\u95f4\u200b\u5185\u200b\u5bfb\u627e\u200b\u6700\u4f18\u200b\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u5728\u200b\\(f_0(x) &lt; t\\)\u200b\u7684\u200b\u533a\u95f4\u200b\u5185\u200b\u5bfb\u627e\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#-","title":"\u51e0\u4f55\u200b\u89c4\u5212\u200b - \u200b\u8f6c\u5316\u200b\u4e3a\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898","text":"<p>\u200b\u5f62\u200b\u5982\u200b\\(\\eqref{5}\\)\u200b\u7684\u200b\u51fd\u6570\u200b\u79f0\u4e3a\u200b\u5355\u9879\u5f0f\u200b\uff0c\u200b\u5355\u9879\u5f0f\u200b\u7684\u200b\u548c\u200b\u79f0\u4e3a\u200b\u6b63\u9879\u200b\u5f0f\u200b\u3002</p> \\[ f(x) = c\\prod_{i=1}^n x_i^{a_i} \\label{5} \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\uff0c\\(x\\in \\bbR^n_{\\plus\\plus}, c &gt; 0\\)\u3002</p> <p>\u200b\u5f62\u200b\u5982\u200b\\(\\eqref{6}\\)\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e3a\u200b\u51e0\u4f55\u200b\u89c4\u5212\u200b\uff08GP\uff09</p> \\[ \\optim{\\min}{f_0(x)}{\\cases{     \\begin{aligned}         &amp; f_i(x)\\leq 1 &amp; i = \\oneto m \\\\         &amp; h_i(x) = 1 &amp; i = \\oneto p     \\end{aligned} }} \\label{6} \\] <p>\u200b\u5176\u4e2d\u200b\\(f_i\\)\u200b\u4e3a\u200b\u6b63\u200b\u9879\u5f0f\u200b\uff0c\\(h_i\\)\u200b\u4e3a\u200b\u5355\u9879\u5f0f\u200b\u3002</p> <p>\u200b\u51e0\u4f55\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u4e0d\u662f\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u4f46\u662f\u200b\u53ef\u4ee5\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u7b49\u4ef7\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_14","title":"\u4e00\u4e9b\u200b\u5e38\u89c1\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898","text":""},{"location":"math/convex-optimization/convex-optimization/#_15","title":"\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898","text":"<p>\u200b\u5982\u679c\u200b\\(\\eqref{4}\\)\u200b\u4e2d\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u548c\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u90fd\u200b\u662f\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\u65f6\u200b\uff0c\u200b\u6b64\u65f6\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e3a\u200b\u7ebf\u6027\u89c4\u5212\u200b\uff08LP\uff09\u3002</p> \\[ \\optim{\\min}{c^\\top x + d}{\\cases{     \\begin{aligned}         &amp; Gx\\preceq h \\\\         &amp; Ax = b     \\end{aligned} }} \\] <p>\u200b\u6d88\u9664\u200b\u7ebf\u6027\u89c4\u5212\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e2d\u200b\u7684\u200b\u5e38\u6570\u200b\u9879\u200b\u3001\u200b\u5f15\u5165\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b\u4ee5\u200b\u6d88\u9664\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u3001\u200b\u4f7f\u7528\u200b\u975e\u8d1f\u200b\u53d8\u91cf\u200b\u8868\u793a\u200b\u65e0\u7ea6\u675f\u200b\u53d8\u91cf\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\uff1a</p> \\[ \\optim{\\min}{c^\\top x}{\\cases{     \\begin{aligned}         &amp; A^\\top x = b \\\\         &amp; x\\succeq 0     \\end{aligned} }} \\label{7} \\] <p>\u200b\u5982\u679c\u200b\\(\\eqref{4}\\)\u200b\u4e2d\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u662f\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\u4e4b\u200b\u6bd4\u200b\uff0c\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u65f6\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\uff0c\u200b\u6b64\u65f6\u200b\u5f97\u5230\u200b\u7ebf\u6027\u200b\u5206\u5f0f\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u3002</p> \\[ \\optim{\\min}{\\frac{c^\\top x + d}{e^\\top x + f}}{\\cases{     \\begin{aligned}         &amp; Gx\\preceq h \\\\         &amp; Ax = b     \\end{aligned} }} \\] <p>\u200b\u7ebf\u6027\u200b\u5206\u5f0f\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u662f\u200b\u62df\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8f6c\u5316\u200b\u5982\u4e0b\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b</p> \\[ \\optim{\\min}{c^\\top y + dz}{\\cases{     \\begin{aligned}         &amp; Gy - hz\\preceq 0 \\\\         &amp; Ay - bz = 0 \\\\         &amp; e^\\top y + fz = 1 \\\\         &amp; z\\geq 0     \\end{aligned} }} \\] <p>\u200b\u7531\u4e8e\u200b\u4e00\u7ec4\u200b\u62df\u51f8\u51fd\u6570\u200b\u7684\u200b\u6700\u5927\u503c\u200b\u4ecd\u200b\u662f\u200b\u62df\u51f8\u51fd\u6570\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u7ebf\u6027\u200b\u5206\u5f0f\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u66ff\u6362\u200b\u4e3a\u200b\u4e00\u200b\u7ec4\u5206\u200b\u5f0f\u200b\u51fd\u6570\u200b\u7684\u200b\u6700\u5927\u503c\u200b\uff0c\u200b\u6b64\u65f6\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4ecd\u7136\u200b\u662f\u200b\u62df\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_16","title":"\u4e8c\u6b21\u200b\u4f18\u5316\u200b\u95ee\u9898","text":"<p>\u200b\u5982\u679c\u200b\\(\\eqref{4}\\)\u200b\u4e2d\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u548c\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u90fd\u200b\u662f\u200b\u51f8\u200b\u4e8c\u6b21\u200b\u578b\u200b\uff0c\u200b\u6b64\u65f6\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e3a\u200b\u4e8c\u6b21\u200b\u7ea6\u675f\u200b\u4e8c\u6b21\u200b\u89c4\u5212\u200b\uff08QCQP\uff09\u200b\u95ee\u9898\u200b\u3002</p> \\[ \\optim{\\min}{\\frac 12x^\\top P_0x + q_0^\\top x + r}{\\cases{     \\begin{aligned}         &amp; \\frac 12x^\\top P_ix + q_i^\\top x + r_i\\leq 0 &amp; i = \\oneto m \\\\         &amp; Ax = b     \\end{aligned} }} \\label{8} \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\\(P_i\\in \\bbS^n_\\plus, q_i\\in \\bbR^n, A\\in \\bbR^{p\\times n}\\)\u3002</p> <p>\u200b\u5982\u679c\u200b\\(\\eqref{8}\\)\u200b\u4e2d\u200b\u7684\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u662f\u200b\u7ebf\u6027\u200b\u7684\u200b\uff0c\u200b\u5219\u200b\u79f0\u4e3a\u200b\u4e8c\u6b21\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\uff08QP\uff09\u3002</p> \\[ \\optim{\\min}{\\frac 12x^\\top Px + q^\\top x + r}{\\cases{     \\begin{aligned}         &amp; Gx\\preceq h \\\\         &amp; Ax = b     \\end{aligned} }} \\label{9} \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\\(G\\in \\bbR^{m\\times n}\\)</p>"},{"location":"math/convex-optimization/convex-optimization/#_17","title":"\u4e8c\u9636\u200b\u9525\u200b\u89c4\u5212","text":"<p>\u200b\u5f53\u200b\\(\\eqref{4}\\)\u200b\u4e2d\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e3a\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\uff0c\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u6784\u6210\u200b\u4e8c\u9636\u200b\u51f8\u9525\u200b\uff0c\u200b\u6b64\u65f6\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u662f\u200b\u4e8c\u9636\u200b\u9525\u200b\u89c4\u5212\u200b\uff08SOCP\uff09\u3002</p> \\[ \\optim{\\min}{f^\\top x}{\\cases{     \\begin{aligned}         &amp; \\Vert A_ix + b\\Vert_2\\leq c_i^\\top x + d_i &amp; i = \\oneto m \\\\         &amp; Fx = g     \\end{aligned} }} \\label{10} \\] <p>\u200b\u79f0\u200b\\(\\Vert A_ix + b\\Vert_2\\leq c_i^\\top x + d_i\\)\u200b\u4e3a\u200b\u4e8c\u9636\u200b\u9525\u200b\u7ea6\u675f\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_18","title":"\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u4e0b\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898","text":"<p>\u200b\u672c\u200b\u8282\u200b\u8ba8\u8bba\u200b\\(f_i: \\bbR^n\\ra\\bbR^{k_i}\\)\u200b\u5728\u200b\\(\\preceq_{K_i}\\)\u200b\u60c5\u5f62\u200b\u4e0b\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_19","title":"\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f","text":"<p>\u200b\u5f62\u200b\u5982\u4e0b\u200b\u5f0f\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u79f0\u4e3a\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p> \\[ \\optim{\\min}{f_0(x)}{\\cases{     \\begin{aligned}         &amp; f_i\\preceq_{K_i} 0 &amp; i = \\oneto m \\\\         &amp; Ax = b     \\end{aligned} }} \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\\(f_0: \\bbR^n\\ra\\bbR\\)\uff0c\\(K_i\\subseteq \\bbR^{k_i}\\)\u200b\u4e3a\u200b\u6b63\u5e38\u200b\u9525\u200b\uff0c\\(f_i: \\bbR^n\\ra\\bbR^{k_i}\\)\u200b\u4e3a\u200b\\(K_i\\)-\u200b\u51f8\u51fd\u6570\u200b\u3002\u200b\u5927\u591a\u6570\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u7ed3\u8bba\u200b\u5728\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u4e0b\u200b\u4ecd\u7136\u200b\u6210\u7acb\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-optimization/#_20","title":"\u9525\u200b\u89c4\u5212","text":"<p>\u200b\u9525\u200b\u89c4\u5212\u200b\u662f\u200b\u5177\u6709\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b</p> \\[ \\optim{\\min}{c^\\top x}{\\cases{     \\begin{aligned}         &amp; Fx + g\\preceq_K 0 \\\\         &amp; Ax = b     \\end{aligned} }} \\] <p>\u200b\u5f53\u200b\\(K = \\bbR^n_{\\plus\\plus}\\)\u200b\u65f6\u200b\uff0c\u200b\u9525\u200b\u89c4\u5212\u200b\u9000\u5316\u200b\u4e3a\u200bLP\u3002\u200b\u4eff\u7167\u200bLP\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\u7684\u200b\u9525\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u3002</p> \\[ \\optim{\\min}{c^\\top x}{\\cases{     \\begin{aligned}         &amp; x\\succeq_K 0 \\\\         &amp; Ax = b     \\end{aligned} }} \\]"},{"location":"math/convex-optimization/convex-optimization/#_21","title":"\u534a\u5b9a\u200b\u89c4\u5212","text":"<p>\u200b\u5f53\u200b\\(K\\in \\bbS^k_{\\plus}\\)\u200b\u65f6\u200b\uff0c\u200b\u76f8\u5e94\u200b\u7684\u200b\u9525\u200b\u89c4\u5212\u200b\u4e3a\u200b\u534a\u5b9a\u200b\u89c4\u5212\u200b\uff08SDP\uff09\u3002</p> \\[ \\optim{\\min}{c^\\top x}{\\cases{     \\begin{aligned}         &amp; \\sum_{i=1}^n x_iF_i + G\\preceq 0 \\\\         &amp; Ax = b     \\end{aligned} }} \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\\(G, F_i\\in \\bbS^k\\)\u3002\u200b\u82e5\u200b\u8fd9\u4e9b\u200b\u77e9\u9635\u200b\u90fd\u200b\u662f\u200b\u5bf9\u89d2\u200b\u77e9\u9635\u200b\uff0cSDP\u200b\u9000\u5316\u200b\u4e3a\u200bLP\u3002\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\\(\\sum_{i=1}^n x_iF_i\\preceq G\\)\u200b\u79f0\u4e3a\u200b\u7ebf\u6027\u200b\u77e9\u9635\u200b\u4e0d\u7b49\u5f0f\u200b\uff08LMI\uff09\uff0c\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u5206\u5757\u200b\u5bf9\u89d2\u200b\u77e9\u9635\u200b\u5c06\u200b\u591a\u4e2a\u200bLMI\u200b\u7ec4\u5408\u200b\u5728\u200b\u4e00\u8d77\u200b\u3002</p> <p>\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\u7684\u200b\u534a\u5b9a\u200b\u89c4\u5212\u200b\u4e3a\u200b</p> \\[ \\optim{\\min}{\\mathbf{tr}(CX)}{\\cases{     \\begin{aligned}         &amp; \\mathbf{tr(A_iX)} = b_i &amp; i = \\oneto p \\\\         &amp; X\\succeq 0     \\end{aligned} }} \\]"},{"location":"math/convex-optimization/convex-optimization/#_22","title":"\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u5305\u542b\u200b\u5173\u7cfb","text":"<ul> <li>\u200b\u5f53\u200bQP\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e2d\u200b\u7684\u200b\\(P = 0\\)\u200b\u65f6\u200b\uff0cQP\u200b\u9000\u5316\u200b\u4e3a\u200bLP\u200b\u95ee\u9898\u200b</li> <li>\u200b\u5f53\u200bQCQP\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u4e2d\u200b\u7684\u200b\\(P_i = 0\\)\u200b\u65f6\u200b\uff0cQCQP\u200b\u9000\u5316\u200b\u4e3a\u200bQP\u200b\u95ee\u9898\u200b</li> <li>\u200b\u5f53\u200bSOCP\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u4e2d\u200b\u7684\u200b\\(c_i = 0\\)\u200b\u65f6\u200b\uff0cSOCP\u200b\u9000\u5316\u200b\u4e3a\u200bQCQP</li> <li>\u200b\u5f62\u200b\u5982\u200b\\(\\eqref{10}\\)\u200b\u7684\u200bSOCP\u200b\u4e0e\u200b\u5982\u4e0b\u200bSDP\u200b\u95ee\u9898\u200b\u7b49\u4ef7\u200b</li> </ul> \\[ \\optim{\\min}{f^\\top x}{\\cases{     \\begin{bmatrix}         (c_i^\\top x + d_i)I &amp; A_ix + b_i \\\\         (A_ix + b_i)I &amp; (c_i^\\top + d_i)     \\end{bmatrix}\\succeq_{\\bbS_\\plus^n} 0 }} \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\text{LP}\\subseteq \\text{QP}\\subseteq \\text{QCQP}\\subseteq \\text{SOCP}\\subseteq \\text{SDP} \\]"},{"location":"math/convex-optimization/convex-set/","title":"\u51f8\u96c6\u200b\u53ca\u5176\u200b\u6027\u8d28","text":"<p>\u200b\u4ece\u200b\u76f4\u89c2\u200b\u610f\u4e49\u200b\u4e0a\u200b\u8bb2\u200b\uff0c\u200b\u51f8\u96c6\u200b\u5c31\u662f\u200b\u6ca1\u6709\u200b\u51f9\u200b\u8fdb\u53bb\u200b\u7684\u200b\u90e8\u5206\u200b\u7684\u200b\u96c6\u5408\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_2","title":"\u5b9a\u4e49","text":""},{"location":"math/convex-optimization/convex-set/#_3","title":"\u76f4\u7ebf","text":"<p>\u200b\u8bbe\u200b\\(x_1 \\not = x_2\\)\u200b\u4e3a\u200b\\(\\mathbb R^n\\)\u200b\u4e0a\u200b\u7684\u200b\u4e24\u4e2a\u200b\u70b9\u200b\uff0c\u200b\u5219\u200b</p> <ul> <li>\\(y = \\theta x_1 + (1 - \\theta) x_2\\) \u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u6761\u200b\u76f4\u7ebf\u200b\u3002</li> <li>\\(y = \\theta x_1 + (1 - \\theta) x_2, 0\\leq\\theta\\leq 1\\) \u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u6761\u200b\u7ebf\u6bb5\u200b\u3002</li> </ul> <p>\u200b\u76f4\u7ebf\u200b\\(y = \\theta x_1 + (1 - \\theta) x_2\\)\u200b\u662f\u200b\u4e00\u6761\u200b\u4ece\u200b\\(x_2\\)\u200b\u51fa\u53d1\u200b\uff0c\u200b\u6cbf\u200b\u5411\u91cf\u200b\\(x_1 - x_2\\)\u200b\u65b9\u5411\u200b\u5ef6\u4f38\u200b\u7684\u200b\u76f4\u7ebf\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_4","title":"\u4eff\u5c04\u200b\u96c6\u5408","text":"<p>\u200b\u5bf9\u4e8e\u200b\\(\\mathbb R^n\\)\u200b\u4e2d\u200b\u7684\u200b\\(m\\)\u200b\u4e2a\u70b9\u200b\\(x_1, \\ldots, x_m\\)\uff0c\u200b\u82e5\u200b\\(\\sum_{i = 1}^m \\theta_i = 1\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\\(\\sum_{i=1}^m \\theta_ix_i\\)\u200b\u4e3a\u200b\\(x_1, \\ldots, x_m\\)\u200b\u7684\u200b\u4eff\u5c04\u200b\u7ec4\u5408\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u4e00\u7ec4\u200b\u5411\u91cf\u200b\\(x_1, \\ldots, x_m\\)\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u5411\u91cf\u200b\u90fd\u200b\u4e0d\u80fd\u200b\u7531\u200b\u5176\u4ed6\u200b\u5411\u91cf\u200b\u7ecf\u200b\u4eff\u5c04\u200b\u7ec4\u5408\u200b\u5f97\u5230\u200b\uff0c\u200b\u79f0\u200b\u8fd9\u7ec4\u200b\u5411\u91cf\u200b\u4eff\u5c04\u200b\u65e0\u5173\u200b\u3002</p> \u200b\u4eff\u5c04\u200b\u65e0\u5173\u200b\u4e0e\u200b\u7ebf\u6027\u200b\u65e0\u5173\u200b <p>\u200b\u4e00\u7ec4\u200b\u5411\u91cf\u200b\\(x_0, x_1, \\ldots, x_m\\)\u200b\u4eff\u5c04\u200b\u65e0\u200b\uff08\u200b\u76f8\u200b\uff09\u200b\u5173\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u5411\u91cf\u200b\\(x_1 - x_0, \\ldots, x_m - x_0\\)\u200b\u7ebf\u6027\u200b\u65e0\u200b\uff08\u200b\u76f8\u200b\uff09\u200b\u5173\u200b\u3002</p> <p>\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(C \\subseteq \\mathbb R^n\\)\uff0c\u200b\u82e5\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u4e2d\u200b\u7684\u200b\u4efb\u610f\u200b\u4e24\u4e2a\u200b\u4e0d\u540c\u70b9\u200b\u6784\u6210\u200b\u7684\u200b\u76f4\u7ebf\u200b\uff08\u200b\u5373\u200b\u5b83\u4eec\u200b\u7684\u200b\u4eff\u5c04\u200b\u7ec4\u5408\u200b\uff09\u200b\u4ecd\u7136\u200b\u4f4d\u4e8e\u200b\u96c6\u5408\u200b\u4e2d\u200b\uff0c\u200b\u5219\u200b\\(C\\)\u200b\u4e3a\u200b\u4eff\u5c04\u200b\u96c6\u5408\u200b\u3002</p> \\[ \\forall x_1, x_2 \\in C, x_1 \\not = x_2, \\forall \\theta\\in \\mathbb R, y = \\theta x_1 + (1 - \\theta) x_2\\in C \\] <p>\u200b\u4eff\u5c04\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ul> <li>\\(C\\)\u200b\u4e2d\u200b\u5305\u542b\u200b\u5176\u4e2d\u200b\u4efb\u610f\u200b\u4e2a\u70b9\u200b\u7684\u200b\u4eff\u5c04\u200b\u7ec4\u5408\u200b\u3002</li> <li>\u200b\u96c6\u5408\u200b\\(V = \\{x - x_0 | x \\in C\\}\\)\u200b\u4e3a\u200b\\(\\mathbb R^n\\)\u200b\u7684\u200b\u5b50\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u4e14\u200b\u5176\u7ef4\u6570\u200b\u4e0e\u200b\u504f\u79fb\u200b\\(x_0\\)\u200b\u65e0\u5173\u200b\u3002\u200b\u5b9a\u4e49\u200b\u4eff\u5c04\u200b\u96c6\u5408\u200b\u7684\u200b\u7ef4\u6570\u200b\u4e3a\u200b\\(V\\)\u200b\u7684\u200b\u7ef4\u6570\u200b\u3002</li> </ul> <p>\u200b\u8bbe\u200b\\(C\\)\u200b\u4e3a\u200b\u4efb\u610f\u200b\u96c6\u5408\u200b\uff0c\u200b\u5219\u200b\\(C\\)\u200b\u4e2d\u200b\u6240\u6709\u200b\u70b9\u200b\u7684\u200b\u4eff\u5c04\u200b\u7ec4\u5408\u200b\u79f0\u4e3a\u200b\\(C\\)\u200b\u7684\u200b\u4eff\u5c04\u200b\u5305\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(\\mathbf{aff} C\\)\uff0c\u200b\u4eff\u5c04\u200b\u5305\u662f\u200b\u5305\u542b\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u7684\u200b\u6700\u5c0f\u200b\uff08\u200b\u5373\u7ef4\u6570\u200b\u6700\u4f4e\u200b\u7684\u200b\uff09\u200b\u4eff\u5c04\u200b\u96c6\u5408\u200b\uff0c\u200b\u5176\u7ef4\u6570\u200b\u79f0\u4e3a\u200b\u4eff\u5c04\u200b\u7ef4\u6570\u200b\u3002</p> <p>\u200b\u5f53\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u7684\u200b\u4eff\u5c04\u200b\u7ef4\u6570\u200b\\(\\dim \\mathbf{aff} C &lt; n\\)\u200b\u65f6\u200b\uff0c\\(C^\\circ = \\varnothing\\)\u3002\u200b\u6b64\u65f6\u200b\u9700\u8981\u200b\u5b9a\u4e49\u200b\u76f8\u5bf9\u200b\u5185\u90e8\u200b\u4e0e\u200b\u76f8\u5bf9\u200b\u8fb9\u754c\u200b\u3002</p> <ul> <li>\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u7684\u200b\u76f8\u5bf9\u200b\u5185\u90e8\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\\(\\mathbf{relint}C = \\{x\\in C | \\exists r\\in \\mathbb R_{ + + }, B(x, r)\\cap \\mathbf{aff} C \\subseteq C\\}\\)</li> <li>\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u7684\u200b\u76f8\u5bf9\u200b\u8fb9\u754c\u200b\u5b9a\u4e49\u200b\u4e3a\u200b\\(\\mathbf{cl} C - \\mathbf{relint}C\\)\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-set/#_5","title":"\u51f8\u96c6","text":"<p>\u200b\u5bf9\u4e8e\u200b\u96c6\u5408\u200b\\(C\\subseteq \\mathbb R^n\\)\u200b\u4e2d\u200b\u7684\u200b\\(m\\)\u200b\u4e2a\u70b9\u200b\\(x_1, \\ldots, x_m\\)\uff0c\u200b\u82e5\u200b\\(\\sum_{i=1}^m \\theta_i = 1, \\theta_i \\geq 0\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\\(\\sum_{i=1}^m \\theta_i x_i\\)\u200b\u4e3a\u200b\\(x_1, \\ldots, x_m\\)\u200b\u7684\u200b\u51f8\u200b\u7ec4\u5408\u200b\u3002</p> <p>\u200b\u82e5\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\u5185\u200b\u4efb\u610f\u200b\u4e24\u70b9\u200b\u7684\u200b\u51f8\u200b\u7ec4\u5408\u200b\u90fd\u200b\u5728\u200b\u8be5\u200b\u96c6\u5408\u200b\u4e2d\u200b\uff0c\u200b\u79f0\u8be5\u200b\u96c6\u5408\u200b\u4e3a\u200b\u51f8\u96c6\u200b\u3002</p> \\[ \\forall x_1, x_2\\in C, \\forall 0\\leq \\theta\\leq 1, \\theta x_1 + (1 - \\theta) x_2\\in C \\] <p>\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u7684\u200b\u6240\u6709\u200b\u70b9\u200b\u7684\u200b\u51f8\u200b\u7ec4\u5408\u200b\u79f0\u4e3a\u200b\u51f8\u5305\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(\\mathbf{conv}C\\)\uff0c\\(C\\)\u200b\u7684\u200b\u51f8\u200b\u5305\u662f\u200b\u5305\u542b\u200b\\(C\\)\u200b\u7684\u200b\u6700\u5c0f\u200b\u51f8\u96c6\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_6","title":"\u9525","text":"<p>\u200b\u5bf9\u4e8e\u200b\u96c6\u5408\u200b\\(C\\)\uff0c\u200b\u82e5\u200b\\(\\forall x\\in C, \\forall \\theta\\geq 0, \\theta x\\in C\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\\(C\\)\u200b\u4e3a\u200b\u9525\u200b\u3002\u200b\u82e5\u200b\\(C\\)\u200b\u8fd8\u662f\u200b\u51f8\u96c6\u200b\uff0c\u200b\u5219\u200b\u79f0\u4e3a\u200b\u51f8\u9525\u200b\u3002\u200b\u51f8\u9525\u200b\u6ee1\u8db3\u200b</p> \\[ \\forall x_1, x_2\\in C, \\forall \\theta_1, \\theta_2 \\geq 0, \\theta_1x_1 + \\theta_2x_2 \\in C \\] <p>\u200b\u5bf9\u4e8e\u200b\u96c6\u5408\u200b\\(C\\subseteq \\mathbb R^n\\)\u200b\u4e2d\u200b\u7684\u200b\\(m\\)\u200b\u4e2a\u70b9\u200b\\(x_1, \\ldots, x_m\\)\uff0c\u200b\u82e5\u200b\\(\\theta_i \\geq 0\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\\(\\sum_{i=1}^m \\theta_i x_i\\)\u200b\u4e3a\u200b\\(x_1, \\ldots, x_m\\)\u200b\u7684\u200b\u9525\u200b\u7ec4\u5408\u200b\u3002</p> <p>\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u4e2d\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u7ec4\u6210\u200b\u7684\u200b\u6240\u6709\u200b\u9525\u200b\u7ec4\u5408\u200b\u79f0\u4e3a\u200b\u9525\u5305\u200b\uff0c\\(C\\)\u200b\u7684\u200b\u9525\u200b\u5305\u662f\u200b\u5305\u542b\u200b\\(C\\)\u200b\u7684\u200b\u6700\u5c0f\u200b\u51f8\u9525\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_7","title":"\u5e38\u89c1\u200b\u7684\u200b\u51f8\u96c6","text":"<p>\u200b\u4ee5\u4e0b\u200b\u662f\u200b\u4e00\u4e9b\u200b\u5e73\u51e1\u200b\u51f8\u96c6\u200b\uff1a</p> <ul> <li>\u200b\u7a7a\u96c6\u200b\\(\\varnothing\\)\u3001\u200b\u5355\u70b9\u200b\u96c6\u200b\\(\\{x_0\\}\\)\u3001\u200b\u5168\u7a7a\u95f4\u200b\\(\\mathbb R^n\\)\u200b\u53ca\u5176\u200b\u5b50\u200b\u7a7a\u95f4\u200b\u90fd\u200b\u662f\u200b\u4eff\u5c04\u200b\u96c6\u200b\uff08\u200b\u4e5f\u200b\u662f\u200b\u51f8\u96c6\u200b\uff09\u3002</li> <li>\u200b\u4efb\u610f\u200b\u76f4\u7ebf\u200b\u90fd\u200b\u662f\u200b\u51f8\u96c6\u200b\uff1b\u200b\u8fc7\u200b\u539f\u70b9\u200b\u7684\u200b\u76f4\u7ebf\u200b\u662f\u200b\u4eff\u5c04\u200b\u96c6\u200b\uff1b\u200b\u7ebf\u6bb5\u200b\u662f\u200b\u51f8\u96c6\u200b\uff1b\u200b\u5c04\u7ebf\u200b\u662f\u200b\u51f8\u96c6\u200b\uff1b\u200b\u8d77\u70b9\u200b\u5728\u200b\u539f\u70b9\u200b\u7684\u200b\u5c04\u7ebf\u200b\u662f\u200b\u51f8\u200b\u9525\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-set/#_8","title":"\u8d85\u5e73\u9762\u200b\u4e0e\u200b\u534a\u200b\u7a7a\u95f4","text":"<p>\u200b\u8bbe\u200b\\(a\\in \\mathbb R^n, a \\not = 0, b\\in \\mathbb R\\)\uff0c\u200b\u65b9\u7a0b\u200b\\(a^\\top x = b\\)\u200b\u786e\u5b9a\u200b\u4e86\u200b\u7ebf\u6027\u200b\u7a7a\u95f4\u200b\\(\\mathbb R^n\\)\u200b\u5185\u200b\u7684\u200b\u4e00\u4e2a\u200b\u8d85\u5e73\u9762\u200b\uff0c\u200b\u8fd9\u4e2a\u200b\u8d85\u5e73\u9762\u200b\u5c06\u200b\u7a7a\u95f4\u200b\u5206\u4e3a\u200b\u4e24\u200b\u90e8\u5206\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u5f00\u534a\u200b\u7a7a\u95f4\u200b\uff0c\u200b\u6bcf\u200b\u4e00\u90e8\u5206\u200b\u52a0\u4e0a\u200b\u8d85\u5e73\u9762\u200b\u79f0\u4e3a\u200b\u534a\u200b\u7a7a\u95f4\u200b\u3002</p> <ul> <li>\u200b\u8d85\u5e73\u9762\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b\\(\\{x | a^\\top x = b\\}\\)</li> <li>\u200b\u534a\u200b\u7a7a\u95f4\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b\\(\\{x | a^\\top x\\leq b\\}\\)</li> </ul> <p>\u200b\u5411\u91cf\u200b\\(a\\)\u200b\u4e3a\u200b\u8d85\u5e73\u9762\u200b\u7684\u200b\u6cd5\u7ebf\u200b\u65b9\u5411\u200b\uff0c\u200b\u8d85\u5e73\u9762\u200b\u4e0a\u200b\u7684\u200b\u6240\u6709\u200b\u70b9\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u7531\u200b\u4e00\u4e2a\u200b\u8d77\u70b9\u200b\\(x_0\\)\u200b\u52a0\u4e0a\u200b\u4e0e\u200b\\(a\\)\u200b\u6b63\u4ea4\u200b\u7684\u200b\u6240\u6709\u200b\u5411\u91cf\u200b\u5f97\u5230\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_9","title":"\u7403","text":"<p>\\(\\mathbb R^n\\)\u200b\u4e2d\u200b\u7684\u200b\u6b27\u6c0f\u200b\u7403\u4e3a\u200b2-\u200b\u8303\u6570\u200b\u4e0b\u200b\u7684\u200b\u7403\u200b\uff0c\u200b\u5373\u200b</p> \\[ B(x, r) = \\{y | \\Vert y - x\\Vert_2\\leq r\\} \\] <p>\u200b\u5176\u4e2d\u200b\\(x\\in \\mathbb R^n\\)\u200b\u4e3a\u200b\u6b27\u6c0f\u200b\u7403\u200b\u7684\u200b\u7403\u5fc3\u200b\uff0c\\(r\\in \\mathbb R_}\\)\u200b\u4e3a\u200b\u5176\u200b\u534a\u5f84\u200b\u3002 <p>\u200b\u692d\u7403\u200b\u662f\u200b\u5404\u4e2a\u200b\u65b9\u5411\u200b\u5ef6\u4f38\u200b\u7a0b\u5ea6\u200b\u4e0d\u540c\u200b\u7684\u200b\u6b27\u6c0f\u200b\u7403\u200b\uff0c\u200b\u5ef6\u4f38\u200b\u7a0b\u5ea6\u200b\u7531\u200b\u77e9\u9635\u200b\\(P\\in \\mathbb S_{\\)\u200b\u786e\u5b9a\u200b\uff0c\u200b\u534a\u8f74\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(\\sqrt{\\lambda_i}\\)\u3002</p> \\[ \\mathcal E = \\{y | (x - y)^\\top P^{-1}(x - y)\\leq 1\\} \\] <p>\u200b\u5f53\u200b\\(P = r^2I\\)\u200b\u65f6\u200b\uff0c\\(\\mathcal E = B(x, r)\\)\u3002</p> <p>\u200b\u5c06\u200b\u6b27\u6c0f\u200b\u7403\u4e2d\u200b\u7684\u200b2-\u200b\u8303\u6570\u200b\u6362\u6210\u200b\u5176\u4ed6\u200b\u7684\u200b\u8303\u6570\u200b\uff0c\u200b\u5c31\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u8303\u6570\u200b\u7403\u200b\uff1a</p> \\[ B_*(x, r) = \\{y | \\Vert y - x\\Vert_* \\leq r\\} \\]"},{"location":"math/convex-optimization/convex-set/#_10","title":"\u9525","text":"<p>\u200b\u65b9\u7a0b\u200b\\(\\Vert x\\Vert_2 \\leq t\\)\u200b\u5728\u200b\\(R^{n+1}\\)\u200b\u4e0a\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u9525\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u4e8c\u9636\u200b\u9525\u200b\u3002\u200b\u9525\u200b\u53ef\u4ee5\u200b\u89c6\u4e3a\u200b\u7531\u200b\u4f4e\u200b\u4e00\u7ef4\u200b\u7684\u200b\u7403\u200b\u5806\u79ef\u200b\u5f97\u5230\u200b\u3002</p> <p>\u200b\u5c06\u200b2-\u200b\u8303\u6570\u200b\u66ff\u6362\u200b\u4e3a\u200b\u5176\u4ed6\u200b\u8303\u6570\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u8303\u6570\u200b\u9525\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_11","title":"\u591a\u9762\u4f53","text":"<p>\u200b\u6709\u9650\u200b\u4e2a\u200b\u7ebf\u6027\u200b\u7b49\u5f0f\u200b\u548c\u200b\u4e0d\u7b49\u5f0f\u200b\u7684\u200b\u89e3\u96c6\u200b\u6784\u6210\u200b\u591a\u9762\u4f53\u200b\u3002</p> \\[ \\begin{aligned}     \\mathcal P &amp;= \\{x | a_i^\\top x\\leq b_i, i = 1, \\ldots, m, c_j^\\top x = d_j, j = 1, \\ldots, p\\} \\\\     &amp;= \\{x | Ax\\preceq b, Cx = d\\} \\end{aligned} \\] <p>\u200b\u591a\u9762\u4f53\u200b\u53ef\u4ee5\u200b\u662f\u200b\u9525\u200b\uff0c\u200b\u5982\u975e\u200b\u8d1f\u200b\u8c61\u9650\u200b\u3002</p> <p>\u200b\u5355\u7eaf\u5f62\u200b\u662f\u200b\u4e00\u79cd\u200b\u7279\u6b8a\u200b\u7684\u200b\u591a\u9762\u4f53\u200b\uff0c\u200b\u4e00\u7ec4\u200b\u4eff\u5c04\u200b\u65e0\u5173\u200b\u7684\u200b\u70b9\u200b\\(x_0, x_1, \\ldots, x_m\\)\u200b\u7684\u200b\u51f8\u200b\u7ec4\u5408\u200b\u786e\u5b9a\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u5355\u7eaf\u5f62\u200b\uff0c\u200b\u7528\u200b\\(\\mathbf{conv}\\{x_0, \\ldots, x_m\\}\\)\u200b\u8868\u793a\u200b\uff0c\u200b\u5176\u200b\u4eff\u5c04\u200b\u7ef4\u5ea6\u200b\u4e3a\u200b\\(m\\)\u3002</p> \\[ \\mathbf{conv}\\{x_0, \\ldots, x_m\\} = \\left\\{\\theta_0x_0 + \\theta_1x_1 + \\cdots + \\theta_mx_m \\middle| \\theta_i \\geq 0, \\sum_{i=1}^m \\theta_i = 1\\right\\} \\] <ul> <li>\u200b\u5355\u4f4d\u200b\u5355\u7eaf\u5f62\u200b\u662f\u200b\u96f6\u200b\u5411\u91cf\u200b\u548c\u200b\u5355\u4f4d\u5411\u91cf\u200b\u786e\u5b9a\u200b\u7684\u200b\\(n\\)\u200b\u7ef4\u200b\u5355\u7eaf\u5f62\u200b\uff0c\u200b\u5373\u200b\\(\\{x | x\\geq 0, \\mathbf{1}^\\top x \\leq 1\\}\\)\u3002</li> <li>\u200b\u6982\u7387\u200b\u5355\u7eaf\u5f62\u200b\u662f\u200b\u5355\u4f4d\u5411\u91cf\u200b\u786e\u5b9a\u200b\u7684\u200b\\(n-1\\)\u200b\u7ef4\u200b\u5355\u7eaf\u5f62\u200b\uff0c\u200b\u5373\u200b\\(\\{x | \\mathbf 1^\\top x = 1\\}\\)\u3002</li> </ul> \u200b\u5355\u7eaf\u5f62\u200b\u4e0e\u200b\u591a\u9762\u4f53\u200b <p>\u200b\u5355\u7eaf\u5f62\u200b\u662f\u200b\u4e00\u79cd\u200b\u7279\u6b8a\u200b\u7684\u200b\u591a\u9762\u4f53\u200b\u3002\u200b\u8bbe\u200b\u5355\u7eaf\u5f62\u200b\\(C\\)\u200b\u7531\u200b\u4eff\u5c04\u200b\u65e0\u5173\u200b\u7684\u200b\u5411\u91cf\u200b\\(x_0, \\ldots, x_m\\)\u200b\u786e\u5b9a\u200b\uff0c\u200b\u5219\u200b</p> \\[ B = \\begin{bmatrix}     x_1 - x_0 &amp; \\cdots &amp; x_m - x_0 \\end{bmatrix} \\] <p>\\(B\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(n\\times m\\)\u200b\u7ef4\u200b\u7684\u200b\u6ee1\u200b\u79e9\u200b\u77e9\u9635\u200b\u3002\u200b\u4ee4\u200b\\(\\theta = \\begin{pmatrix}\\theta_0 &amp; \\theta_1 &amp; \\cdots &amp; \\theta_m\\end{pmatrix}, \\theta' = \\begin{pmatrix}\\theta_1 &amp; \\cdots &amp; \\theta_m\\end{pmatrix}\\)\u3002\u200b\u5f53\u200b\u7a7a\u95f4\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u70b9\u200b\\(x\\)\u200b\u6ee1\u8db3\u200b\\(x\\in C\\)\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\exists \\theta\\succeq 0, \\mathbf 1^\\top \\theta = 1, x = \\sum_{i=0}^m \\theta_i x_i \\Longleftrightarrow \\exists \\theta'\\succeq 0, \\mathbf 1^\\top \\theta' \\leq 1, x = x_0 + B\\theta' \\] <p>\u200b\u800c\u200b\\(B\\)\u200b\u662f\u200b\u6ee1\u200b\u79e9\u200b\u77e9\u9635\u200b\uff0c\u200b\u5219\u200b\u5b58\u5728\u200b\u6ee1\u200b\u79e9\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u4f7f\u5f97\u200b</p> \\[ AB = \\begin{bmatrix}     A_1 \\\\ A_2 \\end{bmatrix}B = \\begin{bmatrix}     I \\\\ 0 \\end{bmatrix} \\] <p>\u200b\u5219\u200b</p> \\[ Ax = \\begin{bmatrix}     A_1 \\\\ A_2 \\end{bmatrix}x = \\begin{bmatrix}     A_1x \\\\ A_2x \\end{bmatrix} = \\begin{bmatrix}     A_1x_0 + \\theta' \\\\ A_2x_0 \\end{bmatrix} \\] <p>\u200b\u8be5\u200b\u5355\u7eaf\u5f62\u200b\u7684\u200b\u591a\u9762\u4f53\u200b\u8868\u793a\u200b\u4e3a\u200b\uff1a</p> \\[ C = \\{x | A_1x\\succeq A_1x_0, A_2x = A_2x_0, \\mathbf 1^\\top A_1x - \\mathbf 1^\\top A_1x_0\\leq 1\\} \\]"},{"location":"math/convex-optimization/convex-set/#_12","title":"\u534a\u200b\u6b63\u5b9a\u200b\u9525","text":"<p>\u200b\u8bbe\u200b\\(\\mathbb S^n_{+}\\)\u200b\u8868\u793a\u200b\u5168\u4f53\u200b\\(n\\)\u200b\u9636\u200b\u5bf9\u79f0\u200b\u534a\u200b\u6b63\u5b9a\u200b\u77e9\u9635\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u5219\u200b\\(\\mathbb S^n_+\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\u51f8\u9525\u200b\u3002\uff08\u200b\u6ce8\u610f\u200b\u6b64\u5904\u200b\u5bf9\u200b\u9525\u200b\u7684\u200b\u7406\u89e3\u200b\u548c\u200b\u8303\u6570\u200b\u9525\u200b\u4e0d\u540c\u200b\uff09\u3002</p> \\[ A\\succeq 0, B\\succeq 0\\Longrightarrow x^\\top (\\theta_1 A + \\theta_2 B)x \\geq 0\\Longrightarrow (\\theta_1A + \\theta_2B)\\succeq 0 \\]"},{"location":"math/convex-optimization/convex-set/#_13","title":"\u4fdd\u51f8\u200b\u8fd0\u7b97","text":"<p>\u200b\u4fdd\u51f8\u200b\u8fd0\u7b97\u200b\u662f\u200b\u5728\u200b\u8fd0\u7b97\u200b\u524d\u540e\u200b\u4fdd\u6301\u200b\u96c6\u5408\u200b\u51f8\u6027\u200b\u7684\u200b\u8fd0\u7b97\u200b\u3002</p> <ol> <li>\u200b\u90e8\u5206\u200b\u548c\u200b\\(\\{(x, y_1 + y_2) | (x, y_1) \\in A, (x, y_2)\\in B\\}\\)\uff0c\u200b\u6709\u200b\u4e24\u79cd\u200b\u6781\u7aef\u200b\u60c5\u51b5\u200b</li> <li>\\(\\dim y_1 = \\dim y_2 = 0\\)\u200b\u65f6\u200b\u9000\u5316\u200b\u4e3a\u4ea4\u200b\u8fd0\u7b97\u200b\\(\\cap\\)\uff1a\u200b\u4e00\u4e2a\u200b\u51f8\u96c6\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b\u4e00\u7cfb\u5217\u200b\u534a\u200b\u7a7a\u95f4\u200b\u7684\u200b\u4ea4\u96c6\u200b\u3002</li> <li>\\(\\dim x = 0\\)\u200b\u65f6\u200b\u9000\u5316\u200b\u4e3a\u200b\u548c\u200b\\(+: A + B = \\{x + y | x\\in A, y\\in B\\}\\)\u3002</li> <li>\u200b\u7ebf\u6027\u53d8\u6362\u200b\\(f: \\mathbb R^n\\rightarrow \\mathbb R^m = Ax + b\\)\u3002</li> <li>\u200b\u76f4\u79ef\u200b\\(\\times, A\\times B = \\{(x, y) | x\\in A, y\\in B\\}\\)\u3002</li> <li>\u200b\u6295\u5f71\u200b\u51fd\u6570\u200b\\(P: \\mathbb R^n\\times \\mathbb R_}\\rightarrow \\mathbb R^n\\)\uff0c\\(P(z, t) = z / t\\)\u3002\u200b\u6295\u5f71\u200b\u51fd\u6570\u200b\u5c06\u200b\\(\\mathbb R^{n}\\times \\mathbb R_{\\)\u200b\u7a7a\u95f4\u200b\u4e2d\u200b\u7684\u200b\u70b9\u200b\u6295\u5f71\u200b\u5230\u200b\u5e73\u9762\u200b\\(x_{n+1} = 1\\)\u200b\u4e0a\u200b\u3002 <li>\u200b\u7ebf\u6027\u53d8\u6362\u200b\u548c\u200b\u6295\u5f71\u200b\u51fd\u6570\u200b\u7684\u200b\u7ec4\u5408\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u7ebf\u6027\u200b\u5206\u5f0f\u200b\u51fd\u6570\u200b\u3002</li>"},{"location":"math/convex-optimization/convex-set/#_14","title":"\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f","text":"<p>\u200b\u5b9a\u4e49\u200b\u6b63\u5e38\u200b\u9525\u4e3a\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u9525\u200b\uff1a</p> <ul> <li>\u200b\u662f\u200b\u51f8\u200b\u9525\u200b\u3002</li> <li>\u200b\u662f\u200b\u95ed\u96c6\u200b\uff08\u200b\u5b58\u5728\u200b\u8fb9\u754c\u200b\uff09</li> <li>\u200b\u5b58\u5728\u200b\u5185\u90e8\u200b\u3002</li> <li>\u200b\u4e0d\u200b\u5305\u542b\u200b\u76f4\u7ebf\u200b\uff08\\(x\\in K, -x\\in K\\Longleftrightarrow x = 0\\)\uff09</li> </ul> <p>\u200b\u6b63\u5e38\u200b\u9525\u200b\\(K\\)\u200b\u53ef\u4ee5\u200b\u5728\u200b\u7a7a\u95f4\u200b\\(\\mathbb R^n\\)\u200b\u4e0a\u200b\u5b9a\u4e49\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\uff1a</p> \\[ \\begin{gathered}     x \\preceq_K y \\triangleq y - x\\in K \\\\     x \\prec_K y \\triangleq y - x\\in K^\\circ \\end{gathered} \\] <ul> <li>\u200b\u82e5\u200b\\(K = \\mathbb R_+\\)\uff0c\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u5b9e\u6570\u200b\u57df\u200b\u4e0a\u200b\u7684\u200b\u4e0d\u7b49\u5f0f\u200b\u3002</li> <li>\u200b\u82e5\u200b\\(K = \\mathbb R^n_+\\)\uff0c\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\\(\\mathbb R^n\\)\u200b\u7a7a\u95f4\u200b\u4e2d\u200b\u5411\u91cf\u200b\u7684\u200b\u5206\u91cf\u200b\u4e0d\u7b49\u5f0f\u200b\u3002</li> <li>\u200b\u82e5\u200b\\(K = \\mathbb S^n_+\\)\uff0c\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u77e9\u9635\u200b\u4e0d\u7b49\u5f0f\u200b\u3002</li> </ul> <p>\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ul> <li>\\(x\\preceq_K y, u\\preceq_K v\\Longrightarrow x + u\\preceq_K y + v\\)\uff0c\u200b\u5bf9\u200b\\(\\prec_K\\)\u200b\u4e5f\u200b\u6210\u7acb\u200b\u3002</li> <li>\\(x\\preceq_K y, y\\preceq_K z\\Longrightarrow x\\preceq_K z\\)\uff0c\u200b\u5bf9\u200b\\(\\prec_K\\)\u200b\u4e5f\u200b\u6210\u7acb\u200b\u3002</li> <li>\\(x\\preceq_K y, \\alpha\\geq 0\\Longrightarrow \\alpha x\\preceq_K \\alpha y\\)\uff0c\u200b\u5f53\u200b\\(\\alpha &gt; 0\\)\u200b\u65f6\u200b\u5bf9\u200b\\(\\prec_K\\)\u200b\u4e5f\u200b\u6210\u7acb\u200b\u3002</li> <li>\\(x\\prec_K y\\Longrightarrow x\\preceq_K y\\)</li> <li>\\(x\\preceq_K x, x\\not\\prec_K x\\)</li> <li>\\(x\\preceq_K y, y\\preceq_K x\\Longrightarrow x = y\\)</li> <li>\\(\\forall i\\in \\mathbb N, x_i\\preceq_K y_i, \\lim_{i\\rightarrow\\infty} x_i = x, \\lim_{i\\rightarrow\\infty} y_i = y\\Longrightarrow x\\preceq_K y\\)</li> <li>\\(x\\preceq_K y\\Longrightarrow \\exists u, \\exists v, x + u\\prec_K y+ v\\)</li> </ul> <p>\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u4e0d\u662f\u200b\u7ebf\u6027\u200b\u7684\u200b\uff0c\u200b\u5373\u200b\\(\\exists x, \\exists y, x\\not \\preceq_K y, y\\not \\preceq_K x\\)\uff0c\u200b\u6b64\u65f6\u200b\u79f0\u200b\\(x, y\\)\u200b\u4e0d\u53ef\u200b\u6bd4\u200b\u3002</p> <p>\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u504f\u5e8f\u200b\u5173\u7cfb\u200b\u540e\u200b\u5c31\u200b\u53ef\u4ee5\u200b\u5728\u200b\u6b64\u57fa\u7840\u200b\u4e0a\u200b\u5b9a\u4e49\u200b\u6781\u5c0f\u503c\u200b\u4e0e\u200b\u6700\u5c0f\u503c\u200b\u3002</p> <ul> <li>\\(x\\in S\\)\u200b\u662f\u200b\u6700\u5c0f\u200b\u5143\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(S\\subseteq \\{y | y\\succeq_K x\\}\\)\uff0c\u200b\u5373\u200b\\(S\\)\u200b\u4e2d\u200b\u5143\u7d20\u200b\u5168\u90e8\u200b\u4e0e\u200b\\(x\\)\u200b\u53ef\u6bd4\u200b\u4e14\u200b\u5927\u4e8e\u200b\u7b49\u4e8e\u200b\\(x\\)\u3002</li> <li>\\(x\\in S\\)\u200b\u662f\u200b\u6781\u5c0f\u200b\u5143\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(S\\cap \\{y | y\\preceq_K x\\} = \\{x\\}\\)\uff0c\u200b\u5373\u200b\\(S\\)\u200b\u4e2d\u200b\u4e0d\u200b\u5b58\u5728\u200b\u548c\u200b\\(x\\)\u200b\u53ef\u6bd4\u200b\u4e14\u200b\u6bd4\u200b\\(x\\)\u200b\u66f4\u200b\u5c0f\u200b\u7684\u200b\u5143\u7d20\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/convex-set/#_15","title":"\u8d85\u5e73\u9762\u200b\u4e0e\u200b\u51f8\u96c6","text":""},{"location":"math/convex-optimization/convex-set/#_16","title":"\u5206\u79bb\u200b\u8d85\u5e73\u9762","text":"<p>\u200b\u7ed9\u5b9a\u200b\u51f8\u96c6\u200b\\(A, B\\)\u200b\u6ee1\u8db3\u200b\\(A\\cap B = \\varnothing\\)\uff0c\u200b\u5219\u200b\u5b58\u5728\u200b\\(a\\in \\mathbb R^n, b\\in \\mathbb R, a\\not = 0\\)\u200b\u4f7f\u5f97\u200b</p> \\[ \\forall x\\in A, \\forall y\\in B, a^\\top x - b\\leq 0, a^\\top y - b\\geq 0 \\] <p>\u200b\u79f0\u200b\\(\\{x | a^\\top x = b\\}\\)\u200b\u4e3a\u200b\u96c6\u5408\u200b\\(A, B\\)\u200b\u7684\u200b\u5206\u79bb\u200b\u8d85\u5e73\u9762\u200b\u3002</p> <p>\u200b\u82e5\u200b\u8d85\u5e73\u9762\u200b\\(\\{x | a^\\top x = b\\}\\)\u200b\u6ee1\u8db3\u200b</p> \\[ \\forall x\\in A, \\forall y\\in B, a^\\top x - b &lt; 0, a^\\top y - b &gt; 0 \\] <p>\u200b\u79f0\u200b\\(\\{x | a^\\top x = b\\}\\)\u200b\u4e25\u683c\u200b\u5206\u79bb\u200b\\(A, B\\)\uff0c\u200b\u4e25\u683c\u200b\u5206\u79bb\u200b\u7684\u200b\u8d85\u5e73\u9762\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u5b58\u5728\u200b\u3002</p> \u200b\u4e0d\u200b\u5b58\u5728\u200b\u4e25\u683c\u200b\u5206\u79bb\u200b\u7684\u200b\u4f8b\u5b50\u200b <p>\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(A = \\{(x, y) | xy\\geq 1\\}, B = \\{(x, y) | x\\leq 0, y\\geq 0\\}\\)\uff0c\u200b\u5219\u200b\u96c6\u5408\u200b\\(A, B\\)\u200b\u88ab\u200b\u8d85\u5e73\u9762\u200b\\(\\{(x, y) | x = 0\\}\\)\u200b\u548c\u200b\\(\\{(x, y) | y = 0\\}\\)\u200b\u5206\u79bb\u200b\uff0c\u200b\u4f46\u200b\u4e0d\u200b\u5b58\u5728\u200b\u4e25\u683c\u200b\u5206\u79bb\u200b\u4e24\u4e2a\u200b\u96c6\u5408\u200b\u7684\u200b\u8d85\u5e73\u9762\u200b\u3002</p> <p></p> <p>\u200b\u540c\u7406\u200b\uff0c\\(\\{(x, y) | y\\leq \\ln x\\}\\)\u200b\u548c\u200b\\(\\{(x, y) | x\\leq 0\\}\\)\u200b\u4e4b\u95f4\u200b\u4e5f\u200b\u4e0d\u200b\u5b58\u5728\u200b\u4e25\u683c\u200b\u5206\u79bb\u200b\u7684\u200b\u8d85\u5e73\u9762\u200b</p> <p>\u200b\u8d85\u5e73\u9762\u200b\u5206\u79bb\u200b\u7684\u200b\u9006\u5b9a\u7406\u200b\u4e0d\u200b\u6210\u7acb\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_17","title":"\u652f\u6491\u200b\u8d85\u5e73\u9762","text":"<p>\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(C\\subseteq \\mathbb R^n\\)\uff0c\u200b\u5bf9\u4e8e\u200b\\(x_0\\in \\partial C\\)\uff0c\u200b\u82e5\u200b\u5b58\u5728\u200b\\(a\\not = 0\\)\u200b\u6ee1\u8db3\u200b</p> \\[ \\forall x\\in C, a^\\top x\\leq a^\\top x_0 \\] <p>\u200b\u79f0\u200b\\(\\{x| a^\\top x = a^\\top x_0\\}\\)\u200b\u4e3a\u200b\\(C\\)\u200b\u5728\u200b\\(x_0\\)\u200b\u5904\u200b\u7684\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u8fb9\u754c\u200b\u4e0a\u200b\u7684\u200b\u67d0\u4e2a\u200b\u70b9\u200b\uff0c\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u552f\u4e00\u200b\u3002</p> <p>\u200b\u82e5\u200b\u96c6\u5408\u200b\\(C\\)\u200b\u662f\u200b\u51f8\u96c6\u200b\uff0c\u200b\u5219\u200b\u5176\u200b\u8fb9\u754c\u200b\u4e0a\u200b\u6240\u6709\u200b\u70b9\u200b\u90fd\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u3002\u200b\u7ed9\u5b9a\u200b\\(C\\)\u200b\u662f\u200b\u95ed\u96c6\u4e14\u200b\\(C^\\circ\\not =\\varnothing\\)\uff0c\u200b\u5219\u200b\u9006\u5b9a\u7406\u200b\u4e5f\u200b\u6210\u7acb\u200b\u3002</p>"},{"location":"math/convex-optimization/convex-set/#_18","title":"\u5bf9\u5076\u200b\u9525","text":"<p>\u200b\u7ed9\u5b9a\u200b\u9525\u200b\\(K\\)\uff0c\u200b\u82e5\u200b\\(K^*\\)\u200b\u6ee1\u8db3\u200b</p> \\[ y\\in K^* \\Longleftrightarrow \\forall x\\in K, x^\\top y\\geq 0 \\] <p>\u200b\u79f0\u200b\\(K^*\\)\u200b\u4e3a\u200b\\(K\\)\u200b\u7684\u200b\u5bf9\u5076\u200b\u9525\u200b\u3002\u200b\u5bf9\u5076\u200b\u9525\u200b\u5fc5\u5b9a\u200b\u662f\u200b\u51f8\u200b\u9525\u200b\u3002</p> \u200b\u5bf9\u5076\u200b\u9525\u200b\u7684\u200b\u51f8\u6027\u200b <p>\u200b\u7ed9\u5b9a\u200b\u9525\u200b\\(K\\)\uff0c\u200b\u6709\u200b\\(\\forall x\\in K, \\forall \\theta\\geq 0, \\theta x \\in K\\)\uff0c\u200b\u5219\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\\(y\\in K^*\\)</p> \\[ \\theta x^\\top y = x^\\top (\\theta y)\\geq 0\\Longrightarrow \\theta y\\in K^* \\] <p>\u200b\u56e0\u6b64\u200b\\(K^*\\)\u200b\u662f\u200b\u9525\u200b\u3002\u200b\u8bbe\u200b\\(y_1, y_2\\in K^*, x\\in K, 0\\leq \\theta\\leq 1\\)\uff0c\u200b\u5219\u200b</p> \\[ x^\\top (\\theta y_1)\\geq 0, x^\\top(1 - \\theta)y_2\\geq 0\\Longrightarrow x^\\top (\\theta y_1 + (1 - \\theta)y_2)\\geq 0 \\] <p>\u200b\u56e0\u6b64\u200b\\(K^*\\)\u200b\u662f\u200b\u51f8\u200b\u7684\u200b\u3002</p> <p>\u200b\u5bf9\u5076\u200b\u9525\u200b\u53ef\u80fd\u200b\u548c\u200b\u539f\u9525\u200b\u76f8\u7b49\u200b\uff0c\u200b\u5982\u200b\\(\\mathbb R^n_{+}, \\mathbb S^n_{+}\\)\u200b\u7684\u200b\u5bf9\u5076\u200b\u5206\u522b\u200b\u662f\u200b\u5176\u200b\u672c\u8eab\u200b\u3002</p> <ul> <li>\\(K^*\\)\u200b\u662f\u200b\u95ed\u51f8\u200b\u9525\u200b\u3002</li> <li>\\(K_1\\subseteq K_2\\Longrightarrow K_2^*\\subseteq K_1^*\\)</li> <li>\u200b\u82e5\u200b\\(K^\\circ\\not = \\varnothing\\)\uff0c\u200b\u5219\u200b\\(K^*\\)\u200b\u662f\u200b\u5c16\u200b\u7684\u200b\u3002</li> <li>\u200b\u82e5\u200b\\(\\mathbf{cl}K\\)\u200b\u662f\u200b\u5c16\u200b\u7684\u200b\uff0c\u200b\u5219\u200b\\((K^*)^\\circ\\not = \\varnothing\\)</li> <li>\\(K^{**} = \\mathbf{cl}K\\)</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u6b63\u5e38\u200b\u9525\u200b\\(K\\)\uff0c\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u5bf9\u5076\u200b\u9525\u200b\\(K^*\\)\uff0c\u200b\u4e24\u8005\u200b\u90fd\u200b\u80fd\u200b\u7528\u4e8e\u200b\u5b9a\u4e49\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\\(\\preceq_K\\)\uff0c\u200b\u79f0\u200b\\(\\preceq_{K^*}\\)\u200b\u4e3a\u200b\\(\\preceq_K\\)\u200b\u7684\u200b\u5bf9\u5076\u200b\u3002</p> <p>\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7684\u200b\u5bf9\u5076\u200b\\(\\preceq_{K^*}\\)\u200b\u6ee1\u8db3\u200b</p> <ul> <li>\\(x\\preceq_K y\\Longleftrightarrow \\forall \\lambda\\succeq_{K^*}0, \\lambda^\\top x\\leq \\lambda^\\top y\\)</li> <li>\\(x\\prec_K y\\Longleftrightarrow \\forall \\lambda\\succeq_{K^*}0, \\lambda\\not = 0, \\lambda^\\top x &lt; \\lambda^\\top y\\)</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u96c6\u5408\u200b\\(S\\)\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\\(x\\in S\\)\uff1a</p> <ul> <li>\\(x\\)\u200b\u5173\u4e8e\u200b\\(\\preceq_K\\)\u200b\u662f\u200b\u6700\u5c0f\u200b\u5143\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\\(\\forall \\lambda \\succeq_{K^\\ast}0\\)\uff0c\\(x\\)\u200b\u662f\u200b\u6781\u5c0f\u200b\u5316\u200b\\(\\lambda^\\top z, z\\in S\\)\u200b\u7684\u200b\u552f\u4e00\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u5373\u200b\\(\\forall \\lambda \\succeq_{K^\\ast}0, \\{z | \\lambda^\\top x = \\lambda^\\top z\\}\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\u4e25\u683c\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u3002</li> <li>\u200b\u82e5\u200b\\(x\\)\u200b\u5173\u4e8e\u200b\\(\\preceq_K\\)\u200b\u662f\u200b\u6781\u5c0f\u200b\u5143\u200b\uff0c\u200b\u5219\u200b\u5b58\u5728\u200b\\(\\lambda\\succ_{K^*} 0\\)\uff0c\\(\\forall z\\in S, \\lambda^\\top z\\geq \\lambda^\\top x\\)\uff0c\u200b\u4e5f\u200b\u5373\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u4e25\u683c\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u3002\u200b\u5f53\u200b\\(S\\)\u200b\u4e3a\u200b\u51f8\u200b\u96c6\u65f6\u200b\uff0c\u200b\u9006\u5b9a\u7406\u200b\u5bf9\u4e8e\u200b\u5b58\u5728\u200b\\(\\lambda\\succeq_{K^*}0\\)\u200b\u6210\u7acb\u200b\u3002</li> </ul>"},{"location":"math/convex-optimization/duality/","title":"\u5bf9\u5076","text":""},{"location":"math/convex-optimization/duality/#_2","title":"\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u200b\u5076\u51fd\u6570","text":"\\[ \\optim{\\min}{f_0(x)}{\\cases{\\begin{aligned}     &amp; f_i(x)\\leq 0 &amp; i = \\oneto m \\\\     &amp; h_i(x) = 0 &amp; i = \\oneto p \\end{aligned}}} \\label{1} \\] <p>\u200b\u5bf9\u4e8e\u200b\u4e00\u4e2a\u200b\u5f62\u200b\u5982\u200b\\(\\eqref{1}\\)\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff08\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u662f\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff09\uff0c\u200b\u5176\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(L: \\bbR^n\\times \\bbR^m\\times \\bbR^p\\)\uff0c\u200b\u5176\u4e2d\u200b\\(\\lambda, \\nu\\)\u200b\u4e3a\u200b\u5411\u91cf\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u4e58\u5b50\u200b\u5411\u91cf\u200b\u3002</p> \\[ L(x, \\lambda, \\nu) = f_0(x) + \\sum_{i=1}^m \\lambda_if_i(x) + \\sum_{i=1}^p\\nu_i h_i(x) \\label{2} \\] <p>\\(\\eqref{2}\\)\u200b\u5173\u4e8e\u200b\\(x\\)\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\\(g(\\lambda, \\nu)\\)\u200b\u79f0\u4e3a\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u200b\u5076\u51fd\u6570\u200b\u3002</p> \\[ g(\\lambda, \\nu) = \\inf_{x\\in \\calD}\\left(f_0(x) + \\sum_{i=1}^m \\lambda_if_i(x) + \\sum_{i=1}^p\\nu_i h_i(x)\\right) \\label{3} \\] <p>\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u200b\u5076\u51fd\u6570\u200b\u662f\u200b\u539f\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u503c\u200b\u7684\u200b\u4e0b\u754c\u200b\uff1a\\(\\forall \\lambda \\succeq 0, \\forall \\nu, g(\\lambda, \\nu)\\leq p^*\\)\uff0c\u200b\u5e76\u4e14\u200b\u662f\u200b\u51f9\u200b\u51fd\u6570\u200b\u3002\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u200b\u5076\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u7531\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\\(f_0\\)\u200b\u7684\u200b\u5171\u8f6d\u200b\u51fd\u6570\u200b\u8868\u793a\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_3","title":"\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u5076\u200b\u95ee\u9898","text":"<p>\u200b\u8bb0\u200b\\(\\eqref{1}\\)\u200b\u7684\u200b\uff08\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\uff09\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e3a\u200b\uff1a</p> \\[ \\optim{\\max}{g(\\lambda, \\nu)}{\\lambda \\succeq 0} \\] <p>\u200b\u5176\u200b\u6700\u4f18\u200b\u89e3\u200b\\((\\lambda^*, \\nu^*)\\)\u200b\u79f0\u4e3a\u200b\u5176\u200b\u5bf9\u5076\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u662f\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u6807\u51c6\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b</p> <p>\u200b\u8003\u8651\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b</p> \\[ \\optim{\\min}{c^\\top x}{\\cases{\\begin{aligned}     &amp; Ax = b \\\\     &amp; x\\succeq 0 \\end{aligned}}} \\] <p>\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u200b\u5076\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ \\begin{aligned}     g(\\lambda, \\nu) &amp;= \\inf_{x}(c^\\top x - \\lambda^\\top x + \\nu^\\top (Ax - b)) \\\\     &amp;= \\inf_{x}(-\\nu^\\top b + (c - \\lambda + A^\\top \\nu)^\\top x) \\\\     &amp;= \\cases{\\begin{aligned}         &amp; -\\nu^\\top b &amp; c - \\lambda + A^\\top\\nu = 0 \\\\         &amp; -\\infty &amp; \\otherwise     \\end{aligned}} \\end{aligned} \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e3a\u200b</p> \\[ \\optim{\\max}{\\nu^\\top b}{\\cases{\\begin{aligned}     &amp; c - \\lambda + A^\\top \\nu = 0 \\\\     &amp; \\lambda \\succeq 0 \\end{aligned}}} \\] <p>\u200b\u5373\u200b</p> \\[ \\optim{\\max}{\\nu^\\top b}{     c + A^\\top \\nu \\succeq 0 } \\] <p>\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u5bf9\u5076\u200b\u7b49\u4ef7\u200b\u4e8e\u539f\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_4","title":"\u5f31\u200b\u5bf9\u5076\u6027","text":"<p>\u200b\u8bb0\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u4e3a\u200b\\(d^*\\)\u3002\u200b\u6839\u636e\u200b\u5bf9\u200b\u5076\u51fd\u6570\u200b\u7684\u200b\u6027\u8d28\u200b\uff0c\u200b\u6709\u200b\\(d^*\\leq p^*\\)\uff0c\u200b\u6b64\u4e3a\u200b\u5f31\u200b\u5bf9\u5076\u6027\u200b\u3002</p> <ul> <li>\u200b\u82e5\u539f\u200b\u95ee\u9898\u200b\u65e0\u200b\u4e0b\u754c\u200b\uff0c\u200b\u5219\u200b\\(p^* = -\\infty\\)\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b\\(d^* = -\\infty\\)\uff0c\u200b\u5373\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e0d\u200b\u53ef\u884c\u200b\u3002</li> <li>\u200b\u82e5\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u65e0\u4e0a\u754c\u200b\uff0c\u200b\u5219\u200b\\(d^* = \\infty\\)\uff0c\u200b\u6b64\u65f6\u200b\\(p^* = \\infty\\)\uff0c\u200b\u5373\u539f\u200b\u95ee\u9898\u200b\u4e0d\u200b\u53ef\u884c\u200b\u3002</li> </ul> <p>\u200b\u8bb0\u200b\\(p^* - d^*\\geq 0\\)\u200b\u4e3a\u200b\u5bf9\u5076\u200b\u6700\u4f18\u200b\u95f4\u9699\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_5","title":"\u5f3a\u200b\u5bf9\u5076\u6027","text":"<p>\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u5373\u200b\\(p^* - d^* = 0\\)\u3002\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u5e76\u200b\u4e0d\u662f\u200b\u603b\u200b\u6210\u7acb\u200b\uff0c\u200b\u9700\u8981\u200b\u5bf9\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u65bd\u52a0\u200b\u7ea6\u675f\u200b\u51c6\u5219\u200b\u3002</p> <p>\u200b\u82e5\u200b\uff08\u200b\u4efb\u610f\u200b\uff09\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u6ee1\u8db3\u200bSlater\u200b\u6761\u4ef6\u200b\uff0c\u200b\u5373\u200b\\(\\exists x\\in \\mathbf{relint} \\calD\\)\u200b\u4f7f\u5f97\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u4e25\u683c\u200b\u6210\u7acb\u200b\uff0c\u200b\u5373\u200b</p> \\[ \\cases{\\begin{aligned}     &amp; f_i(x)\\leq 0 &amp; i = \\oneto m \\\\     &amp; Ax = b \\end{aligned}} \\] <p>\u200b\u65f6\u200b\uff0c\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u539f\u200b\u95ee\u9898\u200b\u4e2d\u200b\u7684\u200b\u67d0\u4e2a\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u662f\u200b\u7ebf\u6027\u200b\u7684\u200b\uff0c\u200b\u5219\u200b\u8fd9\u4e2a\u200b\u7ea6\u675f\u200b\u4e0d\u200b\u9700\u8981\u200b\u6ee1\u8db3\u200b\u4e25\u683c\u200b\u6210\u7acb\u200b\u7684\u200b\u6761\u4ef6\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_6","title":"\u51e0\u4f55\u200b\u89e3\u91ca","text":"<p>\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(\\calG = \\{(f_1(x), \\ldots, f_m(x), h_1(x), \\ldots, h_p(x), f_0(x))\\in \\bbR^m\\times \\bbR^p\\times\\bbR | x\\in \\calD\\}\\)\uff0c\u200b\u5219\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u4e3a\u200b</p> \\[ p^* = \\inf_t\\{(u, v, t)\\in \\calG | u\\preceq 0, v = 0\\} \\] <p>\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u5bf9\u200b\u5076\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b\\(g(\\lambda, \\nu) = \\inf\\{ (\\lambda, \\nu, 1)^\\top (u, v, t) | (u, v, t)\\in \\calG\\}\\leq (\\lambda, \\nu, 1)^\\top (u, v, t)\\)\uff0c\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u4e00\u4e2a\u200b\u8d85\u5e73\u9762\u200b\uff0c\u200b\u5176\u6cd5\u200b\u5411\u91cf\u200b\u4e3a\u200b\\((\\lambda, \\nu, 1)\\)\uff0c\u200b\u4e0e\u200b\u76f4\u7ebf\u200b\\(u=v=0\\)\u200b\u76f8\u4ea4\u200b\u4e8e\u200b\\((0, 0, g(\\lambda, \\nu))\\)\u3002\u200b\u4e5f\u200b\u5373\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u89e3\u200b\u90fd\u200b\u662f\u200b\u96c6\u5408\u200b\\(\\calG\\)\u200b\u7684\u200b\u4e00\u4e2a\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u3002</p> <p>\u200b\u5b9a\u4e49\u200b\\(\\calG\\)\u200b\u7684\u200b\u4e0a\u200b\u5883\u56fe\u200b\\(\\calA = \\calG + \\bbR_\\plus^m\\times\\{0\\}^p\\times \\bbR_\\plus\\)\uff0c\u200b\u5373\u200b</p> \\[ \\calA = \\{(u_0 + x, v_0, t_0 + z) | (u_0, v_0, t_0)\\in \\calG, x\\in \\bbR_\\plus^m, z\\in \\bbR_\\plus\\} \\] <p>\u200b\u5219\u200b\u5bf9\u5e94\u200b\u7684\u200b\u539f\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u503c\u4e3a\u200b</p> \\[ p^* = \\inf\\{t | (0, 0, t)\\in \\calA\\} \\] <p>\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\calA\\)\u200b\u4e2d\u200b\u5b58\u5728\u200b\u4e00\u7ec4\u200b\\((u, v, t)\\in \\mathbf{bd}\\calA\\)\u200b\u4f7f\u5f97\u200b\u5728\u200b\u8be5\u5904\u200b\u7684\u200b\u652f\u6491\u200b\u8d85\u5e73\u9762\u200b\u4e0e\u200b\u5750\u6807\u8f74\u200b\\(u=v=0\\)\u200b\u76f8\u4ea4\u200b\u4e8e\u200b\\((0, 0, p^*)\\)\uff0c\u200b\u4e5f\u200b\u5373\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\calA\\)\u200b\u662f\u200b\u51f8\u96c6\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_7","title":"\u6700\u5927\u200b\u6700\u5c0f\u200b\u89e3\u91ca","text":"<p>\u200b\u8bbe\u200b\\(p = 0\\)\u3002\u200b\u5219\u200b\u539f\u200b\u95ee\u9898\u200b\u53ef\u4ee5\u200b\u5199\u4f5c\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b\u65e0\u7ea6\u675f\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\optim{\\min}{\\sup_{\\lambda\\succeq 0}L(x, \\lambda)}{x\\in \\bbR^n} \\] <p>\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u53ef\u4ee5\u200b\u5199\u4f5c\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\optim{\\max}{\\inf_{x} L(x, \\lambda)}{\\lambda\\succeq 0} \\] <p>\u200b\u56e0\u6b64\u200b\u5f31\u200b\u5bf9\u5076\u6027\u200b\u53ef\u4ee5\u200b\u5199\u4f5c\u200b</p> \\[ \\sup_{\\lambda\\succeq 0}\\inf_{x}L(x, \\lambda) \\leq \\inf_{x}\\sup_{\\lambda\\succeq 0}L(x, \\lambda) \\] <p>\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u53ef\u4ee5\u200b\u5199\u4f5c\u200b</p> \\[ \\sup_{\\lambda\\succeq 0}\\inf_{x}L(x, \\lambda) = \\inf_{x}\\sup_{\\lambda\\succeq 0}L(x, \\lambda) \\] <p>\u200b\u5373\u200b\\(L(x, \\lambda)\\)\u200b\u6ee1\u8db3\u200b\u978d\u70b9\u200b\u6027\u8d28\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_8","title":"\u6700\u4f18\u6027\u200b\u6761\u4ef6","text":"<p>\u200b\u5982\u679c\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u53ef\u884c\u200b\u89e3\u200b\\((\\lambda, \\nu)\\)\uff0c\u200b\u5219\u200b\u8bf4\u660e\u200b\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u200b\u6ee1\u8db3\u200b\\(p^*\\geq g(\\lambda, \\nu)\\)\u3002\u200b\u6b64\u65f6\u200b\u5bf9\u4e8e\u200b\u4e00\u4e2a\u200b\u53ef\u884c\u200b\u89e3\u200b\\(x\\)\uff0c\\(x\\)\u200b\u662f\u200b\u4e00\u4e2a\u200b\\(\\varepsilon = f_0(x) - g(\\lambda, \\nu)\\)-\u200b\u6b21\u4f18\u200b\u89e3\u200b\u3002\u200b\u79f0\u200b\u6b64\u5904\u200b\u7684\u200b\\(f_0(x) - g(\\lambda, \\nu)\\)\u200b\u4e3a\u200b\u5bf9\u5076\u200b\u95f4\u9699\u200b\u3002</p> <p>\u200b\u539f\u200b\u95ee\u9898\u200b\u548c\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u200b\u6ee1\u8db3\u200b</p> \\[ p^*\\in [g(\\lambda, \\nu), f_0(x)]\\qquad d^*\\in [g(\\lambda, \\nu), f_0(x)] \\] <p>\u200b\u53ef\u4ee5\u200b\u76f4\u63a5\u200b\u5f97\u51fa\u200b\u63a8\u8bba\u200b\uff1a\u200b\u82e5\u200b\\(f_0(x) = g(\\lambda, \\nu)\\)\uff0c\u200b\u5219\u200b\u6709\u200b\\(p^*=d^*=f_0(x)\\)</p>"},{"location":"math/convex-optimization/duality/#_9","title":"\u4e92\u8865\u200b\u677e\u5f1b\u200b\u6027","text":"<p>\u200b\u5f53\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\u65f6\u200b\uff0c\u200b\u4e92\u8865\u200b\u677e\u5f1b\u200b\u6027\u6307\u200b\u5bf9\u4e8e\u200b\u539f\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u89e3\u200b\\(x^*\\)\u200b\u53ca\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u89e3\u200b\\(\\lambda^* = (\\lambda_1^*, \\ldots, \\lambda_m^*)\\)\uff0c\u200b\u6ee1\u8db3\u200b</p> \\[ \\lambda_i^* &gt; 0 \\Lora f_i(x^*) = 0 \\] <p>\u200b\u5373\u200b\u5bf9\u5076\u200b\u6700\u4f18\u200b\u89e3\u4e2d\u200b\u7684\u200b\u975e\u200b\u96f6\u200b\u5206\u91cf\u200b\u5bf9\u5e94\u200b\u539f\u200b\u95ee\u9898\u200b\u4e2d\u200b\u7684\u200b\u7d27\u200b\u7ea6\u675f\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#kkt","title":"KKT\u200b\u6700\u4f18\u200b\u6761\u4ef6","text":"<p>\u200b\u8bbe\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u548c\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u5747\u200b\u53ef\u5fae\u200b\uff0c\\(x^*\\)\u200b\u4e3a\u539f\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\\(\\lambda^*, \\nu^*\\)\u200b\u4e3a\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u4e14\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\uff0c\u200b\u5219\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u51fd\u6570\u200b\\(L(x, \\lambda^*, \\nu^*)\\)\u200b\u5728\u200b\\(x^*\\)\u200b\u5904\u200b\u53d6\u5f97\u200b\u6700\u5c0f\u503c\u200b\uff0c\u200b\u5219\u200b</p> \\[ \\nabla L(x^*, \\lambda^*, \\nu^*) = 0 \\] <p>\u200b\u7efc\u5408\u200b\u5176\u5b83\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200bKKT\u200b\u6700\u4f18\u200b\u6761\u4ef6\u200b\u3002</p> \\[ \\cases{ \\begin{aligned}     f_i(x^*) &amp; \\leq 0 &amp; \\text{\uff08\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\uff09} \\\\     h_i(x^*) &amp; = 0 &amp; \\text{\uff08\u200b\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\uff09} \\\\     \\lambda_i^* &amp; \\geq 0 &amp; \\text{\uff08\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7ea6\u675f\u200b\uff09} \\\\     \\lambda_i^* f_i(x^*) &amp;= 0 &amp; \\text{\uff08\u200b\u4e92\u8865\u200b\u677e\u5f1b\u200b\u6027\u200b\uff09} \\\\     \\nabla L(x^*, \\lambda^*, \\nu^*) &amp;= 0 &amp; \\text{\uff08\u200b\u6700\u4f18\u200b\u6761\u4ef6\u200b\uff09}\\\\ \\end{aligned} } \\] <p>\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u82e5\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\uff0c\u200b\u5219\u200b\u6700\u4f18\u200b\u89e3\u200b\u5fc5\u987b\u200b\u6ee1\u8db3\u200bKKT\u200b\u6761\u4ef6\u200b\uff08\u200b\u53cd\u8fc7\u6765\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u6210\u7acb\u200b\uff09\u3002\u200b\u5bf9\u4e8e\u200b\u6ee1\u8db3\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u7684\u200b\u51f8\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\uff0c\u200b\u6ee1\u8db3\u200bKKT\u200b\u6761\u4ef6\u200b\u7684\u200b\u70b9\u200b\u662f\u200b\u539f\u200b\u95ee\u9898\u200b\u548c\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002</p>"},{"location":"math/convex-optimization/duality/#_10","title":"\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\u7684\u200b\u5bf9\u5076\u200b\u95ee\u9898","text":"<p>\u200b\u8bbe\u539f\u200b\u95ee\u9898\u200b\u4e3a\u200b</p> \\[ \\optim{\\min}{f_0(x)}{\\cases{     \\begin{aligned}         &amp; f_i(x)\\preceq_{K_i} 0 \\\\         &amp; h_i(x) = 0     \\end{aligned} }} \\] <p>\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ L(x, \\lambda, \\nu) = f_0(x) + \\sum_{i=1}^m \\lambda_i^\\top f_i(x) + \\sum_{j=1}^p \\nu_j^\\top h_j(x) \\] <p>\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e3a\u200b</p> \\[ \\optim{\\max}{g(\\lambda, \\nu) = \\inf_{x}L(x, \\lambda, \\nu)}{\\lambda_i\\succeq_{K_i^*}0} \\] <p>\u200b\u5728\u200b\u5e7f\u4e49\u200b\u4e0d\u7b49\u5f0f\u200b\u4e0b\u200b\uff0c\u200b\u5f31\u200b\u5bf9\u5076\u6027\u200b\u603b\u662f\u200b\u6210\u7acb\u200b\u3002\u200b\u6ee1\u8db3\u200b\uff08\u200b\u5e7f\u4e49\u200b\uff09Slater\u200b\u6761\u4ef6\u200b\\(\\eqnref{4}\\)\u200b\u65f6\u200b\uff0c\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\u3002</p> \\[ \\forall i=1,\\ldots, m,\\exists x\\in \\mathbf{relint}\\calD, Ax=b, f_i(x)\\prec_{K_i}0 \\label{4} \\] <p>\u200b\u5f3a\u200b\u5bf9\u5076\u6027\u200b\u6210\u7acb\u200b\u65f6\u200b\uff0c\u200b\u4e92\u8865\u200b\u677e\u5f1b\u200b\u6027\u200b\u548c\u200bKKT\u200b\u6761\u4ef6\u200b\u6210\u7acb\u200b\u3002</p>"},{"location":"math/convex-optimization/unconstrained-optimization/","title":"\u65e0\u7ea6\u675f\u200b\u4f18\u5316","text":"<p>\u200b\u65e0\u7ea6\u675f\u200b\u4f18\u5316\u200b\u662f\u200b\u6307\u200b\u5728\u200b\\(\\bbR^n\\)\u200b\u4e0a\u200b\u6700\u5c0f\u5316\u200b\\(f(x)\\)\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u3002\u200b\u4e00\u822c\u6765\u8bf4\u200b\uff0c\\(f: \\bbR^n\\ra \\bbR\\)\u200b\u662f\u200b\u4e8c\u6b21\u200b\u53ef\u5fae\u200b\u7684\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u4e14\u200b\u5728\u200b\\(\\mathbf{dom} f\\)\u200b\u4e0a\u200b\u5b58\u5728\u200b\u6700\u5c0f\u503c\u200b\\(p^*\\)\uff0c\u200b\u4e14\u200b\u552f\u4e00\u200b\u5bf9\u5e94\u200b\u4e00\u4e2a\u200b\u89e3\u4e3a\u200b\\(x^*\\)\u3002\u200b\u6839\u636e\u200b\u6700\u4f18\u6027\u200b\u6761\u4ef6\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\nabla f(x^*) = 0 \\label{1} \\] <p>\\(\\eqref{1}\\)\u200b\u6709\u200b\u4e24\u79cd\u200b\u6c42\u89e3\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5373\u200b\u8fed\u4ee3\u6cd5\u200b\u6c42\u200b\u6570\u503c\u200b\u89e3\u200b\u6216\u200b\u76f4\u63a5\u200b\u6c42\u89e3\u200b\u65b9\u7a0b\u200b\u7684\u200b\u89e3\u6790\u200b\u89e3\u200b\u3002\u200b\u5728\u200b\u90e8\u5206\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\\(\\nabla f\\)\u200b\u53ef\u80fd\u200b\u96be\u4ee5\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u6b64\u65f6\u200b\u9700\u8981\u200b\u4f7f\u7528\u200b\u8fed\u4ee3\u6cd5\u200b\\(\\eqref{2}\\)\u3002\\(x^{(0)}, x^{(1)}, \\ldots\\)\u200b\u79f0\u4e3a\u200b\\(\\eqref{1}\\)\u200b\u7684\u200b\u6781\u5c0f\u200b\u5316\u70b9\u5217\u200b\u3002</p> \\[ x^{(0)}, x^{(1)}, \\ldots \\qquad \\lim_{k\\ra \\infty} f\\left(x^{(k)}\\right) = p^* \\label{2} \\] <p>\u200b\u5bf9\u4e8e\u200b\u8fed\u4ee3\u6cd5\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u4e0d\u4f1a\u200b\u5728\u200b\u6709\u9650\u200b\u65f6\u95f4\u200b\u5185\u200b\u7ed3\u675f\u200b\u3002\u200b\u4f46\u200b\u5f53\u200b\\(f\\left(x^{(k)}\\right) - p^*\\leq \\varepsilon\\)\u200b\u65f6\u200b\uff0c\u200b\u5373\u53ef\u200b\u8ba4\u4e3a\u200b\u8fbe\u5230\u200b\u6700\u4f18\u200b\u503c\u200b\uff0c\u200b\u7b97\u6cd5\u200b\u7ec8\u6b62\u200b\u3002\u200b\u8fed\u4ee3\u200b\u7b97\u6cd5\u200b\u5bf9\u200b\u521d\u59cb\u200b\u70b9\u200b\\(x_0\\)\u200b\u6709\u200b\u4e00\u5b9a\u200b\u8981\u6c42\u200b\uff0c\u200b\u5373\u200b\\(f\\)\u200b\u5728\u200b\\(x_0\\)\u200b\u5904\u200b\u7684\u200b\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u662f\u200b\u95ed\u96c6\u200b\u3002</p>"},{"location":"math/convex-optimization/unconstrained-optimization/#_2","title":"\u5f3a\u51f8\u6027","text":"<p>\u200b\u51fd\u6570\u200b\\(f\\)\u200b\u5982\u679c\u200b\u6ee1\u8db3\u200b\u4e00\u9636\u200b\u6761\u4ef6\u200b</p> \\[ f(y)\\geq f(x) + \\nabla f(x)^\\top (y - x) + \\frac{\\gamma}{2} \\Vert y - x\\Vert^2 \\label{3} \\] <p>\u200b\u6216\u200b\u4e8c\u9636\u200b\u6761\u4ef6\u200b</p> \\[ \\nabla^2 f(x)\\succeq \\gamma I \\label{4} \\] <p>\u200b\u5219\u200b\u79f0\u200b\\(f\\)\u200b\u4e3a\u200b\\(\\gamma\\)-\u200b\u5f3a\u200b\u51f8\u51fd\u6570\u200b\u3002\u200b\u5f3a\u200b\u51f8\u51fd\u6570\u200b\u76f8\u8f83\u200b\u4e8e\u200b\u51f8\u51fd\u6570\u200b\uff0c\u200b\u5728\u200b\u6700\u4f18\u200b\u503c\u5904\u200b\u6709\u200b\u66f4\u597d\u200b\u7684\u200b\u6027\u8d28\u200b\u3002</p> \u200b\u6b21\u4f18\u200b\u6027\u200b\u6761\u4ef6\u200b <p>\u200b\u5bf9\u4e8e\u200b\u7ed9\u5b9a\u200b\\(x\\)\uff0c\\(\\eqref{3}\\)\u200b\u662f\u200b\\(y\\)\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5bf9\u200b\u5176\u6c42\u200b\u68af\u5ea6\u200b\u5f97\u5230\u200b</p> \\[ \\nabla f(y) = \\nabla f(x) + \\gamma(y - x) \\] <p>\u200b\u4ee4\u200b\\(\\nabla f(\\tilde y) = 0\\)\uff0c\u200b\u5219\u200b\\(\\tilde y = x - \\nabla f(x) / \\gamma\\)\u3002\u200b\u6b64\u65f6\u200b\uff0c\u200b\u6709\u200b</p> \\[ f(y)\\geq f(\\tilde y)\\geq f(x) + \\nabla f(x)^\\top (\\tilde y - x) + \\frac{\\gamma}{2} \\Vert\\tilde y - x\\Vert^2 = f(x) - \\frac{1}{2\\gamma}\\Vert \\nabla f(x)\\Vert^2 \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\\(f(x) - p^*\\leq \\frac{1}{2\\gamma}\\Vert \\nabla f(x)\\Vert^2\\)\u3002\u200b\u5373\u200b\u4efb\u4f55\u200b\u68af\u5ea6\u200b\u8db3\u591f\u200b\u5c0f\u200b\u7684\u200b\u70b9\u200b\u90fd\u200b\u662f\u200b\u8fd1\u4f3c\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002</p> \\[ \\Vert\\nabla f(x)\\Vert\\leq \\sqrt{2\\gamma\\varepsilon}\\Lora f(x) - p^*\\leq \\varepsilon \\] <p>\u200b\u6216\u200b</p> \\[ \\Vert x - x^*\\Vert\\leq \\frac{2}{m}\\Vert\\nabla f(x)\\Vert \\] <p>\u200b\u663e\u7136\u200b\uff0c\\(x^*\\)\u200b\u662f\u200b\u552f\u4e00\u200b\u7684\u200b\u3002</p> <p>\u200b\u5728\u200b\u4e0b\u200b\u6c34\u5e73\u200b\u96c6\u200b\\(S\\)\u200b\u4e0a\u200b\uff0c\u200b\u4e8c\u9636\u200b\u68af\u5ea6\u200b\\(\\nabla^2 f\\)\u200b\u5b58\u5728\u200b\u4e0a\u200b\u754c\u200b\\(\\Gamma I\\)\uff0c\u200b\u5b9a\u4e49\u200b\\(\\kappa = \\Gamma / \\gamma\\)\u200b\u4e3a\u200b\\(\\nabla^2 f(x)\\)\u200b\u7684\u200b\u6761\u4ef6\u200b\u6570\u200b\u3002</p>"},{"location":"math/convex-optimization/unconstrained-optimization/#_3","title":"\u4e0b\u964d\u200b\u65b9\u6cd5","text":"<p>\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\\(\\eqref{5}\\)\u200b\u7684\u200b\u65b9\u6cd5\u200b\u6784\u9020\u200b\u70b9\u5217\u200b\uff1a</p> \\[ x^{(k+1)} = x^{(k)} + t^{(k)}\\Delta x^{(k)} \\label{5} \\] <p>\u200b\u5176\u4e2d\u200b\u6b65\u957f\u200b\\(t^{(k)} &gt; 0\\)\uff0c\\(\\Delta x\\)\u200b\u4e3a\u200b\u524d\u8fdb\u65b9\u5411\u200b\u3002\u200b\u82e5\u200b\u6ee1\u8db3\u200b\\(f{(k+1)} &lt; f{(k)}\\)\uff0c\u200b\u5219\u200b\u70b9\u5217\u200b\u6784\u6210\u200b\u4e00\u79cd\u200b\u4e0b\u964d\u200b\u65b9\u6cd5\u200b\u3002\u200b\u6839\u636e\u200b\uff08\u200b\u5f3a\u200b\uff09\u200b\u51f8\u51fd\u6570\u200b\u7684\u200b\u6027\u8d28\u200b\uff0c\\(\\Delta x^{(k)}\\)\u200b\u5fc5\u987b\u200b\u6ee1\u8db3\u200b</p> \\[ \\nabla f(x^{(k)})^\\top \\Delta x^{(k)} &lt; 0 \\] <p>\u200b\u5373\u200b\u524d\u8fdb\u65b9\u5411\u200b\u5fc5\u987b\u200b\u80cc\u5bf9\u200b\u68af\u5ea6\u65b9\u5411\u200b\u3002</p>"},{"location":"math/convex-optimization/unconstrained-optimization/#_4","title":"\u7b97\u6cd5\u200b\u8868\u8ff0","text":""},{"location":"math/discrete-mathematics/","title":"\u79bb\u6563\u6570\u5b66","text":"<p>\u200b\u6839\u636e\u200b\u300a\u200b\u6570\u7406\u903b\u8f91\u200b\u4e0e\u200b\u96c6\u5408\u8bba\u200b\u300b\uff0c\u200b\u77f3\u200b\u7eaf\u4e00\u200b\u7b49\u200b\uff0c\u200b\u6e05\u534e\u5927\u5b66\u51fa\u7248\u793e\u200b\u4e0e\u200b\u6e05\u534e\u5927\u5b66\u200b\u8ba1\u7b97\u673a\u7cfb\u200b\u201c\u200b\u79bb\u6563\u6570\u5b66\u200b\u201d\u200b\u8bfe\u7a0b\u200b\u6574\u7406\u200b\u3002</p> <ul> <li>\u200b\u7b2c\u4e00\u7ae0\u200b \u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u57fa\u672c\u6982\u5ff5\u200b</li> <li>\u200b\u7b2c\u4e8c\u7ae0\u200b \u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u7b49\u503c\u200b\u548c\u200b\u63a8\u7406\u200b\u6f14\u7b97\u200b</li> <li>\u200b\u7b2c\u4e09\u7ae0\u200b \u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u5316\u200b</li> <li>\u200b\u7b2c\u56db\u7ae0\u200b \u200b\u8c13\u8bcd\u200b\u903b\u8f91\u200b\u7684\u200b\u57fa\u672c\u6982\u5ff5\u200b</li> <li>\u200b\u7b2c\u4e94\u7ae0\u200b \u200b\u8c13\u8bcd\u200b\u903b\u8f91\u200b\u7684\u200b\u7b49\u503c\u200b\u548c\u200b\u63a8\u7406\u200b\u6f14\u7b97\u200b</li> <li>\u200b\u7b2c\u4e5d\u7ae0\u200b \u200b\u96c6\u5408\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-1/","title":"\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u57fa\u672c\u6982\u5ff5","text":"<p>\u200b\u547d\u9898\u200b\u662f\u200b\u4e00\u4e2a\u200b\u975e\u771f\u200b\u5373\u200b\u5047\u200b\u7684\u200b\u9648\u8ff0\u53e5\u200b\u3002</p> <ul> <li>\u200b\u547d\u9898\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u975e\u771f\u200b\u975e\u5047\u200b\uff0c\u200b\u4e5f\u200b\u4e0d\u200b\u53ef\u80fd\u200b\u65e2\u200b\u771f\u200b\u53c8\u200b\u5047\u200b</li> <li>\u200b\u547d\u9898\u200b\u7684\u200b\u771f\u5047\u200b\u6027\u200b\u53ef\u4ee5\u200b\u5f85\u5b9a\u200b</li> </ul> <p>\u200b\u547d\u9898\u200b\u7684\u200b\u771f\u5047\u200b\u53ef\u4ee5\u200b\u7528\u200bTrue\u200b\u6216\u200bFalse\u200b\u8868\u793a\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7528\u200b1\u200b\u6216\u200b0\u200b\u8868\u793a\u200b</p> <p>\u200b\u547d\u9898\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u4e0e\u200b\u3001\u200b\u6216\u200b\u3001\u200b\u975e\u200b\u7b49\u200b\u903b\u8f91\u200b\u8054\u7ed3\u8bcd\u200b\u8fde\u63a5\u200b\uff0c\u200b\u5f97\u5230\u200b\u590d\u5408\u200b\u547d\u9898\u200b</p> <p>\u200b\u5f53\u200b\u4e0d\u200b\u6307\u5b9a\u200b\u547d\u9898\u200b\u7684\u200b\u5177\u4f53\u5185\u5bb9\u200b\u65f6\u200b\uff0c\u200b\u79f0\u8be5\u200b\u547d\u9898\u200b\u4e3a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5177\u4f53\u200b\u547d\u9898\u200b\u79f0\u4e3a\u200b\u8be5\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u7684\u200b\u4e00\u4e2a\u200b\u89e3\u91ca\u200b\u3002\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u7684\u200b\u89e3\u91ca\u200b\u53ef\u4ee5\u200b\u662f\u200b\u4efb\u610f\u200b\u547d\u9898\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-1/#_2","title":"\u903b\u8f91\u200b\u8054\u7ed3\u8bcd","text":"<p>\u200b\u547d\u9898\u903b\u8f91\u200b\u4e2d\u200b\u5e38\u7528\u200b\u7684\u200b\u903b\u8f91\u200b\u8054\u7ed3\u8bcd\u200b\u4e3a\u200b\u4e0e\u200b\uff08\\(\\land\\)\uff0c\u200b\u5408\u53d6\u200b\uff09\u3001\u200b\u6216\u200b\uff08\\(\\lor\\)\uff0c\u200b\u6790\u53d6\u200b\uff09\u3001\u200b\u975e\u200b\uff08\\(\\lnot\\)\uff0c\u200b\u5426\u5b9a\u200b\uff09\u3001\u200b\u8574\u542b\u200b\uff08\\(\\rightarrow\\)\uff09\u3001\u200b\u53cc\u200b\u8574\u542b\u200b\uff08\\(\\leftrightarrow\\)\uff09\u200b\u7b49\u200b\u3002\u200b\u9664\u200b\\(\\lnot\\)\u200b\u4e3a\u200b\u4e00\u5143\u200b\u8fd0\u7b97\u7b26\u200b\u4ee5\u5916\u200b\uff0c\u200b\u5176\u4f59\u200b\u903b\u8f91\u200b\u8fd0\u7b97\u7b26\u200b\u5747\u200b\u4e3a\u200b\u4e8c\u5143\u200b\u8fd0\u7b97\u7b26\u200b\u3002\u200b\u8fd9\u4e9b\u200b\u8fd0\u7b97\u7b26\u200b\u7684\u200b\u771f\u503c\u8868\u200b\u5982\u4e0b\u200b\u6240\u200b\u5217\u200b\uff1a</p> <ul> <li>\u200b\u4e0e\u200b</li> </ul> \\(P\\) \\(Q\\) \\(P\\land Q\\) 0 0 0 0 1 0 1 0 0 1 1 1 <ul> <li>\u200b\u6216\u200b</li> </ul> \\(P\\) \\(Q\\) \\(P\\lor Q\\) 0 0 0 0 1 1 1 0 1 1 1 1 <ul> <li>\u200b\u975e\u200b</li> </ul> \\(P\\) \\(\\lnot P\\) 0 1 1 0 <ul> <li>\u200b\u8574\u542b\u200b</li> </ul> \\(P\\) \\(Q\\) \\(P\\rightarrow Q\\) 0 0 1 0 1 1 1 0 0 1 1 1 <ul> <li>\u200b\u53cc\u200b\u8574\u542b\u200b</li> </ul> \\(P\\) \\(Q\\) \\(P\\leftrightarrow Q\\) 0 0 1 0 1 0 1 0 0 1 1 1 <p>\u200b\u6b64\u5916\u200b\uff0c\u200b\u5982\u4e0b\u200b\u903b\u8f91\u200b\u8054\u7ed3\u8bcd\u200b\u4e5f\u200b\u8f83\u4e3a\u200b\u5e38\u7528\u200b\uff1a</p> <ul> <li>\u200b\u4e0e\u975e\u200b\uff1a\\(A\\uparrow B = \\lnot (A\\land B)\\)</li> <li>\u200b\u6216\u975e\u200b\uff1a\\(A\\downarrow B = \\lnot (A\\lor B)\\)</li> <li>\u200b\u5f02\u6216\u200b\uff1a\\(A\\overline \\lor B = (A\\lor B)\\land (\\lnot A\\lor \\lnot B)\\)</li> </ul>"},{"location":"math/discrete-mathematics/chapter-1/#_3","title":"\u5408\u5f0f\u516c\u5f0f","text":"<p>\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff08\u200b\u7b80\u79f0\u200b\u516c\u5f0f\u200b\uff09\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u89c4\u5219\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u7b80\u5355\u200b\u547d\u9898\u200b\u662f\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u82e5\u200b\\(A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\\(\\lnot A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u82e5\u200b\\(A, B\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\\(A\\land B, A\\lor B, A\\rightarrow B, A\\leftrightarrow B\\)\u200b\u5747\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\u7ecf\u8fc7\u200b\u6709\u9650\u200b\u6b21\u200b\u5e94\u7528\u200b\u524d\u200b\u4e09\u6761\u200b\u89c4\u5219\u200b\u6240\u200b\u5f97\u5230\u200b\u7684\u200b\u516c\u5f0f\u200b\u79f0\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> </ul> <p>\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4e2d\u200b\u8fd0\u7b97\u7b26\u200b\u7684\u200b\u4f18\u5148\u7ea7\u200b\u6392\u5217\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ \\lnot &gt; \\land &gt; \\lor &gt; \\rightarrow &gt; \\leftrightarrow \\] <p>\u200b\u5982\u679c\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u5728\u200b\u4efb\u610f\u200b\u89e3\u91ca\u200b\u4e0b\u200b\u90fd\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u79f0\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\u6216\u6c38\u771f\u5f0f\u200b\u3002\u200b\u5982\u679c\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u5728\u200b\u67d0\u4e2a\u200b\u89e3\u91ca\u200b\u4e0b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u79f0\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u53ef\u200b\u6ee1\u8db3\u200b\u3002\u200b\u5982\u679c\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u5728\u200b\u4efb\u610f\u200b\u89e3\u91ca\u200b\u4e0b\u200b\u90fd\u200b\u4e3a\u200b\u5047\u200b\uff0c\u200b\u79f0\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4e3a\u200b\u77db\u76fe\u200b\u5f0f\u200b\u6216\u200b\u6c38\u5047\u5f0f\u200b\u3002\u200b\u4e24\u4e2a\u200b\u91cd\u8a00\u5f0f\u200b\u4f7f\u7528\u200b\\(\\land, \\lor, \\rightarrow, \\leftrightarrow\\)\u200b\u8fde\u63a5\u200b\u540e\u200b\u5f97\u5230\u200b\u7684\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4ecd\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-1/#_4","title":"\u91cd\u8a00\u5f0f\u200b\u7684\u200b\u4ee3\u5165\u200b\u89c4\u5219","text":"<p>\u200b\u82e5\u200b\\(A\\)\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\uff0c\u200b\u5c06\u200b\\(A\\)\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\\(C\\)\u200b\u66ff\u6362\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\\(D\\)\uff0c\u200b\u5219\u200b\u66ff\u6362\u200b\u540e\u200b\u7684\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u4ecd\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\u3002\u200b\u8bb0\u200b\u4f5c\u200b\\(\\frac CD\\)</p> <ul> <li>\u200b\u4ee3\u5165\u200b\u89c4\u5219\u200b\u53ea\u80fd\u200b\u5bf9\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u8fdb\u884c\u200b\u66ff\u4ee3\u200b\uff0c\u200b\u800c\u200b\u4e0d\u80fd\u200b\u66ff\u4ee3\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u200b\u590d\u5408\u200b\u547d\u9898\u200b</li> <li>\u200b\u5bf9\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u67d0\u4e2a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u8fdb\u884c\u200b\u4ee3\u5165\u200b\u65f6\u200b\uff0c\u200b\u5fc5\u987b\u200b\u4ee3\u5165\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u7684\u200b\u6240\u6709\u200b\u547d\u9898\u200b\u53d8\u9879\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-2/","title":"\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u7b49\u503c\u200b\u548c\u200b\u63a8\u7406\u200b\u6f14\u7b97","text":"<p>\u200b\u7b49\u503c\u200b\uff1a\u200b\u8bbe\u200b\\(A, B\\)\u200b\u4e3a\u200b\u516c\u5f0f\u200b\uff0c\\(P_1, \\dots, P_n\\)\u200b\u662f\u200b\u51fa\u73b0\u200b\u5728\u200b\\(A, B\\)\u200b\u4e2d\u200b\u6240\u6709\u200b\u7684\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\uff0c\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u4efb\u610f\u200b\u7684\u200b\u4e00\u7ec4\u200b\\(\\left\\langle P_1, \\dots, P_n\\right\\rangle\\)\uff0c\u200b\u90fd\u200b\u6709\u200b\\(A=B\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u4e0e\u200b\u516c\u5f0f\u200b\\(B\\)\u200b\u7b49\u503c\u200b\uff08\u200b\u7b49\u4ef7\u200b\uff09\uff0c\u200b\u8bb0\u200b\u4e3a\u200b\\(A\\Leftrightarrow B\\)\u200b\u6216\u200b\\(A=B\\)\u3002</p> <p>\u200b\u7b49\u503c\u200b\u5173\u7cfb\u200b\u6ee1\u8db3\u200b\u81ea\u53cd\u6027\u200b\u3001\u200b\u5bf9\u79f0\u6027\u200b\u4e0e\u200b\u4f20\u9012\u6027\u200b\uff1a</p> <ul> <li>\\(A=A\\)</li> <li>\u200b\u82e5\u200b\\(A=B\\)\uff0c\u200b\u5219\u200b\\(B=A\\)</li> <li>\u200b\u82e5\u200b\\(A=B, B=C\\)\uff0c\u200b\u5219\u200b\\(A=C\\)</li> </ul> <p>\\(A=B\\)\u200b\u7684\u200b\u5145\u5206\u200b\u5fc5\u8981\u6761\u4ef6\u200b\u662f\u200b\\(A\\leftrightarrow B\\)\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-2/#_2","title":"\u7b49\u503c\u200b\u516c\u5f0f","text":"<p>\u200b\u57fa\u672c\u200b\u7684\u200b\u7b49\u503c\u200b\u516c\u5f0f\u200b\u5217\u4e3e\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u53cc\u91cd\u200b\u5426\u5b9a\u200b\u5f8b\u200b</li> <li>\\(P = \\lnot \\lnot P\\)</li> <li>\u200b\u7ed3\u5408\u5f8b\u200b</li> <li>\\((P\\lor Q)\\lor R = P \\lor (Q\\lor R)\\)</li> <li>\\((P\\land Q)\\land R = P \\land (Q\\land R)\\)</li> <li>\\((P\\leftrightarrow Q)\\leftrightarrow R = P \\leftrightarrow (Q\\leftrightarrow R)\\)</li> <li>\u200b\u8fd0\u7b97\u7b26\u200b\\(\\rightarrow\\)\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u7ed3\u5408\u5f8b\u200b</li> <li>\u200b\u4ea4\u6362\u5f8b\u200b</li> <li>\\(P\\lor Q = Q\\lor P\\)</li> <li>\\(P\\land Q = Q\\land P\\)</li> <li>\\(P\\leftrightarrow Q = Q\\leftrightarrow P\\)</li> <li>\u200b\u8fd0\u7b97\u7b26\u200b\\(\\rightarrow\\)\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u4ea4\u6362\u5f8b\u200b</li> <li>\u200b\u5206\u914d\u5f8b\u200b</li> <li>\\(P\\lor (Q\\land R) = (P\\lor Q)\\land (P\\lor R)\\)</li> <li>\\(P\\land (Q\\lor R) = (P\\land Q)\\lor (P\\land R)\\)</li> <li>\\(P\\rightarrow (Q\\rightarrow R) = (P\\rightarrow Q)\\rightarrow (P\\rightarrow R)\\)</li> <li>\u200b\u8fd0\u7b97\u7b26\u200b\\(\\leftrightarrow\\)\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u5206\u914d\u5f8b\u200b</li> <li>\u200b\u6052\u7b49\u200b\u5f8b\u200b</li> <li>\\(P\\lor P = P\\)</li> <li>\\(P\\land P = P\\)</li> <li>\\(P\\rightarrow P = T\\)</li> <li>\\(P\\leftrightarrow P = T\\)</li> <li>\u200b\u5438\u6536\u7387\u200b</li> <li>\\(P\\lor (P\\land Q) = P\\)</li> <li>\\(P\\land (P\\lor Q) = P\\)</li> <li>\u200b\u6469\u6839\u200b\u5f8b\u200b</li> <li>\\(\\lnot (P\\lor Q) = \\lnot P\\land \\lnot Q\\)</li> <li>\\(\\lnot (P\\land Q) = \\lnot P\\lor \\lnot Q\\)</li> <li>\\(\\lnot (P\\rightarrow Q) = P\\land \\lnot Q\\)</li> <li>\\(\\lnot (P\\leftrightarrow Q) = (P\\land \\lnot Q) \\lor (\\lnot P\\land Q)\\)</li> <li>\u200b\u540c\u4e00\u5f8b\u200b</li> <li>\\(P\\lor F = P\\)</li> <li>\\(P\\land P = T\\)</li> <li>\\(T\\rightarrow P = P\\)</li> <li>\\(T\\leftrightarrow P = P\\)</li> <li>\\(P\\rightarrow F = \\lnot P\\)</li> <li>\\(P\\leftrightarrow F = \\lnot P\\)</li> <li>\u200b\u96f6\u5f8b\u200b</li> <li>\\(P\\lor T = T\\)</li> <li>\\(P\\land F = F\\)</li> <li>\\(P\\rightarrow T = T\\)</li> <li>\\(F\\rightarrow P = T\\)</li> <li>\u200b\u8865\u4f59\u5f8b\u200b</li> <li>\\(P\\lor \\lnot P = T\\)</li> <li>\\(P\\land \\lnot P = F\\)</li> <li>\\(P\\rightarrow \\lnot P = \\lnot P\\)</li> <li>\\(P\\leftrightarrow \\lnot P = F\\)</li> <li>\u200b\u5176\u4ed6\u200b\u5e38\u7528\u200b\u7684\u200b\u7b49\u503c\u200b\u516c\u5f0f\u200b</li> <li>\\(P\\rightarrow Q = \\lnot Q\\rightarrow \\lnot P\\)\uff08\u200b\u9006\u5426\u547d\u9898\u200b\uff09</li> <li>\\(P\\rightarrow (Q\\rightarrow R) = (P\\land Q)\\rightarrow R\\)</li> <li>\\((P\\rightarrow R)\\land (Q\\rightarrow R) = (P\\lor Q) \\rightarrow R\\)</li> </ul> <p>\u200b\u7531\u200b\u516c\u5f0f\u200b2\u200b\u7684\u200b\u76f4\u63a5\u200b\u63a8\u8bba\u200b\uff1a\\(P\\rightarrow (Q\\rightarrow R) = Q\\rightarrow (P\\rightarrow R)\\)</p> <p>\u200b\u7f6e\u6362\u200b\uff1a\u200b\u5bf9\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u7684\u200b\u5b50\u200b\u516c\u5f0f\u200b\\(B\\)\uff0c\u200b\u4f7f\u7528\u200b\u4e0e\u200b\u4e4b\u200b\u7b49\u503c\u200b\u7684\u200b\u516c\u5f0f\u200b\u8fdb\u884c\u200b\u4ee3\u6362\u200b\u79f0\u4e3a\u200b\u7f6e\u6362\u200b</p> <p>\u200b\u8bbe\u7f6e\u200b\u6362\u540e\u200b\u7684\u200b\u516c\u5f0f\u200b\u4e3a\u200b\\(C\\)\uff0c\u200b\u5219\u200b\u5fc5\u6709\u200b\\(A=C\\)</p>"},{"location":"math/discrete-mathematics/chapter-2/#_3","title":"\u4ece\u200b\u771f\u503c\u8868\u200b\u8ba1\u7b97\u200b\u751f\u6210\u200b\u547d\u9898\u200b\u516c\u5f0f","text":"<p>\u200b\u4ece\u200b\u771f\u503c\u8868\u200b\u751f\u6210\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u6709\u200b\u4e24\u79cd\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u65e2\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u503c\u4e3a\u200b\\(T\\)\u200b\u7684\u200b\u884c\u200b\u751f\u6210\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u503c\u4e3a\u200b\\(F\\)\u200b\u7684\u200b\u884c\u200b\u751f\u6210\u200b</p>"},{"location":"math/discrete-mathematics/chapter-2/#t","title":"\u4ece\u503c\u200b\u4e3a\u200bT\u200b\u7684\u200b\u884c\u200b\u751f\u6210","text":"<p>\u200b\u5bf9\u4e8e\u200b\u5305\u542b\u200b\\(n\\)\u200b\u4e2a\u200b\u547d\u9898\u200b\u53d8\u5143\u200b\\(P_1, \\dots, P_n\\)\u200b\u7684\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5199\u51fa\u200b\u5982\u4e0b\u200b\u5408\u53d6\u200b\u5f0f\u200b\uff1a</p> \\[ P_1\\land P_2\\land \\dots \\land P_n \\] <p>\u200b\u5bf9\u4e8e\u200b\u771f\u503c\u8868\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u53d6\u503c\u200b\u4e3a\u200b\\(T\\)\u200b\u7684\u200b\u884c\u200b\uff0c\u200b\u82e5\u200b\u5bf9\u5e94\u200b\u7684\u200b\\(P_i\\)\u200b\u4e3a\u200b\\(F\\)\uff0c\u200b\u5219\u200b\u5c06\u200b\u5408\u53d6\u200b\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u5bf9\u5e94\u200b\\(P_i\\)\u200b\u66ff\u6362\u200b\u4e3a\u200b\\(\\lnot P_i\\)\u3002\u200b\u5c06\u200b\u6240\u5f97\u200b\u7684\u200b\u516c\u5f0f\u200b\u4f7f\u7528\u200b\u6790\u53d6\u200b\u8fd0\u7b97\u7b26\u200b\u8fde\u63a5\u200b\u5373\u5f97\u200b\u7b26\u5408\u200b\u771f\u503c\u8868\u200b\u7684\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-2/#f","title":"\u4ece\u503c\u200b\u4e3a\u200bF\u200b\u7684\u200b\u884c\u200b\u751f\u6210","text":"<p>\u200b\u5bf9\u4e8e\u200b\u5305\u542b\u200b\\(n\\)\u200b\u4e2a\u200b\u547d\u9898\u200b\u53d8\u5143\u200b\\(P_1, \\dots, P_n\\)\u200b\u7684\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5199\u51fa\u200b\u5982\u4e0b\u200b\u6790\u53d6\u200b\u5f0f\u200b\uff1a</p> \\[ P_1\\lor P_2\\lor \\dots \\lor P_n \\] <p>\u200b\u5bf9\u4e8e\u200b\u771f\u503c\u8868\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u53d6\u503c\u200b\u4e3a\u200b\\(F\\)\u200b\u7684\u200b\u884c\u200b\uff0c\u200b\u82e5\u200b\u5bf9\u5e94\u200b\u7684\u200b\\(P_i\\)\u200b\u4e3a\u200b\\(T\\)\uff0c\u200b\u5219\u200b\u5c06\u200b\u6790\u53d6\u200b\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u5bf9\u5e94\u200b\\(P_i\\)\u200b\u66ff\u6362\u200b\u4e3a\u200b\\(\\lnot P_i\\)\u3002\u200b\u5c06\u200b\u6240\u5f97\u200b\u7684\u200b\u516c\u5f0f\u200b\u4f7f\u7528\u200b\u5408\u53d6\u200b\u8fd0\u7b97\u7b26\u200b\u8fde\u63a5\u200b\u5373\u5f97\u200b\u7b26\u5408\u200b\u771f\u503c\u8868\u200b\u7684\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-2/#_4","title":"\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6","text":"<p>\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(A\\)\u200b\u4e3a\u200b\u7531\u200b\u8054\u7ed3\u8bcd\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u82e5\u200b\u4efb\u610f\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u90fd\u200b\u6709\u200b\u901a\u8fc7\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u4f7f\u7528\u200b\\(A\\)\u200b\u4e2d\u200b\u7684\u200b\u8054\u7ed3\u8bcd\u200b\u7ec4\u5408\u800c\u6210\u200b\u7684\u200b\u516c\u5f0f\u200b\u4e0e\u200b\u4e4b\u200b\u7b49\u503c\u200b\uff0c\u200b\u5219\u200b\\(A\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b</p> <p>\u200b\u901a\u8fc7\u200b\u5982\u4e0a\u200b\u5bf9\u200b\u771f\u503c\u8868\u200b\u7684\u200b\u5206\u6790\u200b\uff0c\u200b\u4efb\u4f55\u200b\u771f\u503c\u8868\u200b\u90fd\u200b\u80fd\u200b\u4f7f\u7528\u200b\\(\\{\\land, \\lor, \\lnot\\}\\)\u200b\u4e09\u4e2a\u200b\u8fd0\u7b97\u7b26\u200b\u8868\u793a\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u3002\u200b\u5047\u8bbe\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u516c\u5f0f\u200b\\(P\\)\uff0c\u200b\u4e0d\u200b\u5b58\u5728\u200b\\(\\{\\land, \\lor, \\lnot\\}\\)\u200b\u7ec4\u6210\u200b\u7684\u200b\u516c\u5f0f\u200b\u4e0e\u200b\u4e4b\u200b\u7b49\u503c\u200b\u3002\u200b\u4f46\u200b\u901a\u8fc7\u200b\\(P\\)\u200b\u7684\u200b\u771f\u503c\u8868\u200b\u53ef\u4ee5\u200b\u6784\u9020\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\\(Q\\)\uff0c\u200b\u5219\u200b\\(Q\\)\u200b\u4e0e\u200b\\(P\\)\u200b\u7684\u200b\u771f\u503c\u8868\u200b\u76f8\u540c\u200b\uff0c\\(P=Q\\)\uff0c\u200b\u77db\u76fe\u200b\u3002\u200b\u56e0\u6b64\u200b\\(\\{\\land, \\lor, \\lnot\\}\\)\u200b\u662f\u200b\u5b8c\u5907\u200b\u7684\u200b\u3002</p> <p>\u200b\u4ee5\u4e0b\u200b\u5217\u51fa\u200b\u4e86\u200b\u4e00\u90e8\u5206\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u6700\u5c0f\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff1a</p> <ul> <li>\\(\\{\\lor, \\lnot\\}\\)</li> <li>\\(\\{\\land, \\lnot\\}\\)</li> <li>\\(\\{\\uparrow\\}\\)</li> <li>\\(\\{\\downarrow\\}\\)</li> <li>\\(\\{\\lnot, \\rightarrow\\}\\)</li> </ul> <p>\u200b\u5982\u679c\u200b\\(A\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff0c\u200b\u96c6\u5408\u200b\\(B\\)\u200b\u4e2d\u200b\u7684\u200b\u8054\u7ed3\u8bcd\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u7ec4\u5408\u200b\u5f97\u5230\u200b\u4e0e\u200b\\(A\\)\u200b\u4e2d\u200b\u8054\u7ed3\u8bcd\u200b\u7b49\u4ef7\u200b\u7684\u200b\u8054\u7ed3\u8bcd\u200b\uff0c\u200b\u5219\u200b\\(B\\)\u200b\u4e5f\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\u3002</p> <p>\u200b\u8bc1\u660e\u200b\uff1a\\(\\{\\lor, \\lnot\\}\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff1a</p> \\[ P\\land Q = \\lnot (\\lnot P\\lor \\lnot Q) \\] <p>\u200b\u8bc1\u660e\u200b\uff1a\\(\\{\\land, \\lnot\\}\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff1a</p> \\[ P\\lor Q = \\lnot (\\lnot P\\land \\lnot Q) \\] <p>\u200b\u8bc1\u660e\u200b\uff1a\\(\\{\\uparrow\\}\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff1a</p> <ul> <li>\u200b\u8bc1\u660e\u200b\\(\\lnot P\\)\u200b\u53ef\u4ee5\u200b\u4ec5\u200b\u4f7f\u7528\u200b\\(\\uparrow\\)\u200b\u8fd0\u7b97\u7b26\u200b\u8868\u793a\u200b\uff1a</li> </ul> <p>$$   \\begin{aligned}   \\lnot P &amp;= \\lnot (P\\land P) \\   &amp;= P\\uparrow P   \\end{aligned}   $$</p> <ul> <li>\u200b\u8bc1\u660e\u200b\\(P\\land Q\\)\u200b\u53ef\u4ee5\u200b\u4ec5\u200b\u4f7f\u7528\u200b\\(\\uparrow\\)\u200b\u8fd0\u7b97\u7b26\u200b\u8868\u793a\u200b\uff1a</li> </ul> <p>$$   \\begin{aligned}   P\\land Q&amp;= \\lnot \\lnot (P\\land Q) \\   &amp;= \\lnot (P\\uparrow Q) \\   &amp;= (P\\uparrow Q) \\uparrow (P\\uparrow Q)   \\end{aligned}   $$</p> <p>\u200b\u8bc1\u660e\u200b\uff1a\\(\\{\\downarrow\\}\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff1a</p> <ul> <li>\u200b\u8bc1\u660e\u200b\\(\\lnot P\\)\u200b\u53ef\u4ee5\u200b\u4ec5\u200b\u4f7f\u7528\u200b\\(\\downarrow\\)\u200b\u8fd0\u7b97\u7b26\u200b\u8868\u793a\u200b\uff1a</li> </ul> <p>$$   \\begin{aligned}   \\lnot P &amp;= \\lnot (P\\lor P) \\   &amp;= P\\downarrow P   \\end{aligned}   $$</p> <ul> <li>\u200b\u8bc1\u660e\u200b\\(P\\lor Q\\)\u200b\u53ef\u4ee5\u200b\u4ec5\u200b\u4f7f\u7528\u200b\\(\\downarrow\\)\u200b\u8fd0\u7b97\u7b26\u200b\u8868\u793a\u200b\uff1a</li> </ul> <p>$$   \\begin{aligned}   P\\lor Q&amp;= \\lnot \\lnot (P\\lor Q) \\   &amp;= \\lnot (P\\downarrow Q) \\   &amp;= (P\\downarrow Q) \\downarrow (P\\downarrow Q)   \\end{aligned}   $$</p> <p>\u200b\u8bc1\u660e\u200b\uff1a\\(\\{\\rightarrow, \\lnot\\}\\)\u200b\u662f\u200b\u8054\u7ed3\u8bcd\u200b\u7684\u200b\u5b8c\u5907\u200b\u96c6\u200b\uff1a</p> <p>\\(P\\lor Q = \\lnot \\lnot P\\lor Q = \\lnot P\\rightarrow Q\\)</p>"},{"location":"math/discrete-mathematics/chapter-2/#_5","title":"\u5bf9\u5076\u200b\u5f0f","text":"<p>\u200b\u8bbe\u200b\\(A\\)\u200b\u4e3a\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5c06\u200b\\(A\\)\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u7684\u200b\\(\\lor, \\land, T, F\\)\u200b\u5206\u522b\u200b\u4f7f\u7528\u200b\\(\\land, \\lor, F, T\\)\u200b\u8fdb\u884c\u200b\u66ff\u6362\u200b\uff0c\u200b\u6240\u200b\u5f97\u5230\u200b\u7684\u200b\u516c\u5f0f\u200b\u79f0\u4e3a\u200b\\(A\\)\u200b\u7684\u200b\u5bf9\u5076\u200b\u5f0f\u200b\uff0c\u200b\u8bb0\u200b\u4e3a\u200b\\(A^*\\)</p> <p>\u200b\u8bbe\u200b\\(A\\)\u200b\u4e3a\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5c06\u200b\\(A\\)\u200b\u4e2d\u200b\u51fa\u73b0\u200b\u7684\u200b\u6240\u6709\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\\(P_1,\\dots, P_n\\)\u200b\u66ff\u6362\u200b\u4e3a\u200b\\(\\lnot P_1, \\dots, \\lnot P_n\\)\uff0c\u200b\u6240\u200b\u5f97\u5230\u200b\u7684\u200b\u516c\u5f0f\u200b\u79f0\u4e3a\u200b\\(A\\)\u200b\u7684\u200b\u5426\u200b\u547d\u9898\u200b\uff0c\u200b\u8bb0\u200b\u4e3a\u200b\\(A^-\\)</p> <p>\u200b\u5219\u200b\u53d6\u200b\u5bf9\u5076\u200b\u5f0f\u200b\u8fd0\u7b97\u200b\u3001\u200b\u53d6\u5426\u200b\u547d\u9898\u200b\u8fd0\u7b97\u200b\u6709\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ol> <li>\\(\\lnot (A^*) = (\\lnot A)^*, \\lnot (A^-) = (\\lnot A)^-\\)</li> <li>\\((A^*)^*=A, (A^-)^- =A\\)</li> <li>\\(A^{*-}=\\lnot A\\)</li> <li>\\(A=B\\Rightarrow A^*=B^*\\)</li> <li>\u200b\u5bf9\u200b\\(A\\rightarrow B\\)\u200b\u53d6\u200b\u5bf9\u5076\u200b\u5f0f\u200b\u53ef\u5f97\u200b\\(A\\rightarrow B=T\\Rightarrow B^*\\rightarrow A^*=T\\)</li> </ol> <p>\u200b\u6839\u636e\u200b\u5bf9\u5076\u200b\u5f0f\u200b\u548c\u200b\u5426\u200b\u547d\u9898\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff0c2\u200b\u5f0f\u200b\u81ea\u7136\u200b\u6210\u7acb\u200b\u30021\u200b\u5f0f\u200b\u548c\u200b3\u200b\u5f0f\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u6570\u5b66\u200b\u5f52\u7eb3\u6cd5\u200b\u8fdb\u884c\u200b\u8bc1\u660e\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-2/#_6","title":"\u8303\u5f0f","text":"<p>\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\u6982\u5ff5\u200b\uff1a</p> <ul> <li>\u200b\u6587\u5b57\u200b\uff1a\u200b\u7b80\u5355\u200b\u547d\u9898\u200b\\(P\\)\u200b\u53ca\u5176\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b</li> <li>\u200b\u5408\u53d6\u200b\u5f0f\u200b\uff1a\u200b\u4e00\u4e9b\u200b\u6587\u5b57\u200b\u7684\u200b\u5408\u53d6\u200b\uff0c\u200b\u5f62\u200b\u5982\u200b\\(P_1\\land \\dots \\land P_n\\)</li> <li>\u200b\u6790\u53d6\u200b\u5f0f\u200b\uff08\u200b\u5b50\u53e5\u200b\uff09\uff1a\u200b\u4e00\u4e9b\u200b\u6587\u5b57\u200b\u7684\u200b\u6790\u53d6\u200b\uff0c\u200b\u5f62\u200b\u5982\u200b\\(P_1\\lor \\dots\\lor P_n\\)</li> <li>\u200b\u4e92\u8865\u200b\u5bf9\u200b\uff1a\u200b\u6587\u5b57\u200b\u4e0e\u5176\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5426\u5b9a\u200b\u5f62\u5f0f\u200b</li> <li>\u200b\u6790\u53d6\u8303\u5f0f\u200b\uff1a\u200b\u5408\u53d6\u200b\u5f0f\u200b\u901a\u8fc7\u200b\u6790\u53d6\u200b\u8054\u7ed3\u8bcd\u200b\u8fde\u63a5\u200b\u7684\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5f62\u200b\u5982\u200b\\(A_1\\lor \\dots\\lor A_n\\)\uff0c\u200b\u5176\u4e2d\u200b\\(A_i\\)\u200b\u4e3a\u200b\u5408\u53d6\u200b\u5f0f\u200b</li> <li>\u200b\u5408\u53d6\u8303\u5f0f\u200b\uff1a\u200b\u6790\u53d6\u200b\u5f0f\u200b\u901a\u8fc7\u200b\u5408\u53d6\u200b\u8054\u7ed3\u8bcd\u200b\u8fde\u63a5\u200b\u7684\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5f62\u200b\u5982\u200b\\(A_1\\land \\dots \\land A_n\\)\uff0c\u200b\u5176\u4e2d\u200b\\(A_i\\)\u200b\u4e3a\u200b\u6790\u53d6\u200b\u5f0f\u200b</li> </ul> <p>\u200b\u7531\u6b64\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5bfc\u51fa\u200b\u8303\u5f0f\u200b\u5b9a\u7406\u200b\uff0c\u200b\u5373\u200b\u4efb\u4e00\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u90fd\u200b\u5b58\u5728\u200b\u6709\u200b\u4e0e\u200b\u4e4b\u200b\u7b49\u503c\u200b\u7684\u200b\u5408\u53d6\u8303\u5f0f\u200b\u4e0e\u200b\u6790\u53d6\u8303\u5f0f\u200b\u3002\u200b\u8303\u5f0f\u200b\u5b9a\u7406\u200b\u7684\u200b\u6b63\u786e\u6027\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u5217\u5199\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u7684\u200b\u771f\u503c\u8868\u200b\u4e0e\u200b\u4ece\u200b\u771f\u503c\u8868\u200b\u5217\u5199\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u8bc1\u660e\u200b\u3002</p> <p>\u200b\u8ba1\u7b97\u200b\u5408\u53d6\u8303\u5f0f\u200b\u4e0e\u200b\u6790\u53d6\u8303\u5f0f\u200b\uff1a</p> <ol> <li>\u200b\u6d88\u200b\u53bb\u200b\u903b\u8f91\u200b\u8054\u7ed3\u8bcd\u200b\\(A\\rightarrow B\\)\u200b\u4e0e\u200b\\(A\\leftrightarrow B\\)\uff0c\u200b\u66ff\u6362\u200b\u4e3a\u200b\\(\\lnot A\\lor B\\)\u200b\u4e0e\u200b\\((\\lnot A\\lor B)\\land (A\\lor \\lnot B)\\)</li> <li>\u200b\u4f7f\u7528\u200b\u6469\u6839\u200b\u5f8b\u200b\u4e0e\u200b\u53cc\u91cd\u200b\u5426\u5b9a\u200b\u5f8b\u200b\u6d88\u9664\u200b\\(\\lor\\)\u200b\u6216\u200b\\(\\land\\)\uff0c\u200b\u5e76\u200b\u6d88\u9664\u200b\u591a\u4f59\u200b\u7684\u200b\\(\\lnot\\)</li> <li>\u200b\u5c06\u200b\u6240\u6709\u200b\u7684\u200b\\(\\lnot\\)\u200b\u79fb\u52a8\u200b\u5230\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\\(P_i\\)\u200b\u524d\u200b</li> <li>\u200b\u4f7f\u7528\u200b\u5206\u914d\u5f8b\u200b\u5c06\u200b\u516c\u5f0f\u5316\u200b\u4e3a\u200b\u8303\u5f0f\u200b</li> </ol>"},{"location":"math/discrete-mathematics/chapter-2/#_7","title":"\u4e3b\u200b\u8303\u5f0f","text":"<p>\u200b\u5bf9\u4e8e\u200b\u516c\u5f0f\u200b\\(Q = Q_1\\land \\dots \\land Q_n\\)\uff0c\u200b\u5f0f\u200b\u4e2d\u200b\\(Q_i=P_i\\)\u200b\u6216\u200b\\(Q_i=\\lnot P_i\\)\u3002\u200b\u5219\u200b\\(Q\\)\u200b\u4e3a\u200b\u6781\u5c0f\u200b\u9879\u200b\uff0c\u200b\u8bb0\u200b\u4e3a\u200b\\(m_i\\)\u3002</p> <ul> <li>\u200b\u6781\u5c0f\u200b\u9879\u4e2d\u200b\u5fc5\u987b\u200b\u51fa\u73b0\u200b\u6240\u6709\u200b\u7684\u200b\u547d\u9898\u200b\u53d8\u9879\u200b</li> <li>\\(n\\)\u200b\u4e2a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u53ef\u4ee5\u200b\u7ec4\u6210\u200b\\(2^n\\)\u200b\u4e2a\u200b\u6781\u5c0f\u200b\u9879\u200b\uff0c\u200b\u4f9d\u6b21\u200b\u8868\u793a\u200b\u4e3a\u200b\\(m_1, \\dots m_{2^n}\\)</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u516c\u5f0f\u200b\\(R = R_1\\lor \\dots \\lor R_n\\)\uff0c\u200b\u5f0f\u200b\u4e2d\u200b\\(R_i = P_i\\)\u200b\u6216\u200b\\(R_i = \\lnot P_i\\)\u3002\u200b\u5219\u200b\\(R\\)\u200b\u4e3a\u200b\u6781\u5927\u200b\u9879\u200b\uff0c\u200b\u8bb0\u200b\u4e3a\u200b\\(M_i\\)</p> <p>\u200b\u4e0e\u200b\u6781\u5c0f\u200b\u9879\u200b\u76f8\u540c\u200b\uff0c\u200b\u6240\u6709\u200b\u7684\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u5fc5\u987b\u200b\u90fd\u200b\u51fa\u73b0\u200b\u5728\u200b\u6781\u5927\u200b\u9879\u4e2d\u200b\uff0c\u200b\u4e14\u200b\\(n\\)\u200b\u4e2a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u53ef\u4ee5\u200b\u7ec4\u6210\u200b\\(2^n\\)\u200b\u4e2a\u200b\u6781\u5927\u200b\u9879\u200b\u3002</p> <p>\u200b\u4efb\u610f\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u7684\u200b\u8303\u5f0f\u200b\u4e2d\u200b\uff0c\u200b\u4ec5\u200b\u7531\u200b\u6781\u5c0f\u200b\u9879\u200b\u6784\u6210\u200b\u7684\u200b\u6790\u53d6\u8303\u5f0f\u200b\u4e3a\u4e3b\u200b\u6790\u53d6\u8303\u5f0f\u200b\uff0c\u200b\u4ec5\u200b\u7531\u200b\u6781\u5927\u200b\u9879\u200b\u6784\u6210\u200b\u7684\u200b\u5408\u53d6\u8303\u5f0f\u200b\u4e3a\u4e3b\u200b\u5408\u53d6\u8303\u5f0f\u200b\u3002\u200b\u4e3b\u200b\u6790\u53d6\u8303\u5f0f\u200b\u4e0e\u200b\u4e3b\u200b\u5408\u53d6\u8303\u5f0f\u200b\u90fd\u200b\u662f\u200b\u552f\u4e00\u200b\u7684\u200b\u3002</p> <p>\u200b\u6781\u5927\u200b\u9879\u200b\u3001\u200b\u6781\u5c0f\u200b\u9879\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ol> <li>\u200b\u6781\u5927\u200b\u9879\u200b\u3001\u200b\u6781\u5c0f\u200b\u9879\u200b\u7684\u200b\u6570\u91cf\u200b\u7b49\u4e8e\u200b\u516c\u5f0f\u200b\u53ef\u80fd\u200b\u7684\u200b\u89e3\u91ca\u200b\u6570\u91cf\u200b\uff0c\u200b\u5373\u200b\\(2^n\\)</li> <li>\u200b\u6bcf\u4e2a\u200b\u6781\u5c0f\u200b\u9879\u4ec5\u200b\u5728\u200b\u516c\u5f0f\u200b\u7684\u200b\u4e00\u4e2a\u200b\u89e3\u91ca\u200b\u4e0b\u200b\u4e3a\u200b\\(T\\)\uff0c\u200b\u6bcf\u4e2a\u200b\u6781\u5927\u200b\u9879\u4ec5\u200b\u5728\u200b\u516c\u5f0f\u200b\u7684\u200b\u4e00\u4e2a\u200b\u89e3\u91ca\u200b\u4e0b\u200b\u4e3a\u200b\\(F\\)</li> <li>\u200b\u6781\u5c0f\u200b\u9879\u200b\u3001\u200b\u6781\u5927\u200b\u9879\u200b\u4e24\u200b\u4e24\u200b\u4e0d\u200b\u7b49\u503c\u200b\uff0c\u200b\u6709\u200b\\(m_i\\land m_j\\leftrightarrow i=j, M_i\\lor M_j\\leftrightarrow i\\not= j\\)\u3002</li> <li>\u200b\u4efb\u4e00\u200b\u542b\u6709\u200b\\(n\\)\u200b\u4e2a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u7684\u200b\u516c\u5f0f\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u7528\u200b\\(k\\leq n\\)\u200b\u4e2a\u200b\u6781\u5c0f\u200b\u9879\u200b\u6216\u200b\\(k\\leq n\\)\u200b\u4e2a\u200b\u6781\u5927\u200b\u9879\u200b\u8868\u793a\u200b\u3002</li> <li>\\(\\bigwedge _{i=0} ^{2^n-1} M_i=F, \\bigvee _{i=0}^{2^n-1} m_i = T\\)</li> </ol>"},{"location":"math/discrete-mathematics/chapter-2/#_8","title":"\u63a8\u7406\u200b\u5f62\u5f0f","text":"<p>\u200b\u5c06\u200b\u81ea\u7136\u200b\u8bed\u53e5\u200b\u63cf\u8ff0\u200b\u7684\u200b\u63a8\u7406\u200b\u5173\u7cfb\u200b\u7ecf\u200b\u62bd\u8c61\u5316\u200b\u540e\u200b\u4f7f\u7528\u200b\u6761\u4ef6\u200b\u5f0f\u200b\u7684\u200b\u8868\u793a\u200b</p> <p>\u200b\u82e5\u200b\\(A\\rightarrow B=T\\)\uff0c\u200b\u79f0\u200b\\(A\\)\u200b\u91cd\u8a00\u200b\u8574\u542b\u200b\\(B\\)\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(A\\Rightarrow B\\)\u3002\u200b\u91cd\u8a00\u200b\u8574\u542b\u200b\u5173\u7cfb\u200b\u5177\u6709\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ol> <li>\u200b\u82e5\u200b\\(A=T\\)\uff0c\u200b\u4e14\u200b\\(A\\Rightarrow B\\)\uff0c\u200b\u5219\u200b\\(B=T\\)</li> <li>\u200b\u82e5\u200b\\(A\\Rightarrow B\\land B\\Rightarrow A\\)\uff0c\u200b\u5219\u200b\\(A=B\\)</li> <li>\u200b\u82e5\u200b\\(A\\Rightarrow B\\land B\\Rightarrow C\\)\uff0c\u200b\u5219\u200b\\(A\\Rightarrow C\\)</li> <li>\u200b\u82e5\u200b\\(A\\Rightarrow B\\land A\\Rightarrow C\\)\uff0c\u200b\u5219\u200b\\(A\\Rightarrow (B\\land C)\\)</li> <li>\u200b\u82e5\u200b\\(A\\Rightarrow C\\land B\\Rightarrow C\\)\uff0c\u200b\u5219\u200b\\((A\\lor B)\\Rightarrow C\\)</li> </ol> <p>\u200b\u8981\u200b\u8bc1\u660e\u200b\u4e00\u4e2a\u200b\u5f0f\u5b50\u200b\u4e3a\u91cd\u200b\u8a00\u200b\u8574\u542b\u200b\u5f0f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8bc1\u660e\u200b\\(A\\rightarrow B\\)\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\u6216\u200b\\(\\lnot A\\land B\\)\u200b\u4e3a\u200b\u77db\u76fe\u200b\u5f0f\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-2/#_9","title":"\u57fa\u672c\u200b\u7684\u200b\u63a8\u7406\u200b\u516c\u5f0f","text":"<p>\u200b\u5982\u4e0b\u200b\u5217\u51fa\u200b\u4e00\u4e9b\u200b\u57fa\u672c\u200b\u7684\u200b\u63a8\u7406\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5f0f\u200b\u4e2d\u200b\\(P,Q,R,S\\)\u200b\u5747\u200b\u4e3a\u200b\u4efb\u610f\u200b\u547d\u9898\u200b\u53d8\u9879\u200b</p> <ul> <li>\\(P\\land Q\\Rightarrow P\\)</li> <li>\\(\\lnot(P\\rightarrow Q)\\Rightarrow P\\)</li> <li>\\(\\lnot(P\\rightarrow Q)\\Rightarrow \\lnot Q\\)</li> <li>\\(P\\Rightarrow P\\lor Q\\)</li> <li>\\(\\lnot P\\Rightarrow P\\rightarrow Q\\)</li> <li>\\(Q\\Rightarrow P\\rightarrow Q\\)</li> <li>\\(\\lnot P \\land(P\\lor Q)\\Rightarrow Q\\)</li> <li>\\(P\\land (P\\rightarrow Q) \\Rightarrow Q\\)\uff0c\u200b\u5206\u79bb\u200b\u89c4\u5219\u200b</li> <li>\\(\\lnot Q\\land (P\\rightarrow Q)\\Rightarrow \\lnot P\\)</li> <li>\\((P\\rightarrow Q)\\land (Q\\rightarrow R)\\Rightarrow P\\rightarrow R\\)\uff0c\u200b\u4e09\u6bb5\u8bba\u200b</li> <li>\\((P\\leftrightarrow Q)\\land (Q\\leftrightarrow R)\\Rightarrow P\\leftrightarrow R\\)</li> <li>\\((P\\rightarrow R)\\land (Q\\rightarrow R)\\land (P\\lor Q)\\Rightarrow R\\)</li> <li>\\((R\\rightarrow Q)\\land (R\\rightarrow S)\\land (P\\lor R)\\Rightarrow Q\\lor S\\)</li> <li>\\((P\\rightarrow Q)\\land (R\\rightarrow S)\\land (\\lnot Q\\lor \\lnot S)\\Rightarrow (\\lnot P\\lor \\lnot R)\\)</li> <li>\\(Q\\rightarrow R\\Rightarrow ((P\\lor Q)\\rightarrow (P\\lor R))\\)</li> <li>\\(Q\\rightarrow R\\Rightarrow ((P\\rightarrow Q)\\rightarrow (P\\rightarrow R))\\)</li> </ul>"},{"location":"math/discrete-mathematics/chapter-2/#_10","title":"\u63a8\u7406\u200b\u6f14\u7b97","text":"<p>\u200b\u63a8\u7406\u200b\u6f14\u7b97\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u89c4\u5219\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\uff1a</p> <ul> <li>\u200b\u524d\u63d0\u200b\u5f15\u5165\u200b\u89c4\u5219\u200b\uff1a\u200b\u63a8\u7406\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u53ef\u4ee5\u200b\u968f\u65f6\u200b\u5f15\u5165\u200b\u524d\u63d0\u200b</li> <li>\u200b\u7ed3\u8bba\u200b\u5f15\u7528\u200b\u89c4\u5219\u200b\uff1a\u200b\u63a8\u7406\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u7684\u200b\u4e2d\u95f4\u200b\u7ed3\u8bba\u200b\u53ef\u4ee5\u200b\u4f5c\u4e3a\u200b\u540e\u7eed\u200b\u63a8\u7406\u200b\u7684\u200b\u524d\u63d0\u200b</li> <li>\u200b\u4ee3\u5165\u200b\u89c4\u5219\u200b\uff1a\u200b\u63a8\u7406\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u5bf9\u200b\u91cd\u8a00\u5f0f\u200b\u7684\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4ee3\u5165\u200b\u89c4\u5219\u200b</li> <li>\u200b\u7f6e\u6362\u200b\u89c4\u5219\u200b\uff1a\u200b\u63a8\u7406\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u4efb\u4f55\u200b\u90e8\u5206\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u7528\u200b\u4e0e\u200b\u4e4b\u200b\u7b49\u503c\u200b\u7684\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u8fdb\u884c\u200b\u66ff\u6362\u200b</li> <li>\u200b\u5206\u79bb\u200b\u89c4\u5219\u200b\uff1a\u200b\u82e5\u200b\\(A\\)\u200b\u4e14\u200b\\(A\\rightarrow B\\)\u200b\u5219\u200b\\(B\\)</li> <li>\u200b\u6761\u4ef6\u200b\u8bc1\u660e\u200b\u89c4\u5219\u200b\uff1a\\(A_1\\land A_2\\Rightarrow B \\Leftrightarrow A_1\\Rightarrow A_2\\rightarrow B\\)</li> </ul>"},{"location":"math/discrete-mathematics/chapter-2/#_11","title":"\u5f52\u7ed3\u200b\u63a8\u7406","text":"<p>\u200b\u601d\u8def\u200b\uff1a\u200b\u8981\u200b\u8bc1\u660e\u200b\\(A\\Rightarrow B\\)\uff0c\u200b\u53ef\u200b\u8bc1\u660e\u200b\\(A\\land \\lnot B=F\\)\u3002</p> <ul> <li>\u200b\u5c06\u200b\\(C = A\\land \\lnot B\\)\u200b\u5316\u4e3a\u200b\u5408\u53d6\u8303\u5f0f\u200b\\(Q_1\\land \\dots \\land Q_n\\)\uff0c\u200b\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u6790\u53d6\u200b\u5f0f\u200b\u6784\u6210\u200b\u5b50\u53e5\u200b\u96c6\u5408\u200b\\(\\{Q_1,\\dots, Q_n\\}\\)</li> <li>\u200b\u6d88\u200b\u53bb\u200b\u5b50\u53e5\u200b\u96c6\u5408\u200b\u4e2d\u200b\u7684\u200b\u4e92\u8865\u200b\u5bf9\u200b\uff0c\u200b\u5982\u200b\\(P\\lor R\\)\u200b\u4e0e\u200b\\(\\lnot P\\lor \\lnot Q\\)\u200b\u5f52\u7ed3\u200b\u5f97\u5230\u200b\\(R\\lor \\lnot Q\\)\u3002\u200b\u4e0d\u65ad\u200b\u91cd\u590d\u200b\u6d88\u200b\u53bb\u200b\u8fc7\u7a0b\u200b\u3002</li> <li>\u200b\u76f4\u5230\u200b\u5b50\u53e5\u200b\u96c6\u5408\u200b\u4e3a\u7a7a\u200b\uff0c\u200b\u5f97\u5230\u200b\u7a7a\u5b50\u200b\u53e5\u200b\\(\\square\\)\uff0c\u200b\u8bc1\u660e\u200b\\(A\\land \\lnot B\\)\u200b\u662f\u200b\u77db\u76fe\u200b\u5f0f\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-3/","title":"\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u5316","text":"<p>\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u5305\u542b\u200b\u5982\u4e0b\u200b\u7ed3\u6784\u200b\uff1a</p> <ul> <li>\u200b\u521d\u59cb\u200b\u7b26\u53f7\u200b\uff1a\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u5141\u8bb8\u200b\u51fa\u73b0\u200b\u7684\u200b\u7b26\u53f7\u200b\u96c6\u5408\u200b</li> <li>\u200b\u5f62\u6210\u200b\u89c4\u5219\u200b\uff1a\u200b\u5b9a\u4e49\u200b\u4e86\u200b\u7b26\u53f7\u200b\u5e8f\u5217\u200b\u7684\u200b\u5408\u6cd5\u6027\u200b</li> <li>\u200b\u516c\u7406\u200b\uff1a\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u6700\u200b\u57fa\u672c\u200b\u7684\u200b\u91cd\u8a00\u5f0f\u200b\uff0c\u200b\u662f\u200b\u63a8\u6f14\u200b\u5176\u4ed6\u200b\u6240\u6709\u200b\u91cd\u8a00\u5f0f\u200b\u7684\u200b\u4f9d\u636e\u200b</li> <li>\u200b\u53d8\u5f62\u200b\u89c4\u5219\u200b\uff1a\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u6240\u200b\u89c4\u5b9a\u200b\u7684\u200b\u63a8\u6f14\u200b\u89c4\u5219\u200b\uff0c\u200b\u516c\u7406\u200b\u901a\u8fc7\u200b\u53d8\u5f62\u200b\u89c4\u5219\u200b\u5f97\u5230\u200b\u7684\u200b\u516c\u5f0f\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u5b9a\u7406\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-3/#_2","title":"\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u200b\u7cfb\u7edf","text":"<p>\u200b\u4f9d\u636e\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u7684\u200b\u7ed3\u6784\u200b\u5efa\u7acb\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u3002\u200b\u5e94\u5f53\u200b\u6ce8\u610f\u200b\u5230\u200b\uff0c\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u4e0d\u662f\u200b\u552f\u4e00\u200b\u7684\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-3/#_3","title":"\u521d\u59cb\u200b\u7b26\u53f7","text":"<p>\u200b\u5927\u5199\u200b\u82f1\u6587\u5b57\u6bcd\u200b\\(A,\\dots ,Z\\)\u200b\u8868\u793a\u200b\u547d\u9898\u200b \\(\\lnot, \\lor\\)\u200b\u8868\u793a\u200b\u8054\u7ed3\u8bcd\u200b \\((, )\\)\u200b\u4e00\u5bf9\u200b\u62ec\u53f7\u200b\u89c4\u5b9a\u200b\u4e86\u200b\u8054\u7ed3\u8bcd\u200b\u8fd0\u7b97\u200b\u7684\u200b\u4f18\u5148\u200b\u987a\u5e8f\u200b \\(\\vdash\\)\u200b\u8868\u660e\u200b\u4e00\u4e2a\u200b\u516c\u5f0f\u200b\u662f\u200b\u6c38\u771f\u5f0f\u200b</p>"},{"location":"math/discrete-mathematics/chapter-3/#_4","title":"\u5f62\u6210\u200b\u89c4\u5219","text":"<p>\u200b\u79f0\u200b\u521d\u59cb\u200b\u7b26\u53f7\u200b\u6784\u6210\u200b\u7684\u200b\u7b26\u53f7\u200b\u5e8f\u5217\u200b\u4e3a\u200b\u516c\u5f0f\u200b\uff0c\u200b\u53ea\u6709\u200b\u7b26\u5408\u200b\u4ee5\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u7b26\u53f7\u200b\u5e8f\u5217\u200b\u662f\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff1a</p> <ol> <li>\u200b\u5355\u4e2a\u200b\u7b26\u53f7\u200b\\(\\pi\\)\uff0c\\(\\pi\\)\u200b\u662f\u200b\u5927\u5199\u200b\u82f1\u6587\u5b57\u6bcd\u200b</li> <li>\u200b\u82e5\u200b\\(A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\\(\\lnot A\\)\u200b\u662f\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u82e5\u200b\\(A, B\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\\((A\\lor B)\\)\u200b\u662f\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> </ol> <p>\u200b\u4e3a\u200b\u7b80\u5316\u200b\u5176\u4ed6\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u7684\u200b\u8868\u8fbe\u200b\uff0c\u200b\u5f15\u5165\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\u7684\u200b\u516c\u5f0f\u200b\uff1a</p> <ul> <li>\\((A\\rightarrow B):= (\\lnot A\\lor B)\\)</li> <li>\\((A\\land B):= \\lnot (\\lnot A\\lor \\lnot B)\\)</li> <li>\\((A\\leftrightarrow B):= ((A\\rightarrow B)\\land (B\\rightarrow A))\\)</li> </ul>"},{"location":"math/discrete-mathematics/chapter-3/#_5","title":"\u516c\u7406","text":"<p>\u200b\u5728\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u5f15\u5165\u200b\u5982\u4e0b\u200b\u516c\u7406\u200b\uff1a</p> <ul> <li>\\(\\vdash ((P\\lor P)\\rightarrow P)\\)</li> <li>\\(\\vdash (P\\rightarrow (P\\lor Q))\\)</li> <li>\\(\\vdash ((P\\lor Q) \\rightarrow (Q\\lor P))\\)</li> <li>\\(\\vdash ((Q\\rightarrow R)\\rightarrow ((P\\lor Q)\\rightarrow (P\\lor R)))\\)</li> </ul>"},{"location":"math/discrete-mathematics/chapter-3/#_6","title":"\u63a8\u7406\u200b\u89c4\u5219","text":"<p>\u200b\u4ee3\u5165\u200b\u89c4\u5219\u200b\u3001\u200b\u5206\u79bb\u200b\u89c4\u5219\u200b\u3001\u200b\u7f6e\u6362\u200b\u89c4\u5219\u200b\u662f\u200b\u547d\u9898\u903b\u8f91\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u7684\u200b\u63a8\u7406\u200b\u89c4\u5219\u200b</p>"},{"location":"math/discrete-mathematics/chapter-3/#_7","title":"\u738b\u6d69\u200b\u7b97\u6cd5","text":""},{"location":"math/discrete-mathematics/chapter-3/#_8","title":"\u521d\u59cb\u200b\u7b26\u53f7","text":"<p>\u200b\u5927\u5199\u5b57\u6bcd\u200b\\(A, \\dots, Z\\)\u200b\u8868\u793a\u200b\u547d\u9898\u200b \\(\\lnot, \\land, \\lor, \\rightarrow, \\leftrightarrow\\)\u200b\u8868\u793a\u200b\u8054\u7ed3\u8bcd\u200b \\((, ), ,\\)\u200b\u8868\u793a\u200b\u5706\u62ec\u53f7\u200b\u4e0e\u200b\u9017\u53f7\u200b \\(\\alpha, \\beta, \\dots, \\omega\\)\u200b\u8868\u793a\u200b\u516c\u5f0f\u200b\u4e32\u200b</p>"},{"location":"math/discrete-mathematics/chapter-3/#_9","title":"\u5f62\u6210\u200b\u89c4\u5219","text":"<p>\u200b\u7b26\u5408\u200b\u4ee5\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u7b26\u53f7\u200b\u5e8f\u5217\u200b\u79f0\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff1a</p> <ol> <li>\u200b\u5355\u4e2a\u200b\u7b26\u53f7\u200b\\(\\pi\\)\uff0c\\(\\pi\\)\u200b\u662f\u200b\u5927\u5199\u200b\u82f1\u6587\u5b57\u6bcd\u200b</li> <li>\u200b\u82e5\u200b\\(A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\u200b\u5219\u200b\\(\\lnot A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u82e5\u200b\\(A, B\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\u200b\u5219\u200b\\((A\\lnot B), (A\\lor B), (A\\rightarrow B), (A\\leftrightarrow B)\\)\u200b\u662f\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> </ol> <p>\u200b\u6ee1\u8db3\u200b\u4ee5\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u79f0\u4e3a\u200b\u516c\u5f0f\u200b\u4e32\u200b\uff1a</p> <ol> <li>\u200b\u7a7a\u200b\u7b26\u53f7\u4e32\u200b\u662f\u200b\u516c\u5f0f\u200b\u4e32\u200b</li> <li>\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u662f\u200b\u516c\u5f0f\u200b\u4e32\u200b</li> <li>\u200b\u5982\u679c\u200b\\(\\alpha\\)\u200b\u548c\u200b\\(\\beta\\)\u200b\u662f\u200b\u516c\u5f0f\u200b\u4e32\u200b\uff0c\u200b\u5219\u200b\\(\\alpha, \\beta\\)\u200b\u662f\u200b\u516c\u5f0f\u200b\u4e32\u200b</li> </ol> <p>\u200b\u516c\u5f0f\u200b\u4e32\u200b\u4e0e\u200b\u7ec4\u6210\u200b\u516c\u5f0f\u200b\u4e32\u200b\u7684\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u7684\u200b\u987a\u5e8f\u200b\u65e0\u5173\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-3/#_10","title":"\u5b9a\u4e49","text":"<ol> <li>\u200b\u5b9a\u4e49\u200b\u76f8\u7ee7\u200b\u5f0f\u200b\u4e3a\u200b\\(\\alpha\\stackrel s\\rightarrow\\beta\\)\uff0c\u200b\u5176\u4e2d\u200b\\(\\alpha, \\beta\\)\u200b\u90fd\u200b\u662f\u200b\u516c\u5f0f\u200b\u4e32\u200b</li> <li>\u200b\u524d\u4ef6\u200b\u4e2d\u200b\u7684\u200b\\(,\\)\u200b\u89e3\u91ca\u200b\u4e3a\u200b\\(\\land\\)\uff0c\u200b\u540e\u4ef6\u200b\u4e2d\u200b\u7684\u200b\\(,\\)\u200b\u89e3\u91ca\u200b\u4e3a\u200b\\(\\lor\\)</li> <li>\u200b\u82e5\u200b\\(\\alpha\\stackrel s\\rightarrow \\beta\\)\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8bb0\u200b\u4f5c\u200b\\(\\alpha\\stackrel s\\Rightarrow \\beta\\)</li> </ol>"},{"location":"math/discrete-mathematics/chapter-3/#_11","title":"\u516c\u7406","text":"<p>\u200b\u82e5\u200b\\(\\alpha\\)\u200b\u548c\u200b\\(\\beta\\)\u200b\u7684\u200b\u516c\u5f0f\u200b\u4ec5\u4e3a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\uff0c\u200b\u4e0d\u200b\u5305\u542b\u200b\u8054\u7ed3\u8bcd\u200b\uff0c\u200b\u5219\u200b\\(\\alpha\\stackrel s\\rightarrow \\beta\\)\u200b\u4e3a\u200b\u91cd\u8a00\u5f0f\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\\(\\alpha\\)\u200b\u4e0e\u200b\\(\\beta\\)\u200b\u4e2d\u200b\u81f3\u5c11\u200b\u542b\u6709\u200b\u4e00\u4e2a\u200b\u76f8\u540c\u200b\u7684\u200b\u547d\u9898\u200b\u53d8\u9879\u200b</p>"},{"location":"math/discrete-mathematics/chapter-3/#_12","title":"\u63a8\u7406\u200b\u89c4\u5219","text":"<p>\u200b\u5bf9\u4e8e\u200b\u5b58\u5728\u200b\u4e8e\u200b\u524d\u4ef6\u200b\u4e2d\u200b\u7684\u200b\u8fd0\u7b97\u7b26\u200b\uff0c\u200b\u63a8\u7406\u200b\u89c4\u5219\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\\(\\lnot\\)\uff1a\u200b\u82e5\u200b\\(\\alpha, \\beta \\stackrel s\\Rightarrow X, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha, \\lnot X, \\beta \\stackrel x\\Rightarrow \\gamma\\)</li> <li>\\(\\land\\)\uff1a\u200b\u82e5\u200b\\(X, Y, \\alpha, \\beta\\stackrel s\\Rightarrow \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha, X\\land Y, \\beta\\stackrel s\\Rightarrow Y\\)</li> <li>\\(\\lor\\)\uff1a\u200b\u82e5\u200b\\(X, \\alpha, \\beta \\stackrel s\\Rightarrow \\gamma\\)\u200b\u4e14\u200b\\(Y, \\alpha, \\beta \\stackrel s\\Rightarrow X, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha, X\\lor Y, \\beta\\stackrel s\\Rightarrow \\gamma\\)</li> <li>\\(\\rightarrow\\)\uff1a\u200b\u82e5\u200b\\(Y, \\alpha, \\beta\\stackrel s\\Rightarrow \\gamma\\)\u200b\u4e14\u200b\\(\\alpha, \\beta\\stackrel s\\Rightarrow X, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha, X\\rightarrow Y, \\beta\\stackrel s\\Rightarrow \\gamma\\)</li> <li>\\(\\leftrightarrow\\)\uff1a\u200b\u82e5\u200b\\(X, Y, \\alpha, \\beta\\stackrel s\\Rightarrow \\gamma\\)\u200b\u4e14\u200b\\(\\alpha, \\beta\\stackrel s\\Rightarrow X, Y, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha, X\\leftrightarrow Y, \\beta\\stackrel s\\Rightarrow \\gamma\\)</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u5b58\u5728\u200b\u4e8e\u200b\u540e\u4ef6\u200b\u4e2d\u200b\u7684\u200b\u8fd0\u7b97\u7b26\u200b\uff0c\u200b\u63a8\u7406\u200b\u89c4\u5219\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\\(\\lnot\\)\uff1a\u200b\u82e5\u200b\\(X, \\alpha\\stackrel s\\Rightarrow \\beta, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha\\stackrel s\\Rightarrow\\beta, \\lnot X, \\gamma\\)</li> <li>\\(\\land\\)\uff1a\u200b\u82e5\u200b\\(\\alpha\\stackrel s\\Rightarrow X, \\beta, \\gamma\\)\u200b\u4e14\u200b\\(\\alpha\\stackrel s\\Rightarrow\\)</li> <li>\\(\\land\\)\uff1a\u200b\u82e5\u200b\\(\\alpha\\stackrel s\\Rightarrow X, \\beta, \\gamma\\)\u200b\u4e14\u200b\\(\\alpha\\stackrel s\\Rightarrow Y, \\beta, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha \\stackrel s\\Rightarrow\\beta, X\\land Y, \\gamma\\)</li> <li>\\(\\lor\\)\uff1a\u200b\u82e5\u200b\\(\\alpha\\stackrel s\\Rightarrow X, Y, \\beta, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha\\stackrel s\\Rightarrow \\beta, X\\lor Y, \\gamma\\)</li> <li>\\(\\rightarrow\\)\uff1a\u200b\u82e5\u200b\\(X, \\alpha\\stackrel s\\Rightarrow Y, \\beta, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha\\stackrel s\\Rightarrow \\beta, X\\rightarrow Y, \\gamma\\)</li> <li>\\(\\leftrightarrow\\)\uff1a\u200b\u82e5\u200b\\(X, \\alpha\\stackrel s\\Rightarrow Y, \\beta, \\gamma\\)\u200b\u4e14\u200b\\(Y, \\alpha\\stackrel s\\Rightarrow X, \\beta, \\gamma\\)\uff0c\u200b\u5219\u200b\\(\\alpha \\stackrel s\\Rightarrow \\beta, X\\leftrightarrow Y, \\gamma\\)</li> </ul> <p>\u200b\u5b9a\u7406\u200b\u7684\u200b\u63a8\u6f14\u200b\u8fc7\u7a0b\u200b\uff1a</p> <ol> <li>\u200b\u5c06\u8981\u200b\u8bc1\u660e\u200b\u7684\u200b\u5b9a\u7406\u200b\u5199\u6210\u200b\u76f8\u7ee7\u200b\u5f0f\u200b\u5f62\u5f0f\u200b</li> <li>\u200b\u4ece\u200b\u76f8\u7ee7\u200b\u5f0f\u200b\u51fa\u53d1\u200b\u4f7f\u7528\u200b\u53d8\u5f62\u200b\u89c4\u5219\u200b\u6d88\u200b\u53bb\u200b\u76f8\u7ee7\u200b\u5f0f\u200b\u4e24\u4fa7\u200b\u7684\u200b\u6240\u6709\u200b\u903b\u8f91\u200b\u8054\u7ed3\u8bcd\u200b</li> <li>\u200b\u82e5\u200b\u5f97\u5230\u200b\u7684\u200b\u6240\u6709\u200b\u76f8\u7ee7\u200b\u5f0f\u200b\u90fd\u200b\u662f\u200b\u516c\u7406\u200b\uff0c\u200b\u5219\u200b\u5b9a\u7406\u200b\u6210\u7acb\u200b</li> </ol>"},{"location":"math/discrete-mathematics/chapter-3/#_13","title":"\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u81ea\u7136\u200b\u6f14\u7ece\u200b\u7cfb\u7edf","text":""},{"location":"math/discrete-mathematics/chapter-3/#_14","title":"\u521d\u59cb\u200b\u7b26\u53f7","text":"<p>\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u81ea\u7136\u200b\u6f14\u7ece\u200b\u7cfb\u7edf\u200b\u7684\u200b\u4e00\u90e8\u5206\u200b\u521d\u59cb\u200b\u7b26\u53f7\u200b\u7ee7\u627f\u200b\u81ea\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u3002\u200b\u6b64\u5916\u200b\uff0c\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u81ea\u7136\u200b\u6f14\u7ece\u200b\u7cfb\u7edf\u200b\u8fd8\u200b\u5305\u542b\u200b\u5982\u4e0b\u200b\u521d\u59cb\u200b\u7b26\u53f7\u200b\uff1a</p> \\[ \\varGamma = \\{A_1, \\dots, A_n\\}=A_1, \\dots, A_n \\] <p>\u200b\u8868\u793a\u200b\u6709\u9650\u200b\u4e2a\u200b\u547d\u9898\u200b\u516c\u5f0f\u200b\u96c6\u5408\u200b\uff0c\u200b\u5b9a\u4e49\u200b\\(\\varGamma \\vdash A\\)\u200b\u8868\u793a\u200b\\(\\varGamma, A\\)\u200b\u4e4b\u95f4\u200b\u6709\u200b\u5f62\u5f0f\u200b\u63a8\u7406\u200b\u5173\u7cfb\u200b\uff0c\\(\\varGamma\\)\u200b\u4e3a\u200b\u5f62\u5f0f\u200b\u524d\u63d0\u200b\uff0c\\(A\\)\u200b\u4e3a\u200b\u5f62\u5f0f\u200b\u7ed3\u8bba\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-3/#_15","title":"\u5f62\u6210\u200b\u89c4\u5219","text":"<p>\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u81ea\u7136\u200b\u6f14\u7ece\u200b\u7cfb\u7edf\u200b\u7684\u200b\u5f62\u6210\u200b\u89c4\u5219\u200b\u4e0e\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u7684\u200b\u5f62\u6210\u200b\u89c4\u5219\u200b\u76f8\u540c\u200b</p>"},{"location":"math/discrete-mathematics/chapter-3/#_16","title":"\u53d8\u5f62\u200b\u89c4\u5219","text":"<p>\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u81ea\u7136\u200b\u6f14\u7ece\u200b\u7cfb\u7edf\u200b\u6709\u200b\u5982\u4e0b\u200b\u53d8\u5f62\u200b\u89c4\u5219\u200b\uff1a</p> <ol> <li>\\(A_1, \\dots, A_n\\vdash A_i(i=1, \\dots, n)\\)\uff0c\u200b\u5373\u200b\u524d\u63d0\u200b\u4e2d\u200b\u7684\u200b\u4efb\u4f55\u200b\u547d\u9898\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u4f5c\u4e3a\u200b\u7ed3\u8bba\u200b\u51fa\u73b0\u200b</li> <li>\u200b\u4f20\u9012\u200b\u5f8b\u200b\uff1a\u200b\u82e5\u200b\\(\\varGamma \\vdash A\\)\u200b\u4e14\u200b\\(A\\vdash B\\)\uff0c\u200b\u5219\u200b\\(\\varGamma\\vdash B\\)</li> <li>\u200b\u53cd\u8bc1\u200b\u5f8b\u200b\uff1a\u200b\u82e5\u200b\\(\\varGamma, \\lnot A\\vdash B\\)\u200b\u4e14\u200b\\(\\varGamma, \\lnot A\\vdash \\lnot B\\)\uff0c\u200b\u5219\u200b\\(\\varGamma \\vdash A\\)\uff08\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u8bc1\u660e\u200b\\(\\varGamma \\land \\lnot A=F\\)\uff09</li> <li>\u200b\u5206\u79bb\u200b\u89c4\u5219\u200b\uff1a\\(A, A\\rightarrow B\\vdash B\\)</li> <li>\u200b\u8574\u542b\u200b\u8bcd\u200b\u5f15\u5165\u200b\uff1a\u200b\u82e5\u200b\\(\\varGamma, A\\vdash B\\)\uff0c\u200b\u5219\u200b\\(\\varGamma\\vdash A\\rightarrow B\\)</li> </ol>"},{"location":"math/discrete-mathematics/chapter-3/#_17","title":"\u5b9a\u7406","text":"<p>\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u81ea\u7136\u200b\u6f14\u7ece\u200b\u7cfb\u7edf\u200b\u5305\u542b\u200b\u7684\u200b\u5b9a\u7406\u200b\u4e0e\u200b\u547d\u9898\u903b\u8f91\u200b\u7684\u200b\u516c\u7406\u200b\u7cfb\u7edf\u200b\u5305\u542b\u200b\u7684\u200b\u5b9a\u7406\u200b\u76f8\u540c\u200b</p>"},{"location":"math/discrete-mathematics/chapter-3/#_18","title":"\u975e\u6807\u51c6\u200b\u903b\u8f91","text":"<ol> <li>\u200b\u591a\u503c\u903b\u8f91\u200b\uff0c\u200b\u82e5\u200b\\(P\\)\u200b\u53d6\u503c\u200b\u4e8e\u200b\\([0,1]\\)\uff0c\u200b\u53ef\u4ee5\u200b\u89e3\u91ca\u200b\u4e3a\u200b<ul> <li>\\(P=0\\)\u200b\u8868\u793a\u200b\\(P\\)\u200b\u771f\u200b</li> <li>\\(P=1\\)\u200b\u8868\u793a\u200b\\(P\\)\u200b\u5047\u200b</li> <li>\\(0&lt;P&lt;1\\)\u200b\u8868\u793a\u200b\\(P\\)\u200b\u4e3a\u200b\u771f\u7684\u200b\u6982\u7387\u200b\u4e3a\u200b\\(1-P\\) </li> </ul> </li> <li>\u200b\u4e09\u503c\u200b\u903b\u8f91\u200b<ul> <li>Kleene\u200b\u903b\u8f91\u200b\uff1a\\(T, F, U\\)\uff0c\\(U\\)\u200b\u8868\u793a\u200b\u672a\u77e5\u200b</li> <li>Lukasiewicz\u200b\u903b\u8f91\u200b\uff1a\\(T, F, I\\)\uff0c\\(I\\)\u200b\u8868\u793a\u200b\u5c06\u6765\u200b\u53ef\u80fd\u200b\uff0c\u200b\u76ee\u524d\u200b\u4e0d\u200b\u5177\u5907\u200b\u771f\u503c\u200b</li> <li>Bochvar\u200b\u903b\u8f91\u200b\uff1a\\(T, F, M\\)\uff0c\\(M\\)\u200b\u8868\u793a\u200b\u6096\u8bba\u200b</li> </ul> </li> </ol>"},{"location":"math/discrete-mathematics/chapter-4/","title":"\u8c13\u8bcd\u200b\u903b\u8f91\u200b\u7684\u200b\u57fa\u672c\u6982\u5ff5","text":"<p>\u200b\u8c13\u8bcd\u200b\u8868\u793a\u200b\u4e86\u200b\u591a\u4e2a\u200b\u4e2a\u4f53\u200b\u8bcd\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5173\u7cfb\u200b\u3002\u200b\u8fdb\u4e00\u6b65\u200b\u5730\u200b\u8bb2\u200b\u662f\u200b\u7ed9\u5b9a\u200b\u7684\u200b\u4e2a\u4f53\u200b\u8bcd\u5230\u200b\u96c6\u5408\u200b\\(\\{T, F\\}\\)\u200b\u4e0a\u200b\u7684\u200b\u4e00\u4e2a\u200b\u6620\u5c04\u200b\uff0c\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5176\u200b\u5199\u200b\u4e3a\u200b\u51fd\u6570\u200b\u7684\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u5982\u200b\\(P(x, y)\\)</p> <p>\u200b\u51fd\u6570\u200b\u8868\u793a\u200b\u4ece\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\u5230\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\u7684\u200b\u6620\u5c04\u200b\uff0c\u200b\u901a\u5e38\u200b\u4f7f\u7528\u200b\u5c0f\u5199\u5b57\u6bcd\u200b\u8fdb\u884c\u200b\u8868\u793a\u200b\u3002</p> <p>\u200b\u6709\u200b\u4e24\u79cd\u200b\u91cf\u8bcd\u200b\uff0c\u200b\u5373\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u4e0e\u200b\u5b58\u5728\u200b\u91cf\u8bcd\u200b\u3002\u200b\u53d7\u5230\u200b\u91cf\u8bcd\u200b\u7ea6\u675f\u200b\u7684\u200b\u53d8\u5143\u200b\u79f0\u4e3a\u200b\u7ea6\u675f\u200b\u53d8\u5143\u200b\uff0c\u200b\u5426\u5219\u200b\u79f0\u4e3a\u200b\u81ea\u7531\u200b\u53d8\u5143\u200b\u3002\u200b\u91cf\u8bcd\u200b\u53ea\u80fd\u200b\u4f5c\u7528\u200b\u4e8e\u200b\u53d8\u5143\u200b\u800c\u200b\u4e0d\u80fd\u200b\u4f5c\u7528\u200b\u4e8e\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u4e0e\u200b\u8c13\u8bcd\u200b\u53d8\u9879\u200b\u3002\u200b\u53d8\u5143\u200b\u4f7f\u7528\u200b\u5c0f\u5199\u5b57\u6bcd\u200b\u8868\u793a\u200b\u3002</p>"},{"location":"math/discrete-mathematics/chapter-4/#_2","title":"\u5408\u5f0f\u516c\u5f0f","text":"<p>\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff1a</p> <ol> <li>\u200b\u547d\u9898\u200b\u5e38\u9879\u200b\u3001\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u548c\u200b\u539f\u5b50\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u90fd\u200b\u662f\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u82e5\u200b\\(A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\u200b\u5219\u200b\\(\\lnot A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> <li>\u200b\u82e5\u200b\\(A, B\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\u200b\u4e14\u200b\\(A, B\\)\u200b\u4e2d\u65e0\u53d8\u200b\u5143\u200b\u5728\u200b\u4e00\u4e2a\u200b\u4e2d\u200b\u7ea6\u675f\u200b\u800c\u200b\u5728\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u4e2d\u200b\u81ea\u7531\u200b\uff0c\u200b\u5219\u200b\\((A\\land B), (A\\lor B), (A\\rightarrow B), (A\\leftrightarrow B)\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\u3002</li> <li>\u200b\u82e5\u200b\\(A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff0c\\(x\\)\u200b\u5728\u200b\\(A\\)\u200b\u4e2d\u4e3a\u200b\u81ea\u7531\u200b\u53d8\u5143\u200b\uff0c\u200b\u5219\u200b\\((\\forall x)A, (\\exists x)A\\)\u200b\u4e3a\u200b\u5408\u5f0f\u516c\u5f0f\u200b</li> </ol> <p>\u200b\u5728\u200b\u6709\u9650\u200b\u8bba\u57df\u200b\u4e0b\u200b\uff0c\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u4e0e\u200b\u7279\u79f0\u200b\u91cf\u8bcd\u200b\u53ef\u200b\u4e0e\u200bi\u200b\u8868\u793a\u200b\u4e3a\u200b\u547d\u9898\u903b\u8f91\u200b\u4e0b\u200b\u7684\u200b\u5408\u5f0f\u516c\u5f0f\u200b\uff1a</p> <ul> <li>\\((\\forall x)P(x) = P(1)\\land P(2)\\land \\dots \\land P(k)\\)</li> <li>\\((\\exists x)P(x) = P(1)\\lor P(2)\\lor \\dots \\lor P(k)\\)\\</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\uff0c\u200b\u82e5\u200b\u5728\u200b\u4efb\u4f55\u200b\u89e3\u91ca\u200b\u4e0b\u200b\u771f\u503c\u200b\u90fd\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u79f0\u8c13\u200b\u8bcd\u200b\u516c\u5f0f\u200b\u4e3a\u200b\u666e\u904d\u200b\u6709\u6548\u200b\u7684\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\uff1b\u200b\u82e5\u200b\u5728\u200b\u67d0\u4e2a\u200b\u89e3\u91ca\u200b\u4e0b\u200b\u771f\u503c\u200b\u4e3a\u200b\u771f\u200b\uff0c\u200b\u79f0\u8c13\u200b\u8bcd\u200b\u516c\u5f0f\u200b\u4e3a\u200b\u53ef\u200b\u6ee1\u8db3\u200b\u7684\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\uff0c\u200b\u5426\u5219\u200b\u79f0\u4e3a\u200b\u4e0d\u53ef\u200b\u6ee1\u8db3\u200b\u7684\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u3002</p> <ul> <li>\u200b\u8c13\u8bcd\u200b\u903b\u8f91\u200b\u662f\u200b\u4e0d\u53ef\u5224\u5b9a\u200b\u7684\u200b</li> <li>\u200b\u53ea\u200b\u542b\u6709\u200b\u4e00\u5143\u200b\u8c13\u8bcd\u200b\u53d8\u9879\u200b\u7684\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u662f\u200b\u53ef\u200b\u5224\u5b9a\u200b\u7684\u200b</li> <li>\u200b\u4e2a\u4f53\u200b\u57df\u200b\u6709\u9650\u200b\u7684\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u662f\u200b\u53ef\u200b\u5224\u5b9a\u200b\u7684\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-5/","title":"\u8c13\u8bcd\u200b\u903b\u8f91\u200b\u7684\u200b\u7b49\u503c\u200b\u548c\u200b\u63a8\u7406\u200b\u6f14\u7b97","text":"<p>\u200b\u5bf9\u4e8e\u200b\u4e24\u4e2a\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\\(A, B\\)\uff0c\u200b\u82e5\u200b\u5728\u200b\\(A, B\\)\u200b\u7684\u200b\u4efb\u4e00\u200b\u89e3\u91ca\u200b\u4e0b\u200b\uff0c\\(A, B\\)\u200b\u90fd\u200b\u6709\u200b\u76f8\u540c\u200b\u7684\u200b\u771f\u503c\u200b\uff0c\u200b\u5373\u200b\\(A\\leftrightarrow B\\)\uff0c\u200b\u79f0\u200b\\(A, B\\)\u200b\u7b49\u503c\u200b\u3002\u200b\u8bb0\u200b\u4e3a\u200b\\(A=B\\)\u200b\u6216\u200b\\(A\\Leftrightarrow B\\)\u3002</p>"},{"location":"math/discrete-mathematics/chapter-5/#_2","title":"\u7b49\u503c\u200b\u5f0f","text":"<ol> <li>\u200b\u5c06\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u4ee3\u5165\u200b\u547d\u9898\u903b\u8f91\u200b\u4e0b\u200b\u7684\u200b\u7b49\u503c\u200b\u516c\u5f0f\u200b\u5373\u53ef\u200b\u5f97\u5230\u200b\u8c13\u8bcd\u200b\u903b\u8f91\u200b\u7684\u200b\u7b49\u503c\u200b\u516c\u5f0f\u200b</li> <li>\u200b\u5426\u5b9a\u200b\u578b\u200b\u7b49\u503c\u200b\u5f0f\u200b</li> <li>\\(\\lnot (\\forall x)P(x) = (\\exists x)\\lnot P(x)\\)</li> <li>\\(\\lnot (\\exists x)P(x) = (\\forall x)\\lnot P(x)\\)</li> <li>\u200b\u91cf\u8bcd\u200b\u5206\u914d\u200b\u7b49\u503c\u200b\u5f0f\u200b\uff08\\(q\\)\u200b\u4e3a\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\uff0c\\(x\\)\u200b\u4e3a\u200b\u53d8\u5143\u200b\uff09</li> <li>\\((\\forall x)(P(x)\\lor q) = (\\forall x)P(x)\\lor q\\)</li> <li>\\((\\exists x)(P(x)\\lor q) = (\\exists x)P(x)\\lor q\\)</li> <li>\\((\\forall x)(P(x)\\land q) = (\\forall x)P(x)\\land q\\)</li> <li>\\((\\exists x)(P(x)\\land q) = (\\forall x)P(x)\\land q\\)</li> <li>\u200b\u5bf9\u200b\\(\\rightarrow\\)\u200b\u7684\u200b\u5206\u914d\u5f8b\u200b</li> <li>\\((\\forall x)(P(x)\\rightarrow q) = (\\exists x)P(x)\\rightarrow q\\)</li> <li>\\((\\exists x)(P(x)\\rightarrow q) = (\\forall x)P(x)\\rightarrow q\\)</li> <li>\\((\\forall x)(p\\rightarrow Q(x)) = p\\rightarrow (\\forall x)Q(x)\\)</li> <li>\\((\\exists x)(p\\rightarrow Q(x)) = p\\rightarrow (\\exists x)Q(x)\\)</li> <li>\u200b\u91cf\u8bcd\u200b\\(\\forall\\)\u200b\u5bf9\u200b\\(\\land\\)\u3001\u200b\u91cf\u8bcd\u200b\\(\\exists\\)\u200b\u5bf9\u200b\\(\\lor\\)\u200b\u7684\u200b\u5206\u914d\u5f8b\u200b</li> <li>\\((\\forall x)(P(x)\\land Q(x)) = (\\forall x)P(x) \\land (\\forall x)Q(x)\\)</li> <li>\\((\\exists x)(P(x)\\lor Q(x)) = (\\exists x)P(x) \\lor (\\exists x)Q(x)\\)</li> <li>\u200b\u53d8\u5143\u200b\u6613\u540d\u200b\u7684\u200b\u5206\u914d\u5f8b\u200b</li> <li>\\((\\forall x)(\\forall y)(P(x)\\lor Q(y)) = (\\forall x)P(x) \\lor (\\forall x)Q(x)\\)</li> <li>\\((\\exists x)(\\exists y)(P(x)\\land Q(y)) = (\\forall x)P(x) \\land (\\forall x)Q(x)\\)</li> </ol>"},{"location":"math/discrete-mathematics/chapter-5/#_3","title":"\u8303\u5f0f","text":"<p>\u200b\u82e5\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u4e2d\u200b\u7684\u200b\u4e00\u5207\u200b\u91cf\u8bcd\u200b\u90fd\u200b\u4f4d\u4e8e\u200b\u516c\u5f0f\u200b\u7684\u200b\u6700\u200b\u5de6\u4fa7\u200b\uff0c\u200b\u4e14\u200b\u91cf\u8bcd\u200b\u7684\u200b\u4f5c\u7528\u57df\u200b\u5ef6\u4f38\u200b\u5230\u200b\u516c\u5f0f\u200b\u7684\u200b\u672b\u7aef\u200b\uff0c\u200b\u5219\u200b\u79f0\u8c13\u200b\u8bcd\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u4e3a\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\u3002\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\u7684\u200b\u4e00\u822c\u200b\u5f62\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ (Q_1x_1)\\dots (Q_nx_n)M(x_1, \\dots, x_n) \\] <p>\u200b\u5176\u4e2d\u200b\\(Q_i\\)\u200b\u4e3a\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\\(\\forall\\)\u200b\u6216\u200b\u7279\u79f0\u200b\u91cf\u8bcd\u200b\\(\\exists\\)\uff0c\\(M\\)\u200b\u79f0\u4e3a\u200b\u6bcd\u5f0f\u200b\uff0c\u200b\u4e0d\u200b\u542b\u6709\u200b\u91cf\u8bcd\u200b\u3002</p> <p>\u200b\u4efb\u4f55\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u90fd\u200b\u6709\u200b\u4e0e\u200b\u4e4b\u200b\u7b49\u503c\u200b\u7684\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\uff0c\u200b\u4f46\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\u4e0d\u200b\u552f\u4e00\u200b\u3002\u200b\u901a\u8fc7\u200b\u5982\u4e0b\u200b\u6d41\u7a0b\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u5316\u200b\u4e3a\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\uff1a</p> <ul> <li>\u200b\u5c06\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\\(\\rightarrow\\)\u200b\u8054\u7ed3\u8bcd\u200b\u4e0e\u200b\\(\\leftrightarrow\\)\u200b\u8054\u7ed3\u8bcd\u200b\u6d88\u200b\u53bb\u200b</li> <li>\u200b\u4f7f\u7528\u200b\u6469\u6839\u200b\u5f8b\u200b\uff0c\u200b\u5c06\u200b\\(\\lnot\\)\u200b\u5185\u79fb\u200b\u5230\u200b\u547d\u9898\u200b\u53d8\u9879\u200b\u6216\u200b\u8c13\u8bcd\u200b\u4e4b\u524d\u200b</li> <li>\u200b\u4f7f\u7528\u200b\u91cf\u8bcd\u200b\u5206\u914d\u200b\u7b49\u503c\u200b\u5f0f\u200b\u5c06\u200b\u91cf\u8bcd\u200b\u79fb\u52a8\u200b\u5230\u200b\u5de6\u4fa7\u200b</li> <li>\u200b\u4f7f\u7528\u200b\u53d8\u5143\u200b\u6613\u540d\u200b\u5206\u914d\u200b\u7b49\u503c\u200b\u5f0f\u200b\u5408\u5e76\u200b\u8c13\u8bcd\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-5/#skolem","title":"Skolem\u200b\u6807\u51c6\u578b","text":"<p>\u200b\u79f0\u200b\u4e00\u4e2a\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u4e3a\u200b\\(\\exists\\)\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u7684\u200b\u5b58\u5728\u200b\u91cf\u8bcd\u200b\u90fd\u200b\u5728\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u7684\u200b\u5de6\u8fb9\u200b\uff0c\u200b\u5e76\u4e14\u200b\u53ef\u4ee5\u200b\u4fdd\u6301\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u81f3\u5c11\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u5b58\u5728\u200b\u91cf\u8bcd\u200b\u3002\u200b\u6bcd\u5f0f\u200b\u4e2d\u200b\u4e0d\u200b\u542b\u6709\u200b\u4efb\u4f55\u200b\u91cf\u8bcd\u200b\uff0c\u200b\u4e5f\u200b\u4e0d\u200b\u542b\u200b\u81ea\u7531\u200b\u4e2a\u4f53\u200b\u53d8\u9879\u200b\u3002 \u200b\u79f0\u200b\u4e00\u4e2a\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u4e3a\u200b\u4ec5\u200b\u4fdd\u7559\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u7684\u200b\u524d\u675f\u200b\u578b\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\u4e2d\u200b\u53ea\u200b\u5305\u542b\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u800c\u200b\u4e0d\u200b\u5305\u542b\u200b\u5b58\u5728\u200b\u91cf\u8bcd\u200b\u3002\u200b\u6bcd\u5f0f\u200b\u4e2d\u200b\u4e0d\u200b\u542b\u6709\u200b\u4efb\u4f55\u200b\u91cf\u8bcd\u200b\uff0c\u200b\u4e5f\u200b\u4e0d\u200b\u542b\u200b\u81ea\u7531\u200b\u4e2a\u4f53\u200b\u53d8\u9879\u200b\u3002</p> <ul> <li>\u200b\u4e00\u4e2a\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u53ef\u4ee5\u200b\u5316\u6210\u200b\u76f8\u5e94\u200b\u7684\u200b\\(\\exists\\)\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\uff0c\\(A\\)\u200b\u666e\u904d\u200b\u6709\u6548\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(\\exists\\)\u200b\u524d\u675f\u200b\u8303\u5f0f\u200b\u666e\u904d\u200b\u6709\u6548\u200b</li> <li>\u200b\u4e00\u4e2a\u200b\u8c13\u8bcd\u200b\u516c\u5f0f\u200b\\(A\\)\u200b\u53ef\u4ee5\u200b\u5316\u6210\u200b\u76f8\u5e94\u200b\u7684\u200b\u4ec5\u200b\u4fdd\u7559\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u7684\u200b\u524d\u675f\u200b\u578b\u200b\uff0c\\(A\\)\u200b\u4e0d\u53ef\u200b\u6ee1\u8db3\u200b\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u4ec5\u200b\u4fdd\u7559\u200b\u5168\u79f0\u200b\u91cf\u8bcd\u200b\u7684\u200b\u524d\u675f\u200b\u578b\u200b\u4e0d\u53ef\u200b\u6ee1\u8db3\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-9/","title":"\u96c6\u5408","text":"<p>\u200b\u96c6\u5408\u200b\u662f\u200b\u4e00\u4e9b\u200b\u786e\u5b9a\u200b\u7684\u200b\u3001\u200b\u53ef\u4ee5\u200b\u533a\u5206\u200b\u7684\u200b\u4e8b\u7269\u200b\u6c47\u805a\u200b\u5728\u200b\u4e00\u8d77\u200b\u7ec4\u6210\u200b\u7684\u200b\u4e00\u4e2a\u200b\u6574\u4f53\u200b\uff0c\u200b\u7ec4\u6210\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u4e8b\u7269\u200b\u79f0\u4e3a\u200b\u8be5\u200b\u96c6\u5408\u200b\u7684\u200b\u4e00\u4e2a\u200b\u5143\u7d20\u200b\u3002</p> <ul> <li>\u200b\u96c6\u5408\u200b\u7684\u200b\u5143\u7d20\u200b\u53ef\u4ee5\u200b\u662f\u200b\u4efb\u4f55\u4e8b\u7269\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u662f\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\uff0c\u200b\u4f46\u200b\u4e0d\u80fd\u200b\u662f\u200b\u81ea\u8eab\u200b</li> <li>\u200b\u96c6\u5408\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u6ee1\u8db3\u200b\u552f\u4e00\u6027\u200b\uff0c\u200b\u4e0d\u4f1a\u200b\u51fa\u73b0\u200b\u76f8\u540c\u200b\u7684\u200b\u5143\u7d20\u200b</li> <li>\u200b\u96c6\u5408\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u6ca1\u6709\u200b\u987a\u5e8f\u200b\u5173\u7cfb\u200b</li> <li>\u200b\u5143\u7d20\u200b\u4e0e\u200b\u96c6\u5408\u200b\u7684\u200b\u5173\u7cfb\u200b\u662f\u200b\u786e\u5b9a\u200b\u7684\u200b\uff0c\u200b\u5373\u200b\u5c5e\u4e8e\u200b\uff08\\(\\in\\)\uff09\u200b\u4e0e\u200b\u4e0d\u200b\u5c5e\u4e8e\u200b\uff08\\(\\not\\in\\)\uff09</li> </ul>"},{"location":"math/discrete-mathematics/chapter-9/#_2","title":"\u96c6\u5408\u200b\u7684\u200b\u8868\u793a","text":"<p>\u200b\u4ee5\u4e0b\u200b\u5217\u4e3e\u200b\u51fa\u200b\u5e38\u89c1\u200b\u7684\u200b\u96c6\u5408\u200b\uff1a</p> <ul> <li>\\(\\mathbf N\\)\u200b\u8868\u793a\u200b\u5168\u4f53\u200b\u81ea\u7136\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b</li> <li>\\(\\mathbf Z\\)\u200b\u8868\u793a\u200b\u5168\u4f53\u200b\u6574\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b</li> <li>\\(\\mathbf Q\\)\u200b\u8868\u793a\u200b\u5168\u4f53\u200b\u6709\u7406\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b</li> <li>\\(\\mathbf R\\)\u200b\u8868\u793a\u200b\u5168\u4f53\u200b\u5b9e\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b</li> <li>\\(\\mathbf C\\)\u200b\u8868\u793a\u200b\u5168\u4f53\u200b\u590d\u6570\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b</li> </ul> <p>\u200b\u53ef\u4ee5\u200b\u7528\u200b\u5916\u5ef6\u200b\u8868\u793a\u6cd5\u200b\u6216\u200b\u5185\u6db5\u200b\u8868\u793a\u6cd5\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\uff1a</p> <ul> <li>\u200b\u5916\u5ef6\u200b\u8868\u793a\u6cd5\u200b\uff1a\u200b\u4e00\u4e00\u5217\u4e3e\u200b\u51fa\u200b\u96c6\u5408\u200b\u7684\u200b\u5168\u4f53\u200b\u5143\u7d20\u200b</li> <li>\u200b\u5185\u6db5\u200b\u8868\u793a\u6cd5\u200b\uff1a\u200b\u4f7f\u7528\u200b\u8c13\u8bcd\u200b\u8868\u8ff0\u200b\u96c6\u5408\u200b\u4e2d\u200b\u5143\u7d20\u200b\u7684\u200b\u6027\u8d28\u200b</li> </ul>"},{"location":"math/discrete-mathematics/chapter-9/#_3","title":"\u96c6\u5408\u200b\u7684\u200b\u8fd0\u7b97","text":""},{"location":"math/discrete-mathematics/chapter-9/#_4","title":"\u96c6\u5408\u200b\u7684\u200b\u5173\u7cfb\u200b\u8fd0\u7b97","text":"<p>\u200b\u96c6\u5408\u200b\u4e0e\u200b\u5143\u7d20\u200b\u5b58\u5728\u200b\u5c5e\u4e8e\u200b\u4e0e\u200b\u4e0d\u200b\u5c5e\u4e8e\u200b\u5173\u7cfb\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(\\in, \\not\\in\\)</p> <p>\u200b\u96c6\u5408\u200b\u4e4b\u95f4\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u5173\u7cfb\u200b\\(=, \\subseteq, \\subset, \\supseteq, \\supset\\)\u3002</p> <ul> <li>\\(A=B\\Leftrightarrow (\\forall x)(x\\in A\\leftrightarrow x\\in B)\\)</li> <li>\\(A\\subseteq B\\Leftrightarrow (\\forall x)(x\\in A\\rightarrow x\\in B)\\)</li> <li>\\(A\\subset B\\Leftrightarrow (\\forall x)(x\\in A\\rightarrow x\\in B)\\land A\\not =B\\)</li> <li>\\(A\\supseteq B\\Leftrightarrow B\\subseteq A\\)</li> <li>\\(A\\supset B\\Leftrightarrow B\\subset A\\)</li> </ul> <p>\\(\\subseteq, \\supseteq, =\\)\u200b\u5173\u7cfb\u200b\u6ee1\u8db3\u200b\u81ea\u53cd\u6027\u200b\u3001\u200b\u4f20\u9012\u6027\u200b\uff0c\\(=\\)\u200b\u5173\u7cfb\u200b\u6ee1\u8db3\u200b\u5bf9\u79f0\u6027\u200b\u3002</p> <p>\\(A\\subseteq B\\land B\\subseteq A\\Leftrightarrow A=B\\)</p> <p>\u200b\u5b9a\u4e49\u200b\u7a7a\u96c6\u200b\u4e3a\u200b\u4e0d\u200b\u542b\u200b\u4efb\u4f55\u200b\u5143\u7d20\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(\\varnothing\\)\uff0c\u200b\u5176\u200b\u5185\u6db5\u200b\u8868\u793a\u200b\u4e3a\u200b\\(\\varnothing = \\{x|x\\not= x\\}\\)\uff0c\u200b\u7a7a\u96c6\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ul> <li>\u200b\u7a7a\u96c6\u200b\u662f\u200b\u4efb\u4f55\u200b\u96c6\u5408\u200b\u7684\u200b\u5b50\u96c6\u200b</li> <li>\u200b\u7a7a\u96c6\u200b\u662f\u200b\u552f\u4e00\u200b\u7684\u200b</li> </ul> <p>\u200b\u5b9a\u4e49\u200b\u5168\u96c6\u200b\u4e3a\u200b\u6240\u6709\u200b\u4e8b\u7269\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(E\\)\uff0c\u200b\u5176\u200b\u5185\u6db5\u200b\u8868\u793a\u200b\u4e3a\u200b\\(E=\\{x|x=x\\}\\)</p>"},{"location":"math/discrete-mathematics/chapter-9/#_5","title":"\u96c6\u5408\u200b\u7684\u200b\u57fa\u672c\u200b\u8fd0\u7b97","text":"<p>\u200b\u96c6\u5408\u200b\u5305\u542b\u200b\u5982\u4e0b\u200b\u57fa\u672c\u200b\u8fd0\u7b97\u200b\uff1a</p> <ul> <li>\u200b\u5e76\u96c6\u200b\uff1a\\(A\\cup B = \\{x\\in A\\lor x\\in B\\}\\)</li> <li>\u200b\u4ea4\u96c6\u200b\uff1a\\(A\\cap B = \\{x\\in A\\land x\\in B\\}\\)</li> <li>\u200b\u5dee\u96c6\u200b\uff1a\\(A-B = \\{x\\in A\\land x\\not \\in B\\}\\)</li> <li>\u200b\u8865\u96c6\u200b\uff1a\\(-A=E-A=\\{x|x\\not\\in A\\}\\)</li> <li>\u200b\u5bf9\u79f0\u200b\u5dee\u200b\uff1a\\(A\\oplus B=(A-B)\\cup (B-A) = \\{x| x\\in A\\overline\\lor x\\in B\\}\\)</li> </ul> <p>\u200b\u57fa\u4e8e\u200b\u5e76\u96c6\u200b\u4e0e\u200b\u4ea4\u96c6\u200b\u8fd0\u7b97\u200b\u53ef\u4ee5\u200b\u62d3\u5c55\u200b\u51fa\u200b\u5e7f\u4e49\u200b\u5e76\u200b\u548c\u200b\u5e7f\u4e49\u200b\u4ea4\u200b\u7684\u200b\u6982\u5ff5\u200b\uff0c\u200b\u8bbe\u200b\u96c6\u5408\u200b\\(A\\)\u200b\u7684\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u90fd\u200b\u662f\u200b\u96c6\u5408\u200b\uff0c\u200b\u5b9a\u4e49\u200b$A\u200b\u7684\u200b\u5e7f\u4e49\u200b\u5e76\u200b\u3001\u200b\u5e7f\u4e49\u200b\u4ea4\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u5e7f\u4e49\u200b\u5e76\u200b\uff1a\\(\\cup A=\\{x| (\\exists z)(z\\in A\\land x\\in z)\\}\\)</li> <li>\u200b\u5e7f\u4e49\u200b\u4ea4\u200b\uff1a\\(\\cap A=\\{x| (\\forall z)(z\\in A\\rightarrow x\\in z)\\}\\)</li> </ul> <p>\u200b\u5373\u200b\uff1a\u200b\u96c6\u5408\u200b\\(A\\)\u200b\u7684\u200b\u5e7f\u4e49\u200b\u5e76\u200b\u662f\u200b\u96c6\u5408\u200b\u4e2d\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u7684\u200b\u5e76\u200b\u96c6\u200b\uff0c\u200b\u96c6\u5408\u200b\\(A\\)\u200b\u7684\u200b\u5e7f\u4e49\u200b\u4ea4\u662f\u200b\u96c6\u5408\u200b\u4e2d\u200b\u6240\u6709\u200b\u5143\u7d20\u200b\u7684\u200b\u4ea4\u96c6\u200b\u3002</p> <p>\u200b\u5b9a\u4e49\u200b\u5e42\u96c6\u200b\u4e3a\u200b\u96c6\u5408\u200b\u6240\u6709\u200b\u5b50\u96c6\u200b\uff08\u200b\u5305\u62ec\u200b\u81ea\u8eab\u200b\uff09\u200b\u7ec4\u6210\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(P(A)\\)</p> <p>\u200b\u5b9a\u4e49\u200b\u7b1b\u5361\u5c14\u200b\u79ef\u200b\\(A\\times B\\)\u200b\u4e3a\u200b\u4e8c\u5143\u200b\u7ec4\u200b\uff08\u200b\u6709\u5e8f\u200b\u5bf9\u200b\uff09\\(\\langle x, y\\rangle\\)\u200b\u7684\u200b\u96c6\u5408\u200b\uff0c\u200b\u5176\u4e2d\u200b\\(x\\in A, y\\in B\\)\u3002\u200b\u6709\u5e8f\u200b\u5bf9\u200b\u7684\u200b\u5b9a\u4e49\u200b\u4e0d\u200b\u4e00\u5b9a\u200b\u552f\u4e00\u200b\uff0c\u200b\u5982\u200b\u96c6\u5408\u200b\\(\\{x, \\{x, y\\}\\}\\)\u200b\u5373\u53ef\u200b\u8868\u793a\u200b\u4e00\u4e2a\u200b\u6709\u5e8f\u200b\u5bf9\u200b\u3002\u200b\u6709\u5e8f\u200b\u5bf9\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ul> <li>\\(x\\not = y\\Rightarrow \\langle x,y\\rangle \\not =\\langle y,x\\rangle\\)</li> <li>\\(\\langle x,y\\rangle = \\langle u,v\\rangle\\Leftrightarrow x=u\\land y=v\\)</li> </ul> <p>\u200b\u53ef\u4ee5\u200b\u5728\u200b\u4e8c\u5143\u200b\u7ec4\u200b\u7684\u200b\u57fa\u7840\u200b\u4e0a\u200b\u62d3\u5c55\u200b\\(n\\)\u200b\u5143\u7ec4\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff1a\\(\\langle x_1, \\dots, x_n\\rangle =\\langle\\dots\\langle x_1, x_2\\rangle, \\dots, x_n\\rangle\\)\u3002\u200b\u7531\u6b64\u200b\\(n\\)\u200b\u9636\u200b\u7b1b\u5361\u5c14\u200b\u79ef\u200b\\(A_1\\times A_2\\times\\dots\\times A_n\\)\u200b\u5373\u200b\u4e3a\u200b\\(\\{\\langle x_1, \\dots, x_n\\rangle|x_1\\in A_1\\land \\dots\\land x_n\\in A_n\\}\\)</p>"},{"location":"math/discrete-mathematics/chapter-9/#_6","title":"\u96c6\u5408\u200b\u8fd0\u7b97\u200b\u7684\u200b\u4f18\u5148\u7ea7","text":"<p>\u200b\u96c6\u5408\u200b\u8fd0\u7b97\u200b\u7684\u200b\u4f18\u5148\u7ea7\u200b\u5217\u4e3e\u5982\u4e0b\u200b\uff1a</p> \\[ \\begin{aligned}     &amp;-A, P(A), \\cap A, \\cup A \\\\     &amp;-, \\cap, \\cup, \\oplus, \\times \\\\     &amp;=, \\subseteq , \\subset, \\in \\\\     &amp;\\lnot \\\\     &amp;\\land, \\lor, \\rightarrow, \\leftrightarrow \\\\     &amp;\\Leftrightarrow, \\Rightarrow \\end{aligned} \\]"},{"location":"math/discrete-mathematics/chapter-9/#_7","title":"\u96c6\u5408\u200b\u8fd0\u7b97\u200b\u7684\u200b\u6027\u8d28","text":"<p>\u200b\u96c6\u5408\u200b\u7684\u200b\u57fa\u672c\u200b\u8fd0\u7b97\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ol> <li>\u200b\u4ea4\u6362\u5f8b\u200b</li> <li>\\(A\\cap B=B\\cap A\\)</li> <li>\\(A\\cup B=B\\cup A\\)</li> <li>\u200b\u7ed3\u5408\u5f8b\u200b</li> <li>\\((A\\cup B)\\cup C = A\\cup (B\\cup C)\\)</li> <li>\\((A\\cap B)\\cap C = A\\cap (B\\cap C)\\)</li> </ol>"},{"location":"math/discrete-mathematics/chapter-9/#_8","title":"\u96c6\u5408\u200b\u7684\u200b\u56fe\u793a\u200b\u6cd5","text":"<ul> <li>\u200b\u5bf9\u4e8e\u200b\u96c6\u5408\u200b\u7684\u200b\u57fa\u672c\u200b\u8fd0\u7b97\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6587\u6c0f\u56fe\u200b\u8868\u793a\u200b</li> <li>\u200b\u5bf9\u4e8e\u200b\u5e42\u96c6\u200b\u8fd0\u7b97\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u7f51\u7edc\u56fe\u200b\u8868\u793a\u200b</li> <li>\u200b\u5bf9\u4e8e\u200b\u7b1b\u5361\u5c14\u200b\u79ef\u200b\u8fd0\u7b97\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u4e8c\u7ef4\u200b\u5750\u6807\u7cfb\u200b\u8868\u793a\u200b</li> </ul>"},{"location":"math/operating-research/","title":"\u8fd0\u7b79\u5b66","text":"<ul> <li>\u200b\u7ebf\u6027\u89c4\u5212\u200b</li> <li>\u200b\u8fd0\u8f93\u200b\u95ee\u9898\u200b</li> <li>\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b</li> </ul>"},{"location":"math/operating-research/chapter-1/","title":"\u7ebf\u6027\u89c4\u5212","text":"<p>\u200b\u5b9a\u4e49\u200b</p> <p>\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\uff1a\u200b\u7ed9\u5b9a\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\\(f(x_1, \\dots, x_n)\\)\u200b\u548c\u200b\u4e00\u7cfb\u5217\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff0c\u200b\u6c42\u200b\u6ee1\u8db3\u200b\u6240\u6709\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u7684\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u6539\u53d8\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\\(x_1, x_2, \\cdots, x_n\\)\u200b\u4f7f\u5f97\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u6240\u200b\u80fd\u200b\u8fbe\u5230\u200b\u7684\u200b\u6700\u5927\u503c\u200b\u6216\u200b\u6700\u5c0f\u503c\u200b\u3002\u200b\u5f53\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u53d6\u200b\u6700\u5927\u200b\uff08\u200b\u6700\u5c0f\u200b\uff09\u200b\u503c\u65f6\u200b\uff0c\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u7684\u200b\u503c\u79f0\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u7684\u200b\u89e3\u200b\u3002</p> <p>\u200b\u7ebf\u6027\u89c4\u5212\u200b\uff1a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u548c\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u5747\u200b\u4e3a\u200b \u200b\u7ebf\u6027\u200b \u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b</p> <p>\u200b\u7ebf\u6027\u89c4\u5212\u200b\u6a21\u578b\u200b\u7684\u200b\u6807\u51c6\u200b\u5f62\u5f0f\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned} &amp;\\max z = c_1x_1 + c_2x_2 + \\cdots + c_nx_n \\\\ s.t. &amp; \\left\\{     \\begin{aligned}         &amp; a_{11}x_1 + a_{12}x_2 + \\cdots + a_{1n}x_n = b_1 \\\\         &amp; a_{21}x_1 + a_{22}x_2 + \\cdots + a_{2n}x_n = b_2 \\\\         &amp; \\vdots \\\\         &amp; a_{m1}x_1 + a_{m2}x_2 + \\cdots + a_{mn}x_n = b_m \\\\         &amp; x_j \\in \\mathbb R \\land x_j \\geq 0, j = 1, \\cdots, n     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5176\u4e2d\u200b\\(a_{ij}, b_i, c_i\\)\u200b\u4e3a\u200b\u5e38\u6570\u200b\uff0c\u200b\u5206\u522b\u200b\u4f7f\u7528\u200b\u77e9\u9635\u200b\\(A, b, c^T\\)\u200b\u8fdb\u884c\u200b\u8868\u793a\u200b\u3002\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\\((x_1, x_2, \\cdots, x_n)^T\\)\u200b\u4f7f\u7528\u200b\u5411\u91cf\u200b\\(x\\)\u200b\u8fdb\u884c\u200b\u8868\u793a\u200b\uff0c\u200b\u5219\u200b\u6807\u51c6\u578b\u200b\u53ef\u4ee5\u200b\u5199\u4f5c\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max z = CX \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; Ax = b \\\\         &amp; x \\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5982\u679c\u200b\\(\\mathrm r(A) &lt; m\\)\uff0c\u200b\u5219\u200b\u90e8\u5206\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u53ef\u4ee5\u200b\u76f8\u4e92\u200b\u62b5\u6d88\u200b\uff0c\u200b\u4ee5\u4e0a\u200b\u53ea\u200b\u8ba8\u8bba\u200b\\(\\mathrm r(A) = m\\)\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002</p> <p>\u200b\u5982\u679c\u200b\u7ea6\u675f\u200b\u4e2d\u200b\u542b\u6709\u200b\u4e0d\u7b49\u5f0f\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u6dfb\u52a0\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b\u4e0e\u200b\u5269\u4f59\u200b\u53d8\u91cf\u200b\u8f6c\u53d8\u200b\u4e3a\u200b\u7b49\u5f0f\u200b</p> <ul> <li>\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b\uff1a\\(a_{i1}x_1 + a_{i2}x_2 + \\cdots + a_{in}x_n \\leq b_i\\)\u200b\u52a0\u5165\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b\\(x_{n+1}\\)\u200b\u540e\u200b\u8f6c\u4e3a\u200b\\(a_{i1}x_1 + a_{i2}x_2 + \\cdots + a_{in}x_n + x_{n+1}= b_i\\)</li> <li>\u200b\u5269\u4f59\u200b\u53d8\u91cf\u200b\uff1a\\(a_{j1}x_1 + a_{j2}x_2 + \\cdots + a_{jn}x_n \\geq b_j\\)\u200b\u52a0\u5165\u200b\u5269\u4f59\u200b\u53d8\u91cf\u200b\\(x_{n+1}\\)\u200b\u540e\u200b\u8f6c\u4e3a\u200b\\(a_{j1}x_1 + a_{j2}x_2 + \\cdots + a_{jn}x_n - x_{n+1}= b_j\\)</li> </ul> <p>\u200b\u82e5\u200b\u67d0\u4e2a\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\\(x_i\\)\u200b\u7684\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u4e3a\u200b\\(\\leq\\)\uff0c\u200b\u5219\u200b\u5c06\u200b\\(x_i\\)\u200b\u66ff\u6362\u200b\u4e3a\u200b\\(x_i' = -x_i\\)\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u7ebf\u6027\u89c4\u5212\u200b\u4e2d\u200b\u7684\u200b\u4e0d\u7b49\u5f0f\u200b\u7ea6\u675f\u200b\uff0c\u200b\u82e5\u200b\u5728\u200b\u6700\u4f18\u200b\u65f6\u8be5\u200b\u7ea6\u675f\u200b\u4e3a\u200b\u7b49\u5f0f\u200b\uff0c\u200b\u5219\u200b\u79f0\u4e3a\u200b\u7d27\u200b\u7ea6\u675f\u200b\uff0c\u200b\u5426\u5219\u200b\u79f0\u4e3a\u200b\u975e\u7d27\u200b\u7ea6\u675f\u200b\u3002</p>"},{"location":"math/operating-research/chapter-1/#_2","title":"\u6c42\u89e3\u200b\u7ebf\u6027\u89c4\u5212","text":"<p>\u200b\u7ebf\u6027\u89c4\u5212\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u56fe\u5f62\u200b\u6cd5\u200b\u3001\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u7b49\u200b\u65b9\u6cd5\u200b\u6c42\u89e3\u200b</p>"},{"location":"math/operating-research/chapter-1/#_3","title":"\u56fe\u5f62\u200b\u6cd5","text":"<p>\u200b\u56fe\u5f62\u200b\u6cd5\u200b\u53ea\u80fd\u200b\u7528\u4e8e\u200b\u6c42\u89e3\u200b\u5305\u542b\u200b\u4e24\u4e2a\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\u3002\u200b\u6b65\u9aa4\u200b\u53ef\u4ee5\u200b\u5217\u4e3e\u5982\u4e0b\u200b\uff1a</p> <ol> <li>\u200b\u5728\u200b\u5750\u6807\u8f74\u200b\u4e0a\u200b\u7ed8\u5236\u200b\u51fa\u200b\u6240\u6709\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff0c\u200b\u786e\u5b9a\u200b\u53ef\u884c\u200b\u57df\u200b</li> <li>\u200b\u5c06\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\\(z = c_1x_1 + c_2x_2\\)\u200b\u6539\u5199\u200b\u4e3a\u200b\\(x_2 = \\frac{z - c_1x_1}{c_2}\\)\uff0c\u200b\u5176\u200b\u659c\u7387\u200b\u4e3a\u200b\\(-\\frac{c_1}{c_2}\\)</li> <li>\u200b\u5728\u200b\u53ef\u884c\u200b\u57df\u200b\u4e2d\u200b\u5bfb\u627e\u200b\u4f7f\u5f97\u200b\u622a\u8ddd\u200b\u6700\u5927\u200b\uff08\u200b\u6700\u5c0f\u200b\uff09\u200b\u4e14\u200b\u659c\u7387\u200b\u4e3a\u200b\\(-\\frac{c_1}{c_2}\\)\u200b\u7684\u200b\u76f4\u7ebf\u200b\u3002\u200b\u8be5\u200b\u622a\u8ddd\u200b\u4e3a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7684\u200b\\(1/c_2\\)<ul> <li>\u200b\u82e5\u200b\u53ef\u884c\u200b\u57df\u200b\u4e3a\u200b\u7a7a\u200b\uff0c\u200b\u5219\u200b\u7ebf\u6027\u89c4\u5212\u200b\u65e0\u200b\u89e3\u200b</li> <li>\u200b\u82e5\u200b\u76f4\u7ebf\u200b\u53ef\u4ee5\u200b\u65e0\u9650\u200b\u5411\u200b\u4f18\u5316\u200b\u65b9\u5411\u200b\u5e73\u79fb\u200b\uff0c\u200b\u5219\u200b\u7ebf\u6027\u89c4\u5212\u200b\u65e0\u200b\u754c\u200b</li> <li>\u200b\u82e5\u200b\u6700\u4f18\u200b\u65f6\u200b\u76f4\u7ebf\u200b\u4e0e\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u7684\u200b\u659c\u7387\u200b\u76f8\u540c\u200b\uff0c\u200b\u5219\u200b\u7ebf\u6027\u89c4\u5212\u200b\u6709\u200b\u65e0\u6570\u200b\u89e3\u200b</li> <li>\u200b\u5426\u5219\u200b\u7ebf\u6027\u89c4\u5212\u200b\u6709\u200b\u552f\u4e00\u200b\u6700\u4f18\u200b\u89e3\u200b</li> </ul> </li> </ol>"},{"location":"math/operating-research/chapter-1/#_4","title":"\u5355\u7eaf\u5f62\u6cd5","text":"<p>\u200b\u5df2\u77e5\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u6ee1\u200b\u79e9\u200b\uff0c\u200b\u6784\u6210\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u7684\u200b\u5217\u200b\u5411\u91cf\u200b\u7684\u200b\u6781\u5927\u200b\u7ebf\u6027\u200b\u65e0\u5173\u200b\u7ec4\u200b\u5305\u542b\u200b\\(m\\)\u200b\u4e2a\u5217\u200b\u5411\u91cf\u200b\u3002\u200b\u56e0\u6b64\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u4f5c\u200b\u5982\u4e0b\u200b\u5212\u5206\u200b\uff1a</p> \\[ A = [B, N] \\] <p>\u200b\u76f8\u4f3c\u200b\u5730\u200b\uff0c\u200b\u5411\u91cf\u200b\\(x\\)\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u4f5c\u200b\u5982\u4e0b\u200b\u5212\u5206\u200b\uff1a</p> \\[ x = \\begin{bmatrix}     x_B \\\\ x_N \\end{bmatrix} \\] <p>\u200b\u5176\u4e2d\u200b\\(B\\)\u200b\u4e3a\u200b\u53ef\u9006\u200b\u77e9\u9635\u200b\u3002\u200b\u82e5\u200b\u4e0d\u200b\u5b58\u5728\u200b\u8fd9\u6837\u200b\u7684\u200b\u77e9\u9635\u200b\uff0c\u200b\u5c06\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u7684\u200b\u5404\u5217\u200b\u8fdb\u884c\u200b\u4ea4\u6362\u200b\u5373\u53ef\u200b\u3002</p> <p>\u200b\u57fa\u200b\u3001\u200b\u57fa\u200b\u5411\u91cf\u200b\u4e0e\u200b\u57fa\u200b\u53d8\u91cf\u200b</p> <p>\u200b\u77e9\u9635\u200b\\(B\\)\u200b\u79f0\u4e3a\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b \u200b\u57fa\u200b \u3002 \u200b\u57fa\u200b \u200b\u4e2d\u200b\u7684\u200b\u5404\u4e2a\u200b\u5217\u200b\u5411\u91cf\u200b\\(P_j\\)\u200b\u4e3a\u200b \u200b\u57fa\u200b\u5411\u91cf\u200b \uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u79f0\u4e3a\u200b \u200b\u57fa\u200b\u53d8\u91cf\u200b \u3002\u200b\u5206\u5272\u200b\u540e\u975e\u200b\u9f50\u6b21\u200b\u7ebf\u6027\u65b9\u7a0b\u7ec4\u200b\\(Ax=b\\)\u200b\u8f6c\u5316\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{bmatrix}     B &amp; N \\end{bmatrix}\\begin{bmatrix}     x_B \\\\ x_N \\end{bmatrix} = Bx_B + Nx_N = b \\] <p>\u200b\u7531\u4e8e\u200b\u65b9\u9635\u200b\\(B\\)\u200b\u6ee1\u200b\u79e9\u200b\uff0c\u200b\u6709\u200b\\(x_B = B^{-1}(b - Nx_N)\\)\u3002\u200b\u5176\u4e2d\u200b\\(x_N\\)\u200b\u4e3a\u200b\u81ea\u7531\u200b\u53d8\u91cf\u200b\u3002\\(-B^{-1}N\\)\u200b\u4e2d\u200b\u7684\u200b\u5217\u200b\u5411\u91cf\u200b\u4e3a\u200b\u65b9\u7a0b\u200b\\(Ax=b\\)\u200b\u7684\u200b\u57fa\u7840\u200b\u89e3\u7cfb\u200b\u3002\u200b\u4e0d\u59a8\u200b\u8bbe\u200b\\(x_N = 0\\)\uff0c\u200b\u5219\u200b\\(x_B = B^{-1}b\\)\u200b\u4e3a\u200b\u7ebf\u6027\u65b9\u7a0b\u7ec4\u200b\u7684\u200b\u4e00\u4e2a\u200b\u89e3\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u57fa\u89e3\u200b\u3002</p> <p>\u200b\u53ef\u884c\u200b\u89e3\u4e0e\u200b\u53ef\u884c\u200b\u57fa\u200b</p> <p>\u200b\u82e5\u200b\\(x_B\\geq 0\\)\uff0c\u200b\u5219\u200b\u89e3\u200b\\(x = \\begin{bmatrix} x_B \\\\ x_N \\end{bmatrix}\\)\u200b\u6ee1\u8db3\u200b\u975e\u8d1f\u200b\u7ea6\u675f\u200b\uff0c\u200b\u4e3a\u200b \u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b \uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u57fa\u200b\u79f0\u4e3a\u200b \u200b\u53ef\u884c\u200b\u57fa\u200b \u3002</p> <p>\u200b\u57fa\u89e3\u200b\u4e0e\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ul> <li>\u200b\u7531\u4e8e\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u5171\u200b\\(n\\)\u200b\u5217\u200b\uff0c\u200b\u4ece\u200b\u5176\u4e2d\u200b\u9009\u53d6\u200b\\(m\\)\u200b\u5217\u200b\uff0c\u200b\u6709\u200b\\(\\binom{n}{m}\\)\u200b\u79cd\u200b\u9009\u53d6\u200b\u65b9\u5f0f\u200b\uff0c\u200b\u800c\u200b\u53ea\u6709\u200b\u7ec4\u6210\u200b\u7ebf\u6027\u200b\u65e0\u5173\u200b\u7684\u200b\\(m\\)\u200b\u4e2a\u200b\u5411\u91cf\u200b\u624d\u80fd\u200b\u6784\u6210\u200b\u4e00\u7ec4\u200b\u57fa\u200b\uff0c\u200b\u56e0\u6b64\u200b\u57fa\u89e3\u200b\u7684\u200b\u6570\u76ee\u200b\u4e0d\u4f1a\u200b\u8d85\u8fc7\u200b\\(\\binom nm\\)\u3002</li> <li>\\(X\\)\u200b\u4e3a\u57fa\u200b\u53ef\u884c\u200b\u89e3\u5f53\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(X\\)\u200b\u7684\u200b\u975e\u200b\u96f6\u200b\u5206\u91cf\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5217\u200b\u5411\u91cf\u200b\u7ebf\u6027\u200b\u65e0\u5173\u200b\u3002\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u662f\u200b\u53ef\u884c\u200b\u57df\u200b\u7684\u200b\u9876\u70b9\u200b\u3002</li> </ul>"},{"location":"math/operating-research/chapter-1/#_5","title":"\u539f\u7406","text":"<p>\u200b\u5982\u4e0b\u200b\u9488\u5bf9\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u6781\u5927\u200b\u5316\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u6781\u5c0f\u200b\u5316\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u53d6\u8d1f\u200b\uff0c\u200b\u5373\u200b\u5f97\u5230\u200b\u9700\u8981\u200b\u6781\u5927\u200b\u5316\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u3002</p> <p>\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u7684\u200b\u601d\u8def\u200b\uff1a</p> <ol> <li>\u200b\u5148\u200b\u627e\u51fa\u200b\u4e00\u4e2a\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b</li> <li>\u200b\u5224\u65ad\u200b\u5f53\u524d\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\u662f\u5426\u200b\u6700\u4f18\u200b<ol> <li>\u200b\u82e5\u200b\u6700\u4f18\u200b\uff0c\u200b\u7ed3\u675f\u200b</li> <li>\u200b\u82e5\u200b\u4e0d\u662f\u200b\u6700\u4f18\u200b\uff0c\u200b\u5219\u200b\u8f6c\u5411\u200b\u76f8\u90bb\u200b\u7684\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b</li> </ol> </li> </ol>"},{"location":"math/operating-research/chapter-1/#_6","title":"\u72b6\u6001\u200b\u8f6c\u79fb","text":"<p>\u200b\u8bbe\u200b\u521d\u59cb\u200b\u57fa\u4e3a\u200b\\(B\\)\u3002\u200b\u5c06\u200b\u65b9\u7a0b\u200b\\(Ax=b\\)\u200b\u5de6\u53f3\u200b\u4e24\u8fb9\u200b\u540c\u65f6\u200b\u5de6\u4e58\u200b\\(B^{-1}\\)\uff0c\u200b\u6709\u200b\\((B^{-1}A)x = B^{-1}b\\)\uff0c\u200b\u5373\u200b\uff1a</p> \\[ \\begin{bmatrix}     I &amp; B^{-1}N \\end{bmatrix}\\begin{bmatrix}     x_B \\\\ x_N \\end{bmatrix} = B^{-1}b \\] <p>\u200b\u8bbe\u200b\\(B^{-1}N = \\{d_{ij}\\}_{m\\times(n-m)}\\)\u3002\u200b\u6839\u636e\u200b\\(x_B = B^{-1}b - B^{-1}Nx_N\\)\uff0c\u200b\u5bf9\u4e8e\u200b\u7b2c\u200b\\(i\\)\u200b\u4e2a\u200b\u5206\u91cf\u200b\uff0c\u200b\u6709\u200b\uff1a</p> \\[ x_i = b_i - \\sum_{j = 1}^{(n - m)}d_{ij}x_j, 1\\leq j\\leq (n-m) \\]"},{"location":"math/operating-research/chapter-1/#_7","title":"\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c","text":"<p>\u200b\u5bf9\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u8fdb\u884c\u200b\u53ef\u9006\u200b\u7ebf\u6027\u53d8\u6362\u200b\uff08\u200b\u53f3\u4e58\u200b\u53ef\u9006\u200b\u77e9\u9635\u200b\uff09\u200b\u4e0d\u4f1a\u200b\u6539\u53d8\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u89e3\u200b\u3002\u200b\u4e3a\u200b\u7b80\u5316\u200b\u8ba1\u7b97\u200b\uff0c\u200b\u4e0d\u59a8\u200b\u5047\u8bbe\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u7684\u200b\u4e00\u4e2a\u200b\u57fa\u200b\u77e9\u9635\u200b\u4e3a\u200b\u5355\u4f4d\u77e9\u9635\u200b\uff0c\u200b\u5373\u200b\uff1a</p> \\[ A = \\begin{bmatrix}     1 &amp; 0 &amp; \\cdots &amp; 0 &amp; a_{1(m + 1)} &amp; a_{1(m + 2)} &amp; \\cdots &amp;  a_{1n} \\\\     0 &amp; 1 &amp; \\cdots &amp; 0 &amp; a_{2(m + 1)} &amp; a_{2(m + 2)} &amp; \\cdots &amp;  a_{2n} \\\\     \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots &amp; \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\     0 &amp; 0 &amp; \\cdots &amp; 1 &amp; a_{m(m + 1)} &amp; a_{m(m + 2)} &amp; \\cdots &amp;  a_{mn} \\\\ \\end{bmatrix} \\] <p>\u200b\u663e\u7136\u200b\uff0c\u200b\u5355\u4f4d\u77e9\u9635\u200b\\(I\\)\u200b\u662f\u200b\\(A\\)\u200b\u7684\u200b\u57fa\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u57fa\u89e3\u200b\\(x_B=b - Nx_N\\)\u3002\u200b\u5c06\u200b\u5411\u91cf\u200b\\(c\\)\u200b\u5206\u4e3a\u200b\\(c_B\\)\u200b\u4e0e\u200b\\(c_N\\)\uff0c\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\\(x_B\\)\u200b\u4e0e\u200b\\(x_N\\)\uff0c\u200b\u6b64\u65f6\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u53ef\u4ee5\u200b\u8ba1\u7b97\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned}     z &amp;= c^Tx \\\\     &amp;= c_B^Tx_B + c_N^T x_N \\\\     &amp;= c_B^Tb + c_N^Tx_N - c_B^TNx_N \\\\     &amp;= c_B^Tb + (c_N - c_B^TN)x_N \\end{aligned} \\] <p>\u200b\u8bbe\u200b\u5411\u91cf\u200b\\(\\lambda = c_N - c_B^TN\\)\uff0c\u200b\u5219\u200b\\(\\frac{\\partial z}{\\partial x_N} = \\lambda\\)\u3002</p> <p>\u200b\u5b9a\u7406\u200b</p> <ul> <li>\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u67d0\u4e2a\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\\(x\\)\uff0c\u200b\u6709\u200b\\(\\lambda &lt; 0\\)\uff0c\u200b\u8bf4\u660e\u200b\\(x\\)\u200b\u4e3a\u200b\u6700\u4f18\u200b\u89e3\u200b</li> <li>\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u67d0\u4e2a\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\\(x\\)\uff0c\u200b\u6709\u200b\\(\\lambda \\leq 0\\)\u200b\u4e14\u200b\\(\\lambda\\)\u200b\u5b58\u5728\u200b\u4e3a\u200b\u96f6\u200b\u7684\u200b\u5206\u91cf\u200b\uff0c\u200b\u8bf4\u660e\u200b\u539f\u200b\u95ee\u9898\u200b\u6709\u200b\u65e0\u6570\u4e2a\u200b\u6700\u4f18\u200b\u89e3\u200b</li> <li>\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u67d0\u4e2a\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\\(x\\)\uff0c\u200b\u5411\u91cf\u200b\\(\\lambda\\)\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\\(\\lambda_{m + i} &gt; 0\\)\uff0c\u200b\u4e14\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5217\u200b\u5411\u91cf\u200b\\(P_{m + i}\\leq 0\\)\uff0c\u200b\u5219\u200b\u539f\u200b\u95ee\u9898\u200b\u65e0\u200b\u754c\u200b</li> </ul>"},{"location":"math/operating-research/chapter-1/#_8","title":"\u57fa\u7684\u200b\u8fed\u4ee3","text":"<p>\u200b\u5728\u200b\u8fed\u4ee3\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u9700\u8981\u200b\u51b3\u5b9a\u200b\u6362\u5165\u200b\u7684\u200b\u57fa\u200b\u4e0e\u200b\u6362\u200b\u51fa\u200b\u7684\u200b\u57fa\u200b\uff1a</p> <ul> <li>\u200b\u4f7f\u7528\u200b\u8d2a\u5fc3\u200b\u7b97\u6cd5\u200b\u9009\u62e9\u200b\u6362\u5165\u200b\u7684\u200b\u57fa\u200b\uff0c\u200b\u9009\u62e9\u200b\u68af\u5ea6\u200b\u6700\u5927\u200b\uff08\u200b\u4e0e\u200b\\(\\lambda\\)\u200b\u6700\u5927\u200b\u6b63\u200b\u5206\u91cf\u200b\u5bf9\u5e94\u200b\uff09\u200b\u7684\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u6362\u5165\u200b\uff0c\\(i = \\arg\\max \\lambda\\)</li> <li>\u200b\u5bf9\u4e8e\u200b\u6362\u5165\u200b\u53d8\u91cf\u200b\\(x_i\\)\uff0c\u200b\u6362\u51fa\u200b\u7684\u200b\u57fa\u200b\\(x_j\\)\u200b\u5e94\u5f53\u200b\u5c3d\u53ef\u80fd\u200b\u4fdd\u8bc1\u200b\u662f\u200b\u53ef\u884c\u200b\u89e3\u200b\u3002\u200b\u6839\u636e\u200b\\(x_B = b - Nx_N\\geq 0\\)\uff0c\u200b\u5f97\u200b\\(Nx_N \\leq b - x_B\\)\uff0c\u200b\u6709\u200b\\(j = \\arg\\min _{j} \\frac{b_j}{a_{ij}}\\)</li> </ul> <p>\u200b\u4e3e\u4f8b\u200b</p> <p>\u200b\u5bf9\u4e8e\u200b\u5982\u4e0b\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max z = 3x_1 - 3x_2 + 5x_4 - x_5 \\\\     &amp;s.t. \\left\\{     \\begin{aligned}         &amp;x_1 &amp; &amp; -2x_3 &amp; +2x_4 &amp; &amp;=12 \\\\         &amp;&amp; x_2 &amp;-2x_3 &amp; &amp; &amp;=1 \\\\         &amp;&amp;&amp; -4x_3 &amp;+3x_4 &amp;+x_5 &amp;=27     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5217\u51fa\u200b\u5355\u7eaf\u5f62\u200b\u8868\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</p> <p></p> <p>\u200b\u5c06\u200b\u53d8\u91cf\u200b\\(x_4\\)\u200b\u8f6c\u5165\u200b\u57fa\u200b\u53d8\u91cf\u200b\uff0c\\(x_1\\)\u200b\u8f6c\u51fa\u200b\u57fa\u200b\u53d8\u91cf\u200b\uff0c\u200b\u5f97\u5230\u200b\u5982\u4e0b\u200b\u5355\u7eaf\u5f62\u200b\u8868\u200b\uff1a</p> <p></p>"},{"location":"math/operating-research/chapter-1/#_9","title":"\u4e24\u200b\u9636\u6bb5\u200b\u6cd5","text":"<p>\u200b\u5f53\u200b\\(A\\)\u200b\u4e0d\u200b\u4fbf\u4e8e\u200b\u8ba1\u7b97\u200b\u51fa\u57fa\u200b\\(B\\)\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5728\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u4e2d\u200b\u5f15\u5165\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u3002\u200b\u7b2c\u4e00\u9636\u6bb5\u200b\u7684\u200b\u5f00\u59cb\u200b\u72b6\u6001\u200b\uff0c\u200b\u6240\u6709\u200b\u7684\u200b\u57fa\u200b\u53d8\u91cf\u200b\u90fd\u200b\u662f\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\uff0c\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e3a\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u4e4b\u200b\u548c\u200b\uff0c\u200b\u5411\u200b\u6700\u5c0f\u503c\u200b\u65b9\u5411\u200b\u4f18\u5316\u200b\u3002\u200b\u901a\u8fc7\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u9010\u6b65\u200b\u5c06\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u6362\u200b\u51fa\u57fa\u200b\u53d8\u91cf\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b</p> <p>\u200b\u5982\u679c\u200b\u67d0\u4e2a\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u5728\u200b\u57fa\u200b\u53d8\u91cf\u200b\u4e2d\u200b\uff0c\u200b\u4f46\u200b\\(\\lambda \\leq 0\\)\uff0c\u200b\u5219\u200b\u7ebf\u6027\u89c4\u5212\u200b\u65e0\u200b\u53ef\u884c\u200b\u89e3\u200b\u3002</p> <p>\u200b\u5c06\u200b\u6240\u6709\u200b\u7684\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u6362\u200b\u51fa\u57fa\u200b\u53d8\u91cf\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6240\u6709\u200b\u7684\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u53d6\u200b\\(0\\)\u3002\u200b\u5220\u53bb\u200b\u4eba\u5de5\u200b\u53d8\u91cf\u200b\u5e76\u200b\u5c06\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u66ff\u6362\u200b\u4e3a\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\uff0c\u200b\u5f97\u5230\u200b\u7b2c\u4e8c\u9636\u6bb5\u200b\u5f00\u59cb\u200b\u72b6\u6001\u200b\u7684\u200b\u5355\u7eaf\u5f62\u200b\u8868\u200b\u3002</p>"},{"location":"math/operating-research/chapter-1/#_10","title":"\u7ebf\u6027\u89c4\u5212\u200b\u89e3\u200b\u7684\u200b\u6027\u8d28","text":"<p>\u200b\u6839\u636e\u200b\u56fe\u89e3\u6cd5\u200b\uff0c\u200b\u6709\u5982\u200b\u4e0b\u7ed3\u8bba\u200b\uff1a</p> <ul> <li>\u200b\u53ef\u884c\u200b\u57df\u200b\u662f\u200b\u4e00\u4e2a\u200b\u51f8\u96c6\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u53ef\u884c\u200b\u57df\u200b\\(X\\)\u200b\u4e2d\u200b\u7684\u200b\u4efb\u610f\u200b\u4e24\u70b9\u200b\\(x = (x_1, x_2, \\cdots, x_n)\\in S\\)\u200b\u4e0e\u200b\\(y = (y_1, y_2, \\cdots, y_n)\\in S\\)\uff0c\u200b\u6709\u200b\\(\\alpha x + (1 - \\alpha) y \\in S, 0\\leq \\alpha \\leq 1\\)<ul> <li>\u200b\u51f8\u200b\u96c6\u4e2d\u200b\u7684\u200b\u4efb\u4f55\u200b\u4e00\u4e2a\u70b9\u200b\u90fd\u200b\u53ef\u4ee5\u200b\u8868\u793a\u200b\u4e3a\u200b\u51f8\u96c6\u200b\u5404\u4e2a\u200b\u9876\u70b9\u200b\u7684\u200b\u51f8\u200b\u7ec4\u5408\u200b</li> </ul> </li> <li>\u200b\u6700\u4f18\u200b\u89e3\u200b\uff08\u200b\u5982\u679c\u200b\u5b58\u5728\u200b\uff09\u200b\u5728\u200b\u53ef\u884c\u200b\u57df\u200b\u7684\u200b\u8fb9\u754c\u200b\u4e0a\u200b\uff0c\u200b\u4e14\u200b\u53ef\u884c\u200b\u57df\u200b\u7684\u200b\u81f3\u5c11\u200b\u4e00\u4e2a\u200b\u9876\u70b9\u200b\u662f\u200b\u6700\u4f18\u200b\u89e3\u200b</li> </ul>"},{"location":"math/operating-research/chapter-1/#_11","title":"\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u5bf9\u5076\u200b\u95ee\u9898","text":"<p>\u200b\u6bcf\u4e2a\u200b\u539f\u200b\u95ee\u9898\u200b\u5bf9\u5e94\u200b\u4e14\u200b\u4ec5\u200b\u5bf9\u5e94\u200b\u4e00\u4e2a\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\uff0c\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u662f\u200b\u539f\u200b\u95ee\u9898\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b\u539f\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max z = c_1x_1 + c_2x_2 + c_3x_3 \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; a_{11}x_1 + a_{12}x_2 + a_{13}x_3 \\leq b_1 \\\\         &amp; a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = b_2 \\\\         &amp; a_{31}x_1 + a_{32}x_2 + a_{33}x_3 \\geq b_3 \\\\         &amp; x_1 \\leq 0 \\\\         &amp; x_3 \\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5176\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\min w = b_1y_1 + b_2y_2 + b_3y_3 \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; a_{11}y_1 + a_{21}y_2 + a_{31}y_3 \\leq c_1 \\\\         &amp; a_{12}y_1 + a_{22}y_2 + a_{32}y_3 = c_2 \\\\         &amp; a_{13}y_1 + a_{23}y_2 + a_{33}y_3 \\geq c_3 \\\\         &amp; y_1 \\geq 0 \\\\         &amp; y_3 \\leq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5047\u8bbe\u200b\u539f\u200b\u95ee\u9898\u200b\u5411\u200b\u6781\u5927\u503c\u200b\u4f18\u5316\u200b\uff0c\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u6709\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ul> <li>\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u5411\u200b\u6781\u5c0f\u503c\u200b\u4f18\u5316\u200b</li> <li>\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7684\u200b\u7cfb\u6570\u200b\u662f\u200b\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u7cfb\u6570\u200b</li> <li>\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u7cfb\u6570\u200b\u662f\u200b\u539f\u200b\u95ee\u9898\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7684\u200b\u7cfb\u6570\u200b</li> <li>\u200b\u539f\u200b\u95ee\u9898\u200b\u4e2d\u200b\\(\\geq 0\\)\u3001\\(\\leq 0\\)\u200b\u7684\u200b\u53d8\u91cf\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e2d\u200b\\(\\geq\\)\u3001\\(\\leq\\)\u200b\u7c7b\u578b\u200b\u7684\u200b\u7ea6\u675f\u200b</li> <li>\u200b\u539f\u200b\u95ee\u9898\u200b\u4e2d\u200b\\(\\geq\\)\u3001\\(\\leq\\)\u200b\u7c7b\u578b\u200b\u7684\u200b\u7ea6\u675f\u200b\u5206\u522b\u200b\u5bf9\u5e94\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e2d\u200b\\(\\leq 0\\)\u3001\\(\\geq 0\\)\u200b\u7684\u200b\u53d8\u91cf\u200b</li> <li>\u200b\u4efb\u4f55\u200b\\(=\\)\u200b\u7c7b\u578b\u200b\u7684\u200b\u7ea6\u675f\u200b\u5bf9\u5e94\u200b\u81ea\u7531\u200b\u53d8\u91cf\u200b\uff0c\u200b\u53cd\u4e4b\u4ea6\u7136\u200b\u3002</li> </ul> <p>\u200b\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7b49\u4e8e\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u3002\u200b\u4e14\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u89e3\u4f1a\u200b\u4ee5\u200b\u8d1f\u503c\u200b\u7684\u200b\u5f62\u5f0f\u200b\u51fa\u73b0\u200b\u5728\u200b\u539f\u200b\u95ee\u9898\u200b\u6700\u4f18\u200b\u5355\u7eaf\u5f62\u200b\u8868\u200b\u7684\u200b\\(\\lambda\\)\u200b\u90e8\u5206\u200b\u3002\u200b\u5373\u200b\uff0c\\(\\lambda = -Y^*\\)</p>"},{"location":"math/operating-research/chapter-1/#_12","title":"\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u5bf9\u5076\u6027","text":"<p>\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b\u539f\u200b\u95ee\u9898\u200b\\((P)\\)\u200b\u4e0e\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\\((D)\\)</p> \\[ \\begin{aligned}     &amp; (P) &amp; \\begin{aligned}         &amp; \\max z=c^Tx \\\\         &amp; s.t. \\left\\{         \\begin{aligned}             &amp; AX \\leq b \\\\             &amp; X\\geq 0         \\end{aligned}         \\right .     \\end{aligned} \\\\     &amp; (D) &amp; \\begin{aligned}         &amp; \\min w=b^Ty \\\\         &amp; s.t. \\left\\{         \\begin{aligned}             &amp; A^Ty \\geq c \\\\             &amp; y\\geq 0         \\end{aligned}         \\right .     \\end{aligned} \\end{aligned} \\]"},{"location":"math/operating-research/chapter-1/#_13","title":"\u5bf9\u5076\u200b\u5b9a\u7406","text":""},{"location":"math/operating-research/chapter-1/#_14","title":"\u5f31\u200b\u5bf9\u5076\u200b\u5b9a\u7406","text":"<p>\u200b\u8bbe\u200b\\((P), (D)\\)\u200b\u7684\u200b\u4e00\u7ec4\u200b\u53ef\u884c\u200b\u89e3\u200b\u5206\u522b\u200b\u4e3a\u200b\\(x, y\\)\uff0c\u200b\u5219\u200b\u6709\u200b\\(c^Tx \\leq b^Ty\\)</p> <p>\u200b\u8bc1\u660e\u200b</p> <p>\u200b\u8003\u8651\u200b\\(y^TAx\\)\uff0c\u200b\u6709\u200b\uff1a</p> <ol> <li>\\(y^TAx = (A^Ty)^TX \\geq c^TX\\)</li> <li>\\(y^TAx \\leq y^Tb = b^T y\\)</li> </ol> <p>\u200b\u6839\u636e\u200b\u5f31\u200b\u5bf9\u5076\u200b\u5b9a\u7406\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5f97\u51fa\u200b\u5982\u4e0b\u200b\u7ed3\u8bba\u200b\uff1a</p> <ol> <li>\u200b\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u662f\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u4e0b\u754c\u200b\uff0c\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u662f\u200b\u539f\u200b\u95ee\u9898\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u4e0a\u754c\u200b</li> <li>\u200b\u82e5\u539f\u200b\u95ee\u9898\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u65e0\u200b\u4e0a\u754c\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u65e0\u200b\u89e3\u200b\uff0c\u200b\u82e5\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u65e0\u200b\u4e0b\u754c\u200b\u7b49\u4ef7\u200b\u4e8e\u539f\u200b\u95ee\u9898\u200b\u65e0\u200b\u89e3\u200b</li> </ol>"},{"location":"math/operating-research/chapter-1/#_15","title":"\u6700\u4f18\u6027\u200b\u5b9a\u7406","text":"<p>\u200b\u8bbe\u200b\\((P), (D)\\)\u200b\u7684\u200b\u4e00\u7ec4\u200b\u53ef\u884c\u200b\u89e3\u200b\u5206\u522b\u200b\u4e3a\u200b\\(x^*, y^*\\)\uff0c\u200b\u82e5\u200b\\(c^Tx^* = b^Ty^*\\)\uff0c\u200b\u5219\u200b\\(x^*, y^*\\)\u200b\u5206\u522b\u200b\u4e3a\u200b\u5404\u81ea\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002</p> <p>\u200b\u8bc1\u660e\u200b</p> <p>\u200b\u6839\u636e\u200b\u5f31\u200b\u5bf9\u5076\u200b\u5b9a\u7406\u200b\uff0c\u200b\u6709\u200b\\(c^Tx\\leq b^Ty^* = c^Tx^*\\leq b^Ty\\)</p>"},{"location":"math/operating-research/chapter-1/#_16","title":"\u6700\u4f18\u200b\u89e3\u200b\u7684\u200b\u5bf9\u5076\u6027","text":"<p>\u200b\u82e5\u200b\\(B\\)\u200b\u4e3a\u200b\\((P)\\)\u200b\u7684\u200b\u6700\u4f18\u200b\u57fa\u200b\uff0c\u200b\u5219\u200b\\((D)\\)\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u4e3a\u200b\\(y^* = c_BB^{-1}\\)\u3002\u200b\u82e5\u200b\\((P)\\)\u200b\u548c\u200b\\((D)\\)\u200b\u5747\u200b\u6709\u200b\u53ef\u884c\u200b\u89e3\u200b\uff0c\u200b\u5219\u200b\u4e24\u8005\u200b\u5747\u200b\u6709\u200b\u6709\u754c\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u5e76\u4e14\u200b\u4e24\u8005\u200b\u7684\u200b\u6700\u4f18\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u76f8\u7b49\u200b\u3002</p>"},{"location":"math/operating-research/chapter-1/#_17","title":"\u4e92\u8865\u200b\u677e\u5f1b\u200b\u5b9a\u7406","text":"<p>\u200b\u5982\u679c\u200b\u539f\u200b\u95ee\u9898\u200b\\((P)\\)\u200b\u4e2d\u200b\u67d0\u200b\u4e00\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5bf9\u5076\u200b\u53d8\u91cf\u503c\u200b\u5927\u4e8e\u200b\\(0\\)\uff0c\u200b\u5219\u200b\u4e0e\u200b\u8be5\u200b\u5bf9\u5076\u200b\u53d8\u91cf\u200b\u5bf9\u5e94\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u53d6\u200b\u7b49\u5f0f\u200b\u3002\u200b\u76f8\u53cd\u200b\uff0c\u200b\u5982\u679c\u200b\u67d0\u4e00\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u53d6\u200b\u4e0d\u7b49\u5f0f\u200b\uff0c\u200b\u5219\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5bf9\u5076\u200b\u53d8\u91cf\u503c\u200b\u4e3a\u200b\\(0\\)\u3002</p> <p>\u200b\u6ce8\u610f\u200b</p> <p>\u200b\u8be5\u200b\u5b9a\u7406\u200b\u7684\u200b\u9006\u547d\u9898\u200b\u4e0d\u200b\u6210\u7acb\u200b\uff0c\u200b\u5373\u200b\u5f53\u539f\u200b\u95ee\u9898\u200b\u4e2d\u200b\u67d0\u200b\u4e00\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u53d6\u200b\u7b49\u5f0f\u200b\u65f6\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5bf9\u5076\u200b\u53d8\u91cf\u503c\u200b\u65e0\u6cd5\u200b\u786e\u5b9a\u200b\u662f\u5426\u200b\u4e3a\u200b\\(0\\)</p> <p>\u200b\u5229\u7528\u200b\u4e92\u8865\u200b\u677e\u5f1b\u200b\u6027\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\u6c42\u89e3\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\u4ece\u800c\u200b\u7b80\u5316\u200b\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u6c42\u89e3\u200b</p>"},{"location":"math/operating-research/chapter-1/#_18","title":"\u5f71\u5b50\u200b\u4ef7\u683c","text":"<p>\u200b\u8003\u8651\u200b\u5f62\u200b\u5982\u200b\\((P)\\)\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\uff0c\u200b\u5f53\u200b\u67d0\u4e2a\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u7684\u200b\u6570\u503c\u200b\u53d8\u52a8\u200b\u4e00\u4e2a\u200b\u5355\u4f4d\u200b\u800c\u200b\u5176\u4ed6\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u4fdd\u6301\u200b\u4e0d\u53d8\u200b\u65f6\u200b\uff0c\u200b\u8be5\u200b\u53d8\u52a8\u200b\u5bf9\u200b\u6700\u4f18\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u9020\u6210\u200b\u7684\u200b\u5f71\u54cd\u200b\u3002\u200b\u5bf9\u5e94\u200b\u7ecf\u6d4e\u5b66\u200b\u4e2d\u200b\u8fb9\u9645\u200b\u6210\u672c\u200b\u7684\u200b\u6982\u5ff5\u200b\u3002</p> <ul> <li>\u200b\u6bcf\u4e2a\u200b\u7ea6\u675f\u200b\u5bf9\u5e94\u200b\u4e00\u4e2a\u200b\u5f71\u5b50\u200b\u4ef7\u683c\u200b</li> <li>\\(y = \\frac{\\partial z}{\\partial b}\\)</li> </ul> <p>\u200b\u5f71\u5b50\u200b\u4ef7\u683c\u200b\u53cd\u6620\u200b\u8d44\u6e90\u200b\u5728\u200b\u7cfb\u7edf\u200b\u5185\u200b\u7684\u200b\u7a00\u7f3a\u200b\u7a0b\u5ea6\u200b\uff0c\u200b\u5f53\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u4e3a\u200b\u4e0d\u200b\u6ee1\u8db3\u200b\u65f6\u200b\uff0c\u200b\u8be5\u200b\u8d44\u6e90\u200b\u7684\u200b\u91cf\u200b\u5728\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u5b9e\u9645\u4e0a\u200b\u4e0d\u200b\u53d7\u7ea6\u675f\u200b\uff0c\u200b\u56e0\u6b64\u200b\u5f71\u5b50\u200b\u4ef7\u683c\u200b\u4e3a\u200b\\(0\\)\u3002</p>"},{"location":"math/operating-research/chapter-1/#_19","title":"\u5bf9\u5076\u200b\u5355\u7eaf\u5f62\u6cd5","text":"<p>\u200b\u6c42\u89e3\u200b\u5f62\u200b\u5982\u200b\\((P)\\)\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\u7684\u200b\u601d\u8def\u200b\u4e3a\u200b\u63a7\u5236\u200b\\(B^{-1}b \\geq 0\\)\u200b\u5bfb\u627e\u200b\u4f7f\u5f97\u200b\\(C-C_{B}B^{-1}A\\leq 0\\)\u200b\u7684\u200b\u57fa\u200b\u3002\u200b\u5bf9\u5076\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u4e0e\u200b\u4e4b\u200b\u76f8\u53cd\u200b\uff0c\u200b\u4fdd\u6301\u200b\\(C-C_{B}B^{-1}A\\leq 0\\)\uff0c\u200b\u5bfb\u627e\u200b\u4f7f\u5f97\u200b\\(B^{-1}b \\geq 0\\)\u200b\u7684\u200b\u4e00\u7ec4\u200b\u57fa\u200b\u3002</p> <p>\u200b\u6362\u5165\u200b\u53d8\u91cf\u200b\u7684\u200b\u786e\u5b9a\u200b\u4e0e\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u76f8\u540c\u200b</p>"},{"location":"math/operating-research/chapter-1/#_20","title":"\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u7075\u654f\u5ea6","text":"<p>\u200b\u7075\u654f\u5ea6\u200b\u8ba8\u8bba\u200b\u7ebf\u6027\u89c4\u5212\u200b\u4e2d\u200b\u53c2\u6570\u200b\u53d8\u5316\u200b\u5bf9\u200b\u7ebf\u6027\u89c4\u5212\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u7684\u200b\u5f71\u54cd\u200b\uff0c\u200b\u5e94\u5f53\u200b\u6ce8\u610f\u200b\uff0c\u200b\u5982\u679c\u200b\u53c2\u6570\u200b\u53d8\u52a8\u200b\u8fc7\u5927\u200b\uff0c\u200b\u53ef\u80fd\u200b\u4f1a\u200b\u5bfc\u81f4\u200b\u7ebf\u6027\u89c4\u5212\u200b\u89e3\u200b\u53d1\u751f\u53d8\u5316\u200b\u3002</p> <p>\u200b\u8003\u8651\u200b\u5f62\u200b\u5982\u4e0b\u200b\u5f0f\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max Z = 5x_1 + 8x_2 + 6x_3 \\\\     s.t. &amp; \\left\\{     \\begin{aligned}         x_1 &amp; + &amp; x_2 + &amp; x_3 + &amp; x_4 &amp; &amp;= &amp; 12 \\\\         x_1 &amp; + &amp; 2x_2 + &amp; 2x_3 + &amp; &amp; x_5 &amp;= &amp; 20 \\\\         x_i &amp;\\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5176\u200b\u6700\u4f18\u200b\u5355\u7eaf\u5f62\u200b\u8868\u4e3a\u200b\uff1a</p> <p> </p> <p>\u200b\u4efb\u4f55\u200b\u7cfb\u6570\u200b\u7684\u200b\u53d8\u5316\u200b\u90fd\u200b\u4f1a\u200b\u9020\u6210\u200b\u4e24\u79cd\u200b\u60c5\u51b5\u200b\uff1a</p> <ol> <li>\u200b\u6700\u4f18\u200b\u89e3\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b</li> <li>\u200b\u6700\u4f18\u200b\u89e3\u200b\u53d1\u751f\u53d8\u5316\u200b</li> </ol>"},{"location":"math/operating-research/chapter-1/#_21","title":"\u76ee\u6807\u200b\u51fd\u6570\u200b\u7cfb\u6570\u200b\u7684\u200b\u7075\u654f\u5ea6","text":"<p>\u200b\u5f53\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7cfb\u6570\u200b\u53d1\u751f\u53d8\u5316\u200b\u65f6\u200b\uff0c\u200b\u5982\u679c\u200b\u6700\u4f18\u200b\u89e3\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u5219\u200b\\(\\frac{\\partial Z}{\\partial C} = X\\)\u3002</p> <ul> <li>\u200b\u4e00\u4e2a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7cfb\u6570\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\u7684\u200b\u5145\u8981\u6761\u4ef6\u200b\u662f\u200b\u53d8\u5316\u200b\u540e\u200b\u7684\u200b\u68c0\u9a8c\u200b\u6570\u200b\u4ecd\u200b\u4e0d\u200b\u8d85\u8fc7\u200b\\(0\\)\u3002</li> <li>\u200b\u591a\u4e2a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7cfb\u6570\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\u7684\u200b\u5145\u5206\u6761\u4ef6\u200b\u662f\u200b\u5404\u4e2a\u200b\u7cfb\u6570\u200b\u53d8\u5316\u200b\u7684\u200b\u5e45\u5ea6\u200b\u6ee1\u8db3\u200b\\(100\\%\\)\u200b\u6cd5\u5219\u200b\u3002</li> </ul> <p>100%\u200b\u6cd5\u5219\u200b</p> <p>\u200b\u5b9a\u4e49\u200b\\(r_i = |\\frac{\\Delta c_i}{u_i}|\\)\uff0c\u200b\u5176\u4e2d\u200b\\(u_i\\)\u200b\u4e3a\u200b\u7cfb\u6570\u200b\\(c_i\\)\u200b\u5141\u8bb8\u200b\u53d8\u5316\u200b\u7684\u200b\u6700\u200b\u5927\u5e45\u5ea6\u200b\u3002\u200b\u82e5\u200b\\(\\sum r_i &lt; 1\\)\uff0c\u200b\u5219\u200b\u6700\u4f18\u200b\u89e3\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\u3002</p>"},{"location":"math/operating-research/chapter-1/#_22","title":"\u7ea6\u675f\u6761\u4ef6\u200b\u7684\u200b\u7075\u654f\u5ea6","text":"<p>\u200b\u5f53\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u53d1\u751f\u53d8\u5316\u200b\u65f6\u200b\uff0c\u200b\u5982\u679c\u200b\u5355\u7eaf\u5f62\u200b\u8868\u4e2d\u200b\u7684\u200b\\(b' = B^{-1}b\\)\u200b\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u5219\u200b\u6700\u4f18\u200b\u57fa\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u5426\u5219\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5bf9\u5076\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u6c42\u200b\u51fa\u65b0\u200b\u7684\u200b\u6700\u4f18\u200b\u57fa\u200b\u3002</p> <p>\u200b\u6ce8\u610f\u200b</p> <p>\u200b\u5728\u200b\u67d0\u4e9b\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\u200b\u5373\u4f7f\u200b\u6700\u4f18\u200b\u57fa\u4e0d\u200b\u53d1\u751f\u53d8\u5316\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u200b\u4e5f\u200b\u4f1a\u200b\u53d1\u751f\u53d8\u5316\u200b\u3002</p> <p>\u200b\u8bbe\u539f\u200b\u95ee\u9898\u200b\u4e2d\u200b\\(b_1\\)\u200b\u5141\u8bb8\u200b\u53d8\u5316\u200b\u7684\u200b\u8303\u56f4\u200b\u4e3a\u200b\\(\\lambda\\)\uff0c\u200b\u5219\u200b\uff1a</p> \\[ b' = B^{-1}b = \\begin{bmatrix}     1 &amp; 1 \\\\ 1 &amp; 2 \\end{bmatrix} ^{-1} \\begin{bmatrix}     \\lambda \\\\ 0 \\end{bmatrix} = \\begin{bmatrix}     4 + 2\\lambda \\\\ 8 - \\lambda \\end{bmatrix} \\] <p>\u200b\u5982\u679c\u200b\u6700\u4f18\u200b\u57fa\u200b\u4e0d\u53d8\u200b\uff0c\u200b\u5219\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\u4e0d\u53d8\u200b\uff0c\u200b\u5373\u200b\u5f71\u5b50\u200b\u4ef7\u683c\u200b\u4e0d\u53d8\u200b\u3002\u200b\u5f71\u5b50\u200b\u4ef7\u683c\u200b\u7684\u200b\u53d8\u5316\u89c4\u5f8b\u200b\u4ecd\u200b\u6ee1\u8db3\u200b\\(100\\%\\)\u200b\u6cd5\u5219\u200b\u3002</p>"},{"location":"math/operating-research/chapter-1/#_23","title":"\u6dfb\u52a0\u200b\u65b0\u200b\u53d8\u91cf\u200b\u6216\u200b\u65b0\u200b\u7ea6\u675f\u200b\u7684\u200b\u7075\u654f\u5ea6","text":"<p>\u200b\u6dfb\u52a0\u200b\u65b0\u200b\u53d8\u91cf\u200b\u65f6\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u65b0\u200b\u53d8\u91cf\u200b\u7684\u200b\u68c0\u9a8c\u200b\u6570\u200b\uff0c\u200b\u5982\u679c\u200b\u68c0\u9a8c\u200b\u6570\u200b\u5c0f\u4e8e\u200b\u7b49\u4e8e\u200b\\(0\\)\uff0c\u200b\u5219\u200b\u65b0\u200b\u52a0\u5165\u200b\u7684\u200b\u53d8\u91cf\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u4e3a\u200b\\(0\\)\uff0c\u200b\u4e0d\u4f1a\u200b\u5bf9\u539f\u200b\u6700\u4f18\u200b\u89e3\u200b\u4ea7\u751f\u200b\u5f71\u54cd\u200b\u3002</p> <p>\u200b\u6dfb\u52a0\u200b\u65b0\u200b\u7ea6\u675f\u200b\u65f6\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u7ea6\u675f\u200b\u5728\u200b\u5f53\u524d\u200b\u7ed3\u679c\u200b\u4e0b\u200b\u662f\u5426\u200b\u6ee1\u8db3\u200b\uff0c\u200b\u5982\u679c\u200b\u7ea6\u675f\u200b\u6ee1\u8db3\u200b\u5219\u200b\u6700\u4f18\u200b\u89e3\u200b\u4e0d\u53d8\u200b\uff0c\u200b\u5426\u5219\u200b\u6700\u4f18\u200b\u89e3\u200b\u6539\u53d8\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u5bf9\u5076\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u8fdb\u884c\u200b\u6c42\u89e3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-1/#_24","title":"\u6539\u53d8\u200b\u7cfb\u6570\u200b\u77e9\u9635\u200b\u7684\u200b\u7075\u654f\u5ea6","text":"<p>\u200b\u8ba1\u7b97\u200b\\(A' = B^{-1}A\\)\uff0c\u200b\u7136\u540e\u200b\u4f7f\u7528\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u6216\u200b\u5bf9\u5076\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u8fdb\u884c\u200b\u6c42\u89e3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/","title":"\u8fd0\u8f93\u200b\u95ee\u9898","text":"<p>\u200b\u8fd0\u8f93\u200b\u95ee\u9898\u200b\u662f\u200b\u4e00\u79cd\u200b\u7279\u6b8a\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u95ee\u9898\u200b\uff0c\u200b\u5177\u6709\u200b\u5982\u4e0b\u200b\u7684\u200b\u6570\u5b66\u200b\u5f62\u5f0f\u200b\uff1a</p> <ul> <li>\u200b\u7269\u8d44\u200b\u7684\u200b\u4ea7\u5730\u200b\u4e3a\u200b\\(A_i,\\; i=1, 2, \\dots m\\)\uff0c\u200b\u4ea7\u91cf\u200b\u4e3a\u200b\\(a_i\\)</li> <li>\u200b\u7269\u8d44\u200b\u7684\u200b\u9500\u5730\u200b\u4e3a\u200b\\(B_j,\\; j=1, 2, \\dots n\\)\uff0c\u200b\u9500\u91cf\u200b\u4e3a\u200b\\(b_j\\)</li> <li>\u200b\u4ece\u200b\\(i\\)\u200b\u5230\u200b\\(j\\)\u200b\u7684\u200b\u5355\u4f4d\u200b\u8fd0\u4ef7\u200b\\(c_{ij}\\)</li> <li>\u200b\u4f18\u5316\u200b\u8fd0\u8f93\u200b\u5b89\u6392\u200b\u4f7f\u5f97\u200b\u603b\u200b\u8fd0\u4ef7\u200b\u6700\u5c0f\u200b</li> </ul> <p>\u200b\u5176\u200b\u7ebf\u6027\u89c4\u5212\u200b\u5f62\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\min Z = \\sum_{i=1}^m \\sum_{i=1}^n c_{ij}x_{ij} \\\\     s.t. &amp; \\left\\{     \\begin{aligned}         &amp; \\sum_{i = 1}^m x_{ij} = b_{j} \\\\         &amp; \\sum_{j = 1}^n x_{ij} = a_{i} \\\\         &amp; x_{ij} \\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max W = \\sum_{i=1}^m a_iu_i + \\sum_{j=1}^n b_jv_j \\\\     s.t. &amp; \\left\\{         \\begin{aligned}             &amp; u_j + v_j \\leq c_{ij} \\\\             &amp; u_i \\geq 0 \\\\             &amp; v_j \\geq 0 \\\\             &amp; i = 1, 2, \\dots m \\\\             &amp; j = 1, 2, \\dots n         \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5982\u679c\u200b\u8fd0\u8f93\u200b\u95ee\u9898\u200b\u7684\u200b\u4ea7\u9500\u200b\u5e73\u8861\u200b\uff0c\u200b\u5373\u200b\\(\\sum a_i = \\sum b_j\\)\uff0c\u200b\u5219\u200b\u8fd0\u8f93\u200b\u95ee\u9898\u200b\u5fc5\u6709\u200b\u6709\u9650\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u4e2d\u57fa\u200b\u53d8\u91cf\u200b\u7684\u200b\u4e2a\u6570\u200b\u4e3a\u200b\\(m + n - 1\\)</p>"},{"location":"math/operating-research/chapter-2/#_2","title":"\u8868\u4e0a\u200b\u4f5c\u4e1a\u200b\u6cd5","text":"<p>\u200b\u8868\u683c\u200b\u7684\u200b\u901a\u5e38\u200b\u683c\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> <p> </p> <ol> <li>\u200b\u8ba1\u7b97\u200b\u521d\u59cb\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\uff0c\u200b\u5373\u200b\u5728\u200b\u8868\u683c\u200b\u4e0a\u200b\u586b\u5165\u200b\\(m + n - 1\\)\u200b\u4e2a\u200b\u6570\u5b57\u200b</li> <li>\u200b\u8ba1\u7b97\u200b\u8868\u4e2d\u200b\u5404\u200b\u7a7a\u683c\u200b\uff08\u200b\u975e\u57fa\u200b\u53d8\u91cf\u200b\uff09\u200b\u7684\u200b\u68c0\u9a8c\u200b\u6570\u200b\uff0c\u200b\u5224\u65ad\u200b\u5f53\u524d\u200b\u89e3\u200b\u662f\u5426\u200b\u6700\u4f18\u200b\u89e3\u200b</li> <li>\u200b\u82e5\u200b\u5f53\u524d\u200b\u89e3\u200b\u4e0d\u662f\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u5219\u200b\u8fed\u4ee3\u200b\u5230\u200b\u4e0b\u200b\u4e00\u4e2a\u200b\u57fa\u200b</li> </ol> <p>\u200b\u6ce8\u610f\u200b</p> <p>\u200b\u57fa\u200b\u53d8\u91cf\u200b\u4e0d\u80fd\u200b\u6784\u6210\u200b\u95ed\u200b\u56de\u8def\u200b\uff0c\u200b\u6784\u6210\u200b\u4e00\u7ec4\u200b\u95ed\u200b\u56de\u8def\u200b\u7684\u200b\u53d8\u91cf\u200b\u4e2d\u200b\u5fc5\u7136\u200b\u5305\u542b\u200b\u81f3\u5c11\u200b\u4e00\u4e2a\u200b\u975e\u57fa\u200b\u53d8\u91cf\u200b\u3002</p> <p>\u200b\u95ed\u200b\u56de\u8def\u200b\u793a\u610f\u56fe\u200b\u5982\u4e0b\u200b\uff1a</p> <p> </p> <p>\u200b\u4e00\u4e2a\u200b\u95ed\u200b\u56de\u8def\u200b\u4e2d\u200b\u542b\u6709\u200b\u5076\u6570\u200b\u4e2a\u200b\u5355\u5143\u683c\u200b\u3002</p> <p>\u200b\u5982\u4e0b\u200b\u4f7f\u7528\u200b \u200b\u7a7a\u683c\u200b \u200b\u8868\u793a\u200b\\(x_{ij} = 0\\)\u200b\u7684\u200b\u5355\u5143\u683c\u200b\uff0c\u200b\u4f7f\u7528\u200b \u200b\u6570\u5b57\u200b\u683c\u200b \u200b\u8868\u793a\u200b\\(x_{ij} &gt; 0\\)\u200b\u7684\u200b\u5355\u5143\u683c\u200b</p>"},{"location":"math/operating-research/chapter-2/#_3","title":"\u786e\u5b9a\u200b\u521d\u59cb\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3","text":"<p>\u200b\u6709\u200b\u4e09\u79cd\u200b\u65b9\u6cd5\u200b\u7528\u4e8e\u200b\u786e\u5b9a\u200b\u521d\u59cb\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\uff1a</p> <ol> <li>\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b\u6cd5\u200b</li> <li>\u200b\u897f\u5317\u89d2\u200b\u6cd5\u200b</li> <li>Vogol\u200b\u6cd5\u200b</li> </ol>"},{"location":"math/operating-research/chapter-2/#_4","title":"\u6700\u5c0f\u200b\u5143\u7d20\u200b\u6cd5","text":"<p>\u200b\u6700\u5c0f\u200b\u5143\u7d20\u200b\u6cd5\u200b\u4f18\u5148\u200b\u5bfb\u627e\u200b\u8fd0\u4ef7\u200b\u8f83\u200b\u4f4e\u200b\u7684\u200b\u53d8\u91cf\u200b\u4f5c\u4e3a\u200b\u57fa\u200b\u53d8\u91cf\u200b\u3002\u200b\u5982\u679c\u200b\u9009\u53d6\u200b\u51fa\u200b\u7684\u200b\\(x_{ij}\\)\u200b\u6b63\u200b\u5206\u91cf\u200b\u4e0d\u200b\u5305\u542b\u200b\u95ed\u200b\u56de\u8def\u200b\uff0c\u200b\u5219\u200b\\(\\{x_{ij}\\}\\)\u200b\u4e3a\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/#_5","title":"\u897f\u5317\u89d2\u200b\u6cd5","text":"<p>\u200b\u897f\u5317\u89d2\u200b\u6cd5\u4ece\u200b\u5de6\u4e0a\u200b\u5f00\u59cb\u200b\uff0c\u200b\u4f18\u5148\u200b\u6700\u5927\u5316\u200b\u6ee1\u8db3\u200b\u5de6\u4e0a\u89d2\u200b\u7684\u200b\u8fd0\u8f93\u200b\u9700\u6c42\u200b\u3002\u200b\u5982\u679c\u200b\u4f9b\u7ed9\u65b9\u200b\u9700\u6c42\u200b\u5df2\u7ecf\u200b\u6ee1\u8db3\u200b\uff0c\u200b\u5219\u200b\u5212\u53bb\u200b\u4f9b\u7ed9\u65b9\u200b\u6240\u5728\u200b\u7684\u200b\u884c\u200b\uff0c\u200b\u5426\u5219\u200b\u5212\u200b\u53bb\u200b\u9700\u6c42\u65b9\u200b\u6240\u5728\u200b\u7684\u200b\u5217\u200b\u3002\u200b\u91cd\u590d\u200b\u4ee5\u4e0a\u200b\u8fc7\u7a0b\u200b\u76f4\u5230\u200b\u53f3\u4e0b\u89d2\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/#vogol","title":"Vogol\u200b\u6cd5","text":"<p>Vogol\u200b\u6cd5\u200b\u8ba1\u7b97\u200b\u5404\u884c\u200b\u5404\u5217\u200b\u4e2d\u200b\u6700\u5c0f\u200b\\(c_{ij}\\)\u200b\u4e0e\u200b\u6b21\u200b\u5c0f\u200b\\(c_{ij}\\)\u200b\u7684\u200b\u5dee\u200b\u4f5c\u4e3a\u200b\u7f5a\u6570\u200b\uff0c\u200b\u4f18\u5148\u200b\u5b89\u6392\u200b\u7f5a\u6570\u200b\u6bd4\u8f83\u200b\u5927\u200b\u7684\u200b\u4ea7\u5730\u200b-\u200b\u9500\u5730\u200b\u3002\u200b\u9009\u53d6\u200b\u7f5a\u6570\u200b\u6700\u5927\u200b\u7684\u200b\u884c\u200b\u6216\u200b\u5217\u200b\uff0c\u200b\u5e76\u200b\u586b\u5165\u200b\u6700\u5c0f\u200b\u7684\u200b\\(c_{ij}\\)\uff0c\u200b\u7136\u540e\u200b\u5212\u200b\u53bb\u200b\u8be5\u884c\u200b\u6216\u5217\u200b\uff0c\u200b\u91cd\u65b0\u200b\u8ba1\u7b97\u200b\u7f5a\u6570\u200b\u3002</p> <p>Vogol\u200b\u6cd5\u200b\u901a\u5e38\u200b\u80fd\u591f\u200b\u5f97\u5230\u200b\u66f4\u4f18\u200b\u7684\u200b\u521d\u59cb\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/#_6","title":"\u6700\u4f18\u6027\u200b\u68c0\u9a8c","text":""},{"location":"math/operating-research/chapter-2/#_7","title":"\u95ed\u200b\u56de\u8def\u200b\u6cd5","text":"<p>\u200b\u5b9a\u7406\u200b</p> <p>\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u7a7a\u683c\u200b\uff0c\u200b\u90fd\u200b\u5b58\u5728\u200b\u4e00\u6761\u200b\u95ed\u200b\u56de\u8def\u200b\uff1a</p> <ol> <li>\u200b\u7a7a\u683c\u200b\u4f4d\u4e8e\u200b\u8be5\u95ed\u200b\u56de\u8def\u200b\u7684\u200b\u62d0\u70b9\u200b\u4e0a\u200b</li> <li>\u200b\u95ed\u200b\u56de\u8def\u200b\u7684\u200b\u5176\u4ed6\u200b\u62d0\u70b9\u200b\u662f\u200b\u6570\u5b57\u200b\u683c\u200b</li> </ol> <p>\u200b\u5c06\u200b\u4ece\u200b\u7a7a\u683c\u200b\u51fa\u53d1\u200b\uff0c\u200b\u5076\u6570\u200b\u5355\u5143\u683c\u200b\u7684\u200b\u8fd0\u4ef7\u200b\u548c\u200b\u4e0e\u200b\u5947\u6570\u200b\u5355\u5143\u683c\u200b\u7684\u200b\u8fd0\u4ef7\u200b\u548c\u200b\u4e4b\u200b\u5dee\u200b\u8bb0\u4e3a\u200b\\(\\lambda_{ij}\\)\uff0c\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u6240\u6709\u200b\u7a7a\u683c\u200b\uff0c\u200b\u90fd\u200b\u6709\u200b\\(\\lambda_{ij} \\leq 0\\)\uff0c\u200b\u5219\u200b\u5f53\u524d\u200b\u7684\u200b\\(\\{x_{ij}\\}\\)\u200b\u4e3a\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/#_8","title":"\u4f4d\u52bf\u200b\u6cd5","text":"<p>\u200b\u8bbe\u200b\\((u_{i}, v_{j})\\)\u200b\u4e3a\u200b\u5bf9\u5076\u200b\u95ee\u9898\u200b\u7684\u200b\u53ef\u884c\u200b\u89e3\u200b\uff0c\\(\\{x_{ij}\\}\\)\u200b\u4e3a\u539f\u200b\u95ee\u9898\u200b\u7684\u200b\u53ef\u884c\u200b\u89e3\u200b\u3002\u200b\u6839\u636e\u200b\u4e92\u8865\u200b\u677e\u5f1b\u200b\u6027\u200b\uff0c\u200b\u82e5\u200b\\(x_{ij}(c_{ij} - u_i - v_j) = 0\\)\uff0c\u200b\u5219\u200b\u4e24\u7ec4\u200b\u89e3\u200b\u5206\u522b\u200b\u4e3a\u200b\u5bf9\u5e94\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002</p> <p>\u200b\u5df2\u77e5\u200b\u5f53\u524d\u200b\u4e00\u7ec4\u200b\u57fa\u200b\u53ef\u884c\u200b\u89e3\u200b\\(\\{x_{ij}\\}\\)\uff1a</p> <ol> <li>\u200b\u5bf9\u4e8e\u200b\u57fa\u200b\u53d8\u91cf\u200b\\(\\{x_{ij}\\}\\)\uff0c\u200b\u5217\u51fa\u200b\u65b9\u7a0b\u200b\\(u_i + v_j = c_{ij}\\)\uff0c\u200b\u7ec4\u5408\u6210\u200b\u65b9\u7a0b\u7ec4\u200b</li> <li>\u200b\u8be5\u200b\u65b9\u7a0b\u7ec4\u200b\u6709\u200b\\(m+n\\)\u200b\u4e2a\u200b\u53d8\u91cf\u200b\u4f46\u200b\u53ea\u6709\u200b\\(m + n - 1\\)\u200b\u4e2a\u200b\u65b9\u7a0b\u200b\uff0c\u200b\u56e0\u6b64\u200b\u6709\u200b\u65e0\u6570\u200b\u89e3\u200b\uff0c\u200b\u4efb\u9009\u200b\u4e00\u7ec4\u200b\u89e3\u200b\u5373\u53ef\u200b</li> <li>\u200b\u5bf9\u4e8e\u200b\u975e\u57fa\u200b\u53d8\u91cf\u200b\\(x_{ij}\\)\uff0c\u200b\u8ba1\u7b97\u200b\\(\\lambda_{ij} = c_{ij} - u_i - v_j\\)</li> <li>\u200b\u82e5\u200b\\(\\lambda _{ij}\\)\u200b\u5168\u90e8\u200b\\(\\geq 0\\)\uff0c\u200b\u5219\u200b\u5f53\u524d\u200b\u89e3\u4e3a\u200b\u6700\u4f18\u200b\u89e3\u200b</li> <li>\u200b\u5426\u5219\u200b\u8f6c\u200b\u51fa\u200b\\((i, j) = \\arg\\max_{(i, j)} -\\lambda_{ij}\\)\uff0c\u200b\u8c03\u6574\u200b\u8be5\u200b\u7a7a\u683c\u200b\u6240\u5728\u200b\u7684\u200b\u95ed\u200b\u56de\u8def\u200b</li> </ol>"},{"location":"math/operating-research/chapter-2/#_9","title":"\u8fd0\u8f93\u200b\u95ee\u9898\u200b\u7684\u200b\u53d8\u79cd","text":""},{"location":"math/operating-research/chapter-2/#_10","title":"\u4e24\u5730\u200b\u65e0\u200b\u8fde\u63a5","text":"<p>\u200b\u5982\u679c\u200b\u6a21\u578b\u200b\u4e2d\u200b\u5b58\u5728\u200b\u4e24\u4e2a\u200b\u5730\u70b9\u200b\u4e4b\u95f4\u200b\u4e0d\u80fd\u200b\u8fd0\u8f93\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u4e24\u5730\u200b\u95f4\u200b\u7684\u200b\u8fd0\u4ef7\u200b\u8bbe\u200b\u4e3a\u200b\u8db3\u591f\u200b\u9ad8\u200b\u7684\u200b\u6b63\u6570\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/#_11","title":"\u4ea7\u9500\u200b\u4e0d\u200b\u5e73\u8861","text":"<p>\u200b\u5f53\u200b\u95ee\u9898\u200b\u4e2d\u200b\u4ea7\u91cf\u200b\u4e0e\u200b\u9500\u91cf\u200b\u4e0d\u200b\u76f8\u7b49\u200b\u65f6\u200b\uff1a</p> <ul> <li>\u200b\u5982\u679c\u200b\u4ea7\u91cf\u200b\u5927\u4e8e\u200b\u9500\u552e\u91cf\u200b\uff0c\u200b\u5219\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u865a\u62df\u200b\u4e70\u65b9\u200b\uff0c\u200b\u589e\u52a0\u200b\\(m\\)\u200b\u4e2a\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b</li> <li>\u200b\u5982\u679c\u200b\u9500\u552e\u91cf\u200b\u5927\u4e8e\u200b\u4ea7\u91cf\u200b\uff0c\u200b\u5219\u200b\u6dfb\u52a0\u200b\u4e00\u4e2a\u200b\u865a\u62df\u200b\u5356\u65b9\u200b\uff0c\u200b\u589e\u52a0\u200b\\(n\\)\u200b\u4e2a\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b</li> </ul>"},{"location":"math/operating-research/chapter-2/#_12","title":"\u9700\u6c42\u200b\u8303\u56f4","text":"<p>\u200b\u5982\u679c\u200b\u4e70\u65b9\u200b\u7684\u200b\u9700\u6c42\u200b\u5728\u200b\u4e00\u4e2a\u200b\u8303\u56f4\u200b\u4e2d\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u5c06\u200b\u5b58\u5728\u200b\u8303\u56f4\u200b\u9700\u6c42\u200b\u7684\u200b\u4e70\u65b9\u200b\u62c6\u200b\u5206\u4e3a\u200b\u4e24\u4e2a\u200b\u90e8\u5206\u200b\uff0c\u200b\u5373\u200b\u5fc5\u987b\u200b\u6ee1\u8db3\u200b\u7684\u200b\u9700\u6c42\u200b\u90e8\u5206\u200b\u4e0e\u200b\u53ef\u9009\u200b\u6ee1\u8db3\u200b\u7684\u200b\u9700\u6c42\u200b\u90e8\u5206\u200b\uff0c\u200b\u4e24\u200b\u90e8\u5206\u200b\u5230\u200b\u5356\u65b9\u200b\u7684\u200b\u8fd0\u4ef7\u200b\u76f8\u540c\u200b\u3002\u200b\u540c\u65f6\u200b\u52a0\u5165\u200b\u4e00\u4e2a\u200b\u865a\u62df\u200b\u5356\u65b9\u200b\uff0c\u200b\u865a\u62df\u200b\u5356\u65b9\u200b\u5230\u200b\u5fc5\u987b\u200b\u6ee1\u8db3\u200b\u7684\u200b\u865a\u62df\u200b\u4e70\u5bb6\u200b\u7684\u200b\u8fd0\u4ef7\u200b\u4e3a\u200b\u8db3\u591f\u200b\u5927\u200b\u7684\u200b\u6b63\u6570\u200b\uff0c\u200b\u865a\u62df\u200b\u5356\u65b9\u200b\u5230\u200b\u53ef\u9009\u200b\u6ee1\u8db3\u200b\u7684\u200b\u865a\u62df\u200b\u4e70\u5bb6\u200b\u7684\u200b\u8fd0\u4ef7\u200b\u4e3a\u200b\\(0\\)\u3002</p>"},{"location":"math/operating-research/chapter-2/#_13","title":"\u4f9b\u5e94\u200b\u95ee\u9898","text":"<p>\u200b\u8003\u8651\u200b\u6309\u200b\u6708\u200b\u751f\u4ea7\u200b\u7684\u200b\u5356\u65b9\u200b\u4e0e\u200b\u6309\u200b\u6708\u200b\u4ea7\u751f\u200b\u9700\u6c42\u200b\u7684\u200b\u4e70\u65b9\u200b\uff0c\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u5ef6\u671f\u200b\u4ea4\u8d27\u200b\u3002\u200b\u5356\u65b9\u200b\u751f\u4ea7\u200b\u7684\u200b\u4ea7\u54c1\u200b\u5f53\u6708\u200b\u53ef\u4ee5\u200b\u4e0d\u200b\u9500\u552e\u200b\u3002\u200b\u5219\u200b\u8be5\u200b\u95ee\u9898\u200b\u53ef\u4ee5\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u8fd0\u8f93\u200b\u95ee\u9898\u200b\u3002</p>"},{"location":"math/operating-research/chapter-2/#_14","title":"\u591a\u7ef4\u200b\u8fd0\u8f93\u200b\u95ee\u9898","text":"<p>\u200b\u5047\u8bbe\u200b\u4e24\u5730\u200b\u4e4b\u95f4\u200b\u5b58\u5728\u200b\\(k\\)\u200b\u4e2a\u200b\u4e2d\u95f4\u200b\u8282\u70b9\u200b\\(c_k\\)\uff0c\u200b\u4ece\u200b\\(A\\)\u200b\u5230\u200b\\(B\\)\u200b\u7684\u200b\u6240\u6709\u200b\u8def\u5f84\u200b\u90fd\u200b\u5fc5\u987b\u200b\u7ecf\u8fc7\u200b\u8282\u70b9\u200b\\(C\\)\uff0c\u200b\u6c42\u200b\u6700\u4f18\u200b\u8fd0\u8f93\u200b\u65b9\u6848\u200b\u3002</p>"},{"location":"math/operating-research/chapter-3/","title":"\u76ee\u6807\u200b\u89c4\u5212","text":"<p>\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u89e3\u51b3\u200b\u73b0\u5b9e\u751f\u6d3b\u200b\u4e2d\u200b\u7684\u200b\u591a\u200b\u76ee\u6807\u200b\u51b3\u7b56\u95ee\u9898\u200b\u3002</p> <p>\u200b\u8bbe\u200b\u7ebf\u6027\u89c4\u5212\u200b\\((P)\\)\uff1a</p> \\[ \\begin{aligned}     &amp; \\max z=c^Tx \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; AX \\leq b \\\\         &amp; X\\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5047\u8bbe\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u7684\u200b\u76ee\u6807\u200b\u8c03\u6574\u200b\u4e3a\u200b\uff1a</p> <p>\u200b\u5bfb\u627e\u200b\u4f7f\u5f97\u200b\\(z\\geq z_0\\)\u200b\u7684\u200b\\(X\\)\uff0c\u200b\u5176\u4e2d\u200b\\(z_0\\)\u200b\u4e3a\u200b\u6307\u5b9a\u200b\u7684\u200b\u76ee\u6807\u200b\u3002\u200b\u5141\u8bb8\u200b\\(z &gt; z_0\\)\u200b\u4f46\u200b\u4e0d\u200b\u5141\u8bb8\u200b\\(z &lt; z_0\\)</p> <p>\u200b\u6b64\u65f6\u200b\u7684\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u5373\u200b\u4e3a\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\uff0c\u200b\u5f15\u5165\u200b\u4e24\u4e2a\u200b\u677e\u5f1b\u200b\u53d8\u91cf\u200b\\(d^+, d^-\\)\uff0c\u200b\u5206\u522b\u200b\u8868\u793a\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u8d85\u51fa\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u90e8\u5206\u200b\u4e0e\u200b\u4e0d\u8db3\u200b\u76ee\u6807\u503c\u200b\u7684\u200b\u90e8\u5206\u200b\uff0c\u200b\u4e3a\u200b\u4fdd\u8bc1\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u8d85\u8fc7\u200b\u76ee\u6807\u503c\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u65b9\u5f0f\u200b\u5efa\u6a21\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max w = d^- \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; c^Tx - d^+ + d^- = z_0 \\\\         &amp; AX \\leq b \\\\         &amp; X\\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u7531\u6b64\u53ef\u89c1\u200b\uff0c\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u76f8\u5f53\u4e8e\u200b\u5728\u200b\u7ebf\u6027\u89c4\u5212\u200b\u4e2d\u200b\u5f15\u5165\u200b\u4e00\u79cd\u200b\u65b0\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff0c\u200b\u5373\u200b\u76ee\u6807\u200b\u7ea6\u675f\u200b\u3002\u200b\u6bcf\u4e2a\u200b\u76ee\u6807\u200b\u7ea6\u675f\u200b\u90fd\u200b\u9700\u8981\u200b\u5f15\u5165\u200b\u4e00\u5bf9\u200b\u53d8\u91cf\u200b\\(d^+\\)\u200b\u4e0e\u200b\\(d^-\\)\u3002\u200b\u5982\u679c\u200b\u4f18\u5316\u200b\u95ee\u9898\u200b\u4e2d\u200b\u5b58\u5728\u200b\u591a\u4e2a\u200b\u76ee\u6807\u200b\u4e14\u200b\u4e4b\u95f4\u200b\u5b58\u5728\u200b\u4f18\u5148\u7ea7\u200b\u5173\u7cfb\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\\(P_i\\)\u200b\u4f5c\u4e3a\u200b\u9700\u8981\u200b\u4f18\u5316\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u7cfb\u6570\u200b\u3002\\(P_i\\)\u200b\u6ee1\u8db3\u200b\\(P_1 \\gg P_2 \\gg \\dots \\gg P_n\\)\u200b\u5e76\u4e14\u200b\u5404\u200b\\(P_i\\)\u200b\u4e3a\u200b\u8db3\u591f\u200b\u5927\u200b\u7684\u200b\u6b63\u6570\u200b\u3002</p> <p>\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e2d\u200b\u53ea\u6709\u200b\u504f\u5dee\u200b\u53d8\u91cf\u200b\uff0c\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u7684\u200b\u4f18\u5316\u200b\u76ee\u6807\u200b\u662f\u200b\u6700\u5c0f\u5316\u200b\u504f\u5dee\u200b\u53d8\u91cf\u200b\u3002\u200b\u5982\u679c\u200b\u6700\u4f18\u200b\u6761\u4ef6\u200b\u4e0b\u200b\\(Z=0\\)\uff0c\u200b\u8bf4\u660e\u200b\u6240\u6709\u200b\u76ee\u6807\u200b\u53ef\u4ee5\u200b\u540c\u65f6\u200b\u6ee1\u8db3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-3/#_2","title":"\u76ee\u6807\u200b\u89c4\u5212\u200b\u7684\u200b\u6c42\u89e3\u200b\u65b9\u6cd5","text":"<p>\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u56fe\u89e3\u6cd5\u200b\u6216\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u5bf9\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u8fdb\u884c\u200b\u6c42\u89e3\u200b</p>"},{"location":"math/operating-research/chapter-3/#_3","title":"\u56fe\u89e3\u6cd5","text":"<p>\u200b\u4f7f\u7528\u200b\u56fe\u89e3\u6cd5\u200b\u6c42\u89e3\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u65f6\u200b\uff0c\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u753b\u51fa\u200b\u76ee\u6807\u200b\u7ea6\u675f\u200b\u4e2d\u200b\u7684\u200b\u786c\u200b\u7ea6\u675f\u200b\uff0c\u200b\u518d\u200b\u6309\u7167\u200b\u76ee\u6807\u200b\u7ea6\u675f\u200b\u7684\u200b\u4f18\u5148\u7ea7\u200b\u753b\u51fa\u200b\u8f6f\u200b\u7ea6\u675f\u200b\uff0c\u200b\u5e76\u200b\u6ce8\u660e\u200b\u7ea6\u675f\u200b\u504f\u5dee\u200b\u65b9\u5411\u200b\u3002\u200b\u6309\u7167\u200b\u7ea6\u675f\u200b\u7684\u200b\u4f18\u5148\u7ea7\u200b\u4f9d\u6b21\u200b\u5212\u5206\u200b\u80fd\u591f\u200b\u6ee1\u8db3\u200b\u7ea6\u675f\u200b\u7684\u200b\u53ef\u884c\u200b\u57df\u200b\u3002\u200b\u76f4\u5230\u200b\u53ef\u884c\u200b\u57df\u200b\u4e3a\u200b\u7a7a\u200b\u6216\u200b\u7ea6\u675f\u200b\u5168\u90e8\u200b\u6ee1\u8db3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-3/#_4","title":"\u5355\u7eaf\u5f62\u6cd5","text":"<p>\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u5c06\u200b\u5355\u7eaf\u5f62\u6cd5\u200b\u4e2d\u200b\u7684\u200b\u786c\u200b\u7ea6\u675f\u200b\u5316\u4e3a\u200b\u6807\u51c6\u578b\u200b\u3002\u200b\u5217\u51fa\u200b\u5355\u7eaf\u5f62\u200b\u8868\u540e\u200b\uff0c\u200b\u9010\u6b65\u200b\u5c06\u200b\u5305\u542b\u200b\u5728\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e2d\u200b\u7684\u200b\u504f\u5dee\u200b\u4ece\u200b\u57fa\u4e2d\u6362\u200b\u51fa\u200b\u3002\u200b\u5e76\u4e14\u200b\u5c3d\u53ef\u80fd\u200b\u4f7f\u5f97\u200b\u975e\u57fa\u200b\u53d8\u91cf\u200b\u7684\u200b\u68c0\u9a8c\u200b\u6570\u4e3a\u200b\u6b63\u200b\uff08\u200b\u56e0\u200b\u76ee\u6807\u200b\u89c4\u5212\u200b\u7684\u200b\u4f18\u5316\u200b\u65b9\u5411\u200b\u4e3a\u200b\u6700\u5c0f\u5316\u200b\uff09\u3002</p>"},{"location":"math/operating-research/chapter-4/","title":"\u6574\u6570\u200b\u89c4\u5212","text":"<p>\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u662f\u200b\u90e8\u5206\u200b\uff08\u200b\u6216\u200b\u5168\u90e8\u200b\uff09\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u5747\u53d6\u200b\u6574\u6570\u200b\u7684\u200b\u7ebf\u6027\u89c4\u5212\u200b\u3002\u200b\u5168\u90e8\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u5747\u53d6\u200b\u6574\u6570\u200b\u7684\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u79f0\u4e3a\u200b\u7eaf\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u3002\u200b\u5982\u679c\u200b\u6574\u6570\u200b\u53d8\u91cf\u200b\u53ea\u80fd\u200b\u53d6\u200b0-1\u200b\u4e24\u4e2a\u200b\u503c\u200b\uff0c\u200b\u5219\u200b\u79f0\u4e3a\u200b0-1\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u3002</p>"},{"location":"math/operating-research/chapter-4/#_2","title":"\u6574\u6570\u200b\u89c4\u5212\u200b\u7684\u200b\u6c42\u89e3\u200b\u65b9\u6cd5","text":"<p>\u200b\u6c42\u89e3\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u76ee\u524d\u200b\u6709\u200b\u5982\u4e0b\u200b\u7b97\u6cd5\u200b\uff1a</p> <ol> <li>\u200b\u5272\u200b\u5e73\u9762\u200b\u6cd5\u200b</li> <li>\u200b\u5206\u652f\u200b\u5b9a\u754c\u200b\u6cd5\u200b</li> <li>\u200b\u9690\u200b\u679a\u4e3e\u6cd5\u200b\uff08\u200b\u9002\u7528\u200b\u4e8e\u200b\u6c42\u89e3\u200b0-1\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\uff09</li> <li>\u200b\u542f\u53d1\u5f0f\u200b\u7b97\u6cd5\u200b</li> </ol> <p>\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b\u7eaf\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max z=c^Tx \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; AX \\leq b \\\\         &amp; X\\geq 0     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u5176\u4e2d\u200b\\(A, b, x\\)\u200b\u5747\u200b\u4e3a\u200b\u6574\u6570\u200b\u3002</p>"},{"location":"math/operating-research/chapter-4/#_3","title":"\u5272\u200b\u5e73\u9762\u200b\u6cd5","text":"<p>\u200b\u79f0\u200b\u6d88\u9664\u200b\u6574\u6570\u200b\u7ea6\u675f\u200b\u7684\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u4e3a\u200b\u677e\u5f1b\u200b\u95ee\u9898\u200b\uff0c\u200b\u5272\u200b\u5e73\u9762\u200b\u6cd5\u200b\u9996\u5148\u200b\u9700\u8981\u200b\u6c42\u89e3\u200b\u677e\u5f1b\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u8bbe\u200b\u4e3a\u200b\\(X^*\\)\u3002\u200b\u901a\u5e38\u200b\u60c5\u51b5\u200b\u4e0b\u200b\uff0c\\(X^*\\)\u200b\u4e0d\u662f\u200b\u6574\u6570\u200b\u6216\u200b\u4e0d\u5168\u662f\u200b\u6574\u6570\u200b\u3002\u200b\u5355\u7eaf\u5f62\u200b\u8868\u200b\u5982\u4e0b\u200b\u6240\u793a\u200b\uff1a</p> <p> </p> <p>\u200b\u8bbe\u200b\u5355\u7eaf\u5f62\u200b\u8868\u4e2d\u200b\u7684\u200b\u7b2c\u200b\\(i\\)\u200b\u884c\u200b\u5bf9\u5e94\u200b\u7684\u200b\\(b'_i\\)\u200b\u4e0d\u200b\u4e3a\u200b\u6574\u6570\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u8fd9\u200b\u4e00\u884c\u200b\uff0c\u200b\u6709\u200b\uff1a</p> \\[ x_i + \\sum_{j=m+1}^n a'_{ij}x_j = b'_i \\] <p>\u200b\u5c06\u200b\\(a'_{ij}, b'_i\\)\u200b\u5212\u5206\u200b\u4e3a\u200b\u6574\u6570\u200b\u90e8\u5206\u200b\u4e0e\u200b\u5c0f\u6570\u200b\u90e8\u5206\u200b\uff0c\u200b\u5373\u200b\uff1a</p> \\[ \\begin{aligned}     a'_{ij} &amp;= \\tilde a_{ij} + \\alpha_{ij}, \\; \\tilde a_{ij} = \\lfloor a'_{ij}\\rfloor \\\\     b'_{i} &amp;= \\tilde b_{i} + \\beta_{i}, \\; \\tilde b_{ij} = \\lfloor b'_{i}\\rfloor \\end{aligned} \\] <p>\u200b\u5c06\u200b\u8fd9\u200b\u4e00\u884c\u200b\u7b49\u5f0f\u200b\u53d8\u5f62\u200b\uff0c\u200b\u5f97\u5230\u200b\uff1a</p> \\[ \\beta_i - \\sum_{j=m+1}^n\\alpha_{ij}y_j = x_i - \\tilde b_i + \\sum_{j=m+1}^n \\tilde \\alpha_{ij}y_j \\] <p>\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u7684\u200b\u6761\u4ef6\u200b\u8981\u6c42\u200b\u7b49\u5f0f\u200b\u53f3\u4fa7\u200b\u4e3a\u200b\u6574\u6570\u200b\uff0c\u200b\u56e0\u6b64\u200b\u7b49\u5f0f\u200b\u5de6\u4fa7\u200b\u540c\u6837\u200b\u9700\u8981\u200b\u4e3a\u200b\u6574\u6570\u200b\uff0c\u200b\u5219\u200b\u6709\u200b\\(\\beta_i - \\sum_{j=m+1}^n\\alpha_{ij}y_j + s_i = 0\\)\uff0c\u200b\u5176\u4e2d\u200b\\(s_i\\)\u200b\u4e3a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b\u3002</p> <p>\u200b\u79f0\u200b\u65b9\u7a0b\u200b</p> \\[ \\beta_i - \\sum_{j=m+1}^n\\alpha_{ij}y_j + s_i = 0 \\] <p>\u200b\u4e3a\u200b\u5272\u200b\u5e73\u9762\u200b\u65b9\u7a0b\u200b\uff0c\u200b\u5272\u200b\u5e73\u9762\u200b\u65b9\u7a0b\u200b\u4f5c\u4e3a\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff0c\u200b\u6d88\u53bb\u200b\u4e86\u200b\u975e\u200b\u6574\u6570\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u7559\u4e0b\u200b\u6574\u6570\u200b\u89e3\u200b\u3002\u200b\u5411\u200b\u677e\u5f1b\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u5355\u7eaf\u5f62\u200b\u8868\u4e2d\u200b\u52a0\u5165\u200b\u5272\u200b\u5e73\u9762\u200b\u65b9\u7a0b\u200b\u4f5c\u4e3a\u200b\u65b0\u200b\u7684\u200b\u7ea6\u675f\u6761\u4ef6\u200b\uff0c\u200b\u7ee7\u7eed\u200b\u6c42\u89e3\u200b\u7ebf\u6027\u89c4\u5212\u200b\u5373\u53ef\u200b\u5f97\u5230\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\u3002</p>"},{"location":"math/operating-research/chapter-4/#_4","title":"\u5206\u652f\u200b\u5b9a\u754c\u200b\u6cd5","text":"<p>\u200b\u5148\u200b\u6c42\u89e3\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u5bf9\u5e94\u200b\u7684\u200b\u677e\u5f1b\u200b\u95ee\u9898\u200b\uff0c\u200b\u5e76\u200b\u4f30\u8ba1\u200b\u4e00\u4e2a\u200b\u975e\u200b\u6700\u4f18\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u4f5c\u4e3a\u200b\u8bb0\u5f55\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u200b\u3002\u200b\u5bf9\u4e8e\u200b\u6700\u4f18\u200b\u89e3\u4e2d\u200b\u7684\u200b\u975e\u200b\u6574\u6570\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\uff0c\u200b\u9009\u62e9\u200b\u5176\u4e2d\u200b\u4e00\u4e2a\u200b\\(x_i\\)\u200b\u8fdb\u884c\u200b\u5206\u652f\u200b\uff1a</p> <ol> <li>\u200b\u6dfb\u52a0\u200b\u7ea6\u675f\u200b\\(x'_i \\leq \\lfloor x_i \\rfloor\\)</li> <li>\u200b\u6dfb\u52a0\u200b\u7ea6\u675f\u200b\\(x'_i \\geq \\lceil x_i \\rceil\\)</li> </ol> <p>\u200b\u6dfb\u52a0\u200b\u7ea6\u675f\u200b\u540e\u200b\u91cd\u65b0\u200b\u6c42\u89e3\u200b\u7ebf\u6027\u89c4\u5212\u200b\uff0c\u200b\u5982\u679c\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u89e3\u200b\u6ee1\u8db3\u200b\u6574\u6570\u200b\u7ea6\u675f\u200b\uff0c\u200b\u5219\u200b\u5c06\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u4e0e\u200b\u8bb0\u5f55\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u76f8\u200b\u6bd4\u8f83\u200b\u3002\u200b\u5982\u679c\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u66f4\u4f18\u200b\uff0c\u200b\u5219\u200b\u66f4\u65b0\u200b\u6700\u4f18\u200b\u503c\u200b\uff0c\u200b\u5426\u5219\u200b\u505c\u6b62\u200b\u5411\u4e0b\u200b\u63a2\u7d22\u200b\uff08\u200b\u526a\u679d\u200b\uff09\u3002</p> <p>\u200b\u5206\u652f\u200b\u53ef\u4ee5\u200b\u91c7\u53d6\u200b\u6df1\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b\u6216\u200b\u5e7f\u5ea6\u200b\u4f18\u5148\u200b\u641c\u7d22\u200b\u3002</p>"},{"location":"math/operating-research/chapter-4/#_5","title":"\u9690\u200b\u679a\u4e3e\u6cd5","text":"<p>\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u5f62\u5f0f\u200b\u7684\u200b0-1\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\uff1a</p> \\[ \\begin{aligned}     &amp; \\max z=c^Tx \\\\     &amp; s.t. \\left\\{     \\begin{aligned}         &amp; AX \\leq b \\\\         &amp; x_i \\in \\{0, 1\\}     \\end{aligned}     \\right . \\end{aligned} \\] <p>\u200b\u9996\u5148\u200b\u8bd5\u63a2\u200b\u51fa\u200b\u4e00\u4e2a\u200b\u53ef\u884c\u200b\u89e3\u200b\uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u4f5c\u4e3a\u200b\u8bb0\u5f55\u200b\u7684\u200b\u6700\u4f18\u200b\u503c\u200b\\(Z_0\\)\uff0c\u200b\u518d\u200b\u679a\u4e3e\u200b\u6240\u6709\u200b\u7684\u200b\u53ef\u884c\u200b\u89e3\u200b\uff0c\u200b\u82e5\u89e3\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u503c\u200b\u8d85\u8fc7\u200b\u6700\u4f18\u200b\u503c\u5219\u200b\u66f4\u65b0\u200b\u6700\u4f18\u200b\u503c\u200b\u3002</p>"},{"location":"math/operating-research/chapter-4/#_6","title":"\u6307\u6d3e\u200b\u95ee\u9898","text":"<p>\u200b\u6307\u6d3e\u200b\u95ee\u9898\u200b\u662f\u200b\u7279\u6b8a\u200b\u7684\u200b0-1\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\uff0c\u200b\u5176\u200b\u76ee\u6807\u200b\u662f\u200b\u5c06\u200b\\(n\\)\u200b\u4e2a\u200b\u5de5\u4eba\u200b\u5728\u200b\\(n\\)\u200b\u4e2a\u200b\u5de5\u4f5c\u200b\u4e2d\u200b\u5206\u914d\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5de5\u4eba\u200b\\(i\\)\u200b\u5728\u200b\u8fdb\u884c\u200b\u5de5\u4f5c\u200b\\(j\\)\u200b\u65f6\u200b\uff0c\u200b\u90fd\u200b\u4f1a\u200b\u4ea7\u751f\u200b\u6210\u672c\u200b\\(c_{ij}\\)\uff0c\u200b\u51b3\u7b56\u95ee\u9898\u200b\u7684\u200b\u76ee\u6807\u200b\u662f\u200b\u6700\u5c0f\u5316\u200b\u603b\u6210\u672c\u200b\u3002\u200b\u8bb0\u200b\u6210\u672c\u200b\u77e9\u9635\u200b\u4e3a\u200b</p> \\[ A = \\begin{bmatrix}     c_{11} &amp; c_{12} &amp; \\cdots &amp; c_{1n} \\\\     c_{21} &amp; c_{22} &amp; \\cdots &amp; c_{2n} \\\\     \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\     c_{n1} &amp; c_{n2} &amp; \\cdots &amp; c_{nn} \\end{bmatrix} \\] <p>\u200b\u6027\u8d28\u200b</p> <p>\u200b\u82e5\u200b\u7cfb\u6570\u200b\u77e9\u9635\u200b\\(A\\)\u200b\u7684\u200b\u67d0\u200b\u4e00\u884c\u200b\u6216\u200b\u67d0\u200b\u4e00\u5217\u200b\u51cf\u53bb\u200b\u5206\u522b\u200b\u51cf\u53bb\u200b\u4e00\u4e2a\u200b\u5e38\u6570\u200b\\(k\\)\uff0c\u200b\u4e0d\u200b\u5f71\u54cd\u200b\u6307\u6d3e\u200b\u95ee\u9898\u200b\u7684\u200b\u89e3\u200b\u3002</p> <p>\u200b\u4f7f\u7528\u200b\u5308\u7259\u5229\u200b\u7b97\u6cd5\u200b\u6c42\u89e3\u200b\u6574\u6570\u200b\u89c4\u5212\u200b\u95ee\u9898\u200b\u3002\u200b\u9996\u5148\u200b\u5bf9\u200b\u7cfb\u6570\u200b\u77e9\u9635\u200b\u5148\u200b\u51cf\u53bb\u200b\u6bcf\u884c\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\uff0c\u200b\u518d\u200b\u51cf\u53bb\u200b\u6bcf\u5217\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b\u3002\u200b\u4ece\u800c\u200b\u6bcf\u884c\u200b\u3001\u200b\u6bcf\u5217\u200b\u4e2d\u200b\u81f3\u5c11\u200b\u51fa\u73b0\u200b\u4e00\u4e2a\u96f6\u200b\u3002\u200b\u5b9a\u4e49\u200b\u201c\u200b\u72ec\u7acb\u200b\u96f6\u200b\u201d\u200b\u4e3a\u200b\u7cfb\u6570\u200b\u77e9\u9635\u200b\u4e2d\u200b\u65e2\u200b\u4e0d\u200b\u5904\u4e8e\u200b\u540c\u4e00\u200b\u884c\u200b\u3001\u200b\u4e5f\u200b\u4e0d\u200b\u5904\u4e8e\u200b\u540c\u4e00\u200b\u5217\u200b\u7684\u200b\\(0\\)\u200b\u7684\u200b\u6700\u5927\u200b\u4e2a\u6570\u200b\u3002</p> <ol> <li>\u200b\u82e5\u200b\u201c\u200b\u72ec\u7acb\u200b\u96f6\u200b\u201d\u200b\u7684\u200b\u4e2a\u6570\u200b\u7b49\u4e8e\u200b\u65b9\u9635\u200b\u7684\u200b\u7ef4\u5ea6\u200b\uff0c\u200b\u5219\u200b\u5f97\u5230\u200b\u6307\u6d3e\u200b\u95ee\u9898\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u200b\u7684\u200b\u6307\u6d3e\u200b\u65b9\u6848\u200b\u4e0e\u200b\u72ec\u7acb\u200b\u96f6\u200b\u7684\u200b\u4f4d\u7f6e\u200b\u5206\u5e03\u200b\u76f8\u540c\u200b\u3002</li> <li>\u200b\u82e5\u200b\u201c\u200b\u72ec\u7acb\u200b\u96f6\u200b\u201d\u200b\u7684\u200b\u4e2a\u6570\u200b\u5c0f\u4e8e\u200b\u65b9\u9635\u200b\u7684\u200b\u7ef4\u5ea6\u200b\u3002\u200b\u9996\u5148\u200b\u627e\u51fa\u200b\u65b9\u9635\u200b\u4e2d\u200b\u80fd\u591f\u200b\u8986\u76d6\u200b\u6240\u6709\u200b\u96f6\u200b\u5143\u7d20\u200b\u7684\u200b\u6700\u5c11\u200b\u76f4\u7ebf\u200b\u6570\u200b\u3002\u200b\u5728\u200b\u4e0d\u200b\u88ab\u200b\u76f4\u7ebf\u200b\u8986\u76d6\u200b\u7684\u200b\u5143\u7d20\u200b\u4e2d\u200b\u5bfb\u627e\u200b\u6700\u5c0f\u503c\u200b\uff0c\u200b\u5c06\u200b\u6240\u5728\u200b\u884c\u6216\u5217\u200b\u51cf\u53bb\u200b\u8fd9\u4e2a\u200b\u6700\u5c0f\u503c\u200b\u3002\u200b\u7136\u540e\u200b\u5bf9\u200b\u4ea7\u751f\u200b\u7684\u200b\u8d1f\u503c\u200b\u8fdb\u884c\u200b\u8c03\u6574\u200b\uff0c\u200b\u76f4\u5230\u200b\u77e9\u9635\u200b\u4e2d\u200b\u4e0d\u200b\u5b58\u5728\u200b\u8d1f\u503c\u200b\u3002\u200b\u7ed3\u675f\u200b\u540e\u200b\u7ee7\u7eed\u200b\u68c0\u67e5\u200b\u72ec\u7acb\u200b\u96f6\u200b\u7684\u200b\u4e2a\u6570\u200b\u3002</li> </ol>"},{"location":"math/operations-mamagement/","title":"\u9ad8\u7b49\u200b\u8fd0\u4f5c\u200b\u7ba1\u7406","text":"<ul> <li>\u200b\u9884\u6d4b\u200b\u6280\u672f\u200b</li> <li>\u200b\u9884\u6d4b\u200b\u4e0e\u200b\u9700\u6c42\u200b\u5efa\u6a21\u200b</li> <li>\u200b\u7efc\u5408\u200b\u751f\u4ea7\u200b\u8ba1\u5212\u200b</li> <li>\u200b\u786e\u5b9a\u200b\u9700\u6c42\u200b\u4e0b\u200b\u7684\u200b\u5e93\u5b58\u200b\u63a7\u5236\u200b</li> <li>\u200b\u968f\u673a\u200b\u9700\u6c42\u200b\u4e0b\u200b\u7684\u200b\u5e93\u5b58\u200b\u63a7\u5236\u200b</li> </ul>"},{"location":"math/operations-mamagement/#todo","title":"TODO","text":"<ul> <li> \u200b\u5468\u671f\u200b\u68c0\u67e5\u200b\u6a21\u578b\u200b\u2014\u2014\\((s, S)\\)\u200b\u7b56\u7565\u200b</li> <li> \u200b\u591a\u7ea7\u200b\u7cfb\u7edf\u200b\u4e0e\u200b\u534f\u8c03\u200b\u8ba2\u8d27\u200b</li> </ul>"},{"location":"math/operations-mamagement/aggregrate-planning/","title":"\u7efc\u5408\u200b\u751f\u4ea7\u200b\u8ba1\u5212","text":""},{"location":"math/operations-mamagement/aggregrate-planning/#_2","title":"\u6a21\u578b\u200b\u8bbe\u5b9a","text":"<p>\u200b\u6a21\u578b\u200b\u4e2d\u200b\u5305\u542b\u200b\u4e00\u4e2a\u200b\u4f01\u4e1a\u200b\uff0c\u200b\u4f01\u4e1a\u200b\u901a\u8fc7\u200b\u96c7\u4f63\u5de5\u4eba\u200b\u751f\u4ea7\u200b\u5e76\u200b\u9500\u552e\u200b\u4ea7\u54c1\u200b\u4f5c\u4e3a\u200b\u6536\u5165\u200b\u3002\u200b\u76f8\u5173\u200b\u7684\u200b\u53c2\u6570\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u6210\u672c\u200b\u53c2\u6570\u200b</li> <li>\u200b\u5e93\u5b58\u200b\u76f8\u5173\u200b\uff1a\u200b\u539f\u6750\u6599\u200b\u6210\u672c\u200b\u3001\u200b\u6301\u6709\u200b\u6210\u672c\u200b\u3001\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b</li> <li>\u200b\u4eba\u5458\u200b\u76f8\u5173\u200b\uff1a\u200b\u96c7\u4f63\u200b\u6210\u672c\u200b\u3001\u200b\u89e3\u96c7\u200b\u6210\u672c\u200b\u3001\u200b\u5de5\u65f6\u200b\u6210\u672c\u200b\u3001\u200b\u95f2\u7f6e\u200b\u6210\u672c\u200b\u4e0e\u200b\u52a0\u73ed\u200b\u6210\u672c\u200b</li> <li>\u200b\u751f\u4ea7\u529b\u200b\u76f8\u5173\u200b\uff1a\u200b\u8f6c\u5305\u200b\u6210\u672c\u200b</li> <li>\u200b\u6536\u76ca\u200b\u53c2\u6570\u200b\uff1a\u200b\u5355\u4f4d\u200b\u4ea7\u54c1\u200b\u552e\u4ef7\u200b</li> <li>\u200b\u751f\u4ea7\u200b\u53c2\u6570\u200b</li> <li>\u200b\u5355\u4f4d\u200b\u4ea7\u54c1\u200b\u751f\u4ea7\u200b\u5de5\u65f6\u200b</li> <li>\u200b\u65e5\u5e38\u200b\u5de5\u4f5c\u200b\u5de5\u65f6\u200b</li> <li>\u200b\u5e93\u5b58\u200b\u53c2\u6570\u200b</li> <li>\u200b\u671f\u521d\u200b\u5b58\u91cf\u200b\u4e0e\u200b\u671f\u672b\u200b\u5b58\u91cf\u200b</li> <li>\u200b\u9700\u6c42\u91cf\u200b</li> </ul> \u200b\u53d8\u91cf\u200b \u200b\u8bf4\u660e\u200b \u200b\u53d8\u91cf\u200b \u200b\u8bf4\u660e\u200b \\(c_F\\) \u200b\u89e3\u96c7\u200b\u6210\u672c\u200b \\(c_H\\) \u200b\u96c7\u4f63\u200b\u6210\u672c\u200b \\(c_I\\) \u200b\u6301\u6709\u200b\u6210\u672c\u200b \\(c_P\\) \u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b \\(c_R\\) \u200b\u65e5\u5e38\u200b\u751f\u4ea7\u6210\u672c\u200b \\(c_O\\) \u200b\u52a0\u73ed\u200b\u751f\u4ea7\u6210\u672c\u200b \\(c_S\\) \u200b\u8f6c\u5305\u200b\u6210\u672c\u200b \\(c_U\\) \u200b\u751f\u4ea7\u529b\u200b\u7684\u200b\u95f2\u7f6e\u200b\u6210\u672c\u200b \\(D_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u9700\u6c42\u200b \\(n_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u5de5\u200b\u65f6\u6570\u200b \\(F_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u89e3\u96c7\u200b\u4eba\u6570\u200b \\(H_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u96c7\u4f63\u200b\u4eba\u6570\u200b \\(K\\) \u200b\u5355\u4f4d\u200b\u5de5\u65f6\u200b\u4ea7\u91cf\u200b \\(W_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u5458\u5de5\u200b\u4eba\u6570\u200b \\(P_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u603b\u5de5\u200b\u65f6\u200b \\(O_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u603b\u200b\u52a0\u73ed\u200b\u5de5\u65f6\u200b \\(U_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u603b\u200b\u95f2\u7f6e\u200b\u5de5\u65f6\u200b \\(S_t\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u5916\u5305\u200b\u4ea7\u91cf\u200b \\(I_t^+\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u591a\u4f59\u200b\u5e93\u5b58\u200b \\(I_t^-\\) \\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u7f3a\u8d27\u200b\u6c34\u5e73\u200b <p>\u200b\u4e0a\u8868\u200b\u4e2d\u200b\\(I_t^\\pm, W_t\\)\u200b\u5206\u522b\u200b\u6307\u7b2c\u200b\\(t\\)\u200b\u4e2a\u200b\u5468\u671f\u200b\u7ed3\u675f\u200b\u65f6\u200b\u7684\u200b\u6570\u503c\u200b\u3002\u200b\u7279\u522b\u200b\u5730\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u521d\u59cb\u72b6\u6001\u200b\uff0c\u200b\u6709\u200b</p> <ul> <li>\\(I_0\\)\u200b\u4e3a\u200b\u521d\u59cb\u200b\u5e93\u5b58\u200b</li> <li>\\(W_0\\)\u200b\u4e3a\u200b\u521d\u59cb\u200b\u5458\u5de5\u200b\u6570\u200b</li> </ul>"},{"location":"math/operations-mamagement/aggregrate-planning/#_3","title":"\u7b56\u7565\u200b\u5206\u7c7b","text":"<p>\u200b\u6839\u636e\u200b\u4e0d\u540c\u200b\u7684\u200b\u6210\u672c\u200b\u7b56\u7565\u200b\u4e0e\u200b\u98ce\u9669\u200b\u504f\u597d\u200b\uff0c\u200b\u7efc\u5408\u200b\u8ba1\u5212\u200b\u7c7b\u578b\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u591a\u79cd\u4e0d\u540c\u200b\u7684\u200b\u8ba1\u5212\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5982\u200b\uff1a</p> <ol> <li>\u200b\u8ffd\u9010\u200b\u7b56\u7565\u200b\uff08\u200b\u6700\u5c0f\u5316\u200b\u5e93\u5b58\u200b\uff09</li> <li>\u200b\u5e73\u7a33\u200b\u7b56\u7565\u200b\uff08\u200b\u6700\u5c0f\u5316\u200b\u5458\u5de5\u200b\u7684\u200b\u53d8\u52a8\u200b\uff09</li> <li>\u200b\u5f39\u6027\u200b\u5de5\u4f5c\u200b\u65f6\u95f4\u200b\u7b56\u7565\u200b</li> <li>\u200b\u6df7\u5408\u7b56\u7565\u200b</li> </ol>"},{"location":"math/operations-mamagement/aggregrate-planning/#_4","title":"\u6c42\u89e3\u200b\uff1a\u200b\u7ebf\u6027\u89c4\u5212","text":"<p>\u200b\u5bf9\u4e8e\u200b\u5e93\u5b58\u200b\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u7ea6\u675f\u200b\uff1a</p> \\[ \\begin{aligned}     I_t &amp;= I_t^+ - I_t^- \\\\     I_t &amp;= I_{t-1} + P_t + S_t - D_t \\end{aligned} \\] <p>\u200b\u5bf9\u4e8e\u200b\u5458\u5de5\u200b\u7684\u200b\u96c7\u4f63\u200b\u4e0e\u200b\u89e3\u96c7\u200b\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u7ea6\u675f\u200b\uff1a</p> \\[ \\begin{aligned}     W_t &amp;= W_{t-1} + H_t - F_t \\\\     0 &amp;= H_t\\cdot F_t \\end{aligned} \\] <p>\u200b\u5bf9\u4e8e\u200b\u5458\u5de5\u200b\u7684\u200b\u751f\u4ea7\u200b\u5de5\u65f6\u200b\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u7ea6\u675f\u200b\uff1a</p> \\[ \\begin{aligned}     P_t &amp;= Kn_t W_t + O_t - U_t \\\\     0 &amp;= O_t \\cdot U_t \\end{aligned} \\] <p>\u200b\u7ebf\u6027\u89c4\u5212\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\u4e3a\u200b\u6700\u5c0f\u5316\u200b\u5404\u9879\u200b\u6210\u672c\u200b\u3002</p>"},{"location":"math/operations-mamagement/fdm/","title":"\u9884\u6d4b\u200b\u4e0e\u200b\u9700\u6c42\u200b\u5efa\u6a21","text":"<p>\u200b\u7ecf\u5178\u200b\u6a21\u578b\u200b\uff08\u200b\u5982\u200b\u6ed1\u52a8\u200b\u5e73\u5747\u200b\u3001\u200b\u6307\u6570\u200b\u5e73\u6ed1\u200b\u7b49\u200b\uff09\u200b\u7684\u200b\u7f3a\u9677\u200b\u5728\u4e8e\u200b\uff1a</p> <ul> <li>\u200b\u7ecf\u5178\u200b\u6a21\u578b\u200b\u6301\u7eed\u200b\u4f9d\u8d56\u4e8e\u200b\u5386\u53f2\u200b\u89c2\u6d4b\u200b\u6570\u636e\u200b</li> <li>\u200b\u90e8\u5206\u200b\u6a21\u578b\u200b\uff08\u200b\u5982\u200b\u6ed1\u52a8\u200b\u5e73\u5747\u200b\uff09\u200b\u4ec5\u200b\u9002\u7528\u200b\u4e8e\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b</li> <li>\u200b\u4ea7\u54c1\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u4e0d\u540c\u200b\uff0c\u200b\u5982\u679c\u200b\u4e00\u4e2a\u200b\u4ea7\u54c1\u200b\u7684\u200b\u751f\u547d\u5468\u671f\u200b\u660e\u663e\u200b\u8f83\u200b\u77ed\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u6ca1\u6709\u200b\u5145\u8db3\u200b\u7684\u200b\u5386\u53f2\u6570\u636e\u200b\u5bf9\u200b\u8be5\u200b\u4ea7\u54c1\u200b\u8fdb\u884c\u200b\u9884\u6d4b\u200b\u3002</li> </ul> <p>\u200b\u672c\u7ae0\u200b\u8ba8\u8bba\u200b\u5982\u4e0b\u200b\u6a21\u578b\u200b\uff0c\u200b\u8fd9\u4e9b\u200b\u6a21\u578b\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u9884\u6d4b\u200b\u751f\u547d\u5468\u671f\u200b\u8f83\u200b\u77ed\u200b\u7684\u200b\u4ea7\u54c1\u200b\u3002</p> <ul> <li>Bass\u200b\u6269\u6563\u200b\u6a21\u578b\u200b</li> <li>\u200b\u9886\u5148\u200b\u6307\u6807\u200b\u6a21\u578b\u200b</li> <li>\u200b\u9009\u62e9\u200b\u6a21\u578b\u200b</li> </ul>"},{"location":"math/operations-mamagement/fdm/#bass","title":"Bass\u200b\u6269\u6563\u200b\u6a21\u578b","text":"<p>Bass\u200b\u6a21\u578b\u200b\u63d0\u51fa\u200b\u4e86\u200b\u6269\u6563\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u6982\u5ff5\u200b\u3002\uff08\u200b\u67d0\u200b\u4e00\u7c7b\u200b\uff09\u200b\u65b0\u200b\u4ea7\u54c1\u200b\u9996\u6b21\u200b\u4e0a\u5e02\u200b\u540e\u200b\uff0c\u200b\u8be5\u200b\u4ea7\u54c1\u200b\u7684\u200b\u8bbe\u8ba1\u200b\u7406\u5ff5\u200b\u7b49\u200b\u5185\u5bb9\u200b\u4f1a\u200b\u5728\u200b\u5e02\u573a\u200b\u4e0a\u200b\u6269\u6563\u200b\uff08\u200b\u88ab\u200b\u6a21\u4eff\u200b\uff09\uff0c\u200b\u9020\u6210\u200b\u540c\u7c7b\u4ea7\u54c1\u200b\u7684\u200b\u4ea7\u91cf\u200b\u4e0a\u5347\u200b\u3002</p> <p>\u200b\u524d\u63d0\u200b\u5047\u8bbe\u200b\uff1a</p> <ul> <li>\u200b\u5e02\u573a\u200b\u4e0a\u200b\u7684\u200b\u7528\u6237\u6570\u91cf\u200b\u4e0d\u53d8\u200b</li> <li>\u200b\u7528\u6237\u200b\u53ef\u4ee5\u200b\u88ab\u200b\u7b80\u5355\u200b\u5206\u4e3a\u200b\u521b\u65b0\u8005\u200b\u4e0e\u200b\u6a21\u4eff\u8005\u200b</li> <li>\u200b\u521b\u65b0\u8005\u200b\u8d2d\u4e70\u200b\u4ea7\u54c1\u200b\u7684\u200b\u51b3\u7b56\u200b\u4e0d\u200b\u53d7\u5230\u200b\u5176\u4ed6\u200b\u4e2a\u4f53\u200b\u7684\u200b\u5f71\u54cd\u200b</li> <li>\u200b\u6a21\u4eff\u8005\u200b\u8d2d\u4e70\u200b\u4ea7\u54c1\u200b\u7684\u200b\u51b3\u7b56\u200b\u53d7\u5230\u200b\u5176\u4ed6\u200b\u4e2a\u4f53\u200b\u7684\u200b\u5f71\u54cd\u200b\uff0c\u200b\u8fd9\u79cd\u200b\u5f71\u54cd\u200b\u65b9\u5f0f\u200b\u901a\u8fc7\u200b\u4e2a\u4f53\u200b\u95f4\u200b\u7684\u200b\u4ea4\u6d41\u200b\u5b9e\u73b0\u200b</li> </ul> <p>\u200b\u5b9a\u4e49\u200b\uff1a\\(P(t)\\)\u200b\u4e3a\u200b\u5e02\u573a\u200b\u4e2d\u200b\u7684\u200b\u4e00\u4e2a\u200b\u987e\u5ba2\u200b\u5728\u200b\\(t\\)\u200b\u65f6\u523b\u200b\u4f5c\u51fa\u200b\u8d2d\u4e70\u51b3\u7b56\u200b\u7684\u200b\u6982\u7387\u200b</p> \\[ P(t) = p + \\frac{q}{m} D(t) \\] <ul> <li>\\(p\\)\u200b\u4e3a\u200b\u521b\u65b0\u200b\u7cfb\u6570\u200b</li> <li>\\(q\\)\u200b\u4e3a\u200b\u6a21\u4eff\u200b\u7cfb\u6570\u200b\uff0c\\(p\\ll q\\)</li> <li>\\(m\\)\u200b\u4e3a\u200b\u6d88\u8d39\u8005\u200b\u7684\u200b\u603b\u91cf\u200b\uff0c\u200b\u4e5f\u200b\u5373\u200b\u5e02\u573a\u200b\u89c4\u6a21\u200b\uff0c\u200b\u5e38\u6570\u200b</li> <li>\\(D(t)\\)\u200b\u4e3a\u200b\u65f6\u95f4\u200b\u4ece\u200b\\(0\\)\u200b\u5230\u200b\\(t\\)\u200b\u7684\u200b\u7d2f\u79ef\u200b\u8d2d\u4e70\u91cf\u200b</li> </ul> <p>\u200b\u5373\u200b\uff0c\u200b\u5e02\u573a\u200b\u4e0a\u200b\u7684\u200b\u7d2f\u8ba1\u200b\u9700\u6c42\u200b\\(D(t)\\)\u200b\u8d8a\u5927\u200b\uff0c\u200b\u6d88\u8d39\u8005\u200b\u8d8a\u200b\u5bb9\u6613\u200b\u4f5c\u51fa\u200b\u8d2d\u4e70\u51b3\u7b56\u200b\u3002</p> <p>\u200b\u4e0d\u59a8\u200b\u8bbe\u200b\\(d(t) = \\frac{\\mathrm dD}{\\mathrm dt}\\)\uff0c\u200b\u5373\u200b\\(t\\)\u200b\u65f6\u523b\u200b\u5e02\u573a\u200b\u4e0a\u200b\u8d2d\u4e70\u200b\u4eba\u6570\u200b\uff0c\u200b\u6839\u636e\u200b\\(P(t)\\)\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u6709\u200b\uff1a</p> \\[ P(t) = \\frac{d(t)}{m - D(t)} \\] <p>\u200b\u8054\u7acb\u200b\u4e8c\u200b\u5f0f\u200b\uff0c\u200b\u5f97\u5230\u200b\uff1a</p> \\[ \\begin{aligned} p + \\frac{q}{m} D(t) &amp;= \\frac{d(t)}{m - D(t)} \\\\ d(t) &amp;= (m-D(t))\\left(p + \\frac qmD(t)\\right) \\\\ d(t) &amp;= -\\frac qmD^2(t) + (q-p)D(t) + pm \\end{aligned} \\] <p>\u200b\u89e3\u200b\u5fae\u5206\u65b9\u7a0b\u200b\\(\\frac{\\mathrm dD}{\\mathrm dt} = -\\frac qmD^2 + (q-p)D + pm\\)\uff0c\u200b\u5f97\u200b\\(D(t)\\)\uff1a</p> \\[ \\boxed{\\begin{aligned}D(t) &amp;= m\\frac{1-e^{-(p+q)t}}{1+\\frac qp e^{-(p+q)t}} \\\\ d(t) &amp;= m\\frac{p(p+q)^2e^{-(p+q)t}}{\\left(p+qe^{-(p+q)t}\\right)^2}\\end{aligned}} \\] <p>\u200b\u5bf9\u200b\\(d(t)\\)\u200b\u8fdb\u4e00\u6b65\u200b\u6c42\u5bfc\u200b\uff0c\u200b\u5f97\u5f53\u200b\u4e14\u200b\u4ec5\u200b\u5f53\u200b\\(t = \\frac{1}{p+q}\\log \\frac{q}{p}\\)\u200b\u65f6\u200b\uff0c\\(d(t)\\)\u200b\u53d6\u5f97\u200b\u6700\u5927\u503c\u200b\u3002</p> <p>\u200b\u6b64\u65f6\u200b\uff1a</p> \\[ \\boxed{\\begin{aligned}d(t) &amp;= \\frac{m(p+q)^2}{4q} \\\\ D(t) &amp;= \\frac{m(q-p)}{2q}\\end{aligned}} \\] <p>\u200b\u7ed3\u679c\u200b\u8bf4\u660e\u200b\uff1a</p> <ul> <li>\u200b\u5f53\u200b\\(p\\)\u200b\u5f88\u200b\u5c0f\u65f6\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u9500\u552e\u91cf\u200b\u7684\u200b\u589e\u957f\u200b\u7f13\u6162\u200b</li> <li>\u200b\u5f53\u200b\\(p, q\\)\u200b\u5f88\u5927\u200b\u65f6\u200b\uff0c\u200b\u610f\u5473\u7740\u200b\u9500\u552e\u91cf\u200b\u7684\u200b\u589e\u957f\u200b\u4e0e\u200b\u4e0b\u8dcc\u200b\u5747\u200b\u8fc5\u901f\u200b</li> <li>\u200b\u4ec5\u5f53\u200b\\(q&gt;p\\)\u200b\u65f6\u200b\uff0c\u200b\u4e0a\u5f0f\u200b\u624d\u200b\u6709\u200b\u610f\u4e49\u200b\uff0c\u200b\u5426\u5219\u200b\u610f\u5473\u7740\u200b\u9500\u552e\u91cf\u200b\u6301\u7eed\u200b\u4e0b\u8dcc\u200b</li> </ul>"},{"location":"math/operations-mamagement/forecast/","title":"\u9884\u6d4b\u200b\u6280\u672f","text":"<p>\u200b\u5e38\u89c1\u200b\u7684\u200b\u5ba2\u89c2\u200b\u9884\u6d4b\u200b\u65b9\u6cd5\u200b\u6709\u200b\u56e0\u679c\u200b\u6a21\u578b\u200b\u4e0e\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u65b9\u6cd5\u200b\u3002\u200b\u8fd9\u4e9b\u200b\u65b9\u6cd5\u200b\u7684\u200b\u524d\u63d0\u200b\u5047\u8bbe\u200b\u662f\u200b\u8fc7\u53bb\u200b\u7684\u200b\u53d8\u91cf\u200b\u53ca\u5176\u200b\u76f8\u4e92\u4f5c\u7528\u200b\u5173\u7cfb\u200b\u5728\u200b\u672a\u6765\u200b\u5c06\u200b\u7ee7\u7eed\u200b\u5b58\u5728\u200b\u5e76\u200b\u53d1\u6325\u4f5c\u7528\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#_2","title":"\u56e0\u679c\u200b\u6a21\u578b","text":"<p>\u200b\u56e0\u679c\u200b\u6a21\u578b\u200b\u5373\u200b\u627e\u5230\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\uff0c\u200b\u4f7f\u7528\u200b\u4e0e\u200b\u9884\u6d4b\u200b\u53d8\u91cf\u200b\u76f8\u5173\u200b\u7684\u200b\u5176\u4ed6\u200b\u53d8\u91cf\u200b\u8fdb\u884c\u200b\u9884\u6d4b\u200b\u3002</p> \\[ y = f(X_1, \\dots, X_n) \\] <p>\u200b\u5176\u4e2d\u200b\\(X_1, \\dots, X_n\\)\u200b\u662f\u200b\u4e0e\u200b\\(y\\)\u200b\u76f8\u5173\u200b\u7684\u200b\u5176\u4ed6\u200b\u53d8\u91cf\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u7ebf\u6027\u200b\u56e0\u679c\u200b\u6a21\u578b\u200b\uff0c\u200b\u51fd\u6570\u200b\\(f\\)\u200b\u662f\u200b\u5173\u4e8e\u200b\\(X_1, \\dots, X_n\\)\u200b\u7684\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\uff0c\u200b\u5373\u200b\uff1a</p> <p>\\(f(X_1, \\dots, X_n) = \\alpha_0 + \\alpha_1X_1 + \\dots + \\alpha_nX_n\\)</p>"},{"location":"math/operations-mamagement/forecast/#_3","title":"\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u65b9\u6cd5","text":"<p>\u200b\u6b64\u5904\u200b\u7684\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u6307\u5f80\u671f\u200b\u5bf9\u200b\u67d0\u200b\u53d8\u91cf\u200b\u89c2\u6d4b\u200b\u503c\u200b\u7684\u200b\u96c6\u5408\u200b\u3002\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u65b9\u6cd5\u200b\u4ec5\u4ec5\u200b\u4f9d\u9760\u200b\u8fd9\u4e9b\u200b\u89c2\u6d4b\u200b\u503c\u200b\u5bf9\u200b\u8be5\u200b\u53d8\u91cf\u200b\u7684\u200b\u672a\u6765\u200b\u8fdb\u884c\u200b\u9884\u6d4b\u200b\u3002</p> <p>\u200b\u5728\u200b\u73b0\u5b9e\u200b\u7684\u200b\u751f\u4ea7\u200b\u8fd0\u4f5c\u200b\u4e2d\u200b\uff0c\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u901a\u5e38\u200b\u4f1a\u200b\u8868\u73b0\u200b\u51fa\u200b\u4ee5\u4e0b\u200b\u6a21\u5f0f\u200b\uff1a</p> <ul> <li>\u200b\u8d8b\u52bf\u200b\uff0c\u200b\u5373\u200b\u89c2\u6d4b\u200b\u503c\u200b\u5728\u200b\u6574\u4e2a\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u6240\u200b\u8868\u73b0\u200b\u51fa\u200b\u7684\u200b\u8d8b\u52bf\u200b</li> <li>\u200b\u5b63\u8282\u200b\uff0c\u200b\u5373\u200b\u89c2\u6d4b\u200b\u503c\u200b\u5728\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u7684\u200b\u4e00\u90e8\u5206\u200b\u6240\u200b\u8868\u73b0\u200b\u51fa\u200b\u7684\u200b\u8d8b\u52bf\u200b</li> <li>\u200b\u5468\u671f\u200b\uff0c\u200b\u5373\u200b\u89c2\u6d4b\u200b\u503c\u200b\u5728\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u4e2d\u200b\u91cd\u590d\u200b\u591a\u6b21\u200b\u51fa\u73b0\u200b\u7684\u200b\u8d8b\u52bf\u200b</li> <li>\u200b\u968f\u673a\u200b\uff0c\u200b\u6307\u200b\u89c2\u6d4b\u200b\u503c\u200b\u65e0\u89c4\u5219\u200b\u7684\u200b\u53d8\u52a8\u200b</li> </ul> <p>\u200b\u5b9e\u9645\u200b\u7684\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u53ef\u80fd\u200b\u540c\u65f6\u200b\u5305\u542b\u200b\u4ee5\u4e0a\u200b\u6240\u6709\u200b\u7279\u5f81\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#_4","title":"\u89c2\u6d4b\u200b\u4e0e\u200b\u9884\u6d4b","text":"<ol> <li>\\(\\{D_t\\}\\)\u200b\u4e3a\u200b\u89c2\u6d4b\u200b\u53d8\u91cf\u200b\u5728\u200b\\(1\\sim t\\)\u200b\u7684\u200b\u65f6\u95f4\u200b\u5185\u200b\u7684\u200b\u89c2\u6d4b\u200b\u503c\u200b\u7ec4\u6210\u200b\u7684\u200b\u5e8f\u5217\u200b\uff1b</li> <li>\\(F_{t-\\tau, t}\\)\u200b\u8868\u793a\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u5728\u200b\\(t-\\tau\\)\u200b\u5468\u671f\u200b\u5185\u200b\u5bf9\u200b\\(t\\)\u200b\u5468\u671f\u200b\u7684\u200b\u89c2\u6d4b\u200b\u503c\u200b\u4f5c\u51fa\u200b\u7684\u200b\u9884\u6d4b\u200b\u3002\u200b\u7279\u6b8a\u200b\u5730\u200b\uff0c\\(F_t=F_{t-1, t}\\)\uff0c\u200b\u5373\u200b\u4e3a\u200b\u63d0\u524d\u200b\u4e00\u4e2a\u200b\u5468\u671f\u200b\u7684\u200b\u9884\u6d4b\u200b\u3002</li> </ol> <p>\u200b\u5982\u4e0b\u200b\u6307\u6807\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u8861\u91cf\u200b\u9884\u6d4b\u200b\u6548\u679c\u200b\uff0c\u200b\u5b9a\u4e49\u200b\u504f\u5dee\u200b\\(e_t = D_t - F_t\\)</p> <ul> <li>\u200b\u5e73\u5747\u200b\u7edd\u5bf9\u200b\u504f\u5dee\u200b\uff08MAD\uff09\uff1a\\(\\frac{1}{n}\\sum_{i=1}^n|e_i|\\)</li> <li>\u200b\u5747\u65b9\u200b\u8bef\u5dee\u200b\uff08MSE\uff09\uff1a\\(\\frac{1}{n}\\sum_{i=1}^ne_i^2\\)</li> <li>\u200b\u5e73\u5747\u200b\u7edd\u5bf9\u200b\u767e\u5206\u200b\u8bef\u5dee\u200b\uff08MAPE\uff09\uff1a\\(\\frac{1}{n}\\sum_{i=1}^n|e_i/D_i|\\times 100\\%\\)</li> </ul>"},{"location":"math/operations-mamagement/forecast/#_5","title":"\u5e73\u7a33\u200b\u5e8f\u5217\u200b\u7684\u200b\u9884\u6d4b","text":"<p>\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b\u662f\u200b\u8054\u5408\u200b\u6982\u7387\u5206\u5e03\u200b\u4e0d\u968f\u200b\u65f6\u95f4\u200b\u6539\u53d8\u200b\uff0c\u200b\u5f62\u200b\u5982\u200b\\(D_t=\\mu + \\epsilon_t\\)\u200b\u7684\u200b\u5e8f\u5217\u200b\u3002</p> <p>\u200b\u5bf9\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5e38\u7528\u200b\u4e24\u79cd\u200b\u9884\u6d4b\u200b\u65b9\u6cd5\u200b\uff0c\u200b\u5373\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\u4e0e\u200b\u6307\u6570\u200b\u5e73\u6ed1\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#_6","title":"\u79fb\u52a8\u200b\u5e73\u5747","text":"<p>\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\uff0c\\(MA(N)\\)\uff1a</p> \\[ F_t = \\frac{1}{N}\\sum_{i=t-N}^{t-1}\\alpha_{i-t+N+1}D_i \\] <p>\u200b\u5f53\u200b\\(\\alpha_i=1\\)\u200b\u65f6\u200b\uff0c\u200b\u79f0\u8be5\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\u4e3a\u200b\u7b80\u5355\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\uff0c\u200b\u5426\u5219\u200b\u4e3a\u200b\u52a0\u6743\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u7b80\u5355\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\uff0c\u200b\u6839\u636e\u200b\u524d\u200b\u4e00\u200b\u5468\u671f\u200b\u7684\u200b\u9884\u6d4b\u503c\u200b\u548c\u200b\u65b0\u200b\u7684\u200b\u89c2\u6d4b\u200b\u503c\u200b\u53ef\u4ee5\u200b\u8ba1\u7b97\u200b\u4e0b\u200b\u4e00\u200b\u5468\u671f\u200b\u7684\u200b\u9884\u6d4b\u503c\u200b\uff1a</p> \\[ \\begin{aligned} F_{t+1} &amp;= \\frac{1}{N}\\sum_{i=t-N+1}^{t}D_i \\\\ &amp;= \\frac{1}{N}\\sum_{i=t-N}^{t-1} - D_{t-N} + D_t \\\\ &amp;= F_{t} + \\frac 1N \\left(D_t - D_{t-N}\\right) \\end{aligned} \\] <p>\u200b\u5bf9\u4e8e\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b\uff0c\u200b\u79fb\u52a8\u200b\u5e73\u5747\u200b\u7684\u200b\u9884\u6d4b\u200b\u901a\u5e38\u200b\u6ede\u540e\u200b\u4e8e\u200b\u8d8b\u52bf\u200b\u7684\u200b\u53d1\u5c55\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#_7","title":"\u6307\u6570\u200b\u5e73\u6ed1","text":"<p>\u200b\u5f53\u524d\u200b\u7684\u200b\u9884\u6d4b\u503c\u200b\u7b49\u4e8e\u200b\u4e0a\u200b\u4e00\u6b21\u200b\u9884\u6d4b\u503c\u200b\u4e0e\u200b\u5f53\u524d\u200b\u9700\u6c42\u200b\u5b9e\u9645\u200b\u503c\u200b\u7684\u200b\u52a0\u6743\u200b\u5e73\u5747\u200b\uff0c\u200b\u5373\u200b\uff1a</p> \\[ \\begin{aligned} F_t &amp;= \\alpha D_{t-1} + (1-\\alpha)F_{t-1} \\\\ &amp;= F_{t-1} + \\alpha e_{t-1} \\end{aligned} \\] <p>\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u7406\u89e3\u200b\u4e3a\u200b\u5bf9\u200b\u4e0a\u200b\u4e00\u6b21\u200b\u9884\u6d4b\u200b\u7684\u200b\u504f\u5dee\u200b\u4f5c\u51fa\u200b\u7cfb\u6570\u200b\u4e3a\u200b\\(\\alpha\\)\u200b\u7684\u200b\u4fee\u6b63\u200b\u3002</p> <p>\u200b\u5df2\u77e5\u200b\\(t=0\\)\u200b\u65f6\u200b\u5bf9\u200b\\(D_1\\)\u200b\u7684\u200b\u9884\u6d4b\u200b\u4e3a\u200b\\(F_1\\)\uff0c\u200b\u5219\u200b\uff1a</p> \\[ \\begin{aligned}     F_2 &amp;= \\alpha D_1 + (1-\\alpha) F_1 \\\\     F_3 &amp;= \\alpha D_2 + (1-\\alpha) F_2 =\\alpha D_2 + (1-\\alpha)\\alpha D_1 + (1-\\alpha) ^ 2 F_1 \\\\     \\vdots \\\\     F_n &amp;= \\sum_{i=1} ^{n-1} \\alpha(1-\\alpha)^{i-1}D_{n-i} + (1-\\alpha)^{n-1}F_1 \\end{aligned} \\]"},{"location":"math/operations-mamagement/forecast/#_8","title":"\u8d8b\u52bf\u200b\u5e8f\u5217","text":"<p>\u200b\u5bf9\u200b\u8d8b\u52bf\u200b\u5e8f\u5217\u200b\u7684\u200b\u9884\u6d4b\u200b\u901a\u5e38\u200b\u4f7f\u7528\u200b\u56de\u5f52\u200b\u5206\u6790\u200b\uff0c\u200b\u5373\u200b\u5047\u8bbe\u200b\u89c2\u6d4b\u200b\u53d8\u91cf\u200b\u4e0e\u200b\u9884\u6d4b\u200b\u53d8\u91cf\u200b\u95f4\u200b\u5b58\u5728\u200b\u51fd\u6570\u200b\u5173\u7cfb\u200b\uff0c\u200b\u5219\u200b\u53ef\u4ee5\u200b\u6839\u636e\u200b\u89c2\u6d4b\u200b\u53d8\u91cf\u200b\u548c\u200b\u9884\u6d4b\u200b\u53d8\u91cf\u200b\u7684\u200b\u89c2\u6d4b\u200b\u503c\u200b\u5bf9\u200b\u9884\u6d4b\u200b\u53d8\u91cf\u200b\u8fdb\u884c\u200b\u9884\u6d4b\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#_9","title":"\u7ebf\u6027\u200b\u56de\u5f52","text":"<p>\u200b\u5f53\u200b\u56de\u5f52\u200b\u4f7f\u7528\u200b\u5f62\u200b\u5982\u200b\\(\\hat Y=a+bX\\)\u200b\u7684\u200b\u7ebf\u6027\u200b\u51fd\u6570\u200b\u65f6\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u7ebf\u6027\u200b\u56de\u5f52\u200b\u3002\u200b\u7531\u6b64\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6700\u5c0f\u200b\u4e8c\u200b\u4e58\u6cd5\u200b\u8fdb\u884c\u200b\u62df\u5408\u200b\u3002</p> <p>\u200b\u6700\u5c0f\u200b\u4e8c\u200b\u4e58\u6cd5\u200b\u4f7f\u5f97\u200b\\(\\sum|Y-\\hat Y|^2\\)\u200b\u6700\u5c0f\u200b\uff0c\u200b\u5373\u200b\u9884\u6d4b\u200b\u70b9\u200b\u548c\u200b\u89c2\u6d4b\u70b9\u200b\u7684\u200b\u8ddd\u79bb\u200b\u5e73\u65b9\u548c\u200b\u6700\u5c0f\u200b\u3002\u200b\u5047\u8bbe\u200b\u89c2\u6d4b\u200b\u53d8\u91cf\u200b\u4e3a\u200b\\(X_1, \\dots, X_m\\)\uff0c\u200b\u9884\u6d4b\u200b\u53d8\u91cf\u200b\u4e3a\u200b\\(y\\)\uff0c\u200b\u5171\u200b\\(n\\)\u200b\u7ec4\u200b\u6570\u636e\u200b\u70b9\u200b\uff0c\u200b\u5219\u200b\u7ebf\u6027\u200b\u56de\u5f52\u200b\u6a21\u578b\u200b\u4e3a\u200b\uff1a</p> \\[ \\hat y = X\\vec a \\] <p>\u200b\u5176\u4e2d\u200b\\(\\vec a, , \\hat y\\)\u200b\u4e3a\u200b\\(m\\)\u200b\u7ef4\u5217\u200b\u5411\u91cf\u200b\uff0c\\(X\\)\u200b\u4e3a\u200b\\(n\\times m\\)\u200b\u77e9\u9635\u200b\u3002</p> <p>\u200b\u5219\u200b\u6b8b\u5dee\u200b\u5e73\u65b9\u548c\u200b\u5373\u200b\u4e3a\u200b\\(f(a; X, y) = (y - \\hat y)^T(y - \\hat y)=(y - Xa)^T(y - Xa)\\)\u3002</p> <p>\u200b\u5bf9\u200b\\(\\vec a, \\vec b\\)\u200b\u6c42\u5bfc\u200b\uff0c\u200b\u5f97\u200b\uff1a</p> \\[     \\frac{\\partial f}{\\partial \\vec a} = 0 \\] <p>\\(\\frac{\\partial f}{\\partial \\vec a} = 0\\)\u200b\u5373\u200b\uff1a</p> \\[ \\begin{aligned}     \\frac{\\partial f}{\\partial \\vec a} &amp;= 2X^T(X\\vec a - y) \\\\     &amp;= 2X^TX\\vec a + 2X^Ty \\\\     &amp;= \\vec 0 \\end{aligned} \\] <p>\u200b\u5f97\u200b\\(\\vec a = (X^TX)^{-1}X^Ty\\)</p>"},{"location":"math/operations-mamagement/forecast/#_10","title":"\u53cc\u91cd\u200b\u6307\u6570\u200b\u5e73\u6ed1","text":"<p>\u200b\u53cc\u91cd\u200b\u6307\u6570\u200b\u5e73\u6ed1\u200b\u53ef\u4ee5\u200b\u7528\u4e8e\u200b\u9884\u6d4b\u200b\u5177\u6709\u200b\u7ebf\u6027\u200b\u8d8b\u52bf\u200b\u7684\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5176\u4e2d\u200b\u4f7f\u7528\u200b\u4e24\u4e2a\u200b\u5e73\u6ed1\u200b\u53c2\u6570\u200b\u5206\u522b\u200b\u9884\u6d4b\u200b\u5e8f\u5217\u200b\u503c\u200b\\(S_t\\)\u200b\u548c\u200b\u8d8b\u52bf\u200b\u503c\u200b\\(G_t\\)</p> <ul> <li>\u200b\u5e8f\u5217\u200b\u503c\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u66f2\u7ebf\u200b\u7684\u200b\u622a\u8ddd\u200b   $$   S_t = \\alpha D_t + (1 - \\alpha)(S_{t-1} + G_{t-1})   $$</li> <li>\u200b\u8d8b\u52bf\u200b\u503c\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\u66f2\u7ebf\u200b\u7684\u200b\u659c\u7387\u200b   $$   G_t = \\beta(S_t - S_{t-1})+(1-\\beta)G_{t-1}   $$</li> <li>\u200b\u5728\u200b\\(t\\)\u200b\u65f6\u523b\u200b\u63d0\u524d\u200b\\(\\tau\\)\u200b\u6b65\u200b\u4f5c\u51fa\u200b\u7684\u200b\u9884\u6d4b\u200b\\(F_{t, t+\\tau} = S_t+\\tau G_t\\)</li> </ul>"},{"location":"math/operations-mamagement/forecast/#_11","title":"\u5b63\u8282\u6027\u200b\u5e8f\u5217","text":"<p>\u200b\u5b63\u8282\u6027\u200b\u5e8f\u5217\u200b\u662f\u200b\u6bcf\u200b\\(N\\)\u200b\u4e2a\u200b\u5468\u671f\u200b\u91cd\u590d\u200b\u4e00\u6b21\u200b\u7684\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5176\u4e2d\u200b\\(N\\)\u200b\u88ab\u200b\u79f0\u4e3a\u200b\u5b63\u200b\u7684\u200b\u957f\u5ea6\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u5b63\u8282\u200b\u4e2d\u200b\u7684\u200b\u6bcf\u4e2a\u200b\u5468\u671f\u200b\uff0c\u200b\u90fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\\(c_t\\)\uff0c\u200b\u6307\u200b\u5e73\u5747\u200b\u610f\u4e49\u200b\u4e0b\u200b\u7b2c\u200b\\(t\\)\u200b\u4e2a\u200b\u5468\u671f\u200b\u7684\u200b\u9700\u6c42\u200b\u4e0e\u200b\u6574\u4f53\u200b\u5e73\u5747\u200b\u9700\u6c42\u200b\u7684\u200b\u6bd4\u503c\u200b\uff0c\u200b\u6709\u200b\\(\\sum c_t = N\\)</p> <ul> <li>\u200b\u8ba1\u7b97\u200b\u6240\u6709\u200b\u6570\u636e\u200b\u7684\u200b\u6837\u672c\u5747\u503c\u200b</li> <li>\u200b\u5c06\u200b\u6bcf\u4e2a\u200b\u89c2\u6d4b\u200b\u503c\u200b\u9664\u4ee5\u200b\u6837\u672c\u5747\u503c\u200b\uff0c\u200b\u5373\u200b\u4e3a\u200b\u6bcf\u4e2a\u200b\u89c2\u6d4b\u200b\u5468\u671f\u200b\u7684\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b</li> <li>\u200b\u5c06\u200b\u6bcf\u5b63\u200b\u4e2d\u200b\u76f8\u540c\u200b\u5468\u671f\u200b\u7684\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\u53d6\u200b\u5747\u503c\u200b\uff0c\u200b\u5f97\u5230\u200b\\(N\\)\u200b\u4e2a\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b</li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u540c\u65f6\u200b\u5177\u6709\u200b\u5b63\u8282\u6027\u200b\u6ce2\u52a8\u200b\u548c\u200b\u8d8b\u52bf\u200b\u7684\u200b\u5b63\u8282\u6027\u200b\u5e8f\u5217\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u9884\u6d4b\u200b\uff1a</p> <ul> <li>\u200b\u8ba1\u7b97\u200b\u5e8f\u5217\u200b\u7684\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b</li> <li>\u200b\u5c06\u200b\u89c2\u6d4b\u200b\u503c\u200b\u9664\u4ee5\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\uff0c\u200b\u5f97\u5230\u200b\u6d88\u9664\u200b\u5b63\u8282\u6027\u200b\u6ce2\u52a8\u200b\u7684\u200b\u5e8f\u5217\u200b</li> <li>\u200b\u7528\u200b\u8d8b\u52bf\u200b\u5e8f\u5217\u200b\u7684\u200b\u9884\u6d4b\u200b\u65b9\u6cd5\u200b\u8fdb\u884c\u200b\u9884\u6d4b\u200b</li> <li>\u200b\u9884\u6d4b\u200b\u7ed3\u679c\u200b\u4e58\u4ee5\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\uff0c\u200b\u5f97\u5230\u200b\u6700\u7ec8\u200b\u7684\u200b\u9884\u6d4b\u503c\u200b</li> </ul> <p>\u200b\u7f3a\u70b9\u200b\uff1a\u200b\u5728\u200b\u83b7\u5f97\u200b\u65b0\u200b\u7684\u200b\u89c2\u6d4b\u200b\u503c\u540e\u200b\uff0c\u200b\u9700\u8981\u200b\u91cd\u65b0\u200b\u8ba1\u7b97\u200b\u6240\u6709\u200b\u7684\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#_12","title":"\u4e09\u91cd\u200b\u6307\u6570\u200b\u5e73\u6ed1","text":"<p>\u200b\u5047\u8bbe\u200b\u9700\u6c42\u200b\u4e3a\u200b\\(D_t = (\\mu +G_t)c_t + \\epsilon_t\\)\uff08\u200b\u4e58\u79ef\u200b\u578b\u200b\uff09\u3002\u200b\u8be5\u200b\u5f62\u5f0f\u200b\u4f7f\u7528\u200b\u4e09\u4e2a\u200b\u6307\u6570\u200b\u5e73\u6ed1\u200b\u5bf9\u200b\u5e8f\u5217\u200b\uff08\u200b\u622a\u8ddd\u200b\uff09\u3001\u200b\u8d8b\u52bf\u200b\uff08\u200b\u659c\u7387\u200b\uff09\u200b\u548c\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\uff08\u200b\u5468\u671f\u200b\uff09\u200b\u5206\u522b\u200b\u8fdb\u884c\u200b\u62df\u5408\u200b\u3002</p> <ul> <li>\u200b\u5e8f\u5217\u200b\uff1a\\(S_t = \\alpha \\frac{D_t}{c_{t-N}} + (1-\\alpha) (S_{t-1} + G_{t-1})\\)\uff0c\\(c_{t-N}\\)\u200b\u7528\u4e8e\u200b\u6d88\u9664\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\u7684\u200b\u5f71\u54cd\u200b\u3002</li> <li>\u200b\u8d8b\u52bf\u200b\uff1a\\(G_t = \\beta(S_t - S_{t-1}) + (1-\\beta) G_{t-1}\\)</li> <li>\u200b\u5b63\u8282\u200b\u56e0\u5b50\u200b\uff1a\\(c_t = \\gamma\\frac{D_t}{S_t} + (1-\\gamma) c_{t-N}\\)</li> </ul>"},{"location":"math/operations-mamagement/forecast/#box-jenkinsarima","title":"Box-Jenkins\u200b\u6a21\u578b\u200b\uff1aARIMA","text":"<p>ARIMA\u200b\u6a21\u578b\u200b\u662f\u200b\u5229\u7528\u200b\u65f6\u95f4\u200b\u5e8f\u5217\u200b\u7684\u200b\u81ea\u200b\u76f8\u5173\u200b\u7ed3\u6784\u200b\uff1a</p> <p>Auto Regressive Integrated Moving Average</p> <p>\u200b\u6784\u5efa\u200b\u6b65\u9aa4\u200b\uff1a</p> <ol> <li>\u200b\u6570\u636e\u200b\u8f6c\u6362\u200b</li> <li>\u200b\u6a21\u578b\u200b\u8bc6\u522b\u200b</li> <li>\u200b\u53c2\u6570\u4f30\u8ba1\u200b</li> <li>\u200b\u9884\u6d4b\u200b</li> <li>\u200b\u8bc4\u4ef7\u200b</li> </ol>"},{"location":"math/operations-mamagement/forecast/#arp","title":"\u81ea\u200b\u56de\u5f52\u200b\u8fc7\u7a0b\u200b\uff1aAR(p)","text":"<p>\u200b\u5bf9\u4e8e\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b\uff1a</p> <ol> <li>\u200b\u95f4\u9694\u200b\u76f8\u540c\u200b\u7684\u200b\u4efb\u610f\u200b\u4e00\u7ec4\u200b\u9700\u6c42\u200b\u7684\u200b\u8054\u5408\u200b\u8fb9\u9645\u200b\u5206\u5e03\u200b\u76f8\u540c\u200b-&gt;\u200b\u534f\u65b9\u5dee\u200b\u76f8\u540c\u200b</li> <li>\u200b\u4efb\u610f\u200b\u4e24\u4e2a\u200b\u89c2\u6d4b\u200b\u7684\u200b\u534f\u65b9\u5dee\u200b\u4ec5\u200b\u53d6\u51b3\u4e8e\u200b\u76f8\u9694\u200b\u7684\u200b\u5468\u671f\u200b\u6570\u200b</li> </ol> \\[ \\mathrm{Cov}(D_{t+m}, D_{t+m+k}) = E(D_{t+m}D_{t+m+k}) - E(D_{t+m})E(D_{t+m+k}) \\] <p>\u200b\u76f8\u5173\u7cfb\u6570\u200b\uff1a\u200b\u6ede\u540e\u200b\\(k\\)\u200b\u671f\u200b\u7684\u200b\u81ea\u200b\u76f8\u5173\u7cfb\u6570\u200b</p> \\[ \\rho_k = \\frac{\\mathrm {Cov}(D_{t+m}, D_{t+m+k})}{\\sqrt{\\mathrm {Var}(D_{t+m})}\\sqrt{\\mathrm {Var}(D_{t+m+k})}} \\] <p>\u200b\u6837\u672c\u200b\u7684\u200b\u81ea\u200b\u76f8\u5173\u200b\u51fd\u6570\u200b\uff1a</p> \\[ r_k = \\frac{\\sum_{t=k+1} ^ n(D_t - \\overline D)(D_{t-k} - \\overline D)}{\\sum_{t=1} ^ n(D_t - \\overline D)^2} \\] \\[ D_t = a_0 + a_1D_{t-1} + \\dots + a_pD_{t-p} + \\epsilon_t \\] <p>\\(p\\)\u200b\u4e3a\u200b\u81ea\u200b\u56de\u5f52\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u9636\u6570\u200b\uff0c\u200b\u4e00\u9636\u81ea\u200b\u56de\u5f52\u200b\u6a21\u578b\u200b\\(AR(1)\\)\uff1a</p> \\[ D_t = a_0 + a_1 D_{t-1} + \\epsilon_t \\] <p>\u200b\u81ea\u200b\u76f8\u5173\u200b\u51fd\u6570\u200b\u4e3a\u200b\uff1a\\(\\rho_j = a_1^j\\)</p> <p>\u200b\u5c06\u200b\u6837\u672c\u200b\u7684\u200b\u81ea\u200b\u76f8\u5173\u200b\u51fd\u6570\u200b\u4e0e\u200b\u7406\u8bba\u200b\u7684\u200b\u81ea\u200b\u76f8\u5173\u200b\u51fd\u6570\u200b\u8fdb\u884c\u200b\u6bd4\u8f83\u200b\uff0c\u200b\u5224\u65ad\u200bARIMA\u200b\u7684\u200b\u62df\u5408\u200b\u6548\u679c\u200b\u3002</p>"},{"location":"math/operations-mamagement/forecast/#maq","title":"\u79fb\u52a8\u200b\u5e73\u5747\u200b\u8fc7\u7a0b\u200b\uff1aMA(q)","text":"\\[ D_t = b_0 - b_1\\epsilon_{t-1} -\\dots - b_q\\epsilon{t-q} +\\epsilon_t \\] <p>\\(AR(1)\\)\u200b\u8fc7\u7a0b\u200b\u672c\u8d28\u200b\u4e0a\u200b\u662f\u200b\\(MA(\\infty)\\)\u200b\u8fc7\u7a0b\u200b\u3002</p> <p>\\(MA(1)\\)\u200b\u8fc7\u7a0b\u200b\uff1a\\(D_t = b_0 - b_1\\epsilon_{t-1} + \\epsilon_t\\)</p> <p>\u200b\u5bf9\u5e94\u200b\u4e8e\u200b\\(AR(1)\\)\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u81ea\u200b\u76f8\u5173\u200b\u51fd\u6570\u200b\uff1a</p> \\[ \\left\\{\\begin{aligned}   \\rho_1 &amp;= \\frac{-b_1}{1+b_1^2} \\\\   \\rho_2 &amp;= \\rho_3 = \\dots = 0 \\end{aligned}\\right. \\]"},{"location":"math/operations-mamagement/forecast/#arma","title":"ARMA\u200b\u6a21\u578b","text":"<p>\\(ARMA(p, q)\\)\u200b\u6a21\u578b\u200b\u662f\u200b\\(AR(p)\\)\u200b\u6a21\u578b\u200b\u4e0e\u200b\\(MA(q)\\)\u200b\u6a21\u578b\u200b\u7684\u200b\u7ed3\u5408\u200b\uff0c\u200b\u5982\u200b\\(ARMA(1, 1)\\)\uff1a</p> \\[ D_t = c+a_1D_{t-1} - b_1\\epsilon_{t-1} + \\epsilon_t \\] <p>\u200b\u5982\u679c\u200b\u5e8f\u5217\u200b\u4e0d\u662f\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b\uff0c\u200b\u800c\u200b\u5b58\u5728\u200b\u8d8b\u52bf\u6027\u200b\u548c\u200b/\u200b\u6216\u200b\u5b63\u8282\u6027\u200b\uff0c\u200b\u9700\u8981\u200b\u5c06\u200b\u975e\u200b\u5e73\u7a33\u200b\u8fc7\u7a0b\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u5e73\u7a33\u200b\u8fc7\u7a0b\u200b\u3002</p> <ul> <li>\u200b\u5f53\u200b\\(D_t\\)\u200b\u5b58\u5728\u200b\u7ebf\u6027\u200b\u8d8b\u52bf\u200b\u65f6\u200b\uff0c\u200b\u4f7f\u7528\u200b\u5dee\u5206\u200b\u65b9\u6cd5\u200b\u6d88\u9664\u200b\u8d8b\u52bf\u6027\u200b\uff1a\\(U_t = D_t - D_{t-1}\\)\uff0c\\(U_t\\)\u200b\u662f\u200b\u5e73\u7a33\u200b\u5e8f\u5217\u200b\u3002</li> <li>\u200b\u5f53\u200b\\(D_t\\)\u200b\u5b58\u5728\u200b\u5b63\u8282\u6027\u200b\u8d8b\u52bf\u200b\u65f6\u200b\uff0c\u200b\u4f7f\u7528\u200b\u5dee\u5206\u200b\u65b9\u6cd5\u200b\u6d88\u9664\u200b\u8d8b\u52bf\u6027\u200b\uff1a\\(U_t = D_t - D_{t-N}\\)</li> <li>\u200b\u5bf9\u200b\\(U_t\\)\u200b\u8fdb\u884c\u200b\u52a0\u200b\u548c\u200b\uff0c\u200b\u5f97\u5230\u200b\u521d\u59cb\u200b\u7684\u200b\\(D_t\\)</li> </ul>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/","title":"\u786e\u5b9a\u200b\u9700\u6c42\u200b\u4e0b\u200b\u7684\u200b\u5e93\u5b58\u200b\u63a7\u5236","text":"<p>\u200b\u5728\u200b\u6301\u6709\u200b\u6210\u672c\u200b\u4e0e\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u4e4b\u95f4\u200b\u5e73\u8861\u200b\u2014\u2014\u200b\u5355\u6b21\u200b\u8ba2\u8d27\u91cf\u200b\\(\\bsQ\\)</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#eoq","title":"EOQ\u200b\u6a21\u578b","text":""},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_2","title":"\u8bb0\u53f7","text":"<ul> <li>\u200b\u9700\u6c42\u200b\u7387\u200b\\(\\lambda\\)\uff0c\u200b\u5468\u671f\u200b\u957f\u5ea6\u200b\\(T\\)</li> <li>\u200b\u6210\u672c\u200b\uff1a\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(K\\)\u3001\u200b\u8fb9\u9645\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(c\\)\u3001\u200b\u8fb9\u9645\u200b\u6301\u6709\u200b\u6210\u672c\u200b\\(h\\)</li> </ul> <p>\u200b\u8fb9\u9645\u200b\u6301\u6709\u200b\u6210\u672c\u200b\\(h\\)\u200b\u901a\u5e38\u200b\u4ee5\u200b\u5229\u7387\u200b\\(I\\)\u200b\u7684\u200b\u5f62\u5f0f\u200b\u8868\u793a\u200b\uff0c\u200b\u6b64\u65f6\u200b\u6709\u200b\\(h = Ic\\)</p> <ul> <li>\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u200b\\(\\tau = 0\\)</li> <li>\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u7f3a\u8d27\u200b</li> </ul>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_3","title":"\u63a8\u5bfc","text":"<p>\u200b\u8003\u8651\u200b\u4e00\u4e2a\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\uff0c\u200b\u7531\u4e8e\u200b\u4e0d\u200b\u8003\u8651\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u91cf\u200b\uff0c\u200b\u8ba2\u8d27\u200b\u540e\u200b\u4ea7\u54c1\u200b\u4f1a\u200b\u7acb\u5373\u200b\u5230\u8fbe\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e3a\u200b\u964d\u4f4e\u200b\u6301\u6709\u200b\u6210\u672c\u200b\uff0c\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u200b\u65b9\u5f0f\u200b\u4e3a\u200b\u5e93\u5b58\u200b\u4e3a\u200b\u96f6\u65f6\u200b\u91cd\u65b0\u200b\u8ba2\u8d27\u200b\u3002\u200b\u56e0\u6b64\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\u6ee1\u8db3\u200b\\(Q = \\lambda T\\)\u3002\u200b\u4e00\u4e2a\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\u4e2d\u200b\u7684\u200b\u6210\u672c\u200b\u4e3a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u548c\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u4e4b\u200b\u548c\u200b\uff0c\u200b\u5373\u200b\\(C = K + cQ + hQT/2\\)\u3002</p> <p></p> <p>\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u7684\u200b\u6210\u672c\u200b</p> \\[ C(Q) = \\frac{C}{T} = \\frac{K\\lambda}{Q} + \\lambda c + \\frac{hQ}{2} \\] <p>\u200b\u5bf9\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u6210\u672c\u200b\u6c42\u5bfc\u200b\uff0c\u200b\u5f97\u5230\u200b</p> \\[ C'(Q) = \\frac{h}{2} - \\frac{K\\lambda}{Q^2} \\] <p>\u200b\u5f53\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u6210\u672c\u200b\u6700\u200b\u5c0f\u65f6\u200b\uff0c\\(C'(Q^\\ast) = 0\\)\uff0c\u200b\u89e3\u5f97\u200b</p> \\[ Q^\\ast = \\sqrt{\\frac{2K\\lambda}{h}} \\] <p>\u200b\u4e0e\u200b\u6b64\u5bf9\u5e94\u200b\u7684\u200b\u6700\u5c0f\u200b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ C(Q^\\ast) = \\frac{K\\lambda}{Q^\\ast} + \\lambda c + \\frac{hQ^\\ast}{2} = \\lambda c + \\sqrt{2K\\lambda h} \\]"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_4","title":"\u6a21\u578b\u200b\u6269\u5145","text":""},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_5","title":"\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f","text":"<p>\u200b\u8bbe\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u4e3a\u200b\\(\\tau &gt; 0\\)\uff0c\u200b\u9996\u5148\u200b\u8003\u8651\u200b\\(\\tau \\leq T\\)\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u7531\u4e8e\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u7f3a\u8d27\u200b\uff0c\u200b\u5728\u200b\u5e93\u5b58\u200b\u63d0\u524d\u200b\u671f\u5185\u200b\u7684\u200b\u6700\u4f4e\u200b\u5e93\u5b58\u200b\u5e94\u5f53\u200b\u4e3a\u200b\u96f6\u200b\uff0c\u200b\u5373\u200b\\(R = \\lambda\\tau\\)\u200b\u65f6\u200b\u5e94\u5f53\u200b\u91cd\u65b0\u200b\u8ba2\u8d27\u200b\uff0c\\(R\\)\u200b\u79f0\u4e3a\u200b\u91cd\u200b\u8ba2\u8d27\u200b\u70b9\u200b\u3002</p> <p>\u200b\u5f53\u200b\\(\\tau &gt; T\\)\u200b\u65f6\u200b\uff0c\u200b\u7531\u4e8e\u200b\u6a21\u578b\u200b\u9762\u5bf9\u200b\u5468\u671f\u6027\u200b\u7684\u200b\u9700\u6c42\u200b\uff0c\u200b\u53ea\u200b\u9700\u200b\u6839\u636e\u200b\u63d0\u524d\u200b\u671f\u200b\\(\\tau' = \\tau\\mod T\\)\u200b\u8ba1\u7b97\u200b\u91cd\u200b\u8ba2\u8d27\u200b\u70b9\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_6","title":"\u654f\u611f\u6027\u200b\u5206\u6790","text":"<p>\u200b\u654f\u611f\u6027\u200b\u5206\u6790\u200b\u6307\u200b\\(Q^*\\)\u200b\u7684\u200b\u4f30\u8ba1\u200b\u504f\u5dee\u200b\u5bf9\u200b\u603b\u6210\u672c\u200b\u7684\u200b\u5f71\u54cd\u200b\u3002\u200b\u504f\u5dee\u200b\u4ee5\u200b\\(C(Q) / C(Q^\\ast)\\)\u200b\u8861\u91cf\u200b\uff0c\u200b\u4e3a\u200b\u4fbf\u4e8e\u200b\u5206\u6790\u200b\uff0c\u200b\u8bbe\u200b\u8fb9\u9645\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(c = 0\\)\uff0c\u200b\u5219\u200b\u603b\u6210\u672c\u200b\u7684\u200b\u504f\u5dee\u200b\u91cf\u200b\\(C(Q) / C(Q^\\ast)\\)\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned}     \\frac{C(Q)}{C(Q^\\ast)} &amp;= \\frac{\\frac{K\\lambda}{Q}  + \\frac{hQ}{2}}{\\sqrt{2K\\lambda h}} \\\\     &amp;= \\sqrt{\\frac{K\\lambda}{2h}}\\frac{1}{Q} + \\sqrt{\\frac{h}{2K\\lambda}}Q \\\\     &amp;= \\frac{1}{2}\\left(\\frac{Q}{Q^\\ast} + \\frac{Q^\\ast}{Q}\\right) \\end{aligned} \\]"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_7","title":"\u5141\u8bb8\u200b\u7f3a\u8d27","text":"<p>\u200b\u5f53\u200b\u5141\u8bb8\u200b\u7f3a\u8d27\u200b\u65f6\u200b\uff0c\u200b\u8bbe\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\u4e3a\u200b\\(p\\)\uff0c\u200b\u6700\u5927\u200b\u7f3a\u8d27\u91cf\u200b\\(xQ\\)\uff0c\u200b\u4e00\u4e2a\u200b\u5468\u671f\u200b\u5185\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u53d8\u5316\u200b\u5982\u4e0b\u200b\u56fe\u200b\u6240\u793a\u200b\uff1a</p> <p></p> <p>\u200b\u6210\u672c\u200b\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ C(Q, x) = \\frac{C}{T} = \\frac{(1-x)^2 Qh + x^2Qp}{2} + \\frac{K\\lambda}{Q} + \\lambda c \\] <p>\u200b\u5206\u522b\u200b\u6c42\u200b\\(C\\)\u200b\u5173\u4e8e\u200b\\(Q, x\\)\u200b\u7684\u200b\u504f\u200b\u5bfc\u6570\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\begin{aligned}   \\frac{\\partial C}{\\partial Q} &amp;= \\frac{(1-x)^2h + x^2p}{2} - \\frac{K\\lambda}{Q^2} \\\\   \\frac{\\partial C}{\\partial x} &amp;= xpQ + (x-1)hQ \\end{aligned} \\] <p>\u200b\u89e3\u65b9\u7a0b\u200b\\(\\partial C/\\partial x =0\\)\u200b\u5f97\u5230\u200b\\(x^* = h/(h+p)\\)\uff0c\u200b\u4ee3\u5165\u200b\u65b9\u7a0b\u200b\\(\\partial C/\\partial Q =0\\)\uff0c\u200b\u89e3\u5f97\u200b\uff1a</p> \\[ Q^* = \\sqrt{\\frac{2K\\lambda}{(1-x)^2h+x^2p}} = \\sqrt{\\frac{2K\\lambda(h+p)}{hp}} \\] <p>\u200b\u82e5\u200b\u5b9a\u4e49\u200b\\(h' = hp/(h+p)\\)\u3002\u200b\u5219\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bEOQ\u200b\u7684\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u5373\u200b\\(\\sqrt{\\frac{2K\\lambda}{h'}}\\)</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_8","title":"\u6709\u9650\u200b\u751f\u4ea7\u7387\u200b\u6a21\u578b","text":"<p>\u200b\u5728\u200b\u6709\u9650\u200b\u751f\u4ea7\u7387\u200b\u6a21\u578b\u200b\u4e2d\u200b\uff0c\u200b\u4ea7\u54c1\u200b\u901a\u8fc7\u200b\u751f\u4ea7\u200b\u800c\u200b\u4e0d\u662f\u200b\u8ba2\u8d2d\u200b\u83b7\u5f97\u200b\u3002\u200b\u4ea7\u54c1\u200b\u7684\u200b\u751f\u4ea7\u7387\u200b\u4e3a\u200b\\(P &gt; \\lambda\\)\uff0c\u200b\u751f\u4ea7\u7ebf\u200b\u7684\u200b\u542f\u52a8\u200b\u6210\u672c\u200b\u4e3a\u200b\\(K\\)\uff0c\u200b\u4ea7\u54c1\u200b\u7684\u200b\u5355\u4f4d\u200b\u751f\u4ea7\u6210\u672c\u200b\u4e3a\u200b\\(c\\)\u3002</p> <p>\u200b\u751f\u4ea7\u6210\u672c\u200b\uff08\u200b\u539f\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\uff09\u200b\u90e8\u5206\u200b\u7684\u200b\u5904\u7406\u200b\u4e0e\u200bEOQ\u200b\u6a21\u578b\u200b\u76f8\u540c\u200b\u3002\u200b\u4f46\u200b\u6709\u9650\u200b\u751f\u4ea7\u7387\u200b\u6a21\u578b\u200b\u7684\u200b\u5e73\u5747\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u4e0e\u200bEOQ\u200b\u6a21\u578b\u200b\u4e0d\u540c\u200b\u3002</p> <ul> <li>\u200b\u4e00\u4e2a\u200b\u5468\u671f\u200b\u4e2d\u200bEOQ\u200b\u6a21\u578b\u200b\u7684\u200b\u5e73\u5747\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\uff1a\\(Q/2\\)</li> <li>\u200b\u4e00\u4e2a\u200b\u5468\u671f\u200b\u4e2d\u200b\u6709\u9650\u200b\u751f\u4ea7\u7387\u200b\u6a21\u578b\u200b\u7684\u200b\u5e73\u5747\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\uff1a\\(Q(P-\\lambda)/2P\\)</li> </ul> <p></p> <p>\u200b\u56e0\u6b64\u200b\u6709\u9650\u200b\u751f\u4ea7\u7387\u200b\u6a21\u578b\u200b\u7684\u200b\u6210\u672c\u200b\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ C(Q) = \\frac{C}{T} = \\frac{K\\lambda}{Q} + \\frac{hQ(P-\\lambda)}{2P} + \\lambda c \\] <p>\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\u4e3a\u200b</p> \\[ Q^\\ast = \\sqrt{\\frac{2PK\\lambda}{h(P-\\lambda)}} \\] <p>\u200b\u82e5\u200b\u5b9a\u4e49\u200b\\(h' = (P-\\lambda) / P\\)\uff0c\u200b\u5219\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u7c7b\u4f3c\u200b\u4e8e\u200bEOQ\u200b\u7684\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u5373\u200b\\(\\sqrt{\\frac{2K\\lambda}{h'}}\\)</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_9","title":"\u6570\u91cf\u200b\u6298\u6263\u200b\u6a21\u578b","text":"<p>\u200b\u5728\u200b\u6570\u91cf\u200b\u6298\u6263\u200b\u6a21\u578b\u200b\u4e2d\u200b\uff0c\u200b\u5f53\u200b\u6bcf\u4e2a\u200b\u5468\u671f\u200b\u7684\u200b\u8ba2\u8d27\u91cf\u200b\u8fbe\u5230\u200b\u4e00\u5b9a\u200b\u6c34\u5e73\u200b\u65f6\u200b\uff0c\u200b\u4f1a\u200b\u83b7\u5f97\u200b\u8fb9\u9645\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u7684\u200b\u6298\u6263\u200b\uff0c\u200b\u6298\u6263\u200b\u5206\u4e3a\u200b\u4e24\u79cd\u200b\u7c7b\u578b\u200b\uff0c\u200b\u5373\u200b\u5168\u91cf\u200b\u6298\u6263\u200b\u4e0e\u200b\u589e\u91cf\u200b\u6298\u6263\u200b\u3002</p> <ul> <li>\u200b\u5168\u91cf\u200b\u6298\u6263\u200b\u6307\u5f53\u200b\u8ba2\u8d2d\u200b\u91cf\u200b\u8d85\u8fc7\u200b\u4e34\u754c\u503c\u200b\u65f6\u200b\uff0c\u200b\u5c06\u200b\u6298\u6263\u200b\u5e94\u7528\u200b\u4e8e\u200b\u5168\u90e8\u200b\u7684\u200b\u8ba2\u8d2d\u200b\u6570\u91cf\u200b\u4e0a\u200b\u3002</li> <li>\u200b\u589e\u91cf\u200b\u6298\u6263\u200b\u6307\u5f53\u200b\u8ba2\u8d2d\u200b\u91cf\u200b\u8d85\u8fc7\u200b\u4e34\u754c\u503c\u200b\u65f6\u200b\uff0c\u200b\u5c06\u200b\u6298\u6263\u200b\u5e94\u7528\u200b\u4e8e\u200b\u8d85\u8fc7\u200b\u4e34\u754c\u503c\u200b\u7684\u200b\u90e8\u5206\u200b\u4e0a\u200b\u3002</li> </ul> <p>\u200b\u6570\u91cf\u200b\u6298\u6263\u200b\u6a21\u578b\u200b\u9488\u5bf9\u200b\u7684\u200b\u662f\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u7684\u200b\u6539\u53d8\u200b\uff0c\u200b\u8fb9\u9645\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u7684\u200b\u6539\u53d8\u200b\u4e0d\u4f1a\u200b\u5f71\u54cd\u200bEOQ\u200b\u6a21\u578b\u200b\u7684\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\u3002</p> <p>\u200b\u901a\u8fc7\u200b\u6570\u91cf\u200b\u6298\u6263\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u8bf1\u5bfc\u200b\u96f6\u552e\u5546\u200b\u63d0\u9ad8\u200b\u8ba2\u8d27\u200b\u6279\u91cf\u200b\uff0c\u200b\u4ece\u800c\u200b\u5b9e\u73b0\u200b\u8d85\u91cf\u200b\u5e93\u5b58\u200b\u7684\u200b\u8f6c\u79fb\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_10","title":"\u5168\u91cf\u200b\u6298\u6263\u200b\u7684\u200b\u6c42\u89e3\u200b\u8fc7\u7a0b","text":"<p>\u200b\u6839\u636e\u200bEOQ\u200b\u6a21\u578b\u200b\u8ba1\u7b97\u200b\u51fa\u200b\u6bcf\u200b\u4e00\u4e2a\u200b\u6298\u6263\u200b\u4ef7\u683c\u200b\\(h_i\\)\u200b\u4e0b\u200b\u5bf9\u5e94\u200b\u7684\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\\(Q^*_i\\)\uff0c\u200b\u786e\u5b9a\u200b\u6700\u5927\u200b\u53ef\u884c\u200b\u7684\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\\(Q^\\ast = \\max Q^\\ast_i\\)\u3002\u200b\u53d6\u8be5\u200b\u6298\u6263\u200b\u70b9\u200b\u4ee5\u53ca\u200b\u9ad8\u4e8e\u200b\u8be5\u200b\u6298\u6263\u200b\u70b9\u200b\u7684\u200b\u6240\u6709\u200b\u6298\u6263\u200b\u70b9\u4e3a\u200b\u5019\u9009\u200b\uff0c\u200b\u5373\u200b\\(I = \\{i | i\\geq \\arg\\max_i Q^\\ast_i\\}\\)\u3002\u200b\u5206\u522b\u200b\u6bd4\u8f83\u200b\u6bcf\u4e2a\u200b\u6298\u6263\u200b\u70b9\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5e73\u5747\u200b\u6210\u672c\u200b\uff0c\u200b\u6210\u672c\u200b\u6700\u4f4e\u200b\u7684\u200b\\(Q\\)\u200b\u5373\u200b\u4e3a\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u200b\u6279\u91cf\u200b\u3002</p> <p></p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_11","title":"\u589e\u91cf\u200b\u6298\u6263\u200b\u7684\u200b\u6c42\u89e3\u200b\u8fc7\u7a0b","text":"<p>\u200b\u6839\u636e\u200bEOQ\u200b\u6a21\u578b\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u51fa\u200b\u6240\u6709\u200b\u6210\u672c\u200b\u66f2\u7ebf\u200b\u6700\u4f4e\u70b9\u200b\u5bf9\u5e94\u200b\u7684\u200b\\(Q\\)\u200b\u503c\u200b\uff0c\u200b\u5e76\u200b\u7b5b\u9009\u200b\u51fa\u200b\u6ee1\u8db3\u200b\u589e\u91cf\u200b\u6298\u6263\u200b\u7ea6\u675f\u200b\u7684\u200b\u8ba2\u8d27\u91cf\u200b\uff0c\u200b\u5e73\u5747\u200b\u6210\u672c\u200b\u6700\u4f4e\u200b\u7684\u200b\\(Q\\)\u200b\u5373\u200b\u4e3a\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u200b\u6279\u91cf\u200b\u3002</p> <p></p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_12","title":"\u77ed\u671f\u200b\u6298\u6263","text":"<p>\u200b\u77ed\u671f\u200b\u6298\u6263\u200b\u5373\u200b\u751f\u4ea7\u5546\u200b\u53ea\u200b\u5728\u200b\u4e00\u4e2a\u200b\u5468\u671f\u200b\u4e2d\u200b\u63d0\u4f9b\u200b\u6298\u6263\u200b\u3002\u200b\u4e3a\u200b\u7b80\u5316\u200b\u5904\u7406\u200b\uff0c\u200b\u6298\u6263\u200b\u65f6\u200b\u96f6\u552e\u5546\u200b\u7684\u200b\u8d2d\u4e70\u91cf\u200b\\(Q_d\\)\u200b\u4e3a\u200bEOQ\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\\(Q^*\\)\u200b\u7684\u200b\u6574\u6570\u200b\\(k\\)\u200b\u500d\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#eoq_1","title":"\u8d44\u6e90\u200b\u7ea6\u675f\u200b\u4e0b\u200b\u7684\u200bEOQ\u200b\u6a21\u578b","text":"<p>EOQ\u200b\u6a21\u578b\u200b\u4ec5\u200b\u9002\u7528\u200b\u4e8e\u200b\u4e00\u79cd\u200b\u4ea7\u54c1\u200b\u7684\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u200b\u60c5\u5f62\u200b\uff0c\u200b\u65e0\u6cd5\u200b\u5904\u7406\u200b\u591a\u79cd\u4ea7\u54c1\u200b\u9700\u8981\u200b\u534f\u8c03\u200b\u8ba2\u8d2d\u200b\u7684\u200b\u95ee\u9898\u200b\u3002\u200b\u5728\u200b\u591a\u79cd\u4ea7\u54c1\u200b\u7684\u200b\u8ba2\u8d2d\u200b\u573a\u666f\u200b\u4e0b\u200b\uff0c\u200b\u4e0d\u540c\u200b\u7684\u200b\u4ea7\u54c1\u200b\u6709\u200b\u4e0d\u540c\u200b\u7684\u200b\u9700\u6c42\u200b\u7387\u200b\uff0c\u200b\u9700\u8981\u200b\u5728\u200b\u4e0d\u540c\u200b\u4ea7\u54c1\u200b\u7684\u200bEOQ\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u91cf\u200b\u8fdb\u884c\u200b\u5e73\u8861\u200b\u3002</p> <p>\u200b\u6a21\u578b\u200b\u5047\u8bbe\u200b\uff1a\u200b\u90e8\u5206\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u4e0e\u200b\u4ea7\u54c1\u79cd\u7c7b\u200b\u6709\u5173\u200b\u2014\u2014\\(K_i\\)\uff0c\u200b\u90e8\u5206\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u4e0e\u200b\u4ea7\u54c1\u79cd\u7c7b\u200b\u65e0\u5173\u200b\u2014\u2014\\(K\\)\u3002</p> <p>\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u5206\u4e3a\u200b\u8d44\u91d1\u200b\u7ea6\u675f\u200b\u4e0e\u200b\u5e93\u5b58\u200b\u7a7a\u95f4\u200b\u7ea6\u675f\u200b\u3002\u200b\u5982\u200b</p> \\[ \\left\\{ \\begin{aligned}   C&amp;\\geq \\sum_{i=1}^n c_iQ_i \\\\   Q_i^{\\mathrm{EOQ}} &amp;= \\sqrt{\\frac{2K_i\\lambda_i}{h_i}} \\end{aligned} \\right. \\] <p>\u200b\u5bf9\u4e8e\u200b\u7d27\u200b\u7ea6\u675f\u200b\u7684\u200b\u60c5\u51b5\u200b\uff0c\u200b\u5c06\u200b\u7ea6\u675f\u200b\u5199\u4f5c\u200b\\(\\sum_{i=1}^n c_iQ_i = C\\)\uff0c\u200b\u5f15\u5165\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u7b97\u5b50\u200b\\(\\theta\\)\uff0c\u200b\u5c06\u200b\u6b8b\u5dee\u200b\u9879\u200b\u4e0e\u200b\u4e58\u200b\u5b50\u9879\u200b\u76f8\u52a0\u200b\uff0c\u200b\u5f97\u5230\u200b\u62c9\u683c\u6717\u200b\u65e5\u200b\u4f18\u5316\u200b\u7684\u200b\u76ee\u6807\u200b\u51fd\u6570\u200b\\(C(Q_1, \\cdots, Q_n, \\theta)\\)</p> \\[ \\min C(Q_1, \\cdots, Q_n, \\theta) = \\sum_{i=1}^n \\left(\\frac{h_iQ_i}{2} + \\frac{K_i\\lambda_i}{Q_i}\\right) + \\theta\\left(\\sum_{i=1}^n c_iQ_i - C\\right) \\] <p>\u200b\u6700\u4f18\u200b\u6761\u4ef6\u200b\u4e0b\u200b\uff0c\u200b\u6709\u200b\\(\\partial C/\\partial \\theta = \\partial C/\\partial Q_i = 0\\)\uff0c\u200b\u5373\u200b</p> \\[ \\begin{aligned}   \\sum_{i=1}^n c_iQ_i^* - C &amp;= 0 \\\\   \\frac{h_i}{2} - \\frac{K_i\\lambda_i}{ {Q_i^*}^2} + \\theta c_i &amp;= 0 \\end{aligned} \\] <p>\u200b\u89e3\u5f97\u200b\\(Q_i^*\\)\uff0c\u200b\u4ee3\u5165\u200b\u7ea6\u675f\u6761\u4ef6\u200b\u5373\u53ef\u200b\u89e3\u5f97\u200b\u6700\u4f18\u200b\u7684\u200b\\(\\theta\\)</p> \\[ \\sum_{i=q}^n c_i\\sqrt{\\frac{2K_i\\lambda_i}{h_i + 2\\theta c_i}} = C \\] <p>\u200b\u4e0d\u59a8\u200b\u8bbe\u200b\\(h_i / c_i = I, \\forall i=1, \\cdots, n\\)\u3002\u200b\u5219\u200b\\(Q_i^* = \\sqrt{\\frac{2K_i\\lambda_i}{h_i}}\\sqrt{\\frac{1}{1 + 2\\theta / I}}\\)\uff0c\u200b\u6ce8\u610f\u200b\u5230\u200b\\(Q_{i}^{\\mathrm{EOQ}} = \\sqrt{\\frac{2K_i\\lambda_i}{h_i}}\\)\uff0c\u200b\u5219\u200b</p> \\[ \\left\\{ \\begin{aligned}   Q_i^* &amp;= \\sqrt{\\frac{1}{1 + 2\\theta / I}}Q_{i}^{\\mathrm{EOQ}} \\\\   C &amp;= \\sum_{i=1}^n c_iQ_i^* \\end{aligned} \\right. \\] <p>\u200b\u89e3\u5f97\u200b</p> \\[ \\theta = \\frac{\\sum_{i=1}^n \\left(c_iQ_i^{\\mathrm{EOQ}}\\right) - c}{2c / I} \\]"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#eoq_2","title":"\u591a\u200b\u4ea7\u54c1\u200bEOQ\u200b\u6a21\u578b","text":"<p>\u200b\u5728\u200b\u591a\u200b\u4ea7\u54c1\u200bEOQ\u200b\u6a21\u578b\u200b\u4e2d\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u9700\u8981\u200b\u9762\u5bf9\u200b\\(U = \\{1, \\cdots, n\\}\\)\u200b\u4e2a\u200b\u4ea7\u54c1\u200b\u7684\u200b\u9700\u6c42\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u4ea7\u54c1\u200b\\(i\\)\u200b\u6709\u200b\u5982\u4e0b\u200b\u7684\u200b\u53c2\u6570\u200b\uff1a</p> <ul> <li>\u200b\u9644\u52a0\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(K_i\\)\u3001\u200b\u8fb9\u9645\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(c_i\\)</li> <li>\u200b\u4ee5\u200b\u5229\u7387\u200b\u5f62\u5f0f\u200b\u8868\u793a\u200b\u7684\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\\(h_i = I_ic_i\\)</li> <li>\u200b\u9700\u6c42\u200b\u7387\u200b\\(\\lambda_i\\)</li> <li>\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u7f3a\u8d27\u200b</li> </ul> <p>\u200b\u5411\u200b\u91cf\u5316\u200b\u8868\u793a\u200b\u7684\u200b\u5404\u4e2a\u200b\u53c2\u6570\u200b\u5206\u522b\u200b\u4e3a\u200b\\(\\boldsymbol{K}, \\boldsymbol{c}, \\boldsymbol{h}, \\boldsymbol{I}, \\boldsymbol{\\lambda}\\)\u3002\u200b\u53e6\u5916\u200b\uff0c\u200b\u6709\u200b\u5168\u5c40\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(K\\)\u3002\u200b\u5728\u200b\u591a\u200b\u4ea7\u54c1\u200bEOQ\u200b\u6a21\u578b\u200b\u4e2d\u200b\uff0c\u200b\u5047\u8bbe\u200b\u67d0\u6b21\u200b\u7684\u200b\u8ba2\u8d27\u91cf\u200b\u4e3a\u200b\\(\\boldsymbol Q = (Q_i)_{n}\\)\uff0c\u200b\u5219\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ \\begin{aligned}   G(\\boldsymbol Q) =&amp; K + \\sum_{i=1}^n K_i\\boldsymbol{1}_{\\{Q_i &gt; 0\\}} &amp; \\text{Setup cost} \\\\   &amp;+\\sum_{i=1}^n c_iQ_i &amp; \\text{Marginal cost} \\end{aligned} \\] <p>\u200b\u73b0\u6709\u200b\u4e09\u79cd\u200b\u8ba2\u8d27\u200b\u65b9\u6cd5\u200b</p> <ol> <li>\u200b\u72ec\u7acb\u200b\u8ba2\u8d27\u200b\u2014\u2014\u200b\u6bcf\u6b21\u200b\u8ba2\u8d2d\u200b\u90fd\u200b\u9700\u8981\u200b\u591a\u200b\u82b1\u8d39\u200b\\(K\\)\u200b\u7684\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b</li> <li>\u200b\u5b8c\u6574\u200b\u8ba2\u8d27\u200b\u2014\u2014\u200b\u5bf9\u4e8e\u200b\u9700\u6c42\u200b\u7387\u4e0d\u9ad8\u200b\u7684\u200b\u4ea7\u54c1\u200b\uff0c\u200b\u8ba2\u8d27\u200b\u8fc7\u4e8e\u200b\u9891\u7e41\u200b\u5bfc\u81f4\u200b\u989d\u5916\u200b\u82b1\u8d39\u200b\u4e86\u200b\\(K_i\\)\u200b\u7684\u200b\u9644\u52a0\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b</li> <li>\u200b\u8054\u5408\u200b\u8ba2\u8d27\u200b\uff1a\u200b\u4e0d\u200b\u9700\u8981\u200b\u8ba2\u8d2d\u200b\u6240\u6709\u200b\u7c7b\u578b\u200b\u7684\u200b\u4ea7\u54c1\u200b</li> </ol> <p>\u200b\u4ee5\u4e0b\u200b\u5bf9\u200b\u4e09\u79cd\u200b\u8ba2\u8d27\u200b\u65b9\u6cd5\u200b\u8fdb\u884c\u200b\u8ba8\u8bba\u200b</p>"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_13","title":"\u72ec\u7acb\u200b\u8ba2\u8d27","text":"<p>\u200b\u72ec\u7acb\u200b\u8ba2\u8d27\u200b\u4e0b\u200b\uff0c\u200b\u6bcf\u79cd\u200b\u4ea7\u54c1\u200b\u6309\u200b\u5176\u200b\u5bf9\u5e94\u200b\u7684\u200bEOQ\u200b\u8ba2\u8d27\u91cf\u200b\u8fdb\u884c\u200b\u8ba2\u8d27\u200b\uff0c\u200b\u5219\u200b\u5e73\u5747\u200b\u6210\u672c\u200b</p> \\[ G^* = \\sum_{i=1}^nG_i^* = \\sum_{i=1}^n\\sqrt{2{K + K_i}\\lambda_i h_i} \\]"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_14","title":"\u5b8c\u6574\u200b\u8ba2\u8d27","text":"<p>\u200b\u5b8c\u6574\u200b\u8ba2\u8d27\u200b\u4e0b\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5468\u671f\u200b\u5185\u200b\u90fd\u200b\u9700\u8981\u200b\u8ba2\u8d2d\u200b\u6240\u6709\u200b\u4ea7\u54c1\u200b\uff0c\u200b\u8bbe\u200b\u8ba2\u8d27\u91cf\u200b\u4e3a\u200b\\(\\boldsymbol{Q} = T\\boldsymbol\\lambda\\)\uff0c\u200b\u7c7b\u6bd4\u200b\u4e8e\u200bEOQ\u200b\u6a21\u578b\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5f97\u5230\u200b\u603b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ G(Q) = \\frac{K + \\sum_{i} K_i}{T} + \\boldsymbol{c}^\\top\\boldsymbol{\\lambda} + \\frac{T\\boldsymbol{h}^\\top\\boldsymbol \\lambda}{2} \\] <p>\u200b\u8f6c\u5316\u200b\u4e3a\u5355\u200b\u4ea7\u54c1\u200bEOQ\u200b\u95ee\u9898\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u4e3a\u200b</p> \\[ \\begin{aligned}   T^* &amp;= \\sqrt{\\frac{2\\left(K + \\sum_{i} K_i\\right)}{\\boldsymbol{h}^\\top \\boldsymbol{\\lambda}}} \\\\   G^* &amp;= \\sqrt{2\\left(K + \\sum_{i} K_i\\right)\\boldsymbol{h}^\\top \\boldsymbol{\\lambda}} \\\\ \\end{aligned} \\]"},{"location":"math/operations-mamagement/inventory-control-deterministic-demand/#_15","title":"\u8054\u5408\u200b\u8ba2\u8d27","text":"<p>\u200b\u8054\u5408\u200b\u8ba2\u8d27\u200b\u6761\u4ef6\u200b\u4e0b\u200b\uff0c\u200b\u6bcf\u6b21\u200b\u8ba2\u8d27\u200b\u4e0d\u9700\u200b\u5305\u542b\u200b\u6240\u6709\u200b\u7c7b\u578b\u200b\u7684\u200b\u4ea7\u54c1\u200b\uff0c\u200b\u4f46\u200b\u9700\u6c42\u91cf\u200b\u6700\u9ad8\u200b\u7684\u200b\u4ea7\u54c1\u200b\u5fc5\u7136\u200b\u5728\u200b\u6bcf\u4e2a\u200b\u8ba2\u5355\u200b\u4e2d\u200b\u90fd\u200b\u51fa\u73b0\u200b\uff0c\u200b\u56e0\u6b64\u200b\uff0c\u200b\u9996\u5148\u200b\u8003\u8651\u200bEOQ\u200b\u6a21\u578b\u200b\u4e0b\u200b\u4ea7\u54c1\u200b\u7684\u200b\u8ba2\u8d27\u200b\u9891\u7387\u200b\uff1a</p> \\[ n_i = \\frac{1}{T_i} = \\sqrt{\\frac{h_i\\lambda_i}{(K + K_i)}} \\] <p>\u200b\u4ee5\u200b\u8ba2\u8d27\u200b\u6700\u200b\u9891\u7e41\u200b\u7684\u200b\u4ea7\u54c1\u200b\\(i^\\ast = \\arg\\max\\limits_{i} n_i\\)\u200b\u5bf9\u5e94\u200b\u7684\u200b\u8ba2\u8d27\u200b\u9891\u7387\u200b\\(\\bar n = n_{i^\\ast}\\)\u3001\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\\(\\bar T = T_{i^\\ast}\\)\u200b\u4e3a\u200b\u57fa\u51c6\u200b\uff0c\u200b\u8003\u8651\u200b\u5176\u4ed6\u200b\u7c7b\u578b\u200b\u4ea7\u54c1\u200b\u7684\u200b\u8ba2\u8d27\u200b\u9891\u7387\u200b\uff1a</p> \\[ m_i = \\left\\lceil\\frac{\\bar n}{n_i}\\right\\rceil \\] <p>\\(m_i\\)\u200b\u8868\u793a\u200b\u4ea7\u54c1\u200b\\(i\\)\u200b\u7684\u200b\u8ba2\u8d2d\u200b\u5468\u671f\u200b\u4e3a\u200b\\(m_i\\bar T_i\\)\uff0c\u200b\u5373\u200b\u6700\u200b\u9891\u7e41\u200b\u8ba2\u5355\u200b\u5468\u671f\u200b\u7684\u200b\\(m_i\\)\u200b\u500d\u200b\uff0c\u200b\u5219\u200b\u4e00\u4e2a\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\u5185\u200b\u7684\u200b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ \\begin{aligned}   G(\\boldsymbol Q) =&amp; \\frac{1}{T}\\left(K + \\sum_i\\frac{K_i}{m_i}\\right) &amp; \\text{Setup cost} \\\\   &amp;+\\frac{h}{2}\\sum_{i}\\lambda_i m_i &amp; \\text{Marginal cost} \\end{aligned} \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u6700\u4f18\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\u4e0e\u200b\u6700\u4f18\u200b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ \\begin{aligned}   T^* &amp;= \\sqrt{\\frac{\\sum_{i} \\lambda_i m_i h}{2\\left(K + \\sum_i\\frac{K_i}{m_i}\\right)}} \\\\   G^* &amp;= \\sqrt{2\\left(\\sum_{i} \\lambda_i m_i h\\right)\\left(K + \\sum_i\\frac{K_i}{m_i}\\right)} \\end{aligned} \\]"},{"location":"math/operations-mamagement/inventory-control-random-demand/","title":"\u968f\u673a\u200b\u9700\u6c42\u200b\u4e0b\u200b\u7684\u200b\u5e93\u5b58\u200b\u63a7\u5236","text":"<p>\u200b\u4f18\u5316\u200b\u76ee\u6807\u200b\uff1a\u200b\u6700\u5c0f\u5316\u200b\u6210\u672c\u200b\u7684\u200b\u6570\u5b66\u200b\u671f\u671b\u200b</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_2","title":"\u5355\u200b\u5468\u671f\u200b\u5355\u7ea7\u200b\u2014\u2014\u200b\u62a5\u7ae5\u200b\u6a21\u578b","text":"<p>\u200b\u62a5\u7ae5\u200b\u6a21\u578b\u200b\u662f\u200b\u5355\u200b\u5468\u671f\u200b\u4e0b\u200b\u7684\u200b\u5e93\u5b58\u200b\u63a7\u5236\u200b\u95ee\u9898\u200b</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_3","title":"\u8bb0\u53f7","text":"<ul> <li>\\(c_o\\)\u200b\u8868\u793a\u200b\u8d85\u989d\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\uff0c\\(c_u\\)\u200b\u8868\u793a\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b</li> <li>\\(f(\\cdot), F(\\cdot)\\)\u200b\u5206\u522b\u200b\u4e3a\u200b\u9700\u6c42\u200b\u7684\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\u4e0e\u200b\u6982\u7387\u5206\u5e03\u200b\u51fd\u6570\u200b</li> <li>\\(Q\\)\u200b\u4e3a\u200b\u8ba2\u8d27\u91cf\u200b</li> <li>\\(x^+ = \\max\\{x, 0\\}\\)</li> </ul>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_4","title":"\u63a8\u5bfc","text":"<p>\u200b\u62a5\u7ae5\u200b\u6a21\u578b\u200b\u7684\u200b\u6210\u672c\u200b\u51fd\u6570\u200b\u4e3a\u200b</p> \\[ G(Q, D) = c_o\\max\\{0, Q-D\\} + c_u\\max\\{0, D-Q\\} \\] <p>\\(G(Q, D)\\)\u200b\u662f\u200b\u968f\u673a\u53d8\u91cf\u200b\\(D\\)\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u56e0\u6b64\u200b\u4e5f\u200b\u662f\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u5bf9\u200b\\(D\\)\u200b\u53d6\u200b\u671f\u671b\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\begin{aligned}     C(Q) &amp;= E(G(Q, D) | Q) \\\\     &amp;= \\int_{0}^\\infty G(Q, D)f(x)\\mathrm dx \\\\     &amp;= \\int_{0}^Q c_o(Q-x)f(x)\\mathrm dx + \\int_Q^\\infty c_u(x-Q)f(x)\\mathrm dx \\\\ \\end{aligned} \\] <p>\u200b\u5f53\u200b\\(C(Q)\\)\u200b\u53d6\u200b\u6781\u5c0f\u503c\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(\\frac{\\mathrm dC}{\\mathrm dD} = 0\\)\uff0c\u200b\u5373\u200b</p> \\[ \\begin{aligned}     \\frac{\\mathrm dC}{\\mathrm dQ} &amp;= c_o\\int_0^Q f(x)\\mathrm dx + c_u\\int_Q^\\infty -f(x)\\mathrm dx \\\\     &amp;= c_oF(Q) - c_u(1-F(Q)) \\\\     &amp;= 0 \\end{aligned} \\] <p>\u200b\u9a8c\u8bc1\u200b\uff1a\\(\\frac{\\mathrm d^2 C}{\\mathrm dQ^2} = (c_o + c_u)f(Q) &gt; 0\\)\uff0c\u200b\u56e0\u6b64\u200b\u4e3a\u200b\u6781\u5c0f\u503c\u200b\u70b9\u200b\u3002\u200b\u6b64\u65f6\u200b\\(F(Q^*) = c_u / (c_o + c_u)\\)\u3002</p> <p>\u200b\u79f0\u200b\\(c_u / (c_o + c_u)\\)\u200b\u4e3a\u200b\u5173\u952e\u200b\u6bd4\u4f8b\u200b\uff08crucial ratio\uff09</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_5","title":"\u5355\u200b\u5468\u671f\u200b\u6a21\u578b\u200b\u6269\u5c55","text":""},{"location":"math/operations-mamagement/inventory-control-random-demand/#_6","title":"\u79bb\u6563\u200b\u9700\u6c42\u200b\u7684\u200b\u6700\u4f18\u200b\u7b56\u7565","text":"<p>\u200b\u786e\u5b9a\u200b\\(\\max_{Q_0}, \\min_{Q_1}\\)\u200b\u6ee1\u8db3\u200b\\(Q_1 &gt; Q_0\\)\u200b\u4e14\u200b\\(F(Q_0) &lt; c_u / (c_u + c_o) &lt; F(Q_1)\\)\uff0c\u200b\u6b64\u65f6\u200b\\(Q_1\\)\u200b\u5373\u200b\u4e3a\u200b\u6700\u4f18\u200b\u7684\u200b\\(Q^*\\)\u3002\u200b\u5373\u200b\u9ad8\u4e8e\u200b\u6700\u4f18\u200b\u5173\u952e\u200b\u6bd4\u4f8b\u200b\u7684\u200b\u6700\u5c0f\u200b\u5173\u952e\u200b\u6bd4\u4f8b\u200b\u5bf9\u5e94\u200b\u7684\u200b\\(Q\\)</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#u","title":"\u5b58\u5728\u200b\u521d\u59cb\u200b\u5e93\u5b58\u200b\\(u\\)","text":"<p>\\(u\\)\u200b\u7684\u200b\u5b58\u5728\u200b\u4e0d\u200b\u5f71\u54cd\u200b\\(Q^*\\)\uff0c\u200b\u56e0\u6b64\u200b\uff0c\u200b\u8ba2\u8d27\u91cf\u200b\\(Q'\\)\u200b\u6ee1\u8db3\u200b</p> \\[ Q^{*\\prime} = \\max\\{Q^*-u, 0\\} \\]"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_7","title":"\u591a\u7ea7\u200b\u7cfb\u7edf","text":"<p>\u200b\u591a\u7ea7\u200b\u5e93\u5b58\u200b\u7cfb\u7edf\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u4e32\u884c\u200b\u7cfb\u7edf\u200b\u3001\u200b\u88c5\u914d\u200b\u7cfb\u7edf\u200b\u548c\u200b\u914d\u9001\u200b\u7cfb\u7edf\u200b\u3002</p> <p>\u200b\u8003\u8651\u200b\u4e00\u4e2a\u200b\u4e32\u884c\u200b\u7684\u200b\u4f9b\u5e94\u94fe\u200b\uff0c\u200b\u4f9b\u5e94\u94fe\u200b\u4e0b\u6e38\u200b\u5411\u200b\u4f9b\u5e94\u94fe\u200b\u4e0a\u6e38\u200b\u91c7\u8d2d\u200b\u4ea7\u54c1\u200b\u3002\u200b\u901a\u5e38\u200b\u91c7\u7528\u200b\u5d4c\u5957\u200b\u7b56\u7565\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u8ba2\u8d27\u200b\uff0c\u200b\u5373\u5f53\u200b\u4e0b\u6e38\u200b\u8ba2\u8d27\u200b\u65f6\u200b\uff0c\u200b\u4e0a\u6e38\u200b\u624d\u200b\u53ef\u80fd\u200b\u8fdb\u884c\u200b\u8ba2\u8d27\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_8","title":"\u5468\u671f\u200b\u68c0\u67e5\u200b\u6a21\u578b","text":"<p>\u200b\u5728\u200b\u5468\u671f\u200b\u68c0\u67e5\u200b\u6a21\u578b\u200b\u4e2d\u200b\uff0c\u200b\u9996\u5148\u200b\u4e0d\u200b\u8003\u8651\u200b\u56fa\u5b9a\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\\(K\\)\u200b\u7684\u200b\u5f71\u54cd\u200b\u3002\u200b\u9700\u6c42\u200b\u5206\u4e3a\u200b\u591a\u4e2a\u200b\u5468\u671f\u200b\uff0c\u200b\u6bcf\u4e2a\u200b\u5468\u671f\u200b\u7684\u200b\u9700\u6c42\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_9","title":"\u8bb0\u53f7","text":"<ul> <li>\u200b\u5269\u4f59\u200b\u5468\u671f\u200b\u6570\u4e3a\u200b\\(n\\)</li> <li>\u200b\u6bcf\u4e2a\u200b\u5468\u671f\u200b\u7684\u200b\u9700\u6c42\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u9700\u6c42\u200b\u7684\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\u4e3a\u200b\\(f(\\cdot)\\)\uff0c\u200b\u6982\u7387\u5206\u5e03\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(F(\\cdot)\\)\u3002</li> <li>\u200b\u5468\u671f\u200b\u4e4b\u95f4\u200b\u7684\u200b\u8d34\u73b0\u7387\u200b\\(\\alpha\\)\u200b\u8868\u793a\u200b</li> <li>\u200b\u4e0d\u200b\u8003\u8651\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u200b\uff0c\\(L=0\\)</li> <li>\\(y_i\\)\u200b\u8868\u793a\u200b\u7b2c\u200b\\(i\\)\u200b\u671f\u200b\u7ed3\u675f\u200b\u65f6\u200b\u7684\u200b\u5e93\u5b58\u200b\uff0c\\(y_0\\)\u200b\u8868\u793a\u200b\u671f\u521d\u200b\u5e93\u5b58\u200b</li> <li>\\(C_n(y_0)\\)\u200b\u8868\u793a\u200b\u7b2c\u200b\\(n\\)\u200b\u671f\u200b\u603b\u200b\u671f\u671b\u200b\u8d34\u73b0\u200b\u6210\u672c\u200b\u7684\u200b\u6700\u5c0f\u503c\u200b</li> </ul> \\[ C_n(y_0) = \\min_{y\\geq y_0}\\left\\{L(y) - cy_0 + \\alpha\\int_{0}^\\infty C_{n-1}[t(y, x)]f(x)\\mathrm dx\\right\\} \\]"},{"location":"math/operations-mamagement/inventory-control-random-demand/#q-r","title":"\u8fde\u7eed\u200b\u68c0\u67e5\u200b\u6a21\u578b\u200b\uff1a(Q, R)\u200b\u6a21\u578b","text":"<p>\u200b\u5728\u200b\u8fde\u7eed\u200b\u68c0\u67e5\u200b\u6a21\u578b\u200b\u4e2d\u200b\uff0c\u200b\u9700\u8981\u200b\u533a\u5206\u200b\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b\u4e0e\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u7684\u200b\u6982\u5ff5\u200b</p> <ul> <li>\u200b\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b\uff08Inventory Position\uff09\uff0c\u200b\u6307\u200b\u76ee\u524d\u200b\u6301\u6709\u200b\u7684\u200b\u5e93\u5b58\u200b\u603b\u91cf\u200b\u52a0\u4e0a\u200b\u5904\u4e8e\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u200b\u7684\u200b\u5e93\u5b58\u91cf\u200b</li> <li>\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\uff08Inventory Level\uff09\uff0c\u200b\u6307\u200b\u76ee\u524d\u200b\u6301\u6709\u200b\u7684\u200b\u5e93\u5b58\u200b\u603b\u91cf\u200b</li> </ul>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_10","title":"\u6a21\u578b\u200b\u8bbe\u5b9a\u200b\u53ca\u200b\u63a8\u5bfc","text":"<p>\\((Q, R)\\)\u200b\u6a21\u578b\u200b\u6709\u200b\u5982\u4e0b\u200b\u8bbe\u5b9a\u200b</p> <ol> <li>\u200b\u6bcf\u6b21\u200b\u8ba2\u8d27\u200b\u7684\u200b\u56fa\u5b9a\u6210\u672c\u200b\u4e3a\u200b\\(K\\)\uff0c\u200b\u8fb9\u9645\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\u4e3a\u200b\\(c\\)\uff0c\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u4e3a\u200b\\(\\tau\\)\uff1b</li> <li>\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u5185\u200b\u7684\u200b\u9700\u6c42\u200b\\(D\\)\u200b\u5728\u200b\u65f6\u95f4\u200b\u4e0a\u200b\u8fde\u7eed\u200b\uff0c\u200b\u5728\u200b\u6570\u91cf\u200b\u4e0a\u200b\u670d\u4ece\u200b\u5747\u503c\u200b\u4e3a\u200b\\(\\mu\\)\u3001\u200b\u6807\u51c6\u5dee\u200b\u4e3a\u200b\\(\\sigma\\)\u200b\u7684\u200b\u67d0\u4e2a\u200b\u6982\u7387\u5206\u5e03\u200b\\(f(\\cdot), F(\\cdot)\\)\uff1b\\(\\lambda\\)\u200b\u4e3a\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u7684\u200b\u671f\u671b\u200b\u9700\u6c42\u200b\u7387\u200b\uff1b</li> <li>\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u7684\u200b\u8fb9\u9645\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u4e3a\u200b\\(h = Ic\\)\uff08\u200b\u5f0f\u200b\u4e2d\u200b\\(I\\)\u200b\u4e3a\u200b\u5229\u7387\u200b\uff09</li> <li>\u200b\u5141\u8bb8\u200b\u7f3a\u8d27\u200b\uff0c\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\\(p\\)\uff08backorder\uff09\uff1b</li> <li>\u200b\u6a21\u578b\u200b\u7684\u200b\u51b3\u7b56\u200b\u53d8\u91cf\u200b\u4e3a\u91cd\u200b\u8ba2\u8d27\u200b\u70b9\u200b\\(R\\)\u3001\u200b\u4e0e\u200b\u8ba2\u8d27\u91cf\u200b\\(Q\\)\uff1a\u200b\u5f53\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u964d\u4f4e\u200b\u81f3\u200b\\(R\\)\u200b\u65f6\u200b\uff0c\u200b\u8ba2\u8d2d\u200b\\(Q\\)\u200b\u5355\u4f4d\u200b\u7684\u200b\u5e93\u5b58\u200b\u3002\u200b\u53d7\u200b\u8ba2\u8d27\u91cf\u200b\\(Q\\)\u200b\u7684\u200b\u5f71\u54cd\u200b\uff0c\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\\(T\\)\u200b\u6ee1\u8db3\u200b\\(Q = \\lambda T\\)</li> <li> <p>\u200b\u6a21\u578b\u200b\u4e2d\u200b\u4f7f\u7528\u200b\u7684\u200b\u5176\u4ed6\u200b\u8bb0\u53f7\u200b\u6709\u200b\uff1a</p> <ul> <li>\\(\\mathrm{IP}\\)\uff1a\u200b\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b</li> <li>\\(\\mathrm{IL}\\)\uff1a\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b</li> <li>\\(\\mathrm{SS} = R - \\mu\\)\uff1a\u200b\u5b89\u5168\u200b\u5e93\u5b58\u200b</li> </ul> </li> </ol> <p>\\((Q, R)\\)\u200b\u6a21\u578b\u200b\u8003\u8651\u200b\u5982\u4e0b\u200b\u6765\u6e90\u200b\u7684\u200b\u6210\u672c\u200b\uff1a</p> <ol> <li> <p>\u200b\u8ba2\u8d27\u200b\u6210\u672c\u200b\uff1a\u200b\u6bcf\u6b21\u200b\u8ba2\u8d2d\u200b\\(Q\\)\u200b\u5355\u4f4d\u200b\u7684\u200b\u5e93\u5b58\u200b\uff0c\u200b\u603b\u6210\u672c\u200b\u4e3a\u200b\\(K + cQ\\)\uff1b</p> \\[ \\frac{K + cQ}{T} = \\frac{K\\lambda}{Q} + c\\lambda \\] </li> <li> <p>\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\uff1a\u200b\u9996\u5148\u200b\u8ba1\u7b97\u200b\u6bcf\u4e2a\u200b\u5468\u671f\u200b\u5185\u200b\u7684\u200b\u671f\u671b\u200b\u7f3a\u8d27\u91cf\u200b\\(n(R)\\)\uff0c\u200b\u6709\u200b</p> \\[ n(R) = E(\\max(D - R, 0)) = \\int_R^\\infty (x-R)f(x)\\mathrm dx \\] <p>\u200b\u6839\u636e\u200b\\(n(R)\\)\uff0c\u200b\u5373\u53ef\u200b\u5f97\u5230\u200b\u4e00\u4e2a\u200b\u8ba2\u8d27\u200b\u5468\u671f\u200b\\(T\\)\u200b\u65f6\u95f4\u200b\u7684\u200b\u5e73\u5747\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\\(\\lambda n(R)p/Q\\)</p> </li> <li> <p>\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\uff1a\u200b\u4e00\u4e2a\u200b\u5468\u671f\u200b\u5185\u200b\u671f\u671b\u200b\u7684\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u53d8\u5316\u200b\u4e3a\u200b\u7ebf\u6027\u200b\uff0c\u200b\u5f53\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u964d\u200b\u81f3\u200b\\(R\\)\u200b\u540e\u200b\u91cd\u65b0\u200b\u8ba2\u8d27\u200b\uff0c\u200b\u671f\u671b\u200b\u5e93\u5b58\u200b\u7ee7\u7eed\u200b\u4e0b\u964d\u200b\uff0c\u200b\u76f4\u81f3\u200b\u964d\u4f4e\u200b\u81f3\u200b\\(R - \\lambda\\tau\\)\u200b\u65f6\u200b\u8ba2\u5355\u200b\u5230\u8fbe\u200b\uff0c\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u5185\u200b\u7684\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ h\\left(\\frac{Q}{2} + R - \\lambda\\tau\\right) \\] </li> </ol> <p>\u200b\u6839\u636e\u200b\u5982\u200b\u4e0a\u200b\u5206\u6790\u200b\uff0c\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u5185\u200b\u7684\u200b\u671f\u671b\u200b\u6210\u672c\u200b\u4e3a\u200b</p> \\[ \\begin{aligned} G(Q, R) &amp;= h\\left(\\frac{Q}{2} + R - \\lambda\\tau\\right) + \\frac{K\\lambda}{Q} + \\frac{p\\lambda n(R)}{Q} \\\\ &amp;= \\frac{hQ}{2} + \\lambda(K + pn(R))\\frac{1}{Q} + hR - h\\lambda\\tau \\end{aligned} \\] <p>\\((Q, R)\\)\u200b\u6a21\u578b\u200b\u7684\u200b\u76ee\u6807\u200b\u5373\u200b\u4e3a\u200b\u6c42\u89e3\u200b\u51fa\u200b\u4f7f\u5f97\u200b\\(G(Q, R)\\)\u200b\u6700\u5c0f\u200b\u7684\u200b\\(Q, R\\)\u3002\u200b\u82e5\u200b\u8868\u793a\u200b\u4e3a\u200b\\((s, S)\\)\u200b\u7b56\u7565\u200b\u7684\u200b\u5f62\u5f0f\u200b\uff0c\u200b\u5373\u200b\u6709\u200b\\(s = R, S = Q + R\\)</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_11","title":"\u6c42\u89e3","text":"<p>\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u8fed\u4ee3\u200b\u7684\u200b\u65b9\u5f0f\u200b\u903c\u8fd1\u200b\\((Q, R)\\)\u200b\u6a21\u578b\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\uff0c\u200b\u5206\u522b\u200b\u5bf9\u200b\\(G\\)\u200b\u6c42\u200b\\(Q, R\\)\u200b\u7684\u200b\u504f\u200b\u5bfc\u6570\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\begin{aligned}     \\frac{\\partial G}{\\partial Q} &amp;= \\frac{h}{2} - \\lambda (K + pn(R))\\frac{1}{Q^2} \\\\     \\frac{\\partial G}{\\partial R} &amp;= h + \\frac{\\lambda p n'(R)}{Q} \\\\     &amp;= h - \\frac{\\lambda p(1-F(R))}{Q} \\end{aligned} \\] <p>\u200b\u8fed\u4ee3\u200b\u8fc7\u7a0b\u200b\u4ece\u200bEOQ\u200b\u6a21\u578b\u200b\u7684\u200b\u6700\u4f18\u200b\u89e3\u200b\\(Q_{\\mathrm{EOQ}}^* = \\sqrt{\\frac{2K\\lambda}{h}}\\)\u200b\u5f00\u59cb\u200b\uff0c\u200b\u6c42\u200b\u51fa\u200b\u4f7f\u5f97\u200b\\(G\\)\u200b\u6700\u4f18\u200b\u7684\u200b\\(R\\)\u200b\u503c\u200b\uff0c\u200b\u518d\u200b\u5c06\u200b\\(R\\)\u200b\u4ee3\u5165\u200b\u6c42\u200b\u51fa\u200b\u6700\u4f18\u200b\u7684\u200b\\(Q\\)\u200b\u503c\u200b\uff0c\u200b\u5982\u6b64\u200b\u5faa\u73af\u200b\uff0c\\(Q, R\\)\u200b\u5373\u200b\u903c\u8fd1\u200b\u6700\u4f18\u200b\u7684\u200b\\(Q^*, R^*\\)\u200b\u53d6\u503c\u200b\uff0c\u200b\u6700\u4f18\u200b\u89e3\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u65b9\u7a0b\u200b\uff1a</p> \\[ \\begin{aligned}     Q &amp;= \\sqrt{\\frac{2\\lambda (K + pn(R))}{h}} \\\\     F(R) &amp;= 1 - \\frac{hQ}{\\lambda p} \\end{aligned} \\]"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_12","title":"\u670d\u52a1\u6c34\u5e73","text":"<p>\u200b\u7c7b\u4f3c\u200b\u4e8e\u200b\u5468\u671f\u200b\u68c0\u67e5\u200b\u6a21\u578b\u200b\uff0c\u200b\u5728\u200b\u8fde\u7eed\u200b\u68c0\u67e5\u200b\u6a21\u578b\u200b\u4e2d\u200b\u4e5f\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u7b2c\u4e00\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\u4e0e\u200b\u7b2c\u4e8c\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\u7b49\u200b\u6307\u6807\u200b\u3002\u200b\u82e5\u200b\u5047\u8bbe\u200b\u6bcf\u4e2a\u200b\u63d0\u524d\u200b\u671f\u5185\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u5728\u200b\u9014\u200b\u8ba2\u5355\u200b\uff1a</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_13","title":"\u7b2c\u4e00\u7c7b\u200b\u670d\u52a1\u6c34\u5e73","text":"<p>\u200b\u7b2c\u4e00\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\u6307\u5728\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u4e2d\u200b\u4e0d\u200b\u53d1\u751f\u200b\u7f3a\u8d27\u200b\u7684\u200b\u6982\u7387\u200b\uff0c\u200b\u7528\u200b\\(\\alpha\\)\u200b\u8868\u793a\u200b\u3002\u200b\u5219\u200b\u6709\u200b\\(\\alpha = F(R)\\)\u3002\u200b\u6c42\u89e3\u200b\u6ee1\u8db3\u200b\u7b2c\u4e00\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\u6761\u4ef6\u200b\u7684\u200b\\((Q, R)\\)\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u6b65\u9aa4\u200b\u8fdb\u884c\u200b</p> <ol> <li>\u200b\u6c42\u89e3\u200b\u6ee1\u8db3\u200b\\(F(R) = \\alpha\\)\u200b\u7684\u200b\\(R\\)\u200b\u503c\u200b</li> <li>\\(Q\\)\u200b\u503c\u53d6\u200bEOQ\u200b\u6700\u4f18\u200b\u503c\u200b\\(\\sqrt{\\frac{2K\\lambda}{h}}\\)\u200b\u5373\u53ef\u200b</li> </ol>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#fill-rate","title":"\u7b2c\u4e8c\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\uff1aFill rate","text":"<p>\u200b\u7b2c\u4e8c\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\u6307\u200b\u6ee1\u8db3\u200b\u7684\u200b\u9700\u6c42\u200b\u5360\u200b\u603b\u9700\u6c42\u200b\u7684\u200b\u6bd4\u4f8b\u200b\u6bd4\u4f8b\u200b\uff0c\u200b\u7528\u200b\\(\\beta\\)\u200b\u8868\u793a\u200b\uff0c\u200b\u7531\u4e8e\u200b\\(n(R)\\)\u200b\u4e3a\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u4e2d\u200b\u7684\u200b\u671f\u671b\u200b\u7f3a\u8d27\u91cf\u200b\uff0c\u200b\u5219\u200b\u6709\u200b\\(n(R)/Q = 1 - \\beta\\)\u3002\u200b\u6839\u636e\u200b\\(F(R) = 1 - \\frac{hQ}{\\lambda p}\\)\uff0c\u200b\u53ef\u200b\u5f97\u200b</p> \\[ p = \\frac{Qh}{\\lambda(1 - F(R))} \\] <p>\u200b\u5c06\u200b\u5176\u200b\u4ee3\u5165\u200b\\(Q = \\sqrt{\\frac{2\\lambda (K + pn(R))}{h}}\\)\uff0c\u200b\u6574\u7406\u200b\u5f97\u5230\u200b</p> \\[ hQ^2 - \\frac{2hn(R)}{1-F(R)}Q - 2K\\lambda = 0 \\] <p>\u200b\u89e3\u5f97\u200b</p> \\[ Q = \\frac{n(R)}{1 - F(R)} + \\sqrt{\\frac{2K\\lambda}{h} + \\left(\\frac{n(R)}{1-F(R)}\\right)^2} \\] <p>\u200b\u53c8\u200b\u6839\u636e\u200b\u7b2c\u4e8c\u7c7b\u200b\u670d\u52a1\u6c34\u5e73\u200b\uff0c\u200b\u6709\u200b</p> \\[n(R) = (1-\\beta)Q\\] <p>\u200b\u8054\u200b\u7acb\u5373\u200b\u5f97\u200b\uff0c\u200b\u4e5f\u200b\u53ef\u200b\u4f7f\u7528\u200b\u8fed\u4ee3\u200b\u65b9\u5f0f\u200b\u6c42\u89e3\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_14","title":"\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b\u4e0e\u200b\u5e93\u5b58\u200b\u6c34\u5e73","text":"<p>\u200b\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b\u4e0e\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u7684\u200b\u5173\u7cfb\u200b\uff1a</p> \\[ \\mathrm{IL}(t+L) = \\mathrm{IP}(t) - D(t, t+L) \\] <p>\u200b\u5f0f\u200b\u4e2d\u200b\\(D(t, t+L)\\)\u200b\u4e3a\u200b\\([t, t+L]\\)\u200b\u65f6\u95f4\u200b\u5185\u200b\u7684\u200b\u9700\u6c42\u200b\u3002\u200b\u56e0\u6b64\u200b\uff0c\u200b\u82e5\u200b\u5df2\u77e5\u200b\\(\\mathrm{IP}(t), D(t, t+L)\\)\u200b\u7684\u200b\u5206\u5e03\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u636e\u6b64\u200b\u8ba1\u7b97\u200b\\(\\mathrm{IL}(t+L)\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u7a33\u6001\u200b\uff0c\u200b\u5728\u200b\u4efb\u610f\u200b\u65f6\u523b\u200b\\(t\\)\uff0c\u200b\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b\\(\\mathrm{IP}(t), \\mathrm{IL}(t+L), D(t, t+L)\\)\u200b\u5206\u522b\u200b\u5bf9\u200b\u4e0d\u540c\u200b\u7684\u200b\\(t\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\u3002\u200b\u7531\u6b64\u200b\uff0c\u200b\u8f6c\u5316\u200b\u4e3a\u200b\u968f\u673a\u53d8\u91cf\u200b\\(\\mathrm{IP}, \\mathrm{IL}, D(L)\\)\u200b\u4e4b\u95f4\u200b\u7684\u200b\u5206\u5e03\u200b\u5173\u7cfb\u200b\u3002\u200b\u5df2\u77e5\u200b\\(\\mathrm{IP}\\)\u200b\u4e3a\u200b\u5747\u5300\u5206\u5e03\u200b\uff0c\u200b\u8bbe\u200b\\(D(L)\\)\u200b\u7684\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\u4e3a\u200b\\(f(x;L)\\)\uff0c\u200b\u5219\u200b</p> \\[ \\begin{aligned}     F_{\\mathrm{IL}}(x) &amp;= P(\\mathrm{IL}\\leq x) \\\\     &amp;= \\int_{R}^{R+Q}P(\\mathrm{IP} = u)P(\\mathrm{IP} - \\mathrm{IL} \\geq u-x) \\mathrm du \\\\     &amp;= \\frac{1}{Q}\\int_{R}^{R+Q} P(D \\geq u-x) \\mathrm du \\\\     &amp;= \\frac{1}{Q}\\int_{R}^{R+Q} (1 - F(u-x;L)) \\mathrm du \\end{aligned} \\]"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_15","title":"\u6a21\u578b\u200b\u6269\u5c55","text":""},{"location":"math/operations-mamagement/inventory-control-random-demand/#_16","title":"\u6b63\u6001\u5206\u5e03\u200b\u9700\u6c42","text":"<p>\u200b\u5f53\u200b\u63d0\u524d\u200b\u671f\u5185\u200b\u9700\u6c42\u200b\\(D\\)\u200b\u670d\u4ece\u200b\u6b63\u6001\u5206\u5e03\u200b\\(N(\\mu, \\sigma)\\)\u200b\u65f6\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b\u6b63\u6001\u5206\u5e03\u200b\u7684\u200b\u6807\u51c6\u200b\u635f\u5931\u200b\u51fd\u6570\u200b\\(L(z)\\)\u200b\u7b80\u5316\u200b\\(n(R)\\)\u200b\u7684\u200b\u8ba1\u7b97\u200b</p> \\[ n(R) = \\sigma L\\left(\\frac{R-\\mu}{\\sigma}\\right) = \\sigma L(z) \\] <p>\\(L(z)\\)\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\\(\\phi(z), \\varPhi(z)\\)\u200b\u8ba1\u7b97\u200b\u5f97\u5230\u200b\uff1a</p> \\[ L(z) = \\phi(z) - z + z\\varPhi(z) \\] <p>\u200b\u5c06\u200b\\(D(L) \\sim N(\\mu, \\sigma)\\)\u200b\u4ee3\u5165\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u7684\u200b\u5206\u5e03\u200b\uff0c\u200b\u5f97\u5230\u200b</p> \\[ F_{\\mathrm{IL}}(x) = \\frac{1}{Q}\\int_{R}^{R+Q} \\left[1 - \\varPhi\\left(\\frac{u-x-\\mu}{\\sigma}\\right)\\right] \\mathrm du \\] <p>\u200b\u7531\u6b64\u200b\u53ef\u200b\u5f97\u200b\uff0c\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u7684\u200b\u6982\u7387\u5206\u5e03\u200b\\(f_{\\mathrm{IL}}(x)\\)\u200b\u4e3a\u200b</p> \\[ f_{\\mathrm{IL}}(x) = \\frac{1}{Q}\\left[\\varPhi\\left(\\frac{R+Q-x-\\mu}{\\sigma}\\right) - \\varPhi\\left(\\frac{R-x-\\mu}{\\sigma}\\right)\\right] \\]"},{"location":"math/operations-mamagement/inventory-control-random-demand/#lost-sales","title":"Lost-sales","text":"<p>\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u7f3a\u8d27\u200b\uff0c\u200b\u5219\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u4e0d\u200b\u4f4e\u4e8e\u200b0\uff0c\u200b\u6b64\u65f6\u200b\u5e73\u5747\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u53d8\u4e3a\u200b</p> \\[ \\frac{Q}{2} + \\max(R-\\mu, 0) \\] <p>\u200b\u5176\u4f59\u200b\u6210\u672c\u200b\u4e0d\u53d8\u200b\u3002</p>"},{"location":"math/operations-mamagement/inventory-control-random-demand/#_17","title":"\u968f\u673a\u200b\u63d0\u524d\u200b\u671f","text":""},{"location":"math/operations-mamagement/inventory-control-random-demand/#one-for-one-policy","title":"One-for-one policy","text":"<p>\u200b\u5f53\u200b\u8ba2\u8d27\u200b\u6279\u91cf\u200b\\(Q = 1\\)\u200b\u65f6\u200b\uff0c\u200b\u79f0\u4e3a\u200bOne-for-one policy\uff0c\u200b\u901a\u5e38\u200b\u9002\u7528\u200b\u4e8e\u200b\u987e\u5ba2\u200b\u9700\u6c42\u200b\u7387\u200b\u8f83\u200b\u4f4e\u200b\u800c\u200b\u4ea7\u54c1\u200b\u4ef7\u503c\u200b\u8f83\u200b\u9ad8\u200b\uff0c\u200b\u5982\u200b\u6c7d\u8f66\u200b\u7b49\u200b\u5927\u5b97\u200b\u5546\u54c1\u200b\u3002\u200b\u8be5\u200b\u7b56\u7565\u200b\u4e00\u200b\u51fa\u73b0\u200b\u9700\u6c42\u200b\u5c31\u200b\u8ba2\u8d27\u200b\uff0c\u200b\u4ece\u800c\u200b\u4f7f\u5f97\u200b\u5e93\u5b58\u200b\u4f4d\u7f6e\u200b\u59cb\u7ec8\u200b\u5904\u4e8e\u200b\\(S = Q + R\\)\u3002\u200b\u4ee5\u4e0b\u200b\u8003\u8651\u200b\u9700\u6c42\u200b\u79bb\u6563\u200b\u7684\u200b\u60c5\u51b5\u200b\uff1a</p> <ol> <li>\u200b\u9700\u6c42\u200b\u670d\u4ece\u200b\u53c2\u6570\u200b\u4e3a\u200b\\(\\lambda\\)\u200b\u7684\u200b\u6cca\u677e\u200b\u5206\u5e03\u200b</li> <li>\u200b\u5141\u8bb8\u200b\u5ef6\u671f\u200b\u4ea4\u8d27\u200b</li> <li>\u200b\u5355\u4f4d\u200b\u65f6\u95f4\u200b\u7684\u200b\u5355\u4f4d\u200b\u6301\u6709\u200b\u6210\u672c\u200b\u4e0e\u200b\u5355\u4f4d\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\u4e3a\u200b\\(h, p\\)</li> </ol> <p>\u200b\u5f53\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u4e3a\u200b\u5e38\u6570\u200b\\(\\tau\\)\u200b\u65f6\u200b\uff0c\u200b\u5206\u6790\u200b\u6301\u6709\u200b\u6210\u672c\u200b\u4e0e\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\u3002\u200b\u9996\u5148\u200b\u8003\u8651\u200b\u5e93\u5b58\u200b\u6c34\u5e73\u200b\u5904\u4e8e\u200b\u4e0d\u540c\u200b\u72b6\u6001\u200b\u7684\u200b\u6982\u7387\u200b\uff1a</p> \\[ P(\\mathrm{IL} = i) = P(D = S - i) = \\frac{e^{-\\lambda t}(\\lambda t)^{S-i}}{(S-i)!} \\] <p>\u200b\u5df2\u77e5\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u4e3a\u200b\\(i\\)\uff0c\u200b\u5bf9\u5e94\u200b\u7684\u200b\u5b58\u8d27\u91cf\u200b\u4e3a\u200b\\(\\max\\{S - D, 0\\}\\)\uff0c\u200b\u7f3a\u8d27\u91cf\u200b\u4e3a\u200b\\(\\max\\{D-S, 0\\}\\)\uff0c\u200b\u56e0\u6b64\u200b\u5e93\u5b58\u200b\u6210\u672c\u200b\u4e0e\u200b\u7f3a\u8d27\u200b\u6210\u672c\u200b\u5206\u522b\u200b\u4e3a\u200b</p> \\[ \\begin{aligned}     c_h &amp;= h\\sum_{D=0}^S (S-D)\\frac{e^{-\\lambda t}(\\lambda t)^{D}}{D!} \\\\     c_u &amp;= p\\sum_{D=S}^\\infty (D-S)\\frac{e^{-\\lambda t}(\\lambda t)^{D}}{D!} \\end{aligned} \\] <p>\u200b\u5f53\u200b\u8ba2\u8d27\u200b\u63d0\u524d\u200b\u671f\u200b\u968f\u673a\u200b\uff0c\u200b\u4e0d\u540c\u200b\u8ba2\u5355\u200b\u7684\u200b\u63d0\u524d\u200b\u671f\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\u65f6\u200b\uff0c\u200b\u4e3a\u200b\u7b80\u5316\u200b\u5904\u7406\u200b\uff0c\u200b\u8bbe\u200b\u63d0\u524d\u200b\u671f\u200b\u670d\u4ece\u200b\u5747\u503c\u200b\u4e3a\u200b\\(\\tau\\)\u200b\u7684\u200b\u6307\u6570\u5206\u5e03\u200b\uff0c\u200b\u5219\u200b\u7cfb\u7edf\u200b\u7684\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\u56fe\u4e3a\u200b</p> <p></p> <p>\u200b\u8be5\u200b\u7cfb\u7edf\u200b\u4e3a\u751f\u200b\u706d\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u5bf9\u4e8e\u200b\u6bcf\u4e2a\u200b\u72b6\u6001\u200b\\(i &lt; S\\)\uff0c\u200b\u6709\u200b</p> \\[ \\left(\\lambda + \\frac{S-i}{\\tau}\\right)P(\\mathrm{IL} = i) = \\lambda P(\\mathrm{IL} = i+1) + \\frac{S-i+1}{\\tau} P(\\mathrm{IL} = i-1) \\] <p>\u200b\u5f53\u200b\\(i = S\\)\u200b\u65f6\u200b\uff0c\\(\\lambda P(\\mathrm{IL}  = S) = \\frac{1}{\\tau}P(\\mathrm{IL} = S-1)\\)\u3002\u200b\u7531\u6b64\u200b\u53ef\u4ee5\u200b\u63a5\u200b\u7684\u200b</p> \\[ \\begin{aligned}     P(\\mathrm{IL} = S) &amp;= e^{-\\lambda\\tau} \\\\     P(\\mathrm{IL} = i) &amp;= \\frac{(\\lambda\\tau)^{S-i}}{(S-i)!}P(\\mathrm{IL} = S) \\\\     &amp;= \\frac{(\\lambda\\tau)^{S-i}e^{-\\lambda\\tau}}{(S-i)!} \\end{aligned} \\] <p>\u200b\u5f53\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u5ef6\u671f\u200b\u4ea4\u8d27\u200b\u65f6\u200b\uff0c\\(\\mathrm{IL}\\geq 0\\)\uff0c\u200b\u60c5\u51b5\u200b\u4e0e\u200b\u5141\u8bb8\u200b\u5ef6\u671f\u200b\u4ea4\u8d27\u200b\u65f6\u200b\u53c8\u200b\u6709\u200b\u4e0d\u540c\u200b\uff0c\u200b\u4ece\u200b\u65e0\u9650\u200b\u72b6\u6001\u200b\u9a6c\u6c0f\u94fe\u200b\u8f6c\u6362\u200b\u4e3a\u200b\u6709\u9650\u200b\u72b6\u6001\u200b\u9a6c\u6c0f\u94fe\u200b\u3002</p>"},{"location":"math/queue-theory/","title":"\u5bf9\u7b56\u200b\u3001\u200b\u51b3\u7b56\u200b\u4e0e\u200b\u6392\u961f\u200b\u8bba","text":"<p>\u200b\u8bfe\u7a0b\u200b\u76ee\u5f55\u200b\uff1a</p> <ol> <li>Chapter 1\uff0c</li> <li>Chapter 2\uff0c</li> <li>Chapter 3\uff0c</li> <li>Chapter 4\uff0c</li> <li>Chapter 5\uff0c</li> <li>Chapter 6\uff0c</li> <li>Chapter 7\uff0c</li> <li>Chapter 8\uff0c</li> <li>Chapter 9\uff0c</li> <li>Chapter 10\uff0c</li> <li>Chapter 11\uff0c</li> <li>Chapter 12\uff0c\u200b\u5e38\u89c1\u200b\u7684\u200b\u6392\u961f\u6a21\u578b\u200b</li> </ol>"},{"location":"math/queue-theory/chapter-12/","title":"\u5e38\u89c1\u200b\u6392\u961f\u6a21\u578b","text":"<p>\u200b\u4e00\u4e2a\u200b\u6392\u961f\u6a21\u578b\u200b\u4e3b\u8981\u200b\u7531\u200b\u4ee5\u4e0b\u200b\u90e8\u5206\u200b\u7ec4\u6210\u200b</p> <ul> <li>\u200b\u987e\u5ba2\u200b\uff08\u200b\u5b9e\u4f53\u200b\uff09\u200b\u5904\u7406\u200b</li> <li>\u200b\u5b9e\u4f53\u200b\u6392\u961f\u200b\u7b49\u5f85\u200b\u5904\u7406\u200b\uff08\u200b\u7cfb\u7edf\u200b\u7684\u200b\u6392\u961f\u200b\u7b56\u7565\u200b\u53ef\u80fd\u200b\u4e0d\u540c\u200b\uff0c\u200b\u635f\u5931\u200b\u5236\u200b\u4e0d\u200b\u5141\u8bb8\u200b\u6392\u961f\u200b\u3001\u200b\u6df7\u5408\u5236\u200b\u5141\u8bb8\u200b\u6392\u200b\u6709\u9650\u200b\u957f\u5ea6\u200b\u7684\u200b\u961f\u5217\u200b\uff09</li> <li>\u200b\u5b9e\u4f53\u200b\u5904\u7406\u5b8c\u6bd5\u200b\uff0c\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b</li> </ul>"},{"location":"math/queue-theory/chapter-12/#_2","title":"\u6392\u961f\u6a21\u578b\u200b\u7684\u200b\u63cf\u8ff0","text":""},{"location":"math/queue-theory/chapter-12/#_3","title":"\u8fdb\u5165\u200b\u7cfb\u7edf","text":"<p>\u200b\u5047\u8bbe\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u5b9e\u4f53\u200b\u6570\u91cf\u200b\u65e0\u9650\u200b\u3002\u200b\u8003\u5bdf\u200b\u76f8\u90bb\u200b\u4e24\u4e2a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u65f6\u95f4\u5dee\u200b\uff1a</p> <ol> <li>\u200b\u5bf9\u4e8e\u200b\u5b9a\u200b\u957f\u200b\u5206\u5e03\u200b\uff0c\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u65f6\u95f4\u5dee\u200b\u4e3a\u200b\u5e38\u91cf\u200b\\(t\\)\uff0c\\(t\\)\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\u4e3a\u200b\uff1a</li> </ol> \\[ f(t; \\alpha)=\\left\\{ \\begin{aligned}     &amp; 1 &amp; t=\\alpha \\\\     &amp; 0 &amp; t\\not =\\alpha \\end{aligned} \\right . \\] <ol> <li>Poisson \u200b\u6d41\u200b\uff0c\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u65f6\u95f4\u5dee\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u670d\u4ece\u200b\u53c2\u6570\u200b\u4e3a\u200b\\(\\lambda\\)\u200b\u7684\u200b\u6307\u6570\u5206\u5e03\u200b</li> </ol> \\[ f_{\\xi_i}(t; \\lambda)=\\left\\{ \\begin{aligned}     &amp; \\lambda e^{-\\lambda t} &amp; t\\geq 0 \\\\     &amp; 0 &amp; t &lt; 0 \\end{aligned} \\right . \\] <ol> <li>Erlang \u200b\u5206\u5e03\u200b\uff0c\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u65f6\u95f4\u5dee\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u670d\u4ece\u200b\u5982\u4e0b\u200b\u516c\u5f0f\u200b\u6240\u793a\u200b\u7684\u200b\u5206\u5e03\u200b\u3002\u200b\u82e5\u200b\u67d0\u4e2a\u200b\u7cfb\u7edf\u200b\u4e2d\u6709\u200b\\(K\\)\u200b\u4e2a\u200b\u5e76\u8054\u200b\u7684\u200b\u670d\u52a1\u53f0\u200b\uff0c\u200b\u8f93\u5165\u200b\u5b9e\u4f53\u200b\u4e3a\u200b Poisson \u200b\u6d41\u200b\uff0c\u200b\u5219\u200b\u7b2c\u200b\\(K\\)\u200b\u4e2a\u200b\u670d\u52a1\u53f0\u200b\u7684\u200b\u987e\u5ba2\u200b\u6d41\u4e3a\u200b Erlang \u200b\u6d41\u200b\uff0c\u200b\u524d\u9762\u200b\u6240\u6709\u200b\\(K - 1\\)\u200b\u4e2a\u200b\u670d\u52a1\u53f0\u200b\u5728\u200b\u7b2c\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u5230\u8fbe\u200b\u4ee5\u540e\u200b\u7684\u200b\u8f93\u5165\u200b\u6d41\u200b\u540c\u6837\u200b\u4e3a\u200b Erlang \u200b\u6d41\u200b\u3002</li> </ol> \\[ f(t; \\lambda, K)=\\left\\{ \\begin{aligned}     &amp; \\frac{\\lambda(\\lambda t)^{K-1}}{(K-1)!} &amp; t\\geq 0 \\\\     &amp; 0 &amp; t &lt; 0 \\end{aligned} \\right . \\]"},{"location":"math/queue-theory/chapter-12/#_4","title":"\u5b9e\u4f53\u200b\u6392\u961f\u200b\u4e0e\u200b\u5904\u7406","text":"<p>\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u53ef\u80fd\u200b\u6709\u200b\u4e00\u4e2a\u200b\u6216\u200b\u591a\u4e2a\u200b\u670d\u52a1\u53f0\u200b\u3002\u200b\u5728\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u7684\u200b\u6240\u6709\u200b\u670d\u52a1\u53f0\u200b\u90fd\u200b\u5904\u4e8e\u200b\u8fd0\u884c\u200b\u72b6\u6001\u200b\u65f6\u200b\uff0c\u200b\u65b0\u200b\u8fdb\u5165\u200b\u7684\u200b\u5b9e\u4f53\u200b\u65e0\u6cd5\u200b\u7acb\u5373\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b\uff0c\u200b\u800c\u662f\u200b\u8fdb\u5165\u200b\u961f\u5217\u200b\u6216\u200b\u76f4\u63a5\u200b\u79bb\u5f00\u200b\uff0c\u200b\u53d6\u51b3\u4e8e\u200b\u7cfb\u7edf\u200b\u7684\u200b\u6392\u961f\u200b\u7b56\u7565\u200b\u3002\u200b\u6392\u961f\u200b\u7b56\u7565\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u635f\u5931\u200b\u5236\u200b\u3001\u200b\u7b49\u5f85\u200b\u5236\u200b\u4e0e\u200b\u6df7\u5408\u5236\u200b\u3002  </p> <p>\u200b\u5728\u200b\u635f\u5931\u200b\u5236\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\uff0c\u200b\u5982\u679c\u200b\u65b0\u200b\u8fdb\u5165\u200b\u7684\u200b\u5b9e\u4f53\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u7acb\u5373\u200b\u5904\u7406\u200b\uff0c\u200b\u5219\u200b\u4f1a\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b\u3002 \u200b\u5728\u200b\u7b49\u5f85\u200b\u5236\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\uff0c\u200b\u5982\u679c\u200b\u65b0\u200b\u8fdb\u5165\u200b\u7684\u200b\u5b9e\u4f53\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u7acb\u5373\u200b\u5904\u7406\u200b\uff0c\u200b\u5219\u200b\u4f1a\u200b\u5728\u200b\u961f\u5217\u200b\u4e2d\u200b\u7b49\u5f85\u200b\uff0c\u200b\u76f4\u5230\u200b\u88ab\u200b\u5904\u7406\u200b\u3002 \u200b\u5728\u200b\u6df7\u5408\u5236\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\uff0c\u200b\u5982\u679c\u200b\u65b0\u200b\u8fdb\u5165\u200b\u7684\u200b\u5b9e\u4f53\u200b\u4e0d\u80fd\u200b\u88ab\u200b\u7acb\u5373\u200b\u5904\u7406\u200b\u4e14\u200b\u6b64\u65f6\u200b\u7684\u200b\u961f\u5217\u200b\u957f\u5ea6\u200b\u5927\u4e8e\u200b\u67d0\u4e00\u5e38\u200b\u6570\u503c\u200b\uff0c\u200b\u5219\u200b\u4f1a\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u5426\u5219\u200b\u4f1a\u200b\u8fdb\u5165\u200b\u961f\u5217\u200b\u7b49\u5f85\u200b\u3002</p> <p>\u200b\u5b9e\u4f53\u200b\u5728\u200b\u6392\u961f\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u7684\u200b\u884c\u4e3a\u200b\u53ef\u4ee5\u200b\u5206\u4e3a\u200b\u5982\u4e0b\u200b\u7c7b\u578b\u200b\uff1a</p> <ul> <li>\u200b\u79bb\u5f00\u200b\uff1a\u200b\u5f53\u200b\u5b9e\u4f53\u200b\u5f97\u77e5\u200b\u9700\u8981\u200b\u7b49\u5f85\u200b\u6216\u200b\u6ca1\u6709\u200b\u8010\u5fc3\u200b\u7ee7\u7eed\u200b\u7b49\u5f85\u200b\u65f6\u200b\uff0c\u200b\u5b9e\u4f53\u200b\u53ef\u80fd\u200b\u9000\u51fa\u200b\u961f\u5217\u200b</li> <li>\u200b\u53d8\u6362\u200b\uff1a\u200b\u5373\u200b\u5b9e\u4f53\u200b\u4e3a\u200b\u51cf\u5c11\u200b\u6392\u961f\u200b\u65f6\u95f4\u200b\u800c\u200b\u5728\u200b\u4e0d\u540c\u200b\u670d\u52a1\u53f0\u200b\u7684\u200b\u961f\u5217\u200b\u4e4b\u95f4\u200b\u79fb\u52a8\u200b</li> </ul> <p>\u200b\u670d\u52a1\u53f0\u200b\u4f1a\u200b\u6309\u7167\u200b\u5982\u4e0b\u200b\u53ef\u80fd\u200b\u7684\u200b\u65b9\u5f0f\u200b\u8fdb\u884c\u200b\u670d\u52a1\u200b\uff1a</p> <ul> <li>FCFS\uff1a\u200b\u5148\u200b\u8fdb\u5165\u200b\u961f\u5217\u200b\u7684\u200b\u5148\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b</li> <li>LCFS\uff1a\u200b\u540e\u200b\u8fdb\u5165\u200b\u961f\u5217\u200b\u7684\u200b\u5148\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b</li> <li>\u200b\u968f\u673a\u200b\u5904\u7406\u200b</li> <li>PS\uff1a\u200b\u4f18\u5148\u7ea7\u200b\u9ad8\u200b\u7684\u200b\u5b9e\u4f53\u200b\u5148\u200b\u8fdb\u884c\u200b\u5904\u7406\u200b</li> <li>\u200b\u65ad\u7eed\u200b\u5904\u7406\u200b\uff1a\u200b\u5728\u200b\u5904\u7406\u200b\u67d0\u4e2a\u200b\u5b9e\u4f53\u200b\u4e00\u6bb5\u65f6\u95f4\u200b\u540e\u200b\uff0c\u200b\u8f6c\u800c\u200b\u5904\u7406\u200b\u53e6\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\uff0c\u200b\u4e0d\u65ad\u200b\u5faa\u73af\u200b\u76f4\u5230\u200b\u5b9e\u4f53\u200b\u5904\u7406\u200b\u5b8c\u6210\u200b</li> </ul> <p>\u200b\u5728\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u7684\u200b\u670d\u52a1\u53f0\u200b\u7684\u200b\u670d\u52a1\u200b\u65f6\u95f4\u200b\u662f\u200b\u4e00\u4e2a\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u6982\u7387\u5206\u5e03\u200b\u4e3a\u200b\u5e38\u6570\u200b\u3001\u200b\u6307\u6570\u5206\u5e03\u200b\u3001K\u200b\u7ea7\u200b Erlang \u200b\u5206\u5e03\u200b\u3001\u200b\u4e00\u822c\u200b\u5206\u5e03\u200b\u7b49\u200b\u3002</p>"},{"location":"math/queue-theory/chapter-12/#_5","title":"\u7cfb\u7edf\u200b\u7684\u200b\u7b26\u53f7\u200b\u8868\u793a","text":"<p>Kendall\u200b\u8bb0\u53f7\u200b\uff0c\u200b\u683c\u5f0f\u200b\u4e3a\u200b\\(X/Y/Z/A/B/C\\)</p> <ul> <li>\\(X\\)\u200b\u8868\u793a\u200b\u5b9e\u4f53\u200b\u5230\u8fbe\u200b\u95f4\u9694\u65f6\u95f4\u200b\u7684\u200b\u5206\u5e03\u200b<ul> <li>\\(M\\)\u200b\u8868\u793a\u200b\u5230\u8fbe\u200b\u8fc7\u7a0b\u200b\u4e3a\u200b Poisson \u200b\u8fc7\u7a0b\u200b\uff08\u200b\u95f4\u9694\u65f6\u95f4\u200b\u670d\u4ece\u200b\u6307\u6570\u5206\u5e03\u200b\uff09\u200b\u6216\u8d1f\u200b\u6307\u6570\u5206\u5e03\u200b</li> <li>\\(D\\)\u200b\u8868\u793a\u200b\u95f4\u9694\u65f6\u95f4\u200b\u4e3a\u200b\u5e38\u6570\u200b</li> <li>\\(E_k\\)\u200b\u8868\u793a\u200b\u95f4\u9694\u65f6\u95f4\u200b\u670d\u4ece\u200b\\(k\\)\u200b\u9636\u200b Erlang \u200b\u5206\u5e03\u200b</li> <li>\\(G\\)\u200b\u8868\u793a\u200b\u95f4\u9694\u65f6\u95f4\u200b\u670d\u4ece\u200b\u4e00\u822c\u200b\u3001\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\u7684\u200b\u968f\u673a\u200b\u5206\u5e03\u200b</li> </ul> </li> <li>\\(Y\\)\u200b\u8868\u793a\u200b\u670d\u52a1\u53f0\u200b\u5355\u6b21\u200b\u670d\u52a1\u200b\u65f6\u95f4\u200b\u7684\u200b\u5206\u5e03\u200b\uff0c\u200b\u8bb0\u53f7\u200b\u4e0e\u200b\\(X\\)\u200b\u7684\u200b\u542b\u4e49\u200b\u76f8\u540c\u200b</li> <li>\\(Z\\)\u200b\u8868\u793a\u200b\u670d\u52a1\u53f0\u200b\u4e2a\u6570\u200b</li> <li>\\(A\\)\u200b\u8868\u793a\u200b\u7cfb\u7edf\u200b\u7684\u200b\u7b49\u5f85\u200b\u7a7a\u95f4\u200b\u5bb9\u91cf\u200b\uff0c\\(0\\)\u200b\u4e3a\u200b\u635f\u5931\u200b\u5236\u200b\u7cfb\u7edf\u200b\uff0c\\(\\infty\\)\u200b\u4e3a\u200b\u7b49\u5f85\u200b\u5236\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u5426\u5219\u200b\u4e3a\u200b\u6df7\u5408\u5236\u200b\u7cfb\u7edf\u200b\u3002</li> <li>\\(B\\)\u200b\u8868\u793a\u200b\u5b9e\u4f53\u200b\u7684\u200b\u603b\u200b\u6570\u91cf\u200b\uff0c\\(\\infty\\)\u200b\u8868\u793a\u200b\u5b9e\u4f53\u200b\u6e90\u200b\u65e0\u9650\u200b</li> <li>\\(C\\)\u200b\u8868\u793a\u200b\u670d\u52a1\u200b\u89c4\u5219\u200b</li> </ul> <p>\\(A, B, C\\)\u200b\u53c2\u6570\u200b\u53ef\u4ee5\u200b\u7701\u7565\u200b\u3002\u200b\u5206\u522b\u200b\u7b49\u4ef7\u200b\u4e8e\u200b\\(\\infty, \\infty, \\mathit{FCFS}\\)</p>"},{"location":"math/queue-theory/chapter-12/#_6","title":"\u7cfb\u7edf\u200b\u7684\u200b\u89c2\u6d4b\u200b\u6307\u6807","text":"<p>\u200b\u8861\u91cf\u200b\u7cfb\u7edf\u200b\u6027\u80fd\u200b\u7684\u200b\u5e38\u7528\u200b\u89c2\u6d4b\u200b\u6307\u6807\u200b\u4e3a\u200b\uff1a</p> <ul> <li>\\(N(L)\\)\uff1a\u200b\u7a33\u6001\u200b\u7cfb\u7edf\u200b\u7684\u200b\u961f\u957f\u200b\uff08\u200b\u6307\u200b\u7cfb\u7edf\u200b\u5185\u200b\u6240\u6709\u200b\u7684\u200b\u5b9e\u4f53\u200b\u6570\u91cf\u200b\uff09</li> <li>\\(N_q(L_q)\\)\u200b\u7a33\u6001\u200b\u7cfb\u7edf\u200b\u7684\u200b\u6392\u961f\u200b\u957f\u200b</li> <li>\\(T(W)\\)\uff1a\u200b\u987e\u5ba2\u200b\u5728\u200b\u7a33\u6001\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u7684\u200b\u505c\u7559\u65f6\u95f4\u200b\uff08\u200b\u5904\u7406\u200b+\u200b\u7b49\u5f85\u200b\uff09</li> <li>\\(T_q(W_q)\\)\uff1a\u200b\u987e\u5ba2\u200b\u5728\u200b\u7a33\u6001\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u7684\u200b\u7b49\u5f85\u65f6\u95f4\u200b</li> <li>\\(p_n\\)\uff1a\u200b\u7a33\u6001\u200b\u7cfb\u7edf\u200b\u4efb\u610f\u200b\u65f6\u523b\u200b\u72b6\u6001\u200b\u4e3a\u200b\\(n\\)\u200b\u7684\u200b\u6982\u7387\u200b\uff0c\\(p_n=P(N=n)=\\lim_{t\\rightarrow \\infty}P(N(t) = n)\\)</li> <li>\\(\\rho\\)\uff1a\u200b\u5229\u7528\u7387\u200b</li> <li>\\(p_D\\)\uff1a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u65f6\u200b\u9700\u8981\u200b\u7b49\u5f85\u200b\u7684\u200b\u6982\u7387\u200b</li> </ul> <p>\u200b\u82e5\u200b\u5c06\u200b\u7cfb\u7edf\u200b\u89c6\u4e3a\u200b\u7ba1\u9053\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u5b9e\u4f53\u200b\u7684\u200b\u6570\u91cf\u200b\u7b49\u4e8e\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u901f\u5ea6\u200b\u4e0e\u200b\u7cfb\u7edf\u200b\u5e73\u5747\u200b\u54cd\u5e94\u200b\uff08\u200b\u5904\u7406\u200b\uff09\u200b\u65f6\u95f4\u200b\u4e4b\u79ef\u200b\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\\(M/M/s\\)\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u4ee4\u200b\\(\\lambda\\)\u200b\u4e3a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\u7684\u200b\u901f\u5ea6\u200b\uff0c\\(\\mu\\)\u200b\u4e3a\u200b\u670d\u52a1\u53f0\u200b\u5904\u7406\u200b\u5b9e\u4f53\u200b\u7684\u200b\u901f\u5ea6\u200b\uff0c\\(s\\)\u200b\u4e3a\u200b\u670d\u52a1\u53f0\u200b\u7684\u200b\u6570\u91cf\u200b\uff0c\u200b\u5219\u200b\uff1a</p> \\[ \\begin{aligned} L &amp;= \\lambda_e W \\\\ L_q &amp;= \\lambda_e W_q \\end{aligned} \\]"},{"location":"math/queue-theory/chapter-12/#_7","title":"\u968f\u673a\u200b\u8fc7\u7a0b","text":"<p>\u200b\u5e38\u89c1\u200b\u7684\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b\u6709\u200b Markov \u200b\u8fc7\u7a0b\u200b\u3001\u200b\u751f\u706d\u200b\u8fc7\u7a0b\u200b\u3001 Poisson \u200b\u8fc7\u7a0b\u200b\u7b49\u200b\u3002\u200b\u6240\u6709\u200b\u7684\u200b Poisson \u200b\u8fc7\u7a0b\u200b\u90fd\u200b\u662f\u200b\u751f\u706d\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u6240\u6709\u200b\u7684\u200b\u751f\u706d\u200b\u8fc7\u7a0b\u200b\u90fd\u200b\u662f\u200b Markov \u200b\u8fc7\u7a0b\u200b\u3002</p>"},{"location":"math/queue-theory/chapter-12/#markov","title":"Markov \u200b\u8fc7\u7a0b","text":"<p>\u200b\u7279\u70b9\u200b\uff1a\\(t_0\\)\u200b\u65f6\u523b\u200b\u7684\u200b\u72b6\u6001\u200b\u4e0e\u200b\u4efb\u610f\u200b\\(t&lt;t_0\\)\u200b\u65f6\u200b\u7684\u200b\u72b6\u6001\u200b\u65e0\u5173\u200b</p> <p>Markov \u200b\u94fe\u200b\u7531\u200b\u9a6c\u5c14\u53ef\u592b\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u7684\u200b\u79bb\u6563\u200b\u72b6\u6001\u200b\u7ec4\u5408\u200b\u3002\u200b\u5bf9\u4e8e\u200b Markov \u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u65e0\u200b\u8bb0\u5fc6\u6027\u200b\uff0c\u200b\u5f53\u524d\u200b\u7684\u200b\u72b6\u6001\u200b\u4e0d\u4f1a\u200b\u5bf9\u200b\u540e\u9762\u200b\u7684\u200b\u72b6\u6001\u200b\u4ea7\u751f\u200b\u5f71\u54cd\u200b\u3002</p> <p>\\(M/M/s\\)\u200b\u7cfb\u7edf\u200b\u53ef\u4ee5\u200b\u4f7f\u7528\u200b Markov \u200b\u8fc7\u7a0b\u200b\u5efa\u6a21\u200b\u3002\u200b\u961f\u5217\u200b\u4e2d\u200b\u5b9e\u4f53\u200b\u7684\u200b\u6570\u91cf\u200b\u662f\u200b\u4e00\u4e2a\u200b Markov \u200b\u94fe\u200b\u3002</p>"},{"location":"math/queue-theory/chapter-12/#_8","title":"\u751f\u706d\u200b\u8fc7\u7a0b","text":"<p>\u200b\u751f\u706d\u200b\u8fc7\u7a0b\u200b\u6309\u200b\u5982\u4e0b\u200b\u56fe\u200b\u8fdb\u884c\u200b\u63cf\u8ff0\u200b\uff08\u200b\u72b6\u6001\u200b\u8f6c\u79fb\u200b\u56fe\u200b\uff09\uff1a</p> <p></p> <p>\u200b\u5047\u8bbe\u200b\u5b9e\u4f53\u200b\u79bb\u6563\u200b\u8fdb\u5165\u200b\u7684\u200b\\(M/M/s\\)\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u961f\u5217\u200b\u4e2d\u200b\u7684\u200b\u5b9e\u4f53\u200b\u6570\u91cf\u200b\u670d\u4ece\u200b\u751f\u706d\u200b\u8fc7\u7a0b\u200b</p>"},{"location":"math/queue-theory/chapter-12/#poisson","title":"Poisson \u200b\u8fc7\u7a0b","text":"<p>\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7684\u200b\u65f6\u95f4\u5dee\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\u4e14\u200b\u670d\u4ece\u200b\u540c\u4e00\u200b\u6307\u6570\u5206\u5e03\u200b\u7684\u200b\u8fc7\u7a0b\u200b</p>"},{"location":"math/queue-theory/chapter-12/#_9","title":"\u751f\u706d\u200b\u8fc7\u7a0b","text":"<p>\u200b\u5bf9\u4e8e\u200b\u751f\u706d\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u6709\u200b\u5982\u4e0b\u200b\u5047\u8bbe\u200b</p> <ul> <li>\u200b\u6bcf\u4e2a\u200b\u65f6\u523b\u200b\u53ea\u6709\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b</li> <li>\u200b\u5b9a\u4e49\u200b\u7cfb\u7edf\u200b\u7684\u200b\u72b6\u6001\u200b\u4e3a\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u7684\u200b\u5b9e\u4f53\u200b\u6570\u91cf\u200b\\(n\\)</li> <li>\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u7684\u200b\u72b6\u6001\u200b\u7531\u200b\\(n-1\\)\u200b\u8f6c\u53d8\u200b\u4e3a\u200b\\(n\\)</li> <li>\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u7684\u200b\u72b6\u6001\u200b\u7531\u200b\\(n\\)\u200b\u8f6c\u53d8\u200b\u4e3a\u200b\\(n+1\\)</li> <li>\u200b\u7cfb\u7edf\u200b\u5904\u4e8e\u200b\u72b6\u6001\u200b\\(n\\)\u200b\u65f6\u200b\uff0c\u200b\u5b9a\u4e49\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7684\u200b\u901f\u5ea6\u200b\u4e3a\u200b\\(\\lambda _n\\)\uff0c\u200b\u7cfb\u7edf\u200b\u7684\u200b\u5904\u7406\u901f\u5ea6\u200b\u4e3a\u200b\\(\\mu_n\\)\u3002\u200b\u5bf9\u4e8e\u200b\\(\\lambda_n\\)\uff0c\\(n\\geq 0\\)\uff1b\u200b\u5bf9\u4e8e\u200b\\(\\mu_n\\)\uff0c\\(n&gt;0\\)</li> </ul> <p>\u200b\u95ee\u9898\u200b\uff1a\u200b\u7cfb\u7edf\u200b\u5904\u4e8e\u200b\u7a33\u5b9a\u200b\u72b6\u6001\u200b\u65f6\u200b\uff0c\u200b\u751f\u706d\u200b\u7cfb\u7edf\u200b\u5904\u4e8e\u200b\u72b6\u6001\u200b\\(n\\)\u200b\u7684\u200b\u6982\u7387\u200b\\(p_n\\)\uff1f</p> <p>\u200b\u5bf9\u4e8e\u200b\u5904\u4e8e\u200b\u72b6\u6001\u200b\\(j\\)\u200b\u7684\u200b\u7cfb\u7edf\u200b\uff0c\u200b\u8003\u5bdf\u200b\u4e00\u4e2a\u200b\u6781\u77ed\u200b\u7684\u200b\u65f6\u95f4\u200b\u95f4\u9694\u200b\\(\\Delta t\\)\uff0c\u200b\u7531\u4e8e\u200b\\(\\Delta t\\)\u200b\u6781\u77ed\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5ffd\u7565\u200b\\(\\Delta t\\)\u200b\u65f6\u95f4\u200b\u5185\u200b\u4e24\u4e2a\u200b\u5b9e\u4f53\u200b\u540c\u65f6\u200b\u8fdb\u5165\u200b/\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b\u6216\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u3001\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002\\(\\Delta t\\)\u200b\u65f6\u95f4\u200b\u5185\u200b\u7cfb\u7edf\u200b\u53d1\u751f\u53d8\u5316\u200b\u7684\u200b\u6982\u7387\u200b\u5982\u4e0b\u200b\uff1a</p> <ul> <li>\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u8fdb\u5165\u200b\u7cfb\u7edf\u200b\uff1a\\(P(n(t+\\Delta t)=j+1 | n(t) = j)=\\lambda_j\\Delta t\\)</li> <li>\u200b\u4e00\u4e2a\u200b\u5b9e\u4f53\u200b\u79bb\u5f00\u200b\u7cfb\u7edf\u200b\uff1a\\(P(n(t+\\Delta t)=j-1| n(t) = j) = \\mu_j\\Delta t\\)</li> <li>\u200b\u7cfb\u7edf\u200b\u5728\u200b\\(\\Delta t\\)\u200b\u65f6\u95f4\u200b\u5185\u200b\u6ca1\u6709\u200b\u53d1\u751f\u200b\u72b6\u6001\u200b\u6539\u53d8\u200b\uff1a\\(P(n(t+\\Delta t) = j | n(t)=j) = 1 - (\\lambda_j+\\mu_j)\\Delta t\\)</li> </ul> <p>\u200b\u6839\u636e\u200b\u5047\u8bbe\u200b\uff0c\u200b\u5904\u4e8e\u200b\u72b6\u6001\u200b\\(i\\)\u200b\u7684\u200b\u7cfb\u7edf\u200b\u5728\u200b\\(t+\\Delta t\\)\u200b\u65f6\u200b\u53ea\u200b\u53ef\u80fd\u200b\u5904\u4e8e\u200b\\(i-1, i, i+1\\)\u200b\u4e09\u79cd\u200b\u72b6\u6001\u200b\u4e2d\u200b\u7684\u200b\u4e00\u79cd\u200b\u3002\u200b\u4e5f\u200b\u5373\u200b\uff0c\u200b\u5904\u4e8e\u200b\u72b6\u6001\u200b\\(i\\)\u200b\u7684\u200b\u7cfb\u7edf\u200b\u5728\u200b\\(t-\\Delta t\\)\u200b\u65f6\u200b\u53ea\u200b\u53ef\u80fd\u200b\u5904\u4e8e\u200b\\(i-1, i, i+1\\)\u200b\u4e09\u79cd\u200b\u72b6\u6001\u200b\u4e2d\u200b\u7684\u200b\u4e00\u79cd\u200b\u3002\u200b\u7531\u6b64\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u5217\u5f0f\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ \\begin{aligned}   p_0(t+\\Delta t) &amp;= (1-\\lambda_0\\Delta t)p_0(t) + \\mu_1\\Delta tp_1(t) \\\\   p_j(t+\\Delta t) &amp;= \\lambda_{j-1}\\Delta tp_{j-1}(t)+(1-\\mu_j\\Delta t-\\lambda_j\\Delta t)p_j(t)+\\mu_{j+1}\\Delta tp_{j+1}(t) \\end{aligned} \\] <p>\u200b\u6574\u7406\u200b\u540e\u53d6\u200b\u6781\u9650\u200b\\(\\Delta t\\rightarrow 0\\)\uff0c\u200b\u4e0a\u200b\u5f0f\u200b\u53d8\u4e3a\u200b\u5bfc\u6570\u200b\u5f62\u5f0f\u200b\uff1a</p> \\[ \\begin{aligned}   &amp;\\lim_{\\Delta t\\rightarrow 0} \\frac{p_j(t+\\Delta t)-p_j(t)}{\\Delta t}=\\lambda_{j-1}p_{j-1}(t)+\\mu_{j+1}p_{j+1}(t) - (\\mu_j+\\lambda_j)p_j(t) \\\\   \\Rightarrow &amp;\\frac{\\mathrm dp_j(t)}{\\mathrm dt}=\\lambda_{j-1}p_{j-1}(t)+\\mu_{j+1}p_{j+1}(t) - (\\mu_j+\\lambda_j)p_j(t) \\end{aligned} \\] <p>\u200b\u5f53\u200b\\(t\\rightarrow \\infty\\)\u200b\u65f6\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u8d8b\u4e8e\u200b\u7a33\u6001\u200b\uff0c\u200b\u5219\u200b\u6709\u200b\\(\\lim_{t\\rightarrow \\infty}p_j(t) = p_j\\)\uff0c\u200b\u6536\u655b\u200b\u7684\u200b\u5145\u5206\u200b\u5fc5\u8981\u6761\u4ef6\u200b\u4e3a\u200b\\(\\lim_{t\\rightarrow\\infty} \\frac{\\mathrm dp_j(t)}{\\mathrm dt}=0\\)\u3002\u200b\u5373\u200b\\(\\lambda_{j-1}p_{j-1}(t)+\\mu_{j+1}p_{j+1}(t) - (\\mu_j+\\lambda_j)p_j(t)=0\\)\u3002</p> <p>\u200b\u89e3\u200b\\(p_{j+1}\\)\uff0c\u200b\u89e3\u5f97\u200b\uff1a</p> \\[ p_{j+1} = \\left(\\frac{\\mu_j+\\lambda_j}{\\mu_{j+1}}\\right)p_j-\\frac{\\lambda_{j-1}}{\\mu_{j+1}}p_{j-1} \\] <p>\u200b\u5bf9\u4e8e\u200b\\(p_1\\)\uff0c\u200b\u6709\u200b\\(p_1=\\frac{\\lambda_0}{\\mu_1}p_0\\)\u3002</p> <p>\u200b\u6839\u636e\u200b\u9012\u63a8\u200b\u516c\u5f0f\u200b\uff0c\u200b\u53ef\u4ee5\u200b\u7528\u200b\\(p_0\\)\u200b\u8868\u793a\u200b\u51fa\u200b\\(p_j\\)\uff1a</p> \\[ \\begin{aligned}   p_2&amp;=\\left(\\frac{\\mu_1+\\lambda_1}{\\mu_2}\\right)p_1-\\frac{\\lambda _0}{\\mu_2}p_0 = \\frac{\\lambda_0(\\mu_1+\\lambda_1)}{\\mu_1\\mu_2}p_0-\\frac{\\lambda_0}{\\mu_2}p_0=\\frac{\\lambda_0\\lambda_1}{\\mu_1\\mu_2}p_0 \\\\   p_3&amp;=\\left(\\frac{\\mu_2+\\lambda_2}{\\mu_3}\\right)p_2-\\frac{\\lambda_1}{\\mu_3}p_1=\\frac{\\lambda_0\\lambda_1(\\mu_2+\\lambda_2)}{\\mu_1\\mu_2\\mu_3}p_0-\\frac{\\lambda_0\\lambda_1}{\\mu_1\\mu_3}p_0=\\frac{\\lambda_0\\lambda_1\\lambda_2}{\\mu_1\\mu_2\\mu_3}p_0 \\\\   &amp;\\vdots \\\\   &amp;p_j = \\frac{\\lambda_0\\lambda_1\\cdots\\lambda_{j-1}}{\\mu_1\\mu_2\\cdots\\mu_j}p_0=p_0\\prod_{i = 1}^{j}\\frac{\\lambda_{i-1}}{\\mu_i} \\end{aligned} \\] <p>\u200b\u7531\u4e8e\u200b\u7cfb\u7edf\u200b\u7684\u200b\u72b6\u6001\u200b\u53ea\u200b\u53ef\u80fd\u200b\u53d6\u200b\\(0\\sim\\infty\\)\u200b\u4e4b\u95f4\u200b\u7684\u200b\u6574\u6570\u200b\u503c\u200b\uff0c\u200b\u6982\u7387\u200b\u4e4b\u200b\u548c\u200b\u4e3a\u200b\\(1\\)\uff0c\u200b\u53ef\u4ee5\u200b\u8ba1\u7b97\u200b\u51fa\u200b\\(p_0\\)\uff1a</p> \\[ p_0=\\frac{1}{1+\\sum_{n=1}^\\infty \\prod_{i=0}^{n}\\frac{\\lambda_{i-1}}{\\mu_i}} \\]"},{"location":"math/queue-theory/chapter-12/#mm1","title":"M/M/1\u200b\u961f\u5217","text":"<p>\u200b\u5bf9\u4e8e\u200b\\(M/M/1\\)\u200b\u7cfb\u7edf\u200b\uff0c\\(\\lambda_0=\\lambda_1=\\cdots=\\lambda_n=\\lambda, \\mu_1=\\cdots=\\mu_n=\\mu\\)\u3002</p> <p>\u200b\u7531\u6b64\u200b\uff0c\u200b\u8ba1\u7b97\u200b\u51fa\u200b\\(M/M/1\\)\u200b\u7cfb\u7edf\u200b\u5728\u200b\u7a33\u6001\u200b\u65f6\u200b\u5904\u4e8e\u200b\\(p_0\\)\u200b\u7684\u200b\u6982\u7387\u200b\uff1a</p> \\[ p_0=\\frac{1}{1+\\sum_{n=1}^\\infty \\prod_{i=0}^{n}\\frac{\\lambda}{\\mu}} = \\frac{1}{1+\\sum_{n=1}^\\infty \\rho^n}=\\frac{1}{\\sum_{n=0}^\\infty \\rho^n}=1-\\rho \\] <p>\u200b\u6982\u7387\u5206\u5e03\u200b\u4e3a\u200b\uff1a\\(P(N=n)=\\rho^n(1-\\rho)\\)</p>"},{"location":"math/queue-theory/chapter-12/#mms","title":"M/M/s\u200b\u961f\u5217","text":"<p>\u200b\u5bf9\u4e8e\u200b\\(M/M/s\\)\u200b\u7cfb\u7edf\u200b\uff0c\\(\\lambda_i, \\mu_i\\)\u200b\u670d\u4ece\u200b\u5982\u4e0b\u200b\u89c4\u5f8b\u200b\uff1a</p> <ul> <li>\\(\\lambda_i = \\lambda \\qquad i=0, 1\\cdots, \\infty\\)</li> <li>\\(\\mu_i = \\max\\{i, s\\}\\mu \\qquad i=1, 2, \\cdots, \\infty\\)</li> </ul> <p>\u200b\u8ba1\u7b97\u200b\u51fa\u200b\\(M/M/s\\)\u200b\u7cfb\u7edf\u200b\u5728\u200b\u7a33\u6001\u200b\u65f6\u200b\u5904\u4e8e\u200b\\(p_0\\)\u200b\u7684\u200b\u6982\u7387\u200b\uff0c\u200b\u4ee4\u200b\\(\\rho_0=\\frac{\\lambda}{\\mu}\\)\uff1a</p> \\[ \\begin{aligned}   p_0&amp;=\\frac{1}{1+\\sum_{n=1}^\\infty \\prod_{i=0}^{n}\\frac{\\lambda}{\\max\\{i, s\\}\\mu}} \\\\   &amp;= \\frac{1}{\\sum_{i=0}^{s-1}\\frac{\\rho_0^i}{i!}+\\sum_{i=s}^\\infty \\frac{\\rho_0^i}{s!s^{i-s}}} \\\\   &amp;=\\frac{1}{\\sum_{i=0}^{s-1}\\frac{\\rho_0^i}{i!}+\\frac{\\rho_0^s}{s!}\\sum_{i=0}^\\infty \\frac{\\rho_0^i}{s^i}} \\\\   &amp;=\\frac{1}{\\sum_{i=0}^{s-1}\\frac{\\rho_0^i}{i!}+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}} \\end{aligned} \\] <p>\u200b\u8bbe\u200b\\(\\sum_{i=0}^{s-1}\\frac{\\rho_0^i}{i!}=T\\)\uff0c\\(p_i\\)\u200b\u53ef\u4ee5\u200b\u901a\u8fc7\u200b\\(p_0\\)\u200b\u63a8\u5bfc\u200b\u5f97\u200b\uff1a</p> \\[ p_i=\\frac{\\lambda_0\\lambda_1\\cdots\\lambda_{i-1}}{\\mu_1\\mu_2\\cdots\\mu_i}p_0=\\left\\{ \\begin{aligned} &amp; \\frac{\\rho_0^i}{i!}p_0=\\frac{\\rho_0^i}{i!\\left(T+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}\\right)} &amp; i\\leq s \\\\ &amp; \\frac{\\rho_0^i}{s!s^{i-s}}p_0=\\frac{\\rho_0^i}{s!s^{i-s}\\left(T+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}\\right)} &amp; i &gt; s \\end{aligned} \\right. \\] <p>\u200b\u5f53\u200b\\(n \\geq s\\)\u200b\u65f6\u200b\uff0c\u200b\u987e\u5ba2\u200b\u5230\u8fbe\u200b\u7cfb\u7edf\u200b\u65f6\u200b\u9700\u8981\u200b\u7b49\u5f85\u200b\uff0c\u200b\u5219\u200b\u6709\u200b\uff1a</p> \\[ p_D=\\sum_{i=s}^\\infty p_i= \\sum_{i=s}^\\infty \\frac{\\rho^i_0}{s!s^{i-s}}p_0 = \\frac{\\rho^s_0p_0}{(s-1)!(s-\\rho_0)}=\\frac{\\rho^s_0}{(s-1)!(s-\\rho_0)T+\\rho_0^s} \\] <p>\u200b\u5f53\u200b\\(N\\leq s\\)\u200b\u65f6\u200b\uff0c\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u6ca1\u6709\u200b\u961f\u5217\u200b\uff0c\\(L_q=0\\)\uff0c\u200b\u5f53\u200b\\(N&gt;s\\)\u200b\u65f6\u200b\uff0c\u200b\u961f\u5217\u200b\u957f\u5ea6\u200b\u4e3a\u200b\\(N-s\\)\uff0c\u200b\u56e0\u6b64\u200b\u7a33\u6001\u200b\u65f6\u200b\u7684\u200b\u5e73\u5747\u200b\u6392\u961f\u200b\u957f\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned}   L_q&amp;=\\sum_{i=s + 1}^\\infty (i - s)p_i \\\\   &amp;=\\sum_{i=s}^\\infty (i-s)\\frac{\\rho^i_0}{s!s^{i-s}}p_0 \\\\   &amp;= \\frac{\\rho^s_0p_0}{s!}\\sum_{i=0}^\\infty \\frac{i\\rho_0^i}{s^i} \\\\   &amp;= \\frac{\\rho^s_0p_0}{s!}\\sum_{i=1}^\\infty \\left(\\frac{\\rho_0}{s}\\right)^i\\frac{s}{s-\\rho_0} \\\\   &amp;= \\frac{\\rho^s_0p_0}{(s-1)!(s-\\rho)}\\frac{\\rho_0/s}{1-\\rho_0/s} \\\\   &amp;= \\frac{\\rho^{s+1}_0p_0}{(s-1)!(s-\\rho_0)^2} \\end{aligned} \\] <p>\u200b\u72b6\u6001\u200b\u4e3a\u200b\\(n\\)\u200b\u7684\u200b\u7a33\u6001\u200b\u65f6\u200b\u7cfb\u7edf\u200b\u63a5\u53d7\u200b\u670d\u52a1\u200b\u7684\u200b\u5b9e\u4f53\u200b\u6570\u91cf\u200b\u4e3a\u200b\uff1a</p> \\[ \\left\\{ \\begin{aligned} &amp;n &amp; n\\leq s \\\\ &amp;s &amp; n&gt; s \\end{aligned} \\right . \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\u200b\u7a33\u6001\u200b\u7cfb\u7edf\u200b\u4e2d\u200b\u63a5\u53d7\u200b\u670d\u52a1\u200b\u7684\u200b\u5b9e\u4f53\u200b\u5e73\u5747\u200b\u6570\u91cf\u200b\u4e3a\u200b\uff1a</p> \\[ \\begin{aligned}   L-L_q&amp;=\\sum_{i=0}^s ip_i+\\sum_{i=s+1}^\\infty sp_i \\\\   &amp;=\\sum_{i=0}^s \\frac{\\rho_0^i}{(i-1)!}p_0+s\\sum_{i=s+1}^\\infty \\frac{\\rho_0^i}{s!s^{i-s}}p_0 \\\\   &amp;= \\sum_{i=0}^s \\frac{\\rho_0^i}{(i-1)!}p_0 + \\frac{\\rho_0^sp_0}{(s-1)!}\\sum_{i=1}^\\infty \\frac{\\rho_0^i}{s^i} \\\\   &amp;= \\sum_{i=0}^s \\frac{\\rho_0^i}{(i-1)!}p_0 + \\frac{\\rho_0^{s+1}p_0}{(s-1)!(s-\\rho_0)} \\\\   &amp;= \\frac{\\sum_{i=0}^s \\frac{\\rho_0^i}{(i-1)!}}{\\left(T+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}\\right)} + \\frac{\\rho_0^{s+1}p_0}{(s-1)!(s-\\rho_0)} \\\\   &amp;= \\frac{\\rho_0 T+1}{T+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}} + \\frac{\\rho_0^{s+1}}{(s-1)!(s-\\rho_0)T+\\rho_0^s} \\\\   &amp;= \\frac{(\\rho_0 T+1)(s-1)!(s-\\rho_0) + \\rho_0^{s+1}}{(s-1)!(s-\\rho_0)T+\\rho_0^s}\\\\   &amp;= \\rho_0 + \\frac{(s-1)!(s-\\rho_0)}{(s-1)!(s-\\rho_0)T+\\rho_0^s} \\end{aligned} \\] <p>\u200b\u987e\u5ba2\u200b\u7684\u200b\u5e73\u5747\u200b\u9017\u7559\u200b\u65f6\u95f4\u200b\\(W=\\frac{L}{\\lambda}\\)\uff1a</p> \\[ \\begin{aligned}   W&amp;=\\frac{L}{\\lambda} \\\\   &amp;= \\frac{\\sum_{i=0}^\\infty ip_i}{\\lambda} \\\\   &amp;= \\frac{\\sum_{i=1}^s \\left(\\frac{\\rho_0^i}{(i-1)!\\left(T+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}\\right)}\\right) + \\sum_{i=s+1}^\\infty \\left(i\\frac{\\rho_0^i}{s!s^{i-s}\\left(T+\\frac{\\rho_0^s}{(s-1)!(s-\\rho_0)}\\right)}\\right) }{\\lambda} \\\\   &amp;= \\frac{\\frac{T(s-1)!(s-\\rho_0)}{T(s-1)!(s-\\rho_0) + \\rho_0^s} + \\frac{\\rho_0^{s+1}(s-\\rho_0+1)/(s-\\rho_0)}{T(s-1)!(s-\\rho_0) + \\rho_0^s}}{\\lambda} \\\\   &amp;= \\frac{T(s-1)!(s-\\rho_0) + \\rho_0^{s+1}(s-\\rho_0+1)/(s-\\rho_0)}{\\lambda(T(s-1)!(s-\\rho_0) + \\rho_0^s)} \\end{aligned} \\]"},{"location":"math/random-process/","title":"\u968f\u673a\u200b\u8fc7\u7a0b","text":"<ul> <li>\u200b\u9884\u5907\u200b\u77e5\u8bc6\u200b</li> <li>\u200b\u6cca\u677e\u200b\u8fc7\u7a0b\u200b</li> <li>\u200b\u8bb0\u53f7\u200b</li> </ul>"},{"location":"math/random-process/basics/","title":"\u9884\u5907\u200b\u77e5\u8bc6","text":""},{"location":"math/random-process/basics/#_2","title":"\u53ef\u6d4b\u200b\u7a7a\u95f4\u200b\u3001\u200b\u6982\u7387\u200b\u7a7a\u95f4\u200b\u4e0e\u200b\u6982\u7387\u200b\u6d4b\u5ea6","text":"<p>\\(\\renewcommand{\\geq}{\\geqslant}\\renewcommand{\\leq}{\\leqslant}\\) \u200b\u5f53\u200b\u4e00\u4e2a\u200b\u8bd5\u9a8c\u200b\u7684\u200b\u7ed3\u679c\u200b\u65e0\u6cd5\u200b\u9884\u5148\u786e\u5b9a\u200b\u65f6\u200b\uff0c\u200b\u79f0\u8be5\u200b\u5b9e\u9a8c\u200b\u4e3a\u200b\u968f\u673a\u200b\u8bd5\u9a8c\u200b\u3002\u200b\u968f\u673a\u200b\u8bd5\u9a8c\u200b\u53ef\u80fd\u200b\u51fa\u73b0\u200b\u7684\u200b\u6240\u6709\u200b\u7ed3\u679c\u200b\u96c6\u5408\u200b\u6784\u6210\u200b\u6837\u672c\u7a7a\u95f4\u200b\\(\\Omega\\)\uff0c\u200b\u6bcf\u4e2a\u200b\u53ef\u80fd\u200b\u7684\u200b\u7ed3\u679c\u200b\u79f0\u4e3a\u200b\u6837\u672c\u200b\u70b9\u200b\\(\\omega\\)\u3002\\(\\Omega\\)\u200b\u7684\u200b\u5b50\u96c6\u200b\u6784\u6210\u200b\u7684\u200b\u96c6\u5408\u200b\u79f0\u4e3a\u200b\u96c6\u7c7b\u200b\\(\\mathcal F\\)\u3002</p> <p>\u200b\u53ef\u6d4b\u200b\u7a7a\u95f4\u200b</p> <p>\u200b\u8bbe\u200b\\(\\mathcal F\\)\u200b\u4e3a\u200b\u7531\u200b\\(\\Omega\\)\u200b\u7684\u200b\u67d0\u4e9b\u200b\u5b50\u96c6\u200b\u6784\u6210\u200b\u7684\u200b\u975e\u200b\u7a7a\u96c6\u200b\u7c7b\u200b\uff0c\u200b\u82e5\u200b\u6ee1\u8db3\u200b\u4ee5\u4e0b\u200b\u6761\u4ef6\u200b\uff1a</p> <ol> <li>\u200b\u82e5\u200b\\(A\\in \\mathcal F\\)\uff0c\u200b\u5219\u200b\\(A^C = \\Omega - A\\in \\mathcal F\\)\u3002</li> <li>\u200b\u82e5\u200b\\(A_n\\in \\mathcal F\\)\uff0c\u200b\u5219\u200b\\(\\bigcup\\limits_{n=1}^\\infty A_n \\in \\mathcal F\\)</li> </ol> <p>\u200b\u79f0\u200b\\(\\mathcal F\\)\u200b\u4e3a\u200b\\(\\sigma\\)\u200b\u57df\u200b\uff0c\\((\\Omega, \\mathcal F)\\)\u200b\u4e3a\u200b\u53ef\u6d4b\u200b\u7a7a\u95f4\u200b\u3002</p> <p>\\(\\sigma\\)\u200b\u57df\u200b\\(\\mathcal F\\)\u200b\u5bf9\u200b\\(\\cap, \\cup, -\\)\u200b\u8fd0\u7b97\u200b\u5c01\u95ed\u200b\uff0c\u200b\u4efb\u4f55\u200b\u5143\u7d20\u200b\u7ecf\u8fc7\u200b\u53ef\u5217\u6b21\u200b\u8fd0\u7b97\u200b\u540e\u200b\u4ecd\u200b\u5c5e\u4e8e\u200b\\(\\mathcal F\\)\u3002</p> <p>\u200b\u5bf9\u4e8e\u200b\u96c6\u7c7b\u200b\\(\\mathcal A\\)\uff0c\u200b\u5305\u542b\u200b\\(\\mathcal A\\)\u200b\u7684\u200b\\(\\sigma\\)\u200b\u57df\u200b\u7684\u200b\u4ea4\u200b\u79f0\u4e3a\u200b\\(\\mathcal A\\)\u200b\u751f\u6210\u200b\u7684\u200b\\(\\sigma\\)\u200b\u57df\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(\\sigma(\\mathcal A)\\)\u3002\u200b\u5982\u200b\uff1a\\(\\sigma(\\{\\varnothing, A, \\Omega\\}) = \\{\\varnothing, A, A^C, \\Omega\\}\\)\u3002\u200b\u7279\u6b8a\u200b\u5730\u200b\uff0c\u200b\u8bb0\u200b\\(\\mathcal B = \\sigma(\\{(-\\infty, \\alpha], \\forall \\alpha \\in \\mathbb R\\})\\)\u200b\u4e3a\u200bBorel\u200b\u57df\u200b\u3002Borel\u200b\u57df\u200b\u89e3\u51b3\u200b\u4e86\u200b\u6837\u672c\u7a7a\u95f4\u200b\u5728\u200b\\(\\mathbb R\\)\u200b\u4e0a\u200b\u8fde\u7eed\u200b\u7684\u200b\u95ee\u9898\u200b\u3002\u200b\u53ef\u4ee5\u200b\u8bc1\u660e\u200b\uff0c\\(\\forall a &lt; b, [a, b]\\in \\mathcal B, (a, b]\\in \\mathcal B, [a, b)\\in \\mathcal B, (a, b)\\in \\mathcal B\\)\u3002\u200b\u5b9a\u4e49\u200b\\(\\mathcal B[a, b]\\)\u200b\u4e3a\u200b\u9650\u5236\u200b\u5728\u200b\\([a, b]\\)\u200b\u4e0a\u200b\u7684\u200bBorel\u200b\u57df\u200b\u3002</p> <p>\u200b\u6982\u7387\u200b\u6d4b\u5ea6\u200b\u4e0e\u200b\u6982\u7387\u200b\u7a7a\u95f4\u200b</p> <p>\u200b\u8bbe\u200b\\((\\Omega, \\mathcal F)\\)\u200b\u4e3a\u200b\u53ef\u6d4b\u200b\u7a7a\u95f4\u200b\uff0c\\(P: \\mathcal F\\rightarrow [0, 1]\\)\u200b\u4e3a\u200b\u5b9a\u4e49\u200b\u5728\u200b\\(\\mathcal F\\)\u200b\u4e0a\u200b\u7684\u200b\u96c6\u200b\u51fd\u6570\u200b\u3002\u200b\u4e14\u200b\\(P\\)\u200b\u6ee1\u8db3\u200b</p> <ol> <li>\u200b\u975e\u200b\u8d1f\u6027\u200b\uff1a\\(\\forall A\\in \\mathcal F, P(A)\\geq 0\\)</li> <li>\u200b\u89c4\u4e00\u6027\u200b\uff1a\\(P(\\Omega) = 1\\)</li> <li> <p>\u200b\u53ef\u5217\u200b\u53ef\u52a0\u6027\u200b\uff1a\u200b\u82e5\u200b\\(\\forall i\\in \\mathbb N, A_i\\in \\mathcal F\\)\uff0c\u200b\u4e14\u200b\\(\\forall i\\not = j, A_i\\cap A_j=\\varnothing\\)\uff0c\u200b\u5219\u200b</p> \\[ P\\left(\\bigcup_{n=0}^\\infty A_n\\right) = \\sum_{n=0}^\\infty P(A_n) \\] </li> </ol> <p>\u200b\u79f0\u200b\\(P\\)\u200b\u4e3a\u200b\u53ef\u6d4b\u200b\u7a7a\u95f4\u200b\\((\\Omega, \\mathcal F)\\)\u200b\u4e0a\u200b\u7684\u200b\u6982\u7387\u200b\u6d4b\u5ea6\u200b\uff0c\\((\\Omega, \\mathcal F, P)\\)\u200b\u4e3a\u200b\u6982\u7387\u200b\u7a7a\u95f4\u200b\u3002\\(\\mathcal F\\)\u200b\u4e3a\u200b\u4e8b\u4ef6\u200b\u57df\u200b\uff0c\\(A\\in \\mathcal F\\)\u200b\u4e3a\u200b\uff08\u200b\u968f\u673a\u200b\uff09\u200b\u4e8b\u4ef6\u200b</p> <p>\u200b\u6982\u7387\u200b\u6d4b\u5ea6\u200b\\(P\\)\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ol> <li>\u200b\u6709\u9650\u200b\u53ef\u52a0\u6027\u200b\uff1a\u200b\u53ef\u4ee5\u200b\u4ee4\u200b\\(A_{n+1} = A_{n+2} = \\cdots = \\varnothing\\)\uff0c\u200b\u7ed3\u5408\u200b\u53ef\u5217\u200b\u53ef\u52a0\u6027\u200b\u63a8\u51fa\u200b</li> <li>\\(P(A^C) = 1 - P(A)\\)<ul> <li>\\(P(\\varnothing) = 0\\)</li> </ul> </li> <li>\u200b\u96c6\u5408\u200b\u7684\u200b\u5305\u542b\u200b\u5173\u7cfb\u200b\uff1a\\(A\\subset B\\Rightarrow P(A)\\leq P(B)\\)</li> <li> <p>\u200b\u5bb9\u65a5\u200b\u539f\u7406\u200b\uff1a</p> \\[ P\\left(\\bigcup_{i=1}^n A_i\\right) = \\sum_{i=1}^n P(A_i) - \\sum_{1\\leq i &lt; j\\leq n}P(A_i\\cap A_j) + \\cdots + (-1)^{n+1}P(A_1\\cap \\cdots \\cap A_n) \\] <ul> <li>\\(P(A\\cup B) = P(A) + P(B) - P(A\\cap B)\\)</li> <li>\\(P(A - B) = P(A\\cup B) - P(B) = P(A) - P(A\\cap B)\\)</li> <li>\\(P\\left(\\bigcup_{i=1}^n A_i\\right)\\leq \\sum_{i=1}^n P(A_i)\\)</li> </ul> </li> </ol> <p>\u200b\u6ee1\u8db3\u200b\\(A_n\\subset A_{n+1}\\)\u200b\u7684\u200b\u4e8b\u4ef6\u200b\u5217\u200b\\(\\{A_n, n\\geq 1\\}\\)\u200b\u79f0\u4e3a\u200b\u5355\u8c03\u200b\u589e\u200b\u5e8f\u5217\u200b\uff0c\u200b\u6ee1\u8db3\u200b\\(A_n\\supset A_{n+1}\\)\u200b\u7684\u200b\u4e8b\u4ef6\u200b\u5217\u200b\\(\\{A_n, n\\geq 1\\}\\)\u200b\u79f0\u4e3a\u200b\u5355\u8c03\u200b\u51cf\u200b\u5e8f\u5217\u200b\u3002\u200b\u7531\u6b64\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u4e8b\u4ef6\u200b\u5217\u200b\u7684\u200b\u6781\u9650\u200b\uff1a</p> <ol> <li>\u200b\u82e5\u200b\\(\\{A_n, n\\geq 1\\}\\)\u200b\u4e3a\u200b\u5355\u8c03\u200b\u589e\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5219\u200b\\(\\lim\\limits_{n\\rightarrow \\infty}A_n = \\bigcup_{i=1}^\\infty A_i\\)</li> <li>\u200b\u82e5\u200b\\(\\{A_n, n\\geq 1\\}\\)\u200b\u4e3a\u200b\u5355\u8c03\u200b\u51cf\u200b\u5e8f\u5217\u200b\uff0c\u200b\u5219\u200b\\(\\lim\\limits_{n\\rightarrow \\infty}A_n = \\bigcap_{i=1}^\\infty A_i\\)</li> </ol> <p>\u200b\u4e8b\u4ef6\u200b\u5217\u200b\u7684\u200b\u6781\u9650\u200b\u6ee1\u8db3\u200b</p> \\[ \\lim_{n\\rightarrow\\infty}P(A_n) = P\\left(\\lim_{n\\rightarrow\\infty} A_n\\right) \\] <p>\u200b\u53ef\u4ee5\u200b\u8bc1\u660e\u200b\uff0c\u200b\u5355\u70b9\u200b\u96c6\u200b\\(\\{a\\}\\)\u200b\u4e3a\u200b\u4e8b\u4ef6\u200b\u5217\u200b\\(\\{[a, a+1/n], n &gt; 0\\}\\)\u200b\u7684\u200b\u6781\u9650\u200b\u3002</p> <p>Borel-Cantelli\u200b\u5f15\u7406\u200b</p> <p>\u200b\u8bbe\u200b\\(\\{A_n, n\\geq 1\\}\\)\u200b\u4e3a\u200b\u4e8b\u4ef6\u200b\u5e8f\u5217\u200b\uff0c\u200b\u6ee1\u8db3\u200b\\(\\sum_{i=1}^\\infty P(A_i) &lt; \\infty\\)\uff0c\u200b\u5219\u200b</p> \\[ P\\left(\\lim_{i\\rightarrow\\infty}\\sup A_i\\right) \\triangleq P\\left(\\bigcap_{n=1}^\\infty \\bigcup_{i=n}^\\infty A_i\\right) = 0 \\] <p>\u200b\u5b9a\u4e49\u200b\u6761\u4ef6\u200b\u6982\u7387\u200b</p> \\[ P(A|B) = \\frac{P(A\\cap B)}{P(B)} \\] <p>\u200b\u82e5\u200b\u4e8b\u4ef6\u200b\\(A, B\\)\u200b\u6ee1\u8db3\u200b\\(P(A\\cap B) = P(A)P(B)\\)\uff0c\u200b\u5219\u200b\u79f0\u200b\u4e8b\u4ef6\u200b\\(A, B\\)\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\uff0c\u200b\u4e14\u200b\\(P(A|B) = P(A)\\)\u3002\u200b\u540c\u7406\u200b\u53ef\u4ee5\u200b\u63a8\u5e7f\u200b\u81f3\u200b\\(n\\)\u200b\u4e2a\u200b\u4e8b\u4ef6\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\u7684\u200b\u60c5\u51b5\u200b\u3002\u200b\u8bbe\u200b\\(A_1, \\cdots, A_n\\in \\mathcal F\\)\uff0c\u200b\u82e5\u200b\u5bf9\u4e8e\u200b\u5176\u4e2d\u200b\u4efb\u610f\u200b\\(k\\)\u200b\u4e2a\u200b\u4e8b\u4ef6\u200b\uff0c\u200b\u90fd\u200b\u6709\u200b</p> \\[ P(A_{i_1}\\cap\\cdots\\cap A_{i_k}) = P(A_{i_1})\\cdots P(A_{i_k}) \\] <p>\u200b\u79f0\u200b\\(A_1, \\cdots, A_n\\)\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\u3002</p> <p>\u200b\u82e5\u200b\\(\\{A_n, n\\geq 1\\}\\)\u200b\u4e3a\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\u7684\u200b\u4e8b\u4ef6\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4e14\u200b\\(\\sum_{n=1}^\\infty P(A_n) = \\infty\\)\uff0c\u200b\u5219\u200b</p> \\[ P\\left(\\lim_{i\\rightarrow\\infty}\\sup A_i\\right) \\triangleq P\\left(\\bigcap_{n=1}^\\infty \\bigcup_{i=n}^\\infty A_i\\right) = 1 \\]"},{"location":"math/random-process/basics/#_3","title":"\u968f\u673a\u53d8\u91cf\u200b\u4e0e\u200b\u5206\u5e03\u200b\u51fd\u6570","text":"<p>\u200b\u8bbe\u200b\u6837\u672c\u7a7a\u95f4\u200b\u4e3a\u200b\\(\\Omega\\)\u3002</p>"},{"location":"math/random-process/basics/#_4","title":"\u968f\u673a\u53d8\u91cf","text":"<p>\u200b\u8bbe\u200b\\((\\Omega, \\mathcal F, P)\\)\u200b\u4e3a\u200b\u6982\u7387\u200b\u7a7a\u95f4\u200b\uff0c\\(X(\\omega)\\)\u200b\u4e3a\u200b\u5b9a\u4e49\u200b\u5728\u200b\\(\\Omega\\)\u200b\u4e0a\u200b\u7684\u200b\u5355\u503c\u200b\u5b9e\u200b\u51fd\u6570\u200b\uff0c\u200b\u5373\u200b\\(X: \\Omega\\rightarrow \\mathbb R\\)\uff0c\u200b\u82e5\u200b\\(\\forall a\\in \\mathbb R\\)\uff0c\u200b\u6709\u200b\\(\\{\\omega:  X(\\omega) \\leq a\\}\\in \\mathcal F\\)\uff0c\u200b\u79f0\u200b\\(X(\\omega)\\)\u200b\u4e3a\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u7b80\u8bb0\u200b\u4e3a\u200b\\(X\\)\u3002\u200b\u5b9a\u4e49\u200b\\(F(x) = P(X\\leq x) = P(X\\in (-\\infty, X])\\)\u200b\u4e3a\u200b\\(X\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u51fd\u6570\u200b\u3002</p> <ul> <li>\u200b\u79bb\u6563\u200b\u578b\u200b\u968f\u673a\u53d8\u91cf\u200b\uff1a\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X\\)\u200b\u7684\u200b\u53ef\u80fd\u200b\u53d6\u503c\u200b\u7684\u200b\u5168\u4f53\u200b\u662f\u200b\u53ef\u5217\u96c6\u200b\u6216\u8005\u200b\u6709\u9650\u200b\u96c6\u200b\u3002</li> <li> <p>\u200b\u8fde\u7eed\u578b\u200b\u968f\u673a\u53d8\u91cf\u200b\uff1a\u200b\u82e5\u200b\\(\\forall B\\in \\mathcal F\\)\uff0c\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\u51fd\u6570\u200b\\(f(x)\\)\u200b\u6ee1\u8db3\u200b</p> \\[ P(X\\in B) = \\int_B f(x)\\mathrm dx \\] <p>\u200b\u79f0\u200b\\(X\\)\u200b\u4e3a\u200b\u8fde\u7eed\u578b\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\\(f(x)\\)\u200b\u4e3a\u200b\\(X\\)\u200b\u7684\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\u3002</p> </li> </ul> <p>\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\u4e0e\u200b\u6982\u7387\u5206\u5e03\u200b\u51fd\u6570\u200b</p> <p>\u200b\u6839\u636e\u200b\u6982\u7387\u5206\u5e03\u200b\u51fd\u6570\u200b\u7684\u200b\u5b9a\u4e49\u200b\uff0c\u200b\u6709\u200b</p> \\[ P(x &lt; X\\leq x + h) = F(x + h) - F(x) = \\int_{x}^{x + h} f(x)\\mathrm dx = f(x)h + o(h) \\] <p>\u200b\u4ee4\u200b\\(h\\rightarrow 0\\)\u200b\u5e76\u53d6\u200b\u6781\u9650\u200b\uff0c\u200b\u5f97\u5230\u200b</p> \\[ \\frac{\\mathrm dF(x)}{\\mathrm dx} = \\lim_{h\\rightarrow 0} \\frac{F(x + h) - F(x)}{h} = f(x) \\] <p>\u200b\u5bf9\u4e8e\u200b\u4e8c\u7ef4\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X, Y\\)\uff0c\u200b\u5b9a\u4e49\u200b\u8054\u5408\u200b\u5206\u5e03\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(F(x, y) = P(X\\leq x, Y\\leq y)\\)\uff0c\u200b\u8fb9\u7f18\u200b\u5206\u5e03\u200b\u4e3a\u200b\\(F_X(x) = P(X\\leq x), F_Y = P(Y\\leq y)\\)\u3002\u200b\u540c\u7406\u200b\u53ef\u4ee5\u200b\u5b9a\u4e49\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\\(f(x, y)\\)\uff1a</p> \\[ F(x, y) = \\int_{-\\infty}^x\\int_{-\\infty}^y f(v, u)\\mathrm du\\mathrm dv \\] <p>\u200b\u82e5\u200b\\(F(x, y) = F_X(x)F_Y(y)\\)\uff0c\u200b\u79f0\u200b\\(X\\)\u200b\u4e0e\u200b\\(Y\\)\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\u3002</p>"},{"location":"math/random-process/basics/#_5","title":"\u6570\u5b57\u200b\u7279\u5f81","text":"<p>\u200b\u8bbe\u200b\\(X\\)\u200b\u4e3a\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u5206\u5e03\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(F(x)\\)\uff0c\u200b\u82e5\u200b\\(\\int_{-\\infty}^\\infty |x|\\mathrm dF(x)\\)\u200b\u5b58\u5728\u200b\uff0c\u200b\u5219\u200b\u5b9a\u4e49\u200b\\(X\\)\u200b\u7684\u200b\u671f\u671b\u200b\\(E(X)\\)\u200b\u4e3a\u200b</p> \\[ E(X) = \\int_{-\\infty}^\\infty x\\mathrm dF(x) \\] <p>\u200b\u6570\u5b66\u200b\u671f\u671b\u200b\\(E(X)\\)\u200b\u6ee1\u8db3\u200b</p> <ul> <li> <p>\u200b\u53ef\u52a0\u6027\u200b</p> \\[ E\\left(\\sum_{i=1}^n c_iX_i\\right) = \\sum_{i=1}^n c_iE(X_i) \\] </li> <li> <p>\u200b\u51fd\u6570\u200b</p> \\[ E(g(x)) = \\int_{-\\infty}^\\infty g(x)\\mathrm dF(x) \\] </li> </ul> <p>\u200b\u5bf9\u4e8e\u200b\u79bb\u6563\u200b\u578b\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u6709\u200b</p> \\[ E(X) = \\sum_{i=1}^\\infty x_nP(X = x_n) \\] <p>\u200b\u5bf9\u4e8e\u200b\u8fde\u7eed\u578b\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u6709\u200b</p> \\[ E(X) = \\int_{-\\infty}^\\infty xf(x)\\mathrm dx \\] <p>\u200b\u82e5\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X, Y\\)\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\uff0c\u200b\u5219\u200b\u6709\u200b</p> \\[ \\begin{aligned}     E(XY) &amp;= \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty xyf(x, y)\\mathrm dx\\mathrm dy \\\\     &amp;= \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty xyf_X(x)f_Y(y)\\mathrm dx\\mathrm dy \\\\     &amp;= \\int_{-\\infty}^\\infty xf_X(x)\\mathrm dx\\int_{-\\infty}^\\infty yf_Y(y)\\mathrm dy \\\\     &amp;= E(X)E(Y) \\end{aligned} \\] <p>\u200b\u5b9a\u4e49\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X\\)\u200b\u7684\u200b\u65b9\u5dee\u200b\u4e3a\u200b\\(\\sigma_X^2 = D(X) = E(X - E(X)) = E(X^2) - E^2(X)\\)\uff0c\u200b\u5b9a\u4e49\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X, Y\\)\u200b\u7684\u200b\u534f\u65b9\u5dee\u200b\u4e3a\u200b\\(\\mathrm{cov}(X, Y) = E(XY) - E(X)E(Y)\\)\uff0c\u200b\u76f8\u5173\u7cfb\u6570\u200b\u4e3a\u200b\\(\\rho = \\mathrm{cov}(X, Y) / (\\sigma_X\\sigma_Y)\\)\uff0c\\(k\\)\u200b\u9636\u77e9\u200b\u4e3a\u200b\\(E(X^k)\\)</p> <p>\u200b\u5b9a\u4e49\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X\\)\u200b\u7684\u200b\u77e9\u6bcd\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(\\psi(t) = E\\left(e^{tX}\\right)\\)\uff0c\u200b\u7279\u5f81\u51fd\u6570\u200b\u4e3a\u200b\\(\\phi(t) = E\\left(e^{\\mathbf itX}\\right)\\)</p> <p>\u200b\u77e9\u6bcd\u200b\u51fd\u6570\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ol> <li>\\(E\\left(X^k\\right) = \\psi^{(k)}(0)\\)</li> <li>\u200b\u8bbe\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X, Y\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(F_X(t), F_Y(t)\\)\uff0c\u200b\u77e9\u6bcd\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(\\psi_X(t), \\psi_Y(t)\\)\uff0c\u200b\u5219\u200b\\(\\psi_X(t) = \\psi_Y(t) \\Leftrightarrow F_X(t) = F_Y(t)\\)</li> </ol> <p>\u200b\u7279\u5f81\u51fd\u6570\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b</p> <ol> <li>\u200b\u8bbe\u200b\u968f\u673a\u53d8\u91cf\u200b\\(X, Y\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u51fd\u6570\u200b\u4e3a\u200b\\(F_X(t), F_Y(t)\\)\uff0c\u200b\u7279\u5f81\u51fd\u6570\u200b\u4e3a\u200b\\(\\phi_X(t), \\phi_Y(t)\\)\uff0c\u200b\u5219\u200b\\(\\phi_X(t) = \\phi_Y(t) \\Leftrightarrow F_X(t) = F_Y(t)\\)</li> <li>\\(i^kE(X^k) = \\phi^{(k)}(0)\\)</li> <li>\u200b\u82e5\u200b\\(X, Y\\)\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\uff0c\u200b\u6709\u200b\\(\\phi_{X+Y}(t) = \\phi_{X}(t) + \\phi_{Y}(t)\\)</li> </ol>"},{"location":"math/random-process/basics/#_6","title":"\u5e38\u89c1\u200b\u968f\u673a\u53d8\u91cf\u200b\u7684\u200b\u5206\u5e03","text":"<ol> <li> <p>\u200b\u4e8c\u9879\u5206\u5e03\u200b\uff1a\\(X\\sim B(n, p)\\)</p> \\[ P(X = k) = \\binom{n}{k} p^k(1 - p)^{n - k} \\] </li> <li> <p>\u200b\u6cca\u677e\u200b\u5206\u5e03\u200b\uff1a\\(X\\sim P(\\lambda)\\)</p> \\[ P(X = k) = \\frac{\\lambda^k}{k!}e^{-\\lambda} \\] </li> <li> <p>\u200b\u51e0\u4f55\u200b\u5206\u5e03\u200b\uff1a\\(X\\sim G(p)\\)</p> \\[ P(X = k) = (1 - p)^{k - 1}p \\] </li> <li> <p>\u200b\u5747\u5300\u5206\u5e03\u200b\uff1a\\(X\\sim U(a, b)\\)</p> \\[ f(x) = \\left\\{\\begin{aligned}     &amp; \\frac{1}{b-a} &amp; a &lt; x &lt; b \\\\     &amp; 0 &amp; \\text{otherwise} \\end{aligned}\\right. \\] </li> <li> <p>\u200b\u6b63\u6001\u5206\u5e03\u200b\uff1a\\(X\\sim N(\\mu, \\sigma^2)\\)</p> \\[ f(x) = \\frac{1}{\\sqrt{2\\pi}\\sigma} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}} \\] </li> <li> <p>\u200b\u6307\u6570\u5206\u5e03\u200b\uff1a\\(X\\sim E(\\lambda)\\)</p> \\[ f(x) = \\left\\{\\begin{aligned}     &amp; \\lambda e^{-\\lambda x} &amp; x\\geq 0 \\\\     &amp; 0 &amp; x &lt; 0 \\end{aligned}\\right. \\] </li> </ol>"},{"location":"math/random-process/basics/#_7","title":"\u6761\u4ef6\u200b\u6570\u5b66\u200b\u671f\u671b","text":"<p>\u200b\u5bf9\u4e8e\u200b\u8fde\u7eed\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u6761\u4ef6\u200b\u6570\u5b66\u200b\u671f\u671b\u200b\\(E(X|Y = y)\\)\u200b\u7684\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ E(X|Y = y) = \\int_{-\\infty}^\\infty x\\frac{f(x, y)}{f_Y(y)}\\mathrm dx \\] <p>\u200b\u5b9a\u4e49\u200b\\(E(X|Y)\\)\u200b\u4e3a\u200b\\(X\\)\u200b\u5173\u4e8e\u200b\\(Y\\)\u200b\u7684\u200b\u6761\u4ef6\u200b\u6570\u5b66\u200b\u671f\u671b\u200b</p> \\[ E(X|Y) = \\sum_j \\boldsymbol 1_{\\{Y = y_j\\}}(\\omega)E(X|Y=y_j) \\] <p>\u200b\u5bf9\u4e8e\u200b\u79bb\u6563\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u6761\u4ef6\u200b\u6570\u5b66\u200b\u671f\u671b\u200b\\(E(X|Y = y)\\)\u200b\u7684\u200b\u5b9a\u4e49\u200b\u5982\u4e0b\u200b\uff1a</p> \\[ E(X|Y = y) = \\sum_i x_iP(X=x_i | Y=y) \\] <p>\u200b\u663e\u7136\u200b\uff0c\\(E(X|Y = y)\\)\u200b\u662f\u200b\\(y\\)\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5b9a\u4e49\u200b\u968f\u673a\u53d8\u91cf\u200b\\(E(X|Y)\\)\u200b\u4e3a\u200b\\(X\\)\u200b\u5173\u4e8e\u200b\\(Y\\)\u200b\u7684\u200b\u6761\u4ef6\u671f\u671b\u200b\uff0c\u200b\u82e5\u200b\\(E(X|Y)\\)\u200b\u6ee1\u8db3\u200b</p> <ol> <li>\\(E(X|Y)\\)\u200b\u662f\u200b\\(Y\\)\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u5f53\u200b\\(Y = y\\)\u200b\u65f6\u200b\uff0c\\(E(X|Y) = E(X|Y = y)\\)</li> <li> <p>\\(\\forall D\\in \\mathcal B\\)\uff0c\u200b\u6709\u200b</p> \\[ E[E(X|Y)|Y\\in D] = E(X|Y\\in D) \\] </li> </ol> <p>\u200b\u6761\u4ef6\u200b\u6570\u5b66\u200b\u671f\u671b\u200b\u7684\u200b\u6027\u8d28\u200b</p> <p>\u200b\u6761\u4ef6\u200b\u6570\u5b66\u200b\u671f\u671b\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6027\u8d28\u200b\uff1a</p> <ol> <li>\\(E(E(X|Y)) = EX\\)</li> <li>\u200b\u82e5\u200b\\(X, Y\\)\u200b\u76f8\u4e92\u200b\u72ec\u7acb\u200b\uff0c\u200b\u5219\u200b\\(E(X|Y) = EX\\)</li> <li>\\(E[g(X)h(Y)|Y] = h(Y)E(g(X)|Y)\\quad \\text{a.s.}\\)</li> </ol>"},{"location":"math/random-process/basics/#_8","title":"\u968f\u673a\u200b\u8fc7\u7a0b","text":"<p>\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b\u662f\u200b\u4e00\u65cf\u200b\u65e0\u7a77\u200b\u591a\u4e2a\u200b\uff0c\u200b\u76f8\u4e92\u200b\u6709\u5173\u200b\u7684\u200b\u968f\u673a\u53d8\u91cf\u200b\u3002\u200b\u8bbe\u200b\\(X(t, \\omega)\\)\u200b\u4e3a\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u5176\u4e2d\u200b\\(t\\in T\\subset \\mathbb R\\)\u200b\u4e3a\u200b\u53c2\u6570\u200b\u3002\u200b\u79f0\u200b\\(X_T = \\{X(t, \\omega), t\\in T\\}\\)\u200b\u4e3a\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b\u3002</p> <ol> <li>\\(X(t, \\omega)\\)\u200b\u53ef\u4ee5\u200b\u7b80\u8bb0\u200b\u4e3a\u200b\\(X(t)\\)\uff1b</li> <li>\u200b\u5f53\u200b\\(T\\)\u200b\u4e3a\u200b\u53ef\u5217\u200b\u96c6\u65f6\u200b\uff0c\u200b\u79f0\u200b\\(X_T\\)\u200b\u4e3a\u200b\u968f\u673a\u200b\u5e8f\u5217\u200b\uff1b</li> <li>\\(X_T\\)\u200b\u7684\u200b\u53d6\u503c\u200b\u8303\u56f4\u200b\u79f0\u4e3a\u200b\u72b6\u6001\u200b\u7a7a\u95f4\u200b\\(S\\)\uff1b</li> <li>\\(X(\\cdot, \\omega), \\omega\\in \\Omega\\)\u200b\u662f\u200b\u5173\u4e8e\u200b\\(t\\)\u200b\u7684\u200b\u51fd\u6570\u200b\uff0c\u200b\u79f0\u4e3a\u200b\u8f68\u9053\u200b\u3002</li> </ol>"},{"location":"math/random-process/notations/","title":"\u8bb0\u53f7","text":"<p>\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b\u4e2d\u200b\u5e38\u7528\u200b\u7684\u200b\u8bb0\u53f7\u200b\u8868\u793a\u200b</p>"},{"location":"math/random-process/notations/#_2","title":"\u9884\u5907\u200b\u77e5\u8bc6","text":"<p>\u200b\u53ef\u5217\u96c6\u200b</p> <p>\u200b\u82e5\u200b\u4e00\u4e2a\u200b\u96c6\u5408\u200b\\(A\\)\u200b\u4e2d\u200b\u7684\u200b\u5143\u7d20\u200b\u53ef\u4ee5\u200b\u4ee5\u200b\u5e8f\u5217\u200b\\(\\{a_n\\}\\)\u200b\u7684\u200b\u5f62\u5f0f\u200b\u8868\u793a\u200b\uff0c\u200b\u5373\u200b\u5b58\u5728\u200b\u4e00\u4e2a\u200b\\(\\mathbb N_+\\rightarrow A\\)\u200b\u7684\u200b\u53cc\u5c04\u200b\uff0c\u200b\u79f0\u200b\u96c6\u5408\u200b\\(A\\)\u200b\u4e3a\u200b\u53ef\u5217\u96c6\u200b\u3002</p> <p>\u200b\u51e0\u4e4e\u200b\u5904\u5904\u200b</p> <p>\u200b\u8bbe\u200b\\(Z_1, Z_2\\)\u200b\u4e3a\u200b\u968f\u673a\u53d8\u91cf\u200b\uff0c\u200b\u4e14\u200b\\(P(Z_1 = Z_2) = 1\\)\uff0c\u200b\u79f0\u200b\\(Z_1, Z_2\\)\u200b\u51e0\u4e4e\u200b\u5904\u5904\u200b\u76f8\u7b49\u200b\uff0c\u200b\u8bb0\u200b\u4f5c\u200b\\(Z_1 = Z_2\\quad \\text{a.s.}\\)</p>"},{"location":"math/random-process/notations/#_3","title":"\u6cca\u677e\u200b\u8fc7\u7a0b","text":"<p>\u200b\u8ba1\u6570\u200b\u8fc7\u7a0b\u200b</p> <p>\u200b\u82e5\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b\\(\\{N(t), t\\geq 0\\}\\)\u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\uff0c\u200b\u79f0\u200b\\(N(t)\\)\u200b\u4e3a\u200b\u8ba1\u6570\u200b\u8fc7\u7a0b\u200b\u3002</p> <ol> <li>\\(N(t)\\in \\mathbb N\\)\uff0c\u200b\u5373\u200b\\(N(t)\\)\u200b\u53d6\u503c\u200b\u4e3a\u200b\u975e\u8d1f\u200b\u6574\u6570\u200b</li> <li>\\(\\forall t &gt; s, N(t)\\geq N(s)\\)</li> </ol>"},{"location":"math/random-process/poisson/","title":"\u6cca\u677e\u200b\u8fc7\u7a0b","text":"<p>\\(\\renewcommand{\\geq}{\\geqslant}\\renewcommand{\\leq}{\\leqslant}\\) \u200b\u6ee1\u8db3\u200b\u5982\u4e0b\u200b\u6761\u4ef6\u200b\u7684\u200b\u968f\u673a\u200b\u8fc7\u7a0b\u200b\\(\\{N(t), t\\geq 0\\}\\)\u200b\u4e3a\u200b\u65f6\u200b\u9f50\u6cca\u677e\u200b\u8fc7\u7a0b\u200b\u3002</p> <ol> <li>\\(\\{N(t), t\\geq 0\\}\\)\u200b\u4e3a\u200b\u8ba1\u6570\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u4e14\u200b\\(N(0) = 0\\)\uff1b</li> <li>\\(\\{N(t), t\\geq 0\\}\\)\u200b\u4e3a\u200b\u72ec\u7acb\u200b\u589e\u91cf\u200b\u8fc7\u7a0b\u200b\uff1b</li> <li>\\(\\{N(t), t\\geq 0\\}\\)\u200b\u5177\u5907\u200b\u589e\u91cf\u200b\u5e73\u7a33\u6027\u200b\uff0c\u200b\u5373\u200b\\(N(t) - N(s)\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u4ec5\u200b\u4e0e\u200b\\(t - s\\)\u200b\u6709\u5173\u200b</li> <li> <p>\u200b\u5bf9\u4e8e\u200b\u5145\u5206\u200b\u5c0f\u200b\u7684\u200b\\(\\Delta t &gt; 0\\)\uff0c\u200b\u6709\u200b</p> \\[ \\left\\{ \\begin{aligned}     P(N(t + \\Delta t) - N(t) = 1) &amp;= \\lambda \\Delta t + o(\\Delta t) \\\\     P(N(t + \\Delta t) - N(t) \\geq 2) &amp;= o(\\Delta t) \\end{aligned} \\right. \\] </li> </ol>"},{"location":"math/random-process/poisson/#_2","title":"\u589e\u91cf\u200b\u7684\u200b\u6982\u7387\u5206\u5e03","text":"<p>\u200b\u6cca\u677e\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u589e\u91cf\u200b\\(N(t + s) - N(s)\\)\u200b\u670d\u4ece\u200b\u53c2\u6570\u200b\u4e3a\u200b\\(\\lambda t\\)\u200b\u7684\u200b\u6cca\u677e\u200b\u5206\u5e03\u200b\uff0c\u200b\u5373\u200b</p> \\[ P[N(t + s) - N(s) = k] = \\frac{(\\lambda t)^k}{k!}e^{-\\lambda t} \\] \u200b\u8bc1\u660e\u200b <p>\u200b\u6839\u636e\u200b\u6cca\u677e\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u589e\u91cf\u200b\u5e73\u7a33\u6027\u200b\uff0c\u200b\u6709\u200b</p> \\[ P(N(s + t) - N(s) = n) = P(N(t) = n) \\] <ol> <li> <p>\\(n = 0\\)\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(\\{N(h + t) = 0, h &gt; 0\\} = \\{N(h+ t) - N(t) = 0, N(t) = 0, h &gt; 0\\}\\)\uff0c\u200b\u6839\u636e\u200b\u589e\u91cf\u200b\u72ec\u7acb\u6027\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\begin{aligned}     P(N(h + t) = 0) &amp;= P(N(h + t) - N(t) = 0) P(N(t) = 0) \\\\     &amp;= P(N(h) = 0) P(N(t) = 0) \\end{aligned} \\] <p>\u200b\u6839\u636e\u200b\u6cca\u677e\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u6027\u8d28\u200b\\(P(N(t + \\Delta t) - N(t) = 1) = \\lambda \\Delta t + o(\\Delta t)\\)\uff0c\u200b\u6709\u200b\uff1a</p> \\[ \\begin{aligned}     \\lim_{h\\rightarrow 0} \\frac{P(t + h) - P(t)}{h} &amp;= \\lim_{h\\rightarrow 0} \\frac{P(t)(1 - \\lambda h - o(h)) - P(t)}{h} \\\\     &amp;= -\\lambda P(t) - \\lim_{h\\rightarrow 0} \\frac{o(h)}{h} \\\\     &amp;= -\\lambda P(t) \\end{aligned} \\] <p>\u200b\u89e3\u200b\u5fae\u5206\u65b9\u7a0b\u200b\\(P_0'(t) = -\\lambda P_0(t)\\)\uff0c\u200b\u89e3\u5f97\u200b\\(P_0(t) = e^{-\\lambda t + C}\\)\uff0c\u200b\u53c8\u200b\u56e0\u4e3a\u200b\\(N(0) = 0, P_0(N(0) = 0) = 1\\)\uff0c\u200b\u5f97\u200b\\(C = 0\\)\uff0c\u200b\u4ece\u800c\u200b\u6709\u200b</p> \\[ P_0(t) = e^{-\\lambda t} \\] </li> <li> <p>\\(n &gt; 0\\)\u200b\u65f6\u200b\uff0c\\(\\{N(t + h) = n\\} = \\bigcup_{k = 0} ^ n \\{N(t) = k, N(t + h) - N(t) = (n - k)\\}\\)\uff0c\u200b\u800c\u200b\\(n - k \\geq 2\\)\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(P(N(t + h) - N(t) = n - k) = o(h)\\)\uff0c\u200b\u7531\u200b\u589e\u91cf\u200b\u72ec\u7acb\u6027\u200b</p> \\[ \\begin{aligned}     P(N(t + h) = n) &amp;= \\sum_{k = 0} ^ n P(N(t) = k) P(N(t + h) - N(t) = n - k) \\\\     &amp;= P(N(t) = n)(1 - \\lambda t - o(h)) + P(N(t) = n - 1)(\\lambda t + o(h)) \\\\     &amp;+ \\sum_{k = 2}^n P(N(t) = n - k)o(h) \\end{aligned} \\] <p>\u200b\u7531\u6b64\u200b</p> \\[ \\begin{aligned}     &amp;\\lim_{h\\rightarrow 0} \\frac{P(N(t + h) = n) - P(N(t) = n)}{h} \\\\     =&amp; \\lim_{h\\rightarrow 0}\\frac{P(N(t) = n)(\\sum_{k = 2}^n P(N(t) = n - k)o(h) - P(N(t) = n)}{h} \\\\     +&amp; \\lim_{h\\rightarrow 0}\\frac{\\sum_{k = 2}^n P(N(t) = n - k)o(h) - P(N(t) = n)}{h} \\\\     =&amp; \\lim_{h\\rightarrow 0}\\frac{P(N(t) = n)(- \\lambda t) + P(N(t) = n - 1)(\\lambda t)}{h} \\\\ \\end{aligned} \\] <p>\u200b\u5f97\u5230\u200b</p> \\[ P_n'(t) = -\\lambda P_n(t) + \\lambda P_{n-1}(t) \\] <p>\u200b\u8003\u8651\u200b\u7ea6\u675f\u6761\u4ef6\u200b\\(P_n(0) = 0\\)\uff0c\u200b\u89e3\u5f97\u200b</p> \\[ \\frac{\\mathrm d}{\\mathrm dt}[e^{\\lambda t} P_n(t)] = \\lambda e^{\\lambda t}P_{n - 1}(t) \\] <ol> <li>\\(n = 0\\)\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(P_n(t) = e^{-\\lambda t} = \\frac{(\\lambda t)^0}{0!}e^{-\\lambda t}\\)</li> <li> <p>\u200b\u8bbe\u200b\u5bf9\u4e8e\u200b\\(n &lt; k\\)\u200b\u65f6\u200b\u547d\u9898\u200b\u6210\u7acb\u200b\uff0c\u200b\u53ea\u200b\u9700\u200b\u8bc1\u660e\u200b\\(n = k\\)\u200b\u65f6\u200b\u547d\u9898\u200b\u6210\u7acb\u200b\u5373\u53ef\u200b\uff0c\u200b\u6709\u200b</p> \\[ \\begin{aligned}     P_k(t) &amp;= e^{-\\lambda t}\\int \\frac{\\mathrm d}{\\mathrm dt}[e^{\\lambda t} P_k(t)] \\mathrm dt \\\\     &amp;= e^{-\\lambda t}\\int \\lambda e^{\\lambda t}P_{k - 1}(t) \\mathrm dt \\\\     &amp;= e^{-\\lambda t}\\int \\lambda \\frac{(\\lambda t)^{k-1}}{(k-1)!} \\mathrm dt \\\\     &amp;= \\frac{(\\lambda t)^k}{k!}e^{-\\lambda t} \\end{aligned} \\] </li> </ol> </li> </ol>"},{"location":"math/random-process/poisson/#_3","title":"\u76f8\u90bb\u200b\u4e8b\u4ef6\u200b\u65f6\u95f4\u200b\u95f4\u9694\u200b\u7684\u200b\u6982\u7387\u5206\u5e03","text":"<p>\u200b\u8bbe\u200b\\(\\{N(t), t \\geq 0\\}\\)\u200b\u4e3a\u200b\u8ba1\u6570\u200b\u8fc7\u7a0b\u200b\uff0c\u200b\u8bbe\u200b\\(S_n\\)\u200b\u8868\u793a\u200b\u7b2c\u200b\\(n\\)\u200b\u4e2a\u200b\u65f6\u95f4\u200b\u53d1\u751f\u200b\u7684\u200b\u65f6\u523b\u200b\uff0c\u200b\u5373\u200b\\(S_n = \\inf\\{t, t &gt; S_{n - 1}, N(t) = n\\}\\)\uff0c\u200b\u7279\u522b\u200b\u5730\u200b\uff0c\u200b\u4ee4\u200b\\(S_0 = 0\\)\u3002\\(S_n, N(t)\\)\u200b\u7684\u200b\u4e8b\u4ef6\u200b\u6ee1\u8db3\u200b</p> \\[ \\begin{aligned}     \\{S_n \\leq t\\} &amp;\\equiv \\{N(t) \\geq n\\} \\\\     \\{N(t) = n\\} &amp;\\equiv \\{S_n\\leq t &lt; S_{n + 1}\\} \\equiv \\{S_n\\leq t\\} - \\{S_{n+1}\\leq t\\} \\end{aligned} \\] <p>\u200b\u7531\u4e8e\u200b\\(\\{S_n \\leq t\\} \\equiv \\{N(t) \\geq n\\}\\)\uff0c\u200b\u53ef\u4ee5\u200b\u8ba1\u7b97\u200b\\(S_n\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u51fd\u6570\u200b</p> \\[ P(S_n\\leq t) = P(N(t)\\geq n) = 1 - e^{-\\lambda t}\\sum_{k=0}^{n-1} \\frac{(\\lambda t)^{k}}{k!} \\] <p>\u200b\u7279\u522b\u200b\u5730\u200b\uff0c\u200b\u5f53\u200b\\(n = 1\\)\u200b\u65f6\u200b\uff0c\u200b\u6709\u200b\\(P(X_1\\leq t) = P(S_1\\leq t) = (1 - e^{-\\lambda t}) I_{\\{t\\geq 0\\}}\\)</p> <p>\u200b\u8ba1\u6570\u200b\u8fc7\u7a0b\u200b\\(\\{N(t), t\\geq 0\\}\\)\u200b\u662f\u200b\u6cca\u677e\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u5145\u8981\u6761\u4ef6\u200b\u662f\u200b\\(\\{X_n = S_{n} - S_{n-1}, n\\geq 1\\}\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u5206\u5e03\u200b\uff0c\u200b\u670d\u4ece\u200b\u7684\u200b\u5206\u5e03\u200b\u4e3a\u200b\u53c2\u6570\u200b\\(\\lambda\\)\u200b\u7684\u200b\u6307\u6570\u5206\u5e03\u200b\u3002</p> \u200b\u8bc1\u660e\u200b <ol> <li> <p>\\(\\Leftarrow\\)\uff1a\u200b\u5fc5\u8981\u6027\u200b\u8bc1\u660e\u200b\uff0c\u200b\u5148\u6c42\u200b\\(S_1, \\cdots, S_n\\)\u200b\u7684\u200b\u8054\u5408\u200b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u200b\uff0c\u200b\u8bbe\u200b\\(t_1 &lt; t_2 &lt; \\cdots &lt; t_n\\)\uff0c\u200b\u8bbe\u200b\\(h\\)\u200b\u4e3a\u200b\u5145\u5206\u200b\u5c0f\u200b\u7684\u200b\u6b63\u6570\u200b\uff0c\u200b\u6ee1\u8db3\u200b</p> \\[ t_1 + \\frac h2 &lt; t_2 - \\frac h2 &lt; t_2 + \\frac h2 &lt; \\cdots &lt; t_n - \\frac h2 \\] <p>\u200b\u8003\u8651\u200b\u4e8b\u4ef6\u200b\\(A\\)</p> \\[ \\begin{aligned}     A&amp; \\\\     &amp;\\equiv \\left\\{t_1 - \\frac h2 &lt; S_1 \\leq t_1 + \\frac h2, t_2 - \\frac h2 &lt; S_2 \\leq t_2 + \\frac h2, \\cdots, t_n - \\frac h2 &lt; S_n \\leq t_n + \\frac h2\\right\\} \\\\     \\equiv &amp; \\left\\{N\\left(t_1 - \\frac h2\\right) = 0, N\\left(t_{i+1} - \\frac h2\\right) - N\\left(t_i + \\frac h2\\right) = 0, N\\left(t_{i} + \\frac h2\\right) - N\\left(t_{i} - \\frac h2\\right) = 1\\right\\} \\\\     &amp; 0\\leq i\\leq n - 1 \\end{aligned} \\] <p>\u200b\u800c\u200b\u6839\u636e\u200b\u6cca\u677e\u200b\u8fc7\u7a0b\u200b\u7684\u200b\u6027\u8d28\u200b</p> \\[ \\begin{aligned}     P\\left(N\\left(t_n + \\frac h2\\right) - N\\left(t_n - \\frac h2\\right)= 1\\right) &amp;= \\lambda h + o(h) \\\\     P\\left(N\\left(t_n + \\frac h2\\right) - N\\left(t_n - \\frac h2\\right)\\geq 2\\right) &amp;= o(h) \\\\ \\end{aligned} \\] <p>\u200b\u6709\u200b</p> \\[ P(A) = (\\lambda h)^ne^{-\\lambda (t_n + h/2)} + o\\left(h^n\\right) = \\lambda ^nh^ne^{-\\lambda t_n} + o\\left(h^n\\right) \\] <p>\u200b\u56e0\u6b64\u200b\uff0c\\(\\left(S_1, \\cdots, S_n\\right)\\)\u200b\u7684\u200b\u6982\u7387\u5bc6\u5ea6\u200b\u4e3a\u200b</p> \\[ f(t_1, \\cdots, t_n) = \\left\\{ \\begin{aligned}     &amp; \\lambda^n e^{-\\lambda t_n} &amp; 0 &lt; t_1 &lt; \\cdots &lt; t_n \\\\     &amp; 0 &amp; \\mathrm{otherwise} \\end{aligned} \\right. \\] <p>\u200b\u800c\u200b\\(X_n = S_n - S_{n-1}\\)\uff0c\u200b\u8bbe\u200b\\(x_n = t_n - t_{n - 1}\\)\uff0c\u200b\u5219\u200b\\(t_n = \\sum_{i=1}^n x_n\\)\uff0c\u200b\u4ece\u800c\u200b\u53d8\u6362\u200b\u7684\u200bJacob\u200b\u77e9\u9635\u200b\u4e3a\u200b</p> \\[ J = \\frac{\\partial (t_1, \\cdots, t_n)}{\\partial (x_1, \\cdots, x_n)} = \\begin{vmatrix}     1 &amp; 0 &amp; \\cdots &amp; 0 \\\\     1 &amp; 1 &amp; \\cdots &amp; 0 \\\\     \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\     1 &amp; 1 &amp; \\cdots &amp; 1 \\end{vmatrix} \\] <p>\u200b\u56e0\u6b64\u200b\\(X_1, \\cdots, X_n\\)\u200b\u7684\u200b\u6982\u7387\u5bc6\u5ea6\u200b\u4e3a\u200b</p> \\[ g(x_1, \\cdots, x_n) = \\lambda^n e^{-\\lambda (x_1 + \\cdots + x_n)} = \\prod_{i=1}^n \\left(\\lambda e^{-\\lambda x_i}\\right), x_i \\geq 0 \\] <p>\u200b\u56e0\u6b64\u200b\\(X_1, \\cdots, X_n\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u6307\u6570\u5206\u5e03\u200b\u3002</p> </li> <li> <p>\\(\\Rightarrow\\)\uff1a\u200b\u5145\u5206\u6027\u200b\u8bc1\u660e\u200b\uff0c\u200b\u8bbe\u200b\\(\\{X_k, k\\geq 1\\}\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u6307\u6570\u5206\u5e03\u200b\uff0c\u200b\u8bbe\u200b\\(S_0 = 0, S_n = S_0 + \\sum_{i = 1}^n X_i\\)\u3002\u200b\u5b9a\u4e49\u200b</p> \\[ N(t) = \\sum_{n=1}^\\infty I_{\\{S_n\\leq t\\}} \\] <ul> <li>\u200b\u7531\u4e8e\u200b\\(I_{\\{S_n\\leq t\\}}\\in \\mathbb N\\)\uff0c\u200b\u5219\u200b\\(N(t)\\in \\mathbb N\\)\u3002</li> <li> <p>\u200b\u5bf9\u4e8e\u200b\\(t &gt; s\\)\uff0c\u200b\u6709\u200b</p> \\[ N(t) - N(s) = \\sum_{n = 1}^\\infty I_{\\{S_n\\leq t\\}}- I_{\\{S_n\\leq s\\}} \\] <p>\u200b\u800c\u200b\\(S_n - S_{n-1} = X_n &gt; 0\\)\uff0c\u200b\u56e0\u6b64\u200b\\(S_n\\)\u200b\u4e3a\u200b\u5355\u8c03\u200b\u589e\u200b\u5e8f\u5217\u200b\uff0c\u200b\u4ece\u800c\u200b\\(\\{S_n\\leq s\\}\\subset \\{S_n\\leq t\\}\\)\uff0c\u200b\u5373\u200b\\(I_{\\{S_n\\leq t\\}}- I_{\\{S_n\\leq s\\}} \\geq 0\\)\uff0c\u200b\u56e0\u6b64\u200b\\(N(t) - N(s)\\)</p> </li> </ul> <p>\u200b\u53ef\u200b\u5f97\u200b\\(N(t)\\)\u200b\u4e3a\u200b\u8ba1\u6570\u200b\u8fc7\u7a0b\u200b\u3002\u200b\u4ee5\u4e0b\u200b\u8ba1\u7b97\u200b\\(S_n\\)\u200b\u7684\u200b\u5206\u5e03\u200b\u3002\u200b\u5df2\u77e5\u200b\\(X_1, \\cdots, X_n\\)\u200b\u72ec\u7acb\u200b\u540c\u200b\u6307\u6570\u5206\u5e03\u200b\uff0c\u200b\u8bbe\u200b\\(t_n = \\sum_{i=1}^n x_i\\)\uff0c\u200b\u5219\u200b\\(x_n = t_n - t_{n-1}\\)\uff0c\u200b\u4ece\u800c\u200bJacob\u200b\u77e9\u9635\u200b\u4e3a\u200b</p> \\[ J = \\frac{\\partial (x_1, \\cdots, x_n)}{\\partial (t_1, \\cdots, t_n)} = \\begin{vmatrix}     1 &amp; 0 &amp; 0 &amp; \\cdots &amp; 0 \\\\     -1 &amp; 1 &amp; 0 &amp; \\cdots &amp; 0 \\\\     0 &amp; -1 &amp; 1 &amp; \\cdots &amp; 0 \\\\     \\vdots &amp; \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\     0 &amp; 0 &amp; 0 &amp; \\cdots &amp; 1 \\end{vmatrix} \\] <p>\u200b\u800c\u200b\\(X_1, \\cdots, X_n\\)\u200b\u7684\u200b\u8054\u5408\u200b\u6982\u7387\u5bc6\u5ea6\u200b\u4e3a\u200b</p> \\[ g(x_1, \\cdots, x_n) = \\lambda^n e^{-\\lambda (x_1 + \\cdots + x_n)} = \\prod_{i=1}^n \\left(\\lambda e^{-\\lambda x_i}\\right), x_i \\geq 0 \\] <p>\u200b\u56e0\u6b64\u200b\\(S_1, \\cdots, S_n\\)\u200b\u7684\u200b\u8054\u5408\u200b\u6982\u7387\u5bc6\u5ea6\u200b\u4e3a\u200b</p> \\[ f(t_1, \\cdots, t_n) = \\left\\{ \\begin{aligned}     &amp; \\lambda^n e^{-\\lambda t_n} &amp; 0 &lt; t_1 &lt; \\cdots &lt; t_n \\\\     &amp; 0 &amp; \\mathrm{otherwise} \\end{aligned} \\right. \\] </li> </ol>"},{"location":"notes/2020/","title":"\u4e00\u4e9b\u200b\u5f88\u200b\u559c\u6b22\u200b\u7684\u200b\u63a8\u9001","text":"<p>\u300a\u200b\u6628\u665a\u200b\u300b by \u200b\u8001\u9f20\u200b\u4ec0\u4e48\u200b\u90fd\u200b\u77e5\u9053\u200b</p> <p>\u300a\u200b\u6740\u6b7b\u200b\u65e0\u804a\u200b\u7684\u200b\u540c\u65f6\u200b\uff0c\u200b\u4f60\u200b\u4e5f\u200b\u5728\u200b\u6740\u6b7b\u200b\u81ea\u5df1\u200b\u300b by \u200b\u4eba\u7269\u200b</p> <p>\u300a\u200b\u75db\u82e6\u200b\u7684\u200b\u4e00\u767e\u5f20\u200b\u5e95\u7247\u200b\u300b by \u200b\u4eba\u7269\u200b</p> <p>\u300a\u200b\u603b\u200b\u4e0d\u80fd\u200b\u4e27\u6c14\u200b\u592a\u4e45\u200b\u5427\u200b\u3002\u300b by \u200b\u6982\u7387\u8bba\u200b</p> <p>\u300a\u200b\u6211\u4eec\u200b\u628a\u200b\u8fd9\u573a\u200b\u521d\u96ea\u200b\u505a\u6210\u200b\u4e86\u200b\u7535\u5f71\u6d77\u62a5\u200b\u300b by \u200b\u6e05\u534e\u5927\u5b66\u200b\u6e05\u65b0\u200b\u65f6\u62a5\u200b</p> <p>\u300a\u200b\u56f0\u4e8e\u200b\u4eba\u8138\u8bc6\u522b\u200b\u300b by \u200b\u4eba\u7269\u200b</p> <p>\u300a\u200b\u65b0\u200b\u7684\u200b\u5341\u5e74\u200b\uff0c\u200b\u4eba\u751f\u200b\u7684\u200b\u5947\u5e7b\u200b\u6f02\u6d41\u200b\u300b by \u200b\u4eba\u7269\u200b</p> <p>\u300a\u200b\u6587\u827a\u200b\u662f\u200b\u4e00\u573a\u200b\u9a97\u5c40\u200b\u5417\u200b\uff1f\u300b by \u200b\u4eba\u7269\u200b</p> <p>\u300a\u200b\u4e00\u7897\u200b\u7c73\u200b\u8c46\u8150\u200b\uff0c\u200b\u7a7f\u8d8a\u200b\u5c71\u6d77\u200b\u300b by \u200b\u4eba\u7269\u200b</p> <p>\u300a\u200b\u4fdd\u7814\u200b\u5931\u8d25\u200b\u4ee5\u540e\u200b\uff0c\u200b\u6211\u200b\u53c8\u200b\u8003\u200b\u4e86\u200b\u6b21\u200b\u6e05\u534e\u200b\u300b by \u200b\u6e05\u534e\u200b\u7814\u8bfb\u200b\u95f4\u200b</p> <p>\u300a\u200b\u4eca\u5929\u200b\u662f\u200b2020\u200b\u5e74\u200b\u4f60\u200b\u5728\u200b\u6e05\u534e\u200b\u7206\u809d\u200b\u7684\u200b\u7b2c\u200b366\u200b\u5929\u200b\u300b by \u200b\u6e05\u534e\u5927\u5b66\u200b\u6e05\u65b0\u200b\u65f6\u62a5\u200b</p>"},{"location":"notes/2021/","title":"\u4e00\u4e9b\u200b\u5f88\u200b\u559c\u6b22\u200b\u7684\u200b\u63a8\u9001","text":"<p>\u300a2020 \u200b\u6700\u200b\u820d\u4e0d\u5f97\u200b\u5220\u200b\u7684\u200b\u7167\u7247\u200b\u300b by \u200b\u6e05\u534e\u5927\u5b66\u200b</p> <p>\u200b\u665a\u5b89\u200b\u6d77\u62a5\u200b by \u200b\u90d1\u5dde\u200b\u53d1\u5e03\u200b</p> <p>\u200b\u77e5\u4e4e\u200b\u5b87\u5b99\u200b\u91cc\u200b\u7684\u200b\u90a3\u4e9b\u200b\u8d22\u5bcc\u200b\u5bc6\u7801\u200b by \u200b\u4e3a\u200b\u4f60\u200b\u5199\u200b\u4e00\u4e2a\u200b\u6545\u4e8b\u200b</p> <p>\u200b\u540c\u5b66\u200b\uff0c\u200b\u8fd8\u200b\u8bb0\u5f97\u200b\u5f53\u5e74\u200b\u90a3\u9053\u200b\u9ad8\u8003\u9898\u200b\u5417\u200b\uff1f by \u200b\u6e05\u534e\u5927\u5b66\u200b</p> <p>\u200b\u7ea2\u200b\u4e86\u200b\u7ea2\u200b\u4e86\u200b\uff0c\u200b\u8fd9\u6b21\u200b\u771f\u7684\u200b\u7ea2\u200b\u4e86\u200b by GQ\u200b\u5b9e\u9a8c\u5ba4\u200b</p> <p>\u200b\u751f\u6d3b\u200b\u5c0f\u7a8d\u95e8\u200b \u200b\u4e4b\u200b \u200b\u5982\u4f55\u200b\u6577\u884d\u200b\u5973\u751f\u8282\u200b by \u200b\u55e8\u200b\u76ae\u200b\u8fd0\u8425\u90e8\u200b</p> <p>\u200b\u6c99\u5c18\u66b4\u200b\u5bf9\u200b\u6e05\u534e\u56ed\u200b\u8bf4\u200b\uff0c\u200b\u4eca\u5929\u200b\u7ed9\u200b\u4f60\u200b\u70b9\u200b\u989c\u8272\u200b\u770b\u770b\u200b by \u200b\u5bf9\u200b\u725b\u200b\u8c08\u6e05\u200b</p>"},{"location":"notes/2022/","title":"\u4e00\u4e9b\u200b\u5f88\u200b\u559c\u6b22\u200b\u7684\u200b\u63a8\u9001","text":"<p>\u200b\u665a\u5b89\u200b\u6d77\u62a5\u200b by \u200b\u90d1\u5dde\u200b\u53d1\u5e03\u200b</p> <p>\u200b\u6e05\u534e\u200b\u5bf9\u200b\u4f60\u200b\u8bf4\u200b\u665a\u5b89\u200b\uff01 by \u200b\u6e05\u534e\u5927\u5b66\u200b</p> <p>\u200b\u5c0f\u533a\u200b\u88ab\u200b\u5c01\u200b\u4e94\u5929\u200b\u540e\u200b\uff0c\u200b\u6211\u200b\u6e10\u6e10\u200b\u61c2\u200b\u4e86\u200b by \u200b\u96f7\u53d4\u200b\u5199\u200b\u6545\u4e8b\u200b</p> <p>\u201c\u200b\u6211\u4eec\u200b\u7ec8\u200b\u4f1a\u200b\u627e\u5230\u200b\u610f\u4e49\u200b\u3002\u201d by \u200b\u6e05\u534e\u5927\u5b66\u200b</p> <p>40\u200b\u4e2a\u200b\u975e\u5fc5\u8981\u200b\u751f\u6d3b\u200b\u4e0b\u200b\u7684\u200b\u5e78\u798f\u200b\u77ac\u95f4\u200b by \u200b\u5168\u5143\u200b\u5149\u6ed1\u200b</p> <p>\u200b\u7ba1\u200b\u4ed6\u200b\u51e0\u5c81\u200b\uff0c\u200b\u9752\u6625\u200b\u4e07\u5c81\u200b by \u200b\u5c0f\u6797\u200b</p> <p>\u300c\u200b\u65f6\u4ee3\u200b\u4e4b\u97f3\u200b\u300d\u200b\u53bb\u200b\u54ea\u513f\u200b\u4e86\u200b\uff1f by \u200b\u4eba\u7269\u200b</p> <p>\u200b\u5e94\u200b \u200b\u6da6\u200b \u200b\u5c3d\u200b \u200b\u6da6\u200b by \u200b\u5168\u5143\u200b\u5149\u6ed1\u200b</p> <p>\u200b\u6bd5\u4e1a\u200b\u7279\u520a\u200b | \u200b\u634c\u200b\u6765\u200b\u65e0\u6059\u200b\uff0c\u200b\u540e\u4f1a\u6709\u671f\u200b by \u200b\u6e05\u534e\u5927\u5b66\u200b\u6e05\u65b0\u200b\u65f6\u62a5\u200b</p> <p>\u200b\u6bd5\u4e1a\u200b\u65f6\u200b\uff0c\u200b\u4f60\u200b\u6700\u200b\u60f3\u200b\u5e26\u8d70\u200b\u7684\u200b\u4e00\u4ef6\u200b\u4e1c\u897f\u200b\u662f\u200b\uff1f by \u200b\u6e05\u534e\u5927\u5b66\u200b</p> <p>\u200b\u6bd5\u4e1a\u200b\u5feb\u4e50\u200b | \u200b\u7247\u523b\u200b\u6210\u4e3a\u200b\u6c38\u6052\u200b by \u200b\u957f\u6b4c\u200b\u695a\u5929\u200b\u78a7\u200b</p> <p>\u200b\u4fdd\u7814\u200b\u5931\u8d25\u200b\u4e86\u200b\uff0c\u200b\u82df\u4f4f\u200b\u5c31\u200b\u597d\u200b by\u200b\u4e28\u200bJust Lose It\u200b\u4e28\u200b</p> <p>10\u200b\u6708\u200b\uff0c\u200b\u5728\u200b\u51ac\u5929\u200b\u5230\u6765\u200b\u4e4b\u524d\u200b\uff0c\u200b\u7559\u4e0b\u200b\u70b9\u200b\u4ec0\u4e48\u200b by \u200b\u4eba\u7269\u200b</p>"}]}
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