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+# Kruskal's Algorithm for Minimum Spanning Tree (MST)
+
+## Problem Statement
+Kruskal's algorithm is a **greedy algorithm** used to find the **Minimum Spanning Tree (MST)** of a connected, undirected graph. The MST is a subset of edges that connects all vertices with the minimum possible total edge weight and no cycles.
+
+### Definitions
+- **Graph**: Consists of vertices and weighted edges.
+- **Minimum Spanning Tree (MST)**: A subset of edges that connects all vertices with minimum total edge weight.
+- **Objective**: Find the MST for the graph.
+
+## Input
+- **V**: Number of vertices in the graph.
+- **E**: List of edges, where each edge includes two vertices and a weight `(u, v, w)`.
+
+## Output
+- A list of edges that form the MST and the total minimum weight.
+
+## Example
+Given:
+- Vertices: `V = {A, B, C, D}`
+- Edges: `[(A, B, 1), (B, C, 2), (C, D, 3), (A, D, 4)]`
+
+The MST would include edges `(A, B)`, `(B, C)`, and `(C, D)` with a total weight of **6**.
+
+## Approach
+1. **Sort the Edges**: Sort all edges in ascending order of their weights.
+2. **Union-Find Structure**: Use the Union-Find data structure to detect cycles and help connect components.
+3. **Add Edges Greedily**: Add edges to the MST in sorted order, skipping any that form a cycle.
+4. **Stop Condition**: Stop once there are `V-1` edges in the MST.
+
+### Steps:
+1. Sort edges by weight.
+2. Initialize MST as an empty set.
+3. For each edge in sorted edges:
+   - If adding the edge doesn’t form a cycle, add it to the MST.
+4. Output the MST and its total weight.
+
+## Complexity
+- **Time Complexity**: \(O(E \log E + E \log V)\), where \(E\) is the number of edges and \(V\) is the number of vertices.
+- **Space Complexity**: \(O(V)\) for storing subsets in Union-Find.
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+class DisjointSet:
+    def __init__(self, vertices):
+        self.parent = {v: v for v in vertices}  # Each vertex is its own parent (makes a set for each vertex)
+        self.rank = {v: 0 for v in vertices}  # To keep track of the "depth" of trees for efficient union operations
+    
+    def find_set(self, v):
+        if self.parent[v] != v:
+            self.parent[v] = self.find_set(self.parent[v])  # Path compression
+        return self.parent[v]
+
+    def union_set(self, u, v):
+        root1 = self.find_set(u)
+        root2 = self.find_set(v)
+
+        if root1 != root2:
+            if self.rank[root1] > self.rank[root2]:
+                self.parent[root2] = root1
+            else:
+                self.parent[root1] = root2
+                if self.rank[root1] == self.rank[root2]:
+                    self.rank[root2] += 1
+
+def kruskal(graph):
+    vertices = list(graph.keys())  # Get a list of all vertices from the graph
+    # If we remove "if u < v" condition then edges will be bidirectional means (a,b,6) & (b,a,6) both will appear
+    edges = [(u, v, w) for u in graph for v, w in graph[u].items() if u < v]
+    # Sort edges based on the given weights
+    edges.sort(key = lambda x: x[2])  # x = (u,v,w)
+
+    # Initialize the disjoint set with all vertices
+    ds = DisjointSet(vertices)
+
+    # Print the initial sets (only once)
+    print("Make sets:", end = " ")
+    for v in vertices:
+        print(f"{{ {v} }}", end = ", ")
+    print()
+
+    # For storing edges of minimum spanning tree
+    mst = []
+    for u, v, w in edges:
+        if ds.find_set(u) != ds.find_set(v):  # Check if u and v belong to different sets
+            print(f"Selected edge: ({u}, {v}, {w})")
+
+            # Union operation and print the sets after union
+            ds.union_set(u, v)
+
+            # Find and print sets after edge selection
+            print("Make sets after selection:", end=" ")
+            sets = {}
+            for vertex in vertices:
+                root = ds.find_set(vertex)   # Find the root of each vertex to know the current set
+                if root not in sets:
+                    sets[root] = []
+                sets[root].append(vertex)  # Group vertices by their root (which set they belong to)
+            for root, group in sets.items():
+                print(f"{{ {' '.join(sorted(group))} }}", end = ", ")  # Print the current sets after the union operation
+            print()
+
+            mst.append((u, v, w))  # Add the selected edge to the MST
+
+    # Return list of selected edges
+    return mst
+
+# Function to take user input for the graph
+def input_graph():
+    # is_directed = input("Is the graph directed? (yes/no): ").lower() == "yes"
+    n = int(input("Enter the number of vertices: "))
+    m = int(input("Enter the number of edges: "))
+    
+    graph = {}
+    
+    print("Enter the edges in the format (u v w), where u and v are vertices, and w is the weight:")
+    for _ in range(m):
+        u, v, w = input().split()
+        w = int(w)  # Convert the weight to an integer
+        
+        if u not in graph:
+            graph[u] = {}   # Initialize an empty dictionary for vertex u if it's not already in the graph
+        if v not in graph:
+            graph[v] = {}  # Same for vertex v.
+        
+        # Since it's an undirected graph, we add both u -> v and v -> u
+        # For Undirected graph
+        graph[u][v] = w
+        graph[v][u] = w
+
+        # Add only the directed edge u -> v
+        # For Directed graph
+        # graph[u][v] = w
+    
+    return graph
+
+# Take graph input from the user
+graph = input_graph()
+
+# Find the MST using Kruskal's algorithm
+mst = kruskal(graph)
+
+# Print the final MST
+total_cost = 0
+print("Minimum Spanning Tree:")
+for u, v, w in mst:
+    print(f"({u}, {v}, {w})")
+    total_cost += w  # Add the weight of each edge to the total cost
+
+print(f"Total cost of MST: {total_cost}")
+"""
+graph = {
+    'a': {'b': 6, 'c': 4},
+    'b': {'a': 6, 'h': 10, 'c': 5, 'e': 14},
+    'c': {'a': 4, 'b': 5, 'f': 2, 'd': 9},
+    'd': {'c': 9},
+    'e': {'b': 14, 'h': 3},
+    'f': {'c': 2, 'g': 15, 'h': 8},
+    'g': {'f': 15},
+    'h': {'b': 10, 'e': 3, 'f': 8}
+}
+
+Points to Remember:-
+
+● Time Complexity = O(nlogn) + O(n) = O(nlogn)
+    For sorting it takes O(nlogn) where n is the no of items
+    For 'for' loop it takes O(n) 
+
+● Space Complexity = O(V) + O(E) = O(V+E)
+    For storing edges of mst which would be 'V-1' it takes O(V)
+    For sorting edges it requires O(E)
+
+● Both Krushkal's and Prim's algorithm is used only for undirected graph and for non-negative & zero weights.
+● Edmonds' algorithm is used to find a minimum spanning tree (MST) in a directed graph.
+"""
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