Let (u, v) be a minimum-weight edge in a graph G. Show that (u, v) belongs to some minimum spanning tree of G.
过程GENERIC-MST的第一步.我们可以选择这样一条割,u在割的一边,v在另一边.此时,u-v就是一条通过割的轻边,把它加进来是安全的.
In the first step of GENERIC-MST, we could choose such a cut, node u is on one side, node v is on another side. Then(u, v) is a light-edge through this cut. So, it is safe to add (u, v)
Professor Sabatier conjectures the following converse of Theorem 23.1. Let G = (V, E) be a connected, undirected graph with a real-valued weight function w defined on E. Let A be a subset of E that is included in some minimum spanning tree for G, let (S, V - S) be any cut of G that respects A, and let (u, v) be a safe edge for A crossing (S, V - S). Then, (u, v) is a light edge for the cut. Show that the professor's conjecture is incorrect by giving a counterexample.
对于该 cut 来说, 虽然 (A, C) 是安全的, 但不是最轻边.
For this cut, although (A, C) is safe, but is it not the lightest.
Show that if an edge (u, v) is contained in some minimum spanning tree, then it is a light edge crossing some cut of the graph.
在这个MST里边,我们先去掉u-v这条边.然后画一个只穿过u-v的割(这个割肯定存在).此时,我们的策略是选择一条轻边,既然u-v之前在这个MST里边,那么u-v就是一条轻边.
In this MST, we remove (u,v) and draw a cut cross (u,v). Now, our strategy is choose a light-weight edge, because (u,v) is in this MST originally, then (u,v) is a light edge.
Give a simple example of a graph such that the set of edges {(u, v) : there exists a cut (S, V - S) such that (u, v) is a light edge crossing (S, V - S)} does not form a minimum spanning tree.
当三角形三条边权重相同时,每条边在某种 cut 中均是最轻,即结果中存在环,所以不是最小生成树.
For example, if we have a triangle with equal weight. In each cut, there will be two equal edge e1 and e2. If we choose e1 into MST, though e2 is also a light weight but it is not in the MST.
Let e be a maximum-weight edge on some cycle of G = (V, E). Prove that there is a minimum spanning tree of G′ = (V, E -{e}) that is also a minimum spanning tree of G. That is, there is a minimum spanning tree of G that does not include e.
因为e是该回路上的最大权边,因此选择这个回路上的其他边都优于选择e.所以G′和G肯定会有一个不含e的MST.
Becase e is the weightest edge in this circle. So other edges in the circle are all better than e. As a result, G and G' must have a MST not containging e.
Show that a graph has a unique minimum spanning tree if, for every cut of the graph, there is a unique light edge crossing the cut. Show that the converse is not true by giving a counterexample.
假设存在两个最小生成树 T 和 T'. 对任意一条边 e 属于 T, 如果从 T 中移除 e, 则 T 变得不连通, 形成 cut (S, V - S), 根据练习 23.1-3 可知, e 是穿过 cut(S, V - S) 最轻边. 假设边 x 属于 T', 并穿过 cut (S, V - S), 则 x 同样是最轻边. 由于穿过 cut(S, V - S) 的最轻边唯一. 既 e 和 x 是同一条边. 所以 e 也属于 T', 由于我们选择 e 是任意的, 所有在 T 中的边, 同样在 T' 中. 即最小生成树唯一.
Assuming there are two MSTs called T and T'. For any edge e in T, if we refove e from T, then T becomes unconnected and we have a cut(S, V - S). According to exercise 23.1-3, e is the light edge through cut(S, V - S). If edge x is in T' and through cut(S, V - S), then x is also a light weight. Because the light edge is unique. So e and x is the same edge, e is also in T'. Because we choose e at random, of all edges in T, also in T'. As a result, the MST is unique.
将条件和结论调换则不成立, 如下.
If inverse, then does not hold. See the picture.
Argue that if all edge weights of a graph are positive, then any subset of edges that connects all vertices and has minimum total weight must be a tree. Give an example to show that the same conclusion does not follow if we allow some weights to be nonpositive.
假设边的子集 T 中存在环, 则某两点之间存在多条通路, 移除其中一条通路, 子集 A' 仍然连通所有点. 因为边的权重为正, 既 w(A') < w(A), 结论与条件矛盾, 所以 T 是树.
Assuming any subset of edges contain circles, then there must be points u,v, the path from u to v is not unique. If we remove a path, subset A' also connects all the points. Because the weight is positive, w(A') < w(A). There is a confliction, because we can produce smaller graph. So no circles, it must be a tree.
如果边的权重准许为负, 则子集 T 不一定是树, 图中三条边总权重最小, 如下.
If some weights could be nonpositive, see picture below. It is a graph with total minimum weights.
Let T be a minimum spanning tree of a graph G, and let L be the sorted list of the edge weights of T. Show that for any other minimum spanning tree T′ of G, the list L is also the sorted list of edge weights of T′.
假设最小生成树有 n 条边, 存在两个最小生成树 T 和 T', 用 w(e) 表示边的权值. T 权值递增排列 w(a1) <= w(a2) <= ... w(an) T' 权值递增排列 w(b1) <= w(b2) <= ... w(bn) 假设 i 是两个列表中, 第一次出现边不同的位置, 既 ai ≠ bi, 先假定 w(ai) >= w(bi).
情况1, 如果 T 中含有边 bi, 由于 ai 和 bi 在列表 i 位置之前都是相同的, 若含有 bi 则一定在 i 位置后, 既有 j > i 使得 w(aj) = w(bi). 得到 w(bi) = w(aj) >= w(ai) >= w(bi), 既 w(bi) = w(aj) = w(ai), 故 i 位置处边的权值相同.
情况2, 如果 T 不包含边 bi, 则把 bi 加到 T 中, 会在某处形成一个圈. 由于 T 是最小生成树, 圈内任何一条边的权值都小于等于 w(bi), 另外这个圈中必定存在 aj 不在 T' 中, 得出 w(aj) <= w(bi) 且 j > i. 因此 w(bi) <= w(ai) <= w(aj) <= w(bi), 既 w(bi) = w(aj) = w(ai), 故 i 位置处边的权值仍相同.
Assuming MST contains n edges, there existing two MST T and T', w(e) stand for the weight of edge e.
For T w(a1) <= w(a2) <= ... w(an)
For T' w(b1) <= w(b2) <= ... w(bn)
Also assuming i is the first occuring index where ai ≠ bi, let's assume w(ai) >= w(bi).
Condition 1 : If bi is in T, because the preceeding edges befor ai and bi are same, if T containing bi then bi must after ai. So there existing j > i, w(aj) = w(bi). Which produce w(bi) = w(aj) >= w(ai) >= w(bi). So w(ai) = w(bi).
Condition 2 : If bi is not in T, then adding bi to T will produce a circle. Because T is MST, any weights of nodes in this circle will be less or equal than w(bi). Besides, in this circles, there must exiting aj which not in T' and have w(aj) <= w(bi), j > i. So, w(bi) <= w(ai) <= w(aj) <= w(bi), w(bi) = w(aj) = w(ai).
Let T be a minimum spanning tree of a graph G = (V, E), and let V′ be a subset of V. Let T′ be the subgraph of T induced by V′, and let G′ be the subgraph of G induced by V′. Show that if T′ is connected, then T′ is a minimum spanning tree of G′.
用 cut (V', V - V') 分割图 G, 该 cut 一定不影响 T', 且 T' 是 T 的子集, 所以 T' 对于 G' 是安全的. 如果 T' 是连通的, 则 T' 一定是 G' 的最小生成树.
We use cut(V', V - V') to cut graph. This cut will not influence T' and T' is the subset of T, so to G', T' is safe. if T′ is connected, then T′ is a minimum spanning tree of G.
Given a graph G and a minimum spanning tree T , suppose that we decrease the weight of one of the edges in T . Show that T is still a minimum spanning tree for G. More formally, let T be a minimum spanning tree for G with edge weights given by weight function w. Choose one edge (x, y) ∈ T and a positive number k, and define the weight function w' by
Show that T is a minimum spanning tree for G with edge weights given by w′.
We prove by cut. Originally, (x,y) is the light edge in a certain cut(V1, V2). Decreasing the weight of (x,y), (x,y) is still a light edge. So T is a minimum spanning tree for G with edge weights given by w′.
Given a graph G and a minimum spanning tree T , suppose that we decrease the weight of one of the edges not in T . Give an algorithm for finding the minimum spanning tree in the modified graph.
假设 (u, v) 不在最小生成树 T 中, 减小 (u, v) 权值后, 形成新的最小生成树 T'. 可能的情况是 T' 包含 (u, v) 或者 T' = T 保持不变. 算法只需寻找 T 中 u -> v 路径中权值最重边 x, 如果该边权值大于 (u, v), 则 T' = T - x + (u, v). 如果 (u, v) 权值大于 x, 则 T' = T. 路径可用 DFS 算法求得, 从 u 开始 v 结束. 因为 T 是最小生成树, 所以路径唯一, 时间 O(V+E).
If(u,v) is not in MST, decrease the weight of(u, v), we may form a new MST T'.
Condition 1 : if the weightest edge e in path from u->v is greater than edge(u,v).Then we can replace e with (u,v).
Condition 2 : if the weightest edge e in path from u->v is less or equal than edge(u,v).Then we need not change,
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