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2009 年 10 月 2 日  星期五   晴天


Brief Proof on Bottleneck Spanning Tree 分類: Programming Impossib...

Brief Proof on Bottleneck Spanning Tree

[Homework for Introduction to Algorithms, Exercise 23-3]

 phoeagon

 

[Definition]

A bottleneck spanning tree (T) of an undirected graph G is a spanning tree of G whose largest edge weight is the minimum one over all spanning trees of G.  We say that the value of the bottleneck spanning tree is the weight of the maximum-weight edge in T.

[Theorem]

A Minimum Spanning Tree (MST) is a Bottleneck Spanning Tree, and the one with the smallest cost.

[Brief Proof]

Consider a Minimum Spanning Tree (A) of Graph G, whose set of sub graphs contain another spanning tree B that is a bottleneck spanning tree, while MST (A) isn't.

First, without lost of generality, suppose e(u, v) in A is the edge with the largest weight in the MST with no ties, whose weight is W(u, v). Then, consider deleting e(u, v) from A, separating the tree into two trees, namely U and V respectively.

Then we construct a set of edges, S, that for each edge (ui, vi), where ui is a vertex in U and vi in V, suffices (ui, vi)S. One may notice that there exists at least one edge (uj, vj)S which is in the Bottleneck Spanning Tree (B). (Otherwise B is not a spanning tree.)

So, as B is the Bottleneck Spanning Tree while A isn’t, each edge e(x, y) in B suffices e(x, y) < W(u, v). Suppose we have edge e(x,y) in S,  replace it with edge e(u,v) in A will make A a Spanning Tree with a even smaller sum of weights. Here we find a contradiction.

 






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