hw2 - k ′ red edges and there is a spanning tree T ′′...

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AMS/MBA 546 (Fall, 2010) Estie Arkin Network Flows Homework Set # 2 Due Tuesday, September 21, 2010. Suggested reading Chapter 13, and Section 3.4 of the text(AMO). 1). (AMO 13.19) Consider the following reverse greedy algorithm: Sort the edges in decreasing order of their weights w e 1 w e 2 ≥ ··· ≥ w e m . Let G := G For k = 1 to m do: If G \ e k is connected then set G := G \ e k end Prove that at the termination of this algorithm G is a minimum spanning tree. 2). Suppose that you are given an undirected connected graph, with each edge coloured either red or green. (a). Describe a method that ±nds a spanning tree with the maximum number of red edges. (b). Suppose that there is a spanning tree T with
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Unformatted text preview: k ′ red edges, and there is a spanning tree T ′′ with k ′′ red edges. Show that for all k , k ′ ≤ k ≤ k ′′ there is a spanning tree with exactly k red edges. 3). (AMO 13.34) We are given an undirected, connected graph G = ( V,E ) with (nonnegative) weightss on the edges w ij . A balanced spanning tree is de±ned as a spanning tree of G if from among all spanning trees of G the di²erence between the maximum edge weight and the minimum edge weight is as small as possible. Describe a polynomial time algorithm for ±nding a balanced spanning tree. Make sure to state the running time of your algorithm....
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