# HW4 - at all clear to them whether there even exists a...

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Homework #4 1) Suppose you are a consultant for the networking company Clunet, and they have the following problem. The network that they are currently working on is modeled by a connected graph G=(V, E) with n nodes. Each edge e is a fiber-optic cable that is owned by one of two companies- creatively named X and Y- and leased to Clunet. Their plan is to choose a spanning tree T of G, and upgrade the links corresponding to the edges of T. Their business relations people have already concluded an agreement with companies X and Y stipulating a number k so that in the tree T that is chosen, k of the edges will be owned by X and n-k-1 of the edges will be owned by Y. Clunet management now faces the following problem: It is not
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Unformatted text preview: at all clear to them whether there even exists a spanning tree meeting these conditions, and how to find one if it exists. So this is the problem they put to you: give a polynomial time algorithm that takes G, with each edges labeled X or Y, and either (i) returns a spanning tree with exactly k edges labeled X, or (ii) reports correctly that no such tree exists. 2) Let us say that a graph G = (V, E) is a near-tree if it is connected and has at most n + 8 edges, where n = |V|. Give an algorithm with running time O(n) that takes a near-tree G with costs on its edges, and returns the minimum spanning tree of G. You may assume that all edge costs are distinct....
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