hwk2 - edges(all costs> 0 Now we construct a new graph G...

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Spring 2011 CMSC 451: Homework 1 Clyde Kruskal Due at the start of class Thursday, October 13, 2011. Problem 1. Do Exercise 3 on pages 189-190 of Kleinberg and Tardos. But use “our favorite method” for the proof. Problem 2. (a) Show that Prim’s algorithm is correct using “our favorite method”. (b) Show that Prim’s algorithm is correct using mathematical induction. Warning: The two solutions should look very similar to each other. Problem 3. Suppose we are given a graph G (connected, undirected) with costs on the
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Unformatted text preview: edges (all costs > 0). Now we construct a new graph G ′ , which is the same as G , except for the costs. The cost of an edge e is deFned to be 1 /c e where c e is the cost of e in G . 1. Is the Minimum cost spanning tree in G , the Maximum cost spanning tree in G ′ ? Prove or disprove. 2. Suppose that P is the shortest path from s to v in G . Is P the longest simple path from s to v in G ′ ? Prove, or disprove. Problem 4. Do Exercise 16 on pages 196-197 of Kleinberg and Tardos. 1...
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