HW8_Sol.pdf - CSCI 570 Spring 2018 HW 8 due 1 Graded...

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CSCI 570 - Spring 2018 - HW 8 due MARCH 23, 2018 1 Graded Problems 1. The edge connectivity of an undirected graph is the minimum number of edges whose removal disconnects the graph. Describe an algorithm to com- pute the edge connectivity of an undirected graph with n vertices and m edges in O ( m 2 n ) time. For a cut ( S, ¯ S ), let c ( S, ¯ S ) denote the number of edges crossing the cut. By definition, the edge connectivity k = min S V c ( S, ¯ S ) Fix a vertex u V . For every cut ( S, ¯ S ), there is a vertex v V such that u and v are on either side of the cut. Let C u,v denote the value of the min u - v cut. Thus k = min v V,v 6 = u C u,v For each v 6 = u , C u,v can be determined by computing the max flow from u to v . Since G is undirected, we need to implement an undirected variant of the max flow algorithm. Set all edge capacities to 1. During each step of the flow computation, search for an undirected augmenting path using breadth first search and send a flow of 1 through this path. There are n - 1 flow computations and each flow computation takes at most O ( m 2 ) time. 2. Solve Kleinberg and Tardos, Chapter 7, Exercise 7. Let s be a source vertex and t a sink vertex. Introduce a vertex for every base station and introduce a vertex for every client.
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