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# hw5 - Cs445 Homework#5 All pairs shortest path Network Flow...

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Cs445 — Homework #5 All pairs shortest path, Network Flow, and Matching in Bipartite Graphs Due: 4/19/2005 during class meeting. 1. Let G ( V, E ) be a graph, with a weight assigned to its of its edges. The weight is either positive or negative. Explain how to modify Johnson’s algorithm so the output of the algorithm is an n × n matrix that specifies for every pairs of vertices u, v V , the first edge in the shortest path from u to v . The running time of the algorithm is O ( | E || V | log | E | ) 2. Assume G ( V, E ) is a graph where the weights of all edges are positive. You have ran Johnson algorithm on this graph. What are the values h ( v ) and ˆ w ( u, v ) given by the algorithm for each pairs of vertices u, v V ? Prove. 3. Let G ( V, E ) be a graph with positive and negative weights given to its edges. Assume that for each vertex v V you are also given a value h ( v ) with the property that for every edge ( u, v ) E , w ( u, v ) + h ( u ) - h ( v ) 0. (Note - the values h ( v ) are given to you — you do not need to compute them). Assume that

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