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# hw4sol - Advanced Analysis of Algorithms Homework...

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Advanced Analysis of Algorithms - Homework IV (Solutions) K. Subramani LCSEE, West Virginia University, Morgantown, WV { [email protected] } 1 Problems 1. Problem 26 . 2(9) on Page 664 of [CLRS01]. Solution: Let G = ( V,E ) denote the undirected graph and let s V denote an arbitrary vertex. Transform G into a flow network G as follows: (i) Replace each undirected edge ( u,v ) E with two directed edges ( u,v ) and ( v,u ) , each with capacity 1 . (ii) Let s be the source vertex of the flow network. Run the max-flow algorithm treating each vertex u V − { s } as the sink t , recording the value of the maximum flow ( mf u ). The edge-connectivity of G is then min u V −{ s } mf u . The crucial observation is that any cut of G , the vertex s is always on one side and hence it suffices to consider only the cases in which s is the source vertex in G . It is straightforward to see that a directed cut in G corresponds to a cut in G with the same capacity and vice versa. a50 2. Problem 26 . 3(3) on Page 668 of [CLRS01]. Solution: Without loss of generality, assume that | L | ≤ | R | . Any augmenting path has the following structure: s L R L... R t In other words, the path enters L from s

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hw4sol - Advanced Analysis of Algorithms Homework...

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