hw6 - UC BerkeleyCS 170 Problem Set 6 Lecturer: Tom...

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Unformatted text preview: UC BerkeleyCS 170 Problem Set 6 Lecturer: Tom Henzinger Due on October 17 at 4:00 p.m. Problem Set 6 for CS 170 Note When asked for an algorithm you must give (1) a brief informal description of the algorithm, (2) a precise description using pseudo-code, (3) an argument for termination and correctness of the algorithm, and (4) an analysis of the running time of the algorithm. Be clear about what the input to the algorithm is, how you measure the size of the input, and what constitutes a step in your running-time analysis. Problem 1. [Graph Cuts] (6 points) Let G = ( V, E ) be a connected undirected graph. A cut of G is a partition { S, V- S } of the vertices into two disjoint, non-empty subsets. An edge { v, w } E is a cut edge if v S and w ( V- S ) (since G is undirected, we write { v, w } rather than ( v, w ) for the edge between v and w ). (a) A cut C 1 contains a cut C 2 if every cut edge of C 2 is a cut edge of C 1 . Does each cut of a graph contain a minimum cut? Prove or give a counterexample. (b) Let K n be the complete graph with n vertices. What is the cost of a minimum cut of K n ? How many different minimum cuts are there? Explain. (c) A contraction sequence on K n does n- 2 contractions and produces a cut. For each n 3, give an example of a contraction sequence on K n that produces a minimum cut....
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hw6 - UC BerkeleyCS 170 Problem Set 6 Lecturer: Tom...

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