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ho5_new_1 - CS237. Practice Problems Set 5: Randomized Min...

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Unformatted text preview: CS237. Practice Problems Set 5: Randomized Min Cut Algorithm. Random Variables Ungraded. Please complete by Thurs Feb 24. March 8, 2011 Reading. M&U Chapter 1.4, L&L Notes on Random Variables. (See also Schaum’s Chapter 5.2,5.3,5.10,5.11, but the other books are more important this week.) Optional Extra Practice. M&U Ex 1.23-1.25 Exercise 1. Consider the min-cut algorithm given in class and M&U 1.4. 1. Prove by giving an explicit counterexample that a cut set C for graph G , does not have to be a cut set for the graph obtained in the i th iteration of the algorithm. 2. The opposite however is true: every cut-set of a graph in an intermediate iteration is a cut-set of the original graph. Solution: Consider the graph G = ( V,E ) , where V = { 1 , 2 , 3 , 4 , 5 } and E = { 1- 3 , 1- 2 , 2- 4 , 2- 3 , 3- 4 , 3- 5 , 4- 5 } . Consider the graph obtained by contracting 3 and 4 and contracting 1 and 3 . Consider further a cut-set C = { 1- 3 , 1- 4 , 1- 2 } . Then, C is a cut for G but not the for the intermediate graph since it does not contain the edge 1- 3 at all. For the second part, consider a graph with some nodes contracted and let C be its cut-set. Let us remove the edges from the graph that are in C . We get at least two components of the intermediate graph. Rollback the contraction operations. We will get at least two components of the original graph G . Therefore, C is a cut-set of G as well. Exercise 2. Consider the following variation of the randomized min cut algorithm discussed in class. Start with a graph of n vertices; contract it down to m vertices. Make k copies of this m vertex graph and now run the randomized algorithm on each copy of the smaller graph, each with independent randomness. Finally, output the smallest cut set (found from one copy of the smaller graph). Bound the probability of finding the minimum cut-set.graph)....
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This note was uploaded on 04/14/2011 for the course CS 237 taught by Professor Goldberg during the Spring '11 term at BU.

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ho5_new_1 - CS237. Practice Problems Set 5: Randomized Min...

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