This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 1.231.25 Exercise 1. Consider the mincut 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 cutset of a graph in an intermediate iteration is a cutset 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 cutset 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 cutset. 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 cutset 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 cutset.graph)....
View
Full
Document
This note was uploaded on 04/14/2011 for the course CS 237 taught by Professor Goldberg during the Spring '11 term at BU.
 Spring '11
 Goldberg

Click to edit the document details