sol10 - Introduction to Algorithms CS 482 Spring 2008...

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Introduction to Algorithms Solution Set 9 CS 482, Spring 2008 (1) There are many correct solutions; here is one. The graph has two vertices s,t and n triples of vertices ( u i ,v i ,w i ) n i =1 , for a total of 3 n + 2 vertices. There are 5 n edges, specified as follows. For all i , the graph contains edges ( s,u i ) , ( u i ,v i ) , ( v i ,t ) , ( u i ,w i ) , ( w i ,t ) . The subgraph consisting of vertices s,t,u i ,v i ,w i and the five edges between them will be denoted by G i . The unique minimum s - t cut is the one that separates s from all other vertices. This cut survives the analogue of Karger’s algorithm if and only if none of the edges ( s,u i ) is ever contracted. For i = 1 , 2 ,...,n , let E i denote the event: “the algorithm picks edge ( s,u i ) before it picks any other edge of G i .” We observe the following facts. 1. Pr( E i ) = 1 / 5 . 2. The events {E i | i = 1 , 2 ,...,n } are mutually independent. 3. The algorithm succeeds only if
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This note was uploaded on 10/02/2008 for the course CS 482 taught by Professor Kleinberg during the Spring '08 term at Cornell.

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sol10 - Introduction to Algorithms CS 482 Spring 2008...

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