hw3(2) - CS 373: Combinatorial Algorithms, Spring 2001...

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CS 373: Combinatorial Algorithms, Spring 2001 Homework 3 (due Thursday, March 8, 2001 at 11:59.99 p.m.) Name: Net ID: Alias: U 3 / 4 1 Name: Net ID: Alias: U 3 / 4 1 Name: Net ID: Alias: U 3 / 4 1 Starting with Homework 1, homeworks may be done in teams of up to three people. Each team turns in just one solution, and every member of a team gets the same grade. Since 1-unit graduate students are required to solve problems that are worth extra credit for other students, 1-unit grad students may not be on the same team as 3/4-unit grad students or undergraduates. Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes above. Please also tell us whether you are an undergraduate, 3/4-unit grad student, or 1-unit grad student by circling U, 3 / 4 , or 1, respectively. Staple this sheet to the top of your homework. Required Problems 1. Hashing: A hash table of size m is used to store n items with n m/ 2. Open addressing is used for collision resolution. (a) Assuming uniform hashing, show that for i = 1 , 2 ,... ,n , the probability that the i th insertion requires strictly more than k probes is at most 2 k . (b) Show that for i = 1 , 2 ,... ,n , the probability that the i th insertion requires more than 2lg n probes is at most 1 /n 2 . Let the random variable X i denote the number of probes required by the i th insertion. You have shown in part (b) that Pr { X i > 2lg n } ≤ 1 /n 2 . Let the random variable X = max 1 i n X i denote the maximum number of probes required by any of the n insertions. (c) Show that Pr { X > 2lg n } ≤ 1 /n . (d) Show that the expected length of the longest probe sequence is E [ X ] = O (lg n ).
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CS 373 Homework 3 (due 3/8/2001) Spring 2001 2. Reliable Network: Suppose you are given a graph of a computer network G = ( V,E ) and a function r ( u,v ) that gives a reliability value for every edge ( u,v ) E such that 0 r ( u,v ) 1. The reliability value gives the probability that the network connection corresponding to that edge will not fail. Describe and analyze an algorithm to Fnd the most reliable path from a given source
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hw3(2) - CS 373: Combinatorial Algorithms, Spring 2001...

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