Hw2b10fall - CS 4540 Advanced Algorithms Homework 2 Fri Due Fri Problem 1 Motwani and Raghavan Problem 4.1 page 97 Note The purpose of this problem

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Unformatted text preview: CS 4540, Advanced Algorithms Homework 2 Fri, Sept 10, 2010 Due Fri, Sept 17, 2010 Problem 1 Motwani and Raghavan, Problem 4.1, page 97. Note: The purpose of this problem is to familiarize you with the use of Chernoff bounds. You may use any of the foffowing forms of Theorems 1 through 5 given at the Class Notes of 09-08-10 (see Class Notes on the class web site). Problem 2 Let 0 ≤ y 1 ≤ y 2 ≤ ... ≤ y 2 N ≤ 1 be 2 N real numbers, with y 2 N +1- j = 1- y j , for 1 ≤ j ≤ N . Let X 1 ,X 2 ,...,X n be independent random variables such that, for 1 ≤ i ≤ n and for 1 ≤ j ≤ N , Pr[ X i = y j ] = p ij and Pr[ X i = y 2 N +1- j ] = Pr[ X i = y j ]. Let p i = E [ X i ] = ∑ 2 N j =1 y j p ij , where <p i < 1. Finally, let X = ∑ n i =1 X n and let μ = E [ X ]. (a) Prove that, for any δ > 0, Pr[ X > (1 + δ ) μ ] < h e δ (1+ δ ) 1+ δ i μ . (b) Prove that, for any 1 > δ > 0, Pr[ X < (1- δ ) μ ] < h e- δ (1- δ ) 1- δ i μ ....
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This note was uploaded on 10/23/2011 for the course CS 4540 taught by Professor Staff during the Fall '10 term at Georgia Institute of Technology.

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