CMPSCI 611: Advanced Algorithms Micah Adler Summary: CMPSCI 611: Advanced Algorithms Micah Adler Problem Set 4 Out: November 14, 2000 Due: November 21, 2000 1. (Based on [MR95], problem 4.1): Suppose you are given a biased coin that has Pr[HEADS] = h, for a  h  b, for some xed a and b. You are not given any other information about h (i.e., you can not assume that it is chosen randomly). (a) Using Chebyshev's inequality, devise a procedure for estimating h by a value ^ h such that you can guarantee that Pr[jh ^ hj > h] < , for any choice of the constants 0 < a; b; ;  < 1. Let N be the number of times you need to ip the biased coin to obtain the estimate, where N is a function of a; b; , and . What is the smallest value of N for which you can still give this guarantee? (b) In Lecture 16, we saw the following two Cherno bounds: Pr[B(n; p)  (1 Æ)np]  e Æ 2 np=2 ; and Pr[B(n; p)  (1 + Æ)np]  e Æ 2 np=3 ; where B(n; p) is a random variable representing the number of heads seen in n tosses of a coin that is heads with probability p. Using these bounds, what is the smallest value of N for which you can still give the guarantee that Pr[jh ^ hj > h] < ? 2. CLR, Problem 36-1 (Page 961). 3. The decision version of the clique problem can be stated as follows: Collections: Computer Technologies and Information Sciences