 
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 361 (Page 961).
3. The decision version of the clique problem can be stated as follows:
