MODp‐tests, almost independence and small probability spaces

Abstract

In this paper, we consider approximations to probability distributions over Z. We present an approach to estimate the quality of approximations to probability distributions towards the construction of small probability spaces. These small spaces are used to derandomize algorithms. In contrast to results by Even, Goldreich, Luby, Nisan, and Veličković [EGLNV], the methods which are used here are simple, and we get smaller sample spaces. Our investigations are motivated by recent work of Azar, Motwani, and Naor [AMN]. They considered the problem to construct in time respective space polynomial in n a good approximation to the joint probability distribution of the mutually independent random variables X1, X2,…,Xn. Each Xi has values in {0, 1} and satisfies Xi=0 with probability q and Xi=1 with probability 1−q where q∈[0, 1] is arbitrary. Our considerations improve on results in [EGLNV] and [AMN] for q=1/p and p a prime. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 293–313, 2000