We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called ffl-biased random variables. The number of random bits needed to generate the random variables is O(log n + log 1 ffl ). Thus, if ffl is polynomially small, then the size of the sample space is also polynomial. Random variables that are ffl-biased can be used to construct "almost" k-wise independent random variables where ffl is a function of k. These probability spaces have various applications: 1. Derandomization of algorithms: many randomized algorithms that require only k- wise independence of their random bits (where k is bounded by O(log n)), can be derandomized by using ffl-biased random variables. 2. Reducing the number of random bits required by certain randomized algorithms, e.g., verification of matrix multiplication. 3. Exhaustive testing of combinatorial circui...

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We formulate the notion of a "good approximation" to a probability distribution over a finite abelian group. The approximate distribution is characterized by a parameter ffl, the quality of the approximation, which is a bound on the difference between corresponding Fourier coefficients of the two distributions. It is also required that the sample space of the approximate distribution be of size polynomial in the representation length of the group elements as well as 1=ffl. Such approximations are useful in reducing or eliminating the use of randomness in randomized algorithms. We demonstrate the existence of such good approximations to arbitrary distributions. In the case of n random variables distributed uniformly and independently over the range f0; : : : ; d \Gamma 1g, we provide an efficient construction of a good approximation. The constructed approximation has the property that any linear combination of the random variables (modulo d) has essentially the same behavior under the ...

In this paper, we consider approximations of probability distributions over ZZ n p . We present an approach to estimate the quality of approximations of probability distributions towards the construction of small probability spaces. These are used to derandomize algorithms. In contrast to results by Even, Goldreich, Luby, Nisan and Velickovi'c [EGLNV], our methods are simple, and for reasonably small p, we get smaller sample spaces. Our considerations are motivated by a problem which was mentioned in recent work of Azar, Motwani and Naor [AMN], namely, how to construct in time polynomial in n a good approximation to the joint probability distribution of the random variables X 1 ; X 2 ; : : : ; Xn where each X i has values in f0; 1g and satisfies X i = 0 with probability q and X i = 1 with probability 1 \Gamma q where q is arbitrary. Our considerations improve on results by [EGLNV] and [AMN]. 1 Introduction During the last years, techniques have been developed to minimize the number...