MOD_p-tests, Almost Independence and Small Probability Spaces

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 Velickovich [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 X1;X2; : : :;Xn where each Xi has values in f0; 1g and satises Xi = 0 with probability q and Xi = 1 with probability

Testing non-uniform k-wise independent distributions over product spaces (extended abstract)

A distribution D over Σ1× ⋯ ×Σ n is called (non-uniform) k-wise independent if for any set of k indices {i 1, ..., i k } and for any z1zki1ik, PrXD[Xi1Xik=z1zk]=PrXD[Xi1=z1]PrXD[Xik=zk]. We study the problem of testing (non-uniform) k-wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from the set of k-wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only for the binary field. For the non-uniform case, we give a new characterization of distributions being k-wise independent and further show that such a characterization is robust. These greatly generalize the results of Alon et al. [1] on uniform k-wise independence over the binary field to non-uniform k-wise independence over product spaces. Our results yield natural testing algorithms for k-wise independence with time and sample complexity sublinear in terms of the support size when k is a constant. The main technical tools employed include discrete Fourier transforms and the theory of linear systems of congruences.National Science Foundation (U.S.) (NSF grant 0514771)National Science Foundation (U.S.) (grant 0728645)National Science Foundation (U.S.) (Grant 0732334)Marie Curie International Reintegration Grants (Grant PIRG03-GA-2008-231077)Israel Science Foundation (Grant 1147/09)Israel Science Foundation (Grant 1675/09)Massachusetts Institute of Technology (Akamai Presidential Fellowship

MODp-tests, almost independence and small probability spaces. Random Structures Algorithms

In this paper, we consider approximations of probability distributions over Z 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 Velickovic [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, howto construct in time polynomial in n a good approximation to the joint probability distribution of the random variables X1;X2;:::;Xn where each Xi has values in f0; 1g and satis es Xi = 0 with probability q and Xi = 1 with probability 1,q where q is arbitrary. Our considerations improve on results by [EGLNV] and [AMN].