Sample spaces uniform on neighborhoods

Abstract

Let a universe of m elements be given, along with a family of subsets of the universe (neighborhoods), each of size at most k. We describe methods for assigning the m elements to points in a small-dimensional vector space (over GF(2)), in such a way that the elements in each neighborhood are assigned to an independent set of vectors. Such constructions lead, through a standard correspondence between linear and statistical independence, to the construction of small sample spaces which restrict to the uniform distribution in each neighborhood. (The sample space is a uniformly-weighted family of binary m-vectors). The size of such a sample space will be a function of the number of neighborhoods; and for sparse families, will be substantially smaller than any space which restricts to the uniform distribution in all k-sets. Previous work on sample spaces with limited independence focused on providing independence or near-independence in every k-set of the universe. We show how to construct the sample spaces efficiently both sequentially and in parallel. In case there are polynomially many (inm) neighborhoods, each of size O (log to), the parallel construction is in NC. These spaces provide a new derandomization technique for algorithms; particularly, algorithms related to the Lovász local lemma. We also describe applications to the exhaustive testing of VLSI circuits, and to coding for burst errors on noisy channels.