Abstract

We show that several discrepancy-like problems can be solved in NC 2 nearly achieving the corresponding sequential bounds. For example, given a set system (X; S), where X is a ground set and S ` 2 X , a set R ` X can be computed in NC 2 so that, for each S 2 S, the discrepancy jjR " Sj \Gamma jR " Sjj is O( p jSj log jSj). Previous NC algorithms could only achieve O( p jSj 1+ffl log jSj), while ours matches the probabilistic bound achieved sequentially by the method of conditional probabilities within a multiplicative factor 1 + o(1). Other problems whose NC solution we improve are lattice approximation, ffl-approximations of range spaces of bounded VC-exponent, sampling in geometric configuration spaces, and approximation of integer linear programs. 1 Introduction Problem and previous work. Discrepancy is an important concept in combinatorics, see e.g. [1, 5], and theoretical computer science, see e.g. [27, 23, 9]. It attempts to capture the idea of a good sample from ...

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