Discrete Low-Discrepancy Sequences

Holroyd and Propp used Hall’s marriage theorem to show that, given a probability distribution π on a finite set S, there exists an infinite sequence s1, s2, . . . in S such that for all integers k ≥ 1 and all s in S, the number of i in [1, k] with si = s differs from k π(s) by at most 1. We prove a generalization of this result using a simple explicit algorithm. A special case of this algorithm yields an extension of Holroyd and Propp’s result to the case of discrete probability distributions on infinite sets. Recently there has been an upsurge of interest in non-random processes that mimic interesting aspects of random processes, where the fidelity of the mimicry is a consequence of discrepancy constraints built into the constructions (for a general survey of discrepancy theory, see [1]). A recent example is the work of Friedrich, Gairing and Sauerwald [7] on load-balancing; other examples, linked by their use of the “rotor-router mechanism”, are the work of Cooper, Doerr, Friedrich, Spencer, and Tardos [2, 3, 4, 5, 6] on derandomized random walk on grids (“P -machines”), the work of Landau, Levine and Peres [9, 11, 12, 13] on derandomized internal diffusion-limited aggregation on grids and trees, and the work of Holroyd and Propp [8] on derandomized Markov chains. Here we focus on derandomizing something even more fundamental to probability theory: the notion of an independent sequence of discrete random variables.