Set membership with a few bit probes

Abstract

We consider the bit-probe complexity of the set membership problem, where a set S of size at most n from a universe of size m is to be represented as a short bit vector in order to answer membership queries of the form "Is x in 5?" by adaptively probing the bit vector at t places. Let s (m, n, t) be the minimum number of bits of storage needed for such a scheme. Buhrman, Miltersen, Radhakrishnan, and Srinivasan [4] and Alon and Feige [2] investigated s (m, n, t) for various ranges of the parameter t. We show the following. For two probes (t-2): (a) There is a constant C > 0, such that for all large m, s (m, n, 2) ≤ C·m1-1/4n+1. This improves on a result of Alon and Feige that states that for n ≥ logm, s (m, n, 2)=0 (mn log ( (1og m)/n)/log m). (b) There is a constant D > 0, such that for 4 ≤ n and all large m, we have s (m, n, 2) ≥ D · m1-[1/4n+1]. Thus, s (m, n, 2)=ω (m) for n ≥ logm, that is, the requirement n ≤ logm in the upper bound of Alon and Feige is essential for any asymptotic improvement over the characteristic vector representation. For three probes (t=3): s (m, n, 3)=ω (√mn/log m). This improves on a result of Alon and Feige that states that s (m, n, 2)-O (m2/3n1/3). In general: We show that for t ≥ 3, s (m, n, t)=ω (m1/t-1). This lower bound improves on (for n ≤ logm) the lower bound s (m, n, 3)=ω (√mn/log m) (valid only for n ≥ 16logm and for non-adaptive schemes) due to Alon and Feige; for small values of n, it also improves on the lower bound s (m, n, t)=ω, (tm1/t n1-1/t) due to Buhrman et al. [4].