Set membership with non-adaptive bit probes

TL;DR

This paper investigates the minimal storage size for set membership data structures using non-adaptive bit probes, presenting improved bounds and schemes for various probe counts.

Contribution

It introduces new schemes and bounds for non-adaptive and adaptive set membership data structures, improving upon previous theoretical limits.

Findings

Existence of schemes for a range of probe counts that surpass previous bounds.

Improved upper bounds for non-adaptive schemes with three probes.

Enhanced lower bounds for non-adaptive schemes with three probes.

Abstract

We consider the non-adaptive 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 S?" by non-adaptively probing the bit vector at t places. Let s_N(m,n,t) be the minimum number of bits of storage needed for such a scheme. In this work, we show existence of non-adaptive and adaptive schemes for a range of t that improves an upper bound of Buhrman, Miltersen, Radhakrishnan and Srinivasan (2002) on s_N(m,n,t). For three non-adaptive probes, we improve the previous best lower bound on s_N(m,n,3) by Alon and Feige (2009).