Set Membership with a Few Bit Probes

TL;DR

This paper studies the minimal storage size for set membership data structures that answer queries with a limited number of adaptive bit probes, improving bounds established in prior research.

Contribution

It provides improved bounds on the minimal storage size for set membership schemes with few bit probes, advancing previous theoretical limits.

Findings

Improved bounds on s(m,n,t) for certain parameter ranges.

Enhanced understanding of the trade-off between storage size and probe complexity.

Refined theoretical limits compared to prior works by Buhrman et al. and Alon and Feige.

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 S?" 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. Several recent works investigate s(m,n,t) for various ranges of the parameter; we obtain improvements over some of the bounds shown by Buhrman, Miltersen, Radhakrishnan, and Srinivasan (2002) and Alon and Feige (2009).