Pseudo-random graphs and bit probe schemes with one-sided error

TL;DR

This paper introduces a new probabilistic bit-probe scheme with one-sided error for membership queries, using auxiliary memory to improve space efficiency over previous two-sided error schemes based on expanders.

Contribution

It presents a novel bit-probe scheme with one-sided error that employs auxiliary memory, reducing space requirements compared to prior two-sided error schemes.

Findings

Scheme achieves $O(n ext{log}^2 m)$ space with one cached word.

Uses auxiliary memory to improve space efficiency.

Outperforms non-cached schemes for certain parameters.

Abstract

We study probabilistic bit-probe schemes for the membership problem. Given a set A of at most n elements from the universe of size m we organize such a structure that queries of type "Is x in A?" can be answered very quickly. H.Buhrman, P.B.Miltersen, J.Radhakrishnan, and S.Venkatesh proposed a bit-probe scheme based on expanders. Their scheme needs space of $O(n\log m)$ bits, and requires to read only one randomly chosen bit from the memory to answer a query. The answer is correct with high probability with two-sided errors. In this paper we show that for the same problem there exists a bit-probe scheme with one-sided error that needs space of $O(n\log^2 m+\poly(\log m))$ bits. The difference with the model of Buhrman, Miltersen, Radhakrishnan, and Venkatesh is that we consider a bit-probe scheme with an auxiliary word. This means that in our scheme the memory is split into two parts of different size: the main storage of $O(n\log^2 m)$ bits and a short word of $\log^{O(1)}m$ bits that is pre-computed once for the stored set A and `cached'. To answer a query "Is x in A?" we allow to read the whole cached word and only one bit from the main storage. For some reasonable values of parameters our space bound is better than what can be achieved by any scheme without cached data.