Improved Bounds for Two Query Adaptive Bitprobe Schemes Storing Five Elements

TL;DR

This paper improves the theoretical lower bounds and proposes a more space-efficient scheme for two-query adaptive bitprobe schemes that store five elements, advancing understanding in data structure efficiency.

Contribution

It establishes a new lower bound of m^{3/4} and introduces a scheme with O(m^{5/6}) space for storing five elements, improving previous bounds.

Findings

Lower bound improved to m^{3/4}

Proposed scheme uses O(m^{5/6}) space

Advances bounds for adaptive bitprobe schemes

Abstract

In this paper, we study two-bitprobe adaptive schemes storing five elements. For these class of schemes, the best known lower bound is m^{1/2} due to Alon and Feige [SODA 2009]. Recently, it was proved by Kesh [FSTTCS 2018] that two-bitprobe adaptive schemes storing three elements will take at least m^{2/3} space, which also puts a lower bound on schemes storing five elements. In this work, we have improved the lower bound to m^{3/4}. We also present a scheme for the same that takes O(m^{5/6}) space. This improves upon the O(m^{18/19})-scheme due to Garg [Ph.D. Thesis] and the O(m^{10/11})-scheme due to Baig et al. [WALCOM 2019].