Lower Bounds for Restricted Schemes in the Two-Adaptive Bitprobe Model

TL;DR

This paper extends lower bound results for restricted schemes in the two-adaptive bitprobe model, showing that space complexity grows significantly with subset size and generalizes previous bounds.

Contribution

It generalizes existing lower bounds for restricted schemes from subsets of size 2 to arbitrary sizes, providing a broader understanding of space requirements.

Findings

Lower bounds for restricted schemes with arbitrary subset sizes

Generalization of previous bounds from size 2 to size n

Space complexity grows with subset size and total universe

Abstract

In the adaptive bitprobe model answering membership queries in two bitprobes, we consider the class of restricted schemes as introduced by Kesh and Sharma (Discrete Applied Mathematics 2021). In that paper, the authors showed that such restricted schemes storing subsets of size 2 require $\Omega(m^\frac{2}{3})$ space. In this paper, we generalise the result to arbitrary subsets of size $n$, and prove that the space required for such restricted schemes will be $\Omega(\left(\frac{m}{n}\right)^{1 - \frac{1}{\lfloor n / 4 \rfloor + 2}})$.