A Tight Lower Bound of $\Omega(\log n)$ for the Estimation of the Number of Defective Items

TL;DR

This paper establishes a tight lower bound of Omega(log n) tests for non-adaptive randomized algorithms estimating the number of defective items in group testing, matching known upper bounds and resolving an open problem.

Contribution

The paper introduces a novel method for deriving tight lower bounds in non-adaptive randomized group testing, proving the Omega(log n) bound for estimating defective items.

Findings

Any such algorithm requires at least Omega(log n) tests.

The bound matches the known upper bound of O(log n).

The result resolves an open problem in the field.

Abstract

Let $X$ be a set of items of size $n$ , which may contain some defective items denoted by $I$, where $I \subseteq X$. In group testing, a {\it test} refers to a subset of items $Q \subset X$. The test outcome is $1$ (positive) if $Q$ contains at least one defective item, i.e., $Q\cap I \neq \emptyset$, and $0$ (negative) otherwise. We give a novel approach to obtaining tight lower bounds in non-adaptive randomized group testing. Employing this new method, we can prove the following result. Any non-adaptive randomized algorithm that, for any set of defective items $I$, with probability at least $2/3$, returns an estimate of the number of defective items $|I|$ to within a constant factor requires at least $\Omega({\log n})$ tests. Our result matches the upper bound of $O(\log n)$ and solves the open problem posed by Damaschke and Sheikh Muhammad.