Bounds for the Number of Tests in Non-Adaptive Randomized Algorithms for Group Testing

TL;DR

This paper establishes tight bounds on the minimum number of tests needed in non-adaptive randomized group testing algorithms, improving upon previous bounds that were often loose or only upper bounds.

Contribution

It provides new analyses that derive tight bounds for the asymptotic measures of tests required across all known models of non-adaptive randomized group testing.

Findings

Derived tight bounds for $c_{\

Provided comprehensive analysis for all known models.

Improved understanding of the minimum tests needed in group testing.

Abstract

We study the group testing problem with non-adaptive randomized algorithms. Several models have been discussed in the literature to determine how to randomly choose the tests. For a model ${\cal M}$, let $m_{\cal M}(n,d)$ be the minimum number of tests required to detect at most $d$ defectives within $n$ items, with success probability at least $1-\delta$, for some constant $\delta$. In this paper, we study the measures $$c_{\cal M}(d)=\lim_{n\to \infty} \frac{m_{\cal M}(n,d)}{\ln n} \mbox{ and } c_{\cal M}=\lim_{d\to \infty} \frac{c_{\cal M}(d)}{d}.$$ In the literature, the analyses of such models only give upper bounds for $c_{\cal M}(d)$ and $c_{\cal M}$, and for some of them, the bounds are not tight. We give new analyses that yield tight bounds for $c_{\cal M}(d)$ and $c_{\cal M}$ for all the known models~${\cal M}$.