On Detecting Some Defective Items in Group Testing

TL;DR

This paper investigates the problem of detecting a subset of defective items in group testing, providing tight bounds on the number of tests needed in various adaptive and non-adaptive scenarios, with and without prior knowledge of defectives.

Contribution

It establishes new upper and lower bounds on the number of tests required for identifying defective items in different settings, including adaptive and non-adaptive, with known or unknown number of defectives.

Findings

Non-adaptive deterministic algorithms require (n) tests.

Adaptive algorithms can achieve (ll \, ( / ll)) tests.

Randomized adaptive algorithms require (ll \, (n / )) tests.

Abstract

Group testing is an approach aimed at identifying up to $d$ defective items among a total of $n$ elements. This is accomplished by examining subsets to determine if at least one defective item is present. In our study, we focus on the problem of identifying a subset of $\ell\leq d$ defective items. We develop upper and lower bounds on the number of tests required to detect $\ell$ defective items in both the adaptive and non-adaptive settings while considering scenarios where no prior knowledge of $d$ is available, and situations where an estimate of $d$ or at least some non-trivial upper bound on $d$ is available. When no prior knowledge on $d$ is available, we prove a lower bound of $ \Omega(\frac{\ell \log^2n}{\log \ell +\log\log n})$ tests in the randomized non-adaptive settings and an upper bound of $O(\ell \log^2 n)$ for the same settings. Furthermore, we demonstrate that any non-adaptive deterministic algorithm must ask $\Theta(n)$ tests, signifying a fundamental limitation in this scenario. For adaptive algorithms, we establish tight bounds in different scenarios. In the deterministic case, we prove a tight bound of $\Theta(\ell\log{(n/\ell)})$. Moreover, in the randomized settings, we derive a tight bound of $\Theta(\ell\log{(n/d)})$. When $d$, or at least some non-trivial estimate of $d$, is known, we prove a tight bound of $\Theta(d\log (n/d))$ for the deterministic non-adaptive settings, and $\Theta(\ell\log(n/d))$ for the randomized non-adaptive settings. In the adaptive case, we present an upper bound of $O(\ell \log (n/\ell))$ for the deterministic settings, and a lower bound of $\Omega(\ell\log(n/d)+\log n)$. Additionally, we establish a tight bound of $\Theta(\ell \log(n/d))$ for the randomized adaptive settings.