Bounds for the Number of Tests in Non-adaptive Randomized Algorithms for 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 $\mathcal{M}$, let $m_\mathcal{M}(n,d)$ 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

$$c_\mathcal{M}(d)=\lim _{n\rightarrow \infty } \frac{m_\mathcal{M}(n,d)}{\ln n} \text{ and } \ c_\mathcal{M}=\lim _{d\rightarrow \infty } \frac{c_\mathcal{M}(d)}{d}.$$

In the literature, the analyses of such models only give upper bounds for $c_\mathcal{M}(d)$ and $c_\mathcal{M}$, and for some of them, the bounds are not tight. We give new analyses that yield tight bounds for $c_\mathcal{M}(d)$ and $c_\mathcal{M}$ for all the known models $\mathcal{M}$.