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 M, let mM(n, d) be the minimum number of tests required to detect at most d defectives within n items, with success probability at least 1 − δ, for some constant δ. In this paper, we study the measures cM(d) = lim n→∞ mM(n, d) ln n and cM = lim d→∞ cM(d) d .In the literature, the analyses of such models only give upper bounds for cM(d) and cM, and for some of them, the bounds are not tight. We give new analyses that yield tight bounds for cM(d) and cM for all the known models M.