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

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 mcal 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-δ, for some constant δ. In this paper, we study the measures ccal M(d)=limn→ ∞ fracmcal M(n,d)ln n mbox and ccal M=limd→ ∞ fracccal M(d)d. In the literature, the analyses of such models only give upper bounds for ccal M(d) and ccal M, and for some of them, the bounds are not tight. We give new analyses that yield tight bounds for ccal M(d) and ccal M for all the known models~cal M.