Group Testing With Random Pools: Optimal Two-Stage Algorithms

We study the group testing of a set of N items each of which is defective with probability p. We focus on the double limit of small defect probability, p ≪ 1, and large number of variables, N ≫ 1, taking either p → 0 after N → ∝ or p = 1/N^β with β ∈ (0,1/2). In both settings the optimal number of tests which are required to identify with certainty the defectives via a two-stage procedure, T̅(N, p), is known to scale as Np |log p|. Here we determine the sharp asymptotic value of T̅(N,p)/(Np|log p|) and construct a class of two-stage algorithms over which this optimal value is attained. This is done by choosing a proper bipartite regular graph (of tests and variable nodes) for the first stage of the detection. Furthermore we prove that this optimal value is also attained on average over a random bipartite graph where all variables have the same degree and the tests connected to a given variable are randomly chosen with uniform distribution among all tests. Finally, we improve the existing upper and lower bounds for the optimal number of tests in the case p = 1/N^β with β ∈ [1/2,1).