Group testing with random pools : Optimal two-stage algorithms
Résumé
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 -> infinity or p = 1/N-beta with beta is an element of (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, (T) over bar (N, p), is known to scale as Np vertical bar log p vertical bar. Here we determine the sharp asymptotic value of (T) over bar (N, p)/(Np vertical bar log p vertical bar) 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 (beta) with beta is an element of (1/2, 1).