Randomized group testing both query-optimal and minimal adaptive
SOFSEM'12 Proceedings of the 38th international conference on Current Trends in Theory and Practice of Computer Science
Distributed sensor failure detection in sensor networks
Signal Processing
Hi-index | 754.84 |
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/Nβ 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/Nβ with β ∈ [1/2,1).