We consider the nonadaptive group testing with @math items, of which @math are defective. We study a test design in which each item appears in nearly the same number of tests. For each item, we independently pick @math tests uniformly at random with replacement and place the item in those tests. We analyze the performance of these designs with simple and practical decoding algorithms in a range of sparsity regimes and show that the performance is consistently improved in comparison with standard Bernoulli designs. We show that our new design requires roughly 23 fewer tests than a Bernoulli design when paired with the simple decoding algorithms known as combinatorial orthogonal matching pursuit and definite defectives (DD). This gives the best known nonadaptive group testing performance for @math and the best proven performance with a practical decoding algorithm for all @math . We also give a converse result showing that the DD algorithm is optimal with respect to our randomized design when @math . We complement our theoretical results with simulations that show a notable improvement over Bernoulli designs in both sparse and dense regimes.
The problem of Group Testing is to identify defective items out of a set of objects by means of pool queries of the form “Does the pool contain at least a defective?”. The aim is of course to perform detection with the fewest possible queries, a problem which has relevant practical applications in different fields including molecular biology and computer science. Here we study GT in the probabilistic setting focusing on the regime of small defective probability and large number of objects, p→0 and N→∞. We construct and analyze one-stage algorithms for which we establish the occurrence of a non-detection detection phase transition resulting in a sharp threshold, ( M ) , for the number of tests. By optimizing the pool design we construct algorithms whose detection threshold follows the optimal scaling ( M Np| p| ) . Then we consider two-stages algorithms and analyze their performance for different choices of the first stage pools. In particular, via a proper random choice of the pools, we construct algorithms which attain the optimal value (previously determined in (Mezard and Toninelli, arXiv:0706.3104)) for the mean number of tests required for complete detection. We finally discuss the optimal pool design in the case of finite p.