Group Testing with Random Pools: optimal two-stage algorithms

TL;DR

This paper determines the precise asymptotic number of tests needed in probabilistic group testing with small defect probability, and constructs optimal two-stage algorithms using bipartite graphs.

Contribution

It establishes the sharp asymptotic value of the tests needed and constructs algorithms that attain this optimality using bipartite graph designs.

Findings

Optimal two-stage algorithms achieve the asymptotic test count.

Random bipartite graphs with fixed degrees also attain optimality.

Improved bounds for the case p=1/N^β with β in [1/2,1).

Abstract

We study Probabilistic 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 $N\to\infty$ or $p=1/N^{\beta}$ with $\beta\in(0,1/2)$. In both settings the optimal number of tests which are required to identify with certainty the defectives via a two-stage procedure, $\bar T(N,p)$, is known to scale as $Np|\log p|$. Here we determine the sharp asymptotic value of $\bar T(N,p)/(Np|\log p|)$ 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, while the tests have Poisson-distributed degrees. Finally, we improve the existing upper and lower bound for the optimal number of tests in the case $p=1/N^{\beta}$ with $\beta\in[1/2,1)$.