Group testing with Random Pools: Phase Transitions and Optimal Strategy

TL;DR

This paper investigates probabilistic group testing with random pools, identifying phase transitions and designing optimal algorithms that minimize the number of tests needed for detection, especially in the small defective probability regime.

Contribution

It introduces one- and two-stage algorithms with optimal scaling and pool designs, achieving minimal tests for detection in probabilistic group testing.

Findings

Existence of a sharp phase transition in detection probability.

Optimal algorithms scale as Np|log p| in the number of tests.

Two-stage algorithms can attain the minimal tests for complete detection.

Abstract

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, $p \to 0$ and $N \to \infty$. 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, $\bar M$, for the number of tests. By optimizing the pool design we construct algorithms whose detection threshold follows the optimal scaling $\bar M\propto Np|\log p|$. 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 Ref. [16]) for the mean number of tests required for complete detection. We finally discuss the optimal pool design in the case of finite $p$.