Fast Splitting Algorithms for Sparsity-Constrained and Noisy Group Testing

TL;DR

This paper introduces fast, near-optimal splitting algorithms for sparsity-constrained and noisy group testing problems, achieving efficient recovery with low storage options and broad applicability in various fields.

Contribution

The paper presents novel splitting algorithms for group testing that are both fast and near-optimal under sparsity and noise constraints, including low-storage variants.

Findings

Algorithms achieve near-optimal number of tests and decoding time.

Low-storage variants based on hashing maintain recovery guarantees.

Algorithms are applicable to practical group testing scenarios with constraints.

Abstract

In group testing, the goal is to identify a subset of defective items within a larger set of items based on tests whose outcomes indicate whether at least one defective item is present. This problem is relevant in areas such as medical testing, DNA sequencing, communication protocols, and many more. In this paper, we study (i) a sparsity-constrained version of the problem, in which the testing procedure is subjected to one of the following two constraints: items are finitely divisible and thus may participate in at most $\gamma$ tests; or tests are size-constrained to pool no more than $\rho$ items per test; and (ii) a noisy version of the problem, where each test outcome is independently flipped with some constant probability. Under each of these settings, considering the for-each recovery guarantee with asymptotically vanishing error probability, we introduce a fast splitting algorithm and establish its near-optimality not only in terms of the number of tests, but also in terms of the decoding time. While the most basic formulations of our algorithms require $\Omega(n)$ storage for each algorithm, we also provide low-storage variants based on hashing, with similar recovery guarantees.