Non-Adaptive Randomized Algorithm for Group Testing

TL;DR

This paper introduces a new semi-disjunction property for non-adaptive group testing algorithms in the RID model, enabling linear-time decoding with near-optimal test counts, outperforming existing methods especially for small defect counts.

Contribution

The paper proposes the semi-disjunction property, allowing linear-time decoding in non-adaptive group testing and achieving near-optimal test numbers in the RID model.

Findings

Number of tests converges to the optimal as defect count increases.

Semi-disjunction property enables linear-time decoding.

Algorithm outperforms disjunction-based methods for small defect counts.

Abstract

We study the problem of group testing with a non-adaptive randomized algorithm in the random incidence design (RID) model where each entry in the test is chosen randomly independently from $\{0,1\}$ with a fixed probability $p$. The property that is sufficient and necessary for a unique decoding is the separability of the tests, but unfortunately no linear time algorithm is known for such tests. In order to achieve linear-time decodable tests, the algorithms in the literature use the disjunction property that gives almost optimal number of tests. We define a new property for the tests which we call semi-disjunction property. We show that there is a linear time decoding for such test and for $d\to \infty$ the number of tests converges to the number of tests with the separability property and is therefore optimal (in the RID model). Our analysis shows that, in the RID model, the number of tests in our algorithm is better than the one with the disjunction property even for small $d$.