Non-adaptive Quantitative Group Testing Using Irregular Sparse Graph Codes

TL;DR

This paper introduces a non-adaptive Quantitative Group Testing scheme using irregular sparse graph codes, achieving near-optimal test efficiency and low computational complexity in the sub-linear defect regime.

Contribution

It proposes a novel non-adaptive QGT scheme with irregular bipartite graphs and BCH codes, improving test efficiency over existing methods in the sub-linear regime.

Findings

Achieves identification of defective items with high probability using fewer tests.

Provides algorithms with low computational complexity for t ≤ 4.

Outperforms recent non-adaptive QGT schemes in the sub-linear regime.

Abstract

This paper considers the problem of Quantitative Group Testing (QGT) where there are some defective items among a large population of $N$ items. We consider the scenario in which each item is defective with probability $K/N$, independently from the other items. In the QGT problem, the goal is to identify all or a sufficiently large fraction of the defective items by testing groups of items, with the minimum possible number of tests. In particular, the outcome of each test is a non-negative integer which indicates the number of defective items in the tested group. In this work, we propose a non-adaptive QGT scheme for the underlying randomized model for defective items, which utilizes sparse graph codes over irregular bipartite graphs with optimized degree profiles on the left nodes of the graph as well as binary $t$-error-correcting BCH codes. We show that in the sub-linear regime, i.e., when the ratio $K/N$ vanishes as $N$ grows unbounded, the proposed scheme with ${m=c(t,d)K(t\log (\frac{\ell N}{c(t,d)K}+1)+1)}$ tests can identify all the defective items with probability approaching $1$, where $d$ and $\ell$ are the maximum and average left degree, respectively, and $c(t,d)$ depends only on $t$ and $d$ (and does not depend on $K$ and $N$). For any $t\leq 4$, the testing and recovery algorithms of the proposed scheme have the computational complexity of $\mathcal{O}(N\log \frac{N}{K})$ and $\mathcal{O}(K\log \frac{N}{K})$, respectively. The proposed scheme outperforms two recently proposed non-adaptive QGT schemes for the sub-linear regime, including our scheme based on regular bipartite graphs and the scheme of Gebhard et al., in terms of the number of tests required to identify all defective items with high probability.