Efficient designs for threshold group testing without gap

TL;DR

The paper addresses threshold group testing without gap, where a test is positive if a subset contains at least $u$ defectives. It introduces deterministic, non-adaptive and 2-stage designs built from single selectors and disjunct matrices to recover the $d$ defective items among $n$, with decoding times that scale polynomially in $d$ and $u$ and logarithmically in $n$. For constant $u$, it achieves $t=O(d^3\log^2 n\log(n/d))$ tests and decoding $O(d^3\log^2 n\log(n/d)+d^2\log n\log^3(n/d))$, and for $u=2$ it yields $t=O(d^3\log n\log(n/d))$ with decoding $O(d^2(\log n+\log^2(n/d)))$, plus a 2-stage design with $O(d^2\log^2(n/d))$ tests. The results substantially improve upon previous non-adaptive schemes by achieving decoding times that are polynomial in $d$ and $u$ and provide explicit constructions under the deterministic, combinatorial setting. These designs have potential implications for rapid, scalable identification tasks in large populations where threshold-based group testing is appropriate.

Abstract

Given $d$ defective items in a population of $n$ items with $d \ll n$, in threshold group testing without gap, the outcome of a test on a subset of items is positive if the subset has at least $u$ defective items and negative otherwise, where $1 \leq u \leq d$. The basic goal of threshold group testing is to quickly identify the defective items via a small number of tests. In non-adaptive design, all tests are designed independently and can be performed in parallel. The decoding time in the non-adaptive state-of-the-art work is a polynomial of $(d/u)^u (d/(d-u))^{d - u}, d$, and $\log{n}$. In this work, we present a novel design that significantly reduces the number of tests and the decoding time to polynomials of $\min\{u^u, (d - u)^{d - u}\}, d$, and $\log{n}$. In particular, when $u$ is a constant, the number of tests and the decoding time are $O(d^3 (\log^2{n}) \log{(n/d)} )$ and $O\big(d^3 (\log^2{n}) \log{(n/d)} + d^2 (\log{n}) \log^3{(n/d)} \big)$, respectively. For a special case when $u = 2$, with non-adaptive design, the number of tests and the decoding time are $O(d^3 (\log{n}) \log{(n/d)} )$ and $O(d^2 (\log{n} + \log^2{(n/d)}) )$, respectively. Moreover, with 2-stage design, the number of tests and the decoding time are $O(d^2 \log^2{(n/d)} )$.

Figures (2)

• Figure 1: Illustration of threshold group testing without gap for $u = 4$ versus standard group testing.
• Figure 2: Illustration of the encoding procedure for $u = 3, d = 5$, and $P = \{ j_1, j_2, j_3, j_4, j_5 \}$.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Definition 1
• Theorem 3
• Theorem 4
• Definition 2
• Lemma 1
• proof
• Lemma 2
• Definition 3