Quickly-Decodable Group Testing with Fewer Tests: Price-Scarlett and Cheraghchi-Nakos's Nonadaptive Splitting with Explicit Scalars

TL;DR

This paper presents a modified nonadaptive group testing method that nearly achieves the optimal number of tests with sub-linear decoding complexity, improving efficiency in identifying infected individuals.

Contribution

It introduces a modified binary splitting approach that reduces the number of tests and decoding complexity, nearly matching the best known scalar and sub-$n$ complexity.

Findings

Tests required are proportional to $k \, ext{ln} \, n$ with a specific constant.

Decoding complexity is $O( ext{ln}^{-2} \, n)$, nearly sublinear.

Achieves a balance between the number of tests and decoding complexity.

Abstract

We modify Cheraghchi-Nakos [CN20] and Price-Scarlett's [PS20] fast binary splitting approach to nonadaptive group testing. We show that, to identify a uniformly random subset of $k$ infected persons among a population of $n$, it takes only $\ln(2 - 4\varepsilon) ^{-2} k \ln n$ tests and decoding complexity $O(\varepsilon^{-2} k \ln n)$, for any small $\varepsilon > 0$, with vanishing error probability. In works prior to ours, only two types of group testing schemes exist. Those that use $\ln(2)^{-2} k \ln n$ or fewer tests require linear-in-$n$ complexity, sometimes even polynomial in $n$; those that enjoy sub-$n$ complexity employ $O(k \ln n)$ tests, where the big-$O$ scalar is implicit, presumably greater than $\ln(2)^{-2}$. We almost achieve the best of both worlds, namely, the almost-$\ln(2)^{-2}$ scalar and the sub-$n$ decoding complexity. How much further one can reduce the scalar $\ln(2)^{-2}$ remains an open problem.