Noisy Nonadaptive Group Testing with Binary Splitting: New Test Design and Improvement on Price-Scarlett-Tan's Scheme
Xiaxin Li, Arya Mazumdar
TL;DR
This work extends fast binary-splitting nonadaptive probabilistic group testing to noisy settings by introducing NEON, a framework that delivers noise-resilient, near-optimal test designs and decoding. It presents two main strands: (i) a noiseless and ρ-False Positive Channel (FPC) NEON achieving $M=\mathcal{O}(\epsilon^{-1} K\log N)$ and $D=\mathcal{O}(\epsilon^{-2} K\log N)$, and (ii) a sublinear-sparse regime NEON under ρ-BSC achieving $D=\mathcal{O}(\epsilon^{-2} K^{1+\epsilon})$ with $M=\mathcal{O}(\epsilon^{-1} K\log N)$, and a $(\rho,\rho')$-BAC NEON attaining $M=D=\mathcal{O}(\epsilon^{-1} K\log N)$ under suitable constraints. By leveraging a structured combination of local decoding, sub-tree analysis, and carefully tuned constants, the authors reduce decoding complexity by up to factors of $({\log N})^{1+\epsilon}$ relative to prior binary-splitting approaches while preserving asymptotically optimal test counts in several noise regimes. The results advance practical nonadaptive GT design with robust performance in realistic noisy environments, and set the stage for broader extensions to BAC/BSC models and tighter constants.
Abstract
In Group Testing, the objective is to identify $K$ defective items out of $N$, $K\ll N$, by testing pools of items together and using the least amount of tests possible. Recently, a fast decoding method based on binary splitting (Price and Scarlett, 2020) has been proposed that simultaneously achieve optimal number of tests and decoding complexity for Non-Adaptive Probabilistic Group Testing (NAPGT). However, the method works only when the test results are noiseless. In this paper, we further study the binary splitting method and propose (1) A NAPGT scheme that generalizes the original binary splitting method from the noiseless case into tests with $ρ$ proportion of false positives (the $ρ$-False Positive Channel), where $ρ$ is a constant, with asymptotically-optimal number of tests and decoding complexity, i.e. $\mathcal{O}(K\log N)$, and (2) A NAPGT scheme in the presence of both false positives and false negatives in test outcomes, improving and generalizing the work of Price, Scarlett and Tan~\cite{price2023fast} in two ways: First, under $ρ$-proportion of test results flipped ($ρ$-Binary Symmetric Channel) and within the general sublinear regime $K=Θ(N^α)$ where $0<α<1$, our algorithm has a decoding complexity of $\mathcal{O}(ε^{-2}K^{1+ε})$ where $ε>0$ is a constant parameter. Second, when the false negative flipping probability $ρ'$ satisfies $ρ'=\mathcal{O}(K^{-ε})$ and the false positive flipping probability $ρ$ is a constant, we can simultaneously achieve $\mathcal{O}(ε^{-1}K\log N)$ for both the number of tests and the decoding complexity. It remains open to achieve these optimals under the general BSC.