A Fast Binary Splitting Approach to Non-Adaptive Group Testing

TL;DR

This paper introduces a fast, non-adaptive group testing algorithm that achieves near-optimal test and runtime complexity, improving efficiency over previous methods by using a binary splitting approach without requiring adaptivity.

Contribution

The paper presents a novel non-adaptive group testing algorithm with $O(k \,\log n)$ tests and runtime, inspired by binary splitting, and includes a low-storage variant using hashing.

Findings

Achieves $O(k \log n)$ tests and runtime for probabilistic group testing.

Improves previous runtime bounds from $O(k^2 \log k \log n)$ to near-linear.

Provides a low-storage version with similar guarantees.

Abstract

In this paper, we consider the problem of noiseless non-adaptive group testing under the for-each recovery guarantee, also known as probabilistic group testing. In the case of $n$ items and $k$ defectives, we provide an algorithm attaining high-probability recovery with $O(k \log n)$ scaling in both the number of tests and runtime, improving on the best known $O(k^2 \log k \cdot \log n)$ runtime previously available for any algorithm that only uses $O(k \log n)$ tests. Our algorithm bears resemblance to Hwang's adaptive generalized binary splitting algorithm (Hwang, 1972); we recursively work with groups of items of geometrically vanishing sizes, while maintaining a list of "possibly defective" groups and circumventing the need for adaptivity. While the most basic form of our algorithm requires $\Omega(n)$ storage, we also provide a low-storage variant based on hashing, with similar recovery guarantees.