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In this paper, we consider the problem of noiseless non-adaptive grouptesting under the for-each recovery guarantee, also known as probabilisticgroup testing. In the case of $n$ items and $k$ defectives, we provide analgorithm attaining high-probability recovery with $O(k \log n)$ scaling inboth 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 adaptivegeneralized binary splitting algorithm (Hwang, 1972); we recursively work withgroups of items of geometrically vanishing sizes, while maintaining a list of"possibly defective" groups and circumventing the need for adaptivity. Whilethe most basic form of our algorithm requires $\Omega(n)$ storage, we alsoprovide a low-storage variant based on hashing, with similar recoveryguarantees.

A Fast Binary Splitting Approach to Non-Adaptive Group Testing