Sublinear-Time Non-Adaptive Group Testing with O(k n) Tests via Bit-Mixing Coding
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The group testing problem consists of determining a small set of defective items from a larger set of items based on tests on groups of items, and is relevant in applications such as medical testing, communication protocols, pattern matching, and many more. While rigorous group testing algorithms have long been known with polynomial runtime, approaches permitting a decoding time sublinear in the number of items have only arisen more recently. In this paper, we introduce a new approach to sublinear-time non-adaptive group testing called bit mixing coding (BMC). We show that BMC achieves asymptotically vanishing error probability with O(k n) tests and O(k^2 · k · n) runtime, where n is the number of items and k is the number of defectives, in the limit as n →∞ (with k having an arbitrary dependence on n). This closes a key open problem of simultaneously achieving poly(k n) decoding time using O(k n) tests without any assumptions on k. In addition, we show that the same scaling laws can be attained in a commonly-considered noisy setting, in which each test outcome is flipped with constant probability.
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