On the Optimality of the Kautz-Singleton Construction in Probabilistic Group Testing

• Published in: IEEE transactions on information theory Vol. 65; no. 9; pp. 5592 - 5603
• Main Authors: Inan, Huseyin A., Kairouz, Peter, Wootters, Mary, Ozgur, Ayfer
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.09.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Atmospheric measurements; Combinatorial analysis; Complexity theory; Construction; Decoding; efficient decoding; explicit constructions; group testing; Noise measurement; Optimization; Particle measurements; Probabilistic logic; Recursive methods; Statistical analysis; Testing
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2019.2902397

We consider the probabilistic group testing problem where <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> random defective items in a large population of <inline-formula> <tex-math notation="LaTeX">N </tex-math></inline-formula> items are identified with high probability by applying binary tests. It is known that the <inline-formula> <tex-math notation="LaTeX">\Theta (d \log N) </tex-math></inline-formula> tests are necessary and sufficient to recover the defective set with vanishing probability of error when <inline-formula> <tex-math notation="LaTeX">d = O(N^{\alpha }) </tex-math></inline-formula> for some <inline-formula> <tex-math notation="LaTeX">\alpha \in (0, 1) </tex-math></inline-formula>. However, to the best of our knowledge, there is no explicit (deterministic) construction achieving <inline-formula> <tex-math notation="LaTeX">\Theta (d \log N) </tex-math></inline-formula> tests in general. In this paper, we show that a famous construction introduced by Kautz and Singleton for the combinatorial group testing problem (which is known to be suboptimal for combinatorial group testing for moderate values of <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula>) achieves the order optimal <inline-formula> <tex-math notation="LaTeX">\Theta (d \log N) </tex-math></inline-formula> tests in the probabilistic group testing problem when <inline-formula> <tex-math notation="LaTeX">d = \Omega (\log ^{2}\,\,N) </tex-math></inline-formula>. This provides a strongly explicit construction achieving the order optimal result in the probabilistic group testing setting for a wide range of values of <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula>. To prove the order-optimality of Kautz and Singleton's construction in the probabilistic setting, we provide a novel analysis of the probability of a non-defective item being covered by a random defective set directly, rather than arguing from combinatorial properties of the underlying code, which has been the main approach in the literature. Furthermore, we use a recursive technique to convert this construction into one that can also be efficiently decoded with only a log-log factor increase in the number of tests.