Efficiently Decodable Non-Adaptive Threshold Group Testing

Thach V. Bui, Minoru Kuribayashi, Mahdi Cheraghchi, Isao Echizen

Abstract: We consider non-adaptive threshold group testing for identification of up to d defective items in a set of n items, where a test is positive if it contains at least 2 ≤ u ≤ d defective items, and negative otherwise. The defective items can be identified using t=O(( [d/u]) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sup> ([d/(d-u)]) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d-u</sup> (ulog[d/u]+log[1/(ε)])d <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> logn) tests with probability at least 1-ε for any or t = O(([b/u]) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sup> ([d/(d-u)]) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d-u</sup> ·d <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> logn ·log[n/d]) tests with probability 1. The decoding time is t× poly (d <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> logn). This result significantly improves the best known results for decoding non-adaptive threshold group testing: O(n logn+nlog[1/(ε)]) for probabilistic decoding, where , and O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sup> logn) for deterministic decoding.