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> (u log d/u + log 1/∈)·d <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> log n) tests with probability at least 1 - ∈ for any ∈ > 0 or 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> d <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> log n · log d/n) 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> log n). This result significantly improves the best known results for decoding non-adaptive threshold group testing: O(n log n + n log 1/∈) for probabilistic decoding, where ∈ > 0, and O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sup> log n) for deterministic decoding.
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