Efficiently Decodable Non-Adaptive Threshold Group Testing
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The basic goal of group testing is to identify at most very specific d defective items among N items, where d is usually much smaller than N. Because testing each item to identify whether it is defective takes time and is expensive, pooling subsets of N items is more preferable. Normally, if there is at least one defective item in a subset, the test outcome of that subset would be positive, and otherwise. However, in many biological applications, it needs more than one defective item in a test in order to get a positive outcome. In this paper, we consider non-adaptive threshold group testing in which a test would be positive if it contains at least u ≤ d defective items, and negative otherwise, and all tests are designed in advance. In this model, at most d defective items can be identified using t = O ( d^2/d^2 - u^2( e (d - u)/u)^u (u d/u + 1/ϵ) · d^2 N) tests with probability at least 1 - ϵ for any ϵ > 0 or t = O ( d^2/d^2 - u^2( e (d-u)/u)^u ·( dN/d + ud/u) · d^2 N) tests with probability of 1. The decoding time is t/O(d^2 N)× poly(d^2 N), where poly(·) is a polynomial of the input. This result significantly improves the best known results on decoding non-adaptive threshold group testing, which are O(NN + N 1/ϵ) for probabilistic decoding, where ϵ > 0, and O(N^u N) for deterministic decoding.
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