A Tight Lower Bound of Ω(log n) for the Estimation of the Number of Defective Items

09/18/2023

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by   Nader H. Bshouty, et al.

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Let X be a set of items of size n , which may contain some defective items denoted by I, where I ⊆ X. In group testing, a test refers to a subset of items Q ⊂ X. The test outcome is 1 (positive) if Q contains at least one defective item, i.e., Q∩ I ≠∅, and 0 (negative) otherwise. We give a novel approach to obtaining tight lower bounds in non-adaptive randomized group testing. Employing this new method, we can prove the following result. Any non-adaptive randomized algorithm that, for any set of defective items I, with probability at least 2/3, returns an estimate of the number of defective items |I| to within a constant factor requires at least Ω(log n) tests. Our result matches the upper bound of O(log n) and solves the open problem posed by Damaschke and Sheikh Muhammad.