Improved Non-Adaptive Algorithms for Threshold Group Testing With a Gap

The basic goal of threshold group testing is to identify up to <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> defective items among a population of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> items, where <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> is usually much smaller than <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. The outcome of a test on a subset of items is positive if the subset has at least <inline-formula> <tex-math notation="LaTeX">$u$ </tex-math></inline-formula> defective items, negative if it has up to <inline-formula> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> defective items, where <inline-formula> <tex-math notation="LaTeX">$0 \leq \ell < u$ </tex-math></inline-formula>, and arbitrary otherwise. This is called threshold group testing. The parameter <inline-formula> <tex-math notation="LaTeX">$g = u - \ell - 1$ </tex-math></inline-formula> is called <italic>the gap</italic>. In this paper, we focus on the case <inline-formula> <tex-math notation="LaTeX">$g > 0$ </tex-math></inline-formula>, i.e., threshold group testing with a gap. Note that the results presented here are also applicable to the case <inline-formula> <tex-math notation="LaTeX">$g = 0$ </tex-math></inline-formula>; however, the results are not as efficient as those in related work. Currently, a few reported studies have investigated test designs and decoding algorithms for identifying defective items. Most of the previous studies have not been feasible because there are numerous constraints on their problem settings or the decoding complexities of their proposed schemes are relatively large. Therefore, it is compulsory to reduce the number of tests as well as the decoding complexity, i.e., the time for identifying the defective items, for achieving practical schemes. The work presented here makes five contributions. The first is a more accurate theorem for a non-adaptive algorithm for threshold group testing proposed by Chen and Fu. The second is an improvement in the construction of disjunct matrices, which are the main tools for tackling (threshold) group testing and other tasks such as constructing cover-free families or learning hidden graphs. Specifically, we present a better exact upper bound on the number of tests for disjunct matrices compared with that in related work. The third and fourth contributions are a reduced exact upper bound on the number of tests and a reduced asymptotic bound on the decoding time for identifying defective items in a noisy setting on test outcomes. The fifth contribution is a simulation on the number of tests of the resulting improvements for previous work and the proposed theorems.