The basic goal of threshold group testing is to identify up to $d$ defective items among a population of $n$ items, where $d$ is usually much smaller than $n$. The outcome of a test on a subset of items is positive if the subset has at least $u$ defective items, negative if it has up to $\ell$ defective items, where $0\leq\ell<u$, and arbitrary otherwise. This is called threshold group testing. The parameter $g=u-\ell-1$ is called \textit{the gap}. In this paper, we focus on the case $g>0$, i.e., threshold group testing with a gap. Note that the results presented here are also applicable to the case $g = 0$; 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. 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.
翻译:阈值组测试的基本目标是在以美元计价的物品中找出最高达美元有缺陷的物品, 美元通常比美元少得多。 如果子组的物品至少有美元有缺陷, 则测试的结果是肯定的。 如果子组的物品有美元有缺陷, 则结果是否定的, 如果它有美元有缺陷的物品, 美元= leq\ ell < u美元, 或者任意的。 这被称为阈值组测试。 参数 $= u\ ell-1$ 被称为 textitit{ 差距} 。 在本文中, 我们侧重于 $g= 0美元, 也就是说, 门槛组的测试结果是肯定的。 这里所报道的几项研究对测试的测试设计和解析算算方法进行了调查。 先前的研究大多不可行, 因为他们的问题组改进或拟议方案的解码复杂程度存在许多限制 。 因此, 开始的临界值组测试是免费的, 隐藏的组群组的测试结果是默认的。 测试过程的难度是精确性, 完成的组的测试过程是精确性, 。 完成的组的试验的进度是精确性测试, 。