In this paper, we consider the problem of noiseless non-adaptive probabilistic group testing, in which the goal is high-probability recovery of the defective set. We show that in the case of $n$ items among which $k$ are defective, the smallest possible number of tests equals $\min\{ C_{k,n} k \log n, n\}$ up to lower-order asymptotic terms, where $C_{k,n}$ is a uniformly bounded constant (varying depending on the scaling of $k$ with respect to $n$) with a simple explicit expression. The algorithmic upper bound follows from a minor adaptation of an existing analysis of the Definite Defectives (DD) algorithm, and the algorithm-independent lower bound builds on existing works for the regimes $k \le n^{1-\Omega(1)}$ and $k = \Theta(n)$. In sufficiently sparse regimes (including $k = o\big( \frac{n}{\log n} \big)$), our main result generalizes that of Coja-Oghlan {\em et al.} (2020) by avoiding the assumption $k \le n^{1-\Omega(1)}$, whereas in sufficiently dense regimes (including $k = \omega\big( \frac{n}{\log n} \big)$), our main result shows that individual testing is asymptotically optimal for any non-zero target success probability, thus strengthening an existing result of Aldridge (2019) in terms of both the error probability and the assumed scaling of $k$.
翻译:在本文中,我们考虑的是无噪音的非适应性概率组测试问题,在测试中,目标是对有缺陷的一组进行高概率回收。我们表明,在美元的项目中,如果有缺陷,最低的可能测试数量等于$min ⁇ C ⁇ k,n}k\log n,n ⁇ $最高等于低排序无效力值,美元等于低排序无效力值为统一约束常数(取决于美元相对于美元比值的美元比值),且不言而喻的表达方式非常明确。算法上限是因为对目前对低性感染者(DD)算法的分析进行了微小的调整,而依赖的算法较低约束值则等于$k\le n ⁇ 1\\\\\\\\\\\\\\\\\\\\odga(1)}美元,而美元在足够稀少的制度中(包括美元=美元比值比值比值(n\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\