几何分组检验

摘要：

分组测试涉及在一组$m$项目中识别$t$缺陷项目，其中每个测试报告项目的特定子集是否包含至少一个缺陷。在非自适应分组测试中，待测子集是预先确定的。通过一次测试多个项目，所需的测试次数可以比$m$少得多。事实上，对于$t \in \mathcal{O}(1)$，（非自适应）测试的最佳数量已知为$Θ(\log{m})$。本文研究了几何环境下的非自适应群体测试问题，其中测试项目是$d$维欧氏空间中的点，测试项目是轴平行的盒子（超矩形）.我们提出的上限和下限所需的测试次数在此几何约束下。与一般的组合情况相比，我们的几何设置中的边界是$m$中的多项式。例如，我们的研究结果意味着，在一组$m$点在平面上识别一个有缺陷的对总是需要$Ω(m^{3/5})$测试，并存在配置的$m$点，其中$\mathcal{O}(m^{2/3})$测试是足够的，而要识别一个单一的缺陷点在平面上，$Θ(m^{1/2})$测试总是必要的，有时是足够的。

英文摘要：

Group testing is concerned with identifying $t$ defective items in a set of $m$ items, where each test reports whether a specific subset of items contains at least one defective. In non-adaptive group testing, the subsets to be tested are fixed in advance. By testing multiple items at once, the required number of tests can be made much smaller than $m$. In fact, for $t \in \mathcal{O}(1)$, the optimal number of (non-adaptive) tests is known to be $Θ(\log{m})$. In this paper, we consider the problem of non-adaptive group testing in a geometric setting, where the items are points in $d$-dimensional Euclidean space and the tests are axis-parallel boxes (hyperrectangles). We present upper and lower bounds on the required number of tests under this geometric constraint. In contrast to the general, combinatorial case, the bounds in our geometric setting are polynomial in $m$. For instance, our results imply that identifying a defective pair in a set of $m$ points in the plane always requires $Ω(m^{3/5})$ tests, and there exist configurations of $m$ points for which $\mathcal{O}(m^{2/3})$ tests are sufficient, whereas to identify a single defective point in the plane, $Θ(m^{1/2})$ tests are always necessary and sometimes sufficient.

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