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 $\Theta(\log{m})$. 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 $\Omega(m^{3/5})$ 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, $\Theta(m^{1/2})$ tests are always necessary and sometimes sufficient.
翻译:组测试涉及在一组美元项目中确定有缺陷的物品, 每份测试报告某一特定组项是否至少含有一个缺陷。 在非适应性组测试中, 要测试的子项是事先固定的。 通过一次测试多个项目, 所需的测试数量可以大大小于百万美元。 事实上, $t $@ in\ mathcal{O}(1)美元, 最佳( 非适应性) 测试数量( 美元) 。 在本文中, 我们考虑到在几何设置中进行非适应性组测试的问题, 而在非适应性组测试中, 需要用美元进行 euclidean 空间和测试的子项是轴- parel 框( 节点 ) 。 我们在此几何限制下显示所需测试数量的上下界限 。 与一般的、 cominal 案例相比, 我们的单项设置的界限总是需要美元。 例如, 我们的结果意味着, 在某平方平方平方平方平面的测试中, 一定的平方平方平方平面测试点 。