具有固定大小样本的成组检验
Group Testing with Pools of Fixed Size
摘要：
在经典的组合（自适应）组测试问题中，给定两个整数\(d\)和\(n\)，其中\(0\le d\le n\)和\(n\)项的总体，确切地说，其中\(d\)已知是有缺陷的。问题是设计一个最佳的顺序算法，在每一步，测试人口的子集，并确定这样的子集是否被污染（即包含有缺陷的项目）或其他。只有在识别出\(d\)缺陷项目后，问题才能解决。最佳顺序算法通常（即在最坏情况下）解决问题所需的最小步骤数由\(M(d, n)\)表示。\(M(d, n)\)的计算似乎是非常困难的，并且仅已知\(d = 1\)的通用公式。我们在这里考虑一个变种的原始问题，其中要测试的子集的大小被限制为一个固定的正整数\(k\)。用\(M^{\lbrack k\rbrack}(d, n)\)表示顺序优化算法的相应最小测试次数。在本文中，我们开始调查的功能\(M^{\lbrack k\rbrack}(d, n)\)。
英文摘要：
In the classical combinatorial (adaptive) group testing problem, one is given two integers \(d\) and \(n\), where \(0\le d\le n\), and a population of \(n\) items, exactly \(d\) of which are known to be defective. The question is to devise an optimal sequential algorithm that, at each step, tests a subset of the population and determines whether such subset is contaminated (i.e. contains defective items) or otherwise. The problem is solved only when the \(d\) defective items are identified. The minimum number of steps that an optimal sequential algorithm takes in general (i.e. in the worst case) to solve the problem is denoted by \(M(d, n)\). The computation of \(M(d, n)\) appears to be very difficult and a general formula is known only for \(d = 1\). We consider here a variant of the original problem, where the size of the subsets to be tested is restricted to be a fixed positive integer \(k\). The corresponding minimum number of tests by a sequential optimal algorithm is denoted by \(M^{\lbrack k\rbrack}(d, n)\). In this paper we start the investigation of the function \(M^{\lbrack k\rbrack}(d, n)\).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
