Modifications of Competitive Group Testing

Abstract

Many fault-detection problems fall into the following model: There is a set of $n$ items, some of which are defective. The goal is to identify the defective items by using the minimum number of tests. Each test is on a subset of items and tells whether the subset contains a defective item or not. Let $M_{\alpha}(d, n) (M_{\alpha}(d\,|\,n))$ denote the maximum number of tests for an algorithm $\alpha$ to identify $d$ defectives from a set of $n$ items provided that $d$, the number of defective items, is known (unknown) before the testing. Let $M(d, n) = \min_{\alpha} M_{\alpha}(d, n)$. An algorithm $\alpha$ is called a {\it competitive algorithm\/} if there exist constants $c$ and $a$ such that for all $n > d > 0$, $M_{\alpha}(d\,|\,n) \leq cM(d, n) + a$. This paper confirms a recent conjecture that there exists a bisecting algorithm $A$ such that $M_A(d\,|\,n) \leq 2M(d, n) + 1$. Also, an algorithm $B$ such that $M_B(d\,|\,n) \leq 1.65M(d, n) + 10$ is presented.