Modifications of Competitive Group Testing

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 Ma(d, n)(M(d In)) denote the maximum number of tests for an algorithm ot 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) mina Ma(d, n). An algorithm c is called a competitive algorithm if there exist constants c and a such that for all n > d > 0, Ma(d In) _< cM(d, n) + a. This paper confirms a recent conjecture that there exists a bisecting algorithm A such that MA(d In) < 2M(d, n) + 1. Also, an algorithm B such that Me(din) <_ 1.65M(d, n) + 10 is presented. Key words, group testing, competitive algorithm AMS subject classification. 68P10