An improved zig zag approach for competitive group testing

In many fault detection problems, we want to identify defective items from a set of $n$ items using the minimum number of tests. Group testing is for the scenario where each test is on a subset of items, and tells whether the subset contains at least one defective item or not. In practice, the number $d$ of defective items is often unknown in advance. In this paper, we improve the previously best algorithm for a central problem in combinatorial group testing with unknown number of defectives (Cheng et al., 2014), and prove that the number of tests used by our new algorithm is no more than $dlog\frac{n}{d}+(5-log5)d+O\left({log}^{2}d\right)$, where $log$ is of base 2.