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Group testing (GT) was originally proposed during the World War II in an attempt to minimize the \emph{cost} and \emph{waiting time} in performing identical blood tests of the soldiers for a low-prevalence disease. Formally, the GT problem asks to find d≪n \emph{defective} elements out of n elements by querying subsets (pools) for the presence of defectives.By the information-theoretic lower bound, essentially dlog2n queries are needed in the worst-case.An \emph{adaptive} strategy proceeds sequentially by performing one query at a time, and it can achieve the lower bound. In various applications, nothing is known about d beforehand and a strategy for this scenario is called \emph{competitive}. Such strategies are usually adaptive and achieve query optimality within a constant factor called the \emph{competitive ratio}. In many applications, queries are time-consuming. Therefore, \emph{minimal-adaptive} strategies which run in a small number s of stages of parallel queries are favorable. This work is mainly devoted to the design of minimal-adaptive strategies combined with other demands of both theoretical and practical interest. First we target unknown d and show that actually competitive GT is possible in as few as 2 stages only. The main ingredient is our randomized estimate of a previously unknown d using nonadaptive queries. In addition, we have developed a systematic approach to obtain optimal competitive ratios for our strategies.When d is a known upper bound,we propose randomized GT strategies which asymptotically achieve query optimality in just 2, 3 or 4 stages depending upon the growth of d versus n.Inspired by application settings, such as at American Red Cross, where in most cases GT is applied to small instances, \textit{e.g.}, n=16. We extended our study of query-optimal GT strategies to solve a given problem instance with fixed values n, d and s. We also considered the situation whenelements to test cannot be divided physically (electronic devices), thus the pools must be disjoint. For GT with \emph{disjoint} simultaneous pools, we show that Θ(sd(n/d)1/s) tests are sufficient, and also necessary for certain ranges of the parameters
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