In the group testing problem the aim is to identify a small set of $ksim n^theta$ infected individuals out of a population size $n$, $0<theta<1$. We avail ourselves of a test procedure capable of testing groups of individuals, with the test returning a positive result iff at least one individual in the group is infected. The aim is to devise a test design with as few tests as possible so that the set of infected individuals can be identified correctly with high probability. We establish an explicit sharp information-theoretic/algorithmic phase transition $minf$ for non-adaptive group testing, where all tests are conducted in parallel. Thus, with more than $minf$ tests the infected individuals can be identified in polynomial time whp, while learning the set of infected individuals is information-theoretically impossible with fewer tests. In addition, we develop an optimal adaptive scheme where the tests are conducted in two stages.
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