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            <title xml:lang="en">Group Testing with Random Pools: optimal two-stage algorithms</title>
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                <title xml:lang="en">Group Testing with Random Pools: optimal two-stage algorithms</title>
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              <p>We study Probabilistic Group Testing of a set of N items each of which is defective with probability p. We focus on the double limit of small defect probability, p&lt;&gt;1, taking either p-&gt;0 after $N\to\infty$ or $p=1/N^{\beta}$ with $\beta\in(0,1/2)$. In both settings the optimal number of tests which are required to identify with certainty the defectives via a two-stage procedure, $\bar T(N,p)$, is known to scale as $Np|\log p|$. Here we determine the sharp asymptotic value of $\bar T(N,p)/(Np|\log p|)$ and construct a class of two-stage algorithms over which this optimal value is attained. This is done by choosing a proper bipartite regular graph (of tests and variable nodes) for the first stage of the detection. Furthermore we prove that this optimal value is also attained on average over a random bipartite graph where all variables have the same degree, while the tests have Poisson-distributed degrees. Finally, we improve the existing upper and lower bound for the optimal number of tests in the case $p=1/N^{\beta}$ with $\beta\in[1/2,1)$.</p>
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