Optimal NonAdaptive Probabilistic Group Testing in General Sparsity Regimes
Abstract
In this paper, we consider the problem of noiseless nonadaptive probabilistic group testing, in which the goal is highprobability recovery of the defective set. We show that in the case of $n$ items among which $k$ are defective, the smallest possible number of tests equals $\min\{ C_{k,n} k \log n, n\}$ up to lowerorder asymptotic terms, where $C_{k,n}$ is a uniformly bounded constant (varying depending on the scaling of $k$ with respect to $n$) with a simple explicit expression. The algorithmic upper bound follows from a minor adaptation of an existing analysis of the Definite Defectives (DD) algorithm, and the algorithmindependent lower bound builds on existing works for the regimes $k \le n^{1\Omega(1)}$ and $k = \Theta(n)$. In sufficiently sparse regimes (including $k = o\big( \frac{n}{\log n} \big)$), our main result generalizes that of CojaOghlan {\em et al.} (2020) by avoiding the assumption $k \le n^{1\Omega(1)}$, whereas in sufficiently dense regimes (including $k = \omega\big( \frac{n}{\log n} \big)$), our main result shows that individual testing is asymptotically optimal for any nonzero target success probability, thus strengthening an existing result of Aldridge (2019) in terms of both the error probability and the assumed scaling of $k$.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.01325
 Bibcode:
 2020arXiv200601325B
 Keywords:

 Computer Science  Information Theory;
 Mathematics  Probability
 EPrint:
 Approximate recovery results are in v1 only