Sub-linear Time Stochastic Threshold Group Testing via Sparse-Graph Codes

The group testing problem is to identify a population of K defective items in a set of n items using the results of a small number of measurements or tests. In this paper, we study the stochastic threshold group testing problem where the result of each test is positive if the testing pool contains at least u defective items, and the result is negative if the pool contains at most ${(\ell\lt u)}$ defective items. The result is random if the number of defectives lies in the interval $(\ell,u)$. We leverage tools and techniques from sparse-graph codes and propose a fast decoding algorithm for stochastic threshold group testing. We consider the asymptotic regime that n gets large, $K = \omega (1)$ grows with n and $K = \mathcal{O}(n^{\beta})$ for some constant $0 \lt \beta \lt 1$, and $u = o(K)$. In this regime, the proposed algorithm requires $\Theta(\sqrt{u}K \log^{3}n)$ tests and recovers all the K defectives with a vanishing error probability. Moreover, our algorithm has a decoding complexity of $\mathcal{O}(u^{3/2}K \log^{4}n)$. This is the first algorithm that solves the stochastic threshold group testing problem with decoding complexity that grows only linearly in K and poly-logarithmically in n.