An efficient algorithm for group testing with runlength constraints

Abstract

In this paper, we provide an efficient algorithm to construct almost optimal $(k,n,d)$-superimposed codes with runlength constraints. A $(k,n,d)$-superimposed code of length $t$ is a $t\times n$ binary matrix such that any two 1’s in each column are separated by a run of at least $d$ 0’s, and such that for any column $c$ and any other $k-1$ columns, there exists a row where $c$ has 1 and all the remaining $k-1$ columns have 0. These combinatorial structures were introduced by Agarwal et al. (2020), in the context of Non-Adaptive Group Testing algorithms with runlength constraints.

By using Moser and Tardos’ constructive version of the Lovász Local Lemma, we provide an efficient randomized Las Vegas algorithm of complexity $\Theta \left(t{n}^2\right)$ for the construction of $(k,n,d)$-superimposed codes of length $t=O(dklogn+k^{2}logn)$. We also show that the length of our codes is shorter, for $n$ sufficiently large, than that of the codes whose existence was proved in Agarwal et al. (2020).