An Efficient Algorithm for Group Testing with Runlength Constraints

TL;DR

This paper introduces an efficient randomized algorithm for constructing almost optimal superimposed codes with runlength constraints, improving upon previous code lengths and applicable to non-adaptive group testing.

Contribution

It presents a novel randomized Las Vegas algorithm for constructing superimposed codes with runlength constraints, achieving shorter code lengths than prior existence proofs.

Findings

Code length is $O(dk ext{log} n + k^2 ext{log} n)$ for large $n$

Algorithm complexity is $ heta(t n^2)$

Constructed codes are shorter than previous known codes for large $n$

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 $\mathbf{c}$ and any other $k-1$ columns, there exists a row where $\mathbf{c}$ has $1$ and all the remaining $k-1$ columns have $0$. These combinatorial structures were introduced by Agarwal et al. [1], in the context of Non-Adaptive Group Testing algorithms with runlength constraints. By using Moser and Tardos' constructive version of the Lov\'asz Local Lemma, we provide an efficient randomized Las Vegas algorithm of complexity $\Theta(t n^2)$ for the construction of $(k,n,d)$-superimposed codes of length $t=O(dk\log n +k^2\log n)$. 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 [1].