Superselectors: Efficient Constructions and Applications

TL;DR

This paper introduces superselectors, a new combinatorial structure that generalizes existing ones and provides efficient algorithms for their construction, with applications in group testing, compressed sensing, and data security.

Contribution

The paper defines superselectors, establishes bounds on their size, and offers the first efficient deterministic algorithms for constructing these structures in various applications.

Findings

Superselectors unify multiple combinatorial structures used in key applications.

Bounds on superselector sizes are close to optimal, matching known bounds for special cases.

Provides the first efficient deterministic algorithms for constructing superselectors in relevant parameter ranges.

Abstract

We introduce a new combinatorial structure: the superselector. We show that superselectors subsume several important combinatorial structures used in the past few years to solve problems in group testing, compressed sensing, multi-channel conflict resolution and data security. We prove close upper and lower bounds on the size of superselectors and we provide efficient algorithms for their constructions. Albeit our bounds are very general, when they are instantiated on the combinatorial structures that are particular cases of superselectors (e.g., (p,k,n)-selectors, (d,\ell)-list-disjunct matrices, MUT_k(r)-families, FUT(k, a)-families, etc.) they match the best known bounds in terms of size of the structures (the relevant parameter in the applications). For appropriate values of parameters, our results also provide the first efficient deterministic algorithms for the construction of such structures.