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Sparsity-Constrained Community-Based Group Testing

Sarthak Jain, Martina Cardone, Soheil Mohajer

TL;DR

A probabilistic group testing algorithm that can identify the infected population with a vanishing probability of error and provide an upper-bound on the number of tests is proposed and can be applied to the classical dilution model, where it outperforms existing noise-level-independent schemes in the literature.

Abstract

In this work, we consider the sparsity-constrained community-based group testing problem, where the population follows a community structure. In particular, the community consists of $F$ families, each with $M$ members. A number $k_f$ out of the $F$ families are infected, and a family is said to be infected if $k_m$ out of its $M$ members are infected. Furthermore, the sparsity constraint allows at most $ρ_T$ individuals to be grouped in each test. For this sparsity-constrained community model, we propose a probabilistic group testing algorithm that can identify the infected population with a vanishing probability of error and we provide an upper-bound on the number of tests. When $k_m = Θ(M)$ and $M \gg \log(FM)$, our bound outperforms the existing sparsity-constrained group testing results trivially applied to the community model. If the sparsity constraint is relaxed, our achievable bound reduces to existing bounds for community-based group testing. Moreover, our scheme can also be applied to the classical dilution model, where it outperforms existing noise-level-independent schemes in the literature.

Sparsity-Constrained Community-Based Group Testing

TL;DR

A probabilistic group testing algorithm that can identify the infected population with a vanishing probability of error and provide an upper-bound on the number of tests is proposed and can be applied to the classical dilution model, where it outperforms existing noise-level-independent schemes in the literature.

Abstract

In this work, we consider the sparsity-constrained community-based group testing problem, where the population follows a community structure. In particular, the community consists of families, each with members. A number out of the families are infected, and a family is said to be infected if out of its members are infected. Furthermore, the sparsity constraint allows at most individuals to be grouped in each test. For this sparsity-constrained community model, we propose a probabilistic group testing algorithm that can identify the infected population with a vanishing probability of error and we provide an upper-bound on the number of tests. When and , our bound outperforms the existing sparsity-constrained group testing results trivially applied to the community model. If the sparsity constraint is relaxed, our achievable bound reduces to existing bounds for community-based group testing. Moreover, our scheme can also be applied to the classical dilution model, where it outperforms existing noise-level-independent schemes in the literature.
Paper Structure (19 sections, 4 theorems, 42 equations, 2 tables)

This paper contains 19 sections, 4 theorems, 42 equations, 2 tables.

Table of Contents

  1. Introduction
  2. System Model
  3. Preliminaries and Related Results
  4. Combinatorial GT
  5. Sparsity-Constrained Combinatorial GT
  6. Community-Based GT Without Sparsity Constraints
  7. Incorporating Sparsity in the Two-Stage Algorithm
  8. The Proposed GT Scheme
  9. First Stage: Identifying Infected Families
  10. Second Stage: Identifying All the Infected Members
  11. Analysis and Comparison
  12. Proof of Theorem \ref{['thm1']}
  13. Proof of Corollary \ref{['corollary:AnalysisT']}
  14. Proof of Proposition \ref{['prop:prereq']}
  15. Proof of Proposition \ref{['prop:prereq']}(i)
  16. ...and 4 more sections

Key Result

Theorem 1

There exists a choice of the parameters $\left(\rho, d \right)$ such that the $d$-threshold decoder requires at most

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Remark 1
  • Corollary 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof