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Fundamental Limits of Non-Adaptive Group Testing with Markovian Correlation

Aditya Narayan Ravi, Ilan Shomorony

TL;DR

Focusing on a very sparse infections regime, this work proposes a non adaptive testing strategy with an efficient decoding scheme that achieves asymptotically vanishing error with a number of tests that is within a multiplicative factor of the fundamental entropy bound.

Abstract

We study a correlated group testing model where items are infected according to a Markov chain, which creates bursty binfection patterns. Focusing on a very sparse infections regime, we propose a non adaptive testing strategy with an efficient decoding scheme that is nearly optimal. Specifically, it achieves asymptotically vanishing error with a number of tests that is within a $1/\ln(2) \approx 1.44$ multiplicative factor of the fundamental entropy bound a result that parallels the independent group testing setting. We show that the number of tests reduces with an increase in the expected burst length of infected items, quantifying the advantage of exploiting correlation in test design.

Fundamental Limits of Non-Adaptive Group Testing with Markovian Correlation

TL;DR

Focusing on a very sparse infections regime, this work proposes a non adaptive testing strategy with an efficient decoding scheme that achieves asymptotically vanishing error with a number of tests that is within a multiplicative factor of the fundamental entropy bound.

Abstract

We study a correlated group testing model where items are infected according to a Markov chain, which creates bursty binfection patterns. Focusing on a very sparse infections regime, we propose a non adaptive testing strategy with an efficient decoding scheme that is nearly optimal. Specifically, it achieves asymptotically vanishing error with a number of tests that is within a multiplicative factor of the fundamental entropy bound a result that parallels the independent group testing setting. We show that the number of tests reduces with an increase in the expected burst length of infected items, quantifying the advantage of exploiting correlation in test design.
Paper Structure (10 sections, 6 theorems, 107 equations, 2 figures)

This paper contains 10 sections, 6 theorems, 107 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. Main Results
  4. Proof of Theorem \ref{['thm:achievability']}
  5. Testing and Decoding Scheme
  6. Error Analysis
  7. Conclusion
  8. Proof of Lemma \ref{['lem:non-increasing']}
  9. Proof of Lemma \ref{['lem:finaltestbound']}
  10. Proof of Theorem \ref{['thm:converse']}

Key Result

Theorem 2

Under the Markovian correlation model, the minimal group testing rate $\tau^{*}$ satisfies where $\beta$ is the inverse of the expected infection burst length.

Figures (2)

  • Figure 1: The $2$-state Markov chain that generates the infection vector $U^n$
  • Figure 2: Grouped testing which selects groups with Bern$(p_1)$. Within selected groups, each item is selected with Ber$(p_2)$.

Theorems & Definitions (10)

  • Definition 1
  • Theorem 2: Converse
  • Theorem 3: Achievability
  • Lemma 1
  • Lemma 2
  • Lemma 2
  • proof
  • Lemma 2
  • proof
  • proof