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Small-Error Cascaded Group Testing

Daniel McMorrow, Nikhil Karamchandani, Sidharth Jaggi

TL;DR

This work analyzes cascaded group testing, where each test outputs the first defective item in an ordered pool, producing richer feedback than binary tests. It develops non-adaptive and adaptive schemes achieving tight small-error guarantees for exact recovery, subset recovery, and k estimation, with test counts that scale with the defectives $k$ and iterated logarithms rather than the population size $n$. The constructions rely on random permutation-based designs and coupon-collector–style analyses, plus a robust mean estimator for unknown $k$, showing substantial improvements over classical group testing in regimes where cascade feedback is available. The results illuminate how order-based observations can dramatically reduce testing requirements and suggest avenues for further lower-bound development.

Abstract

Group testing concerns itself with the accurate recovery of a set of "defective" items from a larger population via a series of tests. While most works in this area have considered the classical group testing model, where tests are binary and indicate the presence of at least one defective item in the test, we study the cascaded group testing model. In cascaded group testing, tests admit an ordering, and test outcomes indicate the first defective item in the test under this ordering. Under this model, we establish various achievability bounds for several different recovery criteria using both non-adaptive and adaptive (with "few" stages) test designs.

Small-Error Cascaded Group Testing

TL;DR

This work analyzes cascaded group testing, where each test outputs the first defective item in an ordered pool, producing richer feedback than binary tests. It develops non-adaptive and adaptive schemes achieving tight small-error guarantees for exact recovery, subset recovery, and k estimation, with test counts that scale with the defectives and iterated logarithms rather than the population size . The constructions rely on random permutation-based designs and coupon-collector–style analyses, plus a robust mean estimator for unknown , showing substantial improvements over classical group testing in regimes where cascade feedback is available. The results illuminate how order-based observations can dramatically reduce testing requirements and suggest avenues for further lower-bound development.

Abstract

Group testing concerns itself with the accurate recovery of a set of "defective" items from a larger population via a series of tests. While most works in this area have considered the classical group testing model, where tests are binary and indicate the presence of at least one defective item in the test, we study the cascaded group testing model. In cascaded group testing, tests admit an ordering, and test outcomes indicate the first defective item in the test under this ordering. Under this model, we establish various achievability bounds for several different recovery criteria using both non-adaptive and adaptive (with "few" stages) test designs.
Paper Structure (10 sections, 4 theorems, 17 equations)

This paper contains 10 sections, 4 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Contributions
  3. Problem Setup
  4. Main Results
  5. Proofs
  6. Proof of Theorem \ref{['thm:non_adaptive_exact']} (Non-Adaptive Exact Recovery)
  7. Proof of Theorem \ref{['thm:non_adaptive_approximate']} (Non-Adaptive Subset Recovery)
  8. Proof of Theorem \ref{['thm:mstage_adaptive_exact']} (Adaptive Exact Recovery with a Limited Number of Stages)
  9. Proof of Theorem \ref{['thm:k_estimate']} (Estimating $k$)
  10. Conclusion

Key Result

Theorem 1

(Non-Adaptive Exact Recovery) Suppose ${\lvert {\mathcal{K}} \rvert = k}$, and let $T = k \log(k/\delta)$ for some $\delta \in (0,1)$. Then, there exists a non-adaptive test design and a decoder such that $\mathbb{P}(\hat{{\mathcal{K}}} \neq {\mathcal{K}}) \leq \delta$.

Theorems & Definitions (7)

  • Example 1
  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Theorem 3
  • Theorem 4