Estimating the number of defectives with group testing

Falahatgar, Moein; Jafarpour, Ashkan; Orlitsky, Alon; Pichapati, Venkatadheeraj; Suresh, Ananda Theertha

doi:10.1109/isit.2016.7541524

Cited by 31 publications

(30 citation statements)

References 24 publications

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“…Many of these protocols require nonadaptive testing, since tests may be time-consuming -for example, one may need to place a group of possibly infected insects with a plant, and wait to see if the plant becomes infected. A recent paper [74] gives a detailed analysis of an adaptive algorithm that estimates the number of defectives. We review the question of counting defectives using group testing in more detail in Section 5.3.…”

Section: Biologymentioning

confidence: 99%

“…However, as argued by Falahatgar et al [74], the requirement to know the number of defectives exactly may be unnecessarily restrictive. They developed a four-stage adaptive algorithm, for which they proved that that an O(log log k) expected number of tests achieves an approximate recovery criterion with high probability [74,Theorem 15]. The algorithm works by successively refining estimates, with each stage creating a better estimate with a certain probability of error.…”

Section: Adaptive Testingmentioning

confidence: 99%

“…Recovery guarantee #Tests Adaptive? [38] Exact O(k log k) yes [74,29] ( Table 5.3: Summary of recovery guarantees for counting defectives, depending on the recovery criteria and availability of adaptive testing. The results here correspond to the case of a constant non-zero error probability; the precise dependencies on the error probability can be found in the above theorem statements.…”

Section: Referencesmentioning

confidence: 99%

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Group Testing: An Information Theory Perspective

Aldridge¹,

Johnson²,

Scarlett³

2019

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The group testing problem concerns discovering a small number of defective items within a large population by performing tests on pools of items. A test is positive if the pool contains at least one defective, and negative if it contains no defectives. This is a sparse inference problem with a combinatorial flavour, with applications in medical testing, biology, telecommunications, information technology, data science, and more.In this monograph, we survey recent developments in the group testing problem from an information-theoretic perspective. We cover several related developments: efficient algorithms with practical storage and computation requirements, achievability bounds for optimal decoding methods, and algorithmindependent converse bounds. We assess the theoretical guarantees not only in terms of scaling laws, but also in terms of the constant factors, leading to the notion of the rate of group testing, indicating the amount of information learned per test. Considering both noiseless and noisy settings, we identify several regimes where existing algorithms are provably optimal or near-optimal, as well as regimes where there remains greater potential for improvement.In addition, we survey results concerning a number of variations on the standard group testing problem, including partial recovery criteria, adaptive algorithms with a limited number of stages, constrained test designs, and sublineartime algorithms.S 0 , S 1 partition of the defective set (Equation (4.4)) ı information density (Equation (4.6)) X 0,τ , X 1,τ sub-matrices of X corresponding to (S 0 , S 1 ) with |S 0 | = τ (Equation (4.14))τ ) (Equation (4.16)) 2. If the preceding test is negative, A contains no defective items, and we halt. If the test is positive, continue. 3. If A consists of a single item, then that item is defective, and we halt. Otherwise, pick half of the items in A, and call this set B. Perform a single test of the pool B. 4. If the test is positive, set A := B. If the test is negative, set A := A \ B. Return to Step 3. A BRIEF REVIEW OF ZERO-ERROR GROUP TESTINGObserve that S(i) is the set of tests containing item i, while S(K) is the set of positive tests when the defective set is K.Definition 1.10. A matrix X is called k-separable if the support unions S(L) are distinct over all subsets L ⊆ {1, 2, . . . , n} of size |L| = k.A matrix X is calledk-separable if the support unions S(L) are distinct over all subsets L ⊆ {1, 2, . . . , n} of size |L| ≤ k.Clearly, using a k-separable matrix as a test design ensures that group testing will provide different outcomes for each possible defective set of size k; thus,

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Section: Biologymentioning

confidence: 99%

Section: Adaptive Testingmentioning

confidence: 99%

Section: Referencesmentioning

confidence: 99%

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Group Testing: An Information Theory Perspective

Aldridge¹,

Johnson²,

Scarlett³

2019

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“…Note that for a high probability bound, which is what we focus on in this paper, the algorithm would naively require O(log n • log log m • poly(1/ε)) queries to achieve success probability at least 1 − 1/n. Falahatgar, Jafarpour, Orlitsky, Pichapati, and Suresh [12] gave an improved algorithm that estimates m up to a factor of (1 + ε) with probability 1 − δ using 2 log log m + O (1/ε 2 ) log(1/δ) subset queries. Nearly matching lower bounds are also known for subset queries [21,22,19,12].…”

Section: Introductionmentioning

confidence: 99%

“…Falahatgar, Jafarpour, Orlitsky, Pichapati, and Suresh [12] gave an improved algorithm that estimates m up to a factor of (1 + ε) with probability 1 − δ using 2 log log m + O (1/ε 2 ) log(1/δ) subset queries. Nearly matching lower bounds are also known for subset queries [21,22,19,12]. Ron and Tsur [19] also study a restriction of subset queries, called interval queries, where the universe U is ordered and the subsets are intervals of elements.…”

Section: Introductionmentioning

confidence: 99%

Edge Estimation with Independent Set Oracles

Beame

Har-Peled

Ramamoorthy

et al. 2020

ACM Trans. Algorithms

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We study the problem of estimating the number of edges in a graph with access to only an independent set oracle. Independent set queries draw motivation from group testing and have applications to the complexity of decision versus counting problems. We give two algorithms to estimate the number of edges in an n-vertex graph: one that uses only polylog(n) bipartite independent set queries, and another one that uses n 2/3 • polylog(n) independent set queries.

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Optimal Deterministic Group Testing Algorithms to Estimate the Number of Defectives

Bshouty

Haddad-Zaknoon

2020

Combinatorial Optimization and Applications

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Estimating the number of defectives with group testing

Cited by 31 publications

References 24 publications

Group Testing: An Information Theory Perspective

Group Testing: An Information Theory Perspective

Edge Estimation with Independent Set Oracles

Optimal Deterministic Group Testing Algorithms to Estimate the Number of Defectives

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