Optimal Deterministic Group Testing Algorithms to Estimate the Number of Defectives

Bshouty, Nader H.; Haddad-Zaknoon, Catherine A.

doi:10.1007/978-3-030-64843-5_27

Cited by 3 publications

(3 citation statements)

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“…It is worth noting that testing k-juntas only requires determining whether a given f has this property or is far from having this property [24], which is distinct from learning k-juntas, i.e., either determining the k inputs that f depends on or estimating f itself. Further studies of the k-junta problem vary according to whether the inputs x are chosen by the tester ('membership queries') [29], uniformly at random by nature [150], or according to some quantum state [14,19]. In this sense, group testing with a combinatorial prior is a special case of the k-junta learning problem, where we are sure that the function is an OR of the k inputs.…”

Section: Data Sciencementioning

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: Data Sciencementioning

confidence: 99%

“…By tuning the performance of the algorithm of [74], Bshouty et al [29] improved the performance guarantee by a constant factor, obtaining the following result [29,Theorem 8].…”

Section: Adaptive Testingmentioning

confidence: 99%

Group Testing: An Information Theory Perspective

Aldridge¹,

Johnson²,

Scarlett³

2019

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“…. , n} (and similarly for subsets of items), so that we do not need to assume a priori knowledge regarding k. While numerous works have given algorithms for estimating k in the noiseless setting [31]- [33], analogous results for the noisy setting appear to be very limited. The following result is based on a simple approach that can likely be improved, but is sufficient for our purposes.…”

Section: Estimating the Number Of Defectivesmentioning

confidence: 99%

Noisy Adaptive Group Testing via Noisy Binary Search

Teo¹,

Scarlett²

2021

Preprint

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The group testing problem consists of determining a small set of defective items from a larger set of items based on a number of possibly-noisy tests, and has numerous practical applications. One of the defining features of group testing is whether the tests are adaptive (i.e., a given test can be chosen based on all previous outcomes) or non-adaptive (i.e., all tests must be chosen in advance). In this paper, building on the success of binary splitting techniques in noiseless group testing (Hwang, 1972), we introduce noisy group testing algorithms that apply noisy binary search as a subroutine. We provide three variations of this approach with increasing complexity, culminating in an algorithm that succeeds using a number of tests that matches the best known previously (Scarlett, 2019), while overcoming fundamental practical limitations of the existing approach, and more precisely capturing the dependence of the number of tests on the error probability. We provide numerical experiments demonstrating that adaptive group testing strategies based on noisy binary search can be highly effective in practice, using significantly fewer tests compared to state-of-the-art non-adaptive strategies.

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Adaptive Surveillance Testing for Efficient Infection Rate Estimation

Feng

Yang

2021

2021 IEEE International Symposium on Information Theory (ISIT)

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Optimal Deterministic Group Testing Algorithms to Estimate the Number of Defectives

Cited by 3 publications

References 24 publications

Group Testing: An Information Theory Perspective

Group Testing: An Information Theory Perspective

Noisy Adaptive Group Testing via Noisy Binary Search

Adaptive Surveillance Testing for Efficient Infection Rate Estimation

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