319 research outputs found

    GROTESQUE: Noisy Group Testing (Quick and Efficient)

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    Group-testing refers to the problem of identifying (with high probability) a (small) subset of DD defectives from a (large) set of NN items via a "small" number of "pooled" tests. For ease of presentation in this work we focus on the regime when D = \cO{N^{1-\gap}} for some \gap > 0. The tests may be noiseless or noisy, and the testing procedure may be adaptive (the pool defining a test may depend on the outcome of a previous test), or non-adaptive (each test is performed independent of the outcome of other tests). A rich body of literature demonstrates that Θ(Dlog(N))\Theta(D\log(N)) tests are information-theoretically necessary and sufficient for the group-testing problem, and provides algorithms that achieve this performance. However, it is only recently that reconstruction algorithms with computational complexity that is sub-linear in NN have started being investigated (recent work by \cite{GurI:04,IndN:10, NgoP:11} gave some of the first such algorithms). In the scenario with adaptive tests with noisy outcomes, we present the first scheme that is simultaneously order-optimal (up to small constant factors) in both the number of tests and the decoding complexity (\cO{D\log(N)} in both the performance metrics). The total number of stages of our adaptive algorithm is "small" (\cO{\log(D)}). Similarly, in the scenario with non-adaptive tests with noisy outcomes, we present the first scheme that is simultaneously near-optimal in both the number of tests and the decoding complexity (via an algorithm that requires \cO{D\log(D)\log(N)} tests and has a decoding complexity of {O(D(logN+log2D)){\cal O}(D(\log N+\log^{2}D))}. Finally, we present an adaptive algorithm that only requires 2 stages, and for which both the number of tests and the decoding complexity scale as {O(D(logN+log2D)){\cal O}(D(\log N+\log^{2}D))}. For all three settings the probability of error of our algorithms scales as \cO{1/(poly(D)}.Comment: 26 pages, 5 figure

    Constraining the Number of Positive Responses in Adaptive, Non-Adaptive, and Two-Stage Group Testing

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    Group testing is a well known search problem that consists in detecting the defective members of a set of objects O by performing tests on properly chosen subsets (pools) of the given set O. In classical group testing the goal is to find all defectives by using as few tests as possible. We consider a variant of classical group testing in which one is concerned not only with minimizing the total number of tests but aims also at reducing the number of tests involving defective elements. The rationale behind this search model is that in many practical applications the devices used for the tests are subject to deterioration due to exposure to or interaction with the defective elements. In this paper we consider adaptive, non-adaptive and two-stage group testing. For all three considered scenarios, we derive upper and lower bounds on the number of "yes" responses that must be admitted by any strategy performing at most a certain number t of tests. In particular, for the adaptive case we provide an algorithm that uses a number of "yes" responses that exceeds the given lower bound by a small constant. Interestingly, this bound can be asymptotically attained also by our two-stage algorithm, which is a phenomenon analogous to the one occurring in classical group testing. For the non-adaptive scenario we give almost matching upper and lower bounds on the number of "yes" responses. In particular, we give two constructions both achieving the same asymptotic bound. An interesting feature of one of these constructions is that it is an explicit construction. The bounds for the non-adaptive and the two-stage cases follow from the bounds on the optimal sizes of new variants of d-cover free families and (p,d)-cover free families introduced in this paper, which we believe may be of interest also in other contexts

    Efficient Probabilistic Group Testing Based on Traitor Tracing

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    Inspired by recent results from collusion-resistant traitor tracing, we provide a framework for constructing efficient probabilistic group testing schemes. In the traditional group testing model, our scheme asymptotically requires T ~ 2 K ln N tests to find (with high probability) the correct set of K defectives out of N items. The framework is also applied to several noisy group testing and threshold group testing models, often leading to improvements over previously known results, but we emphasize that this framework can be applied to other variants of the classical model as well, both in adaptive and in non-adaptive settings.Comment: 8 pages, 3 figures, 1 tabl

    Noisy Non-Adaptive Group Testing: A (Near-)Definite Defectives Approach

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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 is relevant in applications such as medical testing, communication protocols, pattern matching, and many more. We study the noisy version of the problem, where the output of each standard noiseless group test is subject to independent noise, corresponding to passing the noiseless result through a binary channel. We introduce a class of algorithms that we refer to as Near-Definite Defectives (NDD), and study bounds on the required number of tests for vanishing error probability under Bernoulli random test designs. In addition, we study algorithm-independent converse results, giving lower bounds on the required number of tests under Bernoulli test designs. Under reverse ZZ-channel noise, the achievable rates and converse results match in a broad range of sparsity regimes, and under ZZ-channel noise, the two match in a narrower range of dense/low-noise regimes. We observe that although these two channels have the same Shannon capacity when viewed as a communication channel, they can behave quite differently when it comes to group testing. Finally, we extend our analysis of these noise models to the symmetric noise model, and show improvements over the best known existing bounds in broad scaling regimes.Comment: Submitted to IEEE Transactions on Information Theor

    Nearly Optimal Sparse Group Testing

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    Group testing is the process of pooling arbitrary subsets from a set of nn items so as to identify, with a minimal number of tests, a "small" subset of dd defective items. In "classical" non-adaptive group testing, it is known that when dd is substantially smaller than nn, Θ(dlog(n))\Theta(d\log(n)) tests are both information-theoretically necessary and sufficient to guarantee recovery with high probability. Group testing schemes in the literature meeting this bound require most items to be tested Ω(log(n))\Omega(\log(n)) times, and most tests to incorporate Ω(n/d)\Omega(n/d) items. Motivated by physical considerations, we study group testing models in which the testing procedure is constrained to be "sparse". Specifically, we consider (separately) scenarios in which (a) items are finitely divisible and hence may participate in at most γo(log(n))\gamma \in o(\log(n)) tests; or (b) tests are size-constrained to pool no more than ρo(n/d)\rho \in o(n/d)items per test. For both scenarios we provide information-theoretic lower bounds on the number of tests required to guarantee high probability recovery. In both scenarios we provide both randomized constructions (under both ϵ\epsilon-error and zero-error reconstruction guarantees) and explicit constructions of designs with computationally efficient reconstruction algorithms that require a number of tests that are optimal up to constant or small polynomial factors in some regimes of n,d,γ,n, d, \gamma, and ρ\rho. The randomized design/reconstruction algorithm in the ρ\rho-sized test scenario is universal -- independent of the value of dd, as long as ρo(n/d)\rho \in o(n/d). We also investigate the effect of unreliability/noise in test outcomes. For the full abstract, please see the full text PDF

    Near-Optimal Noisy Group Testing via Separate Decoding of Items

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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 tests, and is relevant in applications such as medical testing, communication protocols, pattern matching, and more. In this paper, we revisit an efficient algorithm for noisy group testing in which each item is decoded separately (Malyutov and Mateev, 1980), and develop novel performance guarantees via an information-theoretic framework for general noise models. For the special cases of no noise and symmetric noise, we find that the asymptotic number of tests required for vanishing error probability is within a factor log20.7\log 2 \approx 0.7 of the information-theoretic optimum at low sparsity levels, and that with a small fraction of allowed incorrectly decoded items, this guarantee extends to all sublinear sparsity levels. In addition, we provide a converse bound showing that if one tries to move slightly beyond our low-sparsity achievability threshold using separate decoding of items and i.i.d. randomized testing, the average number of items decoded incorrectly approaches that of a trivial decoder.Comment: Submitted to IEEE Journal of Selected Topics in Signal Processin
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