319 research outputs found
GROTESQUE: Noisy Group Testing (Quick and Efficient)
Group-testing refers to the problem of identifying (with high probability) a
(small) subset of defectives from a (large) set of 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 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 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 {}. 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 {}. 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
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
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
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
-channel noise, the achievable rates and converse results match in a broad
range of sparsity regimes, and under -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
Group testing is the process of pooling arbitrary subsets from a set of
items so as to identify, with a minimal number of tests, a "small" subset of
defective items. In "classical" non-adaptive group testing, it is known
that when is substantially smaller than , 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 times, and most tests
to incorporate 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 tests; or (b) tests are
size-constrained to pool no more than 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 -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 and . The randomized design/reconstruction
algorithm in the -sized test scenario is universal -- independent of the
value of , as long as . 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
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 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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