8,513 research outputs found
Explicit Non-Adaptive Combinatorial Group Testing Schemes
Group testing is a long studied problem in combinatorics: A small set of
ill people should be identified out of the whole ( people) by using only
queries (tests) of the form "Does set X contain an ill human?". In this paper
we provide an explicit construction of a testing scheme which is better
(smaller) than any known explicit construction. This scheme has \bigT{\min[r^2
\ln n,n]} tests which is as many as the best non-explicit schemes have. In our
construction we use a fact that may have a value by its own right: Linear
error-correction codes with parameters meeting the
Gilbert-Varshamov bound may be constructed quite efficiently, in \bigT{q^km}
time.Comment: 15 pages, accepted to ICALP 200
Noise-Resilient Group Testing: Limitations and Constructions
We study combinatorial group testing schemes for learning -sparse Boolean
vectors using highly unreliable disjunctive measurements. We consider an
adversarial noise model that only limits the number of false observations, and
show that any noise-resilient scheme in this model can only approximately
reconstruct the sparse vector. On the positive side, we take this barrier to
our advantage and show that approximate reconstruction (within a satisfactory
degree of approximation) allows us to break the information theoretic lower
bound of that is known for exact reconstruction of
-sparse vectors of length via non-adaptive measurements, by a
multiplicative factor .
Specifically, we give simple randomized constructions of non-adaptive
measurement schemes, with measurements, that allow efficient
reconstruction of -sparse vectors up to false positives even in the
presence of false positives and false negatives within the
measurement outcomes, for any constant . We show that, information
theoretically, none of these parameters can be substantially improved without
dramatically affecting the others. Furthermore, we obtain several explicit
constructions, in particular one matching the randomized trade-off but using measurements. We also obtain explicit constructions
that allow fast reconstruction in time \poly(m), which would be sublinear in
for sufficiently sparse vectors. The main tool used in our construction is
the list-decoding view of randomness condensers and extractors.Comment: Full version. A preliminary summary of this work appears (under the
same title) in proceedings of the 17th International Symposium on
Fundamentals of Computation Theory (FCT 2009
Derandomization and Group Testing
The rapid development of derandomization theory, which is a fundamental area
in theoretical computer science, has recently led to many surprising
applications outside its initial intention. We will review some recent such
developments related to combinatorial group testing. In its most basic setting,
the aim of group testing is to identify a set of "positive" individuals in a
population of items by taking groups of items and asking whether there is a
positive in each group.
In particular, we will discuss explicit constructions of optimal or
nearly-optimal group testing schemes using "randomness-conducting" functions.
Among such developments are constructions of error-correcting group testing
schemes using randomness extractors and condensers, as well as threshold group
testing schemes from lossless condensers.Comment: Invited Paper in Proceedings of 48th Annual Allerton Conference on
Communication, Control, and Computing, 201
Construction of Almost Disjunct Matrices for Group Testing
In a \emph{group testing} scheme, a set of tests is designed to identify a
small number of defective items among a large set (of size ) of items.
In the non-adaptive scenario the set of tests has to be designed in one-shot.
In this setting, designing a testing scheme is equivalent to the construction
of a \emph{disjunct matrix}, an matrix where the union of supports
of any columns does not contain the support of any other column. In
principle, one wants to have such a matrix with minimum possible number of
rows (tests). One of the main ways of constructing disjunct matrices relies on
\emph{constant weight error-correcting codes} and their \emph{minimum
distance}. In this paper, we consider a relaxed definition of a disjunct matrix
known as \emph{almost disjunct matrix}. This concept is also studied under the
name of \emph{weakly separated design} in the literature. The relaxed
definition allows one to come up with group testing schemes where a
close-to-one fraction of all possible sets of defective items are identifiable.
Our main contribution is twofold. First, we go beyond the minimum distance
analysis and connect the \emph{average distance} of a constant weight code to
the parameters of an almost disjunct matrix constructed from it. Our second
contribution is to explicitly construct almost disjunct matrices based on our
average distance analysis, that have much smaller number of rows than any
previous explicit construction of disjunct matrices. The parameters of our
construction can be varied to cover a large range of relations for and .Comment: 15 Page
Efficiently Decodable Non-Adaptive Threshold Group Testing
We consider non-adaptive threshold group testing for identification of up to
defective items in a set of items, where a test is positive if it
contains at least defective items, and negative otherwise.
The defective items can be identified using tests with
probability at least for any or tests with probability 1. The decoding time is
. This result significantly improves the
best known results for decoding non-adaptive threshold group testing:
for probabilistic decoding, where
, and for deterministic decoding
- …