4,634 research outputs found
Pooling spaces associated with finite geometry
AbstractMotivated by the works of Ngo and Du [H. Ngo, D. Du, A survey on combinatorial group testing algorithms with applications to DNA library screening, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 55 (2000) 171–182], the notion of pooling spaces was introduced [T. Huang, C. Weng, Pooling spaces and non-adaptive pooling designs, Discrete Mathematics 282 (2004) 163–169] for a systematic way of constructing pooling designs; note that geometric lattices are among pooling spaces. This paper attempts to draw possible connections from finite geometry and distance regular graphs to pooling spaces: including the projective spaces, the affine spaces, the attenuated spaces, and a few families of geometric lattices associated with the orbits of subspaces under finite classical groups, and associated with d-bounded distance-regular graphs
Efficient Two-Stage Group Testing Algorithms for Genetic Screening
Efficient two-stage group testing algorithms that are particularly suited for
rapid and less-expensive DNA library screening and other large scale biological
group testing efforts are investigated in this paper. The main focus is on
novel combinatorial constructions in order to minimize the number of individual
tests at the second stage of a two-stage disjunctive testing procedure.
Building on recent work by Levenshtein (2003) and Tonchev (2008), several new
infinite classes of such combinatorial designs are presented.Comment: 14 pages; to appear in "Algorithmica". Part of this work has been
presented at the ICALP 2011 Group Testing Workshop; arXiv:1106.368
Lower bounds for identifying subset members with subset queries
An instance of a group testing problem is a set of objects \cO and an
unknown subset of \cO. The task is to determine by using queries of
the type ``does intersect '', where is a subset of \cO. This
problem occurs in areas such as fault detection, multiaccess communications,
optimal search, blood testing and chromosome mapping. Consider the two stage
algorithm for solving a group testing problem. In the first stage a
predetermined set of queries are asked in parallel and in the second stage,
is determined by testing individual objects. Let n=\cardof{\cO}. Suppose that
is generated by independently adding each x\in \cO to with
probability . Let () be the number of queries asked in the
first (second) stage of this algorithm. We show that if
, then \Exp(q_2) = n^{1-o(1)}, while there
exist algorithms with and \Exp(q_2) =
o(1). The proof involves a relaxation technique which can be used with
arbitrary distributions. The best previously known bound is q_1+\Exp(q_2) =
\Omega(p\log(n)). For general group testing algorithms, our results imply that
if the average number of queries over the course of ()
independent experiments is , then with high probability
non-singleton subsets are queried. This
settles a conjecture of Bill Bruno and David Torney and has important
consequences for the use of group testing in screening DNA libraries and other
applications where it is more cost effective to use non-adaptive algorithms
and/or too expensive to prepare a subset for its first test.Comment: 9 page
Non-adaptive Group Testing on Graphs
Grebinski and Kucherov (1998) and Alon et al. (2004-2005) study the problem
of learning a hidden graph for some especial cases, such as hamiltonian cycle,
cliques, stars, and matchings. This problem is motivated by problems in
chemical reactions, molecular biology and genome sequencing.
In this paper, we present a generalization of this problem. Precisely, we
consider a graph G and a subgraph H of G and we assume that G contains exactly
one defective subgraph isomorphic to H. The goal is to find the defective
subgraph by testing whether an induced subgraph contains an edge of the
defective subgraph, with the minimum number of tests. We present an upper bound
for the number of tests to find the defective subgraph by using the symmetric
and high probability variation of Lov\'asz Local Lemma
Group Testing with Probabilistic Tests: Theory, Design and Application
Identification of defective members of large populations has been widely
studied in the statistics community under the name of group testing. It
involves grouping subsets of items into different pools and detecting defective
members based on the set of test results obtained for each pool.
In a classical noiseless group testing setup, it is assumed that the sampling
procedure is fully known to the reconstruction algorithm, in the sense that the
existence of a defective member in a pool results in the test outcome of that
pool to be positive. However, this may not be always a valid assumption in some
cases of interest. In particular, we consider the case where the defective
items in a pool can become independently inactive with a certain probability.
Hence, one may obtain a negative test result in a pool despite containing some
defective items. As a result, any sampling and reconstruction method should be
able to cope with two different types of uncertainty, i.e., the unknown set of
defective items and the partially unknown, probabilistic testing procedure.
In this work, motivated by the application of detecting infected people in
viral epidemics, we design non-adaptive sampling procedures that allow
successful identification of the defective items through a set of probabilistic
tests. Our design requires only a small number of tests to single out the
defective items. In particular, for a population of size and at most
defective items with activation probability , our results show that tests is sufficient if the sampling procedure should
work for all possible sets of defective items, while
tests is enough to be successful for any single set of defective items.
Moreover, we show that the defective members can be recovered using a simple
reconstruction algorithm with complexity of .Comment: Full version of the conference paper "Compressed Sensing with
Probabilistic Measurements: A Group Testing Solution" appearing in
proceedings of the 47th Annual Allerton Conference on Communication, Control,
and Computing, 2009 (arXiv:0909.3508). To appear in IEEE Transactions on
Information Theor
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
Group testing problems in experimental molecular biology
In group testing, the task is to determine the distinguished members of a set
of objects L by asking subset queries of the form ``does the subset Q of L
contain a distinguished object?'' The primary biological application of group
testing is for screening libraries of clones with hybridization probes. This is
a crucial step in constructing physical maps and for finding genes. Group
testing has also been considered for sequencing by hybridization. Another
important application includes screening libraries of reagents for useful
chemically active zones. This preliminary report discusses some of the
constrained group testing problems which arise in biology.Comment: 7 page
Optimal Nested Test Plan for Combinatorial Quantitative Group Testing
We consider the quantitative group testing problem where the objective is to
identify defective items in a given population based on results of tests
performed on subsets of the population. Under the quantitative group testing
model, the result of each test reveals the number of defective items in the
tested group. The minimum number of tests achievable by nested test plans was
established by Aigner and Schughart in 1985 within a minimax framework. The
optimal nested test plan offering this performance, however, was not obtained.
In this work, we establish the optimal nested test plan in closed form. This
optimal nested test plan is also order optimal among all test plans as the
population size approaches infinity. Using heavy-hitter detection as a case
study, we show via simulation examples orders of magnitude improvement of the
group testing approach over two prevailing sampling-based approaches in
detection accuracy and counter consumption. Other applications include anomaly
detection and wideband spectrum sensing in cognitive radio systems
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