646 research outputs found
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
Efficiently decodable non-adaptive group testing
We consider the following "efficiently decodable" non-adaptive
group testing problem. There is an unknown string
x 2 f0; 1gn [x is an element of set {0,1} superscript n] with at most d ones in it. We are allowed to test
any subset S [n] [S subset [n] ]of the indices. The answer to the test
tells whether xi = 0 [x subscript i = 0] for all i 2 S [i is an element of S] or not. The objective
is to design as few tests as possible (say, t tests) such that
x can be identifi ed as fast as possible (say, poly(t)-time).
Efficiently decodable non-adaptive group testing has applications
in many areas, including data stream algorithms and
data forensics.
A non-adaptive group testing strategy can be represented
by a t x n matrix, which is the stacking of all the
characteristic vectors of the tests. It is well-known that if
this matrix is d-disjunct, then any test outcome corresponds
uniquely to an unknown input string. Furthermore, we know
how to construct d-disjunct matrices with t = O(d2 [d superscript 2] log n)
efficiently. However, these matrices so far only allow for a
"decoding" time of O(nt), which can be exponentially larger
than poly(t) for relatively small values of d.
This paper presents a randomness efficient construction
of d-disjunct matrices with t = O(d2 [d superscript 2] log n) that can be decoded
in time poly(d) [function composed of] t log2 t + O(t2) [t log superscript 2 t and O (t superscript 2)]. To the best of our
knowledge, this is the first result that achieves an efficient decoding
time and matches the best known O(d2 log n) [O (d superscript 2 log n)] bound
on the number of tests. We also derandomize the construction,
which results in a polynomial time deterministic construction
of such matrices when d = O(log n= log log n).
A crucial building block in our construction is the
notion of (d,l)-list disjunct matrices, which represent the
more general "list group testing" problem whose goal is to
output less than d + l positions in x, including all the (at
most d) positions that have a one in them. List disjunct
matrices turn out to be interesting objects in their own right
and were also considered independently by [Cheraghchi,
FCT 2009]. We present connections between list disjunct
matrices, expanders, dispersers and disjunct matrices. List
disjunct matrices have applications in constructing (d,l)-
sparsity separator structures [Ganguly, ISAAC 2008] and in
constructing tolerant testers for Reed-Solomon codes in the
data stream model.
1 IntroductionDavid & Lucile Packard FoundationCenter for Massive Data Algorithmics (MADALGO)National Science Foundation (U.S.) (Grant CCF-0728645)National Science Foundation (U.S.) (Grant CCF-0347565)National Science Foundation (U.S.) (CAREER Award CCF-0844796
Linear-time list recovery of high-rate expander codes
We show that expander codes, when properly instantiated, are high-rate list
recoverable codes with linear-time list recovery algorithms. List recoverable
codes have been useful recently in constructing efficiently list-decodable
codes, as well as explicit constructions of matrices for compressive sensing
and group testing. Previous list recoverable codes with linear-time decoding
algorithms have all had rate at most 1/2; in contrast, our codes can have rate
for any . We can plug our high-rate codes into a
construction of Meir (2014) to obtain linear-time list recoverable codes of
arbitrary rates, which approach the optimal trade-off between the number of
non-trivial lists provided and the rate of the code. While list-recovery is
interesting on its own, our primary motivation is applications to
list-decoding. A slight strengthening of our result would implies linear-time
and optimally list-decodable codes for all rates, and our work is a step in the
direction of solving this important problem
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
Some Applications of Coding Theory in Computational Complexity
Error-correcting codes and related combinatorial constructs play an important
role in several recent (and old) results in computational complexity theory. In
this paper we survey results on locally-testable and locally-decodable
error-correcting codes, and their applications to complexity theory and to
cryptography.
Locally decodable codes are error-correcting codes with sub-linear time
error-correcting algorithms. They are related to private information retrieval
(a type of cryptographic protocol), and they are used in average-case
complexity and to construct ``hard-core predicates'' for one-way permutations.
Locally testable codes are error-correcting codes with sub-linear time
error-detection algorithms, and they are the combinatorial core of
probabilistically checkable proofs
A framework for generalized group testing with inhibitors and its potential application in neuroscience
The main goal of group testing with inhibitors (GTI) is to efficiently
identify a small number of defective items and inhibitor items in a large set
of items. A test on a subset of items is positive if the subset satisfies some
specific properties. Inhibitor items cancel the effects of defective items,
which often make the outcome of a test containing defective items negative.
Different GTI models can be formulated by considering how specific properties
have different cancellation effects. This work introduces generalized GTI
(GGTI) in which a new type of items is added, i.e., hybrid items. A hybrid item
plays the roles of both defectives items and inhibitor items. Since the number
of instances of GGTI is large (more than 7 million), we introduce a framework
for classifying all types of items non-adaptively, i.e., all tests are designed
in advance. We then explain how GGTI can be used to classify neurons in
neuroscience. Finally, we show how to realize our proposed scheme in practice
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