9 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
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
Efficient (nonrandom) construction and decoding for non-adaptive group testing
The task of non-adaptive group testing is to identify up to defective
items from items, where a test is positive if it contains at least one
defective item, and negative otherwise. If there are tests, they can be
represented as a measurement matrix. We have answered the question
of whether there exists a scheme such that a larger measurement matrix, built
from a given measurement matrix, can be used to identify up to
defective items in time . In the meantime, a
nonrandom measurement matrix with can
be obtained to identify up to defective items in time .
This is much better than the best well-known bound, . For the special case , there exists an efficient
nonrandom construction in which at most two defective items can be identified
in time using tests. Numerical results show
that our proposed scheme is more practical than existing ones, and experimental
results confirm our theoretical analysis. In particular, up to
defective items can be identified in less than s even for
Generalized Group Testing
In the problem of classical group testing one aims to identify a small subset
(of size ) diseased individuals/defective items in a large population (of
size ). This process is based on a minimal number of suitably-designed group
tests on subsets of items, where the test outcome is positive iff the given
test contains at least one defective item. Motivated by physical
considerations, we consider a generalized setting that includes as special
cases multiple other group-testing-like models in the literature. In our
setting, which subsumes as special cases a variety of noiseless and noisy
group-testing models in the literature, the test outcome is positive with
probability , where is the number of defectives tested in a pool, and
is an arbitrary monotonically increasing (stochastic) test function.
Our main contributions are as follows.
1. We present a non-adaptive scheme that with probability
identifies all defective items. Our scheme requires at most tests, where is a suitably
defined "sensitivity parameter" of , and is never larger than , but may be substantially smaller for many
.
2. We argue that any testing scheme (including adaptive schemes) needs at
least
tests to ensure reliable recovery. Here is a suitably defined
"concentration parameter" of .
3. We prove that for a variety of
sparse-recovery group-testing models in the literature, and for any other test function
Efficiently decodable non-adaptive threshold group testing
We consider non-adaptive threshold group testing for identification of up to d defective items in a set of n items, where a test is positive if it contains at least 2leq uleq d defective items, and negative otherwise. The defective items can be identified using t=O(( frac d u) u(frac d d-u) d-u(ulogfrac d u+logfrac 1 Ο΅)d 2log n) tests with probability at least 1-Ο΅ for any Ο΅ > 0 or t= Oleft(left(frac b uright) uleft(frac d d-uright) d-ucdot d 3log n cdot log frac n dright) tests with probability 1. The decoding time is ttimes poly (d 2log n). This result significantly improves the best known results for decoding non-adaptive threshold group testing: O(n log n+nlogfrac 1 Ο΅) for probabilistic decoding, where Ο΅ > 0, and O(n ulog n) for deterministic decoding