162,030 research outputs found
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
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
Improved Combinatorial Group Testing Algorithms for Real-World Problem Sizes
We study practically efficient methods for performing combinatorial group
testing. We present efficient non-adaptive and two-stage combinatorial group
testing algorithms, which identify the at most d items out of a given set of n
items that are defective, using fewer tests for all practical set sizes. For
example, our two-stage algorithm matches the information theoretic lower bound
for the number of tests in a combinatorial group testing regimen.Comment: 18 pages; an abbreviated version of this paper is to appear at the
9th Worksh. Algorithms and Data Structure
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
Recommended from our members
Extended two-stage adaptive designswith three target responses forphase II clinical trials
We develop a nature-inspired stochastic population-based algorithm and call it discrete particle swarm optimization tofind extended two-stage adaptive optimal designs that allow three target response rates for the drug in a phase II trial.Our proposed designs include the celebrated Simon’s two-stage design and its extension that allows two target responserates to be specified for the drug. We show that discrete particle swarm optimization not only frequently outperformsgreedy algorithms, which are currently used to find such designs when there are only a few parameters; it is also capableof solving design problems posed here with more parameters that greedy algorithms cannot solve. In stage 1 of ourproposed designs, futility is quickly assessed and if there are sufficient responders to move to stage 2, one tests one ofthe three target response rates of the drug, subject to various user-specified testing error rates. Our designs aretherefore more flexible and interestingly, do not necessarily require larger expected sample size requirements thantwo-stage adaptive designs. Using a real adaptive trial for melanoma patients, we show our proposed design requires onehalf fewer subjects than the implemented design in the study
- …