7 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
New Constructions for Competitive and Minimal-Adaptive Group Testing
Group testing (GT) was originally proposed during the World War II in an attempt to minimize the \emph{cost} and \emph{waiting time} in performing identical blood tests of the soldiers for a low-prevalence disease.
Formally, the GT problem asks to find \emph{defective} elements out of elements by querying subsets (pools) for the presence of defectives.
By the information-theoretic lower bound, essentially queries are needed in the worst-case.
An \emph{adaptive} strategy proceeds sequentially by performing one query at a time, and it can achieve the lower bound. In various applications, nothing is known about beforehand and a strategy for this scenario is called \emph{competitive}. Such strategies are usually adaptive and achieve query optimality within a constant factor called the \emph{competitive ratio}.
In many applications, queries are time-consuming. Therefore, \emph{minimal-adaptive} strategies which run in a small number of stages of parallel
queries are favorable.
This work is mainly devoted to the design of minimal-adaptive strategies combined with other demands of both theoretical and practical interest. First we target unknown and show that actually competitive GT is possible in as few as stages only.
The main ingredient is our randomized estimate of a previously unknown using nonadaptive queries.
In addition, we have developed a systematic approach to obtain optimal competitive ratios for our strategies.
When is a known upper bound,
we propose randomized GT strategies which asymptotically achieve query optimality in just , or stages depending upon the growth of versus .
Inspired by application settings, such as at American Red Cross, where in most cases GT is applied to small instances, \textit{e.g.}, . We extended our study of query-optimal GT strategies to solve a given problem instance with fixed values , and . We also considered the situation when
elements to test cannot be divided physically (electronic devices), thus the pools must be disjoint. For GT with \emph{disjoint} simultaneous pools, we show that tests are sufficient, and also necessary for certain ranges of the parameters
Superimposed Codes and Threshold Group Testing
D’yachkov A, Rykov V, Deppe C, Lebedev V. Superimposed Codes and Threshold Group Testing. In: Aydinian H, Cicalese F, Deppe C, eds. Information Theory, Combinatorics, and Search Theory. Lecture Notes in Computer Science. Vol 7777. Berlin, Heidelberg: Springer Berlin Heidelberg; 2013: 509-533
Superimposed Codes and Threshold Group Testing
D'yachkov A, Rykov V, Deppe C, Lebedev V. Superimposed Codes and Threshold Group Testing. 2014.We will discuss superimposed codes and non-adaptive group testing designsarising from the potentialities of compressed genotyping models in molecularbiology. The given paper was motivated by the 30th anniversary ofD'yachkov-Rykov recurrent upper bound on the rate of superimposed codespublished in 1982. We were also inspired by recent results obtained fornon-adaptive threshold group testing which develop the theory of superimposedcode