# Engineering Competitive and Query-Optimal Minimal-Adaptive Randomized Group Testing Strategies Licentiatavhandling, 2011

Suppose that given is a collection of $n$ elements where $d$ of them are \emph{defective}. We can query an arbitrarily chosen subset of elements which returns Yes if the subset contains at least one defective and No if the subset is free of defectives. The problem of group testing is to identify the defectives with a minimum number of such queries. By the information-theoretic lower bound at least $\log_2 \binom {n}{d} \approx d\log_2 (\frac{n}{d}) \approx d\log_2 n$ queries are needed. Using adaptive group testing, i.e., asking one query at a time, the lower bound can be easily achieved. However, strategies are preferred that work in a fixed small number of stages, where queries in a stage are asked in parallel. A group testing strategy is called \emph{competitive} if it works for completely unknown $d$ and requires only $O(d\log_2 n)$ queries. Usually competitive group testing is based on sequential queries. We have shown that actually competitive group testing with expected $O(d\log_2 n)$ queries is possible in only $2$ or $3$ stages. Then we have focused on minimizing the hidden constant factor in the query number and proposed a systematic approach for this purpose. Another main result is related to the design of query-optimal and minimal-adaptive strategies. We have shown that a $2$-stage randomized strategy with prescribed success probability can asymptotically achieve the information-theoretic lower bound for $d \ll n$ and growing much slower than $n$. Similarly, we can approach the entropy lower bound in $4$ stages when $d=o(n)$.

learning by queries

combinatorial search

competitive ratio

randomization

pooling design

group testing

linear program

room EA, EDIT building, Chalmers University of Technology, Gothenburg.
Opponent: Prof. Dr. Eberhard Triesch, Chair for Mathematics,Technical University (RWTH) Aachen, Germany.

## Författare

Chalmers, Data- och informationsteknik, Datavetenskap

#### Competitive group testing and learning hidden vertex covers with minimum adaptivity

Lecture Notes in Computer Science,; Vol. 5699(2009)p. 84-95

Paper i proceeding

#### Bounds for nonadaptive group tests to estimate the amount of defectives

Lecture Notes in Computer Science,; Vol. 6509(2010)p. 117-130

Paper i proceeding

#### Competitive group testing and learning hidden vertex covers with minimum adaptivity

Discrete Mathematics, Algorithms and Applications,; Vol. 2(2010)p. 291-311

Artikel i vetenskaplig tidskrift

#### Styrkeområden

Informations- och kommunikationsteknik

#### Drivkrafter

Hållbar utveckling

#### Fundament

Grundläggande vetenskaper

#### Ämneskategorier

Bioinformatik och systembiologi

Datavetenskap (datalogi)

Technical report L - Department of Computer Science and Engineering, Chalmers University of Technology and Göteborg University: 81L

#### Utgivare

Chalmers tekniska högskola

room EA, EDIT building, Chalmers University of Technology, Gothenburg.

Opponent: Prof. Dr. Eberhard Triesch, Chair for Mathematics,Technical University (RWTH) Aachen, Germany.

2018-07-06