# New Constructions for Competitive and Minimal-Adaptive Group Testing Doktorsavhandling, 2013

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 $d\ll n$ \emph{defective} elements out of $n$ elements by querying subsets (pools) for the presence of defectives. By the information-theoretic lower bound, essentially $d\log_2 n$ 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 $d$ 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 $s$ 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 $d$ and show that actually competitive GT is possible in as few as $2$ stages only. The main ingredient is our randomized estimate of a previously unknown $d$ using nonadaptive queries. In addition, we have developed a systematic approach to obtain optimal competitive ratios for our strategies. When $d$ is a known upper bound, we propose randomized GT strategies which asymptotically achieve query optimality in just $2$, $3$ or $4$ stages depending upon the growth of $d$ versus $n$. Inspired by application settings, such as at American Red Cross, where in most cases GT is applied to small instances, \textit{e.g.}, $n=16$. We extended our study of query-optimal GT strategies to solve a given problem instance with fixed values $n$, $d$ and $s$. 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 $\Theta (sd(n/d)^{1/s})$ tests are sufficient, and also necessary for certain ranges of the parameters.

strict group testing

disjoint pools

randomization

competitive group testing

exact bounds

competitive ratio

Room EF, EDIT building, Chalmers.
Opponent: Prof. Dr. Ingo Althöfer

## Författare

Chalmers, Data- och informationsteknik, Datavetenskap

#### Two new perspectives on multi-stage group testing

Algorithmica,; Vol. 67(2013)p. 324-354

Artikel i vetenskaplig tidskrift

#### Randomized group testing both query-optimal and minimal adaptive

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics),; Vol. 7147(2012)p. 214-225

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

#### A toolbox for provably optimal multistage strict group testing strategies

19th International Computing and Combinatorics Conference COCOON 2013, Lecture Notes in Computer Science,; Vol. 7936(2013)p. 446-457

Paper i proceeding

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

Discrete Mathematics, Algorithms and Applications,; Vol. 3(2011)p. 517-536

Artikel i vetenskaplig tidskrift

#### Styrkeområden

Informations- och kommunikationsteknik

#### Drivkrafter

Hållbar utveckling

#### Fundament

Grundläggande vetenskaper

#### Ämneskategorier

Datavetenskap (datalogi)

#### ISBN

978-91-7385-913-4

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 3594

Technical report D - Department of Computer Science and Engineering, Chalmers University of Technology and Göteborg University: 98D

Room EF, EDIT building, Chalmers.

Opponent: Prof. Dr. Ingo Althöfer

2017-10-08