New Constructions for Competitive and Minimal-Adaptive Group Testing
Doctoral thesis, 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


competitive group testing

minimal-adaptive group testing

exact bounds

competitive ratio

Room EF, EDIT building, Chalmers.
Opponent: Prof. Dr. Ingo Althöfer


Muhammad Azam Sheikh

Chalmers, Computer Science and Engineering (Chalmers), Computing Science (Chalmers)

Two new perspectives on multi-stage group testing

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

Journal article

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 in proceeding

Competitive group testing and learning hidden vertex covers with minimum adaptivity

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

Journal article

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 in proceeding

Bounds for nonadaptive group tests to estimate the amount of defectives

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

Journal article

Areas of Advance

Information and Communication Technology

Driving Forces

Sustainable development


Basic sciences

Subject Categories

Computer Science



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 Althöfer

More information