Combinatorial group testing and its applications

259 Pages·1993·9.142 MB·English

by  Dingzhu Du, Frank Hwang

COMBINATORIAL
GEOUP TESTING
ATOIfSAPPUCATIONS
SERIES ON APPLIED MATHEMATICS
Editor-in-Chief: Frank Hwang
Associate Editors-in-Chief: Zhong-ci Shi and Kunio Tanabe
Vol. 1 International Conference on Scientific Computation
eds. T. Chan and Z.-C. Shi
Vol. 2 Network Optimization Problems — Algorithms, Applications and
Complexity
eds. D.-Z. Du and P. M. Pandalos
Vol. 3 Combinatorial Group Testing and Its Applications
by D.-Z. Du and F. K. Hwang
Vol. 4 Computation of Differential Equations and Dynamical Systems
eds. K. Feng and Z.-C. Shi
Vol. 5 Numerical Mathematics
eds. Z.-C. Shi and T. Ushijima
Series on
Applied Mathematics
Volume 3
Ding-Zhu Du
Department of Computer Science
University of Minnesota
and
Institute of Applied Mathematics
Academia Sinica, Beijing
Frank K. Hwang
AT&T Bell Laboratories
Murray Hill
V fe World Scientific
«• SSininggaappoorere • •N Neeww J Jeersrseeyy •L L ondon • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
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Library of Congress Cataloging-in-Publication Data
Du, Dingzhu.
Combinatorial group testing and its applications / Ding-Zhu Du,
Frank K. Hwang.
p. cm. — (Series on applied mathematics; vol. 3)
Includes bibliographical references and index.
ISBN 9810212933
1. Combinatorial group theory. I. Hwang, Frank. II. Title.
III. Series: Series on applied mathematics v. 3.
QA182.5.D8 1993
512'.2-dc20 93-26812
CIP
Copyright © 1993 by World Scientific Publishing Co. Pte. Ltd.
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Preface
Group testing has been around for fifty years. It started as an idea to do large scale
blood testing economically. When such needs subsided, group testing stayed dormant
for many years until it was revived with needs for new industrial testing. Later,
group testing also emerged from many nontesting situations, such as experimental
designs, multiaccess communication, coding theory, clone library screening, nonlinear
optimization, computational complexity, etc.. With a potential world-wide outbreak
of AIDS, group testing just might go the full cycle and becomes an effective tool in
blood testing again. Another fertile area for application is testing zonal environmental
pollution.
Group testing literature can be generally divided into two types, probabilistic and
combinatorial. In the former, a probability model is used to describe the distribution
of defectives, and the goal is to minimize the expected number of tests. In the latter,
a deterministic model is used and the goal is usually to minimize the number of tests
under a worst-case scenario. While both types are important, we will focus on the
second type in this book because of the different flavors for these two types of results.
To find optimal algorithms for combinatorial group testing is difficult, and there
are not many optimal results in the existing literature. In fact, the computational
complexity of combinatorial group testing has not been determined. We suspect that
the general problem is hard in some complexity class, but do not know which class.
(It has been known that the problem belongs to the class PSPACE, but seems not
PSPACE-complete.) The difficulty is that the input consists of two or more integers,
which is too simple for complexity analysis. However, even if a proof of hardness will
eventually be given, this does not spell the end of the subject, since the subject has
many, many branches each posing a different set of challenging problems.
This book is not only the first attempt to collect all theory and applications about
combinatorial group testing in one place, but it also carries the personal perspective
of the authors who have worked on this subject for a quarter of a century. We hope
that this book will provide a forum and focus for further research on this subject,
and also be a source for references and publications. Finally, we thank E. Barillot,
A.T. Borchers, R.V. Book, G.J. Chang, F.R.K. Chung, A.G. Dyachkov, D. Kelley,
K.-I Ko, M. Parnes, D. Raghavarao, M. Ruszinko, V.V. Rykov, J. Spencer, M. Sobel,
U. Vaccaro, and A.C. Yao for giving us encouragements and helpful discussions at
various stage of the formation of this book. Of course, the oversights and errors are
our sole responsibility.
v
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Contents
Preface v
Chapter 1 Introduction 1
1.1 The History of Group Testing 1
1.2 The Binary Tree Representation of a Group Testing Algorithm
and the Information Lower Bound 5
1.3 The Structure of Group Testing 7
1.4 Number of Group Testing Algorithms 10
1.5 A Prototype Problem and Some Basic Inequalities 12
1.6 Variations of the Prototype Problem 17
References 18
Chapter 2 General Algorithms 19
2.1 Li's s-Stage Algorithm 19
2.2 Hwang's Generalized Binary Splitting Algorithm 20
2.3 The Nested Class 23
2.4 (d, n) Algorithms and Merging Algorithms 27
2.5 Some Practical Considerations 30
2.6 An Application to Clone Screenings 34
References 36
Chapter 3 Algorithms for Special Cases 38
3.1 Two Disjoint Sets Each Containing Exactly One Defective 38
3.2 An Application to Locating Electrical Shorts 43
3.3 The 2-Defective Case 48
3.4 The 3-Defective Case 53
3.5 When is Individual Testing Minimax? 56
3.6 Identifying a Single Defective with Parallel Tests 59
References 60
vii
viii Contents
Chapter 4 Nonadaptive Algorithms and Binary Superimposed Codes . 62
4.1 The Matrix Representation 62
4.2 Basic Relations and Bounds 63
4.3 Constant Weight Matrices and Random Codes 68
4.4 General Constructions 73
4.5 Special Constructions 78
References 87
Chapter 5 Multiaccess Channels and Extensions 91
5.1 Multiaccess Channels 92
5.2 Nonadaptive Algorithms 96
5.3 Two Variations 99
5.4 The k-Channel 101
5.5 Quantitative Channels 105
References 105
Chapter 6 Some Other Group Testing Models 107
6.1 Symmetric Group Testing 107
6.2 Some Additive Models 109
6.3 A Maximum Model 115
6.4 Some Models for d = 2 118
References 123
Chapter 7 Competitive Group Testing 126
7.1 The First Competitiveness 126
7.2 Bisecting 128
7.3 Doubling 132
7.4 Jumping 134
7.5 The Second Competitiveness 138
7.6 Digging 140
7.7 Tight Bound 143
References 148
Chapter 8 Unreliable Tests 149
8.1 Ulam's Problem 149
8.2 Geperal Lower and Upper Bounds 155
8.3 Linearly Bounded Lies (1) 160
Contents ix
8.4 The Chip Game 164
8.5 Linearly Bounded Lies (2) 168
8.6 Other Restrictions on Lies 172
References 175
Chapter 9 Optimal Search in One Variable 177
9.1 Midpoint Strategy 177
9.2 Fibonacci Search 179
9.3 Minimum Root Identification 183
References 190
Chapter 10 Unbounded Search 193
10.1 Introduction 193
10.2 Bentley-Yao Algorithms 195
10.3 Search with Lies 199
10.4 Unbounded Fibonacci Search 200
References 202
Chapter 11 Group Testing on Graphs 203
11.1 On Bipartite Graphs 203
11.2 On Graphs 205
11.3 On Hypergraphs 207
11.4 On Trees 212
11.5 Other Constraints 216
References 217
Chapter 12 Membership Problems 218
12.1 Examples 218
12.2 Polyhedral Membership 220
12.3 Boolean Formulas and Decision Trees 222
12.4 Recognition of Graph Properties 226
References 229
Chapter 13 Complexity Issues 231
13.1 General Notions 231
13.2 The Prototype Problem is in PSPACE 233
x Contents
13.3 Consistency 234
13.4 Determinacy 236
13.5 On Sample Space S(n) 237
13.6 Learning by Examples 243
References 244
Index 245