Advances in Randomized Parallel Computing

307 Pages·1999·21.829 MB·English

by  Aviad Cohen, Yuri Rabinovich, Assaf Schuster (auth.), Panos M. Pardalos, Sanguthevar Rajasekaran (eds.)

Advances in Randomized Parallel Computing
COMBINATORIAL OPTIMIZATION
VOLUME 5
Through monographs and contributed works the objective of the series is to publish state of
the art expository research covering all topics in the field of combinatorial optimization. In
addition, the series will include books which are suitable for graduate level courses in com
puter science, engineering, business, applied mathematics, and operations research.
Combinatoria (or discrete) optimization problems arise in various applications, including
communications network design, VLSI design, machine vision, airline crew scheduling, cor
porate planning, computer-aided design and manufacturing, database query design, cellular
telephone frequency assignment, constraint directed reasoning, and computational biology.
The topics of the books will cover complexity analysis and algorithm design (parallel and
serial), computational experiments and applications in science and engineering.
Series Editors:
Ding-Zhu Du, University of Minnesota
Panos M. Pardalos, University of Florida
Advisory Editorial Board:
Alfonso Ferreira, CNRS-LIP ENS Lyon
Jun Gu, University of Galgary
D. Frank Hsu, Fordham University
David S. Johnson, AT&T Research
James B. Orlin, M.l. T.
Christos H. Papadimitriou, University of California at Berkely
Fred S. Roberts, Rutgers University
The titles published in this series are listed at the end of this volume.
Advances in
Randomized
Parallel Computing
edited by
Panos M. Pardalos
University of Florida
Gainesville, Florida, U.S.A.
and
Sanguthevar Rajasekaran
University of Florida
Gainesville, Florida, U.S.A.
"~.
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-13: 978-1-4613-3284-8 e-ISBN-13: 978-1-4613-3282-4
001: 10.1007/978-1-4613-3282-4
Published by Kluwer Academic Publishers,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
Sold and distributed in North, Central and South America
by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers,
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Printed on acid-free paper
All Rights Reserved
© 1999 Kluwer Academic Publishers
Softcover reprint of the hardcover 1st editon 1999
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Preface IX
Contributing Authors XXI
1
Optimal Bounds on Tail Probabilities: A Study of an Approach 1
Aviad Cohen, Yuri Rabinovich, Assaf Schuster, and Hadas Shachnai
1.1 Introduction 2
1.2 Bounding Tail Probabilities with the Laplace Transform. 3
1.3 When only the Mean is given: The Hoeffding Bound 6
1.4 When the Mean and the Variance are given: A Simple Proof of the Bennett
Bound 9
1.5 When the First n Moments are Given: A Glimpse of the General Theory 11
1.6 An Application: Improved Bounds for the List Update Problem 20
References 23
2
Parallelism in Comparison Problems 25
Danny Krizanc
2.1 Introduction 25
2.2 Selection 26
2.3 Merging 31
2.4 Sorting 35
2.5 Conclusions 37
References 37
3
Random Sa:,'pling 41
Rajeev Raman
3.1 Introduction 41
3.2 Preliminaries 42
3.3 Partitioning I: Sorting 46
3.4 Partitioning II: List Ranking 49
v
vi ADVANCES IN RANDOMIZED PARALLEL COMPUTING
3.5 Pruning I: Selection 51
3.6 Pruning II: Row maxima of monotone matrices 53
3.7 Pruning III: Graph Connected Components 57
3.8 Other examples 61
3.9 Bibliographic Notes 61
References 62
4
Randomized Algorithms on the Mesh 67
Lata Narayanan
4.1 Introduction 67
4.2 Preliminaries 68
4.3 Routing on the mesh 71
4.4 Sorting on the mesh 73
4.5 Selection on the mesh 76
References 80
5
Efficient Randomized Algorithms 85
David S. L. Wei and Kishirasagar Naik
5.1 Introduction 86
5.2 Preliminaries 86
5.3 Randomized Routing 89
5.4 Randomized Selection 93
5.5 Randomized Sorting 97
5.6 Randomized PRAM Emulation 99
5.7 Selection and Sorting Schemes for Processing Large Distributed Files 101
5.8 Conclusions 108
References 108
6
Ultrafast Randomized Parallel Algorithms for Spanning Forests 113
Anders Des.~mark, Carsten Dorgerloh, Andrzej Lingas, and Jurgen Wirtgen
6.1 Introduction 113
6.2 Ultrafast Parallel Algorithms 116
6.3 Dense Instances 118
6.4 Ultrafast Algorithms for Spanning Forests 123
6.5 Open Problems and Further Research 129
References 129
7
Parallel Randomized Techniques for Some Fundamental Geometric Problems 133
Suneeta Ramaswami
7.1 In 1:c')d uction 133
7.2 Pr~:iminaries and Definitions 135
7.3 The Use of Randomization in Computational Geometry 141
Contents vii
7.4 Applications to Fundamental Geometric Problems 145
7.5 Summary 153
References 154
8
Capturing the Connectivity of High-Dimensional Geometric Spaces 159
David Hsu, Jean-Claude Latombe, Rajeev Motwani, and Lydia E. Kavraki
8.1 Introduction 160
8.2 Basic Probabilistic Roadmap Planner 164
8.3 Other Sampling Strategies 166
8.4 Roadmap Coverage 167
8.5 Roadmap Connectedness 170
8.6 Current and Future Work 173
Appendix: A. Proof of Theorem 1 175
Appendix: B. Proof of Theorem 2 176
Appendix: C. Proof of Theorem 3 176
Appendix: D. Proof of Theorem 4 179
References 180
9
Randomized Parallel Prefetching and Buffer Management 183
Mahesh Kali'lhalla and Peter J. Varman
9.1 Introduction 184
9.2 Definitions 187
9.3 Read-Once Reference Strings 190
9.4 Read-Many Reference Strings 194
9.5 Concluding Remarks 206
References 207
10
DFA Problems 209
B. Ravikumar
10.1 Introduction 210
10.2 Membership Problem 212
10.3 Containment and Equivalence Problems 213
10.4 Ranking and Related Problems 216
10.5 Coarsest Partition Problems 218
10.6 Automata Testing Problems 225
10.7 Conversion from Regular Expression to NFA 231
10.8 Applications 234
10.9 Open Problems 235
References 236
11
LAPACK90 241
Jack Dongarra and Jerzy Wasniewski
viii ADVANCES IN RANDOMIZED PARALLEL COMPUTING
11.1 Inti .:>duction 242
11.2 Interface Blocks for LAPACK 17 246
11.3 Interface Blocks for LAPACK 90 248
11.4 Code of LAPACK90 Routines 250
11.5 LAPACK90 Documentation 251
11.6 LAPACK90 Test Programs 251
11.7 LAPACK90 User Callable Routines 252
References 252
Appendix: A, Generic Interfaces 254
A.1 LAPACK17 Generic Interface Blocks 254
A.2 LAPACK90 Generic Interface Blocks 257
Appendix: B, Interface Subroutines 259
B.1 LA_GESV and LA_GETRI subroutines 259
B.2 Auxiliary Routines 261
Appendix: C 263
C1 Documentation of LA_GESV 263
Appendix: 0 267
0.1 The LA_GESV test results 267
Appendix: E 268
E.1 LAPACK90 User Callable Routines 268
INDEX 217
Preface
The technique of randomization has been employed to solve numerous prob
lems of computing both sequentially and in parallel. Examples of randomized
algorithms that are asymptotically better than their deterministic counterparts
in solving various fundamental problems abound. Randomized algorithms have
the advantages of simplicity and better performance both in theory and often
in practice. This book is a collection of articles written by renowned experts
in the area of randomized parallel computing.
A brief introduction to randomized algorithms
In the aflalysis of algorithms, at least three different measures of performance
can be used: the best case, the worst case, and the average case. Often, the
average case run time of an algorithm is much smaller than the worst case.
For instance, the worst case run time of Hoare's quicksort is O(n2), whereas its
average case run time is only O( n log n). The average case analysis is conducted
with an assumption on the input space. The assumption made to arrive at the
O( n log n) average run time for quicksort is that each input permutation is
equally likely. Clearly, any average case analysis is only as good as how valid
the assumption made on the input space is.
Randomized algorithms achieve superior performances without making any
assumptions on the inputs by making coin flips within the algorithm. Any
analysis done of randomized algorithms will be valid for all p0:.sible inputs.
A randomized algorithm can be thought of as one wherein certain decisions
are made based on the outcomes of coin-flips made in the algorithm. A random
ized algorithm with one possible sequence of outcomes for the coin flips can be
considered as being different from the same algorithm with a different sequence
of outcomes for the coin flips. Thus a randomized algorithm can be conceived
of as a family of algorithms. For a given input, some of the algorithms in this
family might have a 'poor performance'. We must ensure that the number of
such bad algorithms in the family is only a small fraction of the total number
of algorithms. If for any input we can find at least (1 - 10) (10 being very close
to 0) portion of algorithms in the family that will have a 'good performance'
IX