Distributed Algorithms (The Morgan Kaufmann Series in Data Management Systems)

899 Pages·1996·38.99 MB·English

by  Nancy A. Lynch

Distributed
Algorithms
The Morgan Kaufmann Series in Data Management Systems
Series Editor, Jim Gray
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Distributed
Algorithms
Nancy Lynch
A.
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Library of Congress Cataloging-in-Publication Data
Lynch, Nancy A. (Nancy Ann), 1948-
Distributed algorithms / Nancy A. Lynch
p. cm.
Includes bibliographical references and index.
ISBN-13:978-1-55860-348-6 ISBN-10:1-55860-348-4
1. Computer algorithms. 2. Electonic data
processing--Distributed processing I. Title
QA76.9.A43L96 1997
005.2' 76--dc21 97-413 7 5
ISBN- 13: 978-1-55860-348-6
ISBN-10:1-55860-348-4
To Dennis, Patrick, and Mary
This Page Intentionally Left Blank
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . six
1 Introduction 1
1.1 The Subject Matter . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Our Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Overview of Chapters 2-25 . . . . . . . . . . . . . . . . . . . . . 6
1.4 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Part I Synchronous Network Algorithms 15
2 Modelling I: Synchronous Network Model 17
2.1 Synchronous Network Systems . . . . . . . . . . . . . . . . . . . 17
2.2 Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Inputs and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Executions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Proof Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Complexity Measures . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Randomization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.8 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Leader Election in a Synchronous Ring 25
3.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Impossibility Result for Identical Processes ............ 27
3.3 A Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 An Algorithm with O (nlogn) Communication Complexity . . . 31
3.5 Non-Comparison-Based Algorithms . . . . . . . . . . . . . . . . 35
3.5.1 The TimeSlice Algorithm . . . . . . . . . . . . . . . . . . 35
3.5.2 The VariableSpeeds Algorithm . . . . . . . . . . . . . . . 36
3.6 Lower Bound for Comparison-Based Algorithms ......... 38
viii CONTENTS
3.7 Lower Bound for Non-Comparison-Based Algorithms* ...... 44
3.8 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Algorithms in General Synchronous Networks 51
4.1 Leader Election in a General Network . . . . . . . . . . . . . . . 52
4.1.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1.2 A Simple Flooding Algorithm . . . . . . . . . . . . . . . . 52
4.1.3 Reducing the Communication Complexity ......... 54
4.2 Breadth-First Search . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.2 A Basic Breadth-First Search Algorithm .......... 58
4.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Shortest Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Minimum Spanning Tree . . . . . . . . . . . . . . . . . . . . . . 63
4.4.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.2 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4.3 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Maximal Independent Set . . . . . . . . . . . . . . . . . . . . . . 71
4.5.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5.2 A Randomized Algorithm . . . . . . . . . . . . . . . . . . 71
4.5.3 Analysis* . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5 Distributed Consensus with Link Failures 81
5.1 The Coordinated Attack Problem--Deterministic Version .... 82
5.2 The Coordinated Attack Problem--Randomized Version ..... 86
5.2.1 Formal Modelling . . . . . . . . . . . . . . . . . . . . . . 87
5.2.2 An Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.3 A Lower Bound on Disagreement . . . . . . . . . . . . . . 93
5.3 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6 Distributed Consensus with Process Failures 99
6.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2 Algorithms for Stopping Failures . . . . . . . . . . . . . . . . . . 102
6.2.1 A Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . 103
6.2.2 Reducing the Communication . . . . . . . . . . . . . . . . 105
6.2.3 Exponential Information Gathering Algorithms ...... 108
CONTENTS ix
6.2.4 Byzantine Agreement with Authentication ......... 115
6.3 Algorithms for Byzantine Failures . . . . . . . . . . . . . . . . . 116
6.3.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3.2 EIG Algorithm for Byzantine Agreement ......... 119
6.3.3 General Byzantine Agreement Using Binary Byzantine
Agreement . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3.4 Reducing the Communication Cost ............. 125
6.4 Number of Processes for Byzantine Agreement .......... 129
6.5 Byzantine Agreement in General Graphs .............. 135
6.6 Weak Byzantine Agreement . . . . . . . . . . . . . . . . . . . . . 139
6.7 Number of Rounds with Stopping Failures ............. 142
6.8 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7 More Consensus Problems 161
7.1 k-Agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.1.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.1.2 An Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 162
7.1.3 Lower Bound* . . . . . . . . . . . . . . . . . . . . . . . . 164
7.2 Approximate Agreement . . . . . . . . . . . . . . . . . . . . . . . 177
7.3 The Commit Problem . . . . . . . . . . . . . . . . . . . . . . . . 182
7.3.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.3.2 Two-Phase Commit . . . . . . . . . . . . . . . . . . . . . 184
7.3.3 Three-Phase Commit . . . . . . . . . . . . . . . . . . . . 185
7.3.4 Lower Bound on the Number of Messages ......... 189
7.4 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Part II Asynchronous Algorithms 197
8 Modelling II: Asynchronous System Model 199
8.1 I/O Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
8.2 Operations on Automata . . . . . . . . . . . . . . . . . . . . . . . 206
8.2.1 Composition . . . . . . . . . . . . . . . . . . . . . . . . . 207
8.2.2 Hiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.3 Fairness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.4 Inputs and Outputs for Problems . . . . . . . . . . . . . . . . . . 215
8.5 Properties and Proof Methods . . . . . . . . . . . . . . . . . . . 216
8.5.1 Invariant Assertions . . . . . . . . . . . . . . . . . . . . . 216