Fundamentals of Algorithmics

546 Pages·1995·27.61 MB·English

by  Gilles Brassard

FUNDAMENTALS
OF ALGORITHMICS

FUNDAMENTALS
OF ALGORITHMICS
Gilles Brassard and Paul Bratley
Dopartement d'informatiquee t de recherche opgrationelle
UniversitW de Montreal
PRENTICE HALL
Englewood Cliffs, New Jersey 07632
Library of Congress Cataloging-in-Publication Data
BRASSARD, GILLES
Fundamentals of Algorithmics / Gilles Brassard and Paul Bratley
p. cm.
Includes bibliographical references and index.
ISBN 0-13-335068-1
1. Algorithms. I. Bratley, Paul. H. Title
QA9.58.B73 1996 95-45581
511'.8-dc2O CU'
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Production editor: Irwin Zucker
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© 1996 by Prentice-Hall, Inc.
A Simon & Schuster Company
Englewood Cliffs, New Jersey 07632
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efforts include the development, research, and testing of the theories and formulas to determine
their effectiveness. The author and publisher shall not be liable in any event for incidental or
consequential damages in connection with, or arising out of, the furnishing, performance, or use
of these formulas.
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Contents
PREFACE xv
* 1 PRELIMINARIES
1.1 Introduction 1
1.2 What is an algorithm? 1
1.3 Notation for programs 6
1.4 Mathematical notation 7
1.4.1 Propositional calculus 7
1.4.2 Set theory 8
1.4.3 Integers, reals and intervals 8
1.4.4 Functions and relations 9
1.4.5 Quantifiers 10
1.4.6 Sums and products 11
1.4.7 Miscellaneous 12
1.5 Proof technique 1 - Contradiction 13
1.6 Proof technique 2 - Mathematical induction 16
1.6.1 The principle of mathematical induction 18
1.6.2 A horse of a different colour 23
1.6.3 Generalized mathematical induction 24
1.6.4 Constructive induction 27
1.7 Some reminders 31
1.7.1 Limits 31
1.7.2 Simple series 34
1.7.3 Basic combinatorics 38
1.7.4 Elementary probability 41
Vii
viii Contents
1.8 Problems 48
1.9 References and further reading 55
*' 2 ELEMENTARY ALGORITHMICS 57
2.1 Introduction 57
2.2 Problems and instances 58
2.3 The efficiency of algorithms 59
2.4 Average and worst-case analyses 61
2.5 What is an elementary operation? 64
2.6 Why look for efficiency? 66
2.7 Some examples 67
2.7.1 Calculating determinants 68
2.7.2 Sorting 68
2.7.3 Multiplication of large integers 70
2.7.4 Calculating the greatest common divisor 71
2.7.5 Calculating the Fibonacci sequence 72
2.7.6 Fourier transforms 73
2.8 When is an algorithm specified? 74
2.9 Problems 74
2.10 References and further reading 78
* 3 ASYMPTOTIC NOTATION 79
3.1 Introduction 79
3.2 A notation for "the order of" 79
3.3 Other asymptotic notation 85
3.4 Conditional asymptotic notation 88
3.5 Asymptotic notation with several parameters 91
3.6 Operations on asymptotic notation 91
3.7 Problems 92
3.8 References and further reading 97
* 4 ANALYSIS OF ALGORITHMS 98
4.1 Introduction 98
4.2 Analysing control structures 98
4.2.1 Sequencing 98
4.2.2 "For" loops 99
4.2.3 Recursive calls 101
4.2.4 "While" and "repeat" loops 102
Contents ix
4.3 Using a barometer 104
4.4 Supplementary examples 106
4.5 Average-case analysis 111
4.6 Amortized analysis 112
4.7 Solving recurrences 116
4.7.1 Intelligent guesswork 116
4.7.2 Homogeneous recurrences 118
4.7.3 Inhomogeneous recurrences 123
4.7.4 Change of variable 130
4.7.5 Range transformations 136
4.7.6 Asymptotic recurrences 137
4.8 Problems 139
4.9 References and further reading 146
* 5 SOME DATA STRUCTURES 147
5.1 Arrays, stacks and queues 147
5.2 Records and pointers 150
5.3 Lists 151
5.4 Graphs 152
5.5 Trees 154
5.6 Associative tables 159
5.7 Heaps 162
5.8 Binomial heaps 170
5.9 Disjoint set structures 175
5.10 Problems 181
5.11 References and further reading 186
* 6 GREEDYALGORITHMS 187
6.1 Making change (1) 187
6.2 General characteristics of greedy algorithms 188
6.3 Graphs: Minimum spanning trees 190
6.3.1 Kruskal's algorithm 193
6.3.2 Prim's algorithm 196
6.4 Graphs: Shortest paths 198
6.5 The knapsack problem (1) 202
6.6 Scheduling 205
6.6.1 Minimizing time in the system 205
6.6.2 Scheduling with deadlines 207