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FUNDAMENTALS OF
COMPUTER
ALGORITHMS
COMPUTER SCIENCE PRESS
FUNDAMENTALS OF
COMPUTER
ALGORITHMS
COMPUTER SOFTWARE ENGINEERING SERIES
ELLIS HOROWITZ, EDITOR
University of Southern California
WAYNE AMSBURY
Structured BASIC and Beyond
JEAN-LOUP BAER
Computer Systems Architecture
PETER CALINGAERT
Assemblers, Compilers, and Program Translation
M. S. CARBERRY, H. M. KHALIL, J. F. LEATHRUM, J. S. LEVY
Foundations of Computer Science
SHIMON EVEN
Graph Algorithms
W. FINDLAY and D. A. WATT
Pascal: An Introduction to Methodical Programming, Second Edition
ELLIS HOROWITZ and SARTA J SAHNI
Fundamentals of Computer Algorithms
ELLIS HOROWITZ and SARTAJ SAHNI
Fundamentals of Data Structures
ELLIS HOROWITZ and SARTAJ SAHNI
Fundamentals of Data Structures in Pascal
ELLIS HOROWITZ
Fundamentals of Programming Languages, Second Edition
ELLIS HOROWITZ, EDITOR
Programming Languages: A Grand Tour
TOM LOGSDON
Computers and Social Controversy
IRA POHL and ALAN SHAW
The Nature of Computation: An Introduction to Computer Science
DONALD D. SPENCER
Computers in Number Theory
JEFFREY D. ULLMAN
Principles of Database Systems, Second Edition
~ ~--------
FUNDAMENTALS OF
COMPUTER
ALGORITHMS
ELLIS HOROWITZ
University of Southern California
SARTAJ SAHNI
University of Minnesota
COMPUTER SCIENCE PRESS
Copyright © 1978 Computer Science Press, Inc.
Printed in the United States of America
All rights reserved. No part of this work may be reproduced, transmitted,
or stored in any form or by any means, without the prior written consent of
the Publisher.
Computer Science Press
1803 Research Blvd.
Rockville, Maryland 20850
IO 11 12 Printing Year 89 88 87 86
This book represents a joint effort of both authors who fully col·
laborated on all chapters. The names of the authors of this text
have been listed in alphabetical order. This does not imply any
senior-junior relationship. Dr. Sartaj Sahni was the primary author
of Chapters 4, 5, 6, 8, 11, and 12 and Dr. Ellis Horowitz was the
primary author of Chapters 1, 2, 9, and JO.
Library of Congress Cataloging in Publication Data
Horowitz, Ellis.
Fundamentals of computer algorithms.
(Computer software engineering series)
Includes index.
1. Electronic digital computers- -Programming.
2. Algorithms I. Sahni, Sartaj, joint author.
II. Title.
QA76.6.H67 519.4 78·14735
ISBN 0-914894-22-6
dedicated to our wives
Neeta Sahni
Maryanne Horowitz
PREFACE
If we try to identify those contributions of computer science which will be
long lasting, surely one of these will be the refinement of the concept called
algorithm. Ever since man invented the idea of a machine which could per
form basic mathematical operations, the study of what can be computed
and how it can be done well was launched. This study, inspired by the
computer, has led to the discovery of many important and clever algorithms.
The discipline called computer science has embraced the study of algo
rithms as its own. It is the purpose of this book to organize what is known
about them in a coherent fashion so that students and practitioners can
learn to devise and analyze new algorithms for themselves.
But a book which contains every algorithm ever invented would be ex
ceedingly large, and traditionally algorithms books have proceeded by ex
amining only a small number of problem areas in depth. For each specific
problem the most efficient algorithm for its solution is usually presented
and analyzed. Having taught courses in this way for several years we were
well aware that this approach has one major flaw. Though the student sees
many fast algorithms and may master the tools of analysis, he remains un
confident about how to devise good algorithms in the first place.
The missing ingredient is a lack of emphasis on design techniques. A
knowledge of design will certainly help one to create good algorithms, yet
without the tools of analysis there is no way to determine the quality of
the result. This observation that design should be taught on a par with
analysis led us to a more promising line of approach: namely to organize
our courses, and subsequently this book, around some fundamental strat
egies of algorithm design. The number of basic design strategies is reason
ably small. Moreover all of the algorithms one would typically wish to study
can easily be fit into these categories; for example, mergesort and quick
sort are perfect examples of the divide-and-conquer strategy while Kruskal's
minimum spanning tree algorithm and Dijkstra's single source shortest
path algorithm are straightforward examples of the greedy strategy. An
understanding of these strategies is an essential first step towards acquiring
the skills of design.
vii
viii Preface
Though we both strongly feel that the emphasis on design as well as
analysis is the appropriate way to organize the study of algorithms, a cau
tionary remark is in order. First, we have not included every known design
principle. One example is linear programming which is one of the most
successful techniques, but is often discussed in a course of its own. Secondly,
the student should be inhibited from taking a cookbook approach to algo
rithm design by assuming that each algorithm must derive from only a
single technique. This is not so.
A major portion of this book, chapters 3 through 9, deal with the dif
ferent design strategies. First each strategy is described in general terms.
Typically a "program abstraction" is given which outlines the form that
the computation will take if this strategy can be applied. Following this
there are a succession of examples which reveal the intricacies and varieties
of the general strategy. The examples are somewhat loosely ordered in terms
of increasing complexity. The type of complexity may arise in several ways.
Usually we begin with a problem which is very simple to understand and
requires no data structures other than a one-dimensional array. For this
problem it is usually obvious that the given design strategy yields a correct
solution. Later examples may require a proof that an algorithm based on
this design technique does work. Or, the later algorithms may require more
sophisticated data structures (e.g. trees or graphs) and their analyses may
be more complex. The major goal of this organization is to emphasize the
arts of synthesis and analysis of algorithms. Auxiliary goals are to expose
the student to good program structure and to proofs of algorithm correct
ness.
One of the most energetic areas of computer science research today is
called computational complexity. That name denotes the study of what
makes functions intrinsically difficult to compute. Two products of com
putational complexity have been the development of algorithms with the
lowest asymptotic computing time and facts concerning the minimum num
ber of operations required to compute a given function. Many of these re
sults can be found here. However our decision in writing this book was to
emphasize algorithms which were not only of theoretical interest but which
are practical to use. Unfortunately many of the "best" algorithms, from
an asymptotic point of view, are quite hard to program and require such
a great amount of overhead that their practical value is limited. We have
avoided lengthy presentations of such algorithms and contented ourselves
with pointing to the available literature.
The algorithms presented here are written in SP ARKS, the name we
have given to our ALGOL/PASCAL-like language which we first introduced
in Fundamentals of Data Structures. The syntax of some statements has
Preface ix
been improved, but the changes are such that the meanings of all state
ments is still immediately discernible. Chapter one presents the precise
semantics of each statement via flowcharts and gives some drill in the art
of program structuring. We hope that by studying well-written programs
the student will apply these same principles to his or her own program
composition. Another important aspect of this book is program testing.
Though computer science still lacks an adequate formal treatment of this
subject, for some algorithms we show how to devise a range of data sets
which can be used for debugging and performance measurement. Also we
have felt obliged to provide programs which are essentially complete in all
details. Though this may complicate the presentation of the algorithms, it
has as its virtue the fact that each algorithm can be quickly programmed
and executed. Of course, subroutines are used to improve clarity.
The material in this book does not correspond to any existing course
within ACM's recommended Curriculum '68. However it does seem likely
that the IEEE Computer Society will include an algorithms course within
its new recommendations. As the course structure of many computer science
programs is now firmly established, it has become harder to introduce new
courses. Nevertheless we are confident that these subjects are of sufficient
merit that many computer science educators will attempt to cover this ma
terial. Thus we offer the arguments we used to get our own departments
to adopt a course on The Design and Analysis of Algorithms. First and
foremost, we argued that "algorithm" is a fundamental concept of com
puter science and hence there should be a course devoted to its study. Sec
ondly the skills of algorithm synthesis and analysis will improve both the
students basic knowledge and his or her ability to comprehend more sophis
ticated algorithms in later courses. Finally there now exists some important
theoretical results (e.g. NP-Completeness which is discussed in Chapter 11)
which deserve to be covered.
We view the material presented here as ideal for a one semester or two
quarter course given to juniors, seniors or graduate students. It does re
quire prior experience with programming in a higher level language but
everything else is self-contained. Practically speaking, it seems that a course
on data structures is helpful, if only for the fact that the students have
greater programming maturity. For a school on the quarter system, the
first quarter might cover the basic design techniques as given in chapters 3
through 8: divide-and-conquer, the greedy method, dynamic programming,
search and traversal, backtracking, and branch-and-bound. The second
quarter would cover the more theoretical subjects of chapters 10 through
12: lower bound theory, NP-Completeness and approximation methods.
For a semester schedule where the student has already encountered data
