Godrich, tamassia, mount data structures and algorithms in c++

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Data Structures and
A
lgorithms in C
++
Second Edition
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Data Structures and
A
lgorithms in C
++
Second Edition

Michael T. Goodrich
Department of Computer Science
University of California, Irvine
Roberto Tamassia
Department of Computer Science
Brown University
David M. Mount
Department of Computer Science
Un
iversity of Maryland
John Wiley & Sons, Inc.
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ACQUISITIONS EDITOR B
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To Karen, Paul, Anna, and Jack
– Mi
chael T. Goodrich
To Isabel
– Roberto Tamassia

To Jeanine
– David M. Mount
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Preface
T
his second edition of Data Structures and Algorithms in C++ is designed to pro-
vide an introduction to data structures and algorithms, including their design, analy-
sis, and implementation. In terms of curricula based on the IEEE/ACM 2001 Com-
puting Curriculum, this book is appropriate for use in the courses CS102 (I/O/B
versions), CS103 (I/O/B versions), CS111 (A version), and CS112 (A/I/O/F/H ver-
sions). We discuss its use for such courses in more detail later in this preface.
The major changes in the second edition are the following:
• We added more examples of data structure and algorithm analysi
s.
• We enhanced consistency with the C++ Standard Template Library (STL).
• We incorporated STL data structures into many of our data structures.
• We added a chapter on arrays, linked lists, and iterators (Chapter 3).
• We added a chapter on memory management and B-trees (Chapter 14).
• We enhanced the discussion of algorithmic design techniques, like dynamic
programming and the greedy method.
• We simplified and reorganized the presentation of code fragments.

• We have introduced STL-style iterators into our container classes, and have
presented C++ implementations for these iterators, even for complex struc-
tures such as hash tables and binary search trees.
• We have modified our priority-queue interface to use STL-style comparator
objects.
• We expanded and revised exercises, continuing our approach of
dividing
them into reinforcement, creativity, and project exercises.
This book is related to the following books:
• M.T. Goodrich and R. Tamassia, Data Structures and Algorithms in Java,
John Wiley & Sons, Inc. This book has a similar overall structure to the
present book, but uses Java as the underlying language (with some modest,
but necessary pedagogical differences required by this approach).
• M.T. Goodrich and R. Tamassia, Algorithm Design: Foundations, Analysis,
and Internet Examples, John Wiley & Sons, Inc. This is a textbook for a more
advanced algorithms and data structures course, such as CS210 (T/W/C/S
versions) in the IEEE/ACM 2001 curriculum.
While this book retains the same pedagogical approach and general structure
as Data Structures and Algorithms in Java, the code fragments have been com-
pletely redesigned. We have been careful to make full use of C++’s capabilities and
design code in a manner that is consistent with modern C++ usage. In particular,
whenever appropriate, we make extensive use of C++ elements that are not part of
Java, including the C++ Standard Template Library (STL), C++ memory allocation
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viii
Preface
and deallocation (and the associated issues of destructors), virtual functions, stream
input and output, operator overloading, and C++’s safe run-time casting.
Use as a Textbook
T
he design and analysis of efficient data structures has long been recognized as a
vital subject in computing, because the study of data structures is part of the core
of every collegiate computer science and computer engineering major program we
are familiar with. Typically, the introductory courses are presented as a two- or
three-course sequence. Elementary data structures are often briefly introduced in
the first programming course or in an introduction to computer science course and
this is followed by a more in-depth introduction to data structures in the courses that
follow after this. Furthermore, this course sequence is typically followed at a later
point in the curriculum by a more in-depth study of data structures and algorithms.
We feel that the central role of data structure design and analysis in the curriculum
is fully justified, given the importance of efficient data structures in most software
systems, including the Web, operating systems, databases, compilers, and scientific
simulation systems.
With the emergence of the object-oriented paradigm as the framework of choice
for building robust and reusable software, we have tried to take a consistent object-
oriented viewpoint throughout this text. One of the main ideas behind the object-
oriented approach is that data should be presented as being encapsulated with the
methods that access and modify them. That is, rather than simply viewing data
as a collection of bytes and addresses, we think of data objects as instances of an
abstract data type (ADT), which includes a repertoire of methods for performing
operations on data objects of this type. Likewise, object-oriented solutions are often

organized utilizing common design patterns, which facilitate software reuse and
robustness. Thus, we present each data structure using ADTs and their respective
implementations and we introduce important design patterns as a way to organize
those implementations into classes, methods, and objects.
For most of the ADTs presented in this book, we provide a description of the
public interface in C++. Also, concrete data structures realizing the ADTs are
discussed and we often give concrete C++ classes implementing these interfaces.
We also give C++ implementations of fundamental algorithms, such as sorting and
graph searching. Moreover, in addition to providing techniques for using data struc-
tures to implement ADTs, we also give sample applications of data structures, such
as HTML tag matching and a simple system to maintain a play list for a digital
audio system. Due to space limitations, however, we only show code fragments of
some of the implementations in this book and make additional source code avail-
able on the companion web site.
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ix
Online Resour ces
T
his book is accompanied by an extensive set of online resources, which can be
found at the following web site:
www.wiley.com/college/goodrich

Included on this Web site is a collection of educational aids th
at augment the
topics of this book, for both students and instructors. Students are encouraged to
use this site along with the book, to help with exercises and increase understand-
ing of the subject. Instructors are likewise welcome to use the site to help plan,
organize, and present their course materials. Because of their added value, some of
these online resources are password protected.
For the Student
For all readers, and especially for students, we include the following resources:
• All the C++ source code presented in this book.
• PDF handouts of Powerpoint slides (four-per-page) provided to instructors.
• A database of hints to all exercises, indexed by problem number.
• An online study guide, which includes solutions to selected exercises.
The hints should be of considerable use to anyone needing a little help getting
started on certain exercises, and the solutions should help anyone wishing to see
completed exercises. Students who have purchased a new copy of this book will
get password access to the hints and other password-protected online resources at
no extra charge. Other readers can purchase password access for a nominal fee.
For the Instructor
For instructors using this book, we include the following addi
tional teaching aids:
• Solutions to over 200 of the book’s exercises.
• A database of additional exercises, suitable for quizes and exams.
• Additional C++ source code.
• Slides in Powerpoint and PDF (one-per-page) format.
• Self-contained, special-topic supplements, including discussions on convex
hulls, range trees, and orthogonal segment intersection.
The slides are fully editable, so as to allow an instructor using this book full free-
dom in customizing his or her presentations. All the online resources are provided
at no extra charge to any instructor adopting this book for his or her course.

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Preface
A Resour ce for Teaching Data Structures and Algorithms
T
his book contains many C++-code and pseudo-code fragments, and hundreds of
exercises, which are divided into roughly 40% reinforcement exercises, 40% cre-
ativity exercises, and 20% programming projects.
This book can be used for the CS2 course, as described in the 1978 ACM Com-
puter Science Curriculum, or in courses CS102 (I/O/B versions), CS103 (I/O/B ver-
sions), CS111 (A version), and/or CS112 (A/I/O/F/H versions), as described in the
IEEE/ACM 2001 Computing Curriculum, with instructional units as outlined in
Table 0.1.
Instructional Unit Relevant Material
PL1. Overview of Programming Languages Chapters 1 and 2
PL2. Virtual Machines Sections 14.1.1 and 14.1.2
PL3. Introduction to Language Translation Section 1.7
PL4. Declarations and Types Sections 1.1.2, 1.1.3, and 2.2.5
PL5. Abstraction Mechanisms Sections 2.2.5, 5.1–5.3, 6.1.1, 6.2.1, 6.3,
7
.1, 7.3.1, 8.1, 9.1, 9.5, 11.4, and 13.1.1
PL6. Object-Oriented Programming Chapters 1 and 2 and Sections 6.2.1,

7
.3.7, 8.1.2, and 13.3.1
PF1. Fundamental Programming Constructs Chapters 1 and 2
PF2. Algorithms and Problem-Solving Sections 1.7 and 4.2
PF3. Fundamental Data Structures Sections 3.1, 3.2, 5.1–5.3, 6.1–6.3, 7.1,
7
.3, 8.1, 8.3, 9.1–9.4, 10.1, and 13.1.1
PF4. Recursion Section 3.5
SE1. Software Design Chapter 2 and Sections 6.2.1, 7.3.7,
8
.1.2, and 13.3.1
SE2. Using APIs Sections 2.2.5, 5.1–5.3, 6.1.1, 6.2.1, 6.3,
7
.1, 7.3.1, 8.1, 9.1, 9.5, 11.4, and 13.1.1
AL1. Basic Algorithmic Analysis Chapter 4
AL2. Algorithmic Strategies Sections 11.1.1, 11.5.1, 12.2, 12.3.1, and
1
2.4.2
AL3. Fundamental Computing Algorithms Sections 8.1.5, 8.2.2, 8.3.5, 9.2, and
9
.3.1, and Chapters 11, 12, and 13
DS1. Functions, Relations, and Sets Sections 4.1, 8.1, and 11.4
DS3. Proof Techniques Sections 4.3, 6.1.3, 7.3.3, 8.3, 10.2–10.5,
1
1.2.1, 11.3.1, 11.4.3, 13.1.1, 13.3.1,
13.4, and 13.5
DS4. Basics of Counting Sections 2.2.3 and 11.1.5
DS5. Graphs and Trees Chapters 7, 8, 10, and 13
DS6. Discrete Probability Appendix A and Sections 9.2, 9.4.2,
1

1.2.1, and 11.5
Table 0.1: Material for units in the IEEE/ACM 2001 Computing Curriculum.
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xi
Contents and Organization
T
he chapters for this course are organized to provide a pedagog
ical path that starts
with the basics of C++ programming and object-oriented design. We provide an
early discussion of concrete structures, like arrays and linked lists, in order to pro-
vide a concrete footing to build upon when constructing other data structures. We
then add foundational techniques like recursion and algorithm analysis, and, in the
main portion of the book, we present fundamental data structures and algorithms,
concluding with a discussion of memory management (that is, the architectural
underpinnings of data structures). Specifically, the chapters for this book are orga-
nized as follows:
1. A C++ Primer
2. Object-Oriented Design
3. Arrays, Linked Lists, and Recursion
4. Analysis Tools
5. Stacks, Queues, and Deques

6. List and Iterator ADTs
7. Trees
8. Heaps and Priority Queues
9. Hash Tables, Maps, and Skip Lists
10. Search Trees
11. Sorting, Sets, and Selection
12. Strings and Dynamic Programming
13. Graph Algorithms
14. Memory Management and B-Trees
A. Useful Mathematical Facts
A more detailed listing of the contents of this book can be found
in the table of
contents.
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xii
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Prerequisites
W
e have written this book assuming that the reader comes to it with certain knowl-
edge. We assume that the reader is at least vaguely familiar with a high-level pro-
gramming language, such as C, C++, Python, or Java, and that he or she understands
the main constructs from such a high-level language, including:

• Variables and expressions.
• Functions (also known as methods or procedures).
• Decision structures (such as if-statements and switch-state
ments).
• Iteration structures (for-loops and while-loops).
For readers who are familiar with these concepts, but not with how they are ex-
pressed in C++, we provide a primer on the C++ language in Chapter 1. Still, this
book is primarily a data structures book, not a C++ book; hence, it does not provide
a comprehensive treatment of C++. Nevertheless, we do not assume that the reader
is necessarily familiar with object-oriented design or with linked structures, such
as linked lists, since these topics are covered in the core chapters of this book.
In terms of mathematical background, we assume the reader is somewhat famil-
iar with topics from high-school mathematics. Even so, in Chapter 4, we discuss
the seven most-important functions for algorithm analysis. In fact, sections that use
something other than one of these seven functions are considered optional, and are
indicated with a star (⋆). We give a summary of other useful mathematical facts,
including elementary probability, in Appendix A.
About the Authors
P
rofessors Goodrich, Tamassia, and Mount are well-recognized researchers in al-
gorithms and data structures, having published many papers in this field, with ap-
plications to Internet computing, information visualization, computer security, and
geometric computing. They have served as principal investigators in several joint
projects sponsored by the National Science Foundation, the Army Research Of-
fice, the Office of Naval Research, and the Defense Advanced Research Projects
Agency. They are also active in educational technology research.
Michael Goodrich received his Ph.D. in Computer Science from Purdue Uni-
versity in 1987. He is currently a Chancellor’s Professor in the Department of Com-
puter Science at University of California, Irvine. Previously, he was a professor at
Johns Hopkins University. He is an editor for a number of journals in computer

science theory, computational geometry, and graph algorithms. He is an ACM Dis-
tinguished Scientist, a Fellow of the American Association for the Advancement of
Science (AAAS), a Fulbright Scholar, and a Fellow of the IEEE. He is a recipient of
the IEEE Computer Society Technical Achievement Award, the ACM Recognition
of Service Award, and the Pond Award for Excellence in Undergraduate Teaching.
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xiii
Roberto Tamassia received his Ph.D. in Electrical and Computer Engineering
from the University of Illinois at Urbana-Champaign in 1988. He is the Plastech
Professor of Computer Science and the Chair of the Department of Computer Sci-
ence at Brown University. He is also the Director of Brown’s Center for Geometric
Computing. His research interests include information security, cryptography, anal-
ysis, design, and implementation of algorithms, graph drawing, and computational
geometry. He is an IEEE Fellow and a recipient of the Technical Achievement
Award from the IEEE Computer Society for pioneering the field of graph drawing.
He is an editor of several journals in geometric and graph algorithms. He previously
served on the editorial board of IEEE Transactions on Computers.
David Mount received his Ph.D. in Computer Science from Purdue University
in 1983. He is currently a professor in the Department of Computer Science at
the University of Maryland with a joint appointment in the University of Mary-
land’s Institute for Advanced Computer Studies. He is an associate editor for ACM

Transactions on Mathematical Software and the International Journal of Compu-
tational Geometry and Applications. He is the recipient of two ACM Recognition
of Service Awards.
In addition to their research accomplishments, the authors also have extensive
experience in the classroom. For example, Dr. Goodrich has taught data structures
and algorithms courses, including Data Structures as a freshman-sophomore level
course and Introduction to Algorithms as an upper-level course. He has earned sev-
eral teaching awards in this capacity. His teaching style is to involve the students in
lively interactive classroom sessions that bring out the intuition and insights behind
data structuring and algorithmic techniques. Dr. Tamassia has taught Data Struc-
tures and Algorithms as an introductory freshman-level course since 1988. One
thing that has set his teaching style apart is his effective use of interactive hyper-
media presentations integrated with the Web. Dr. Mount has taught both the Data
Structures and the Algorithms courses at the University of Maryland since 1985.
He has won a number of teaching awards from Purdue University, the University of
Maryland, and the Hong Kong University of Science and Technology. His lecture
notes and homework exercises for the courses that he has taught are widely used as
supplementary learning material by students and instructors at other universities.
Acknowledgments
T
here are a number of individuals who have made contributions to this book.
We are grateful to all our research collaborators and teaching assistants, who
provided feedback on early drafts of chapters and have helped us in developing
exercises, software, and algorithm animation systems. There have been a number of
friends and colleagues whose comments have lead to improvements in the text. We
are particularly thankful to Michael Goldwasser for his many valuable suggestions.
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We are also grateful to Karen Goodrich, Art Moorshead, Scott Smith, and Ioannis
Tollis for their insightful comments.
We are also truly indebted to the outside reviewers and readers for their co-
pious comments, emails, and constructive criticism, which were extremely use-
ful in writing this edition. We specifically thank the following reviewers for their
comments and suggestions: Divy Agarwal, University of California, Santa Bar-
bara; Terry Andres, University of Manitoba; Bobby Blumofe, University of Texas,
Austin; Michael Clancy, University of California, Berkeley; Larry Davis, Univer-
sity of Maryland; Scott Drysdale, Dartmouth College; Arup Guha, University of
Central Florida; Chris Ingram, University of Waterloo; Stan Kwasny, Washington
University; Calvin Lin, University of Texas at Austin; John Mark Mercer, McGill
University; Laurent Michel, University of Connecticut; Leonard Myers, California
Polytechnic State University, San Luis Obispo; David Naumann, Stevens Institute
of Technology; Robert Pastel, Michigan Technological University; Bina Rama-
murthy, SUNY Buffalo; Ken Slonneger, University of Iowa; C.V. Ravishankar,
University of Michigan; Val Tannen, University of Pennsylvania; Paul Van Ar-
ragon, Messiah College; and Christopher Wilson, University of Oregon.
We are grateful to our editor, Beth Golub, for her enthusiastic support of this
project. The team at Wiley has been great. Many thanks go to Mike Berlin, Lil-
ian Brady, Regina Brooks, Paul Crockett, Richard DeLorenzo, Jen Devine, Simon
Durkin, Micheline Frederick, Lisa Gee, Katherine Hepburn, Rachael Leblond, An-
dre Legaspi, Madelyn Lesure, Frank Lyman, Hope Miller, Bridget Morrisey, Chris
Ruel, Ken Santor, Lauren Sapira, Dan Sayre, Diana Smith, Bruce Spatz, Dawn

Stanley, Jeri Warner, and Bill Zobrist.
The computing systems and excellent technical support staff in the departments
of computer science at Brown University, University of California, Irvine, and Uni-
versity of Maryland gave us reliable working environments. This manuscript was
prepared primarily with the L
A
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E
X typesetting package.
F
inally, we would like to warmly thank Isabel Cruz, Karen Goodrich, Jeanine
Mount, Giuseppe Di Battista, Franco Preparata, Ioannis Tollis, and our parents for
providing advice, encouragement, and support at various stages of the preparation
of this book. We also thank them for reminding us that there are things in life
beyond writing books.
Michael T. Goodrich
Roberto Tamassia
David M. Mount
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Contents
1
A C++ Primer 1

1.1 Basic C++ Programming Elements . . . . . . . . . . . . . . . 2
1.1.1 A Simple C++ Program . . . . . . . . . . . . . . . . . . 2
1.1.2 Fundamental Types . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Pointers, Arrays, and Structures . . . . . . . . . . . . . 7
1.1.4 Named Constants, Scope, and Namespaces . . . . . . . 13
1.2 Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.1 Changing Types through Casting . . . . . . . . . . . . . 20
1.3 Control Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4.1 Argument Passing . . . . . . . . . . . . . . . . . . . . . 28
1.4.2 Overloading and Inlining . . . . . . . . . . . . . . . . . 30
1.5 Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.5.1 Class Structure . . . . . . . . . . . . . . . . . . . . . . 33
1.5.2 Constructors and Destructors . . . . . . . . . . . . . . . 37
1.5.3 Classes and Memory Allocation . . . . . . . . . . . . . . 40
1.5.4 Class Friends and Class Members . . . . . . . . . . . . . 43
1.5.5 The Standard Template Library . . . . . . . . . . . . . . 45
1.6 C++ Program and File Organization . . . . . . . . . . . . . . 47
1.6.1 An Example Program . . . . . . . . . . . . . . . . . . . 48
1.7 Writing a C++ Program . . . . . . . . . . . . . . . . . . . . . 53
1.7.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.7.2 Pseudo-Code . . . . . . . . . . . . . . . . . . . . . . . 54
1.7.3 Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.7.4 Testing and Debugging . . . . . . . . . . . . . . . . . . 57
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2 Object-Oriented Design 65
2.1 Goals, Principles, and Patterns . .
. . . . . . . . . . . . . . 66
2.1.1 Object-Oriented Design Goals . . . . . . . . . . . . . . 66
2.1.2 Object-Oriented Design Principles . . . . . . . . . . . . 67

2.1.3 Design Patterns . . . . . . . . . . . . . . . . . . . . . . 70
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Contents
2.2 Inheritance and Polymorphism . . . . . . . . . . . . . . . . . 71
2.2.1 Inheritance in C++ . . . . . . . . . . . . . . . . . . . . . 71
2.2.2 Polymorphism . . . . . . . . . . . . . . . . . . . . . . . 78
2.2.3 Examples of Inheritance in C++ . . . . . . . . . . . . . . 79
2.2.4 Multiple Inheritance a nd Class Casting . . . . . . . . . . 84
2.2.5 Interfaces and Abstract Classes . . . . . . . . . . . . . . 87
2.3 Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.3.1 Function Templates . . . . . . . . . . . . . . . . . . . . 90
2.3.2 Class Templates . . . . . . . . . . . . . . . . . . . . . . 91
2.4 Exceptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.4.1 Exception Objects . . . . . . . . . . . . . . . . . . . . . 93
2.4.2 Throwing and Catching Exceptions . . . . . . . . . . . . 94
2.4.3 Exception Specification . . . . . . . . . . . . . . . . . . 96
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3 Arrays, Linked Lists, and Recursion 103
3.1 Using Arrays . .
. . . . . . . . . . . . . . . . . . . . . . . . . 104

3.1.1 Storing Game Entries i n an Array . . . . . . . . . . . . . 104
3.1.2 Sorting an Array . . . . . . . . . . . . . . . . . . . . . . 109
3.1.3 Two-Dimensional Arrays and Positional Games . . . . . 111
3.2 Singly Linked Lists . . . . . . . . . . . . . . . . . . . . . . . . 117
3.2.1 Implementing a S ingly Linked List . . . . . . . . . . . . 117
3.2.2 Insertion to the Front of a Singly Linked Lis t . . . . . . 119
3.2.3 Removal from the Front of a Singly Linked List . . . . . 119
3.2.4 Implementing a Generic Singly Linked List . . . . . . . . 121
3.3 Doubly Linked Lists . . . . . . . . . . . . . . . . . . . . . . . 123
3.3.1 Insertion into a Doubly Linked List . . . . . . . . . . . . 123
3.3.2 Removal from a Doubly Linked List . . . . . . . . . . . 124
3.3.3 A C++ Implementation . . . . . . . . . . . . . . . . . . 125
3.4 Circularly Linked Lists a nd List Reversal . . . . . . . . . . . 129
3.4.1 Circularly Linked Lists . . . . . . . . . . . . . . . . . . . 129
3.4.2 Reversing a Linked Lis t . . . . . . . . . . . . . . . . . . 133
3.5 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.5.1 Linear Recursion . . . . . . . . . . . . . . . . . . . . . . 140
3.5.2 Binary Recursion . . . . . . . . . . . . . . . . . . . . . 144
3.5.3 Multiple Recursion . . . . . . . . . . . . . . . . . . . . 147
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4 Analysis Tools 153
4.1 The Seven Functions Used in This Book . .
. . . . . . . . . 154
4.1.1 The Constant Function . . . . . . . . . . . . . . . . . . 154
4.1.2 The Logarithm Function . . . . . . . . . . . . . . . . . 154
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4.1.3 The Linear Function . . . . . . . . . . . . . . . . . . . . 156
4.1.4 The N-Log-N Function . . . . . . . . . . . . . . . . . . 156
4.1.5 The Quadratic Function . . . . . . . . . . . . . . . . . . 156
4.1.6 The Cubic Function and Other Polynomials . . . . . . . 158
4.1.7 The Exponential Function . . . . . . . . . . . . . . . . . 159
4.1.8 Comparing Growth Rates . . . . . . . . . . . . . . . . . 161
4.2 Analysis of Algorithms . . . . . . . . . . . . . . . . . . . . . 162
4.2.1 Experimental Studies . . . . . . . . . . . . . . . . . . . 163
4.2.2 Primitive Operations . . . . . . . . . . . . . . . . . . . 164
4.2.3 Asymptotic Notation . . . . . . . . . . . . . . . . . . . 166
4.2.4 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . 170
4.2.5 Using the B ig-Oh Notation . . . . . . . . . . . . . . . . 172
4.2.6 A Recursive Al gorithm for Computing Powers . . . . . . 176
4.2.7 Some More Examples of Algorithm Analysis . . . . . . . 177
4.3 Simple Justification Techniques . . . . . . . . . . . . . . . . 181
4.3.1 By Example . . . . . . . . . . . . . . . . . . . . . . . . 181
4.3.2 The “Contra” Attack . . . . . . . . . . . . . . . . . . . 181
4.3.3 Induction and Loop Invariants . . . . . . . . . . . . . . 182
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5 Stacks, Queues, and Deques 193
5.1 Stacks . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.1.1 The Stack Abstract Data Type . . . . . . . . . . . . . . 195
5.1.2 The STL St ack . . . . . . . . . . . . . . . . . . . . . . 196

5.1.3 A C++ Stack Interface . . . . . . . . . . . . . . . . . . 196
5.1.4 A Simple Array-Based Stack Implementation . . . . . . 198
5.1.5 Implementing a S tack with a Generic Linked List . . . . 202
5.1.6 Reversing a Vector Using a Stack . . . . . . . . . . . . . 203
5.1.7 Matching Parentheses and HTML Tags . . . . . . . . . 204
5.2 Queues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
5.2.1 The Queue Abs tract Data Ty pe . . . . . . . . . . . . . 208
5.2.2 The STL Queue . . . . . . . . . . . . . . . . . . . . . . 209
5.2.3 A C++ Queue Interface . . . . . . . . . . . . . . . . . . 210
5.2.4 A Simple Array-Based Implementation . . . . . . . . . . 211
5.2.5 Implementing a Queue with a Circularly Linked List . . . 213
5.3 Double-Ended Queues . . . . . . . . . . . . . . . . . . . . . . 217
5.3.1 The Deque Abstract Data Type . . . . . . . . . . . . . 217
5.3.2 The STL Deque . . . . . . . . . . . . . . . . . . . . . . 218
5.3.3 Implementing a Deque with a Doubly Linked List . . . . 218
5.3.4 Adapters and the Adapter Design Pattern . . . . . . . . 220
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
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6 List and Iterator ADTs 227
6.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

6.1.1 The Vector Abstract Data Type . . . . . . . . . . . . . 228
6.1.2 A Simple Array-Based Implementation . . . . . . . . . . 229
6.1.3 An Extendable Array Implementation . . . . . . . . . . . 231
6.1.4 STL Vectors . . . . . . . . . . . . . . . . . . . . . . . . 236
6.2 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
6.2.1 Node-Based Operations and Iterators . . . . . . . . . . . 238
6.2.2 The List Abs tract Data Ty pe . . . . . . . . . . . . . . . 240
6.2.3 Doubly Linked List Implementation . . . . . . . . . . . . 242
6.2.4 STL Lists . . . . . . . . . . . . . . . . . . . . . . . . . 247
6.2.5 STL Containers and Iterators . . . . . . . . . . . . . . . 248
6.3 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
6.3.1 The Sequence Abstra ct Data Type . . . . . . . . . . . . 255
6.3.2 Implementing a S equence with a Doubly Linked Lis t . . 255
6.3.3 Implementing a S equence with an Array . . . . . . . . . 257
6.4 Case Study: Bubble-Sort on a Sequence . . . . . . . . . . . 259
6.4.1 The Bubble-Sort Algorithm . . . . . . . . . . . . . . . . 259
6.4.2 A Sequence-Based Analysis of Bubble-Sort . . . . . . . . 260
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
7 Trees 267
7.1 General Trees . .
. . . . . . . . . . . . . . . . . . . . . . . . . 268
7.1.1 Tree Definitions and Properties . . . . . . . . . . . . . . 269
7.1.2 Tree Functions . . . . . . . . . . . . . . . . . . . . . . . 272
7.1.3 A C++ Tree Interface . . . . . . . . . . . . . . . . . . . 273
7.1.4 A Linked Structure for General Trees . . . . . . . . . . . 274
7.2 Tree Traversal Algorithms . . . . . . . . . . . . . . . . . . . 275
7.2.1 Depth and Height . . . . . . . . . . . . . . . . . . . . . 275
7.2.2 Preorder Trav ersal . . . . . . . . . . . . . . . . . . . . . 278
7.2.3 Postorder Traversal . . . . . . . . . . . . . . . . . . . . 281
7.3 Binary Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

7.3.1 The Binary Tree ADT . . . . . . . . . . . . . . . . . . . 285
7.3.2 A C++ Binary Tree Interface . . . . . . . . . . . . . . . 286
7.3.3 Properties of Binary Trees . . . . . . . . . . . . . . . . 287
7.3.4 A Linked Structure for Binary Trees . . . . . . . . . . . 289
7.3.5 A Vector-Based Structure for Binary Trees . . . . . . . . 295
7.3.6 Traversals of a Binary Tree . . . . . . . . . . . . . . . . 297
7.3.7 The Template Function Pattern . . . . . . . . . . . . . 303
7.3.8 Representing General Trees with Binary Trees . . . . . . 309
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
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8 Heaps and Priority Queues 321
8.1 The Priority Queue Abstract Data Type . . . . . . . . . . . 322
8.1.1 Keys, Priorities, and Total Order Relations . . . . . . . . 322
8.1.2 Comparators . . . . . . . . . . . . . . . . . . . . . . . . 324
8.1.3 The Priority Queue ADT . . . . . . . . . . . . . . . . . 327
8.1.4 A C++ Priority Queue Interface . . . . . . . . . . . . . . 328
8.1.5 Sorting with a Priority Queue . . . . . . . . . . . . . . . 329
8.1.6 The STL priority
queue Class . . . . . . . . . . . . . . . 3
30

8.2 Implementing a Priority Queue with a List . . . . . . . . . . 331
8.2.1 A C++ Priority Queue Implementation using a List . . . 333
8.2.2 Selection-Sort and Insertion-Sort . . . . . . . . . . . . . 335
8.3 Heaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
8.3.1 The Heap D ata Structure . . . . . . . . . . . . . . . . . 337
8.3.2 Complete Binary Trees and Their Representation . . . . 340
8.3.3 Implementing a Priority Queue with a Heap . . . . . . . 344
8.3.4 C++ Implementation . . . . . . . . . . . . . . . . . . . 349
8.3.5 Heap-Sort . . . . . . . . . . . . . . . . . . . . . . . . . 351
8.3.6 Bottom-Up Heap Construction ⋆ . . . . . . . . . . . . . 353
8.4 Adaptable Priority Queues . . . . . . . . . . . . . . . . . . . 357
8.4.1 A List-Based Implementation . . . . . . . . . . . . . . . 358
8.4.2 Location-Aware Entries . . . . . . . . . . . . . . . . . . 360
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
9 Hash Tables, Maps, and Skip Lists 367
9.1 Maps . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
9.1.1 The Map ADT . . . . . . . . . . . . . . . . . . . . . . 369
9.1.2 A C++ Map Interface . . . . . . . . . . . . . . . . . . . 371
9.1.3 The STL map Class . . . . . . . . . . . . . . . . . . . . 372
9.1.4 A Simple List-Based Map Implementation . . . . . . . . 374
9.2 Hash Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
9.2.1 Bucket Arrays . . . . . . . . . . . . . . . . . . . . . . . 375
9.2.2 Hash Functions . . . . . . . . . . . . . . . . . . . . . . 376
9.2.3 Hash Codes . . . . . . . . . . . . . . . . . . . . . . . . 376
9.2.4 Compression Functions . . . . . . . . . . . . . . . . . . 380
9.2.5 Collision-Handling Schemes . . . . . . . . . . . . . . . . 382
9.2.6 Load Factors and Rehashing . . . . . . . . . . . . . . . 386
9.2.7 A C++ Hash Table Implementation . . . . . . . . . . . . 387
9.3 Ordered Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 394

9.3.1 Ordered Search Tables and Binary Search . . . . . . . . 395
9.3.2 Two Applications of Ordered Maps . . . . . . . . . . . . 399
9.4 Skip Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
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9.4.1 Search and Update Operations in a Skip List . . . . . . 404
9.4.2 A Probabilistic Analysis of Skip Lists ⋆ . . . . . . . . . 408
9.5 Dictionaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
9.5.1 The Dictionary ADT . . . . . . . . . . . . . . . . . . . 411
9.5.2 A C++ Dictionary Implementation . . . . . . . . . . . . 413
9.5.3 Implementations with Location-Aware Entries . . . . . . 415
9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
10 Search Trees 423
10.1 Binary Search Trees . .
. . . . . . . . . . . . . . . . . . . . . 424
10.1.1 Searching . . . . . . . . . . . . . . . . . . . . . . . . . 426
10.1.2 Update Operations . . . . . . . . . . . . . . . . . . . . 428
10.1.3 C++ Implementation of a Binary Search Tree . . . . . . 432
10.2 AVL Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
10.2.1 Update Operations . . . . . . . . . . . . . . . . . . . . 440
10.2.2 C++ Implementation of an AVL Tree . . . . . . . . . . . 446

10.3 Splay Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
10.3.1 Splaying . . . . . . . . . . . . . . . . . . . . . . . . . . 450
10.3.2 When to Splay . . . . . . . . . . . . . . . . . . . . . . . 454
10.3.3 Amortized Analysis of Splaying ⋆ . . . . . . . . . . . . 456
10.4 (2,4) Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
10.4.1 Multi-Way Search Trees . . . . . . . . . . . . . . . . . . 461
10.4.2 Update Operations for (2,4) Trees . . . . . . . . . . . . 467
10.5 Red-Black Trees . . . . . . . . . . . . . . . . . . . . . . . . . 473
10.5.1 Update Operations . . . . . . . . . . . . . . . . . . . . 475
10.5.2 C++ Implementation of a Red-Black Tree . . . . . . . . 488
10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
11 Sorting, Sets, and Selection 499
11.1 Merge-Sort . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 500
11.1.1 Divide-and-Conquer . . . . . . . . . . . . . . . . . . . . 500
11.1.2 Merging Arrays and Lists . . . . . . . . . . . . . . . . . 505
11.1.3 The Running Time of Merge-Sort . . . . . . . . . . . . 508
11.1.4 C++ Implementations of Merge-Sort . . . . . . . . . . . 509
11.1.5 Merge-Sort and Recurrence Equations ⋆ . . . . . . . . . 511
11.2 Quick-Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
11.2.1 Randomized Quick-Sort . . . . . . . . . . . . . . . . . . 521
11.2.2 C++ Implementations and O ptimizations . . . . . . . . . 523
11.3 St udying Sorting through an Algorithmic Lens . . . . . . . 526
11.3.1 A Lower Bound for Sorting . . . . . . . . . . . . . . . . 526
11.3.2 Linear-Time Sorting: Bucket-Sort and Radix-Sort . . . . 528
11.3.3 Comparing Sorting Algorithms . . . . . . . . . . . . . . 531
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11.4 Sets and Union/Find Structures . . . . . . . . . . . . . . . . 533
11.4.1 The Set A DT . . . . . . . . . . . . . . . . . . . . . . . 533
11.4.2 Mergable Sets and the Template Method Pattern . . . . 534
11.4.3 Partitions with Union-Find Operations . . . . . . . . . . 538
11.5 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
11.5.1 Prune-and-Search . . . . . . . . . . . . . . . . . . . . . 542
11.5.2 Randomized Quick-Select . . . . . . . . . . . . . . . . . 543
11.5.3 Analyzing Randomized Quick-Select . . . . . . . . . . . 544
11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
12 Strings and Dynamic Programming 553
12.1 St ring Operations . .
. . . . . . . . . . . . . . . . . . . . . . 554
12.1.1 The STL String Class . . . . . . . . . . . . . . . . . . . 555
12.2 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . 557
12.2.1 Matrix Chain-Product . . . . . . . . . . . . . . . . . . . 557
12.2.2 DNA and Text Sequence Alignment . . . . . . . . . . . 560
12.3 Pattern Matching Algorithms . . . . . . . . . . . . . . . . . 564
12.3.1 Brute Force . . . . . . . . . . . . . . . . . . . . . . . . 564
12.3.2 The Boyer-Moore Algorithm . . . . . . . . . . . . . . . 566
12.3.3 The Knuth-Morris-Pratt Algorithm . . . . . . . . . . . . 570
12.4 Text Compression and the Greedy Method . . . . . . . . . 575
12.4.1 The Huffman-Coding Algorithm . . . . . . . . . . . . . 576
12.4.2 The Greedy Method . . . . . . . . . . . . . . . . . . . . 577

12.5 Tries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
12.5.1 Standard Tries . . . . . . . . . . . . . . . . . . . . . . . 578
12.5.2 Compressed Tri es . . . . . . . . . . . . . . . . . . . . . 582
12.5.3 Suffix Tries . . . . . . . . . . . . . . . . . . . . . . . . 584
12.5.4 Search Engines . . . . . . . . . . . . . . . . . . . . . . 586
12.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
13 Graph Algorithms 593
13.1 Graphs . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
13.1.1 The Graph ADT . . . . . . . . . . . . . . . . . . . . . . 599
13.2 Data Structures for Graphs . . . . . . . . . . . . . . . . . . . 600
13.2.1 The Edge List Structure . . . . . . . . . . . . . . . . . . 600
13.2.2 The Adjacency List Structure . . . . . . . . . . . . . . . 603
13.2.3 The Adjacency Matrix Structure . . . . . . . . . . . . . 605
13.3 Graph Traversals . . . . . . . . . . . . . . . . . . . . . . . . . 607
13.3.1 Depth-First Search . . . . . . . . . . . . . . . . . . . . 607
13.3.2 Implementing Depth-First Search . . . . . . . . . . . . . 611
13.3.3 A Generic DFS Implementation in C++ . . . . . . . . . . 613
13.3.4 Polymorphic Objects and Decorator Values ⋆ . . . . . . 62
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13.3.5 Breadth-First Search . . . . . . . . . . . . . . . . . . . 623
13.4 Directed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 626
13.4.1 Traversing a Digraph . . . . . . . . . . . . . . . . . . . 628
13.4.2 Transitive Closure . . . . . . . . . . . . . . . . . . . . . 630
13.4.3 Directed Acyclic Graphs . . . . . . . . . . . . . . . . . . 633
13.5 Shortest Paths . . . . . . . . . . . . . . . . . . . . . . . . . . 637
13.5.1 Weighted Graphs . . . . . . . . . . . . . . . . . . . . . 637
13.5.2 Dijkstra’s Algorithm . . . . . . . . . . . . . . . . . . . . 639
13.6 Minimum Spanning Trees . . . . . . . . . . . . . . . . . . . . 645
13.6.1 Kruskal’s Algorithm . . . . . . . . . . . . . . . . . . . . 647
13.6.2 The Prim-Jarn´ık Algorithm . . . . . . . . . . . . . . . . 651
13.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654
14 Memory M anagement and B-Trees 665
14.1 Memory Management . .
. . . . . . . . . . . . . . . . . . . . 666
14.1.1 Memory Allocation in C++ . . . . . . . . . . . . . . . . 669
14.1.2 Garbage Collection . . . . . . . . . . . . . . . . . . . . 671
14.2 External Memory and C aching . . . . . . . . . . . . . . . . . 673
14.2.1 The Memory Hierarchy . . . . . . . . . . . . . . . . . . 673
14.2.2 Caching Strat egies . . . . . . . . . . . . . . . . . . . . 674
14.3 External Searching and B-Trees . . . . . . . . . . . . . . . . 679
14.3.1 (a,b) Trees . . . . . . . . . . . . . . . . . . . . . . . . 680
14.3.2 B-Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 682
14.4 External-Memory Sorting . . . . . . . . . . . . . . . . . . . . 683
14.4.1 Multi-Way Merging . . . . . . . . . . . . . . . . . . . . 684
14.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
A Useful Mathematical Facts 689
Bi
bliography 697

Index 702
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Chapter
1
A C++ P
rimer
Contents
1.1 Basic C
++ Programming Elements . . . . . . . . . . . 2
1.1.1 A Simple C++ Program . . . . . . . . . . . . . . . . 2
1.1.2 Fundamental Typ e s . . . . . . . . . . . . . . . . . . 4
1.1.3 Pointers, Arrays, and Structures . . . . . . . . . . . 7
1.1.4 Named Constants, Scope, and Namespaces . . . . . 13
1.2 Expressions . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.1 Changing Types through Casting . . . . . . . . . . . 20
1.3 Control Flow . . . . . . . . . . . . . . . . . . . . . . . 23
1.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4.1 Argument Passing . . . . . . . . . . . . . . . . . . . 28
1.4.2 Overloadin g and Inlining . . . . . . . . . . . . . . . 30
1.5 Classes . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.5.1 Class Structure . . . . . . . . . . . . . . . . . . . . 33
1.5.2 Con structors and Destructors . . . . . . . . . . . . . 37

1.5.3 Class e s and Memory Allocation . . . . . . . . . . . . 40
1.5.4 Class Friends and Class Memb e rs . . . . . . . . . . . 43
1.5.5 T h e Standard Template Library . . . . . . . . . . . . 45
1.6 C++ Program and File Organization . . . . . . . . . . 47
1.6.1 An Example Program . . . . . . . . . . . . . . . . . 48
1.7 Writing a C++ Program . . . . . . . . . . . . . . . . . 53
1.7.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.7.2 Pseudo-Code . . . . . . . . . . . . . . . . . . . . . 54
1.7.3 Codin g . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.7.4 Testing and Debugging . . . . . . . . . . . . . . . . 57
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 60