Java Primer: Data Types, Operators, and Input/Output, Study Guides, Projects, Research of Data Structures and Algorithms

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ISBN: 978-1-118-77133-4 (paperback)

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

To Karen, Paul, Anna, and Jack

• Michael T. Goodrich

To Isabel

• Roberto Tamassia

To Susan, Calista, and Maya

• Michael H. Goldwasser

Preface to the Sixth Edition

Data Structures and Algorithms in Java provides an introduction to data structures

and algorithms, including their design, analysis, and implementation. The major

changes in this sixth edition include the following:

• We redesigned the entire code base to increase clarity of presentation and

consistency in style and convention, including reliance on type inference , as

introduced in Java 7, to reduce clutter when instantiating generic types.

• We added 38 new figures, and redesigned 144 existing figures.
• We revised and expanded exercises, bringing the grand total to 794 exercises!

We continue our approach of dividing them into reinforcement, creativity,

and project exercises. However, we have chosen not to reset the number-

ing scheme with each new category, thereby avoiding possible ambiguity

between exercises such as R-7.5, C-7.5, P-7.5.

• The introductory chapters contain additional examples of classes and inheri-

tance, increased discussion of Java’s generics framework, and expanded cov-

erage of cloning and equivalence testing in the context of data structures.

• A new chapter, dedicated to the topic of recursion, provides comprehensive

coverage of material that was previously divided within Chapters 3, 4, and

9 of the fifth edition, while newly introducing the use of recursion when

processing file systems.

• We provide a new empirical study of the efficiency of Java’s StringBuilder

class relative to the repeated concatenation of strings, and then discuss the

theoretical underpinnings of its amortized performance.

• We provide increased discussion of iterators, contrasting between so-called

lazy iterators and snapshot iterators , with examples of both styles of imple-

mentation for several data structures.

• We have increased the use of abstract base classes to reduce redundancy

when providing multiple implementations of a common interface, and the

use of nested classes to provide greater encapsulation for our data structures.

• We have included complete Java implementations for many data structures

and algorithms that were only described with pseudocode in earlier editions.

These new implementations include both array-based and linked-list-based

queue implementations, a heap-based adaptable priority queue, a bottom-up

heap construction, hash tables with either separate chaining or linear probing,

splay trees, dynamic programming for the least-common subsequence prob-

lem, a union-find data structure with path compression, breadth-first search

of a graph, the Floyd-Warshall algorithm for computing a graph’s transitive

closure, topological sorting of a DAG, and both the Prim-Jarn´ık and Kruskal

algorithms for computing a minimum spanning tree.

v

vi Preface

Prerequisites

We assume that the reader is at least vaguely familiar with a high-level program-

ming 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
• Methods (also known as functions or procedures)
• Decision structures (such as if-statements and switch-statements)
• Iteration structures (for-loops and while-loops)

For readers who are familiar with these concepts, but not with how they are ex-

pressed in Java, we provide a primer on the Java language in Chapter 1. Still, this

book is primarily a data structures book, not a Java book; hence, it does not provide

a comprehensive treatment of Java. Nevertheless, we do not assume that the reader

is necessarily familiar with object-oriented design or with linked structures, such

as linked lists, for 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 (⋆).

Online Resources

This book is accompanied by an extensive set of online resources, which can be

found at the following website:

www.wiley.com/college/goodrich

Included on this website is a collection of educational aids that augment the topics

of this book, for both students and instructors. For all readers, and especially for

students, we include the following resources:

• All Java source code presented in this book
• An appendix of useful mathematical facts
• PDF handouts of PowerPoint slides (four-per-page)
• A study guide with hints to exercises, indexed by problem number

For instructors using this book, we include the following additional teaching aids:

• Solutions to hundreds of the book’s exercises
• Color versions of all figures and illustrations from the book
• Slides in PowerPoint and PDF (one-per-page) format

The slides are fully editable, so as to allow an instructor using this book full free-

dom in customizing his or her presentations.

viii Preface

About the Authors

Michael Goodrich received his Ph.D. in Computer Science from Purdue University

in 1987. He is currently a Chancellor’s Professor in the Department of Computer

Science at University of California, Irvine. Previously, he was a professor at Johns

Hopkins University. He is a Fulbright Scholar and a Fellow of the American As-

sociation for the Advancement of Science (AAAS), Association for Computing

Machinery (ACM), and Institute of Electrical and Electronics Engineers (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.

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 a Fellow of the American Association for the Advancement of

Science (AAAS), Association for Computing Machinery (ACM) and Institute for

Electrical and Electronic Engineers (IEEE). He is a recipient of the IEEE Computer

Society Technical Achievement Award.

Michael Goldwasser received his Ph.D. in Computer Science from Stanford

University in 1997. He is currently Professor and Director of the Computer Science

program in the Department of Mathematics and Computer Science at Saint Louis

University. He was previously a faculty member in the Department of Computer

Science at Loyola University Chicago. His research interests focus on the design

and implementation of algorithms, having published work involving approximation

algorithms, online computation, computational biology, and computational geom-

etry. He is also active in the computer science education community.

Additional Books by These Authors

• Di Battista, Eades, Tamassia, and Tollis, Graph Drawing , Prentice Hall
• Goodrich, Tamassia, and Goldwasser, Data Structures and Algorithms in Python ,

Wiley

• Goodrich, Tamassia, and Mount, Data Structures and Algorithms in C++ , Wiley
• Goodrich and Tamassia, Algorithm Design: Foundations, Analysis, and Internet

Examples , Wiley

• Goodrich and Tamassia, Introduction to Computer Security , Addison-Wesley
• Goldwasser and Letscher, Object-Oriented Programming in Python , Prentice

Hall

Preface ix

Acknowledgments

There are so many individuals who have made contributions to the development of

this book over the past decade, it is difficult to name them all. We wish to reit-

erate our thanks to the many research collaborators and teaching assistants whose

feedback shaped the previous versions of this material. The benefits of those con-

tributions carry forward to this book.

For the sixth edition, we are indebted to the outside reviewers and readers for

their copious comments, emails, and constructive criticisms. We therefore thank the

following people for their comments and suggestions: Sameer O. Abufardeh (North

Dakota State University), Mary Boelk (Marquette University), Frederick Crabbe

(United States Naval Academy), Scot Drysdale (Dartmouth College), David Eisner,

Henry A. Etlinger (Rochester Institute of Technology), Chun-Hsi Huang (Univer-

sity of Connecticut), John Lasseter (Hobart and William Smith Colleges), Yupeng

Lin, Suely Oliveira (University of Iowa), Vincent van Oostrom (Utrecht Univer-

sity), Justus Piater (University of Innsbruck), Victor I. Shtern (Boston University),

Tim Soethout, and a number of additional anonymous reviewers.

There have been a number of friends and colleagues whose comments have led

to improvements in the text. We are particularly thankful to Erin Chambers, Karen

Goodrich, David Letscher, David Mount, and Ioannis Tollis for their insightful

comments. In addition, contributions by David Mount to the coverage of recursion

and to several figures are gratefully acknowledged.

We appreciate the wonderful team at Wiley, including our editor, Beth Lang

Golub, for her enthusiastic support of this project from beginning to end, and the

Product Solutions Group editors, Mary O’Sullivan and Ellen Keohane, for carrying

the project to its completion. The quality of this book is greatly enhanced as a result

of the attention to detail demonstrated by our copyeditor, Julie Kennedy. The final

months of the production process were gracefully managed by Joyce Poh.

Finally, we would like to warmly thank Karen Goodrich, Isabel Cruz, Susan

Goldwasser, Giuseppe Di Battista, Franco Preparata, Ioannis Tollis, and our parents

for providing advice, encouragement, and support at various stages of the prepa-

ration of this book, and Calista and Maya Goldwasser for offering their advice

regarding the artistic merits of many illustrations. More importantly, we thank all

of these people for reminding us that there are things in life beyond writing books.

Michael T. Goodrich

Roberto Tamassia

Michael H. Goldwasser

Chapter

Java Primer

• 1 Java Primer
  • 1.1 Getting Started
    • 1.1.1 Base Types
  • 1.2 Classes and Objects
    • 1.2.1 Creating and Using Objects
    • 1.2.2 Defining a Class
  • 1.3 Strings, Wrappers, Arrays, and Enum Types
  • 1.4 Expressions
    • 1.4.1 Literals
    • 1.4.2 Operators
    • 1.4.3 Type Conversions
  • 1.5 Control Flow
    • 1.5.1 The If and Switch Statements
    • 1.5.2 Loops
    • 1.5.3 Explicit Control-Flow Statements
  • 1.6 Simple Input and Output
  • 1.7 An Example Program
  • 1.8 Packages and Imports
  • 1.9 Software Development
    • 1.9.1 Design
    • 1.9.2 Pseudocode
    • 1.9.3 Coding
    • 1.9.4 Documentation and Style
    • 1.9.5 Testing and Debugging
  • 1.10 Exercises
• 2 Object-Oriented Design
  • 2.1 Goals, Principles, and Patterns
    • 2.1.1 Object-Oriented Design Goals
    • 2.1.2 Object-Oriented Design Principles
    • 2.1.3 Design Patterns
  • 2.2 Inheritance
    • 2.2.1 Extending the CreditCard Class
    • 2.2.2 Polymorphism and Dynamic Dispatch
    • 2.2.3 Inheritance Hierarchies
  • 2.3 Interfaces and Abstract Classes
    • 2.3.1 Interfaces in Java
    • 2.3.2 Multiple Inheritance for Interfaces
    • 2.3.3 Abstract Classes
  • 2.4 Exceptions
    • 2.4.1 Catching Exceptions
    • 2.4.2 Throwing Exceptions
    • 2.4.3 Java’s Exception Hierarchy
  • 2.5 Casting and Generics
    • 2.5.1 Casting Contents xi
    • 2.5.2 Generics
  • 2.6 Nested Classes
  • 2.7 Exercises
• 3 Fundamental Data Structures
  • 3.1 Using Arrays
    • 3.1.1 Storing Game Entries in an Array
    • 3.1.2 Sorting an Array
    • 3.1.3 java.util Methods for Arrays and Random Numbers
    • 3.1.4 Simple Cryptography with Character Arrays
    • 3.1.5 Two-Dimensional Arrays and Positional Games
  • 3.2 Singly Linked Lists
    • 3.2.1 Implementing a Singly Linked List Class
  • 3.3 Circularly Linked Lists
    • 3.3.1 Round-Robin Scheduling
    • 3.3.2 Designing and Implementing a Circularly Linked List
  • 3.4 Doubly Linked Lists
    • 3.4.1 Implementing a Doubly Linked List Class
  • 3.5 Equivalence Testing
    • 3.5.1 Equivalence Testing with Arrays
    • 3.5.2 Equivalence Testing with Linked Lists
  • 3.6 Cloning Data Structures
    • 3.6.1 Cloning Arrays
    • 3.6.2 Cloning Linked Lists
  • 3.7 Exercises
• 4 Algorithm Analysis
  • 4.1 Experimental Studies
    • 4.1.1 Moving Beyond Experimental Analysis
  • 4.2 The Seven Functions Used in This Book
    • 4.2.1 Comparing Growth Rates
  • 4.3 Asymptotic Analysis
    • 4.3.1 The “Big-Oh” Notation
    • 4.3.2 Comparative Analysis
    • 4.3.3 Examples of Algorithm Analysis
  • 4.4 Simple Justification Techniques
    • 4.4.1 By Example
    • 4.4.2 The “Contra” Attack
    • 4.4.3 Induction and Loop Invariants
  • 4.5 Exercises
• 5 Recursion
  • 5.1 Illustrative Examples
    • 5.1.1 The Factorial Function
    • 5.1.2 Drawing an English Ruler
    • 5.1.3 Binary Search
    • 5.1.4 File Systems xii Contents
  • 5.2 Analyzing Recursive Algorithms
  • 5.3 Further Examples of Recursion
    • 5.3.1 Linear Recursion
    • 5.3.2 Binary Recursion
    • 5.3.3 Multiple Recursion
  • 5.4 Designing Recursive Algorithms
  • 5.5 Recursion Run Amok
    • 5.5.1 Maximum Recursive Depth in Java
  • 5.6 Eliminating Tail Recursion
  • 5.7 Exercises
• 6 Stacks, Queues, and Deques
  • 6.1 Stacks
    • 6.1.1 The Stack Abstract Data Type
    • 6.1.2 A Simple Array-Based Stack Implementation
    • 6.1.3 Implementing a Stack with a Singly Linked List
    • 6.1.4 Reversing an Array Using a Stack
    • 6.1.5 Matching Parentheses and HTML Tags
  • 6.2 Queues
    • 6.2.1 The Queue Abstract Data Type
    • 6.2.2 Array-Based Queue Implementation
    • 6.2.3 Implementing a Queue with a Singly Linked List
    • 6.2.4 A Circular Queue
  • 6.3 Double-Ended Queues
    • 6.3.1 The Deque Abstract Data Type
    • 6.3.2 Implementing a Deque
    • 6.3.3 Deques in the Java Collections Framework
  • 6.4 Exercises
• 7 List and Iterator ADTs
  • 7.1 The List ADT
  • 7.2 Array Lists
    • 7.2.1 Dynamic Arrays
    • 7.2.2 Implementing a Dynamic Array
    • 7.2.3 Amortized Analysis of Dynamic Arrays
    • 7.2.4 Java’s StringBuilder class
  • 7.3 Positional Lists
    • 7.3.1 Positions
    • 7.3.2 The Positional List Abstract Data Type
    • 7.3.3 Doubly Linked List Implementation
  • 7.4 Iterators
    • 7.4.1 The Iterable Interface and Java’s For-Each Loop
    • 7.4.2 Implementing Iterators
  • 7.5 The Java Collections Framework
    • 7.5.1 List Iterators in Java
    • 7.5.2 Comparison to Our Positional List ADT
    • 7.5.3 List-Based Algorithms in the Java Collections Framework Contents xiii
  • 7.6 Sorting a Positional List
  • 7.7 Case Study: Maintaining Access Frequencies
    • 7.7.1 Using a Sorted List
    • 7.7.2 Using a List with the Move-to-Front Heuristic
  • 7.8 Exercises
• 8 Trees
  • 8.1 General Trees
    • 8.1.1 Tree Definitions and Properties
    • 8.1.2 The Tree Abstract Data Type
    • 8.1.3 Computing Depth and Height
  • 8.2 Binary Trees
    • 8.2.1 The Binary Tree Abstract Data Type
    • 8.2.2 Properties of Binary Trees
  • 8.3 Implementing Trees
    • 8.3.1 Linked Structure for Binary Trees
    • 8.3.2 Array-Based Representation of a Binary Tree
    • 8.3.3 Linked Structure for General Trees
  • 8.4 Tree Traversal Algorithms
    • 8.4.1 Preorder and Postorder Traversals of General Trees
    • 8.4.2 Breadth-First Tree Traversal
    • 8.4.3 Inorder Traversal of a Binary Tree
    • 8.4.4 Implementing Tree Traversals in Java
    • 8.4.5 Applications of Tree Traversals
    • 8.4.6 Euler Tours
  • 8.5 Exercises
• 9 Priority Queues
  • 9.1 The Priority Queue Abstract Data Type
    • 9.1.1 Priorities
    • 9.1.2 The Priority Queue ADT
  • 9.2 Implementing a Priority Queue
    • 9.2.1 The Entry Composite
    • 9.2.2 Comparing Keys with Total Orders
    • 9.2.3 The AbstractPriorityQueue Base Class
    • 9.2.4 Implementing a Priority Queue with an Unsorted List
    • 9.2.5 Implementing a Priority Queue with a Sorted List
  • 9.3 Heaps
    • 9.3.1 The Heap Data Structure
    • 9.3.2 Implementing a Priority Queue with a Heap
    • 9.3.3 Analysis of a Heap-Based Priority Queue
    • 9.3.4 Bottom-Up Heap Construction ⋆
    • 9.3.5 Using the java.util.PriorityQueue Class
  • 9.4 Sorting with a Priority Queue
    • 9.4.1 Selection-Sort and Insertion-Sort
    • 9.4.2 Heap-Sort
  • 9.5 Adaptable Priority Queues xiv Contents
    • 9.5.1 Location-Aware Entries
    • 9.5.2 Implementing an Adaptable Priority Queue
  • 9.6 Exercises
• 10 Maps, Hash Tables, and Skip Lists
  • 10.1 Maps
    • 10.1.1 The Map ADT
    • 10.1.2 Application: Counting Word Frequencies
    • 10.1.3 An AbstractMap Base Class
    • 10.1.4 A Simple Unsorted Map Implementation
  • 10.2 Hash Tables
    • 10.2.1 Hash Functions
    • 10.2.2 Collision-Handling Schemes
    • 10.2.3 Load Factors, Rehashing, and Efficiency
    • 10.2.4 Java Hash Table Implementation
  • 10.3 Sorted Maps
    • 10.3.1 Sorted Search Tables
    • 10.3.2 Two Applications of Sorted Maps
  • 10.4 Skip Lists
    • 10.4.1 Search and Update Operations in a Skip List
    • 10.4.2 Probabilistic Analysis of Skip Lists ⋆
  • 10.5 Sets, Multisets, and Multimaps
    • 10.5.1 The Set ADT
    • 10.5.2 The Multiset ADT
    • 10.5.3 The Multimap ADT
  • 10.6 Exercises
• 11 Search Trees
  • 11.1 Binary Search Trees
    • 11.1.1 Searching Within a Binary Search Tree
    • 11.1.2 Insertions and Deletions
    • 11.1.3 Java Implementation
    • 11.1.4 Performance of a Binary Search Tree
  • 11.2 Balanced Search Trees
    • 11.2.1 Java Framework for Balancing Search Trees
  • 11.3 AVL Trees
    • 11.3.1 Update Operations
    • 11.3.2 Java Implementation
  • 11.4 Splay Trees
    • 11.4.1 Splaying
    • 11.4.2 When to Splay
    • 11.4.3 Java Implementation
    • 11.4.4 Amortized Analysis of Splaying ⋆
  • 11.5 (2,4) Trees
    • 11.5.1 Multiway Search Trees
    • 11.5.2 (2,4)-Tree Operations
  • 11.6 Red-Black Trees Contents xv
    • 11.6.1 Red-Black Tree Operations
    • 11.6.2 Java Implementation
  • 11.7 Exercises
• 12 Sorting and Selection
  • 12.1 Merge-Sort
    • 12.1.1 Divide-and-Conquer
    • 12.1.2 Array-Based Implementation of Merge-Sort
    • 12.1.3 The Running Time of Merge-Sort
    • 12.1.4 Merge-Sort and Recurrence Equations ⋆
    • 12.1.5 Alternative Implementations of Merge-Sort
  • 12.2 Quick-Sort
    • 12.2.1 Randomized Quick-Sort
    • 12.2.2 Additional Optimizations for Quick-Sort
  • 12.3 Studying Sorting through an Algorithmic Lens
    • 12.3.1 Lower Bound for Sorting
    • 12.3.2 Linear-Time Sorting: Bucket-Sort and Radix-Sort
  • 12.4 Comparing Sorting Algorithms
  • 12.5 Selection
    • 12.5.1 Prune-and-Search
    • 12.5.2 Randomized Quick-Select
    • 12.5.3 Analyzing Randomized Quick-Select
  • 12.6 Exercises
• 13 Text Processing
  • 13.1 Abundance of Digitized Text
    • 13.1.1 Notations for Character Strings
  • 13.2 Pattern-Matching Algorithms
    • 13.2.1 Brute Force
    • 13.2.2 The Boyer-Moore Algorithm
    • 13.2.3 The Knuth-Morris-Pratt Algorithm
  • 13.3 Tries
    • 13.3.1 Standard Tries
    • 13.3.2 Compressed Tries
    • 13.3.3 Suffix Tries
    • 13.3.4 Search Engine Indexing
  • 13.4 Text Compression and the Greedy Method
    • 13.4.1 The Huffman Coding Algorithm
    • 13.4.2 The Greedy Method
  • 13.5 Dynamic Programming
    • 13.5.1 Matrix Chain-Product
    • 13.5.2 DNA and Text Sequence Alignment
  • 13.6 Exercises
• 14 Graph Algorithms xvi Contents
  • 14.1 Graphs
    • 14.1.1 The Graph ADT
  • 14.2 Data Structures for Graphs
    • 14.2.1 Edge List Structure
    • 14.2.2 Adjacency List Structure
    • 14.2.3 Adjacency Map Structure
    • 14.2.4 Adjacency Matrix Structure
    • 14.2.5 Java Implementation
  • 14.3 Graph Traversals
    • 14.3.1 Depth-First Search
    • 14.3.2 DFS Implementation and Extensions
    • 14.3.3 Breadth-First Search
  • 14.4 Transitive Closure
  • 14.5 Directed Acyclic Graphs
    • 14.5.1 Topological Ordering
  • 14.6 Shortest Paths
    • 14.6.1 Weighted Graphs
    • 14.6.2 Dijkstra’s Algorithm
  • 14.7 Minimum Spanning Trees
    • 14.7.1 Prim-Jarn´ık Algorithm
    • 14.7.2 Kruskal’s Algorithm
    • 14.7.3 Disjoint Partitions and Union-Find Structures
  • 14.8 Exercises
• 15 Memory Management and B-Trees
  • 15.1 Memory Management
    • 15.1.1 Stacks in the Java Virtual Machine
    • 15.1.2 Allocating Space in the Memory Heap
    • 15.1.3 Garbage Collection
  • 15.2 Memory Hierarchies and Caching
    • 15.2.1 Memory Systems
    • 15.2.2 Caching Strategies
  • 15.3 External Searching and B-Trees
    • 15.3.1 ( a , b ) Trees
    • 15.3.2 B-Trees
  • 15.4 External-Memory Sorting
    • 15.4.1 Multiway Merging
  • 15.5 Exercises
• Bibliography
• Index
• 1.1 Getting Started Contents
  • 1.1.1 Base Types
• 1.2 Classes and Objects
  • 1.2.1 Creating and Using Objects
  • 1.2.2 Defining a Class
• 1.3 Strings, Wrappers, Arrays, and Enum Types
• 1.4 Expressions
  • 1.4.1 Literals
  • 1.4.2 Operators
  • 1.4.3 Type Conversions
• 1.5 Control Flow
  • 1.5.1 The If and Switch Statements
  • 1.5.2 Loops
  • 1.5.3 Explicit Control-Flow Statements
• 1.6 Simple Input and Output
• 1.7 An Example Program
• 1.8 Packages and Imports
• 1.9 Software Development
  • 1.9.1 Design
  • 1.9.2 Pseudocode
  • 1.9.3 Coding
  • 1.9.4 Documentation and Style
  • 1.9.5 Testing and Debugging
• 1.10 Exercises

2 Chapter 1. Java Primer

1.1 Getting Started

Building data structures and algorithms requires that we communicate detailed in-

structions to a computer. An excellent way to perform such communication is

using a high-level computer language, such as Java. In this chapter, we provide an

overview of the Java programming language, and we continue this discussion in the

next chapter, focusing on object-oriented design principles. We assume that readers

are somewhat familiar with an existing high-level language, although not necessar-

ily Java. This book does not provide a complete description of the Java language

(there are numerous language references for that purpose), but it does introduce all

aspects of the language that are used in code fragments later in this book.

We begin our Java primer with a program that prints “Hello Universe!” on the

screen, which is shown in a dissected form in Figure 1.1.

Figure 1.1: A “Hello Universe!” program.

In Java, executable statements are placed in functions, known as methods , that

belong to class definitions. The Universe class, in our first example, is extremely

simple; its only method is a static one named main, which is the first method to be

executed when running a Java program. Any set of statements between the braces

“{” and “}” define a program block. Notice that the entire Universe class definition

is delimited by such braces, as is the body of the main method.

The name of a class, method, or variable in Java is called an identifier , which

can be any string of characters as long as it begins with a letter and consists of let-

ters, numbers, and underscore characters (where “letter” and “number” can be from

any written language defined in the Unicode character set). We list the exceptions

to this general rule for Java identifiers in Table 1.1.