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《A Practical Introduction to Data Structures and Algorithm Analy...

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《A Practical Introduction to Data Structures and Algorithm Analysis》

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A Practical Introduction to

Data Structures and Algorithm

Analysis

Edition 3.2 (C++ Version)

Clifford A. Shaffer

Department of Computer Science

Virginia Tech

Blacksburg, VA 24061

May 20, 2011

This document is made freely available for educational and other

non-commercial use.

You may make copies of this ﬁle and redistribute it without charge.

You may extract portions of this document provided that the front page,

including the title, author, and this notice are included.

Any commercial use of this document requires the written consent of the

author.

The author can be reached at shaffer@cs.vt.edu.

Further information about this text is available at

http://people.cs.vt.edu/

shaffer/Book/

Contents

Preface xiii

I Preliminaries 1

1 Data Structures and Algorithms 3

1.1 A Philosophy of Data Structures 4

1.1.1 The Need for Data Structures 4

1.1.2 Costs and Beneﬁts 6

1.2 Abstract Data Types and Data Structures 8

1.3 Design Patterns 12

1.3.1 Flyweight 13

1.3.2 Visitor 13

1.3.3 Composite 14

1.3.4 Strategy 15

1.4 Problems, Algorithms, and Programs 16

1.5 Further Reading 18

1.6 Exercises 20

2 Mathematical Preliminaries 25

2.1 Sets and Relations 25

2.2 Miscellaneous Notation 29

2.3 Logarithms 31

2.4 Summations and Recurrences 32

2.5 Recursion 36

2.6 Mathematical Proof Techniques 38

iii

iv Contents

2.6.1 Direct Proof 39

2.6.2 Proof by Contradiction 39

2.6.3 Proof by Mathematical Induction 40

2.7 Estimation 46

2.8 Further Reading 47

2.9 Exercises 48

3 Algorithm Analysis 55

3.1 Introduction 55

3.2 Best, Worst, and Average Cases 61

3.3 A Faster Computer, or a Faster Algorithm? 62

3.4 Asymptotic Analysis 65

3.4.1 Upper Bounds 65

3.4.2 Lower Bounds 67

3.4.3 Θ Notation 68

3.4.4 Simplifying Rules 69

3.4.5 Classifying Functions 70

3.5 Calculating the Running Time for a Program 71

3.6 Analyzing Problems 76

3.7 Common Misunderstandings 77

3.8 Multiple Parameters 79

3.9 Space Bounds 80

3.10 Speeding Up Your Programs 82

3.11 Empirical Analysis 85

3.12 Further Reading 86

3.13 Exercises 86

3.14 Projects 90

II Fundamental Data Structures 93

4 Lists, Stacks, and Queues 95

4.1 Lists 96

4.1.1 Array-Based List Implementation 100

4.1.2 Linked Lists 103

4.1.3 Comparison of List Implementations 112

Contents v

4.1.4 Element Implementations 114

4.1.5 Doubly Linked Lists 115

4.2 Stacks 120

4.2.1 Array-Based Stacks 121

4.2.2 Linked Stacks 124

4.2.3 Comparison of Array-Based and Linked Stacks 125

4.2.4 Implementing Recursion 125

4.3 Queues 129

4.3.1 Array-Based Queues 129

4.3.2 Linked Queues 134

4.3.3 Comparison of Array-Based and Linked Queues 134

4.4 Dictionaries 134

4.5 Further Reading 145

4.6 Exercises 145

4.7 Projects 149

5 Binary Trees 151

5.1 Deﬁnitions and Properties 151

5.1.1 The Full Binary Tree Theorem 153

5.1.2 A Binary Tree Node ADT 155

5.2 Binary Tree Traversals 155

5.3 Binary Tree Node Implementations 160

5.3.1 Pointer-Based Node Implementations 160

5.3.2 Space Requirements 166

5.3.3 Array Implementation for Complete Binary Trees 168

5.4 Binary Search Trees 168

5.5 Heaps and Priority Queues 178

5.6 Huffman Coding Trees 185

5.6.1 Building Huffman Coding Trees 186

5.6.2 Assigning and Using Huffman Codes 192

5.6.3 Search in Huffman Trees 195

5.7 Further Reading 196

5.8 Exercises 196

5.9 Projects 200

6 Non-Binary Trees 203

vi Contents

6.1 General Tree Deﬁnitions and Terminology 203

6.1.1 An ADT for General Tree Nodes 204

6.1.2 General Tree Traversals 205

6.2 The Parent Pointer Implementation 207

6.3 General Tree Implementations 213

6.3.1 List of Children 214

6.3.2 The Left-Child/Right-Sibling Implementation 215

6.3.3 Dynamic Node Implementations 215

6.3.4 Dynamic “Left-Child/Right-Sibling” Implementation 218

6.4 K-ary Trees 218

6.5 Sequential Tree Implementations 219

6.6 Further Reading 223

6.7 Exercises 223

6.8 Projects 226

III Sorting and Searching 229

7 Internal Sorting 231

7.1 Sorting Terminology and Notation 232

7.2 Three Θ(n

) Sorting Algorithms 233

7.2.1 Insertion Sort 233

7.2.2 Bubble Sort 235

7.2.3 Selection Sort 237

7.2.4 The Cost of Exchange Sorting 238

7.3 Shellsort 239

7.4 Mergesort 241

7.5 Quicksort 244

7.6 Heapsort 251

7.7 Binsort and Radix Sort 252

7.8 An Empirical Comparison of Sorting Algorithms 259

7.9 Lower Bounds for Sorting 261

7.10 Further Reading 265

7.11 Exercises 265

7.12 Projects 269

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