**Chapter 1 Programming: A General Overview 1**

1.1 What’s This Book About? 1

1.2 Mathematics Review 2

1.2.1 Exponents 3

1.2.2 Logarithms 3

1.2.3 Series 4

1.2.4 Modular Arithmetic 5

1.2.5 The *P *Word 6

1.3 A Brief Introduction to Recursion 8

1.4 C++ Classes 12

1.4.1 Basic class Syntax 12

1.4.2 Extra Constructor Syntax and Accessors 13

1.4.3 Separation of Interface and Implementation 16

1.4.4 vector and string 19

1.5 C++ Details 21

1.5.1 Pointers 21

1.5.2 Lvalues, Rvalues, and References 23

1.5.3 Parameter Passing 25

1.5.4 Return Passing 27

1.5.5 std::swap and std::move 29

1.5.6 The Big-Five: Destructor, Copy Constructor, Move Constructor, Copy Assignment operator=, Move Assignment operator= 30

1.5.7 C-style Arrays and Strings 35

1.6 Templates 36

1.6.1 Function Templates 37

1.6.2 Class Templates 38

1.6.3 Object, Comparable, and an Example 39

1.6.4 Function Objects 41

1.6.5 Separate Compilation of Class Templates 44

1.7 Using Matrices 44

1.7.1 The Data Members, Constructor, and Basic Accessors 44

1.7.2 operator[] 45

1.7.3 Big-Five 46

Summary 46

Exercises 46

References 48

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**Chapter 2 Algorithm Analysis 51**

2.1 Mathematical Background 51

2.2 Model 54

2.3 What to Analyze 54

2.4 Running-Time Calculations 57

2.4.1 A Simple Example 58

2.4.2 General Rules 58

2.4.3 Solutions for the Maximum Subsequence Sum Problem 60

2.4.4 Logarithms in the Running Time 66

2.4.5 Limitations of Worst Case Analysis 70

Summary 70

Exercises 71

References 76

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**Chapter 3 Lists, Stacks, and Queues 77**

3.1 Abstract Data Types (ADTs) 77

3.2 The List ADT 78

3.2.1 Simple Array Implementation of Lists 78

3.2.2 Simple Linked Lists 79

3.3 vector and list in the STL 80

3.3.1 Iterators 82

3.3.2 Example: Using erase on a List 83

3.3.3 const_iterators 84

3.4 Implementation of vector 86

3.5 Implementation of list 91

3.6 The Stack ADT 103

3.6.1 Stack Model 103

3.6.2 Implementation of Stacks 104

3.6.3 Applications 104

3.7 The Queue ADT 112

3.7.1 Queue Model 113

3.7.2 Array Implementation of Queues 113

3.7.3 Applications of Queues 115

Summary 116

Exercises 116

** **

**Chapter 4 Trees 121**

4.1 Preliminaries 121

4.1.1 Implementation of Trees 122

4.1.2 Tree Traversals with an Application 123

4.2 Binary Trees 126

4.2.1 Implementation 128

4.2.2 An Example: Expression Trees 128

4.3 The Search Tree ADT–Binary Search Trees 132

4.3.1 contains 134

4.3.2 findMin and findMax 135

4.3.3 insert 136

4.3.4 remove 139

4.3.5 Destructor and Copy Constructor 141

4.3.6 Average-Case Analysis 141

4.4 AVL Trees 144

4.4.1 Single Rotation 147

4.4.2 Double Rotation 149

4.5 Splay Trees 158

4.5.1 A Simple Idea (That Does Not Work) 158

4.5.2 Splaying 160

4.6 Tree Traversals (Revisited) 166

4.7 B-Trees 168

4.8 Sets and Maps in the Standard Library 173

4.8.1 Sets 173

4.8.2 Maps 174

4.8.3 Implementation of set and map 175

4.8.4 An Example That Uses Several Maps 176

Summary 181

Exercises 182

References 189

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**Chapter 5 Hashing 193**

5.1 General Idea 193

5.2 Hash Function 194

5.3 Separate Chaining 196

5.4 Hash Tables without Linked Lists 201

5.4.1 Linear Probing 201

5.4.2 Quadratic Probing 202

5.4.3 Double Hashing 207

5.5 Rehashing 208

5.6 Hash Tables in the Standard Library 210

5.7 Hash Tables with Worst-Case *O*(1) Access 212

5.7.1 Perfect Hashing 213

5.7.2 Cuckoo Hashing 215

5.7.3 Hopscotch Hashing 224

5.8 Universal Hashing 230

5.9 Extendible Hashing 233

Summary 236

Exercises 238

References 242

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**Chapter 6 Priority Queues (Heaps) 245**

6.1 Model 245

6.2 Simple Implementations 246

6.3 Binary Heap 247

6.3.1 Structure Property 247

6.3.2 Heap-Order Property 248

6.3.3 Basic Heap Operations 249

6.3.4 Other Heap Operations 252

6.4 Applications of Priority Queues 257

6.4.1 The Selection Problem 258

6.4.2 Event Simulation 259

6.5 *d*-Heaps 260

6.6 Leftist Heaps 261

6.6.1 Leftist Heap Property 261

6.6.2 Leftist Heap Operations 262

6.7 Skew Heaps 269

6.8 Binomial Queues 271

6.8.1 Binomial Queue Structure 271

6.8.2 Binomial Queue Operations 271

6.8.3 Implementation of Binomial Queues 276

6.9 Priority Queues in the Standard Library 283

Summary 283

Exercises 283

References 288

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**Chapter 7 Sorting 291**

7.1 Preliminaries 291

7.2 Insertion Sort 292

7.2.1 The Algorithm 292

7.2.2 STL Implementation of Insertion Sort 293

7.2.3 Analysis of Insertion Sort 294

7.3 A Lower Bound for Simple Sorting Algorithms 295

7.4 Shellsort 296

7.4.1 Worst-Case Analysis of Shellsort 297

7.5 Heapsort 300

7.5.1 Analysis of Heapsort 301

7.6 Mergesort 304

7.6.1 Analysis of Mergesort 306

7.7 Quicksort 309

7.7.1 Picking the Pivot 311

7.7.2 Partitioning Strategy 313

7.7.3 Small Arrays 315

7.7.4 Actual Quicksort Routines 315

7.7.5 Analysis of Quicksort 318

7.7.6 A Linear-Expected-Time Algorithm for Selection 321

7.8 A General Lower Bound for Sorting 323

7.8.1 Decision Trees 323

7.9 Decision-Tree Lower Bounds for Selection Problems 325

7.10 Adversary Lower Bounds 328

7.11 Linear-Time Sorts: Bucket Sort and Radix Sort 331

7.12 External Sorting 336

7.12.1 Why We Need New Algorithms 336

7.12.2 Model for External Sorting 336

7.12.3 The Simple Algorithm 337

7.12.4 Multiway Merge 338

7.12.5 Polyphase Merge 339

7.12.6 Replacement Selection 340

Summary 341

Exercises 341

References 347

** **

**Chapter 8 The Disjoint Sets Class 351**

8.1 Equivalence Relations 351

8.2 The Dynamic Equivalence Problem 352

8.3 Basic Data Structure 353

8.4 Smart Union Algorithms 357

8.5 Path Compression 360

8.6 Worst Case for Union-by-Rank and Path Compression 361

8.6.1 Slowly Growing Functions 362

8.6.2 An Analysis by Recursive Decomposition 362

8.6.3 An *O*( *M *log **N *) Bound 369

8.6.4 An *O*( *M α*(*M, N*) ) Bound 370

8.7 An Application 372

Summary 374

Exercises 375

References 376

** **

**Chapter 9 Graph Algorithms 379**

9.1 Definitions 379

9.1.1 Representation of Graphs 380

9.2 Topological Sort 382

9.3 Shortest-Path Algorithms 386

9.3.1 Unweighted Shortest Paths 387

9.3.2 Dijkstra’s Algorithm 391

9.3.3 Graphs with Negative Edge Costs 400

9.3.4 Acyclic Graphs 400

9.3.5 All-Pairs Shortest Path 404

9.3.6 Shortest Path Example 404

9.4 Network Flow Problems 406

9.4.1 A Simple Maximum-Flow Algorithm 408

9.5 Minimum Spanning Tree 413

9.5.1 Prim’s Algorithm 414

9.5.2 Kruskal’s Algorithm 417

9.6 Applications of Depth-First Search 419

9.6.1 Undirected Graphs 420

9.6.2 Biconnectivity 421

9.6.3 Euler Circuits 425

9.6.4 Directed Graphs 429

9.6.5 Finding Strong Components 431

9.7 Introduction to NP-Completeness 432

9.7.1 Easy vs. Hard 433

9.7.2 The Class NP 434

9.7.3 NP-Complete Problems 434

Summary 437

Exercises 437

References 445

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**Chapter 10 Algorithm Design Techniques 449**

10.1 Greedy Algorithms 449

10.1.1 A Simple Scheduling Problem 450

10.1.2 Huffman Codes 453

10.1.3 Approximate Bin Packing 459

10.2 Divide and Conquer 467

10.2.1 Running Time of Divide-and-Conquer Algorithms 468

10.2.2 Closest-Points Problem 470

10.2.3 The Selection Problem 475

10.2.4 Theoretical Improvements for Arithmetic Problems 478

10.3 Dynamic Programming 482

10.3.1 Using a Table Instead of Recursion 483

10.3.2 Ordering Matrix Multiplications 485

10.3.3 Optimal Binary Search Tree 487

10.3.4 All-Pairs Shortest Path 491

10.4 Randomized Algorithms 494

10.4.1 Random-Number Generators 495

10.4.2 Skip Lists 500

10.4.3 Primality Testing 503

10.5 Backtracking Algorithms 506

10.5.1 The Turnpike Reconstruction Problem 506

10.5.2 Games 511

Summary 518

Exercises 518

References 527

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**Chapter 11 Amortized Analysis 533**

11.1 An Unrelated Puzzle 534

11.2 Binomial Queues 534

11.3 Skew Heaps 539

11.4 Fibonacci Heaps 541

11.4.1 Cutting Nodes in Leftist Heaps 542

11.4.2 Lazy Merging for Binomial Queues 544

11.4.3 The Fibonacci Heap Operations 548

11.4.4 Proof of the Time Bound 549

11.5 Splay Trees 551

Summary 555

Exercises 556

References 557

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**Chapter 12 Advanced Data Structures and Implementation 559**

12.1 Top-Down Splay Trees 559

12.2 Red-Black Trees 566

12.2.1 Bottom-Up Insertion 567

12.2.2 Top-Down Red-Black Trees 568

12.2.3 Top-Down Deletion 570

12.3 Treaps 576

12.4 Suffix Arrays and Suffix Trees 579

12.4.1 Suffix Arrays 580

12.4.2 Suffix Trees 583

12.4.3 Linear-Time Construction of Suffix Arrays and Suffix Trees 586

12.5 *k*-d Trees 596

12.6 Pairing Heaps 602

Summary 606

Exercises 608

References 612

** **

**Appendix A Separate Compilation of Class Templates 615**

A.1 Everything in the Header 616

A.2 Explicit Instantiation 616

Index 619