Foundations of Algorithms Using Java Pseudocode

, by Neapolitan, Richard E.; Naimipour, Kumarass

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ISBN: 9780763721299 | 0763721298
Cover: Hardcover
Copyright: 7/1/2002

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Foundations of Algorithms Using Java Pseudocode offers a well-balanced presentation on designing algorithms, complexity analysis of algorithms, and computational complexity that is accessible to mainstream computer science students who have a background in college algebra and discrete structures. To support their approach, The authors present mathematical concepts using standard English and a simpler notation than is found in most texts. A review of essential mathematical concepts is presented in three appendices. In addition, they reinforce the explanations with numerous concrete examples to help students grasp theoretical concepts.

Preface

CHAPTER 1 Algorithms: Efficiency, Analysis, and Order

(46)

1.1 Algorithms

(7)

1.2 The Importance of Developing Efficient Algorithms

(8)

1.2.1 Sequential Search versus Binary Search

(3)

1.2.2 The Fibonacci Sequence

(5)

1.3 Analysis of Algorithms

(8)

1.3.1 Complexity Analysis

(7)

1.3.2 Applying the Theory

(1)

1.3.3 Analysis of Correctness

(1)

1.4 Order

(16)

1.4.1 An Intuitive Introduction to Order

(2)

1.4.2 A Rigorous Introduction to Order

(12)

1.4.3 Using a Limit to Determine Order

(2)

1.5 Outline of This Book

(6)

Exercises

(5)

CHAPTER 2 Divide-and-Conquer

(44)

2.1 Binary Search

(5)

2.2 Mergesort

(6)

2.3 The Divide-and-Conquer Approach

(1)

2.4 Quicksort (Partition Exchange Sort)

(7)

2.5 Strassen's Matrix Multiplication Algorithm

(5)

2.6 Arithmetic with Large Integers

(6)

2.6.1 Representation of Large Integers: Addition and Other Linear-Time Operations

(1)

2.6.2 Multiplication of Large Integers

(6)

2.7 Determining Thresholds

(4)

2.8 When Not to Use Divide-and-Conquer

(9)

Exercises

(8)

CHAPTER 3 Dynamic Programming

(46)

3.1 The Binomial Coefficient

(5)

3.2 Floyd's Algorithm for Shortest Paths

(8)

3.3 Dynamic Programming and Optimization Problems

105

(2)

3.4 Chained Matrix Multiplication

107

(9)

3.5 Optimal Binary Search Trees

116

(9)

3.6 The Traveling Salesperson Problem

125

(12)

Exercises

133

(4)

CHAPTER 4 The Greedy Approach

137

(50)

4.1 Minimum Spanning Trees

140

(16)

4.1.1 Prim's Algorithm

144

(6)

4.1.2 Kruskal's Algorithm

150

(5)

4.1.3 Comparing Prim's Algorithm with Kruskal's Algorithm

155

(1)

4.1.4 Final Discussion

155

(1)

4.2 Dijkstra's Algorithm for Single-Source Shortest Paths

156

(3)

4.3 Scheduling

159

(10)

4.3.1 Minimizing Total Time in the System

160

(2)

4.3.2 Scheduling with Deadlines

162

(7)

4.4 Huffman Code

169

(6)

4.4.1 Prefix Codes

170

(1)

4.4.2 Huffman's Algorithm

171

(4)

4.5 The Greedy Approach versus Dynamic Programming: The Knapsack Problem

175

(12)

4.5.1 A Greedy Approach to the 0-1 Knapsack Problem

175

(2)

4.5.2 A Greedy Approach to the Fractional Knapsack Problem

177

(1)

4.5.3 A Dynamic Programming Approach to the 0-1 Knapsack Problem

177

(1)

4.5.4 A Refinement of the Dynamic Programming Algorithm for the 0-1 Knapsack Problem

178

(3)

Exercises

181

(6)

CHAPTER 5 Backtracking

187

(46)

5.1 The Backtracking Technique

188

(8)

5.2 The n-Queens Problem

196

(4)

5.3 Using a Monte Carlo Algorithm to Estimate the Efficiency of a Backtracking Algorithm

200

(4)

5.4 The Sum-of-Subsets Problem

204

(5)

5.5 Graph Coloring

209

(5)

5.6 The Hamiltonian Circuits Problem

214

(3)

5.7 The 0-1 Knapsack Problem

217

(16)

5.7.1 A Backtracking Algorithm for the 0-1 Knapsack Problem

217

(10)

5.7.2 Comparing the Dynamic Programming Algorithm and the Backtracking Algorithm for the 0-1 Knapsack Problem

227

(1)

Exercises

227

(6)

CHAPTER 6 Branch-and-Bound

233

(34)

6.1 Illustrating Branch-and-Bound with the 0-1 Knapsack Problem

235

(11)

6.1.1 Breadth-First Search with Branch-and-Bound Pruning

235

(6)

6.1.2 Best-First Search with Branch-and-Bound Pruning

241

(5)

6.2 The Traveling Salesperson Problem

246

(9)

6.3 Abductive Inference (Diagnosis)

255

(9)

Exercises

264

(3)

CHAPTER 7 Introduction to Computational Complexity: The Sorting Problem

267

(52)

7.1 Computational Complexity

268

(2)

7.2 Insertion Sort and Selection Sort

270

(5)

7.3 Lower Bounds for Algorithms that Remove at Most One Inversion per Comparison

275

(2)

7.4 Mergesort Revisited

277

(6)

7.5 Quicksort Revisited

283

(2)

7.6 Heapsort

285

(11)

7.6.1 Heaps and Basic Heap Routines

285

(4)

7.6.2 An Implementation of Heapsort

289

(7)

7.7 Comparison of Mergesort, Quicksort, and Heapsort

296

(1)

7.8 Lower Bounds for Sorting Only by Comparison of Keys

297

(11)

7.8.1 Decision Trees for Sorting Algorithms

297

(3)

7.8.2 Lower Bounds for Worst-Case Behavior

300

(3)

7.8.3 Lower Bounds for Average-Case Behavior

303

(5)

7.9 Sorting by Distribution (Radix Sort)

308

(11)

Exercises

312

(7)

CHAPTER 8 More Computational Complexity: The Searching Problem

319

(56)

8.1 Lower Bounds for Searching Only by Comparisons of Keys

320

(10)

8.1.1 Lower Bounds for Worst-Case Behavior

322

(2)

8.1.2 Lower Bounds for Average-Case Behavior

324

(6)

8.2 Interpolation Search

330

(3)

8.3 Searching in Trees

333

(6)

8.3.1 Binary Search Trees

334

(4)

8.3.2 B-Trees

338

(1)

8.4 Hashing

339

(5)

8.5 The Selection Problem: Introduction to Adversary Arguments

344

(31)

8.5.1 Finding the Largest Key

344

(2)

8.5.2 Finding Both the Smallest and Largest Keys

346

(7)

8.5.3 Finding the Second-Largest Key

353

(5)

8.5.4 Finding the kappath-Smallest Key

358

(8)

8.5.5 A Probabilistic Algorithm for the Selection Problem

366

(4)

Exercises

370

(5)

CHAPTER 9 Computational Complexity and Intractability: An Introduction to the Theory of NP

375

(44)

9.1 Intractability

376

(2)

9.2 Input Size Revisited

378

(4)

9.3 The Three General Problems

382

(2)

9.3.1 Problems for Which Polynomial-Time Algorithms Have Been Found

382

(1)

9.3.2 Problems That Have Been Proven to Be Intractable

382

(1)

9.3.3 Problems That Have Not Been Proven to Be Intractable but for Which Polynomial-Time Algorithms Have Never Been Found

383

(1)

9.4 The Theory of NP

384

(22)

9.4.1 The Sets P and NP

386

(4)

9.4.2 NP-Complete Problems

390

(12)

9.4.3 NP-Hard, NP-Easy, and NP-Equivalent Problems

402

(4)

9.5 Handling NP-Hard Problems

406

(10)

9.5.1 An Approximation Algorithm for the Traveling Salesperson Problem

407

(4)

9.5.2 An Approximation Algorithm for the Bin-Packing Problem

411

(5)

Exercises

416

(3)

CHAPTER 10 Number-Theoretic Algorithms

419

(66)

10.1 Number Theory Review

420

(7)

10.1.1 Composite and Prime Numbers

420

(1)

10.1.2 Greatest Common Divisor

421

(3)

10.1.3 Prime Factorization

424

(3)

10.1.4 Least Common Multiple

427

(1)

10.2 Computing the Greatest Common Divisor

427

(7)

10.2.1 Euclid's Algorithm

428

(4)

10.2.2 An Extension to Euclid's Algorithm

432

(2)

10.3 Modular Arithmetic Review

434

(14)

10.3.1 Group Theory

434

(2)

10.3.2 Congruency Modulo n

436

(6)

10.3.3 Subgroups

442

(6)

10.4 Solving Modular Linear Equations

448

(6)

10.5 Computing Modular Powers

454

(2)

10.6 Finding Large Prime Numbers

456

(20)

10.6.1 Searching for a Large Prime

457

(1)

10.6.2 Checking Whether a Number Is Prime

458

(18)

10.7 The RSA Public-Key Cryptosystem

476

(9)

10.7.1 Public-Key Cryptosystems

476

(1)

10.7.2 The RSA Cryptosystem

477

(3)

Exercises

480

(5)

CHAPTER 11 Introduction to Parallel Algorithms

485

(26)

11.1 Parallel Architectures

488

(7)

11.1.1 Control Mechanism

488

(2)

11.1.2 Address-Space Organization

490

(1)

11.1.3 Interconnection Networks

491

(4)

11.2 The PRAM Model

495

(16)

11.2.1 Designing Algorithms for the CREW PRAM Model

497

(8)

11.2.2 Designing Algorithms for the CRCW PRAM Model

505

(3)

Exercises

508

(3)

APPENDIX A Review of Necessary Mathematics

511

(38)

A.1 Notation

511

(2)

A.2 Functions

513

(1)

A.3 Mathematical Induction

514

(7)

A.4 Theorems and Lemmas

521

(1)

A.5 Logarithms

522

(4)

A.5.1 Definition and Properties of Logarithms

522

(2)

A.5.2 The Natural Logarithm

524

(2)

A.6 Sets

526

(2)

A.7 Permutations and Combinations

528

(3)

A.8 Probability

531

(18)

A.8.1 Randomness

536

(4)

A.8.2 The Expected Value

540

(2)

Exercises

542

(7)

APPENDIX B Solving Recurrence Equations: With Applications to Analysis of Recursive Algorithms

549

(40)

B.1 Solving Recurrences Using Induction

549

(4)

B.2 Solving Recurrences Using the Characteristic Equation

553

(18)

B.2.1 Homogeneous Linear Recurrences

553

(9)

B.2.2 Nonhomogeneous Linear Recurrences

562

(6)

B.2.3 Change of Variables (Domain Transformations)

568

(3)

B.3 Solving Recurrences by Substitution

571

(2)

B.4 Extending Results for n, a Power of a Positive Constant b, to n in General

573

(6)

B.5 Proofs of Theorems

579

(10)

Exercises

582

(7)

APPENDIX C Data Structures for Disjoint Sets

589

(10)

REFERENCES

599

(6)

INDEX

605

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Foundations of Algorithms Using Java Pseudocode

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