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Algorithms in a Nutshell
Table of Contents
Copyright..................................................................................................... 1
Preface........................................................................................................ 2
Part I: I....................................................................................................... 9
Chapter 1. Algorithms Matter....................................................................................................................................................... 10
Section 1.1. Understand the Problem......................................................................................................................................... 11
Section 1.2. Experiment if Necessary........................................................................................................................................ 12
Section 1.3. Side Story................................................................................................................................................................ 16
Section 1.4. The Moral of the Story............................................................................................................................................ 17
Section 1.5. References.............................................................................................................................................................. 18
Chapter 2. The Mathematics of Algorithms.................................................................................................................................. 19
Section 2.1. Size of a Problem Instance..................................................................................................................................... 19
Section 2.2. Rate of Growth of Functions.................................................................................................................................. 21
Section 2.3. Analysis in the Best, Average, and Worst Cases................................................................................................... 25
Section 2.4. Performance Families........................................................................................................................................... 29
Section 2.5. Mix of Operations.................................................................................................................................................. 42
Section 2.6. Benchmark Operations......................................................................................................................................... 43
Section 2.7. One Final Point...................................................................................................................................................... 45
Section 2.8. References............................................................................................................................................................. 45
Chapter 3. Patterns and Domains................................................................................................................................................. 46
Section 3.1. Patterns: A Communication Language................................................................................................................. 46
Section 3.2. Algorithm Pattern Format.................................................................................................................................... 48
Section 3.3. Pseudocode Pattern Format.................................................................................................................................. 49
Section 3.4. Design Format....................................................................................................................................................... 50
Section 3.5. Empirical Evaluation Format................................................................................................................................ 51
Section 3.6. Domains and Algorithms...................................................................................................................................... 53
Section 3.7. Floating-Point Computations................................................................................................................................ 54
Section 3.8. Manual Memory Allocation................................................................................................................................... 57
Section 3.9. Choosing a Programming Language..................................................................................................................... 60
Section 3.10. References............................................................................................................................................................ 61
Part II: II................................................................................................... 62
Chapter 4. Sorting Algorithms...................................................................................................................................................... 63
Section 4.1. Overview................................................................................................................................................................ 63
Section 4.2. Insertion Sort........................................................................................................................................................ 69
Section 4.3. Median Sort........................................................................................................................................................... 73
Section 4.4. Quicksort............................................................................................................................................................... 84
Section 4.5. Selection Sort......................................................................................................................................................... 91
Section 4.6. Heap Sort............................................................................................................................................................... 92
Section 4.7. Counting Sort......................................................................................................................................................... 97
Section 4.8. Bucket Sort............................................................................................................................................................ 99
Section 4.9. Criteria for Choosing a Sorting Algorithm.......................................................................................................... 105
Section 4.10. References.......................................................................................................................................................... 109
Chapter 5. Searching.................................................................................................................................................................... 111
Section 5.1. Overview................................................................................................................................................................ 111
Section 5.2. Sequential Search................................................................................................................................................. 112
Section 5.3. Binary Search....................................................................................................................................................... 118
Section 5.4. Hash-based Search.............................................................................................................................................. 122
Section 5.5. Binary Tree Search............................................................................................................................................... 135
Chapter 6. Graph Algorithms...................................................................................................................................................... 142
Section 6.1. Overview............................................................................................................................................................... 142
Section 6.2. Depth-First Search.............................................................................................................................................. 148
Section 6.3. Breadth-First Search............................................................................................................................................ 155
Section 6.4. Single-Source Shortest Path................................................................................................................................ 159
Section 6.5. All Pairs Shortest Path.......................................................................................................................................... 171
Section 6.6. Minimum Spanning Tree Algorithms................................................................................................................. 175
Section 6.7. References............................................................................................................................................................ 177
Chapter 7. Path Finding in AI...................................................................................................................................................... 178
Algorithms in a Nutshell
Section 7.1. Overview............................................................................................................................................................... 178
Section 7.2. Depth-First Search............................................................................................................................................... 187
Section 7.3. Breadth-First Search............................................................................................................................................ 196
Section 7.4. A*Search.............................................................................................................................................................. 200
Section 7.5. Comparison.......................................................................................................................................................... 210
Section 7.6. Minimax............................................................................................................................................................... 213
Section 7.7. NegMax................................................................................................................................................................ 219
Section 7.8. AlphaBeta............................................................................................................................................................ 223
Section 7.9. References........................................................................................................................................................... 230
Chapter 8. Network Flow Algorithms......................................................................................................................................... 232
Section 8.1. Overview.............................................................................................................................................................. 232
Section 8.2. Maximum Flow................................................................................................................................................... 235
Section 8.3. Bipartite Matching.............................................................................................................................................. 245
Section 8.4. Reflections on Augmenting Paths...................................................................................................................... 248
Section 8.5. Minimum Cost Flow............................................................................................................................................ 252
Section 8.6. Transshipment.................................................................................................................................................... 252
Section 8.7. Transportation..................................................................................................................................................... 253
Section 8.8. Assignment.......................................................................................................................................................... 254
Section 8.9. Linear Programming........................................................................................................................................... 255
Section 8.10. References......................................................................................................................................................... 256
Chapter 9. Computational Geometry.......................................................................................................................................... 257
Section 9.1. Overview............................................................................................................................................................... 257
Section 9.2. Convex Hull Scan................................................................................................................................................ 266
Section 9.3. LineSweep............................................................................................................................................................ 274
Section 9.4. Nearest Neighbor Queries.................................................................................................................................. 286
Section 9.5. Range Queries..................................................................................................................................................... 298
Section 9.6. References........................................................................................................................................................... 304
Part III: III.............................................................................................. 305
Chapter 10. When All Else Fails................................................................................................................................................. 306
Section 10.1. Variations on a Theme....................................................................................................................................... 306
Section 10.2. Approximation Algorithms............................................................................................................................... 307
Section 10.3. Offline Algorithms............................................................................................................................................. 307
Section 10.4. Parallel Algorithms........................................................................................................................................... 308
Section 10.5. Randomized Algorithms................................................................................................................................... 308
Section 10.6. Algorithms That Can Be Wrong, but with Diminishing Probability................................................................. 315
Section 10.7. References.......................................................................................................................................................... 318
Chapter 11. Epilogue.................................................................................................................................................................... 319
Section 11.1. Overview.............................................................................................................................................................. 319
Section 11.2. Principle: Know Your Data................................................................................................................................. 319
Section 11.3. Principle: Decompose the Problem into Smaller Problems.............................................................................. 320
Section 11.4. Principle: Choose the Right Data Structure....................................................................................................... 321
Section 11.5. Principle: Add Storage to Increase Performance.............................................................................................. 322
Section 11.6. Principle: If No Solution Is Evident, Construct a Search.................................................................................. 323
Section 11.7. Principle: If No Solution Is Evident, Reduce Your Problem to Another Problem That Has a Solution.......... 323
Section 11.8. Principle: Writing Algorithms Is Hard—Testing Algorithms Is Harder........................................................... 324
Part IV: IV............................................................................................... 326
Appendix A. Benchmarking........................................................................................................................................................ 327
Section A.1. Statistical Foundation......................................................................................................................................... 327
Section A.2. Hardware............................................................................................................................................................ 328
Section A.3. Reporting............................................................................................................................................................. 337
Section A.4. Precision.............................................................................................................................................................. 338
About the Authors................................................................................... 340
Colophon................................................................................................ 340
Algorithms in a Nutshell
Page 1
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Algorithms in a Nutshell
by George T. Heineman, Gary Pollice, and Stanley Selkow
Copyright © 2009 George Heineman, Gary Pollice, and Stanley Selkow. All rights reserved.
Printed in the United States of America.
Published by O’Reilly Media, Inc., 1005 Gravenstein Highway North, Sebastopol, CA 95472.
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While every precaution has been taken in the preparation of this book, the publisher and
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ISBN: 978-0-596-51624-6
[M]
Algorithms in a Nutshell
Page 2
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Chapter 2
Preface
As Trinity states in the movie
The Matrix:
It’s the question that drives us, Neo. It’s the question that brought you here.
You know the question, just as I did.
As authors of this book, we answer the question that has led you here:
Can I use algorithm
X
to solve my problem? If so, how do I implement it?
You likely do not need to understand the reasons why an algorithm is correct—if
you do, turn to other sources, such as the 1,180-page bible on algorithms,
Intro-
duction to Algorithms,
Second Edition, by Thomas H. Cormen et al. (2001). There
you will find lemmas, theorems, and proofs; you will find exercises and step-by-step
examples showing the algorithms as they perform. Perhaps surprisingly, however,
you will not find any real code, only fragments of “pseudocode,” the device used by
countless educational textbooks to present a high-level description of algorithms.
These educational textbooks are important within the classroom, yet they fail the
software practitioner because they assume it will be straightforward to develop real
code from pseudocode fragments.
We intend this book to be used frequently by experienced programmers looking
for appropriate solutions to their problems. Here you will find solutions to the
problems you must overcome as a programmer every day. You will learn what
decisions lead to an improved performance of key algorithms that are essential for
the success of your software applications. You will find real code that can be
adapted to your needs and solution methods that you can learn.
All algorithms are fully implemented with test suites that validate the correct
implementation of the algorithms. The code is fully documented and available as
a code repository addendum to this book. We rigorously followed a set of princi-
ples as we designed, implemented, and wrote this book. If these principles are
meaningful to you, then you will find this book useful.
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