The Algorithm Design Manual

The Algorithm Design Manual

(Parte 1 de 6)

The Algorithm Design Manual Second Edition

Steven S. Skiena

The Algorithm Design Manual Second Edition

Steven S. Skiena Department of Computer Science State University of New York at Stony Brook

New York, USA

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library

Library of Congress Control Number: 2008931136 c© Springer-Verlag London Limited 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.

Printed on acid-free paper

Springer Science+Business Media


Most professional programmers that I’ve encountered are not well prepared to tackle algorithm design problems. This is a pity, because the techniques of algorithm design form one of the core practical technologies of computer science. Designing correct, efficient, and implementable algorithms for real-world problems requires access to two distinct bodies of knowledge:

• Techniques – Good algorithm designers understand several fundamental algorithm design techniques, including data structures, dynamic programming, depth-first search, backtracking, and heuristics. Perhaps the single most important design technique is modeling, the art of abstracting a messy real-world application into a clean problem suitable for algorithmic attack.

• Resources – Good algorithm designers stand on the shoulders of giants.

Rather than laboring from scratch to produce a new algorithm for every task, they can figure out what is known about a particular problem. Rather than re-implementing popular algorithms from scratch, they seek existing implementations to serve as a starting point. They are familiar with many classic algorithmic problems, which provide sufficient source material to model most any application.

This book is intended as a manual on algorithm design, providing access to combinatorial algorithm technology for both students and computer professionals. It is divided into two parts: Techniques and Resources. The former is a general guide to techniques for the design and analysis of computer algorithms. The Resources section is intended for browsing and reference, and comprises the catalog of algorithmic resources, implementations, and an extensive bibliography.


To the Reader

I have been gratified by the warm reception the first edition of The Algorithm Design Manual has received since its initial publication in 1997. It has been recognized as a unique guide to using algorithmic techniques to solve problems that often arise in practice. But much has changed in the world since the The Algorithm Design Manual was first published over ten years ago. Indeed, if we date the origins of modern algorithm design and analysis to about 1970, then roughly 30% of modern algorithmic history has happened since the first coming of The Algorithm Design Manual.

Three aspects of The Algorithm Design Manual have been particularly beloved: (1) the catalog of algorithmic problems, (2) the war stories, and (3) the electronic component of the book. These features have been preserved and strengthened in this edition:

• The Catalog of Algorithmic Problems – Since finding out what is known about an algorithmic problem can be a difficult task, I provide a catalog of the 75 most important problems arising in practice. By browsing through this catalog, the student or practitioner can quickly identify what their problem is called, what is known about it, and how they should proceed to solve it. To aid in problem identification, we include a pair of “before” and “after” pictures for each problem, illustrating the required input and output specifications. One perceptive reviewer called my book “The Hitchhiker’s Guide to Algorithms” on the strength of this catalog.

The catalog is the most important part of this book. To update the catalog for this edition, I have solicited feedback from the world’s leading experts on each associated problem. Particular attention has been paid to updating the discussion of available software implementations for each problem.

• War Stories – In practice, algorithm problems do not arise at the beginning of a large project. Rather, they typically arise as subproblems when it becomes clear that the programmer does not know how to proceed or that the current solution is inadequate.

To provide a better perspective on how algorithm problems arise in the real world, we include a collection of “war stories,” or tales from our experience with real problems. The moral of these stories is that algorithm design and analysis is not just theory, but an important tool to be pulled out and used as needed.

This edition retains all the original war stories (with updates as appropriate) plus additional new war stories covering external sorting, graph algorithms, simulated annealing, and other topics.

• Electronic Component – Since the practical person is usually looking for a program more than an algorithm, we provide pointers to solid implementations whenever they are available. We have collected these implementations

PREFACE vii at one central website site∼algorith for easy retrieval. We have been the number one “Algorithm” site on Google pretty much since the initial publication of the book.

Further, we provide recommendations to make it easier to identify the correct code for the job. With these implementations available, the critical issue in algorithm design becomes properly modeling your application, more so than becoming intimate with the details of the actual algorithm. This focus permeates the entire book.

Equally important is what we do not do in this book. We do not stress the mathematical analysis of algorithms, leaving most of the analysis as informal arguments. You will not find a single theorem anywhere in this book. When more details are needed, the reader should study the cited programs or references. The goal of this manual is to get you going in the right direction as quickly as possible.

To the Instructor

This book covers enough material for a standard Introduction to Algorithms course. We assume the reader has completed the equivalent of a second programming course, typically titled Data Structures or Computer Science I.

A full set of lecture slides for teaching this course is available online at . Further, I make available online audio and video lectures using these slides to teach a full-semester algorithm course. Let me help teach your course, by the magic of the Internet!

This book stresses design over analysis. It is suitable for both traditional lecture courses and the new “active learning” method, where the professor does not lecture but instead guides student groups to solve real problems. The “war stories” provide an appropriate introduction to the active learning method.

I have made several pedagogical improvements throughout the book. Textbookoriented features include:

• More Leisurely Discussion – The tutorial material in the first part of the book has been doubled over the previous edition. The pages have been devoted to more thorough and careful exposition of fundamental material, instead of adding more specialized topics.

• False Starts – Algorithms textbooks generally present important algorithms as a fait accompli, obscuring the ideas involved in designing them and the subtle reasons why other approaches fail. The war stories illustrate such development on certain applied problems, but I have expanded such coverage into classical algorithm design material as well.

• Stop and Think – Here I illustrate my thought process as I solve a topicspecific homework problem—false starts and all. I have interspersed such viii PREFACE problem blocks throughout the text to increase the problem-solving activity of my readers. Answers appear immediately following each problem.

• More and Improved Homework Problems – This edition of The Algorithm

Design Manual has twice as many homework exercises as the previous one. Exercises that proved confusing or ambiguous have been improved or replaced. Degree of difficulty ratings (from 1 to 10) have been assigned to all problems.

• Self-Motivating Exam Design – In my algorithms courses, I promise the students that all midterm and final exam questions will be taken directly from homework problems in this book. This provides a “student-motivated exam,” so students know exactly how to study to do well on the exam. I have carefully picked the quantity, variety, and difficulty of homework exercises to make this work; ensuring there are neither too few or too many candidate problems.

• Take-Home Lessons – Highlighted “take-home” lesson boxes scattered throughout the text emphasize the big-picture concepts to be gained from the chapter.

• Links to Programming Challenge Problems – Each chapter’s exercises will contain links to 3-5 relevant “Programming Challenge” problems from These can be used to add a programming component to paper-and-pencil algorithms courses.

• More Code, Less Pseudo-code – More algorithms in this book appear as code (written in C) instead of pseudo-code. I believe the concreteness and reliability of actual tested implementations provides a big win over less formal presentations for simple algorithms. Full implementations are available for study at .

• Chapter Notes – Each tutorial chapter concludes with a brief notes section, pointing readers to primary sources and additional references.


Updating a book dedication after ten years focuses attention on the effects of time. Since the first edition, Renee has become my wife and then the mother of our two children, Bonnie and Abby. My father has left this world, but Mom and my brothers Len and Rob remain a vital presence in my life. I dedicate this book to my family, new and old, here and departed.

I would like to thank several people for their concrete contributions to this new edition. Andrew Gaun and Betson Thomas helped in many ways, particularly in building the infrastructure for the new∼algorithand dealing with a variety of manuscript preparation issues. David Gries offered valuable feedback well beyond the call of duty. Himanshu Gupta and Bin Tang bravely

PREFACE ix taught courses using a manuscript version of this edition. Thanks also to my Springer-Verlag editors, Wayne Wheeler and Allan Wylde.

A select group of algorithmic sages reviewed sections of the Hitchhiker’s guide, sharing their wisdom and alerting me to new developments. Thanks to:

Ami Amir, Herve Bronnimann, Bernard Chazelle, Chris Chu, Scott Cotton, Yefim Dinitz, Komei Fukuda, Michael Goodrich, Lenny Heath, Cihat Imamoglu, Tao Jiang, David Karger, Giuseppe Liotta, Albert Mao, Silvano Martello, Catherine McGeoch, Kurt Mehlhorn, Scott A. Mitchell, Naceur Meskini, Gene Myers, Gonzalo Navarro, Stephen North, Joe O’Rourke, Mike Paterson, Theo Pavlidis, Seth Pettie, Michel Pocchiola, Bart Preneel, Tomasz Radzik, Edward Reingold, Frank Ruskey, Peter Sanders, Joao Setubal, Jonathan Shewchuk, Robert Skeel, Jens Stoye, Torsten Suel, Bruce Watson, and Uri Zwick.

Several exercises were originated by colleagues or inspired by other texts. Reconstructing the original sources years later can be challenging, but credits for each problem (to the best of my recollection) appear on the website.

It would be rude not to thank important contributors to the original edition.

Ricky Bradley and Dario Vlah built up the substantial infrastructure required for the W site in a logical and extensible manner. Zhong Li drew most of the catalog figures using xfig. Richard Crandall, Ron Danielson, Takis Metaxas, Dave Miller, Giri Narasimhan, and Joe Zachary all reviewed preliminary versions of the first edition; their thoughtful feedback helped to shape what you see here.

Much of what I know about algorithms I learned along with my graduate students. Several of them (Yaw-Ling Lin, Sundaram Gopalakrishnan, Ting Chen, Francine Evans, Harald Rau, Ricky Bradley, and Dimitris Margaritis) are the real heroes of the war stories related within. My Stony Brook friends and algorithm colleagues Estie Arkin, Michael Bender, Jie Gao, and Joe Mitchell have always been a pleasure to work and be with. Finally, thanks to Michael Brochstein and the rest of the city contingent for revealing a proper life well beyond Stony Brook.


It is traditional for the author to magnanimously accept the blame for whatever deficiencies remain. I don’t. Any errors, deficiencies, or problems in this book are somebody else’s fault, but I would appreciate knowing about them so as to determine who is to blame.

Steven S. Skiena

Department of Computer Science

Stony Brook University

Stony Brook, NY 11794-40∼skiena April 2008


I Practical Algorithm Design 1

1.1 Robot Tour Optimization5
1.2 Selecting the Right Jobs9
1.3 Reasoning about Correctness1
1.4 Modeling the Problem19
1.5 About the War Stories2
1.6 War Story: Psychic Modeling23
1.7 Exercises27

1 Introduction to Algorithm Design 3

2.1 The RAM Model of Computation31
2.2 The Big Oh Notation34
2.3 Growth Rates and Dominance Relations37
2.4 Working with the Big Oh40
2.5 Reasoning About Efficiency41
2.6 Logarithms and Their Applications46
2.7 Properties of Logarithms50
2.8 War Story: Mystery of the Pyramids51
2.9 Advanced Analysis (*)54
2.10 Exercises57
3.2 Stacks and Queues71
3.3 Dictionaries72
3.4 Binary Search Trees7
3.5 Priority Queues83
3.6 War Story: Stripping Triangulations85
3.7 Hashing and Strings89
3.8 Specialized Data Structures93
3.9 War Story: String ’em Up94
3.10 Exercises98


4.1 Applications of Sorting104
4.2 Pragmatics of Sorting107
4.3 Heapsort: Fast Sorting via Data Structures108
4.4 War Story: Give me a Ticket on an Airplane118
4.5 Mergesort: Sorting by Divide-and-Conquer120
4.6 Quicksort: Sorting by Randomization123
4.7 Distribution Sort: Sorting via Bucketing129
4.8 War Story: Skiena for the Defense131
4.9 Binary Search and Related Algorithms132
4.10 Divide-and-Conquer135
4.1 Exercises139

4 Sorting and Searching 103

5.1 Flavors of Graphs146
5.2 Data Structures for Graphs151
5.3 War Story: I was a Victim of Moore’s Law155
5.4 War Story: Getting the Graph158
5.5 Traversing a Graph161
5.6 Breadth-First Search162
5.7 Applications of Breadth-First Search166
5.8 Depth-First Search169
5.9 Applications of Depth-First Search172
5.10 Depth-First Search on Directed Graphs178
5.1 Exercises184
6.1 Minimum Spanning Trees192
6.2 War Story: Nothing but Nets202
6.3 Shortest Paths205
6.4 War Story: Dialing for Documents212
6.5 Network Flows and Bipartite Matching217
6.6 Design Graphs, Not Algorithms2


7.1 Backtracking231
7.2 Search Pruning238
7.3 Sudoku239
7.4 War Story: Covering Chessboards244
7.5 Heuristic Search Methods247
7.6 War Story: Only it is Not a Radio260
7.7 War Story: Annealing Arrays263
7.8 Other Heuristic Search Methods266
7.9 Parallel Algorithms267
7.10 War Story: Going Nowhere Fast268
7.1 Exercises270

7 Combinatorial Search and Heuristic Methods 230

8.1 Caching vs. Computation274
8.2 Approximate String Matching280
8.3 Longest Increasing Sequence289
8.4 War Story: Evolution of the Lobster291
8.5 The Partition Problem294
8.6 Parsing Context-Free Grammars298
8.7 Limitations of Dynamic Programming: TSP301
8.8 War Story: What’s Past is Prolog304
8.9 War Story: Text Compression for Bar Codes307
8.10 Exercises310
9.1 Problems and Reductions317
9.2 Reductions for Algorithms319
9.3 Elementary Hardness Reductions323
9.4 Satisfiability328
9.5 Creative Reductions330
9.6 The Art of Proving Hardness334
9.7 War Story: Hard Against the Clock337
9.8 War Story: And Then I Failed339
9.9 P vs. NP341
9.10 Dealing with NP-complete Problems344
9.1 Exercises350

9 Intractable Problems and Approximation Algorithms 316 10 How to Design Algorithms 356

I The Hitchhiker’s Guide to Algorithms 361 1 A Catalog of Algorithmic Problems 363 xiv CONTENTS

12.1 Dictionaries367
12.2 Priority Queues373
12.3 Suffix Trees and Arrays377
12.4 Graph Data Structures381
12.5 Set Data Structures385
12.6 Kd-Trees389

12 Data Structures 366

13.1 Solving Linear Equations395
13.2 Bandwidth Reduction398
13.3 Matrix Multiplication401
13.4 Determinants and Permanents404
13.5 Constrained and Unconstrained Optimization407
13.6 Linear Programming411
13.7 Random Number Generation415
13.8 Factoring and Primality Testing420
13.9 Arbitrary-Precision Arithmetic423
13.10 Knapsack Problem427
13.1 Discrete Fourier Transform431

13 Numerical Problems 393

14.1 Sorting436
14.2 Searching441
14.3 Median and Selection445
14.4 Generating Permutations448
14.5 Generating Subsets452
14.6 Generating Partitions456
14.7 Generating Graphs460
14.8 Calendrical Calculations465
14.9 Job Scheduling468
14.10 Satisfiability472

14 Combinatorial Problems 434

15.1 Connected Components477
15.2 Topological Sorting481
15.3 Minimum Spanning Tree484
15.4 Shortest Path489
15.5 Transitive Closure and Reduction495
15.6 Matching498
15.7 Eulerian Cycle/Chinese Postman502
15.8 Edge and Vertex Connectivity505
15.9 Network Flow509
15.1 Drawing Trees517
15.12 Planarity Detection and Embedding520


16.1 Clique525
16.2 Independent Set528
16.3 Vertex Cover530
16.4 Traveling Salesman Problem533
16.5 Hamiltonian Cycle538
16.6 Graph Partition541
16.7 Vertex Coloring544
16.8 Edge Coloring548
16.9 Graph Isomorphism550
16.10 Steiner Tree5
16.1 Feedback Edge/Vertex Set559

16 Graph Problems: Hard Problems 523

17.1 Robust Geometric Primitives564
17.2 Convex Hull568
17.3 Triangulation572
17.4 Voronoi Diagrams576
17.5 Nearest Neighbor Search580
17.6 Range Search584
17.7 Point Location587
17.8 Intersection Detection591
17.9 Bin Packing595
17.10 Medial-Axis Transform598
17.1 Polygon Partitioning601
17.12 Simplifying Polygons604
17.13 Shape Similarity607
17.14 Motion Planning610
17.15 Maintaining Line Arrangements614
17.16 Minkowski Sum617
18.1 Set Cover621
18.2 Set Packing625
18.3 String Matching628
18.4 Approximate String Matching631
18.5 Text Compression637
18.6 Cryptography641
18.7 Finite State Machine Minimization646
18.8 Longest Common Substring/Subsequence650


19.1 Software Systems657
19.2 Data Sources663
19.3 Online Bibliographic Resources663
19.4 Professional Consulting Services664

19 Algorithmic Resources 657

1 Introduction to Algorithm Design

What is an algorithm? An algorithm is a procedure to accomplish a specific task. An algorithm is the idea behind any reasonable computer program.

To be interesting, an algorithm must solve a general, well-specified problem.A n algorithmic problem is specified by describing the complete set of instances it must work on and of its output after running on one of these instances. This distinction, between a problem and an instance of a problem, is fundamental. For example, the algorithmic problem known as sorting is defined as follows:

Problem: Sorting

Input: A sequence of n keys a1,...,a n.

Output: The permutation (reordering) of the input sequence such that a′1 ≤ a′2 ≤

An instance of sorting might be an array of names, like {Mike, Bob, Sally, Jill,

(Parte 1 de 6)