ALGORITHMS INTRODUCTION TO

A L G O R I T H M S

I N T R O D U C T I O N T O

T H I R D E D I T I O N

C H A R L E S E.

R O N A L D L .

C L I F F O R D S T E I N R I V E S T

L E I S E R S O N

Third Edition

Charles E. Leiserson Ronald L. Rivest Clifford Stein

Introduction to Algorithms

Third Edition

The MIT Press

Cambridge, Massachusetts London, England

(including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher.

For information about special quantity discounts, please email special sales@mitpress.mit.edu.

This book was set in Times Roman and Mathtime Pro 2 by the authors.

Printed and bound in the United States of America.

Library of Congress Cataloging-in-Publication Data

Introduction to algorithms / Thomas H. Cormen . . . [et al.].—3rd ed.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-262-03384-8 (hardcover : alk. paper)—ISBN 978-0-262-53305-8 (pbk. : alk. paper) 1. Computer programming. 2. Computer algorithms. I. Cormen, Thomas H.

QA76.6.I5858 2009 005.1—dc22

2009008593

10 9 8 7 6 5 4 3 2

Preface xiii

I Foundations

Introduction 3

1 The Role of Algorithms in Computing 5 1.1 Algorithms 5

1.2 Algorithms as a technology 11 2 Getting Started 16

2.1 Insertion sort 16

2.2 Analyzing algorithms 23 2.3 Designing algorithms 29 3 Growth of Functions 43

3.1 Asymptotic notation 43

3.2 Standard notations and common functions 53 4 Divide-and-Conquer 65

4.1 The maximum-subarray problem 68

4.2 Strassen’s algorithm for matrix multiplication 75 4.3 The substitution method for solving recurrences 83 4.4 The recursion-tree method for solving recurrences 88 4.5 The master method for solving recurrences 93

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5 Probabilistic Analysis and Randomized Algorithms 114 5.1 The hiring problem 114

5.2 Indicator random variables 118 5.3 Randomized algorithms 122

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II Sorting and Order Statistics Introduction 147 6 Heapsort 151

6.1 Heaps 151

6.2 Maintaining the heap property 154 6.3 Building a heap 156

6.4 The heapsort algorithm 159 6.5 Priority queues 162

7 Quicksort 170

7.1 Description of quicksort 170 7.2 Performance of quicksort 174

7.3 A randomized version of quicksort 179 7.4 Analysis of quicksort 180

8 Sorting in Linear Time 191 8.1 Lower bounds for sorting 191 8.2 Counting sort 194

8.3 Radix sort 197 8.4 Bucket sort 200

9 Medians and Order Statistics 213 9.1 Minimum and maximum 214

9.2 Selection in expected linear time 215 9.3 Selection in worst-case linear time 220 III Data Structures

Introduction 229

10 Elementary Data Structures 232 10.1 Stacks and queues 232 10.2 Linked lists 236

10.3 Implementing pointers and objects 241 10.4 Representing rooted trees 246

11 Hash Tables 253

11.1 Direct-address tables 254 11.2 Hash tables 256

11.3 Hash functions 262 11.4 Open addressing 269

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12 Binary Search Trees 286

12.1 What is a binary search tree? 286 12.2 Querying a binary search tree 289 12.3 Insertion and deletion 294

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13.1 Properties of red-black trees 308 13.2 Rotations 312

13.3 Insertion 315 13.4 Deletion 323

14 Augmenting Data Structures 339 14.1 Dynamic order statistics 339

14.2 How to augment a data structure 345 14.3 Interval trees 348

IV Advanced Design and Analysis Techniques Introduction 357

15 Dynamic Programming 359 15.1 Rod cutting 360

15.2 Matrix-chain multiplication 370

15.3 Elements of dynamic programming 378 15.4 Longest common subsequence 390 15.5 Optimal binary search trees 397 16 Greedy Algorithms 414

16.1 An activity-selection problem 415 16.2 Elements of the greedy strategy 423 16.3 Huffman codes 428

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17.1 Aggregate analysis 452 17.2 The accounting method 456 17.3 The potential method 459 17.4 Dynamic tables 463

V Advanced Data Structures

Introduction 481 18 B-Trees 484

18.1 Definition of B-trees 488 18.2 Basic operations on B-trees 491 18.3 Deleting a key from a B-tree 499 19 Fibonacci Heaps 505

19.1 Structure of Fibonacci heaps 507 19.2 Mergeable-heap operations 510

19.3 Decreasing a key and deleting a node 518 19.4 Bounding the maximum degree 523 20 van Emde Boas Trees 531

20.1 Preliminary approaches 532 20.2 A recursive structure 536 20.3 The van Emde Boas tree 545 21 Data Structures for Disjoint Sets 561

21.1 Disjoint-set operations 561

21.2 Linked-list representation of disjoint sets 564 21.3 Disjoint-set forests 568

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Introduction 587

22 Elementary Graph Algorithms 589 22.1 Representations of graphs 589 22.2 Breadth-first search 594 22.3 Depth-first search 603 22.4 Topological sort 612

22.5 Strongly connected components 615 23 Minimum Spanning Trees 624

23.1 Growing a minimum spanning tree 625 23.2 The algorithms of Kruskal and Prim 631

24 Single-Source Shortest Paths 643 24.1 The Bellman-Ford algorithm 651

24.2 Single-source shortest paths in directed acyclic graphs 655 24.3 Dijkstra’s algorithm 658

24.4 Difference constraints and shortest paths 664 24.5 Proofs of shortest-paths properties 671 25 All-Pairs Shortest Paths 684

25.1 Shortest paths and matrix multiplication 686 25.2 The Floyd-Warshall algorithm 693

25.3 Johnson’s algorithm for sparse graphs 700 26 Maximum Flow 708

26.1 Flow networks 709

26.2 The Ford-Fulkerson method 714 26.3 Maximum bipartite matching 732

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Introduction 769

27 Multithreaded Algorithms 772

27.1 The basics of dynamic multithreading 774 27.2 Multithreaded matrix multiplication 792 27.3 Multithreaded merge sort 797

28 Matrix Operations 813

28.1 Solving systems of linear equations 813 28.2 Inverting matrices 827

28.3 Symmetric positive-definite matrices and least-squares approximation 832

29 Linear Programming 843

29.1 Standard and slack forms 850

29.2 Formulating problems as linear programs 859 29.3 The simplex algorithm 864

29.4 Duality 879

29.5 The initial basic feasible solution 886

30 Polynomials and the FFT 898 30.1 Representing polynomials 900 30.2 The DFT and FFT 906

30.3 Efficient FFT implementations 915 31 Number-Theoretic Algorithms 926

31.1 Elementary number-theoretic notions 927 31.2 Greatest common divisor 933

31.3 Modular arithmetic 939

31.4 Solving modular linear equations 946 31.5 The Chinese remainder theorem 950 31.6 Powers of an element 954

31.7 The RSA public-key cryptosystem 958

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32.1 The naive string-matching algorithm 988 32.2 The Rabin-Karp algorithm 990

32.3 String matching with finite automata 995

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33.1 Line-segment properties 1015

33.2 Determining whether any pair of segments intersects 1021 33.3 Finding the convex hull 1029

33.4 Finding the closest pair of points 1039 34 NP-Completeness 1048

34.1 Polynomial time 1053

34.2 Polynomial-time verification 1061 34.3 NP-completeness and reducibility 1067 34.4 NP-completeness proofs 1078

34.5 NP-complete problems 1086 35 Approximation Algorithms 1106

35.1 The vertex-cover problem 1108 35.2 The traveling-salesman problem 1111 35.3 The set-covering problem 1117

35.4 Randomization and linear programming 1123 35.5 The subset-sum problem 1128

VIII Appendix: Mathematical Background Introduction 1143 A Summations 1145

A.1 Summation formulas and properties 1145 A.2 Bounding summations 1149

B Sets, Etc. 1158 B.1 Sets 1158 B.2 Relations 1163 B.3 Functions 1166 B.4 Graphs 1168 B.5 Trees 1173

C Counting and Probability 1183 C.1 Counting 1183

C.2 Probability 1189

C.3 Discrete random variables 1196

C.4 The geometric and binomial distributions 1201

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D.1 Matrices and matrix operations 1217 D.2 Basic matrix properties 1222

Bibliography 1231 Index 1251

Before there were computers, there were algorithms. But now that there are com- puters, there are even more algorithms, and algorithms lie at the heart of computing.

This book provides a comprehensive introduction to the modern study of com- puter algorithms. It presents many algorithms and covers them in considerable depth, yet makes their design and analysis accessible to all levels of readers. We have tried to keep explanations elementary without sacrificing depth of coverage or mathematical rigor.

Each chapter presents an algorithm, a design technique, an application area, or a related topic. Algorithms are described in English and in a pseudocode designed to be readable by anyone who has done a little programming. The book contains 244 figures—many with multiple parts—illustrating how the algorithms work. Since we emphasize efficiency as a design criterion, we include careful analyses of the running times of all our algorithms.

The text is intended primarily for use in undergraduate or graduate courses in algorithms or data structures. Because it discusses engineering issues in algorithm design, as well as mathematical aspects, it is equally well suited for self-study by technical professionals.

In this, the third edition, we have once again updated the entire book. The changes cover a broad spectrum, including new chapters, revised pseudocode, and a more active writing style.

To the teacher

We have designed this book to be both versatile and complete. You should find it useful for a variety of courses, from an undergraduate course in data structures up through a graduate course in algorithms. Because we have provided considerably more material than can fit in a typical one-term course, you can consider this book to be a “buffet” or “smorgasbord” from which you can pick and choose the material that best supports the course you wish to teach.

You should find it easy to organize your course around just the chapters you need. We have made chapters relatively self-contained, so that you need not worry about an unexpected and unnecessary dependence of one chapter on another. Each chapter presents the easier material first and the more difficult material later, with section boundaries marking natural stopping points. In an undergraduate course, you might use only the earlier sections from a chapter; in a graduate course, you might cover the entire chapter.

We have included 957 exercises and 158 problems. Each section ends with exer- cises, and each chapter ends with problems. The exercises are generally short ques- tions that test basic mastery of the material. Some are simple self-check thought exercises, whereas others are more substantial and are suitable as assigned home- work. The problems are more elaborate case studies that often introduce new ma- terial; they often consist of several questions that lead the student through the steps required to arrive at a solution.

Departing from our practice in previous editions of this book, we have made publicly available solutions to some, but by no means all, of the problems and ex- ercises. Our Web site, http://mitpress.mit.edu/algorithms/, links to these solutions.

You will want to check this site to make sure that it does not contain the solution to an exercise or problem that you plan to assign. We expect the set of solutions that we post to grow slowly over time, so you will need to check it each time you teach the course.

We have starred (?) the sections and exercises that are more suitable for graduate students than for undergraduates. A starred section is not necessarily more diffi- cult than an unstarred one, but it may require an understanding of more advanced mathematics. Likewise, starred exercises may require an advanced background or more than average creativity.

To the student

We hope that this textbook provides you with an enjoyable introduction to the field of algorithms. We have attempted to make every algorithm accessible and interesting. To help you when you encounter unfamiliar or difficult algorithms, we describe each one in a step-by-step manner. We also provide careful explanations of the mathematics needed to understand the analysis of the algorithms. If you already have some familiarity with a topic, you will find the chapters organized so that you can skim introductory sections and proceed quickly to the more advanced material.

This is a large book, and your class will probably cover only a portion of its material. We have tried, however, to make this a book that will be useful to you now as a course textbook and also later in your career as a mathematical desk reference or an engineering handbook.

What are the prerequisites for reading this book?

 You should have some programming experience. In particular, you should un- derstand recursive procedures and simple data structures such as arrays and linked lists.

 You should have some facility with mathematical proofs, and especially proofs by mathematical induction. A few portions of the book rely on some knowledge of elementary calculus. Beyond that, Parts I and VIII of this book teach you all the mathematical techniques you will need.

We have heard, loud and clear, the call to supply solutions to problems and exercises. Our Web site, http://mitpress.mit.edu/algorithms/, links to solutions for a few of the problems and exercises. Feel free to check your solutions against ours.

We ask, however, that you do not send your solutions to us.

To the professional

The wide range of topics in this book makes it an excellent handbook on algo- rithms. Because each chapter is relatively self-contained, you can focus in on the topics that most interest you.

Most of the algorithms we discuss have great practical utility. We therefore address implementation concerns and other engineering issues. We often provide practical alternatives to the few algorithms that are primarily of theoretical interest.

If you wish to implement any of the algorithms, you should find the transla- tion of our pseudocode into your favorite programming language to be a fairly straightforward task. We have designed the pseudocode to present each algorithm clearly and succinctly. Consequently, we do not address error-handling and other software-engineering issues that require specific assumptions about your program- ming environment. We attempt to present each algorithm simply and directly with- out allowing the idiosyncrasies of a particular programming language to obscure its essence.

We understand that if you are using this book outside of a course, then you might be unable to check your solutions to problems and exercises against solutions provided by an instructor. Our Web site, http://mitpress.mit.edu/algorithms/, links to solutions for some of the problems and exercises so that you can check your work. Please do not send your solutions to us.

To our colleagues

We have supplied an extensive bibliography and pointers to the current literature.

Each chapter ends with a set of chapter notes that give historical details and ref- erences. The chapter notes do not provide a complete reference to the whole field

of algorithms, however. Though it may be hard to believe for a book of this size, space constraints prevented us from including many interesting algorithms.

Despite myriad requests from students for solutions to problems and exercises, we have chosen as a matter of policy not to supply references for problems and exercises, to remove the temptation for students to look up a solution rather than to find it themselves.

Changes for the third edition

What has changed between the second and third editions of this book? The mag- nitude of the changes is on a par with the changes between the first and second editions. As we said about the second-edition changes, depending on how you look at it, the book changed either not much or quite a bit.

A quick look at the table of contents shows that most of the second-edition chap- ters and sections appear in the third edition. We removed two chapters and one section, but we have added three new chapters and two new sections apart from these new chapters.

We kept the hybrid organization from the first two editions. Rather than organiz- ing chapters by only problem domains or according only to techniques, this book has elements of both. It contains technique-based chapters on divide-and-conquer, dynamic programming, greedy algorithms, amortized analysis, NP-Completeness, and approximation algorithms. But it also has entire parts on sorting, on data structures for dynamic sets, and on algorithms for graph problems. We find that although you need to know how to apply techniques for designing and analyzing al- gorithms, problems seldom announce to you which techniques are most amenable to solving them.

Here is a summary of the most significant changes for the third edition:

 We added new chapters on van Emde Boas trees and multithreaded algorithms, and we have broken out material on matrix basics into its own appendix chapter.

 We revised the chapter on recurrences to more broadly cover the divide-and- conquer technique, and its first two sections apply divide-and-conquer to solve two problems. The second section of this chapter presents Strassen’s algorithm for matrix multiplication, which we have moved from the chapter on matrix operations.

 We removed two chapters that were rarely taught: binomial heaps and sorting networks. One key idea in the sorting networks chapter, the 0-1 principle, ap- pears in this edition within Problem 8-7 as the 0-1 sorting lemma for compare- exchange algorithms. The treatment of Fibonacci heaps no longer relies on binomial heaps as a precursor.

 We revised our treatment of dynamic programming and greedy algorithms. Dy- namic programming now leads off with a more interesting problem, rod cutting, than the assembly-line scheduling problem from the second edition. Further- more, we emphasize memoization a bit more than we did in the second edition, and we introduce the notion of the subproblem graph as a way to understand the running time of a dynamic-programming algorithm. In our opening exam- ple of greedy algorithms, the activity-selection problem, we get to the greedy algorithm more directly than we did in the second edition.

 The way we delete a node from binary search trees (which includes red-black trees) now guarantees that the node requested for deletion is the node that is actually deleted. In the first two editions, in certain cases, some other node would be deleted, with its contents moving into the node passed to the deletion procedure. With our new way to delete nodes, if other components of a program maintain pointers to nodes in the tree, they will not mistakenly end up with stale pointers to nodes that have been deleted.

 The material on flow networks now bases flows entirely on edges. This ap- proach is more intuitive than the net flow used in the first two editions.

 With the material on matrix basics and Strassen’s algorithm moved to other chapters, the chapter on matrix operations is smaller than in the second edition.

 We have modified our treatment of the Knuth-Morris-Pratt string-matching al- gorithm.

 We corrected several errors. Most of these errors were posted on our Web site of second-edition errata, but a few were not.

 Based on many requests, we changed the syntax (as it were) of our pseudocode.

We now use “D ” to indicate assignment and “= =” to test for equality, just as C, C++, Java, and Python do. Likewise, we have eliminated the keywords do and then and adopted “//” as our comment-to-end-of-line symbol. We also now use dot-notation to indicate object attributes. Our pseudocode remains procedural, rather than object-oriented. In other words, rather than running methods on objects, we simply call procedures, passing objects as parameters.

 We added 100 new exercises and 28 new problems. We also updated many bibliography entries and added several new ones.

 Finally, we went through the entire book and rewrote sentences, paragraphs, and sections to make the writing clearer and more active.

Web site

You can use our Web site, http://mitpress.mit.edu/algorithms/, to obtain supple- mentary information and to communicate with us. The Web site links to a list of known errors, solutions to selected exercises and problems, and (of course) a list explaining the corny professor jokes, as well as other content that we might add.

The Web site also tells you how to report errors or make suggestions.

How we produced this book

Like the second edition, the third edition was produced in L^ATEX 2". We used the Times font with mathematics typeset using the MathTime Pro 2 fonts. We thank Michael Spivak from Publish or Perish, Inc., Lance Carnes from Personal TeX, Inc., and Tim Tregubov from Dartmouth College for technical support. As in the previous two editions, we compiled the index using Windex, a C program that we wrote, and the bibliography was produced with BIBTEX. The PDF files for this book were created on a MacBook running OS 10.5.

We drew the illustrations for the third edition using MacDraw Pro, with some of the mathematical expressions in illustrations laid in with the psfrag package for L^ATEX 2". Unfortunately, MacDraw Pro is legacy software, having not been marketed for over a decade now. Happily, we still have a couple of Macintoshes that can run the Classic environment under OS 10.4, and hence they can run Mac- Draw Pro—mostly. Even under the Classic environment, we find MacDraw Pro to be far easier to use than any other drawing software for the types of illustrations that accompany computer-science text, and it produces beautiful output.¹ Who knows how long our pre-Intel Macs will continue to run, so if anyone from Apple is listening: Please create an OS X-compatible version of MacDraw Pro!

Acknowledgments for the third edition

We have been working with the MIT Press for over two decades now, and what a terrific relationship it has been! We thank Ellen Faran, Bob Prior, Ada Brunstein, and Mary Reilly for their help and support.

We were geographically distributed while producing the third edition, working in the Dartmouth College Department of Computer Science, the MIT Computer

1We investigated several drawing programs that run under Mac OS X, but all had significant short- comings compared with MacDraw Pro. We briefly attempted to produce the illustrations for this book with a different, well known drawing program. We found that it took at least five times as long to produce each illustration as it took with MacDraw Pro, and the resulting illustrations did not look as good. Hence the decision to revert to MacDraw Pro running on older Macintoshes.

Science and Artificial Intelligence Laboratory, and the Columbia University De- partment of Industrial Engineering and Operations Research. We thank our re- spective universities and colleagues for providing such supportive and stimulating environments.

Julie Sussman, P.P.A., once again bailed us out as the technical copyeditor. Time and again, we were amazed at the errors that eluded us, but that Julie caught. She also helped us improve our presentation in several places. If there is a Hall of Fame for technical copyeditors, Julie is a sure-fire, first-ballot inductee. She is nothing short of phenomenal. Thank you, thank you, thank you, Julie! Priya Natarajan also found some errors that we were able to correct before this book went to press. Any errors that remain (and undoubtedly, some do) are the responsibility of the authors (and probably were inserted after Julie read the material).

The treatment for van Emde Boas trees derives from Erik Demaine’s notes, which were in turn influenced by Michael Bender. We also incorporated ideas from Javed Aslam, Bradley Kuszmaul, and Hui Zha into this edition.

The chapter on multithreading was based on notes originally written jointly with Harald Prokop. The material was influenced by several others working on the Cilk project at MIT, including Bradley Kuszmaul and Matteo Frigo. The design of the multithreaded pseudocode took its inspiration from the MIT Cilk extensions to C and by Cilk Arts’s Cilk++ extensions to C++.

We also thank the many readers of the first and second editions who reported errors or submitted suggestions for how to improve this book. We corrected all the bona fide errors that were reported, and we incorporated as many suggestions as we could. We rejoice that the number of such contributors has grown so great that we must regret that it has become impractical to list them all.

Finally, we thank our wives—Nicole Cormen, Wendy Leiserson, Gail Rivest, and Rebecca Ivry—and our children—Ricky, Will, Debby, and Katie Leiserson;

Alex and Christopher Rivest; and Molly, Noah, and Benjamin Stein—for their love and support while we prepared this book. The patience and encouragement of our families made this project possible. We affectionately dedicate this book to them.

THOMAS H. CORMEN Lebanon, New Hampshire

CHARLESE. LEISERSON Cambridge, Massachusetts

RONALDL. RIVEST Cambridge, Massachusetts

CLIFFORD STEIN New York, New York

February 2009

Third Edition

This part will start you thinking about designing and analyzing algorithms. It is intended to be a gentle introduction to how we specify algorithms, some of the design strategies we will use throughout this book, and many of the fundamental ideas used in algorithm analysis. Later parts of this book will build upon this base.

Chapter 1 provides an overview of algorithms and their place in modern com- puting systems. This chapter defines what an algorithm is and lists some examples.

It also makes a case that we should consider algorithms as a technology, along- side technologies such as fast hardware, graphical user interfaces, object-oriented systems, and networks.

In Chapter 2, we see our first algorithms, which solve the problem of sorting a sequence of n numbers. They are written in a pseudocode which, although not directly translatable to any conventional programming language, conveys the struc- ture of the algorithm clearly enough that you should be able to implement it in the language of your choice. The sorting algorithms we examine are insertion sort, which uses an incremental approach, and merge sort, which uses a recursive tech- nique known as “divide-and-conquer.” Although the time each requires increases with the value of n, the rate of increase differs between the two algorithms. We determine these running times in Chapter 2, and we develop a useful notation to express them.

Chapter 3 precisely defines this notation, which we call asymptotic notation. It starts by defining several asymptotic notations, which we use for bounding algo- rithm running times from above and/or below. The rest of Chapter 3 is primarily a presentation of mathematical notation, more to ensure that your use of notation matches that in this book than to teach you new mathematical concepts.

Chapter 4 delves further into the divide-and-conquer method introduced in Chapter 2. It provides additional examples of divide-and-conquer algorithms, in- cluding Strassen’s surprising method for multiplying two square matrices. Chap- ter 4 contains methods for solving recurrences, which are useful for describing the running times of recursive algorithms. One powerful technique is the “mas- ter method,” which we often use to solve recurrences that arise from divide-and- conquer algorithms. Although much of Chapter 4 is devoted to proving the cor- rectness of the master method, you may skip this proof yet still employ the master method.

Chapter 5 introduces probabilistic analysis and randomized algorithms. We typ- ically use probabilistic analysis to determine the running time of an algorithm in cases in which, due to the presence of an inherent probability distribution, the running time may differ on different inputs of the same size. In some cases, we assume that the inputs conform to a known probability distribution, so that we are averaging the running time over all possible inputs. In other cases, the probability distribution comes not from the inputs but from random choices made during the course of the algorithm. An algorithm whose behavior is determined not only by its input but by the values produced by a random-number generator is a randomized algorithm. We can use randomized algorithms to enforce a probability distribution on the inputs—thereby ensuring that no particular input always causes poor perfor- mance—or even to bound the error rate of algorithms that are allowed to produce incorrect results on a limited basis.

Appendices A–D contain other mathematical material that you will find helpful as you read this book. You are likely to have seen much of the material in the appendix chapters before having read this book (although the specific definitions and notational conventions we use may differ in some cases from what you have seen in the past), and so you should think of the Appendices as reference material.

On the other hand, you probably have not already seen most of the material in Part I. All the chapters in Part I and the Appendices are written with a tutorial flavor.

What are algorithms? Why is the study of algorithms worthwhile? What is the role of algorithms relative to other technologies used in computers? In this chapter, we will answer these questions.

1.1 Algorithms

Informally, an algorithm is any well-defined computational procedure that takes some value, or set of values, as input and produces some value, or set of values, as output. An algorithm is thus a sequence of computational steps that transform the input into the output.

We can also view an algorithm as a tool for solving a well-specified computa- tional problem. The statement of the problem specifies in general terms the desired input/output relationship. The algorithm describes a specific computational proce- dure for achieving that input/output relationship.

For example, we might need to sort a sequence of numbers into nondecreasing order. This problem arises frequently in practice and provides fertile ground for introducing many standard design techniques and analysis tools. Here is how we formally define the sorting problem:

Input: A sequence of n numbers ha1; a2; : : : ; ani.

Output: A permutation (reordering) ha⁰₁; a₂⁰; : : : ; a_n⁰i of the input sequence such that a⁰₁ a⁰₂     a_n⁰.

For example, given the input sequence h31; 41; 59; 26; 41; 58i, a sorting algorithm returns as output the sequence h26; 31; 41; 41; 58; 59i. Such an input sequence is called an instance of the sorting problem. In general, an instance of a problem consists of the input (satisfying whatever constraints are imposed in the problem statement) needed to compute a solution to the problem.

Because many programs use it as an intermediate step, sorting is a fundamental operation in computer science. As a result, we have a large number of good sorting algorithms at our disposal. Which algorithm is best for a given application depends on—among other factors—the number of items to be sorted, the extent to which the items are already somewhat sorted, possible restrictions on the item values, the architecture of the computer, and the kind of storage devices to be used: main memory, disks, or even tapes.

An algorithm is said to be correct if, for every input instance, it halts with the correct output. We say that a correct algorithm solves the given computational problem. An incorrect algorithm might not halt at all on some input instances, or it might halt with an incorrect answer. Contrary to what you might expect, incorrect algorithms can sometimes be useful, if we can control their error rate. We shall see an example of an algorithm with a controllable error rate in Chapter 31 when we study algorithms for finding large prime numbers. Ordinarily, however, we shall be concerned only with correct algorithms.

An algorithm can be specified in English, as a computer program, or even as a hardware design. The only requirement is that the specification must provide a precise description of the computational procedure to be followed.

What kinds of problems are solved by algorithms?

Sorting is by no means the only computational problem for which algorithms have been developed. (You probably suspected as much when you saw the size of this book.) Practical applications of algorithms are ubiquitous and include the follow- ing examples:

 The Human Genome Project has made great progress toward the goals of iden- tifying all the 100,000 genes in human DNA, determining the sequences of the 3 billion chemical base pairs that make up human DNA, storing this informa- tion in databases, and developing tools for data analysis. Each of these steps requires sophisticated algorithms. Although the solutions to the various prob- lems involved are beyond the scope of this book, many methods to solve these biological problems use ideas from several of the chapters in this book, thereby enabling scientists to accomplish tasks while using resources efficiently. The savings are in time, both human and machine, and in money, as more informa- tion can be extracted from laboratory techniques.

 The Internet enables people all around the world to quickly access and retrieve large amounts of information. With the aid of clever algorithms, sites on the Internet are able to manage and manipulate this large volume of data. Examples of problems that make essential use of algorithms include finding good routes on which the data will travel (techniques for solving such problems appear in

Chapter 24), and using a search engine to quickly find pages on which particular information resides (related techniques are in Chapters 11 and 32).

 Electronic commerce enables goods and services to be negotiated and ex- changed electronically, and it depends on the privacy of personal informa- tion such as credit card numbers, passwords, and bank statements. The core technologies used in electronic commerce include public-key cryptography and digital signatures (covered in Chapter 31), which are based on numerical algo- rithms and number theory.

 Manufacturing and other commercial enterprises often need to allocate scarce resources in the most beneficial way. An oil company may wish to know where to place its wells in order to maximize its expected profit. A political candidate may want to determine where to spend money buying campaign advertising in order to maximize the chances of winning an election. An airline may wish to assign crews to flights in the least expensive way possible, making sure that each flight is covered and that government regulations regarding crew schedul- ing are met. An Internet service provider may wish to determine where to place additional resources in order to serve its customers more effectively. All of these are examples of problems that can be solved using linear programming, which we shall study in Chapter 29.

Although some of the details of these examples are beyond the scope of this book, we do give underlying techniques that apply to these problems and problem areas. We also show how to solve many specific problems, including the following:

 We are given a road map on which the distance between each pair of adjacent intersections is marked, and we wish to determine the shortest route from one intersection to another. The number of possible routes can be huge, even if we disallow routes that cross over themselves. How do we choose which of all possible routes is the shortest? Here, we model the road map (which is itself a model of the actual roads) as a graph (which we will meet in Part VI and Appendix B), and we wish to find the shortest path from one vertex to another in the graph. We shall see how to solve this problem efficiently in Chapter 24.

 We are given two ordered sequences of symbols, X D hx1; x2; : : : ; xmi and Y D hy1; y2; : : : ; yni, and we wish to find a longest common subsequence of X and Y . A subsequence of X is just X with some (or possibly all or none) of its elements removed. For example, one subsequence of hA; B; C; D; E; F; Gi would be hB; C; E; Gi. The length of a longest common subsequence of X and Y gives one measure of how similar these two sequences are. For example, if the two sequences are base pairs in DNA strands, then we might consider them similar if they have a long common subsequence. If X has m symbols and Y has n symbols, then X and Y have 2^m and 2ⁿ possible subsequences,

respectively. Selecting all possible subsequences of X and Y and matching them up could take a prohibitively long time unless m and n are very small.

We shall see in Chapter 15 how to use a general technique known as dynamic programming to solve this problem much more efficiently.

 We are given a mechanical design in terms of a library of parts, where each part may include instances of other parts, and we need to list the parts in order so that each part appears before any part that uses it. If the design comprises n parts, then there are nŠ possible orders, where nŠ denotes the factorial function.

Because the factorial function grows faster than even an exponential function, we cannot feasibly generate each possible order and then verify that, within that order, each part appears before the parts using it (unless we have only a few parts). This problem is an instance of topological sorting, and we shall see in Chapter 22 how to solve this problem efficiently.

 We are given n points in the plane, and we wish to find the convex hull of these points. The convex hull is the smallest convex polygon containing the points. Intuitively, we can think of each point as being represented by a nail sticking out from a board. The convex hull would be represented by a tight rubber band that surrounds all the nails. Each nail around which the rubber band makes a turn is a vertex of the convex hull. (See Figure 33.6 on page 1029 for an example.) Any of the 2ⁿ subsets of the points might be the vertices of the convex hull. Knowing which points are vertices of the convex hull is not quite enough, either, since we also need to know the order in which they appear. There are many choices, therefore, for the vertices of the convex hull.

Chapter 33 gives two good methods for finding the convex hull.

These lists are far from exhaustive (as you again have probably surmised from this book’s heft), but exhibit two characteristics that are common to many interest- ing algorithmic problems:

1. They have many candidate solutions, the overwhelming majority of which do not solve the problem at hand. Finding one that does, or one that is “best,” can present quite a challenge.

2. They have practical applications. Of the problems in the above list, finding the shortest path provides the easiest examples. A transportation firm, such as a trucking or railroad company, has a financial interest in finding shortest paths through a road or rail network because taking shorter paths results in lower labor and fuel costs. Or a routing node on the Internet may need to find the shortest path through the network in order to route a message quickly. Or a person wishing to drive from New York to Boston may want to find driving directions from an appropriate Web site, or she may use her GPS while driving.

Not every problem solved by algorithms has an easily identified set of candidate solutions. For example, suppose we are given a set of numerical values represent- ing samples of a signal, and we want to compute the discrete Fourier transform of these samples. The discrete Fourier transform converts the time domain to the fre- quency domain, producing a set of numerical coefficients, so that we can determine the strength of various frequencies in the sampled signal. In addition to lying at the heart of signal processing, discrete Fourier transforms have applications in data compression and multiplying large polynomials and integers. Chapter 30 gives an efficient algorithm, the fast Fourier transform (commonly called the FFT), for this problem, and the chapter also sketches out the design of a hardware circuit to compute the FFT.

Data structures

This book also contains several data structures. A data structure is a way to store and organize data in order to facilitate access and modifications. No single data structure works well for all purposes, and so it is important to know the strengths and limitations of several of them.

Technique

Although you can use this book as a “cookbook” for algorithms, you may someday encounter a problem for which you cannot readily find a published algorithm (many of the exercises and problems in this book, for example). This book will teach you techniques of algorithm design and analysis so that you can develop algorithms on your own, show that they give the correct answer, and understand their efficiency.

Different chapters address different aspects of algorithmic problem solving. Some chapters address specific problems, such as finding medians and order statistics in Chapter 9, computing minimum spanning trees in Chapter 23, and determining a maximum flow in a network in Chapter 26. Other chapters address techniques, such as divide-and-conquer in Chapter 4, dynamic programming in Chapter 15, and amortized analysis in Chapter 17.

Hard problems

Most of this book is about efficient algorithms. Our usual measure of efficiency is speed, i.e., how long an algorithm takes to produce its result. There are some problems, however, for which no efficient solution is known. Chapter 34 studies an interesting subset of these problems, which are known as NP-complete.

Why are NP-complete problems interesting? First, although no efficient algo- rithm for an NP-complete problem has ever been found, nobody has ever proven

that an efficient algorithm for one cannot exist. In other words, no one knows whether or not efficient algorithms exist for NP-complete problems. Second, the set of NP-complete problems has the remarkable property that if an efficient algo- rithm exists for any one of them, then efficient algorithms exist for all of them. This relationship among the NP-complete problems makes the lack of efficient solutions all the more tantalizing. Third, several NP-complete problems are similar, but not identical, to problems for which we do know of efficient algorithms. Computer scientists are intrigued by how a small change to the problem statement can cause a big change to the efficiency of the best known algorithm.

You should know about NP-complete problems because some of them arise sur- prisingly often in real applications. If you are called upon to produce an efficient algorithm for an NP-complete problem, you are likely to spend a lot of time in a fruitless search. If you can show that the problem is NP-complete, you can instead spend your time developing an efficient algorithm that gives a good, but not the best possible, solution.

As a concrete example, consider a delivery company with a central depot. Each day, it loads up each delivery truck at the depot and sends it around to deliver goods to several addresses. At the end of the day, each truck must end up back at the depot so that it is ready to be loaded for the next day. To reduce costs, the company wants to select an order of delivery stops that yields the lowest overall distance traveled by each truck. This problem is the well-known “traveling-salesman problem,” and it is NP-complete. It has no known efficient algorithm. Under certain assumptions, however, we know of efficient algorithms that give an overall distance which is not too far above the smallest possible. Chapter 35 discusses such “approximation algorithms.”

Parallelism

For many years, we could count on processor clock speeds increasing at a steady rate. Physical limitations present a fundamental roadblock to ever-increasing clock speeds, however: because power density increases superlinearly with clock speed, chips run the risk of melting once their clock speeds become high enough. In order to perform more computations per second, therefore, chips are being designed to contain not just one but several processing “cores.” We can liken these multicore computers to several sequential computers on a single chip; in other words, they are a type of “parallel computer.” In order to elicit the best performance from multicore computers, we need to design algorithms with parallelism in mind. Chapter 27 presents a model for “multithreaded” algorithms, which take advantage of multiple cores. This model has advantages from a theoretical standpoint, and it forms the basis of several successful computer programs, including a championship chess program.

Exercises

1.1-1

Give a real-world example that requires sorting or a real-world example that re- quires computing a convex hull.

1.1-2

Other than speed, what other measures of efficiency might one use in a real-world setting?

1.1-3

Select a data structure that you have seen previously, and discuss its strengths and limitations.

1.1-4

How are the shortest-path and traveling-salesman problems given above similar?

How are they different?

1.1-5

Come up with a real-world problem in which only the best solution will do. Then come up with one in which a solution that is “approximately” the best is good enough.

1.2 Algorithms as a technology

Suppose computers were infinitely fast and computer memory was free. Would you have any reason to study algorithms? The answer is yes, if for no other reason than that you would still like to demonstrate that your solution method terminates and does so with the correct answer.

If computers were infinitely fast, any correct method for solving a problem would do. You would probably want your implementation to be within the bounds of good software engineering practice (for example, your implementation should be well designed and documented), but you would most often use whichever method was the easiest to implement.

Of course, computers may be fast, but they are not infinitely fast. And memory may be inexpensive, but it is not free. Computing time is therefore a bounded resource, and so is space in memory. You should use these resources wisely, and algorithms that are efficient in terms of time or space will help you do so.

Efficiency

Different algorithms devised to solve the same problem often differ dramatically in their efficiency. These differences can be much more significant than differences due to hardware and software.

As an example, in Chapter 2, we will see two algorithms for sorting. The first, known as insertion sort, takes time roughly equal to c1n²to sort n items, where c1

is a constant that does not depend on n. That is, it takes time roughly proportional to n². The second, merge sort, takes time roughly equal to c2n lg n, where lg n stands for log₂n and c2is another constant that also does not depend on n. Inser- tion sort typically has a smaller constant factor than merge sort, so that c1 < c2. We shall see that the constant factors can have far less of an impact on the running time than the dependence on the input size n. Let’s write insertion sort’s running time as c1n  n and merge sort’s running time as c₂n  lg n. Then we see that where insertion sort has a factor of n in its running time, merge sort has a factor of lg n, which is much smaller. (For example, when n D 1000, lg n is approximately 10, and when n equals one million, lg n is approximately only 20.) Although insertion sort usually runs faster than merge sort for small input sizes, once the input size n becomes large enough, merge sort’s advantage of lg n vs. n will more than com- pensate for the difference in constant factors. No matter how much smaller c1 is than c2, there will always be a crossover point beyond which merge sort is faster.

For a concrete example, let us pit a faster computer (computer A) running inser- tion sort against a slower computer (computer B) running merge sort. They each must sort an array of 10 million numbers. (Although 10 million numbers might seem like a lot, if the numbers are eight-byte integers, then the input occupies about 80 megabytes, which fits in the memory of even an inexpensive laptop com- puter many times over.) Suppose that computer A executes 10 billion instructions per second (faster than any single sequential computer at the time of this writing) and computer B executes only 10 million instructions per second, so that com- puter A is 1000 times faster than computer B in raw computing power. To make the difference even more dramatic, suppose that the world’s craftiest programmer codes insertion sort in machine language for computer A, and the resulting code requires 2n² instructions to sort n numbers. Suppose further that just an average programmer implements merge sort, using a high-level language with an inefficient compiler, with the resulting code taking 50n lg n instructions. To sort 10 million numbers, computer A takes

2  .10⁷/²instructions

10¹⁰instructions/second D 20,000 seconds (more than 5.5 hours) ; while computer B takes

50  10⁷lg 10⁷instructions

10⁷ instructions/second  1163 seconds (less than 20 minutes) :

By using an algorithm whose running time grows more slowly, even with a poor compiler, computer B runs more than 17 times faster than computer A! The advan- tage of merge sort is even more pronounced when we sort 100 million numbers:

where insertion sort takes more than 23 days, merge sort takes under four hours.

In general, as the problem size increases, so does the relative advantage of merge sort.

Algorithms and other technologies

The example above shows that we should consider algorithms, like computer hard- ware, as a technology. Total system performance depends on choosing efficient algorithms as much as on choosing fast hardware. Just as rapid advances are being made in other computer technologies, they are being made in algorithms as well.

You might wonder whether algorithms are truly that important on contemporary computers in light of other advanced technologies, such as

 advanced computer architectures and fabrication technologies,

 easy-to-use, intuitive, graphical user interfaces (GUIs),

 object-oriented systems,

 integrated Web technologies, and

 fast networking, both wired and wireless.

The answer is yes. Although some applications do not explicitly require algorith- mic content at the application level (such as some simple, Web-based applications), many do. For example, consider a Web-based service that determines how to travel from one location to another. Its implementation would rely on fast hardware, a graphical user interface, wide-area networking, and also possibly on object ori- entation. However, it would also require algorithms for certain operations, such as finding routes (probably using a shortest-path algorithm), rendering maps, and interpolating addresses.

Moreover, even an application that does not require algorithmic content at the application level relies heavily upon algorithms. Does the application rely on fast hardware? The hardware design used algorithms. Does the application rely on graphical user interfaces? The design of any GUI relies on algorithms. Does the application rely on networking? Routing in networks relies heavily on algorithms.

Was the application written in a language other than machine code? Then it was processed by a compiler, interpreter, or assembler, all of which make extensive use

of algorithms. Algorithms are at the core of most technologies used in contempo- rary computers.

Furthermore, with the ever-increasing capacities of computers, we use them to solve larger problems than ever before. As we saw in the above comparison be- tween insertion sort and merge sort, it is at larger problem sizes that the differences in efficiency between algorithms become particularly prominent.

Having a solid base of algorithmic knowledge and technique is one characteristic that separates the truly skilled programmers from the novices. With modern com- puting technology, you can accomplish some tasks without knowing much about algorithms, but with a good background in algorithms, you can do much, much more.

Exercises

1.2-1

Give an example of an application that requires algorithmic content at the applica- tion level, and discuss the function of the algorithms involved.

1.2-2

Suppose we are comparing implementations of insertion sort and merge sort on the same machine. For inputs of size n, insertion sort runs in 8n²steps, while merge sort runs in 64n lg n steps. For which values of n does insertion sort beat merge sort?

1.2-3

What is the smallest value of n such that an algorithm whose running time is 100n² runs faster than an algorithm whose running time is 2ⁿon the same machine?

Problems

1-1 Comparison of running times

For each function f .n/ and time t in the following table, determine the largest size n of a problem that can be solved in time t , assuming that the algorithm to solve the problem takes f .n/ microseconds.

1 1 1 1 1 1 1

second minute hour day month year century

lg np n n n lg n

n² n³ 2ⁿ nŠ

Chapter notes

There are many excellent texts on the general topic of algorithms, including those by Aho, Hopcroft, and Ullman [5, 6]; Baase and Van Gelder [28]; Brassard and Bratley [54]; Dasgupta, Papadimitriou, and Vazirani [82]; Goodrich and Tamassia [148]; Hofri [175]; Horowitz, Sahni, and Rajasekaran [181]; Johnsonbaugh and Schaefer [193]; Kingston [205]; Kleinberg and Tardos [208]; Knuth [209, 210, 211]; Kozen [220]; Levitin [235]; Manber [242]; Mehlhorn [249, 250, 251]; Pur- dom and Brown [287]; Reingold, Nievergelt, and Deo [293]; Sedgewick [306];

Sedgewick and Flajolet [307]; Skiena [318]; and Wilf [356]. Some of the more practical aspects of algorithm design are discussed by Bentley [42, 43] and Gonnet [145]. Surveys of the field of algorithms can also be found in the Handbook of The- oretical Computer Science, Volume A [342] and the CRC Algorithms and Theory of Computation Handbook [25]. Overviews of the algorithms used in computational biology can be found in textbooks by Gusfield [156], Pevzner [275], Setubal and Meidanis [310], and Waterman [350].

This chapter will familiarize you with the framework we shall use throughout the book to think about the design and analysis of algorithms. It is self-contained, but it does include several references to material that we introduce in Chapters 3 and 4.

(It also contains several summations, which Appendix A shows how to solve.) We begin by examining the insertion sort algorithm to solve the sorting problem introduced in Chapter 1. We define a “pseudocode” that should be familiar to you if you have done computer programming, and we use it to show how we shall specify our algorithms. Having specified the insertion sort algorithm, we then argue that it correctly sorts, and we analyze its running time. The analysis introduces a notation that focuses on how that time increases with the number of items to be sorted.

Following our discussion of insertion sort, we introduce the divide-and-conquer approach to the design of algorithms and use it to develop an algorithm called merge sort. We end with an analysis of merge sort’s running time.

2.1 Insertion sort

Our first algorithm, insertion sort, solves the sorting problem introduced in Chap- ter 1:

Input: A sequence of n numbers ha1; a2; : : : ; ani.

Output: A permutation (reordering) ha⁰₁; a⁰₂; : : : ; a_n⁰i of the input sequence such that a⁰₁ a⁰₂     a_n⁰.

The numbers that we wish to sort are also known as the keys. Although conceptu- ally we are sorting a sequence, the input comes to us in the form of an array with n elements.

In this book, we shall typically describe algorithms as programs written in a pseudocode that is similar in many respects to C, C++, Java, Python, or Pascal. If you have been introduced to any of these languages, you should have little trouble

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Figure 2.1 Sorting a hand of cards using insertion sort.

reading our algorithms. What separates pseudocode from “real” code is that in pseudocode, we employ whatever expressive method is most clear and concise to specify a given algorithm. Sometimes, the clearest method is English, so do not be surprised if you come across an English phrase or sentence embedded within a section of “real” code. Another difference between pseudocode and real code is that pseudocode is not typically concerned with issues of software engineering.

Issues of data abstraction, modularity, and error handling are often ignored in order to convey the essence of the algorithm more concisely.

We start with insertion sort, which is an efficient algorithm for sorting a small number of elements. Insertion sort works the way many people sort a hand of playing cards. We start with an empty left hand and the cards face down on the table. We then remove one card at a time from the table and insert it into the correct position in the left hand. To find the correct position for a card, we compare it with each of the cards already in the hand, from right to left, as illustrated in Figure 2.1. At all times, the cards held in the left hand are sorted, and these cards were originally the top cards of the pile on the table.

We present our pseudocode for insertion sort as a procedure called INSERTION- SORT, which takes as a parameter an array AŒ1 : : n containing a sequence of length n that is to be sorted. (In the code, the number n of elements in A is denoted by A: length.) The algorithm sorts the input numbers in place: it rearranges the numbers within the array A, with at most a constant number of them stored outside the array at any time. The input array A contains the sorted output sequence when the INSERTION-SORT procedure is finished.

1 2 3 4 5 6

5 2 4 6 1 3

(a)

1 2 3 4 5 6

2 5 4 6 1 3

(b)

1 2 3 4 5 6

2 4 5 6 1 3

(c)

1 2 3 4 5 6

2 4 5 6 1 3

(d)

1 2 3 4 5 6

2 4 5 6

1 3

(e)

1 2 3 4 5 6

2 4 5 6

1 3

(f)

Figure 2.2 The operation of INSERTION-SORTon the array A D h5; 2; 4; 6; 1; 3i. Array indices appear above the rectangles, and values stored in the array positions appear within the rectangles.

(a)–(e) The iterations of the for loop of lines 1–8. In each iteration, the black rectangle holds the key taken from AŒj , which is compared with the values in shaded rectangles to its left in the test of line 5. Shaded arrows show array values moved one position to the right in line 6, and black arrows indicate where the key moves to in line 8. (f) The final sorted array.

INSERTION-SORT.A/

1 forj D 2 to A:length 2 key D AŒj 

3 //Insert AŒj  into the sorted sequence AŒ1 : : j  1.

4 i D j  1

5 whilei > 0 and AŒi  > key

6 AŒi C 1 D AŒi 

7 i D i  1

8 AŒi C 1 D key

Loop invariants and the correctness of insertion sort

Figure 2.2 shows how this algorithm works for A D h5; 2; 4; 6; 1; 3i. The in- dex j indicates the “current card” being inserted into the hand. At the beginning of each iteration of the for loop, which is indexed by j , the subarray consisting of elements AŒ1 : : j  1 constitutes the currently sorted hand, and the remaining subarray AŒj C 1 : : n corresponds to the pile of cards still on the table. In fact, elements AŒ1 : : j  1 are the elements originally in positions 1 through j  1, but now in sorted order. We state these properties of AŒ1 : : j  1 formally as a loop invariant:

At the start of each iteration of the for loop of lines 1–8, the subarray AŒ1 : : j 1 consists of the elements originally in AŒ1 : : j  1, but in sorted order.

We use loop invariants to help us understand why an algorithm is correct. We must show three things about a loop invariant:

Initialization: It is true prior to the first iteration of the loop.

Maintenance: If it is true before an iteration of the loop, it remains true before the next iteration.

Termination: When the loop terminates, the invariant gives us a useful property that helps show that the algorithm is correct.

When the first two properties hold, the loop invariant is true prior to every iteration of the loop. (Of course, we are free to use established facts other than the loop invariant itself to prove that the loop invariant remains true before each iteration.) Note the similarity to mathematical induction, where to prove that a property holds, you prove a base case and an inductive step. Here, showing that the invariant holds before the first iteration corresponds to the base case, and showing that the invariant holds from iteration to iteration corresponds to the inductive step.

The third property is perhaps the most important one, since we are using the loop invariant to show correctness. Typically, we use the loop invariant along with the condition that caused the loop to terminate. The termination property differs from how we usually use mathematical induction, in which we apply the inductive step infinitely; here, we stop the “induction” when the loop terminates.

Let us see how these properties hold for insertion sort.

Initialization: We start by showing that the loop invariant holds before the first loop iteration, when j D 2.¹ The subarray AŒ1 : : j  1, therefore, consists of just the single element AŒ1, which is in fact the original element in AŒ1.

Moreover, this subarray is sorted (trivially, of course), which shows that the loop invariant holds prior to the first iteration of the loop.

Maintenance: Next, we tackle the second property: showing that each iteration maintains the loop invariant. Informally, the body of the for loop works by moving AŒj  1, AŒj  2, AŒj  3, and so on by one position to the right until it finds the proper position for AŒj  (lines 4–7), at which point it inserts the value of AŒj  (line 8). The subarray AŒ1 : : j  then consists of the elements originally in AŒ1 : : j , but in sorted order. Incrementing j for the next iteration of the for loop then preserves the loop invariant.

A more formal treatment of the second property would require us to state and show a loop invariant for the while loop of lines 5–7. At this point, however,

1When the loop is a for loop, the moment at which we check the loop invariant just prior to the first iteration is immediately after the initial assignment to the loop-counter variable and just before the first test in the loop header. In the case of INSERTION-SORT, this time is after assigning 2 to the variable j but before the first test of whether j  A: length.