个 AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS SECOND E DIT I O N ROBERT SEDGEWICK PHILIPPE FLAJOLET www.it-ebooks.info
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FOREWORD P EOPLE who analyze algorithms have double happiness.First of all they experience the sheer beauty of elegant mathematical patterns that sur- round elegant computational procedures.Then they receive a practical payoff when their theories make it possible to get other jobs done more quickly and more economically. Mathematical models have been a crucial inspiration for all scientific activity,even though they are only approximate idealizations of real-world phenomena.Inside a computer,such models are more relevant than ever be- fore,because computer programs create artificial worlds in which mathemat- ical models often apply precisely.I think that's why I got hooked on analysis of algorithms when I was a graduate student,and why the subject has been my main life's work ever since. Until recently,however,analysis of algorithms has largely remained the preserve of graduate students and post-graduate researchers.Its concepts are not really esoteric or difficult,but they are relatively new,so it has taken awhile to sort out the best ways of learning them and using them. Now,after more than 40 years of development,algorithmic analysis has matured to the point where it is ready to take its place in the standard com- puter science curriculum.The appearance of this long-awaited textbook by Sedgewick and Flajolet is therefore most welcome.Its authors are not only worldwide leaders of the field,they also are masters of exposition.I am sure that every serious computer scientist will find this book rewarding in many ways. D.E.Knuth www.it-ebooks.info
F O R EW O R D P EOPLE who analyze algorithms have double happiness. First of all they experience the sheer beauty of elegant mathematical patterns that surround elegant computational procedures. Ļen they receive a practical payoff when their theories make it possible to get other jobs done more quickly and more economically. Mathematical models have been a crucial inspiration for all scientiŀc activity, even though they are only approximate idealizations of real-world phenomena. Inside a computer, such models are more relevant than ever before, because computer programs create artiŀcial worlds in which mathematical models often apply precisely. I think that’s why I got hooked on analysis of algorithms when I was a graduate student, and why the subject has been my main life’s work ever since. Until recently, however, analysis of algorithms has largely remained the preserve of graduate students and post-graduate researchers. Its concepts are not really esoteric or difficult, but they are relatively new, so it has taken awhile to sort out the best ways of learning them and using them. Now, after more than 40 years of development, algorithmic analysis has matured to the point where it is ready to take its place in the standard computer science curriculum. Ļe appearance of this long-awaited textbook by Sedgewick and Flajolet is therefore most welcome. Its authors are not only worldwide leaders of the ŀeld, they also are masters of exposition. I am sure that every serious computer scientist will ŀnd this book rewarding in many ways. D. E. Knuth www.it-ebooks.info
PREFACE ookbe thoughvvof th niques used in the mathematical analysis of algorithms.The material covered draws from classical mathematical topics,including discrete mathe- matics,elementary real analysis,and combinatorics,as well as from classical computer science topics,including algorithms and data structures.The focus ison“average-case”or“probabilistic”"analysis,though the basic mathematical tools required for“worst-case”or“complexity”analysis are covered as well.. We assume that the reader has some familiarity with basic concepts in both computer science and real analysis.In a nutshell,the reader should be able to both write programs and prove theorems.Otherwise,the book is intended to be self-contained. The book is meant to be used as a textbook in an upper-level course on analysis of algorithms.It can also be used in a course in discrete mathematics for computer scientists,since it covers basic techniques in discrete mathemat- ics as well as combinatorics and basic properties of important discrete struc- tures within a familiar context for computer science students.It is traditional to have somewhat broader coverage in such courses,but many instructors may find the approach here to be a useful way to engage students in a substantial portion of the material.The book also can be used to introduce students in mathematics and applied mathematics to principles from computer science related to algorithms and data structures. Despite the large amount of literature on the mathematical analysis of algorithms,basic information on methods and models in widespread use has not been directly accessible to students and researchers in the field.This book aims to address this situation,bringing together a body of material intended to provide readers with both an appreciation for the challenges of the field and the background needed to learn the advanced tools being developed to meet these challenges.Supplemented by papers from the literature,the book can serve as the basis for an introductory graduate course on the analysis of algo- rithms,or as a reference or basis for self-study by researchers in mathematics or computer science who want access to the literature in this field. Preparation.Mathematical maturity equivalent to one or two years'study at the college level is assumed.Basic courses in combinatorics and discrete mathematics may provide useful background(and may overlap with some www.it-ebooks.info
P R E F A C E T HIS book is intended to be a thorough overview of the primary techniques used in the mathematical analysis of algorithms. Ļe material covered draws from classical mathematical topics, including discrete mathematics, elementary real analysis, and combinatorics, as well as from classical computer science topics, including algorithms and data structures. Ļe focus is on “average-case” or “probabilistic” analysis, though the basic mathematical tools required for “worst-case” or “complexity” analysis are covered as well. We assume that the reader has some familiarity with basic concepts in both computer science and real analysis. In a nutshell, the reader should be able to both write programs and prove theorems. Otherwise, the book is intended to be self-contained. Ļe book is meant to be used as a textbook in an upper-level course on analysis of algorithms. It can also be used in a course in discrete mathematics for computer scientists, since it covers basic techniques in discrete mathematics as well as combinatorics and basic properties of important discrete structures within a familiar context for computer science students. It is traditional to have somewhat broader coverage in such courses, but many instructors may ŀnd the approach here to be a useful way to engage students in a substantial portion of the material. Ļe book also can be used to introduce students in mathematics and applied mathematics to principles from computer science related to algorithms and data structures. Despite the large amount of literature on the mathematical analysis of algorithms, basic information on methods and models in widespread use has not been directly accessible to students and researchers in the ŀeld. Ļis book aims to address this situation, bringing together a body of material intended to provide readers with both an appreciation for the challenges of the ŀeld and the background needed to learn the advanced tools being developed to meet these challenges. Supplemented by papers from the literature, the book can serve as the basis for an introductory graduate course on the analysis of algorithms, or as a reference or basis for self-study by researchers in mathematics or computer science who want access to the literature in this ŀeld. Preparation. Mathematical maturity equivalent to one or two years’ study at the college level is assumed. Basic courses in combinatorics and discrete mathematics may provide useful background (and may overlap with some www.it-ebooks.info
viii PREFACE material in the book),as would courses in real analysis,numerical methods, or elementary number theory.We draw on all of these areas,but summarize the necessary material here,with reference to standard texts for people who want more information. Programming experience equivalent to one or two semesters'study at the college level,including elementary data structures,is assumed.We do not dwell on programming and implementation issues,but algorithms and data structures are the central object of our studies.Again,our treatment is complete in the sense that we summarize basic information,with reference to standard texts and primary sources. Related books.Related texts include The Art of Computer Programming by Knuth;Algorithms,Fourth Edition,by Sedgewick and Wayne;Introduction to Algorithms by Cormen,Leiserson,Rivest,and Stein;and our own Analytic Combinatorics.This book could be considered supplementary to each of these. In spirit,this book is closest to the pioneering books by Knuth.Our fo- cus is on mathematical techniques of analysis,though,whereas Knuth's books are broad and encyclopedic in scope,with properties of algorithms playing a primary role and methods of analysis a secondary role.This book can serve as basic preparation for the advanced results covered and referred to in Knuth's books.We also cover approaches and results in the analysis of algorithms that have been developed since publication of Knuth's books. We also strive to keep the focus on covering algorithms of fundamen- tal importance and interest,such as those described in Sedgewick's Algorithms (now in its fourth edition,coauthored by K.Wayne).That book surveys classic algorithms for sorting and searching,and for processing graphs and strings. Our emphasis is on mathematics needed to support scientific studies that can serve as the basis of predicting performance of such algorithms and for com- paring different algorithms on the basis of performance. Cormen,Leiserson,Rivest,and Stein's Introduction to Algorithms has emerged as the standard textbook that provides access to the research litera- ture on algorithm design.The book (and related literature)focuses on design and the theory of algorithms,usually on the basis of worst-case performance bounds.In this book,we complement this approach by focusing on the aral- ysis of algorithms,especially on techniques that can be used as the basis for scientific studies (as opposed to theoretical studies).Chapter 1 is devoted entirely to developing this context. www.it-ebooks.info
viii P Ş ő Œ ō ŏ ő material in the book), as would courses in real analysis, numerical methods, or elementary number theory. We draw on all of these areas, but summarize the necessary material here, with reference to standard texts for people who want more information. Programming experience equivalent to one or two semesters’ study at the college level, including elementary data structures, is assumed. We do not dwell on programming and implementation issues, but algorithms and data structures are the central object of our studies. Again, our treatment is complete in the sense that we summarize basic information, with reference to standard texts and primary sources. Related books. Related texts include Ļe Art of Computer Programming by Knuth; Algorithms, Fourth Edition, by Sedgewick and Wayne; Introduction to Algorithms by Cormen, Leiserson, Rivest, and Stein; and our own Analytic Combinatorics. Ļis book could be considered supplementary to each of these. In spirit, this book is closest to the pioneering books by Knuth. Our focus is on mathematical techniques of analysis, though, whereas Knuth’s books are broad and encyclopedic in scope, with properties of algorithms playing a primary role and methods of analysis a secondary role. Ļis book can serve as basic preparation for the advanced results covered and referred to in Knuth’s books. We also cover approaches and results in the analysis of algorithms that have been developed since publication of Knuth’s books. We also strive to keep the focus on covering algorithms of fundamental importance and interest, such as those described in Sedgewick’s Algorithms (now in its fourth edition, coauthored by K.Wayne). Ļat book surveys classic algorithms for sorting and searching, and for processing graphs and strings. Our emphasis is on mathematics needed to support scientiŀc studies that can serve as the basis of predicting performance of such algorithms and for comparing different algorithms on the basis of performance. Cormen, Leiserson, Rivest, and Stein’s Introduction to Algorithms has emerged as the standard textbook that provides access to the research literature on algorithm design. Ļe book (and related literature) focuses on design and the theory of algorithms, usually on the basis of worst-case performance bounds. In this book, we complement this approach by focusing on the analysis of algorithms, especially on techniques that can be used as the basis for scientiŀc studies (as opposed to theoretical studies). Chapter 1 is devoted entirely to developing this context. www.it-ebooks.info
PREFACE ix This book also lays the groundwork for our Analytic Combinatorics,a general treatment that places the material here in a broader perspective and develops advanced methods and models that can serve as the basis for new research,not only in the analysis of algorithms but also in combinatorics and scientific applications more broadly.A higher level of mathematical matu- rity is assumed for that volume,perhaps at the senior or beginning graduate student level.Of course,careful study of this book is adequate preparation. It certainly has been our goal to make it sufficiently interesting that some readers will be inspired to tackle more advanced material! How to use this book.Readers of this book are likely to have rather diverse backgrounds in discrete mathematics and computer science.With this in mind,it is useful to be aware of the implicit structure of the book:nine chap- ters in all,an introductory chapter followed by four chapters emphasizing mathematical methods,then four chapters emphasizing combinatorial struc- tures with applications in the analysis of algorithms,as follows: INTRODUCTION ONE ANALYSIS OF ALGORITHMS DISCRETE MATHEMATICAL METHODS TWO RECURRENCE RELATIONS THREE GENERATING FUNCTIONS FOUR ASYMPTOTIC APPROXIMATIONS FIVE ANALYTIC COMBINATORICS ALGORITHMS AND COMBINATORIAL STRUCTURES SIX TREES SEVEN PERMUTATIONS EIGHT STRINGS AND TRIES NINE WORDS AND MAPPINGS Chapter 1 puts the material in the book into perspective,and will help all readers understand the basic objectives of the book and the role of the re- maining chapters in meeting those objectives.Chapters 2 through 4 cover www.it-ebooks.info
P Ş ő Œ ō ŏ ő ix Ļis book also lays the groundwork for our Analytic Combinatorics, a general treatment that places the material here in a broader perspective and develops advanced methods and models that can serve as the basis for new research, not only in the analysis of algorithms but also in combinatorics and scientiŀc applications more broadly. A higher level of mathematical maturity is assumed for that volume, perhaps at the senior or beginning graduate student level. Of course, careful study of this book is adequate preparation. It certainly has been our goal to make it sufficiently interesting that some readers will be inspired to tackle more advanced material! How to use this book. Readers of this book are likely to have rather diverse backgrounds in discrete mathematics and computer science. With this in mind, it is useful to be aware of the implicit structure of the book: nine chapters in all, an introductory chapter followed by four chapters emphasizing mathematical methods, then four chapters emphasizing combinatorial structures with applications in the analysis of algorithms, as follows: ANALYSIS OF ALGORITHMS RECURRENCE RELATIONS GENERATING FUNCTIONS ASYMPTOTIC APPROXIMATIONS ANALYTIC COMBINATORICS TREES PERMUTATIONS STRINGS AND TRIES WORDS AND MAPPINGS INTRODUCTION DISCRETE MATHEMATICAL METHODS ALGORITHMS AND COMBINATORIAL STRUCTURES ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE Chapter 1 puts the material in the book into perspective, and will help all readers understand the basic objectives of the book and the role of the remaining chapters in meeting those objectives. Chapters 2 through 4 cover www.it-ebooks.info
x PREFACE methods from classical discrete mathematics,with a primary focus on devel- oping basic concepts and techniques.They set the stage for Chapter 5,which is pivotal,as it covers analytic combinatorics,a calculus for the study of large discrete structures that has emerged from these classical methods to help solve the modern problems that now face researchers because of the emergence of computers and computational models.Chapters 6 through 9 move the fo- cus back toward computer science,as they cover properties of combinatorial structures,their relationships to fundamental algorithms,and analytic results. Though the book is intended to be self-contained,this structure sup- ports differences in emphasis when teaching the material,depending on the background and experience of students and instructor.One approach,more mathematically oriented,would be to emphasize the theorems and proofs in the first part of the book,with applications drawn from Chapters 6 through 9. Another approach,more oriented towards computer science,would be to briefly cover the major mathematical tools in Chapters 2 through 5 and em- phasize the algorithmic material in the second half of the book.But our primary intention is that most students should be able to learn new mate- rial from both mathematics and computer science in an interesting context by working carefully all the way through the book. Supplementing the text are lists of references and several hundred ex- ercises,to encourage readers to examine original sources and to consider the material in the text in more depth. Our experience in teaching this material has shown that there are nu- merous opportunities for instructors to supplement lecture and reading ma- terial with computation-based laboratories and homework assignments.The material covered here is an ideal framework for students to develop exper- tise in a symbolic manipulation system such as Mathematica,MAPLE,or SAGE.More important,the experience of validating the mathematical stud- ies by comparing them against empirical studies is an opportunity to provide valuable insights for students that should not be missed. Booksite.An important feature of the book is its relationship to the booksite aofa.cs.princeton.edu.This site is freely available and contains supple- mentary material about the analysis of algorithms,including a complete set of lecture slides and links to related material,including similar sites for A/go- rithms and Analytic Combinatorics.These resources are suitable both for use by any instructor teaching the material and for self-study. www.it-ebooks.info
x P Ş ő Œ ō ŏ ő methods from classical discrete mathematics, with a primary focus on developing basic concepts and techniques. Ļey set the stage for Chapter 5, which is pivotal, as it covers analytic combinatorics, a calculus for the study of large discrete structures that has emerged from these classical methods to help solve the modern problems that now face researchers because of the emergence of computers and computational models. Chapters 6 through 9 move the focus back toward computer science, as they cover properties of combinatorial structures, their relationships to fundamental algorithms, and analytic results. Ļough the book is intended to be self-contained, this structure supports differences in emphasis when teaching the material, depending on the background and experience of students and instructor. One approach, more mathematically oriented, would be to emphasize the theorems and proofs in the ŀrst part of the book, with applications drawn from Chapters 6 through 9. Another approach, more oriented towards computer science, would be to brieły cover the major mathematical tools in Chapters 2 through 5 and emphasize the algorithmic material in the second half of the book. But our primary intention is that most students should be able to learn new material from both mathematics and computer science in an interesting context by working carefully all the way through the book. Supplementing the text are lists of references and several hundred exercises, to encourage readers to examine original sources and to consider the material in the text in more depth. Our experience in teaching this material has shown that there are numerous opportunities for instructors to supplement lecture and reading material with computation-based laboratories and homework assignments. Ļe material covered here is an ideal framework for students to develop expertise in a symbolic manipulation system such as Mathematica, MAPLE, or SAGE. More important, the experience of validating the mathematical studies by comparing them against empirical studies is an opportunity to provide valuable insights for students that should not be missed. Booksite. An important feature of the book is its relationship to the booksite aofa.cs.princeton.edu. Ļis site is freely available and contains supplementary material about the analysis of algorithms, including a complete set of lecture slides and links to related material, including similar sites for Algorithms and Analytic Combinatorics. Ļese resources are suitable both for use by any instructor teaching the material and for self-study. www.it-ebooks.info
PREFACE i Acknowledgments.We are very grateful to INRIA,Princeton University, and the National Science Foundation,which provided the primary support for us to work on this book.Other support has been provided by Brown Uni- versity,European Community(Alcom Project),Institute for Defense Anal- yses,Ministere de la Recherche et de la Technologie,Stanford University, Universite Libre de Bruxelles,and Xerox Palo Alto Research Center.This book has been many years in the making,so a comprehensive list of people and organizations that have contributed support would be prohibitively long, and we apologize for any omissions. Don Knuth's influence on our work has been extremely important,as is obvious from the text. Students in Princeton,Paris,and Providence provided helpful feedback in courses taught from this material over the years,and students and teach- ers all over the world provided feedback on the first edition.We would like to specifically thank Philippe Dumas,Mordecai Golin,Helmut Prodinger, Michele Soria,Mark Daniel Ward,and Mark Wilson for their help. Corfu,September 1995 Ph.F.and R.S. Paris,December 2012 R.S. www.it-ebooks.info
P Ş ő Œ ō ŏ ő xi Acknowledgments. We are very grateful to INRIA, Princeton University, and the National Science Foundation, which provided the primary support for us to work on this book. Other support has been provided by Brown University, European Community (Alcom Project), Institute for Defense Analyses, Ministère de la Recherche et de la Technologie, Stanford University, Université Libre de Bruxelles, and Xerox Palo Alto Research Center. Ļis book has been many years in the making, so a comprehensive list of people and organizations that have contributed support would be prohibitively long, and we apologize for any omissions. Don Knuth’s inłuence on our work has been extremely important, as is obvious from the text. Students in Princeton, Paris, and Providence provided helpful feedback in courses taught from this material over the years, and students and teachers all over the world provided feedback on the ŀrst edition. We would like to speciŀcally thank Philippe Dumas, Mordecai Golin, Helmut Prodinger, Michele Soria, Mark Daniel Ward, and Mark Wilson for their help. Corfu, September 1995 Ph. F. and R. S. Paris, December 2012 R. S. www.it-ebooks.info
NOTE ON THE SECOND EDITION N March 2011,I was traveling with my wife Linda in a beautiful but some- what rmo reof the word.Carchinguth mye offline,I found the shocking news that my friend and colleague Philippe had passed away,suddenly,unexpectedly,and far too early.Unable to travel to Paris in time for the funeral,Linda and I composed a eulogy for our dear friend that I would now like to share with readers of this book. Sadly,I am writing from a distant part of the world to pay my respects to my longtime friend and colleague,Philippe Flajolet.I am very sorry not to be there in person,but I know that there will be many opportunities to bonor Pbilippe in the future and expect to be fully and personally involved on these occasions. Brilliant,creative,inquisitive,and indefatigable,yet generous and charming. Philippe's approach to life was contagious.He changed many lives,including my own.As our research papers led to a survey paper,then to a monograph,then to a book,then to two books,then to a life's work,I learned,as many students and collaborators around the world have learned,that working with Philippe was based on a genuine and beartfelt camaraderie.We met and worked together in cafes,bars,luncbrooms,and lounges all around the world.Philippe's routine was always the same.We would discuss something amusing that happened to one friend or another and then get to work.After a wink,a hearty but quick laugh, a puff of smoke,another sip of a beer,a few bites of steak frites,and a drawn out"Well..."we could proceed to solve the problem or prove the theorem.For so many of us,these moments are frozen in time. The world has lost a brilliant and productive mathematician.Philippe's un- timely passing means that many things may never be known.But bis legacy is a coterie offollowers passionately devoted to Philippe and his mathematics who will carry on.Our conferences will include a toast to him,our research will build upon his work,our papers will include the inscription"Dedicated to the memory of Philippe Flajolet,"and we will teach generations to come.Dear friend,we miss you so very much,but rest assured that your spirit will live on in our work. This second edition of our book An Introduction to the Analysis of Algorithms was prepared with these thoughts in mind.It is dedicated to the memory of Philippe Flajolet,and is intended to teach generations to come. Jamestown RI,October 2012 R.S. www.it-ebooks.info
N O T E O N T H E S E C O N D E D I T I O N I N March 2011, I was traveling with my wife Linda in a beautiful but somewhat remote area of the world. Catching up with my mail after a few days offline, I found the shocking news that my friend and colleague Philippe had passed away, suddenly, unexpectedly, and far too early. Unable to travel to Paris in time for the funeral, Linda and I composed a eulogy for our dear friend that I would now like to share with readers of this book. Sadly, I am writing from a distant part of the world to pay my respects to my longtime friend and colleague, Philippe Flajolet. I am very sorry not to be there in person, but I know that there will be many opportunities to honor Philippe in the future and expect to be fully and personally involved on these occasions. Brilliant, creative, inquisitive, and indefatigable, yet generous and charming, Philippe’s approach to life was contagious. He changed many lives, including my own. As our research papers led to a survey paper, then to a monograph, then to a book, then to two books, then to a life’s work, I learned, as many students and collaborators around the world have learned, that working with Philippe was based on a genuine and heartfelt camaraderie. We met and worked together in cafes, bars, lunchrooms, and lounges all around the world. Philippe’s routine was always the same. We would discuss something amusing that happened to one friend or another and then get to work. After a wink, a hearty but quick laugh, a puff of smoke, another sip of a beer, a few bites of steak frites, and a drawn out “Well...” we could proceed to solve the problem or prove the theorem. For so many of us, these moments are frozen in time. Ļe world has lost a brilliant and productive mathematician. Philippe’s untimely passing means that many things may never be known. But his legacy is a coterie of followers passionately devoted to Philippe and his mathematics who will carry on. Our conferences will include a toast to him, our research will build upon his work, our papers will include the inscription “Dedicated to the memory of Philippe Flajolet ,” and we will teach generations to come. Dear friend, we miss you so very much, but rest assured that your spirit will live on in our work. Ļis second edition of our book An Introduction to the Analysis of Algorithms was prepared with these thoughts in mind. It is dedicated to the memory of Philippe Flajolet, and is intended to teach generations to come. Jamestown RI, October 2012 R. S. www.it-ebooks.info
TABLE OF CONTENTS CHAPTER ONE:ANALYSIS OF ALGORITHMS 3 1.1 Why Analyze an Algorithm? 3 1.2 Theory of Algorithms 6 1.3 Analysis of Algorithms 13 1.4 Average-Case Analysis 16 1.5 Example:Analysis of Quicksort 18 1.6 Asymptotic Approximations 27 1.7 Distributions 30 1.8 Randomized Algorithms 33 CHAPTER TWO:RECURRENCE RELATIONS 41 2.1 Basic Properties 43 2.2 First-Order Recurrences 48 2.3 Nonlinear First-Order Recurrences 52 2.4 Higher-Order Recurrences 55 2.5 Methods for Solving Recurrences 61 2.6 Binary Divide-and-Conquer Recurrences and Binary 70 Numbers 2.7 General Divide-and-Conquer Recurrences 80 CHAPTER THREE:GENERATING FUNCTIONS 91 3.1 Ordinary Generating Functions 92 3.2 Exponential Generating Functions 97 3.3 Generating Function Solution of Recurrences 101 3.4 Expanding Generating Functions 111 3.5 Transformations with Generating Functions 114 3.6 Functional Equations on Generating Functions 117 3.7 Solving the Quicksort Median-of-Three Recurrence 120 with OGFs 3.8 Counting with Generating Functions 123 3.9 Probability Generating Functions 129 3.10 Bivariate Generating Functions 132 3.11 Special Functions 140 www.it-ebooks.info
T A B L E O F C O N T E N T S CŔōŜŠőŞ OŚő: AŚōŘťş ŕş śŒ AŘœśŞ ŕŠŔřş 3 1.1 Why Analyze an Algorithm? 3 1.2 Ļeory of Algorithms 6 1.3 Analysis of Algorithms 13 1.4 Average-Case Analysis 16 1.5 Example: Analysis of Quicksort 18 1.6 Asymptotic Approximations 27 1.7 Distributions 30 1.8 Randomized Algorithms 33 CŔōŜŠőŞ Tţś: RőŏšŞŞőŚŏő RőŘōŠ ඦş 41 2.1 Basic Properties 43 2.2 First-Order Recurrences 48 2.3 Nonlinear First-Order Recurrences 52 2.4 Higher-Order Recurrences 55 2.5 Methods for Solving Recurrences 61 2.6 Binary Divide-and-Conquer Recurrences and Binary 70 Numbers 2.7 General Divide-and-Conquer Recurrences 80 CŔōŜŠőŞ TŔŞőő: GőŚőŞōŠ ŕŚœ FšŚŏŠ ඦş 91 3.1 Ordinary Generating Functions 92 3.2 Exponential Generating Functions 97 3.3 Generating Function Solution of Recurrences 101 3.4 Expanding Generating Functions 111 3.5 Transformations with Generating Functions 114 3.6 Functional Equations on Generating Functions 117 3.7 Solving the Quicksort Median-of-Ļree Recurrence 120 with OGFs 3.8 Counting with Generating Functions 123 3.9 Probability Generating Functions 129 3.10 Bivariate Generating Functions 132 3.11 Special Functions 140 xv www.it-ebooks.info
xvi TABLE OF CONTENTS CHAPTER FOUR:ASYMPTOTIC APPROXIMATIONS 151 4.1 Notation for Asymptotic Approximations 153 4.2 Asymptotic Expansions 160 4.3 Manipulating Asymptotic Expansions 169 4.4 Asymptotic Approximations of Finite Sums 176 4.5 Euler-Maclaurin Summation 179 4.6 Bivariate Asymptotics 187 4.7 Laplace Method 203 4.8“Normal'"Examples from the Analysis of Algorithms 207 4.9“Poisson"Examples from the Analysis of Algorithms 211 CHAPTER FIVE:ANALYTIC COMBINATORICS 219 5.1 Formal Basis 220 5.2 Symbolic Method for Unlabelled Classes 221 5.3 Symbolic Method for Labelled Classes 229 5.4 Symbolic Method for Parameters 241 5.5 Generating Function Coefficient Asymptotics 247 CHAPTER SIX:TREES 257 6.1 Binary Trees 258 6.2 Forests and Trees 261 6.3 Combinatorial Equivalences to Trees and Binary Trees 264 6.4 Properties of Trees 272 6.5 Examples of Tree Algorithms 277 6.6 Binary Search Trees 281 6.7 Average Path Length in Catalan Trees 287 6.8 Path Length in Binary Search Trees 293 6.9 Additive Parameters of Random Trees 297 6.10 Height 302 6.11 Summary of Average-Case Results on Properties of Trees 310 6.12 Lagrange Inversion 312 6.13 Rooted Unordered Trees 315 6.14 Labelled Trees 327 6.15 Other Types of Trees 331 www.it-ebooks.info
xvi T ō Ŏ Ř ő ś Œ C ś Ś Š ő Ś Š ş CŔōŜŠőŞ FśšŞ: AşťřŜŠśŠ ŕŏ AŜŜŞśŤ ŕřōŠ ඦş 151 4.1 Notation for Asymptotic Approximations 153 4.2 Asymptotic Expansions 160 4.3 Manipulating Asymptotic Expansions 169 4.4 Asymptotic Approximations of Finite Sums 176 4.5 Euler-Maclaurin Summation 179 4.6 Bivariate Asymptotics 187 4.7 Laplace Method 203 4.8 “Normal” Examples from the Analysis of Algorithms 207 4.9 “Poisson” Examples from the Analysis of Algorithms 211 CŔōŜŠőŞ F ŕŢő: AŚōŘťŠ ŕŏ CśřŎ ŕŚōŠśŞ ŕŏş 219 5.1 Formal Basis 220 5.2 Symbolic Method for Unlabelled Classes 221 5.3 Symbolic Method for Labelled Classes 229 5.4 Symbolic Method for Parameters 241 5.5 Generating Function Coefficient Asymptotics 247 CŔōŜŠőŞ S ŕŤ: TŞőőş 257 6.1 Binary Trees 258 6.2 Forests and Trees 261 6.3 Combinatorial Equivalences to Trees and Binary Trees 264 6.4 Properties of Trees 272 6.5 Examples of Tree Algorithms 277 6.6 Binary Search Trees 281 6.7 Average Path Length in Catalan Trees 287 6.8 Path Length in Binary Search Trees 293 6.9 Additive Parameters of Random Trees 297 6.10 Height 302 6.11 Summary of Average-Case Results on Properties of Trees 310 6.12 Lagrange Inversion 312 6.13 Rooted Unordered Trees 315 6.14 Labelled Trees 327 6.15 Other Types of Trees 331 www.it-ebooks.info