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An Introduction to the Analysis of Algorithms, Second Edition

Book Description

Despite growing interest, basic information on methods and models for mathematically analyzing algorithms has rarely been directly accessible to practitioners, researchers, or students. An Introduction to the Analysis of Algorithms, Second Edition, organizes and presents that knowledge, fully introducing primary techniques and results in the field.

Robert Sedgewick and the late Philippe Flajolet have drawn from both classical mathematics and computer science, integrating discrete mathematics, elementary real analysis, combinatorics, algorithms, and data structures. They emphasize the mathematics needed to support scientific studies that can serve as the basis for predicting algorithm performance and for comparing different algorithms on the basis of performance.

Techniques covered in the first half of the book include recurrences, generating functions, asymptotics, and analytic combinatorics. Structures studied in the second half of the book include permutations, trees, strings, tries, and mappings. Numerous examples are included throughout to illustrate applications to the analysis of algorithms that are playing a critical role in the evolution of our modern computational infrastructure.

Improvements and additions in this new edition include

  • Upgraded figures and code

  • An all-new chapter introducing analytic combinatorics

  • Simplified derivations via analytic combinatorics throughout

The book’s thorough, self-contained coverage will help readers appreciate the field’s challenges, prepare them for advanced results—covered in their monograph Analytic Combinatorics and in Donald Knuth’s The Art of Computer Programming books—and provide the background they need to keep abreast of new research.

"[Sedgewick and Flajolet] 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."

—From the Foreword by Donald E. Knuth

Table of Contents

  1. Title Page
  2. Copyright Page
  3. Foreword
  4. Preface
    1. Preparation
    2. Related books
    3. How to use this book
    4. Booksite
    5. Acknowledgments
  5. Note on the Second Edition
  6. Table of Contents
  7. Notation
  8. Chapter One. Analysis of Algorithms
    1. 1.1. Why Analyze an Algorithm?
    2. 1.2. Theory of Algorithms
    3. 1.3. Analysis of Algorithms
    4. 1.4. Average-Case Analysis
    5. 1.5. Example: Analysis of Quicksort
    6. 1.6. Asymptotic Approximations
    7. 1.7. Distributions
    8. 1.8. Randomized Algorithms
    9. References
  9. Chapter Two. Recurrence Relations
    1. 2.1. Basic Properties
    2. 2.2. First-Order Recurrences
    3. 2.3. Nonlinear First-Order Recurrences
    4. 2.4. Higher-Order Recurrences
    5. 2.5. Methods for Solving Recurrences
    6. 2.6. Binary Divide-and-Conquer Recurrences and Binary Numbers
    7. 2.7. General Divide-and-Conquer Recurrences
    8. References
  10. Chapter Three. Generating Functions
    1. 3.1. Ordinary Generating Functions
    2. 3.2. Exponential Generating Functions
    3. 3.3. Generating Function Solution of Recurrences
    4. 3.4. Expanding Generating Functions
    5. 3.5. Transformations with Generating Functions
    6. 3.6. Functional Equations on Generating Functions
    7. 3.7. Solving the Quicksort Median-of-Three Recurrence with OGFs
    8. 3.8. Counting with Generating Functions
    9. 3.9. Probability Generating Functions
    10. 3.10. Bivariate Generating Functions
    11. 3.11. Special Functions
    12. References
  11. Chapter Four. Asymptotic Approximations
    1. 4.1. Notation for Asymptotic Approximations
    2. 4.2. Asymptotic Expansions
    3. 4.3. Manipulating Asymptotic Expansions
    4. 4.4. Asymptotic Approximations of Finite Sums
    5. 4.5. Euler-Maclaurin Summation
    6. 4.6. Bivariate Asymptotics
    7. 4.7. Laplace Method
    8. 4.8. “Normal” Examples from the Analysis of Algorithms
    9. 4.9. “Poisson” Examples from the Analysis of Algorithms
    10. References
  12. Chapter Five. Analytic Combinatorics
    1. 5.1. Formal Basis
    2. 5.2. Symbolic Method for Unlabelled Classes
    3. 5.3. Symbolic Method for Labelled Classes
    4. 5.4. Symbolic Method for Parameters
    5. 5.5. Generating Function Coefficient Asymptotics
    6. References
  13. Chapter Six. Trees
    1. 6.1. Binary Trees
    2. 6.2. Forests and Trees
    3. 6.3. Combinatorial Equivalences to Trees and Binary Trees
    4. 6.4. Properties of Trees
    5. 6.5. Examples of Tree Algorithms
    6. 6.6. Binary Search Trees
    7. 6.7. Average Path Length in Random Catalan Trees
    8. 6.8. Path Length in Binary Search Trees
    9. 6.9. Additive Parameters of Random Trees
    10. 6.10. Height
    11. 6.11. Summary of Average-Case Results on Properties of Trees
    12. 6.12. Lagrange Inversion
    13. 6.13. Rooted Unordered Trees
    14. 6.14. Labelled Trees
    15. 6.15. Other Types of Trees
    16. References
  14. Chapter Seven. Permutations
    1. 7.1. Basic Properties of Permutations
    2. 7.2. Algorithms on Permutations
    3. 7.3. Representations of Permutations
    4. 7.4. Enumeration Problems
    5. 7.5. Analyzing Properties of Permutations with CGFs
    6. 7.6. Inversions and Insertion Sorts
    7. 7.7. Left-to-Right Minima and Selection Sort
    8. 7.8. Cycles and In Situ Permutation
    9. 7.9. Extremal Parameters
    10. References
  15. Chapter Eight. Strings and Tries
    1. 8.1. String Searching
    2. 8.2. Combinatorial properties of bitstrings
    3. 8.3. Regular Expressions
    4. 8.4. Finite-State Automata and the Knuth-Morris-Pratt Algorithm
    5. 8.5. Context-Free Grammars
    6. 8.6. Tries
    7. 8.7. Trie Algorithms
    8. 8.8. Combinatorial Properties of Tries
    9. 8.9. Larger Alphabets
    10. References
  16. Chapter Nine. Words and Mappings
    1. 9.1. Hashing with Separate Chaining
    2. 9.2. The Balls-and-Urns Model and Properties of Words
    3. 9.3. Birthday Paradox and Coupon Collector Problem
    4. 9.4. Occupancy Restrictions and Extremal Parameters
    5. 9.5. Occupancy Distributions
    6. 9.6. Open Addressing Hashing
    7. 9.7. Mappings
    8. 9.8. Integer Factorization and Mappings
    9. References
  17. List of Theorems
  18. List of Tables
  19. List of Figures
  20. Index
  21. Ad Page