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