本书为全英文。它全面介绍了算法的数学分析中使用的基本方法，所涉及的内容来自经典的数学素材(包括离散数学、初等实分析、组合数学)，以及经典的计算机科学素材(包括算法和数据结构)。虽然书中论述了“最坏情形”和“复杂性问题”分析所需的基本数学工具，但是重点还是讨论“平均情形”或“概率”分析。论题涉及递归、生成函数、渐近性、树、串、映射等内容，以及对排序、树查找、串查找和散列诸算法的分析。

本书全面介绍了算法的数学分析中使用的基本方法，所涉及的内容来自经典的数学素材(包括离散数学、初等实分析、组合数学)，以及经典的计算机科学素材(包括算法和数据结构)。虽然书中论述了“最坏情形”和“复杂性问题”分析所需的基本数学工具，但是重点还是讨论“平均情形”或“概率”分析。论题涉及递归、生成函数、渐近性、树、串、映射等内容，以及对排序、树查找、串查找和散列诸算法的分析。

尽管人们极为关注算法的数学分析，但是广泛使用的方法和模型方面的基本信息尚不能为该领域的工作和研究所直接使用。作者在本书中处理这种需求，把该领域出现的挑战以及为跟上新的研究以迎接这些挑战所必需的背景资料完美地结合在一起。

CHAPTER ONE：ANALYSIS OF ALGORITHMS

  1.1 Why Analyze an Algorithm?

  1.2 Computational Complexity

  1.3 Analysis of Algorithms

  1.4 Average-Case Analysis

  1.5 Example: Analysis of Quieksort

  1.6 Asymptotic Approximations

  1.7 Distributions

  1.8 Probabilistic Algorithms

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

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 The Symbolic Method

  3.10 Lagrange Inversion

  3.11 Probability Generating Functions

  3.12 Bivariate Generating Functions

  3.13 Special Functions

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

  4.10 Generating Function Asymptotics

CHAPTER FIVE: TREES

  5.1 Binary Trees

  5.2 Trees and Forests

  5.3 Properties of Trees

  5.4 Tree Algorithms

  5.5 Binary Search Trees

  5.6 Average Path Length in Catalan Trees

  5.7 Path Length in Binary Search Trees

  5.8 Additive Parameters of Random Trees

  5.9 Height

  5.10 Summary of Average-Case Results on Properties of Trees

  5.11 Representations of Trees and Binary Trees

  5.12 Unordered Trees

  5.13 Labelled Trees

  5.14 Other Types of Trees

CHAPTER SIX: PERMUTATIONS

  6.1 Basic Properties of Permutations

  6.2 Algorithms on Permutations

  6.3 Representations of Permutations

  6.4 Enumeration Problems

  6.5 Analyzing Properties of Permutations with CGFs

  6.6 Inversions and Insertion Sorts

  6.7 Left-to-Right Minima and Selection Sort

  6.8 Cycles and In Situ Permutation

  6.9 Extremal Parameters

CHAPTER SEVEN：STRINGS AND TRIES

  7.1 String Searching

  7.2 Combinatorial Properties of Bitstrings

  7.3 Regular Expressions

  7.4 Finite-State Automata and the Knuth-Morris-Pratt

   Algorithm

  7.5 Context-Free Grammars

  7.6 Tries

  7.7 Trie Algorithms

  7.8 Combinatorial Properties of Tries

  7.9 Larger Alphabets

CHAPTER EIGHT: WORDS AND MAPS

  8.1 Hashing with Separate Chaining

  8.2 Basic Properties of Words

  8.3 Birthday Paradox and Coupon Collector Problem

  8.4 Occupancy Restrictions and Extremal Parameters

  8.5 Occupancy Distributions

  8.6 Open Addressing Hashing

  8.7 Maps

  8.8 Integer Facterization and Maps

   List of Theorems

   Index