This book is a thorough overview of the primary techniques and models used in the mathematical analysis of algorithms. The first half of the book draws upon classical mathematical material from discrete mathematics, elementary real analysis, and combinatorics; the second half discusses properties of discrete structures and covers the analysis of a variety of classical sorting, searching, and string processing algorithms.
Analysis of Algorithms.
Example: Analysis of Quicksort.
2. Recurrence Relations.
Nonlinear First-Order Recurrences.
Methods for Solving Recurrences.
Binary Divide-and-Conquer Recurrences and Binary Numbers.
General Divide-and-Conquer Recurrences.
3. Generating Functions.
Exponential Generating Functions.
Generating Function Solution of Recurrences.
Expanding Generating Functions.
Transformations with Generating Functions.
Functional Equations on Generating Functions.
Solving the Quicksort Median-of-Three.
Recurrence with OGFs.
Counting with Generating Functions.
The Symbolic Method.
Probability Generating Functions.
Bivariate Generating Functions.
4. Asymptotic Approximations.
Manipulating Asymptotic Expansions.
Asymptotic Approximations of Finite Sums.
“Normal” Examples from the Analysis of Algorithms.
“Poisson” Examples from the Analysis of Algorithms.
Generating Function Asymptotics.
Trees and Forests.
Properties of Trees.
Binary Search Trees.
Average Path Length in Catalan Trees.
Path Length in Binary Search Trees.
Additive Parameters of Random Trees.
Summary of Average-Case Results on Properties of Trees.
Representations of Trees and Binary Trees.
Other Types of Trees.
Algorithms on Permutations.
Representations of Permutations.
Analyzing Properties of Permutations with CGFs.
Inversions and Insertion Sorts.
Left-to-Right Minima and Selection Sort.
Cycles and In Situ Permutation.
7. Strings and Tries.
Combinatorial Properties of Bitstrings.
Finite-State Automata and Knuth-Morris-Pratt Algorithm.
Combinatorial Properties of Tries.
8. Words and Maps.
Basic Properties of Words.
Birthday Paradox and Coupon Collector Problem.
Occupancy Restrictions and Extremal Parameters.
Open Addressing Hashing.
Integer Factorization and Maps. 020140009XT04062001