Introduction to Algorithms | Table of Contents

Introduction to Algorithms | Table of Contents

I Foundations

1 The Role of Algorithms in Computing

1.1 Algorithms
1.2 Algorithms as a technology

2 Getting Started

2.1 Insertion sort
2.2 Analyzing algorithms
2.3 Designing Algorithms

3 Growth of Functions

3.1 Asymptotic notation
3.2 Standard notations and common functions

4 Recurrences

4.1 The substitution method
4.2 The recursion-tree method
4.3 The master method
4.4 Proof of the master theorem

5 Probabilistic Analysis and Randomized Algorithms

5.1 The hiring problem
5.2 Indicator random variables
5.3 Randomized algorithms
5.4 Probabilistic analysis and further uses of indicator random variables

II Sorting and Order Statistics

6 Heapsort

6.1 Heaps
6.2 Maintaining the heap property
6.3 Building a heap
6.4 The heapsort algorithm
6.5 Priority queues

7 Quicksort

7.1 Description of quicksort
7.2 Performance of quicksort
7.3 Randomized versions of quicksort
7.4 Analysis of quicksort

8 Sorting in Linear Time

8.1 Lower bounds for sorting
8.2 Counting sort
8.3 Radix sort
8.4 Bucket sort

9 Medians and Order Statistics

9.1 Minimum and maximum
9.2 Selection in expected linear time
9.3 Selection in worst-case linear time

III Data Structures

10 Elementary Data Structures

10.1 Stacks and queues
10.2 Linked lists
10.3 Implementing pointers and objects
10.4 Representing rooted trees

11 Hash Tables

11.1 Direct-address tables
11.2 Hash tables
11.3 Hash functions
11.4 Open addressing
11.5 Perfect hashing

12 Binary Search Trees

12.1 What is a binary search tree?
12.2 Querying a binary search tree
12.3 Insertion and deletion
12.4 Randomly built binary search trees

13 Red-Black Trees

13.1 Properties of red-black trees
13.2 Rotations
13.3 Insertion
13.4 Deletion

14 Augmenting Data Structures

14.1 Dynamic order statistics
14.2 How to augment a data structure
14.3 Interval trees

IV Advanced Design and Analysis Technique

15 Dynamic Programming

15.1 Assembly-line scheduling
15.2 Matrix-chain multiplication
15.3 Elements of dynamic programming
15.4 Longest common subsequence
15.5 Optimal binary search trees

16 Greedy Algorithms

16.1 An activity-selection problem
16.2 Elements of the greedy strategy
16.3 Huffman codes
16.4 Theoretical foundations for greedy methods
16.5 A task-scheduling problem

17 Amortized Analysis

17.1 Aggregate analysis
17.2 The accounting method
17.3 The potential method
17.4 Dynamic tables

V Advanced Data Structures

18 B-Trees

18.1 Definition of B-trees
18.2 Basic operations on B-trees
18.3 Deleting a key from a B-tree

19 Binomial Heaps

19.1 Binomial trees and binomial heaps
19.2 Operations on binomial heaps

20 Fibonacci Heaps

20.1 Structure of Fibonacci heaps
20.2 Mergeable-heap operations
20.3 Decreasing a key and deleting a node
20.4 Bounding the maximum degree

21 Data Structures for Disjoint Sets

21.1 Disjoint-set operations
21.2 Linked-list representation of disjoint sets
21.3 Disjoint-set forests
21.4 Analysis of union by rank with path compression

VI Graph Algorithms

22 Elementary Graph Algorithms

22.1 Representations of graphs
22.2 Breadth-first search
22.3 Depth-first search
22.4 Topological sort
22.5 Strongly connected components

23 Minimum Spanning Trees

23.1 Growing a minimum spanning tree
23.2 The algorithms of Kruskal and Prim

24 Single-Source Shortest Paths

24.1 The Bellman-Ford algorithm
24.2 Single-source shortest paths in directed acyclic graphs
24.3 Dijkstra's algorithm
24.4 Difference constraints and shortest paths
24.5 Proofs of shortest-paths properties

25 All-Pairs Shortest Paths

25.1 Shortest paths and matrix multiplication
25.2 The Floyd-Warshall algorithm
25.3 Johnson's algorithm for sparse graphs

26 Maximum Flow

26.1 Flow networks
26.2 The Ford-Fulkerson method
26.3 Maximum bipartite matching
26.4 Push-relabel algorithms
26.5 The relabel-to-front algorithm

VII Selected Topics

27 Sorting Networks

27.1 Comparison networks
27.2 The zero-one principle
27.3 A bitonic sorting network
27.4 A merging network
27.5 A sorting network

28 Matrix Operations

28.1 Properties of matrices
28.2 Strassen's algorithm for matrix multiplication
28.3 Solving systems of linear equations
28.4 Inverting matrices
28.5 Symmetric positive-definite matrices and least-squares approximation

29 Linear Programming

29.1 Standard and slack forms
29.2 Formulating problems as linear programs
29.3 The simplex algorithm
29.4 Duality
29.5 The initial basic feasible solution

30 Polynomials and the FFT

30.1 Representation of polynomials
30.2 The DFT and FFT
30.3 Efficient FFT implementations

31 Number-Theoretic Algorithms

31.1 Elementary number-theoretic notions
31.2 Greatest common divisor
31.3 Modular arithmetic
31.4 Solving modular linear equations
31.5 The Chinese remainder theorem
31.6 Powers of an element
31.7 The RSA public-key cryptosystem
31.8 Primality testing
31.9 Integer factorization

32 String Matching

32.1 The naive string-matching algorithm
32.2 The Rabin-Karp algorithm
32.3 String matching with finite automata
32.4 The Knuth-Morris-Pratt algorithm

33 Computational Geometry

33.1 Line-segment properties
33.2 Determining whether any pair of segments intersects
33.3 Finding the convex hull
33.4 Finding the closest pair of points

34 NP-Completeness

34.1 Polynomial time
34.2 Polynomial-time verification
34.3 NP-completeness and reducibility
34.4 NP-completeness proofs
34.5 NP-complete problems

35 Approximation Algorithms

35.1 The vertex-cover problem
35.2 The traveling-salesman problem
35.3 The set-covering problem
35.4 Randomization and linear programming
35.4 The subset-sum problem

VIII Appendix: Mathematical Background

A Summations

A.1 Summation formulas and properties
A.2 Bounding summations

B Sets, Etc.

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