Introduction to Algorithms

Contents
Preface
Part I Foundations
	Introduction
	1 The Role of Algorithms in Computing
		1.1 Algorithms
		1.2 Algorithms as a technology
		Problems
		Chapter notes
	2 Getting Started
		2.1 Insertion sort
		2.2 Analyzing algorithms
		2.3 Designing algorithms
		Problems
		Chapter notes
	3 Characterizing Running Times
		3.1 O-notation, Ω-notation, and Θ-notation
		3.2 Asymptotic notation: formal definitions
		3.3 Standard notations and common functions
		Problems
		Chapter notes
	4 Divide-and-Conquer
		4.1 Multiplying square matrices
		4.2 Strassen’s algorithm for matrix multiplication
		4.3 The substitution method for solving recurrences
		4.4 The recursion-tree method for solving recurrences
		4.5 The master method for solving recurrences
		4.6 Proof of the continuous master theorem
		4.7 Akra-Bazzi recurrences
		Problems
		Chapter notes
	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
		Problems
		Chapter notes
Part II Sorting and Order Statistics
	Introduction
	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
		Problems
		Chapter notes
	7 Quicksort
		7.1 Description of quicksort
		7.2 Performance of quicksort
		7.3 A randomized version of quicksort
		7.4 Analysis of quicksort
		Problems
		Chapter notes
	8 Sorting in Linear Time
		8.1 Lower bounds for sorting
		8.2 Counting sort
		8.3 Radix sort
		8.4 Bucket sort
		Problems
		Chapter notes
	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
		Problems
		Chapter notes
Part III Data Structures
	Introduction
	10 Elementary Data Structures
		10.1 Simple array-based data structures: arrays, matrices, stacks, queues
		10.2 Linked lists
		10.3 Representing rooted trees
		Problems
		Chapter notes
	11 Hash Tables
		11.1 Direct-address tables
		11.2 Hash tables
		11.3 Hash functions
		11.4 Open addressing
		11.5 Practical considerations
		Problems
		Chapter notes
	12 Binary Search Trees
		12.1 What is a binary search tree?
		12.2 Querying a binary search tree
		12.3 Insertion and deletion
		Problems
		Chapter notes
	13 Red-Black Trees
		13.1 Properties of red-black trees
		13.2 Rotations
		13.3 Insertion
		13.4 Deletion
		Problems
		Chapter notes
Part IV Advanced Design and Analysis Techniques
	Introduction
	14 Dynamic Programming
		14.1 Rod cutting
		14.2 Matrix-chain multiplication
		14.3 Elements of dynamic programming
		14.4 Longest common subsequence
		14.5 Optimal binary search trees
		Problems
		Chapter notes
	15 Greedy Algorithms
		15.1 An activity-selection problem
		15.2 Elements of the greedy strategy
		15.3 Huffman codes
		15.4 Offline caching
		Problems
		Chapter notes
	16 Amortized Analysis
		16.1 Aggregate analysis
		16.2 The accounting method
		16.3 The potential method
		16.4 Dynamic tables
		Problems
		Chapter notes
Part V Advanced Data Structures
	Introduction
	17 Augmenting Data Structures
		17.1 Dynamic order statistics
		17.2 How to augment a data structure
		17.3 Interval trees
		Problems
		Chapter notes
	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
		Problems
		Chapter notes
	19 Data Structures for Disjoint Sets
		19.1 Disjoint-set operations
		19.2 Linked-list representation of disjoint sets
		19.3 Disjoint-set forests
		19.4 Analysis of union by rank with path compression
		Problems
		Chapter notes
Part VI Graph Algorithms
	Introduction
	20 Elementary Graph Algorithms
		20.1 Representations of graphs
		20.2 Breadth-first search
		20.3 Depth-first search
		20.4 Topological sort
		20.5 Strongly connected components
		Problems
		Chapter notes
	21 Minimum Spanning Trees
		21.1 Growing a minimum spanning tree
		21.2 The algorithms of Kruskal and Prim
		Problems
		Chapter notes
	22 Single-Source Shortest Paths
		22.1 The Bellman-Ford algorithm
		22.2 Single-source shortest paths in directed acyclic graphs
		22.3 Dijkstra’s algorithm
		22.4 Difference constraints and shortest paths
		22.5 Proofs of shortest-paths properties
		Problems
		Chapter notes
	23 All-Pairs Shortest Paths
		23.1 Shortest paths and matrix multiplication
		23.2 The Floyd-Warshall algorithm
		23.3 Johnson’s algorithm for sparse graphs
		Problems
		Chapter notes
	24 Maximum Flow
		24.1 Flow networks
		24.2 The Ford-Fulkerson method
		24.3 Maximum bipartite matching
		Problems
		Chapter notes
	25 Matchings in Bipartite Graphs
		25.1 Maximum bipartite matching (revisited)
		25.2 The stable-marriage problem
		25.3 The Hungarian algorithm for the assignment problem
		Problems
		Chapter notes
Part VII Selected Topics
	Introduction
	26 Parallel Algorithms
		26.1 The basics of fork-join parallelism
		26.2 Parallel matrix multiplication
		26.3 Parallel merge sort
		Problems
		Chapter notes
	27 Online Algorithms
		27.1 Waiting for an elevator
		27.2 Maintaining a search list
		27.3 Online caching
		Problems
		Chapter notes
	28 Matrix Operations
		28.1 Solving systems of linear equations
		28.2 Inverting matrices
		28.3 Symmetric positive-definite matrices and least-squares approximation
		Problems
		Chapter notes
	29 Linear Programming
		29.1 Linear programming formulations and algorithms
		29.2 Formulating problems as linear programs
		29.3 Duality
		Problems
		Chapter notes
	30 Polynomials and the FFT
		30.1 Representing polynomials
		30.2 The DFT and FFT
		30.3 FFT circuits
		Problems
		Chapter notes
	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
		Problems
		Chapter notes
	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
		32.5 Suffix arrays
		Problems
		Chapter notes
	33 Machine-Learning Algorithms
		33.1 Clustering
		33.2 Multiplicative-weights algorithms
		33.3 Gradient descent
		Problems
		Chapter notes
	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
		Problems
		Chapter notes
	35 Approximation Algorithms
		35.1 The vertex-cover problem
		35.2 The traveling-salesperson problem
		35.3 The set-covering problem
		35.4 Randomization and linear programming
		35.5 The subset-sum problem
		Problems
		Chapter notes
Part VIII Appendix: Mathematical Background
	Introduction
	A Summations
		A.1 Summation formulas and properties
		A.2 Bounding summations
		Problems
		Appendix notes
	B Sets, Etc.
		B.1 Sets
		B.2 Relations
		B.3 Functions
		B.4 Graphs
		B.5 Trees
		Problems
		Appendix notes
	C Counting and Probability
		C.1 Counting
		C.2 Probability
		C.3 Discrete random variables
		C.4 The geometric and binomial distributions
		C.5 The tails of the binomial distribution
		Problems
		Appendix notes
	D Matrices
		D.1 Matrices and matrix operations
		D.2 Basic matrix properties
		Problems
		Appendix notes
Bibliography
Index