Introduction to Algorithms Third Edition

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1 Thomas H. Cormen Charles E. Leiserson Ronald L. Rivest Clifford Stein Introduction to Algorithms Third Edition The MIT Press Cambridge, Massachusetts London, England

2 Preface xiü I Foundations Introduction 3 1 The Role of Algorithms in Computing Algorithms Algorithms as a technology 11 2 Getting Started Insertion sort Analyzing algorithms Designing algorithms 29 3 Growth of Functions Asymptotic notation Standard notations and common functions 53 4 Divide-and-Conquer The maximum-subarray problem Strassen's algorithm for matrix multiplication The substitution method for solving recurrences The recursion-tree method for solving recurrences The master method for solving recurrences 93 * 4.6 Proof of the master theorem 97 5 Probabilistic Analysis and Randomized Algorithms The hiring problem Indicator random variables Randomized algorithms 122 * 5.4 Probabilistic analysis and further uses of indicator random variables 130

3 VI Contents II Sorting and Order Statistics Introduction Heapsort Heaps Maintaining the heap property Building a heap The heapsort algorithm Priority queues Quicksort Description of quicksort Performance of quicksort A randomized version of quicksort Analysis of quicksort Sorting in Linear Time Lower bounds for sorting Counting sort Radix sort Bucket sort Medians and Order Statistics Minimum and maximum Selection in expected linear time Selection in worst-case linear time 220 HI Data Structures Introduction Elementary Data Structures Stacks and queues Linked lists Implementing pointers and objects Representing rooted trees Hash Tables Direct-address tables Hash tables Hash functions Open addressing Perfect hashing 277

4 12 Binary Search Trees What is a binary search tree? Querying a binary search tree Insertion and deletion 294 * 12.4 Randomly built binary search trees Red-Black Trees Properties of red-black trees Rotations Insertion Deletion Augmenting Data Structures Dynamic order statistics How to augment a data structure Interval trees 348 TV Advanced Design and Analysis Techniques Introduction Dynamic Programming Rod cutting Matrix-chain multiplication Elements of dynamic programming Longest common subsequence Optimal binary search trees Greedy Algorithms An activity-selection problem Elements of the greedy strategy Huffman codes 428 * 16.4 Matroids and greedy methods 437 * 16.5 A task-scheduling problem as a matroid Amortized Analysis Aggregate analysis The accounting method The potential method Dynamic tables 463

5 viii Contents V Advanced Data Structures Introduction B-Trees Definition of B-trees Basic operations on B-trees Deleting a key from а В-tree Fibonacci Heaps Structure of Fibonacci heaps Mergeable-heap operations Decreasing a key and deleting a node Bounding the maximum degree van Emde Boas Trees Preliminary approaches A recursive structure The van Emde Boas tree Data Structures for Disjoint Sets Disjoint-set operations Linked-list representation of disjoint sets Disjoint-set forests 568 * 21.4 Analysis of union by rank with path compression 573 VI Graph Algorithms Introduction Elementary Graph Algorithms Representations of graphs Breadth-first search Depth-first search Topological sort Strongly connected components Minimum Spanning Trees Growing a minimum spanning tree The algorithms of Kruskal and Prim 631

6 ix 24 Single-Source Shortest Paths The Bellman-Ford algorithm Single-source shortest paths in directed acyclic graphs Dijkstra's algorithm Difference constraints and shortest paths Proofs of shortest-paths properties АН-Pairs Shortest Paths Shortest paths and matrix multiplication The Floyd-Warshall algorithm Johnson's algorithm for sparse graphs Maximum Flow Flow networks The Ford-Fulkerson method Maximum bipartite matching 732 к 26А Push-relabel algorithms The relabel-to-front algorithm 748 VII Selected Topics Introduction Multithreaded Algorithms Л The basics of dynamic multithreading Multithreaded matrix multiplication ,3 Multithreaded merge sort Matrix Operations Solving systems of linear equations Inverting matrices Symmetric positive-definite matrices and least-squares approximation Linear Programming Standard and slack forms Formulating problems as linear programs The simplex algorithm Duality The initial basic feasible solution 886

7 30 Polynomials and the FFT Representing polynomials The DFT and FFT Efficient FFT implementations Number-Theoretic Algorithms Elementary number-theoretic notions Greatest common divisor Modular arithmetic Solving modular linear equations The Chinese remainder theorem Powers of an element The RS A public-key cryptosystem Primality testing Integer factorization String Matching The naive string-matching algorithm The Rabin-Karp algorithm String matching with finite automata The Knuth-Morris-Pratt algorithm Computational Geometry Line-segment properties Determining whether any pair of segments intersects 33.3 Finding the convex hull Finding the closest pair of points NP-Completeness Polynomial time Polynomial-time verification NP-completeness and reducibility NP-completeness proofs NP-complete problems Approximation Algorithms The vertex-cover problem The traveling-salesman problem The set-covering problem Randomization and linear programming The subset-sum problem 7728

8 VIII Appendix: Mathematical Background Introduction 1143 A Summations 1145 A.l Summation formulas and properties 1145 A.2 Bounding summations 7749 В Sets, Etc B.l Sets 7758 B.2 Relations 1163 B.3 Functions 7766 B.4 Graphs 7768 B.5 Trees 7773 С Counting and Probability 1183 C.l Counting 7785 C.2 Probability 7789 C.3 Discrete random variables 7796 C.4 The geometric and binomial distributions 7207 * C.5 The tails of the binomial distribution 7208 D Matrices 1217 D.l Matrices and matrix operations 7277 D.2 Basic matrix properties 7222 Bibliography 1231 Index 1251

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