算法导论24.4-8

doctorweb 2010-08-16 10:10:52

设Ax<=b是关于n个未知量的m个约束条件的差分约束系统。证明对其相应的约束图运行Bellman-Ford算法，可以求得满足Ax<=b,并且对所有xi，有xi<=0,式∑i=1 n xi的最大值。

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zhoupin 2010-08-27

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很强大

air_snake 2010-08-26

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这个是network flow么?算法导论的lecture manual上也没有。
这个东西真不是很懂。一般也不喜欢证明，呵呵。

绿色夹克衫 2010-08-20

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就这块儿没看，还被问到了，唉，学得太不扎实了！

boYwell 2010-08-20

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我正在看，还没看到这里。

kuzalid 2010-08-17

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我想，应该是的~~
我看过算法导论，看不下去，太难了啊。。

doctorweb 2010-08-17

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没人对这些感兴趣吗？

中文名: 算法导论 原名: Introduction to Algorithms 作者: Thomas H. Cormen Ronald L. Rivest Charles E. Leiserson Clifford Stein 资源格式: PDF 版本: 文字版出版社: The MIT Press书号: 978-0262033848发行时间: 2009年09月30日地区: 美国语言: 英文简介: 内容介绍: Editorial Reviews Review "In light of the explosive growth in the amount of data and the diversity of computing applications, efficient algorithms are needed now more than ever. This beautifully written, thoughtfully organized book is the definitive introductory book on the design and analysis of algorithms. The first half offers an effective method to teach and study algorithms; the second half then engages more advanced readers and curious students with compelling material on both the possibilities and the challenges in this fascinating field." —Shang-Hua Teng, University of Southern California "Introduction to Algorithms, the 'bible' of the field, is a comprehensive textbook covering the full spectrum of modern algorithms: from the fastest algorithms and data structures to polynomial-time algorithms for seemingly intractable problems, from classical algorithms in graph theory to special algorithms for string matching, computational geometry, and number theory. The revised third edition notably adds a chapter on van Emde Boas trees, one of the most useful data structures, and on multithreaded algorithms, a topic of increasing importance." —Daniel Spielman, Department of Computer Science, Yale University "As an educator and researcher in the field of algorithms for over two decades, I can unequivocally say that the Cormen book is the best textbook that I have ever seen on this subject. It offers an incisive, encyclopedic, and modern treatment of algorithms, and our department will continue to use it for teaching at both the graduate and undergraduate levels, as well as a reliable research reference." —Gabriel Robins, Department of Computer Science, University of Virginia Product Description Some books on algorithms are rigorous but incomplete; others cover masses of material but lack rigor. Introduction to Algorithms uniquely combines rigor and comprehensiveness. The book covers a broad range of algorithms in depth, yet makes their design and analysis accessible to all levels of readers. Each chapter is relatively self-contained and can be used as a unit of study. The algorithms are described in English and in a pseudocode designed to be readable by anyone who has done a little programming. The explanations have been kept elementary without sacrificing depth of coverage or mathematical rigor. The first edition became a widely used text in universities worldwide as well as the standard reference for professionals. The second edition featured new chapters on the role of algorithms, probabilistic analysis and randomized algorithms, and linear programming. The third edition has been revised and updated throughout. It includes two completely new chapters, on van Emde Boas trees and multithreaded algorithms, and substantial additions to the chapter on recurrences (now called "Divide-and-Conquer"). It features improved treatment of dynamic programming and greedy algorithms and a new notion of edge-based flow in the material on flow networks. Many new exercises and problems have been added for this edition. As of the third edition, this textbook is published exclusively by the MIT Press. About the Author Thomas H. Cormen is Professor of Computer Science and former Director of the Institute for Writing and Rhetoric at Dartmouth College. Charles E. Leiserson is Professor of Computer Science and Engineering at the Massachusetts Institute of Technology. Ronald L. Rivest is Andrew and Erna Viterbi Professor of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology. Clifford Stein is Professor of Industrial Engineering and Operations Research at Columbia University. 目录: Introduction 3 1 The Role of Algorithms in Computing 5 1.1 Algorithms 5 1.2 Algorithms as a technology 11 2 Getting Started 16 2.1 Insertion sort 16 2.2 Analyzing algorithms 23 2.3 Designing algorithms 29 3 Growth of Functions 43 3.1 Asymptotic notation 43 3.2 Standard notations and common functions 53 4 Divide-and-Conquer 65 4.1 The maximum-subarray problem 68 4.2 Strassen's algorithm for matrix multiplication 75 4.3 The substitution method for solving recurrences 83 4.4 The recursion-tree method for solving recurrences 88 4.5 The master method for solving recurrences 93 4.6 Proof of the master theorem 97 5 Probabilistic Analysis and Randomized Algorithms 114 5.1 The hiring problem 114 5.2 Indicator random variables 118 5.3 Randomized algorithms 122 5.4 Probabilistic analysis and further uses of indicator random variables 130 II Sorting and Order Statistics Introduction 147 6 Heapsort 151 6.1 Heaps 151 6.2 Maintaining the heap property 154 6.3 Building a heap 156 6.4 The heapsort algorithm 159 6.5 Priority queues 162 7 Quicksort 170 7.1 Description of quicksort 170 7.2 Performance of quicksort 174 7.3 A randomized version of quicksort 179 7.4 Analysis of quicksort 180 8 Sorting in Linear Time 191 8.1 Lower bounds for sorting 191 8.2 Counting sort 194 8.3 Radix sort 197 8.4 Bucket sort 200 9 Medians and Order Statistics 213 9.1 Minimum and maximum 214 9.2 Selection in expected linear time 215 9.3 Selection in worst-case linear time 220 III Data Structures Introduction 229 10 Elementary Data Structures 232 10.1 Stacks and queues 232 10.2 Linked lists 236 10.3 Implementing pointers and objects 241 10.4 Representing rooted trees 246 11 Hash Tables 253 11.1 Direct-address tables 254 11.2 Hash tables 256 11.3 Hash functions 262 11.4 Open addressing 269 11.5 Perfect hashing 277 12 Binary Search Trees 286 12.1 What is a binary search tree? 286 12.2 Querying a binary search tree 289 12.3 Insertion and deletion 294 12.4 Randomly built binary search trees 299 13 Red-Black Trees 308 13.1 Properties of red-black trees 308 13.2 Rotations 312 13.3 Insertion 315 13.4 Deletion 323 14 Augmenting Data Structures 339 14.1 Dynamic order statistics 339 14.2 How to augment a data structure 345 14.3 Interval trees 348 IV Advanced Design and Analysis Techniques Introduction 357 15 Dynamic Programming 359 15.1 Rod cutting 360 15.2 Matrix-chain multiplication 370 15.3 Elements of dynamic programming 378 15.4 Longest common subsequence 390 15.5 Optimal binary search trees 397 16 Greedy Algorithms 414 16.1 An activity-selection problem 415 16.2 Elements of the greedy strategy 423 16.3 Huffman codes 428 16.4 Matroids and greedy methods 437 16.5 A task-scheduling problem as a matroid 443 17 Amortized Analysis 451 17.1 Aggregate analysis 452 17.2 The accounting method 456 17.3 The potential method 459 17.4 Dynamic tables 463 V Advanced Data Structures Introduction 481 18 B-Trees 484 18.1 Definition of B-trees 488 18.2 Basic operations on B-trees 491 18.3 Deleting a key from a B-tree 499 19 Fibonacci Heaps 505 19.1 Structure of Fibonacci heaps 507 19.2 Mergeable-heap operations 510 19.3 Decreasing a key and deleting a node 518 19.4 Bounding the maximum degree 523 20 van Emde Boas Trees 531 20.1 Preliminary approaches 532 20.2 A recursive structure 536 20.3 The van Emde Boas tree 545 21 Data Structures for Disjoint Sets 561 21.1 Disjoint-set operations 561 21.2 Linked-list representation of disjoint sets 564 21.3 Disjoint-set forests 568 21.4 Analysis of union by rank with path compression 573 VI Graph Algorithms Introduction 587 22 Elementary Graph Algorithms 589 22.1 Representations of graphs 589 22.2 Breadth-first search 594 22.3 Depth-first search 603 22.4 Topological sort 612 22.5 Strongly connected components 615 23 Minimum Spanning Trees 624 23.1 Growing a minimum spanning tree 625 23.2 The algorithms of Kruskal and Prim 631 24 Single-Source Shortest Paths 643 24.1 The Bellman-Ford algorithm 651 24.2 Single-source shortest paths in directed acyclic graphs 655 24.3 Dijkstra's algorithm 658 24.4 Difference constraints and shortest paths 664 24.5 Proofs of shortest-paths properties 671 25 All-Pairs Shortest Paths 684 25.1 Shortest paths and matrix multiplication 686 25.2 The Floyd-Warshall algorithm 693 25.3 Johnson's algorithm for sparse graphs 700 26 Maximum Flow 708 26.1 Flow networks 709 26.2 The Ford-Fulkerson method 714 26.3 Maximum bipartite matching 732 26.4 Push-relabel algorithms 736 26.5 The relabel-to-front algorithm 748 VII Selected Topics Introduction 769 27 Multithreaded Algorithms Sample Chapter - Download PDF (317 KB) 772 27.1 The basics of dynamic multithreading 774 27.2 Multithreaded matrix multiplication 792 27.3 Multithreaded merge sort 797 28 Matrix Operations 813 28.1 Solving systems of linear equations 813 28.2 Inverting matrices 827 28.3 Symmetric positive-definite matrices and least-squares approximation 832 29 Linear Programming 843 29.1 Standard and slack forms 850 29.2 Formulating problems as linear programs 859 29.3 The simplex algorithm 864 29.4 Duality 879 29.5 The initial basic feasible solution 886 30 Polynomials and the FFT 898 30.1 Representing polynomials 900 30.2 The DFT and FFT 906 30.3 Efficient FFT implementations 915 31 Number-Theoretic Algorithms 926 31.1 Elementary number-theoretic notions 927 31.2 Greatest common divisor 933 31.3 Modular arithmetic 939 31.4 Solving modular linear equations 946 31.5 The Chinese remainder theorem 950 31.6 Powers of an element 954 31.7 The RSA public-key cryptosystem 958 31.8 Primality testing 965 31.9 Integer factorization 975 32 String Matching 985 32.1 The naive string-matching algorithm 988 32.2 The Rabin-Karp algorithm 990 32.3 String matching with finite automata 995 32.4 The Knuth-Morris-Pratt algorithm 1002 33 Computational Geometry 1014 33.1 Line-segment properties 1015 33.2 Determining whether any pair of segments intersects 1021 33.3 Finding the convex hull 1029 33.4 Finding the closest pair of points 1039 34 NP-Completeness 1048 34.1 Polynomial time 1053 34.2 Polynomial-time verification 1061 34.3 NP-completeness and reducibility 1067 34.4 NP-completeness proofs 1078 34.5 NP-complete problems 1086 35 Approximation Algorithms 1106 35.1 The vertex-cover problem 1108 35.2 The traveling-salesman problem 1111 35.3 The set-covering problem 1117 35.4 Randomization and linear programming 1123 35.5 The subset-sum problem 1128 VIII Appendix: Mathematical Background Introduction 1143 A Summations 1145 A.1 Summation formulas and properties 1145 A.2 Bounding summations 1149 B Sets, Etc. 1158 B.1 Sets 1158 B.2 Relations 1163 B.3 Functions 1166 B.4 Graphs 1168 B.5 Trees 1173 C Counting and Probability 1183 C.1 Counting 1183 C.2 Probability 1189 C.3 Discrete random variables 1196 C.4 The geometric and binomial distributions 1201 C.5 The tails of the binomial distribution 1208 D Matrices 1217 D.1 Matrices and matrix operations 1217 D.2 Basic matrix properties 122

算法导论，英文【本书目录】 I Foundations Introduction 3 l The Role of Algorithms in Computing 5 l.l Algorithms 5 l.2 Algorithms as a technology 10 2 Getting Started I5 2.l Insertion sort 15 2.2 Analyzing algorithms 21 2.3 Designing algorithms 27 3 Growth of Functions 41 3.l Asymptotic notation 41 3.2 Standard notations and common functions 51 4 Recurrences 62 4.l The substitution method 63 4.2 The recursion-tree method 67 4.3 The master method 73 4.4 Proof of the master theorem 76 5 Probabilistic Analysis and Randomized Algorithms 5.l The hiring problem 91 5.2 Indicator random variables 94 5.3 Randomized algorithms 99 5.4 Probabi1istic analysis and further uses of indicator 106 II Sorting and Order Statistics Introduction 123 6 Heapsort 127 6.l Heaps I27 6.2 Maintaining the heap property 130 6.3 Building a heap 132 6.4 The heapsort algorithm 135 6.5 Priority queues 138 7 Quicksort 145 7.l Description of quicksort 145 7.2 Performance ofquicksort 149 7.3 A randomized version of quicksort 153 7.4 Analysis ofquicksort 55 8 Sorting in Linear Time 165 8.l Lower bounds for sorting 165 8.2 Counting sort i68 8.3 Radix sort 170 8.4 Bucket sort 174 9 Medians and Order Statistics 183 9.1 Minimum and maximum 184 9.2 Selection in expected linear time 185 9.3 Selection in worst-case linear time 189 III Data Structures Introduction 197 10 Elementary Data Structures 200 l0.l Stacks and queues 200 l0.2 Linked lists 204 l0.3 Implementing pointers and objects 209 l0.4 Representing rooted trees 214 11 Hash Tables 221 ll.l Direct-address tables 222 11.2 Hash tables 224 ll.3 Hash functions 229 ll.4 Open addressing 237 ll.5 Perfect hashing 24S l2 Binary Search Trees 253 l2.l What is a binary search tree? 2S3 l2.2 Querying a binary search tree 2S6 l2.3 Insertion and deletion 261 l2.4 Randoinly built binary search trees 265 13 Red-Black Thees 273 l3.l Properties of red-black trees 273 l3.2 Rotations 277 l3.3 Insertion 280 l3.4 Deletion 288 14 Augmenting Data Structures 302 l4.l Dynamic order statistics 302 l4.2 How to augment a data structure 308 l4.3 Interval trees 311 IV Advanced Desthe and Analysis Techniques Introduction 321 15 Dynamic Programming J2J l5.l Assembly--line scheduling 324 l5.2 Matrix-chain multiplication 331 l5.3 Elements of dynamic programming 339 15.4 Longest common subsequence 350 l5.5 Optimal binary search trees 356 l6 Greedy Algorithms 370 l6.l An activity-selection problem 371 l6.2 Elements of the greedy strategy 379 l6.3 Huffman codes 385 l6.4 Theoretical foundations for greedy methods 393 16.5 A task-scheduling problem 399 17 Amortized Analysis 405 l7.1 Aggregate analysis 406 17.2 The accounting method 410 17.3 The potential method 412 l7.4 Dynamic tables 416 V Advanced Data Structures Introduction 431 18 B-Trees 434 18.l Definition of B--trees 438 l8.2 Basic operations on B-trees 44j l8.3 Deleting a key from a B--tree 449 19 Binomial Heaps 455 l9.l Binomial trees and binomial heaps 457 19.2 Operations on binomial heaps 461 20 Fibonacci Heaps 476 20.l Structure of Fibonacci heaps 477 20.2 Mergeable-heap operations 479 20.3 Decreasing a key and deleting a node 489 20.4 Bounding the maximum degree 493 21 Data Structures for Disjoint Sets 498 2l.l Disjoint--set operations 498 2l.2 Linked-list representation of disjoint sets 501 2l.3 Disjoint--set forests 505 2l.4 Analysis of union by rank with path compression 50 VI Graph Algorithms Introduction 525 22 Elementary Graph Algorithms 527 22.l Representations of graphs 527 22.2 Breadth-first search 531 22.3 Depth-first search 540 22.4 Topological sort 549 22.5 Strongly connected components 552 23 Minimum Spanning Trees 561 23.l Growing a minimum spanning tree 562 23.2 The algorithms of Kruskal and Prim 567 24 Single-Source Shortest Paths 580 24.l The Bellman-Ford algorithm 588 24.2 Single-source shortest paths in directed acyclic graphs 24.3 Dijkstra's algorithm 595 24.4 Difference constraints and shortest paths 601 24.5 Proofs of shortest-paths properties 607 25 All-Pairs Shortest Paths 620 25.l Shortest paths and matrix multiplication 622 25.2 The Floyd-Warshall a1gorithm 629 25.3 Johnson's algorithm for sparse graphs 636 26 Maximum Flow d43 26.l Flow networks 644 26.2 The Ford-Fulkerson method 651 26.3 Maximum bipartite matching 664 26.4 Push--relabel algorithms 669 26.5 The relabel--to-front a1gorithm 68I VII Selected Topics Introduction 701 27 Sorting Networks 704 27.l Comparison networks 704 27.2 The zero-one principle 709 27.3 A bitonic sorting network 712 27.4 A merging network 716 27.5 A sorting network 719 28 Matrix Operations 725 28.l Properties of matrices 725 28.2 Strassen's algorithm for matrix multiplication 735 28.3 Solving systems of linear equations 742 28.4 Inverting matrices 7S5 28.5 Symmetric positive-definite matrices and least-squares approximation760 29 Linear Programming 770 29.1 Standard and slack forms 777 29.2 Formulating problems as linear programs 785 29.3 The simplex algorithm 790 29.4 Duality 804 29.5 The initial basic feasible solution 811 30 Polynomials and the FFT 822 30.l Representation of polynomials 824 30.2 The DFT and FFT 830 30.3 Efficient FFT implementations 839 31 Number-Theoretic Algorithms 849 3l.l E1ementary numbertheoretic notions 850 31.2 Greatest common divisor 856 3l.3 Modular arithmetic 862 3l.4 Solving modular linear equations 869 3l.5 The Chinese remainder theorem 873 3l.6 Powers of an element 876 3l.7 The RSA public-key cryptosystem 881 3l.8 Primality testing 887 3l.9 Integer factorization 896 32 String Matching 906 32.l The naive string-matching algorithm 909 32.2 The Rabin-Karp algorithm 911 32.3 String matching with finite automata 916 32.4 The Knuth-Morris-Pratt algorithm 923 33 Computational Geometry 933 33.l Line--segment properties 934 33.2 Determining whether any pair of segments intersects 940 33.3 Finding the convex hull 947 33.4 Finding the c1osest pair of points 957 34 NP-Completeness 966 34.1 Polynomial time 971 34.2 Polynomial-time verification 979 34.3 NP-completeness and reducibility 984 34.4 NP--completeness proofs 995 34.5 NP-complete problems 1003 35 Approximation Algorithms 1022 35.l The vertex-cover problem 1024 35.2 The traveling-salesman problem 1027 35.3 The set-covering problem 1033 35.4 Randomization and linear programming ]039 35.5 The subset-sum problem 1043 VH APPendir: Mathematical Background Introduction 1057 A Summations 1058 A.l Summation formulas and properties 1058 A.2 Bounding summations 1062 B Sets, Etc. 1070 B.1 Sets 1070 B.2 Relations 1075 B.3 Functions 1077 B.4 Graphs 1080 B.5 Trees 1085 C Counting and Probability 1094 C.l Counting 1094 C.2 Probability 1100 C.3 Discrete random variables 1106 C.4 The geometric and binomial distributions 1112 C.5 The tails of the binomial distribution 1118 Bibliography 1127 Index 1145

算法导论英文版，非图片版。 I Foundations Introduction 3 1 The Role of Algorithms in Computing 5 1.1 Algorithms 5 1.2 Algorithms as a technology 11 2 Getting Started 16 2.1 Insertion sort 16 2.2 Analyzing algorithms 23 2.3 Designing algorithms 29 3 Growth of Functions 43 3.1 Asymptotic notation 43 3.2 Standard notations and common functions 53 4 Divide-and-Conquer 65 4.1 The maximum-subarray problem 68 4.2 Strassen’s algorithm for matrix multiplication 75 4.3 The substitution method for solving recurrences 4.4 The recursion-tree method for solving recurrences 88 4.5 The master method for solving recurrences 93 ? 4.6 Proof of the master theorem 97 5 Probabilistic Analysis and Randomized Algorithms 114 5.1 The hiring problem 114 5.2 Indicator random variables 118 5.3 Randomized algorithms 122 ? 5.4 Probabilistic analysis and further uses of indicator random variables 130 83 vi Contents II Sorting and Order Statistics Introduction 147 6 Heapsort 151 6.1 Heaps 151 6.2 Maintaining the heap property 154 6.3 Building a heap 156 6.4 The heapsort algorithm 159 6.5 Priority queues 162 7 Quicksort 170 7.1 Description of quicksort 170 7.2 Performance of quicksort 174 7.3 A randomized version of quicksort 179 7.4 Analysis of quicksort 180 8 Sorting in Linear Time 191 8.1 Lower bounds for sorting 191 8.2 Counting sort 194 8.3 Radix sort 197 8.4 Bucket sort 200 9 Medians and Order Statistics 213 9.1 Minimum and maximum 214 9.2 Selection in expected linear time 215 9.3 Selection in worst-case linear time 220 III Data Structures Introduction 229 10 Elementary Data Structures 232 10.1 Stacks and queues 232 10.2 Linked lists 236 10.3 Implementing pointers and objects 241 10.4 Representing rooted trees 246 11 Hash Tables 253 11.1 Direct-address tables 254 11.2 Hash tables 256 11.3 Hash functions 262 11.4 Open addressing 269 ? 11.5 Perfect hashing 277 Contents vii 12 Binary Search Trees 286 12.1 What is a binary search tree? 286 12.2 Querying a binary search tree 289 12.3 Insertion and deletion 294 ? 12.4 Randomly built binary search trees 299 13 Red-Black Trees 308 13.1 Properties of red-black trees 308 13.2 Rotations 312 13.3 Insertion 315 13.4 Deletion 323 14 Augmenting Data Structures 339 14.1 Dynamic order statistics 339 14.2 How to augment a data structure 345 14.3 Interval trees 348 IV Advanced Design and Analysis Techniques Introduction 357 15 Dynamic Programming 359 15.1 Rod cutting 360 15.2 Matrix-chain multiplication 370 15.3 Elements of dynamic programming 378 15.4 Longest common subsequence 390 15.5 Optimal binary search trees 397 16 Greedy Algorithms 414 16.1 An activity-selection problem 16.2 Elements of the greedy strategy 423 16.3 Huffman codes 428 415 ? 16.4 Matroids and greedy methods 437 ? 16.5 A task-scheduling problem as a matroid 443 17 Amortized Analysis 451 17.1 Aggregate analysis 452 17.2 The accounting method 456 17.3 The potential method 459 17.4 Dynamic tables 463 viii Contents V Advanced Data Structures Introduction 481 18 B-Trees 484 18.1 Definition of B-trees 488 18.2 Basic operations on B-trees 491 18.3 Deleting a key from a B-tree 499 19 Fibonacci Heaps 505 19.1 Structure of Fibonacci heaps 507 19.2 Mergeable-heap operations 510 19.3 Decreasing a key and deleting a node 518 19.4 Bounding the maximum degree 523 20 van Emde Boas Trees 531 20.1 Preliminary approaches 532 20.2 A recursive structure 536 20.3 The van Emde Boas tree 545 21 Data Structures for Disjoint Sets 561 21.1 Disjoint-set operations 561 21.2 Linked-list representation of disjoint sets 564 21.3 Disjoint-set forests 568 ? 21.4 Analysis of union by rank with path compression 573 VI Graph Algorithms Introduction 587 22 Elementary Graph Algorithms 589 22.1 Representations of graphs 589 22.2 Breadth-first search 594 22.3 Depth-first search 22.4 Topological sort 612 22.5 Strongly connected components 615 23 Minimum Spanning Trees 624 23.1 Growing a minimum spanning tree 625 23.2 The algorithms of Kruskal and Prim 631 603 Contents ix 24 Single-Source Shortest Paths 643 24.1 The Bellman-Ford algorithm 651 24.2 Single-source shortest paths in directed acyclic graphs 24.3 Dijkstra’s algorithm 658 24.4 Difference constraints and shortest paths 664 24.5 Proofs of shortest-paths properties 671 655 25 All-Pairs Shortest Paths 684 25.1 Shortest paths and matrix multiplication 25.2 The Floyd-Warshall algorithm 693 25.3 Johnson’s algorithm for sparse graphs 686 700 26 Maximum Flow 708 26.1 Flow networks 709 26.2 The Ford-Fulkerson method 714 26.3 Maximum bipartite matching 732 ? 26.4 Push-relabel algorithms 736 ? 26.5 The relabel-to-front algorithm 748 VII Selected Topics Introduction 769 27 Multithreaded Algorithms 772 27.1 The basics of dynamic multithreading 27.2 Multithreaded matrix multiplication 792 27.3 Multithreaded merge sort 797 28 Matrix Operations 813 28.1 Solving systems of linear equations 813 28.2 Inverting matrices 827 28.3 Symmetric positive-definite matrices and least-squares approximation 832 29 Linear Programming 843 29.1 Standard and slack forms 850 29.2 Formulating problems as linear programs 859 29.3 The simplex algorithm 864 29.4 Duality 879 29.5 The initial basic feasible solution 886 774 x Contents 30 Polynomials and the FFT 898 30.1 Representing polynomials 900 30.2 The DFT and FFT 906 30.3 Efficient FFT implementations 915 31 Number-Theoretic Algorithms 926 31.1 Elementary number-theoretic notions 927 31.2 Greatest common divisor 933 31.3 Modular arithmetic 939 31.4 Solving modular linear equations 946 31.5 The Chinese remainder theorem 950 31.6 Powers of an element 954 31.7 The RSA public-key cryptosystem ? 31.8 Primality testing 965 ? 31.9 Integer factorization 975 958 32 String Matching 985 32.1 The naive string-matching algorithm 988 32.2 The Rabin-Karp algorithm 990 32.3 String matching with finite automata 995 ? 32.4 The Knuth-Morris-Pratt algorithm 1002 33 Computational Geometry 1014 33.1 Line-segment properties 1015 33.2 Determining whether any pair of segments intersects 1021 33.3 Finding the convex hull 1029 33.4 Finding the closest pair of points 1039 34 NP-Completeness 1048 34.1 Polynomial time 1053 34.2 Polynomial-time verification 1061 34.3 NP-completeness and reducibility 1067 34.4 NP-completeness proofs 1078 34.5 NP-complete problems 1086 35 Approximation Algorithms 1106 35.1 The vertex-cover problem 1108 35.2 The traveling-salesman problem 1111 35.3 The set-covering problem 1117 35.4 Randomization and linear programming 1123 35.5 The subset-sum problem 1128 Contents xi VIII Appendix: Mathematical Background Introduction 1143 A Summations 1145 A.1 Summation formulas and properties 1145 A.2 Bounding summations 1149 B Sets, Etc. 1158 B.1 Sets 1158 B.2 Relations 1163 B.3 Functions 1166 B.4 Graphs 1168 B.5 Trees 1173 C Counting and Probability 1183 C.1 Counting 1183 C.2 Probability 1189 C.3 Discrete random variables 1196 C.4 The geometric and binomial distributions 1201 D Matrices 1217 D.1 Matrices and matrix operations 1217 D.2 Basic matrix properties 1222 Bibliography 1231 Index 1251

在有关算法的书中，有一些叙述非常严谨，但不够全面；另一些涉及了大量的题材，但又缺乏严谨性。本书将严谨性和全面性融为一体，深入讨论各类算法，并着力使这些算法的设计和分析能为各个层次的读者接受。全书各章自成体系，可以作为独立的学习单元；算法以英语和伪代码的形式描述，具备初步程序设计经验的人就能看懂；说明和解释力求浅显易懂，不失深度和数学严谨性。 --------------------------------------------------------------- 目录 Introduction to Algorithms, Third Edition 出版者的话译者序前言第一部分　基础知识第1章　算法在计算中的作用　1.1　算法　1.2　作为一种技术的算法　思考题　本章注记第2章　算法基础　2.1　插入排序　2.2　分析算法　2.3　设计算法　　2.3.1　分治法　　2.3.2　分析分治算法　思考题　本章注记第3章　函数的增长　3.1　渐近记号　3.2　标准记号与常用函数　思考题　本章注记第4章　分治策略　4.1　最大子数组问题　4.2　矩阵乘法的Strassen算法　4.3　用代入法求解递归式　4.4　用递归树方法求解递归式　4.5　用主方法求解递归式　4.6　证明主定理　　4.6.1　对b的幂证明主定理　　4.6.2　向下取整和向上取整　思考题　本章注记第5章　概率分析和随机算法　5.1　雇用问题　5.2　指示器随机变量　5.3　随机算法　?5.4　概率分析和指示器随机变量的进一步使用　　5.4.1　生日悖论　　5.4.2　球与箱子　　5.4.3　特征序列　　5.4.4　在线雇用问题　思考题　本章注记第二部分　排序和顺序统计量第6章　堆排序　6.1　堆　6.2　维护堆的性质　6.3　建堆　6.4　堆排序算法　6.5　优先队列　思考题　本章注记第7章　快速排序　7.1　快速排序的描述　7.2　快速排序的性能　7.3　快速排序的随机化版本　7.4　快速排序分析　　7.4.1　最坏情况分析　　7.4.2　期望运行时间　思考题　本章注记第8章　线性时间排序　8.1　排序算法的下界　8.2　计数排序　8.3　基数排序　8.4　桶排序　思考题　本章注记第9章　中位数和顺序统计量　9.1　最小值和最大值　9.2　期望为线性时间的选择算法　9.3　最坏情况为线性时间的选择算法　思考题　本章注记第三部分　数据结构第10章　基本数据结构　10.1　栈和队列　10.2　链表　10.3　指针和对象的实现　10.4　有根树的表示　思考题　本章注记第11章　散列表　11.1　直接寻址表　11.2　散列表　11.3　散列函数　　11.3.1　除法散列法　　11.3.2　乘法散列法　　11.3.3　全域散列法　11.4　开放寻址法　11.5　完全散列　思考题　本章注记第12章　二叉搜索树　12.1　什么是二叉搜索树　12.2　查询二叉搜索树　12.3　插入和删除　12.4　随机构建二叉搜索树　思考题　本章注记第13章　红黑树　13.1　红黑树的性质　13.2　旋转　13.3　插入　13.4　删除　思考题　本章注记第14章　数据结构的扩张　14.1　动态顺序统计　14.2　如何扩张数据结构　14.3　区间树　思考题　本章注记第四部分　高级设计和分析技术第15章　动态规划　15.1　钢条切割　15.2　矩阵链乘法　15.3　动态规划原理　15.4　最长公共子序列　15.5　最优二叉搜索树　思考题　本章注记第16章　贪心算法　16.1　活动选择问题　16.2　贪心算法原理　16.3　赫夫曼编码　16.4　拟阵和贪心算法　16.5　用拟阵求解任务调度问题　思考题　本章注记第17章　摊还分析　17.1　聚合分析　17.2　核算法　17.3　势能法　17.4　动态表　　17.4.1　表扩张　　17.4.2　表扩张和收缩　思考题　本章注记第五部分　高级数据结构第18章　B树　18.1　B树的定义　18.2　B树上的基本操作　18.3　从B树中删除关键字　思考题　本章注记第19章　斐波那契堆　19.1　斐波那契堆结构　19.2　可合并堆操作　19.3　关键字减值和删除一个结点　19.4　最大度数的界　思考题　本章注记第20章　van Emde Boas树　20.1　基本方法　20.2　递归结构　　20.2.1　原型van Emde Boas结构　　20.2.2　原型van Emde Boas结构上的操作　20.3　van Emde Boas树及其操作　　20.3.1　van Emde Boas树　　20.3.2　van Emde Boas树的操作　思考题　本章注记第21章　用于不相交集合的数据结构　21.1　不相交集合的操作　21.2　不相交集合的链表表示　21.3　不相交集合森林　*21.4　带路径压缩的按秩合并的分析　思考题　本章注记第六部分　图算法第22章　基本的图算法　22.1　图的表示　22.2　广度优先搜索　22.3　深度优先搜索　22.4　拓扑排序　22.5　强连通分量　思考题　本章注记第23章　最小生成树　23.1　最小生成树的形成　23.2　Kruskal算法和Prim算法　思考题　本章注记第24章　单源最短路径　24.1　Bellman?Ford算法　24.2　有向无环图中的单源最短路径问题　24.3　Dijkstra算法　24.4　差分约束和最短路径　24.5　最短路径性质的证明　思考题　本章注记第25章　所有结点对的最短路径问题　25.1　最短路径和矩阵乘法　25.2　Floyd?Warshall算法　25.3　用于稀疏图的Johnson算法　思考题　本章注记第26章　最大流　26.1　流网络　26.2　Ford\Fulkerson方法　26.3　最大二分匹配　26.4　推送重贴标签算法　26.5　前置重贴标签算法　思考题　本章注记第七部分　算法问题选编第27章　多线程算法　27.1　动态多线程基础　27.2　多线程矩阵乘法　27.3　多线程归并排序　思考题　本章注记第28章　矩阵运算　28.1　求解线性方程组　28.2　矩阵求逆　28.3　对称正定矩阵和最小二乘逼近　思考题　本章注记第29章　线性规划　29.1　标准型和松弛型　29.2　将问题表达为线性规划　29.3　单纯形算法　29.4　对偶性　29.5　初始基本可行解　思考题　本章注记第30章　多项式与快速傅里叶变换　30.1　多项式的表示　30.2　DFT与FFT 　30.3　高效FFT实现　思考题　本章注记第31章　数论算法　31.1　基础数论概念　31.2　最大公约数　31.3　模运算　31.4　求解模线性方程　31.5　中国余数定理　31.6　元素的幂　31.7　RSA公钥加密系统　31.8　素数的测试　31.9　整数的因子分解　思考题　本章注记第32章　字符串匹配　32.1　朴素字符串匹配算法　32.2　Rabin\Karp算法　32.3　利用有限自动机进行字符串匹配　32.4　Knuth?Morris?Pratt算法　思考题　本章注记第33章　计算几何学　33.1　线段的性质　33.2　确定任意一对线段是否相交　33.3　寻找凸包　33.4　寻找最近点对　思考题　本章注记第34章　NP完全性　34.1　多项式时间　34.2　多项式时间的验证　34.3　NP完全性与可归约性　34.4　NP完全性的证明　34.5　NP完全问题　　34.5.1　团问题　　34.5.2　顶点覆盖问题　　34.5.3　哈密顿回路问题　　34.5.4　旅行商问题　　34.5.5　子集和问题　思考题　本章注记第35章　近似算法　35.1　顶点覆盖问题　35.2　旅行商问题　35.2.1　满足三角不等式的旅行商问题　35.2.2　一般旅行商问题　35.3　集合覆盖问题　35.4　随机化和线性规划　35.5　子集和问题　思考题　本章注记第八部分　附录：数学基础知识附录A　求和　A.1　求和公式及其性质　A.2　确定求和时间的界　思考题　附录注记附录B　集合等离散数学内容　B.1　集合　B.2　关系　B.3　函数　B.4　图　B.5　树　　B.5.1　自由树　　B.5.2　有根树和有序树　　B.5.3　二叉树和位置树　思考题　附录注记附录C　计数与概率　C.1　计数　C.2　概率 C.3　离散随机变量　C.4　几何分布与二项分布　*C.5　二项分布的尾部　思考题　附录注记附录D　矩阵　D.1　矩阵与矩阵运算　D.2　矩阵基本性质　思考题　附录注记

一、本书的内容目前，市面上有关计算机算法的书很多，有些叙述严谨但不全面，另外一些则是容量很大但不够严谨。本书将叙述的严谨性以及内容的深度和广度有机地结合了起来。第1版推出后，即在世界范围内受到了广泛的欢迎，被各高等院校用作多种课程的教材和业界的标准参考资料。它深入浅出地介绍了大量的算法及相关的数据结构，以及用于解决一些复杂计算问题的高级策略(如动态规划、贪心算法、平摊分析等)，重点在于算法的分析和设计。对于每一个专题，作者都试图提供目前最新的研究成果及样例解答，并通过清晰的图示来说明算法的执行过程。. 本书是原书的第2版，在第1版的基础之上增加了一些新的内容，涉及算法的作用、概率分析和随机化算法、线性规划，以及对第1版中详尽的、几乎涉及到每一小节的修订。这些修订看似细微，实际上非常重要。书中引入了“循环不变式”，并贯穿始终地用来证明算法的正确性。在不改动数学和分析重点的前提下，作者将第1版中的许多数学基础知识从第一部分移到了附录中。二、本书的特点本书在进行算法分析的过程中，保持了很好的数学严谨性。书中的分析和设计可以被具有各种水平的读者所理解。相对来说，每一章都可以作为一个相对独立的单元来教授或学习。书中的算法以英语加伪代码的形式给出，只要有一点程序设计经验的人都能读懂，并可以用任何计算机语言(如C／C++和Java等)方便地实现。在书中，作者将算法的讨论集中在一些比较现代的例子上，它们来自分子生物学(如人类基因项目)、商业和工程等领域。每一小节通常以对相关历史素材的讨论结束，讨论了在每一算法领域的原创研究。本书的特点可以概括为以下几个方面： 1．概念清晰，广度、深度兼顾。本书收集了现代计算机常用的数据结构和算法，并作了系统而深入的介绍。对涉及的概念和背景知识都作了清晰的阐述，有关的定理给出了完整的证明。 2．“五个一”的描述方法。本书以相当的深度介绍了许多常用的数据结构和有效的算法。编写上采用了“五个一”，即一章介绍一个算法、一种设计技术、一个应用领域和一个相关话题。.. 3．图文并茂，可读性强。书中的算法均以通俗易懂的语言进行说明，并采用了大量插图来说明算法是如何工作的，易于理解。 4．算法的“伪代码”形式简明实用。书中的算法均以非常简明的“伪代码”形式来设计，可以很容易地把它转化为计算机程序，直接应用。注重算法设计的效率，对所有的算法进行了仔细、精确的运行时间分析，有利于进一步改进算法。三、本书的用法本书对内容进行了精心的设计和安排，尽可能考虑到所有水平的读者。即使是初学计算机算法的人，也可以在本书中找到所需的材料。每一章都是独立的，读者只需将注意力集中到最感兴趣的章节阅读。 1．适合作为教材或教学参考书。本书兼顾通用性与系统性，覆盖了许多方面的内容。本书不但阐述通俗、严谨，而且提供了大量练习和思考题。针对每一节的内容，都给出了数量和难度不等的练习题。练习题用于考察对基本内容的掌握程度，思考题有一定的难度，需进行精心的研究，有时还通过思考题介绍一些新的知识。前言回到顶部↑本书提供了对当代计算机算法研究的一个全面、综合性的介绍。书中给出了多个算法，并对它们进行了较为深入的分析，使得这些算法的设计和分析易于被各个层次的读者所理解。力求在不牺牲分析的深度和数学严密性的前提下，给出深入浅出的说明。. 书中每一章都给出了一个算法、一种算法设计技术、一个应用领域或一个相关的主题。算法是用英语和一种“伪代码”来描述的，任何有一点程序设计经验的人都能看得懂。书中给出了230多幅图，说明各个算法的工作过程。我们强调将算法的效率作为一种设计标准，对书中的所有算法，都给出了关于其运行时间的详细分析。本书主要供本科生和研究生的算法或数据结构课程使用。因为书中讨论了算法设计中的工程问题及其数学性质，因此，本书也可以供专业技术人员自学之用。本书是第2版。在这个版本里，我们对全书进行了更新。所做的改动从新增了若干章，到个别语句的改写。致使用本书的教师本书的设计目标是全面、适用于多种用途。它可用于若干课程，从本科生的数据结构课程到研究生的算法课程。由于书中给出的内容比较多，只讲一学期一般讲不完，因此，教师们应该将本书看成是一种“缓存区”或“瑞典式自助餐”，从中挑选出能最好地支持自己希望教授的课程的内容。教师们会发现，要围绕自己所需的各个章节来组织课程是比较容易的。书中的各章都是相对独立的，因此，你不必担心意想不到的或不必要的各章之间的依赖关系。每一章都是以节为单位，内容由易到难。如果将本书用于本科生的课程，可以选用每一章的前面几节内容；在研究生课程中，则可以完整地讲授每一章。全书包含920多个练习题和140多个思考题。每一节结束时给出练习题，每一章结束时给出一些

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