算法导论动态规划章节中rod-cutting的一个问题

dracularking 2016-05-04 10:03:31

为了计算最优的原木切割法，公式是这样的：

可后来就变形成这样了，没明白这样怎么也可以（左手段换成了p）：

p：不切割，直接卖的价值
r：优化方式切割后的总价值

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wshcdr 2016-12-29

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公式2 是做了简化，从左边切下长度为i的一段，然后只对右边长度为n-i长度的钢条进行切割，对左边的不再切割了

dracularking 2016-05-09

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谢谢回答，确实是这样，不管怎样分割，只要最优分割后，总能从最后的分割结果中，找出一个p[i]（未必是单价最高），重新组合成这样： r[n] = p[i] + r[n-i] (1<=i<=n) 说明这种分割方式总能组成最优解，是不是这样？

Tiger_Zhao 2016-05-04

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r 是1个或多个 p 组成的集合。

怎么推导不知道，不过可以反过来论证：
1)假定已经按公式1找到了最优的r[n]；
2)那么从里面拿走单价最高的一条p[k]
2.a)如果k=n，直接符合公式2
2.b)如果k<n，剩下集合必定是最优r[n-k]，否则用最优的r[n-k]和p[k]可以组成更优的r[n]了。

所以公式2是正确的：
要么是 p[n] 直接组成最优的 r[n]；
要么是某个最优的　r[n-i] 加上 p[n] 组成最优的 r[n]。

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

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Introduction to Algorithms Third Editio Contents 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 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 vi Contents II Sorting and Order Statistics Introduction 147 6Heapsort151 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

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

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