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download Introduction to Algorithms


this book talking about algorighms

The Role of Algorithms in Computing 5
1.1 Algorithms 5
1.2 Algorithms as a technology 10
2 Getting Started 15
2.1 Insertion sort 15
2.2 Analyzing algorithms 21
2.3 Designing algorithms 27
3 Growth of Functions 41
3.1 Asymptotic notation 41
3.2 Standard notations and common functions 51
4 Recurrences 62
4.1 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 91
5.1 The hiring problem 91
5.2 Indicator random variables 94
5.3 Randomized algorithms 99
⋆ 5.4 Probabilistic analysis and further uses of indicator random variables
105


vi Contents
II Sorting and Order Statistics
Introduction 123
6 Heapsort 127
6.1 Heaps 127
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
7.1 Description of quicksort 145
7.2 Performance of quicksort 149
7.3 A randomized version of quicksort 153
7.4 Analysis of quicksort 155
8 Sorting in Linear Time 165
8.1 Lower bounds for sorting 165
8.2 Counting sort 168
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
10.1 Stacks and queues 200
10.2 Linked lists 204
10.3 Implementing pointers and objects 209
10.4 Representing rooted trees 214
11 Hash Tables 221
11.1 Direct-address tables 222
11.2 Hash tables 224
11.3 Hash functions 229
11.4 Open addressing 237
⋆ 11.5 Perfect hashing 245
Contents vii
12 Binary Search Trees 253
12.1 What is a binary search tree? 253
12.2 Querying a binary search tree 256
12.3 Insertion and deletion 261
⋆ 12.4 Randomly built binary search trees 264
13 Red-Black Trees 273
13.1 Properties of red-black trees 273
13.2 Rotations 277
13.3 Insertion 280
13.4 Deletion 288
14 Augmenting Data Structures 302
14.1 Dynamic order statistics 302
14.2 How to augment a data structure 308
14.3 Interval trees 311
IV Advanced Design and Analysis Techniques
Introduction 321
15 Dynamic Programming 323
15.1 Assembly-line scheduling 324
15.2 Matrix-chain multiplication 331
15.3 Elements of dynamic programming 339
15.4 Longest common subsequence 350
15.5 Optimal binary search trees 356
16 Greedy Algorithms 370
16.1 An activity-selection problem 371
16.2 Elements of the greedy strategy 379
16.3 Huffman codes 385
⋆ 16.4 Theoretical foundations for greedy methods 393
⋆ 16.5 A task-scheduling problem 399
17 Amortized Analysis 405
17.1 Aggregate analysis 406
17.2 The accounting method 410
17.3 The potential method 412
17.4 Dynamic tables 416
viii Contents
V Advanced Data Structures
Introduction 431
18 B-Trees 434
18.1 Definition of B-trees 438
18.2 Basic operations on B-trees 441
18.3 Deleting a key from a B-tree 449
19 Binomial Heaps 455
19.1 Binomial trees and binomial heaps 457
19.2 Operations on binomial heaps 461
20 Fibonacci Heaps 476
20.1 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
21.1 Disjoint-set operations 498
21.2 Linked-list representation of disjoint sets 501
21.3 Disjoint-set forests 505
⋆ 21.4 Analysis of union by rank with path compression 509
VI Graph Algorithms
Introduction 525
22 Elementary Graph Algorithms 527
22.1 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
23.1 Growing a minimum spanning tree 562
23.2 The algorithms of Kruskal and Prim 567
24 Single-Source Shortest Paths 580
24.1 The Bellman-Ford algorithm 588
24.2 Single-source shortest paths in directed acyclic graphs 592
24.3 Dijkstra’s algorithm 595
24.4 Difference constraints and shortest paths 601
24.5 Proofs of shortest-paths properties 607
Contents ix
25 All-Pairs Shortest Paths 620
25.1 Shortest paths and matrix multiplication 622
25.2 The Floyd-Warshall algorithm 629
25.3 Johnson’s algorithm for sparse graphs 636
26 Maximum Flow 643
26.1 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 algorithm 681
VII Selected Topics
Introduction 701
27 Sorting Networks 704
27.1 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
28.1 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 755
28.5 Symmetric positive-definite matrices and least-squares approximation
760
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
30.1 Representation of polynomials 824
30.2 The DFT and FFT 830
30.3 Efficient FFT implementations 839
x Contents
31 Number-Theoretic Algorithms 849
31.1 Elementary number-theoretic notions 850
31.2 Greatest common divisor 856
31.3 Modular arithmetic 862
31.4 Solving modular linear equations 869
31.5 The Chinese remainder theorem 873
31.6 Powers of an element 876
31.7 The RSA public-key cryptosystem 881
⋆ 31.8 Primality testing 887
⋆ 31.9 Integer factorization 896
32 String Matching 906
32.1 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
33.1 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 closest 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
35.1 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 1039
35.5 The subset-sum problem 1043
VIII Appendix: Mathematical Background
Introduction 1057
A Summations 1058
A.1 Summation formulas and properties 1058
A.2 Bounding summations 1062
Contents xi
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.1 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

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