Discrete Algorithmic Mathematics

793 Pages·2005·148.85 MB·English

by  Stephen B. Maurer

.-- ISCRETE
LGORITHMIC
ATHEMATICS
Stephen B Ma1urer
Anthony Ralston
( )
List of Algorithms
This table lists all the numbered algorithms in the book. All algorithms displayed in
boxes in the text are numbered, as well as the more important algorithrns displayed
in boxes in the problems. In addition, there are numerous algorithm fragn1ents
both in the text and in the problems. The fragrr1ents and unnumbered algorithms
illustrate some point and are generally not important algorithms in themselves.
Number Name Output or Purpose Page
1.1 Po,verA Computes integer powers of a number 74
1.2 PowerB Speed-up of 1.1 for some powers 75
1.3 Po,verC Speed-up of 1.1 75
1.4 PrimeNum First K prime numbers 77
1.5 FiberNet Minimum cost links in a network 80
1.6 Euclid Greatest common divisor (gcd) of two integers 84,123,170
1.7 Euclidl Possibly faster variant of 1.6 87
1.8 Euclid-RecFunc Recursive function version of 1.6 96,163
1.9 Euclid-RecPro Recursive procedure version of 1.6 97,165
1.10 Hanoi Towers of Hanoi 101
1.11 StringProc Process a string of characters 109
1.12 MaxNumber Maximum of n numbers 119,587
1.13 SeqS earch Sequential search of a list 125
1.14 BinSearch Binary search of a list 127, 172
1.15 Euclid2 Possibly faster variant of 1.6 132
2.1 Hanoi Slight rewrite of 1.10 160
2.2 TOH Variant recursive procedure for Hanoi 166
2.3 EuclidRound-Func Possibly faster variant of 1.8 167
2.4 EuclidRound-Pro Possibly faster variant of 1.9 167
2.5 Euclid-Round Up Possibly faster variant of 1.6 167
2.6 Iterative-TOH Non-recursive Towers of Hanoi 179
2.7 Sum-Search Find if one number is the sum of some others 213
2.8 EuclidZ Version of 1.6 that also \Vorks for negative integers 214
3.1 BuildTree Binary tree of a list of words 223
3.2 Warshall Path matrix of a graph 244
3.3 Pathgrow Grows a path in a graph 255
3.4 Ecycle An Eulerian cycle in a graph 256
Number Name Output or Purpose Page
3.5 HamPath A Hamiltonian path in a complete graph 261
3.6 Dijkstra Shortest path in a weighted graph 272
3.7 BreadthFirstSearch Breadth first search of a connected graph 279
3.8 PathLength Lengths of all paths from a given vertex 281
3.9 DepthFirstSearch Depth first search of a connected graph 282
3.10 Bipartite Determine if a graph is 2-colorable 288
3.11 K-Colorable A proper coloring of a graph 291
3.13 Color-by-Vertices Variant of 3.11 294
3.13 One-Color-at-a-Time Different approach to coloring a graph 294
3.14 Span'I'ree Spanning tree of a graph 297
3.15 MinSpanTree Minimurn weight spanning tree of a graph 299
3.16 Binary'I'reeTraversal Three ways of traversing a binary tree 306
3.17 Ford Shortest path in a ,veighted graph 314
4.1 PermV alue The permutation number P(n , r) 343
4.2 PermValue2 The permutation number P(n , r) 347
4.3 Permute-1 Randorn permutation of (n] 388, 590
4.4 Permute-2 Random permutation of (n] 390,592
4.5 Permute-3 Randorn permutation of (n] 391
4.6 AllPerm All permutations of [n] 394
4.7 Monotonic Longest monotonic subsequence 402
5.1 Straightfor,vard Polynomial evaluation 463
5.2 Horner Faster polynomial evaluation 464
5.3 FibA Fibonacci numbers 466
5.4 FibB Fibonacci numbers by recursion 466
5.5 Po,verPoly Polynomial evaluation 475
6.1 BigNext Two largest of n numbers 593
7.1 Mult Integer multi plication 615
7.2 Insertion Insertion sort of a list 678
E.l StringAlign-R.ec Optimal alignment of DNA strings 697
E.2 StringAlign Iterative Optimal Alignment 699
E.3 StringAlign2 Variant of E. 2 704
E.4 ProjectSched Minin1um completion time 709
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or1 I C
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a ICS
Third Edition
Stephen B Maurer
Swarthmore College
Anthony Ralston
State University of New York at Buffalo
AK Peters, Ltd.
Wellesley, Massachusetts
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2005 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Inform a business
No claim to original U.S. Government works
Version Date: 20130204
International Standard Book Number-13: 978-1-4398-6375-6 (eBook - PDF)
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Contents
List of Algorithn1s inside front cover
.
Instructor's Preface
lX
.
Student's Preface
XIV
Chapter Sumrnaries
XVll
Pathways Through the Book xx
..
Problem Difficulty Rating
XXll
Symbols, Notation, Abbreviations arid Conventions
XXlll
PROLOGUE • What Is Discrete Algorithmic Mathematics? l
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CHAPTER 0 Mathematical Preliminaries 9
0.1 Sets, Relations, and Functions 9
0.2 Some Important Functions 20
0.3 Growth Rates and Order Notation 30
0.4 Sumrnation and ProdU1ct Notation 40
0.5 Matrix Algebra 50
0.6 The Language and Methods of Reasoning 56
Supplementary Problems 69
CHAPTER l • Algorithms 73
l .1 Some Examples of Algorithms 73
l .2 Aspects of AL 88
1.3 Recursive Algorithms 94
l .4 Algorith1nic Language: Procedures and Functions 105
l .5 The Analysis of Algorithrns 117
Supplementary Problems 131
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CHAPTER2 Mathematical Induction 135
2.1 Introduction 135
2.2 First Exa1nples 137
V
vi Contents
2.3 Strong Induction and Other Variants 151
2.4 Ir1duction and Recursive Algorithms 158
2.5 Ir1duction and Iterative Algorithms 168
2.6 How to Conjecture What to Prove 180
2.7 Inductive Defir1itions 194
2.8 Faulty Inductions 201
Supplementary Problems 210
CHAPTERJ • Graphs and Trees 217
3.1 Examples and Terminology 217
3.2 Paths, Cycles and the Adjacency Matrix 236
3.3 Eulerian and Hamiltonian Paths and Cycles 250
3.4 Shortest Paths 266
3.5 Breadth First Search and Depth First Search 278
3.6 Coloring Problems 285
3.7 Trees 296
Supplementary Problems 311
CHAPTER 4 • Fundamental Counting Methods 321
4.1 Introduction 321
4.2 First Exa1nples: The Sum and Product Rules 322
4.3 Subtler Examples and Further Rules 329
4.4 Permutations and Combinations 341
4.5 Cornbinatorial Identities and Combinatorial Arguments 348
4.6 The Binomial Theorem 355
4.7 Four Comn1on Problems with Balls and Urns 365
4.8 Generating Functions 375
4.9 Combinatorial Algorithms 385
4.10 Algorith1nic Pigeonholes 399
Supplementary Problems 406
CHAPTER 5 • Difference Equations 411
5.1 Introduction 411
5.2 Modeling with Difference Equations 413
5.3 Getting Inforrnation frorn Difference Equations 426
5.4 Terminology and a Fundamental Theorem 433
5.5 Constant Coefficient Homogeneous Linear Difference Equations 439
5.6 Qualitative Analysis 451
5.7 Nonhomogeneous Difference Equations 457
5.8 Applications to Algorithms 462
Contents vii
5.9 Variable Coefficients, Surns, and Recent Advances in Computer
Algebra 479
5.10 Nonlinear Difference Equations 484
5. 11 Finite Differences 495
Supplementary Problems 508
CHAPTER 6 • Probability 513
6.1 Introduction 513
6.2 Probability Space 517
6.3 Conditional Probability, l11dependence, and Bayes' Theorem 527
6.4 Random Variables and Distributions 539
6.5 Expected Value 557
6.6 Variance 569
6. 7 Statistical Esti1nation 577
6.8 Applications to Algorithms: Proofs of Prior Claims 586
6.9 Recursive Methods in Probability 595
Supplementary Proble1ns 606
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CHAPTER 7 An Introduction to Mathematical Logic 611
7.1 Introduction and Terminology 611
7.2 The Propositional Calculus 616
7.3 Validity and Tautology 630
7 .4 Algorithm Verification 637
7.5 Boolean Algebra 643
7.6 The Predicate Calculus 663
7.7 Algorithm Verification Using the Predicate Calculus 676
7.8 Theorems and Proofs 682
Supplementary Proble1ns 689
EPILOGUE • Coming Full Circle with Biology and Minimax Theorems 691
E. l DNA Matching 691
E.2 Algorithmic lVIathematics and Minimax Theorems 707
Final Pro ble ms 717
References 723
Appendix: Limits 729
Hints and Answers 733
Index 761