Methods in algorithmic analysis
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CRC Press/Taylor & Francis
2010
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| Schriftenreihe: | Chapman & Hall/CRC computer and information science series
A Chapman & Hall book |
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Datensatz im Suchindex
| _version_ | 1819679223125639168 |
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| adam_text | METHODS IN ALGORITHMIC ANALYSIS VLADIMIR A. DOBRUSHKIN BROWN UNIVERSITY
PROVIDENCE, RHODE ISLAND, U.S.A. (G) CRC PRESS TAYLOR & FRANCIS CROUP
BOCA RATON LONDON NEW YORK CRC PRESS IS AN IMPRINT OF THE TAYLOR &
FRANCIS GROUP, AN INFORMA BUSINESS A CHAPMAN & HALL BOOK CONTENTS
PREFACE XIII LIST OF SYMBOLS XVII ABBREVIATIONS XIX 1 PRELIMINARIES 1
1.1 WHY DO WE ANALYZE ALGORITHMS? 2 1.1.1 COST OF ALGORITHMS 2 1.1.2 ONE
PROBLEM - SEVERAL SOLUTIONS 3 1.1.3 MATRIX MULTIPLICATION 8 1.2 PROOFS
10 1.2.1 PROOF BY COUNTING 12 1.2.2 INDUCTION 14 1.3 ITERATION AND
RECURSION 22 2 COMBINATORICS 25 2.1 PROPERTIES OF SUMMATION 26 2.1.1
INDEX TRANSFORMATIONS 28 2.2 MULTIPLE SUMS 34 2.2.1 CHANGING ORDER OF
SUMMATION 34 2.2.2 SUMMATIONS AND FINITE DIFFERENCES 36 2.2.3 SUMMATION
BY PARTS 40 2.3 PRINCIPLES OF COUNTING 42 2.4 PERMUTATIONS AND
COMBINATIONS 44 2.4.1 COMBINATIONS AND LATTICE PATHS 48 2.5 BINOMIAL
COEFFICIENTS 56 2.5.1 DEFINITIONS AND PROPERTIES 56 2.5.2
TRANSFORMATIONS AND BASIC SUMS 58 2.5.3 INVERSE RELATIONS 64 2.5.4
VANDERMONDE CONVOLUTION 66 2.6 BINOMIAL COEFFICIENTS AND HYPERGEOMETRIC
FUNCTIONS 72 2.6.1 ABEL S IDENTITY 75 2.7 STIRLING APPROXIMATION 77 3
PROBABILITY 79 3.1 SET OPERATIONS 80 3.2 SAMPLE SPACE AND RANDOM
VARIABLES 82 3.3 CALCULATING PROBABILITIES 86 3.4 RANDOM VARIABLES 94
3.4.1 PROBABILITY MASS FUNCTION 94 3.4.2 EXPECTED VALUES 99 3.4.3
VARIANCE AND MOMENTS 112 3.4.4 FUNCTIONS OF RANDOM VARIABLES 119 3.5
CONDITIONAL PROBABILITIES 119 3.6 INDEPENDENCE 123 3.7 JOINT
DISTRIBUTIONS 127 3.8 DEPENDENT RANDOM VARIABLES 130 4 MORE ABOUT
PROBABILITY 135 4.1 SPECIAL DISTRIBUTIONS 135 4.1.1 BERNOULLI VARIABLES
AND THE BINOMIAL DISTRIBUTION 136 4.1.2 THE MULTINOMIAL DISTRIBUTION 140
4.1.3 THE GEOMETRIC DISTRIBUTION 141 4.1.4 THE NEGATIVE-BINOMIAL
DISTRIBUTION 148 4.1.5 THE HYPERGEOMETRIC DISTRIBUTION 151 4.1.6 THE
POISSON DISTRIBUTION 154 4.1.7 THE NORMAL DISTRIBUTION 157 4.2 TYPES OF
PROBABILISTIC CONVERGENCE 158 4.3 THE THEOREM OF TOTAL PROBABILITY 162
4.4 BAYES THEOREM 170 4.5 CONVOLUTION 174 4.6 ORDER STATISTICS 183 4.7
CHEBYSHEV INEQUALITIES 188 4.8 SUNDRY EXAMPLES 192 5 RECURRENCES OR
DIFFERENCE EQUATIONS 199 5.1 HOW DO DIFFERENCE EQUATIONS ARISE? 200 5.2
PROPERTIES OF DIFFERENCE EQUATIONS 208 5.3 FIRST ORDER LINEAR DIFFERENCE
EQUATIONS 215 5.4 RECURRENCES WITH INTEGER FUNCTIONS 221 5.4.1
DIVIDE-AND-CONQUER RECURRENCES 221 5.4.2 BINARY SEARCH RECURRENCE 224
5.4.3 MERGESORT RECURRENCE 227 5.5 QUICKSORT RECURRENCE 229 5.5.1 SOME
FULL-HISTORY RECURRENCES 233 5.6 RECURRENCES IN NUMERICAL ANALYSIS 236
5.7 CONTINUED FRACTIONS 242 5.8 PARTIAL DIFFERENCE EQUATIONS 251 5.8.1
THE RELATIONS WITH CONTINUOUS CALCULATIONS 254 5.8.2 COUNTING
ARRANGEMENTS*WITH AND WITHOUT REPETITION 256 5.8.3 STIRLING NUMBERS 261
5.9 SOME APPLICATIONS 263 5.9.1 BOUNDS FROM RECURRENCES 263 5.9.2
RECURRENCES AND FINITE DIFFERENCES 266 6 INTRODUCTION TO GENERATING
FUNCTIONS 271 6.1 GENERATING FUNCTIONS * DEFINITIONS 272 6.1.1 ORDINARY
GENERATING FUNCTIONS 272 6.1.2 EXPONENTIAL GENERATING FUNCTIONS 277
6.1.3 SOME GENERALIZATIONS 280 6.1.4 MULTIVARIATE GENERATING FUNCTIONS
283 6.1.5 MULTISECTION OF SERIES 285 6.2 EXTRACTION OF COEFFICIENTS 287
6.2.1 TRANSFORMATIONS BETWEEN THE GENERATING FUNCTIONS 293 6.2.2
MULTIVARIATE GENERATING FUNCTIONS 294 6.2.3 RECURRENCES FROM GENERATING
FUNCTIONS 296 6.3 COUNTING BINARY TREES 299 6.4 SOLVING RECURRENCES 305
6.4.1 ORDINARY RECURRENCE RELATIONS 305 6.4.2 VECTOR RECURRENCE
RELATIONS 315 6.4.3 PARTIAL DIFFERENCE EQUATIONS 319 6.4.4 WALKS ON THE
INTEGER GRID 323 6.5 SNAKE OIL SUMMATION 327 6.6 APPLICATIONS IN
PROBABILITY 331 6.6.1 DEFINITION OF GENERATING FUNCTIONS USED IN
PROBABILITY 331 6.6.2 EXAMPLES AND PROBLEMS 335 6.6.3 CONVOLUTION 337
6.6.4 QUICKSORT AND BINARY SEARCH ANALYSIS 340 6.7 THE LAGRANGE
INVERSION THEOREM 347 7 ENUMERATION WITH GENERATING FUNCTIONS 355 7.1
DEFINITION OF ENUMERATORS 355 7.2 SUM AND PRODUCT RULES 359 7.2.1 THE
SUM RULE 359 7.2.2 THE PRODUCT RULE 360 7.3 COUNTING COMPOSITIONS OF
INTEGERS 364 7.3.1 HOMOGENEOUS COMPOSITIONS 364 7.3.2 INHOMOGENEOUS
COMPOSITIONS 367 7.3.3 COMPOSITIONS WITH RESTRICTIONS 369 7.3.4
HETEROGENEOUS COMPONENTS 372 7.3.5 LIMITED SELECTION 375 7.4 FURTHER SET
OPERATIONS 381 7.4.1 SUBSTITUTION 383 7.4.2 MARKING 388 7.4.3 POWER SET
390 7.4.4 MULTISET OPERATION 391 7.5 PARTITIONS OF INTEGERS 392 7.6
EXPONENTIAL ENUMERATORS 403 7.6.1 THE SUM AND PRODUCT OF LABELED
STRUCTURES 404 7.6.2 PERMUTATIONS AND CYCLES 408 7.6.3 SHUFFLE PRODUCT
413 7.6.4 THE BIRTHDAY PROBLEMS 416 8 FURTHER ENUMERATION METHODS 423
8.1 ENUMERATION OF TREES 423 8.1.1 UNLABELED TREES 424 8.1.2 LABELED
TREES 429 8.1.3 COUNTING ALTERNATING PERMUTATIONS 430 8.2 OCCUPANCY
ENUMERATION 434 8.2.1 DISTRIBUTION OF IDENTICAL BALLS INTO
DISTINGUISHABLE BINS 435 8.2.2 DISTRIBUTION OF DISTINCT OBJECTS INTO
ORDERED CELLS 439 8.2.3 DISTRIBUTION OF IDENTICAL OBJECTS INTO IDENTICAL
CELLS 445 8.2.4 DISTRIBUTION OF DISTINCT OBJECTS INTO IDENTICAL CELLS
445 8.3 THE PRINCIPLE OF INCLUSION AND EXCLUSION (PIE) 446 8.3.1 THE PIE
FOR HOMOGENEOUS PROPERTIES 455 8.4 EXTENSIONS AND FURTHER APPLICATIONS
OF THE PIE 460 8.4.1 THE PIE VIA THE SYMBOLIC METHOD 464 8.5
PROBABILISTIC INCLUSION - EXCLUSION PRINCIPLE 468 8.6 RUNS IN
PERMUTATIONS 479 8.6.1 COUNTING PERMUTATIONS OF [1.. N] WITH * ASCENTS
480 8.6.2 COUNTING PERMUTATIONS OF [L..N] WITH RUNS OF ASCENTS OF LENGTH
R . . 482 8.6.3 COUNTING PERMUTATIONS OF [L..N] WITH M MAXIMAL RUNS OF
ASCENTS . . 483 8.7 SPECIAL TOPICS 483 9 COMBINATORICS OF STRINGS 489 9.
1 OPERATIONS ON LANGUAGES 490 9.2 REGULAR LANGUAGES 494 9.2.1
DEFINITIONS 495 9.2.2 FINITE STATE AUTOMATA 497 9.2.3 FINITE STATE
AUTOMATA AND REGULAR LANGUAGES 499 9.3 COUNTING REGULAR LANGUAGES 503
9.3.1 WORD EQUATIONS 503 9.3.2 COUNTING REGULAR LANGUAGES 505 9.3.3
ADMISSIBILITY CONSIDERATIONS 514 9.4 WAITING TIME PROBABILISTIC PROBLEMS
517 9.5 ALGORITHMS AND MARKOV CHAINS 527 10 INTRODUCTION TO ASYMPTOTICS
545 10.1 ASYMPTOTIC NOTATIONS AND APPLICATIONS 546 10.1.1 PROPERTIES OF
THE BIG OH NOTATION 547 10.1.2 ASYMPTOTIC EXPANSIONS 550 10.1.3 LIMITS
FOR INDETERMINATE FORMS 559 10.2 THE CRITICAL RANGE METHOD 563 10.3
RICE S METHOD 569 10.4 THE EULER SUMMATION FORMULA 579 10.4.1
ALTERNATING SERIES 589 10.5 FINDING PRIMES 591 10.6 ASYMPTOTICS FROM
RECURRENCES 598 10.6.1 DIVIDE-AND-CONQUER ASYMPTOTICS 598 10.6.2 DIRECT
ASYMPTOTICS 604 10.7 LIMIT LAWS IN PROBABILITY 609 10.7.1 LAWS OF LARGE
NUMBERS 610 10.7.2 THE CENTRAL LIMIT THEOREM 616 10.7.3 RANDOM WALKS 623
11 ASYMPTOTICS AND GENERATING FUNCTIONS 629 11.1 ELEMENTARY BOUNDS FROM
GENERATING FUNCTIONS 629 11.1.1 THE LAGRANGE INVERSION FORMULA 632 11.2
ESTIMATES FROM SINGULARITIES 634 11.2.1 POLES 634 11.2.2 DIFFERENCE
EQUATIONS: ASYMPTOTICS FROM GFS 641 11.2.3 REMOVAL OF SINGULARITIES 644
11.2.4 DARBOUX THEOREM AND ITS RELATION TO THE BINOMIAL THEOREM 646
11.2.5 LOGARITHMIC SINGULARITIES 648 11.3 ESTIMATES FROM ENTIRE
FUNCTIONS 649 11.3.1 INTEGRATION BY PARTS 649 11.3.2 ASYMPTOTICS OF
HYPERGEOMETRIC SERIES 653 11.3.3 THE LAPLACE METHOD 656 11.4 EXAMPLES
AND EXERCISES 658 12 REVIEW OF ANALYTIC TECHNIQUES 661 12.1 COMPLEX
NUMBERS 661 12.2 REVIEW OF POWER SERIES 667 12.2.1 THE TAYLOR SERIES 667
12.2.2 OPERATIONS ON POWER SERIES 673 12.3 FUNCTIONS OF A COMPLEX
VARIABLE: BASIC CONCEPTS 677 12.3.1 DIAGONALIZATION OF SERIES 683 12.4
DIFFERENTIAL OPERATORS 686 12.5 PARTIAL FRACTION DECOMPOSITION 692 12.6
SOME SPECIAL FUNCTIONS 694 12.7 STIELTJES INTEGRALS 696 APPENDICES 699
APPENDIX A: BINOMIAL COEFFICIENTS 699 APPENDIX *: THE BERNOULLI NUMBERS
706 APPENDIX C: STIRLING AND EULER/EULERIAN NUMBERS 708 APPENDIX D:
FIBONACCI NUMBERS 715 APPENDIX E: HARMONIC NUMBERS 719 APPENDIX F:
MISCELLANEOUS FORMULAS 721 APPENDIX G: THE GAMMA FUNCTION 725 APPENDIX
H: RANDOM VARIABLES AND DISTRIBUTIONS 728 APPENDIX I: COMBINATORICS OF
PERMUTATIONS 730 APPENDIX J: CONTINUED FRACTIONS 732 APPENDIX K:
OCCUPANCY ENUMERATION 738 APPENDIX L: GENERATING FUNCTIONS 740 APPENDIX
M: RECURRENCES 749 ANSWERS/HINTS TO SELECTED PROBLEMS 753 BIBLIOGRAPHY
772 INDEX 783
|
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| id | DE-604.BV035989488 |
| illustrated | Illustrated |
| indexdate | 2024-12-23T22:56:29Z |
| institution | BVB |
| isbn | 1420068296 9781420068290 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018882227 |
| oclc_num | 633015193 |
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| owner_facet | DE-634 DE-11 DE-824 DE-92 |
| physical | XIX, 804 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | CRC Press/Taylor & Francis |
| record_format | marc |
| series2 | Chapman & Hall/CRC computer and information science series A Chapman & Hall book |
| spellingShingle | Dobrushkin, Vladimir A. Methods in algorithmic analysis Computer science / Mathematics Computer algorithms Algorithms Informatik Mathematik Algorithmus (DE-588)4001183-5 gnd Numerische Mathematik (DE-588)4042805-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4001183-5 (DE-588)4042805-9 (DE-588)4128130-5 (DE-588)4123623-3 |
| title | Methods in algorithmic analysis |
| title_auth | Methods in algorithmic analysis |
| title_exact_search | Methods in algorithmic analysis |
| title_full | Methods in algorithmic analysis Vladimir A. Dobrushkin |
| title_fullStr | Methods in algorithmic analysis Vladimir A. Dobrushkin |
| title_full_unstemmed | Methods in algorithmic analysis Vladimir A. Dobrushkin |
| title_short | Methods in algorithmic analysis |
| title_sort | methods in algorithmic analysis |
| topic | Computer science / Mathematics Computer algorithms Algorithms Informatik Mathematik Algorithmus (DE-588)4001183-5 gnd Numerische Mathematik (DE-588)4042805-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Computer science / Mathematics Computer algorithms Algorithms Informatik Mathematik Algorithmus Numerische Mathematik Numerisches Verfahren Lehrbuch |
| url | http://www.gbv.de/dms/bowker/toc/9781420068290.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018882227&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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