Probability and computing: randomization and probabilistic techniques in algorithms and data analysis
Gespeichert in:
| Hauptverfasser: | , |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge; New York ; Port Melbourne ; Delhi ; Singapore
Cambridge University Press
2017
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| Ausgabe: | Second edition |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | xx, 467 Seiten Diagramme |
| ISBN: | 9781107154889 110715488X |
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Datensatz im Suchindex
| _version_ | 1804177447768293376 |
|---|---|
| adam_text | Titel: Probability and computing
Autor: Mitzenmacher, Michael
Jahr: 2017
Contents
Preface to the Second Edition page xv
Preface to the First Edition xvii
1 Events and Probability 1
1.1 Application: Verifying Polynomial Identities 1
1.2 Axioms of Probability 3
1.3 Application: Verifying Matrix Multiplication 8
1.4 Application: Naive Bayesian Classifier 12
1.5 Application: A Randomized Min-Cut Algorithm 15
1.6 Exercises 17
2 Discrete Random Variables and Expectation 23
2.1 Random Variables and Expectation 23
2.1.1 Linearity of Expectations 25
2.1.2 Jensen s Inequality 26
2.2 The Bernoulli and Binomial Random Variables 27
2.3 Conditional Expectation 29
2.4 The Geometric Distribution 33
2.4.1 Example: Coupon Collector s Problem 35
2.5 Application: The Expected Run-Time of Quicksort 37
2.6 Exercises 40
3 Moments and Deviations 47
3.1 Markov s Inequality 47
3.2 Variance and Moments of a Random Variable 48
3.2.1 Example: Variance of a Binomial Random Variable 51
Vll
CONTENTS
3.3 Chebyshev s Inequality 51
3.3.1 Example: Coupon Collector s Problem 53
3.4 Median and Mean 55
3.5 Application: A Randomized Algorithm for Computing the Median 57
3.5.1 The Algorithm 58
3.5.2 Analysis of the Algorithm 59
3.6 Exercises 62
4 ChernofF and Hoeffding Bounds 66
4.1 Moment Generating Functions 66
4.2 Deriving and Applying Chernoff Bounds 68
4.2.1 Chernoff Bounds for the Sum of Poisson Trials 68
4.2.2 Example: Coin Flips 72
4.2.3 Application: Estimating a Parameter 72
4.3 Better Bounds for Some Special Cases 73
4.4 Application: Set Balancing 76
4.5 The Hoeffding Bound 77
4.6* Application: Packet Routing in Sparse Networks 79
4.6.1 Permutation Routing on the Hypercube 80
4.6.2 Permutation Routing on the Butterfly 85
4.7 Exercises 90
5 Balls, Bins, and Random Graphs 97
5.1 Example: The Birthday Paradox 97
5.2 Balls into Bins 99
5.2.1 The Balls-and-Bins Model 99
5.2.2 Application: Bucket Sort 101
5.3 The Poisson Distribution 101
5.3.1 Limit of the Binomial Distribution 105
5.4 The Poisson Approximation 107
5.4.1* Example: Coupon Collector s Problem, Revisited 111
5.5 Application: Hashing 113
5.5.1 Chain Hashing 113
5.5.2 Hashing: Bit Strings 114
5.5.3 Bloom Filters 116
5.5.4 Breaking Symmetry 118
5.6 Random Graphs 119
5.6.1 Random Graph Models 119
5.6.2 Application: Hamiltonian Cycles in Random Graphs 121
5.7 Exercises 127
5.8 An Exploratory Assignment 133
6 The Probabilistic Method 135
6.1 The Basic Counting Argument 135
viii
CONTENTS
6.2 The Expectation Argument 137
6.2.1 Application: Finding a Large Cut 138
6.2.2 Application: Maximum Satisfiability 139
6.3 Derandomization Using Conditional Expectations 140
6.4 Sample and Modify 142
6.4.1 Application: Independent Sets 142
6.4.2 Application: Graphs with Large Girth 143
6.5 The Second Moment Method 143
6.5.1 Application: Threshold Behavior in Random Graphs 144
6.6 The Conditional Expectation Inequality 145
6.7 The Lovâsz Local Lemma 147
6.7.1 Application: Edge-Disjoint Paths 150
6.7.2 Application: Satisfiability 151
6.8* Explicit Constructions Using the Local Lemma 152
6.8.1 Application: A Satisfiability Algorithm 152
6.9 Lovâsz Local Lemma: The General Case 155
6.10* The Algorithmic Lovâsz Local Lemma 158
6.11 Exercises 162
7 Markov Chains and Random Walks 168
7.1 Markov Chains: Definitions and Representations 168
7.1.1 Application: A Randomized Algorithm for 2-Satisfiability 171
7.1.2 Application: A Randomized Algorithm for 3-Satisfiability 174
7.2 Classification of States 178
7.2.1 Example: The Gambler s Ruin 181
7.3 Stationary Distributions 182
7.3.1 Example: A Simple Queue 188
7.4 Random Walks on Undirected Graphs 189
7.4.1 Application: An s-t Connectivity Algorithm 192
7.5 Parrondo s Paradox 193
7.6 Exercises 198
8 Continuous Distributions and the Poisson Process 205
8.1 Continuous Random Variables 205
8.1.1 Probability Distributions in M. 205
8.1.2 Joint Distributions and Conditional Probability 208
8.2 The Uniform Distribution 210
8.2.1 Additional Properties of the Uniform Distribution 211
8.3 The Exponential Distribution 213
8.3.1 Additional Properties of the Exponential Distribution 214
8.3.2* Example: Balls and Bins with Feedback 216
8.4 The Poisson Process 218
8.4.1 Interarrivai Distribution 221
ix
CONTENTS
8.4.2 Combining and Splitting Poisson Processes 222
8.4.3 Conditional Arrival Time Distribution 224
8.5 Continuous Time Markov Processes 226
8.6 Example: Markovian Queues 229
8.6.1 M/M/1 Queue in Equilibrium 230
8.6.2 M/M/1 /K Queue in Equilibrium 233
8.6.3 The Number of Customers in an M/M/oo Queue 233
8.7 Exercises 236
9 The Normal Distribution 242
9.1 The Normal Distribution 242
9.1.1 The Standard Normal Distribution 242
9.1.2 The General Univariate Normal Distribution 243
9.1.3 The Moment Generating Function 246
9.2* Limit of the Binomial Distribution 247
9.3 The Central Limit Theorem 249
9.4* Multivariate Normal Distributions 252
9.4.1 Properties of the Multivariate Normal Distribution 255
9.5 Application: Generating Normally Distributed Random Values 256
9.6 Maximum Likelihood Point Estimates 258
9.7 Application: EM Algorithm For a Mixture of Gaussians 261
9.8 Exercises 265
10 Entropy, Randomness, and Information 269
10.1 The Entropy Function 269
10.2 Entropy and Binomial Coefficients 272
10.3 Entropy: A Measure of Randomness 274
10.4 Compression 278
10.5* Coding: Shannon s Theorem 281
10.6 Exercises 290
11 The Monte Carlo Method 297
11.1 The Monte Carlo Method 297
11.2 Application: The DNF Counting Problem 300
11.2.1 The Naive Approach 300
11.2.2 A Fully Polynomial Randomized Scheme for DNF Counting 302
11.3 From Approximate Sampling to Approximate Counting 304
11.4 The Markov Chain Monte Carlo Method 308
11.4.1 The Metropolis Algorithm 310
11.5 Exercises 312
11.6 An Exploratory Assignment on Minimum Spanning Trees 315
CONTENTS
12 Coupling of Markov Chains 317
12.1 Variation Distance and Mixing Time 317
12.2 Coupling 320
12.2.1 Example: Shuffling Cards 321
12.2.2 Example: Random Walks on the Hypercube 322
12.2.3 Example: Independent Sets of Fixed Size 323
12.3 Application: Variation Distance Is Nonincreasing 324
12.4 Geometric Convergence 327
12.5 Application: Approximately Sampling Proper
Colorings 328
12.6 Path Coupling 332
12.7 Exercises 336
13 Martingales 341
13.1 Martingales 341
13.2 Stopping Times 343
13.2.1 Example: A Ballot Theorem 345
13.3 Wald s Equation 346
13.4 Tail Inequalities for Martingales 349
13.5 Applications of the Azuma-Hoeffding Inequality 351
13.5.1 General Formalization 351
13.5.2 Application: Pattern Matching 353
13.5.3 Application: Balls and Bins 354
13.5.4 Application: Chromatic Number 355
13.6 Exercises 355
14 Sample Complexity, VC Dimension, and Rademacher
Complexity 361
14.1 The Learning Setting 362
14.2 VC Dimension 363
14.2.1 Additional Examples of VC Dimension 365
14.2.2 Growth Function 366
14.2.3 VC dimension component bounds 368
14.2.4 e-nets and e-samples 369
14.3 The e-net Theorem 370
14.4 Application: PAC Learning 374
14.5 The e-sample Theorem 377
14.5.1 Application: Agnostic Learning 379
14.5.2 Application: Data Mining 380
14.6 Rademacher Complexity 382
14.6.1 Rademacher Complexity and Sample Error 385
XI
CONTENTS
14.6.2 Estimating the Rademacher Complexity 387
14.6.3 Application: Agnostic Learning of a Binary Classification 388
14.7 Exercises 389
15 Pairwise Independence and Universal Hash Functions 392
15.1 Pairwise Independence 392
15.1.1 Example: A Construction of Pairwise Independent Bits 393
15.1.2 Application: Derandomizing an Algorithm for Large Cuts 394
15.1.3 Example: Constructing Pairwise Independent Values Modulo
a Prime 395
15.2 Chebyshev s Inequality for Pairwise Independent Variables 396
15.2.1 Application: Sampling Using Fewer Random Bits 397
15.3 Universal Families of Hash Functions 399
15.3.1 Example: A 2-Universal Family of Hash Functions 401
15.3.2 Example: A Strongly 2-Universal Family of Hash Functions 402
15.3.3 Application: Perfect Hashing 404
15.4 Application: Finding Heavy Hitters in Data Streams 407
15.5 Exercises 411
16 Power Laws and Related Distributions 415
16.1 Power Law Distributions: Basic Definitions and Properties 416
16.2 Power Laws in Language 418
16.2.1 Zipf s Law and Other Examples 418
16.2.2 Languages via Optimization 419
16.2.3 Monkeys Typing Randomly 419
16.3 Preferential Attachment 420
16.3.1 A Formal Version 422
16.4 Using the Power Law in Algorithm Analysis 425
16.5 Other Related Distributions 427
16.5.1 Lognormal Distributions 427
16.5.2 Power Law with Exponential Cutoff 428
16.6 Exercises 429
17 Balanced Allocations and Cuckoo Hashing 433
17.1 The Power of Two Choices 433
17.1.1 The Upper Bound 433
17.2 Two Choices: The Lower Bound 438
17.3 Applications of the Power of Two Choices 441
17.3.1 Hashing 441
17.3.2 Dynamic Resource Allocation 442
17.4 Cuckoo Hashing 442
17.5 Extending Cuckoo Hashing 452
17.5.1 Cuckoo Hashing with Deletions 452
xii
CONTENTS
17.5.2 Handling Failures 453
17.5.3 More Choices and Bigger Bins 454
17.6 Exercises 456
Further Reading 463
Index 464
Note: Asterisks indicate advanced material for this chapter.
xiu
|
| any_adam_object | 1 |
| author | Mitzenmacher, Michael 1969- Upfal, Eli 1954- |
| author_GND | (DE-588)140232281 (DE-588)140232338 |
| author_facet | Mitzenmacher, Michael 1969- Upfal, Eli 1954- |
| author_role | aut aut |
| author_sort | Mitzenmacher, Michael 1969- |
| author_variant | m m mm e u eu |
| building | Verbundindex |
| bvnumber | BV044266884 |
| classification_rvk | SK 800 ST 130 ST 230 |
| classification_tum | DAT 530f MAT 606f |
| ctrlnum | (OCoLC)1002235503 (DE-599)BVBBV044266884 |
| discipline | Informatik Mathematik |
| edition | Second edition |
| format | Book |
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| id | DE-604.BV044266884 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:48:14Z |
| institution | BVB |
| isbn | 9781107154889 110715488X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029671630 |
| oclc_num | 1002235503 |
| open_access_boolean | |
| owner | DE-83 DE-703 DE-20 DE-29T DE-739 DE-384 DE-355 DE-BY-UBR DE-11 |
| owner_facet | DE-83 DE-703 DE-20 DE-29T DE-739 DE-384 DE-355 DE-BY-UBR DE-11 |
| physical | xx, 467 Seiten Diagramme |
| publishDate | 2017 |
| publishDateSearch | 2017 |
| publishDateSort | 2017 |
| publisher | Cambridge University Press |
| record_format | marc |
| spelling | Mitzenmacher, Michael 1969- Verfasser (DE-588)140232281 aut Probability and computing randomization and probabilistic techniques in algorithms and data analysis Michael Mitzenmacher, Eli Upfal Second edition Cambridge; New York ; Port Melbourne ; Delhi ; Singapore Cambridge University Press 2017 © 2017 xx, 467 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Wahrscheinlichkeit (DE-588)4137007-7 gnd rswk-swf Algorithmentheorie (DE-588)4200409-3 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Algorithmus (DE-588)4001183-5 gnd rswk-swf Randomisierter Algorithmus (DE-588)4176929-6 gnd rswk-swf Algorithms Probabilities Stochastic analysis Algorithmentheorie (DE-588)4200409-3 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Stochastische Analysis (DE-588)4132272-1 s DE-604 Algorithmus (DE-588)4001183-5 s Wahrscheinlichkeit (DE-588)4137007-7 s Randomisierter Algorithmus (DE-588)4176929-6 s Upfal, Eli 1954- Verfasser (DE-588)140232338 aut https://www.loc.gov/catdir/enhancements/fy1618/2016041654-t.html Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029671630&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mitzenmacher, Michael 1969- Upfal, Eli 1954- Probability and computing randomization and probabilistic techniques in algorithms and data analysis Wahrscheinlichkeit (DE-588)4137007-7 gnd Algorithmentheorie (DE-588)4200409-3 gnd Stochastische Analysis (DE-588)4132272-1 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Algorithmus (DE-588)4001183-5 gnd Randomisierter Algorithmus (DE-588)4176929-6 gnd |
| subject_GND | (DE-588)4137007-7 (DE-588)4200409-3 (DE-588)4132272-1 (DE-588)4079013-7 (DE-588)4001183-5 (DE-588)4176929-6 |
| title | Probability and computing randomization and probabilistic techniques in algorithms and data analysis |
| title_auth | Probability and computing randomization and probabilistic techniques in algorithms and data analysis |
| title_exact_search | Probability and computing randomization and probabilistic techniques in algorithms and data analysis |
| title_full | Probability and computing randomization and probabilistic techniques in algorithms and data analysis Michael Mitzenmacher, Eli Upfal |
| title_fullStr | Probability and computing randomization and probabilistic techniques in algorithms and data analysis Michael Mitzenmacher, Eli Upfal |
| title_full_unstemmed | Probability and computing randomization and probabilistic techniques in algorithms and data analysis Michael Mitzenmacher, Eli Upfal |
| title_short | Probability and computing |
| title_sort | probability and computing randomization and probabilistic techniques in algorithms and data analysis |
| title_sub | randomization and probabilistic techniques in algorithms and data analysis |
| topic | Wahrscheinlichkeit (DE-588)4137007-7 gnd Algorithmentheorie (DE-588)4200409-3 gnd Stochastische Analysis (DE-588)4132272-1 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Algorithmus (DE-588)4001183-5 gnd Randomisierter Algorithmus (DE-588)4176929-6 gnd |
| topic_facet | Wahrscheinlichkeit Algorithmentheorie Stochastische Analysis Wahrscheinlichkeitstheorie Algorithmus Randomisierter Algorithmus |
| url | https://www.loc.gov/catdir/enhancements/fy1618/2016041654-t.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029671630&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT mitzenmachermichael probabilityandcomputingrandomizationandprobabilistictechniquesinalgorithmsanddataanalysis AT upfaleli probabilityandcomputingrandomizationandprobabilistictechniquesinalgorithmsanddataanalysis |
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