The probabilistic method:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
New York [u.a.]
Wiley
2000
|
| Edition: | 2. ed. |
| Series: | Wiley-interscience series in discrete mathematics and optimization
A Wiley-Interscience publication |
| Subjects: | |
| Online Access: | FRO01 FUBA1 UBR01 Volltext Inhaltsverzeichnis |
| Item Description: | Erscheinungsjahr des Ebooks: [2004?] |
| Physical Description: | 1 Online-Ressource |
| ISBN: | 0471370460 9780471370468 9780471722151 9780471653981 |
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Record in the Search Index
| _version_ | 1804136566268887040 |
|---|---|
| adam_text | Contents
Dedication v
Preface vii
Acknowledgments ix
Parti METHODS
1 The Basic Method 1
1.1 The Probabilistic Method ]
1.2 Graph Theory 3
1.3 Combinatorics 6
1.4 Combinatorial Number Theory 8
1.5 Disjoint Pairs 9
1.6 Exercises 10
The Probabilistic Lens: The Erdos Ko Rado Theorem 12
2 Linearity of Expectation 13
2.1 Basics 13
xi
X/7 CONTENTS
2.2 Splitting Graphs 14
2.3 Two Quickies 16
2.4 Balancing Vectors 17
2.5 Unbalancing Lights 18
2.6 Without Coin Flips 20
2.7 Exercises 20
The Probabilistic Lens: Bregman s Theorem 22
3 Alterations 25
3.1 Ramsey Numbers 25
3.2 Independent Sets 27
3.3 Combinatorial Geometry 28
3.4 Packing 29
3.5 Recoloring 30
3.6 Continuous Time 33
3.7 Exercises 37
The Probabilistic Lens: High Girth and High
Chromatic Number 38
4 The Second Moment 41
4.1 Basics 41
4.2 Number Theory 42
4.3 More Basics 45
4.4 Random Graphs 47
4.5 Clique Number 50
4.6 Distinct Sums 52
4.7 The Rodl Nibble 53
4.8 Exercises 58
The Probabilistic Lens: Hamiltonian Paths 60
5 The Local Lemma 63
5.1 The Lemma 63
5.2 Property B and Multicolored Sets of Real Numbers 65
5.3 Lower Bounds for Ramsey Numbers 67
5.4 A Geometric Result 68
5.5 The Linear Arboricity of Graphs 69
CONTENTS Xiii
5.6 Latin Transversals 73
5.7 The Algorithmic Aspect 74
5.8 Exercises 77
The Probabilistic Lens: Directed Cycles 78
6 Correlation Inequalities 81
6.1 The Four Functions Theorem of Ahlswede
and Daykin 82
6.2 The FKG Inequality 84
6.3 Monotone Properties 86
6.4 Linear Extensions of Partially Ordered Sets 88
6.5 Exercises 90
The Probabilistic Lens: Turdn s Theorem 91
7 Martingales and Tight Concentration 93
7.1 Definitions 93
7.2 Large Deviations 95
7.3 Chromatic Number 96
7.4 Two General Settings 99
7.5 Four Illustrations 103
7.6 Talagrand s Inequality 105
7.7 Applications of Talagrand s Inequality 108
7.8 Kim Vu Polynomial Concentration 110
7.9 Exercises 112
The Probabilistic Lens: Weierstrass Approximation
Theorem 113
8 The Poisson Paradigm 115
8.1 The Janson Inequalities 115
8.2 The Proofs 117
8.3 Brun s Sieve 119
8.4 Large Deviations 122
8.5 Counting Extensions 123
8.6 Counting Representations 125
8.7 Further Inequalities 128
8.8 Exercises 129
Xiv CONTENTS
The Probabilistic Lens: Local Coloring 130
9 Pseudorandomness 133
9.1 The Quadratic Residue Tournaments 134
9.2 Eigenvalues and Expanders 137
9.3 Quasi Random Graphs 142
9.4 Exercises 148
The Probabilistic Lens: Random Walks 150
Part II TOPICS
10 Random Graphs 155
10.1 Subgraphs 156
10.2 Clique Number 158
10.3 Chromatic Number 160
10.4 Branching Processes 161
10.5 The Giant Component 165
10.6 Inside the Phase Transition 168
10.7 Zero One Laws 171
10.8 Exercises 178
The Probabilistic Lens: Counting Subgraphs 180
11 Circuit Complexity 183
11.1 Preliminaries 183
11.2 Random Restrictions and Bounded Depth Circuits 185
11.3 More on Bounded Depth Circuits 189
11.4 Monotone Circuits 191
11.5 Formulae 194
11.6 Exercises 196
The Probabilistic Lens: Maximal Antichains 197
12 Discrepancy 199
12.1 Basics 199
12.2 Six Standard Deviations Suffice 201
CONTENTS XV
12.3 Linear and Hereditary Discrepancy 204
12.4 Lower Bounds 207
12.5 The Beck Fiala Theorem 209
12.6 Exercises 210
The Probabilistic Lens: Unbalancing Lights 212
13 Geometry 215
13.1 The Greatest Angle among Points
in Euclidean Spaces 216
13.2 Empty Triangles Determined by Points in the Plane 217
13.3 Geometrical Realizations of Sign Matrices 218
13.4 e Nets and VC Dimensions of Range Spaces 220
13.5 Dual Shatter Functions and Discrepancy 225
13.6 Exercises 228
The Probabilistic Lens: Efficient Packing 229
14 Codes, Games and Entropy 231
14.1 Codes 231
14.2 Liar Game 233
14.3 Tenure Game 236
14.4 Balancing Vector Game 237
14.5 Nonadaptive Algorithms 239
14.6 Entropy 240
14.7 Exercises 245
The Probabilistic Lens: An Extremal Graph 247
15 Derandomization 249
15.1 The Method of Conditional Probabilities 249
15.2 d Wise Independent Random Variables in Small
Sample Spaces 253
15.3 Exercises 257
The Probabilistic Lens: Crossing Numbers, Incidences, Sums
and Products 259
Appendix A: Bounding of Large Deviations 263
XVi CONTENTS
A.I Bounding of Large Deviations 263
A.2 Exercises 271
The Probabilistic Lens: Triangle free Graphs Have
Large Independence Numbers 272
Appendix B: Paul Erdos 275
B.I Papers 275
B.2 Conjectures 277
B.3 On Erdos 278
B.4 Uncle Paul 279
References 283
Subject Index 295
Author Index 299
|
| adam_txt |
Contents
Dedication v
Preface vii
Acknowledgments ix
Parti METHODS
1 The Basic Method 1
1.1 The Probabilistic Method ]
1.2 Graph Theory 3
1.3 Combinatorics 6
1.4 Combinatorial Number Theory 8
1.5 Disjoint Pairs 9
1.6 Exercises 10
The Probabilistic Lens: The Erdos Ko Rado Theorem 12
2 Linearity of Expectation 13
2.1 Basics 13
xi
X/7 CONTENTS
2.2 Splitting Graphs 14
2.3 Two Quickies 16
2.4 Balancing Vectors 17
2.5 Unbalancing Lights 18
2.6 Without Coin Flips 20
2.7 Exercises 20
The Probabilistic Lens: Bregman 's Theorem 22
3 Alterations 25
3.1 Ramsey Numbers 25
3.2 Independent Sets 27
3.3 Combinatorial Geometry 28
3.4 Packing 29
3.5 Recoloring 30
3.6 Continuous Time 33
3.7 Exercises 37
The Probabilistic Lens: High Girth and High
Chromatic Number 38
4 The Second Moment 41
4.1 Basics 41
4.2 Number Theory 42
4.3 More Basics 45
4.4 Random Graphs 47
4.5 Clique Number 50
4.6 Distinct Sums 52
4.7 The Rodl Nibble 53
4.8 Exercises 58
The Probabilistic Lens: Hamiltonian Paths 60
5 The Local Lemma 63
5.1 The Lemma 63
5.2 Property B and Multicolored Sets of Real Numbers 65
5.3 Lower Bounds for Ramsey Numbers 67
5.4 A Geometric Result 68
5.5 The Linear Arboricity of Graphs 69
CONTENTS Xiii
5.6 Latin Transversals 73
5.7 The Algorithmic Aspect 74
5.8 Exercises 77
The Probabilistic Lens: Directed Cycles 78
6 Correlation Inequalities 81
6.1 The Four Functions Theorem of Ahlswede
and Daykin 82
6.2 The FKG Inequality 84
6.3 Monotone Properties 86
6.4 Linear Extensions of Partially Ordered Sets 88
6.5 Exercises 90
The Probabilistic Lens: Turdn 's Theorem 91
7 Martingales and Tight Concentration 93
7.1 Definitions 93
7.2 Large Deviations 95
7.3 Chromatic Number 96
7.4 Two General Settings 99
7.5 Four Illustrations 103
7.6 Talagrand's Inequality 105
7.7 Applications of Talagrand's Inequality 108
7.8 Kim Vu Polynomial Concentration 110
7.9 Exercises 112
The Probabilistic Lens: Weierstrass Approximation
Theorem 113
8 The Poisson Paradigm 115
8.1 The Janson Inequalities 115
8.2 The Proofs 117
8.3 Brun's Sieve 119
8.4 Large Deviations 122
8.5 Counting Extensions 123
8.6 Counting Representations 125
8.7 Further Inequalities 128
8.8 Exercises 129
Xiv CONTENTS
The Probabilistic Lens: Local Coloring 130
9 Pseudorandomness 133
9.1 The Quadratic Residue Tournaments 134
9.2 Eigenvalues and Expanders 137
9.3 Quasi Random Graphs 142
9.4 Exercises 148
The Probabilistic Lens: Random Walks 150
Part II TOPICS
10 Random Graphs 155
10.1 Subgraphs 156
10.2 Clique Number 158
10.3 Chromatic Number 160
10.4 Branching Processes 161
10.5 The Giant Component 165
10.6 Inside the Phase Transition 168
10.7 Zero One Laws 171
10.8 Exercises 178
The Probabilistic Lens: Counting Subgraphs 180
11 Circuit Complexity 183
11.1 Preliminaries 183
11.2 Random Restrictions and Bounded Depth Circuits 185
11.3 More on Bounded Depth Circuits 189
11.4 Monotone Circuits 191
11.5 Formulae 194
11.6 Exercises 196
The Probabilistic Lens: Maximal Antichains 197
12 Discrepancy 199
12.1 Basics 199
12.2 Six Standard Deviations Suffice 201
CONTENTS XV
12.3 Linear and Hereditary Discrepancy 204
12.4 Lower Bounds 207
12.5 The Beck Fiala Theorem 209
12.6 Exercises 210
The Probabilistic Lens: Unbalancing Lights 212
13 Geometry 215
13.1 The Greatest Angle among Points
in Euclidean Spaces 216
13.2 Empty Triangles Determined by Points in the Plane 217
13.3 Geometrical Realizations of Sign Matrices 218
13.4 e Nets and VC Dimensions of Range Spaces 220
13.5 Dual Shatter Functions and Discrepancy 225
13.6 Exercises 228
The Probabilistic Lens: Efficient Packing 229
14 Codes, Games and Entropy 231
14.1 Codes 231
14.2 Liar Game 233
14.3 Tenure Game 236
14.4 Balancing Vector Game 237
14.5 Nonadaptive Algorithms 239
14.6 Entropy 240
14.7 Exercises 245
The Probabilistic Lens: An Extremal Graph 247
15 Derandomization 249
15.1 The Method of Conditional Probabilities 249
15.2 d Wise Independent Random Variables in Small
Sample Spaces 253
15.3 Exercises 257
The Probabilistic Lens: Crossing Numbers, Incidences, Sums
and Products 259
Appendix A: Bounding of Large Deviations 263
XVi CONTENTS
A.I Bounding of Large Deviations 263
A.2 Exercises 271
The Probabilistic Lens: Triangle free Graphs Have
Large Independence Numbers 272
Appendix B: Paul Erdos 275
B.I Papers 275
B.2 Conjectures 277
B.3 On Erdos 278
B.4 Uncle Paul 279
References 283
Subject Index 295
Author Index 299 |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed. |
| format | Electronic eBook |
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| id | DE-604.BV022476771 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T17:46:49Z |
| indexdate | 2024-07-09T20:58:26Z |
| institution | BVB |
| isbn | 0471370460 9780471370468 9780471722151 9780471653981 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015684181 |
| oclc_num | 85820345 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-861 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-861 DE-188 |
| physical | 1 Online-Ressource |
| psigel | ZDB-35-WIC ebook FHR_PDA_WIC UBG_PDA_WIC ZDB-35-WIC FRO_PDA_WIC ZDB-35-WIC ZDB-35-WIC 2020 |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Wiley |
| record_format | marc |
| series2 | Wiley-interscience series in discrete mathematics and optimization A Wiley-Interscience publication |
| spelling | Alon, Noga Verfasser (DE-588)170483320 aut The probabilistic method Noga Alon ; Joel H. Spencer 2. ed. New York [u.a.] Wiley 2000 1 Online-Ressource txt rdacontent c rdamedia cr rdacarrier Wiley-interscience series in discrete mathematics and optimization A Wiley-Interscience publication Erscheinungsjahr des Ebooks: [2004?] Combinatorial analysis Probabilities Kombinatorik (DE-588)4031824-2 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd rswk-swf Kombinatorische Analysis (DE-588)4164746-4 gnd rswk-swf Wahrscheinlichkeit (DE-588)4137007-7 gnd rswk-swf Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 s DE-604 Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Wahrscheinlichkeit (DE-588)4137007-7 s Kombinatorische Analysis (DE-588)4164746-4 s Kombinatorik (DE-588)4031824-2 s 1\p DE-604 Spencer, Joel H. 1946- Verfasser (DE-588)133091546 aut https://onlinelibrary.wiley.com/doi/book/10.1002/0471722154 Verlag Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015684181&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Alon, Noga Spencer, Joel H. 1946- The probabilistic method Combinatorial analysis Probabilities Kombinatorik (DE-588)4031824-2 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd Kombinatorische Analysis (DE-588)4164746-4 gnd Wahrscheinlichkeit (DE-588)4137007-7 gnd |
| subject_GND | (DE-588)4031824-2 (DE-588)4064324-4 (DE-588)4132446-8 (DE-588)4164746-4 (DE-588)4137007-7 |
| title | The probabilistic method |
| title_auth | The probabilistic method |
| title_exact_search | The probabilistic method |
| title_exact_search_txtP | The probabilistic method |
| title_full | The probabilistic method Noga Alon ; Joel H. Spencer |
| title_fullStr | The probabilistic method Noga Alon ; Joel H. Spencer |
| title_full_unstemmed | The probabilistic method Noga Alon ; Joel H. Spencer |
| title_short | The probabilistic method |
| title_sort | the probabilistic method |
| topic | Combinatorial analysis Probabilities Kombinatorik (DE-588)4031824-2 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd Kombinatorische Analysis (DE-588)4164746-4 gnd Wahrscheinlichkeit (DE-588)4137007-7 gnd |
| topic_facet | Combinatorial analysis Probabilities Kombinatorik Wahrscheinlichkeitsrechnung Kombinatorische Wahrscheinlichkeitstheorie Kombinatorische Analysis Wahrscheinlichkeit |
| url | https://onlinelibrary.wiley.com/doi/book/10.1002/0471722154 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015684181&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT alonnoga theprobabilisticmethod AT spencerjoelh theprobabilisticmethod |