Extremal combinatorics: with applications in computer science
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin ; Heidelberg
Springer
[2011]
|
| Ausgabe: | Second edition |
| Schriftenreihe: | Texts in theoretical computer science
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XXIII, 411 Seiten Illustrationen, Diagramme |
| ISBN: | 9783642173639 |
Internformat
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Datensatz im Suchindex
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IMAGE 1
CONTENTS
NOTATION XIII
PART I THE CLASSICS
1 COUNTING 3
1.1 THE BINOMIAL THEOREM 3
1.2 SELECTION WITH REPETITIONS 6
1.3 PARTITIONS 7
1.4 DOUBLE COUNTING 8
1.5 THE AVERAGING PRINCIPLE 11
1.6 THE INCLUSION-EXCLUSION PRINCIPLE 13
EXERCISES 16
2 ADVANCED COUNTING 23
2.1 BOUNDS ON INTERSECTION SIZE 23
2.2 GRAPHS WITH NO 4-CYCLES 24
2.3 GRAPHS WITH NO INDUCED 4-CYCLES 26
2.4 ZARANKIEWICZ'S PROBLEM 29
2.5 DENSITY OF 0-1 MATRICES 33
2.6 THE LOVASZ-STEIN THEOREM 34
2.6.1 COVERING DESIGNS 36
EXERCISES 37
3 PROBABILISTIC COUNTING 41
3.1 PROBABILISTIC PRELIMINARIES 41
3.2 TOURNAMENTS 44
3.3 UNIVERSAL SETS 45
3.4 COVERING BY BIPARTITE CLIQUES 46
3.5 2-COLORABLE FAMILIES 47
3.6 THE CHOICE NUMBER OF GRAPHS 49
EXERCISES 50
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1007777672
DIGITALISIERT DURCH
IMAGE 2
XVUEI CONTENTS
4 THE PIGEONHOLE PRINCIPLE 53
4.1 SOME QUICKIES 53
4.2 THE ERDOES-SZEKERES THEOREM 55
4.3 MANTEL'S THEOREM 56
4.4 TURAN'S THEOREM 58
4.5 DIRICHLET'S THEOREM 59
4.6 SWELL-COLORED GRAPHS 60
4.7 THE WEIGHT SHIFTING ARGUMENT 61
4.8 SCHUR'S THEOREM 63
4.9 RAMSEYAN THEOREMS FOR GRAPHS 65
4.10 RAMSEY'S THEOREM FOR SETS 68
EXERCISES 70
5 SYSTEMS OF DISTINCT REPRESENTATIVES 77
5.1 THE MARRIAGE THEOREM 77
5.2 TWO APPLICATIONS 79
5.2.1 LATIN RECTANGLES 79
5.2.2 DECOMPOSITION OF DOUBLY STOCHASTIC MATRICES 80 5.3 MIN-MAX
THEOREMS 81
5.4 MATCHINGS IN BIPARTITE GRAPHS 82
EXERCISES 85
PART II EXTREMAL SET THEORY
6 SUNFLOWERS 89
6.1 THE SUNFLOWER LEMMA 89
6.2 MODIFICATIONS 91
6.2.1 RELAXED CORE 91
6.2.2 RELAXED DISJOINTNESS 92
6.3 APPLICATIONS 93
6.3.1 THE NUMBER OF MINTERMS 93
6.3.2 SMALL DEPTH FORMULAS 94
EXERCISES 96
7 INTERSECTING FAMILIES 99
7.1 ULTRAFILTERS AND HELLY PROPERTY 99
7.2 THE ERDOES-KO-RADO THEOREM 100
7.3 FISHER'S INEQUALITY 101
7.4 MAXIMAL INTERSECTING FAMILIES 102
7.5 CROSS-INTERSECTING FAMILIES 104
EXERCISES 105
IMAGE 3
CONTENTS XIX
8 CHAINS AND ANTICHAINS 107
8.1 DECOMPOSITION IN CHAINS AND ANTICHAINS 108
8.2 APPLICATION: THE MEMORY ALLOCATION PROBLEM 110
8.3 SPERNER'S THEOREM I LL
8.4 THE BOLLOBAS THEOREM 112
8.5 STRONG SYSTEMS OF DISTINCT REPRESENTATIVES 115
8.6 UNION-FREE FAMILIES 116
EXERCISES 117
9 BLOCKING SETS AND THE DUALITY 119
9.1 DUALITY 119
9.2 THE BLOCKING NUMBER 121
9.3 HELLY-TYPE THEOREMS 122
9.4 BLOCKING SETS AND DECISION TREES 123
9.5 BLOCKING SETS AND MONOTONE CIRCUITS 126
EXERCISES 132
10 DENSITY AND UNIVERSALITY 135
10.1 DENSE SETS 135
10.2 HEREDITARY SETS 136
10.3 MATROIDS AND APPROXIMATION 139
10.4 THE KRUSKAL-KATONA THEOREM 143
10.5 UNIVERSAL SETS 148
10.6 PALEY GRAPHS 149
10.7 FULL GRAPHS 151
EXERCISES 153
11 WITNESS SETS AND ISOLATION 155
11.1 BONDY'S THEOREM 155
11.2 AVERAGE WITNESSES 156
11.3 THE ISOLATION LEMMA 159
11.4 ISOLATION IN POLITICS: THE DICTATOR PARADOX 160
EXERCISES 162
12 DESIGNS 165
12.1 REGULARITY 166
12.2 FINITE LINEAR SPACES 167
12.3 DIFFERENCE SETS 168
12.4 PROJECTIVE PLANES 169
12.4.1 THE CONSTRUCTION 171
12.4.2 BRUEN'S THEOREM 172
12.5 RESOLVABLE DESIGNS 173
12.5.1 AFFINE PLANES 174
EXERCISES 175
IMAGE 4
CONTENTS
PART III THE LINEAR ALGEBRA METHOD
13 THE BASIC METHOD 179
13.1 THE LINEAR ALGEBRA BACKGROUND 179
13.2 GRAPH DECOMPOSITIONS 185
13.3 INCLUSION MATRICES 186
13.4 DISJOINTNESS MATRICES 187
13.5 TWO-DISTANCE SETS 189
13.6 SETS WITH FEW INTERSECTION SIZES 190
13.7 CONSTRUCTIVE RAMSEY GRAPHS 191
13.8 ZERO-PATTERNS OF POLYNOMIALS 192
EXERCISES 193
14 ORTHOGONALITY AND RANK ARGUMENTS 197
14.1 ORTHOGONAL CODING 197
14.2 BALANCED PAIRS 198
14.3 HADAMARD MATRICES 200
14.4 MATRIX RANK AND RAMSEY GRAPHS 203
14.5 LOWER BOUNDS FOR BOOLEAN FORMULAS 205
14.5.1 REDUCTION TO SET-COVERING 205
14.5.2 THE RANK LOWER BOUND 207
EXERCISES 210
15 EIGENVALUES AND GRAPH EXPANSION 213
15.1 EXPANDER GRAPHS 213
15.2 SPECTRAL GAP AND THE EXPANSION 214
15.2.1 RAMANUJAN GRAPHS 218
15.3 EXPANDERS AND DERANDOMIZATION 220
EXERCISES 221
16 THE POLYNOMIAL METHOD 223
16.1 DEMILLO-LIPTON-SCHWARTZ-ZIPPEL LEMMA 223
16.2 SOLUTION OF KAKEYA'S PROBLEM IN FINITE FIELDS 226
16.3 COMBINATORIAL NULLSTELLENSATZ 228
16.3.1 THE PERMANENT LEMMA 230
16.3.2 COVERING CUBE BY AFFINE HYPERPLANES 231
16.3.3 REGULAR SUBGRAPHS 231
16.3.4 SUM-SETS 232
16.3.5 ZERO-SUM SETS 233
EXERCISES 235
IMAGE 5
CONTENTS XXI
17 COMBINATORICS OF CODES 237
17.1 ERROR-CORRECTING CODES 237
17.2 BOUNDS ON CODE SIZE 239
17.3 LINEAR CODES 243
17.4 UNIVERSAL SETS FROM LINEAR CODES 245
17.5 SPANNING DIAMETER 245
17.6 EXPANDER CODES 247
17.7 EXPANSION OF RANDOM GRAPHS 250
EXERCISES 251
PART IV THE PROBABILISTIC METHOD
18 LINEARITY OF EXPECTATION 255
18.1 HAMILTON PATHS IN TOURNAMENTS 255
18.2 SUM-FREE SETS 256
18.3 DOMINATING SETS 257
18.4 THE INDEPENDENCE NUMBER 258
18.5 CROSSINGS AND INCIDENCES 259
18.5.1 CROSSING NUMBER 259
18.5.2 THE SZEMEREDI-TROTTER THEOREM 261
18.6 FAR AWAY STRINGS 262
18.7 LOW DEGREE POLYNOMIALS 264
18.8 MAXIMUM SATISFIABILITY 265
18.9 HASH FUNCTIONS 267
18.10 DISCREPANCY 268
18.11 LARGE DEVIATION INEQUALITIES 273
EXERCISES 276
19 THE LOVASZ SIEVE 279
19.1 THE LOVASZ LOCAL LEMMA 279
19.2 DISJOINT CYCLES 283
19.3 COLORINGS 284
19.4 THE FC-SAT PROBLEM 287
EXERCISES 290
20 THE DELETION METHOD 293
20.1 EDGE CLIQUE COVERING 293
20.2 INDEPENDENT SETS 294
20.3 COLORING LARGE-GIRTH GRAPHS 295
20.4 POINT SETS WITHOUT OBTUSE TRIANGLES 296
20.5 AFFINE CUBES OF INTEGERS 298
EXERCISES 301
IMAGE 6
XXII CONTENTS
21 THE SECOND MOMENT METHOD 303
21.1 THE METHOD 303
21.2 DISTINCT SUMS 304
21.3 PRIME FACTORS 305
21.4 SEPARATORS 307
21.5 THRESHOLD FOR CLIQUES 309
EXERCISES 311
22 THE ENTROPY FUNCTION 313
22.1 QUANTIFYING INFORMATION 313
22.2 LIMITS TO DATA COMPRESSION 314
22.3 SHANNON ENTROPY 318
22.4 SUBADDITIVITY 320
22.5 COMBINATORIAL APPLICATIONS 322
EXERCISES 325
23 RANDOM WALKS 327
23.1 THE SATISFIABILITY PROBLEM 327
23.1.1 PAPADIMITRIOU'S ALGORITHM FOR 2-SAT 328 23.1.2 SCHOENING'S
ALGORITHM FOR 3-SAT 329
23.2 RANDOM WALKS IN LINEAR SPACES 331
23.2.1 SMALL FORMULAS FOR COMPLICATED FUNCTIONS 333 23.3 RANDOM WALKS
AND DERANDOMIZATION 336
EXERCISES 339
24 DERANDOMIZATION 341
24.1 THE METHOD OF CONDITIONAL PROBABILITIES 341
24.1.1 A GENERAL FRAME 342
24.1.2 SPLITTING GRAPHS 343
24.1.3 MAXIMUM SATISFIABILITY: THE ALGORITHMIC ASPECT 344 24.2 THE
METHOD OF SMALL SAMPLE SPACES 345
24.2.1 REDUCING THE NUMBER OF RANDOM BITS 346
24.2.2 FC-WISE INDEPENDENCE 347
24.3 SUM-FREE SETS: THE ALGORITHMIC ASPECT 350
EXERCISES 352
PART V FRAGMENTS OF RAMSEY THEORY
25 RAMSEYAN THEOREMS FOR NUMBERS 357
25.1 ARITHMETIC PROGRESSIONS 357
25.2 SZEMEREDI'S CUBE LEMMA 360
25.3 SUM-FREE SETS 362
25.3.1 KNESER'S THEOREM 363
25.4 SUM-PRODUCT SETS 365
EXERCISES 368
IMAGE 7
CONTENTS XXIII
26 THE HALES-JEWETT THEOREM 371
26.1 THE THEOREM AND ITS CONSEQUENCES 371
26.1.1 VAN DER WAERDEN'S THEOREM 373
26.1.2 GALLAI-WITT'S THEOREM 373
26.2 SHELAH'S PROOF OF HJT 374
EXERCISES 377
27 APPLICATIONS IN COMMUNICATION COMPLEXITY 379
27.1 MULTI-PARTY COMMUNICATION 379
27.2 THE HYPERPLANE PROBLEM 381
27.3 THE PARTITION PROBLEM 383
27.4 LOWER BOUNDS VIA DISCREPANCY 385
27.5 MAKING NON-DISJOINT COVERINGS DISJOINT 388
EXERCISES 389
REFERENCES 393
INDEX 407 |
| any_adam_object | 1 |
| author | Jukna, Stasys 1953- |
| author_GND | (DE-588)17409146X |
| author_facet | Jukna, Stasys 1953- |
| author_role | aut |
| author_sort | Jukna, Stasys 1953- |
| author_variant | s j sj |
| building | Verbundindex |
| bvnumber | BV039663548 |
| classification_rvk | SK 890 |
| classification_tum | MAT 050f |
| ctrlnum | (OCoLC)756376503 (DE-599)DNB1007777672 |
| dewey-full | 511.6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.6 |
| dewey-search | 511.6 |
| dewey-sort | 3511.6 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
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| id | DE-604.BV039663548 |
| illustrated | Illustrated |
| indexdate | 2024-07-21T00:14:10Z |
| institution | BVB |
| isbn | 9783642173639 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024512865 |
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| physical | XXIII, 411 Seiten Illustrationen, Diagramme |
| publishDate | 2011 |
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| publisher | Springer |
| record_format | marc |
| series2 | Texts in theoretical computer science |
| spelling | Jukna, Stasys 1953- Verfasser (DE-588)17409146X aut Extremal combinatorics with applications in computer science Stasys Jukna Second edition Berlin ; Heidelberg Springer [2011] XXIII, 411 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Texts in theoretical computer science Kombinatorik (DE-588)4031824-2 gnd rswk-swf Extrembedingung (DE-588)4502205-7 gnd rswk-swf Extremalproblem (DE-588)4439315-5 gnd rswk-swf Kombinatorik (DE-588)4031824-2 s Extremalproblem (DE-588)4439315-5 s DE-604 Extrembedingung (DE-588)4502205-7 s 1\p DE-604 Erscheint auch als Online-Ausgabe 978-3-642-17364-6 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3552168&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024512865&sequence=000001&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 | Jukna, Stasys 1953- Extremal combinatorics with applications in computer science Kombinatorik (DE-588)4031824-2 gnd Extrembedingung (DE-588)4502205-7 gnd Extremalproblem (DE-588)4439315-5 gnd |
| subject_GND | (DE-588)4031824-2 (DE-588)4502205-7 (DE-588)4439315-5 |
| title | Extremal combinatorics with applications in computer science |
| title_auth | Extremal combinatorics with applications in computer science |
| title_exact_search | Extremal combinatorics with applications in computer science |
| title_full | Extremal combinatorics with applications in computer science Stasys Jukna |
| title_fullStr | Extremal combinatorics with applications in computer science Stasys Jukna |
| title_full_unstemmed | Extremal combinatorics with applications in computer science Stasys Jukna |
| title_short | Extremal combinatorics |
| title_sort | extremal combinatorics with applications in computer science |
| title_sub | with applications in computer science |
| topic | Kombinatorik (DE-588)4031824-2 gnd Extrembedingung (DE-588)4502205-7 gnd Extremalproblem (DE-588)4439315-5 gnd |
| topic_facet | Kombinatorik Extrembedingung Extremalproblem |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3552168&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024512865&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT juknastasys extremalcombinatoricswithapplicationsincomputerscience |