Geometric approximation algorithms

378 Pages·2011·2.32 MB·English

by  Sariel Har-peled

Mathematical
Surveys
and
Monographs
Volume 173
Geometric Approximation
Algorithms
Sariel Har-Peled
American Mathematical Society

surv-173-har-peled-cov.indd 1 4/14/11 1:05 PM
Geometric Approximation
Algorithms

Mathematical
Surveys
and
Monographs
Volume 173
Geometric Approximation
Algorithms
Sariel Har-Peled
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Ralph L. Cohen, Chair Michael A. Singer
Eric M. Friedlander Benjamin Sudakov
Michael I. Weinstein
2010 Mathematics Subject Classification. Primary 68U05, 68W25;
Secondary 68P05, 52Cxx.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-173
Library of Congress Cataloging-in-Publication Data
Har-Peled,Sariel,1971–
Geometricapproximationalgorithms/SarielHar-Peled.
p.cm. —(Mathematicalsurveysandmonographs;v.173)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4911-8(alk.paper)
1. Approximation algorithms. 2. Geometry—Data processing. 3. Computer graphics.
4.Discretegeometry. I.Title.
QA448.D38H377 2011
516(cid:2).11—dc22
2011002940
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Contents
Preface xi
Chapter1. ThePowerofGrids–ClosestPairandSmallestEnclosingDisk 1
1.1. Preliminaries 1
1.2. Closestpair 1
1.3. Aslow2-approximationalgorithmforthek-enclosingdisk 5
1.4. Alineartime2-approximationforthek-enclosingdisk 6
1.5. Bibliographicalnotes 10
1.6. Exercises 11
Chapter2. Quadtrees–HierarchicalGrids 13
2.1. Quadtrees–asimplepoint-locationdata-structure 13
2.2. Compressedquadtrees 15
2.3. Dynamicquadtrees 20
2.4. Bibliographicalnotes 24
2.5. Exercises 26
Chapter3. Well-SeparatedPairDecomposition 29
3.1. Well-separatedpairdecomposition(WSPD) 29
3.2. ApplicationsofWSPD 33
3.3. Semi-separatedpairdecomposition(SSPD) 40
3.4. Bibliographicalnotes 43
3.5. Exercises 44
Chapter4. Clustering–DefinitionsandBasicAlgorithms 47
4.1. Preliminaries 47
4.2. Onk-centerclustering 49
4.3. Onk-medianclustering 51
4.4. Onk-meansclustering 57
4.5. Bibliographicalnotes 57
4.6. Exercises 59
Chapter5. OnComplexity,Sampling,andε-Netsandε-Samples 61
5.1. VCdimension 61
5.2. Shatteringdimensionandthedualshatteringdimension 64
5.3. Onε-netsandε-sampling 70
5.4. Discrepancy 75
5.5. Proofoftheε-nettheorem 80
5.6. Abetterboundonthegrowthfunction 82
5.7. Bibliographicalnotes 83
5.8. Exercises 84
v
vi CONTENTS
Chapter6. ApproximationviaReweighting 87
6.1. Preliminaries 87
6.2. Computingaspanningtreewithlowcrossingnumber 88
6.3. Geometricsetcover 94
6.4. Geometricsetcovervialinearprogramming 98
6.5. Bibliographicalnotes 100
6.6. Exercises 100
Chapter7. YetEvenMoreonSampling 103
7.1. Introduction 103
7.2. Applications 106
7.3. ProofofTheorem7.7 109
7.4. Bibliographicalnotes 119
7.5. Exercises 119
Chapter8. SamplingandtheMomentsTechnique 121
8.1. Verticaldecomposition 121
8.2. Generalsettings 125
8.3. Applications 128
8.4. Boundsontheprobabilityofaregiontobecreated 130
8.5. Bibliographicalnotes 131
8.6. Exercises 133
Chapter9. DepthEstimationviaSampling 135
9.1. Theatmostk-levels 135
9.2. Thecrossinglemma 136
9.3. Ageneralboundfortheatmostk-weight 140
9.4. Bibliographicalnotes 142
9.5. Exercises 143
Chapter10. ApproximatingtheDepthviaSamplingandEmptiness 145
10.1. Fromemptinesstoapproximaterangecounting 145
10.2. Application: Halfplaneandhalfspacerangecounting 148
10.3. Relativeapproximationviasampling 149
10.4. Bibliographicalnotes 150
10.5. Exercises 150
Chapter11. RandomPartitionviaShifting 151
11.1. Partitionviashifting 151
11.2. Hierarchicalrepresentationofapointset 155
11.3. LowqualityANNsearch 158
11.4. Bibliographicalnotes 160
11.5. Exercises 160
Chapter12. GoodTriangulationsandMeshing 163
12.1. Introduction–goodtriangulations 163
12.2. Triangulationsandfattriangulations 164
12.3. Analysis 168
12.4. Theresult 175
12.5. Bibliographicalnotes 176
CONTENTS vii
Chapter13. ApproximatingtheEuclideanTravelingSalesmanProblem(TSP) 177
13.1. TheTSPproblem–introduction 177
13.2. Whentheoptimalsolutionisfriendly 178
13.3. TSPapproximationviaportalsandslidingquadtrees 182
13.4. Bibliographicalnotes 190
13.5. Exercises 190
Chapter14. ApproximatingtheEuclideanTSPUsingBridges 191
14.1. Overview 191
14.2. Cutsandbridges 192
14.3. Thedynamicprogramming 198
14.4. Theresult 202
14.5. Bibliographicalnotes 202
Chapter15. LinearProgramminginLowDimensions 203
15.1. Linearprogramming 203
15.2. Low-dimensionallinearprogramming 205
15.3. Linearprogrammingwithviolations 208
15.4. Approximatelinearprogrammingwithviolations 209
15.5. LP-typeproblems 210
15.6. Bibliographicalnotes 213
15.7. Exercises 215
Chapter16. Polyhedrons,Polytopes,andLinearProgramming 217
16.1. Preliminaries 217
16.2. Propertiesofpolyhedrons 219
16.3. Verticesofapolytope 227
16.4. Linearprogrammingcorrectness 230
16.5. Bibliographicalnotes 232
16.6. Exercises 232
Chapter17. ApproximateNearestNeighborSearchinLowDimension 233
17.1. Introduction 233
17.2. Theboundedspreadcase 233
17.3. ANN–theunboundedgeneralcase 236
17.4. LowqualityANNsearchviatheringseparatortree 238
17.5. Bibliographicalnotes 240
17.6. Exercises 242
Chapter18. ApproximateNearestNeighborviaPoint-Location 243
18.1. ANNusingpoint-locationamongballs 243
18.2. ANNusingpoint-locationamongapproximateballs 250
18.3. ANNusingpoint-locationamongballsinlowdimensions 252
18.4. ApproximateVoronoidiagrams 253
18.5. Bibliographicalnotes 255
18.6. Exercises 256
Chapter19. DimensionReduction–TheJohnson-Lindenstrauss(JL)Lemma 257
19.1. TheBrunn-Minkowskiinequality 257
19.2. Measureconcentrationonthesphere 260
viii CONTENTS
19.3. ConcentrationofLipschitzfunctions 263
19.4. TheJohnson-Lindenstrausslemma 263
19.5. Bibliographicalnotes 266
19.6. Exercises 267
Chapter20. ApproximateNearestNeighbor(ANN)SearchinHighDimensions 269
20.1. ANNonthehypercube 269
20.2. LSHandANNinEuclideanspace 274
20.3. Bibliographicalnotes 276
Chapter21. ApproximatingaConvexBodybyanEllipsoid 279
21.1. Ellipsoids 279
21.2. Bibliographicalnotes 282
Chapter22. ApproximatingtheMinimumVolumeBoundingBoxofaPointSet 283
22.1. Somegeometry 283
22.2. Approximatingtheminimumvolumeboundingbox 284
22.3. Exactalgorithmfortheminimumvolumeboundingbox 286
22.4. Approximatingtheminimumvolumeboundingboxinthreedimensions 288
22.5. Bibliographicalnotes 289
22.6. Exercises 289
Chapter23. Coresets 291
23.1. Coresetfordirectionalwidth 291
23.2. Approximatingtheextentoflinesandothercreatures 297
23.3. Extentofpolynomials 301
23.4. Rootsofpolynomials 302
23.5. Bibliographicalnotes 306
23.6. Exercises 306
Chapter24. ApproximationUsingShellSets 307
24.1. Coveringproblems,expansion,andshellsets 307
24.2. Coveringbycylinders 310
24.3. Bibliographicalnotes 313
24.4. Exercises 313
Chapter25. Duality 315
25.1. Dualityoflinesandpoints 315
25.2. Higherdimensions 318
25.3. Bibliographicalnotes 319
25.4. Exercises 321
Chapter26. FiniteMetricSpacesandPartitions 323
26.1. Finitemetricspaces 323
26.2. Randompartitions 325
26.3. Probabilisticembeddingintotrees 327
26.4. EmbeddinganymetricspaceintoEuclideanspace 329
26.5. Bibliographicalnotes 332
26.6. Exercises 333
Chapter27. SomeProbabilityandTailInequalities 335
CONTENTS ix
27.1. Someprobabilitybackground 335
27.2. Tailinequalities 338
27.3. Hoeffding’sinequality 342
27.4. Bibliographicalnotes 344
27.5. Exercises 344
Chapter28. MiscellaneousPrerequisite 347
28.1. Geometryandlinearalgebra 347
28.2. Calculus 348
Bibliography 349
Index 357