Algorithms in combinatorial geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1987
|
| Series: | European Association for Theoretical Computer Science: EATCS monographs on theoretical computer science
10. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [381] - 394 |
| Physical Description: | XV, 423 S. graph. Darst. |
| ISBN: | 354013722X 038713722X |
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| 245 | 1 | 0 | |a Algorithms in combinatorial geometry |c Herbert Edelsbrunner |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1987 | |
| 300 | |a XV, 423 S. |b graph. Darst. | ||
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| 490 | 1 | |a European Association for Theoretical Computer Science: EATCS monographs on theoretical computer science |v 10. | |
| 500 | |a Literaturverz. S. [381] - 394 | ||
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| 650 | 7 | |a Combinatorische meetkunde |2 gtt | |
| 650 | 4 | |a Géométrie - Informatique | |
| 650 | 4 | |a Géométrie combinatoire | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Combinatorial geometry | |
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| adam_text | TABLE OF CONTENTS
PART I COMBINATORIAL GEOMETRY 1
CHAPTER 1 Fundamental Concepts in Combinatorial Geometry 3
1.1. Arrangements of Hyperplanes 4
1.2. Counting Faces and Incidences 6
1.3. Combinatorial Equivalence 10
1.4. Configurations of Points 12
1.5. Sylvester s Problem 15
1.6. Convex Polytopes and Convex Polyhedra 16
1.7. Zonotopes 20
1.8. Voronoi Diagrams 23
1.9. Exercises and Research Problems 25
1.10. Bibliographic Notes 27
CHAPTER 2 Permutation Tables 29
2.1. Circular Sequences 29
2.2. Encoding Arrangements and Configurations 32
2.3. A Circularly Non Realizable 5 Sequence 35
2.4. Arrangements of Pseudo Lines 37
2.5. Some Combinatorial Problems in the Plane 39
2.6. Exercises and Research Problems 41
2.7. Bibliographic Notes 44
CHAPTER 3 Semispaces of Configurations 45
3.1. Semispaces and Arrangements 46
3.2. fc Sets and Levels in Arrangements 47
3.3. A Lower Bound on the Number of Bisections in the Plane 50
3.4. Lower Bounds on the Number of fc Sets in the Plane 52
3.5. Extensions to Three and Higher Dimensions 54
3.6. Semispaces and Circular Sequences 55
3.7. An Upper Bound on the Number of fc Sets in the Plane 58
3.8. Exercises and Research Problems 60
3.9. Bibliographic Notes 61
XII
CHAPTER 4 Dissections of Point Sets 63
4.1. Centerpoints 63
4.2. Ham Sandwich Cuts 66
4.3. Erasing Subdivisions in the Plane 70
4.4. Simultaneous Four Section in Three Dimensions 73
4.5. Erasing Cell Complexes in Three Dimensions 77
4.6. Generalizations to Higher Dimensions 78
4.7. Exercises and Research Problems 79
4.8. Bibliographic Notes 81
CHAPTER 5 Zones in Arrangements 83
5.1. Supported Cells, Zones, and Duality 84
5.2. Sweeping a Simple Arrangement 86
5.3. Tight Bounds in the Plane 89
5.4. Asymptotically Tight Bounds in d Dimensions 93
5.5. An Implication of the Result on Zones 94
5.6. Exercises and Research Problems 95
5.7. Bibliographic Notes 96
CHAPTER 6 The Complexity of Families of Cells 97
6.1. Definitions and Preliminary Results 98
6.2. The Complexity of a Polytope 99
6.2.1. Cyclic Polytopes 100
6.2.2. Euler s Relation 102
6.2.3. The Dehn Sommerville Relations 104
6.2.4. An Asymptotic Version of the Upper Bound Theorem 106
6.3. The Complexity of a Few Cells in Two Dimensions 107
6.4. Lower Bounds for Moderately Many Cells 109
6.5. Lower Bounds for Many Cells Ill
6.6. Upper Bounds for Many Cells 114
6.7. Exercises and Research Problems 116
6.8. Bibliographic Notes 118
PART II FUNDAMENTAL GEOMETRIC ALGORITHMS 121
CHAPTER 7 Constructing Arrangements 123
7.1. Representing an Arrangement in Storage 123
7.2. The Incremental Approach 125
7.3. Initiating the Construction 126
7.4. Geometric Preliminaries 128
7.5. Incrementing the Arrangement 130
7.6. The Analysis of the Algorithm 134
7.7. Exercises and Research Problems 136
XIII
7.8. Bibliographic Notes 137
CHAPTER 8 Constructing Convex Hulls 139
8.1. Convex Hulls and Duality 140
8.2. The Incidence Graph of a Convex Polytope 141
8.3. Two Algorithms in Two Dimensions 142
8.3.1. The Beneath Beyond Method 143
8.3.2. Using Divide and Conquer 145
8.4. The Beneath Beyond Method in d Dimensions 147
8.4.1. Geometric Preliminaries 148
8.4.2. Towards the Incrementation of the Convex Hull 152
8.4.3. Pyramidal Updates 153
8.4.4. Non Pyramidal Updates 154
8.4.5. The Analysis of the Algorithm 156
8.5. Divide and Conquer in Three Dimensions 158
8.5.1. Coping with Degenerate Cases 159
8.5.2. The Upgraded Incidence Graph 160
8.5.3. Geometric Preliminaries 162
8.5.4. Wrapping Two Convex Polytopes 167
8.5.5. The Analysis of the Algorithm 172
8.6. Exercises and Research Problems 173
8.7. Bibliographic Notes 175
CHAPTER 9 Skeletons in Arrangements 177
9.1. Skeletons and Eulerian Tours 178
9.2. Towards the Construction of a Skeleton 181
9.3. Constructing a Skeleton in a Simple Arrangement 183
9.4. Simulating Simplicity 185
9.4.1. A Conceptual Perturbation 186
9.4.2. Simulating the Perturbation 188
9.4.3. Computing the Sign of a Determinant of Polynomials 190
9.5. Penetration Search and Extremal Queries 192
9.5.1. Extremal Queries in the Plane 193
9.5.2. Extremal Queries in Three Dimensions: the Data Structure 195
9.5.3. Extremal Queries in Three Dimensions: the Query Algorithm 201
9.5.4. Dynamic Extremal Search 202
9.6. Exercises and Research Problems 205
9.7. Bibliographic Notes 208
CHAPTER 10 Linear Programming 209
10.1. The Solution to a Linear Program 210
10.2. Linear Programming and Duality 212
10.3. Linear Programming in Two Dimensions 214
XIV
10.3.1. Prune: Eliminate Redundant Half Planes 215
10.3.2. Bisect: Decrease the Range of the Linear Program 217
10.3.3. Find_Test: Determine the Median 219
10.3.4. Assembling the Algorithm 220
10.4. Linear Programming in Three and Higher Dimensions 223
10.4.1. The Geometry of Pruning 224
10.4.2. The Geometry of Bisecting 225
10.4.3. Searching Lines in the Plane 228
10.4.4. The Geometry of Searching 230
10.4.5. The Overall Algorithm 234
10.5. Exercises and Research Problems 236
10.6. Bibliographic Notes 238
CHAPTER 11 Planar Point Location Search 241
11.1. Euler s Relation for Planar Graphs 242
11.2. The Geometry of Monotone Subdivisions 244
11.3. A Tree of Separators for Point Location 218
11.4. Representing a Plane Subdivision 251
11.5. Constructing a Family of Separators 252
11.6. Optimal Search by Connecting Separators 256
11.7. Constructing the Layered DAG 258
11.8. Refining Non Monotone Subdivisions 260
11.9. Exercises and Research Problems 262
11.10. Bibliographic Notes 265
PART HI GEOMETRIC AND ALGORITHMIC APPLICATIONS 269
CHAPTER 12 Problems for Configurations and Arrangements 271
12.1. Largest Convex Subsets 272
12.2. The Visibility Graph for Line Segments 275
12.3. Degeneracies in Configurations 278
12.4. Minimum Measure Simplices 282
12.5. Computing Ranks: Sorting in d Dimensions? 284
12.6. A Vector Sum Maximization Problem 286
12.7. Exercises and Research Problems 288
12.8. Bibliographic Notes 290
CHAPTER 13 Voronoi Diagrams 293
13.1. Classical Voronoi Diagrams 294
13.2. Applications in the Plane 298
13.2.1. The Post Office Problem 298
13.2.2. Triangulating Point Sets 299
13.2.3. Delaunay Triangulations from Convex Hulls 303
XV
13.2.4. Finding Closest Neighbors 306
13.2.5. Minimum Spanning Trees 306
13.2.6. Shapes of Point Sets 309
13.3. Higher Order Voronoi Diagrams 315
13.4. The Complexity of Higher Order Voronoi Diagrams 319
13.5. Constructing Higher Order Voronoi Diagrams 324
13.6. Power Diagrams 327
13.7. Exercises and Research Problems 328
13.8. Bibliographic Notes 332
CHAPTER 14 Separation and Intersection in the Plane 335
14.1. Constructing Ham Sandwich Cuts in Two Dimensions 336
14.1.1. Ham Sandwich Cuts and Duality 336
14.1.2. Testing a Line 339
14.1.3. Finding Test Lines and Pruning 341
14.1.4. The Overall Algorithm 343
14.2. Answering Line Queries 345
14.2.1. The Ham Sandwich Tree 345
14.2.2. Point Location in Arrangements 348
14.3. A Self Dual Intersection Problem 349
14.4. Triangular Range Queries 351
14.5. Exercises and Research Problems 354
14.6. Bibliographic Notes 356
CHAPTER 15 Paradigmatic Design of Algorithms 359
15.1. The Problem: Stabbing Line Segments in the Plane 359
15.2. Geometric Transformation 361
15.3. Combinatorial Analysis 363
15.4. Divide and Conquer 365
15.5. Incremental Construction 367
15.6. Prune and Search 370
15.7. The Locus Approach 371
15.8. Dynamization by Decomposition 373
15.9. Exercises and Research Problems 376
i 15.10. Bibliographic Notes 378
REFERENCES 381
APPENDIX A Definitions 395
APPENDIX B Notational Conventions 409
INDEX 417
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| language | English |
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| series2 | European Association for Theoretical Computer Science: EATCS monographs on theoretical computer science |
| spelling | Edelsbrunner, Herbert 1958- Verfasser (DE-588)111356377 aut Algorithms in combinatorial geometry Herbert Edelsbrunner Berlin [u.a.] Springer 1987 XV, 423 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier European Association for Theoretical Computer Science: EATCS monographs on theoretical computer science 10. Literaturverz. S. [381] - 394 Algoritmen gtt Combinatorische meetkunde gtt Géométrie - Informatique Géométrie combinatoire Datenverarbeitung Combinatorial geometry Geometry Data processing Kombinatorische Geometrie (DE-588)4140733-7 gnd rswk-swf Algorithmus (DE-588)4001183-5 gnd rswk-swf Kombinatorische Geometrie (DE-588)4140733-7 s Algorithmus (DE-588)4001183-5 s DE-604 European Association for Theoretical Computer Science: EATCS monographs on theoretical computer science 10. (DE-604)BV005600727 10 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000481214&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Edelsbrunner, Herbert 1958- Algorithms in combinatorial geometry European Association for Theoretical Computer Science: EATCS monographs on theoretical computer science Algoritmen gtt Combinatorische meetkunde gtt Géométrie - Informatique Géométrie combinatoire Datenverarbeitung Combinatorial geometry Geometry Data processing Kombinatorische Geometrie (DE-588)4140733-7 gnd Algorithmus (DE-588)4001183-5 gnd |
| subject_GND | (DE-588)4140733-7 (DE-588)4001183-5 |
| title | Algorithms in combinatorial geometry |
| title_auth | Algorithms in combinatorial geometry |
| title_exact_search | Algorithms in combinatorial geometry |
| title_full | Algorithms in combinatorial geometry Herbert Edelsbrunner |
| title_fullStr | Algorithms in combinatorial geometry Herbert Edelsbrunner |
| title_full_unstemmed | Algorithms in combinatorial geometry Herbert Edelsbrunner |
| title_short | Algorithms in combinatorial geometry |
| title_sort | algorithms in combinatorial geometry |
| topic | Algoritmen gtt Combinatorische meetkunde gtt Géométrie - Informatique Géométrie combinatoire Datenverarbeitung Combinatorial geometry Geometry Data processing Kombinatorische Geometrie (DE-588)4140733-7 gnd Algorithmus (DE-588)4001183-5 gnd |
| topic_facet | Algoritmen Combinatorische meetkunde Géométrie - Informatique Géométrie combinatoire Datenverarbeitung Combinatorial geometry Geometry Data processing Kombinatorische Geometrie Algorithmus |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000481214&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV005600727 |
| work_keys_str_mv | AT edelsbrunnerherbert algorithmsincombinatorialgeometry |