Computational geometry: an introduction through randomized algorithms
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Upper Saddle River, NJ
Prentice Hall
2002
|
| Ausgabe: | Transferred to digital print on demand |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | XV, 447 S. Ill., graph. Darst. |
| ISBN: | 0133363635 |
Internformat
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| 100 | 1 | |a Mulmuley, Ketan |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Computational geometry |b an introduction through randomized algorithms |c Ketan Mulmuley |
| 250 | |a Transferred to digital print on demand | ||
| 264 | 1 | |a Upper Saddle River, NJ |b Prentice Hall |c 2002 | |
| 300 | |a XV, 447 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Datensatz im Suchindex
| _version_ | 1804141167946760192 |
|---|---|
| adam_text | Contents
Preface
ix
Notation xvi
I Basics
1
1
Quick-sort and search
3
1.1
Quick-sort
............................. 4
1.2
Another view of quick-sort
.................... 5
1.3
Randomized binary trees
..................... 7
1.3.1
Dynamization
....................... 11
1.3.2
Another interpretation of rotations
........... 13
1.4
Skip lists
............................. 18
2
What is computational geometry?
26
2.1
Range queries
........................... 28
2.2
Arrangements
........................... 29
2.3
Trapezoidal decompositions
................... 32
2.4
Convex polytopes
......................... 36
2.4.1
Duality
.......................... 40
2.5
Voronoi diagrams
......................... 46
2.6
Hidden surface removal
..................... 51
2.7
Numerical precision and degeneracies
.............. 52
2.8
Early deterministic algorithms
................. 55
2.8.1
Planar convex hulls
.................... 56
2.8.2
Arrangements of lines
.................. 58
2.8.3
Trapezoidal decompositions
............... 61
2.8.4
Planar Voronoi diagrams
................ 67
2.8.5
Planar point location
................... 74
2.9
Deterministic vs. randomized algorithms
............ 78
vi
CONTENTS
2.10
The model of randomness
.................... 78
3
Incremental algorithms
81
3.1
Trapezoidal decompositions
................... 84
3.1.1
History
.......................... 90
3.1.2
Planar graphs
....................... 94
3.1.3
Simple polygons*
..................... 94
3.2
Convex polytopes
......................... 96
3.2.1
Conflict maintenance
................... 99
3.2.2
History
.......................... 103
3.2.3
On-line linear programming
............... 104
3.3
Voronoi diagrams
......................... 106
3.4
Configuration spaces
.......................
Ill
3.5
Tail estimates*
.......................... 120
4
Dynamic algorithms
126
4.1
Trapezoidal decompositions
................... 129
4.1.1
Point location
....................... 132
4.2
Voronoi diagrams
......................... 135
4.2.1
Conflict search
...................... 137
4.3
History and configuration spaces
................ 140
4.3.1
Expected performance*
................. 142
4.4
Rebuilding history
........................ 149
4.5
Deletions in history
........................ 151
4.5.1 3D
convex polytopes
................... 158
4.5.2
Trapezoidal decompositions
............... 162
4.6
Dynamic shufBing
........................ 167
5
Random sampling
173
5.1
Configuration spaces with bounded valence
.......... 176
5.2
Top-down sampling
....................... 181
5.2.1
Arrangements of
Unes
.................. 182
5.3
Bottom-up sampling
....................... 184
5.3.1
Point location in arrangements
............. 185
5.4
Dynamic sampling
........................ 192
5.4.1
Point location in arrangements
............. 193
5.5
Average conflict size
....................... 197
5.6
More dynamic algorithms
.................... 201
5.6.1
Point location in trapezoidal decompositions
..... 204
5.6.2
Point location in Voronoi diagrams
........... 210
5.7
Range spaces and «-nets
..................... 215
5.7.1
VC dimension of a range space
............. 221
CONTENTS
vii
5.8
Comparisons
........................... 223
II Applications
227
6
Arrangements of
hyperplanes 229
6.1
Incremental construction
..................... 232
6.2
Zone Theorem
.......................... 234
6.3
Canonical
triangulations
..................... 238
6.3.1
Cutting Lemma
...................... 241
6.4
Point location and ray shooting
................. 242
6.4.1
Static setting
....................... 242
6.4.2
Dynamization
....................... 248
6.5
Point location and range queries
................ 250
6.5.1
Dynamic maintenance
.................. 253
7
Convex polytopes
260
7.1
Linear programming
....................... 262
7.2
The number of faces
....................... 267
7.2.1
Dehn-Sommerville relations
............... 268
7.2.2
Upper bound theorem: Asymptotic form
........ 271
7.2.3
Upper bound theorem: Exact form
........... 271
7.2.4
Cyclic polytopes
..................... 274
7.3
Incremental construction
..................... 276
7.3.1
Conflicts and linear programming
............ 278
7.3.2
Conflicts and history
................... 279
7.4
The expected structural and conflict change
.......... 280
7.5
Dynamic maintenance
...................... 284
7.6
Voronoi diagrams
......................... 285
7.7
Search problems
......................... 286
7.7.1
Vertical ray shooting
................... 288
7.7.2
Half-space range queries
................. 289
7.7.3
Nearest fc-neighbor queries
............... 291
7.7.4
Dynamization
....................... 292
7.8
Levels and Voronoi diagrams of order
к
............ 294
7.8.1
Incremental construction
................ 301
7.8.2
Single level
........................ 306
8
Range search
311
8.1
Orthogonal intersection search
................. 311
8.1.1
Randomized segment tree
................ 312
8.1.2
Arbitrary dimension
................... 317
viii CONTENTS
8.2
Nonintersecting
segments
in the
plane............. 322
8.3 Dynamic
point location
..................... 327
8.4 Simplex
range search
....................... 328
8.4.1
Partition theorem
.................... 332
8.4.2
Preprocessing time
.................... 336
8.5
Half-space range queries
..................... 338
8.5.1
Partition theorem
.................... 341
8.5.2
Half-space emptiness
................... 343
8.6
Decomposable search problems
................. 345
8.6.1
Dynamic range queries
.................. 348
8.7
Parametric search
......................... 349
9
Computer graphics
358
9.1
Hidden surface removal
..................... 361
9.1.1
Analysis
.......................... 367
9.2
Binary Space Partitions
..................... 372
9.2.1
Dimension two
...................... 372
9.2.2
Dimension three
..................... 379
9.3
Moving viewpoint
......................... 383
9.3.1
Construction of a cylindrical partition
......... 391
9.3.2
Randomized incremental construction
......... 392
10
How crucial is randomness?
398
10.1
Pseudo-random sources
..................... 399
10.2
Derandomization
......................... 410
10.2.1
є
-approximations
.....................
412
10.2.2
The method of conditional probabilities
........ 413
10.2.3
Divide and conquer
.................... 417
A Tail estimates
422
A.I Chernoff s technique
....................... 423
A.I.I Binomial distribution
.................. 424
A.1.2 Geometric distribution
.................. 426
A.1.3 Harmonic distribution
.................. 428
A.2 Chebychev s technique
...................... 429
Bibliography 431
Index
Computational
Geometry
An Introduction Through
Randomized Algorithms
Ketan Mulmuley
This textbook gives a concise and up-to-date introduction to computa¬
tional geometry with emphasis on randomized algorithms. It begins
with simple two-dimensional problems and their deterministic solutions
and shifts to randomized solutions as the problems become more
complex. Since the simplest solutions to a large number of problems in
computational geometry are randomized, this approach should enable
readers to get a wide perspective of the field quickly.
Another feature of randomized methods, which makes them ideal for
the purpose of introduction, is that they are all based on a few basic
principies
which can be applied systematically to a large number of
apparently dissimilar problems. This book strives to give an account
that focuses on simplicity and unity of randomized algorithms as well
as their depth.
The book is organized into two parts. In the first part, the basic princi¬
ples are formulated and illustrated with the help of simple problems
in planar geometry. Higher dimensional applications are given in the
second part Thus, this book is useful to beginning graduate students
as well as to specialists.
PRENTICE HALL
Upper Saddle River, NJ
07458
ISBN
а-із-ззьзьз
-s
о
3
3636
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| spelling | Mulmuley, Ketan Verfasser aut Computational geometry an introduction through randomized algorithms Ketan Mulmuley Transferred to digital print on demand Upper Saddle River, NJ Prentice Hall 2002 XV, 447 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Randomisierter Algorithmus (DE-588)4176929-6 gnd rswk-swf Algorithmische Geometrie (DE-588)4130267-9 gnd rswk-swf Algorithmische Geometrie (DE-588)4130267-9 s Randomisierter Algorithmus (DE-588)4176929-6 s 1\p DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018990113&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018990113&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Mulmuley, Ketan Computational geometry an introduction through randomized algorithms Randomisierter Algorithmus (DE-588)4176929-6 gnd Algorithmische Geometrie (DE-588)4130267-9 gnd |
| subject_GND | (DE-588)4176929-6 (DE-588)4130267-9 |
| title | Computational geometry an introduction through randomized algorithms |
| title_auth | Computational geometry an introduction through randomized algorithms |
| title_exact_search | Computational geometry an introduction through randomized algorithms |
| title_full | Computational geometry an introduction through randomized algorithms Ketan Mulmuley |
| title_fullStr | Computational geometry an introduction through randomized algorithms Ketan Mulmuley |
| title_full_unstemmed | Computational geometry an introduction through randomized algorithms Ketan Mulmuley |
| title_short | Computational geometry |
| title_sort | computational geometry an introduction through randomized algorithms |
| title_sub | an introduction through randomized algorithms |
| topic | Randomisierter Algorithmus (DE-588)4176929-6 gnd Algorithmische Geometrie (DE-588)4130267-9 gnd |
| topic_facet | Randomisierter Algorithmus Algorithmische Geometrie |
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