Space-filling curves: an introduction with applications in scientific computing
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2013
|
| Schriftenreihe: | Texts in computational science and engineering
9 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XIII, 278 S. Ill., graph. Darst. |
| ISBN: | 3642310451 9783642310454 9783642310461 |
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| 245 | 1 | 0 | |a Space-filling curves |b an introduction with applications in scientific computing |c Michael Bader |
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Datensatz im Suchindex
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IMAGE 1
CONTENTS
1 TWO MOTIVATING EXAMPLES: SEQUENTIAL ORDERS ON QUADTREES AND
MULTIDIMENSIONAL DATA STRUCTURES 1
1.1 MODELLING COMPLICATED GEOMETRIES WITH QUADTREES, OCTREES, AND
SPACETREES 1
1.1.1 QUADTREES AND OCTREES 3
1.1.2 A SEQUENTIAL ORDER ON QUADTREE CELLS 3
1.1.3 A MORE LOCAL SEQUENTIAL ORDER ON QUADTREE CELLS 6
1.2 NUMERICAL SIMULATION: SOLVING A SIMPLE HEAT EQUATION 7
1.3 SEQUENTIALISATION O F MULTIDIMENSIONAL DATA 9
1.3.1 REQUIREMENTS FOR EFFICIENT SEQUENTIAL ORDERS 11
1.3.2 ROW-MAJOR AND COLUMN-MAJOR SEQUENTIALISATION 12
2 HOW TO CONSTRUCT SPACE-FILLING CURVES 15
2.1 TOWARDS A BIJECTIVE MAPPING O F THE UNIT INTERVAL TO THE UNIT SQUARE
15
2.2 CONTINUOUS MAPPINGS AND (SPACE-FILLING) CURVES 17
2.3 THE HILBERT CURVE 18
2.3.1 ITERATIONS OF THE HILBERT CURVE 18
2.3.2 APPROXIMATING POLYGONS 19
2.3.3 DEFINITION O F THE HILBERT CURVE 20
2.3.4 PROOF: H DEFINES A SPACE-FILLING CURVE 22
2.3.5 CONTINUITY O F THE HILBERT CURVE 23
2.3.6 MOORE'S VERSION OF THE HILBERT CURVE 24
2.4 PEANO CURVE 25
2.5 SPACE-FILLING CURVES: REQUIRED ALGORITHMS 27
3 GRAMMAR-BASED DESCRIPTION O F SPACE-FILLING CURVES 31
3.1 DESCRIPTION O F THE HILBERT CURVE USING GRAMMARS 31
3.2 A TRAVERSAL ALGORITHM FOR 2D DATA 3 4
3.3 GRAMMAR-BASED DESCRIPTION OF THE PEANO CURVE 37
3.4 A GRAMMAR FOR TURTLE GRAPHICS 39
HTTP://D-NB.INFO/1022401912
IMAGE 2
X C O N T E N T S
4 ARITHMETIC REPRESENTATION O F SPACE-FILLING CURVES 47
4.1 ARITHMETIC REPRESENTATION O F THE HILBERT MAPPING 47
4.2 CALCULATING THE VALUES O F H 49
4.3 UNIQUENESS OF THE HILBERT MAPPING 52
4.4 COMPUTATION OF THE INVERSE: HILBERT INDICES 55
4.5 ARITHMETISATION OF THE PEANO CURVE 57
4.6 EFFICIENT COMPUTATION O F SPACE-FILLING MAPPINGS 59
4.6.1 COMPUTING HILBERT MAPPINGS VIA RECURSION UNROLLING 60
4.6.2 FROM RECURSION UNROLLING TO STATE DIAGRAMS 61
5 APPROXIMATING POLYGONS 67
5.1 APPROXIMATING POLYGONS O F THE HILBERT AND PEANO CURVE 67
5.2 MEASURING CURVE LENGTHS WITH APPROXIMATING POLYGONS 69
5.3 FRACTAL CURVES AND THEIR LENGTH 7 0
5.4 A QUICK EXCURSION ON FRACTAL CURVES 72
6 SIERPINSKI CURVES 77
6.1 THE SIERPINSKI-KNOPP CURVE 77
6.1.1 CONSTRUCTION OF THE SIERPINSKI CURVE 77
6.1.2 GRAMMAR-BASED DESCRIPTION O F THE SIERPINSKI CURVE. 79 6.1.3
ARITHMETISATION OF THE SIERPINSKI CURVE 80
6.1.4 COMPUTATION O F THE SIERPINSKI MAPPING 81
6.2 GENERALISED SIERPINSKI CURVES 82
6.2.1 BISECTING TRIANGLES ALONG TAGGED EDGES 83
6.2.2 CONTINUITY AND LOCALITY O F GENERALISED SIERPINSKI CURVES 85
6.2.3 FILLING TRIANGLES WITH CURVED EDGES 87
7 FURTHER SPACE-FILLING CURVES 93
7.1 CHARACTERISATION O F SPACE-FILLING CURVES 93
7.2 LEBESGUE CURVE AND MORTON ORDER 95
7.3 THE //-INDEX 99
7.4 THE ^^2-CURVE 101
7.5 THE GOSPER FLOWSNAKE 104
8 SPACE-FILLING CURVES IN 3D 109
8.1 3D HILBERT CURVES . . . , 109
8.1.1 POSSIBILITIES TO CONSTRUCT A 3D HILBERT CURVE 109
8.1.2 ARITHMETISATION O F THE 3D HILBERT CURVE 113
8.1.3 A 3D HILBERT GRAMMAR WITH MINIMAL NUMBER OF NON-TERMINALS 114
8.2 3D PEANO CURVES 116
8.2.1 A DIMENSION-RECURSIVE GRAMMAR TO CONSTRUCT A 2D PEANO CURVE 116
8.2.2 EXTENSION OF THE DIMENSION-RECURSIVE GRAMMAR TO CONSTRUCT 3D PEANO
CURVES 117
IMAGE 3
C O N T E N T S XI
8.2.3 PEANO CURVES BASED ON 5 X 5 OR 7 X 7 REFINEMENT 119
8.2.4 TOWARDS PEANO'S ORIGINAL CONSTRUCTION 122
8.3 A 3D SIERPINSKI CURVE 123
9 REFINEMENT TREES AND SPACE-FILLING CURVES 129
9.1 SPACETREES AND REFINEMENT TREES 129
9.1.1 NUMBER O F GRID CELLS FOR THE NORM CELL SCHEME AND FOR A QUADTREE
131
9.2 USING SPACE-FILLING CURVES TO SEQUENTIALISE SPACETREE GRIDS 132
9.2.1 ADAPTIVELY REFINED SPACETREES 134
9.2.2 A GRAMMAR FOR ADAPTIVE HILBERT ORDERS 135
9.2.3 REFINEMENT INFORMATION AS BITSTREAMS 137
9.3 SEQUENTIALISATION O F ADAPTIVE GRIDS USING SPACE-FILLING CURVES 138
10 PARALLELISATION WITH SPACE-FILLING CURVES 1 43
10.1 PARALLEL COMPUTATION O F THE HEAT DISTRIBUTION ON A METAL PLATE 143
10.2 PARTITIONING WITH SPACE-FILLING CURVES 146
10.3 PARTITIONING AND LOAD-BALANCING BASED ON REFINEMENT TREES AND
SPACE-FILLING CURVES 149
10.4 SUBTREE-BASED LOAD DISTRIBUTION 150
10.5 PARTITIONING ON SEQUENTIALISED REFINEMENT TREES 153
10.5.1 MODIFIED DEPTH-FIRST TRAVERSALS FOR PARALLELISATION 153 10.5.2
REFINEMENT TREES FOR PARALLEL GRID PARTITIONS 155
10.6 DATA EXCHANGE BETWEEN PARTITIONS DEFINED VIA SPACE-FILLING CURVES
157
10.6.1 REFINEMENT-TREE PARTITIONS USING GHOST CELLS 159
10.6.2 NON-OVERLAPPING REFINEMENT-TREE PARTITIONS 160
11 LOCALITY PROPERTIES O F SPACE-FILLING CURVES 1 67
11.1 HOLDER CONTINUITY OF SPACE-FILLING CURVES 167
11.1.1 HOLDER CONTINUITY O F THE 3D HILBERT CURVE 168
11.1.2 HOLDER CONTINUITY AND PARALLELISATION 168
11.1.3 DISCRETE LOCALITY MEASURES FOR ITERATIONS OF SPACE-FILLING CURVES
171
11.2 GRAPH-RELATED LOCALITY MEASURES 172
11.2.1 THE EDGE CUT AND THE SURFACE O F PARTITION BOUNDARIES 173
11.2.2 CONNECTEDNESS OF PARTITIONS 174
12 SIERPINSKI CURVES ON TRIANGULAR AND TETRAHEDRAL MESHES 181
12.1 TRIANGULAR MESHES AND QUASI-SIERPINSKI CURVES 181
12.1.1 TRIANGULAR MESHES USING RED-GREEN REFINEMENT 181 12.1.2
TWO-DIMENSIONAL QUASI-SIERPINSKI CURVES 182
12.1.3 RED-GREEN CLOSURE FOR QUASI-SIERPINSKI ORDERS 184
IMAGE 4
XII C O N T E N T S
12.2 TETRAHEDRAL GRIDS AND 3D SIERPINSKI CURVES 184
12.2.1 BISECTION-BASED TETRAHEDRAL GRIDS 184
12.2.2 SPACE-FILLING ORDERS ON TETRAHEDRAL MESHES 188
13 CASE STUDY: CACHE EFFICIENT ALGORITHMS FOR MATRIX OPERATIONS 195 13.1
CACHE EFFICIENT ALGORITHMS AND LOCALITY PROPERTIES 195
13.2 CACHE OBLIVIOUS MATRIX-VECTOR MULTIPLICATION 199
13.3 MATRIX MULTIPLICATION USING PEANO CURVES 201
13.3.1 BLOCK-RECURSIVE PEANO MATRIX MULTIPLICATION 204
13.3.2 MEMORY ACCESS PATTERNS DURING THE PEANO MATRIX MULTIPLICATION 205
13.3.3 LOCALITY PROPERTIES AND CACHE EFFICIENCY 206
13.3.4 CACHE MISSES ON AN IDEAL CACHE 208
13.3.5 MULTIPLYING MATRICES OF ARBITRARY SIZE 210
13.4 SPARSE MATRICES AND SPACE-FILLING CURVES 211
14 CASE STUDY: NUMERICAL SIMULATION ON SPACETREE GRIDS USING
SPACE-FILLING CURVES 215
14.1 CACHE-OBLIVIOUS ALGORITHMS FOR ELEMENT-ORIENTED TRAVERSALS . 215
14.1.1 ELEMENT-BASED TRAVERSALS ON SPACETREE GRIDS 217
14.1.2 TOWARDS STACK-BASED TRAVERSALS 219
14.2 IMPLEMENTATION O F THE ELEMENT-ORIENTED TRAVERSALS 221
14.2.1 GRAMMARS FOR STACK COLOURING 222
14.2.2 INPUT/OUTPUT STACKS VERSUS COLOUR STACKS 222
14.2.3 ALGORITHM FOR SIERPINSKI TRAVERSAL 225
14.2.4 ADAPTIVITY: AN ALGORITHM FOR CONFORMING REFINEMENT 226
14.2.5 A MEMORY-EFFICIENT SIMULATION APPROACH FOR DYNAMICALLY ADAPTIVE
GRIDS 227
14.3 WHERE IT WORKS: AND WHERE IT DOESN'T 228
14.3.1 THREE-DIMENSIONAL HILBERT TRAVERSALS 229
14.3.2 A LOOK AT MORTON ORDER 229
15 FURTHER APPLICATIONS O F SPACE-FILLING CURVES: REFERENCES AND
READINGS 235
A SOLUTIONS TO SELECTED EXERCISES 239
A. 1 TWO MOTIVATING EXAMPLES 239
A.2 SPACE-FILLING CURVES 239
A.3 GRAMMAR-BASED DESCRIPTION OF SPACE-FILLING CURVES 240
A.4 ARITHMETIC REPRESENTATION OF SPACE-FILLING CURVES 241
A.5 APPROXIMATING POLYGONS 243
A.6 SIERPINSKI CURVES 246
A.7 FURTHER SPACE-FILLING CURVES 246
A.8 SPACE-FILLING CURVES IN 3D 247
A.9 REFINEMENT TREES AND SPACE-FILLING CURVES 248
IMAGE 5
C O N T E N T S XIII
A. 10 PARALLELISATION WITH SPACE-FILLING CURVES 249
A. 11 LOCALITY PROPERTIES O F SPACE-FILLING CURVES 251
A.12 SIERPINSKI CURVES ON TRIANGULAR AND TETRAHEDRAL MESHES 251 A. 13
CACHE EFFICIENT ALGORITHMS FOR MATRIX OPERATIONS 253
A. 14 NUMERICAL SIMULATION ON SPACETREE GRIDS USING SPACE-FILLING CURVES
253
REFERENCES 257
INDEX 27 1 |
| any_adam_object | 1 |
| author | Bader, Michael 1971- |
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| dewey-sort | 3516.5 |
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| discipline | Mathematik |
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| language | English |
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| physical | XIII, 278 S. Ill., graph. Darst. |
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| publisher | Springer |
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| series | Texts in computational science and engineering |
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| spelling | Bader, Michael 1971- Verfasser (DE-588)133983404 aut Space-filling curves an introduction with applications in scientific computing Michael Bader Berlin [u.a.] Springer 2013 XIII, 278 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in computational science and engineering 9 Raumfüllende Kurve (DE-588)4374972-0 gnd rswk-swf Raumfüllende Kurve (DE-588)4374972-0 s DE-604 Texts in computational science and engineering 9 (DE-604)BV016971315 9 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4040639&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=025700496&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bader, Michael 1971- Space-filling curves an introduction with applications in scientific computing Texts in computational science and engineering Raumfüllende Kurve (DE-588)4374972-0 gnd |
| subject_GND | (DE-588)4374972-0 |
| title | Space-filling curves an introduction with applications in scientific computing |
| title_auth | Space-filling curves an introduction with applications in scientific computing |
| title_exact_search | Space-filling curves an introduction with applications in scientific computing |
| title_full | Space-filling curves an introduction with applications in scientific computing Michael Bader |
| title_fullStr | Space-filling curves an introduction with applications in scientific computing Michael Bader |
| title_full_unstemmed | Space-filling curves an introduction with applications in scientific computing Michael Bader |
| title_short | Space-filling curves |
| title_sort | space filling curves an introduction with applications in scientific computing |
| title_sub | an introduction with applications in scientific computing |
| topic | Raumfüllende Kurve (DE-588)4374972-0 gnd |
| topic_facet | Raumfüllende Kurve |
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