theory practice vs · 2016-07-01 · locate by walk talk outline basic incremental algorithm locate...

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Delaunay triangulation,

Theory vs practice

Olivier Devillers

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http://www.inria.fr/sophia/members/Olivier.Devillers/EuroCG2012/

Delaunay triangulation,

Theory vs practice

Bibliographical notes

[References, given in the abstract]

Benchmarks (src code)

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Everybody in this room know what Delaunay is!

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empty circle property

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empty sphere property

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A (partial) history of Delaunay algorithms(and of computational geometry)

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Earlier algorithms [1970. . . ]

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Earlier algorithms

Incremental

Gift wrapping

2D/3D

2D

[1970. . . ]

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Earlier algorithms

Incremental

Gift wrapping

2D/3D

2D

Non optimal

”simple”

actually coded

linked to applications: meshes, reconstruction

[1970. . . ]

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Worst case algorithms [1980. . . ]

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Divide & conquer

Plane sweep

2D

2D

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Earlier algorithms

Non optimal

”simple”

actually coded

linked to applications: meshes, reconstruction

[1970. . . ]


Divide & conquer

Plane sweep

2D

2D

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Randomized algorithms [1990. . . ]

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Randomized algorithms

Delaunay tree

Clarkson & Shor

Dynamic updates (history graph)

Delaunay hierarchy

Spatial sorting (BRIO)

[1990. . . ]

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[1995. . . ]Robustness issues

[1990. . . ]

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[1995. . . ]Robustness issues

[1990. . . ]

Properties checking

Exact computation paradigm

VRONI

TRIANGLE

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Talk outline

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Locate by walk

Talk outline

Basic incremental algorithm

Locate using randomized data structures

Vertex removal in 2D

Remarks on CGAL programming

Conclusion

One word on robustness issues

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Locate by walk





Conclusion


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The exact computation paradigm

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Compute predicates approximately

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Certify result

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Certify result

Run exact computation

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Certify result

Run exact computation

Extra cost

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Certify result

Extra cost

Delaunay 2D 10M points

Cartesian<double>

Exact predicates inexact constructions kernel

10.6 seconds

9.7 seconds

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Certify result

Extra cost


Cartesian<double>


3D

10.6 seconds

9.7 seconds

82 seconds

75 seconds

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Certify result

Extra cost


Cartesian<double>


3D

10.6 seconds

9.7 seconds

82 seconds

75 secondsmay loop (or crash)

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Certify result

Extra cost


Cartesian<double>


3D

10.6 seconds

9.7 seconds

82 seconds

75 seconds

All benchmarks

2.3 GHz, 16 GByte workstation


3.9 (Release mode)

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Locate by walk




Conclusion



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Locate

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Find conflicts

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Remove triangles

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Fill the hole

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Vertex removal in 2DRemarks on CGAL programmingConclusion

One word on robustness issuesBasic incremental algorithm

Locate by walk

Straight walk

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Locate by walk - straight walk

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Exit edge ?One orientation predicate

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Exit edge ?One orientation predicate

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End of walk ?A second orientation predicate

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End of walk ?A second orientation predicate

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Locate by walk

Straight walkVisibility walk

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Locate by walk - visibility walk

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Triangle with two exitsOne orientation predicate

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Triangle with one exit1.5 orientation predicate

One predicate

Two predicates

if this neighbor tried first

if this neighbor tried first

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1.25 orientation predicate ?

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Visibility vs straight walk

Locate by walk

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less predicates per crossed edge

similar number of crossed edges

2D and 3D

Locate by walk

experimental / theoretical

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Speed improvement ?

Locate by walk

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Speed improvement ?

Walk in Delaunay 1 Mpoints

Straight: 324 µs

Visibility: 285 µs

3D: 97 µs

Locate by walk

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Speed improvement ?

Much easier to code


Straight: 324 µs

Visibility: 285 µs

3D: 97 µs

Locate by walk

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Locate by walk

Straight walkVisibility walkStructural filtering

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Locate by visibility walk - structural filtering

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Walk may loop (not in Delaunay)

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Robustness issue:

Non certified arithmetic

Rounding errors

Wrong decisionsduring walk

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Robustness issue:

even in Delaunay

may loop

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Robustness issue:

even in Delaunay

But only in very

special configurations

may loop

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Orientation predicates

Certify all along the walk

Certify after a while, just in case

285 µseconds

220 µseconds

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Orientation predicates

Certify all along the walk

Certify after a while, just in case

285 µseconds

220 µseconds

97 µseconds

81 µseconds

2D 3D


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Locate by walk

Straight walkVisibility walkStructural filteringWalk shape

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Locate by visibility walk - walk shape

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Rightmost

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Leftmost

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In between

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Turn counterclockwise from previous

Rightmost

1

2

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Leftmost

Turn clockwise from previous

1

2

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first with proba 13

first with proba 23

Balance left and right turns

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220 µseconds


Leftmost

188 µsecondsBalanced

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Locate by visibility walk

220 µsecondsStructural filtering

188 µsecondsBalanced walk

Straight walk

Visibility walk

324 µseconds

285 µseconds


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Locate by walk


Delaunay tree

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Data structures to locate - the Delaunay tree

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locate based on incircle predicate

] triangles in the Delaunay tree

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How many triangles created by the last point ?

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= 6n (randomized)

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Locate by walk


Delaunay tree

Delaunay hierarchy

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Data structures to locate - the Delaunay hierarchy

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Nearest Neighbor

Query

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The Delaunay tree



= 6n (randomized)

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The Delaunay tree



= 6n (randomized)

based on orientation predicate

] triangles in the hierarchy

= 1.03× 2n (expected)

can be chosen

The Delaunay hierarchy

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The Delaunay tree



= 6n (randomized)




can be chosen

O(n log n)


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The Delaunay tree



= 6n (randomized)




can be chosen

17 seconds2.3 seconds

50000 random points (original benchmarks in 2000).


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Locate by walk


Delaunay tree

Biased randomized insertion order

Delaunay hierarchy

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Data structures to locate - biased random insertion order

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Locate is easy if you know a vertex nearby

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Locate is easy if you know a vertex nearby

Natural idea: sort the points, locate from previous

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x-order

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Hilbert order

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Biased order (Spatial sorting)

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random order

x-order

Hilbert order


locate using Delaunay hierarchy

0.7 seconds

157 seconds

3 seconds

0.8 seconds

6 seconds

Delaunay 2D 1M random points

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random order

x-order

Hilbert order


locate using Delaunay hierarchy

Delaunay 2D 100K parabola points

128 seconds

632 seconds

46 seconds

0.3 seconds

0.3 seconds

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10.6 secondsBiased random order

Delaunay tree

Delaunay hierarchy

∼ 10 mn (estimate)

90 seconds

Construction of Delaunay 10 M random points

Data structures to locate

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Remarks on CGAL programmingConclusion


Locate by walk



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Vertex removal

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Locate by walk



Various algorithmsBoundary expansion

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release 3.5, 2D implementation

Vertex removal - boundary expansion

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hole boundary = queue


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find new incident triangle in linear time


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Locate by walk



Various algorithmsBoundary expansionTriangulate and sew

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current 3D implementation

Vertex removal - triangulate and sew

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Delaunay of neighbors


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delete extra triangles and sew


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delete extra triangles and sew

not implemented in 2D


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Locate by walk



Various algorithmsBoundary expansionTriangulate and sewFlip the hole

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Vertex removal - flip the hole

34triangulate from any vertex


34queue of edges to be checked


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a litle bit faster


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Locate by walk



Various algorithms

Low degree optimization

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degree 3

Vertex removal - low degree optimization

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degree 3

almost nothing to do


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degree 4


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degree 4

just one incircle test to decide


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degree 5


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degree 5

”star” the pentagon from the right vertex


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degree 5

Decision tree


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degree 5

Decision tree3∈012

4∈0234∈013

yesno

4∈1234∈012

v0v2 v1

v3

v4v4

v0

v2v1

v3v4


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degree 6


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degree 6

14 results


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degree 6Decision tree


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1∈230

4∈235

1∈2341∈235

4∈0135∈1241∈345star5

4∈0135∈013 5∈014antiN0N2

antiN0

5∈014

5∈014 star1N1star4 antiN1

star1N1

diamond1

4∈503

star3 antiN0

4∈235

4∈2305∈230

4∈012star05∈0124∈503

4∈1254∈502antiN2N0 star5star0

5∈012

diamond0

N2 4∈015

star2 star4N2 antiN1

v0

v1

v2v3

v5

v4


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1∈230

4∈235

1∈2341∈235

4∈0135∈1241∈345star5

4∈0135∈013 5∈014antiN0N2

antiN0

5∈014

5∈014 star1N1star4 antiN1

star1N1

diamond1

4∈503

star3 antiN0

4∈235

4∈2305∈230

4∈012star05∈0124∈503

4∈1254∈502antiN2N0 star5star0

5∈012

diamond0

N2 4∈015

star2 star4N2 antiN1

v0

v1

v2v3

v5

v4

6 incircle predicates


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symmetric tree



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symmetric tree


10 incircle predicates


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degree 3 4 5 6 7 8? 9] results 1 2 5 14 42 132 429] leaves 1 2 6 24 130 '500dlog2 ]resultse 0 1 3 4 6 8 9tree height 0 1 3 6 10 '14] lines of code 30 40 90 280 700 '2500? not implemented. The sizes of the tree and the code are estimated


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Remarks on implementation

limited memory allocation, use old faces ”in place”

re-use as many neighbor links as possible


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Remarks on implementation

limited memory allocation, use old faces ”in place”

re-use as many neighbor links as possible

tree implementationif incircle(...)

if incircle(...)

if incircle(...) use_this_shape(face0,face1,face2...)

else use_other_shape(face2,face3,face4...)

......


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degree

small degrees

deletion time per vertex

3 4 5 6 7 8 9 10 11 ≥ 12

10µs

Vertex removal

Boundary expansionFlip the hole

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degree

3.5

small degrees


3 4 5 6 7 8 9 10 11 ≥ 12

10µs

Vertex removal


3.6

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degreeinit (load memory)

3.5

small degrees


3 4 5 6 7 8 9 10 11 ≥ 12

10µs

Vertex removal


3.6

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degree

] points

30%

init (load memory)

3.5

small degrees


degree distribution

3 4 5 6 7 8 9 10 11 ≥ 12

10µs

Vertex removal


3.6

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Conclusion


Locate by walk




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Algorithmic choices

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Algorithmic choices

Theoretical efficiency

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Algorithmic choices


Practical efficiency

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Algorithmic choices



Robustness issues

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Algorithmic choices



Robustness issues

Modularitytraits classes

data structuresgeometry

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Algorithmic choices



Robustness issues

Modularitytraits classes

data structuresgeometry

Minimal requirements

e.g. do not use strange predicates

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Delaunay triangulation

Geometric traits Triangulation data structure

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Geometric traits Triangulation data structureIR2

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IR3 projection

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IR2 other metric

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Periodic triangulation

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' 1µs per point

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' 8µs per point

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Locate by walk




Conclusion

Delaunay challenges

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Delaunay challenges

Practical vs worst case size of Delaunay 3D

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Known results

Delaunay challenges


Θ(n2) worst case

Θ(n) random in ball

Ω(n)O(n log n) random on polyhedron

O(n log n) good sample of smooth generic surface

Θ(n log n) random on cylinder

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Known results

Delaunay challenges


Θ(n2) worst case

Θ(n) random in ball

Ω(n)O(n log n) random on polyhedron

O(n log n) good sample of smooth generic surface

Θ(n log n) random on cylinder

Find good models of practical data

(Smooth analysis)

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Delaunay challenges


Better algorithm for 3D deletion

10 µs to insert

100 µs to delete

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Delaunay challenges


Better algorithm for 3D deletion

One billion points

Needs memory efficient algorithms

Cache effects are already important

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Questions

theory practice vs · 2016-07-01 · locate by walk talk outline basic incremental algorithm locate...

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