streaming construction of delaunay triangulationsjrs/papers/dtstalk.pdf · 2007-09-04 · streaming...

University of California at BerkeleyJonathan Shewchuk

Delaunay TriangulationsStreaming Construction of

Joint work with Martin Isenburg, Yuanxin Liu, & Jack Snoeyink

Delaunay TriangulationsStreaming Construction of

Or, ‘‘How we compute supercomputer−sized Delaunay triangulations on anordinary laptop computer.’’

Every point set has a Delaunay triangulation. Think ofit as a function that takes a set of points and outputs atriangulation.

The Delaunay Triangulation


...is a triangulation whose triangles are all Delaunay.

An triangle is Delaunay if it has an empty circumscribingcircle − one that encloses no vertex.

The circumcircle of every Delaunay triangle is empty.


Terrain Databases and

Contouring

Geographical Information Systems

Applications of Triangulations

Rendering

Elevation Collection by LIDAR

Related Work

Amenta, Choi, & Rote [2003] use

triangulation construction.

randomized insertion ordersbiased

to improvevirtual memory performance of 3D Delaunay

Related Work




Agarwal, Arge, and Yi [2005] implemented

(1 billion triangles on desktop machine.)


an external memory 2D triangulator basedon randomized divide−and−conquer.

Related Work




Agarwal, Arge, and Yi [2005] implemented

(1 billion triangles on desktop machine.)

Blandford, Blelloch, and Kadow [2006]implemented a 3D parallel triangulator.(10 billion tetrahedra on 64 processors.)


an external memory 2D triangulator basedon randomized divide−and−conquer.

Streaming Delaunay

Streaming Computation

A restricted version of out−of−core computation.

Algorithm makes one (or a few) pass(es)over input stream, and writes output stream.




Processes data in memory buffer muchsmaller than the stream sizes.




Processes data in memory buffer muchsmaller than the stream sizes.

Nothing is stored temporarily to disk.

Advantages of Streaming

Streaming tools can run concurrently ina pipeline.

Often much faster than other out−of−corealgorithms!

Our Accomplishments

9 billion triangles in 6.6 hours on a laptop.

Our Accomplishments


12 times faster than Agarwal−Arge−Yi.

Our Accomplishments


800 million tetrahedra in under 3 hours.

12 times faster than Agarwal−Arge−Yi.

Algorithm Choice

Theconstructing Delaunay triangulations.

incremental insertion algorithm for

Algorithm Choice



We have little control over the order of pointsin the input stream.

Why?

Algorithm Choice



We have little control over the order of pointsin the input stream.

We modified existing, sequential Delaunaytriangulation codes (2D & 3D) for streaming.

Why?

Insert one vertex at a time.

Bowyer−Watson Algorithm

Insert one vertex at a time.Remove all triangles/tetrahedra that are no longer Delaunay.


Insert one vertex at a time.Remove all triangles/tetrahedra that are no longer Delaunay.Retriangulate the cavity with a fan around the new vertex.


Spatial Finalization


finalization regions.Subdivide space into


finalization regions.

spatial finalization tagsthat indicate ‘‘there are nomore points in this region.’’

Point stream

Finalized point stream

Inject into the stream

Subdivide space into

Spatial Finalization: Why?

Triangles whosecircumscribingcircles lie entirelyin finalized spacecan be written todisk immediately.

Regions not yet finalized

Finalized space

Triangles whosecircumscribing


Finalized space

circles intersectan unfinalizedregion remainin memory.

Streaming Delaunay Pipeline

TriangulatorFinalizer

The Finalizer

How to Finalize

Computebounding box.

1

How to Finalize

Computebounding box.Finalization grid.

1

2

1

1 1

2 2

3 4

How to Finalize

Computebounding box.Finalization grid.Count points.

1

2

4

1 1

11

1

3

4 2

4

5

5

5

How to Finalize


1

2

1111

4 1

2 1246

5

5

5

6

7

1

6

6 7

5

How to Finalize


1

2

2

2 11

3

4

25

2

21

54

32

3

5

63

77

78

9

7

6

1 6

6 6

7

5

How to Finalize


1

2

2

1454

5

7

4 3 4 2 3

5

5

9

4 6 6 7 6

2776

63

77

78

9

7

6

1 6

6 6

7

5

7 4

How to Finalize


1

2

5

761

6

7

6

8

6

7 3 6

2776779

4 6 6 6 7 6 5

8

46

67

66

58

87

9

5

5

7 4

How to Finalize


1

2

9

6

75

761

666

7 8

9

4 6

6

7

5

2

67

7

66

8 5 7 6 8

466875

67

37

7

7 4

How to Finalize

Computebounding box.Finalization grid.Count points.Store ‘‘sprinklepoints.’’

1

2

466687

86

5

2

5

7

4

6

7

7589

6

6 7664

7

73

1

79

77

6

7

6 6

7

5 7

6

6

How to Finalize


Output finalizedpoints.

Count points.Store ‘‘sprinklepoints.’’

Release chunks.Inject finalizationtags.

1

2

3

466687

86

5

2

5

7

4

6

7

7589

6

6 7664

7

73

1

79

776

5 5 6

5

2 3

4

6

How to Finalize





1

2

3

5 2

466687

86

5

2

5

7

4

6

7

7589

6

6 7664

8 6 7

736

5

44

42

1 1 2

5

How to Finalize





1

2

3

5

761

66

7

2

466687

86

5

2

5

7

46

6

7

6

7589

553

6

1

634

3

1

2

6

4

How to Finalize





1

2

3

5

761

666

7 8 7

79 7

3 6

2

4

1

66687

4

86545

5

7

5

24

23

5

5

1

232

How to Finalize





1

2

3

9

5

761

666

7 8 7

79

4 6

7

3 6 8 3

7 2

67

7

6

5

6

2

434253

2 7

3

211

1

How to Finalize





1

2

3

9

6

5

761

666

7 8 7

79

4 6

7

3 6 8 3

7

5

2

67

7

66

8 5 7 6 8

466875

6

2

How to Finalize





1

2

3

9

6

75

761

666

7 8 7

79

4 6

7

3 6 8 3

7

5

2

67

7

66

8 5 7 6 8

466875

6

How to Finalize





1

2

3

pointsofchunk

pointsofchunk

Streaming Point Format# npoints 22514# bb_min_f 0 0 192.25

# datatype SP_FLOATv 0 3 193.1v 0.4375 3.5 193.11v 0.59375 4 193.09v 0.6875 4.5 193.11v 0.90625 2.5 193.13v 0.90625 4.5 193.07x cell 2732v 0 7.5 192.91v 0.21875 9 192.93v 0.53125 9 192.99v 0.8125 7.5 192.96v 0.9375 9 193.04v 1.03125 9.5 193.06v 1.65625 8.5 193.04x cell 2740v 0.21875 0 193.06

# bb_max_f 122.5 158.5 215.7

finalization tag

finalization tag

Spatial CoherenceThe correlation between the proximity of points inspace and their proximity in the stream.

Danger: Potential Quadratic Time

Point Reordering by Finalizer

Chunking:all points arrive, then releasing them together.

Buffering points in a region until

Points in a chunk are randomly reordered.

Point Reordering by Finalizer

Chunking:all points arrive, then releasing them together.

Buffering points in a region until

Points in a chunk are randomly reordered.

Sprinkling: Promoting a small random sampleof ‘‘sprinkle points’’ to earlier in the stream.

Circumvents danger of quadratic behavior.

The Triangulator

Streaming Delaunay Triangulation

When point arrives:

triangle.Locate enclosing

triangulation.Update


Finalized space


When point arrives:



When tag arrives:

triangles.Identify final

Write them out &free their memory.



When point arrives:


Start at newesttriangle.


When point arrives:


Start at newesttriangle.Walk to point.


When point arrives:


Start at newesttriangle.Walk to point.If that fails,restart fromcell triangle.If that fails,do exhaustivesearch.


When point arrives:




When tag arrives:


Only these triangles

New triangles(since last tag).

need to be checked:

for this tag.Triangles waiting


When tag arrives:


Checked trianglesthat aren’t final areassigned a tag towait for.

Results

Results

Us: 12 min finalize + 36 min triang. = 48 min.

Agarwal−Arge−Yi:3 hr sort + 7.5 hr triang. = 10.5 hr.

Results

107 min finalize + 278 min triang. = 6 hr 25 min.425 MB finalize + 11 MB triang. = 436 MB.

tetrahedra

& 795 MB.in 2:41 hrs

points.)finalized(Pre−

815 million

3D Streaming Delaunay

for surface scans.Circumspheresare too big.


Does not work

Delaunay triangulation of a random sample.


Future work: spherical finalization regions from

Conclusions

sorting doesn’t help.

Conclusions

Most real−world points sets have lots ofspatial coherence


Conclusions


Like aikido: Use your opponent’s spatialcoherence against him.


Conclusions



Not an external memory algorithm!No temporary storage to disk.


Conclusions




12 times faster than best previous 2D approach.


Conclusions




Streaming triangulation can be piped directlyto another streaming application.

12 times faster than best previous 2D approach.

Streaming Digital Elevation Maps

Streaming Contour Extraction

Thanks

Kevin Yi at Duke supplied the Neuse RiverBasin data.

Martin Isenburg & Yuanxin ‘‘Leo’’ Liu did mostof the programming.

streaming construction of delaunay triangulationsjrs/papers/dtstalk.pdf · 2007-09-04 · streaming...

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