Walking Algorithms for Point Location

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Walking Algorithms for Point Location

• Roman Soukal
• Ivana Kolingerová

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Point Location

• Fundamental problem in Computational Geometry
• Triangulations, Tetrahedronizations,
  Pseudotriangulations, Voronoi Diagrams etc.
• A lot of applications
• Geographic Information Systems (GIS)
• Computer Aided Design (CAD)
• Computer Graphics
• Motion Planning
• Etc.

Roman Soukal Walking Algorithms for Point
Location
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The most effective solutions

• Expected computation complexity O(log n) where n
  is a number of elements in a mesh
• Sophisticated structures are required
• DAG, skip list, quadtree, octree, random sampling
  
• Disadvantages
• Memory demands (usually O(n))
• Difficult implementation (especially
  implementation of modifications)
• Etc.
• Alternative solutions
• Walking location algorithms
• Computational complexity to
  De01

Roman Soukal Walking Algorithms for Point
Location
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Walking algorithms

• Location of an element (the element which
  contains the query point) using walking through
  neighboring elements
• No extra data structures it needs only
  information about connectivity
• Types of walking algorithms
• Visibility walk
• Straight walk
• Orthogonal walk

Figure 2 Point location in triangulation using
walking algorithms
Roman Soukal Walking Algorithms for Point
Location
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Performed research

• Triangulation
• Visibility walk
• Straight walk
• Orthogonal walk
• Tetrahedronisation
• Visibility walk
• Pseudotriangulation
• Special modification of Visibility walk Tr08
• Choice of the first elementMu99

Roman Soukal Walking Algorithms for Point
Location
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Visibility walk in E2

• Fixed orientation of ?
• Determinant test
• Simple implementation

Figure 3 Point location in triangulation using
Visibility walk algorithm
Roman Soukal Walking Algorithms for Point
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Remembering Stochastic walk in E2

• Stochastic otherwise only Delaunay
  Triangulation
• Remembering

Figure 5 Visibility walk may loop on a
non-Delaunay triangulation
Roman Soukal Walking Algorithms for Point
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Improvement of Remembering Stochastic walk by I.
Kolingerova

• Remembering Stochastic walk
• 1 or 2 edge tests per triangle
• Average 1,4
• Improvement
• 1 edge test per triangle (about 40 fewer edge
  tests)
• How to find the query triangle?

Roman Soukal Walking Algorithms for Point
Location
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Improvement of Remembering Stochastic walk by
Ko06

• Existed solutions for looping
• Ko06 Using standard Remembering Stochastic
  walk algorithm for final location
• Flag Needs modification of structure
• Solution
• Distance control after A iterations where A is a
  constant depended on N
• New possible solutions?

Roman Soukal Walking Algorithms for Point
Location
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Original Straight walk

• Two steps of algorithm
• Initialization step
• Straight walk step
• Determinant orientation test is used
• Goals
• Simplify initialization step
• Use faster tests than determinant orientation
  tests

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Straight walk example

• Initialization step - 1 Determinant orientation
  test
• Straight walk step 2 Determinant orientation
  tests

Figure 7 Original Straight walk
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Modified Straight walk

• The initialization step is simplified to constant
  number of operations
• Implicit line equation test is used

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Modified Straight walk Example

• Initialization step
• Implicit line equation test for pq
• Problem and solution

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Length of Remembering Stochastic Walk in Straight
walk
Table 1 The properties of final location with
Remembering Stochastic walk
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Solution

• Remembering Stochastic walk for final location
• 2 solutions
• Distance
• Normal
• Normal is faster

Figure 9 Normal Straight walk
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Orthogonal walk
Starting point of walking for ax X

• Best result
• Why?
• Practical application
• Dynamic Delaunay triangulation for kinetic data
• Terrain deformation for virtual reality
• Faster up to 15

Ax of walking in X
Ax of walking in Y
Last triangle of walking in X and starting
triangle of walking in Y
Path of Orthogonal walk
Query point
Path of Remembering Stochastic walk
Figure 10 Point location in triangulation using
Orthogonal walk
Roman Soukal Walking Algorithms for Point
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How to choose the first element?

• Mu99 We choose the closest element from the
  set
• The set contains some randomly selected elements
  from mesh
• How big must be this set?
• Depends on the speed of the algorithm
• The size of the set

Roman Soukal Walking Algorithms for Point
Location
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Is there a speedup on all types of datasets?

• Tests on DT made on different datasets
• Random generated points in square
• Clusters
• Grid
• Gauss
• Arc
• 100, 200, 500, 1000, 2000, 5000, 104, 2104,
  5104, 105, 2105, 5105, 106, 2106 points in
  dataset

Roman Soukal Walking Algorithms for Point
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Results

• Speedup up to 350-550

Table 2 Speedup with the optimal choice of the
first element (time of location of 10000
points) Intel Pentium D 3GHz 2x2MB L2 Cache, 3GB
RAM
Roman Soukal Walking Algorithms for Point
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Remembering Stochastic walk in E3

• Determinant test

Figure 12 Walking in tetrahedra mesh
Roman Soukal Walking Algorithms for Point
Location
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Conclusion

• E2
• The best is Orthogonal walk
• Expected computational complexity down to
• E3
• The best is Remembering Stochastic walk for E3
• Expected computational complexity down to
• Pseudotriangulations

Figure 13 Results of chosen algorithms
Roman Soukal Walking Algorithms for Point
Location
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Future work

• Tests of properties on non-Delaunay
  triangulations (regular, greedy, etc.)
• Tests of hierarchical structures

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The most important references

• De01 Devillers, O., Pion, S., Teillaud, M. ,
  2001. Walking in a Triangulation. International
  Journal of Foundations of Computer Science.
• Fl91 L. D. Floriani, B. Falcidieno, G. Nagy, C.
  Pienovi On sorting triangles in Delaunay
  tessellation, Algorithmica 6, 1991
• Mu99 Mücke, E. P., Saias, I., Zhu, B., 1999.
  Fast Randomized Point Location without
  Preprocessing in Two- and Three-dimensional
  Delaunay Triangulations. Computational Geometry
  Theory and Applications.
• Ko06 Kolingerová, I., 2006. A Small Improvement
  in the Walking Algorithm for Point Location in a
  Triangulation. 22nd European Workshop on
  Computational Geometry.
• Tr08 Trcka, J., Kolingerová, I., 2008. The
  Stochastic Walk Algorithm for Pseudotriangulations
  . Draft.

Roman Soukal Walking Algorithms for Point
Location
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Thank you for your attention
Questions?
Roman Soukal Walking Algorithms for Point
Location