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Walking in random Delaunay triangulation Nicolas Broutin Olivier - - PowerPoint PPT Presentation
Walking in random Delaunay triangulation Nicolas Broutin Olivier - - PowerPoint PPT Presentation
Walking in random Delaunay triangulation Nicolas Broutin Olivier Devillers Ross Hemsley 1 The problem random points 2 The problem Delaunay triangulation 2 Visibility walk The problem 2 Visibility walk The problem 2 Visibility walk
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Delaunay triangulation
The problem
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Visibility walk The problem
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Visibility walk The problem
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Visibility walk The problem
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Visibility walk The problem
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Visibility walk The problem
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Straight walk The problem
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Straight walk The problem
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Vertex walk The problem
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The problem How many visited triangles ?
Average on point distribution Worst case on walk choices Worst case on start and query
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The problem Cone walk Results
P
- |walk| ≥ cst(|length|√n + log6 n)
- ≤ 1
n
|stretch factor| ≤ 3.7
Th.
|worst walk| = O(√n)
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Visibility walk The problem Results
P
- |walk| ≥ cst(|length|√n + log3 n)
- |worst walk| = O(√n)
≤ e−cst·log
3 2 n
Th. Cor.
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The difficulty
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The difficulty
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The difficulty
At a given step in the walk already some knowledge of unexplored part
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The difficulty
At a given step in the walk already some knowledge of unexplored part Delicate dependencies to manage
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Straight walk
[Devroye Lemaire Moreau, Bose Devroye]
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Straight walk
[Devroye Lemaire Moreau, Bose Devroye]
Is edge p1p2 Delaunay and part of the walk ?
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Straight walk
[Devroye Lemaire Moreau, Bose Devroye]
Is edge p1p2 Delaunay and part of the walk ? p1 p2 Half-moon graph give an upper bound
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Straight walk
[Devroye Lemaire Moreau, Bose Devroye]
Is edge p1p2 Delaunay and part of the walk ? p1 p2 Half-moon graph give an upper bound Does not depend too much on other points
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Straight walk
[Devroye Lemaire Moreau, Bose Devroye]
Is edge p1p2 Delaunay and part of the walk ? p1 p2 Half-moon graph give an upper bound Does not depend too much on other points |worst walk| = O(√n)
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Cone walk
[Broutin, Devillers, Hemsley. AofA’14]
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Vertex path in Delaunay
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z q z′ y
If q far enough, cones do not overlap
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z q z′ y
If q far enough, cones do not overlap One step is independant from previous ones
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q y y′ z
Knowledge of previous step may influence ♯ points in disk
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q y y′ z
Knowledge of previous step may influence ♯ points in disk But it can only goes down
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Cone walk
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Cone walk A lot of technical probability
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Cone walk
♯ substeps in a step is expected constant
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Cone walk
♯ steps is proportional to length
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Cone walk
♯ neighbors is ok
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Cone walk
dealing with boundary conditions
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Cone walk Results
P
- |walk| ≥ cst(|length|√n + log6 n)
- ≤ 1
n
|stretch factor| ≤ 3.7
Th.
|worst walk| = O(√n)
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Visibility walk
[Devillers, Hemsley]
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Results
P
- |walk| ≥ cst(|length|√n + log3 n)
- |worst walk| = O(√n)
≤ e−cst·log
3 2 n
Th. Cor.
Visibility walk
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First trick: progress measure by power
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First trick: progress measure by power
Change in circle power
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First trick: progress measure by power
d α ℓ Change in circle power
= 2dℓ sin α
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First trick: progress measure by power
d α ℓ Change in circle power
= 2dℓ sin α
If d and α are not small then there is measurable progress
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First trick: progress measure by power
d α ℓ Change in circle power
= 2dℓ sin α
If d and α are not small then there is measurable progress call this a good edge
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Second trick: make boxes
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Second trick: make boxes
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Second trick: make boxes
definition of good edge Choose grid size
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Second trick: make boxes
definition of good edge Choose grid size P(∃ a bad edge in cell 0) is small
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Second trick: make boxes Few bad cells
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Second trick: make boxes Few bad cells
not independant between neighboring cells
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Third trick: color boxes
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Third trick: color boxes
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Third trick: color boxes
Being bad for cells of same color is independant with very high probability
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Percolation
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Percolation
Walk Bad cells
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Percolation
Walk Bad cells
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Walk Bad cells
look at the worst color
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Walk
Percolation
Bad cells
percolation ⇒ there is a linear number on good cells look at the worst color
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Is there a long walk in Delaunay triangulation ?
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Is there a long walk in Delaunay triangulation ? Long walk in Delaunay triangulation → long walk in lattice → many bad cells in lattice → below percolation threshold
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Is there a long walk in Delaunay triangulation ? Long walk in Delaunay triangulation → long walk in lattice → many bad cells in lattice → below percolation threshold Impossible by good choice of parameters
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Is there a long walk in Delaunay triangulation ? Long walk in Delaunay triangulation → long walk in lattice → many bad cells in lattice → below percolation threshold Impossible by good choice of parameters Pretty bad constants in O
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