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 The problem 2

Visibility walk The problem 2

Visibility walk The problem 2

Straight walk The problem 2

Straight walk The problem 2

Vertex walk The problem 2

The problem How many visited triangles ? Average on point distribution Worst case on walk choices Worst case on start and query 2

Cone walk The problem Results | walk | ≥ cst ( | length |√ n + log 6 n ) ≤ 1 � � Th. P n | worst walk | = O ( √ n ) | stretch factor | ≤ 3 . 7 2

Visibility walk The problem Results | walk | ≥ cst ( | length |√ n + log 3 n ) � � Th. P 3 2 n ≤ e − cst · log | worst walk | = O ( √ n ) Cor. 2

The difficulty 3

The difficulty 3

The difficulty At a given step in the walk already some knowledge of unexplored part 3

The difficulty At a given step in the walk already some knowledge of unexplored part Delicate dependencies to manage 3

Straight walk [Devroye Lemaire Moreau, Bose Devroye] 4

Straight walk [Devroye Lemaire Moreau, Bose Devroye] Is edge p 1 p 2 Delaunay and part of the walk ? 4

Straight walk [Devroye Lemaire Moreau, Bose Devroye] Is edge p 1 p 2 Delaunay and part of the walk ? Half-moon graph give an upper bound p 1 p 2 4

Straight walk [Devroye Lemaire Moreau, Bose Devroye] Is edge p 1 p 2 Delaunay and part of the walk ? Half-moon graph give an upper bound Does not depend too much on other points p 1 p 2 4

Straight walk [Devroye Lemaire Moreau, Bose Devroye] Is edge p 1 p 2 Delaunay and part of the walk ? Half-moon graph give an upper bound Does not depend too much on other points | worst walk | = O ( √ n ) p 1 p 2 4

Cone walk [Broutin, Devillers, Hemsley. AofA’14] 5

6

6

6

6

6

6

6

6

6

Vertex path in Delaunay 6

If q far enough, cones do not overlap y q z z ′ 7

If q far enough, cones do not overlap y q z z ′ One step is independant from previous ones 7

Knowledge of previous step may influence ♯ points in disk y z q y ′ 8

Knowledge of previous step may influence ♯ points in disk But it can only goes down y z q y ′ 8

Cone walk 9

Cone walk A lot of technical probability 9

Cone walk ♯ substeps in a step is expected constant 9

Cone walk ♯ steps is proportional to length 9

Cone walk ♯ neighbors is ok 9

Cone walk dealing with boundary conditions 9

Cone walk Results | walk | ≥ cst ( | length |√ n + log 6 n ) ≤ 1 � � Th. P n | worst walk | = O ( √ n ) | stretch factor | ≤ 3 . 7 10

Visibility walk [Devillers, Hemsley] 11

Visibility walk Results | walk | ≥ cst ( | length |√ n + log 3 n ) � � Th. P 3 2 n ≤ e − cst · log | worst walk | = O ( √ n ) Cor. 12

First trick: progress measure by power 13

First trick: progress measure by power Change in circle power 13

First trick: progress measure by power Change in circle power = 2 dℓ sin α d α ℓ 13

First trick: progress measure by power Change in circle power = 2 dℓ sin α If d and α are not small then there is measurable progress d α ℓ 13

First trick: progress measure by power Change in circle power = 2 dℓ sin α If d and α are not small then there is measurable progress d call this a good edge α ℓ 13

Second trick: make boxes 14

Second trick: make boxes 14

Second trick: make boxes Choose definition of good edge grid size 14

Second trick: make boxes Choose definition of good edge grid size P ( ∃ a bad edge in cell 0 ) is small 14

Second trick: make boxes Few bad cells 14

Second trick: make boxes not independant between neighboring cells Few bad cells 14

Third trick: color boxes 15

Third trick: color boxes 15

Third trick: color boxes Being bad for cells of same color is independant with very high probability 15

Percolation 16

Percolation Bad cells Walk 16

Percolation Bad cells Walk 16

Bad cells look at the worst color Walk 16

Percolation Bad cells look at the worst color percolation ⇒ there is a linear number on good cells Walk 16

Is there a long walk in Delaunay triangulation ? 17

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 17

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 17

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 17

Thank you 18