Walking in Poisson Delaunay triangulations Olivier Devillers [D. - - PowerPoint PPT Presentation

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Walking in Poisson Delaunay triangulations Olivier Devillers [D. - - PowerPoint PPT Presentation

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Walking in Poisson Delaunay triangulations Olivier Devillers [D. & Hemsley, 2016] [Chenavier & D.,2018] [de Castro & D., 2018] [D. & Noizet, 2018] 1 Walking in Delaunay triangulations Straight walk 2 - 1 Walking in Delaunay

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SLIDE 1

1 Olivier Devillers

Walking in Poisson Delaunay triangulations

[D. & Hemsley, 2016] [Chenavier & D.,2018] [de Castro & D., 2018] [D. & Noizet, 2018]

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SLIDE 2

2 - 1 Straight walk

Walking in Delaunay triangulations

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2 - 2 Straight walk

Walking in Delaunay triangulations

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2 - 3 Straight walk

Walking in Delaunay triangulations

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2 - 4 Straight walk

Walking in Delaunay triangulations

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SLIDE 6

2 - 5 Straight walk

Walking in Delaunay triangulations

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SLIDE 7

2 - 6 Straight walk

Walking in Delaunay triangulations

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SLIDE 8

2 - 7 Straight walk Exit edge ? One orientation predicate

Walking in Delaunay triangulations

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SLIDE 9

2 - 8 Straight walk End of walk ? A second orientation predicate

Walking in Delaunay triangulations

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SLIDE 10

2 - 9 Straight walk Two orientation predicates per edge

Walking in Delaunay triangulations

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SLIDE 11

3 - 1 Visibility walk

Walking in Delaunay triangulations

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3 - 2 Visibility walk

Walking in Delaunay triangulations

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3 - 3 Visibility walk

Walking in Delaunay triangulations

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3 - 4 Visibility walk

Walking in Delaunay triangulations

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3 - 5 Visibility walk

Walking in Delaunay triangulations

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3 - 6 Visibility walk

Walking in Delaunay triangulations

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3 - 7 Visibility walk Triangle with two exits One orientation predicate

Walking in Delaunay triangulations

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3 - 8 Visibility walk Triangle with one exit 1.5 orientation predicate

One predicate Two predicates if this neighbor tried first if this neighbor tried first

Walking in Delaunay triangulations

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3 - 9 Visibility walk 1.25 orientation predicate per edge ?

Walking in Delaunay triangulations

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4 - 1

Walking in Delaunay triangulations

Straight walk Visibility walk How many edges crossed ?

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SLIDE 21

4 - 2

Walking in Delaunay triangulations

Straight walk Visibility walk How many edges crossed ? Worst case in a triangulation (non Delaunay) 2n

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SLIDE 22

4 - 3

Walking in Delaunay triangulations

Straight walk Visibility walk How many edges crossed ? Worst case in a triangulation (non Delaunay) 2n ∞

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SLIDE 23

4 - 4

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SLIDE 24

4 - 5 May loop

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SLIDE 25

4 - 6

Walking in Delaunay triangulations

Straight walk Visibility walk How many edges crossed ? Worst case in a triangulation (non Delaunay) 2n ∞ ≥ 2

3

√n randomized

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SLIDE 26

4 - 7 p p random choice

[D., Pion, & Teillaud 2002]

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SLIDE 27

4 - 8 p p random choice

[D., Pion, & Teillaud 2002]

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SLIDE 28

4 - 9

Walking in Delaunay triangulations

Straight walk Visibility walk How many edges crossed ? Worst case in a Delaunay triangulation 2n 2n

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SLIDE 29

4 - 10 Not Delaunay May loop

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SLIDE 30

4 - 11 Green power Red power < Power decreases

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SLIDE 31

4 - 12

Walking in Delaunay triangulations

Straight walk Visibility walk How many edges crossed ? Worst case in a Delaunay triangulation 2n 2n O(√n) random

[Devroye, Lemaire, & Moreau, 2004]

64 3π2

√n + O( 1

√n) ≃ 2.16√n

O(√n)

uniform in domain Stretch in infinite Poisson distribution Stretch in infinite Poisson distribution

[D. & Hemsley, 2016]

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SLIDE 32

5 - 1 s t Walk between vertices

Walking in Delaunay triangulations

Walk between vertices

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SLIDE 33

5 - 2 s t Walk between vertices

Walking in Delaunay triangulations

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SLIDE 34

5 - 3

Shortest path

s t Walk between vertices

Walking in Delaunay triangulations

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SLIDE 35

5 - 4

Upper path

s t Walk between vertices

Walking in Delaunay triangulations

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SLIDE 36

5 - 5

Compass walk

s t Walk between vertices

Walking in Delaunay triangulations

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SLIDE 37

5 - 6

Voronoi path

s t Walk between vertices

Walking in Delaunay triangulations

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SLIDE 38

5 - 7

Voronoi path with shortcuts

s t Walk between vertices

Walking in Delaunay triangulations

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SLIDE 39

5 - 8

Shortest path Upper path Voronoi path with shortcuts Compass walk

s t Walk between vertices

Walking in Delaunay triangulations

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6 - 1 Walk between vertices, worst case

Walking in Delaunay triangulations

Shortest path

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SLIDE 41

6 - 2 Walk between vertices, worst case

Walking in Delaunay triangulations

Shortest path

Search this ”subgraph”

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SLIDE 42

6 - 3 Walk between vertices, worst case

Walking in Delaunay triangulations

Shortest path

Search this ”subgraph”

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SLIDE 43

6 - 4 Walk between vertices, worst case

Walking in Delaunay triangulations

Shortest path

Search this ”subgraph”

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SLIDE 44

6 - 5 Walk between vertices, worst case

Walking in Delaunay triangulations

Shortest path

Search this ”subgraph” Dobkin, Friedman, and Supowit 1987 Keil and Gutwin 1989 Xia 2011 5.08

1+ √ 5 2

π 2.42 1.998

2π 3 cos(π/6)

Upper bound

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SLIDE 45

6 - 6 Walk between vertices, worst case

Walking in Delaunay triangulations

Shortest path

Search this ”subgraph” Chew 1989

Bose, Devroye, L¨

  • ffler, Snoeyink, & Verma 2011

Xia & Zhang 2011 1.5708

π 2

1.5846 1.5932 Lower bound

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SLIDE 46

7 - 1 Walk between vertices, worst case

Walking in Delaunay triangulations

Voronoi path

Unbounded

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7 - 2 Walk between vertices, worst case

Walking in Delaunay triangulations

Voronoi path

Unbounded

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SLIDE 48

7 - 3 Walk between vertices, worst case

Walking in Delaunay triangulations

Voronoi path

Unbounded

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SLIDE 49

8 - 1 Walk between vertices, worst case

Walking in Delaunay triangulations

Upper path

Unbounded

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SLIDE 50

8 - 2 Walk between vertices, worst case

Walking in Delaunay triangulations

Upper path

Unbounded

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SLIDE 51

8 - 3 Walk between vertices, worst case

Walking in Delaunay triangulations

Upper path

Unbounded

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SLIDE 52

9 - 1 Walk between vertices, worst case

Walking in Delaunay triangulations

Compass walk

Unbounded

[Bose & Morin 2004]

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SLIDE 53

9 - 2 Walk between vertices, worst case

Walking in Delaunay triangulations

Compass walk

Unbounded

[Bose & Morin 2004]

α α

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SLIDE 54

9 - 3 Walk between vertices, worst case

Walking in Delaunay triangulations

Compass walk

Unbounded

[Bose & Morin 2004]

α α −ǫ −2ǫ

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SLIDE 55

9 - 4 Walk between vertices, worst case

Walking in Delaunay triangulations

Compass walk

Unbounded

[Bose & Morin 2004]

α α −ǫ −2ǫ

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SLIDE 56

10 Walk between vertices

Walking in Delaunay triangulations

Upper path Voronoi path shortcuts Shortest path Compass walk

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SLIDE 57

11 - 1

Shortest path Upper path Voronoi path Shortened V. path Compass walk

Walking in Delaunay triangulations

Expected length (experiments) Euclidean length 1 1.04 1.07 1.16 1.18 1.27

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SLIDE 58

11 - 2

Shortest path Upper path Voronoi path Shortened V. path Compass walk

Walking in Delaunay triangulations

Expected length (experiments) Euclidean length 1 1.04 1.07 1.16 1.18 1.27 theory ≥ 1 + 10−11 1.16

numerical integration

35 3π2 ≃ 1.18 4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

[ C h e n a v i e r & D . , 2 1 8 ] [ D . & N

  • i

z e t , 2 1 8 ]

[ C h e n a v i e r & D . , 2 1 8 ]

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SLIDE 59

11 - 3

Shortest path Upper path Voronoi path Shortened V. path Compass walk

Walking in Delaunay triangulations

Expected length (experiments) Euclidean length 1 1.04 1.07 1.16 1.18 1.27 theory ≥ 1 + 10−11 1.16

numerical integration

35 3π2 ≃ 1.18 4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

[ C h e n a v i e r & D . , 2 1 8 ] [ D . & N

  • i

z e t , 2 1 8 ]

[ C h e n a v i e r & D . , 2 1 8 ]

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SLIDE 60

12 - 1

E [length] = E  

  • triangle∈X3

n

1[triangle is Delaunay]1[first edge above st]length(first edge)  

Expected length of upper path

Poisson Delaunay triangulation, rate n

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12 - 2

E [length] = E  

  • triangle∈X3

n

1[triangle is Delaunay]1[first edge above st]length(first edge)  

Expected length of upper path

Poisson Delaunay triangulation, rate n

= n3

  • (R2)3 P [triangle is Delaunay] 1[first edge above st]length(first edge)dtriangle

Slivnyak-Mecke

s t

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SLIDE 62

12 - 3

E [length] = E  

  • triangle∈X3

n

1[triangle is Delaunay]1[first edge above st]length(first edge)  

Expected length of upper path

Poisson Delaunay triangulation, rate n

= n3

  • (R2)3 P [triangle is Delaunay] 1[first edge above st]length(first edge)dtriangle

= n3 ∞

r=0

1

xz=0

r

yz=−r

  • [0,2π]3 e−nπr21[first edge above st]2r sin α1 − α2

2 r32A(triangle)dα1:3dyzdxzdr

Blaschke-Petkantschin

s t α1 α2 α3 r z

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SLIDE 63

12 - 4

E [length] = E  

  • triangle∈X3

n

1[triangle is Delaunay]1[first edge above st]length(first edge)  

Expected length of upper path

Poisson Delaunay triangulation, rate n

= n3

  • (R2)3 P [triangle is Delaunay] 1[first edge above st]length(first edge)dtriangle

= n3 ∞

r=0

1

xz=0

r

yz=−r

  • [0,2π]3 e−nπr21[first edge above st]2r sin α1 − α2

2 r32A(triangle)dα1:3dyzdxzdr = 4n3 ∞

r=0

e−nπr2r5dr

  • ·

1

h= yz

r =−1

  • [0,2π]3 1[first edge above st] sin α1 − α2

2 A(triangle)dα1:3dh

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SLIDE 64

12 - 5

E [length] = E  

  • triangle∈X3

n

1[triangle is Delaunay]1[first edge above st]length(first edge)  

Expected length of upper path

Poisson Delaunay triangulation, rate n

= n3

  • (R2)3 P [triangle is Delaunay] 1[first edge above st]length(first edge)dtriangle

= n3 ∞

r=0

1

xz=0

r

yz=−r

  • [0,2π]3 e−nπr21[first edge above st]2r sin α1 − α2

2 r32A(triangle)dα1:3dyzdxzdr = 4n3 · 1 π3n3 · 35π 12 = 35 3π2 = 4n3 ∞

r=0

e−nπr2r5dr

  • ·

1

h= yz

r =−1

  • [0,2π]3 1[first edge above st] sin α1 − α2

2 A(triangle)dα1:3dh

  • s

t 1 h α1 α2 α3 arcsin h

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SLIDE 65

13

Shortest path Upper path Voronoi path Shortened V. path Compass walk

Walking in Delaunay triangulations

Expected length (experiments) Euclidean length 1 1.04 1.07 1.16 1.18 1.27 theory ≥ 1 + 10−11 1.16

numerical integration

35 3π2 ≃ 1.18 4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

[ D . & N

  • i

z e t , 2 1 8 ]

[ C h e n a v i e r & D . , 2 1 8 ]

[ C h e n a v i e r & D . , 2 1 8 ]

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SLIDE 66

14 - 1

Shortest path

Walking in Delaunay triangulations

E [length(shortest path)]

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SLIDE 67

14 - 2

Shortest path

Walking in Delaunay triangulations

E [length(shortest path)] limit density → ∞

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SLIDE 68

14 - 3

Shortest path

Walking in Delaunay triangulations

E [length(shortest path)] limit density → ∞ by subadditivity remove start and target from point set look at shortest path between closest neighbor = between path from start to target negligible

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15 Bad edge = almost horizontal edge P [bad] = small constant difficult dependencies to handle Many bad edges ⇐ length close to 1

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16 - 1 Bad cell = cell with enough bad edges Many bad cells ⇐ length close to 1 Still dependencies Make a grid

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16 - 2 Split cells in 4 families Many bad cells in 1 family ⇐ length close to 1 No more dependencies Make a grid

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16 - 3 Make a grid If E [♯ ∈ cell] = constant ♯ possible paths = 4n (too big)

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16 - 4 Make a grid If E [♯ ∈ cell] = constant ♯ possible paths = 4n (too big) big cells √n × √n 4

√n × n

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SLIDE 74

17 - 1 A good cell ?

No Delaunay circles go outside

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17 - 2 A good cell ?

No Delaunay circles go outside No short path from left to right

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17 - 3 A good cell ?

No Delaunay circles go outside No short path from left to right Not enough edges to make a short path

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SLIDE 77

17 - 4 A good cell ?

No Delaunay circles go outside No short path from left to right Not enough edges to make a short path Choose ♯ points in cell, Choose what ”short path” means

P

  • length ≥ 1 + 2.5 × 10−11

≤ O

  • 1

√n

  • 153

1 + 10−10

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SLIDE 78

18

Shortest path Upper path Shortened V. path Compass walk

Walking in Delaunay triangulations

Expected length (experiments) Euclidean length 1 1.04 1.07 1.16 1.18 1.27 theory ≥ 1 + 10−11 1.16

numerical integration

35 3π2 ≃ 1.18 4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

[ D . & N

  • i

z e t , 2 1 8 ]

[ C h e n a v i e r & D . , 2 1 8 ]

[ C h e n a v i e r & D . , 2 1 8 ]

Voronoi path in higher dimension

[de Castro & D., 2018]

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SLIDE 79

19 - 1

Voronoi path

Walking in Delaunay triangulations

1.27

4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

s t

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SLIDE 80

19 - 2

Voronoi path

Walking in Delaunay triangulations

1.27

4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

Voronoi path in higher dimension

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SLIDE 81

19 - 3

Voronoi path

Walking in Delaunay triangulations

1.27

4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

E [ℓ(V PX)] = 1

2

∞

−∞

∞

  • (Sd−1)2 P [B((x,0,...,0), r) ∩ X = ∅] 1[(x,0,...,0)∈[st]]

·ru1u2| det(JΦ)|dα1,1 . . . dα1,d−1dα2,1 . . . dα2,d−1drdx

Integral form

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SLIDE 82

19 - 4

Voronoi path

Walking in Delaunay triangulations

1.27

4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

E [ℓ(V PX)] = 1

2

∞

−∞

∞

  • (Sd−1)2 P [B((x,0,...,0), r) ∩ X = ∅] 1[(x,0,...,0)∈[st]]

·ru1u2| det(JΦ)|dα1,1 . . . dα1,d−1dα2,1 . . . dα2,d−1drdx

Integral form Use Taylor expansion to be able to integrate

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SLIDE 83

19 - 5

Voronoi path

Walking in Delaunay triangulations

1.27

4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

Γ d

2

4 24d−5d π2(2d − 2)!

  • 1 −

d − 1 4d2 − 1 √ 2 ≤ E [ℓ(V PX)] ≤ Γ d

2

4 24d−5d π2(2d − 2)!

  • 1 +

1 4d − 2 √ 2

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SLIDE 84

19 - 6

Voronoi path

Walking in Delaunay triangulations

1.27

4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

asymptotic behavior

  • 2d

π

− 1 4 √ 2dπ + O(d

3 2 )

+ 3 4 √ 2dπ + O(d

3 2 )

between

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SLIDE 85

19 - 7

Voronoi path

Walking in Delaunay triangulations

1.27

4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

asymptotic behavior

  • 2d

π

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SLIDE 86

19 - 8

Voronoi path

Walking in Delaunay triangulations

1.27

4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

d k lower bound correct upper bound ≃ value ≃ 3 41

788984278470257640690697143 745000536337515228912680960

√ 2 1.49770 1.500

4523370364712510658076963509 4264485828690604413776035840

√ 2 1.50007 4 7

102494570 8729721 √ 2 π2

1.6823 1.698

121774997 10270260 √ 2 π2

1.6990 5 3

135 104

√ 2 1.8357 1.875

21305 16016

√ 2 1.8812 6 1

3014656 225225 √ 2 π2

1.9179 2.04

753664 51975 √ 2 π2

2.0778 7 1

210 143

√ 2 2.0768 2.2

225 143

√ 2 2.2252 8 1

2080374784 134008875 √ 2 π2

2.2244 2.3

130023424 7882875 √ 2 π2

2.3635

numerical integration

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SLIDE 87

20

[ D . & N

  • i

z e t , 2 1 8 ]

Shortest path Upper path Voronoi path Shortened V. path Compass walk

Walking in Delaunay triangulations

Expected length (experiments) Euclidean length 1 1.04 1.07 1.16 1.18 1.27 theory ≥ 1 + 10−11 1.16

numerical integration

35 3π2 ≃ 1.18 4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

[ C h e n a v i e r & D . , 2 1 8 ]

[ C h e n a v i e r & D . , 2 1 8 ] [ O n g

  • i

n g w

  • r

k ]

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SLIDE 88

21 - 1

Compass walk

Walking in Delaunay triangulations

Complicated dependencies Stretch factor

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SLIDE 89

21 - 2

Compass walk

Walking in Delaunay triangulations

Complicated dependencies Stretch factor = ℓ1 + ℓ2 + . . . ℓk x1 + x2 + . . . xk edge lengths length of horizontal projections

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SLIDE 90

21 - 3

Compass walk

Walking in Delaunay triangulations

Complicated dependencies Stretch factor = ℓ1 + ℓ2 + . . . ℓk x1 + x2 + . . . xk If independent: E [ℓ] E [x] = 175π 512 ≃ 1.0738

slide-91
SLIDE 91

21 - 4

Compass walk

Walking in Delaunay triangulations

Complicated dependencies Stretch factor = ℓ1 + ℓ2 + . . . ℓk x1 + x2 + . . . xk If independent: E [ℓ] E [x] = 175π 512 ≃ 1.0738 Experimental: There is a positive bias 1.06777

slide-92
SLIDE 92

21 - 5

Compass walk

Walking in Delaunay triangulations

Complicated dependencies Stretch factor = ℓ1 + ℓ2 + . . . ℓk x1 + x2 + . . . xk If independent: E [ℓ] E [x] = 175π 512 ≃ 1.0738 Experimental: There is a positive bias 1.06777 Experimental:

E [ℓ] E [x] ≃ 1.061 0.988 ≃ 1.0738

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SLIDE 93

21 - 6

Compass walk

Walking in Delaunay triangulations

Complicated dependencies Stretch factor = ℓ1 + ℓ2 + . . . ℓk x1 + x2 + . . . xk If independent: E [ℓ] E [x] = 175π 512 ≃ 1.0738 Experimental: There is a positive bias 1.06777 Experimental:

E [ℓ] E [x] ≃ 1.061 0.988 ≃ 1.0738 E [ℓ] E [x] ≃ 1.116 1.045 ≃ 1.0681 E [ℓ] E [x] ≃ 1.117 1.046 ≃ 1.0678

at depth 1 at depth 2

slide-94
SLIDE 94

22 - 1

[ D . & N

  • i

z e t , 2 1 8 ]

Shortest path Upper path Voronoi path Shortened V. path Compass walk

Walking in Delaunay triangulations

Expected length (experiments) Euclidean length 1 1.04 1.07 1.16 1.18 1.27 theory ≥ 1 + 10−11 1.16

35 3π2 ≃ 1.18 4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

[ C h e n a v i e r & D . , 2 1 8 ]

[ C h e n a v i e r & D . , 2 1 8 ]

slide-95
SLIDE 95

22 - 2

[ D . & N

  • i

z e t , 2 1 8 ]

Shortest path Upper path Voronoi path Shortened V. path Compass walk

Walking in Delaunay triangulations

Expected length (experiments) Euclidean length 1 1.04 1.07 1.16 1.18 1.27 theory ≥ 1 + 10−11 1.16

35 3π2 ≃ 1.18 4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

[ C h e n a v i e r & D . , 2 1 8 ]

[ C h e n a v i e r & D . , 2 1 8 ]

Locally defined path ”easy” to analyze

(computation may be difficult)

slide-96
SLIDE 96

22 - 3

[ D . & N

  • i

z e t , 2 1 8 ]

Shortest path Upper path Voronoi path Shortened V. path Compass walk

Walking in Delaunay triangulations

Expected length (experiments) Euclidean length 1 1.04 1.07 1.16 1.18 1.27 theory ≥ 1 + 10−11 1.16

35 3π2 ≃ 1.18 4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

[ C h e n a v i e r & D . , 2 1 8 ]

[ C h e n a v i e r & D . , 2 1 8 ]

Locally defined path ”easy” to analyze

(computation may be difficult)

Incrementally defined path dependency issues the tree can be analyzed

slide-97
SLIDE 97

22 - 4

[ D . & N

  • i

z e t , 2 1 8 ]

Shortest path Upper path Voronoi path Shortened V. path Compass walk

Walking in Delaunay triangulations

Expected length (experiments) Euclidean length 1 1.04 1.07 1.16 1.18 1.27 theory ≥ 1 + 10−11 1.16

35 3π2 ≃ 1.18 4 π ≃ 1.27

[ B a c c e l l i e t a l . , 2 ]

[ C h e n a v i e r & D . , 2 1 8 ]

[ C h e n a v i e r & D . , 2 1 8 ]

Locally defined path ”easy” to analyze

(computation may be difficult)

Incrementally defined path dependency issues the tree can be analyzed Incrementally defined path also dependency issues no idea about tight bounds

slide-98
SLIDE 98

23 Thank you