Walking in Poisson Delaunay triangulations

Transcription

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

2 Walking in Delaunay triangulations Straight walk 2-1

3 Walking in Delaunay triangulations Straight walk 2-2

4 Walking in Delaunay triangulations Straight walk 2-3

5 Walking in Delaunay triangulations Straight walk 2-4

6 Walking in Delaunay triangulations Straight walk 2-5

7 Walking in Delaunay triangulations Straight walk 2-6

8 2-7 Walking in Delaunay triangulations Straight walk Exit edge? One orientation predicate

9 2-8 Walking in Delaunay triangulations Straight walk End of walk? A second orientation predicate

10 2-9 Walking in Delaunay triangulations Straight walk Two orientation predicates per edge

11 Walking in Delaunay triangulations Visibility walk 3-1

12 Walking in Delaunay triangulations Visibility walk 3-2

13 Walking in Delaunay triangulations Visibility walk 3-3

14 Walking in Delaunay triangulations Visibility walk 3-4

15 Walking in Delaunay triangulations Visibility walk 3-5

16 Walking in Delaunay triangulations Visibility walk 3-6

17 3-7 Walking in Delaunay triangulations Visibility walk Triangle with two exits One orientation predicate

18 Walking in Delaunay triangulations Visibility walk Triangle with one exit 1.5 orientation predicate One predicate if this neighbor tried first 3-8 Two predicates if this neighbor tried first

19 3-9 Walking in Delaunay triangulations Visibility walk 1.25 orientation predicate per edge?

20 4-1 Walking in Delaunay triangulations How many edges crossed? Straight walk Visibility walk

21 Walking in Delaunay triangulations How many edges crossed? Straight walk 2n Visibility walk 4-2 Worst case in a triangulation (non Delaunay)

22 Walking in Delaunay triangulations How many edges crossed? Straight walk 2n Visibility walk 4-3 Worst case in a triangulation (non Delaunay)

23 4-4

24 4-5 May loop

25 Walking in Delaunay triangulations How many edges crossed? Straight walk 2n Visibility walk 2 3 n randomized 4-6 Worst case in a triangulation (non Delaunay)

26 random choice p p 4-7 [D., Pion, & Teillaud 2002]

27 random choice p p 4-8 [D., Pion, & Teillaud 2002]

28 Walking in Delaunay triangulations How many edges crossed? Straight walk 2n Visibility walk 2n 4-9 Worst case in a Delaunay triangulation

29 4-10 May loop Not Delaunay

30 4-11 Green power < Red power Power decreases

31 Walking in Delaunay triangulations How many edges crossed? Straight walk 2n Visibility walk 2n O( n) random [Devroye, Lemaire, & Moreau, 2004] 64 3π n + O( n 1 ) 2.16 n 2 O( n) uniform in domain Stretch in infinite Poisson distribution [D. & Hemsley, 2016] Stretch in infinite Poisson distribution 4-12 Worst case in a Delaunay triangulation

32 5-1 Walking in Delaunay triangulations Walk between vertices s Walk between vertices t

33 5-2 Walking in Delaunay triangulations Walk between vertices s t

34 5-3 Walking in Delaunay triangulations Walk between vertices s t Shortest path

35 5-4 Walking in Delaunay triangulations Walk between vertices s t Upper path

36 5-5 Walking in Delaunay triangulations Walk between vertices s t Compass walk

37 5-6 Walking in Delaunay triangulations Walk between vertices s t Voronoi path

38 5-7 Walking in Delaunay triangulations Walk between vertices s t Voronoi path with shortcuts

39 5-8 Walking in Delaunay triangulations Walk between vertices s t Shortest path Upper path Compass walk Voronoi path with shortcuts

40 Walking in Delaunay triangulations Walk between vertices, worst case Shortest path 6-1

41 Walking in Delaunay triangulations Walk between vertices, worst case Shortest path Search this subgraph 6-2

42 Walking in Delaunay triangulations Walk between vertices, worst case Shortest path Search this subgraph 6-3

43 Walking in Delaunay triangulations Walk between vertices, worst case Shortest path Search this subgraph 6-4

44 6-5 Walking in Delaunay triangulations Walk between vertices, worst case Shortest path Search this subgraph Upper bound Dobkin, Friedman, and Supowit 1987 Keil and Gutwin 1989 Xia π π 3 cos(π/6)

45 6-6 Walking in Delaunay triangulations Walk between vertices, worst case Shortest path Search this subgraph Lower bound Chew 1989 Bose, Devroye, Löffler, Snoeyink, & Verma 2011 Xia & Zhang π

46 Walking in Delaunay triangulations Walk between vertices, worst case Voronoi path Unbounded 7-1

47 Walking in Delaunay triangulations Walk between vertices, worst case Voronoi path Unbounded 7-2

48 Walking in Delaunay triangulations Walk between vertices, worst case Voronoi path Unbounded 7-3

49 Walking in Delaunay triangulations Walk between vertices, worst case Upper path Unbounded 8-1

50 Walking in Delaunay triangulations Walk between vertices, worst case Upper path Unbounded 8-2

51 Walking in Delaunay triangulations Walk between vertices, worst case Upper path Unbounded 8-3

52 Walking in Delaunay triangulations Walk between vertices, worst case Compass walk Unbounded [Bose & Morin 2004] 9-1

53 Walking in Delaunay triangulations Walk between vertices, worst case Compass walk Unbounded α [Bose & Morin 2004] α 9-2

54 Walking in Delaunay triangulations Walk between vertices, worst case Compass walk Unbounded α ɛ [Bose & Morin 2004] α 2ɛ 9-3

55 Walking in Delaunay triangulations Walk between vertices, worst case Compass walk Unbounded α ɛ [Bose & Morin 2004] α 2ɛ 9-4

56 Walking in Delaunay triangulations Walk between vertices Shortest path Upper path shortcuts Compass walk Voronoi path 10

57 Walking in Delaunay triangulations Expected length (experiments) Euclidean length 1 Shortest path 1.04 Compass walk 1.07 Shortened V. path 1.16 Upper path 1.18 Voronoi path

58 11-2 Walking in Delaunay triangulations Expected length (experiments) Euclidean length 1 Shortest path 1.04 Compass walk 1.07 Shortened V. path 1.16 Upper path 1.18 Voronoi path 1.27 theory [Chenavier & D.,2018] numerical integration π π [D. & Noizet, 2018] [Chenavier & D.,2018] [Baccelli et al., 2000]

59 11-3 Walking in Delaunay triangulations Expected length (experiments) Euclidean length 1 Shortest path 1.04 Compass walk 1.07 Shortened V. path 1.16 Upper path 1.18 Voronoi path 1.27 theory [Chenavier & D.,2018] numerical integration π π [D. & Noizet, 2018] [Chenavier & D.,2018] [Baccelli et al., 2000]

60 12-1 Expected length of upper path Poisson Delaunay triangulation, rate n E [length] = E triangle X 3 n 1 [triangle is Delaunay] 1 [first edge above length(first edge) st]

61 12-2 Expected length of upper path s Poisson Delaunay triangulation, rate n E [length] = E triangle X 3 n 1 [triangle is Delaunay] 1 [first edge above length(first edge) st] t = n (R 3 P [triangle is Delaunay] 1 [first edge above length(first edge)dtriangle 2 ) 3 st] Slivnyak-Mecke

62 12-3 Expected αlength of upper path 2 s Poisson Delaunay triangulation, rate n E [length] = E triangle X 3 n α 3 z r α 1 1 [triangle is Delaunay] 1 [first edge above length(first edge) st] t = n (R 3 P [triangle is Delaunay] 1 [first edge above length(first edge)dtriangle 2 ) 3 st] = n 3 1 r r=0 x z =0 y z = r [0,2π] 3 e nπr2 1 [first edge above st] 2r sin α 1 α 2 r 3 2A(triangle)dα 1:3 dy z dx z dr 2 Blaschke-Petkantschin

63 12-4 Expected length of upper path Poisson Delaunay triangulation, rate n E [length] = E triangle X 3 n 1 [triangle is Delaunay] 1 [first edge above length(first edge) st] = n (R 3 P [triangle is Delaunay] 1 [first edge above length(first edge)dtriangle 2 ) 3 st] = n 3 r=0 1 x z =0 r y z = r [0,2π] 3 e nπr2 1 [first edge above st] 2r sin α 1 α 2 r 3 2A(triangle)dα 1:3 dy z dx z dr 2 ( ) ( 1 ) = 4n 3 e nπr2 r 5 dr 1 r=0 h= y z [first edge above st] sin α 1 α 2 A(triangle)dα 1:3 dh r = 1 [0,2π] 2 3

64 Expected length of upper path s Poisson Delaunay triangulation, rate n E [length] = E α 2 triangle X 3 n α 1 arcsin h α 3 1 [triangle is Delaunay] 1 [first edge above length(first edge) st] 1 h t = n (R 3 P [triangle is Delaunay] 1 [first edge above length(first edge)dtriangle 2 ) 3 st] = n 3 = 4n r=0 1 x z =0 r y z = r 1 π 3 n 35π 3 12 = 35 3π 2 [0,2π] 3 e nπr2 1 [first edge above st] 2r sin α 1 α 2 r 3 2A(triangle)dα 1:3 dy z dx z dr 2 ( ) ( 1 ) = 4n 3 e nπr2 r 5 dr 1 r=0 h= y z [first edge above st] sin α 1 α 2 A(triangle)dα 1:3 dh r = 1 [0,2π] 2 3

65 13 Walking in Delaunay triangulations Expected length (experiments) Euclidean length 1 Shortest path 1.04 Compass walk 1.07 Shortened V. path 1.16 Upper path 1.18 Voronoi path 1.27 theory [Chenavier & D.,2018] numerical integration π π [D. & Noizet, 2018] [Chenavier & D.,2018] [Baccelli et al., 2000]

66 14-1 Walking in Delaunay triangulations Shortest path E [length(shortest path)]

67 14-2 Walking in Delaunay triangulations Shortest path E [length(shortest path)] density limit

68 14-3 Walking in Delaunay triangulations Shortest path E [length(shortest path)] density limit by subadditivity remove start and target from point set look at shortest path between closest neighbor between path from start to target negligible

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

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

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

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

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

74 17-1 A good cell? No Delaunay circles go outside

75 17-2 A good cell? No Delaunay circles go outside No short path from left to right

76 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

77 A good cell? No Delaunay circles go outside No short path from left to right Not enough edges to make a short path 17-4 Choose points in cell, 153 Choose what short path means P [ length ] ( 10 1 O n )

78 Walking in Delaunay triangulations Expected length (experiments) theory 18 Euclidean length 1 Shortest path Compass walk Shortened V. path Upper path Voronoi path in higher dimension numerical integration π π [de Castro & D., 2018] [Chenavier & D.,2018] [D. & Noizet, 2018] [Chenavier & D.,2018] [Baccelli et al., 2000]

79 19-1 Walking in Delaunay triangulations Voronoi path π 1.27 [Baccelli et al., 2000] s t

80 19-2 Walking in Delaunay triangulations Voronoi path π 1.27 Voronoi path in higher dimension [Baccelli et al., 2000]

81 19-3 Walking in Delaunay triangulations Voronoi path π 1.27 [Baccelli et al., 2000] E [l(v P X )] = (S d 1 ) 2 P [B((x,0,...,0), r) X = ] 1 [(x,0,...,0) [st]] r u 1 u 2 det(j Φ ) dα 1,1... dα 1,d 1 dα 2,1... dα 2,d 1 drdx Integral form

82 19-4 Walking in Delaunay triangulations Voronoi path π 1.27 [Baccelli et al., 2000] E [l(v P X )] = (S d 1 ) 2 P [B((x,0,...,0), r) X = ] 1 [(x,0,...,0) [st]] r u 1 u 2 det(j Φ ) dα 1,1... dα 1,d 1 dα 2,1... dα 2,d 1 drdx Integral form Use Taylor expansion to be able to integrate

83 19-5 Walking in Delaunay triangulations Voronoi path π 1.27 [Baccelli et al., 2000] Γ ( d 2 ) 4 2 4d 5 d π 2 (2d 2)! ( 1 d 1 ) 2 E [l(v PX 4d 2 )] Γ ( ) d d 5 d 1 π 2 (2d 2)! ( ) 2 4d 2

84 19-6 Walking in Delaunay triangulations Voronoi path π 1.27 [Baccelli et al., 2000] asymptotic behavior 2d π between 1 4 2dπ + O(d 3 2 ) dπ + O(d 3 2 )

85 19-7 Walking in Delaunay triangulations Voronoi path π 1.27 [Baccelli et al., 2000] asymptotic behavior 2d π

86 Walking in Delaunay triangulations Voronoi path π 1.27 [Baccelli et al., 2000] d k lower bound correct upper bound value π π π π π π numerical integration

87 20 Walking in Delaunay triangulations Expected length (experiments) theory Euclidean length 1 Shortest path Compass walk Shortened V. path 1.16 Upper path Voronoi path [Ongoing work] 1.16 [Chenavier & D.,2018] [D. & Noizet, 2018] numerical integration π π [Chenavier & D.,2018] [Baccelli et al., 2000]

88 21-1 Walking in Delaunay triangulations Compass walk Complicated dependencies Stretch factor

89 21-2 Walking in Delaunay triangulations Compass walk Complicated dependencies edge lengths Stretch factor = l 1 + l l k x 1 + x x k length of horizontal projections

90 21-3 Walking in Delaunay triangulations Compass walk Complicated dependencies Stretch factor = l 1 + l l k x 1 + x x k If independent: E [l] E [x] = 175π

91 Walking in Delaunay triangulations Compass walk Complicated dependencies Stretch factor = l 1 + l l k x 1 + x x k If independent: Experimental: E [l] E [x] = 175π There is a positive bias

92 Walking in Delaunay triangulations Compass walk Experimental: Complicated dependencies E [l] E [x] Stretch factor = l 1 + l l k x 1 + x x k If independent: E [l] E [x] = 175π Experimental: There is a positive bias 21-5

93 Walking in Delaunay triangulations Compass walk Experimental: Complicated dependencies at depth 1 Stretch factor = l 1 + l l k E [l] at depth x x x k E [l] E [x] E [l] E [x] E [x] If independent: Experimental: E [l] E [x] = 175π There is a positive bias

94 22-1 Walking in Delaunay triangulations Expected length (experiments) theory Euclidean length 1 Shortest path Compass walk Shortened V. path 1.16 Upper path Voronoi path [Chenavier & D.,2018] [D. & Noizet, 2018] π π [Chenavier & D.,2018] [Baccelli et al., 2000]

95 22-2 Walking in Delaunay triangulations Expected length (experiments) theory Euclidean length 1 Shortest path Compass walk Shortened V. path Upper path Voronoi path Locally defined path [Chenavier & D.,2018] [D. & Noizet, 2018] π 2 4 easy to analyze π 1.27 [Chenavier & D.,2018] (computation may be difficult) [Baccelli et al., 2000]

96 22-3 Walking in Delaunay triangulations Expected length (experiments) theory Euclidean length 1 Shortest path Compass walk Shortened V. path Upper path Voronoi path [Chenavier & D.,2018] Incrementally defined path Locally dependency defined path issues [D. & Noizet, 2018] π 2 4 the easy tree to cananalyze be analyzed π 1.27 [Chenavier & D.,2018] (computation may be difficult) [Baccelli et al., 2000]

97 22-4 Walking in Delaunay triangulations Expected length (experiments) theory Euclidean length 1 Shortest path Compass walk Shortened V. path Upper path Voronoi path [Chenavier & D.,2018] Incrementally defined path Incrementally defined path Locally also dependency defined path issues [D. & Noizet, 2018] π 2 4 the easy tree to cananalyze no idea about tight be bounds analyzed π 1.27 [Chenavier & D.,2018] (computation may be difficult) [Baccelli et al., 2000]

98 23 Thank you