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

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 triangulations Straight walk 2 - 2

Walking in Delaunay triangulations Straight walk 2 - 3

Walking in Delaunay triangulations Straight walk 2 - 4

Walking in Delaunay triangulations Straight walk 2 - 5

Walking in Delaunay triangulations Straight walk 2 - 6

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

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

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

Walking in Delaunay triangulations Visibility walk 3 - 1

Walking in Delaunay triangulations Visibility walk 3 - 2

Walking in Delaunay triangulations Visibility walk 3 - 3

Walking in Delaunay triangulations Visibility walk 3 - 4

Walking in Delaunay triangulations Visibility walk 3 - 5

Walking in Delaunay triangulations Visibility walk 3 - 6

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

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

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

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

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

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

4 - 4

May loop 4 - 5

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

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

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

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

May loop Not Delaunay 4 - 10

Green power Red power < Power decreases 4 - 11

Walking in Delaunay triangulations How many edges crossed ? Straight walk O ( √ n ) 2 n [Devroye, Lemaire, & Moreau, 2004] uniform in domain √ n + O ( 1 √ n ) ≃ 2 . 16 √ n 64 3 π 2 Stretch in infinite Poisson distribution Visibility walk O ( √ n ) [D. & Hemsley, 2016] 2 n random Stretch in infinite Poisson distribution Worst case in a Delaunay triangulation 4 - 12

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

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

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

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

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

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

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

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

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

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

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

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

Walking in Delaunay triangulations Walk between vertices, worst case Shortest path Search this ”subgraph” Upper bound √ 1+ 5 5.08 π Dobkin, Friedman, and Supowit 1987 2 2 π Keil and Gutwin 1989 2.42 3 cos( π/ 6) Xia 2011 1.998 6 - 5

Walking in Delaunay triangulations Walk between vertices, worst case Shortest path Search this ”subgraph” Lower bound π 1.5708 Chew 1989 2 1.5846 Bose, Devroye, L¨ offler, Snoeyink, & Verma 2011 Xia & Zhang 2011 1.5932 6 - 6

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

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

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

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

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

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

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

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

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

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

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

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 11 - 1

Walking in Delaunay triangulations Expected length (experiments) theory ] 8 1 0 2 , . D & Euclidean length 1 r e i v a n e h C [ ≥ 1 + 10 − 11 Shortest path 1.04 ] 8 1 0 2 , t e Compass walk 1.07 z i o N & . D [ numerical integration Shortened V. path 1.16 1.16 35 Upper path 1.18 3 π 2 ≃ 1 . 18 8 ] 1 0 2 , . D & r e i v a n e h C [ 4 Voronoi path 1.27 π ≃ 1 . 27 ] 0 0 0 2 , . l a t e i l l e c c a B [ 11 - 2

Walking in Delaunay triangulations Expected length (experiments) theory ] 8 1 0 2 , . D & Euclidean length 1 r e i v a n e h C [ ≥ 1 + 10 − 11 Shortest path 1.04 ] 8 1 0 2 , t e Compass walk 1.07 z i o N & . D [ numerical integration Shortened V. path 1.16 1.16 35 Upper path 1.18 3 π 2 ≃ 1 . 18 8 ] 1 0 2 , . D & r e i v a n e h C [ 4 Voronoi path 1.27 π ≃ 1 . 27 ] 0 0 0 2 , . l a t e i l l e c c a B [ 11 - 3

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

Expected length of upper path s t Poisson Delaunay triangulation, rate n   � E [ length ] = E 1 [ triangle is Delaunay ] 1 [ first edge above st ] length(first edge)   triangle ∈ X 3 n � = n 3 ( R 2 ) 3 P [ triangle is Delaunay ] 1 [ first edge above st ] length(first edge) d triangle Slivnyak-Mecke 12 - 2

Expected length of upper path α 2 α 1 α 3 z s t Poisson Delaunay triangulation, rate n r   � E [ length ] = E 1 [ triangle is Delaunay ] 1 [ first edge above st ] length(first edge)   triangle ∈ X 3 n � = n 3 ( R 2 ) 3 P [ triangle is Delaunay ] 1 [ first edge above st ] length(first edge) d triangle � ∞ � 1 � r [0 , 2 π ] 3 e − nπr 2 1 [ first edge above st ] 2 r sin α 1 − α 2 � = n 3 r 3 2 A ( triangle )d α 1:3 d y z d x z d r 2 Blaschke-Petkantschin r =0 x z =0 y z = − r 12 - 3

Expected length of upper path Poisson Delaunay triangulation, rate n   � E [ length ] = E 1 [ triangle is Delaunay ] 1 [ first edge above st ] length(first edge)   triangle ∈ X 3 n � = n 3 ( R 2 ) 3 P [ triangle is Delaunay ] 1 [ first edge above st ] length(first edge) d triangle � ∞ � 1 � r [0 , 2 π ] 3 e − nπr 2 1 [ first edge above st ] 2 r sin α 1 − α 2 � = n 3 r 3 2 A ( triangle )d α 1:3 d y z d x z d r 2 r =0 x z =0 y z = − r �� ∞ �� 1 � � [0 , 2 π ] 3 1 [ first edge above st ] sin α 1 − α 2 � e − nπr 2 r 5 d r = 4 n 3 · A ( triangle )d α 1:3 d h 2 h = yz r =0 r = − 1 12 - 4