Connections between Theta-Graphs, TD-Delaunay Triangulations, and Orthogonal Surfaces

1. Connections between Theta-Graphs, TD-Delaunay Triangulations, and Orthogonal Surfaces Nicolas Bonichon, Cyril Gavoille Nicolas Hanusse, David Ilcinkas LaBRI University of Bordeaux France WG 2010

2. H G 4 4 4 5 5 3 3 4 Spanner a Let G be a weighted graph, and let H be a spanning subgraph of G. H is an s-spanner of G if, for all u,v dH(u,v) ≤ sdG(u,v) s is the stretch of H b e c d a 4 4 b e Ex: dG(b,d)=5, dH(b,d)=7 dG(b,e)=4, dH(b,e)=8 H is a 2-spanner of G. 3 3 c d 4

3. Geometric Spanners Let (E,d)be a metric space. Let S be a set of points of E. G(S)is the complete graph. The length of (u,v) is d(u,v). In this talk (E,d)is the Euclidean plane Goals: Small stretch s Few edges Small max degree Routable Planar … www.2m40.com Accidents: - 26 in 2009 - 10 in 2010 - Last one: June 22nd

4. Delaunay Triangulation Voronoï cell: Delaunay triangulation: si is a neighbor of sj iff [Dobkin et al. 90] Delaunay T. is a plane 5.08-spanner [Keil & Gutwin 92] Delaunay T. is a plane 2.42-spanner Stretch > 1.414 for any plane spanner [Chew 89] Stretch > 1.416 for Delaunay triangulations [Mulzner 04]

5. Triangular Distance Delaunay Triangulation Triangular “distance”: TD(u,v) = size of the smallest equilateral triangle centred at u touching v. [ TD(u,v) ≠ TD(v,u) in general ] v u TD(u,v) [Chew 89] TD-Delaunay is a plane 2-spanner

6. θk-graph[Clarkson 87][Keil 88] Vertex set of θk-Graph is S Space around each vertex of S is split into k cones of angle θk = 2/k. Edge set ofθk-Graph: for eachvertex u and each cone C, add an edge toward vertexv in C with the projection on the bisector that is closest to u. No bounds on the stretch are known to be tight.

7. Half-θk-graph Half-θk-Graph(S): Like a θk-Graph(S) but one preserves edges from half of the cones only. For k=6: Theorem 1: Half-θ6-Graph(S) = TD-Delaunay(S) Corollary: • Half-θ6-Graph(S) is a plane 2-spanner • θ6-Graph(S) is a 2-spanner (optimal stretch)

8. Proof: contact between 2 triangles • Whenever two triangles touch, it’s a tip that touches a side. • v touches north tip of u’s triangle iff v belongs to the north cone of u. • Let v be a vertex in the north cone of u. The time when both triangles touch is y(v)-y(u). • There is an edge between u and v iff v’s triangle is the first to touch the tip of u’s triangle. QED v u

9. z y x Orthogonal Surface[Miller 02] [Felsner 03] [Felsner & Zickfeld 08] Coplanar if all points of S are in (P): x+y+z=cste General position: no two points with same x,y, or z.

10. Geodesic Embedding[Miller 02] [Felsner 03] [Felsner & Zickfeld 08] Properties[Felsner et al.] : The geodesicembedding of every orthogonal surface of coplanar point set Sis a plane triangulation. Every plane triangulation is the geodesicembedding of orthogonal surface of somecoplanar point set S. Theorem 2: TD-Delaunay(S)  GeoEmbedding(S) Corollary: Every plane triangulation is TD-Delaunay realizable

11. TD-Voronoï  Coplanar Orthogonal Surface Proof: growing 2D triangles viewed as 3D cones

12. TD-Delaunay  Geodesic Embedding

13. Delaunay Realizability • A graph G is Delaunay realizable if there exists S such that G=Delaunay(S). • [Dillencourt & Smith 96]: some sufficient conditions, and some necessary conditions. No characterization known. Decision problem: in PSPACE, NP-hard? • But, trivial for TD-Delaunay realizability: Every plane triangulation is TD-Delaunay realizable (S constructible in O(|V(G)|) time). Graphs that are non Delaunay realizable

14. Thank You!