Post on 03-Jan-2016
description
Connections between Theta-Graphs, TD-Delaunay Triangulations, and
Orthogonal Surfaces
WG 2010
Nicolas Bonichon, Cyril GavoilleNicolas Hanusse, David Ilcinkas
LaBRIUniversity of Bordeaux
France
Spanner
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Let G be a weighted graph, andlet H be a spanning subgraph of G.
H is an s-spanner of G if, for all u,v
dH(u,v) ≤ s dG(u,v)
s is the stretch of H
Ex: dG(b,d)=5, dH(b,d)=7dG(b,e)=4, dH(b,e)=8
H is a 2-spanner of G.
Geometric Spanners
In this talk (E,d) is the Euclidean plane
www.2m40.com
Accidents:
- 26 in 2009
- 10 in 2010
- Last one: June 22nd
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).
Goals:– Small stretch s– Few edges– Small max degree– Routable– Planar– …
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]
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 ]
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TD(u,v)
[Chew 89] TD-Delaunay is a plane 2-spanner
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 of θk-Graph: for each vertex u and each cone C, add an edge toward vertex v in C
with the projection on the bisector that is closest to u.
No bounds on the stretch are known to be tight.
k Stretch
< 9 ???
9 < 8.11
10 < 4.50
… …
14 < 2.14
15 < 1.98
Half-θk-graph
Half-θk-Graph(S):
Like a θk-Graph(S) but one preserves edges from half of the cones only.
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)
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)
For k=6:
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
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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.
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Geodesic Embedding [Miller 02] [Felsner 03] [Felsner & Zickfeld 08]
Properties [Felsner et al.] :
1.The geodesic embedding of every orthogonal surface of coplanar point set S is a plane triangulation.
2.Every plane triangulation is the geodesic embedding of orthogonal surface of some coplanar point set S.
Theorem 2: TD-Delaunay(S) GeoEmbedding(S)
Corollary: Every plane triangulation is TD-Delaunay realizable
Theorem 2: TD-Delaunay(S) GeoEmbedding(S)
Corollary: Every plane triangulation is TD-Delaunay realizable
TD-Voronoï Coplanar Orthogonal Surface
Proof: growing 2D triangles viewed as 3D cones
TD-Delaunay Geodesic Embedding
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 nonDelaunay realizable
Thank You!