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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. H. G. 4. 4. 4. 5. 5. 3. 3. 4. Spanner. a. Let G be a weighted graph, and - PowerPoint PPT Presentation
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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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4
3 3
4 4
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a
b
c d
eG
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3 3
4 4
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c d
eH
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 ]
u
v
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
u
v
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.
x
z
y
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!