Improved upper bound on the stretch factor of delaunay triangulations

2011 ◽  
Author(s):  
Ge Xia
Keyword(s):  
Upper Bound ◽  
Delaunay Triangulations ◽  
Stretch Factor

scholarly journals Stretch factor in a planar Poisson–Delaunay triangulation with a large intensity

2018 ◽  
Vol 50 (01) ◽  
pp. 35-56 ◽  
Author(s):  
Nicolas Chenavier ◽  
Olivier Devillers
Keyword(s):  
Lower Bound ◽  
Point Process ◽  
Delaunay Triangulation ◽  
Shortest Path ◽  
Upper Bound ◽  
Poisson Point Process ◽  
Large Intensity ◽  
Stretch Factor ◽  
Expected Length

Abstract Let X := X n ∪ {(0, 0), (1, 0)}, where X n is a planar Poisson point process of intensity n. We provide a first nontrivial lower bound for the distance between the expected length of the shortest path between (0, 0) and (1, 0) in the Delaunay triangulation associated with X when the intensity of X n goes to ∞. Simulations indicate that the correct value is about 1.04. We also prove that the expected length of the so-called upper path converges to 35 / 3π2, yielding an upper bound for the expected length of the smallest path.

YAO GRAPHS SPAN THETA GRAPHS

2012 ◽  
Vol 04 (02) ◽  
pp. 1250024 ◽  
Author(s):  
MIRELA DAMIAN ◽  
KRISTIN RAUDONIS
Keyword(s):  
Orthogonal Projection ◽  
Upper Bound ◽  
Euclidean Distance ◽  
Nearest Neighbor ◽  
Edge Length ◽  
Fixed Integer ◽  
Equal Angle ◽  
Stretch Factor ◽  
Point Set ◽  
The Difference

Yao and Theta graphs are defined for a given point set and a fixed integer k > 0. The space around each point is divided into k cones of equal angle, and each point is connected to a nearest neighbor in each cone. The difference between Yao and Theta graphs is in the way the nearest neighbor is defined: Yao graphs minimize the Euclidean distance between a point and its neighbor, and Theta graphs minimize the Euclidean distance between a point and the orthogonal projection of its neighbor on the bisector of the hosting cone. We prove that, corresponding to each edge of the Theta graph Θ6, there is a path in the Yao graph Y6 whose length is at most 8.82 times the edge length. Combined with the result of Bonichon et al., who prove an upper bound of 2 on the stretch factor of Θ6, we obtain an upper bound of 17.64 on the stretch factor of Y6.

scholarly journals On the stretch factor of Delaunay triangulations of points in convex position

2011 ◽  
Vol 44 (2) ◽  
pp. 104-109 ◽  
Author(s):  
Shiliang Cui ◽  
Iyad A. Kanj ◽  
Ge Xia
Keyword(s):  
Delaunay Triangulations ◽  
Stretch Factor ◽  
Convex Position

scholarly journals An upper bound for conforming Delaunay triangulations

1993 ◽  
Vol 10 (2) ◽  
pp. 197-213 ◽  
Author(s):  
Herbert Edelsbrunner ◽  
Tiow Seng Tan
Keyword(s):  
Upper Bound ◽  
Delaunay Triangulations

The Stretch Factor of L 1- and L  ∞ -Delaunay Triangulations

2012 ◽  
pp. 205-216 ◽  
Author(s):  
Nicolas Bonichon ◽  
Cyril Gavoille ◽  
Nicolas Hanusse ◽  
Ljubomir Perković
Keyword(s):  
Delaunay Triangulations ◽  
Stretch Factor

scholarly journals Almost all Delaunay triangulations have stretch factor greater than π/2

2011 ◽  
Vol 44 (2) ◽  
pp. 121-127 ◽  
Author(s):  
Prosenjit Bose ◽  
Luc Devroye ◽  
Maarten Löffler ◽  
Jack Snoeyink ◽  
Vishal Verma
Keyword(s):  
Delaunay Triangulations ◽  
Stretch Factor ◽  
Almost All
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