6,201 research outputs found
Essential Constraints of Edge-Constrained Proximity Graphs
Given a plane forest of points, we find the minimum
set of edges such that the edge-constrained minimum spanning
tree over the set of vertices and the set of constraints contains .
We present an -time algorithm that solves this problem. We
generalize this to other proximity graphs in the constraint setting, such as
the relative neighbourhood graph, Gabriel graph, -skeleton and Delaunay
triangulation. We present an algorithm that identifies the minimum set
of edges of a given plane graph such that for , where is the
constraint -skeleton over the set of vertices and the set of
constraints. The running time of our algorithm is , provided that the
constrained Delaunay triangulation of is given.Comment: 24 pages, 22 figures. A preliminary version of this paper appeared in
the Proceedings of 27th International Workshop, IWOCA 2016, Helsinki,
Finland. It was published by Springer in the Lecture Notes in Computer
Science (LNCS) serie
Connected Spatial Networks over Random Points and a Route-Length Statistic
We review mathematically tractable models for connected networks on random
points in the plane, emphasizing the class of proximity graphs which deserves
to be better known to applied probabilists and statisticians. We introduce and
motivate a particular statistic measuring shortness of routes in a network.
We illustrate, via Monte Carlo in part, the trade-off between normalized
network length and in a one-parameter family of proximity graphs. How close
this family comes to the optimal trade-off over all possible networks remains
an intriguing open question. The paper is a write-up of a talk developed by the
first author during 2007--2009.Comment: Published in at http://dx.doi.org/10.1214/10-STS335 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition
A geometric graph is angle-monotone if every pair of vertices has a path
between them that---after some rotation---is - and -monotone.
Angle-monotone graphs are -spanners and they are increasing-chord
graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in
2014 and proved that Gabriel triangulations are angle-monotone graphs. We give
a polynomial time algorithm to recognize angle-monotone geometric graphs. We
prove that every point set has a plane geometric graph that is generalized
angle-monotone---specifically, we prove that the half--graph is
generalized angle-monotone. We give a local routing algorithm for Gabriel
triangulations that finds a path from any vertex to any vertex whose
length is within times the Euclidean distance from to .
Finally, we prove some lower bounds and limits on local routing algorithms on
Gabriel triangulations.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
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