79 research outputs found
Upper and Lower Bounds for Competitive Online Routing on Delaunay Triangulations
Consider a weighted graph G where vertices are points in the plane and edges
are line segments. The weight of each edge is the Euclidean distance between
its two endpoints. A routing algorithm on G has a competitive ratio of c if the
length of the path produced by the algorithm from any vertex s to any vertex t
is at most c times the length of the shortest path from s to t in G. If the
length of the path is at most c times the Euclidean distance from s to t, we
say that the routing algorithm on G has a routing ratio of c.We present an
online routing algorithm on the Delaunay triangulation with competitive and
routing ratios of 5.90. This improves upon the best known algorithm that has
competitive and routing ratio 15.48. The algorithm is a generalization of the
deterministic 1-local routing algorithm by Chew on the L1-Delaunay
triangulation. When a message follows the routing path produced by our
algorithm, its header need only contain the coordinates of s and t. This is an
improvement over the currently known competitive routing algorithms on the
Delaunay triangulation, for which the header of a message must additionally
contain partial sums of distances along the routing path.We also show that the
routing ratio of any deterministic k-local algorithm is at least 1.70 for the
Delaunay triangulation and 2.70 for the L1-Delaunay triangulation. In the case
of the L1-Delaunay triangulation, this implies that even though there exists a
path between two points x and y whose length is at most 2.61|[xy]| (where
|[xy]| denotes the length of the line segment [xy]), it is not always possible
to route a message along a path of length less than 2.70|[xy]|. From these
bounds on the routing ratio, we derive lower bounds on the competitive ratio of
1.23 for Delaunay triangulations and 1.12 for L1-Delaunay triangulations
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
The Stretch Factor of - and -Delaunay Triangulations
In this paper we determine the stretch factor of the -Delaunay and
-Delaunay triangulations, and we show that this stretch is
. Between any two points of such
triangulations, we construct a path whose length is no more than
times the Euclidean distance between and , and this
bound is best possible. This definitively improves the 25-year old bound of
by Chew (SoCG '86). To the best of our knowledge, this is the first
time the stretch factor of the well-studied -Delaunay triangulations, for
any real , is determined exactly
Competitive online routing in geometric graphs
AbstractWe consider online routing algorithms for finding paths between the vertices of plane graphs. Although it has been shown in Bose et al. (Internat. J. Comput. Geom. 12(4) (2002) 283) that there exists no competitive routing scheme that works on all triangulations, we show that there exists a simple online O(1)-memory c-competitive routing strategy that approximates the shortest path in triangulations possessing the diamond property, i.e., the total distance travelled by the algorithm to route a message between two vertices is at most a constant c times the shortest path. Our results imply a competitive routing strategy for certain classical triangulations such as the Delaunay, greedy, or minimum-weight triangulation, since they all possess the diamond property. We then generalize our results to show that the O(1)-memory c-competitive routing strategy works for all plane graphs possessing both the diamond property and the good convex polygon property
Stabbing line segments with disks: complexity and approximation algorithms
Computational complexity and approximation algorithms are reported for a
problem of stabbing a set of straight line segments with the least cardinality
set of disks of fixed radii where the set of segments forms a straight
line drawing of a planar graph without edge crossings. Close
geometric problems arise in network security applications. We give strong
NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel
graphs and other subgraphs (which are often used in network design) for and some constant where and
are Euclidean lengths of the longest and shortest graph edges
respectively. Fast -time -approximation algorithm is
proposed within the class of straight line drawings of planar graphs for which
the inequality holds uniformly for some constant
i.e. when lengths of edges of are uniformly bounded from above by
some linear function of Comment: 12 pages, 1 appendix, 15 bibliography items, 6th International
Conference on Analysis of Images, Social Networks and Texts (AIST-2017
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