40 research outputs found
The Stretch Factor of the Delaunay Triangulation Is Less Than 1.998
Let be a finite set of points in the Euclidean plane. Let be a
Delaunay triangulation of . The {\em stretch factor} (also known as {\em
dilation} or {\em spanning ratio}) of is the maximum ratio, among all
points and in , of the shortest path distance from to in
over the Euclidean distance . Proving a tight bound on the stretch
factor of the Delaunay triangulation has been a long standing open problem in
computational geometry.
In this paper we prove that the stretch factor of the Delaunay triangulation
of a set of points in the plane is less than , improving the
previous best upper bound of 2.42 by Keil and Gutwin (1989). Our bound 1.998 is
better than the current upper bound of 2.33 for the special case when the point
set is in convex position by Cui, Kanj and Xia (2009). This upper bound breaks
the barrier 2, which is significant because previously no family of plane
graphs was known to have a stretch factor guaranteed to be less than 2 on any
set of points.Comment: 41 pages, 16 figures. A preliminary version of this paper appeared in
the Proceedings of the 27th Annual Symposium on Computational Geometry (SoCG
2011). This is a revised version of the previous preprint [v1
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
There are Plane Spanners of Maximum Degree 4
Let E be the complete Euclidean graph on a set of points embedded in the
plane. Given a constant t >= 1, a spanning subgraph G of E is said to be a
t-spanner, or simply a spanner, if for any pair of vertices u,v in E the
distance between u and v in G is at most t times their distance in E. A spanner
is plane if its edges do not cross.
This paper considers the question: "What is the smallest maximum degree that
can always be achieved for a plane spanner of E?" Without the planarity
constraint, it is known that the answer is 3 which is thus the best known lower
bound on the degree of any plane spanner. With the planarity requirement, the
best known upper bound on the maximum degree is 6, the last in a long sequence
of results improving the upper bound. In this paper we show that the complete
Euclidean graph always contains a plane spanner of maximum degree at most 4 and
make a big step toward closing the question. Our construction leads to an
efficient algorithm for obtaining the spanner from Chew's L1-Delaunay
triangulation
Bounded-degree Plane Geometric Spanners: Connecting the Dots Between Theory and Practice
The construction of bounded-degree plane geometric spanners has been a focus of interest since 2002 when Bose, Gudmundsson, and Smid proposed the first algorithm to construct such spanners. To date, eleven algorithms have been designed with various trade-offs in degree and stretch factor. We have implemented these sophisticated algorithms in C++ using the CGAL library and experimented with them using large synthetic and real-world pointsets. Our experiments have revealed their practical behavior and real-world efficacy. We share the implementations via GitHub for broader uses and future research.
We present a simple practical algorithm, named AppxStretchFactor, that can estimate stretch factors (obtains lower bounds on the exact stretch factors) of geometric spanners – a challenging problem for which no practical algorithm is known yet. In our experiments with bounded-degree plane geometric spanners, we find that AppxStretchFactor estimates stretch factors almost precisely. Further, it gives linear runtime performance in practice for the pointset distributions considered in this work, making it much faster than the naive Dijkstra-based algorithm for calculating stretch factors
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
Lower bounds on the dilation of plane spanners
(I) We exhibit a set of 23 points in the plane that has dilation at least
, improving the previously best lower bound of for the
worst-case dilation of plane spanners.
(II) For every integer , there exists an -element point set
such that the degree 3 dilation of denoted by in the domain of plane geometric spanners. In the
same domain, we show that for every integer , there exists a an
-element point set such that the degree 4 dilation of denoted by
The
previous best lower bound of holds for any degree.
(III) For every integer , there exists an -element point set
such that the stretch factor of the greedy triangulation of is at least
.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2
table