290 research outputs found
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
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
The Stretch Factor of Hexagon-Delaunay Triangulations
The problem of computing the exact stretch factor (i.e., the tight bound on the worst case stretch factor) of a Delaunay triangulation is one of the longstanding open problems in computational geometry. Over the years, a series of upper and lower bounds on the exact stretch factor have been obtained but the gap between them is still large. An alternative approach to solving the problem is to develop techniques for computing the exact stretch factor of "easier" types of Delaunay triangulations, in particular those defined using regular-polygons instead of a circle. Tight bounds exist for Delaunay triangulations defined using an equilateral triangle and a square. In this paper, we determine the exact stretch factor of Delaunay triangulations defined using a regular hexagon: It is 2. We think that the main contribution of this paper are the two techniques we have developed to compute tight upper bounds for the stretch factor of Hexagon-Delaunay triangulations
Degree Four Plane Spanners: Simpler and Better
Let P be a set of n points embedded in the plane, and let C be the complete Euclidean graph whose point-set is P. Each edge in C between two points p, q is realized as the line segment [pq], and is assigned a weight equal to the Euclidean distance |pq|. In this paper, we show how to construct in O(nlg{n}) time a plane spanner of C of maximum degree at most 4 and of stretch factor at most 20. This improves a long sequence of results on the construction of bounded degree plane spanners of C. Our result matches the smallest known upper bound of 4 by Bonichon et al. on the maximum degree while significantly improving their stretch factor upper bound from 156.82 to 20. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner, and reveals useful structural properties of the Delaunay triangulations defined with respect to the equilateral-triangle distance
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
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