Lower bounds on the dilation of plane spanners

(I) We exhibit a set of 23 points in the plane that has dilation at least $1.4308$, improving the previously best lower bound of $1.4161$ for the worst-case dilation of plane spanners. (II) For every integer $n\geq13$, there exists an $n$-element point set $S$ such that the degree 3 dilation of $S$ denoted by $\delta_0(S,3) \text{ equals } 1+\sqrt{3}=2.7321\ldots$ in the domain of plane geometric spanners. In the same domain, we show that for every integer $n\geq6$, there exists a an $n$-element point set $S$ such that the degree 4 dilation of $S$ denoted by $\delta_0(S,4) \text{ equals } 1 + \sqrt{(5-\sqrt{5})/2}=2.1755\ldots$ The previous best lower bound of $1.4161$ holds for any degree. (III) For every integer $n\geq6$, there exists an $n$-element point set $S$ such that the stretch factor of the greedy triangulation of $S$ is at least $2.0268$.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2 table

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

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

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

Oriented Spanners

Given a point set $P$ in the Euclidean plane and a parameter $t$, we define an \emph{oriented $t$-spanner} as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest cycle in $G$ through those points is at most a factor $t$ longer than the shortest oriented cycle in the complete bi-directed graph. We investigate the problem of computing sparse graphs with small oriented dilation. As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a $1$-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in $O(n^8)$ time for $n$ points, and a greedy algorithm that computes a $5$-spanner in $O(n\log n)$ time. Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in an oriented $O(1)$-spanner.Comment: conference version: ESA '2