Results 1 
2 of
2
Sparse geometric graphs with small dilation. Computational Geometry: Theory and Application. Article in press. doi:10.1016/j.comgeo.2007.07.004
"... Given a set S of n points in R D, and an integer k such that 0 � k < n, we show that a geometric graph with vertex set S, at most n − 1 + k edges, maximum degree five, and dilation O(n/(k + 1)) can be computed in time O(n log n). For any k, we also construct planar npoint sets for which any geom ..."
Abstract

Cited by 14 (5 self)
 Add to MetaCart
(Show Context)
Given a set S of n points in R D, and an integer k such that 0 � k < n, we show that a geometric graph with vertex set S, at most n − 1 + k edges, maximum degree five, and dilation O(n/(k + 1)) can be computed in time O(n log n). For any k, we also construct planar npoint sets for which any geometric graph with n − 1 + k edges has dilation Ω(n/(k + 1)); a slightly weaker statement holds if the points of S are required to be in convex position. 1 Preliminaries and introduction A geometric network is an undirected graph whose vertices are points in R D. Geometric networks, especially geometric networks of points in the plane, arise in many applications. Road networks, railway networks, computer networks—any collection of objects that have some connections between them can be modeled as a geometric network. A natural and widely studied type of geometric network is the Euclidean network, where the weight of an edge is simply the Euclidean distance between its two endpoints. Such networks for points in R D form the topic of study of our paper. When designing a network for a given set S of points, several criteria have to be taken into account. In particular, in many applications it is important to ensure a short connection between every two points in S.
Experimental Study of Geometric tSpanners
, 2009
"... The construction of tspanners of a given point set has received a lot of attention, especially from a theoretical perspective. In this article, we experimentally study the performance and quality of the most common construction algorithms for points in the Euclidean plane. We implemented the most w ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
(Show Context)
The construction of tspanners of a given point set has received a lot of attention, especially from a theoretical perspective. In this article, we experimentally study the performance and quality of the most common construction algorithms for points in the Euclidean plane. We implemented the most wellknown tspanner algorithms and tested them on a number of different point sets. The experiments are discussed and compared to the theoretical results, and in several cases, we suggest modifications that are implemented and evaluated. The measures of quality that we consider are the number of edges, the weight, the maximum degree, the spanner diameter, and the number of crossings. This is the first time an extensive comparison has been made between the running times of construction algorithms of tspanners and the quality of the generated spanners.