Consider a weighted graph G whose vertices are points in the plane and edges are line segments between pairs of points whose weight is the Euclidean distance between its endpoints. A routing algorithm on G sends a message from any vertex s to any vertex t in G. The algorithm has a competitive ratio of c if the length of the path taken by the message is at most c times the length of the shortest...

In this paper, we complete the study of the length of a longest Delaunay edge for points randomly distributed in multidimensional Euclidean spaces carried out in [1]. The Delaunay graph is defined over a set of n points distributed uniformly at random in a ddimensional body of unit volume, assuming that the probability that those points are not in general position is negligible [6]. The motivat...

Higher order Delaunay triangulations are a generalization of the Delaunay triangulation which provides a class of well-shaped triangulations, over which extra criteria can be optimized. A triangulation is order-k Delaunay if the circumcircle of each triangle of the triangulation contains at most k points. In this paper we study lower and upper bounds on the number of higher order Delaunay trian...

Let P be a set of n points in R2, and let DT(P) denote its Euclidean Delaunay triangulation. We introduce the notion of an edge of DT(P) being stable. Defined in terms of a parameter α > 0, a Delaunay edge pq is called α-stable, if the (equal) angles at which p and q see the corresponding Voronoi edge epq are at least α. A subgraph G of DT(P) is called (cα, α)-stable Delaunay graph (SDG in shor...

We study affine invariant 2D triangulation methods. That is, methods that produce the same for a point set S any (unknown) transformation of S. Our work is based on method by Nielson (1993) uses inverse covariance matrix to define an norm, denoted AS, and triangulation, DTAS[S]. revisit AS-norm from geometric perspective, show DTAS[S] can be seen as standard Delaunay transformed prove it retain...

Delaunay has shown that the Delaunay complex of a finite set of points P of Euclidean space R triangulates the convex hull of P , provided that P satisfies a mild genericity property. Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds. However, Delaunay’s genericity assumption no longer guarantees that the Delaunay complex will yield a triangulation; stron...

In the Euclidean plane, a Delaunay triangulation can be characterized by the requirement that the circumcircle of each triangle be empty of vertices of all other triangles. For triangulating a surface S in R3, the Delaunay paradigm has typically been employed in the form of the restricted Delaunay triangulation, where the empty circumcircle property is defined by using the Euclidean metric in R...

For n + 1 disjoint flats of dimension k in H, we produce a Delaunay cell which is a generalization of the Delaunay simplex associated to n + 1 points in H. Combinatorially, these Delaunay cells resemble truncated n-dimensional simplices. For certain classes of arrangements of flats in H, we prove that these Delaunay cells can be glued together to form a Delaunay complex, with the result that al...

In the Euclidean plane, a Delaunay triangulation can be characterized by the requirement that the circumcircle of each triangle be empty of vertices of all other triangles. For triangulating a surface S in R3, the Delaunay paradigm has typically been employed in the form of the restricted Delaunay triangulation, where the empty circumcircle property is defined by using the Euclidean metric in R...