This paper considers the problem of embedding trees into the hyperbolic plane. We show that any tree can be realized as the Delaunay graph of its embedded vertices. Particularly, a weighted tree can be embedded such that the weight on each edge is realized as the hyperbolic distance between its embedded vertices. Thus the embedding preserves the metric information of the tree along with its top...

Oblivious online routing (OOR) algorithms are important for the applications with only local information available to make routing decisions. This paper gives two new OOR algorithms for a class of geometric graphs called Delaunay triangulations (DTs): the Midpoint Routing algorithm and the Compass Midpoint algorithm. Moreover, the former is generalized into a set of OOR algorithms that use the ...

SUMMARY Ka-me is a Voronoi image analyzer that allows users to analyze any image with a convex polygonal tessellation or any spatial point distribution by fitting Voronoi polygons and their dual, Delaunay triangulations, to the pattern. The analytical tools include a variety of graph theoretic and geometric tools that summarize the distribution of the numbers of edges per face, areas, perimeter...

In work with faculty Wei Wang, Jans Prins, Alex Tropsha, and Jack Snoeyink, they tested different graph representations of protein structures, including Delaunay tessellation graphs, contact maps, and chose almost-Delaunay graphs as being robust in the presence of coordinate perturbations. They developed a fast method to generate maximal frequent subgraphs directly using a tree representation, ...

We present a new definition of an implicit surface over a noisy point cloud, based on the weighted least squares approach. It can be evaluated very fast, but artifacts are significantly reduced. We propose to use a different kernel function that approximates geodesic distances on the surface by utilizing a geometric proximity graph. From a variety of possibilities, we have examined the Delaunay...

In this work we study the degree distribution, the maximum vertex and edge flow in non-uniform random Delaunay triangulations when geodesic routing is used. We also investigate the vertex and edge flow in Erdös-Renyi random graphs, geometric random graphs, expanders and random k–regular graphs. Moreover we show that adding a random matching to the original graph can considerably reduced the max...

We use here the results on the influence graph[1] to adapt them for particular cases where additional information is available. In some cases, it is possible to improve the expected randomized complexity of algorithms from O(n log n) to O(n log n). This technique applies in the following applications : triangulation of a simple polygon, skeleton of a simple polygon, Delaunay triangulation of po...

The paper addresses the problem of the 3D meshing of the complex free form Boundary Representation geometry with multiple domains. It introduces a generic graph based method of the CAD model analyses with optimal boundary recovery. A three stage approach is used: firstly, an underlying geometry is repaired to provide a conformal model without gaps and overlaps. Secondly, the model is analyzed a...

The authors have subjected several sets of real and simplified model protein structures to Delaunay tessellation. The system of contacts defined by residues joined with simplex edges in the tessellation can be thought of as a graph or network. Properties like the graph distances between residues and clustering coefficients can be computed to investigate the nature of such contact networks. Usin...