Let P be a set of n points in the plane. A geometric proximity graph on P is a graph where two points are connected by a straight-line segment if they satisfy some prescribed proximity rule. We consider four classes of higher order proximity graphs, namely, the k-nearest neighbor graph, the k-relative neighborhood graph, the k-Gabriel graph and the k-Delaunay graph. For k = 0 (k = 1 in the case...

Several localized routing protocols [1] guarantee the delivery of the packets when the underlying network topology is the Delaunay triangulation of all wireless nodes. However, it is expensive to construct the Delaunay triangulation in a distributed manner. Given a set of wireless nodes, we more accurately model the network as a unit-disk graph UDG , in which a link in between two nodes exist o...

Various graph-theoretic definitions of neighborhood have been used in the past in computer vision and clustering applications. In this paper we study the properties of a set of four related closest-point graphs using Monte Carlo methods: (i) the Delaunay Triangulation (DT) and its dual, Voronoi tessellation, (ii) the Gabriel Graph (GG), (iii) the Relative Neighborhood Graph (RNG), and (iv) the ...

This thesis describes our novel methods for data clustering, graph characterizing and image matching. In Chapter 3, our main contribution is the M1NN agglomerative clustering method with a new parallel merging algorithm. A cluster characterizing quantity is derived from the path-based dissimilarity measure. In Chapter 4, our main contribution is the modified log-likelihood model for quantitativ...

Several localized routing protocols guarantee the delivery of the packets when the underlying network topology is a planar graph. Typically, relative neighborhood graph (RNG) or Gabriel graph (GG) is used as such planar structure. However, it is well-known that the spanning ratios of these two graphs are not bounded by any constant (even for uniform randomly distributed points). Bose et al. [11...

In this work we study the order-k Delaunay graph, which is formed by edges pq having a circle through p and q and containing no more than k sites. We study the combinatorial structure of the set of triangulations that can be constructed with edges of this graph and show that it is connected under the flip operation if k ≤ 1 and for every k if points are in convex position. We also study the ham...

A b s t r a c t . Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of a...

We present two new Delaunay refinement algorithms, second an extension of the first. For a given input domain (a set of points or a planar straight line graph), and a threshold angle α, the Delaunay refinement algorithms compute triangulations that have all angles at least α. Our algorithms have the same theoretical guarantees as the previous Delaunay refinement algorithms. The original Delauna...