Consider an n-point metric space M = (V, δ), and a transmission range assignment r : V → R+ that maps each point v ∈ V to the disk of radius r(v) around it. The symmetric disk graph (henceforth, SDG) that corresponds to M and r is the undirected graph over V whose edge set includes an edge (u, v) if both r(u) and r(v) are no smaller than δ(u, v). SDGs are often used to model wireless communicat...

All triangulations of euclidean oriented matroids are of the same PLhomeomorphism type by a result of Anderson. That means all triangulations of euclidean acyclic oriented matroids are PL-homeomorphic to PL-balls and that all triangulations of totally cyclic oriented matroids are PL-homeomorphic to PLspheres. For non-euclidean oriented matroids this question is wide open. One key point in the p...

In this paper we propose an encoding scheme and ad hoc operators for a genetic approach to hierarchical graph clustering. Given a connected graph whose vertices correspond to points within a Euclidean space and a ®tness function, a hierarchy of graphs in which each vertex corresponds to a connected subgraph of the graph below is generated. Both the number of clustering levels and the number of ...

Given a planar graph G, the graph theoretic dilation of G is defined as the maximum ratio of the shortestpath distance and the Euclidean distance between any two vertices of G. Given a planar point set S, a triangulation of S that achieves minimum graph theoretic dilation is called a minimum dilation triangulation of S. In this paper, we show that a simple exclusion region for an edge e of 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...

Given a Euclidean graph G in Rd with n vertices and m edges, we consider the problem of adding an edge to G such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a graph with positive edge weights runs in O(nm+n2 log n) time, resulting in a trivial O(n3m+n4 log n) time algorithm for computing the optimal edge. Firs...

We formulate the problem of bipartite graph inference as a supervised learning problem, and propose a new method to solve it from the viewpoint of distance metric learning. The method involves the learning of two mappings of the heterogeneous objects to a unified Euclidean space representing the network topology of the bipartite graph, where the graph is easy to infer. The algorithm can be form...

We report on an adapted version of the Bentley-Ottmann algorithm for computing all the intersection points among the edges of the projection of a three-dimensional graph. This graph is given as a set of vertices together with their space Euclidean coordinates, and a set of edges connecting them. More precisely, the three-dimensional graph represents the approximation of a closed and smooth impl...

A graph-based classification method is proposed both for semi-supervised learning in the case of Euclidean data and for classification in the case of graph data. Our manifold learning technique is based on a convex optimization problem involving a convex regularization term and a concave loss function with a trade-off parameter carefully chosen so that the objective function remains convex. As ...

We introduce the notion of generalized Delaunay triangulation of a planar straight-line graph G = (V, E) in the Euclidean plane and present some characterizations of the triangulation. It is shown that the generalized Delaunay triangulation has the property that the minimum angle of the triangles in the triangulation is maximum among all possible triangulations of the graph. A general algorithm...