Improving shortest paths in the Delaunay triangulation

We study a problem about shortest paths in Delaunay triangulations. Given two nodes s; t in the Delaunay triangulation of a point set P, we look for a new point p that can be added, such that the shortest path from s to t in the Delaunay triangulation of P u{p} improves as much as possible. We study properties of the problem and give efficient algorithms to find such a point when the graph-distance used is Euclidean and for the link-distance. Several other variations of the problem are also discussed.

Approximations on minimum weight pseudo-triangulation problem using ant colony optimization

In this work, we consider the Minimum Weight Pseudo-Triangulation (MWPT) problem of a given set of n points in the plane. Globally optimal pseudo-triangulations with respect to the weight, as optimization criteria, are difficult to be found by deterministic methods, since no polynomial algorithm is known. We show how the Ant Colony Optimization (ACO) metaheuristic can be used to find high quality pseudo-triangulations of minimum weight. We present the experimental and statistical study based on our own set of instances since no reference to benchmarks for these problems were found in the literature. Throughout the experimental evaluation, we appraise the ACO metaheuristic performance for MWPT problem.

On the approximate parametrization problem of algebraic curves.

The problem of parameterizing approximately algebraic curves and surfaces is an active research field, with many implications in practical applications. The problem can be treated locally or globally. We formally state the problem, in its global version for the case of algebraic curves (planar or spatial), and we report on some algorithms approaching it, as well as on the associated error distance analysis.