Optimal Higher Order Delaunay Triangulations of Polygons

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Abstract

This paper presents an algorithm to triangulate polygons optimally using order-k Delaunay triangulations, for a number of quality measures. The algorithm uses properties of higher order Delaunay triangulations to improve the O(n 3) running time required for normal triangulations to O(k 2 n logk + knlogn) expected time, where n is the number of vertices of the polygon. An extension to polygons with points inside is also presented, allowing to compute an optimal triangulation of a polygon with h ≥ 1 components inside in O(knlogn) + O(k) h + 2 n expected time. Furthermore, through experimental results we show that, in practice, it can be used to triangulate point sets optimally for small values of k. This represents the first practical result on optimization of higher order Delaunay triangulations for k > 1.

This research has been partially funded by the Netherlands Organisation for Scientific Research (NWO) under the project GOGO.