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LATIN 2008: Theoretical Informatics

Volume 4957 of the series Lecture Notes in Computer Science pp 133-145

Optimal Higher Order Delaunay Triangulations of Polygons

  • Rodrigo I. SilveiraAffiliated withDepartment of Information and Computing Sciences, Utrecht University
  • , Marc van KreveldAffiliated withDepartment of Information and Computing Sciences, Utrecht University

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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.