Optimal higher order Delaunay triangulations of polygons

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^2nlogk+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^+^2n 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.

[1]  Günter Rote,et al.  Minimum-weight triangulation is NP-hard , 2006, JACM.

[2]  Maarten Löffler,et al.  Optimization for First Order Delaunay Triangulations , 2007, WADS.

[3]  Andrzej Lingas,et al.  Fast algorithms for greedy triangulation , 1992, BIT Comput. Sci. Sect..

[4]  Peter Gilbert.  New Results on Planar Triangulations. , 1979 .

[5]  David Eppstein,et al.  Edge insertion for optimal triangulations , 1993, Discret. Comput. Geom..

[6]  Joachim Gudmundsson,et al.  Higher order Delaunay triangulations , 2000, Comput. Geom..

[7]  Maarten Löffler,et al.  Generating realistic terrains with higher-order Delaunay triangulations , 2005, Comput. Geom..

[8]  Marian Neamtu,et al.  Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay , 2007 .

[9]  Alexander Wolff,et al.  Constructing interference-minimal networks , 2005, EuroCG.

[10]  J. Mark Keil,et al.  Algorithms for optimal area triangulations of a convex polygon , 2006, Comput. Geom..

[11]  Tiow Seng Tan,et al.  A quadratic time algorithm for the minmax length triangulation , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[12]  Goos Kant,et al.  Augmenting Outerplanar Graphs , 1996, J. Algorithms.

[13]  János Komlós,et al.  Storing a sparse table with O(1) worst case access time , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[14]  Joachim Gudmundsson,et al.  Constrained higher order Delaunay triangulations , 2005, Comput. Geom..

[15]  Timothy J. Baker,et al.  A Comparison Of Triangle Quality Measures , 2001, IMR.

[16]  L. Paul Chew,et al.  Constrained Delaunay triangulations , 1987, SCG '87.

[17]  Christos Levcopoulos,et al.  The greedy triangulation can be computed from the Delaunay triangulation in linear time , 1999, Comput. Geom..

[18]  Maarten Löffler,et al.  Optimization for first order Delaunay triangulations , 2007, Comput. Geom..

[19]  Subhash Suri,et al.  A pedestrian approach to ray shooting: shoot a ray, take a walk , 1993, SODA '93.

[20]  M. G. Anderson.  Modelling geomorphological systems , 1988 .

[21]  Leila De Floriani,et al.  Delaunay-based representation of surfaces defined over arbitrarily shaped domains , 1985, Comput. Vis. Graph. Image Process..

[22]  Marc J. van Kreveld,et al.  Towards a Definition of Higher Order Constrained Delaunay Triangulations , 2009, CCCG.

[23]  G. Klincsek.  Minimal Triangulations of Polygonal Domains , 1980 .

[24]  D. Eppstein,et al.  MESH GENERATION AND OPTIMAL TRIANGULATION , 1992 .