Preprocessing Imprecise Points and Splitting Triangulations

Given a triangulation of a set of n points in the plane, each colored red or blue, we show how to compute a triangulation of just the blue points in time O(n). We apply this result to show that one can preprocess a set of disjoint regions (representing "imprecise points") in the plane having total complexity n in O(n logn) time so that if one point per region is specified with precise coordinates, a triangulation of the n points can be computed in O(n) time.

[1]  Timothy M. Chan.  Three problems about simple polygons , 2006, Comput. Geom..

[2]  Maarten Löffler,et al.  Largest Bounding Box, Smallest Diameter, and Related Problems on Imprecise Points , 2007, WADS.

[3]  Joseph S. B. Mitchell,et al.  Triangulating input-constrained planar point sets , 2008, Inf. Process. Lett..

[4]  D. Salesin,et al.  Constructing strongly convex approximate hulls with inaccurate primitives , 1990, Algorithmica.

[5]  Manuel Abellanas,et al.  Structural Tolerance and Delaunay Triangulation , 1999, Inf. Process. Lett..

[6]  Bernard Chazelle,et al.  Splitting a Delaunay Triangulation in Linear Time , 2002, Algorithmica.

[7]  Michel Pocchiola,et al.  Pseudo-triangulations: theory and applications , 1996, SCG '96.

[8]  Maarten Löffler,et al.  Delaunay triangulation of imprecise points in linear time after preprocessing , 2010, Comput. Geom..

[9]  Anthony P. Leclerc,et al.  Correct Delaunay Triangulation in the Presence of Inexact Inputs and Arithmetic , 2000, Reliab. Comput..

[10]  Maarten Löffler,et al.  Largest and Smallest Convex Hulls for Imprecise Points , 2010, Algorithmica.

[11]  Frank Weller.  Stability of voronoi neighborship under perturbations of the sites , 1997, CCCG.

[12]  Bernard Chazelle,et al.  Computing hereditary convex structures , 2009, SCG '09.

[13]  Andrzej Lingas,et al.  On computing Voronoi diagrams for sorted point sets , 1995, Int. J. Comput. Geom. Appl..

[14]  Azriel Rosenfeld,et al.  Fuzzy Geometry: An updated Overview , 1998, Inf. Sci..

[15]  Abbas Edalat,et al.  Computability of Partial Delaunay Triangulation and Voronoi Diagram , 2002, CCA.

[16]  William S. Evans,et al.  Guaranteed Voronoi Diagrams of Uncertain Sites , 2008, CCCG.

[17]  Francis Y. L. Chin,et al.  Finding the Constrained Delaunay Triangulation and Constrained Voronoi Diagram of a Simple Polygon in Linear Time , 1999, SIAM J. Comput..

[18]  Abbas Edalat,et al.  Computing Delaunay Triangulation with Imprecise Input Data , 2003, CCCG.

[19]  Xiaojun Shen,et al.  Covering convex sets with non-overlapping polygons , 1990, Discret. Math..

[20]  Maarten Löffler,et al.  Approximating largest convex hulls for imprecise points , 2007, J. Discrete Algorithms.

[21]  M Pocchiola,et al.  On polygonal covers , 1998 .

[22]  Yin-Feng Xu,et al.  On Constrained Minimum Pseudotriangulations , 2003, COCOON.

[23]  Jack Snoeyink,et al.  Almost-Delaunay simplices: nearest neighbor relations for imprecise points , 2004, SODA '04.

[24]  Maarten Löffler,et al.  Delaunay triangulations of imprecise pointsin linear time after preprocessing , 2008, SCG '08.

[25]  R. Seidel.  A Method for Proving Lower Bounds for Certain Geometric Problems , 1984 .