Dense point sets have sparse Delaunay triangulations
نویسنده
چکیده
The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R3 with spread ∆ has complexity O(∆3). This bound is tight in the worst case for all ∆ = O( √ n). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of k-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any n and ∆ = O(n), we construct a regular triangulation of complexity Ω(n∆) whose n vertices have spread ∆. ∗Portions of this work were done while the author was visiting The Ohio State University. This research was partially supported by an Sloan Research Fellowship, by NSF CAREER grant CCR-0093348, and by NSF ITR grants DMR-0121695 and CCR-0219594. An extended abstract of this paper was presented at the 13th Annual ACM-SIAM Symposium on Discrete Algorithms [53]. See http://www.cs.uiuc.edu/ ̃jeffe/pubs/screw.html for the most recent version of this paper. Dense Point Sets Have Sparse Delaunay Triangulations 1
منابع مشابه
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The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R3 with spread ∆ has complexity O(∆3). This bound is tight in the worst case for all ∆ = O( √ n). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to re...
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عنوان ژورنال:
- CoRR
دوره cs.CG/0110030 شماره
صفحات -
تاریخ انتشار 2001