## Dense Point Sets Have Sparse Delaunay Triangulations

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Citations: | 30 - 2 self |

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@MISC{Erickson_densepoint,

author = {Jeff Erickson},

title = {Dense Point Sets Have Sparse Delaunay Triangulations },

year = {}

}

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### Abstract

Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearest-neighbor searching, clustering, finite-element mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since three-dimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worst-case running time \Omega (n2). However, this behavior is almost never observed in practice except for highly-contrived inputs. For all practical purposes, three-dimensional Delaunay triangulations appear to have linear complexity. This frustrating