Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams

Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams

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Article ID: iaor19981489
Country: United States
Volume: 18
Issue: 4
Start Page Number: 433
End Page Number: 454
Publication Date: Dec 1997
Journal: Discrete and Computational Geometry
Authors: , ,
Keywords: geometry
Abstract:

In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E4. Our algorithm runs in O((n + f) log2 f) time and uses O(n + f) space. This is the first algorithm within a polylogarithmic factor of optimal O(n log f + f) time over the whole range of f. By a standard lifting map, we obtain output-sensitive algorithms for the Voronoi diagram or Delaunay triangulation in E3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the ‘ultimate convex hull algorithm’ of Kirkpatrick and Seidel in E2 and also leads to improved output-sensitive results on constructing convex hulls in Ed for any even constant d > 4.

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