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

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 E^d for any even constant d > 4.