, Volume 10, Issue 1, pp 377-409

An optimal convex hull algorithm in any fixed dimension

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Abstract

We present a deterministic algorithm for computing the convex hull ofn points inE d in optimalO(n logn+n ⌞d/2⌟ ) time. Optimal solutions were previously known only in even dimension and in dimension 3. A by-product of our result is an algorithm for computing the Voronoi diagram ofn points ind-space in optimalO(n logn+n ⌜d/2⌝ ) time.

This research was supported in part by the National Science Foundation under Grant CCR-9002352 and The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc. A preliminary version of this paper has appeared in “An optimal convex hull algorithm and new results on cuttings”,Proceedings of the 32nd Annual IEEE Symposium on the Foundations of Computer Science, October 1991, pp. 29–38. The convex hull algorithm given in the present paper, although similar in spirit, is considerably simpler than the one given in the proceedings.