Abstract
It is well known that, using standard models of computation, Ω(n logn) time is required to build a Voronoi diagram forn point sites. This follows from the fact that a Voronoi diagram algorithm can be used to sort. However, if the sites are sorted before we start, can the Voronoi diagram be built any faster? We show that for certain interesting, although nonstandard, types of Voronoi diagrams, sorting helps. These nonstandard types of Voronoi diagrams use a convex distance function instead of the standard Euclidean distance. A convex distance function exists for any convex shape, but the distance functions based on polygons (especially triangles) lead to particularly efficient Voronoi diagram algorithms. Specifically, a Voronoi diagram using a convex distance function based on a triangle can be built inO (n log logn) time after initially sorting then sites twice. Convex distance functions based on other polygons require more initial sorting.
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Communicated by B. Chazelle.
This work was supported by the Advanced Research Projects Agency of the Department of Defense under ONR Contract N00014-92-J-1989, and by ONR Contract N0014-92-J-1839, NSF Contract IRI-9006137, and AFOSR Contract AFOSR-91-0328.
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Chew, L.P., Fortune, S. Sorting helps for voronoi diagrams. Algorithmica 18, 217–228 (1997). https://doi.org/10.1007/BF02526034
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DOI: https://doi.org/10.1007/BF02526034