2009 50th Annual IEEE Symposium on Foundations of Computer Science (2009)
Oct. 25, 2009 to Oct. 27, 2009
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2009.53
We present several results about Delaunay triangulations (DTs) and convex hulls in transdichotomous and hereditary settings: (i) the DT of a planar point set can be computed in expected time O(sort(n)) on a word RAM, where sort(n)is the time to sort n numbers. We assume that the word RAM supports the shuffle-operation in constant time; (ii) if we know the ordering of a planar point set in x- and in y-direction, its DT can be found by a randomized algebraic computation tree of expected linear depth;(iii) given a universe U of points in the plane, we construct a data structure D for Delaunay queries: for any subset P of U, D can find the DT of P in time O(|P| loglog |U|); (iv) given a universe U of points in 3-space in general convex position, there is a data structure D for convex hull queries: for any subset P of U, D can find the convex hull of P in time O(|P| (log log |U|)^2);(v) given a convex polytope in 3-space with n vertices which are colored with k > 2 colors, we can split it into the convex hulls of the individual color classes in time O(n (log log n)^2).The results (i)--(iii) generalize to higher dimensions. We need a wide range of techniques. Most prominently, we describe a reduction from DTs to nearest-neighbor graphs that relies on a new variant of randomized incremental constructions using dependent sampling.
Delaunay triangulation, word RAM, transdichotomous algorithm, hereditary algorithm
W. Mulzer and K. Buchin, "Delaunay Triangulations in O(sort(n)) Time and More," 2009 50th Annual IEEE Symposium on Foundations of Computer Science(FOCS), Atlanta, Georgia, 2009, pp. 139-148.