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2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06) (2006)
Berkeley, California
Oct. 21, 2006 to Oct. 24, 2006
ISSN: 0272-5428
ISBN: 0-7695-2720-5
pp: 333-344
Timothy M. Chan , University of Waterloo, Canada
ABSTRACT
Given n points in the plane with integer coordinates bounded by U \leqslant 2^{w}, we show that the Voronoi diagram can be constructed in O(min{n logn/loglogn, n\sqrt {\log U}) expected time by a randomized algorithm on the unit-cost RAM with word size w. Similar results are also obtained for many other fundamental problems in computational geometry, such as constructing the convex hull of a 3-dimensional point set, computing the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. <p>These are the first results to beat the \Omega(nlogn) algebraic-decision-tree lower bounds known for these problems. The results are all derived from a new twodimensional version of fusion trees that can answer point location queries in O(min{logn/loglogn, \sqrt {\log U}) time with linear space. Higher-dimensional extensions and applications are also mentioned in the paper.</p>
INDEX TERMS
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CITATION

T. M. Chan, "Point Location in o(log n) Time, Voronoi Diagrams in o(n log n) Time, and Other Transdichotomous Results in Computational Geometry," 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)(FOCS), Berkeley, California, 2006, pp. 333-344.
doi:10.1109/FOCS.2006.62
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