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

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2006.62

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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