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Optimal in-place algorithms for 3-D convex hulls and 2-D segment intersection

Published: 08 June 2009 Publication History

Abstract

We describe the first optimal randomized in-place algorithm for the basic 3-d convex hull problem (and, in particular, for 2-d Voronoi diagrams). The algorithm runs in O(n log n) expected time using only O(1) extra space; this improves the previous O(n log3 n) bound by Bronnimann, Chan, and Chen [SoCG'04]. The same approach leads to an optimal randomized in-place algorithm for the 2-d line segment intersection problem, with O(n log n+K) expected running time for output size K, improving the previous O(n log2 n + K) bound by Vahrenhold [WADS'05]. As a bonus, we also point out a simplification of a known optimal cache-oblivious (non-in-place) algorithm by Kumar and Ramos (2002) for 3-d convex hulls, and observe its applicability to 2-d segment intersection, extending a recent result for red/blue segment intersection by Arge, Molhave, and Zeh [ESA'08]. Our results are all obtained by standard random sampling techniques, with some interesting twists.

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  1. Optimal in-place algorithms for 3-D convex hulls and 2-D segment intersection

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    cover image ACM Conferences
    SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry
    June 2009
    426 pages
    ISBN:9781605585017
    DOI:10.1145/1542362
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    Published: 08 June 2009

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    Author Tags

    1. cache-oblivious algorithms
    2. convex hulls
    3. in-place algorithms
    4. segment intersection
    5. voronoi diagrams

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