Almost tight bounds for eliminating depth cycles in three dimensions

Boris Aronov, Micha Sharir

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    Given n non-vertical lines in 3-space, their vertical depth (above/below) relation can contain cycles. We show that the lines can be cut into O(n3/2 polylog n) pieces, such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. As a consequence, we deduce that the number of pairwise nonoverlapping cycles, namely, cycles whose xy-projections do not overlap, is O(n3/2 polylog n); this bound too is almost tight in the worst case. Previous results on this topic could only handle restricted cases of the problem (such as handling only triangular cycles, by Aronov, Koltun, and Sharir, or only cycles in grid-like patterns, by Chazelle et al.), and the bounds were considerably weaker-much closer to the trivial quadratic bound. Our proof uses a recent variant of the polynomial partitioning technique, due to Guth, and some simple tools from algebraic geometry. It is much more straightforward than the previous "purely combinatorial" methods. Our approach extends to eliminating all cycles in the depth relation among segments, and among constant-degree algebraic arcs. We hope that a suitable extension of this technique could be used to handle the much more difficult case of pairwise-disjoint triangles as well. Our results almost completely settle a long-standing (35 years old) open problem in computational geometry, motivated by hidden-surface removal in computer graphics.

    Original languageEnglish (US)
    Title of host publicationSTOC 2016 - Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing
    EditorsYishay Mansour, Daniel Wichs
    PublisherAssociation for Computing Machinery
    Pages1-8
    Number of pages8
    ISBN (Electronic)9781450341325
    DOIs
    StatePublished - Jun 19 2016
    Event48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016 - Cambridge, United States
    Duration: Jun 19 2016Jun 21 2016

    Publication series

    NameProceedings of the Annual ACM Symposium on Theory of Computing
    Volume19-21-June-2016
    ISSN (Print)0737-8017

    Other

    Other48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016
    CountryUnited States
    CityCambridge
    Period6/19/166/21/16

    Keywords

    • Computational geometry
    • Depth order
    • Lines in space
    • Painter's algorithm
    • Polynomial partitioning

    ASJC Scopus subject areas

    • Software

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