On the number of regular vertices of the union of Jordan Regions

Boris Aronov, Alon Efrat, Dan Halperin, Micha Sharir

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

    Abstract

    Let C be a collection of n Jordan regions in the plane in general position, such that each pair of their boundaries intersect in at most s points, where s is a constant. If the boundaries of two sets in C cross exactly twice, then their intersection points are called regular vertices of the arrangement A(C). Let R(C) denote the set of regular vertices on the boundary of the union of C. We present several bounds on |R(C)|, determined by the type of the sets of C. (i) If each set of C is convex, then [R(C)[ = O(n1.5+ε) for any ε > 0.4 (ii) If C consists of two collections C1 and C2 where C1 is a collection of n convex pseudo-disks in the plane (closed Jordan regions with the property that the boundaries of any two of them intersect at most twice), and C2 is a collection of polygons with a total of n sides, then |R(C)| = O(n4/3), and this bound is tight in the worst case. (iii) If no further assumptions are made on the sets of C, then we show that there is a positive integer t that depends only on s such that |R(C)| = O(n2-1/t).

    Original languageEnglish (US)
    Title of host publicationAlgorithm Theory — SWAT 1998 - 6th Scandinavian Workshop on Algorithm Theory, Proceedings
    EditorsStefan Arnborg, Lars Ivansson
    PublisherSpringer Verlag
    Pages322-334
    Number of pages13
    ISBN (Print)3540646825, 9783540646822
    DOIs
    StatePublished - 1998
    Event6th Scandinavian Workshop on Algorithm Theory, SWAT 1998 - Stockholm, Sweden
    Duration: Jul 8 1998Jul 10 1998

    Publication series

    NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume1432
    ISSN (Print)0302-9743
    ISSN (Electronic)1611-3349

    Other

    Other6th Scandinavian Workshop on Algorithm Theory, SWAT 1998
    CountrySweden
    CityStockholm
    Period7/8/987/10/98

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Computer Science(all)

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  • Cite this

    Aronov, B., Efrat, A., Halperin, D., & Sharir, M. (1998). On the number of regular vertices of the union of Jordan Regions. In S. Arnborg, & L. Ivansson (Eds.), Algorithm Theory — SWAT 1998 - 6th Scandinavian Workshop on Algorithm Theory, Proceedings (pp. 322-334). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1432). Springer Verlag. https://doi.org/10.1007/BFb0054379