No Quadrangulation is extremely odd

Prosenjit Bose, Godfried Toussaint

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

    Abstract

    Given a set S of n points in the plane, a quadrangulation of S is a planar subdivision whose vertices are the points of S, whose outer face is the convex hull of S, and every face of the subdivision (except possibly the outer face) is a quadrilateral. We show that S admits a quadrangulation if and only if S does not have an odd number of extreme points. If S admits a quadrangulation, we present an algorithm that computes a quadrangulation of S in O(nlogn) time, which is optimal, even in the presence of collinear points. If S does not admit a quadrangulation, then our algorithm can quadrangulate S with the addition of one extra point, which is optimal. Finally, our results imply that a fc-angulation of a set of points can be achieved with the addition of at most k — 3 extra points within the same time bound.

    Original languageEnglish (US)
    Title of host publicationAlgorithms and Computations - 6th International Symposium, ISAAC 1995, Proceedings
    PublisherSpringer-Verlag
    Pages372-381
    Number of pages10
    ISBN (Print)3540605738, 9783540605737
    StatePublished - Jan 1 1995
    Event6th International Symposium on Algorithms and Computations, ISAAC 1995 - Cairns, Australia
    Duration: Dec 4 1995Dec 6 1995

    Publication series

    NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume1004
    ISSN (Print)0302-9743
    ISSN (Electronic)1611-3349

    Other

    Other6th International Symposium on Algorithms and Computations, ISAAC 1995
    CountryAustralia
    CityCairns
    Period12/4/9512/6/95

    Fingerprint

    Quadrangulation
    Odd
    Face
    Subdivision
    Odd number
    Collinear
    Extreme Points
    Convex Hull
    Set of points
    If and only if
    Imply

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Computer Science(all)

    Cite this

    Bose, P., & Toussaint, G. (1995). No Quadrangulation is extremely odd. In Algorithms and Computations - 6th International Symposium, ISAAC 1995, Proceedings (pp. 372-381). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1004). Springer-Verlag.

    No Quadrangulation is extremely odd. / Bose, Prosenjit; Toussaint, Godfried.

    Algorithms and Computations - 6th International Symposium, ISAAC 1995, Proceedings. Springer-Verlag, 1995. p. 372-381 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1004).

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

    Bose, P & Toussaint, G 1995, No Quadrangulation is extremely odd. in Algorithms and Computations - 6th International Symposium, ISAAC 1995, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1004, Springer-Verlag, pp. 372-381, 6th International Symposium on Algorithms and Computations, ISAAC 1995, Cairns, Australia, 12/4/95.
    Bose P, Toussaint G. No Quadrangulation is extremely odd. In Algorithms and Computations - 6th International Symposium, ISAAC 1995, Proceedings. Springer-Verlag. 1995. p. 372-381. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
    Bose, Prosenjit ; Toussaint, Godfried. / No Quadrangulation is extremely odd. Algorithms and Computations - 6th International Symposium, ISAAC 1995, Proceedings. Springer-Verlag, 1995. pp. 372-381 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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