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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