### 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 language | English (US) |
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Title of host publication | Algorithms and Computations - 6th International Symposium, ISAAC 1995, Proceedings |

Publisher | Springer-Verlag |

Pages | 372-381 |

Number of pages | 10 |

ISBN (Print) | 3540605738, 9783540605737 |

State | Published - Jan 1 1995 |

Event | 6th International Symposium on Algorithms and Computations, ISAAC 1995 - Cairns, Australia Duration: Dec 4 1995 → Dec 6 1995 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1004 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 6th International Symposium on Algorithms and Computations, ISAAC 1995 |
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Country | Australia |

City | Cairns |

Period | 12/4/95 → 12/6/95 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

TY - GEN

T1 - No Quadrangulation is extremely odd

AU - Bose, Prosenjit

AU - Toussaint, Godfried

PY - 1995/1/1

Y1 - 1995/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=21844504391&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21844504391&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:21844504391

SN - 3540605738

SN - 9783540605737

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 372

EP - 381

BT - Algorithms and Computations - 6th International Symposium, ISAAC 1995, Proceedings

PB - Springer-Verlag

ER -