Article

Random triangulations of planar point sets

Authors:

Emo WelzlAuthors Info & Claims

SCG '06: Proceedings of the twenty-second annual symposium on Computational geometry

Pages 273 - 281

https://doi.org/10.1145/1137856.1137898

Published: 05 June 2006 Publication History

Abstract

Let S be a finite set of n+3 points in general position in the plane, with 3 extreme points and n interior points. We consider triangulations drawn uniformly at random from the set of all triangulations of S, and investigate the expected number, v_i, of interior points of degree i in such a triangulation. We provide bounds that are linear in n on these numbers. In particular, n/43≤v₃≤ (2n+3)/5.Moreover, we relate these results to the question about the maximum and minimum possible number of triangulations in such a set S, and show that the number of triangulations of any set of n points in the plane is at most 43ⁿ, thereby improving on a previous bound by Santos and Seidel.

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Cited By

Rutschmann DWettstein M(2023)Chains, Koch Chains, and Point Sets with Many TriangulationsJournal of the ACM10.1145/358553570:3(1-26)Online publication date: 23-May-2023
https://dl.acm.org/doi/10.1145/3585535
Kupavskii AVolostnov AYarovikov Y(2023)Minimum number of partial triangulationsEuropean Journal of Combinatorics10.1016/j.ejc.2022.103636108(103636)Online publication date: Feb-2023
https://doi.org/10.1016/j.ejc.2022.103636
Caputo PMartinelli FSinclair AStauffer A(2015)Random lattice triangulations: Structure and algorithmsThe Annals of Applied Probability10.1214/14-AAP103325:3Online publication date: 1-Jun-2015
https://doi.org/10.1214/14-AAP1033
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cover image ACM Conferences

SCG '06: Proceedings of the twenty-second annual symposium on Computational geometry

June 2006

500 pages

ISBN:1595933409

DOI:10.1145/1137856

Program Chairs:
Nina Amenta
University of California at Davis
,
Otfried Cheong
Korea Advanced Institute of Science and Technology

Copyright © 2006 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 05 June 2006

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SoCG06: 22nd Annual Symposium on Computational Geometry

June 5 - 7, 2006

Arizona, Sedona, USA

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Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

Rutschmann DWettstein M(2023)Chains, Koch Chains, and Point Sets with Many TriangulationsJournal of the ACM10.1145/358553570:3(1-26)Online publication date: 23-May-2023
https://dl.acm.org/doi/10.1145/3585535
Kupavskii AVolostnov AYarovikov Y(2023)Minimum number of partial triangulationsEuropean Journal of Combinatorics10.1016/j.ejc.2022.103636108(103636)Online publication date: Feb-2023
https://doi.org/10.1016/j.ejc.2022.103636
Caputo PMartinelli FSinclair AStauffer A(2015)Random lattice triangulations: Structure and algorithmsThe Annals of Applied Probability10.1214/14-AAP103325:3Online publication date: 1-Jun-2015
https://doi.org/10.1214/14-AAP1033
Francke ATóth CCheng SDevillers O(2014)A Census of Plane Graphs with Polyline EdgesProceedings of the thirtieth annual symposium on Computational geometry10.1145/2582112.2582116(242-250)Online publication date: 8-Jun-2014
https://dl.acm.org/doi/10.1145/2582112.2582116
Ackerman EAllen MBarequet GLöffler MMermelstein JSouvaine DTóth C(2014)The Flip Diameter of Rectangulations and Convex SubdivisionsLATIN 2014: Theoretical Informatics10.1007/978-3-642-54423-1_42(478-489)Online publication date: 2014
https://doi.org/10.1007/978-3-642-54423-1_42
Agryzkov TOliver JTortosa LVicent J(2014)A Method to Triangulate a Set of Points in the PlaneComputational Science and Its Applications – ICCSA 201410.1007/978-3-319-09129-7_25(330-341)Online publication date: 2014
https://doi.org/10.1007/978-3-319-09129-7_25
Caputo PMartinelli FSinclair AStauffer ABoneh DRoughgarden TFeigenbaum J(2013)Random lattice triangulationsProceedings of the forty-fifth annual ACM symposium on Theory of Computing10.1145/2488608.2488685(615-624)Online publication date: 1-Jun-2013
https://dl.acm.org/doi/10.1145/2488608.2488685
Sharir MSheffer AWelzl E(2013)Counting plane graphsJournal of Combinatorial Theory Series A10.1016/j.jcta.2013.01.002120:4(777-794)Online publication date: 1-May-2013
https://dl.acm.org/doi/10.1016/j.jcta.2013.01.002
Ben-Ner MSchulz ASheffer A(2013)On numbers of pseudo-triangulationsComputational Geometry: Theory and Applications10.1016/j.comgeo.2012.11.00246:6(688-699)Online publication date: 1-Aug-2013
https://dl.acm.org/doi/10.1016/j.comgeo.2012.11.002
Dumitrescu ATóth C(2012)Computational geometry column 54ACM SIGACT News10.1145/2421119.242113643:4(90-97)Online publication date: 19-Dec-2012
https://dl.acm.org/doi/10.1145/2421119.2421136
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