research-article

A simple aggregative algorithm for counting triangulations of planar point sets and related problems

Authors:

Victor Alvarez,

Raimund SeidelAuthors Info & Claims

SoCG '13: Proceedings of the twenty-ninth annual symposium on Computational geometry

Pages 1 - 8

https://doi.org/10.1145/2462356.2462392

Published: 17 June 2013 Publication History

Abstract

We give an algorithm that determines the number (S) of straight line triangulations of a set S of n points in the plane in worst case time O(n² 2ⁿ). This is the the first algorithm that is provably faster than enumeration, since (S) is known to be Ω(2.43ⁿ) for any set S of n points. Our algorithm requires exponential space.

The algorithm generalizes to counting all triangulations of S that are constrained to contain a given set of edges. It can also be used to compute an optimal triangulation of S (unconstrained or constrained) for a reasonably wide class of optimality criteria (that includes e.g. minimum weight triangulations). Finally, the approach can also be used for the random generation of triangulations of S according to the perfect uniform distribution.

The algorithm has been implement and is substantially faster than existing methods on a variety of inputs.

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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
Klein RDriemel AHaverkort HKlein RDriemel AHaverkort H(2022)Voronoi-DiagrammeAlgorithmische Geometrie10.1007/978-3-658-37711-3_5(250-296)Online publication date: 21-Jun-2022
https://doi.org/10.1007/978-3-658-37711-3_5
Eppstein D(2020)Counting Polygon Triangulations is HardDiscrete & Computational Geometry10.1007/s00454-020-00251-7Online publication date: 13-Oct-2020
https://doi.org/10.1007/s00454-020-00251-7
Show More Cited By

Index Terms

A simple aggregative algorithm for counting triangulations of planar point sets and related problems
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Conferences

SoCG '13: Proceedings of the twenty-ninth annual symposium on Computational geometry

June 2013

472 pages

ISBN:9781450320313

DOI:10.1145/2462356

General Chairs:
Guilherme D. da Fonseca
UNIRIO
,
Thomas Lewiner
PUC-Rio
,
Luis Peñaranda
IMPA
,
Program Chairs:
Timothy Chan
University of Waterloo
,
Rolf Klein
University of Bonn

Copyright © 2013 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: 17 June 2013

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SoCG '13: Symposium on Computational Geometry 2013

June 17 - 20, 2013

Rio de Janeiro, Brazil

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SoCG '13 Paper Acceptance Rate 48 of 137 submissions, 35%;

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
Klein RDriemel AHaverkort HKlein RDriemel AHaverkort H(2022)Voronoi-DiagrammeAlgorithmische Geometrie10.1007/978-3-658-37711-3_5(250-296)Online publication date: 21-Jun-2022
https://doi.org/10.1007/978-3-658-37711-3_5
Eppstein D(2020)Counting Polygon Triangulations is HardDiscrete & Computational Geometry10.1007/s00454-020-00251-7Online publication date: 13-Oct-2020
https://doi.org/10.1007/s00454-020-00251-7
Alvarez VBringmann KRay SSeidel R(2018)Counting triangulations and other crossing-free structures approximatelyComputational Geometry: Theory and Applications10.1016/j.comgeo.2014.12.00648:5(386-397)Online publication date: 29-Dec-2018
https://dl.acm.org/doi/10.1016/j.comgeo.2014.12.006
DUMITRESCU ATÓTH C(2017)Convex Polygons in Geometric TriangulationsCombinatorics, Probability and Computing10.1017/S096354831700014126:05(641-659)Online publication date: 30-May-2017
https://doi.org/10.1017/S0963548317000141
Wettstein M(2017)Trapezoidal Diagrams, Upward Triangulations, and Prime Catalan NumbersDiscrete & Computational Geometry10.1007/s00454-017-9896-558:3(505-525)Online publication date: 1-Oct-2017
https://dl.acm.org/doi/10.1007/s00454-017-9896-5
Tanigawa S(2016)Enumeration of Non-crossing Geometric GraphsEncyclopedia of Algorithms10.1007/978-1-4939-2864-4_729(638-640)Online publication date: 22-Apr-2016
https://doi.org/10.1007/978-1-4939-2864-4_729
Alvarez VBringmann KCurticapean RRay S(2015)Counting Triangulations and Other Crossing-Free Structures via Onion LayersDiscrete & Computational Geometry10.1007/s00454-015-9672-353:4(675-690)Online publication date: 1-Jun-2015
https://dl.acm.org/doi/10.1007/s00454-015-9672-3
Karpinski MLingas ASledneu D(2015)A QPTAS for the Base of the Number of Crossing-Free Structures on a Planar Point SetAutomata, Languages, and Programming10.1007/978-3-662-47672-7_64(785-796)Online publication date: 20-Jun-2015
https://doi.org/10.1007/978-3-662-47672-7_64
Dumitrescu ATóth C(2015)Convex Polygons in Geometric TriangulationsAlgorithms and Data Structures10.1007/978-3-319-21840-3_24(289-300)Online publication date: 28-Jul-2015
https://doi.org/10.1007/978-3-319-21840-3_24
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