### Abstract

We consider the problem of counting straight-edge triangulations of a given set P of npoints in the plane. Until very recently it was not known whether the exact number of triangulations of P can be computed asymptotically faster than by enumerating all triangulations. We now know that the number of triangulations of P can be computed in O∗(2^{n}) time [9], which is less than the lower bound of Ω(2.43^{n}) on the number of triangulations of any point set [30]. In this paper we address the question of whether one can approximately count triangulations in sub-exponential time. We present an algorithm with sub-exponential running time and sub-exponential approximation ratio, that is, denoting by Λ the output of our algorithm and by cn the exact number of triangulations of P, for some positive constant c, we prove that c^{n} ≤ Λ ≤ c^{n}· 2°^{(n)}. This is the first algorithm that in sub-exponential time computes a (1 + o(1))-approximation of the base of the number of triangulations, more precisely, c ≤ Λ_{n-}^{1}≤ (1 + o(1))c. Our algorithm can be adapted to approximately count other crossing-free structures on P, keeping the quality of approximation and running time intact. In this paper we show how to do this for matchings and spanning trees.

Original language | English (US) |
---|---|

Pages (from-to) | 386-397 |

Number of pages | 12 |

Journal | Computational Geometry: Theory and Applications |

Volume | 48 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 2015 |

### Fingerprint

### Keywords

- Algorithmic geometry
- Approximation algorithms
- Counting algorithms
- Crossing-free structures
- Triangulations

### ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

*Computational Geometry: Theory and Applications*,

*48*(5), 386-397. https://doi.org/10.1016/j.comgeo.2014.12.006

**Counting triangulations and other crossing-free structures approximately.** / Alvarez, Victor; Bringmann, Karl; Ray, Saurabh; Seidel, Raimund.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 48, no. 5, pp. 386-397. https://doi.org/10.1016/j.comgeo.2014.12.006

}

TY - JOUR

T1 - Counting triangulations and other crossing-free structures approximately

AU - Alvarez, Victor

AU - Bringmann, Karl

AU - Ray, Saurabh

AU - Seidel, Raimund

PY - 2015/1/1

Y1 - 2015/1/1

N2 - We consider the problem of counting straight-edge triangulations of a given set P of npoints in the plane. Until very recently it was not known whether the exact number of triangulations of P can be computed asymptotically faster than by enumerating all triangulations. We now know that the number of triangulations of P can be computed in O∗(2n) time [9], which is less than the lower bound of Ω(2.43n) on the number of triangulations of any point set [30]. In this paper we address the question of whether one can approximately count triangulations in sub-exponential time. We present an algorithm with sub-exponential running time and sub-exponential approximation ratio, that is, denoting by Λ the output of our algorithm and by cn the exact number of triangulations of P, for some positive constant c, we prove that cn ≤ Λ ≤ cn· 2°(n). This is the first algorithm that in sub-exponential time computes a (1 + o(1))-approximation of the base of the number of triangulations, more precisely, c ≤ Λn-1≤ (1 + o(1))c. Our algorithm can be adapted to approximately count other crossing-free structures on P, keeping the quality of approximation and running time intact. In this paper we show how to do this for matchings and spanning trees.

AB - We consider the problem of counting straight-edge triangulations of a given set P of npoints in the plane. Until very recently it was not known whether the exact number of triangulations of P can be computed asymptotically faster than by enumerating all triangulations. We now know that the number of triangulations of P can be computed in O∗(2n) time [9], which is less than the lower bound of Ω(2.43n) on the number of triangulations of any point set [30]. In this paper we address the question of whether one can approximately count triangulations in sub-exponential time. We present an algorithm with sub-exponential running time and sub-exponential approximation ratio, that is, denoting by Λ the output of our algorithm and by cn the exact number of triangulations of P, for some positive constant c, we prove that cn ≤ Λ ≤ cn· 2°(n). This is the first algorithm that in sub-exponential time computes a (1 + o(1))-approximation of the base of the number of triangulations, more precisely, c ≤ Λn-1≤ (1 + o(1))c. Our algorithm can be adapted to approximately count other crossing-free structures on P, keeping the quality of approximation and running time intact. In this paper we show how to do this for matchings and spanning trees.

KW - Algorithmic geometry

KW - Approximation algorithms

KW - Counting algorithms

KW - Crossing-free structures

KW - Triangulations

UR - http://www.scopus.com/inward/record.url?scp=84932620216&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84932620216&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2014.12.006

DO - 10.1016/j.comgeo.2014.12.006

M3 - Article

AN - SCOPUS:84932620216

VL - 48

SP - 386

EP - 397

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 5

ER -