### 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 - 2015 |

### 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

## Fingerprint Dive into the research topics of 'Counting triangulations and other crossing-free structures approximately'. Together they form a unique fingerprint.

## Cite this

*Computational Geometry: Theory and Applications*,

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