### Abstract

We consider the problem of counting straight-edge triangulations of a given set P of n points 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 [2], which is less than the lower bound of Ω(2.43^{n}) on the number of triangulations of any point set [11]. 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 subexponential approximation ratio, that is, if we denote by c^{n} 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^{o}(^{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 1/n ≥ (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. The algorithm may be useful in guessing, through experiments, the right constants c1 and c2 such that the number of triangulations of any set of n points is between c^{n}_{1} and c^{n}_{2} . Currently there is a large gap between c_{1} and c_{2}, we know that c1 ≥ 2.43 and c2 ≤ 30.

Original language | English (US) |
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Pages | 85-89 |

Number of pages | 5 |

State | Published - Jan 1 2013 |

Event | 25th Canadian Conference on Computational Geometry, CCCG 2013 - Waterloo, Canada Duration: Aug 8 2013 → Aug 10 2013 |

### Other

Other | 25th Canadian Conference on Computational Geometry, CCCG 2013 |
---|---|

Country | Canada |

City | Waterloo |

Period | 8/8/13 → 8/10/13 |

### Fingerprint

### ASJC Scopus subject areas

- Geometry and Topology
- Computational Mathematics

### Cite this

*Counting triangulations approximately*. 85-89. Paper presented at 25th Canadian Conference on Computational Geometry, CCCG 2013, Waterloo, Canada.

**Counting triangulations approximately.** / Alvarez, Victor; Bringmann, Karl; Ray, Saurabh; Seidel, Raimund.

Research output: Contribution to conference › Paper

}

TY - CONF

T1 - Counting triangulations approximately

AU - Alvarez, Victor

AU - Bringmann, Karl

AU - Ray, Saurabh

AU - Seidel, Raimund

PY - 2013/1/1

Y1 - 2013/1/1

N2 - We consider the problem of counting straight-edge triangulations of a given set P of n points 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 [2], which is less than the lower bound of Ω(2.43n) on the number of triangulations of any point set [11]. 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 subexponential approximation ratio, that is, if we denote by cn 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 · 2o(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 1/n ≥ (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. The algorithm may be useful in guessing, through experiments, the right constants c1 and c2 such that the number of triangulations of any set of n points is between cn1 and cn2 . Currently there is a large gap between c1 and c2, we know that c1 ≥ 2.43 and c2 ≤ 30.

AB - We consider the problem of counting straight-edge triangulations of a given set P of n points 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 [2], which is less than the lower bound of Ω(2.43n) on the number of triangulations of any point set [11]. 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 subexponential approximation ratio, that is, if we denote by cn 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 · 2o(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 1/n ≥ (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. The algorithm may be useful in guessing, through experiments, the right constants c1 and c2 such that the number of triangulations of any set of n points is between cn1 and cn2 . Currently there is a large gap between c1 and c2, we know that c1 ≥ 2.43 and c2 ≤ 30.

UR - http://www.scopus.com/inward/record.url?scp=84926066084&partnerID=8YFLogxK

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M3 - Paper

AN - SCOPUS:84926066084

SP - 85

EP - 89

ER -