Counting triangulations approximately

Victor Alvarez, Karl Bringmann, Saurabh Ray, Raimund Seidel

Research output: Contribution to conferencePaper

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∗(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.

Original languageEnglish (US)
Pages85-89
Number of pages5
StatePublished - Jan 1 2013
Event25th Canadian Conference on Computational Geometry, CCCG 2013 - Waterloo, Canada
Duration: Aug 8 2013Aug 10 2013

Other

Other25th Canadian Conference on Computational Geometry, CCCG 2013
CountryCanada
CityWaterloo
Period8/8/138/10/13

Fingerprint

Triangulation
Counting
Exponential time
Count
Approximation
Point Sets
Straight
Lower bound
Denote
Output

ASJC Scopus subject areas

  • Geometry and Topology
  • Computational Mathematics

Cite this

Alvarez, V., Bringmann, K., Ray, S., & Seidel, R. (2013). 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.

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

Research output: Contribution to conferencePaper

Alvarez, V, Bringmann, K, Ray, S & Seidel, R 2013, 'Counting triangulations approximately' Paper presented at 25th Canadian Conference on Computational Geometry, CCCG 2013, Waterloo, Canada, 8/8/13 - 8/10/13, pp. 85-89.
Alvarez V, Bringmann K, Ray S, Seidel R. Counting triangulations approximately. 2013. Paper presented at 25th Canadian Conference on Computational Geometry, CCCG 2013, Waterloo, Canada.
Alvarez, Victor ; Bringmann, Karl ; Ray, Saurabh ; Seidel, Raimund. / Counting triangulations approximately. Paper presented at 25th Canadian Conference on Computational Geometry, CCCG 2013, Waterloo, Canada.5 p.
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