Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
10327368 | Computational Geometry | 2015 | 12 Pages |
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 Oâ(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 Î the output of our algorithm and by cn the exact number of triangulations of P, for some positive constant c, we prove that cnâ¤Îâ¤cnâ
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, câ¤Î1nâ¤(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.
Related Topics
Physical Sciences and Engineering
Computer Science
Computational Theory and Mathematics
Authors
Victor Alvarez, Karl Bringmann, Saurabh Ray, Raimund Seidel,