2,127 research outputs found
A simple and less slow method for counting triangulations and for related problems
We present a simple dynamic programming based method for counting straight-edge triangulations of planar point sets. This method can be adapted to solve related problems such as nding the best triangulation of a point set according to certain optimality criteria, or generating a triangulation of a point set uniformly at random. We have implemented our counting method. It appears to be substantially less slow than previous methods:
instances with 20 points, which used to take minutes, can now be handled in less than a second, and instances with 30 points, which used to be solvable only by employing several workstations in parallel over a substantial amount of time, an now be solved in about one minute on a single standard workstation.International Max Planck Research Schoo
A QPTAS for the Base of the Number of Triangulations of a Planar Point Set
The number of triangulations of a planar n point set is known to be ,
where the base lies between and The fastest known algorithm
for counting triangulations of a planar n point set runs in time.
The fastest known arbitrarily close approximation algorithm for the base of the
number of triangulations of a planar n point set runs in time subexponential in
We present the first quasi-polynomial approximation scheme for the base of
the number of triangulations of a planar point set
A better upper bound on the number of triangulations of a planar point set
We show that a point set of cardinality in the plane cannot be the vertex
set of more than straight-edge triangulations of its convex
hull. This improves the previous upper bound of .Comment: 6 pages, 1 figur
Counting Triangulations and other Crossing-Free Structures Approximately
We consider the problem of counting straight-edge triangulations of a given
set of points in the plane. Until very recently it was not known
whether the exact number of triangulations of can be computed
asymptotically faster than by enumerating all triangulations. We now know that
the number of triangulations of can be computed in time,
which is less than the lower bound of on the number of
triangulations of any point set. 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 the output of our
algorithm, and by the exact number of triangulations of , for some
positive constant , we prove that . This is the first algorithm that in sub-exponential time computes a
-approximation of the base of the number of triangulations, more
precisely, . Our algorithm can be
adapted to approximately count other crossing-free structures on , keeping
the quality of approximation and running time intact. In this paper we show how
to do this for matchings and spanning trees.Comment: 19 pages, 2 figures. A preliminary version appeared at CCCG 201
Uniform Infinite Planar Triangulations
The existence of the weak limit as n --> infinity of the uniform measure on
rooted triangulations of the sphere with n vertices is proved. Some properties
of the limit are studied. In particular, the limit is a probability measure on
random triangulations of the plane.Comment: 36 pages, 4 figures; Journal revised versio
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