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research
Counting Triangulations and other Crossing-Free Structures Approximately
Authors
Victor Alvarez
Karl Bringmann
Saurabh Ray
Raimund Seidel
Publication date
1 April 2014
Publisher
Doi
Cite
View
on
arXiv
Abstract
We consider the problem of counting straight-edge triangulations of a given set
P
P
P
of
n
n
n
points in the plane. Until very recently it was not known whether the exact number of triangulations of
P
P
P
can be computed asymptotically faster than by enumerating all triangulations. We now know that the number of triangulations of
P
P
P
can be computed in
O
β
(
2
n
)
O^{*}(2^{n})
O
β
(
2
n
)
time, which is less than the lower bound of
Ξ©
(
2.4
3
n
)
\Omega(2.43^{n})
Ξ©
(
2.4
3
n
)
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
Ξ
\Lambda
Ξ
the output of our algorithm, and by
c
n
c^{n}
c
n
the exact number of triangulations of
P
P
P
, for some positive constant
c
c
c
, we prove that
c
n
β€
Ξ
β€
c
n
β
2
o
(
n
)
c^{n}\leq\Lambda\leq c^{n}\cdot 2^{o(n)}
c
n
β€
Ξ
β€
c
n
β
2
o
(
n
)
. This is the first algorithm that in sub-exponential time computes a
(
1
+
o
(
1
)
)
(1+o(1))
(
1
+
o
(
1
))
-approximation of the base of the number of triangulations, more precisely,
c
β€
Ξ
1
n
β€
(
1
+
o
(
1
)
)
c
c\leq\Lambda^{\frac{1}{n}}\leq(1 + o(1))c
c
β€
Ξ
n
1
β
β€
(
1
+
o
(
1
))
c
. Our algorithm can be adapted to approximately count other crossing-free structures on
P
P
P
, 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
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