44 research outputs found
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
Rainbow Cycles in Flip Graphs
The flip graph of triangulations has as vertices all triangulations of a convex n-gon, and an edge between any two triangulations that differ in exactly one edge. An r-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly r times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of r-rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex n-gon, the flip graph of plane spanning trees on an arbitrary set of n points, and the flip graph of non-crossing perfect matchings on a set of n points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of {1,2,...,n } and the flip graph of k-element subsets of {1,2,...,n }. In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of r, n and k
Rainbow cycles in flip graphs
The flip graph of triangulations has as vertices all triangulations of a convex -gon, and an edge between any two triangulations that differ in exactly one edge. An -rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly ~times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of -rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex -gon, the flip graph of plane trees on an arbitrary set of ~points, and the flip graph of non-crossing perfect matchings on a set of ~points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of and the flip graph of -element subsets of . In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of~, and~
Counting and Enumerating Crossing-free Geometric Graphs
We describe a framework for counting and enumerating various types of
crossing-free geometric graphs on a planar point set. The framework generalizes
ideas of Alvarez and Seidel, who used them to count triangulations in time
where is the number of points. The main idea is to reduce the
problem of counting geometric graphs to counting source-sink paths in a
directed acyclic graph.
The following new results will emerge. The number of all crossing-free
geometric graphs can be computed in time for some .
The number of crossing-free convex partitions can be computed in time
. The number of crossing-free perfect matchings can be computed in
time . The number of convex subdivisions can be computed in time
. The number of crossing-free spanning trees can be computed in time
for some . The number of crossing-free spanning cycles
can be computed in time for some .
With the same bounds on the running time we can construct data structures
which allow fast enumeration of the respective classes. For example, after
time of preprocessing we can enumerate the set of all crossing-free
perfect matchings using polynomial time per enumerated object. For
crossing-free perfect matchings and convex partitions we further obtain
enumeration algorithms where the time delay for each (in particular, the first)
output is bounded by a polynomial in .
All described algorithms are comparatively simple, both in terms of their
analysis and implementation