A simple aggregative algorithm for counting triangulations of planar point sets and related problems

Abstract

We give an algorithm that determines the number tr(S) of straight line triangulations of a set S of n points in the plane in worst case time O(n 2 2 n ). This is the the first algorithm that is provably faster than enumeration, since tr(S) is known to be Ω(2.43 n ) for any set S of n points. Our algorithm requires exponential space.The algorithm generalizes to counting all triangulations of S that are constrained to contain a given set of edges. It can also be used to compute an optimal triangulation of S (unconstrained or constrained) for a reasonably wide class of optimality criteria (that includes e.g. minimum weight triangulations). Finally, the approach can also be used for the random generation of triangulations of S according to the perfect uniform distribution.The algorithm has been implement and is substantially faster than existing methods on a variety of inputs.

Counting and Enumerating Crossing-free Geometric Graphs

“…Theorem 5 (Triangulations; c.f. Theorem 3 in [6]). There exists a combination graph over T P of size O(2 n n 3 ) that represents C sd (T P ).…”

“…After removing all dead ends in Γ, which is also possible in time linear in the size of Γ, enumeration of the corresponding set C requires time at most linear in the length of the longest ⊥-⊤ path per enumerated object. We will abstain from describing the required algorithms in detail, and instead refer to [6] for an example.…”

“…We also define the set Cš t P , which contains all crossing-free combinations C of S st P with the following additional properties. 6 In what follows, the out-degree of a point p ∈ P in C denotes the number of segments in C that have p as an endpoint and are directed away from p.…”

“…. , 6 that are assigned to the points in P for any given combination C in Cš c P . Color 1 means that the corresponding point p has degree 2 and exposes no drain.…”

“…However, for a long time no counting algorithm was always provably faster than enumerating the set whose size was to be determined. For triangulations this changed with a remarkable paper by Alvarez and Seidel [6], who showed how to compute the number tr(P ) in time O(2 n n 2 ). This is always exponentially faster than enumeration because tr(P ) = Ω * (2.43 n ) holds for all sets P , see Table 1.…”