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

We give an algorithm that determines the number (S) of straight line triangulations of a set S of n points in the plane in worst case time O(n2 2n). This is the the first algorithm that is provably faster than enumeration, since (S) is known to be Ω(2.43n) 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.

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