Locally minimal triangulations.

Abstract

A triangulation of a set of points in the plane is a maximal set of segments connecting those points such that no two segments cross each other, forming a set of triangles. The weight of a triangulation is defined as the sum of the Euclidean lengths of the segments. Minimizing this weight is a goal in some applications, for example when the segments indicate the locations for interpolation of data. The triangulation of minimal weight is called the Minimum Weight Triangulation (MWT). Although the MWT can be computed in polynomial time for some point sets, such as points that are on the boundary of a convex polygon, no known algorithm is guaranteed to compute the MWT for general point sets in polynomial time. This leaves two areas for study, both of which are examined: attempts at quick computation of the MWT and exploration of approximations. Approximation methods that are examined here rely on local properties that are viewed as minimal. In particular, pairs of triangles must form the MWT of a quadrilateral. Triangulations can be formed by sets of these pairs, and several of these are examined. Algorithms that produce particular sets are run on many input sets, producing empirical evidence that shows that some appear to have weight within a factor of 3 of the weight of the MWT. The best of these triangulations is one achieved by a greedy algorithm, which has been called the Greedy Triangulation (GT) in the literature. Faster algorithms for computing the GT are presented, with expected runtimes O( n) for the points taken from a uniform distribution and worst-case runtime O(n2 log n). Also presented is a polynomial-time algorithm that produces the “triangle-based LMT skeleton”, a subset of the MWT. For many input sets, the portion that is not triangulated by this skeleton is a set of simple polygons, which can be finished in polynomial time, hence computing the MWT in polynomial time. In particular, this method finds the MWT of a point set for which the algorithm previously considered to be the best (the LMT-skeleton) was unable to complete the MWT.