Minimum weight pseudo-triangulations

We consider the problem of computing a minimum weight pseudo-triangulation of a set S of n points in the plane. We first present an O(nlogn)-time algorithm that produces a pseudo-triangulation of weight O(logn⋅wt(M(S))) which is shown to be asymptotically worst-case optimal, i.e., there exists a point set S for which every pseudo-triangulation has weight Ω(logn⋅wt(M(S))), where wt(M(S)) is the weight of a minimum weight spanning tree of S. We also present a constant factor approximation algorithm running in cubic time. In the process we give an algorithm that produces a minimum weight pseudo-triangulation of a simple polygon.