Minimum Weight Pseudo-Triangulations

Abstract

We consider the problem of computing a minimum weight pseudo-triangulation of a set ${\mathcal S}$ of n points in the plane. We first present an $\mathcal O(n {\rm log} n)$-time algorithm that produces a pseudo-triangulation of weight $O(wt(\mathcal M(\mathcal S)).{\rm log} n)$ which is shown to be asymptotically worst-case optimal, i.e., there exists a point set ${\mathcal S}$ for which every pseudo-triangulation has weight $\Omega({\rm log} n.wt(\mathcal M(\mathcal S))$, where $wt(\mathcal M(\mathcal S))$ is the weight of a minimum spanning tree of ${\mathcal S}$. 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.