Algorithmic Issues

Abstract

In previous chapters, we have not been directly concerned with how to actually carry on calculations required to investigate triangulations. For example, how can we efficiently verify whether a set of simplices forms a triangulation? How can we verify whether a triangulation is regular? We have already counted triangulations via formulas for a few well-structured instances, but we have not discussed how one could actually list or enumerate all triangulations of arbitrary dimensional configurations (though we did some of that in dimension two). Similarly, we have not addressed the issue of finding optimal or special triangulations explicitly. For instance, given a d-dimensional point configuration (e.g., the vertices of a d-cube), how can we find a triangulation with the smallest number of d-simplices? In this chapter, we deal with the computation and complexity of carrying through some natural computations on the space of triangulations.