Enumerating triangulations for products of two simplices and for arbitrary configurations of points

Abstract

We propose algorithms to enumerate

• (1) classes of regular triangulations in respect of symmetry for products of two simplices and

• (2) all triangulations, regular or not, for arbitrary configurations of points. There are many results for triangulations in two dimension, but little is known for higher dimensions. Both objects we enumerate in this paper are for general dimensions.

Products of two simplices, our first object, are polytopes rather simple, but their triangulations are not yet well understood. Since these polytopes are highly symmetric, counting all triangulations naively is inefficient: we may count the “same” triangulation many times. Our first algorithm enumerates the classes of regular triangulations, a subset of triangulations, with respect to the symmetry. We use reverse search technique, utilizing the symmetric structure of the polytope. This enables time complexity linear to the number of these classes, and space complexity of the size of several triangulations.

Even for the polytope of this product, a nonregular triangulation was found. Though algorithms to enumerate regular triangulations are studied well, no efficient algorithm to enumerate all triangulations, including nonregular ones, has been known. Our second algorithm handles this problem for arbitrary configurations of points. It formulates triangulations as maximal independent sets of the intersection graph, and applies a general maximal independent set enumeration algorithm. The intersection graph here is the graph with all maximal dimensional simplices the vertices and edges between those intersecting improperly. This algorithm works in time proportional to the number of maximal independent sets. We last apply this second algorithm to the polytope of the product.