Discrete & Computational Geometry

, Volume 30, Issue 4, pp 509–528

Asymptotically Efficient Triangulations of the d-Cube

Article

DOI: 10.1007/s00454-003-2845-5

Cite this article as:
Orden, D. & Santos, F. Discrete Comput Geom (2003) 30: 509. doi:10.1007/s00454-003-2845-5

Abstract

Let $P$ and $Q$ be polytopes, the first of “low” dimension and the second of “high” dimension. We show how to triangulate the product $P \times Q$ efficiently (i.e., with few simplices) starting with a given triangulation of $Q$. Our method has a computational part, where we need to compute an efficient triangulation of $P \times \Delta^m$, for a (small) natural number $m$ of our choice. $\Delta^m$ denotes the $m$-simplex. Our procedure can be applied to obtain (asymptotically) efficient triangulations of the cube $I^n$: We decompose $I^n = I^k \times I^{n-k}$, for a small $k$. Then we recursively assume we have obtained an efficient triangulation of the second factor and use our method to triangulate the product. The outcome is that using $k=3$ and $m=2$, we can triangulate $I^n$ with $O(0.816^{n} n!)$ simplices, instead of the $O(0.840^{n} n!)$ achievable before.

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Estad’istica y Computación, Universidad de Cantabria, Av. Los Castros s/n, E-39005 Santander, CantabriaSpain