Asymptotically efficient triangulations of the d-cube

Details Asymptotically efficient triangulations of the d-cube Asymptotically efficient triangulations of the d-cube

2002-04-11

arxiv.org/abs/math/0204157v3

Discrete Comput. Geom., 30:4 (2003), 509-528.

Mathematics

Combinatorics

19 pages, 6 figures. Only minor changes from previous versions, some suggested by anonymous referees. Paper accepted in "Discr

Scientific paper

10.1007/s00454-003-2845-5

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.

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