Abstract
We prove that, for any constant e>0, the complexity of the vertical decomposition of a set of n triangles in three-dimensional space is O(n2+e+K), where K is the complexity of the arrangement of the triangles. For a single cell the complexity of the vertical decomposition is shown to be O(n2+e). These bounds are almost tight in the worst case.We also give a deterministic output-sensitive algorithm for computing the vertical decomposition that runs in O(n2logn+Vlogn) time, where V is the complexity of the decomposition. The algorithm is reasonably simple (in particular, it tries to perform as much of the computation in two-dimensional spaces as possible) and thus is a good candidate for efficient implementations.
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