Improved construction of vertical decompositions of three-dimensional arrangements

Abstract

We present new results concerning the refinement of three-dimensional arrangements by vertical decompositions. First, we describe a new output-sensitive algorithm for computing the vertical decomposition of arrangements of n triangles in O(nlog2 n+Vlog n) time, where V is the complexity of the decomposition. This improves significantly over the best previously known algorithms. Next, we propose an alternative sparser refinement, which we call the partial vertical decomposition and has the advantages that it produces fewer cells and requires lower degree constructors. We adapt the output-sensitive algorithm to efficiently compute the partial decomposition as well. We implemented algorithms that construct the full and the partial decompositions and we compare the two types theoretically and experimentally. The improved output-sensitive construction extends to the case of arrangements of n well-behaved surfaces with the same asymptotic running time. We also extended the implementation to the case of polyhedral surfaces---this can serve as the basis for robust implementation of approximations of arrangements of general surfaces.