Decomposing the Complement of the Union of Cubes and Boxes in Three Dimensions

Abstract

Let \(\mathcal {C}\) be a set of n axis-aligned cubes of arbitrary sizes in \({\mathbb R}^3\) in general position. Let \(\mathcal {U}:=\mathcal {U}(\mathcal {C})\) be their union, and let \(\kappa \) be the number of vertices on \(\partial \mathcal {U}\) ; \(\kappa \) can vary between O(1) and \(\Theta (n^2)\) . We present a partition of \(\mathop {\textrm{cl}}({\mathbb R}^3\setminus \mathcal {U})\) into \(O(\kappa \log ^4 n)\) axis-aligned boxes with pairwise-disjoint interiors that can be computed in \(O(n \log ^2 n + \kappa \log ^6 n)\) time if the faces of \(\partial \mathcal {U}\) are pre-computed. We also show that a partition of size \(O(\sigma \log ^4 n + \kappa \log ^2 n)\) , where \(\sigma \) is the number of input cubes that appear on \(\partial \mathcal {U}\) , can be computed in \(O(n \log ^2 n + \sigma \log ^8 n + \kappa \log ^6 n)\) time if the faces of \(\partial \mathcal {U}\) are pre-computed. The complexity and runtime bounds improve to \(O(n\log n)\) if all cubes in \(\mathcal {C}\) are congruent and the faces of \(\partial \mathcal {U}\) are pre-computed. Finally, we show that if \(\mathcal {C}\) is a set of arbitrary axis-aligned boxes in \({\mathbb R}^3\) , then a partition of \(\mathop {\textrm{cl}}({\mathbb R}^3\setminus \mathcal {U})\) into \(O(n^{3/2}+\kappa )\) boxes can be computed in time \(O((n^{3/2}+\kappa )\log n)\) , where \(\kappa \) is, as above, the number of vertices in \(\mathcal {U}(\mathcal {C})\) , which now can vary between O(1) and \(\Theta (n^3)\) .