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

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