## Abstract

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(nlog^{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(nlog^{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(nlogn) 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)logn), where κ is, as above, the number of vertices in U(C), which now can vary between O(1) and Θ(n^{3}).

Original language | English |
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Journal | Discrete and Computational Geometry |

DOIs | |

State | Accepted/In press - 2024 |

## Keywords

- 52C45
- 68U05
- Decomposition
- Fat boxes
- Free space
- Union complexity