Binary Space Partitions for Axis-Aligned Fat Rectangles

Abstract

It is shown that for any n disjoint axis-aligned fat rectangles in three-space there is a binary space partition (BSP) of $O(n\log^8 n)$ size and $O(\log^5 n)$ height and it can be constructed in $O(n \,\mathrm{polylog}\, n)$ time. This improves earlier bounds of Agarwal et al. [SIAM J. Comput., 29 (2000), pp. 1422–1448]. On the other hand, for every $n\in \mathbb{N}$, there are n disjoint axis-aligned fat rectangles in $\mathbb{R}^3$ such that their smallest axis-aligned BSP has $\Omega(n\log n)$ size.