Binary plane partitions for disjoint line segments
Proceedings of the twenty-fifth annual symposium on Computational geometry
Persistent predecessor search and orthogonal point location on the word RAM
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Persistent Predecessor Search and Orthogonal Point Location on the Word RAM
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
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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.