Binary space partitions of orthogonal subdivisions

We consider the problem of constructing binary space partitions (BSPs) for orthogonal subdivisions (space filling packings of boxes) in <i>d</i>-space. We show that a subdivision with <i>n</i> boxes can be refined into a BSP of size <i>O</i>(<i>n</i> d+1/3), for all <i>d</i> ≥ 3, and that such a partition can be computed in time <i>O</i>(<i>K</i> log <i>n</i>), where <i>K</i> is the size of the BSP produced. Our upper bound on the BSP size is tight for 3-dimensional subdivisions in higher dimensions, this is the first nontrivial result for general full-dimensional boxes. We also present a lower bound construction for a subdivision of <i>n</i> boxes in <i>d</i>-space that requires a BSP of size Ω(<i>n</i>^946;(d)), where <i>β</i>(<i>d</i>) converges to (1+ √5 )/2 as <i>d</i> → ∞.