Binary space partitions for 3D subdivisions

We consider the following question: Given a subdivision of space into <i>n</i> convex polyhedral cells, what is the worst-case complexity of a binary space partition (BSP) for the subdivision? We show that if the subdivision is rectangular and axis-aligned, then the worstcase complexity of an axis-aligned BSP is Ω(<i>n</i><sup>4/3</sup>) and <i>O</i>(<i>n</i><sup>α</sup> log<sup>2</sup> <i>n</i>), where α = 1 + log<inf>2</inf>(4/3 ) = 1.4150375 .... By contrast, it is known that the BSP of a collection of <i>n</i> rectangular cells not forming a subdivision has worstcase complexity Θ(<i>n</i><sup>3/2</sup>). We also show that the worstcase complexity of a BSP for a general convex polyhedral subdivision of total complexity <i>O</i>(<i>n</i>) is Ω(<i>n</i><sup>3/2</sup>).

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