Binary space partitions for line segments with a limited number of directions

This paper considers binary space partitions (BSP) for <i>n</i> disjoint line segments in the plane. It is known that there is a BSP of size at most <i>O</i>(<i>n</i> log <i>n</i>), in general, and the smallest BSP can be as big as Ω(<i>n</i> log <i>n</i>/log log <i>n</i>) in the worst case. It is shown that there exists a BSP of size <i>O</i>(<i>kn</i>) if the line segments have at most <i>k</i> different orientations. No linear upper bound was known, so far, for any fixed <i>k</i> > 2.

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