We provide a variety of new results, including upper and lower bounds, as well as simpler proof techniques for the efficient construction of binary space partitions (BSP's) of axis-parallel segments, rectangles, and hyperrectangles. (a) A consequence of the analysis in  is that any set of n axis-parallel and pairwise-disjoint line segments in the plane admits a binary space partition of size at most 2n - 1. We establish a worst-case lower bound of 2n - o(n) for the size of such a BSP, thus showing that this bound is almost tight in the worst case. (b) We give an improved worst-case lower bound of 9/4n - o(n) on the size of a BSP for isothetic pairwise disjoint rectangles. (c) We present simple methods, with equally simple analysis, for constructing BSP's for axisparallel segments in higher dimensions, simplifying the technique of  and improving the constants. (d) We obtain an alternative construction (to that in ) of BSP's for collections of axis-parallel rectangles in 3-space. (e) We present a construction of BSP's of size O(n5/3) for n axis-parallel pairwise disjoint 2-rectangles in R4, and give a matching worstcase lower bound of Ω(n5/3) for the size of such a BSP. (f) We extend the results of  to axis-parallel k-dimensional rectangles in Rd, for k < d/2, and obtain a worst-case tight bound of Θ(nd/(d - k)) for the size of a BSP of n rectangles. Both upper and lower bounds also hold for d/2 ≤ k ≤ d - 1 if we allow the rectangles to intersect.
|Number of pages||10|
|State||Published - 2001|
|Event||17th Annual Symposium on Computational Geometry (SCG'01) - Medford, MA, United States|
Duration: 3 Jun 2001 → 5 Jun 2001
|Conference||17th Annual Symposium on Computational Geometry (SCG'01)|
|Period||3/06/01 → 5/06/01|