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Binary Space Partitions for Axis-Parallel Segments, Rectangles, and Hyperrectangles

By Adrian Dumitrescu, Joseph S. B. Mitchell and Micha Sharir

Abstract

We provide a variety of new results, including upper and lower bounds, as well as simpler proof techniques for the ecient construction of binary space partitions (BSP's) of axis-parallel segments, rectangles, and hyperrectangles. (a) A consequence of the analysis in [1] 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 4 n 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 [9] and improving the constants. (d) We obtain an alternative construction (to that in [9]) of BSP's for collections of axis-parallel rectangles in 3-space. (e) We present a construction of BSP's of size O(n 5=3 ) for n axis-parallel pairwise disjoint 2-rectangles in R 4 , and give a matching worstcase lower bound of n 5=3 ) for the size of such a BSP. (f) We extend the results of [9] to axis-parallel k-dimensional rectangles in R d , for k < d=2, and obtain a worst-case tight bound of (n d=(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

Publisher: ACM Press
Year: 2001
DOI identifier: 10.1145/378583.378649
OAI identifier: oai:CiteSeerX.psu:10.1.1.22.1060
Provided by: CiteSeerX
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