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

  • Authors:

  • Adrian Dumitrescu;Joseph S. B. Mitchell;Micha Sharir

  • Affiliations:

  • Department of Electrical Engineering and Computer Science, University of Wisconsin–Milwaukee, Milwaukee, WI 53211, USA;Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794, USA;School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA

  • Venue:
  • Discrete & Computational Geometry
  • Year:
  • 2004

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Abstract

We provide a variety of new upper and lower bounds and simpler proof techniques for the efficient construction of binary space partitions (BSPs) of axis-parallel rectangles of various dimensions. (a) We construct a set of $n$ disjoint axis-parallel segments in the plane such that any binary space auto-partition has size at least $2n-o(n)$, almost matching an upper bound of d’Amore and Franciosa. (b) We establish a similar lower bound of $7n/3-o(n)$ for disjoint rectangles in the plane. (c) We simplify and improve BSP constructions of Paterson and Yao for disjoint segments in $\reals^d$ and disjoint rectangles in $\reals^3$. (d) We derive a worst-case bound of $\Theta(n^{5/3})$ for the size of BSPs of disjoint $2$-rectangles in $4$-space. (e) For disjoint $k$-rectangles in $d$-space, we prove the worst-case bound $\Theta(n^{d/(d-k)})$, for any $k