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Discrete & Computational Geometry

, Volume 43, Issue 2, pp 221–241 | Cite as

Cuttings for Disks and Axis-Aligned Rectangles in Three-Space

  • Eynat Rafalin
  • Diane L. Souvaine
  • Csaba D. Tóth
Article

Abstract

We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r∈ℕ, a \(\frac{1}{r}\) -cutting is a covering of the space with simplices such that the interior of each simplex intersects at most n/r objects. For n pairwise disjoint disks in ℝ3 and a parameter r∈ℕ, we construct a \(\frac{1}{r}\) -cutting of size O(r 2). For n axis-aligned rectangles in ℝ3, we construct a \(\frac{1}{r}\) -cutting of size O(r 3/2).

As an application related to multi-point location in three-space, we present tight bounds on the cost of spanning trees across barriers. Given n points and a finite set of disjoint disk barriers in ℝ3, the points can be connected with a straight line spanning tree such that every disk is stabbed by at most \(O(\sqrt{n})\) edges of the tree. If the barriers are axis-aligned rectangles, then there is a straight line spanning tree such that every rectangle is stabbed by O(n 1/3) edges. Both bounds are best possible.

Keywords

Simplicial partition Binary space partition Spanning tree 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Eynat Rafalin
    • 1
  • Diane L. Souvaine
    • 2
  • Csaba D. Tóth
    • 3
  1. 1.Google Inc.Mountain ViewUSA
  2. 2.Department of Computer ScienceTufts UniversityMedfordUSA
  3. 3.Department of MathematicsUniversity of CalgaryCalgaryCanada

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