Halving Balls in Deterministic Linear Time

  • Michael Hoffmann
  • Vincent Kusters
  • Tillmann Miltzow
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8737)

Abstract

Let \({\mathcal{D}}\) be a set of n pairwise disjoint unit balls in ℝ d and P the set of their center points. A hyperplane \({\mathcal{H}}\) is an m-separator for \({\mathcal{D}}\) if each closed halfspace bounded by \({\mathcal{H}}\) contains at least m points from P. This generalizes the notion of halving hyperplanes (n/2-separators). The analogous notion for point sets has been well studied. Separators have various applications, for instance, in divide-and-conquer schemes. In such a scheme any ball that is intersected by the separating hyperplane may still interact with both sides of the partition. Therefore it is desirable that the separating hyperplane intersects a small number of balls only.

We present three deterministic algorithms to bisect or approximately bisect a given set of n disjoint unit balls by a hyperplane: firstly, a linear-time algorithm to construct an αn-separator in ℝ d , for 0 < α < 1/2, that intersects at most cn (d − 1)/d balls, where c depends on d and α. The number of balls intersected is best possible up to the constant c. Secondly, we present a near-linear time algorithm to find an (n/2 − o(n))-separator in ℝ d that intersects o(n) balls. Finally, we give a linear-time algorithm to construct a halving line in ℝ2 that intersects O(n (5/6) + ε ) disks.

Our results improve the runtime of a disk sliding algorithm by Bereg, Dumitrescu and Pach. In addition, our results improve and derandomize an algorithm to construct a space decomposition used by Löffler and Mulzer to construct an onion decomposition for imprecise points.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Katchalski, M., Pulleyblank, W.R.: Cutting disjoint disks by straight lines. Discrete & Computational Geometry 4, 239–243 (1989)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Atkin, A.O.L., Bernstein, D.J.: Prime sieves using binary quadratic forms. Math. Comput. 73(246), 1023–1030 (2004)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Barequet, G.: A lower bound for Heilbronn’s triangle problem in d dimensions. SIAM Journal on Discrete Mathematics 14(2), 230–236 (2001)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bereg, S., Dumitrescu, A., Pach, J.: Sliding disks in the plane. International Journal of Computational Geometry & Applications 18(05), 373–387 (2008)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Blum, M., Floyd, R.W., Pratt, V., Rivest, R.L., Tarjan, R.E.: Time bounds for selection. Journal of Computer and System Sciences 7(4), 448–461 (1973)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Esposito, L., Ferone, V., Kawohl, B., Nitsch, C., Trombetti, C.: The longest shortest fence and sharp Poincaré–sobolev inequalities. Archive for Rational Mechanics and Analysis 206(3), 821–851 (2012)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Held, M., Mitchell, J.S.: Triangulating input-constrained planar point sets. Information Processing Letters 109(1), 54–56 (2008)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Lefmann, H.: On Heilbronn’s problem in higher dimension. Combinatorica 23(4), 669–680 (2003)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Lo, C.-Y., Matoušek, J., Steiger, W.L.: Algorithms for ham-sandwich cuts. Discrete & Computational Geometry 11, 433–452 (1994)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Löffler, M., Mulzer, W.: Unions of onions: preprocessing imprecise points for fast onion layer decomposition. In: Algorithms and Data Structures, pp. 487–498. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Martini, H., Schöbel, A.: Median hyperplanes in normed spaces – a survey. Discrete Applied Mathematics 89(1), 181–195 (1998)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Matoušek, J.: Efficient partition trees. Discrete & Computational Geometry 8(1), 315–334 (1992)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Pach, J., Sharir, M.: Combinatorial geometry and its algorithmic applications: The Alcalá lectures. Mathematical Surveys and Monographs, vol. 152. Amer. Math. Soc. (2009)Google Scholar
  14. 14.
    Roth, K.F.: On a problem of Heilbronn. J. London Math. Soc. 26(3), 198–204 (1951)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Tverberg, H.: A seperation property of plane convex sets. Mathematica Scandinavica 45, 255–260 (1979)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Michael Hoffmann
    • 1
  • Vincent Kusters
    • 1
  • Tillmann Miltzow
    • 2
  1. 1.Department of Computer ScienceETH ZürichSwitzerland
  2. 2.Institute of Computer ScienceFreie Universität BerlinGermany

Personalised recommendations