Document Open Access Logo

On the Complexity of the k-Level in Arrangements of Pseudoplanes

Authors Micha Sharir, Chen Ziv



PDF
Thumbnail PDF

File

LIPIcs.SoCG.2019.62.pdf
  • Filesize: 1.45 MB
  • 15 pages

Document Identifiers

Author Details

Micha Sharir
  • School of Computer Science, Tel Aviv University, Tel Aviv, Israel
Chen Ziv
  • School of Computer Science, Tel Aviv University, Tel Aviv, Israel

Cite AsGet BibTex

Micha Sharir and Chen Ziv. On the Complexity of the k-Level in Arrangements of Pseudoplanes. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 62:1-62:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.62

Abstract

A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in R^d (vertices with exactly k of the hyperplanes passing below them). This is essentially a dual version of the k-set problem, which, in a primal setting, seeks bounds for the maximum number of k-sets determined by n points in R^d, where a k-set is a subset of size k that can be separated from its complement by a hyperplane. The k-set problem is still wide open even in the plane. In three dimensions, the best known upper and lower bounds are, respectively, O(nk^{3/2}) [M. Sharir et al., 2001] and nk * 2^{Omega(sqrt{log k})} [G. Tóth, 2000]. In its dual version, the problem can be generalized by replacing hyperplanes by other families of surfaces (or curves in the planes). Reasonably sharp bounds have been obtained for curves in the plane [M. Sharir and J. Zahl, 2017; H. Tamaki and T. Tokuyama, 2003], but the known upper bounds are rather weak for more general surfaces, already in three dimensions, except for the case of triangles [P. K. Agarwal et al., 1998]. The best known general bound, due to Chan [T. M. Chan, 2012] is O(n^{2.997}), for families of surfaces that satisfy certain (fairly weak) properties. In this paper we consider the case of pseudoplanes in R^3 (defined in detail in the introduction), and establish the upper bound O(nk^{5/3}) for the number of k-level vertices in an arrangement of n pseudoplanes. The bound is obtained by establishing suitable (and nontrivial) extensions of dual versions of classical tools that have been used in studying the primal k-set problem, such as the Lovász Lemma and the Crossing Lemma.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Theory of computation → Computational geometry
Keywords
  • k-level
  • pseudoplanes
  • arrangements
  • three dimensions
  • k-sets

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. P. K. Agarwal, B. Aronov, T. M. Chan, and M. Sharir. On levels in arrangements of lines, segments, planes, and triangles. Discrete Comput. Geom., 19:315-331, 1998. Google Scholar
  2. P. K. Agarwal and M. Sharir. Davenport-Schinzel Sequences and Their Geometric Applications, pages 216-217. Cambridge University Press, NY, USA, 1995. Google Scholar
  3. P. K. Agarwal and M. Sharir. Pseudo-line arrangements: duality, algorithms, and appliations. SIAM J. Comput., pages 34:526-552, 2005. Google Scholar
  4. M. Ajtai, V. Chvátal, M. M. Newborn, and E. Szemerédi. Crossing-free subgraphs, pages 9-12. North-Holland Mathematics Studies, Amsterdam, 1982. Google Scholar
  5. I. Bárány, Z. Füredi, and L. Lovász. On the number of halving planes. Combinatorica, 10:175-183, 1990. Google Scholar
  6. T. M. Chan. On the bichromatic k-set problem. ACM Transactions on Algorithms, 6(4):62:1-62:20, 2010. Google Scholar
  7. T. M. Chan. On levels in arrangements of surfaces in three dimensions. Discrete Comput. Geom., 48:1-18, 2012. Google Scholar
  8. T. K. Dey. Improved bounds on planar k-sets and related problems. Discrete Comput. Geom., 19:373-382, 1998. Google Scholar
  9. T. K. Dey and H. Edelsbrunner. Counting triangle crossing and halving planes. Discrete Comput. Geom., 12:281-289, 1994. Google Scholar
  10. H. Edelsbrunner. Algorithms in Combinatorial Geometry, pages 271-291. Springer Verlag, Heidelberg, 1987. Google Scholar
  11. H. Edelsbrunner and L. Guibas. Topologically sweeping an arrangement. J. Computer Systems Sci., 38:165-194, 1989. Google Scholar
  12. J. Hershberger and J. Snoeyink. Sweeping arrangements of curves. Proc. 5th ACM Sympos. on Computational Geometry, pages 354-363, 1989. Google Scholar
  13. G. Nivasch. An improved, simple construction of many halving edges. Contemporary Mathematics, 453:299-306, 2008. Google Scholar
  14. M. Sharir. An improved bound for k-sets in four dimensions. Combinatorics, Probability and Computing, 20:119-129, 2011. Google Scholar
  15. M. Sharir, S. Smorodinsky, and G. Tardos. An improved bound for k-sets in three dimensions. Discrete Comput. Geom., 26:195-204, 2001. Google Scholar
  16. M. Sharir and J. Zahl. Cutting algebraic curves into pseudo-segments and applications. J. Combinat. Theory, Ser. A, pages 150:37-42, 2017. Google Scholar
  17. M. Sharir and C. Ziv. On the complexity of the k-level in arrangements of pseudoplanes, 2019. URL: http://arxiv.org/abs/1903.07196.
  18. H. Tamaki and T. Tokuyama. A characterization of planar graphs by pseudo-line arrangements. Algorithmica, 35:269-285, 2003. Google Scholar
  19. G. Tóth. Point sets with many k-sets. Proc. 16th Annual Sympos. on Computational Geometry, 35:37-42, 2000. Google Scholar
  20. C. Ziv. On the Complexity of the k-Level in Arrangements of Pseudoplanes. Master’s thesis, School of Computer Science, Tel Aviv University, 2019. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail