Skip to main content
SpringerLink
Log in

Decomposing Arrangements of Hyperplanes: VC-Dimension, Combinatorial Dimension, and Point Location

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

This work is motivated by several basic problems and techniques that rely on space decomposition of arrangements of hyperplanes in high-dimensional spaces, most notably Meiser’s 1993 algorithm (Meiser in Inf Comput 106(2):286–303, 1993) for point location in such arrangements. A standard approach to these problems is via random sampling, in which one draws a random sample of the hyperplanes, constructs a suitable decomposition of its arrangement, and recurses within each cell of the decomposition with the subset of hyperplanes that cross the cell. The efficiency of the resulting algorithm depends on the quality of the sample, which is controlled by various parameters. One of these parameters is the classical VC-dimension, and its associated primal shatter dimension, of a suitably defined corresponding range space. Another parameter, which we refer to here as the combinatorial dimension, is the maximum number of hyperplanes that are needed to define a cell that can arise in the decomposition of some sample of the input hyperplanes; this parameter arises in Clarkson’s (and later Clarkson and Shor’s) random sampling technique. We re-examine these parameters for the two main space decomposition techniques—bottom-vertex triangulation, and vertical decomposition, including their explicit dependence on the dimension d, and discover several unexpected phenomena, which show that, in both techniques, there are large gaps between the VC-dimension (and primal shatter dimension), and the combinatorial dimension. Our main application is to point location in an arrangement of n hyperplanes is \(\mathbb {R}^d\), in which we show that the query cost in Meiser’s algorithm can be improved if one uses vertical decomposition instead of bottom-vertex triangulation, at the cost of some increase in the preprocessing cost and storage (which seem to be stated incorrectly, and are not worked out, in Meiser’s work). Our improved bounds rely on establishing several new structural properties and improved complexity bounds for vertical decomposition, which are of independent interest, and which we expect to find additional applications.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. In most applications, the VC-dimension and primal shatter dimension are equivalent up to a logarithmic factor. However, when the leading constant in the “growth function” (see later in this section for the definition) is considerably large, it may increase the gap between these two measures—see Sect. 4.3 for such a scenario.

  2. This large dependence of the storage on d was missed in the analysis in [29]; it can be reduced by increasing (significantly) the query time—see Sect. 5 for full details.

  3. A follow-up work of Kane et al. [22] extends their technique to arbitrary hyperplanes. It offers a simpler space decomposition technique, but results in worse query time than the one derived here and in [15].

  4. Kane et al. use a more general notion of “low inference dimension”, and show that low-complexity hyperplanes do have low inference dimension.

  5. The general statement is that if \(\rho \ge 2x\ln x\) then \(\rho \ge x\ln \rho \). We will use this property again, in the proof of Lemma 3.7.

  6. We will shortly argue that the actual number of hyperplanes is only at most \(n-1\).

  7. Any single point q is in fact described by only D / 2 variables, but we extend this polynomial to \(\mathbb {R}^D\), making it independent of the additional coordinates.

  8. Specifically, Milnor and Thom proved [4, Chap. 7] a bound on the sum of the Betti numbers of algebraic sets (presented by the zero set of a collection of polynomials). Using these bounds, Warren [36] proved a bound on the number of connected components of the realizations of strict sign conditions of a family of polynomials (namely, the cells of the complement of the union of the zero sets of these polynomials), see [4, Chap. 7] for more details.

  9. Here the notation \({\widetilde{O}}\) hides a polylogarithmic factor in d.

  10. The number of distinct cells is only \(O(\rho ^d)\), but because of their possible repetitions in the trees \(Q_C\), we simply multiply by d the number of leaves.

  11. Since the bounds on the complexity of vertical decomposition are not known to be tight, it is conceivable that we do not pay anything extra for storage using this technique.

  12. This \((d-1)\)-dimensional sign pattern can also be computed and stored during preprocessing, but computing it on the fly, as we do here, does not affect the asymptotic cost of the query, and somewhat simplifies the preprocessing and storage.

  13. These arbitrary choices of affine representations of projective quantities should give us some flexibility in the algorithm that follows. However, we do not see how to exploit this flexibility; neither does the machinery in [23].

  14. To traverse a compact sign pattern in \(T_R\) in constant time per hyperplane, we store the children of each node in \(T_R\) in a hash table.

References

  1. Agarwal, P.K., Matoušek, J., Schwarzkopf, O.: Computing many faces in arrangements of lines and segments. SIAM J. Comput. 27(2), 491–505 (1998)

    MathSciNet  MATH  Google Scholar 

  2. Agarwal, P.K., Sharir, M.: Arrangements and their applications. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 49–119. North-Holland, Amsterdam (2000)

    Google Scholar 

  3. Basu, S.: Algorithms in real algebraic geometry: a survey. In: Monnier, J.-P. (ed.) Real Algebraic Geometry. Panoramas and Synthèses, vol. 51, pp. 107–153. Société Mathématique de France, Paris (2017). arXiv:1409.1534

  4. Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 10, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  5. Blumer, A., Ehrenfeucht, A., Haussler, D., Warmuth, M.K.: Learnability and the Vapnik–Chervonenkis dimension. J. Assoc. Comput. Mach. 36(4), 929–965 (1989)

    MathSciNet  MATH  Google Scholar 

  6. Chazelle, B.: Cuttings. In: Mehta, D.P., Sahni, S. (eds.) Handbook of Data Structures and Applications. Chapman and Hall / CRC, Boca Raton (2005)

    Google Scholar 

  7. Chazelle, B., Edelsbrunner, H., Guibas, L.J., Sharir, M.: A singly-exponential stratification scheme for real semi-algebraic varieties and its applications. Theoret. Comput. Sci. 84, 77–105 (1991). Also in 16th Int. Colloq. on Automata, Languages and Programming, pp. 179–193 (1989)

  8. Chazelle, B., Friedman, J.: A deterministic view of random sampling and its use in geometry. Combinatorica 10(3), 229–249 (1990)

    MathSciNet  MATH  Google Scholar 

  9. Clarkson, K.L.: New applications of random sampling in computational geometry. Discrete Comput. Geom. 2(2), 195–222 (1987)

    MathSciNet  MATH  Google Scholar 

  10. Clarkson, K.L.: A randomized algorithm for closest-point queries. SIAM J. Comput. 17(4), 830–847 (1988)

    MathSciNet  MATH  Google Scholar 

  11. Clarkson, K.L., Mehlhorn, K., Seidel, R.: Four results on randomized incremental constructions. Comput. Geom. 3(4), 185–212 (1993)

    MathSciNet  MATH  Google Scholar 

  12. Clarkson, K.L., Shor, P.W.: Applications of random sampling in computational geometry II. Discrete Comput. Geom. 4(5), 387–421 (1989)

    MathSciNet  MATH  Google Scholar 

  13. de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Berlin (2008)

    MATH  Google Scholar 

  14. de Berg, M., Schwarzkopf, O.: Cuttings and applications. Int. J. Comput. Geom. Appl. 5(4), 343–355 (1995)

    MathSciNet  MATH  Google Scholar 

  15. Ezra, E., Sharir, M.: A nearly quadratic bound for point-location in hyperplane arrangements, in the linear decision tree model. Discrete Comput. Geom. 61(4), 735–755 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Gärtner, B., Welzl, E.: Linear programming: randomization and abstract frameworks. In: Proceedings of the 13th Annual Symposium on Theoretical Aspects of Computer Science. Springer Lecture Notes in Computer Science, vol. 1046, pp. 669–687. Springer, Berlin (1996)

  17. Guibas, L.J., Halperin, D., Matoušek, J., Sharir, M.: Vertical decomposition of arrangements of hyperplanes in four dimensions. Discrete Comput. Geom. 14(2), 113–122 (1995)

    MathSciNet  MATH  Google Scholar 

  18. Har-Peled, S.: Constructing planar cuttings in theory and practice. SIAM J. Comput. 29(6), 2016–2039 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Har-Peled, S.: Geometric Approximation Algorithms. Mathematical Surveys and Monographs, vol. 173. American Mathematical Society, Providence (2011)

    MATH  Google Scholar 

  20. Har-Peled, S.: Shortest path in a polygon using sublinear space. J. Comput. Geom. 7(2), 19–45 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Haussler, D., Welzl, E.: \(\varepsilon \)-Nets and simplex range queries. Discrete Comput. Geom. 2(2), 127–151 (1987)

    MathSciNet  MATH  Google Scholar 

  22. Kane, D.M., Lovett, S., Moran, S.: Generalized comparison trees for point-location problems. In: Chatzigiannakis, C. et al. (eds.) Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP). LIPIcs. Leibniz International Proceedings in Informatics, vol. 107, pp. 82:1–82:13. Schloss Dagstuhl. Leibniz-Zentrum für Informatik, Wadern (2018)

  23. Kane, D.M., Lovett, S., Moran, S.: Near-optimal linear decision trees for k-SUM and related problems. In: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, (STOC), pp. 554–563. ACM, New York (2018)

  24. Kane, D.M., Lovett, S., Moran, S., Zhang, J.: Active classification with comparison queries. In: Proceedings of the 58th IEEE Annual Symposium on Foundations of Computer Science, (FOCS), pp. 355–366. IEEE Computer Society, Los Alamitos (2017)

  25. Koltun, V.: Sharp bounds for vertical decompositions of linear arrangements in four dimensions. Discrete Comput. Geom. 31(3), 435–460 (2004)

    MathSciNet  MATH  Google Scholar 

  26. Komlós, J., Pach, J., Woeginger, G.: Almost tight bounds for \(\varepsilon \)-nets. Discrete Comput. Geom. 7(2), 163–173 (1992)

    MathSciNet  MATH  Google Scholar 

  27. Liu, D.: A note on point location in arrangements of hyperplanes. Inf. Process. Lett. 90(2), 93–95 (2004)

    MathSciNet  MATH  Google Scholar 

  28. Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)

    Google Scholar 

  29. Meiser, S.: Point location in arrangements of hyperplanes. Inf. Comput. 106(2), 286–303 (1993)

    MathSciNet  MATH  Google Scholar 

  30. Meyer auf der Heide, F.: A polynomial linear search algorithm for the \(n\)-dimensional knapsack problem. J. Assoc. Comput. Mach. 31(3), 668–676 (1984)

    MathSciNet  MATH  Google Scholar 

  31. Pach, J., Agarwal, P.K.: Combinatorial Geometry. Wiley, New York (1995)

    MATH  Google Scholar 

  32. Sharir, M., Agarwal, P.K.: Davenport–Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York (1995)

    MATH  Google Scholar 

  33. Spanier, E.H.: Algebraic Topology. Springer, New York (1966)

    MATH  Google Scholar 

  34. Vapnik, V.N., Chervonenkis, A.Ya.: On the uniform convergence of the frequencies of occurrence of events to their probabilities. In: Empirical Inference, pp. 7–12. Springer, Berlin (2013)

  35. Vapnik, V.N., Chervonenkis, A.Ya.: On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16, 264–280 (1971)

  36. Warren, H.E.: Lower bound for approximation by nonlinear manifolds. Trans. Am. Math. Soc. 133(1), 167–178 (1968)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank an anonymous referee for his/her useful comments.

Author information

Authors and Affiliations

  1. Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel

    Esther Ezra

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Esther Ezra

  3. Department of Computer Science, University of Illinois, 201 N. Goodwin Avenue, Urbana, IL, 61801, USA

    Sariel Har-Peled

  4. Blavatnik School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Haim Kaplan & Micha Sharir

Authors
  1. Esther Ezra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sariel Har-Peled
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Haim Kaplan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Micha Sharir
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Esther Ezra.

Additional information

Editor in Charge: János Pach

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Work by Esther Ezra was partially supported by NSF CAREER under grant CCF:AF-1553354 and by Grant 824/17 from the Israel Science Foundation. Work by Sariel Har-Peled was partially supported by NSF AF awards CCF-1421231 and CCF-1217462. Work by Haim Kaplan was supported by Grant 1841/14 from the Israel Science Fundation and by Grant 1161/2011 from the German Israeli Science Fundation (GIF). Work by Micha Sharir has been supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, and by Grant 892/13 from the Israel Science Foundation. Work by Haim Kaplan and Micha Sharir was also supported by the Israeli Centers for Research Excellence (I-CORE) program (center no. 4/11), by the Blavatnik Computer Science Research Fund at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ezra, E., Har-Peled, S., Kaplan, H. et al. Decomposing Arrangements of Hyperplanes: VC-Dimension, Combinatorial Dimension, and Point Location. Discrete Comput Geom 64, 109–173 (2020). https://doi.org/10.1007/s00454-019-00141-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-019-00141-7

Keywords

Mathematics Subject Classification

Search

Navigation