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A Nearly Quadratic Bound for Point-Location in Hyperplane Arrangements, in the Linear Decision Tree Model

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We consider the point location problem in an arrangement of n arbitrary hyperplanes in any dimension d, in the linear decision tree model, in which we only count linear comparisons involving the query point, and all other operations do not explicitly access the query and are for free. We mainly consider the simpler variant (which arises in many applications) where we only want to determine whether the query point lies on some input hyperplane. We present an algorithm that performs a point location query with \(O(d^2\log n)\) linear comparisons, improving the previous best result by about a factor of d. Our approach is a variant of Meiser’s technique for point location (Inf Comput 106(2):286–303, 1993) (see also Cardinal et al. in: Proceedings of the 24th European symposium on algorithms, 2016), and its improved performance is due to the use of vertical decompositions in an arrangement of hyperplanes in high dimensions, rather than bottom-vertex triangulation used in the earlier approaches. The properties of such a decomposition, both combinatorial and algorithmic (in the standard real RAM model), are developed in a companion paper (Ezra et al. arXiv:1712.02913, 2017), and are adapted here (in simplified form) for the linear decision tree model. Several applications of our algorithm are presented, such as the k-SUM problem and the Knapsack and SubsetSum problems. However, these applications have been superseded by the more recent result of Kane et al. (in: Proceedings of the 50th ACM symposium on theory of computing, 2018), obtained after the original submission (and acceptance) of the conference version of our paper (Ezra and Sharir in: Proceedings of the 33rd international symposium on computational geometry, 2017). This result only applies to ‘low-complexity’ hyperplanes (for which the \(\ell _1\)-norm of their coefficient vector is a small integer), which arise in the aforementioned applications. Still, our algorithm has currently the best performance for arbitrary hyperplanes.

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  1. 1.

    As shown in the companion paper [14], this is also true, in a certain sense, in the real RAM model.

  2. 2.

    On the other hand, their algorithm uses simpler queries that involve considerably fewer, and simpler looking auxiliary hyperplanes.

  3. 3.

    If we care about the complexity of the resulting decomposition, in terms of its dependence on n, which is an irrelevant issue in our model, it is better to stop the recursion earlier, as done in previous works. The terminal dimension is \(d=2\) or \(d=3\) in (the two respective versions of) [7], and \(d=4\) in [24].

  4. 4.

    Note that in degenerate situations, although they may not be unique, we can choose the two new defining hyperplanes arbitrarily. This does not violate the representation of the final prism.

  5. 5.

    In the original studies (see, e.g., [6]), these subcells were taken to be simplices, obtained from the bottom-vertex triangulation of \(\mathcal {A}(H)\) [27], although both forms have been used in the literature by now.

  6. 6.

    The probability approaches 1 when we increase c.

  7. 7.

    By this we mean that they do not compute any explicit expression that depends on the coordinates of \(\mathbf{x}\); recall the discussion in the introduction.


  1. 1.

    Agarwal, P.K., Sharir, M.: Arrangements and their applications. In: Sack, J., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 973–1027. North-Holland, Amsterdam (2000)

  2. 2.

    Ailon, N., Chazelle, B.: Lower bounds for linear degeneracy testing. J. ACM 52(2), 157–171 (2005)

  3. 3.

    Cardinal, J., Iacono, J., Ooms, A.: Solving \(k\)-SUM using few linear queries. In: Sankowski, P., Zaroliagis, C. (eds.) Proceedings of the 24th European Symposium on Algorithms. LIPIcs. Leibniz International Proceedings in Informatics, vol. 57, pp. 25:1–25:17. Schloss Dagstuhl. Leibniz-Zentrum für Informatik, Wadern (2016)

  4. 4.

    Chan, T.M.: More logarithmic-factor speedups for 3SUM, (median,+)-convolution, and some geometric 3SUM-hard problems. In: Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’18), pp. 881–897. SIAM, Philadelphia (2018)

  5. 5.

    Chan, T.M., Lewenstein, M.: Clustered integer 3SUM via additive combinatorics. In: Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC’15), pp. 31–40. ACM, New York (2015)

  6. 6.

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

  7. 7.

    Chazelle, B., Edelsbrunner, H., Guibas, L.J., Sharir, M.: A singly exponential stratification scheme for real semi-algebraic varieties and its applications. Theor. Comput. Sci. 84(1), 77–105 (1991). Also in: Proceedings of the 16th International Colloquium on Automata, Languages and Programming (ICALP’89), pp. 179–193 (1989)

  8. 8.

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

  9. 9.

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

  10. 10.

    Dobkin, D., Lipton, R.J.: A lower bound of \(n^2/2\) on linear search programs for the knapsack problem. J. Comput. Syst. Sci. 16(3), 413–417 (1978)

  11. 11.

    Dobkin, D., Lipton, R.J.: On the complexity of computations under varying set of primitives. J. Comput. Syst. Sci. 18(1), 86–91 (1979)

  12. 12.

    Erickson, J.: Lower bounds for linear satisfiability problems. In: Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 388–395. San Francisco, California (1995)

  13. 13.

    Ezra, E., Sharir, M.: A nearly quadratic bound for the decision tree complexity of \(k\)-SUM. In: Aronov, B., Katz, M.J. (eds.) Proceedings of the 33rd International Symposium on Computational Geometry (SoCG’17). Leibniz International Proceedings in Informatics, vol. 77, pp. 41:1–41:15. Schloss Dagstuhl. Leibniz-Zentrum für Informatik, Wadern (2017)

  14. 14.

    Ezra, E., Har-Peled, S., Kaplan, H., Sharir, M.: Decomposing arrangements of hyperplanes: VC-dimension, combinatorial dimension, and point location (2017). arXiv:1712.02913

  15. 15.

    Freund, A.: Improved subquadratic 3SUM. Algorithmica 77(2), 440–458 (2017)

  16. 16.

    Gajentaan, A., Overmars, M.H.: On a class of \({O}(n^2)\) problems in computational geometry. Comput. Geom. 5(3), 165–185 (1995)

  17. 17.

    Gold, O., Sharir, M.: Improved bounds for 3SUM, \(k\)-SUM, and linear degeneracy. In: Pruhs, K., Sohler, C. (eds.) Proceedings of the 25th European Symposium on Algorithms (ESA’17). LIPIcs. Leibniz International Proceedings in Informatics, vol. 87, pp. 42:1–42:13. Schloss Dagstuhl. Leibniz-Zentrum für Informatik, Wadern (2017). Also in arXiv:1512.05279v2

  18. 18.

    Grønlund, A., Pettie, S.: Threesomes, degenerates, and love triangles. In: Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS’14), pp. 621–630. IEEE, Los Alamitos (2014)

  19. 19.

    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)

  20. 20.

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

  21. 21.

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

  22. 22.

    Kane, D.M., Lovett, S., Moran, S.: Near-optimal linear decision trees for \(k\)-SUM and related problems. In: Proceedings of the 50th ACM Symposium on Theory of Computing (STOC’15), pp. 554–563. ACM, New York (2018)

  23. 23.

    Kane, D.M., Lovett, S., Moran, S.: Generalized comparison trees for point-location problems. In: Proceedings of the 45th International Colloquium on Automata, Languages and Programming (ICALP’18), vol. 82:1–82:13 (2018)

  24. 24.

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

  25. 25.

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

  26. 26.

    Matoušek, J.: Cutting hyperplane arrangements. Discrete Comput. Geom. 6(5), 385–406 (1991)

  27. 27.

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

  28. 28.

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

  29. 29.

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

  30. 30.

    VanArsdale, D.: Homogeneous transformation matrices for computer graphics. Comput. Graphics 18(2), 177–191 (1994)

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The authors would like to thank Shachar Lovett, Sariel Har-Peled, and Haim Kaplan for many useful discussions. We also acknowledge the collaboration with Sariel and Haim on the companion paper [14], from which many of the ideas and techniques presented in this work have emerged. We also acknowledge useful comments by the referees, which have led to a major revision, hopefully for the better, of the paper from its original conference version.

Author information

Correspondence to Esther Ezra.

Additional information

Work on this paper by Esther Ezra was supported by NSF CAREER under Grant CCF:AF 1553354 and by Grant 824/17 from the Israel Science Foundation. Work on this paper by Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation, by Grant 2012/229 from the U.S.—Israel Binational Science Foundation, by the Blavatnik Research Fund in Computer Science at Tel Aviv University, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. A preliminary version of this paper appeared in Proc. 33rd Int. Sympos. Computational Geometry, 2017 [13].

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Ezra, E., Sharir, M. A Nearly Quadratic Bound for Point-Location in Hyperplane Arrangements, in the Linear Decision Tree Model. Discrete Comput Geom 61, 735–755 (2019). https://doi.org/10.1007/s00454-018-0043-8

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  • Point location in geometric arrangements
  • k-SUM and k-LDT
  • Linear decision tree model
  • Epsilon-cuttings
  • Vertical decomposition of geometric arrangements

Mathematics Subject Classification

  • 52C99
  • 52C45
  • 68Q87
  • 68Q25