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

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Abstract

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 However, these applications have been superseded by the more recent result of Kane et al.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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Notes

  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.

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Acknowledgements

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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Keywords

  • 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