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Subquadratic Algorithms for Some 3Sum-Hard Geometric Problems in the Algebraic Decision Tree Model

Authors Boris Aronov , Mark de Berg , Jean Cardinal , Esther Ezra , John Iacono , Micha Sharir

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Computational geometry
  • Algebraic decision-tree model
  • Polynomial partitioning
  • Primal-dual range searching
  • Order types
  • Point location
  • Hierarchical partitions

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Abstract

We present subquadratic algorithms in the algebraic decision-tree model for several 3Sum-hard geometric problems, all of which can be reduced to the following question: Given two sets A, B, each consisting of n pairwise disjoint segments in the plane, and a set C of n triangles in the plane, we want to count, for each triangle Δ ∈ C, the number of intersection points between the segments of A and those of B that lie in Δ. The problems considered in this paper have been studied by Chan (2020), who gave algorithms that solve them, in the standard real-RAM model, in O((n²/log²n) log^O(1) log n) time. We present solutions in the algebraic decision-tree model whose cost is O(n^{60/31+ε}), for any ε > 0.
Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl (2020).
A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the order type of the lines, a "handicap" that turns out to be beneficial for speeding up our algorithm.

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Boris Aronov, Mark de Berg, Jean Cardinal, Esther Ezra, John Iacono, and Micha Sharir. Subquadratic Algorithms for Some 3Sum-Hard Geometric Problems in the Algebraic Decision Tree Model. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ISAAC.2021.3

Author Details

Boris Aronov
  • Tandon School of Engineering, New York University, Brooklyn, NY, USA
Mark de Berg
  • Eindhoven University of Technology, The Netherlands
Jean Cardinal
  • Université libre de Bruxelles (ULB), Brussels, Belgium
Esther Ezra
  • School of Computer Science, Bar Ilan University, Ramat Gan, Israel
John Iacono
  • Université libre de Bruxelles (ULB), Brussels, Belgium
  • Tandon School of Engineering, New York University, Brooklyn, NY, USA
Micha Sharir
  • School of Computer Science, Tel Aviv University, Israel

Funding

  • Aronov, Boris: Partially supported by NSF Grants CCF-15-40656 and CCF-20-08551, and by Grant 2014/170 from the US-Israel Binational Science Foundation.
  • de Berg, Mark: Partially supported by the Dutch Research Council (NWO) through Gravitation Grant NETWORKS (project no. 024.002.003).
  • Cardinal, Jean: Partially supported by the F.R.S.-FNRS (Fonds National de la Recherche Scientifique) under CDR Grant J.0146.18.
  • Ezra, Esther: Partially supported by NSF CAREER under Grant CCF:AF-1553354 and by Grant 824/17 from the Israel Science Foundation.
  • Iacono, John: Partially supported by Fonds National de la Recherche Scientifique (FNRS) under Grant no. MISU F.6001.1.
  • Sharir, Micha: Partially supported by ISF Grant 260/18, by Grant 1367/2016 from the German-Israeli Science Foundation (GIF), and by Blavatnik Research Fund in Computer Science at Tel Aviv University.

Acknowledgements

We thank Zuzana Patáková for helpful discussions on multilevel polynomial partitioning.

References

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