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Geometric Pattern Matching Reduces to k-SUM

Authors Boris Aronov , Jean Cardinal

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Author Details

Boris Aronov
  • Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY ,USA
Jean Cardinal
  • Université libre de Bruxelles (ULB), Belgium

Funding

  • Aronov, Boris: Partially supported by NSF grant CCF-15-40656 and by grant 2014/170 from the US-Israel Binational Science Foundation. Work by B.A. on this paper has been partially carried out while visiting ULB in November-December 2019, with support from ULB and F.R.S.-FNRS (Fonds National de la Recherche Scientifique).
  • Cardinal, Jean: Supported by the F.R.S.-FNRS (Fonds National de la Recherche Scientifique) under CDR Grant J.0146.18.

Cite As Get BibTex

Boris Aronov and Jean Cardinal. Geometric Pattern Matching Reduces to k-SUM. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 32:1-32:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ISAAC.2020.32

Abstract

We prove that some exact geometric pattern matching problems reduce in linear time to o k-SUM when the pattern has a fixed size k. This holds in the real RAM model for searching for a similar copy of a set of k ≥ 3 points within a set of n points in the plane, and for searching for an affine image of a set of k ≥ d+2 points within a set of n points in d-space.
As corollaries, we obtain improved real RAM algorithms and decision trees for the two problems. In particular, they can be solved by algebraic decision trees of near-linear height.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Computational geometry
Keywords
  • Geometric pattern matching
  • k-SUM problem
  • Linear decision trees

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