Geometric Pattern Matching Reduces to k-SUM

Authors Boris Aronov , Jean Cardinal

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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

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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)


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
  • Geometric pattern matching
  • k-SUM problem
  • Linear decision trees


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