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Geometric Pattern Matching Reduces to k -SUM

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Abstract

We prove that some exact geometric pattern matching problems reduce in linear time to 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 \(k\ge 3\) points within a set of n points in the plane, and for searching for an affine image of a set of \(k\ge d+2\) 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.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, 11201, USA

    Boris Aronov

  2. Université Libre de Bruxelles (ULB), 1050, Brussels, Belgium

    Jean Cardinal

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  1. Boris Aronov
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  2. Jean Cardinal
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Correspondence to Jean Cardinal.

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Boris Aronov: Partially supported by NSF grant CCF-15-40656 and by grant 2014/170 from the US-Israel Binational Science Foundation. Work 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).

Jean Cardinal: Supported by the F.R.S.-FNRS (Fonds National de la Recherche Scientifique) under CDR Grant J.0146.18.

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Aronov, B., Cardinal, J. Geometric Pattern Matching Reduces to k -SUM. Discrete Comput Geom 68, 850–859 (2022). https://doi.org/10.1007/s00454-021-00324-1

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  • DOI: https://doi.org/10.1007/s00454-021-00324-1

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