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.
@InProceedings{aronov_et_al:LIPIcs.ISAAC.2020.32, author = {Aronov, Boris and Cardinal, Jean}, title = {{Geometric Pattern Matching Reduces to k-SUM}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {32:1--32:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.32}, URN = {urn:nbn:de:0030-drops-133760}, doi = {10.4230/LIPIcs.ISAAC.2020.32}, annote = {Keywords: Geometric pattern matching, k-SUM problem, Linear decision trees} }
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