Geometric Pattern Matching Reduces to k-SUM
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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≥ 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.
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