Geometric Pattern Matching Reduces to k-SUM

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.