Geometric Pattern Matching Reduces to k -SUM
Boris S. Aronov, Jean Cardinal · Discrete & Computational Geometry · 2021
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.