Finding Largest Common Point Sets

Abstract

Let [Formula: see text] and [Formula: see text] be two discrete point sets in [Formula: see text] of sizes [Formula: see text] and [Formula: see text], respectively, and let [Formula: see text] be a given input threshold. The largest common point set problem (LCP) seeks the largest subsets [Formula: see text] and [Formula: see text] such that [Formula: see text] and there exists a transformation [Formula: see text] that makes the bottleneck distance between [Formula: see text] and [Formula: see text] at most [Formula: see text]. We present two algorithms that solve a relaxed version of this problem under translations in [Formula: see text] and under rigid motions in the plane, and that takes an additional input parameter [Formula: see text]. Let [Formula: see text] be the largest subset size achievable for the given [Formula: see text]. Our algorithm finds subsets [Formula: see text] and [Formula: see text] of size [Formula: see text] and a transformation [Formula: see text] such that the bottleneck distance between [Formula: see text] and [Formula: see text] is at most [Formula: see text]. For rigid motions in the plane, the running time is [Formula: see text]. For translations in [Formula: see text], the running time is [Formula: see text], where [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text].