Finding largest common point sets

Abstract

Let P and Q be two discrete point sets in ϵ>0d of sizes m and n, respectively, and let > 0 be a given input threshold. The largest common point set problem (LCP) seeks the largest subsets A ⊆P and B⊆Q such that |A| = |B| and there exists a transformation Φthat makes the bottleneck distance between Φ(A) and B at mostϵ. We present two algorithms that solve a relaxed version of this problem under translations in Rd and under rigid motions in the plane, and that takes an additional input parameter• > 0. Let ℓbe the largest subset size achievable for the given . Our algorithm finds subsets A ⊆P and B ⊆ Q of size |A| = |B|≥ ℓand a transformation Φsuch that the bottleneck distance between Ï•(A) and B is at most (1 + n). For rigid motions in the plane, the running time is O(n2m2/2(n + m)log n). For translations inRd, the running time is O(nm\n(n + m)1.5log n), where κ= 1 for d = 2 and κ= 2d-1 for d ≥ 3.