Bipartite Diameter and Other Measures Under Translation

Let A and B be two sets of points in R^d, where | A| = | B| = n and the distance between them is defined by some bipartite measure dist(A,B). We study several problems in which the goal is to translate the set B, so that dist(A,B) is minimized. The main measures that we consider are (i) the diameter in two and higher dimensions, that is diam(A,B)=max{d(a,b)∣a∈A,b∈B}, where d(a, b) is the Euclidean distance between a and b, (ii) the uniformity in the plane, that is uni(A,B)=diam(A,B)-d(A,B), where d(A,B)=min{d(a,b)∣a∈A,b∈B}, and (iii) the union width in two and three dimensions, that is union_width(A,B)=width(A∪B). For each of these measures, we present efficient algorithms for finding a translation of B that minimizes the distance: For diameter we present near-linear-time algorithms in R² and R³ and a subquadratic algorithm in R^d for any fixed d≥ 4 , for uniformity we describe a roughly O(n^{9 / 4}) -time algorithm in the plane, and for union width we offer a near-linear-time algorithm in R² and a quadratic-time one in R³.