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 ℝ^2 and ℝ^3 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 ℝ^2 and a quadratic-time one in ℝ^3 .