Bipartite Diameter and Other Measures Under Translation

Abstract

Let A and B be two sets of points in $\mathbb {R}^d$, where $|A|=|B|=n$ and the distance between them is defined by some bipartite measure ${{\,\mathrm{dist}\,}}(A, B)$. We study several problems in which the goal is to translate the set B, so that ${{\,\mathrm{dist}\,}}(A,B)$ is minimized. The main measures that we consider are (i) the diameter in two and higher dimensions, that is ${{\,\mathrm{diam}\,}}(A,B)=\max {\{d(a,b)\mid a\in A,\,b \in B\}}$, where d(a, b) is the Euclidean distance between a and b, (ii) the uniformity in the plane, that is ${{\,\mathrm{uni}\,}}(A,B) = {{\,\mathrm{diam}\,}}(A,B)-d(A,B)$, where $d(A,B)=\min {\{d(a,b)\mid a\in A,\,b\in B\}}$, and (iii) the union width in two and three dimensions, that is ${{\,\mathrm{union\_width}\,}}(A,B) = {{\,\mathrm{width}\,}}(A \cup B)$. 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 $\mathbb {R}^2$ and $\mathbb {R}^3$ and a subquadratic algorithm in $\mathbb {R}^d$ for any fixed $d\ge 4$, 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 $\mathbb {R}^2$ and a quadratic-time one in $\mathbb {R}^3$.