Bipartite Diameter and Other Measures Under Translation

Thu, Nov 21, 2019 12:30 CET

Speakers: Boris Aronov

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 three dimensions, that is diam(A,B) = max {d(a,b) : 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 uni(A,B) = diam(A,B) - d(A,B), where d(A,B)=min {d(a,b) : a in A, b in B}, and

(iii) the union width in two and three dimensions, that is uniwidth(A,B) = 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 R^2 and R^3, for uniformity we describe a roughly O(n^{9/4})-time algorithm, and for union width we offer a near-linear-time algorithm in R^2 and a quadratic-time one in R^3.

This is joint work with Omrit Filtser, Matthew J. Katz, and Khadijeh Sheikhan.