Bipartite Diameter and Other Measures Under Translation

Boris Aronov, Omrit Filtser, Matthew J. Katz, Khadijeh Sheikhan

Research output: Contribution to journalArticlepeer-review

Abstract

Let A and B be two sets of points in Rd, 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 Rd for any fixed d≥ 4 , for uniformity we describe a roughly O(n9 / 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.

Original languageEnglish
Article number3
Pages (from-to)647-663
Number of pages17
JournalDiscrete and Computational Geometry
Volume68
Issue number3
DOIs
StatePublished - Oct 2022

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Geometric optimization
  • Minimum-width annulus
  • Translation-invariant similarity measures

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