## Abstract

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}.

Original language | English |
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Article number | 3 |

Pages (from-to) | 647-663 |

Number of pages | 17 |

Journal | Discrete and Computational Geometry |

Volume | 68 |

Issue number | 3 |

DOIs | |

State | Published - Oct 2022 |

### Bibliographical note

Funding Information:Work on this paper by Khadijeh Sheikhan was performed while she was a graduate student at the Tandon School of Engineering, NYU. Her work was supported by NSF Grant CCF-12-18791.

Funding Information:

Work on this paper by Omrit Filtser was supported by the Eric and Wendy Schmidt Fund for Strategic Innovation, by the Council for Higher Education of Israel, and by Ben-Gurion University of the Negev.

Funding Information:

Work on this paper by Matthew Katz was supported by Grant 1884/16 from the Israel Science Foundation, and by Grant 2014/170 from the US-Israel Binational Science Foundation.

Funding Information:

Work on this paper by Boris Aronov was supported by NSF Grants CCF-11-17336, CCF-12-18791, and CCF-15-40656, and by Grant 2014/170 from the US-Israel Binational Science Foundation.

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