Bipartite Diameter and Other Measures Under Translation

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

doi:10.1007/s00454-022-00433-5

Bipartite Diameter and Other Measures Under Translation

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

Computer Science discipline

Research output: Contribution to journal › Article › peer-review

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 R² and R³ 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 R² and a quadratic-time one in R³.

Original language	English
Article number	3
Pages (from-to)	647-663
Number of pages	17
Journal	Discrete and Computational Geometry
Volume	68
Issue number	3
DOIs	https://doi.org/10.1007/s00454-022-00433-5
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

Access to Document

10.1007/s00454-022-00433-5

Cite this

@article{f07c03141272489f8bff4f5b6f406b79,

title = "Bipartite Diameter and Other Measures Under Translation",

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.",

keywords = "Geometric optimization, Minimum-width annulus, Translation-invariant similarity measures",

author = "Boris Aronov and Omrit Filtser and Katz, {Matthew J.} and Khadijeh Sheikhan",

note = "Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.",

year = "2022",

month = oct,

doi = "10.1007/s00454-022-00433-5",

language = "אנגלית",

volume = "68",

pages = "647--663",

journal = "Discrete and Computational Geometry",

issn = "0179-5376",

publisher = "Springer New York",

number = "3",

}

TY - JOUR

T1 - Bipartite Diameter and Other Measures Under Translation

AU - Aronov, Boris

AU - Filtser, Omrit

AU - Katz, Matthew J.

AU - Sheikhan, Khadijeh

PY - 2022/10

Y1 - 2022/10

N2 - 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.

AB - 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.

KW - Geometric optimization

KW - Minimum-width annulus

KW - Translation-invariant similarity measures

UR - http://www.scopus.com/inward/record.url?scp=85137816295&partnerID=8YFLogxK

U2 - 10.1007/s00454-022-00433-5

DO - 10.1007/s00454-022-00433-5

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85137816295

SN - 0179-5376

VL - 68

SP - 647

EP - 663

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 3

M1 - 3

ER -

Bipartite Diameter and Other Measures Under Translation

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this