TY - JOUR

T1 - Bipartite Diameter and Other Measures Under Translation

AU - Aronov, Boris

AU - Filtser, Omrit

AU - Katz, Matthew J.

AU - Sheikhan, Khadijeh

N1 - 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.

PY - 2022/10/1

Y1 - 2022/10/1

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 - Article

AN - SCOPUS:85137816295

VL - 68

SP - 647

EP - 663

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 3

ER -