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Bipartite Diameter and Other Measures Under Translation

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

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Author Details

Boris Aronov
  • Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY 11201, USA
Omrit Filtser
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
Matthew J. Katz
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
Khadijeh Sheikhan
  • Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY 11201, USA

Funding

  • Aronov, Boris: 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.
  • Filtser, Omrit: Work on this paper by Omrit Filtser was supported by the Israel Ministry of Science, Technology & Space, and by the Lynn and William Frankel Center.
  • Katz, Matthew J.: 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.
  • Sheikhan, Khadijeh: Work on this paper by Khadijeh Sheikhan was supported by NSF Grant CCF-12-18791.

Acknowledgements

We would like to thank Pankaj K. Agarwal, Mark de Berg, and Timothy Chan for their assistance in the preparation of this manuscript.

Cite AsGet BibTex

Boris Aronov, Omrit Filtser, Matthew J. Katz, and Khadijeh Sheikhan. Bipartite Diameter and Other Measures Under Translation. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.STACS.2019.8

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 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 union_width(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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Translation-invariant similarity measures
  • Geometric optimization
  • Minimum-width annulus

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