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


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

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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)


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


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