Document Open Access Logo

Bipartite Diameter and Other Measures Under Translation

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

PDF

File

Thumbnail PDF
PDF
LIPIcs.STACS.2019.8.pdf
  • Filesize: 485 kB
  • 14 pages

Document Identifiers

Subject Classification

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

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.

Cite As Get 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

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.

References

  1. Pankaj K. Agarwal, Boris Aronov, Marc van Kreveld, Maarten Löffler, and Rodrigo Silveira. Computing Correlation between Piecewise-Linear Functions. SIAM J. Comput., 42:1867-1887, 2013. Google Scholar
  2. Pankaj K. Agarwal, Sariel Har-Peled, Micha Sharir, and Yusu Wang. Hausdorff distance under translation for points and balls. In Proc. 19th Ann. Symp. Comput. Geometry, pages 282-291. ACM, 2003. Google Scholar
  3. Pankaj K. Agarwal and Micha Sharir. Efficient randomized algorithms for some geometric optimization problems. Discr. Comput. Geometry, 16(4):317-337, 1996. Google Scholar
  4. Pankaj K. Agarwal, Micha Sharir, and Sivan Toledo. Applications of Parametric Searching in Geometric Optimization. J. Algorithms, 17(3):292-318, 1994. URL: http://dx.doi.org/10.1006/jagm.1994.1038.
  5. Hee-Kap Ahn, Peter Brass, and Chan-Su Shin. Maximum overlap and minimum convex hull of two convex polyhedra under translations. Computational Geometry, 40(2):171-177, 2008. Google Scholar
  6. Helmut Alt, Kurt Mehlhorn, Hubert Wagener, and Emo Welzl. Congruence, similarity, and symmetries of geometric objects. Discr. Comput. Geometry, 3(3):237-256, 1988. Google Scholar
  7. Rinat Ben-Avraham, Matthias Henze, Rafel Jaume, Balázs Keszegh, Orit E. Raz, Micha Sharir, and Igor Tubis. Minimum partial-matching and Hausdorff RMS-distance under translation: Combinatorics and algorithms. In European Symp. Algorithms, pages 100-111. Springer, 2014. Google Scholar
  8. Arijit Bishnu, Sandip Das, Subhas C. Nandy, and Bhargab B. Bhattacharya. Simple algorithms for partial point set pattern matching under rigid motion. Pattern Recognition, 39(9):1662-1671, 2006. Google Scholar
  9. K. Q. Brown. Geometric Transforms for Fast Geometric Algorithms. Ph.D. thesis, Carnegie-Mellon University, Pittsburgh, PA, 1980. Google Scholar
  10. Bernard Chazelle, Richard Cole, Franco P. Preparata, and Chee Yap. New upper bounds for neighbor searching. Information and Control, 68(1-3):105-124, 1986. Google Scholar
  11. Bernard Chazelle, Herbert Edelsbrunner, Leonidas J. Guibas, and Micha Sharir. Algorithms for bichromatic line segment problems and polyhedral terrains. Algorithmica, 11:116-132, 1994. Google Scholar
  12. Kenneth L. Clarkson. Las Vegas algorithms for linear and integer programming when the dimension is small. J. ACM, 42:488-499, 1995. Google Scholar
  13. Kenneth L. Clarkson and Peter W. Shor. Applications of random sampling in computational geometry, II. Discr. Comput. Geometry, 4(1):387-421, 1989. Google Scholar
  14. M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, third edition, 2008. Google Scholar
  15. Alon Efrat, Alon Itai, and Matthew J. Katz. Geometry helps in bottleneck matching and related problems. Algorithmica, 31(1):1-28, 2001. Google Scholar
  16. Leonidas J. Guibas and Raimund Seidel. Computing convolutions by reciprocal search. Discr. Comput. Geometry, 2(2):175-193, 1987. Google Scholar
  17. Paul J. Heffernan and Stefan Schirra. Approximate decision algorithms for point set congruence. Computational Geometry, 4(3):137-156, 1994. Google Scholar
  18. M. E. Houle and G. T. Toussaint. Computing the width of a set. IEEE Trans. Pattern Anal. Mach. Intell., PAMI-10(5):761-765, 1988. Google Scholar
  19. Daniel P. Huttenlocher, Klara Kedem, and Micha Sharir. The Upper Envelope of Voronoi Surfaces and Its Applications. Discr. Comput. Geometry, 9:267-291, 1993. URL: http://dx.doi.org/10.1007/BF02189323.
  20. Piotr Indyk and Suresh Venkatasubramanian. Approximate congruence in nearly linear time. Computational Geometry, 24(2):115-128, 2003. Google Scholar
  21. Nimrod Megiddo. Linear-time algorithms for linear programming in R³ and related problems. SIAM J. Comput., 12(4):759-776, 1983. Google Scholar
  22. Edgar A. Ramos. Deterministic algorithms for 3-D diameter and some 2-D lower envelopes. In Proc. 16th Ann. Symp. Comput. Geometry, pages 290-299. ACM, 2000. Google Scholar
  23. M. I. Shamos. Computational Geometry. PhD thesis, Yale University, 1978. Google Scholar
  24. Emo Welzl. Smallest Enclosing Disks (balls and Ellipsoids). In Results and New Trends in Computer Science, pages 359-370. Springer-Verlag, 1991. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact