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Bottleneck Non-crossing Matching in the Plane

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Algorithms – ESA 2012 (ESA 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7501))

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Abstract

Let P be a set of 2n points in the plane, and let M C (resp., M NC) denote a bottleneck matching (resp., a bottleneck non-crossing matching) of P. We study the problem of computing M NC. We present an O(n 1.5log0.5 n)-time algorithm that computes a non-crossing matching M of P, such that \(bn(M) \le 2\sqrt{10} \cdot bn(M_{\rm NC})\), where bn(M) is the length of a longest edge in M. An interesting implication of our construction is that \(bn(M_{\rm NC})/bn(M_{\rm C}) \le 2\sqrt{10}\). We also show that when the points of P are in convex position, one can compute M NC in O(n 3) time. (In the full version of this paper, we also prove that the problem is NP-hard and does not admit a PTAS.)

Work by A.K. Abu-Affash was partially supported by a fellowship for doctoral students from the Planning & Budgeting Committee of the Israel Council for Higher Education, and by a scholarship for advanced studies from the Israel Ministry of Science and Technology. Work by A.K. Abu-Affash and Y. Trabelsi was partially supported by the Lynn and William Frankel Center for Computer Sciences. Work by P. Carmi was partially supported by grant 680/11 from the Israel Science Foundation. Work by M. Katz was partially supported by grant 1045/10 from the Israel Science Foundation. Work by M. Katz and P. Carmi was partially supported by grant 2010074 from the United States – Israel Binational Science Foundation.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel

    A. Karim Abu-Affash, Paz Carmi, Matthew J. Katz & Yohai Trabelsi

Authors
  1. A. Karim Abu-Affash
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  2. Paz Carmi
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  3. Matthew J. Katz
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  4. Yohai Trabelsi
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Haifa, Mount Carmel, 31905, Haifa, Israel

    Leah Epstein

  2. Dipartimento di Informatica, Università di Pisa, Largo B. Pontecorvo 3, 56100, Pisa, Italy

    Paolo Ferragina

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Abu-Affash, A.K., Carmi, P., Katz, M.J., Trabelsi, Y. (2012). Bottleneck Non-crossing Matching in the Plane. In: Epstein, L., Ferragina, P. (eds) Algorithms – ESA 2012. ESA 2012. Lecture Notes in Computer Science, vol 7501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33090-2_5

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  • DOI: https://doi.org/10.1007/978-3-642-33090-2_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33089-6

  • Online ISBN: 978-3-642-33090-2

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