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Approximating Minimum Manhattan Networks in Higher Dimensions

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Algorithms – ESA 2011 (ESA 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6942))

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Abstract

We consider the minimum Manhattan network problem, which is defined as follows. Given a set of points called terminals in ℝd, find a minimum-length network such that each pair of terminals is connected by a set of axis-parallel line segments whose total length is equal to the pair’s Manhattan (that is, L 1-) distance. The problem is NP-hard in 2D and there is no PTAS for 3D (unless \({\cal P}\!=\!{\cal NP}\)). Approximation algorithms are known for 2D, but not for 3D.

We present, for any fixed dimension d and any \(\ensuremath{\varepsilon} >0\), an \(O(n^\ensuremath{\varepsilon} )\)-approximation. For 3D, we also give a 4(k − 1)-approximation for the case that the terminals are contained in the union of k ≥ 2 parallel planes.

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Author information

Authors and Affiliations

  1. Dept. of Comp. Sci., University of Arizona, Tucson, AZ, U.S.A.

    Aparna Das & Stephen Kobourov

  2. AT&T Labs Research, Florham Park, NJ, U.S.A.

    Emden R. Gansner

  3. Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Germany

    Michael Kaufmann

  4. Institut für Informatik, Universität Würzburg, Germany

    Joachim Spoerhase & Alexander Wolff

Authors
  1. Aparna Das
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  2. Emden R. Gansner
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  3. Michael Kaufmann
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  4. Stephen Kobourov
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  5. Joachim Spoerhase
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  6. Alexander Wolff
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Editor information

Editors and Affiliations

  1. Dipartimento di Informatica e Sistemistica, University of Rome " La Sapienza", via Ariosto 25, 00185, Roma, Italy

    Camil Demetrescu

  2. School of Computer Science, Reykjavik University, Menntavegur 1, 101, Reykjavik, Iceland

    Magnús M. Halldórsson

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Das, A., Gansner, E.R., Kaufmann, M., Kobourov, S., Spoerhase, J., Wolff, A. (2011). Approximating Minimum Manhattan Networks in Higher Dimensions. In: Demetrescu, C., Halldórsson, M.M. (eds) Algorithms – ESA 2011. ESA 2011. Lecture Notes in Computer Science, vol 6942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23719-5_5

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  • DOI: https://doi.org/10.1007/978-3-642-23719-5_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23718-8

  • Online ISBN: 978-3-642-23719-5

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