Algorithmica

, Volume 71, Issue 1, pp 36–52 | Cite as

Approximating Minimum Manhattan Networks in Higher Dimensions

  • Aparna Das
  • Emden R. Gansner
  • Michael Kaufmann
  • Stephen Kobourov
  • Joachim Spoerhase
  • Alexander Wolff
Article

Abstract

We study the minimum Manhattan network problem, which is defined as follows. Given a set of points called terminals in \({\mathbb {R}}^{d}\), 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, L1-) distance. The problem is NP-hard in 2D and there is no PTAS for 3D (unless \({\mathcal{P}} = \mathcal{NP}\)). Approximation algorithms are known for 2D, but not for 3D.

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

Keywords

Approximation algorithms Computational geometry Minimum Manhattan network 

Notes

Acknowledgements

This work was started at the 2009 Bertinoro Workshop on Graph Drawing. We thank the organizers Beppe Liotta and Walter Didimo for creating an inspiring atmosphere. We also thank Steve Wismath, Henk Meijer, Jan Kratochvíl, and Pankaj Agarwal for discussions. We are indebted to Stefan Felsner for pointing us to Soto and Telha’s work [18].

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Aparna Das
    • 1
  • Emden R. Gansner
    • 2
  • Michael Kaufmann
    • 3
  • Stephen Kobourov
    • 1
  • Joachim Spoerhase
    • 4
  • Alexander Wolff
    • 4
  1. 1.Dept. of Comp. Sci.University of ArizonaTucsonUSA
  2. 2.AT&T Labs ResearchFlorham ParkUSA
  3. 3.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany
  4. 4.Lehrstuhl I, Institut für InformatikUniversität WürzburgWürzburgGermany

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