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Approximating Minimum Manhattan Networks in Higher Dimensions

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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, L 1-) 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(n ε)-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.

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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].

Author information

Correspondence to Joachim Spoerhase.

Additional information

A preliminary version of this paper appeared in Proc. 19th Annual European Symposium on Algorithms, Lect. Notes Comput. Sci., vol. 6942, pp. 46–60. Springer, Berlin (2011).

The research of A. Das and S. Kobourov was funded in part by NSF grants CCF-0545743 and CCF-1115971.

M. Kaufmann and A. Wolff acknowledge support by the ESF EuroGIGA project GraDR (DFG grants Ka 812/16-1 and Wo 758/5-1, respectively).

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Das, A., Gansner, E.R., Kaufmann, M. et al. Approximating Minimum Manhattan Networks in Higher Dimensions. Algorithmica 71, 36–52 (2015). https://doi.org/10.1007/s00453-013-9778-z

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Keywords

  • Approximation algorithms
  • Computational geometry
  • Minimum Manhattan network