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The Minimum Manhattan Network Problem: A Fast Factor-3 Approximation

  • Marc Benkert
  • Alexander Wolff
  • Florian Widmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3742)

Abstract

Given a set of nodes in the plane and a constant t ≥ 1, a Euclidean t-spanner is a network in which, for any pair of nodes, the ratio of the network distance and the Euclidean distance of the two nodes is at most t. These networks have applications in transportation or communication network design and have been studied extensively.

In this paper we study 1-spanners under the Manhattan (or L 1-) metric. Such networks are called Manhattan networks. A Manhattan network for a set of nodes can be seen as a set of axis-parallel line segments whose union contains an x- and y-monotone path for each pair of nodes. It is not known whether it is NP-hard to compute minimum Manhattan networks, i.e. Manhattan networks of minimum total length. In this paper we present a factor-3 approximation algorithm for this problem. Given a set of n nodes, our algorithm takes O(n log n) time and linear space.

Keywords

Line Segment Point Pair Input Point Additional Segment Segment Endpoint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    Arya, S., Das, G., Mount, D.M., Salowe, J.S., Smid, M.: Euclidean spanners: Short, thin, and lanky. In: Andersson, S.I. (ed.) Summer University of Southern Stockholm 1993. LNCS, vol. 888, pp. 489–498. Springer, Heidelberg (1995)Google Scholar
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    Benkert, M., Widmann, F., Wolff, A.: The minimum Manhattan network problem: A fast factor-3 approximation. Technical Report 2004-16, Fakultät für Informatik, Universität Karlsruhe (2004), Available at http://www.ubka.uni-karlsruhe.de/cgi-bin/psview?document=/ira/2004/16
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    Gudmundsson, J., Levcopoulos, C., Narasimhan, G.: Approximating a minimum Manhattan network. Nordic J. Comput. 8, 219–232 (2001)zbMATHMathSciNetGoogle Scholar
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    Kato, R., Imai, K., Asano, T.: An improved algorithm for the minimum Manhattan network problem. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 344–356. Springer, Heidelberg (2002)CrossRefGoogle Scholar
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Marc Benkert
    • 1
  • Alexander Wolff
    • 1
  • Florian Widmann
    • 1
  1. 1.Faculty of Computer ScienceKarlsruhe UniversityKarlsruheGermany

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