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Generalised k-Steiner Tree Problems in Normed Planes

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Abstract

The 1-Steiner tree problem, the problem of constructing a Steiner minimum tree containing at most one Steiner point, has been solved in the Euclidean plane by Georgakopoulos and Papadimitriou using plane subdivisions called oriented Dirichlet cell partitions. Their algorithm produces an optimal solution within O(n 2) time. In this paper we generalise their approach in order to solve the k-Steiner tree problem, in which the Steiner minimum tree may contain up to k Steiner points for a given constant k. We also extend their approach further to encompass other normed planes, and to solve a much wider class of problems, including the k-bottleneck Steiner tree problem and other generalised k-Steiner tree problems. We show that, for any fixed k, such problems can be solved in O(n 2k) time.

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Authors and Affiliations

  1. ARC Special Research Centre for Ultra-Broadband Information Networks (CUBIN), Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, Victoria, 3010, Australia

    Marcus Brazil & Charl J. Ras

  2. Department of Mathematics, London School of Economics and Political Science, University of London, London, WC2A 2AE, UK

    Konrad J. Swanepoel

  3. Department of Mechanical Engineering, The University of Melbourne, Parkville, Victoria, 3010, Australia

    Doreen A. Thomas

Authors
  1. Marcus Brazil
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  2. Charl J. Ras
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  3. Konrad J. Swanepoel
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  4. Doreen A. Thomas
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Corresponding author

Correspondence to Marcus Brazil.

Additional information

This research was supported by an ARC Discovery Grant. Part of this paper was written while Konrad Swanepoel was visiting the Department of Mechanical Engineering of the University of Melbourne on a Tewkesbury Fellowship.

CUBIN is an affiliated program of National ICT Australia.

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Brazil, M., Ras, C.J., Swanepoel, K.J. et al. Generalised k-Steiner Tree Problems in Normed Planes. Algorithmica 71, 66–86 (2015). https://doi.org/10.1007/s00453-013-9780-5

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  • DOI: https://doi.org/10.1007/s00453-013-9780-5

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