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Gradient-Constrained Minimum Networks. III. Fixed Topology

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Abstract

The gradient-constrained Steiner tree problem asks for a shortest total length network interconnecting a given set of points in 3-space, where the length of each edge of the network is determined by embedding it as a curve with absolute gradient no more than a given positive value m, and the network may contain additional nodes known as Steiner points. We study the problem for a fixed topology, and show that, apart from a few easily classified exceptions, if the positions of the Steiner points are such that the tree is not minimum for the given topology, then there exists a length reducing perturbation that moves exactly 1 or 2 Steiner points. In the conclusion, we discuss the application of this work to a heuristic algorithm for solving the global problem (across all topologies).

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Acknowledgements

Research supported by the Australian Research Council, the Canadian Research Chairs Program and NSERC.

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Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, The University of Melbourne, Melbourne, Victoria, 3010, Australia

    M. Brazil

  2. Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Victoria, 3010, Australia

    J. H. Rubinstein

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

    D. A. Thomas & J. F. Weng

  4. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    N. Wormald

Authors
  1. M. Brazil
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  2. J. H. Rubinstein
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  3. D. A. Thomas
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  4. J. F. Weng
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  5. N. Wormald
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Corresponding author

Correspondence to M. Brazil.

Additional information

Communicated by Reiner Burkard.

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Brazil, M., Rubinstein, J.H., Thomas, D.A. et al. Gradient-Constrained Minimum Networks. III. Fixed Topology. J Optim Theory Appl 155, 336–354 (2012). https://doi.org/10.1007/s10957-012-0036-3

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  • DOI: https://doi.org/10.1007/s10957-012-0036-3

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