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The Rectilinear Steiner Tree Problem with Given Topology and Length Restrictions

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Computing and Combinatorics (COCOON 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9198))

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Abstract

We consider the problem of embedding the Steiner points of a Steiner tree with given topology into the rectilinear plane. Thereby, the length of the path between a distinguished terminal and each other terminal must not exceed given length restrictions. We want to minimize the total length of the tree.

The problem can be formulated as a linear program and therefore it is solvable in polynomial time. In this paper we analyze the structure of feasible embeddings and give a combinatorial polynomial time algorithm for the problem. Our algorithm combines a dynamic programming approach and binary search and relies on the total unimodularity of a matrix appearing in a sub-problem.

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References

  1. Bartoschek, C., Held, S., Maßberg, J., Rautenbach, D., Vygen, J.: The repeater tree construction problem. Information Processing Letters 110(24), 1079–1083 (2010)

    Article  MathSciNet  Google Scholar 

  2. Victor Cabot, A., Francis, R.L., Stary, M.A.: A network flow solution to a rectilinear distance facility location problem. AIIE Transactions 2(2) 132–141 (1970)

    Google Scholar 

  3. Edmonds, J.: Submodular functions, matroids and certain polyhedra. In: Gordon, Breach (eds.) Combinatorial Structures and Their Applications, New York, pp. 68–87 (1970)

    Google Scholar 

  4. Garey, M.R., Johnson, D.S.: The rectilinear Steiner tree problem is NP-complete. SIAM Journal on Applied Mathematics 32(4), 826–834 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hanan, M.: On Steiner’s problem with rectilinear distance. SIAM Journal on Applied Mathematics 14(2), 255–265 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  6. Jiang, T., Wang, L.: Computing shortest networks with fixed topologies. Advances in Steiner Trees. vol. 6. Combinatorial Optimization, pp. 39–62. Springer, US (2000)

    Google Scholar 

  7. Kortsarz, G., Peleg, D.: Approximating the weight of shallow Steiner trees. Discrete Applied Mathematics 93(2–3), 265–285 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Rao, S.K., Sadayappan, P., Hwang, F.K., Shor, P.W.: The rectilinear Steiner arborescence problem. Algorithmica 7(1–6), 277–288 (1992)

    Google Scholar 

  9. Shi, W., Chen, S.: The rectilinear Steiner arborescence problem is NP-complete. SIAM Journal on Computation 35(3), 729–740 (2005)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Optimierung und Operations Research, Universität Ulm, Ulm, Germany

    Jens Maßberg

Authors
  1. Jens Maßberg
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Correspondence to Jens Maßberg .

Editor information

Editors and Affiliations

  1. Beijing University of Technology, Beijing, China

    Dachuan Xu

  2. University of New Brunswick, Fredericton, Canada

    Donglei Du

  3. University of Texas at Dallas, Richardson, USA

    Dingzhu Du

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Maßberg, J. (2015). The Rectilinear Steiner Tree Problem with Given Topology and Length Restrictions. In: Xu, D., Du, D., Du, D. (eds) Computing and Combinatorics. COCOON 2015. Lecture Notes in Computer Science(), vol 9198. Springer, Cham. https://doi.org/10.1007/978-3-319-21398-9_35

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  • DOI: https://doi.org/10.1007/978-3-319-21398-9_35

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21397-2

  • Online ISBN: 978-3-319-21398-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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