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Steiner Trees in Industry pp 405–426Cite as

Polynomial Time Algorithms for the Rectilinear Steiner Tree Problem

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Part of the book series: Combinatorial Optimization ((COOP,volume 11))

Abstract

The rectilinear Steiner tree problem has been proven to be NP-hard. However, an exact polynomial time algorithm may exist if the given point set has a special configuration (geometric structure). In this paper we survey the known algorithms for such configurations, including small point sets, points on a rectangular boundary, points on parallel lines and sets of points that are constrained to lie on curves.

Supported by a guant from the Australian Research Council.

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References

  1. S. Arora, Polynomial-time approximation schemes for Euclidean TSP and other geometric problems, Proceedings of 37th IEEE Symp. on Foundations of Computer Science, (1996) pp. 2–12.

    Google Scholar 

  2. P.K. Agarwal and M.-T. Shing, Algorithms for the special cases of rectilinear Steiner trees: I. Points on the boundary of a rectilinear rectangle, Networks, Vol.20 (1990) pp. 453–485.

    Article  MathSciNet  MATH  Google Scholar 

  3. A.V. Aho, M.R. Garey and F.K. Hwang, Rectilinear Steiner trees, Efficient special-case algorithms, Networks, Vol 7 (1977) pp. 37–58.

    Article  MathSciNet  MATH  Google Scholar 

  4. M.W. Bern, Faster exact algorithms for Steiner trees in planar networks, Networks, Vol. 20 (1990) pp. 109-120.

    Article  MathSciNet  MATH  Google Scholar 

  5. M.W. Bern and D. Bienstock, Polynomially solvable special cases of the Steiner problem in planar networks, Ann. Oper. Res. Vol. 33 (1991) pp. 405–418.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Brazil, D.A. Thomas and J.F. Weng, Rectilinear Steiner minimum trees on parallel lines, DIMACS Series, Vol. 40 (1998) pp. 27–37.

    Google Scholar 

  7. M. Brazil, D.A. Thomas and J.F. Weng, A polynomial time algorithm for rectilinear Steiner trees with terminals constrained to curves, Networks, Vol 33 (1999) pp. 145–155.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Brazil, D.A. Thomas and J.F. Weng, Minimum networks in uniform orientation metrics, SIAM J. Comput. Vol. 30 (2000), pp. 1579–1593.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Brazil, D.A. Thomas and J.F. Weng, Complexity of the rectilinear Steiner tree problem for parallel lines, preprint.

    Google Scholar 

  10. S-W. Cheng, The Steiner tree problem for terminals on the boundary of a rectilinear polygon, Theoret. Comput. Sci. Vol. 237 (2000) pp. 213–238.

    Article  MathSciNet  MATH  Google Scholar 

  11. S-W. Cheng and C-K. Tang, A fast algorithm for computing optimal rectilinear Steiner trees for extremal point sets, LNCS, Vol. 1004 (1995) pp. 322–331.

    MathSciNet  Google Scholar 

  12. S-W. Cheng, A Lim and C-T. Wu, Optimal Rectilinear Steiner trees for extremal point sets, LNCS, Vol. 762 (1993) pp. 523–532.

    MathSciNet  Google Scholar 

  13. E.J. Cockayne, On the Steiner tree problem, Canad. Math. Bull. Vol. 10 (1967) pp. 431–450.

    Article  MathSciNet  MATH  Google Scholar 

  14. J.H. Cohoon, D.S. Richards and J.S. Salowe, An optimal Steiner tree routing algorithm for a net whose terminals lie on the perimeter of a rectangle, IEEE Trans. on Computer-Aided Design, Vol 9 (1990) 398–407.

    Article  Google Scholar 

  15. D-Z. Du, E. Shragowitz and P-J. Wan, Two special cases for rectilinear Steiner minimum trees, LNEM, Vol 450 (1997) pp. 221–233.

    MathSciNet  Google Scholar 

  16. X. Du, D-Z. Du, B. Gao and L. Qü, A simple proof for a result of Ollerenshaw on Steiner trees, in D.-Z. Du and P.M. Pardalos (eds.) Advances in Optimization and Approximation, (Dordrecht, Kluwer Academic Publisher, 1994) pp. 68–71.

    Google Scholar 

  17. J.L. Ganley and J.P. Cohoon, Improved computation of optimal rectilinear Steiner trees, Int. J. Comput. Geometry Appl. Vol. 7 (1997) pp. 457–472.

    Article  MathSciNet  MATH  Google Scholar 

  18. M.R. Garey and D.S. Johnson, The rectilinear Steiner tree problem is NP-complete, SIAM J. Appl. Math. Vol. 32 (1977) pp. 826–834.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Hanan, On Steiner’s problem with rectilinear distance, SIAM J. Appl. Math. Vol. 30 (1966) pp. 255–265.

    Article  MathSciNet  Google Scholar 

  20. F.K. Hwang, On Steiner minimal trees with rectilinear distance, SIAM J. Appl. Math. Vol. 30 (1976) pp. 104–114.

    Article  MathSciNet  MATH  Google Scholar 

  21. F.K. Hwang, D.S. Richards and P. Winter, The Steiner Tree Problem, (Annals of Discrete Math. Vol. 53, North-Holland, Amsterdam, 1992.)

    MATH  Google Scholar 

  22. M. Kaufmann, S. Gao and K. Thulasiraman, On Steiner minimal trees in grid graphs and its application to VLSI routing, LNCS Vol. 834 (1994) pp. 351–359.

    Google Scholar 

  23. M. Kaufmann, S. Gao and K. Thulasiraman, An algorithm for Steiner trees in grid graphs and its application to homotopic routing, J. Circuits, Sys. and Comput. Vol. 6 (1996) pp. 1–13.

    Article  Google Scholar 

  24. Z.A. Melzak, On the problem of Steiner, Canad. Math. Bull., Vol. 4 (1961), pp. 143–148.

    Article  MathSciNet  MATH  Google Scholar 

  25. K. Ollerenshaw, Minimum networks linking four points in a plane, Inst. Math. appl. Vol. 15 (1978) pp. 208–211.

    MathSciNet  Google Scholar 

  26. J.S. Provan, Convexity and the Steiner tree problem, Networks, Vol. 18 (1988) pp. 55–72.

    Article  MathSciNet  MATH  Google Scholar 

  27. D.S. Richards and J.S. Salowe, A rectilinear Steiner minimal tree algorithm for convex point sets, LNCS, Vol 447 (1990) pp. 201–212.

    MathSciNet  Google Scholar 

  28. D.S. Richards and J.S. Salowe, A simple proof of Hwang’s theorem for rectilinear Steiner minimal trees, Ann. Oper. Res. Vol 33 (1991) pp. 549–556.

    Article  MathSciNet  MATH  Google Scholar 

  29. D.S. Richards and J.S. Salowe, A linear-time algorithm to construct a rectilinear Steiner tree for κ-extremal point sets, Algorithmica, Vol. 7 (1992) pp. 247–276.

    Article  MathSciNet  MATH  Google Scholar 

  30. J.H. Rubinstein, D.A. Thomas and N.C. Wormald, Steiner trees for terminals constrained to curves, SIAM J. Discrete Math., Vol. 10 (1997) pp. 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  31. D.M. Warme, P. Winter and M. Zachariasen, Exact algorithms for plane Steiner tree problems: a computational study, in D.-Z. Du and P.M. Pardalos (eds.) Advances in Steiner Trees, (Boston, Kluwer Academic Publisher, 1998) pp. 81–116.

    Google Scholar 

  32. J.F. Weng, Steiner polygon in the Steiner tree problem, Geometriae Dedicate Vol. 52 (1994) pp. 119–127.

    Article  MATH  Google Scholar 

  33. J.F. Weng, Linear Steiner trees for infinite spirals, SIAM J. Discrete Math., Vol. 10 (1997), pp. 388–398.

    Article  MathSciNet  MATH  Google Scholar 

  34. J.F. Weng, Expansion of linear Steiner trees, Algorithmica, Vol. 19 (1997), pp. 318–330.

    Article  MathSciNet  MATH  Google Scholar 

  35. Y.Y. Yang and O. Wing, Optimal and suboptimal solution algorithms for the wiring problem, Int’l Symp. Circuit Theory, 1972, pp. 154–158.

    Google Scholar 

  36. G.Y. Yan, A. Albrecht, G.H.F. Young and C.K. Wong, The Steiner problem in orientation metrics, J. Comput. System Sci., Vol. 55 (1997), pp. 529–546.

    Article  MathSciNet  MATH  Google Scholar 

  37. M. Zachariasen, Rectilinear full Steiner tree generation, Networks, Vol. 33 (1997), pp. 125–143.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

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

    Doreen A. Thomas & Jia F. Weng

Authors
  1. Doreen A. Thomas
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  2. Jia F. Weng
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN, USA

    Xiu Zhen Cheng  & Ding-Zhu Du  & 

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© 2001 Kluwer Academic Publishers

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Thomas, D.A., Weng, J.F. (2001). Polynomial Time Algorithms for the Rectilinear Steiner Tree Problem. In: Cheng, X.Z., Du, DZ. (eds) Steiner Trees in Industry. Combinatorial Optimization, vol 11. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0255-1_13

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  • DOI: https://doi.org/10.1007/978-1-4613-0255-1_13

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7963-8

  • Online ISBN: 978-1-4613-0255-1

  • eBook Packages: Springer Book Archive

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