Skip to main content
SpringerLink
Log in
Book cover

Advances in Steiner Trees pp 137–162Cite as

Exact Steiner Trees in Graphs and Grid Graphs

  • Chapter

Part of the book series: Combinatorial Optimization ((COOP,volume 6))

Abstract

Given a graph with nonnegative edge lengths and a selected subset of vertices, the Steiner tree problem is to find a tree of minimum length that spans the selected vertices. This problem is also commonly called the graphical Steiner minimal tree problem or GSMT problem for short. We call the selected vertices terminals. In a Steiner tree, any vertex which is not a terminal and has degree at least three is called a Steiner vertex.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P.K. Agarwal and M.T. Shing, Algorithms for 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 

  2. 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 

  3. M. Bern, Network design problems: Steiner trees and spanning k-trees, Ph.D. thesis, Computer Science Division, University of California at Berkeley, 1987.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Bern and R.L. Graham, The shortest-network problem, Scientific American, January (1989) pp. 84–89.

    Google Scholar 

  7. M. Brazil, D.A. Thomas, and J.F. Weng, Rectilinear Steiner minimal trees on parallel lines, in D.Z. Du, J.M. Smith, and J.H. Rubinstein (eds.) Advances in Steiner trees, (Kluwer Academic Publishers, 1998).

    Google Scholar 

  8. M. Brazil, D.A. Thomas, and J.F. Weng, A polynomial time algorithm for rectilinear Steiner trees with terminals constrained to curves, manuscript, 1998.

    Google Scholar 

  9. S.W. Cheng, Steiner tree for terminals on the boundary of a rectilinear polygon, Proceedings of DIMACS workshop on network connectivity and facilities location, April 28–30, 1997, (DIMACS Series in Discrete Mathematics and Theoretical Computer Science 40, 1998) pp. 39–57.

    Google Scholar 

  10. S.W. Cheng, A. Lim, and C.T. Wu, Optimal Steiner trees for extremal point sets, Proceedings of International Symposium on Algorithms and Computation, 1993 pp. 523–532.

    Google Scholar 

  11. S.W. Cheng and C.K. Tang, Fast algorithms for optimal Steiner trees for extremal point sets, Proceedings of International Symposium on Algorithms and Computation, 1995 pp. 322–331.

    Google Scholar 

  12. C. Chiang, M. Sarrafzadeh, and C.K. Wong, An optimal algorithm for rectilinear Steiner trees for channels with obstacles, International Journal of Circuit Theory and Application, Vol.19 (1991) pp. 551–563.

    Article  MATH  Google Scholar 

  13. C. Chiang, M. Sarrafzadeh, and C.K. Wong, An algorithm for exact rectilinear Steiner trees for switchbox with obstacles, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, Vol.39 (1992) pp. 446–454.

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  15. T.H. Cormen, C.E. Leiserson, and R.L. Rivest, Introduction to Algorithms, (Cambridge, Massachusetts, The MIT Press, 1994).

    Google Scholar 

  16. S.E. Dreyfus and R.A. Wagner, The Steiner problem in graphs, Networks, Vol.1 (1972) pp. 195–207.

    Article  MathSciNet  MATH  Google Scholar 

  17. R.E Erickson, C.L. Monma, and A.F. Veinott, Send-and-split method for minimum-concave-cost network flows, Mathematics of Operations Research, Vol.12 (1987) pp. 634–664.

    Article  MathSciNet  MATH  Google Scholar 

  18. M.L. Fredman and R.E. Tarjan, Fibonacci heaps and their uses in improved network optimization algorithms, Journal of ACM, Vol.16 (1987) pp. 1004–1023.

    Google Scholar 

  19. M.R. Garey and D.S. Johnson, Computers and Intractability, (New York, W.H. Freeman and Company, 1979).

    MATH  Google Scholar 

  20. R.I. Greenberg and F.M. Maley, Minimum separation for single-layer channel routing, Information Processing Letters, Vol.43 (1992) pp. 201–205.

    Article  MATH  Google Scholar 

  21. S.L. Hakimi, Steiner’s problem in graphs and its implications, Networks, Vol.l (1971) pp. 113–133.

    Google Scholar 

  22. M. Hanan, On Steiner’s problem with rectilinear distance, SIAM Journal on Applied mathematics, Vol.14 (1966) pp. 255–265.

    Article  MathSciNet  MATH  Google Scholar 

  23. F.K. Hwang, On Steiner minimal trees with rectilinear distance, SIAM Journal on Applied Mathematics, Vol.30 (1976) pp. 104–114.

    Article  MathSciNet  MATH  Google Scholar 

  24. F.K. Hwang and D.S. Richards, Steiner tree problems, Networks, Vol.22 (1992) pp. 55–89.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  26. R.M. Karp, Reducibility among combinatorial problems, in R.E. Miller and J.W. Thatcher (eds.), Complexity of Computer Computations, (New York, Plenum Press, 1972) pp. 85–103.

    Chapter  Google Scholar 

  27. M. Kaufmann, S. Gao, and K. Thulasiraman, An algorithm for S t einer trees in grid graphs and its application to homotopic routing, Journal of Circuits, Systems, and Computers, Vol.6 (1996) pp. 1–13.

    Article  Google Scholar 

  28. C.E. Leiserson and F.M. Maley, Algorithms for routing and testing routability of planar VLSI layouts, Proceedings of the 17th Annual ACM Symposium on the Theory of Computing, 1985 pp. 69–78.

    Google Scholar 

  29. A.J. Levin, Algorithm for the shortest connection of a group of graph vertices, Soviet Math. Doklady, Vol.12 (1971) pp. 1477–1481.

    MATH  Google Scholar 

  30. F.M. Maley, Single-Layer Wire Routing and Compaction, (Cambridge, Massachusetts, The MIT Press, 1990).

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. D.S. Richards and J.S. Salowe, A linear-time algorithm to construct a rectilinear Steiner tree for k-extremal points, Algorithmica, Vol.7 (1992) pp. 246–276.

    Article  MathSciNet  Google Scholar 

  33. P. Winter, Steiner problem in networks: A survey. Networks, Vol.17 (1987) pp. 129–167.

    Article  MathSciNet  MATH  Google Scholar 

  34. A.Z. Zelikovsky, An 11/6-approximation algorithm for the network Steiner problem, Algorithmica, Vol.9 (1993) pp. 463–470.

    Article  MathSciNet  MATH  Google Scholar 

  35. A.Z. Zelikovsky, A faster approximation algorithm for the Steiner tree problem in graphs, Information Processing Letters, Vol.46 (1993) pp. 79–83.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, The Hong Kong University of Science & Technology, Clear Water Bay, Hong Kong

    Siu-Wing Cheng

Authors
  1. Siu-Wing Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Minnesota, USA

    Ding-Zhu Du

  2. Department of Mechanical & Industrial Engineering, University of Massachusetts, USA

    J. M. Smith

  3. Department of Mathematics, University of Melbourne, Australia

    J. H. Rubinstein

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Cheng, SW. (2000). Exact Steiner Trees in Graphs and Grid Graphs. In: Du, DZ., Smith, J.M., Rubinstein, J.H. (eds) Advances in Steiner Trees. Combinatorial Optimization, vol 6. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3171-2_8

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-3171-2_8

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4824-3

  • Online ISBN: 978-1-4757-3171-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation