Skip to main content
SpringerLink
Log in

Steiner Minimal Trees: An Introduction, Parallel Computation, and Future Work

  • Reference work entry
  • First Online:
Handbook of Combinatorial Optimization

Abstract

Given a set of N cities, construct a connected network which has minimum length. The problem is simple enough, but the catch is that you are allowed to add junctions in your network. Therefore, the problem becomes how many extra junctions should be added and where should they be placed so as to minimize the overall network length. This intriguing optimization problem is known as the Steiner minimal tree (SMT) problem, where the junctions that are added to the network are called Steiner points. This chapter presents a brief overview of the problem, presents an approximation algorithm which performs very well, then reviews the computational algorithms implemented for this problem. The foundation of this chapter is a parallel algorithm for the generation of what Pawel Winter termed T_list and its implementation. This generation of T_list is followed by the extraction of the proper answer. When Winter developed his algorithm, the time for extraction dominated the overall computation time. After Cockayne and Hewgill’s work, the time to generate T_list dominated the overall computation time. The parallel algorithms presented here were implemented in a program called PARSTEINER94, and the results show that the time to generate T_list has now been cut by an order of magnitude. So now the extraction time once again dominates the overall computation time. This chapter then concludes with the characterization of SMTs for certain size grids. Beginning with the known characterization of the SMT for a 2 ×m grid, a grammar with rewrite rules is presented for characterizations of SMTs for 3 ×m, 4 ×m, 5 ×m, 6 ×m, and 7 ×m grids.

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

Access this chapter

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 3,400.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
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

Institutional subscriptions

References

  1. A. Aggarwal, B. Chazelle, L. Guibas, C. O’Dunlaing, C. Yap, Parallel computational geometry. Algorithmica 3(3), 293–327 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. M.J. Atallah, M.T. Goodrich, Parallel algorithms for some functions of two convex polygons. Algorithmica 3(4), 535–548 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. J.E. Beasley, Or-library: distributing test problems by electronic mail. J. Oper. Res. Soc. 41(11), 1069–1072 (1990)

    Google Scholar 

  4. J.E. Beasley, Or-library. http://people.brunel.ac.uk/~mastjjb/jeb/info.html. Last Accessed 29 Dec 2010

  5. M.W. Bern, R.L. Graham, The shortest-network problem. Sci. Am. 260(1), 84–89 (1989)

    Article  Google Scholar 

  6. W.M. Boyce, J.R. Seery, STEINER 72 – an improved version of Cockayne and Schiller’s program STEINER for the minimal network problem. Technical Report 35, Bell Labs., Department of Computer Science, 1975

    Google Scholar 

  7. G.X. Chen, The shortest path between two points with a (linear) constraint [in Chinese]. Knowl. Appl. Math. 4, 1–8 (1980)

    Google Scholar 

  8. A. Chow, Parallel Algorithms for Geometric Problems. PhD thesis, University of Illinois, Urbana-Champaign, IL, 1980

    Google Scholar 

  9. F.R.K. Chung, M. Gardner, R.L. Graham, Steiner trees on a checkerboard. Math. Mag. 62, 83–96 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. F.R.K. Chung, R.L. Graham, in Steiner Trees for Ladders, ed. by B. Alspach, P. Hell, D.J. Miller, Annals of Discrete Mathematics, vol. 2 (Elsevier Science Publishers B.V., The Netherlands, 1978), pp. 173–200

    Google Scholar 

  11. E.J. Cockayne, On the Steiner problem. Can. Math. Bull. 10(3), 431–450 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  12. E.J. Cockayne, On the efficiency of the algorithm for Steiner minimal trees. SIAM J. Appl. Math. 18(1), 150–159 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  13. E.J. Cockayne, D.E. Hewgill, Exact computation of Steiner minimal trees in the plane. Info. Process. Lett. 22(3), 151–156 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. E.J. Cockayne, D.E. Hewgill, Improved computation of plane Steiner minimal trees. Algorithmica 7(2/3), 219–229 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. E.J. Cockayne, D.G. Schiller, in Computation of Steiner Minimal Trees, ed. by D.J.A. Welsh, D.R. Woodall, Combinatorics, pp. 52–71, Maitland House, Warrior Square, Southend-on-Sea, Essex SS1 2J4, 1972. Mathematical Institute, Oxford, Inst. Math. Appl.

    Google Scholar 

  16. R. Courant, H. Robbins, What Is Mathematics? An Elementary Approach to Ideas and Methods (Oxford University Press, London, 1941)

    MATH  Google Scholar 

  17. D.Z. Du, F.H. Hwang, A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica 7(2/3), 121–135 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. M.R. Garey, R.L. Graham, D.S Johnson, The complexity of computing Steiner minimal trees. SIAM J. Appl. Math. 32(4), 835–859 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Geist, A. Beguelin, J. Dongarra, W. Jiang, R. Manchek, V. Sunderam, PVM: Parallel Virtual Machine – A User’s Guide and Tutorial for Networked Parallel Computing (MIT Press, Cambridge, MA, 1994)

    Google Scholar 

  20. R. Geist, R. Reynolds, C. Dove, Context sensitive color quantization. Technical Report 91–120, Dept. of Comp. Sci., Clemson Univ., Clemson, SC 29634, July 1991

    Google Scholar 

  21. R. Geist, R. Reynolds, D. Suggs, A markovian framework for digital halftoning. ACM Trans. Graph. 12(2), 136–159 (1993)

    Article  MATH  Google Scholar 

  22. R. Geist, D. Suggs, Neural networks for the design of distributed, fault-tolerant, computing environments, in Proc. 11th IEEE Symp. on Reliable Distributed Systems (SRDS), Houston, Texas, October 1992, pp. 189–195

    Google Scholar 

  23. R. Geist, D. Suggs, R. Reynolds, Minimizing mean seek distance in mirrored disk systems by cylinder remapping, in Proc. 16th IFIP Int. Symp. on Computer Performance Modeling, Measurement, and Evaluation (PERFORMANCE ‘93), Rome, Italy, September 1993, pp. 91–108

    Google Scholar 

  24. R. Geist, D. Suggs, R. Reynolds, S. Divatia, F. Harris, E. Foster, P. Kolte, Disk performance enhancement through Markov-based cylinder remapping, in Proc. of the ACM Southeastern Regional Conf., ed. by C.M. Pancake, D.S. Reeves, Raleigh, North Carolina, April 1992, pp. 23–28. The Association for Computing Machinery, Inc.

    Google Scholar 

  25. G. Georgakopoulos, C. Papadimitriou, A 1-steiner tree problem. J. Algorithm 8(1), 122–130 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  26. E.N. Gilbert, H.O. Pollak, Steiner minimal trees. SIAM J. Appl. Math. 16(1), 1–29 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Grossberg, Nonlinear neural networks: Principles, mechanisms, and architectures. Neural Network 1, 17–61 (1988)

    Article  Google Scholar 

  28. F.C. Harris, Jr, Parallel Computation of Steiner Minimal Trees. PhD thesis, Clemson, University, Clemson, SC 29634, May 1994

    Google Scholar 

  29. F.C. Harris, Jr, A stochastic optimization algorithm for steiner minimal trees. Congr. Numer. 105, 54–64 (1994)

    MathSciNet  Google Scholar 

  30. F.C. Harris, Jr, An introduction to steiner minimal trees on grids. Congr. Numer. 111, 3–17 (1995)

    MathSciNet  MATH  Google Scholar 

  31. F.C. Harris, Jr, Parallel computation of steiner minimal trees, in Proc. of the 7th SIAM Conf. on Parallel Process. for Sci. Comput., ed. by David H. Bailey, Petter E. Bjorstad, John R. Gilbert, Michael V. Mascagni, Robert S. Schreiber, Horst D. Simon, Virgia J. Torczan, Layne T. Watson, San Francisco, California, February 1995. SIAM, pp. 267–272

    Google Scholar 

  32. S. Hedetniemi, Characterizations and constructions of minimally 2-connected graphs and minimally strong digraphs, in Proc. 2 nd Louisiana Conf. on Combinatorics, Graph Theory, and Computing, Louisiana State University, Baton Rouge, Louisiana, March 1971, pages 257–282

    Google Scholar 

  33. J. Hegie, Steiner minimal trees on the gpu. Master’s thesis, University of Nevada, Reno, 2012

    Google Scholar 

  34. Universitat Heidelberg, Tsplib. http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/. Last Accessed 29 Dec 2010

  35. J.J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. 81, 3088–3092 (1984)

    Article  Google Scholar 

  36. F.K. Hwang, J.F. Weng, The shortest network under a given topology. J. Algorithm 13(3), 468–488 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  37. F.K. Hwang, D.S. Richards, Steiner tree problems. Networks 22(1), 55–89 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  38. F.K. Hwang, D.S. Richards, P. Winter, The Steiner Tree Problem, vol. 53 of Ann. Discrete Math. (North-Holland, Amsterdam, 1992)

    MATH  Google Scholar 

  39. F.K. Hwang, G.D. Song, G.Y. Ting, D.Z. Du, A decomposition theorem on Euclidian Steiner minimal trees. Disc. Comput. Geom. 3(4), 367–382 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  40. J. JáJá, An Introduction to Parallel Algorithms (Addison-Wesley, Reading, MA, 1992)

    MATH  Google Scholar 

  41. V. Jarník, O. Kössler, O minimálnich gratech obsahujicich n daných bodu [in Czech]. Casopis Pesk. Mat. Fyr. 63, 223–235 (1934)

    Google Scholar 

  42. S. Kirkpatrick, C. Gelatt, M. Vecchi, Optimization by simulated annealing. Science 220(13), 671–680 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  43. V. Kumar, A. Grama, A. Gupta, G. Karypis, Introduction to Parallel Computing: Design and Analysis of Algorithms (The Benjamin/Cummings Publishing, Redwood City, 1994)

    MATH  Google Scholar 

  44. Z.A. Melzak, On the problem of Steiner. Can. Math. Bull. 4(2), 143–150 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  45. M.K. Molloy, Performance analysis using stochastic petri nets. IEEE Trans. Comput. C-31(9), 913–917 (1982)

    Article  Google Scholar 

  46. Nvidia, Cuda zone. http://www.nvidia.com/object/cuda_home_new.html. Last Accessed 29 Dec 2010

  47. Nvidia, Geforce gtx 580. http://www.nvidia.com/object/product-geforce-gtx-580-us.html. Last Accessed 29 Dec 2010

  48. J.D. Owens, D. Luebke, N. Govindaraju, M. Harris, J. Krger, A.E. Lefohn, T.J. Purcell, A survey of general-purpose computation on graphics hardware. Comput. Graph. Forum 26(1), 80–113 (2007)

    Article  Google Scholar 

  49. J.L. Peterson, Petri Net Theory and the Modeling of Systems (Prentice-Hall, Englewood Cliffs, 1981)

    Google Scholar 

  50. F.P. Preparata, M.I. Shamos, Computational Geometry: An Introduction (Springer, New York, 1988)

    Google Scholar 

  51. M.J. Quinn, Parallel Computing: Theory and Practice (McGraw-Hill, New York, 1994)

    Google Scholar 

  52. M.J. Quinn, N. Deo, An upper bound for the speedup of parallel best-bound branch-and-bound algorithms. BIT 26(1), 35–43 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  53. W.R. Reynolds, A Markov Random Field Approach to Large Combinatorial Optimization Problems. PhD thesis, Clemson, University, Clemson, SC 29634, August 1993

    Google Scholar 

  54. M.I. Shamos, Computational Geometry. PhD thesis, Department of Computer Science, Yale University, New Haven, 1978

    Google Scholar 

  55. J.R. Smith, The Design and Analysis of Parallel Algorithms (Oxford University Press, New York, 1993)

    MATH  Google Scholar 

  56. D. Trietsch, Augmenting Euclidean networks – the Steiner case. SIAM J. Appl. Math. 45, 855–860 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  57. D. Trietsch, F.K. Hwang, An improved algorithm for Steiner trees. SIAM J. Appl. Math. 50, 244–263 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  58. D.M. Warme, P. Winter, M. Zachariasen, Exact algorithms for plane steiner tree problems: a computational study, in Advances in Steiner Trees, ed. by D.-Z. Du, J.M. Smith, J.H. Rubinstein (Kluwer Academic, Boston, 2000), pp. 81–116

    Chapter  Google Scholar 

  59. D.M. Warme, A new exact algorithm for rectilinear steiner trees, in 16th International Symposium on Mathematical Programming. American Mathematical Society, Lausanne, Switzerland, 1997, pp. 357–395

    Google Scholar 

  60. P. Winter, An algorithm for the Steiner problem in the Euclidian plane. Networks 15(3), 323–345 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  61. P. Winter, M. Zachariasen, Large euclidean steiner minimum trees in an hour. Technical Report 96/34, DIKU, Department of Computer Science, University of Copenhagen, 1996

    Google Scholar 

  62. P. Winter, M. Zachariasen, Euclidean Steiner minimum trees: an improved exact algorithm. Networks 30, 149–166 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science & Engineering, University of Nevada, Reno, 89557, NV,USA

    Frederick C. Harris Jr. & Rakhi Motwani

Authors
  1. Frederick C. Harris Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rakhi Motwani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Frederick C. Harris Jr. or Rakhi Motwani .

Editor information

Editors and Affiliations

  1. Department of Industrial and Systems Eng, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Computer Science, University of Texas, Dallas, Richardson, Texas, USA

    Ding-Zhu Du

  3. Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

    Ronald L. Graham

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this entry

Cite this entry

Harris, F.C., Motwani, R. (2013). Steiner Minimal Trees: An Introduction, Parallel Computation, and Future Work. In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_56

Download citation

Publish with us

Policies and ethics

Search

Navigation