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

Reference work entry


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.


Parallel Algorithm Parallel Machine Decomposition Theorem Steiner Point Steiner Tree Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    A. Aggarwal, B. Chazelle, L. Guibas, C. O’Dunlaing, C. Yap, Parallel computational geometry. Algorithmica 3(3), 293–327 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    M.J. Atallah, M.T. Goodrich, Parallel algorithms for some functions of two convex polygons. Algorithmica 3(4), 535–548 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J.E. Beasley, Or-library: distributing test problems by electronic mail. J. Oper. Res. Soc. 41(11), 1069–1072 (1990)Google Scholar
  4. 4.
    J.E. Beasley, Or-library. Last Accessed 29 Dec 2010
  5. 5.
    M.W. Bern, R.L. Graham, The shortest-network problem. Sci. Am. 260(1), 84–89 (1989)CrossRefGoogle Scholar
  6. 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, 1975Google Scholar
  7. 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. 8.
    A. Chow, Parallel Algorithms for Geometric Problems. PhD thesis, University of Illinois, Urbana-Champaign, IL, 1980Google Scholar
  9. 9.
    F.R.K. Chung, M. Gardner, R.L. Graham, Steiner trees on a checkerboard. Math. Mag. 62, 83–96 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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. 173–200Google Scholar
  11. 11.
    E.J. Cockayne, On the Steiner problem. Can. Math. Bull. 10(3), 431–450 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    E.J. Cockayne, On the efficiency of the algorithm for Steiner minimal trees. SIAM J. Appl. Math. 18(1), 150–159 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    E.J. Cockayne, D.E. Hewgill, Exact computation of Steiner minimal trees in the plane. Info. Process. Lett. 22(3), 151–156 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    E.J. Cockayne, D.E. Hewgill, Improved computation of plane Steiner minimal trees. Algorithmica 7(2/3), 219–229 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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. 16.
    R. Courant, H. Robbins, What Is Mathematics? An Elementary Approach to Ideas and Methods (Oxford University Press, London, 1941)zbMATHGoogle Scholar
  17. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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. 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 1991Google Scholar
  21. 21.
    R. Geist, R. Reynolds, D. Suggs, A markovian framework for digital halftoning. ACM Trans. Graph. 12(2), 136–159 (1993)CrossRefzbMATHGoogle Scholar
  22. 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. 189–195Google Scholar
  23. 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. 91–108Google Scholar
  24. 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. 25.
    G. Georgakopoulos, C. Papadimitriou, A 1-steiner tree problem. J. Algorithm 8(1), 122–130 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    E.N. Gilbert, H.O. Pollak, Steiner minimal trees. SIAM J. Appl. Math. 16(1), 1–29 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    S. Grossberg, Nonlinear neural networks: Principles, mechanisms, and architectures. Neural Network 1, 17–61 (1988)CrossRefGoogle Scholar
  28. 28.
    F.C. Harris, Jr, Parallel Computation of Steiner Minimal Trees. PhD thesis, Clemson, University, Clemson, SC 29634, May 1994Google Scholar
  29. 29.
    F.C. Harris, Jr, A stochastic optimization algorithm for steiner minimal trees. Congr. Numer. 105, 54–64 (1994)MathSciNetGoogle Scholar
  30. 30.
    F.C. Harris, Jr, An introduction to steiner minimal trees on grids. Congr. Numer. 111, 3–17 (1995)MathSciNetzbMATHGoogle Scholar
  31. 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. 267–272Google Scholar
  32. 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 257–282Google Scholar
  33. 33.
    J. Hegie, Steiner minimal trees on the gpu. Master’s thesis, University of Nevada, Reno, 2012Google Scholar
  34. 34.
    Universitat Heidelberg, Tsplib. Last Accessed 29 Dec 2010
  35. 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)CrossRefGoogle Scholar
  36. 36.
    F.K. Hwang, J.F. Weng, The shortest network under a given topology. J. Algorithm 13(3), 468–488 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    F.K. Hwang, D.S. Richards, Steiner tree problems. Networks 22(1), 55–89 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    F.K. Hwang, D.S. Richards, P. Winter, The Steiner Tree Problem, vol. 53 of Ann. Discrete Math. (North-Holland, Amsterdam, 1992)zbMATHGoogle Scholar
  39. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    J. JáJá, An Introduction to Parallel Algorithms (Addison-Wesley, Reading, MA, 1992)zbMATHGoogle Scholar
  41. 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. 42.
    S. Kirkpatrick, C. Gelatt, M. Vecchi, Optimization by simulated annealing. Science 220(13), 671–680 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 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)zbMATHGoogle Scholar
  44. 44.
    Z.A. Melzak, On the problem of Steiner. Can. Math. Bull. 4(2), 143–150 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    M.K. Molloy, Performance analysis using stochastic petri nets. IEEE Trans. Comput. C-31(9), 913–917 (1982)CrossRefGoogle Scholar
  46. 46.
    Nvidia, Cuda zone. Last Accessed 29 Dec 2010
  47. 47.
    Nvidia, Geforce gtx 580. Last Accessed 29 Dec 2010
  48. 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)CrossRefGoogle Scholar
  49. 49.
    J.L. Peterson, Petri Net Theory and the Modeling of Systems (Prentice-Hall, Englewood Cliffs, 1981)Google Scholar
  50. 50.
    F.P. Preparata, M.I. Shamos, Computational Geometry: An Introduction (Springer, New York, 1988)Google Scholar
  51. 51.
    M.J. Quinn, Parallel Computing: Theory and Practice (McGraw-Hill, New York, 1994)Google Scholar
  52. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    W.R. Reynolds, A Markov Random Field Approach to Large Combinatorial Optimization Problems. PhD thesis, Clemson, University, Clemson, SC 29634, August 1993Google Scholar
  54. 54.
    M.I. Shamos, Computational Geometry. PhD thesis, Department of Computer Science, Yale University, New Haven, 1978Google Scholar
  55. 55.
    J.R. Smith, The Design and Analysis of Parallel Algorithms (Oxford University Press, New York, 1993)zbMATHGoogle Scholar
  56. 56.
    D. Trietsch, Augmenting Euclidean networks – the Steiner case. SIAM J. Appl. Math. 45, 855–860 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    D. Trietsch, F.K. Hwang, An improved algorithm for Steiner trees. SIAM J. Appl. Math. 50, 244–263 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 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. 81–116CrossRefGoogle Scholar
  59. 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. 357–395Google Scholar
  60. 60.
    P. Winter, An algorithm for the Steiner problem in the Euclidian plane. Networks 15(3), 323–345 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 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, 1996Google Scholar
  62. 62.
    P. Winter, M. Zachariasen, Euclidean Steiner minimum trees: an improved exact algorithm. Networks 30, 149–166 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer Science & EngineeringUniversity of NevadaRenoNV,USA

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