Gradient-Constrained Minimum Interconnection Networks

Reference work entry

Abstract

In two- or three-dimensional space, an embedded network is called gradient constrained if the absolute gradient of any differentiable point on the edges in the network is no more than a given value m. This chapter gives an overview of the properties of gradient-constrained networks that interconnect a given set of points and that minimize some property of the network, such as the total length of edges. The focus is particularly on geometric Steiner and Gilbert networks. These networks are of interest both mathematically and for their applications to the design of underground mines.

Keywords

Zinc Drilling Hull Fermat Harness 

Notes

Acknowledgements

The research for and writing of this chapter was supported by an ARC Linkage Grant. We gratefully acknowledge the contributions of our collaborators to much of the theory described in this chapter, particularly Professor Doreen Thomas at The University of Melbourne.

Recommended Reading

  1. 1.
    A.V. Aho, M.R. Garey, F.K. Hwang, Rectilinear Steiner trees: efficient special-case algorithms. Networks 7, 37–58 (1977)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    C. Alford, M. Brazil, D.H. Lee, Optimisation in underground mining, in Handbook on Operations Research in Natural Resources, ed. by A. Weintraub et al. (Springer, New York, 2007), pp. 561–577Google Scholar
  3. 3.
    C. Bajaj, The algebraic degree of geometric optimization problems. Discret. Comput. Geom. 3(2), 177–191 (1988)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    S. Bhaskaran, F.J.M. Salzborn, Optimal design of gas pipeline networks. J. Oper. Res. Soc. 30(12), 1047–1060 (1979)MATHGoogle Scholar
  5. 5.
    S. Boyd, A. Mutapcic, Subgradient methods, lecture notes of EE364b, Stanford University, Winter Quarter, 2006–2007 (Available at http://www.stanford.edu/class/ee364b/lectures/subgrad_method_notes.pdf)
  6. 6.
    S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, New York, 2004)CrossRefMATHGoogle Scholar
  7. 7.
    M. Brazil, Steiner minimum trees in uniform orientation metrics, in Steiner Trees in Industry, ed. by X. Cheng, D.-Z. Du (Kluwer Academic Publishers, Dordrecht/Boston/London, 2001), pp. 1–28Google Scholar
  8. 8.
    M. Brazil, D.A. Thomas, Network optimization for the design of underground mines. Networks 49(1), 40–50 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    M. Brazil, M. Zachariasen, Steiner trees for fixed orientation metrics. J. Glob. Optim. 43, 141–169 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    M. Brazil, D.A. Thomas, J.F. Weng, Gradient-constrained minimal Steiner trees, in Network Design: Connectivity and Facilities Location, ed. by P.M. Pardalos, D.-Z. Du DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 40 (American Mathematical Society, Providence, 1998), pp. 23–38Google Scholar
  11. 11.
    M. Brazil, D.H. Lee, J.H. Rubinstein, D.A. Thomas, J.F. Weng, N.C. Wormald, Network optimisation of underground mine design. Australas. Inst. Min. Metall. Proc. 305(1), 57–65 (2000)Google Scholar
  12. 12.
    M. Brazil, D.A. Thomas, J.F. Weng, On the complexity of the Steiner problem. J. Comb. Optim. 4, 187–195 (2000)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    M. Brazil, J.H. Rubinstein, D.A. Thomas, J.F. Weng, N.C. Wormald, Gradient-constrained minimum networks. I. Fundamentals. J. Glob. Optim. 21(2), 139–155 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    M. Brazil, D.H. Lee, J.H. Rubinstein, D.A. Thomas, J.F. Weng, N.C. Wormald, A network model to optimise cost in underground mine design. Trans. S. Afr. Inst. Electr. Eng. 93(2), 97–103 (2002)Google Scholar
  15. 15.
    M. Brazil, D.H. Lee, M. van Leuven, J.H. Rubinstein, D.A. Thomas, N.C. Wormald, Optimising declines in underground mines. Min. Technol. 112, 164–170 (2003)CrossRefGoogle Scholar
  16. 16.
    M. Brazil, D.H. Lee, J.H. Rubinstein, D.A. Thomas, J.F. Weng, N.C. Wormald, Optimisation in the design of underground mine access, in Orebody Modelling and Strategic Mine Planning: Uncertainty and Risk Management, ed. by R. Dimitrakopoulos. Spectrum Series, vol. 14 (Australasian Institute of Mining and Metallurgy, Carlton, 2005), pp. 121–124Google Scholar
  17. 17.
    M. Brazil, D.A. Thomas, J.F. Weng, J.H. Rubinstein, D.H. Lee, Cost optimisation for underground mining networks. Optim. Eng. 6(2), 241–256 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    M. Brazil, D.A. Thomas, J.F. Weng, M. Zachariasen, Canonical forms and algorithms for Steiner trees in uniform orientation metrics. Algorithmica 44, 281–300 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    M. Brazil, P.A. Grossman, D.H. Lee, J.H. Rubinstein, D.A. Thomas, N.C. Wormald, Constrained path optimisation for underground mine layout, in International Conference of Applied and Engineering Mathematics, London, July 2007, pp. 856–861Google Scholar
  20. 20.
    M. Brazil, P.A. Grossman, D.H. Lee, J.H. Rubinstein, D.A. Thomas, N.C. Wormald, Decline design in underground mines using constrained path optimisation. Min. Technol. 117, 93–99 (2008)CrossRefGoogle Scholar
  21. 21.
    M. Brazil, D.A. Thomas, J.F. Weng, Gradient-constrained minimum networks. II. Labelled or locally minimal Steiner points. J. Glob. Optim. 42, 23–37 (2008)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    M. Brazil, J.H. Rubinstein, D.A. Thomas, J.F. Weng, N.C. Wormald, Gradient-constrained minimum networks, III. Fixed topology. J. Optim. Theory Appl. 155(1), 336–354 (2012).MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    J. Brimberg, The Fermat-Weber location problem revisited. Math. Program. 71(1), 71–76 (1995)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    J. Brimberg, H. Juel, A. Schöbel, Linear facility location in three dimensions—models and solution methods. Oper. Res. 50(6), 1050–1057 (2002)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    R. Chandrasekaran, A. Tamir, Open questions concerning Weiszfeld’s algorithm for the Fermat-Weber location problem. Math. Program. 44, 293–295 (1989)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    C.L. Cox, Flow-dependent networks: existence and behavior at Steiner points. Networks 31(3), 149–156 (1998)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Z. Drezner, H.W. Hamacher (eds.), Facility Location: Applications and Theory (Springer, Berlin, 2004)Google Scholar
  28. 28.
    D.-Z. Du, F.K. Hwang, Reducing the Steiner problem in a normed space. SIAM J. Comput. 21(6), 1001–1007 (1992)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    D.-Z. Du, B. Gao, R.L. Graham, Z.-C. Liu, P.-J. Wan, Minimum Steiner trees in normed planes. Discret. Comput. Geom. 9(1), 351–370 (1993)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    R. Durier, C. Michelot, Geometrical properties of the Fermat-Weber problem. Eur. J. Oper. Res. 20(3), 332–343 (1985)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    G.A. Ferguson, Mining engineers toolkit. Camborne Sch. Mines Assoc. J. 10–13 (2000)Google Scholar
  32. 32.
    M.R. Garey, D.S. Johnson, The rectilinear Steiner tree problem is NP-complete. SIAM J. Appl. Math. 32(4), 826–834 (1977)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    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)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    E.N. Gilbert, Minimum cost communication networks. Bell Syst. Tech. J. 46, 2209–2227 (1967)CrossRefGoogle Scholar
  35. 35.
    E.N. Gilbert, H.O. Pollak, Steiner minimal trees. SIAM J. Appl. Math. 16(1), 1–29 (1968)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Z. Gligoric, C. Beljic, V. Simeunovic, Shaft location selection at deep multiple orebody deposit by using fuzzy TOPSIS method and network optimization. Expert Syst. Appl. 37(2), 1408–1418 (2010)CrossRefGoogle Scholar
  37. 37.
    F.K. Hwang, On Steiner minimal trees with rectilinear distance. SIAM J. Appl. Math. 30, 104–114 (1976)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    F.K. Hwang, A primer of the Euclidean Steiner problem. Ann. Oper. Res. 33, 73–84 (1991)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    F.K. Hwang, J.F. Weng, The shortest network under a given topology. J. Algorithms 13(3), 468–488 (1992)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    F.K. Hwang, D.S. Richards, P. Winter, The Steiner Tree Problem. Annals of Discrete Mathematics, vol. 53 (Elsevier Science Publishers, North Holland, Amsterdam, 1992)MATHGoogle Scholar
  41. 41.
    H.W. Kuhn, A note on Fermat’s problem. J. Math. Program. 4(1), 98–107 (1973)CrossRefMATHGoogle Scholar
  42. 42.
    H.W. Kuhn, “Steiner’s” problem revisited, in Studies in Optimization, ed. by G.B. Dantzig, B.C. Eaves. Studies in Mathematics, vol. 10 (Mathematical Association of America, Washington, 1974), pp. 52–70Google Scholar
  43. 43.
    H.W. Kuhn, R.E. Kuenne, An efficient algorithm for the numerical solution of the generalized Weber problem in spatial economics. J. Reg. Sci. 4(2), 21–33 (1962)CrossRefGoogle Scholar
  44. 44.
    Y.S. Kupitz, H. Martini, Geometric aspects of the generalized Fermat-Torricelli problem, in Intuitive Geometry, ed. by I. Bárány, K. Boroczky. Bolyai Society Mathematical Studies, vol. 6 (Janos Bolyai Mathematical Society, Budapest, 1997), pp. 55–127Google Scholar
  45. 45.
    D.H. Lee, Low cost drainage networks. Networks, 6, 351–371 (1976)CrossRefMATHGoogle Scholar
  46. 46.
    H. Lerchs, I.F. Grossmann, Optimum design of open-pit mines. Trans. Can. Inst. Min. Metall. LXVIII, 17–24 (1965)Google Scholar
  47. 47.
    Y. Lizotte, J. Elbrond, Optimal layout of underground mining levels. Can. Inst. Min. Metall. Petrol. Bull. 78(873), 41–48 (1985)Google Scholar
  48. 48.
    Z.A. Melzak, On the problem of Steiner. Can. Math. Bull. 4, 143–148 (1961)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    B.K. Nielsen, P. Winter, M. Zachariasen, An exact algorithm for the uniformly-oriented Steiner tree problem, in Proceedings of the 10th European Symposium on Algorithms. Lecture Notes in Computer Science, vol. 2461 (Springer, Berlin, 2002), pp. 760–772Google Scholar
  50. 50.
    K. Prendergast, Steiner ratio for gradient constrained networks. Ph.D. dissertation, Department of Electrical and Electronic Engineering, The University of Melbourne, Australia, 2006Google Scholar
  51. 51.
    K. Prendergast, D.A. Thomas, J.F. Weng, Optimum Steiner ratio for gradient-constrained networks connecting three points in 3-space, part I. Networks 53(2), 212–220 (2009)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    D.S. Richards, J.S. Salowe, A simple proof of Hwang’s theorem for rectilinear Steiner minimal trees. Ann. Oper. Res. 33, 549–556 (1991)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970)MATHGoogle Scholar
  54. 54.
    J.H. Rubinstein, D.A. Thomas, A variational approach to the Steiner network problem. Ann. Oper. Res. 33, 481–499 (1991)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    J.H. Rubinstein, D.A. Thomas, N.C. Wormald, Steiner trees for terminals constrained to curves. SIAM J. Discret. Math. 10, 1–17 (1997)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    J.H. Rubinstein, D.A. Thomas, J.F. Weng, Minimum networks for four points in space. Geom. Dedic. 93, 571–70 (2002)MathSciNetGoogle Scholar
  57. 57.
    N.Z. Shor, Minimization Methods for Non-differentiable Functions (Springer, Berlin/New York, 1985)CrossRefMATHGoogle Scholar
  58. 58.
    W.D. Smith, How to find minimal trees in Euclidean d-space. Algorithmica 7, 137–177 (1992)MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    K.J. Swanepoel, The local Steiner problem in normed planes. Networks 36, 104–113 (2000)MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    D.A. Thomas, J.F. Weng, Minimum cost flow-dependent communication networks. Networks 48(1), 39–46 (2006)MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    D.A. Thomas, J.F. Weng, Gradient-constrained minimum networks: an algorithm for computing Steiner points. J. Discret. Optim. 7, 21–31 (2010)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    D.A. Thomas, M. Brazil, D.H. Lee, N.C. Wormald, Network modelling of underground mine layout: two case studies. Int. Trans. Oper. Res. 14(2), 143–158 (2007)CrossRefMATHGoogle Scholar
  63. 63.
    A.C. Thompson, Minkowski Geometry. Encyclopedia of Mathematics and Its Applications, vol. 63 (Cambridge University Press, Cambridge/New York, 1996)CrossRefMATHGoogle Scholar
  64. 64.
    D. Trietsch, Minimal Euclidean networks with flow dependent costs — the generalized Steiner case, May 1985, discussion paper no. 655, Center for Mathematical Studies in Economics and Management Science, Northwestern University, Evanston, IL, 1985Google Scholar
  65. 65.
    D. Trietsch, J.F. Weng, Pseudo-Gilbert-Steiner trees. Networks 33, 175–178 (1999)MathSciNetCrossRefMATHGoogle Scholar
  66. 66.
    M.G. Volz, Gradient-constrained flow-dependent networks for underground mine design. PhD thesis, Department of Electrical and Electronic Engineering, The University of Melbourne, 2008Google Scholar
  67. 67.
    M.G. Volz, M. Brazil, C.J. Ras, K.J. Swanepoel, D.A. Thomas, The Gilbert arborescence problem Networks (accepted, May 2012)Google Scholar
  68. 68.
    M.G. Volz, M. Brazil, D.A. Thomas, The gradient-constrained Fermat-Weber problem Networks (submitted 2011)Google Scholar
  69. 69.
    D.M. Warme, Spanning trees in hypergraphs with applications to Steiner trees. PhD thesis, Computer Science Department, The University of Virginia, 1998Google Scholar
  70. 70.
    D.M. Warme, P. Winter, M. Zachariasen, Exact solutions to large-scale plane Steiner tree problems, in Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore (1999), pp. 49–53Google Scholar
  71. 71.
    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. Rubenstein (Kluwer Academic Publishers, Boston, 2000), pp. 81–116CrossRefGoogle Scholar
  72. 72.
    D.M. Warme, P. Winter, M. Zachariasen, GeoSteiner 3.1. Department of Computer Science, University of Copenhagen (DIKU) (2001), http://www.diku.dk/geosteiner/
  73. 73.
    A. Weber, Uber den Standort der Industrien (Verlag J.C.B. Mohr, Tubingen, 1909), (Translation by C. J. Friedrich Theory of the Location of Industries (University of Chicago Press, Chicago, 1929)Google Scholar
  74. 74.
    E. Weiszfeld, Sur le point pour lequel la somme des distances de n points donnés est minimum. Tohoku Math. J. 43, 355–386 (1937)Google Scholar
  75. 75.
    P. Widmayer, Y.F. Yu, C.K. Wong, Distance problems in computational geometry with fixed orientations, in Proceedings of the Symposium on Computational Geometry, Baltimore, MD (1985), pp. 186–195Google Scholar
  76. 76.
    P. Widmayer, Y.F. Yu, C.K. Wong, On some distance problems in fixed orientations. SIAM J. Comput. 16(4), 728–746 (1987)MathSciNetCrossRefMATHGoogle Scholar
  77. 77.
    M. Zachariasen, Rectilinear full Steiner tree generation. Networks 338, 125–143 (1999)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Electrical and Electronic EngineeringThe University of MelbourneParkvilleAustralia
  2. 2.TSG ConsultingMelbourneAustralia

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