Skip to main content
SpringerLink
Log in

Steiner Trees with Other Cost Functions and Constraints

  • Chapter
Optimal Interconnection Trees in the Plane

Part of the book series: Algorithms and Combinatorics ((AC,volume 29))

  • 1185 Accesses

Abstract

In this chapter we look at Steiner tree problems that involve other cost functions and constraints (beyond those discussed in the first three chapters) but that still can be solved exactly by exploiting the geometric properties of minimal solutions. We focus particularly on four types of Steiner tree problems: the gradient-constrained Steiner tree problem, which serves as another example of an exactly solvable Steiner tree problem in a Minkowski plane with useful applications; the obstacle-avoiding Steiner tree problem, which is an important variation of the Steiner tree problem with applications in the physical design of microchips; bottleneck and other k-Steiner tree problems, where there is a given bound on the number of Steiner points; and Steiner tree problems optimising a cost associated with flow on the network.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
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

Notes

  1. 1.

    This background on mining, which focusses on hard rock mines containing metallic deposits, is drawn primarily from [8] and [48]. For a more comprehensive introduction to the infrastructure of an underground mine see [186].

  2. 2.

    Some examples of the many heuristic approaches developed for the rectilinear obstacle-avoiding Steiner tree problem include the algorithms of Lin et al. [258], Long et al. [267], Li and Young [254], Liu et al. [262] and Ajwani et al. [5]. The last of these makes use of the FLUTE algorithm, described in Sect. 3.4 Also of note is a graph-based approximation algorithm for the octilinear obstacle-avoiding Steiner tree problem proposed by Müller-Hannemann and Schulze [288].

  3. 3.

    For the equivalent result to Lemma 4.8 in the Euclidean plane, see for example [345].

  4. 4.

    One of the most important early references on the Euclidean problem with polygonal obstacles is the paper of Provan [317] on approximation schemes for this problem. Other early papers relating to the Euclidean problem include [350] and [391]. The key papers on algorithmic approaches to solving the exact problem are those of Winter [403] and Zachariasen and Winter [433].

  5. 5.

    Although numerous heuristics for the rectilinear minimum obstacle-avoiding Steiner tree problem have been developed, the literature on exact solutions is rather sparse. The first substantial results appear to be those of Ganley and Cohoon [163], who developed algorithms for solving instances with up to four terminals, using the so-called escape graph. More recently, Huang et al. [204, 208] presented a GeoSteiner-type algorithm able, in theory, to exactly solve instances of arbitrary size, using an approach similar to the one developed in this section. These methods were further refined and discussed by Juhl [225]. We note, however, that all of these previous results assume that the polygonal obstacles have boundary edges using only legal orientations, whereas we make no restrictions on the orientations of the boundary edges of obstacles in this section.

  6. 6.

    A slightly weaker version of this result has appeared in the literature, but only for the rectilinear Steiner tree problem, with rectilinear obstacles; see [205, 206, 208] and [225].

  7. 7.

    This application was first suggested by David Lee in his unpublished paper, ‘Some industrial case studies of Steiner trees’ presented at the NATO Advanced Research Workshop, ‘Topological Network Design: Analysis and Synthesis’, in Copenhagen, 1989.

  8. 8.

    The bottleneck Steiner tree problem was first proposed by Chiang et al. [96] in the context of Steiner trees in graphs, and was originally known as the Steiner min-max tree problem. There it was shown that this problem has a simple polynomial-time algorithm, in terms of the number of vertices and edges of the entire graph. However, in terms of the cardinality of the terminal set, Berman and Zelikovsky [28] have shown that the problem does not admit any polynomial-time approximation algorithm with performance ratio less than 2 unless P = NP. The Euclidean version of the geometric bottleneck k-Steiner tree problem in the plane, as defined in this chapter, was first studied by Du et al. in [139] and [384]. A survey of the results in this area up to 2008, with a focus on approximation algorithms, can be found in Chapter 6 of [134].

  9. 9.

    The algorithm developed in Sect. 4.3 for the general k-Steiner tree problem can in theory apply to a wider range of norms than the PE norms. The algorithm for these other norms will, however, be difficult to implement in practice. The precise required properties of the widest class of norms for which the algorithm can apply are given in [52].

  10. 10.

    Note that [21] gives a slightly weaker version of condition 3, the bound on the degree of Steiner points; in particular, for ℓ 1 and ℓ ∞ an upper bound of 7 rather than 5 is given. The bounds in [21] are based on the Hadwiger numbers of the unit balls, that is, the largest number of non-overlapping translates of the unit ball that can be brought into contact with it. However, for ℓ 1 and ℓ ∞ (each of which have polygonal unit balls) a tighter bound can be found by using Lemma 4.22 together with the arguments of Monma and Suri in [285].

  11. 11.

    This paper actually predates Gilbert’s seminal paper with Henry Pollak [179] on the Euclidean Steiner tree problem.

  12. 12.

    The problem of constructing a Steiner point in the weighted case (i.e., the weighted Fermat-Torricelli problem) was first solved, for the Euclidean plane, by Weber [390] in 1909. More details on the history and mathematics behind the general weighted Fermat-Torricelli problem (for Steiner points of degree ≥ 3) can be found in the surveys of Wesolowsky [396] and Kupitz and Martini [240]. A recent treatment of the Euclidean weighted Fermat-Torricelli problem for Steiner points of degree 3 can be found in a paper of Jalal and Krarup [222], where the problem is referred to as FERPOS.

  13. 13.

    Ptolemy’s theorem is named after the Greek astronomer and mathematician Claudius Ptolemaeus, who stated and used the result in about AD 150. Jalal and Krarup [222] prove Theorem 4.37 without the use of Ptolemy’s theorem, and are then able to obtain Ptolemy’s theorem as a simple corollary.

  14. 14.

    Note that in [124], condition (W3b), the subadditivity condition, was incorrectly interpreted as concavity of the cost function.

  15. 15.

    The formulation and solution of a hierarchical network design problem in which there are at least two grades of service dates back to a 1986 paper of Current et al. [126]. The first study of the grade of service Steiner tree problem, where the grades of service on the edges are determined by weights on the terminals, was a paper by Colbourn and Xue in 2000 [116], although this was for the Steiner tree problem in graphs. The main study of the geometric version of this problem is the paper of Xue et al. [416]. Note that our formulation of the Euclidean grade of service Steiner tree problem in the plane slightly simplifies that given in [416].

  16. 16.

    The former of these two methods of bounding the number of Steiner points is the most popular in the literature, and is employed in [160, 166, 355] and [28] amongst others. The second method is particularly relevant to applications surrounding the modelling of wireless sensor network deployment; see for example [412] and [413]. Both methods are also discussed in [57].

Bibliography

  1. Abellanas, M., Hurtado, F., Icking, C., Klein, R., Langetepe, E., Ma, L., Palop, B., Sacristán, V.: The farthest color Voronoi diagram and related problems. Technical report, Department of Computer Science I, University of Bonn (2006)

    Google Scholar 

  2. Ajwani, G., Chu, C., Mak, W.-K.: FOARS: FLUTE based obstacle-avoiding rectilinear Steiner tree construction. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 30(2), 194–204 (2011)

    Article  Google Scholar 

  3. Alford, C., Brazil, M., Lee, D.H.: Optimisation in underground mining. In: Weintraub, A., Romero, C., Bjørndal, T., Epstein, R. (eds.) Handbook of Operations Research in Natural Resources, pp. 561–577. Springer, New York (2007)

    Google Scholar 

  4. Asano, T., Asano, T., Guibas, L., Herchberger, J., Imai, H.: Visibility of disjoint polygons. Algorithmica 1, 49–63 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  5. Aurenhammer, F., Klein, R.: Voronoi diagrams. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, chapter 5, pp. 201–290. North-Holland, Amsterdam (2000)

    Chapter  Google Scholar 

  6. Bae, S.W., Choi, S., Lee, C., Tanigawa, S.: Exact algorithms for the bottleneck Steiner tree problem. Algorithmica 61(4), 924–948 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bae, S.W., Lee, C., Choi, S.: On exact solutions to the Euclidean bottleneck Steiner tree problem. Inf. Process. Lett. 110(16), 672–678 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bainbridge, S., Eggeling, D., Page, G.: Lessons from the field – two years of deploying operational wireless sensor networks on the Great Barrier Reef. Sensors 11(7), 6842–6855 (2011)

    Article  Google Scholar 

  9. Berman, P., Zelikovsky, A.: On approximation of the power-p and bottleneck Steiner trees. In: Du, D.-Z., Smith, J.M., Rubinstein, J.H. (eds.) Advances in Steiner Trees. Combinatorial Optimization, vol. 6, pp. 117–135. Springer, New York (2000)

    Chapter  Google Scholar 

  10. Bienstock, D., Brickell, E.F., Monma, C.L.: On the structure of minimum-weight k-connected spanning networks. SIAM J. Discret. Math. 3(3), 320–329 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  11. Borah, M., Owens, R.M., Irwin, M.J.: An edge-based heuristic for Steiner routing. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 13, 1563–1568 (1994)

    Article  Google Scholar 

  12. Brazil, M., Grossman, P.A., Lee, D.H., Rubinstein, J.H., Thomas, D.A., Wormald, N.C.: Constrained path optimisation for underground mine layout. In: World Congress on Engineering 2007, London, pp. 856–861 (2007)

    Google Scholar 

  13. Brazil, M., Lee, D.H., Rubinstein, J.H., Thomas, D.A., Weng, J.F., Wormald, N.C.: Network optimisation of underground mine design. Proc. Australas. Inst. Min. Metall. 305(1), 57–66 (2000)

    Google Scholar 

  14. Brazil, M., Lee, D.H., Van Leuven, M., Rubinstein, J.H., Thomas, D.A., Wormald, N.C.: Optimising declines in underground mines. Min. Technol. 112(3), 164–170 (2003)

    Article  Google Scholar 

  15. Brazil, M., Nielsen, B.K., Winter, P., Zachariasen, M.: Rotationally optimal spanning and Steiner trees in uniform orientation metrics. Comput. Geom. Theory Appl. 29, 251–263 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Brazil, M., Ras, C.J., Swanepoel, K.J., Thomas, D.A.: Generalised k-Steiner tree problems in normed planes. Algorithmica 71(1), 66–86 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  17. Brazil, M., Ras, C.J., Thomas, D.A.: Approximating minimum Steiner point trees in Minkowski planes. Networks 56(4), 244–254 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Brazil, M., Ras, C.J., Thomas, D.A.: The bottleneck 2-connected k-Steiner network problem for k ≤ 2. Discret. Appl. Math. 160(7), 1028–1038 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Brazil, M., Ras, C.J., Thomas, D.A.: Relay augmentation for lifetime extension of wireless sensor networks. IET Wirel. Sens. Syst. 3(2), 145–152 (2013)

    Article  Google Scholar 

  20. Brazil, M., Ras, C.J., Thomas, D.A.: An exact algorithm for the bottleneck 2-connected k-Steiner network problem in L p planes. arXiv preprint arXiv:1111.2105 (2014)

    Google Scholar 

  21. Brazil, M., Ras, C.J., Thomas, D.A.: A flow-dependent quadratic Steiner tree problem in the Euclidean plane. Networks 64(1), 18–28 (2014)

    Article  MathSciNet  Google Scholar 

  22. Brazil, M., Rubinstein, J.H., Thomas, D.A., Weng, J.F., Wormald, N.C.: Gradient-constrained minimum networks. I. Fundamentals. J. Global Optim. 21(2), 139–155 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Brazil, M., Rubinstein, J.H., Thomas, D.A., Weng, J.F., Wormald, N.C.: Gradient-constrained minimum networks. III. Fixed topology. J. Optim. Theory Appl. 155(1), 336–354 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. Brazil, M., Thomas, D.A., Weng, J.F.: Gradient constrained minimal Steiner trees. In: Pardalas, P.M., Du, D.-Z. (eds.) Network Design: Connectivity and Facilities Location (DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 40). American Mathematical Society, Providence, Rhode Island (1998)

    Google Scholar 

  25. Brazil, M., Thomas, D.A., Weng, J.F.: On the complexity of the Steiner problem. J. Comb. Optim. 4, 187–195 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Brazil, M., Thomas, D.A., Weng, J.F.: Gradient-constrained minimum networks (II). Labelled or locally minimal Steiner points. J. Global Optim. 42(1), 23–37 (2008)

    MATH  MathSciNet  Google Scholar 

  27. Brazil, M., Volz, M.G.: Gradient-constrained minimum interconnection networks. In: Pardalos, P.M., Du, D.Z., Graham, R.L. (eds.) Handbook of Combinatorial Optimization, pp. 1459–1510. Springer, New York (2013)

    Chapter  Google Scholar 

  28. Brazil, M., Zachariasen, M.: Steiner trees for fixed orientation metrics. J. Global Optim. 43, 141–169 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Cavendish, J.C.: Automatic triangulation of arbitrary planar domains for the finite element method. Int. J. Numer. Methods Eng. 8(4), 679–696 (1974)

    Article  MATH  Google Scholar 

  30. Chen, D., Du, D.-Z., Hu, X.-D., Lin, G.-H., Wang, L., Xue, G.: Approximations for Steiner trees with minimum number of Steiner points. J. Global Optim. 18(1), 17–33 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  31. Cheng, X., Du, D.-Z., Wang, L., Xu, B.: Relay sensor placement in wireless sensor networks. Wirel. Netw. 14(3), 347–355 (2008)

    Article  Google Scholar 

  32. Chew, L.P., Drysdale III, R.L.: Voronoi diagrams based on convex distance function. In: Proceedings of the First Annual ACM Symposium on Computational Geometry (SCG), New York, pp. 235–244 (1985)

    Google Scholar 

  33. Chiang, C., Sarrafzadeh, M., Wong, C.K.: Global routing based on Steiner min-max trees. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 9, 1318–1325 (1990)

    Article  Google Scholar 

  34. Cockayne, E.J., Melzak, Z.A.: Euclidean constructibility in graph-minimization problems. Math. Mag. 42(4), 206–208 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  35. Colbourn, C.J., Xue, G.: Grade of service Steiner trees in series-parallel networks. In: Du, D.-Z., Smith, J.M., Rubinstein, J.H. (eds) Advances in Steiner Trees, pp. 163–174. Kluwer Academic Publishers, Dordrecht (2000)

    Chapter  Google Scholar 

  36. Colthurst, T., Cox, C., Foisy, J., Howards, H., Kollett, K., Lowy, H., Root, S.: Networks minimizing length plus the number of Steiner points. In: Du, D.-Z., Pardalos, P.M. (eds.) Network Optimization Problems: Algorithms, Applications and Complexity, pp. 23–36. World Scientific, Singapore (1993)

    Chapter  Google Scholar 

  37. Cox, C.L.: Flow-dependent networks: existence and behavior at Steiner points. Networks 31(3), 149–156 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  38. Coxeter, H.S.M.: Introduction to Geometry. Wiley, New York (1969)

    MATH  Google Scholar 

  39. Current, J.R., ReVelle, C.S., Cohon, J.L.: The hierarchical network design problem. Eur. J. Oper. Res. 27(1), 57–66 (1986)

    Article  MATH  Google Scholar 

  40. Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1(1), 269–271 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  41. Drezner, Z., Wesolowsky, G.O.: A new method for the multifacility minimax location problem. J. Oper. Res. Soc. 29(11), 1095–1101 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  42. Du, D.-Z., Hu, X.-D.: Steiner Tree Problems in Computer Communication Networks. World Scientific, Singapore (2008)

    Book  MATH  Google Scholar 

  43. Du, D.-Z., Wang, L., Xu, B.: The Euclidean bottleneck Steiner tree and Steiner tree with minimum number of Steiner points. In: Wang, J. (ed.) Computing and Combinatorics, pp. 509–518. Springer, New York (2001)

    Chapter  Google Scholar 

  44. Elzinga, J., Hearn, D., Randolph, W.D.: Minimax multifacility location with Euclidean distances. Transp. Sci. 10(4), 321–336 (1976)

    Article  MathSciNet  Google Scholar 

  45. Ganley, J.L.: Geometric interconnection and placement algorithms. PhD thesis, The University of Virginia (1995)

    Google Scholar 

  46. Ganley, J.L., Cohoon, J.P.: Routing a multi-terminal critical net: Steiner tree construction in the presence of obstacles. In: IEEE Proceedings of the International Symposium on Circuits and Systems, London, pp. 113–116 (1994)

    Google Scholar 

  47. Ganley, J.L., Salowe, J.S.: Optimal and approximate bottleneck Steiner trees. Oper. Res. Lett. 19, 217–224 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  48. Ganley, J.L., Salowe, J.S.: The power-p Steiner tree problem. Nord. J. Comput. 5(2), 115–127 (1998)

    MATH  MathSciNet  Google Scholar 

  49. Georgakopoulos, G., Papadimitriou, C.H.: The 1-Steiner tree problem. J. Algorithms 8, 122–130 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  50. Ghosh, S.K., Mount, D.M.: An output-sensitive algorithm for computing visibility graphs. SIAM J. Comput. 20(5), 888–910 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  51. Gilbert, E.N.: Minimum cost communication networks. Bell Syst. Tech. J. 46, 2209–2227 (1967)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  53. Hamrin, H.: Underground mining methods and applications. In: Hustrilid, W.A., Bullock, R.L. (eds.) Underground Mining Methods. Society for Mining, Metallurgy, and Exploration, Littleton, Colorado (2001)

    Google Scholar 

  54. Huang, T., Li, L., Young, E.F.Y.: On the construction of optimal obstacle-avoiding rectilinear Steiner minimum trees. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 30(5), 718–731 (2011)

    Article  Google Scholar 

  55. Huang, T., Young, E.F.Y.: Obstacle-avoiding rectilinear Steiner minimum tree construction: an optimal approach. In: Proceedings ACM International Conference on Computer-Aided Design (ICCAD), San Jose, pp. 610–613 (2010)

    Google Scholar 

  56. Huang, T., Young, E.F.Y.: An exact algorithm for the construction of rectilinear Steiner minimum trees among complex obstacles. In: Proceedings of the 48th ACM/IEEE Design Automation Conference (DAC), San Diego, pp. 164–169 (2011)

    Google Scholar 

  57. Huang, T., Young, E.F.Y.: Obsteiner: an exact algorithm for the construction of rectilinear Steiner minimum trees in the presence of complex rectilinear obstacles. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 32(6), 882–893 (2013)

    Article  Google Scholar 

  58. Hwang, F.K., Weng, J.F.: The shortest network under a given topology. J. Algorithms 13, 468–488 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  59. Jalal, G., Krarup, J.: Geometrical solution to the Fermat problem with arbitrary weights. Ann. Oper. Res. 123, 67–104 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  60. Juhl, D.D.: Full Steiner trees for the obstacle-avoiding rectilinear Steiner tree problem. Master’s thesis, Department of Computer Science, University of Copenhagen (2013)

    Google Scholar 

  61. Kahng, A.B., Robins, G.: A new class of iterative Steiner tree heuristics with good performance. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 11(7), 893–902 (1992)

    Article  Google Scholar 

  62. Karl, H., Willig, A.: Protocols and Architectures for Wireless Sensor Networks. Wiley, New York (2007)

    Google Scholar 

  63. Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms. Algorithms and Combinatorics, 4th edn. Springer, Berlin (2008)

    Google Scholar 

  64. Kupitz, Y.S., Martini, H.: Geometric aspects of the generalized Fermat-Torricelli problem. In: Barany, I., Boroczky, K. (eds.) Bolyai Society Mathematical Studies, 6: Intuitive Geometry, pp. 55–127. Janos Bolyai Mathematical Society, Budapest (1997)

    Google Scholar 

  65. Lee, J.: A first course in combinatorial optimization. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  66. Lerchs, H., Grossmann, I.F.: Optimum design of open-pit mines. Trans. Can. Inst. Min. Metall. Pet. 68, 17–24 (1965)

    Google Scholar 

  67. Li, C.-S., Tong, F.F.-K., Georgiou, C.J., Chen, M.: Gain equalization in metropolitan and wide area optical networks using optical amplifiers. In: Proceedings of the 13th IEEE Conference on Computer Communications: Networking for Global Communications (INFOCOM), Toronto, pp. 130–137 (1994)

    Google Scholar 

  68. Li, L., Young, E.F.: Obstacle-avoiding rectilinear Steiner tree construction. In: Proceedings ACM International Conference on Computer-Aided Design (ICCAD), San Jose, pp. 523–528 (2008)

    Google Scholar 

  69. Lin, C.-W., Chen, S.-Y., Li, C.-F., Chang, Y.-W., Yang, C.-L.: Obstacle-avoiding rectilinear Steiner tree construction based on spanning graphs. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 27(4), 643–653 (2008)

    Article  Google Scholar 

  70. Lin, G.-H., Xue, G.: Steiner tree problem with minimum number of Steiner points and bounded edge-length. Inf. Process. Lett. 69(2), 53–57 (1999)

    Article  MathSciNet  Google Scholar 

  71. Liu, C.-H., Kuo, S.-Y., Lee, D.T., Lin, C.-S., Weng, J.-H., Yuan, S.-Y.: Obstacle-avoiding rectilinear Steiner tree construction: a Steiner-point-based algorithm. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 31(7), 1050–1060 (2012)

    Article  Google Scholar 

  72. Long, J., Zhou, H., Memik, S.O.: EBOARST: an efficient edge-based obstacle-avoiding rectilinear Steiner tree construction algorithm. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 27(12), 2169–2182 (2008)

    Article  Google Scholar 

  73. Lu, B., Gu, J., Hu, X., Shragowitz, E.: Wire segmenting for buffer insertion based on RSTP-MSP. Theor. Comput. Sci. 262(1), 257–267 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  74. Măndoiu, I.I., Zelikovsky, A.Z.: A note on the MST heuristic for bounded edge-length Steiner trees with minimum number of Steiner points. Inf. Process. Lett. 75(4), 165–167 (2000)

    Article  MATH  Google Scholar 

  75. Monma, C., Suri, S.: Transitions in geometric minimum spanning trees. Discret. Comput. Geom. 8, 265–293 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  76. Müller-Hannemann, M., Schulze, A.: Hardness and approximation of octilinear Steiner trees. Int. J. Comput. Geom. Appl. 17, 231–260 (2007)

    Article  MATH  Google Scholar 

  77. Newman, A.M., Rubio, E., Caro, R., Weintraub, A., Eurek, K.: A review of operations research in mine planning. Interfaces 40(3), 222–245 (2010)

    Article  Google Scholar 

  78. Nielsen, B.K., Winter, P., Zachariasen, M.: Rectilinear trees under rotation and related problems. In: Proceedings of the 18th European Workshop on Computational Geometry, Warsaw, pp. 18–22 (2002)

    Google Scholar 

  79. Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. Wiley, New York (1995)

    Google Scholar 

  80. Oppermann, F.J., Boano, C.A., Römer, K.: A decade of wireless sensing applications: survey and taxonomy. In: Ammari, H.M. (ed.) The Art of Wireless Sensor Networks, pp. 11–50. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  81. Provan, J.S.: An approximation scheme for finding Steiner trees with obstacles. SIAM J. Comput. 17(5), 920–934 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  82. Ramamurthy, B., Iness, J., Mukherjee, B.: Minimizing the number of optical amplifiers needed to support a multi-wavelength optical LAN/MAN. In: Proceedings of INFOCOM’97. Sixteenth Annual Joint Conference of the IEEE Computer and Communications Societies. Driving the Information Revolution, Kobe, vol. 1, pp. 261–268 (1997)

    Google Scholar 

  83. Rubinstein, J.H., Thomas, D.A., Weng, J.F.: Degree-five Steiner points cannot reduce network costs for planar sets. Networks 22(6), 531–537 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  84. Sarrafzadeh, M., Wong, C.K.: Bottleneck Steiner trees in the plane. IEEE Trans. Comput. 41, 370–374 (1992)

    Article  MathSciNet  Google Scholar 

  85. Segev, A.: The node-weighted Steiner tree problem. Networks 17(1), 1–17 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  86. Shang, S., Hu, X., Jing, T.: Rotational Steiner ratio problem under uniform orientation metrics. In: Discrete Geometry, Combinatorics and Graph Theory. Lecture Notes in Computer Science, vol. 4381, pp. 166–176. Springer, Berlin/Heidelberg (2007)

    Google Scholar 

  87. Sharir, M., Schorr, A.: On shortest paths in polyhedral spaces. SIAM J. Comput. 15(1), 193–215 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  88. Smith, J.M., Liebman, J.S.: Steiner trees, Steiner circuits and the interference problem in building design. Eng. Optim. 4, 15–36 (1979)

    Google Scholar 

  89. Smith, W.D.: How to find Steiner minimal trees in Euclidean d-space. Algorithmica 7(2/3), 137–177 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  90. Soukup, J.: On minimum cost networks with nonlinear costs. SIAM J. Appl. Math. 29(4), 571–581 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  91. Swanepoel, K.J.: The local Steiner problem in normed planes. Networks 36, 104–113 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  92. Thomas, D.A., Brazil, M., Lee, D.H., Wormald, N.C.: Network modelling of underground mine layout: two case studies. Int. Trans. Oper. Res. 14(2), 143–158 (2007)

    Article  MATH  Google Scholar 

  93. Thomas, D.A., Weng, J.F.: Minimum cost flow-dependent communication networks. Networks 48, 39–46 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  94. Underwood, A.: A modified Melzak procedure for computing node-weighted Steiner trees. Networks 27, 73–79 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  95. Volz, M.G.: Gradient-constrained flow-dependent networks for underground mine design. PhD thesis, The University of Melbourne (2008)

    Google Scholar 

  96. Volz, M.G., Brazil, M., Ras, C.J., Swanepoel, K.J., Thomas, D.A.: The Gilbert arborescence problem. Networks 61(3), 238–247 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  97. Wang, F., Wang, D., Liu, J.: Traffic-aware relay node deployment for data collection in wireless sensor networks. In: 6th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks (SECON’09), Rome, pp. 1–9 (2009)

    Google Scholar 

  98. Wang, L., Du, D.-Z.: Approximations for a bottleneck Steiner tree problem. Algorithmica 32(4), 554–561 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  99. Weber, A.: Über den Standort der Industrien. Teil I: Reine Theorie des Standorts. Verlag J. C. B. Mohr (1909). Translated by C. J. Friedrich as Alfred Weber’s Theory of the Location of Industries. Chicago University Press, Chicago, 1929

    Google Scholar 

  100. Welzl, E.: Constructing the visibility graph for n-line segments in O(n 2) time. Inf. Process. Lett. 20, 167–171 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  101. Weng, J.F.: Shortest networks for smooth curves. SIAM J. Optim. 7(4), 1054–1068 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  102. Weng, J.F.: Minimum networks for separating and surrounding objects. In: Cheng, X., Du, D.-Z. (eds.) Steiner Trees in Industry. Combinatorial Optimization, vol. 11, pp. 427–439. Springer, New York (2001)

    Chapter  Google Scholar 

  103. Weng, J.F.: Generalized Melzak’s construction in the Steiner tree problem. Int. J. Comput. Geom. Appl. 12(6), 481–488 (2002)

    Article  MATH  Google Scholar 

  104. Weng, J.F., Smith, J.M.: Steiner minimal trees with one polygonal obstacle. Algorithmica 29(4), 638–648 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  105. Wesolowsky, G.O.: The Weber problem: history and perspectives. Locat. Sci. 1, 5–23 (1993)

    MATH  Google Scholar 

  106. Winter, P.: Euclidean Steiner minimal trees with obstacles and Steiner visibility graphs. Discret. Appl. Math. 47, 187–206 (1993)

    Article  MATH  Google Scholar 

  107. Winter, P., Smith, J.M.: Steiner minimal trees for three points with one convex polygonal obstacle. Ann. Oper. Res. 33, 577–599 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  108. Xin, Y., Guven, T., Shayman, M.: Relay deployment and power control for lifetime elongation in sensor networks. In: IEEE International Conference on Communications, Istanbul, vol. 8, pp. 3461–3466 (2006)

    Google Scholar 

  109. Xu, K., Hassanein, H., Takahara, G., Wang, Q.: Relay node deployment strategies in heterogeneous wireless sensor networks. IEEE Trans. Mob. Comput. 9(2), 145–159 (2010)

    Article  Google Scholar 

  110. Xue, G.: A branch-and-bound algorithm for computing node weighted Steiner minimum trees. In: Jiang, T., Lee, D.T. (eds.) Computing and Combinatorics. Lecture Notes in Computer Science, vol. 1276, pp. 383–392. Springer, Berlin/Heidelberg (1997)

    Google Scholar 

  111. Xue, G., Lin, G.-H., Du, D.-Z.: Grade of service Steiner minimum trees in the Euclidean plane. Algorithmica 31, 479–500 (2004)

    Article  MathSciNet  Google Scholar 

  112. Zachariasen, M., Winter, P.: Obstacle-avoiding Euclidean Steiner trees in the plane: an exact algorithm. In: Workshop on Algorithm Engineering and Experimentation (ALENEX), Baltimore. Lecture Notes in Computer Science 1619, pp. 282–295. Springer, Berlin/Heidelberg (1999)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, Victoria, Australia

    Marcus Brazil

  2. Department of Computer Science (DIKU), University of Copenhagen, Copenhagen, Denmark

    Martin Zachariasen

Authors
  1. Marcus Brazil
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Zachariasen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Brazil, M., Zachariasen, M. (2015). Steiner Trees with Other Cost Functions and Constraints. In: Optimal Interconnection Trees in the Plane. Algorithms and Combinatorics, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-319-13915-9_4

Download citation

Publish with us

Policies and ethics

Search

Navigation