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Archive for History of Exact Sciences

, Volume 68, Issue 3, pp 327–354 | Cite as

On the history of the Euclidean Steiner tree problem

  • Marcus Brazil
  • Ronald L. Graham
  • Doreen A. Thomas
  • Martin Zachariasen
Article

Abstract

The history of the Euclidean Steiner tree problem, which is the problem of constructing a shortest possible network interconnecting a set of given points in the Euclidean plane, goes back to Gergonne in the early nineteenth century. We present a detailed account of the mathematical contributions of some of the earliest papers on the Euclidean Steiner tree problem. Furthermore, we link these initial contributions with results from the recent literature on the problem.

Keywords

Fermat Minimum Span Tree Equilateral Triangle Steiner Tree Steiner Point 
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.

Notes

Acknowledgments

Two of the authors, Marcus Brazil and Doreen Thomas, were partially supported in the writing of this paper by a grant from the Australian Research Council. We would also like to thank: Francois Lauze (University of Copenhagen) and Morgan Tort (The University of Melbourne) for their generous assistance with the French translations required for this paper; Henry Pollak for useful commentary on Gauss’ letters; Pavol Hell (Simon Fraser University) for assistance with translating (Jarník and Kössler 1934); Jakob Krarup (University of Copenhagen) for help with providing some original sources; Donald Knuth for helpful comments and suggestions on an earlier draft of this paper; and Konrad Swanepoel for alerting us to the existence of the paper of Menger (1931).

References

  1. Arora, S. 1998. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM 45(5): 753–782.CrossRefMATHMathSciNetGoogle Scholar
  2. Beardwoord, J., J.H. Halton, and J.M. Hammersley. 1959. The shortest path through many points. Proceedings of Cambridge Philosophical Society 55: 299–327.CrossRefGoogle Scholar
  3. Bopp, K. 1879. Üeber das kürzeste Verbindungssystem zwischen vier Punkten. PhD thesis, Universität Göttingen.Google Scholar
  4. Cavalieri, B. 1647. Exercitationes Geometricae Sex.Google Scholar
  5. Cavalli-Sforza, L.L., and A.W.F. Edwards. 1967. Phylogenetic analysis: Models and estimation procedures. Evolution 21: 550–570.CrossRefGoogle Scholar
  6. Cheng, X., Y. Li, D.Z. Du, and H.Q. Ngo. 2004. Steiner trees in industry. In Handbook of combinatorial optimization, vol. 5, ed. D.Z. Du, and P.M. Pardalos, 193–216. Dordrecht: Kluwer.Google Scholar
  7. Choquet, G. 1938. Étude de certains réseaux de routes. Comptes Rendus Acad Sci 206: 310–313.Google Scholar
  8. Cieslik, D. 2004a. The essential of Steiner’s problem in normed planes. Technical report 8, Ernst-Moritz-Arndt-Universität Greifswald, Preprint-Reihe Mathematik.Google Scholar
  9. Cieslik, D. 2004b. Shortest connectivity—Introduction with applications in phylogeny, combinatorial optimization, vol. 17. New York: Springer.Google Scholar
  10. Cockayne, E.J. 1967. On the Steiner problem. Canadian Mathematical Bulletin 10: 431–450.CrossRefMATHMathSciNetGoogle Scholar
  11. Colbourn, C.J., and C. Huybrechts. 2008. Fully gated graphs: Recognition and convex operations. Discrete Mathematics 308: 5184–5195.CrossRefMATHMathSciNetGoogle Scholar
  12. Courant, R. 1940. Soap film experiments with minimal surfaces. The American Mathematical Monthly 47: 167–174.MathSciNetGoogle Scholar
  13. Courant, R., and H. Robbins. 1941. What is mathematics?. London: Oxford University Press.Google Scholar
  14. Dahan-Dalmedico, A. 1986. Un texte de philosophie mathématique de Gergonne. Revue d’histoire des sciences 39: 97–126.CrossRefMATHMathSciNetGoogle Scholar
  15. de Fermat, P. 1891. Oeuvres, vol. 1.Google Scholar
  16. Descartes, R. 1896. Oeuvres de Descartes, vol. II. Paris: J. Vrin.Google Scholar
  17. Du, D.Z., and W. Wu. 2007. Approximations for Steiner minimum trees. In Handbook of approximation algorithms and metaheuristics. Chapman and Hall/CRC.Google Scholar
  18. Du, D.Z., F.K. Hwang, G.D. Song, and G.Y. Ting. 1987a. Steiner minimal trees on sets of four points. Discrete and Computational Geometry 2: 401–414.CrossRefMATHMathSciNetGoogle Scholar
  19. Du, D.Z., F.K. Hwang, and J.F. Weng. 1987b. Steiner minimal trees for regular polygons. Discrete Computational Geometry 2: 65–84.CrossRefMATHMathSciNetGoogle Scholar
  20. Franksen, O.I., and I. Grattan-Guiness. 1989. The earliest contribution to location theory? Spatio-economic equilibrium with Lamé and Clapeyron, 1829. Mathematics and Computers in Simulation 31: 195–220.CrossRefMATHMathSciNetGoogle Scholar
  21. Gander, M.J., K. Santugini, and A. Steiner. 2008. Shortest road network connecting cities. Bollettino dei docenti di matematica 56: 9–19.Google Scholar
  22. Garey, M.R., R.L. Graham, and D.S. Johnson. 1977. The complexity of computing Steiner minimal trees. SIAM Journal on Applied Mathematics 32(4): 835–859.CrossRefMATHMathSciNetGoogle Scholar
  23. Gergonne, J.D. 1810. Solutions purement géométriques des problèmes de minimis proposés aux pages 196, 232 et 292 de ce volume, et de divers autres problèmes analogues. Annales de Mathématiques pures et appliquées 1: 375–384.Google Scholar
  24. Gilbert, E.N. 1967. Minimum cost communication networks. Bell System Technical Journal 46: 2209–2227.CrossRefGoogle Scholar
  25. Gilbert, E.N., and H.O. Pollak. 1968. Steiner minimal trees. SIAM Journal on Applied Mathematics 16(1): 1–29.CrossRefMATHMathSciNetGoogle Scholar
  26. Guggenbuhl, L. 1959. Gergonne, founder of the Annales de Mathmatiques. The Mathematics Teacher 52(8): 621–629.Google Scholar
  27. Hammersley, J.M. 1961. On Steiner’s network problem. Mathematika 8: 131–132.CrossRefMATHGoogle Scholar
  28. Hanan, M. 1966. On Steiner’s problem with rectilinear distance. SIAM Journal on Applied Mathematics 14(2): 255–265.CrossRefMATHMathSciNetGoogle Scholar
  29. Heinen, F. 1834. Über Systeme von Kräften. G. D. Bädeker.Google Scholar
  30. Hoffmann, E. 1890. Über das kürzeste Verbindungssystem zwischen vier Punkten der Ebene. In Program des Königlichen Gymnasiums zu Wetzlar für das Schuljahr von Ostern 1889 bis Ostern 1890. Schnitzler.Google Scholar
  31. Jarník, V., and M. Kössler. 1934. O minimálních grafeth obeahujících n daných bodú. Cas Pest Mat a Fys 63: 223–235.Google Scholar
  32. Korte, B., and J. Nesetril. 2001. Vojtech Jarnik’s work in combinatorial optimization. Discrete Mathematics 235: 1–17.CrossRefMATHMathSciNetGoogle Scholar
  33. Krarup, J., and S. Vajda. 1997. On Torricelli’s geometrical solution to a problem of Fermat. IMA Journal of Management Mathematics 8(3): 215–224. doi: 10.1093/imaman/8.3.215.CrossRefMATHMathSciNetGoogle Scholar
  34. Kupitz, Y.S., and H. Martini. 1997. Geometric aspects of the generalized Fermat–Torricelli problem. In Bolyai Society Mathematical Studies, vol. 6, ed. I. Barany, and K. Boroczky, 55–127. Budapest: Intuitive Geometry, Janos Bolyai Mathematical Society.Google Scholar
  35. Lamé, G., and B. Clapeyron. 1829. Mémoire sur lapplication de la statique à la solution des problèmes relatifs à la théorie des moindres distances. Journal des Voies et Communications 10: 26–49.Google Scholar
  36. Melzak, Z.A. 1961. On the problem of Steiner. Canad Math Bull 4(2): 143–148.CrossRefMATHMathSciNetGoogle Scholar
  37. Menger, K. 1931. Some applications of point-set methods. Annals of Mathematics, Second Series 32: 739–760.CrossRefMathSciNetGoogle Scholar
  38. Miehle, W. 1958. Link-length minimization in networks. Operations Research 6(2): 232–243.CrossRefMathSciNetGoogle Scholar
  39. Ollerenshaw, D.K. 1978. Minimum networks linking four points in a plane. Bulletin of the Institute of Mathematics and its Applications 15: 208–211.MathSciNetGoogle Scholar
  40. Pollak, H. 1978. Some remarks on the Steiner problem. Journal of Combinatorial Theory, Series A 24(3): 278–295.CrossRefMATHMathSciNetGoogle Scholar
  41. Rubinstein, J.H., D.A. Thomas, and N.C. Wormald. 1997. Steiner trees for terminals constrained to curves. SIAM Journal on Discrete Mathematics 10(1): 1–17.CrossRefMATHMathSciNetGoogle Scholar
  42. Schreiber, P. 1986. Zur Geschichte des sogenannten Steiner-Weber-Problems. Technical report, Wissenschaftliche Zeitschrift der Ernst-Moritz-Arndt-Universität Greifswald, Mathematisch-Naturwissenschaftliche Reihe. 58.Google Scholar
  43. Schumacher, H.C. 1810. Géométrie analitique. Solution analitique d’un problème de géométrie. Annales de Mathématiques pures et appliquées 1: 193–195.MathSciNetGoogle Scholar
  44. Scriba, C.J., and P. Schreiber. 2010. 5000 Jahre Geometrie. Berlin: Springer.CrossRefMATHGoogle Scholar
  45. Simpson, T. 1750. The Doctrine and applications of fluxions. London: John Nourse Publisher.Google Scholar
  46. Steiner, J. 1882. Gesammelte Werke, vol. II. Berlin: Reimer.Google Scholar
  47. Stigler, S.M. 1976. The anonymous Professor Gergonne. Historia Mathematica 3(1): 71–74.CrossRefMATHMathSciNetGoogle Scholar
  48. Tédenat, M. 1810. Questions résolues. Solution du premier des deux problèmes proposés à la page 196 de ce volume. Annales de Mathématiques pures et appliquées 1: 285–291.Google Scholar
  49. Torricelli, E. 1919. De maximis et minimis. In Opere di Evangelista Torricelli, ed. G. Loria, and G. Vassura. Faenza, Italy.Google Scholar
  50. Viviani, V. 1659. De maximis et minimis geometrica divination in quintum Conicorum Apollonii Pergaei, Vol. II. Florence, Italy.Google Scholar
  51. Warme, D.M. 1998. Spanning Trees in Hypergraphs with Applications to Steiner Trees. PhD thesis, Computer Science Department, The University of VirginiaGoogle Scholar
  52. Warme, D.M., P. Winter, and M. Zachariasen. 1999. Exact solutions to large-scale plane Steiner tree problems. In Proceedings of the tenth annual ACM-SIAM symposium on discrete algorithms, 979–980.Google Scholar
  53. Warme, D.M., P. Winter, and M. Zachariasen. 2001. GeoSteiner 3.1. Department of Computer Science, University of Copenhagen (DIKU), http://www.diku.dk/geosteiner/.
  54. Winter, P., and M. Zachariasen. 1997. Euclidean Steiner minimum trees: an improved exact algorithm. Networks 30: 149–166.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Marcus Brazil
    • 1
  • Ronald L. Graham
    • 2
  • Doreen A. Thomas
    • 3
  • Martin Zachariasen
    • 4
  1. 1.Department of Electrical and Electronic EngineeringThe University of MelbourneMelbourneAustralia
  2. 2.Department of Computer Science and EngineeringUC San DiegoLa JollaUSA
  3. 3.Department of Mechanical EngineeringThe University of MelbourneMelbourneAustralia
  4. 4.Department of Computer ScienceUniversity of CopenhagenCopenhagen ØDenmark

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