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On the history of the Euclidean Steiner tree problem

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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.

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Notes

  1. Quoted in an unpublished paper by P. J. H. Piñeyro: “Gergonne; the isoperimetric problem and the Steiners symmetrization”, January 2012.

  2. This appears to be the same sequence as the number of trivially fully gated graphs on \(n\) nodes (Colbourn and Huybrechts 2008); however, the equivalence between these two sequences is still to be established.

  3. It was thought by a number of sources, such as (Kupitz and Martini 1997, p. 101), that the Steiner tree problem had been rediscovered and discussed by Lamé and Clapeyron in 1827 or 1829 in the paper (Lamé and Clapeyron 1829). It is clear, however, from the 1989 paper of Franksen and Grattan-Guiness (1989), which includes a complete translation of Lamé and Clapeyron (1829), that although Lamé and Clapeyron studied a number of generalisations of the Fermat–Torricelli problem, they did not work on any problem that was equivalent to the Steiner tree problem.

  4. Personal record on Karl Bopp obtained from Research Library for the History of Education at the German Institute for International Educational Research (http://bbf.dipf.de).

  5. Personal record on Eduard Hoffmann obtained from Research Library for the History of Education at the German Institute for International Educational Research (http://bbf.dipf.de).

  6. There were at least four earlier papers citing (Jarník and Kössler 1934), mostly in the Czech mathematical literature; however, none of these papers deal directly with the Euclidean Steiner Tree problem.

  7. In particular, Denes König’s seminal textbook on graph theory, “Theorie der Endlichen und Unendlichen Graphen” was not published until 1936, two years after this paper.

  8. In a private communication, when asked why he worked on the minimum spanning tree and Steiner tree problems, Choquet simply replied that when he was young he worked on all kinds of different problems (and that he had nothing more to contribute on these two problems).

  9. This, in fact, corrects a mistake made by Menger (1931), who attempted to establish the same result using minimum spanning trees. It is unknown whether Choquet was aware of Menger’s work.

  10. Kupitz and Martini (1997) argue that this dismissal of the generalised Fermat–Torricelli problem is based on a misreading of Steiner’s paper in pages 729–731 of Steiner (1882).

  11. This is according to one of Melzak’s students, Raymond Booth, in a private communication.

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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).

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Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, The University of Melbourne, Melbourne, VIC, 3010, Australia

    Marcus Brazil

  2. Department of Computer Science and Engineering, UC San Diego, La Jolla, CA, 92093-0404, USA

    Ronald L. Graham

  3. Department of Mechanical Engineering, The University of Melbourne, Melbourne, VIC, 3010, Australia

    Doreen A. Thomas

  4. Department of Computer Science, University of Copenhagen, 2100 , Copenhagen Ø, Denmark

    Martin Zachariasen

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Correspondence to Marcus Brazil.

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Communicated by: Jesper Lützen.

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Brazil, M., Graham, R.L., Thomas, D.A. et al. On the history of the Euclidean Steiner tree problem. Arch. Hist. Exact Sci. 68, 327–354 (2014). https://doi.org/10.1007/s00407-013-0127-z

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