IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v172y2017i3d10.1007_s10957-016-1036-5.html
   My bibliography  Save this article

Approximate Euclidean Steiner Trees

Author

Listed:
  • Charl Ras

    (University of Melbourne)

  • Konrad Swanepoel

    (London School of Economics and Political Science)

  • Doreen Anne Thomas

    (University of Melbourne)

Abstract

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error from $$120^{\circ }$$ 120 ∘ . This notion arises in numerical approximations of minimum Steiner trees. We investigate the worst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology. It has been conjectured that this relative error is at most linear in the maximum error at the angles, independent of the number of terminals. We verify this conjecture for the two-dimensional case as long as the maximum angle error is sufficiently small in terms of the number of terminals. In the two-dimensional case we derive a lower bound for the relative error in length. This bound is linear in terms of the maximum angle error when the angle error is sufficiently small in terms of the number of terminals. We find improved estimates of the relative error in length for larger values of the maximum angle error and calculate exact values in the plane for three and four terminals.

Suggested Citation

  • Charl Ras & Konrad Swanepoel & Doreen Anne Thomas, 2017. "Approximate Euclidean Steiner Trees," Journal of Optimization Theory and Applications, Springer, vol. 172(3), pages 845-873, March.
    • Handle: RePEc:spr:joptap:v:172:y:2017:i:3:d:10.1007_s10957-016-1036-5
    DOI: 10.1007/s10957-016-1036-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-016-1036-5
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-016-1036-5?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:172:y:2017:i:3:d:10.1007_s10957-016-1036-5. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.