Skip to main content
SpringerLink
Log in

Algorithms for Radius-Optimally Augmenting Trees in a Metric Space

  • Conference paper
  • First Online:
Algorithms and Data Structures (WADS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12808))

Included in the following conference series:

Abstract

Let T be a tree with n vertices embedded in a metric space. We consider the problem of adding one edge to T to minimize the radius of the resulting graph.

For the continuous version of the problem where a center may be a point in the interior of an edge of the graph we give a linear time algorithm. In the case when the center is restricted to lie on a vertex, the discrete version, we give an \(O(n \log n)\) expected time algorithm.

Previously linear-time algorithms were known for the special case when the input graph is a path.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight 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

References

  1. Bender, M.A., Farach-Colton, M.: The LCA problem revisited. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 88–94. Springer, Heidelberg (2000). https://doi.org/10.1007/10719839_9

    Chapter  Google Scholar 

  2. Bilò, D.: Almost optimal algorithms for diameter-optimally augmenting trees. In: Proceedings of the 29th International Symposium on Algorithms and Computation (ISAAC), pp. 40:1–40:13 (2018)

    Google Scholar 

  3. Bilò, D., Gualà, L., Proietti, G.: Improved approximability and non-approximability results for graph diameter decreasing problems. Theoret. Comput. Sci. 417, 12–22 (2012)

    Article  MathSciNet  Google Scholar 

  4. Carufel, J.D., Grimm, C., Maheshwari, A., Smid, M.H.M.: Minimizing the continuous diameter when augmenting paths and cycles with shortcuts. In: Proceedings of the 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pp. 27:1–27:14 (2016)

    Google Scholar 

  5. De Carufel, J.-L., Grimm, C., Schirra, S., Smid, M.: Minimizing the continuous diameter when augmenting a tree with a shortcut. In: WADS 2017. LNCS, vol. 10389, pp. 301–312. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-62127-2_26

    Chapter  Google Scholar 

  6. Demaine, E.D., Zadimoghaddam, M.: Minimizing the diameter of a network using shortcut edges. In: Kaplan, H. (ed.) SWAT 2010. LNCS, vol. 6139, pp. 420–431. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13731-0_39

    Chapter  Google Scholar 

  7. Frati, F., Gaspers, S., Gudmundsson, J., Mathieson, L.: Augmenting graphs to minimize the diameter. Algorithmica 72(4), 995–1010 (2015). https://doi.org/10.1007/s00453-014-9886-4

    Article  MathSciNet  MATH  Google Scholar 

  8. Gao, Y., Hare, D.R., Nastos, J.: The parametric complexity of graph diameter augmentation. Discret. Appl. Math. 161(10–11), 1626–1631 (2013)

    Article  MathSciNet  Google Scholar 

  9. Große, U., Gudmundsson, J., Knauer, C., Smid, M., Stehn, F.: Fast algorithms for diameter-optimally augmenting paths. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 678–688. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47672-7_55

    Chapter  MATH  Google Scholar 

  10. Gudmundsson, J., Sha, Y.: Augmenting graphs to minimize the radius. Manuscript (2021)

    Google Scholar 

  11. Johnson, C., Wang, H.: A linear-time algorithm for radius-optimally augmenting paths in a metric space. In: Proceedings of the 15th International Symposium on Algorithms and Data Structures, pp. 466–480 (2019)

    Google Scholar 

  12. Oh, E., Ahn, H.: A near-optimal algorithm for finding an optimal shortcut of a tree. In: Proceedings of the 27th International Symposium on Algorithms and Computation (ISAAC), pp. 59:1–59:12 (2016)

    Google Scholar 

  13. Schoone, A.A., Bodlaender, H.L., van Leeuwen, J.: Diameter increase caused by edge deletion. J. Graph Theory 11(3), 409–427 (1987)

    Article  MathSciNet  Google Scholar 

  14. Wang, H.: An improved algorithm for diameter-optimally augmenting paths in a metric space. Comput. Geom. 75, 11–21 (2018)

    Article  MathSciNet  Google Scholar 

  15. Wang, H., Zhao, Y.: Algorithms for Diameters of Unicycle Graphs and Diameter-optimally Augmenting Trees (2020)

    Google Scholar 

  16. Wang, H., Zhao, Y.: A linear-time algorithm for discrete radius optimally augmenting paths in a metric space (2020)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The University of Sydney, Sydney, NSW, 2006, Australia

    Joachim Gudmundsson & Yuan Sha

Authors
  1. Joachim Gudmundsson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuan Sha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuan Sha .

Editor information

Editors and Affiliations

  1. University of Waterloo, Waterloo, ON, Canada

    Anna Lubiw

  2. University of Alberta, Edmonton, AB, Canada

    Mohammad Salavatipour

  3. Dalhousie University, Halifax, Canada

    Meng He

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gudmundsson, J., Sha, Y. (2021). Algorithms for Radius-Optimally Augmenting Trees in a Metric Space. In: Lubiw, A., Salavatipour, M., He, M. (eds) Algorithms and Data Structures. WADS 2021. Lecture Notes in Computer Science(), vol 12808. Springer, Cham. https://doi.org/10.1007/978-3-030-83508-8_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-83508-8_33

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-83507-1

  • Online ISBN: 978-3-030-83508-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation