Minimizing the Continuous Diameter When Augmenting a Tree with a Shortcut

  • Jean-Lou De Carufel
  • Carsten GrimmEmail author
  • Stefan Schirra
  • Michiel Smid
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)


We augment a tree \(T\) with a shortcut \(pq\) to minimize the largest distance between any two points along the resulting augmented tree \(T+pq\). We study this problem in a continuous and geometric setting where \(T\) is a geometric tree in the Euclidean plane, a shortcut is a line segment connecting any two points along the edges of \(T\), and we consider all points on \(T+pq\) (i.e., vertices and points along edges) when determining the largest distance along \(T+pq\). The continuous diameter is the largest distance between any two points along edges. We establish that a single shortcut is sufficient to reduce the continuous diameter of a geometric tree \(T\) if and only if the intersection of all diametral paths of \(T\) is neither a line segment nor a point. We determine an optimal shortcut for a geometric tree with \(n\) straight-line edges in \(O(n \log n)\) time.


Network augmentation Continuous diameter minimization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cáceres, J., Garijo, D., González, A., Márquez, A., Puertas, M.L., Ribeiro, P.: Shortcut Sets for Plane Euclidean Networks. Electron. Notes Discrete Math. 54, 163–168 (2016)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chepoi, V., Vaxès, Y.: Augmenting Trees to Meet Biconnectivity and Diameter Constraints. Algorithmica 33(2), 243–262 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    De Carufel, J.L., Grimm, C., Maheshwari, A., Smid, M.: Minimizing the continuous diameter when augmenting paths and cycles with shortcuts. In: SWAT 2016, pp. 27:1–27:14 (2016)Google Scholar
  4. 4.
    Farshi, M., Giannopoulos, P., Gudmundsson, J.: Improving the Stretch Factor of a Geometric Network by Edge Augmentation. SIAM J. Comp. 38(1), 226–240 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Frati, F., Gaspers, S., Gudmundsson, J., Mathieson, L.: Augmenting Graphs to Minimize the Diameter. Algorithmica 72(4), 995–1010 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gao, Y., Hare, D.R., Nastos, J.: The Parametric Complexity of Graph Diameter Augmentation. Discrete Appl. Math. 161(10–11), 1626–1631 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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). doi: 10.1007/978-3-662-47672-7_55 CrossRefGoogle Scholar
  8. 8.
    Große, U., Gudmundsson, J., Knauer, C., Smid, M., Stehn, F.: Fast Algorithms for Diameter-Optimally Augmenting Paths and Trees (2016). arXiv:1607.05547
  9. 9.
    Hakimi, S.L.: Optimum locations of switching centers and the absolute centers and medians of a graph. Operations Research 12(3), 450–459 (1964)CrossRefzbMATHGoogle Scholar
  10. 10.
    Li, C.L., McCormick, S., Simchi-Levi, D.: On the Minimum-Cardinality-Bounded-Diameter and the Bounded-Cardinality-Minimum-Diameter Edge Addition Problems. Operations Research Letters 11(5), 303–308 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Luo, J., Wulff-Nilsen, C.: Computing best and worst shortcuts of graphs embedded in metric spaces. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 764–775. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-92182-0_67 CrossRefGoogle Scholar
  12. 12.
    Oh, E., Ahn, H.K.: A near-optimal algorithm for finding an optimal shortcut of a tree. In: ISAAC 2016, pp. 59:1–59:12 (2016)Google Scholar
  13. 13.
    Schoone, A.A., Bodlaender, H.L., van Leeuwen, J.: Diameter Increase Caused by Edge Deletion. Journal of Graph Theory 11(3), 409–427 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Yang, B.: Euclidean Chains and their Shortcuts. Theor. Comput. Sci. 497, 55–67 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Jean-Lou De Carufel
    • 1
  • Carsten Grimm
    • 2
    • 3
    Email author
  • Stefan Schirra
    • 3
  • Michiel Smid
    • 2
  1. 1.School of Electrical Engineering and Computer ScienceUniversity of OttawaOttawaCanada
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
  3. 3.Institut für Simulation und GraphikOtto-von-Guericke-Universität MagdeburgMagdeburgGermany

Personalised recommendations