Efficiently Testing \(T\)-Interval Connectivity in Dynamic Graphs

  • Arnaud CasteigtsEmail author
  • Ralf Klasing
  • Yessin M. Neggaz
  • Joseph G. Peters
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9079)


Many types of dynamic networks are made up of durable entities whose links evolve over time. When considered from a global and discrete standpoint, these networks are often modelled as evolving graphs, i.e. a sequence of static graphs \(\mathcal{{G}}=\{G_1,G_2,...,G_{\delta }\}\) such that \(G_i=(V,E_i)\) represents the network topology at time step \(i\). Such a sequence is said to be \(T\)-interval connected if for any \(t\in [1, \delta -T+1]\) all graphs in \(\{G_t,G_{t+1},...,G_{t+T-1}\}\) share a common connected spanning subgraph. In this paper, we consider the problem of deciding whether a given sequence \(\mathcal{{G}}\) is \(T\)-interval connected for a given \(T\). We also consider the related problem of finding the largest \(T\) for which a given \(\mathcal{{G}}\) is \(T\)-interval connected. We assume that the changes between two consecutive graphs are arbitrary, and that two operations, binary intersection and connectivity testing, are available to solve the problems. We show that \(\varOmega (\delta )\) such operations are required to solve both problems, and we present optimal \(O(\delta )\) online algorithms for both problems.


\(T\)-interval connectivity Dynamic graphs Time-varying graphs 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barjon, M., Casteigts, A., Chaumette, S., Johnen, C., Neggaz, Y.M.: Testing temporal connectivity in sparse dynamic graphs. CoRR abs/1404.7634, 8p (2014). a French version appeared in Proc. of ALGOTEL (2014)Google Scholar
  2. 2.
    Bui-Xuan, B., Ferreira, A., Jarry, A.: Computing shortest, fastest, and foremost journeys in dynamic networks. Int. J. of Foundations of Computer Science 14(2), 267–285 (2003)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Casteigts, A., Chaumette, S., Ferreira, A.: Characterizing topological assumptions of distributed algorithms in dynamic networks. In: Kutten, S., Žerovnik, J. (eds.) SIROCCO 2009. LNCS, vol. 5869, pp. 126–140. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  4. 4.
    Casteigts, A., Flocchini, P., Quattrociocchi, W., Santoro, N.: Time-varying graphs and dynamic networks. Int. J. of Parallel, Emergent and Distributed Systems 27(5), 387–408 (2012)Google Scholar
  5. 5.
    Gibbons, A., Rytter, W.: Efficient parallel algorithms. Cambridge University Press (1988)Google Scholar
  6. 6.
    Jain, S., Fall, K., Patra, R.: Routing in a delay tolerant network. In: Proc. of SIGCOMM, pp. 145–158 (2004)Google Scholar
  7. 7.
    JáJá, J.: An Introduction to Parallel Algorithms. Addison-Wesley (1992)Google Scholar
  8. 8.
    Kuhn, F., Lynch, N., Oshman, R.: Distributed computation in dynamic networks. In: Proc. of STOC, pp. 513–522. ACM, Cambridge (2010)Google Scholar
  9. 9.
    O’Dell, R., Wattenhofer, R.: Information dissemination in highly dynamic graphs. In: Proc. of DIALM-POMC, pp. 104–110. ACM, Cologne (2005)Google Scholar
  10. 10.
    Whitbeck, J., Dias de Amorim, M., Conan, V., Guillaume, J.L.: Temporal reachability graphs. In: Proc. of MOBICOM, pp. 377–388. ACM (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Arnaud Casteigts
    • 1
    Email author
  • Ralf Klasing
    • 1
  • Yessin M. Neggaz
    • 1
  • Joseph G. Peters
    • 2
  1. 1.LaBRI, CNRSUniversity of BordeauxTalenceFrance
  2. 2.School of Computing ScienceSimon Fraser UniversityBurnabyCanada

Personalised recommendations