Efficient Computation of Optimal Temporal Walks Under Waiting-Time Constraints

  • Anne-Sophie HimmelEmail author
  • Matthias Bentert
  • André Nichterlein
  • Rolf Niedermeier
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 882)


Node connectivity plays a central role in temporal network analysis. We provide a comprehensive study of various concepts of walks in temporal graphs, that is, graphs with fixed vertex sets but edge sets changing over time. Importantly, the temporal aspect results in a rich set of optimization criteria for “shortest” walks. Extending and significantly broadening state-of-the-art work of Wu et al. [IEEE TKDE 2016], we provide an algorithm for computing shortest walks that is capable to deal with various optimization criteria and any linear combination of these. It runs in \(O (|V| + |E| \log |E|)\) time where |V| is the number of vertices and |E| is the number of time edges. A central distinguishing factor to Wu et al.’s work is that our model allows to, motivated by real-world applications, respect waiting-time constraints for vertices, that is, the minimum and maximum waiting time allowed in intermediate vertices of a walk. Moreover, other than Wu et al. our algorithm also allows to search for walks that pass multiple edges in one time step, and it can optimize a richer set of optimization criteria. Our experimental studies indicate that our richer modeling can be achieved without significantly worsening the running time when compared to Wu et al.’s algorithms.


Temporal networks Temporal paths Shortest path computation Waiting policies Infectious disease spreading 


  1. 1.
    Barabási, A.L.: Network Science. Cambridge University Press, Cambridge (2016)zbMATHGoogle Scholar
  2. 2.
    Bast, H., Delling, D., Goldberg, A., Müller-Hannemann, M., Pajor, T., Sanders, P., Wagner, D., Werneck, R.F.: Route planning in transportation networks. In: Algorithm Engineering, pp. 19–80. Springer (2016)Google Scholar
  3. 3.
    Casteigts, A., Himmel, A.S., Molter, H., Zschoche, P.: The computational complexity of finding temporal paths under waiting time constraints. arXiv preprint arXiv:1909.06437 (2019)
  4. 4.
    Dean, B.C.: Algorithms for minimum-cost paths in time-dependent networks with waiting policies. Networks 44, 41–46 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Holme, P.: Temporal network structures controlling disease spreading. Phys. Rev. E 94(2), 022305 (2016)CrossRefGoogle Scholar
  6. 6.
    Holme, P., Saramäki, J.: Temporal networks. Phys. Rep. 519(3), 97–125 (2012)CrossRefGoogle Scholar
  7. 7.
    Kivelä, M., Cambe, J., Saramäki, J., Karsai, M.: Mapping temporal-network percolation to weighted, static event graphs. Sci. Rep. 8(1), 12357 (2018)CrossRefGoogle Scholar
  8. 8.
    Leskovec, J., Krevl, A.: SNAP Datasets: stanford large network dataset collection. (2014)
  9. 9.
    Lightenberg, W., Pei, Y., Fletcher, G., Pechenizkiy, M.: Tink: a temporal graph analytics library for Apache Flink. In: Proceedings of WWW 2018, pp. 71–72. International World Wide Web Conferences Steering Committee (2018)Google Scholar
  10. 10.
    Masuda, N., Holme, P.: Predicting and controlling infectious disease epidemics using temporal networks. F1000prime Rep. 5, 6 (2013)CrossRefGoogle Scholar
  11. 11.
    Modiri, A.B., Karsai, M., Kivelä, M.: Efficient limited time reachability estimation in temporal networks. arXiv preprint arXiv:1908.11831 (2019)
  12. 12.
    Newman, M.E.J.: Networks. Oxford University Press, Oxford (2018)CrossRefGoogle Scholar
  13. 13.
    Pan, R.K., Saramäki, J.: Path lengths, correlations, and centrality in temporal networks. Phys. Rev. E 84(1), 016105 (2011)CrossRefGoogle Scholar
  14. 14.
    Salathé, M., Kazandjieva, M., Lee, J.W., Levis, P., Feldman, M.W., Jones, J.H.: A high-resolution human contact network for infectious disease transmission. Proc. Nat. Acad. Sci. 107(51), 22020–22025 (2010)CrossRefGoogle Scholar
  15. 15.
    Wu, H., Cheng, J., Ke, Y., Huang, S., Huang, Y., Wu, H.: Efficient algorithms for temporal path computation. IEEE Trans. Knowl. Data Eng. 28(11), 2927–2942 (2016)CrossRefGoogle Scholar
  16. 16.
    Xuan, B.B., Ferreira, A., Jarry, A.: Computing shortest, fastest, and foremost journeys in dynamic networks. Int. J. Found. Comput. Sci. 14(02), 267–285 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhao, A., Liu, G., Zheng, B., Zhao, Y., Zheng, K.: Temporal paths discovery with multiple constraints in attributed dynamic graphs. World Wide Web, 1–24 (2019) Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Anne-Sophie Himmel
    • 1
    Email author
  • Matthias Bentert
    • 1
  • André Nichterlein
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Faculty IV, Algorithmics and Computational ComplexityTU BerlinBerlinGermany

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