A Deterministic Algorithm for Online Steiner Tree Leasing

  • Marcin Bienkowski
  • Artur Kraska
  • Paweł Schmidt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)


We study the Online Steiner Tree Leasing (OSTL) problem, defined in a weighted undirected graph with a distinguished root node r. There is a known set \(\mathcal {L}\) of available lease types, where each type \(\ell \in \mathcal {L}\) is characterized by its duration \(D_\ell \) and cost factor \(C_\ell \). As an input, an online algorithm is given a sequence of terminals and has to connect them to the root r using leased edges. An edge of length d can be leased using lease type \(\ell \) for cost \(C_\ell \cdot d\) and remains valid for time \(D_\ell \).

The OSTL problem contains the online Steiner tree and the single-source rent-or-buy problems as specific subcases. We present the first deterministic online algorithm for OSTL, whose competitive ratio is \(O(|\mathcal {L}| \cdot \log k)\), where k is the number of different terminals in the input. The currently best randomized algorithm attains the ratio of \(O(\log |\mathcal {L}| \cdot \log n)\), where \(n \ge k\) is the number of nodes in the graph.


Steiner tree Leasing Competitive analysis Online algorithms 


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  1. 1.
    Abshoff, S., Kling, P., Markarian, C., Meyer auf der Heide, F., Pietrzyk, P.: Towards the price of leasing online. Journal of Combinatorial Optimization, 1–20 (2015)Google Scholar
  2. 2.
    Abshoff, S., Markarian, C., Meyer auf der Heide, F.: Randomized online algorithms for set cover leasing problems. In: Zhang, Z., Wu, L., Xu, W., Du, D.-Z. (eds.) COCOA 2014. LNCS, vol. 8881, pp. 25–34. Springer, Cham (2014). doi: 10.1007/978-3-319-12691-3_3 Google Scholar
  3. 3.
    Albers, S., Koga, H.: New on-line algorithms for the page replication problem. Journal of Algorithms 27(1), 75–96 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alon, N., Azar, Y.: On-line Steiner trees in the Euclidean plane. In: Proc. 8th ACM Symp. on Computational Geometry (SoCG), pp. 337–343 (1992)Google Scholar
  5. 5.
    Angelopoulos, S.: On the competitiveness of the online asymmetric and euclidean steiner tree problems. In: Bampis, E., Jansen, K. (eds.) WAOA 2009. LNCS, vol. 5893, pp. 1–12. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-12450-1_1 CrossRefGoogle Scholar
  6. 6.
    Anthony, B.M., Gupta, A.: Infrastructure leasing problems. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 424–438. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-72792-7_32 CrossRefGoogle Scholar
  7. 7.
    Armbrust, M., Fox, A., Griffith, R., Joseph, A.D., Katz, R.H., Konwinski, A., Lee, G., Patterson, D.A., Rabkin, A., Stoica, I., Zaharia, M.: Above the clouds: A Berkeley view of cloud computing. Tech. Rep. UCB/EECS-2009-28, EECS Department, University of California, Berkeley (2009).
  8. 8.
    Awerbuch, B., Azar, Y.: Buy-at-bulk network design. In: Proc. 38th IEEE Symp. on Foundations of Computer Science (FOCS), pp. 542–547 (1997)Google Scholar
  9. 9.
    Awerbuch, B., Azar, Y., Bartal, Y.: On-line generalized Steiner problem. Theoretical Computer Science 324(2–3), 313–324 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Awerbuch, B., Bartal, Y., Fiat, A.: Competitive distributed file allocation. In: Proc. 25th ACM Symp. on Theory of Computing (STOC), pp. 164–173 (1993)Google Scholar
  11. 11.
    Bartal, Y., Fiat, A., Rabani, Y.: Competitive algorithms for distributed data management. Journal of Computer and System Sciences 51(3), 341–358 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Black, D.L., Sleator, D.D.: Competitive algorithms for replication and migration problems. Tech. Rep. CMU-CS-89-201, Department of Computer Science, Carnegie-Mellon University (1989)Google Scholar
  13. 13.
    Chowdhury, N.M.K., Boutaba, R.: A survey of network virtualization. Computer Networks 54(5), 862–876 (2010)CrossRefzbMATHGoogle Scholar
  14. 14.
    Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. Journal of Computer and System Sciences 69(3), 485–497 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fleischer, R., Głazek, W., Seiden, S.S.: New results for online page replication. Theoretical Computer Science 324(2–3), 219–251 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Imase, M., Waxman, B.M.: Dynamic Steiner tree problem. SIAM Journal on Discrete Mathematics 4(3), 369–384 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lund, C., Reingold, N., Westbrook, J., Yan, D.C.K.: Competitive on-line algorithms for distributed data management. SIAM Journal on Computing 28(3), 1086–1111 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Matsubayashi, A.: Non-greedy online steiner trees on outerplanar graphs. In: Jansen, K., Mastrolilli, M. (eds.) WAOA 2016. LNCS, vol. 10138, pp. 129–141. Springer, Cham (2017). doi: 10.1007/978-3-319-51741-4_11 CrossRefGoogle Scholar
  19. 19.
    Meyerson, A.: The parking permit problem. In: Proc. 46th IEEE Symp. on Foundations of Computer Science (FOCS), pp. 274–284 (2005)Google Scholar
  20. 20.
    Nagarajan, C., Williamson, D.P.: Offline and online facility leasing. Discrete Optimization 10(4), 361–370 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Umboh, S.: Online network design algorithms via hierarchical decompositions. In: Proc. 26th ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 1373–1387 (2015)Google Scholar
  22. 22.
    Westbrook, J., Yan, D.C.K.: The performance of greedy algorithms for the on-line Steiner tree and related problems. Mathematical Systems Theory 28(5), 451–468 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wu, W., Huang, Y.: Steiner trees. In: Encyclopedia of Algorithms, pp. 2102–2107 (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Marcin Bienkowski
    • 1
  • Artur Kraska
    • 1
  • Paweł Schmidt
    • 1
  1. 1.Institute of Computer ScienceUniversity of WrocławWrocławPoland

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