Skip to main content
SpringerLink
Log in

Non-Greedy Online Steiner Trees on Outerplanar Graphs

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

This paper addresses the classic online Steiner tree problem on edge-weighted graphs. It is known that a greedy (nearest neighbor) online algorithm has a tight competitive ratio for wide classes of graphs, such as trees, rings, any class including series-parallel graphs, and unweighted graphs with bounded diameter. However, we do not know any greedy or non-greedy tight deterministic algorithm for other classes of graphs. In this paper, we observe that a greedy algorithm is \(\Omega (\log n)\)-competitive on outerplanar graphs, where n is the number of vertices, and propose a 5.828-competitive deterministic algorithm on outerplanar graphs. Our algorithm connects a requested vertex and the tree constructed thus far using a path that is constant times longer than the distance between them. We also present a lower bound of 4 for arbitrary deterministic online Steiner tree algorithms on outerplanar graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Alon, N., Azar, Y.: On-line Steiner trees in the Euclidean plane. Discrete Comput. Geom. 10, 113–121 (1993)

    Article  MathSciNet  Google Scholar 

  2. Angelopoulos, S.: Online priority Steiner tree problems. In: Dehne, F., Gavrilova, M., Sack, J.R., Tóth, C.D. (eds.) WADS 2009. Lecture Notes in Computer Science, vol. 5664, pp. 37–48. Springer, Heidelberg (2009)

    Google Scholar 

  3. Angelopoulos, S.: Parameterized analysis of online Steiner tree problems. In: Adaptive, Output Sensitive, Online and Parameterized Algorithms, Dagstuhl Seminar Proceedings (2009). http://drops.dagstuhl.de/opus/volltexte/2009/2121

  4. Angelopoulos, S.: On the competitiveness of the online asymmetric and Euclidean Steiner tree problems. In: Bampis, E., Jansen, K. (eds.) WAOA 2009. Lecture Notes in Computer Science, vol. 5893, pp. 1–12. Springer, Heidelberg (2010)

    Google Scholar 

  5. Averbuch, B., Azar, Y., Bartal, Y.: On-line generalized Steiner problem. Theor. Comput. Sci. 324, 313–324 (2004)

    Article  MathSciNet  Google Scholar 

  6. Awerbuch, B., Bartal, Y., Fiat, A.: Competitive distributed file allocation. Inf. Comput. 185(1), 1–40 (2003)

    Article  MathSciNet  Google Scholar 

  7. Bartal, Y., Fiat, A., Rabani, Y.: Competitive algorithms for distributed data management. J. Comput. Syst. Sci. 51(3), 341–358 (1995)

    Article  MathSciNet  Google Scholar 

  8. Ben-David, S., Borodin, A., Karp, R., Tardos, G., Wigderson, A.: On the power of randomization in on-line algorithms. Algorithmica 11, 2–14 (1994)

    Article  MathSciNet  Google Scholar 

  9. Berman, P., Coulston, C.: On-line algorithms for Steiner tree problems. In: Proceedings of the 29th ACM Symposium on Theory of Computing, pp. 344–353 (1997)

  10. Chekuri, C., Gupta, A., Newman, I., Rabinovich, Y., Sinclair, A.: Embedding \(k\)-outerplanar graphs into \(\ell _1\). SIAM J. Discrete Math. 20(1), 119–136 (2006)

    Article  MathSciNet  Google Scholar 

  11. Fleischner, H.J., Geller, D.P., Harary, F.: Outerplanar graphs and weak duals. J. Indian Math. Soc. 38, 215–219 (1974)

    MathSciNet  MATH  Google Scholar 

  12. Gupta, A., Newman, I., Rabinovich, Y., Sinclair, A.: Cuts, trees, and \(\ell _1\)-embedding of graphs. Combinatorica 24(2), 233–269 (2004)

    Article  MathSciNet  Google Scholar 

  13. Imase, M., Waxman, B.M.: Dynamic Steiner tree problem. SIAM J. Discrete Math. 4(3), 369–384 (1991)

    Article  MathSciNet  Google Scholar 

  14. Lund, C., Reingold, N., Westbrook, J., Yan, D.: Competitive on-line algorithms for distributed data management. SIAM J. Comput. 28(3), 1086–1111 (1999)

    Article  MathSciNet  Google Scholar 

  15. Naor, J.S., Panigrahi, D., Singh, M.: Online node-weighted Steiner tree and related problems. In: Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, pp. 210–219 (2011)

  16. Westbrook, J., Yan, D.C.K.: The performance of greedy algorithms for the on-line Steiner tree and related problems. Math. Syst. Theory 28, 451–468 (1995)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Electrical Engineering and Computer Science, Kanazawa Univ., Kanazawa, 920-1192, Japan

    Akira Matsubayashi

Authors
  1. Akira Matsubayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akira Matsubayashi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A preliminary version appeared in the proceedings of the 14th International Workshop on Approximation and Online Algorithms (WAOA 2016).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Matsubayashi, A. Non-Greedy Online Steiner Trees on Outerplanar Graphs. Algorithmica 83, 613–640 (2021). https://doi.org/10.1007/s00453-020-00768-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-020-00768-6

Keywords

Search

Navigation