Skip to main content
SpringerLink
Log in

Distributed backup placement in networks

  • Published:
Distributed Computing Aims and scope Submit manuscript

Abstract

We consider the backup placement problem, defined as follows. Some nodes (processors) in a given network have objects (e.g., files, tasks) whose backups should be stored in additional nodes for increased fault resilience. To minimize the disturbance in case of a failure, it is required that a backup copy should be located at a neighbor of the primary node. The goal is to find an assignment of backup copies to nodes which minimizes the maximum load (number or total size of backup copies) over all nodes in the network. It is known that a natural selfish local improvement policy has approximation ratio \(\varOmega (\log n/\log \log n)\); we show that it may take this policy \(\varOmega (\sqrt{n})\) time to reach equilibrium in the distributed setting. Our main result in this paper is a randomized distributed algorithm which finds a placement in polylogarithmic time and achieves approximation ratio \(O\left(\frac{\log n}{\log \log n}\right)\). We obtain this result using a randomized distributed approximation algorithm for f-matching in bipartite graphs that may be of independent interest.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. The reduction works for general k because the target flow value is known in advance (it is nk). The proof of Theorem 2 uses instances that are not fully covered.

References

  1. Amini, O., Peleg, D., Pérennes, S., Sau, I., Saurabh, S.: On the approximability of some degree-constrained subgraph problems. Discret. Appl. Math. 160(12), 1661–1679 (2012). doi:10.1016/j.dam.2012.03.025

    Article  MathSciNet  MATH  Google Scholar 

  2. Awerbuch, B.: Complexity of network synchronization. J. ACM 32(4), 804–823 (1985). doi:10.1145/4221.4227

    Article  MathSciNet  MATH  Google Scholar 

  3. Azar, Y.: On-line load balancing. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms. Lecture Notes in Computer Science, vol. 1442, pp. 178–195. Springer, Berlin (1998). doi:10.1007/BFb0029569

  4. Azar, Y., Naor, J., Rom, R.: The competitiveness of on-line assignments. J. Algorithms 18(2), 221–237 (1995). doi:10.1006/jagm.1995.1008

    Article  MathSciNet  MATH  Google Scholar 

  5. Barenboim, L., Elkin, M., Pettie, S., Schneider, J.: The locality of distributed symmetry breaking. J. ACM 63(3), 20:1–20:45 (2016). doi:10.1145/2903137

  6. Bokal, D., Brešar, B., Jerebic, J.: A generalization of hungarian method and Hall’s theorem with applications in wireless sensor networks. Discret. Appl. Math. 160(4–5), 460–470 (2012). doi:10.1016/j.dam.2011.11.007

    Article  MathSciNet  MATH  Google Scholar 

  7. Czygrinow, A., Hanćkowiak, M., Szymańska, E., Wawrzyniak, W.: Distributed 2-approximation algorithm for the semi-matching problem. In: 26th DISC, pp. 210–222 (2012). doi:10.1007/978-3-642-33651-5_15

  8. Dessmark, A., Garrido, O., Lingas, A.: A note on parallel complexity of maximum f-matching. Inform. Process. Lett. 65(2), 107–109 (1998). doi:10.1016/S0020-0190(97)00196-8

    Article  MathSciNet  MATH  Google Scholar 

  9. Dósa, G., Epstein, L.: The convergence time for selfish bin packing. In: 7th SAGT, pp. 37–48 (2014). doi:10.1007/978-3-662-44803-8_4

  10. Even-Dar, E., Kesselman, A., Mansour, Y.: Convergence time to Nash equilibrium in load balancing. ACM T. Algorithms 3(3), 32 (2007). doi:10.1145/1273340.1273348

  11. Fakcharoenphol, J., Laekhanukit, B., Nanongkai, D.: Faster algorithms for semi-matching problems. ACM Trans. Algorithms 10(3), 14:1–14:3 (2014). doi:10.1145/2601071

    Article  MathSciNet  MATH  Google Scholar 

  12. Gairing, M., Lücking, T., Mavronicolas, M., Monien, B.: The price of anarchy for restricted parallel links. Parallel Process. Lett. 16(1), 117–131 (2006). doi:10.1142/S0129626406002514

    Article  MathSciNet  MATH  Google Scholar 

  13. Galčík, F., Katrenič, J., Semanišin, G.: On computing an optimal semi-matching. In: 37th WG, pp. 250–261 (2011). doi:10.1007/978-3-642-25870-1_23

  14. Garg, N., Kavitha, T., Kumar, A., Mehlhorn, K., Mestre, J.: Assigning papers to referees. Algorithmica 58(1), 119–136 (2010). doi:10.1007/s00453-009-9386-0

    Article  MathSciNet  MATH  Google Scholar 

  15. Halldórsson, M.M., Köhler, S., Rawitz, D.: Distributed approximation of \(k\)-service assignment. In: 19th International Conference on Principles of Distributed Systems, pp. 122–137 (2015)

  16. Hańćkowiak, M., Karoński, M., Panconesi, A.: A faster distributed algorithm for computing maximal matchings deterministically. In: 18th PODC, pp. 219–228 (1999). doi:10.1145/301308.301360

  17. Harvey, N.J.A., Ladner, R.E., Lovász, L., Tamir, T.: Semi-matchings for bipartite graphs and load balancing. J. Algorithms 59(1), 53–78 (2006). doi:10.1016/j.jalgor.2005.01.003

    Article  MathSciNet  MATH  Google Scholar 

  18. Hazan, E., Safra, S., Schwartz, O.: On the complexity of approximating \(k\)-set packing. Comput. Complex. 15(1), 20–39 (2006). doi:10.1007/s00037-006-0205-6

    Article  MathSciNet  MATH  Google Scholar 

  19. Hopcroft, J.E., Karp, R.M.: An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973). doi:10.1137/0202019

    Article  MathSciNet  MATH  Google Scholar 

  20. Kann, V.: Maximum bounded 3-dimensional matching is MAX SNP-complete. Inform. Process. Lett. 37(1), 27–35 (1991). doi:10.1016/0020-0190(91)90246-E

    Article  MathSciNet  MATH  Google Scholar 

  21. Katrenič, J., Semanišin, G.: Maximum semi-matching problem in bipartite graphs. Discuss. Math. Graph Theory 33(3), 559–569 (2013). doi:10.7151/dmgt.1694

    Article  MathSciNet  MATH  Google Scholar 

  22. Köhler, S., Turau, V., Mentges, G.: Self-stabilizing local \(k\)-placement of replicas with minimal variance. In: 14th SSS, pp. 16–30 (2012). doi:10.1007/978-3-642-33536-5_2

  23. Koufogiannakis, C., Young, N.E.: Distributed algorithms for covering, packing and maximum weighted matching. Distrib. Comput. 24(1), 45–63 (2011). doi:10.1007/s00446-011-0127-7

    Article  MATH  Google Scholar 

  24. Lotker, Z., Patt-Shamir, B., Pettie, S.: Improved distributed approximate matching. J. ACM 62(5), 38:1–38:17 (2015). doi:10.1145/2786753

    Article  MathSciNet  Google Scholar 

  25. Low, C.P.: An approximation algorithm for the load-balanced semi-matching problem in weighted bipartite graphs. Inform. Process. Lett. 100(4), 154–161 (2006). doi:10.1016/j.ipl.2006.06.004

    Article  MathSciNet  MATH  Google Scholar 

  26. Panconesi, A., Rizzi, R.: Some simple distributed algorithms for sparse networks. Distrib. Comput. 14(2), 97–100 (2001). doi:10.1007/PL00008932

    Article  Google Scholar 

  27. Patt-Shamir, B., Rawitz, D., Scalosub, G.: Distributed approximation of cellular coverage. J. Parallel Distrib. Comput. 72(3), 402–408 (2012). doi:10.1016/j.jpdc.2011.12.003

    Article  MATH  Google Scholar 

  28. Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. Society for Industrial and Applied Mathematics, Philadelphia (2000)

  29. Shiloach, Y.: Another look at the degree constrained subgraph problem. Inform. Process. Lett. 12(2), 89–92 (1981). doi:10.1016/0020-0190(81)90009-0

    Article  MathSciNet  MATH  Google Scholar 

  30. Vöcking, B.: Selfish load balancing. In: Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V. (eds.) Algorithmic Game Theory, pp. 699–716. Cambridge University Press, Cambridge (2007)

Download references

Author information

Authors and Affiliations

  1. Reykjavik University, Reykjavík, Iceland

    Magnús M. Halldórsson

  2. University of Freiburg, Freiburg, Germany

    Sven Köhler

  3. Tel Aviv University, Tel Aviv, Israel

    Boaz Patt-Shamir

  4. Bar-Ilan University, Ramat Gan, Israel

    Dror Rawitz

Authors
  1. Magnús M. Halldórsson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sven Köhler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Boaz Patt-Shamir
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dror Rawitz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sven Köhler.

Additional information

A preliminary version was presented at the 27th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA) 2015.

Magnús M. Halldórsson supported in part by Icelandic Research Fund (Grants Nos. 120032011 and 152679-051).

Sven Köhler supported in part by the Sustainability Center Freiburg, Germany, which is a cooperation of the Fraunhofer Society and the University of Freiburg and is supported by Grants from the Baden-Württemberg Ministry of Economics and the Baden-Württemberg Ministry of Science, Research and the Arts.

Boaz Patt-Shamir supported in part by the Ministry of Science, Technology and Space, Israel (Grant No. 3-10996) and the Israel Science Foundation (Grants No. 1444/14).

Dror Rawitz supported in part by the Ministry of Science, Technology and Space, Israel (Grant No. 3-10996) and the Israel Science Foundation (Grant No. 497/14).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Halldórsson, M.M., Köhler, S., Patt-Shamir, B. et al. Distributed backup placement in networks. Distrib. Comput. 31, 83–98 (2018). https://doi.org/10.1007/s00446-017-0299-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00446-017-0299-x

Search

Navigation