Skip to main content
SpringerLink
Log in

Four shades of deterministic leader election in anonymous networks

  • Published:
Distributed Computing Aims and scope Submit manuscript

Abstract

Leader election is one of the fundamental problems in distributed computing: a single node, called the leader, must be specified. This task can be formulated either in a weak way, where one node outputs leader and all other nodes output non-leader, or in a strong way, where all nodes must also learn which node is the leader. If the nodes of the network have distinct identifiers, then such an agreement means that all nodes have to output the identifier of the elected leader. For anonymous networks, the strong version of leader election requires that all nodes must be able to find a path to the leader, as this is the only way to identify it. In this paper, we study variants of deterministic leader election in arbitrary anonymous networks. Leader election is impossible in some anonymous networks, regardless of the allocated amount of time, even if nodes know the entire map of the network. This is due to possible symmetries in the network. However, even in networks in which it is possible to elect a leader knowing the map, the task may be still impossible without any initial knowledge, regardless of the allocated time. On the other hand, for any network in which leader election (weak or strong) is possible knowing the map, there is a minimum time, called the election index, in which this can be done. We consider four formulations of leader election discussed in the literature in the context of anonymous networks: one is the weak formulation, and the three others specify three different ways of finding the path to the leader in the strong formulation. Our aim is to compare the amount of initial information needed to accomplish each of these “four shades” of leader election in minimum time. Following the framework of algorithms with advice, this information (a single binary string) is provided to all nodes at the start by an oracle knowing the entire network. The length of this string is called the size of advice. We show that the size of advice required to accomplish leader election in the weak formulation in minimum time is exponentially smaller than that needed for any of the strong formulations. Thus, if the required amount of advice is used as a measure of the difficulty of the task, the weakest version of leader election in minimum time is drastically easier than any version of the strong formulation in minimum time.

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
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Notes

  1. All our results concerning problems PE, PPE and CPPE remain valid if the definitions are changed to additionally require that the simple paths from every node to the leader are branches of a fixed spanning tree rooted at the leader, and, in the case of the model PE, outputted ports lead from a node to its parent. Following such consecutive outputted ports guarantees obtaining a path from every node to the leader.

References

  1. Abiteboul, S., Kaplan, H., Milo, T.: Compact labeling schemes for ancestor queries. In: Proceeding of 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), pp. 547–556

  2. Aho, A.V., Hopcroft, J.E., Ullman, J.D.: Data Structures and Algorithms. Addison-Wesley, Boston (1983)

    MATH  Google Scholar 

  3. Angluin, D.: Local and global properties in networks of processors. In: Proceedings of 12th Annual ACM Symposium on Theory of Computing (STOC 1980), pp. 82–93

  4. Attiya, H., Snir, M.: Better computing on the anonymous Ring. J. Algorithms 12, 204–238 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Attiya, H., Snir, M., Warmuth, M.: Computing on an anonymous ring. J. ACM 35, 845–875 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boldi, P., Shammah, S., Vigna, S., Codenotti, B., Gemmell, P., Simon, J.: Symmetry breaking in anonymous networks: characterizations. In: Proceedings of 4th Israel Symposium on Theory of Computing and Systems, (ISTCS 1996), pp. 16–26

  7. Boldi, P., Vigna, S.: Computing anonymously with arbitrary knowledge. In: Proceedings of 18th ACM Symposium on Principles of Distributed Computing (PODC 1999), pp. 181–188

  8. Burns, J.E.: A formal model for message passing systems, Tech. Report TR-91, Computer Science Department, Indiana University, Bloomington, September 1980

  9. Dereniowski, D., Pelc, A.: Drawing maps with advice. J. Parallel Distribut. Comput. 72, 132–143 (2012)

    Article  MATH  Google Scholar 

  10. Dereniowski, D., Pelc, A.: Leader election for anonymous asynchronous agents in arbitrary networks. Distrib. Comput. 27, 21–38 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dieudonné, Y., Pelc, A.: Impact of knowledge on election time in anonymous networks. Algorithmica 81, 238–288 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dobrev, S., Pelc, A.: Leader election in rings with nonunique labels. Fund. Inform. 59, 333–347 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Emek, Y., Fraigniaud, P., Korman, A., Rosen, A.: Online computation with advice. Theoret. Comput. Sci. 412, 2642–2656 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Flocchini, P., Kranakis, E., Krizanc, D., Luccio, F.L., Santoro, N.: Sorting and election in anonymous asynchronous rings. J. Parallel Distribut. Comput. 64, 254–265 (2004)

    Article  MATH  Google Scholar 

  15. Fraigniaud, P., Gavoille, C., Ilcinkas, D., Pelc, A.: Distributed computing with advice: Information sensitivity of graph coloring. Distrib. Comput. 21, 395–403 (2009)

    Article  MATH  Google Scholar 

  16. Fraigniaud, P., Ilcinkas, D., Pelc, A.: Communication algorithms with advice. J. Comput. Syst. Sci. 76, 222–232 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fraigniaud, P., Ilcinkas, D., Pelc, A.: Tree exploration with advice. Inf. Comput. 206, 1276–1287 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fraigniaud, P., Korman, A., Lebhar, E.: Local MST computation with short advice. Theory Comput. Syst. 47, 920–933 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fredrickson, G.N., Lynch, N.A.: Electing a leader in a synchronous ring. J. ACM 34, 98–115 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fusco, E., Pelc, A.: How much memory is needed for leader election. Distrib. Comput. 24, 65–78 (2011)

    Article  MATH  Google Scholar 

  21. Fusco, E., Pelc, A.: Knowledge, level of symmetry, and time of leader election. Distrib. Comput. 28, 221–232 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fusco, E., Pelc, A.: Trade-offs between the size of advice and broadcasting time in trees. Algorithmica 60, 719–734 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fusco, E., Pelc, A., Petreschi, R.: Topology recognition with advice. Inf. Comput. 247, 254–265 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gavoille, C., Peleg, D., Pérennes, S., Raz, R.: Distance labeling in graphs. J. Algorithms 53, 85–112 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Glacet, C., Miller, A., Pelc, A.: Time vs. information tradeoffs for leader election in anonymous trees. ACM Trans. Algorithms 13, 31:1-31:41 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Haddar, M.A., Kacem, A.H., Métivier, Y., Mosbah, M., Jmaiel, M.: Electing a leader in the local computation model using mobile agents. In: Proceedings of 6th ACS/IEEE International Conference on Computer Systems and Applications (AICCSA 2008), pp. 473–480

  27. Hendrickx, J.: Views in a graph: To which depth must equality be checked? IEEE Trans. Parallel Distrib. Syst. 25, 1907–1912 (2014)

    Article  Google Scholar 

  28. Hirschberg, D.S., Sinclair, J.B.: Decentralized extrema-finding in circular configurations of processes. Commun. ACM 23, 627–628 (1980)

  29. Ilcinkas, D., Kowalski, D., Pelc, A.: Fast radio broadcasting with advice. Theoret. Comput. Sci. 411, 1544–1557 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Jurdzinski, T., Kutylowski, M., Zatopianski, J.: Efficient algorithms for leader election in radio networks. In: Proceedings of 21st ACM Symposium on Principles of Distributed Computing (PODC 2002), pp. 51–57

  31. Katz, M., Katz, N., Korman, A., Peleg, D.: Labeling schemes for flow and connectivity. SIAM J. Comput. 34, 23–40 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Korman, A., Kutten, S., Peleg, D.: Proof labeling schemes. Distrib. Comput. 22, 215–233 (2010)

    Article  MATH  Google Scholar 

  33. Kowalski, D., Pelc, A.: Leader election in ad hoc radio networks: A keen ear helps. J. Comput. Syst. Sci. 79, 1164–1180 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Le Lann, G.: Distributed systems—towards a formal approach. In: Proceedings of IFIP Congress, pp. 155–160, North Holland (1977)

  35. Lynch, N.L.: Distributed Algorithms. Morgan Kaufmann Publ. Inc., San Francisco (1996)

    MATH  Google Scholar 

  36. Miller, A., Pelc, A.: Election vs. selection: How much advice is needed to find the largest node in a graph? In: Proceedings of 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2016), pp. 377–386

  37. Nakano, K., Olariu, S.: Uniform leader election protocols for radio networks. IEEE Trans. Parallel Distrib. Syst. 13, 516–526 (2002)

    Article  Google Scholar 

  38. Nisse, N., Soguet, D.: Graph searching with advice. Theoret. Comput. Sci. 410, 1307–1318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Peleg, D.: Distributed Computing, A Locality-Sensitive Approach, SIAM Monographs on Discrete Mathematics and Applications, Philadelphia (2000)

  40. Peterson, G.L.: An \(O(n \log n)\) unidirectional distributed algorithm for the circular extrema problem. ACM Trans. Program. Lang. Syst. 4, 758–762 (1982)

    Article  MATH  Google Scholar 

  41. Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52, 1–24 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  42. Willard, D.E.: Log-logarithmic selection resolution protocols in a multiple access channel. SIAM J. Comput. 15, 468–477 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yamashita, M., Kameda, T.: Electing a leader when procesor identity numbers are not distinct. In: Proc. 3rd Workshop on Distributed Algorithms (WDAG 1989), LNCS 392, pp. 303–314

  44. Yamashita, M., Kameda, T.: Computing on anonymous networks: part I—characterizing the solvable cases. IEEE Trans. Parallel Distribut. Syst. 7, 69–89 (1996)

    Article  Google Scholar 

Download references

Funding

Barun Gorain acknowledges the support by the Science and Engineering Research Board (SERB), Department of Science and Technology, Govt. of India (Grant Number: CRG/2020/005964). Barun Gorain also acknowledges the support by SERB (Grant Number: MTR/2021/000118). Avery Miller acknowledges the support by NSERC Discovery Grant RGPIN–2017–05936. Andrzej Pelc acknowledges the support by NSERC Discovery Grant RGPIN–2018–03899 and by the Research Chair in Distributed Computing at the Université du Québec en Outaouais.

Author information

Authors and Affiliations

  1. Department of Computer Science, Indian Institute of Technology Bhilai, Raipur, Chhattisgarh, 492015, India

    Barun Gorain

  2. Department of Computer Science, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada

    Avery Miller

  3. Département d’informatique et d’ingénierie, Université du Québec en Outaouais, Gatineau, QC, J8X 3X7, Canada

    Andrzej Pelc

Authors
  1. Barun Gorain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Avery Miller
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrzej Pelc
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All the authors contributed equally in the manuscript.

Corresponding author

Correspondence to Barun Gorain.

Ethics declarations

Conflicts of interest

None of the authors have any competing interests in the manuscript.

Additional information

Publisher's Note

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

A preliminary version of this paper appeared in the Proceedings of the 33rd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2021).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gorain, B., Miller, A. & Pelc, A. Four shades of deterministic leader election in anonymous networks. Distrib. Comput. 36, 419–449 (2023). https://doi.org/10.1007/s00446-023-00451-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00446-023-00451-3

Keywords

Search

Navigation