, Volume 81, Issue 5, pp 1901–1920 | Cite as

Deterministic Leader Election Takes \(\Theta (D + \log n)\) Bit Rounds

  • A. CasteigtsEmail author
  • Y. Métivier
  • J. M. Robson
  • A. Zemmari


Leader election is, together with consensus, one of the most central problems in distributed computing. This paper presents a distributed algorithm, called \(\mathcal{STT}\) , for electing deterministically a leader in an arbitrary network, assuming processors have unique identifiers of size \(O(\log n)\), where n is the number of processors. It elects a leader in \(O(D +\log n)\) rounds, where D is the diameter of the network, with messages of size O(1). Thus it has a bit round complexity of \(O(D +\log n)\). This substantially improves upon the best known algorithm whose bit round complexity is \(O(D\log n)\). In fact, using the lower bound by Kutten et al. (J ACM 62(1):7:1–7:27, 2015) and Kutten et al. (Theor Comput Sci 561:134–143, 2015) and a result of Dinitz and Solomon (Theor Comput Sci 384(2–3):168–183, 2007), we show that the bit round complexity of \(\mathcal{STT}\) is optimal (up to a constant factor), which is a significant step forward in understanding the interplay between time and message optimality for the election problem. Our algorithm requires no knowledge on the graph such as n or D, and the pipelining technique we introduce to break the \(O(D\log n)\) barrier is general.



We thank the anonymous referees for their many helpful comments on an earlier version of this article.


  1. 1.
    Afek, Y., Matias, Y.: Elections in anonymous networks. Inf. Comput. 113(2), 312–330 (1994)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Angluin, D.: Local and global properties in networks of processors. In: Proceedings of the 12th Symposium on Theory of Computing, pp. 82–93 (1980)Google Scholar
  3. 3.
    Attiya, H., Snir, M., Warmuth, M.K.: Computing on an anonymous ring. J. ACM 35(4), 845–875 (1988)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Attiya, H., Welch, J.: Distributed Computing: Fundamentals, Simulations, and Advanced Topics. Wiley, Hoboken (2004)zbMATHGoogle Scholar
  5. 5.
    Awerbuch, B.: Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems (detailed summary). In: Proc. of 19th Symp. on Theory of Computing, 1987, New York, USA, pp. 230–240 (1987)Google Scholar
  6. 6.
    Bar-Noy, A., Naor, J., Naor, M.: One-bit algorithms. Distrib. Comput. 4, 3–8 (1990)MathSciNetGoogle Scholar
  7. 7.
    Bodlaender, H.L., Moran, S., Warmuth, M.K.: The distributed bit complexity of the ring: from the anonymous case to the non-anonymous case. Inf. comput. 114(2), 34–50 (1994)zbMATHGoogle Scholar
  8. 8.
    Bodlaender, H.L., Tel, G.: Bit-optimal election in synchronous rings. Inf. Process. Lett. 36(1), 53–56 (1990)zbMATHGoogle Scholar
  9. 9.
    Boldi, P., Codenotti, B., Gemmell, P., Shammah, S., Simon, J., Vigna, S.: Symmetry breaking in anonymous networks: characterizations. In: Proc. 4th Israeli Symposium on Theory of Computing and Systems, pp. 16–26. IEEE Press (1996)Google Scholar
  10. 10.
    Boldi, P., Vigna, S.: Fibrations of graphs. Discrete Math. 243, 21–66 (2002)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Burns, J.E.: A Formal Model for Message Passing Systems. Computer Science Department, Indiana University, Bloomington (1980)Google Scholar
  12. 12.
    Casteigts, A., Métivier, Y., Robson, J.M., Zemmari, A.: Deterministic leader election in \(O(D + \log n)\) time with messages of size \(O(1)\). In: 30th Int. Symp. on Distributed Computing (DISC) (2016)Google Scholar
  13. 13.
    Chalopin, J.: Local computations on closed unlabelled edges: the election problem and the naming problem (extended abstract). In: Proc. of 31 st Conference on Current Trends in Theory and Practice of Informatics, SOFSEM’04, number 3381 in LNCS, pp. 81–90 (2005)Google Scholar
  14. 14.
    Chalopin, J., Godard, E., Métivier, Y.: Election in partially anonymous networks with arbitrary knowledge in message passing systems. Distrib. Comput. 25(4), 297–311 (2012)zbMATHGoogle Scholar
  15. 15.
    Chalopin, J., Métivier, Y.: An efficient message passing election algorithm based on Mazurkiewicz’s algorithm. Fundam. Inform. 80(1–3), 221–246 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Chalopin, J., Métivier, Y.: On the power of synchronization between two adjacent processes. Distrib. Comput. 23, 177–196 (2010)zbMATHGoogle Scholar
  17. 17.
    Dieudonné, Y., Pelc, A.: Impact of knowledge on election time in anonymous networks. arXiv preprint arXiv:1604.05023 (2016)
  18. 18.
    Dinitz, Y., Moran, S., Rajsbaum, S.: Bit complexity of breaking and achieving symmetry in chains and rings. J. ACM (2008). MathSciNetzbMATHGoogle Scholar
  19. 19.
    Dinitz, Y., Solomon, N.: Two absolute bounds for distributed bit complexity. Theor. Comput. Sci. 384(2–3), 168–183 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Förster, K.-T., Seidel, J., Wattenhofer, R.: Deterministic leader election in multi-hop beeping networks (extended abstract). In: Distributed Computing—28th International Symposium, DISC 2014, Austin, TX, USA, October 12–15, 2014. Proceedings, pp. 212–226 (2014)Google Scholar
  21. 21.
    Fusco, E.G., Pelc, A.: Knowledge, level of symmetry, and time of leader election. Distrib. Comput. 28(4), 221–232 (2015)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gallager, R.G.: Finding a leader in a network with \(o(e + n\log n)\) messages. Technical Report Internal Memo., M.I.T., Cambridge, MA (1979)Google Scholar
  23. 23.
    Gallager, R.G., Humblet, P.A., Spira, PhM: A distributed algorithm for minimum-weight spanning trees. ACM Trans. Program. Lang. Syst. 5(1), 66–77 (1983)zbMATHGoogle Scholar
  24. 24.
    Gilbert, S., Robinson, P., Sourav, S.: Leader election in well-connected graphs. In: Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, pp. 227–236. ACM (2018)Google Scholar
  25. 25.
    Glacet, C., Miller, A., Pelc, A.: Time versus information tradeoffs for leader election in anonymous trees. ACM Trans Algorithms (TALG) 13(3), 31 (2017)zbMATHGoogle Scholar
  26. 26.
    Godard, E., Métivier, Y., Muscholl, A.: Characterization of classes of graphs recognizable by local computations. Theory Comput. Syst. 37, 249–293 (2004)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Itai, A., Rodeh, M.: Symmetry breaking in distributed networks. Inf. Comput. 88(1), 60–87 (1990)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Korach, E., Kutten, S., Moran, S.: A modular technique for the design of efficient distributed leader finding algorithms. ACM Trans. Program. Lang. Syst. 12(1), 84–101 (1990)Google Scholar
  29. 29.
    Kothapalli, K., Onus, M., Scheideler, C., Schindelhauer, C.: Distributed coloring in \({O}(\sqrt{\log n})\) bit rounds. In: 20th Int. Parallel and Distributed Processing Symposium (IPDPS), Rhodes Island, Greece. IEEE (2006)Google Scholar
  30. 30.
    Krzywdziński, K., Kowalski, D.: On the complexity of distributed bfs in ad hoc networks with non-spontaneous wake-ups. Discrete Math. Theor. Comput. Sci. 15, 101–118 (2013)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  32. 32.
    Kutten, S., Pandurangan, G., Peleg, D., Robinson, P., Trehan, A.: On the complexity of universal leader election. J. ACM 62(1), 7:1–7:27 (2015)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Kutten, S., Pandurangan, G., Peleg, D., Robinson, P., Trehan, A.: Sublinear bounds for randomized leader election. Theor. Comput. Sci. 561, 134–143 (2015)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Lavault, C.: Evaluation des Algorithmes Distribués. Hermès, Paris (1995)zbMATHGoogle Scholar
  35. 35.
    LeLann, G.: Distributed systems: towards a formal approach. In: Gilchrist, B. (ed.) Information Processing ’77, pp. 155–160. North-Holland (1977)Google Scholar
  36. 36.
    Lynch, N.A.: Distributed Algorithms. Morgan Kaufman, San Francisco (1996)zbMATHGoogle Scholar
  37. 37.
    Mazurkiewicz, A.: Distributed enumeration. Inf. Process. Lett. 61(5), 233–239 (1997)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Métivier, Y., Robson, J.M., Zemmari, A.: Analysis of fully distributed splitting and naming probabilistic procedures and applications. Theor. Comput. Sci. 584, 115–130 (2015)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Miller, A., Pelc, A.: Election versus selection: How much advice is needed to find the largest node in a graph? In: Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures, pp. 377–386. ACM (2016)Google Scholar
  40. 40.
    Peleg, D.: Time-optimal leader election in general networks. J. Parallel Distrib. Comput. 8(1), 96–99 (1990)Google Scholar
  41. 41.
    Rosen, K.H. (ed.): Handbook of Discrete and Combinatorial Mathematics. CRC Press, Boca Raton (2000)zbMATHGoogle Scholar
  42. 42.
    Santoro, N.: On the message complexity of distributed problems. Int. J. Parallel Program. 13(3), 131–147 (1984)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Santoro, N.: Design and Analysis of Distributed Algorithm. Wiley, Hoboken (2007)zbMATHGoogle Scholar
  44. 44.
    Schieber, B., Snir, M.: Calling names on nameless networks. Inf. Comput. 113(1), 80–101 (1994)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Schneider, J., Wattenhofer, R.: Trading bit, message, and time complexity of distributed algorithms. In: Distributed Computing—25th International Symposium, DISC 2011, Rome, Italy, September 20–22, 2011. Proceedings, pp. 51–65 (2011)Google Scholar
  46. 46.
    Segall, A.: Distributed network protocols. IEEE Trans. Inf. Theory 29(1), 23–34 (1983)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Tanenbaum, A., van Steen, M.: Distributed Systems—Principles and Paradigms. Prentice Hall (2002)Google Scholar
  48. 48.
    Tel, G.: Introduction to Distributed Algorithms. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  49. 49.
    Yamashita, M., Kameda, T.: Computing on an anonymous network. In: PODC, pp. 117–130 (1988)Google Scholar
  50. 50.
    Yamashita, M., Kameda, T.: Computing on anonymous networks: Part i—characterizing the solvable cases. IEEE Trans. Parallel Distrib. Syst. 7(1), 69–89 (1996)Google Scholar
  51. 51.
    Yao, A.C.: Some complexity questions related to distributed computing. In: Proc. of 11th Symp. on Theory of computing (STOC), pp. 209–213. ACM Press (1979)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université de Bordeaux - Bordeaux INP LaBRI, UMR CNRS 5800TalenceFrance

Personalised recommendations