, Volume 81, Issue 1, pp 238–288 | Cite as

Impact of Knowledge on Election Time in Anonymous Networks

  • Yoann DieudonnéEmail author
  • Andrzej Pelc


Leader election is one of the basic problems in distributed computing. This is a symmetry breaking problem: all nodes of a network must agree on a single node, called the leader. If the nodes of the network have distinct labels, then such an agreement means that all nodes have to output the label of the elected leader. For anonymous networks, the task of leader election is formulated as follows: every node v of the network must output a simple path, which is coded as a sequence of port numbers, such that all these paths end at a common node, the leader. In this paper, we study deterministic leader election in arbitrary anonymous networks. It is well known that deterministic leader election is impossible in some networks, regardless of the allocated amount of time, even if nodes know the map of the network. This is due to possible symmetries in it. However, even in networks in which it is possible to elect a leader knowing the map, the task may be still impossible without any knowledge, regardless of the allocated time. On the other hand, for any network in which leader election is possible knowing the map, there is a minimum time, called the election index, in which this can be done. Informally, the election index of a network is the minimum depth at which views of all nodes are distinct. Our aim is to establish tradeoffs between the allocated time \(\tau \) and the amount of information that has to be given a priori to the nodes to enable leader election in time \(\tau \) in all networks for which leader election in this time is at all possible. 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. For a given time \(\tau \) allocated to leader election, we give upper and lower bounds on the minimum size of advice sufficient to perform leader election in time \(\tau \). We focus on the two sides of the time spectrum. For the smallest possible time, which is the election index of the network, we show that the minimum size of advice is linear in the size n of the network, up to polylogarithmic factors. On the other hand, we consider large values of time: larger than the diameter D by a summand, respectively, linear, polynomial, and exponential in the election index; for these values, we prove tight bounds on the minimum size of advice, up to multiplicative constants. We also show that constant advice is not sufficient for leader election in all graphs, regardless of the allocated time.


Leader election Anonymous network Advice Deterministic distributed algorithm Time 


  1. 1.
    Abiteboul, S., Kaplan, H., Milo, T.: Compact labeling schemes for ancestor queries. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), pp. 547–556Google Scholar
  2. 2.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: Data Structures and Algorithms. Addison-Wesley, Boston (1983)zbMATHGoogle Scholar
  3. 3.
    Angluin, D.: Local and global properties in networks of processors. In: Proceedings of the 12th Annual ACM Symposium on Theory of Computing (STOC 1980), pp. 82–93Google Scholar
  4. 4.
    Attiya, H., Snir, M.: Better computing on the anonymous ring. J. Algorithms 12, 204–238 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Attiya, H., Snir, M., Warmuth, M.: Computing on an anonymous ring. J. ACM 35, 845–875 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Boldi, P., Shammah, S., Vigna, S., Codenotti, B., Gemmell, P., Simon, J.: Symmetry breaking in anonymous networks: characterizations. In: Proceedings of the 4th Israel Symposium on Theory of Computing and Systems (ISTCS 1996), pp. 16–26Google Scholar
  7. 7.
    Boldi, P., Vigna, S.: Computing anonymously with arbitrary knowledge. In: Proceedings of the 18th ACM Symposium on Principles of Distributed Computing (PODC 1999), pp. 181–188Google Scholar
  8. 8.
    Burns, J.E.: A formal model for message passing systems. Tech. Report TR-91, Computer Science Department, Indiana University, Bloomington (1980)Google Scholar
  9. 9.
    Casteigts, A., Métivier, Y., Robson, J., Zemmari, A.: Deterministic leader election in O(D + log n) time with messages of size O(1). In: Proceedings of the 30th International Symposium on Distributed Computing (DISC 2016), pp. 16–28Google Scholar
  10. 10.
    Dereniowski, D., Pelc, A.: Drawing maps with advice. J. Parallel Distrib. Comput. 72, 132–143 (2012)CrossRefzbMATHGoogle Scholar
  11. 11.
    Dereniowski, D., Pelc, A.: Leader election for anonymous asynchronous agents in arbitrary networks. Distrib. Comput. 27, 21–38 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Derhab, A., Badache, N.: A self-stabilizing leader election algorithm in highly dynamic ad hoc mobile networks. IEEE Trans. Parallel Distrib. Syst. 19, 926939 (2008)CrossRefzbMATHGoogle Scholar
  13. 13.
    Dobrev, S., Pelc, A.: Leader election in rings with nonunique labels. Fundam. Inform. 59, 333–347 (2004)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Emek, Y., Fraigniaud, P., Korman, A., Rosen, A.: Online computation with advice. Theor. Comput. Sci. 412, 2642–2656 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Flocchini, P., Kranakis, E., Krizanc, D., Luccio, F.L., Santoro, N.: Sorting and election in anonymous asynchronous rings. J. Parallel Distrib. Comput. 64, 254–265 (2004)CrossRefzbMATHGoogle Scholar
  16. 16.
    Fraigniaud, P., Gavoille, C., Ilcinkas, D., Pelc, A.: Distributed computing with advice: information sensitivity of graph coloring. Distrib. Comput. 21, 395–403 (2009)CrossRefzbMATHGoogle Scholar
  17. 17.
    Fraigniaud, P., Ilcinkas, D., Pelc, A.: Communication algorithms with advice. J. Comput. Syst. Sci. 76, 222–232 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fraigniaud, P., Ilcinkas, D., Pelc, A.: Tree exploration with advice. Inf. Comput. 206, 1276–1287 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fraigniaud, P., Korman, A., Lebhar, E.: Local MST computation with short advice. Theory Comput. Syst. 47, 920–933 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fredrickson, G.N., Lynch, N.A.: Electing a leader in a synchronous ring. J. ACM 34, 98–115 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fusco, E., Pelc, A.: Knowledge, level of symmetry, and time of leader election. Distrib. Comput. 28, 221–232 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fusco, E., Pelc, A.: Trade-offs between the size of advice and broadcasting time in trees. Algorithmica 60, 719–734 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fusco, E., Pelc, A., Petreschi, R.: Topology recognition with advice. Inf. Comput. 247, 254–265 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gavoille, C., Peleg, D., Pérennes, S., Raz, R.: Distance labeling in graphs. J. Algorithms 53, 85–112 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 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 the 6th ACS/IEEE International Conference on Computer Systems and Applications (AICCSA 2008), pp. 473–480Google Scholar
  27. 27.
    Hendrickx, J.: Views in a graph: to which depth must equality be checked? IEEE Trans. Parallel Distrib. Syst. 25, 1907–1912 (2014)CrossRefGoogle Scholar
  28. 28.
    Hirschberg, D.S., Sinclair, J.B.: Decentralized extrema-finding in circular configurations of processes. Commun. ACM 23, 627–628 (1980)CrossRefzbMATHGoogle Scholar
  29. 29.
    Ilcinkas, D., Kowalski, D., Pelc, A.: Fast radio broadcasting with advice. Theor. Comput. Sci. 411, 1544–1557 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ingram, R., Radeva, T., Shields, P., Viqar, S., Walter, J.E., Welch, J.L.: A leader election algorithm for dynamic networks with causal clocks. Distrib. Comput. 26, 75–97 (2013)CrossRefzbMATHGoogle Scholar
  31. 31.
    Jurdzinski, T., Kutylowski, M., Zatopianski, J.: Efficient algorithms for leader election in radio networks. In: Proceedings of the 21st ACM Symposium on Principles of Distributed Computing (PODC 2002), pp. 51–57Google Scholar
  32. 32.
    Katz, M., Katz, N., Korman, A., Peleg, D.: Labeling schemes for flow and connectivity. SIAM J. Comput. 34, 23–40 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Korman, A., Kutten, S., Peleg, D.: Proof labeling schemes. Distrib. Comput. 22, 215–233 (2010)CrossRefzbMATHGoogle Scholar
  34. 34.
    Kowalski, D., Pelc, A.: Leader election in ad hoc radio networks: a keen ear helps. J. Comput. Syst. Sci. 79, 1164–1180 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Le Lann, G.: Distributed systems—towards a formal approach. In: Proceedings of the IFIP Congress, North Holland, pp. 155–160 (1977)Google Scholar
  36. 36.
    Lynch, N.L.: Distributed Algorithms. Morgan Kaufmann Publ. Inc., San Francisco (1996)zbMATHGoogle Scholar
  37. 37.
    Miller, A., Pelc, A.: Election vs. selection: two ways of finding the largest node in a graph. In: Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2016), pp. 377–386Google Scholar
  38. 38.
    Nakano, K., Olariu, S.: Uniform leader election protocols for radio networks. IEEE Trans. Parallel Distrib. Syst. 13, 516–526 (2002)CrossRefGoogle Scholar
  39. 39.
    Nisse, N., Soguet, D.: Graph searching with advice. Theor. Comput. Sci. 410, 1307–1318 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  41. 41.
    Peleg, D.: Time-optimal leader election in general networks. J. Parallel Distrib. Comput. 8, 96–99 (1990)CrossRefGoogle Scholar
  42. 42.
    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)CrossRefzbMATHGoogle Scholar
  43. 43.
    Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52, 1–24 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Willard, D.E.: Log-logarithmic selection resolution protocols in a multiple access channel. SIAM J. Comput. 15, 468–477 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Yamashita, M., Kameda, T.: Electing a leader when processor identity numbers are not distinct. In: Proceedings of the 3rd Workshop on Distributed Algorithms (WDAG 1989), LNCS, vol. 392, pp. 303–314Google Scholar
  46. 46.
    Yamashita, M., Kameda, T.: Computing on anonymous networks: part I—characterizing the solvable cases. IEEE Trans. Parallel Distrib. Syst. 7, 69–89 (1996)CrossRefGoogle Scholar

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

Authors and Affiliations

  1. 1.MISUniversité de Picardie Jules Verne AmiensAmiensFrance
  2. 2.Département d’informatiqueUniversité du Québec en OutaouaisGatineauCanada

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