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Impact of Knowledge on Election Time in Anonymous Networks

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Abstract

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.

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References

  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. 547–556

  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 the 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 the 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 the 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 (1980)

  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. 16–28

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  16. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  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 the 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)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. 51–57

  32. 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 

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

    Article  MATH  Google Scholar 

  34. 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 

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

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

    MATH  Google Scholar 

  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. 377–386

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  41. Peleg, D.: Time-optimal leader election in general networks. J. Parallel Distrib. Comput. 8, 96–99 (1990)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. 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 

  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. 303–314

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

    Article  Google Scholar 

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Authors and Affiliations

  1. MIS, Université de Picardie Jules Verne Amiens, Amiens, France

    Yoann Dieudonné

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

    Andrzej Pelc

Authors
  1. Yoann Dieudonné
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  2. Andrzej Pelc
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Correspondence to Yoann Dieudonné.

Additional information

A preliminary version of this paper appeared in Proc. 29th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2017). Yoann Dieudonné is partially supported by the European Regional Development Fund (ERDF) and the Hauts-de-France region under Project TOREDY. Andrzej Pelc is partially supported by NSERC discovery Grant 8136 – 2013, and by the Research Chair in Distributed Computing at the Université du Québec en Outaouais.

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Dieudonné, Y., Pelc, A. Impact of Knowledge on Election Time in Anonymous Networks. Algorithmica 81, 238–288 (2019). https://doi.org/10.1007/s00453-018-0444-3

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