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Leader Election in Well-Connected Graphs

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Abstract

In this paper, we look at the problem of randomized leader election in synchronous distributed networks with a special focus on the message complexity. We provide an algorithm that solves the implicit version of leader election (where non-leader nodes need not be aware of the identity of the leader) in any general network with \(O(\sqrt{n} \log ^{7/2} n \cdot t_{mix})\) messages and in \(O(t_{mix}\log ^2 n)\) time, where n is the number of nodes and \(t_{mix}\) refers to the mixing time of a random walk in the network graph G. For several classes of well-connected networks (that have a large conductance or alternatively small mixing times e.g., expanders, hypercubes, etc), the above result implies extremely efficient (sublinear running time and messages) leader election algorithms. Correspondingly, we show that any substantial improvement is not possible over our algorithm, by presenting an almost matching lower bound for randomized leader election. We show that \(\varOmega (\sqrt{n}/\phi ^{3/4})\) messages are needed for any leader election algorithm that succeeds with probability at least \(1-o(1)\), where \(\phi \) refers to the conductance of a graph. To the best of our knowledge, this is the first work that shows a dependence between the time and message complexity to solve leader election and the connectivity of the graph G, which is often characterized by the graph’s conductance \(\phi \). Apart from the \(\varOmega (m)\) bound in Kutten et al. (J ACM 62(1):7:1–7:27, 2015) (where m denotes the number of edges of the graph), this work also provides one of the first non-trivial lower bounds for leader election in general networks.

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Notes

  1. Throughout the paper, we use the \(\tilde{O}\) notation to hide \(\log n\) factors.

  2. Our lower bound holds even if nodes start with unique IDs.

  3. This range guarantees that the chosen values are unique with high probability [31] (Chapter 4, page 72).

  4. We slightly abuse notation by saying a clique C sends a message when, in fact, some node in C performs the sending action.

References

  1. Afek, Y., Gafni, E.: Time and message bounds for election in synchronous and asynchronous complete networks. In: Proceedings of the Fourth Annual ACM Symposium on Principles of Distributed Computing, PODC ’85, pp. 186–195, New York, NY, USA. ACM (1985)

  2. Angluin, D.: Local and global properties in networks of processors (extended abstract). In: Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing, STOC ’80, pp. 82–93, New York, NY, USA. ACM (1980)

  3. Attiya, H., van Leeuwen, J., Santoro, N., Zaks, S.: Efficient elections in chordal ring networks. Algorithmica 4(1), 437–446 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Attiya, H., Welch, J.: Distributed Computing: Fundamentals, Simulations and Advanced Topics, 2nd edn. Wiley, New York (2004)

    Book  MATH  Google Scholar 

  5. Augustine, J., Pandurangan, G., Robinson, P.: Fast byzantine leader election in dynamic networks. In: Distributed Computing—29th International Symposium, DISC 2015, Tokyo, Japan, Oct 7–9, 2015, Proceedings, pp. 276–291 (2015)

  6. Awerbuch, B.: Optimal distributed algorithms for minimum weight spanning tree, counting, leader election, and related problems. In: Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, STOC ’87, pp. 230–240, New York, NY, USA. ACM (1987)

  7. Bollobás, B.: The isoperimetric number of random regular graphs. Eur. J. Comb. 9(3), 241–244 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bollobás, B.: Random Graphs. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  9. Brunekreef, J., Katoen, J.-P., Koymans, R., Mauw, S.: Design and analysis of dynamic leader election protocols in broadcast networks. Distrib. Comput. 9(4), 157 (1996)

    Article  MATH  Google Scholar 

  10. Casteigts, A., Métivier, Y., Robson, J.M., Zemmari, A.: Deterministic leader election takes \(\theta (d+\log n)\) bit rounds. Algorithmica 81(5), 1901–1920 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chang, E., Roberts, R.: An improved algorithm for decentralized extrema-finding in circular configurations of processes. Commun. ACM 22(5), 281–283 (1979)

    Article  MATH  Google Scholar 

  12. Chatterjee, S., Pandurangan, G., Robinson, P.: The complexity of leader election: a chasm at diameter two. In: Proceedings of the 19th International Conference on Distributed Computing and Networking, ICDCN ’18, pp. 13:1–13:10, New York, NY, USA. ACM (2018)

  13. Dieudonné, Y., Pelc, A.: Impact of knowledge on election time in anonymous networks. Algorithmica 81(1), 238–288 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fetzer, C., Cristian, F.: A highly available local leader election service. IEEE Trans. Softw. Eng. 25(5), 603–618 (1999)

    Article  Google Scholar 

  15. Flocchini, P., Mans, B.: Optimal elections in labeled hypercubes. J. Parallel Distrib. Comput. 33(1), 76–83 (1996)

    Article  Google Scholar 

  16. Frederickson, G.N., Lynch, N.A.: Electing a leader in a synchronous ring. J. ACM 34(1), 98–115 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fusco, E.G., Pelc, A.: Knowledge, level of symmetry, and time of leader election. Distrib. Comput. 28(4), 221–232 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gallager, R.G., Humblet, P.A., Spira, P.M.: A distributed algorithm for minimum-weight spanning trees. ACM Trans. Program. Lang. Syst. 5(1), 66–77 (1983)

    Article  MATH  Google Scholar 

  19. Giakkoupis, G.: Tight bounds for rumor spreading in graphs of a given conductance. In: Proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS), Mar 10–12, pp. 57–68 (2011)

  20. Gilbert, S., Kowalski, D.R.: Distributed agreement with optimal communication complexity. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’10, pp. 965–977, Philadelphia, PA, USA. Society for Industrial and Applied Mathematics (2010)

  21. Gilbert, S., Robinson, P., Sourav, S.: Leader election in well-connected graphs. In: Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, PODC ’18, pp. 227–236, New York, NY, USA. Association for Computing Machinery (2018)

  22. Gupta, I., van Renesse, R., Birman, K.P.: A probabilistically correct leader election protocol for large groups. In: Herlihy, M. (eds.) Distributed Computing, pp. 89–103, Berlin. Springer, Berlin (2000)

  23. Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Am. Math. Soc. 43(04), 439–562 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jerrum, M., Sinclair, A.: Conductance and the rapid mixing property for markov chains: the approximation of permanent resolved. In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing, STOC ’88, pp. 235–244, New York, NY, USA. ACM (1988)

  25. Karp, R., Schindelhauer, C., Shenker, S., Vocking, B.: Randomized rumor spreading. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, FOCS ’00, p. 565, Washington, DC, USA. IEEE Computer Society (2000)

  26. Korach, E., Moran, S., Zaks, S.: The optimality of distributive constructions of minimum weight and degree restricted spanning trees in a complete network of processors. SIAM J. Comput. 16(2), 231–236 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kowalski, D.R., Pelc, A.: Leader election in ad hoc radio networks: a keen ear helps. J. Comput. Syst. Sci. 79(7), 1164–1180 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kutten, S., Pandurangan, G., Peleg, D., Robinson, P., Trehan, A.: Sublinear bounds for randomized leader election. Theor. Comput. Sci. 561(PB), 134–143 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Le Lann, G.: Distributed systems: towards a formal approach. In: IFIP Congress, pp. 155–160 (1977)

  31. Lynch, N.A.: Distributed Algorithms. Morgan Kaufmann Publishers Inc., San Francisco (1996)

    MATH  Google Scholar 

  32. Malkhi, D., Reiter, M.K., Wool, A., Wright, R.N.: Probabilistic quorum systems. Inf. Comput. 170(2), 184–206 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, New York (2005)

    Book  MATH  Google Scholar 

  34. Molla, A.R., Pandurangan, G.: Distributed computation of mixing time. In: Proceedings of the 18th International Conference on Distributed Computing and Networking, ICDCN ’17, pp. 5:1–5:4, New York, NY, USA. ACM (2017)

  35. Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, New York (1995)

    Book  MATH  Google Scholar 

  36. Newport, C.: Leader election in a smartphone peer-to-peer network. In: 2017 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pp. 172–181 (2017)

  37. Overmars, M., Santoro, N.: Improved bounds for electing a leader in a synchronous ring. Algorithmica 18(2), 246–262 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  39. Peleg, D.: Time-optimal leader election in general networks. J. Parallel Distrib. Comput. 8(1), 96–99 (1990)

    Article  Google Scholar 

  40. Peres, Y., Sousi, P.: Mixing times are hitting times of large sets. J. Theor. Probab. 28(2), 488–519 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ramanathan, M.K., Ferreira, R.A., Jagannathan, S., Grama, A., Szpankowski, W.: Randomized leader election. Distrib. Comput. 19(5), 403–418 (2007)

    Article  MATH  Google Scholar 

  42. Ratnasamy, S., Francis, P., Handley, M., Karp, R., Shenker, S.: A scalable content-addressable network. SIGCOMM Comput. Commun. Rev. 31(4), 161–172 (2001)

    Article  MATH  Google Scholar 

  43. Rowstron, A.I.T., Druschel, P.: Pastry: scalable, decentralized object location, and routing for large-scale peer-to-peer systems. In: Proceedings of the IFIP/ACM International Conference on Distributed Systems Platforms Heidelberg, Middleware ’01, pp. 329–350, London, UK. Springer (2001)

  44. Sinclair, A.: Algorithms for Random Generation and Counting. Birkhauser, Boston (1993)

    MATH  Google Scholar 

  45. Singh, G.: Leader election in the presence of link failures. IEEE Trans. Parallel Distrib. Syst. 7(3), 231–236 (1996)

    Article  MathSciNet  Google Scholar 

  46. Sourav, S.: Latency, conductance, and the role of connectivity in graph problems. https://scholarbank.nus.edu.sg/handle/10635/162756 (2019)

  47. Zhao, B.Y., Huang, L., Stribling, J., Rhea, S.C., Joseph, A.D., Kubiatowicz, J.D.: Tapestry: a resilient global-scale overlay for service deployment. IEEE J. Sel. A. Commun. 22(1), 41–53 (2006)

    Article  Google Scholar 

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Acknowledgements

This research was supported by MOE Tier 2 (T2EP20122-0021) project: Data-Driven Distributed Algorithms.

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Authors and Affiliations

  1. National University of Singapore, Singapore, Singapore

    Seth Gilbert

  2. Augusta University, Augusta, USA

    Peter Robinson

  3. Singapore University of Technology and Design, Singapore, Singapore

    Suman Sourav

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  1. Seth Gilbert
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Correspondence to Suman Sourav.

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Preliminary results of this paper appeared in the proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, PODC 2018 [21] and in the academic thesis [46].

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Gilbert, S., Robinson, P. & Sourav, S. Leader Election in Well-Connected Graphs. Algorithmica 85, 1029–1066 (2023). https://doi.org/10.1007/s00453-022-01068-x

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