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

Abstract

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.

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Acknowledgements

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

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Correspondence to A. Casteigts.

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This research has been partially supported by ANR projects DESCARTES and ESTATE (resp. ANR-16-CE40-0023 and ANR-16-CE25-0009-03). A preliminary subset of this work appeared in the proceedings of DISC 2016 [12].

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Casteigts, A., Métivier, Y., Robson, J.M. et al. Deterministic Leader Election Takes \(\Theta (D + \log n)\) Bit Rounds. Algorithmica 81, 1901–1920 (2019). https://doi.org/10.1007/s00453-018-0517-3

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Keywords

  • Leader Election Problem
  • Round Complexity
  • High Identifiers
  • Spanning Tree
  • Deterministic Election