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
Throughout the paper, we use the \(\tilde{O}\) notation to hide \(\log n\) factors.
Our lower bound holds even if nodes start with unique IDs.
This range guarantees that the chosen values are unique with high probability [31] (Chapter 4, page 72).
We slightly abuse notation by saying a clique C sends a message when, in fact, some node in C performs the sending action.
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This research was supported by MOE Tier 2 (T2EP20122-0021) project: Data-Driven Distributed Algorithms.
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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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DOI: https://doi.org/10.1007/s00453-022-01068-x