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Knowledge, level of symmetry, and time of leader election

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Abstract

We study the time needed for deterministic leader election in the \(\mathcal{LOCAL}\) model, where in every round a node can exchange any messages with its neighbors and perform any local computations. The topology of the network is unknown and nodes are unlabeled, but ports at each node have arbitrary fixed labelings which, together with the topology of the network, can create asymmetries to be exploited in leader election. We consider two versions of the leader election problem: strong LE in which exactly one leader has to be elected, if this is possible, while all nodes must terminate declaring that leader election is impossible otherwise, and weak LE, which differs from strong LE in that no requirement on the behavior of nodes is imposed, if leader election is impossible. We show that the time of leader election depends on three parameters of the network: its diameter \(D\), its size \(n\), and its level of symmetry \(\lambda \), which, when leader election is feasible, is the smallest depth at which some node has a unique view of the network. It also depends on the knowledge by the nodes, or lack of it, of parameters \(D\) and \(n\).

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Notes

  1. Notice that there is no solvable graph with \(D=1\) and \(\lambda =0\), because the latter condition, for solvable graphs, means that there is a node of a unique degree, contradicting the requirement \(D=1\), i.e., having a clique of at least 2 nodes.

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Authors and Affiliations

  1. Department of Computer, Control, and Management Engineering “Antonio Ruberti”, Sapienza University of Rome, Via Ariosto 25, 00185, Rome, Italy

    Emanuele G. Fusco

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

    Andrzej Pelc

Authors
  1. Emanuele G. Fusco
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  2. Andrzej Pelc
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Correspondence to Emanuele G. Fusco.

Additional information

Andrzej Pelc was partially supported by NSERC discovery Grant and by the Research Chair in Distributed Computing at the Université du Québec en Outaouais.

A preliminary version of this paper appeared in the Proc. of the 20th Annual European Symposium on Algorithms (ESA 2012), LNCS 7501, 479–490.

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Fusco, E.G., Pelc, A. Knowledge, level of symmetry, and time of leader election. Distrib. Comput. 28, 221–232 (2015). https://doi.org/10.1007/s00446-014-0237-0

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