Knowledge, Level of Symmetry, and Time of Leader Election

TL;DR

This paper analyzes the time complexity of deterministic leader election in anonymous networks within the LOCAL model, revealing how network parameters like size, diameter, and symmetry influence the feasibility and duration of leader election.

Contribution

It introduces a detailed study of how network symmetry and knowledge of parameters affect leader election time in anonymous networks.

Findings

Leader election time depends on network diameter, size, and symmetry level.

Feasibility of leader election is characterized by the network's symmetry level.

Knowledge of network parameters by nodes influences leader election complexity.

Abstract

We study the time needed for deterministic leader election in the ${\cal LOCAL}$ 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 $n$.