Loosely-stabilizing leader election with polylogarithmic convergence time

Abstract

A loosely-stabilizing leader election protocol with polylogarithmic convergence time in the population protocol model is presented in this paper. In the population protocol model, which is a common abstract model of mobile sensor networks, it is known to be impossible to design a self-stabilizing leader election protocol. Thus, in our prior work, we introduced the concept of loose-stabilization, which is weaker than self-stabilization but has similar advantage as selfstabilization in practice. Following this work, several loosely-stabilizing leader election protocols are presented. The loosely-stabilizing leader election guarantees that, starting from an arbitrary configuration, the system reaches a safe configuration with a single leader within a relatively short time, and keeps the unique leader for an sufficiently long time thereafter. The convergence times of all the existing loosely-stabilizing protocols, i.e., the expected time to reach a safe configuration, are polynomial in n where n is the number of nodes (while the holding times to keep the unique leader are exponential in n). In this paper, a loosely-stabilizing protocol with polylogarithmic convergence time is presented. Its holding time is not exponential, but arbitrarily large polynomial in n.

Loosely Stabilizing Leader Election on Arbitrary Graphs in Population Protocols without Identifiers or Random Numbers

“…, we have Pr(∃i ≤ mt epi /2, C i ∈ T one ∩T half ∩V zero ∩L exist ) ≥ 1−o(1). We show the analyses for those four steps as Lemmas 12,14,15,and 16. In total, these lemmas together show that an execution of P AR starting from any configuration reaches S AR within 4(mt epi + mn 2 d log 2 n) interactions with probability 1 − o(1) ⊂ Ω(1), i.e., we obtain (4).…”

“…However, we believe that anonymity is still an important assumption: assigning distinct identifiers to a huge number of agents is not an easy task, and memory corruption may cause conflicts among their identifiers. Actually, many works assume anonymity and agent memory space of O(log n) or more (e.g., [4]- [7], [13], [14], [16], [21], [22]). In this paper, we analyze the time complexities for undirected graphs for simplicity; however, it works on any directed graphs without modifications.…”

“…Each token has an epidemic timer (timer E ). The epidemic timer is decremented by one every time the token moves (Lines [16][17], and thus, it eventually becomes zero. Note that the number of tokens never becomes zero once a token exists because the number of tokens decreases only when two tokens meet at an interaction.…”

Time-Optimal Self-Stabilizing Leader Election on Rings in Population Protocols

“…Consequently, many studies have been devoted to self-stabilizing population protocols [3]- [6], [9]- [12], [14]- [19]. For example, Angluin et al [3] proposed self-stabilizing protocols for a variety of problems, i.e., leader election in rings, token circulation in rings with a pre-selected leader, 2-hop coloring in degreebounded graphs, consistent global orientation in undirected rings, and spanning tree construction in regular graphs.…”

Time-Optimal Leader Election in Population Protocols

“…Agents are strongly anonymous: they do not have identifiers and they cannot distinguish their neighbors with the same states. As with the majority of studies on population protocols [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], we assume that the network of agents is a complete graph and that the scheduler selects an interacting pair of agents at each step uniformly at random.…”