Gathering with a strong team in weakly Byzantine environments

TL;DR

This paper improves gathering algorithms for mobile agents in networks by reducing time complexity when the team has fewer Byzantine agents, using new algorithms that leverage the number of agents and ID lengths.

Contribution

It introduces two new algorithms for gathering with fewer Byzantine agents, achieving faster convergence and simultaneous termination under certain conditions.

Findings

First algorithm achieves non-simultaneous gathering in O((f+|Λ_good|)·X(N)) rounds.

Second algorithm achieves simultaneous gathering in O((f+|Λ_all|)·X(N)) rounds.

Significant reduction in time complexity when |Λ_all|=O(|Λ_good|) and n is known.

Abstract

We study the gathering problem requiring a team of mobile agents to gather at a single node in arbitrary networks. The team consists of $k$ agents with unique identifiers (IDs), and $f$ of them are weakly Byzantine agents, which behave arbitrarily except falsifying their identifiers. The agents move in synchronous rounds and cannot leave any information on nodes. If the number of nodes $n$ is given to agents, the existing fastest algorithm tolerates any number of weakly Byzantine agents and achieves gathering with simultaneous termination in $O(n^4\cdot|\Lambda_{good}|\cdot X(n))$ rounds, where $|\Lambda_{good}|$ is the length of the maximum ID of non-Byzantine agents and $X(n)$ is the number of rounds required to explore any network composed of $n$ nodes. In this paper, we ask the question of whether we can reduce the time complexity if we have a strong team, i.e., a team with a few Byzantine agents, because not so many agents are subject to faults in practice. We give a positive answer to this question by proposing two algorithms in the case where at least $4f^2+9f+4$ agents exist. Both the algorithms take the upper bound $N$ of $n$ as input. The first algorithm achieves gathering with non-simultaneous termination in $O((f+|\Lambda_{good}|)\cdot X(N))$ rounds. The second algorithm achieves gathering with simultaneous termination in $O((f+|\Lambda_{all}|)\cdot X(N))$ rounds, where $|\Lambda_{all}|$ is the length of the maximum ID of all agents. The second algorithm significantly reduces the time complexity compared to the existing one if $n$ is given to agents and $|\Lambda_{all}|=O(|\Lambda_{good}|)$ holds.