Gathering despite a linear number of weakly Byzantine agents

We study the gathering problem to make multiple agents initially scattered in arbitrary networks gather at a single node. There exist $k$ agents with unique identifiers (IDs) in the network, and $f$ of them are weakly Byzantine agents, which behave arbitrarily except for falsifying their IDs. The agents behave in synchronous rounds, and each node does not have any memory like a whiteboard. In the literature, there exists a gathering algorithm that tolerates any number of Byzantine agents, while the fastest gathering algorithm requires $\Omega(f^2)$ non-Byzantine agents. This paper proposes an algorithm that solves the gathering problem efficiently with $\Omega(f)$ non-Byzantine agents since there is a large gap between the number of non-Byzantine agents in previous works. The proposed algorithm achieves the gathering in $O(f\cdot|\Lambda_{good}|\cdot X(N))$ rounds in case of $9f+8\leq k$ and simultaneous startup if $N$ is given to agents, where $|\Lambda_{good}|$ is the length of the largest ID among non-Byzantine agents, and $X(n)$ is the number of rounds required to explore any network composed of $n$ nodes. This algorithm is faster than the most fault-tolerant existing algorithm and requires fewer non-Byzantine agents than the fastest algorithm if $n$ is given to agents, although the guarantees on simultaneous termination and startup delay are not the same. To achieve this property, we propose a new technique to simulate a Byzantine consensus algorithm for synchronous message-passing systems on agent systems.