Self-Stabilizing Weakly Byzantine Perpetual Gathering of Mobile Agents

We study the \emph{Byzantine} gathering problem involving $k$ mobile agents with unique identifiers (IDs), $f$ of which are Byzantine. These agents start the execution of a common algorithm from (possibly different) nodes in an $n$-node network, potentially starting at different times. Once started, the agents operate in synchronous rounds. We focus on \emph{weakly} Byzantine environments, where Byzantine agents can behave arbitrarily but cannot falsify their IDs. The goal is for all \emph{non-Byzantine} agents to eventually terminate at a single node simultaneously. In this paper, we first prove two impossibility results: (1) for any number of non-Byzantine agents, no algorithm can solve this problem without global knowledge of the network size or the number of agents, and (2) no self-stabilizing algorithm exists if $k\le 2f$ even with $n$, $k$, $f$, and the length ${\mathrm{\Lambda }}_g$ of the largest ID among IDs of non-Byzantine agents, where the self-stabilizing algorithm enables agents to gather starting from arbitrary (inconsistent) initial states. Next, based on these results, we introduce a \emph{perpetual gathering} problem and propose a self-stabilizing algorithm for this problem. This problem requires that all non-Byzantine agents always be co-located from a certain time onwards. If the agents know ${\mathrm{\Lambda }}_g$ and upper bounds $N$, $K$, $F$ on $n$, $k$, $f$, the proposed algorithm works in $O(K\cdot F\cdot {\mathrm{\Lambda }}_g\cdot X(N))$ rounds, where $X(n)$ is the time required to visit all nodes in a $n$-nodes network. Our results indicate that while no algorithm can solve the original self-stabilizing gathering problem for any $k$ and $f$ even with \emph{exact} global knowledge of the network size and the number of agents, the self-stabilizing perpetual gathering problem can always be solved with just upper bounds on this knowledge.