A further study on weak Byzantine gathering of mobile agents

The gathering of mobile agents in the presence of Byzantine faults is first studied by Dieudonné et al. Authors provide a polynomial time algorithm handling any number of weak Byzantine agents in the presence of at least one good agent considering start-up delays, i.e., the good agents may not wake up at the same time. Hirose et al. [1] come up with an algorithm considering start-up delays that use a strong team of at least $4f^2+8f+4$ many good agents but runs much faster than that of Dieudonné et al. Later, Hirose et al. [2] provided another polynomial time algorithm for gathering in the presence of at least $7f+7$ good agents. This algorithm works considering start-up delay and achieves simultaneous termination. However, this algorithm depends on the length of the largest ID in the system. We, in this work, provide an algorithm considering start-up delays of the good agents, reducing the number of good agents w.r.t. [1] to $f^2+4f+9$, and good agents achieve simultaneous termination. Our algorithm runs faster than [2] when the ID range of the good agents is significantly smaller in comparison to the ID range of all the agents. We also provide a much faster $O\left(n^2\right)$ time algorithm for trees using $3f+2$ agents handling start-up delays and guaranteeing simultaneous termination on a restricted ID range.