In this paper, we study the problem of gathering distance-1 myopic robots on an infinite triangular grid. We show that the algorithm developed by Goswami et al. (SSS, 2022) is lattice-linear (cf. Gupta and Kulkarni, SRDS 2023). This implies that a distributed scheduler, assumed therein, is not required for this algorithm: it runs correctly in asynchrony. It also implies that the algorithm works correctly even if the robots are equipped with a unidirectional \textit{camera} to see the neighbouring robots (rather than an omnidirectional one, which would be required under a distributed scheduler). Due to lattice-linearity, we can predetermine the point of gathering. We also show that this algorithm converges in $2n$ rounds, which is lower than the complexity ($2.5(n+1)$ rounds) that was shown in Goswami et al.

翻译：本文研究在无限三角形网格上聚集距离为1的近视机器人问题。我们证明了Goswami等人（SSS, 2022）提出的算法具有格线性特性（参见Gupta和Kulkarni, SRDS 2023）。这意味着该算法不需要其中假设的分布式调度器：它在异步条件下也能正确运行。同时，这也表明即便机器人仅配备单向摄像头（而非分布式调度器所需的全向摄像头）来观测相邻机器人，该算法仍能正常工作。由于格线性特性，我们可以预先确定聚集点。此外，我们还证明该算法可在$2n$轮内收敛，低于Goswami等人所证明的复杂度（$2.5(n+1)$轮）。