Lattice Linearity in Assembling Myopic Robots on an Infinite Triangular Grid

07/24/2023

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by   Arya Tanmay Gupta, et al.

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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. 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 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 that (2.5(n+1) rounds) shown in Goswami et al.