Time optimal gathering of myopic robots on an infinite triangular grid

This work deals with the problem of gathering n oblivious mobile entities, called robots, at a point (not known beforehand) placed on an infinite triangular grid. The robots are considered to be myopic, i.e., robots have limited visibility. Earlier works of gathering mostly considered the robots either on a plane or on a circle or on a rectangular grid under both full and limited visibility. In the triangular grid, there are two works to the best of our knowledge. The first one is by Cicerone et al. on arbitrary pattern formation where full visibility is considered. The other one by Shibata et al. which considers seven robots with 2-hop visibility that form a hexagon with one robot in the center of the hexagon in a collision-less environment under a fully synchronous scheduler.

In this work, we first show that gathering on a triangular grid with 1-hop vision of robots is not possible even under a fully synchronous scheduler if the robots do not agree on any axis. So one axis agreement has been considered in this work (i.e., the robots agree on a direction and its orientation). We have also shown that the lower bound for time is $\Omega \left(n\right)$ epochs when n number of robots are gathering on an infinite triangular grid. An algorithm is then presented where a swarm of n number of robots with 1-hop visibility can gather within $O\left(n\right)$ epochs under a semi-synchronous scheduler. So the algorithm presented here is time optimal.