Move optimal and time optimal arbitrary pattern formations by asynchronous robots on infinite grid

The ARBITRARY PATTERN FORMATION (APF) is widely studied in distributed computing for swarm robots. This paper deals with the APF problem in an infinite grid under an asynchronous scheduler. In [Bose K, Adhikary R, Kundu MK, et al. Arbitrary pattern formation on infinite grid by asynchronous oblivious robots. Theor Comput Sci. 2020;815:213-227], the authors proposed an algorithm for APF problem in OBLOT model under an asynchronous scheduler, but the proposed algorithm was neither time optimal nor move optimal. This work provides two algorithms that solve APF problem in an asynchronous scheduler. The first algorithm is move optimal considering OBLOT model and the second algorithm is move and time optimal considering the LUMI model, where each robot has one light having three distinct colours.