Space optimal and asymptotically move optimal Arbitrary Pattern Formation on rectangular grid by asynchronous robot swarm
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Arbitrary pattern formation (Apf) is a well-studied problem in swarm robotics. The problem has been considered in two different settings so far; one is in a plane and another is in an infinite grid. This work deals with the problem in an infinite rectangular grid setting. The previous works in literature dealing with Apf problem in infinite grid had a fundamental issue. These deterministic algorithms use a lot of space in the grid to solve the problem mainly because of maintaining the asymmetry of the configuration or to avoid a collision. These solution techniques can not be useful if there is a space constraint in the application field. In this work, we consider luminous robots (with one light that can take two colors) to avoid symmetry, but we carefully designed a deterministic algorithm that solves the Apf problem using minimal required space in the grid. The robots are autonomous, identical, and anonymous and they operate in Look-Compute-Move cycles under a fully asynchronous scheduler. The Apf algorithm proposed in [WALCOM'2019] by Bose et al. can be modified using luminous robots so that it uses minimal space but that algorithm is not move-optimal. The algorithm proposed in this paper not only uses minimal space but also asymptotically move-optimal. The algorithm proposed in this work is designed for an infinite rectangular grid but it can be easily modified to work in a finite grid as well.
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