Parking problem by oblivious mobile robots in infinite grids

In this paper, the parking problem of a swarm of mobile robots has been studied. The robots are deployed at the nodes of an infinite grid, which has a subset of prefixed nodes marked as parking nodes. A parking node $p_i$ has a capacity of $k_i$, which is given as an input and represents the number of robots it is capable of accommodating. As a solution to the parking problem, in the final configuration, robots need to partition themselves into groups so that each parking node contains a number of robots that are equal to the capacity of the node. It is assumed that the number of robots in the initial configuration equals the sum of the capacities of the parking nodes. The robots are assumed to be autonomous, anonymous, homogeneous, identical and oblivious. They operate under an asynchronous scheduler. They do not have any agreement on the coordinate axes, nor do they agree on a common chirality. All the initial configurations for which the problem is unsolvable have been identified. A deterministic distributed algorithm has been proposed for the remaining configurations, ensuring the solvability of the problem.