Crash tolerant gathering on grid by asynchronous oblivious robots

Consider a system of autonomous mobile robots initially randomly deployed on the nodes of an anonymous finite grid. A gathering algorithm is a sequence of moves to be executed independently by each robot so that all robots meet at a single node after finite time. The robots operate in Look-Compute-Move cycles. In each cycle, a robot takes a snapshot of the current configuration of the grid in terms of occupied nodes (Look), then based on the perceived configuration, decides whether to stay put or to move to an adjacent node (Compute), and in the later case makes an instantaneous move accordingly (Move). The robots have weak multiplicity detection capability, which enables them to detect if a node is empty or occupied by a single robot or by multiple robots. The robots are asynchronous, oblivious, anonymous, can not communicate with each other and execute the same distributed algorithm. In a faulty system, however, any robot can crash, which means that it becomes completely inactive and does not take part in the process any further. In that case a fault-tolerant gathering algorithm is an algorithm that gathers all the non-faulty robots at a single node. This paper considers a faulty system that can have at most one crash fault. With these assumptions deterministic fault-tolerant gathering algorithms are presented that gather all initial configurations that are gatherable in a non-faulty system, except for one specific configuration called the 2S2 configuration.