We propose optimal (w.r.t. the number of robots) solutions for the deterministic terminating exploration (exploration for short) of a grid-shaped network by a team of k asynchronous oblivious robots in the asynchronous non-atomic model, so-called CORDA.In more details, we first consider the ATOM model. We show that it is impossible to explore a grid of at least three nodes with less than three robots. Next, we show that it is impossible to explore a (2, 2)-Grid with less than 4 robots, and a (3, 3)-Grid with less than 5 robots, respectively. The two first results hold for both deterministic and probabilistic settings, while the latter holds only in the deterministic case. ATOM being strictly stronger than CORDA, all these impossibility results also hold in CORDA.Then, we propose deterministic algorithms in CORDA to exhibit the optimal number of robots allowing to explore of a given grid. Our results show that except in two particular cases, 3 robots are necessary and sufficient to deterministically explore a grid of at least three nodes. The optimal number of robots for the two remaining cases is: 4 for the (2, 2)-Grid and 5 for the (3, 3)-Grid, respectively.

In this article, we show that some fundamental self-and snap-stabilizing wave protocols (e.g., token circulation, PIF, etc.) implicitly assume a very light property that we call BreakingIn. We prove that BreakingIn is strictly induced by self-and snap-stabilization. Combined with a transformer, BreakingIn allows to easily turn the non-fault-tolerant versions of those protocols into snap-stabilizing versions. Unlike the previous solutions, the transformed protocols are very efficient and work at least with the same daemon as the initial versions extended to satisfy BreakingIn. Finally, we show how to use an additional property of the transformer to design snap-stabilizing extensions of those fundamental protocols like Mutual Exclusion.

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