For bidirectional rings, there have been proposed self-stabilizing mutual exclusion protocols, which are either time-adaptive (i.e., efficient in recovery) or 1-latent (i.e., efficient in legal execution) but not both. This paper proposes a randomized self-stabilizing mutual exclusion protocol that inherits both of the advantages from them: It is 1-latent in the sense that the privilege is circulated in a linear round (i.e., very intuitively, the privilege is transferred from a process to another by a "step"), provided that the system always stays in legitimate configurations, and is weakly time-adaptive in the sense that the system stabilizes from any configuration c in O(f) steps with a high probability, where f is the number of corrupted processes in c. It also guarantees with a high probability that there is at most one privilege even while converging to a legitimate configuration.
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