The inherent bit complexity of leader election on asynchronous unidirectional rings of processors is examined under various assumptions about global knowledge of the ring. If processors have unique identities with a maximum of $m$ bits, then the expected number of communication bits sufficient to elect a leader with probability 1, on a ring of (unknown) size $n$ is $O(nm)$. If the ring size is known to within a multiple of 2, then the expected number of communication bits sufficient to elect a leader with probability 1 is $O(n \log n)$.
These upper bounds are complemented by lower bounds on the communication complexity of a related problem called solitude verification that reduces to leader election in $O(n)$ bits. If processors have unique identities chosen from a sufficiently large universe of size $s$, then the average, over all choices of identities, of the communication complexity of verifying solitude is $\Omega (n \log s)$ bits. When the ring size is known only approximately, then $\Omega (n \log n)$ bits are required for solitude verification. The lower bounds address the complexity of certifying solitude. This is modelled by tbe best case behaviour of non-deterministic solitude verification algorithms.
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