Probabilistic Solitude Detection II: Ring Size Known Exactly

Karl Abrahamson, Andrew Adler, Lisa Higham and David G. Kirkpatrick
Publishing date
April 1987

Upper and lower bounds that match to within a constant factor are found for the expected bit complexity of a problem on asynchronous unidirectional rings of known size $n$, for algorithms that must reach a correct conclusion with probability at least $1 - \epsilon$ for some small preassigned $\epsilon \geq 0$. The problem is for a nonempty set of contenders to determine whether there is precisely one contender. If distributive termination is required, the expected bit complexity is \( \Theta (n \min ( \log u (n) + \sqrt{\log \log (\frac{1}{\epsilon})}, \sqrt{\log n}, \log \log (\frac{1}{\epsilon}))) \), where $u (n)$ is the least nondivisor of $n$. For nondistributive termination, $ \sqrt{\log \log (\frac{1}{\epsilon})}$ and $\sqrt{\log n}$ are replaced by $\log \log \log(\frac{1}{\epsilon})$ and $\log \log n$ respectively. The lower bounds hold even for probabilistic algorithms that exhibit some nondeterministic features.