Tight Bounds For Distributed MST Verification
Résumé
This paper establishes tight bounds for the Minimum-weight Spanning Tree (MST) verification problem in the distributed setting. Specifically, we provide an MST verification algorithm that achieves {\em simultaneously} $\tilde{O}(|E|)$ messages and $\tilde{O}(\sqrt{n} + D)$ time, where $|E|$ is the number of edges in the given graph $G$ and $D$ is $G$'s diameter. On the negative side, we show that any MST verification algorithm must send $\Omega(|E|)$ messages and incur $\tilde{\Omega}(\sqrt{n} + D)$ time in worst case. Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of $\Omega(|E|)$ messages and $\Omega(\sqrt{n} + D)$ time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves {\em simultaneously} $\tilde{O}(|E|)$ messages and $\tilde{O}(\sqrt{n} + D)$ time. Specifically, the best known time-optimal algorithm (using $\tO(\sqrt{n} + D)$ time) requires $O(|E|+n^{3/2})$ messages, and the best known message-optimal algorithm (using $\tO(|E|)$ messages) requires $O(n)$ time. On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction.