Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification

This paper introduces the notion of distributed verification without preprocessing. It focuses on the Minimum-weight Spanning Tree (MST) verification problem and establishes tight upper and lower bounds for the time and message complexities of this problem. Specifically, we provide an MST verification algorithm that achieves simultaneously Õ messages and Õ(√n + D) time, where m is the number of edges in the given graph G, n is the number of nodes, and D is G's diameter. On the other hand, we show that any MST verification algorithm must send Ω̃(m) messages and incur Ω̃(√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 Ω̃(m) messages and Ω̃(√n + D) time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves simultaneously Ω̃(m) messages and Ω̃(√n + D) time. Specifically, the best known time-optimal algorithm (using Ω̃(√n + D) time) requires O(m+n ^3/2) messages, and the best known message-optimal algorithm (using Ω̃(m + n^3/2) 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.