  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 O(m) messages and O($\sqrt$
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 {\Omega}(m) messages and incur
{\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}(m) messages and
{\Omega}($\sqrt$ n+D time hold, but at the moment there is no known distributed
algorithm that constructs an MST and achieves simultaneously O(m) messages and
O($\sqrt$ n + D) time. Specifically, the best known time-optimal algorithm
(using O($\sqrt$ n + D) time) requires O(m + n 3/2) messages, and the best
known message-optimal algorithm (using O(m) 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.
