  We consider two natural variants of the problem of minimum spanning tree
(MST) of a graph in the parallel setting: MST verification (verifying if a
given tree is an MST) and the sensitivity analysis of an MST (finding the
lowest cost replacement edge for each edge of the MST). These two problems have
been studied extensively for sequential algorithms and for parallel algorithms
in the PRAM model of computation. In this paper, we extend the study to the
standard model of Massive Parallel Computation (MPC).
  It is known that for graphs of diameter $D$, the connectivity problem can be
solved in $O(\log D + \log\log n)$ rounds on an MPC with low local memory (each
machine can store only $O(n^{\delta})$ words for an arbitrary constant $\delta
> 0$) and with linear global memory, that is, with optimal utilization.
However, for the related task of finding an MST, we need $\Omega(\log
D_{\text{MST}})$ rounds, where $D_{\text{MST}}$ denotes the diameter of the
minimum spanning tree. The state of the art upper bound for MST is $O(\log n)$
rounds; the result follows by simulating existing PRAM algorithms. While this
bound may be optimal for general graphs, the benchmark of connectivity and
lower bound for MST suggest the target bound of $O(\log D_{\text{MST}})$
rounds, or possibly $O(\log D_{\text{MST}} + \log\log n)$ rounds. As for now,
we do not know if this bound is achievable for the MST problem on an MPC with
low local memory and linear global memory. In this paper, we show that two
natural variants of the MST problem: MST verification and sensitivity analysis
of an MST, can be completed in $O(\log D_T)$ rounds on an MPC with low local
memory and with linear global memory; here $D_T$ is the diameter of the input
``candidate MST'' $T$. The algorithms asymptotically match our lower bound,
conditioned on the 1-vs-2-cycle conjecture.
