  We present time-efficient distributed algorithms for decomposing graphs with
large edge or vertex connectivity into multiple spanning or dominating trees,
respectively. As their primary applications, these decompositions allow us to
achieve information flow with size close to the connectivity by parallelizing
it along the trees. More specifically, our distributed decomposition algorithms
are as follows:
  (I) A decomposition of each undirected graph with vertex-connectivity $k$
into (fractionally) vertex-disjoint weighted dominating trees with total weight
$\Omega(\frac{k}{\log n})$, in $\widetilde{O}(D+\sqrt{n})$ rounds.
  (II) A decomposition of each undirected graph with edge-connectivity
$\lambda$ into (fractionally) edge-disjoint weighted spanning trees with total
weight $\lceil\frac{\lambda-1}{2}\rceil(1-\varepsilon)$, in
$\widetilde{O}(D+\sqrt{n\lambda})$ rounds.
  We also show round complexity lower bounds of
$\tilde{\Omega}(D+\sqrt{\frac{n}{k}})$ and
$\tilde{\Omega}(D+\sqrt{\frac{n}{\lambda}})$ for the above two decompositions,
using techniques of [Das Sarma et al., STOC'11]. Moreover, our
vertex-connectivity decomposition extends to centralized algorithms and
improves the time complexity of [Censor-Hillel et al., SODA'14] from $O(n^3)$
to near-optimal $\tilde{O}(m)$.
  As corollaries, we also get distributed oblivious routing broadcast with
$O(1)$-competitive edge-congestion and $O(\log n)$-competitive
vertex-congestion. Furthermore, the vertex connectivity decomposition leads to
near-time-optimal $O(\log n)$-approximation of vertex connectivity: centralized
$\widetilde{O}(m)$ and distributed $\tilde{O}(D+\sqrt{n})$. The former moves
toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an $O(m)$
centralized exact algorithm while the latter is the first distributed vertex
connectivity approximation.
