  We present a universally-optimal distributed algorithm for the exact weighted
min-cut. The algorithm is guaranteed to complete in $\widetilde{O}(D +
\sqrt{n})$ rounds on every graph, recovering the recent result of Dory, Efron,
Mukhopadhyay, and Nanongkai~[STOC'21], but runs much faster on structured
graphs. Specifically, the algorithm completes in $\widetilde{O}(D)$ rounds on
(weighted) planar graphs or, more generally, any (weighted) excluded-minor
family.
  We obtain this result by designing an aggregation-based algorithm: each node
receives only an aggregate of the messages sent to it. While somewhat
restrictive, recent work shows any such black-box algorithm can be simulated on
any minor of the communication network. Furthermore, we observe this also
allows for the addition of (a small number of) arbitrarily-connected virtual
nodes to the network. We leverage these capabilities to design a min-cut
algorithm that is significantly simpler compared to prior distributed work. We
hope this paper showcases how working within this paradigm yields
simple-to-design and ultra-efficient distributed algorithms for global
problems.
  Our main technical contribution is a distributed algorithm that, given any
tree $T$, computes the minimum cut that $2$-respects $T$ (i.e., cuts at most
$2$ edges of $T$) in universally near-optimal time. Moreover, our algorithm
gives a \emph{deterministic} $\widetilde{O}(D)$-round 2-respecting cut solution
for excluded-minor families and a \emph{deterministic} $\widetilde{O}(D +
\sqrt{n})$-round solution for general graphs, the latter resolving a question
of Dory, et al.~[STOC'21]
