  Minimum-weight cut (min-cut) is a basic measure of a network's connectivity
strength. While the min-cut can be computed efficiently in the sequential
setting [Karger STOC'96], there was no efficient way for a distributed network
to compute its own min-cut without limiting the input structure or dropping the
output quality: In the standard CONGEST model, existing algorithms with
nearly-optimal time (e.g. [Ghaffari, Kuhn, DISC'13; Nanongkai, Su, DISC'14])
can guarantee a solution that is $(1+\epsilon)$-approximation at best while the
exact $\tilde O(n^{0.8}D^{0.2} + n^{0.9})$-time algorithm [Ghaffari, Nowicki,
Thorup, SODA'20] works only on *simple* networks (no weights and no parallel
edges). Here $n$ and $D$ denote the network's number of vertices and
hop-diameter, respectively. For the weighted case, the best bound was $\tilde
O(n)$ [Daga, Henzinger, Nanongkai, Saranurak, STOC'19].
  In this paper, we provide an *exact* $\tilde O(\sqrt n + D)$-time algorithm
for computing min-cut on *weighted* networks. Our result improves even the
previous algorithm that works only on simple networks. Its time complexity
matches the known lower bound up to polylogarithmic factors. At the heart of
our algorithm are a clever routing trick and two structural lemmas regarding
the structure of a minimum cut of a graph. These two structural lemmas
considerably strengthen and generalize the framework of Mukhopadhyay-Nanongkai
[STOC'20] and can be of independent interest.
