  We design an $n^{2+o(1)}$-time algorithm that constructs a cut-equivalent
(Gomory-Hu) tree of a simple graph on $n$ nodes. This bound is almost-optimal
in terms of $n$, and it improves on the recent $\tilde{O}(n^{2.5})$ bound by
the authors (STOC 2021), which was the first to break the cubic barrier.
Consequently, the All-Pairs Maximum-Flow (APMF) problem has time complexity
$n^{2+o(1)}$, and for the first time in history, this problem can be solved
faster than All-Pairs Shortest Paths (APSP). We further observe that an
almost-linear time algorithm (in terms of the number of edges $m$) is not
possible without first obtaining a subcubic algorithm for multigraphs.
  Finally, we derandomize our algorithm, obtaining the first subcubic
deterministic algorithm for Gomory-Hu Tree in simple graphs, showing that
randomness is not necessary for beating the $n-1$ times max-flow bound from
1961. The upper bound is $\tilde{O}(n^{2\frac{2}{3}})$ and it would improve to
$n^{2+o(1)}$ if there is a deterministic single-pair maximum-flow algorithm
that is almost-linear. The key novelty is in using a ``dynamic pivot''
technique instead of the randomized pivot selection that was central in recent
works.
