  We give a randomized algorithm that finds a minimum cut in an undirected
weighted $m$-edge $n$-vertex graph $G$ with high probability in $O(m \log^2 n)$
time. This is the first improvement to Karger's celebrated $O(m \log^3 n)$ time
algorithm from 1996. Our main technical contribution is a deterministic $O(m
\log n)$ time algorithm that, given a spanning tree $T$ of $G$, finds a minimum
cut of $G$ that 2-respects (cuts two edges of) $T$.
