  Given a signed permutation on $n$ elements, we need to sort it with the
fewest reversals. This is a fundamental algorithmic problem motivated by
applications in comparative genomics, as it allows to accurately model
rearrangements in small genomes. The first polynomial-time algorithm was given
in the foundational work of Hannenhalli and Pevzner [J. ACM'99]. Their approach
was later streamlined and simplified by Kaplan, Shamir, and Tarjan [SIAM J.
Comput.'99] and their framework has eventually led to an algorithm that works
in $\mathcal{O}(n^{3/2}\sqrt{\log n})$ time given by Tannier, Bergeron, and
Sagot [Discr. Appl. Math.'07]. However, the challenge of finding a
nearly-linear time algorithm remained unresolved. In this paper, we show how to
leverage the results on dynamic graph connectivity to obtain a surprisingly
simple $\mathcal{O}(n \log^2 n / \log \log n)$ time algorithm for this problem.
