Sorting Signed Permutations by Reversals in Nearly-Linear Time

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 O(n(3/2)root 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 results on dynamic graph connectivity to obtain a surprisingly simple O(n log(2) n/log log n) time algorithm for this problem.