Sorting Signed Permutations by Reversals in Nearly-Linear Time
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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√(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 𝒪(n log^2 n / loglog n) time algorithm for this problem.
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