RLE edit distance in near optimal time

Abstract

We show that the edit distance between two run-length encoded strings of compressed lengths $m$ and $n$ respectively, can be computed in $\mathcal{O}(mn\log (mn))$ time. This improves the previous record by a factor of $\mathcal{O}(n/\log (mn))$. The running time of our algorithm is within subpolynomial factors of being optimal, subject to the standard SETH-hardness assumption. This effectively closes a line of algorithmic research first started in 1993.