# Edit Distance in Near-Linear Time: it's a Constant Factor

We present an algorithm for approximating the edit distance between two strings of length n in time n^1+ϵ, for any ϵ>0, up to a constant factor. Our result completes the research direction set forth in the recent breakthrough paper [Chakraborty-Das-Goldenberg-Koucky-Saks, FOCS'18], which showed the first constant-factor approximation algorithm with a (strongly) sub-quadratic running time. Several recent results have shown near-linear complexity under different restrictions on the inputs (eg, when the edit distance is close to maximal, or when one of the inputs is pseudo-random). In contrast, our algorithm obtains a constant-factor approximation in near-linear running time for any input strings.

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