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For any T ≥ 1, there are constants R=R(T) ≥ 1 and ζ=ζ(T)>0 and a randomized algorithm that takes as input an integer n and two strings x,y of length at most n, and runs in time O(n1+1/T) and outputs an upper bound U on the edit distance of edit(x,y) that with high probability, satisfies U ≤ R(edit(x,y)+n1−ζ). In particular, on any input with edit(x,y) ≥ n1−ζ the algorithm outputs a constant factor approximation with high probability. A similar result has been proven independently by Brakensiek and Rubinstein (this proceedings).

Constant factor approximations to edit distance on far input pairs in nearly linear time