  For any $T \geq 1$, there are constants $R=R(T) \geq 1$ and
$\zeta=\zeta(T)>0$ 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
$O(n^{1+\frac{1}{T}})$ and outputs an upper bound $U$ on the edit distance
$ED(x,y)$ that with high probability, satisfies $U \leq
R(ED(x,y)+n^{1-\zeta})$. In particular, on any input with $ED(x,y) \geq
n^{1-\zeta}$ the algorithm outputs a constant factor approximation with high
probability.
  A similar result has been proven independently by Brakensiek and Rubinstein
(2019).
