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

Abstract

For any $T\ge 1$, there are constants $R=R(T)\ge 1$ 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 $U\le R(ED(x,y)+n^{1-\zeta })$. In particular, on any input with $ED(x,y)\ge n^{1-\zeta }$ the algorithm outputs a constant factor approximation with high probability. A similar result has been proven independently by Brakensiek and Rubinstein (2019).

Corrected typos. Revised argument in Section 4.9, results unchanged