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

## TL;DR

This paper presents a randomized algorithm that approximates the edit distance between two strings in nearly linear time, providing constant factor guarantees for inputs with sufficiently large edit distance.

## Contribution

It introduces a nearly linear time algorithm that achieves constant factor approximation for edit distance on far input pairs, improving efficiency.

## Key findings

- Runs in time $O(n^{1+1/T})$ for any $T \\geq 1$
- Provides a constant factor approximation when edit distance is large
- Achieves high probability guarantees for the approximation

## Abstract

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).