Polylogarithmic Approximation for Edit Distance and the Asymmetric Query Complexity

TL;DR

This paper introduces a near-linear time algorithm that approximates edit distance within a polylogarithmic factor, significantly improving previous bounds and exploring a new asymmetric query model with matching lower bounds.

Contribution

It presents the first efficient polylogarithmic approximation algorithm for edit distance and establishes a novel lower bound in an asymmetric query model, revealing new hardness results.

Findings

Achieves (log n)^O(1/epsilon) approximation in n^(1+epsilon) time.

Provides the first lower bound showing edit distance hardness for repetitive strings.

Demonstrates a separation between edit distance and Ulam distance.

Abstract

We present a near-linear time algorithm that approximates the edit distance between two strings within a polylogarithmic factor; specifically, for strings of length n and every fixed epsilon>0, it can compute a (log n)^O(1/epsilon) approximation in n^(1+epsilon) time. This is an exponential improvement over the previously known factor, 2^(O (sqrt(log n))), with a comparable running time (Ostrovsky and Rabani J.ACM 2007; Andoni and Onak STOC 2009). Previously, no efficient polylogarithmic approximation algorithm was known for any computational task involving edit distance (e.g., nearest neighbor search or sketching). This result arises naturally in the study of a new asymmetric query model. In this model, the input consists of two strings x and y, and an algorithm can access y in an unrestricted manner, while being charged for querying every symbol of x. Indeed, we obtain our main result by designing an algorithm that makes a small number of queries in this model. We then provide a nearly-matching lower bound on the number of queries. Our lower bound is the first to expose hardness of edit distance stemming from the input strings being "repetitive", which means that many of their substrings are approximately identical. Consequently, our lower bound provides the first rigorous separation between edit distance and Ulam distance, which is edit distance on non-repetitive strings, such as permutations.