Approximation Schemes for Edit Distance and LCS in Quasi-Strongly Subquadratic Time

TL;DR

This paper introduces randomized approximation schemes for Edit Distance and LCS that run faster than classical algorithms, providing new insights into their computational complexity and derandomization hardness.

Contribution

It presents the first quasi-strongly subquadratic time approximation algorithms for ED and LCS, revealing complexity separations and derandomization hardness implications.

Findings

Approximate ED and LCS in time $n^2 / 2^{ ext{polylog}(n)}$

Shows separation between approximate and exact ED complexity

Demonstrates derandomization hardness for LCS approximation

Abstract

We present novel randomized approximation schemes for the Edit Distance (ED) problem and the Longest Common Subsequence (LCS) problem that, for any constant $\epsilon>0$, compute a $(1+\epsilon)$-approximation for ED and a $(1-\epsilon)$-approximation for LCS in time $n^2 / 2^{\log^{\Omega(1)}(n)}$ for two strings of total length at most $n$. This running time improves upon the classical quadratic-time dynamic programming algorithms by a quasi-polynomial factor. Our results yield significant insights into fine-grained complexity: Firstly, for ED, prior work indicates that any exact algorithm cannot be improved beyond a few logarithmic factors without refuting established complexity assumptions [Abboud, Hansen, Vassilevska Williams, Williams, 2016]; our quasi-polynomial speed-up shows a separation the complexity of approximate ED from that of exact ED, even for approximation factor arbitrarily close to $1$. Secondly, for LCS, obtaining similar approximation-time tradeoffs via deterministic algorithms would imply breakthrough circuit lower bounds [Chen, Goldwasser, Lyu, Rothblum, Rubinstein, 2019]; our randomized algorithm demonstrates derandomization hardness for LCS approximation.