Gap Edit Distance via Non-Adaptive Queries: Simple and Optimal

TL;DR

This paper presents a simple, optimal non-adaptive algorithm for approximating edit distance in sublinear time across all parameter ranges, improving previous results and matching known bounds.

Contribution

It introduces a non-adaptive query algorithm with optimal complexity for the Gap Edit Distance problem, covering the entire parameter range and surpassing prior adaptive methods.

Findings

Query complexity of  O(n/k^{c-0.5}) is optimal up to polylog factors.

Achieves optimal time complexity  O(n/k^{c-0.5}) for c   1.5.

Outperforms all previous algorithms in nontrivial cases.

Abstract

We study the problem of approximating edit distance in sublinear time. This is formalized as the $(k,k^c)$-Gap Edit Distance problem, where the input is a pair of strings $X,Y$ and parameters $k,c>1$, and the goal is to return YES if $ED(X,Y)\leq k$, NO if $ED(X,Y)> k^c$, and an arbitrary answer when $k < ED(X,Y) \le k^c$. Recent years have witnessed significant interest in designing sublinear-time algorithms for Gap Edit Distance. In this work, we resolve the non-adaptive query complexity of Gap Edit Distance for the entire range of parameters, improving over a sequence of previous results. Specifically, we design a non-adaptive algorithm with query complexity $\tilde{O}(n/k^{c-0.5})$, and we further prove that this bound is optimal up to polylogarithmic factors. Our algorithm also achieves optimal time complexity $\tilde{O}(n/k^{c-0.5})$ whenever $c\geq 1.5$. For $1<c<1.5$, the running time of our algorithm is $\tilde{O}(n/k^{2c-1})$. In the restricted case of $k^c=\Omega(n)$, this matches a known result [Batu, Erg\"un, Kilian, Magen, Raskhodnikova, Rubinfeld, and Sami; STOC 2003], and in all other (nontrivial) cases, our running time is strictly better than all previous algorithms, including the adaptive ones. However, an independent work of Bringmann, Cassis, Fischer, and Nakos [STOC 2022] provides an adaptive algorithm that bypasses the non-adaptive lower bound, but only for small enough $k$ and $c$.