Faster Sublinear-Time Edit Distance

Abstract: We study the fundamental problem of approximating the edit distance of two strings. After an extensive line of research led to the development of a constant-factor approximation algorithm in almost-linear time, recent years have witnessed a notable shift in focus towards sublinear-time algorithms. Here, the task is typically formalized as the $(k, K)$-gap edit distance problem: Distinguish whether the edit distance of two strings is at most $k$ or more than $K$. Surprisingly, it is still possible to compute meaningful approximations in this challenging regime. Nevertheless, in almost all previous work, truly sublinear running time of $O(n<sup>{1-\varepsilon})$ (for a constant $\varepsilon &gt; 0$) comes at the price of at least polynomial gap $K \ge k \cdot n<sup>{\Omega(\varepsilon)}$. Only recently, [Bringmann, Cassis, Fischer, and Nakos; STOC'22] broke through this barrier and solved the sub-polynomial $(k, k<sup>{1+o(1)})$-gap edit distance problem in time $O(n/k + k<sup>{4+o(1)})$, which is truly sublinear if $$n^{\Omega(1)} \le k \le n<sup>{\frac14-\Omega(1)}$$.The $n/k$ term is inevitable (already for Hamming distance), but it remains an important task to optimize the $\mathrm{poly}(k)$ term and, in general, solve the $(k, k<sup>{1+o(1)})$-gap edit distance problem in sublinear-time for larger values of $k$. In this work, we design an improved algorithm for the $(k, k<sup>{1+o(1)})$-gap edit distance problem in sublinear time $O(n/k + k<sup>{2+o(1)})$, yielding a significant quadratic speed-up over the previous $O(n/k + k<sup>{4+o(1)})$-time algorithm. Notably, our algorithm is unconditionally almost-optimal (up to subpolynomial factors) in the regime where $k \leq n<sup>{\frac13}$ and improves upon the state of the art for $k \leq n<sup>{\frac12-o(1)}$.