Bounded Edit Distance: Optimal Static and Dynamic Algorithms for Small Integer Weights

The edit distance of two strings is the minimum number of insertions, deletions, and substitutions needed to transform one string into the other. The textbook algorithm determines the edit distance of length-$n$ strings in $O(n^2)$ time, which is optimal up to subpolynomial factors under Orthogonal Vectors Hypothesis. In the bounded version of the problem, parameterized by the edit distance $k$, the algorithm of Landau and Vishkin [JCSS'88] achieves $O(n+k^2)$ time, which is optimal as a function of $n$ and $k$. The dynamic version of the problem asks to maintain the edit distance of two strings that change dynamically, with each update modeled as an edit. A folklore approach supports updates in $\tilde O(k^2)$ time, where $\tilde O(\cdot)$ hides polylogarithmic factors. Recently, Charalampopoulos, Kociumaka, and Mozes [CPM'20] showed an algorithm with update time $\tilde O(n)$, which is optimal under OVH in terms of $n$. The update time of $\tilde O(\min{n,k^2})$ raised an exciting open question of whether $\tilde O(k)$ is possible; we answer it affirmatively. Our solution relies on tools originating from weighted edit distance, where the weight of each edit depends on the edit type and the characters involved. The textbook algorithm supports weights, but the Landau-Vishkin approach does not, and a simple $O(nk)$-time procedure long remained the fastest for bounded weighted edit distance. Only recently, Das et al. [STOC'23] provided an $O(n+k^5)$-time algorithm, whereas Cassis, Kociumaka, and Wellnitz [FOCS'23] presented an $\tilde O(n+\sqrt{nk^3})$-time solution and a matching conditional lower bound. In this paper, we show that, for integer edit weights between $0$ and $W$, weighted edit distance can be computed in $\tilde O(n+Wk^2)$ time and maintained dynamically in $\tilde O(W^2k)$ time per update. Our static algorithm can also be implemented in $\tilde O(n+k^{2.5})$ time.