Bounded Weighted Edit Distance: Dynamic Algorithms and Matching Lower Bounds
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866915370434035712 |
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| author | Boneh, Itai Gorbachev, Egor Kociumaka, Tomasz |
| author_facet | Boneh, Itai Gorbachev, Egor Kociumaka, Tomasz |
| contents | The edit distance $ed(X,Y)$ of two strings $X,Y\in Σ^*$ is the minimum number of character edits (insertions, deletions, and substitutions) needed to transform $X$ into $Y$. Its weighted counterpart $ed^w(X,Y)$ minimizes the total cost of edits, which are specified using a function $w$, normalized so that each edit costs at least one. The textbook dynamic-programming procedure, given strings $X,Y\in Σ^{\le n}$ and oracle access to $w$, computes $ed^w(X,Y)$ in $O(n^2)$ time. Nevertheless, one can achieve better running times if the computed distance, denoted $k$, is small: $O(n+k^2)$ for unit weights [Landau and Vishkin; JCSS'88] and $\tilde{O}(n+\sqrt{nk^3})$ for arbitrary weights [Cassis, Kociumaka, Wellnitz; FOCS'23].
In this paper, we study the dynamic version of the weighted edit distance problem, where the goal is to maintain $ed^w(X,Y)$ for strings $X,Y\in Σ^{\le n}$ that change over time, with each update specified as an edit in $X$ or $Y$. Very recently, Gorbachev and Kociumaka [STOC'25] showed that the unweighted distance $ed(X,Y)$ can be maintained in $\tilde{O}(k)$ time per update after $\tilde{O}(n+k^2)$-time preprocessing; here, $k$ denotes the current value of $ed(X,Y)$. Their algorithm generalizes to small integer weights, but the underlying approach is incompatible with large weights.
Our main result is a dynamic algorithm that maintains $ed^w(X,Y)$ in $\tilde{O}(k^{3-γ})$ time per update after $\tilde{O}(nk^γ)$-time preprocessing. Here, $γ\in [0,1]$ is a real trade-off parameter and $k\ge 1$ is an integer threshold fixed at preprocessing time, with $\infty$ returned whenever $ed^w(X,Y)>k$. We complement our algorithm with conditional lower bounds showing fine-grained optimality of our trade-off for $γ\in [0.5,1)$ and justifying our choice to fix $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02548 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded Weighted Edit Distance: Dynamic Algorithms and Matching Lower Bounds Boneh, Itai Gorbachev, Egor Kociumaka, Tomasz Data Structures and Algorithms The edit distance $ed(X,Y)$ of two strings $X,Y\in Σ^*$ is the minimum number of character edits (insertions, deletions, and substitutions) needed to transform $X$ into $Y$. Its weighted counterpart $ed^w(X,Y)$ minimizes the total cost of edits, which are specified using a function $w$, normalized so that each edit costs at least one. The textbook dynamic-programming procedure, given strings $X,Y\in Σ^{\le n}$ and oracle access to $w$, computes $ed^w(X,Y)$ in $O(n^2)$ time. Nevertheless, one can achieve better running times if the computed distance, denoted $k$, is small: $O(n+k^2)$ for unit weights [Landau and Vishkin; JCSS'88] and $\tilde{O}(n+\sqrt{nk^3})$ for arbitrary weights [Cassis, Kociumaka, Wellnitz; FOCS'23]. In this paper, we study the dynamic version of the weighted edit distance problem, where the goal is to maintain $ed^w(X,Y)$ for strings $X,Y\in Σ^{\le n}$ that change over time, with each update specified as an edit in $X$ or $Y$. Very recently, Gorbachev and Kociumaka [STOC'25] showed that the unweighted distance $ed(X,Y)$ can be maintained in $\tilde{O}(k)$ time per update after $\tilde{O}(n+k^2)$-time preprocessing; here, $k$ denotes the current value of $ed(X,Y)$. Their algorithm generalizes to small integer weights, but the underlying approach is incompatible with large weights. Our main result is a dynamic algorithm that maintains $ed^w(X,Y)$ in $\tilde{O}(k^{3-γ})$ time per update after $\tilde{O}(nk^γ)$-time preprocessing. Here, $γ\in [0,1]$ is a real trade-off parameter and $k\ge 1$ is an integer threshold fixed at preprocessing time, with $\infty$ returned whenever $ed^w(X,Y)>k$. We complement our algorithm with conditional lower bounds showing fine-grained optimality of our trade-off for $γ\in [0.5,1)$ and justifying our choice to fix $k$. |
| title | Bounded Weighted Edit Distance: Dynamic Algorithms and Matching Lower Bounds |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2507.02548 |