2026-07-22T06:50:48Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:24513 2025-12-16T14:00:12Z ddc:004 open_access

Bounded Weighted Edit Distance: Dynamic Algorithms and Matching Lower Bounds Boneh, Itai Gorbachev, Egor Kociumaka, Tomasz Edit distance dynamic algorithms conditional lower bounds The edit distance ed(X,Y) of two strings X,Y ∈ 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, where the costs of individual edits, depending on the edit type and the characters involved, are specified using a function w, normalized so that each edit costs at least one. The textbook dynamic-programming procedure, given strings X,Y ∈ Σ^{≤ n} and oracle access to w, computes ed^w(X,Y) in 𝒪(n²) time. Nevertheless, one can achieve better running times if the computed distance, denoted k, is small: 𝒪(n+k²) for unit weights [Landau and Vishkin; JCSS'88] and Õ(n+√{nk³}) 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 ∈ Σ^{≤ 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 Õ(k) time per update after Õ(n+k²)-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 Õ(k^{3-γ}) time per update after Õ(nk^γ)-time preprocessing. Here, γ ∈ [0,1] is a real trade-off parameter and k ≥ 1 is an integer threshold fixed at preprocessing time, with ∞ 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 ∈ [0.5,1) and justifying our choice to fix k. We also generalize our solution to a much more robust setting while preserving the fine-grained optimal trade-off. Our full algorithm maintains X ∈ Σ^{≤ n} subject not only to character edits but also substring deletions and copy-pastes, each supported in Õ(k²) time. Instead of dynamically maintaining Y, it answers queries that, given any string Y specified through a sequence of 𝒪(k) arbitrary edits transforming X into Y, in Õ(k^{3-γ}) time compute ed^w(X,Y) or report that ed^w(X,Y) > k. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Itai Boneh and Egor Gorbachev and Tomasz Kociumaka 2025 Is Part Of LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2025.45 urn:nbn:de:0030-drops-245139 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.45 eng https://creativecommons.org/licenses/by/4.0/legalcode