2026-08-07T09:03:37Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:23122 2025-10-02T12:38:12Z ddc:004 open_access

Faster Approximate Elastic-Degenerate String Matching - Part A Pissis, Solon P. Radoszewski, Jakub Zuba, Wiktor ED string approximate string matching Hamming distance edit distance An elastic-degenerate (ED) string 𝐓 is a sequence 𝐓 = 𝐓[1] ⋯ 𝐓[n] of n finite sets of strings. The cardinality m of 𝐓 is the total number of strings in 𝐓[i], for all i ∈ [1..n]. The size N of 𝐓 is the total length of all m strings of 𝐓. ED strings have been introduced to represent a set of closely-related DNA sequences. Let P = P[1..p] be a pattern of length p and k > 0 be an integer. We consider the problem of k-Approximate ED String Matching (EDSM): searching k-approximate occurrences of P in the language of 𝐓. We call k-Approximate EDSM under the Hamming distance, k-Mismatch EDSM; and we call k-Approximate EDSM under edit distance, k-Edit EDSM. Bernardini et al. (Theoretical Computer Science, 2020) showed a simple 𝒪(k m p + kN)-time algorithm for k-Mismatch EDSM and an 𝒪(k² m p + kN)-time algorithm for k-Edit EDSM. We improve the dependency on k in both results, obtaining an Õ(k^{2/3}mp+√kN)-time algorithm for k-Mismatch EDSM and an Õ(kmp+ kN)-time algorithm for k-Edit EDSM. Bernardini et al. (Theory of Computing Systems, 2024) presented several algorithms for 1-Approximate EDSM working in Õ(np²+N) time. They have also left the possibility to generalize these solutions for k > 1 as an open problem. We improve the runtime of their solution for 1-Mismatch and 1-Edit EDSM from Õ(np²+N) to 𝒪(np²+N). We further show algorithms for k-Approximate EDSM for the Hamming and edit distances working in Õ(np² + N) time, for any constant k > 0. Finally, we show how our techniques can be applied to improve upon the complexity of the k-Approximate ED String Intersection and k-Approximate Doubly EDSM problems that were introduced very recently by Gabory et al. (Information and Computation, 2025). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Solon P. Pissis and Jakub Radoszewski and Wiktor Zuba 2025 Is Part Of LIPIcs, Volume 331, 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CPM.2025.28 urn:nbn:de:0030-drops-231227 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2025.28 eng https://creativecommons.org/licenses/by/4.0/legalcode