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Small-Space Algorithms for the Online Language Distance Problem for Palindromes and Squares

Authors Gabriel Bathie , Tomasz Kociumaka , Tatiana Starikovskaya

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Author Details

Gabriel Bathie
  • DIENS, École normale supérieure de Paris, PSL Research University, France
  • LaBRI, Université de Bordeaux, France
Tomasz Kociumaka
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Tatiana Starikovskaya
  • DIENS, École normale supérieure de Paris, PSL Research University, France

Funding

Partially funded by the grant ANR-20-CE48-0001 from the French National Research Agency.

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Gabriel Bathie, Tomasz Kociumaka, and Tatiana Starikovskaya. Small-Space Algorithms for the Online Language Distance Problem for Palindromes and Squares. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.10

Abstract

We study the online variant of the language distance problem for two classical formal languages, the language of palindromes and the language of squares, and for the two most fundamental distances, the Hamming distance and the edit (Levenshtein) distance. In this problem, defined for a fixed formal language L, we are given a string T of length n, and the task is to compute the minimal distance to L from every prefix of T. We focus on the low-distance regime, where one must compute only the distances smaller than a given threshold k. In this work, our contribution is twofold: 1) First, we show streaming algorithms, which access the input string T only through a single left-to-right scan. Both for palindromes and squares, our algorithms use O(k polylog n) space and time per character in the Hamming-distance case and O(k² polylog n) space and time per character in the edit-distance case. These algorithms are randomised by necessity, and they err with probability inverse-polynomial in n. 2) Second, we show deterministic read-only online algorithms, which are also provided with read-only random access to the already processed characters of T. Both for palindromes and squares, our algorithms use O(k polylog n) space and time per character in the Hamming-distance case and O(k⁴ polylog n) space and amortised time per character in the edit-distance case.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Pattern matching
Keywords
  • Approximate pattern matching
  • streaming algorithms
  • palindromes
  • squares

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