Validating the Knuth-Morris-Pratt Failure Function, Fast and Online View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2013-12-06

AUTHORS

Paweł Gawrychowski, Artur Jeż, Łukasz Jeż

ABSTRACT

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\pi'_{w}$\end{document} denote the failure function of the Knuth-Morris-Pratt algorithm for a word w. In this paper we study the following problem: given an integer array \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A'[1 \mathinner {\ldotp \ldotp }n]$\end{document}, is there a word w over an arbitrary alphabet Σ such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A'[i]=\pi'_{w}[i]$\end{document} for all i? Moreover, what is the minimum cardinality of Σ required? We give an elementary and self-contained \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{O}(n\log n)$\end{document} time algorithm for this problem, thus improving the previously known solution (Duval et al. in Conference in honor of Donald E. Knuth, 2007), which had no polynomial time bound. Using both deeper combinatorial insight into the structure of π′ and advanced algorithmic tools, we further improve the running time to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{O}(n)$\end{document}. More... »

PAGES

337-372

References to SciGraph publications

  • 2009. Counting Parameterized Border Arrays for a Binary Alphabet in LANGUAGE AND AUTOMATA THEORY AND APPLICATIONS
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  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00224-013-9522-8

    DOI

    http://dx.doi.org/10.1007/s00224-013-9522-8

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1000648785


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