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On Approximate Jumbled Pattern Matching in Strings

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Abstract

Given a string s, the Parikh vector of s, denoted p(s), counts the multiplicity of each character in s. Searching for a match of a Parikh vector q in the text s requires finding a substring t of s with p(t)=q. This can be viewed as the task of finding a jumbled (permuted) version of a query pattern, hence the term Jumbled Pattern Matching. We present several algorithms for the approximate version of the problem: Given a string s and two Parikh vectors u,v (the query bounds), find all maximal occurrences in s of some Parikh vector q such that u≤q≤v. This definition encompasses several natural versions of approximate Parikh vector search. We present an algorithm solving this problem in sub-linear expected time using a wavelet tree of s, which can be computed in time O(n) in a preprocessing phase. We then discuss a Scrabble-like variation of the problem, in which a weight function on the letters of s is given and one has to find all occurrences in s of a substring t with maximum weight having Parikh vector p(t)≤v. For the case of a binary alphabet, we present an algorithm which solves the decision version of the Approximate Jumbled Pattern Matching problem in constant time, by indexing the string in subquadratic time.

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Authors and Affiliations

  1. Department of Computer Algebra, Eötvös Loránd University, Budapest, Hungary

    Péter Burcsi

  2. Dipartimento di Informatica ed Applicazioni, University of Salerno, Salerno, Italy

    Ferdinando Cicalese

  3. I3S, UMR6070, CNRS et Université de Nice-Sophia, Antipolis, France

    Gabriele Fici

  4. AG Genominformatik, Technische Fakultät, Bielefeld University, Bielefeld, Germany

    Zsuzsanna Lipták

Authors
  1. Péter Burcsi
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  2. Ferdinando Cicalese
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  3. Gabriele Fici
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  4. Zsuzsanna Lipták
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Corresponding author

Correspondence to Ferdinando Cicalese.

Additional information

Some of the results in this paper appeared in preliminary form in the proceedings of FUN 2010 [8].

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Burcsi, P., Cicalese, F., Fici, G. et al. On Approximate Jumbled Pattern Matching in Strings. Theory Comput Syst 50, 35–51 (2012). https://doi.org/10.1007/s00224-011-9344-5

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  • DOI: https://doi.org/10.1007/s00224-011-9344-5

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