Extracting powers and periods in a word from its runs structure

  • Authors:

  • M. Crochemore;C. S. Iliopoulos;M. Kubica;J. Radoszewski;W. Rytter;T. Waleń

  • Affiliations:

  • Kings College London, London WC2R 2LS, UK and Université Paris-Est, France;Kings College London, London WC2R 2LS, UK and Digital Ecosystems & Business Intelligence Institute, Curtin University of Technology, Perth, WA 6845, Australia;Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland;Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland;Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland and Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. C ...;Laboratory of Bioinformatics and Protein Engineering, International Institute of Molecular and Cell Biology in Warsaw, ul. Ks. Trojdena 4, 02-109 Warsaw, Poland and Faculty of Mathematics, Informa ...

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2014

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Abstract

A breakthrough in the field of text algorithms was the discovery of the fact that the maximal number of runs in a word of length n is O(n) and that they can all be computed in O(n) time. We study some applications of this result. New simpler O(n) time algorithms are presented for classical textual problems: computing all distinct k-th word powers for a given k, in particular squares for k=2, and finding all local periods in a given word of length n. Additionally, we present an efficient algorithm for testing primitivity of factors of a word and computing their primitive roots. Applications of runs, despite their importance, are underrepresented in existing literature (approximately one page in the paper of Kolpakov and Kucherov, 1999 [25,26]). In this paper we attempt to fill in this gap. We use Lyndon words and introduce the Lyndon structure of runs as a useful tool when computing powers. In problems related to periods we use some versions of the Manhattan skyline problem.