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URN: urn:nbn:de:0030-drops-39376
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Recompression: a simple and powerful technique for word equations

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Abstract

We present an application of a local recompression technique, previously developed by the author in the context of compressed membership problems and compressed pattern matching, to word equations. The technique is based on local modification of variables (replacing X by aX or Xa) and replacement of pairs of letters appearing in the equation by a 'fresh' letter, which can be seen as a bottom-up compression of the solution of the given word equation, to be more specific, building an SLP (Straight-Line Programme) for the solution of the word equation. Using this technique we give new self-contained proofs of many known results for word equations: the presented nondeterministic algorithm runs in O(n log n) space and in time polynomial in log N and n, where N is the size of the length-minimal solution of the word equation. It can be easily generalised to a generator of all solutions of the word equation. A further analysis of the algorithm yields a doubly exponential upper bound on the size of the length-minimal solution. The presented algorithm does not use exponential bound on the exponent of periodicity. Conversely, the analysis of the algorithm yields a new proof of the exponential bound on exponent of periodicity. For O(1) variables with arbitrary many appearances it works in linear space.

BibTeX - Entry

@InProceedings{jez:LIPIcs:2013:3937,
  author =	{Artur Jez},
  title =	{{Recompression: a simple and powerful technique for word equations}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{233--244},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-50-7},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{20},
  editor =	{Natacha Portier and Thomas Wilke},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2013/3937},
  URN =		{urn:nbn:de:0030-drops-39376},
  doi =		{10.4230/LIPIcs.STACS.2013.233},
  annote =	{Keywords: Word equations, exponent of periodicity, string unification}
}

Keywords: Word equations, exponent of periodicity, string unification
Seminar: 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)
Issue date: 2013
Date of publication: 2013


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