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#### Recompression: A Simple and Powerful Technique for Word Equations

##### Locator

http://drops.dagstuhl.de/opus/volltexte/2013/3937/

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##### Citation

Jeż, A. (2013). Recompression: A Simple and Powerful Technique for Word Equations.
In N. Portier, & T. Wilke (*30th International Symposium
on Theoretical Aspects of Computer Science* (pp. 233-244). Wadern: Schloss Dagstuhl. doi:10.4230/LIPIcs.STACS.2013.233.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0015-3F2C-7

##### 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.