  Functional transductions realized by two-way transducers (equivalently, by
streaming transducers and by MSO transductions) are the natural and standard
notion of "regular" mappings from words to words. It was shown recently
(LICS'13) that it is decidable if such a transduction can be implemented by
some one-way transducer, but the given algorithm has non-elementary complexity.
We provide an algorithm of different flavor solving the above question, that
has double exponential space complexity. We further apply our technique to
decide whether the transduction realized by a two-way transducer can be
implemented by a sweeping transducer, with either known or unknown number of
passes.
