  We consider polar codes for memoryless sources with side information and show
that the blocklength, construction, encoding and decoding complexities are
bounded by a polynomial of the reciprocal of the gap between the compression
rate and the conditional entropy. This extends the recent results of Guruswami
and Xia to a slightly more general setting, which in turn can be applied to (1)
sources with non-binary alphabets, (2) key generation for discrete and Gaussian
sources, and (3) Slepian-Wolf coding and multiple accessing. In each of these
cases, the complexity scaling with respect to the number of users is also
controlled. In particular, we construct coding schemes for these multi-user
information theory problems which achieve optimal rates with an overall
polynomial complexity.
