  The mixing time of the Glauber dynamics for spin systems on trees is closely
related to reconstruction problem. Martinelli, Sinclair and Weitz established
this correspondence for a class of spin systems with soft constraints bounding
the log-Sobolev constant by a comparison with the block dynamics. However, when
there are hard constraints, the block dynamics may be reducible.
  We introduce a variant of the block dynamics extending these results to a
wide class of spin systems with hard constraints. This applies for essentially
any spin system that has non-reconstruction provided that on average the root
is not locally frozen in a large neighborhood. In particular we prove that the
mixing time of the Glauber dynamics for colorings on the regular tree is
$O(n\log n)$ in the entire known non-reconstruction regime.
