We apply recent results on extracting randomness from independent
sources to "extract" Kolmogorov complexity. For any alpha,
epsilon > 0, given a string x with K(x) > alpha|x|, we show
how to use a constant number of advice bits to efficiently
compute another string y, |y|=Omega(|x|), with K(y) >
(1-epsilon)|y|. This result holds for both classical and
space-bounded Kolmogorov complexity.

We use the extraction procedure for space-bounded complexity to
establish zero-one laws for polynomial-space strong dimension.  Our
results include:

1. If Dim_pspace(E) > 0, then Dim_pspace(E/O(1)) = 1.
2. Dim(E/O(1)|ESPACE) is either 0 or 1.
3. Dim(E/poly|ESPACE) is either 0 or 1.

In other words, from a dimension standpoint and with respect to a
small amount of advice, the exponential-time class E is either
minimally complex (dimension 0) or maximally complex (dimension 1)
within ESPACE.