We apply results on extracting randomness from independent sources to "extract" Kolmogorov complexity. For any α, ε > 0, given a string x with K(x) > α|x|, we show how to use a constant number of advice bits to efficiently compute another string y, |y|=Ω(|x|), with K(y) > (1-ε)|y|. This result holds for both unbounded and space-bounded Kolmogorov complexity.
We use the extraction procedure for space-bounded complexity to establish zero-one laws for the strong dimension of complexity classes within ESPACE. For example, Dim(E | ESPACE) is either 0 or 1.
The extraction procedure for unbounded complexity yields a similar result for constructive strong dimension: for any sequence S with Dim(S) > 0, there is a sequence T ≤T S with Dim(T) arbitrarily close to 1.