  Je\v{r}\'abek showed that cuts in classical propositional logic proofs in
deep inference can be eliminated in quasipolynomial time. The proof is indirect
and it relies on a result of Atserias, Galesi and Pudl\'ak about monotone
sequent calculus and a correspondence between that system and cut-free
deep-inference proofs. In this paper we give a direct proof of Je\v{r}\'abek's
result: we give a quasipolynomial-time cut-elimination procedure for classical
propositional logic in deep inference. The main new ingredient is the use of a
computational trace of deep-inference proofs called atomic flows, which are
both very simple (they only trace structural rules and forget logical rules)
and strong enough to faithfully represent the cut-elimination procedure.
