Deterministic protocols are well-known tools to obtain extended formulations, with many applications to polytopes arising in combinatorial optimization. Although constructive, those tools are not output-efficient, since the time needed to produce the extended formulation also depends on the size of the slack matrix (hence, of the exact description in the original space). We give general sufficient conditions under which those tools can be implemented as to be output-efficient. In particular we apply this to Yannakakis' extended formulation for the stable set polytope of perfect graphs, for which, to the best of our knowledge, an efficient construction was previously not known.