  Typing of lambda-terms in Elementary and Light Affine Logic (EAL, LAL, resp.)
has been studied for two different reasons: on the one hand the evaluation of
typed terms using LAL (EAL, resp.) proof-nets admits a guaranteed polynomial
(elementary, resp.) bound; on the other hand these terms can also be evaluated
by optimal reduction using the abstract version of Lamping's algorithm. The
first reduction is global while the second one is local and asynchronous. We
prove that for LAL (EAL, resp.) typed terms, Lamping's abstract algorithm also
admits a polynomial (elementary, resp.) bound. We also show its soundness and
completeness (for EAL and LAL with type fixpoints), by using a simple geometry
of interaction model (context semantics).
