Let B be a basis of a Euclidean lattice, and \tilde{B} an approximation thereof. We give a sufficient condition on the closeness between \tilde{B} and B so that an LLL-reducing transformation U for \tilde{B} remains valid for B. Further, we analyse an efficient reduction algorithm when B is itself a small deformation of an LLL-reduced basis. Applications include speeding-up reduction by keeping only the most significant bits of B, reducing a basis that is only approximately known, and effi- ciently batching LLL reductions for closely related inputs.