Previous works on lexicographic optimal chains have shown that they provide meaningful geometric homology representatives while being easier to compute than their l 1-norm optimal counterparts. This work presents a novel algorithm to efficiently compute lexicographic optimal chains with a given boundary in a triangulation of 3-space, by leveraging Lefschetz duality and an augmented version of the disjoint-set data structure. Furthermore, by observing that lexicographic minimization is a linear operation, we define a canonical basis of lexicographic optimal chains, called critical basis, and show how to compute it. In applications, the presented algorithms offer new promising ways of efficiently reconstructing open surfaces in difficult acquisition scenarios.