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. We present a novel algorithm to efficiently compute lexicographic optimal chains with a given boundary in a triangulation of 3-space, by leveraging a Lefschetz duality at the chain level and an augmented version of the classical disjoint-set data structure. We also show that the space of lexicographic optimal cycles forms a vector space isomorphic to the homology groups of the complex, a property suggesting a parallel with \(l^2\) -norm optimal chains and Hodge theory. A canonical basis for this space of lexicographic optimal chains can be defined, called critical basis, and we show how to compute it using standard matrix reduction algorithms. In applications, we show how both computing optimal chains with a given boundary and critical bases offer new promising ways of efficiently reconstructing open surfaces in difficult acquisition scenarios.
