We introduce an algorithm to decompose any finite-type persistence module
with coefficients in a field into what we call an {\em interval basis}. This
construction yields both the standard persistence pairs of Topological Data
Analysis (TDA), as well as a special set of generators inducing the interval
decomposition of the Structure theorem. The computation of this basis can be
distributed over the steps in the persistence module. This construction works
for general persistence modules on a field F, not necessarily
deriving from persistent homology. We subsequently provide a parallel algorithm
to build a persistent homology module over R by leveraging the Hodge
decomposition, thus providing new motivation to explore the interplay between
TDA and the Hodge Laplacian.Comment: 37 pages, 6 figure