In Topological Data Analysis, filtered chain complexes enter the persistence pipeline between the initial filtering of data and the final persistence invariants extraction. It is known that they admit a tame class of indecomposables, called interval spheres. In this paper, we provide an algorithm to decompose filtered chain complexes into such interval spheres. This algorithm provides geometric insights into various aspects of the standard persistence algorithm and two of its runtime optimizations. Moreover, since it works for any filtered chain complexes, our algorithm can be applied in more general cases. As an application, we show how to decompose filtered kernels with it.

We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal.