Open Collections

Description

Computation of the interleaving distance between persistence modules is a central task in topological data analysis. For $1$-D persistence modules, thanks to the isometry theorem, this can be done by computing the bottleneck distance with known efficient algorithms. The question is open for most $n$-D persistence modules, $n>1$, because of the well recognized complications of the indecomposables. Here, we consider a reasonably complicated class called {\em $2$-D interval decomposable} modules whose indecomposables may have a description of non-constant complexity. We present a polynomial time algorithm to compute the interleaving distance between two such indecomposables. This leads to a polynomial time algorithm for computing the bottleneck distance between two $2$-D interval decomposable modules, which bounds their interleaving distance from above. We give another algorithm to compute a new distance called {\em dimension distance} that bounds it from below.