Computing Generalized Ranks of Persistence Modules via Unfolding to Zigzag Modules

For a P-indexed persistence module M, the (generalized) rank of M is defined as the rank of the limit-to-colimit map for M over the poset P. For $2$-parameter persistence modules, recently a zigzag persistence based algorithm has been proposed that takes advantage of the fact that generalized rank for $2$-parameter modules is equal to the number of full intervals in a zigzag module defined on the boundary of the poset. Analogous definition of boundary for d-parameter persistence modules or general P-indexed persistence modules does not seem plausible. To overcome this difficulty, we first unfold a given P-indexed module M into a zigzag module MZZ and then check how many full interval modules in a decomposition of MZZ can be folded back to remain full in M. This number determines the generalized rank of M. For special cases of degree-d homology for d-complexes, we obtain a more efficient algorithm including a linear time algorithm for degree-$1$ homology in graphs.