For a $P$-indexed persistence module ${\sf M}$, the (generalized) rank of ${\sf M}$ is defined as the rank of the limit-to-colimit map for the diagram of vector spaces of ${\sf M}$ 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 ${\sf M}$ into a zigzag module ${\sf M}_{ZZ}$ and then check how many full interval modules in a decomposition of ${\sf M}_{ZZ}$ can be folded back to remain full in a decomposition of ${\sf M}$. This number determines the generalized rank of ${\sf M}$. 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.

翻译：对于$P$索引持久模${\sf M}$，其（广义）秩定义为该模在偏序集$P$上的向量空间图中极限到余极限映射的秩。针对二参数持久模，近期提出了一种基于之字形持久性的算法，该算法利用了二参数模的广义秩等于定义在偏序集边界上的之字形模中完全区间数目这一特性。对于$d$参数持久模或一般$P$索引持久模，其边界的类似定义似乎并不可行。为克服这一困难，我们首先将给定的$P$索引模${\sf M}$展开为之字形模${\sf M}_{ZZ}$，然后检验${\sf M}_{ZZ}$分解中的完全区间模有多少个可折叠回${\sf M}$分解中保持完全性。该数量决定了${\sf M}$的广义秩。针对$d$复形的$d$阶同调这一特例，我们获得了一种更高效的算法，其中包括图上一阶同调的线性时间算法。