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 ${\sf M}$ 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 ${\sf M}$. 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$P$索引的持久模${\sf M}$，其广义秩定义为${\sf M}$在偏序集$P$上的极限到余极限映射的秩。针对双参数持久模，近期提出了一种基于Zigzag持久性的算法，该算法利用了双参数模的广义秩等于定义在偏序集边界上的Zigzag模中完全区间数目这一性质。然而，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Zigzag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回为${\sf M}$中的完全区间模。此数目决定了${\sf M}$的广义秩。对于$d$-复形的$d$阶同调的特殊情形，我们获得了更高效的算法，其中包括针对图中1阶同调的线性时间算法。