When persistence diagrams are formalized as the Mobius inversion of the birth-death function, they naturally generalize to the multi-parameter setting and enjoy many of the key properties, such as stability, that we expect in applications. The direct definition in the 2-parameter setting, and the corresponding brute-force algorithm to compute them, require $\Omega(n^4)$ operations. But the size of the generalized persistence diagram, $C$, can be as low as linear (and as high as cubic). We elucidate a connection between the 2-parameter and the ordinary 1-parameter settings, which allows us to design an output-sensitive algorithm, whose running time is in $O(n^3 + Cn)$.

翻译：当持久性图形式化为出生-死亡函数的莫比乌斯反演时，它们自然推广至多参数设定，并保留了许多关键性质，例如应用中至关重要的稳定性。在二参数设定下，直接定义对应的暴力计算算法需要$\Omega(n^4)$次运算。但广义持久性图的大小$C$可低至线性（最高可达立方）。我们阐明了二参数与普通一参数设定之间的联系，从而设计出一种输出敏感算法，其运行时间处于$O(n^3 + Cn)$。