It is well known that ordinary persistence on graphs can be computed more efficiently than the general persistence. Recently, it has also been shown that zigzag persistence on graphs also exhibits similar behavior. Motivated by these results, we revisit graph persistence and propose efficient algorithms especially for local updates on filtrations, similar to what is done in ordinary persistence for computing the vineyard. We show that, for a filtration of length $m$ (i) switches (transpositions) in ordinary graph persistence can be done in $O(\log^4 m)$ amortized time; (ii) zigzag persistence on graphs can be computed in $O(m\log m)$ time, which improves a recent $O(m\log^4n)$ time algorithm assuming $n$, the size of the union of all graphs in the filtration, satisfies $n\in\Omega({m^\varepsilon})$ for any fixed $0<\varepsilon<1$; (iii) open-closed, closed-open, and closed-closed bars in dimension $0$ for graph zigzag persistence can be updated in $O(\log^4m)$ amortized time, whereas the open-open bars in dimension $0$ and closed-closed bars in dimension $1$ can be done in $O(m)$ time.

翻译：众所周知，图上的普通持续同调比一般持续同调能更高效地计算。近期研究也表明，图上的锯齿形持续同调展现出类似特性。受这些结果启发，我们重新审视图持续同调，并提出针对滤流局部更新的高效算法，这与普通持续同调中计算vineyard的方法类似。我们证明，对于长度为$m$的滤流：（i）普通图持续同调中的交换（转置）操作可在均摊$O(\log^4 m)$时间内完成；（ii）图上的锯齿形持续同调可在$O(m\log m)$时间内计算，这改进了近期一个$O(m\log^4 n)$时间的算法，其中$n$为滤流中所有图并集的大小，且对任意固定$0<\varepsilon<1$均满足$n\in\Omega({m^\varepsilon})$；（iii）图锯齿形持续同调中维度$0$的开-闭、闭-开及闭-闭条可在均摊$O(\log^4 m)$时间内更新，而维度$0$的开-开条与维度$1$的闭-闭条可在$O(m)$时间内完成。