重新审视图持久性：更新与效率
Revisiting Graph Persistence for Updates and Efficiency
AI总结：
本文重新审视图持久性，提出高效算法支持过滤上的局部更新，将普通图持久性切换时间降至对数级，之字形持久性计算改进至$O(m\log m)$，并优化各类条目的更新复杂度。
摘要：
众所周知，图上的普通持久性比一般持久性计算更高效。最近研究表明，图上的之字形持久性也表现出类似行为。受这些结果启发，我们重新审视图持久性，并提出高效算法，特别针对过滤上的局部更新，类似于普通持久性中计算葡萄园的做法。我们证明，对于长度为$m$的过滤，(i)普通图持久性中的切换（转置）可以在$O(\log m)$时间内完成；(ii)图上的之字形持久性可以在$O(m\log m)$时间内计算，这改进了最近一个$O(m\log^4 n)$时间的算法，假设$n$（过滤中所有图的并集大小）满足$n\in\Omega({m^\varepsilon})$对于任意固定$0<\varepsilon<1$；(iii)维度$0$的图之字形持久性中，开-闭、闭-开和闭-闭条可以在$O(\log m)$时间内更新，而维度$0$的开-开条和维度$1$的闭-闭条可以在$O(\sqrt{m}\\,\log m)$时间内完成。
英文摘要：
It is well known that ordinary persistence on graphs can be computed more efficiently than the general persistence. Recently, it has 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 $O(\log m)$ 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 $n\inΩ({m^\varepsilon})$ 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 $O(\log m)$ time, whereas the open-open bars in dimension $0$ and closed-closed bars in dimension $1$ can be done in $O(\sqrt{m}\,\log m)$ time.
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