可分图上的动态有效阻力与近似Schur补
Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs
摘要：
我们考虑的问题，动态保持（近似）所有对有效电阻可分图，这是那些承认一个$n^{c}$ -分离定理，一些$c<1$。我们给出了一个完全动态的算法，该算法保持$n$ -顶点图$G$的所有对有效电阻的$(1+\varepsilon)$ -近似，该图经历边插入和删除，具有$\tilde{O}(\sqrt{n}/\varepsilon^2)$最坏情况更新时间和$\tilde{O}(\sqrt{n}/\varepsilon^2)$最坏情况查询时间，如果$G$保证是$\sqrt{n}$ -可分离的（即，它取自满足$\sqrt{n}$ -分离器定理的类），并且它的分离器可以在$\tilde{O}(n)$时间内计算。我们的算法是建立在一个动态的算法，用于维护\emph{approximate Schur complement}，大约保留成对的有效电阻之间的一组终端可分离的图形，这可能是独立的利益。我们补充我们的结果证明，对于任何两个固定的顶点$s$和$t$，没有增量或减量算法可以保持$s-t$有效电阻$\sqrt{n}$ -可分离的图与最坏情况下的更新时间$O(n^{1/2-δ})$和查询时间$O(n^{1-δ})$的任何$δ>0$，除非在线矩阵向量乘法（OMv）猜想是假的。我们进一步表明，对于\emph{general}图，没有增量或减量算法可以保持$s-t$有效电阻问题的最坏情况下的更新时间$O(n^{1-δ})$和查询时间$O(n^{2-δ})$的任何$δ>0$，除非OMv猜想是假的。
英文摘要：
We consider the problem of dynamically maintaining (approximate) all-pairs effective resistances in separable graphs, which are those that admit an $n^{c}$-separator theorem for some $c<1$. We give a fully dynamic algorithm that maintains $(1+\varepsilon)$-approximations of the all-pairs effective resistances of an $n$-vertex graph $G$ undergoing edge insertions and deletions with $\tilde{O}(\sqrt{n}/\varepsilon^2)$ worst-case update time and $\tilde{O}(\sqrt{n}/\varepsilon^2)$ worst-case query time, if $G$ is guaranteed to be $\sqrt{n}$-separable (i.e., it is taken from a class satisfying a $\sqrt{n}$-separator theorem) and its separator can be computed in $\tilde{O}(n)$ time. Our algorithm is built upon a dynamic algorithm for maintaining \emph{approximate Schur complement} that approximately preserves pairwise effective resistances among a set of terminals for separable graphs, which might be of independent interest. We complement our result by proving that for any two fixed vertices $s$ and $t$, no incremental or decremental algorithm can maintain the $s-t$ effective resistance for $\sqrt{n}$-separable graphs with worst-case update time $O(n^{1/2-δ})$ and query time $O(n^{1-δ})$ for any $δ>0$, unless the Online Matrix Vector Multiplication (OMv) conjecture is false. We further show that for \emph{general} graphs, no incremental or decremental algorithm can maintain the $s-t$ effective resistance problem with worst-case update time $O(n^{1-δ})$ and query-time $O(n^{2-δ})$ for any $δ>0$, unless the OMv conjecture is false.
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