一种估计扩张图有效电阻和计算生成树的新方法
A New Approach to Estimating Effective Resistances and Counting Spanning Trees in Expander Graphs
摘要：
我们证明，对于扩展图，对于所有的$ε> 0,$存在一个数据结构的大小$\widetilde{O}(nε^{-1})$，它可以用来返回$(1 + ε)$ -近似有效的电阻$\widetilde{O}(1)$时间每个查询。在存储所有有效电阻的情况下，以前的最佳方法可以通过存储每个顶点的Johnson-Lindenstrauss向量来实现每个查询的$\widetilde{O}(nε^{-2})$大小和$\widetilde{O}(ε^{-2})$时间，或者通过存储光谱草图来实现每个查询的$\widetilde{O}(nε^{-1})$大小和$\widetilde{O}(nε^{-1})$时间。我们的构造基于两个关键思想：1）$ε^{-1}$ -稀疏，$ε$ -添加剂近似$DL^+1_u$的所有$u,$可以用来恢复$(1 + ε)$ -近似的有效电阻，2）在扩展图中，只有$\widetilde{O}(ε^{-1})$坐标的矢量类似于$DL^+1_u$大于$ε.$我们给出了一个有效的结构，这样的数据结构在$\widetilde{O}(m + nε^{-2})$时间通过随机游走。这导致了一种用于计算在$\widetilde{O}(m + nε^{-2} + s)$时间内运行的扩展器中的$s$顶点对的$(1+ε)$ -近似有效电阻的算法，改进了先前已知的用于$s = ω(nε^{-0.5}).$的$m^{1 + o(1)} + (n + s)n^{o(1)}ε^{-1.5}$的运行时间。我们采用上述算法来计算扩展器图中生成树的数量的$(1+δ)$ -近似，或者等效地，在$\widetilde{O}(m + n^{1.5}δ^{-1})$时间内近似其拉普拉斯算子的（伪）行列式。这改进了以前最好的已知结果的$m^{1+o(1)} + n^{1.875+o(1)}δ^{-1.75}$时间，并匹配最好的已知大小的行列式稀疏。
英文摘要：
We demonstrate that for expander graphs, for all $ε> 0,$ there exists a data structure of size $\widetilde{O}(nε^{-1})$ which can be used to return $(1 + ε)$-approximations to effective resistances in $\widetilde{O}(1)$ time per query. Short of storing all effective resistances, previous best approaches could achieve $\widetilde{O}(nε^{-2})$ size and $\widetilde{O}(ε^{-2})$ time per query by storing Johnson-Lindenstrauss vectors for each vertex, or $\widetilde{O}(nε^{-1})$ size and $\widetilde{O}(nε^{-1})$ time per query by storing a spectral sketch. Our construction is based on two key ideas: 1) $ε^{-1}$-sparse, $ε$-additive approximations to $DL^+1_u$ for all $u,$ can be used to recover $(1 + ε)$-approximations to the effective resistances, 2) In expander graphs, only $\widetilde{O}(ε^{-1})$ coordinates of a vector similar to $DL^+1_u$ are larger than $ε.$ We give an efficient construction for such a data structure in $\widetilde{O}(m + nε^{-2})$ time via random walks. This results in an algorithm for computing $(1+ε)$-approximate effective resistances for $s$ vertex pairs in expanders that runs in $\widetilde{O}(m + nε^{-2} + s)$ time, improving over the previously best known running time of $m^{1 + o(1)} + (n + s)n^{o(1)}ε^{-1.5}$ for $s = ω(nε^{-0.5}).$ We employ the above algorithm to compute a $(1+δ)$-approximation to the number of spanning trees in an expander graph, or equivalently, approximating the (pseudo)determinant of its Laplacian in $\widetilde{O}(m + n^{1.5}δ^{-1})$ time. This improves on the previously best known result of $m^{1+o(1)} + n^{1.875+o(1)}δ^{-1.75}$ time, and matches the best known size of determinant sparsifiers.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
