We provide new algorithms and conditional hardness for the problem of estimating effective resistances in $n$-node $m$-edge undirected, expander graphs. We provide an $\widetilde{O}(m\epsilon^{-1})$-time algorithm that produces with high probability, an $\widetilde{O}(n\epsilon^{-1})$-bit sketch from which the effective resistance between any pair of nodes can be estimated, to $(1 \pm \epsilon)$-multiplicative accuracy, in $\widetilde{O}(1)$-time. Consequently, we obtain an $\widetilde{O}(m\epsilon^{-1})$-time algorithm for estimating the effective resistance of all edges in such graphs, improving (for sparse graphs) on the previous fastest runtimes of $\widetilde{O}(m\epsilon^{-3/2})$ [Chu et. al. 2018] and $\widetilde{O}(n^2\epsilon^{-1})$ [Jambulapati, Sidford, 2018] for general graphs and $\widetilde{O}(m + n\epsilon^{-2})$ for expanders [Li, Sachdeva 2022]. We complement this result by showing a conditional lower bound that a broad set of algorithms for computing such estimates of the effective resistances between all pairs of nodes require $\widetilde{\Omega}(n^2 \epsilon^{-1/2})$-time, improving upon the previous best such lower bound of $\widetilde{\Omega}(n^2 \epsilon^{-1/13})$ [Musco et. al. 2017]. Further, we leverage the tools underlying these results to obtain improved algorithms and conditional hardness for more general problems of sketching the pseudoinverse of positive semidefinite matrices and estimating functions of their eigenvalues.

翻译：我们针对$n$节点$m$边无向扩展图中有效电阻估计问题，给出新算法与条件困难性结果。我们提出一个$\widetilde{O}(m\epsilon^{-1})$时间复杂度的算法，能以高概率生成$\widetilde{O}(n\epsilon^{-1})$比特的草图，由此可在$\widetilde{O}(1)$时间内以$(1 \pm \epsilon)$乘法精度估计任意节点对间的有效电阻。相应地，我们获得一个$\widetilde{O}(m\epsilon^{-1})$时间复杂度的算法，用于估计此类图中所有边的有效电阻（针对稀疏图），改进此前一般图上的最快运行时间$\widetilde{O}(m\epsilon^{-3/2})$ [Chu等, 2018]和$\widetilde{O}(n^2\epsilon^{-1})$ [Jambulapati, Sidford, 2018]，以及扩展图上的$\widetilde{O}(m + n\epsilon^{-2})$ [Li, Sachdeva 2022]。我们通过证明条件下界对该结果进行补充：一类广泛的计算所有节点对有效电阻估计的算法需要$\widetilde{\Omega}(n^2 \epsilon^{-1/2})$时间，改进了此前$\widetilde{\Omega}(n^2 \epsilon^{-1/13})$的最佳下界 [Musco等, 2017]。此外，我们利用支撑这些结果的工具，在半正定矩阵伪逆草图及特征值函数估计等更一般问题上，获得了改进算法与条件困难性结果。