  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.
