A New Approach to Estimating Effective Resistances and Counting Spanning Trees in Expander Graphs

Lawrence Li, Sushant Sachdeva

Abstract: We demonstrate that for expander graphs, for all ε > 0, there exists a data structure of size Õ(nε-1) which can be used to return (1 + ε)-approximations to effective resistances in Õ(1) time per query. Short of storing all effective resistances, previous best approaches could achieve Õ(nε-2) size and Õ (ε-2) time per query by storing Johnson-Lindenstrauss vectors for each vertex, or Õ (nε-1) size and Õ (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 σu for all u, vectors similar to DL+ 1u, can be used to recover (1 + ε)-approximations to the effective resistances, 2) In expander graphs, only Õ(ε-1) coordinates of σu are larger than ε. We give an efficient construction for such a data structure in Õ(m + nε-2) time via random walks. This results in an algorithm on expander graphs for computing (1 + ε)-approximate effective resistances for s vertex pairs that runs in Õ (m + nε-2 + s) time, improving over the previously best known running time of m1+o(1) + (n + s)no(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 Õ(m + n1.5δ-1) time. This improves on the previously best known result of m1+o(1) + n1.875+o(1)δ-1.75 time, and matches the best known size of determinant sparsifiers. * This research is supported by an NSERC Discovery grant awarded to Sushant Sachdeva.