A New Approach to Estimating Effective Resistances and Counting Spanning Trees in Expander Graphs
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We demonstrate that for expander graphs, for all ϵ > 0, there exists a data structure of size O(nϵ^-1) which can be used to return (1 + ϵ)-approximations to effective resistances in O(1) time per query. Short of storing all effective resistances, previous best approaches could achieve O(nϵ^-2) size and O(ϵ^-2) time per query by storing Johnson-Lindenstrauss vectors for each vertex, or O(nϵ^-1) size and 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 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 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 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 O(m + n^1.5δ^-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.
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