2022/11/02 by Lawrence Li, Sushant Sachdeva, Li, Lawrence +1 · 4 citations
Computer Science · Engineering · Materials Science · #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Graphene research and applications #VLSI and FPGA Design Techniques
paper · pdf · doi:10.48550/arxiv.2211.01468
openalex publication_date 2022/11/02 · openalex created_date 2022/11/09 · openalex updated_date 2026/07/28
We demonstrate that for expander graphs, for all ε> 0, there exists a data structure of size \widetildeO(nε-1) which can be used to return (1 + ε)-approximations to effective resistances in \widetildeO(1) time per query. Short of storing all effective resistances, previous best approaches could achieve \widetildeO(nε-2) size and \widetildeO(ε-2) time per query by storing Johnson-Lindenstrauss vectors for each vertex, or \widetildeO(nε-1) size and \widetildeO(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+1u for all u, can be used to recover (1 + ε)-approximations to the effective resistances, 2) In expander graphs, only \widetildeO(ε-1) coordinates of a vector similar to DL+1u are larger than ε. We give an efficient construction for such a data structure in \widetildeO(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 \widetildeO(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 \widetildeO(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.