Sparsified Block Elimination for Directed Laplacians
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We show that the sparsified block elimination algorithm for solving undirected Laplacian linear systems from [Kyng-Lee-Peng-Sachdeva-Spielman STOC'16] directly works for directed Laplacians. Given access to a sparsification algorithm that, on graphs with n vertices and m edges, takes time 𝒯_ S(m) to output a sparsifier with 𝒩_ S(n) edges, our algorithm solves a directed Eulerian system on n vertices and m edges to ϵ relative accuracy in time O(𝒯_ S(m) + 𝒩_ S(n)lognlog(n/ϵ)) + Õ(𝒯_ S(𝒩_ S(n)) log n), where the Õ(·) notation hides loglog(n) factors. By previous results, this implies improved runtimes for linear systems in strongly connected directed graphs, PageRank matrices, and asymmetric M-matrices. When combined with slower constructions of smaller Eulerian sparsifiers based on short cycle decompositions, it also gives a solver that runs in O(n log^5n log(n / ϵ)) time after O(n^2 log^O(1) n) pre-processing. At the core of our analyses are constructions of augmented matrices whose Schur complements encode error matrices.
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