# Sparsified Block Elimination for Directed Laplacians

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 ๐ฏ_ S(m) to output a sparsifier with ๐ฉ_ S(n) edges, our algorithm solves a directed Eulerian system on n vertices and m edges to ฯต relative accuracy in time O(๐ฏ_ S(m) + ๐ฉ_ S(n)lognlog(n/ฯต)) + ร(๐ฏ_ S(๐ฉ_ S(n)) log n), where the ร(ยท) 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 / ฯต)) 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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