Abstract
We present an efficient algorithm for solving local linear systems with a boundary condition using the Green’s function of a connected induced subgraph related to the system. We introduce the method of using the Dirichlet heat kernel pagerank<sup>1</sup> vector to approximate local solutions to linear systems in the graph Laplacian, satisfying given boundary conditions over a particular subset of vertices. With an efficient algorithm for approximating Dirichlet heat kernel pagerank, our Local Linear Solver algorithm computes an approximate local solution with multiplicative and additive error ε by performing _O_(ε<sup>−5</sup>_s_<sup>3</sup>log (_s_<sup>3</sup>ε<sup>−1</sup>)log _n_) random walk steps, where _n_ is the number of vertices in the full graph, and _s_ is the size of the local system on the induced subgraph.
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