# A Distributed Laplacian Solver and its Applications to Electrical Flow and Random Spanning Tree Computation

We present a distributed solver for a large and important class of Laplacian systems that we call "one-sink" Laplacian systems. Specifically, our solver can produce solutions for systems of the form Lx = b where exactly one of the coordinates of b is negative. Our solver is an organically distributed algorithm that takes O(n/λ_2^L) rounds for bounded degree graphs to produce an approximate solution where λ_2^L is the second smallest eigenvalue of the Laplacian matrix of graph, also known as the Fiedler value or algebraic connectivity of graph. The class of one-sink Laplacians includes the important voltage computation problem and allows us to compute the effective resistance between nodes in a distributed manner. As a result, our Laplacian solver can be used to adapt the approach by Kelner and Mądry (2009) to give the first distributed algorithm to compute approximate random spanning trees efficiently. Our solver, which we call "Distributed Random Walk-based Laplacian Solver" (DRW-LSolve) works by approximating the stationary distribution of a multi-dimensional Markov chain. This chain describes the evolution of a "data collection" process where each node v for which b_v > 0 generates data packets with a rate proportional to b_v and the node for which b_v < 0 acts as a sink. The nodes of the graph relay the packets, staging them in their queues and transmitting one at a time. We show that when this multidimensional chain is ergodic the vector whose vth coordinate is proportional to the probability at stationarity of the queue at v being non-empty is a solution to Lx = b.

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