We present a distributed algorithm for solving Laplacian systems on strongly connected directed graphs, the first that can be analyzed in terms of the underlying graph's parameters. Our distributed solver works for a large and important class of Laplacian systems that we call “one-sink” Laplacian systems, which includes the important electrical flow computation problem. Specifically, given $D_{\text{out}}$ as the diagonal out-degree matrix and $\overrightarrow{A}$ as the adjacency matrix of the directed graph with $L=D_{\text{out}}-\overrightarrow{A}$, our solver can produce solutions for systems of the form ${\mathit{x}}^{T}L={\mathit{b}}^T$, where exactly one of the coordinates of b is negative. Our solver takes $\tilde{O}(t_{\text{hit}}{d}_{\max }^{\text{in}})$ time (where $\tilde{O}$ hides $\text{poly}\log n$ factors) to produce an approximate solution where $t_{\text{hit}}$ is the worst-case hitting time of the random walk on the graph, which is $\mathrm{\Theta }(n)$ for a large set of important graphs and $d_{\max }^{\text{in}}$ is the maximum in-degree of the graph.

我们提出了一种用于在强连通有向图上求解拉普拉斯系统的分布式算法，该算法可以根据基础图的参数进行分析。我们的分布式求解器可用于一大类重要的拉普拉斯系统，我们称之为“单沉”拉普拉斯系统，其中包括重要的电流计算问题。具体来说，给定$d_{\text{出}}$ 作为对角度矩阵 $\overrightarrow{\mathrm{一种}}$ 作为有向图的邻接矩阵 $\mathrm{大号}=d_{\text{出}}-\overrightarrow{\mathrm{一种}}$，我们的求解器可以为以下形式的系统提供解决方案 ${\mathit{X}}^{Ť}\mathrm{大号}={\mathit{b}}^{Ť}$，其中b的坐标之一恰好为负。我们的求解器需要$\stackrel{〜}{Ø}（{Ť}_{\text{击中}}{d}_{\mathrm{最高}}^{\text{在}}）$ 时间（其中 $\stackrel{〜}{Ø}$ 皮 $\text{聚}\mathrm{日志}ñ$ 因子）以生成近似解 ${Ť}_{\text{击中}}$ 是图上随机游走的最坏情况的命中时间，即 $\mathrm{\Theta }（ñ）$ 对于大量的重要图和 $d_{\mathrm{最高}}^{\text{在}}$ 是图形的最大度数。