A symmetric matrix is called a Laplacian if it has nonpositive off-diagonal entries and zero row sums. Since the seminal work of Spielman and Teng (2004) on solving Laplacian linear systems in nearly linear time, several algorithms have been designed for the task. Yet, the work of Kyng and Sachdeva (2016) remains the simplest and most practical sequential solver. They presented a solver purely based on random sampling and without graph-theoretic constructions such as low-stretch trees and sparsifiers. In this work, we extend the result of Kyng and Sachdeva to a simple parallel Laplacian solver with $O(m \log^3 n \log\log n)$ or $O((m + n\log^5 n)\log n \log\log n)$ work and $O(\log^2 n \log\log n)$ depth using the ideas of block Cholesky factorization from Kyng et al. (2016). Compared to the best known parallel Laplacian solvers that achieve polylogarithmic depth due to Lee et al. (2015), our solver achieves both better depth and, for dense graphs, better work.

翻译：对称矩阵若具有非正非对角元且行和为零，则称为拉普拉斯矩阵。自Spielman与Teng（2004）提出近线性时间求解拉普拉斯线性系统的开创性工作以来，研究者已针对该问题设计了多种算法。然而，Kyng与Sachdeva（2016）的工作仍是目前最简单且最实用的顺序求解器。他们提出的求解器纯粹基于随机采样，无需依赖低伸展树和稀疏化等图论构造。本文基于Kyng等人（2016）的块Cholesky分解思想，将Kyng与Sachdeva的结果推广至一种简单的并行拉普拉斯求解器，其工作复杂度为$O(m \log^3 n \log\log n)$或$O((m + n\log^5 n)\log n \log\log n)$，深度为$O(\log^2 n \log\log n)$。与Lee等人（2015）实现多对数深度且为目前最优的并行拉普拉斯求解器相比，我们的求解器在深度上更优，且对于稠密图在计算工作量方面亦具优势。