  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.
