Before revision:  It adapts the high-level algorithmic framework of  Laplacian system solvers, but requires several new tools: vertex elimination under constraints, a new family of graph ultra-sparsifiers, and accelerated proximal gradient methods with inexact proximal mapping computation.
After revision:  It adapts the high-level algorithmic framework of nearly linear time Laplacian system solvers, but requires several new tools: vertex elimination under constraints, a new family of graph ultra-sparsifiers, and accelerated proximal gradient methods with inexact proximal mapping computation.