Key points are not available for this paper at this time.
We present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an n×n symmetric diagonally dominant matrix A with m non-zero entries and a vector b such that Ax̅ = b for some (unknown) vector x̅, our algorithm computes a vector x such that ∥x-x̅∥ A ≤ϵ∥x̅∥ A 1 in time Õ (m log n log (1/ϵ)) 2 . The solver utilizes in a standard way a 'preconditioning' chain of progressively sparser graphs. To claim the faster running time we make a two-fold improvement in the algorithm for constructing the chain. The new chain exploits previously unknown properties of the graph sparsification algorithm given in Koutis,Miller,Peng, FOCS 2010, allowing for stronger preconditioning properties.We also present an algorithm of independent interest that constructs nearly-tight low-stretch spanning trees in time Õ (m log n), a factor of O (log n) faster than the algorithm in Abraham,Bartal,Neiman, FOCS 2008. This speedup directly reflects on the construction time of the preconditioning chain.
Koutis et al. (Sat,) studied this question.