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2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (2011)
Palm Springs, California USA
Oct. 22, 2011 to Oct. 25, 2011
ISSN: 0272-5428
ISBN: 978-0-7695-4571-4
pp: 590-598
We present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an $n\times n$ symmetric diagonally dominant matrix $A$ with $m$ non-zero entries and a vector $b$ such that $A\bar{x} = b$ for some (unknown) vector $\bar{x}$, our algorithm computes a vector $x$ such that $||{x}-\bar{x}||_A1 in time. O tiled (m log n log (1/epsilon))^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 Otiled (mlog 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.
algorithms, spectral graph theory, linear systems, combinatorial preconditioning

R. Peng, I. Koutis and G. L. Miller, "A Nearly-m log n Time Solver for SDD Linear Systems," 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science(FOCS), Palm Springs, California USA, 2011, pp. 590-598.
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