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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
ABSTRACT
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.
INDEX TERMS
algorithms, spectral graph theory, linear systems, combinatorial preconditioning
CITATION

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.
doi:10.1109/FOCS.2011.85
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