TY - GEN
T1 - A nearly-m log n time solver for SDD linear systems
AU - Koutis, Ioannis
AU - Miller, Gary L.
AU - Peng, Richard
PY - 2011
Y1 - 2011
N2 - We present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an n x n symmetric diagonally dominant matrix A with m non-zero entries and a vector b such that Ax̄ = b for some (unknown) vector x̄, our algorithm computes a vector x such that ∥x-x̄∥ A<ε∥x̄∥ A in time. Õ (m log n log (1/ε)). 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(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.
AB - We present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an n x n symmetric diagonally dominant matrix A with m non-zero entries and a vector b such that Ax̄ = b for some (unknown) vector x̄, our algorithm computes a vector x such that ∥x-x̄∥ A<ε∥x̄∥ A in time. Õ (m log n log (1/ε)). 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(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.
KW - algorithms
KW - combinatorial preconditioning
KW - linear systems
KW - spectral graph theory
UR - http://www.scopus.com/inward/record.url?scp=84863329239&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84863329239&partnerID=8YFLogxK
U2 - 10.1109/FOCS.2011.85
DO - 10.1109/FOCS.2011.85
M3 - Conference contribution
AN - SCOPUS:84863329239
SN - 9780769545714
T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
SP - 590
EP - 598
BT - Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011
T2 - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011
Y2 - 22 October 2011 through 25 October 2011
ER -