Solving Sparse, Symmetric, Diagonally-Dominant Linear Systems in Time O (m^{1.31})

Authors: Daniel A. Spielman and Shang-Hua Teng

Bibliographic Information: To appear in the Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science.


We present a linear-system solver that, given an $n$-by-$n$ symmetric positive semi-definite, diagonally dominant matrix $A$ with $m$ non-zero entries and an $n$-vector $b $, produces a vector $\tilde{x}$ within relative distance $\epsilon$ of the solution to $A x = b$ in time $O (m^{1.31} \log (n \kappa_{f} (A)/\epsilon )^{O (1)} )$, where $\kappa_{f} (A)$ is the log of the ratio of the largest to smallest non-zero eigenvalue of $A$. In particular, $\log (\kappa_{f} (A)) = O (b \log n)$, where $b $ is the logarithm of the ratio of the largest to smallest non-zero entry of $A$. If the graph of $A$ has genus $m^{2\theta }$ or does not have a $K_{m^{\theta }} $ minor, then the exponent of $m$ can be improved to the minimum of $1 + 5 \theta $ and $(9/8) (1+\theta )$. The key contribution of our work is an extension of Vaidya's techniques for constructing and analyzing combinatorial preconditioners.

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This version includes proofs that were omitted from the FOCS version, and cleans up some numerical details.
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Daniel A. Spielman
Last modified: Wed Mar 31 17:36:01 EST 2004