A nearly-mlogn time solver for SDD linear systems (1102.4842v4)

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}||_A < \epsilon ||\bar{x}||_A $ {$||\cdot||_A$ denotes the A-norm} in time $${\tilde O}(m\log n \log (1/\epsilon)).$$ 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 $\tilde{O}(m\log{n})$, 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.

• The paper presents an enhanced solver for SDD linear systems with a time complexity of \( \tilde O(m \log n \log(1/\epsilon)) \), improving upon previous algorithms.
• The algorithm utilizes improved graph sparsification techniques and constructs a chain of sparser graphs for enhanced preconditioning.
• The algorithm's speedup provides significant practical benefits for solving large-scale SDD linear systems used in many computational problems.

Overview of a Nearly-$m \log n$ Time Solver for SDD Linear Systems

The paper presents an optimized algorithm for solving symmetric diagonally dominant (SDD) linear systems, crucial in many computational problems such as spectral graph theory and optimization. These systems arise naturally when using the finite element method for partial differential equations, flow problems, and numerous applications within computer science, including network analysis and computer graphics.

Key Contributions

1. Algorithm Improvements: The authors introduce an enhanced solver that computes approximate solutions to $(n \times n)$ SDD linear systems efficiently, with a time complexity of $\tilde O(m \log n \log(1/\epsilon))$, where $m$ is the number of non-zero entries in the matrix $A$, and $n$ is the matrix size. This runtime improves upon the $\tilde O(m \log^2 n \log(1/\epsilon))$ from previous algorithms.
2. Graph Sparsification: The algorithm utilizes a 'preconditioning' strategy by constructing a chain of progressively sparser graphs, specifically employing improved graph sparsification techniques. Leveraging unknown properties from earlier work, the authors were able to achieve stronger preconditioning, leading to significant performance gains. The developed graph sparsification using low-stretch spanning trees enables faster processing.
3. Improved Low-Stretch Spanning Trees: The paper introduces a new algorithm that constructs low-stretch spanning trees achieving nearly optimal construction time, precisely $\tilde{O}(m\log{n})$, thus improving the efficiency of the preconditioning chain. This enhancement is especially pertinent as it addresses a previously acknowledged bottleneck in constructing these trees.

Theoretical and Practical Implications

The impact of this work extends both theoretically and practically. Theoretically, it provides concrete evidence against the suspected suboptimality of prior SDD solvers, enhancing our understanding of linear time approximation techniques. The practical benefits are considerable, given that SDD solvers are foundational components in a variety of large-scale computational problems, where the resultant speedup can lead to significant savings in computational resources.

Future Developments in AI

This refined algorithm suggests promising directions for further research into sparsification techniques and their roles in effective preconditioning for broader categories of sparse problems. As AI applications increasingly handle substantial datasets represented as graph structures, improvements in computational graph theories, like those introduced by this paper, will be critical in maintaining feasible execution times.

Overall, this paper contributes substantial advancements to the toolkit for solving SDD systems, paving the way for more efficient linear algebraic computations in both existing and future applications.