Solving SDD linear systems in nearly mlog1/2n time

We show an algorithm for solving symmetric diagonally dominant (SDD) linear systems with m non-zero entries to a relative error of ε in O(m log1/2 n logc n log(1/ε)) time. Our approach follows the recursive preconditioning framework, which aims to reduce graphs to trees using iterative methods. We improve two key components of this framework: random sampling and tree embeddings. Both of these components are used in a variety of other algorithms, and our approach also extends to the dual problem of computing electrical flows. We show that preconditioners constructed by random sampling can perform well without meeting the standard requirements of iterative methods. In the graph setting, this leads to ultra-sparsifiers that have optimal behavior in expectation. The improved running time makes previous low stretch embedding algorithms the running time bottleneck in this framework. In our analysis, we relax the requirement of these embeddings to snowflake spaces. We then obtain a two-pass approach algorithm for constructing optimal embeddings in snowflake spaces that runs in O(m log log n) time. This algorithm is also readily parallelizable.