Using petal-decompositions to build a low stretch spanning tree
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
A fast solver for a class of linear systems
Communications of the ACM
Algorithms, graph theory, and the solution of laplacian linear equations
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
Runtime guarantees for regression problems
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Efficient preconditioning of laplacian matrices for computer graphics
ACM Transactions on Graphics (TOG) - SIGGRAPH 2013 Conference Proceedings
A new approach to computing maximum flows using electrical flows
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
A simple, combinatorial algorithm for solving SDD systems in nearly-linear time
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Spectral sparsification of graphs: theory and algorithms
Communications of the ACM
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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.