A new approach to the maximum-flow problem
Journal of the ACM (JACM)
Approximating s-t minimum cuts in Õ(n2) time
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Graph Embeddings and Laplacian Eigenvalues
SIAM Journal on Matrix Analysis and Applications
Pass efficient algorithms for approximating large matrices
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Database-friendly random projections: Johnson-Lindenstrauss with binary coins
Journal of Computer and System Sciences - Special issu on PODS 2001
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Fast monte-carlo algorithms for finding low-rank approximations
Journal of the ACM (JACM)
Fast Monte Carlo Algorithms for Matrices II: Computing a Low-Rank Approximation to a Matrix
SIAM Journal on Computing
Fast computation of low-rank matrix approximations
Journal of the ACM (JACM)
Sampling from large matrices: An approach through geometric functional analysis
Journal of the ACM (JACM)
IEEE Transactions on Knowledge and Data Engineering
Genetic clustering of social networks using random walks
Computational Statistics & Data Analysis
Proceedings of the forty-first annual ACM symposium on Theory of computing
Graph partitioning using single commodity flows
Journal of the ACM (JACM)
Spectral Sparsification of Graphs
SIAM Journal on Computing
A fast random sampling algorithm for sparsifying matrices
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Sparsification and sampling of networks for collective classification
SBP'13 Proceedings of the 6th international conference on Social Computing, Behavioral-Cultural Modeling and Prediction
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 a nearly linear time algorithm that produces high-quality spectral sparsifiers of weighted graphs. Given as input a weighted graph $G=(V,E,w)$ and a parameter $\epsilon0$, we produce a weighted subgraph $H=(V,\tilde{E},\tilde{w})$ of $G$ such that $|\tilde{E}|=O(n\log n/\epsilon^2)$ and all $x\in\mathbb{R}^V$ satisfy $(1-\epsilon)\sum_{uv\in E}\,(x(u)-x(v))^2w_{uv}\leq\sum_{uv\in\tilde{E}}\,(x(u)-x(v))^2\tilde{w}_{uv}\leq(1+\epsilon)\sum_{uv\in E}\,(x(u)-x(v))^2w_{uv}$. This improves upon the spectral sparsifiers constructed by Spielman and Teng, which had $O(n\log^{c}n)$ edges for some large constant $c$, and upon the cut sparsifiers of Benczúr and Karger, which only satisfied these inequalities for $x\in\{0,1\}^V$. A key ingredient in our algorithm is a subroutine of independent interest: a nearly linear time algorithm that builds a data structure from which we can query the approximate effective resistance between any two vertices in a graph in $O(\log n)$ time.