Graph Sparsification by Effective Resistances

We present a nearly-linear time algorithm that produces high-quality sparsifiers of weighted graphs. Given as input a weighted graph $G=(V,E,w)$ and a parameter $\epsilon>0$, we produce a weighted subgraph $H=(V,\tilde{E},\tilde{w})$ of $G$ such that $|\tilde{E}|=O(n\log n/\epsilon^2)$ and for all vectors $x\in\R^V$ $(1-\epsilon)\sum{uv\in E}(x(u)-x(v))^2w{uv}\le \sum{uv\in\tilde{E}}(x(u)-x(v))^2\tilde{w}{uv} \le (1+\epsilon)\sum{uv\in E}(x(u)-x(v))^2w{uv}. ()$ This improves upon the sparsifiers constructed by Spielman and Teng, which had $O(n\log^c n)$ edges for some large constant $c$, and upon those of Bencz\'ur and Karger, which only satisfied () 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.