Graph Sparsification by Effective Resistances

We present a nearly-linear time algorithm that produces high-quality sparsifiers of weighted graphs. Given is input a weighted graph G = (V, E, w) and a parameter epsilon > 0, we produce a weighted subgraph H = (V, (E) over tilde, (w) over tilde) of G such that vertical bar(E) over tilde vertical bar = O(n log n/epsilon(2)) and for all vectors x is an element of R-V (1 - epsilon) Sigma(uv is an element of E) (x(u) - x(v))(2)w(uv) <= Sigma(uv is an element of(E) over tilde) (x(u) - x(v))(2)(w) over tilde (uv) <= (1 + epsilon) Sigma(uv is an element of E) (x(u) - x(v))(2)w(uv). (1) 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 Benczur and Karger, which only satisfied (1) for x is an element of {0, 1}(V). We conjecture the existence of sparsifiers with O(n) edges, noting that these would generalize the notion of expander graphs, which are constant-degree sparsifiers for the complete graph. A key ingredient in our algorithm is a subroutine of independent interest: a nearly-linear time algorithm that builds a data structure front which we can query the approximate effective resistance between any two vertices in a graph in O(log n) time.