Graph sparsification by effective resistances 论文

摘要

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 ε>0, we produce a weighted subgraph H=(V,~E,~w) of G such that |~E|=O(n log n/ε2) and for all vectors x in RV. (1-ε) ∑uv ∈ E (x(u)-x(v))2wuv≤ ∑uv in ~E(x(u)-x(v))2~wuv ≤ (1+ε)∑uv ∈ E(x(u)-x(v))2wuv. This improves upon the sparsifiers constructed by Spielman and Teng, which had O(n logc n) edges for some large constant c, and upon those of Benczur and Karger, which only satisfied (1) for x in {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 from which we can query the approximate effective resistance between any two vertices in a graph in O(log n) time.