Papers › Graph Sparsification by Effective Resistances
Graph Sparsification by Effective Resistances
Daniel A. Spielman, Nikhil Srivastava
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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,Ẽ,w̃) of G such that |Ẽ|=O(nlogn/ϵ²) and for all vectors x∈ⱽ (1-ϵ)∑_(uv∈E)(x(u)-x(v))²wᵤᵥ≤∑_(uv∈Ẽ)(x(u)-x(v))²w̃ᵤᵥ ≤(1+ϵ)∑_(uv∈E)(x(u)-x(v))²wᵤᵥ. (*) This improves upon the sparsifiers constructed by Spielman and Teng, which had O(nlogᶜ n) edges for some large constant c, and upon those of Bencz\'ur and Karger, which only satisfied (*) for x∈{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(logn) time.
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