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Graph sparsification by effective resistances
| Venue: | SIAM J. Comput |
| Citations: | 143 - 9 self |
BibTeX
@ARTICLE{Spielman_graphsparsification,
author = {Daniel A. Spielman and Nikhil Srivastava},
title = {Graph sparsification by effective resistances},
journal = {SIAM J. Comput},
year = {},
pages = {2011}
}
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Abstract
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 ǫ> 0, we produce a weighted subgraph H = (V, ˜ E, ˜w) of G such that | ˜ E | = O(n log n/ǫ 2) and for all vectors x ∈ R V (1 − ǫ) ∑ (x(u) − x(v)) 2 wuv ≤ ∑ (x(u) − x(v)) 2 ˜wuv ≤ (1 + ǫ) ∑ (x(u) − x(v)) 2 wuv. (1) uv∈E uv ∈ ˜ E 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 Benczúr and Karger, which only satisfied (1) for x ∈ {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. uv∈E
Keyphrases
effective resistance graph sparsification weighted graph nearly-linear time algorithm data structure complete graph independent interest approximate effective resistance weighted subgraph expander graph large constant constant-degree sparsifiers key ingredient high-quality sparsifiers uv uv
