The paper constructs a for-each spectral sparsifier in O-tilde(n/ε) streaming space, breaking the Ω(n/ε^2) for-all sparsifier barrier, and uses it for near-optimal minimum cut and effective resistance algorithms.
Woodruff, and Samson Zhou
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Space Complexity of Minimum Cut Problems in Single-Pass Streams
The paper constructs a for-each spectral sparsifier in O-tilde(n/ε) streaming space, breaking the Ω(n/ε^2) for-all sparsifier barrier, and uses it for near-optimal minimum cut and effective resistance algorithms.