Before revision:  We provide an O(m)-time randomized algorithm for the \ell_2-norm flow diffusion problem, obtaining the approximation factor of 1+1/\mathrm{poly local cut structure, we give an alternative approach for finding locally-biased low conductance cuts .
After revision:  We design an O(m)-time randomized algorithm for the \ell_2-norm flow diffusion problem, obtaining the approximation factor of 1+1/\mathrm{poly local cut structure, we give an alternative approach for finding locally-biased low conductance cuts .