Before revision:  The UGA algorithm^n, we propose a nonnegative UGA algorithm which needs O(mn^2+ n^3/\epsilon^2) time to find a 1+\epsilon/\beta{1-\epsilon/\beta}-spectral sparsifier with positive coefficients with sparsity }\le\lceil{\epsilon^2}}\rceil can output a \frac{(1+\epsilon)^2{(1-\epsilon)^2}} {1-O(\epsilon)}} -spectral sparsifier with \lceiln{\epsilon^2}\rceil edges in O(m+n^2/\epsilon^2) time  with m edges and n vertices for a general random graph satisfying a mild sufficient condition .
After revision:  For a general spectral sparsification problem, e.g., positive subset selection problem from a set of m vectors from \mathbb{R^n, we propose a nonnegative UGA algorithm which needs O(mn^2+ n^3/\epsilon^2) time to find a 1+\epsilon/\beta{1-\epsilon/\beta}-spectral sparsifier with positive coefficients with sparsity }\le\lceil{\epsilon^2}}\rceil can output a \frac{(1+\epsilon)^2{(1-\epsilon)^2}} {1-O(\epsilon)}} -spectral sparsifier with \lceiln{\epsilon^2}\rceil edges in O(m+n^2/\epsilon^2) time with m edges and n vertices for a general random graph satisfying a mild sufficient condition .