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:  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, where \beta is the ratio between the smallest length and largest length of the vectors. The convergence of the nonnegative UGA algorithm will be established. For the graph sparsification problem, another UGA algorithm will be proposed which 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 .