Before revision:  Let ndenote the }  number of sets in \mathcal{S number of elements in the universe and the maximum frequency of an element across all the updates.
After revision:  Let ndenote the } , a challenge is to minimize the update time while guaranteeing close to the optimal \min(O(\log n), f) approximation factor. (Throughout, m, n, f, and C are parameters denoting the maximum number of sets in \mathcal{S number of elements in the universe and the maximum frequency of an element across all the updates.