Before revision:  In a subsequent algorithm, we establish the trade-off between approximation quality 2 (1+\frac{1{k}) and update time O(k^2\log n) for k\in \mathbb{N}.
After revision:  In a subsequent algorithm, we establish the trade-off between approximation quality 2 {k}) and update time O(k^2\log n) for k\in \mathbb{N}.