Before revision:  In this case , the running time of our algorithms reduces to O(n ) using core-sets that support: streaming, dynamic, and distributed parallel computations (e.g. on the cloud) in poly-logarithmic update time.
After revision:  If \pi is given , the running time of our algorithms reduces to O(n ) using core-sets that support: streaming, dynamic, and distributed parallel computations (e.g. on the cloud) in poly-logarithmic update time.