Before revision:  Our first result is an algorithm that maintains (1+\epsilon) -approximate distances from a set of s sources in (sm) total update time , assuming that s= n^{\Omega(1), \epsilon = \Omega(1) and |E|= n^{1+\Omega(1)}.
After revision:  Our first result is an algorithm that maintains (1+\epsilon) -approximate distances from a set of s sources in edges such that for any pair of nodes u, v \in V, there is a path in G \cup H of at most \beta hops, whose length is within a (1+\epsilon) factor from the distance between u and v in G. We provide a decremental algorithm for maintaining hopsets with a polylogarithmic hopbound, with a total update time , assuming that s= n^{\Omega(1), \epsilon = \Omega(1) and |E|= n^{1+\Omega(1)}.