 - ( 1+\epsilon)-approximate shortest paths from a set of s sources in (s) amortized update time, assuming that s= n^{\Omega(1), and |E|= n^{1+\Omega(1)}.
<meaning-changed>  - ( 1+\epsilon)-approximate shortest paths from a set of s sources in (s) amortized update time, assuming that s= n^{\Omega(1), and |E|= n^{1+\Omega(1)}.
 By applying our decremental hopset construction, we get improved or near optimal bounds for several distance problems. Most importantly, we show how to decrementally maintain ( 1+\epsilon)-approximate shortest paths from a set of s sources in (s) amortized update time, assuming that s= n^{\Omega(1), and |E|= n^{1+\Omega(1)}.