Before revision:  with n vertices , m edgesand nonnegative edge weights , an s-sized source vertex set S\subseteq V and a positive integer k , there exists an algorithm , in time O(ms^{1/k}\log^5n) , with high probability constructing an S-source-wise round-trip spanner of stretch O(k\log n) and size O(ns^{1/k}\log^2n) . Compared with the state of the art}\hspace{0pt}%DIFAUXCMD , our algorithm improve the running time and the number of edges in the spanner when k is super-constant.
After revision:  with n vertices , m edgesand nonnegative edge weights , an s-sized source vertex set S\subseteq V and a positive integer k , there exists an algorithm , in time O(ms^{1/k}\log^5n) , with high probability constructing an S-source-wise round-trip spanner of stretch O(k\log n) and size O(ns^{1/k}\log^2n) edges with high probability. Compared to the fast algorithms for constructing all-pairs round-trip spanners \mbox{%DIFAUXCMD \cite{PRS+18,CLR+20}\hspace{0pt}%DIFAUXCMD , our algorithm improve the running time and the number of edges in the spanner when k is super-constant.