Before revision:  (where \omega \in [2,2.373) and 2.373 is the matrix multiplication exponent) to nearly linear O(ms^{1/k}\log^5n), while still keeping a spanner stretch O(k \log n) and size O(ns^{1/k\log^2n), asymptotically similar to their stretch 2k+\epsilon and size O((k^2/\epsilon)ns^{1/k}\log(nw)), respectively}  }\hspace{0pt}%DIFAUXCMD can be adapted to the source-wise setting} .
After revision:  (where \omega \in [2,2.373) and 2.373 is the matrix multiplication exponent) to nearly linear O(ms^{1/k}\log^5n), while still keeping a spanner stretch O(k \log n) and size O(ns^{1/k\log^2n), asymptotically similar to their stretch 2k+\epsilon and size O((k^2/\epsilon)ns^{1/k}\log(nw)), respectively} in the stretch. As an important building block of the algorithm, we develop a graph partitioning algorithm to partition G into clusters of bounded radius and prove that for every u,v\in S\times V at small round-trip distance, the probability of separating them in different clusters is small. The algorithm takes the size of S as input and does not need the knowledge of S. With the algorithm and a reachability vertex size estimation algorithm, we show that the recursive algorithm for constructing standard round-trip spanners \mbox{%DIFAUXCMD \cite{PRS+18 }\hspace{0pt}%DIFAUXCMD can be adapted to the source-wise setting} .