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), at the expense of paying an extra O( \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} .