  In this paper, we study the problem of fast constructions of source-wise
round-trip spanners in weighted directed graphs. For a source vertex set
$S\subseteq V$ in a graph $G(V,E)$, an $S$-sourcewise round-trip spanner of $G$
of stretch $k$ is a subgraph $H$ of $G$ such that for every pair of vertices
$u,v\in S\times V$, their round-trip distance in $H$ is at most $k$ times of
their round-trip distance in $G$. We show that for a graph $G(V,E)$ with $n$
vertices and $m$ edges, an $s$-sized source vertex set $S\subseteq V$ and an
integer $k>1$, there exists an algorithm that in time $O(ms^{1/k}\log^5n)$
constructs an $S$-sourcewise round-trip spanner of stretch $O(k\log n)$ and
$O(ns^{1/k}\log^2n)$ edges with high probability. Compared to the fast
algorithms for constructing all-pairs round-trip spanners \cite{PRS+18,CLR+20},
our algorithm improve the running time and the number of edges in the spanner
when $k$ is super-constant. Compared with the existing algorithm for
constructing source-wise round-trip spanners \cite{ZL17}, our algorithm
significantly improves their construction time $\Omega(\min\{ms,n^\omega\})$
(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
$O(\log n)$ 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
\cite{PRS+18} can be adapted to the source-wise setting.
