  A roundtrip spanner of a directed graph $G$ is a subgraph of $G$ preserving
roundtrip distances approximately for all pairs of vertices. Despite extensive
research, there is still a small stretch gap between roundtrip spanners in
directed graphs and undirected graphs. For a directed graph with real edge
weights in $[1,W]$, we first propose a new deterministic algorithm that
constructs a roundtrip spanner with $(2k-1)$ stretch and $O(k n^{1+1/k}\log
(nW))$ edges for every integer $k> 1$, then remove the dependence of size on
$W$ to give a roundtrip spanner with $(2k-1)$ stretch and $O(k n^{1+1/k}\log
n)$ edges. While keeping the edge size small, our result improves the previous
$2k+\epsilon$ stretch roundtrip spanners in directed graphs [Roditty, Thorup,
Zwick'02; Zhu, Lam'18], and almost matches the undirected $(2k-1)$-spanner with
$O(n^{1+1/k})$ edges [Alth\"ofer et al. '93] when $k$ is a constant, which is
optimal under Erd\"os conjecture.
