  Roundtrip spanners are the analog of spanners in directed graphs, where the
roundtrip metric is used as a notion of distance. Recent works have shown
existential results of roundtrip spanners nearly matching the undirected case,
but the time complexity for constructing roundtrip spanners is still widely
open.
  This paper focuses on developing fast algorithms for roundtrip spanners and
related problems. For any $n$-vertex directed graph $G$ with $m$ edges (with
non-negative edge weights), our results are as follows:
  - 3-roundtrip spanner faster than APSP: We give an
$\tilde{O}(m\sqrt{n})$-time algorithm that constructs a roundtrip spanner of
stretch $3$ and optimal size $O(n^{3/2})$. Previous constructions of roundtrip
spanners of the same size either required $\Omega(nm)$ time [Roditty, Thorup,
Zwick SODA'02; Cen, Duan, Gu ICALP'20], or had worse stretch $4$ [Chechik and
Lifshitz SODA'21].
  - Optimal roundtrip emulator in dense graphs: For integer $k\ge 3$, we give
an $O(kn^2\log n)$-time algorithm that constructs a roundtrip \emph{emulator}
of stretch $(2k-1)$ and size $O(kn^{1+1/k})$, which is optimal for constant $k$
under Erd\H{o}s' girth conjecture. Previous work of [Thorup and Zwick STOC'01]
implied a roundtrip emulator of the same size and stretch, but it required
$\Omega(nm)$ construction time. Our improved running time is near-optimal for
dense graphs.
  - Faster girth approximation in sparse graphs: We give an
$\tilde{O}(mn^{1/3})$-time algorithm that $4$-approximates the girth of a
directed graph. This can be compared with the previous $2$-approximation
algorithm in $\tilde{O}(n^2, m\sqrt{n})$ time by [Chechik and Lifshitz
SODA'21]. In sparse graphs, our algorithm achieves better running time at the
cost of a larger approximation ratio.
