for constant factor girth approximation in directed graphs and in turn show several improvements on the girth approximation algorithms and roundtrip spanner constructions in the work of Pachocki [*et al.*]{} [@PRSTV18]. Here and throughout the remainder of the paper we use $\tilde{O}(\cdot)$ notation to hide factors polylogarithmic in $n$, where $n$ is the number of vertices in the graph.

In \[sec:constantapprox\] we consider obtaining constant approximations to the girth. In particular we provide a randomized algorithm that obtains a $3$-approximation to the girth on graphs with non-negative integer edge weights in ${\tilde{O}}(m \sqrt{n})$ time. Up to logarithmic factors this matches the runtime that would be predicted from the fact that $(2k - 1)$-undirected spanners with ${\tilde{O}}(n^{1+1/k})$ edges can be constructed in ${\tilde{O}}(mn^{1/k})$ time for $k = 2$. Further, we show that this procedure can be used to with high probability obtain constant multiplicative roundtrip spanners in directed graphs with arbitrary edge weights in ${\tilde{O}}(m \sqrt{n})$ time.

\[thm:3girth\] For any directed graph $G$ with $n$ vertices, $m$ edges, integer non-negative edge weights, and unknown girth $g$ we can compute in $\tilde{O}(m \sqrt{n})$ time an estimate $g'$ such that $g \le g' \le 3g$ with high probability in $n$.

\[thm:const\_spanner\] For any 