time would be a major breakthrough.

This has motivated the study of efficient *approximation* algorithms for the girth. There has been extensive work on approximating the girth in *undirected* graphs [@IR77; @LL09; @RW12; @DKMS17]. Many such algorithms use the concept of a $\alpha$-*spanner* of a graph $G$, a fundamental combinatorial object which was introduced by Chew [@Chew89]. An $\alpha$-spanner of a graph $G$ is a subgraph of $G$ which multiplicatively preserves distances up to a factor of $\alpha$. It is well-known that $(2k-1)$-spanners with $O(n^{1+ 1/k})$ edges exist for any undirected weighted graph [@ADDJ93], and work on the efficient construction of such spanners [@TZ05; @LTZ05; @BS03] implies a $O(mn^\frac1k)$ time algorithm for $(2k-1)$-multiplicative girth approximation in undirected graphs. There has also been work on improved spanner constructions in the case of undirected unweighted graphs [@LL09; @RW12], and these algorithms also immediately imply algorithms for girth approximation in undirected unweighted graphs.

Therefore, in order to obtain efficient constant factor girth approximations in directed graphs, it is natural to study an analog of spanners in directed graphs. Unfortunately, approximately computing all pairs distances in directed graphs is a notoriously difficult problem and while sparse spanners do exist in all undirected graphs, they 