between $u$ and $v$ is at most $R$ in $G$, then their roundtrip distance is at most $O(Rk\log\log n)$ in $H$.

The key insight of guiding our algorithm is the following: instead of partitioning the graph into disjoint pieces and recursing (as is done in [@PRSTV18]), we instead allow the pieces to overlap on the boundaries. This is justified by the following observation. Consider a subgraph $W$ of $G$, and let $W'$ denote the subgraph consisting of all vertices within distance $R$ of $W$. Then if we recursively build a roundtrip spanner on $W'$, then we are guaranteed that we can delete $W$ from our graph. Indeed, if $u \in W$ and the roundtrip distance between $u$ and $v$ is at most $R$, then $u, v \in W'$. This simple observation allows us to overcome the critical challenge in [@PRSTV18], arguing that that graph can be broken apart, while nevertheless preserving roundtrip distance.

This observation also forms the basis of an optimal spanner construction on unweighted undirected graphs, which appears in a book of Peleg (exercise 3 on page 188 in [@Peleg00]). Specifically, for any integer $k \ge 1$, we can construct a $(2k-1)$-spanner with $O(n^{1 + 1/k})$ edges in 