Roundtrip Spanners with $(2k-1)$ Stretch

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 spanners. 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 $k\ge 1$, then remove the dependence of size on $W$ to give a roundtrip spanner with $(2k-1+o(1))$ 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 match the undirected $(2k-1)$-spanner with $O(k n^{1+1/k})$ edges [Alth\"ofer et al. '93] which is optimal under Erd\"os conjecture.