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 graphs. 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(kn1+1/klog(nW)) edges for every integer k>1, then remove the dependence of size on
W to give a roundtrip spanner with (2k−1) stretch and O(kn1+1/klogn) edges. While keeping the edge size small, our result improves the previous
2k+ϵ stretch roundtrip spanners in directed graphs [Roditty, Thorup,
Zwick'02; Zhu, Lam'18], and almost matches the undirected (2k−1)-spanner with
O(n1+1/k) edges [Alth\"ofer et al. '93] when k is a constant, which is
optimal under Erd\"os conjecture.Comment: 12 page