Roundtrip Spanners with (2k-1) Stretch
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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/klog (nW)) edges for every integer k> 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/klog n) 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 match the undirected (2k-1)-spanner with O(k n^1+1/k) edges [Althöfer et al. '93] which is optimal under Erdös conjecture.
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