A Fast Algorithm for Source-Wise Round-Trip Spanners

In this paper, we study the problem of efficiently constructing source-wise round-trip spanners in weighted directed graphs. For a source vertex set $S\subseteq V$ in a digraph $G(V,E)$, an $S$-source-wise round-trip spanner of $G$ of stretch $k$ is a subgraph $H$ of $G$ such that for every $u\in S,v\in V$, the round-trip distance between $u$ and $v$ in $H$ is at most $k$ times of the original distance in $G$. We show that, for a digraph $G(V,E)$ with $n$ vertices, $m$ edges and nonnegative edge weights, an $s$-sized source vertex set $S\subseteq V$ and a positive integer $k$, there exists an algorithm, in time $O(ms^{1/k}\log^5n)$, with high probability constructing an $S$-source-wise round-trip spanner of stretch $O(k\log n)$ and size $O(ns^{1/k}\log^2n)$. Compared with the state of the art for constructing source-wise round-trip spanners, our algorithm significantly improves their construction time $\Omega(\min\{ms,n^\omega\})$ (where $\omega \in [2,2.373)$ and 2.373 is the matrix multiplication exponent) to nearly linear $O(ms^{1/k}\log^5n)$, while still keeping a spanner stretch $O(k\log n)$ and size $O(ns^{1/k}\log^2n)$, asymptotically similar to their stretch $2k+\epsilon$ and size $O((k^2/\epsilon)ns^{1/k}\log(nw))$, respectively.