In this paper, we study the problem of fast constructions of source-wise round-trip spanners in weighted directed graphs. For a source vertex set $S\subseteq V$ in a graph $G(V,E)$, an $S$-sourcewise round-trip spanner of $G$ of stretch $k$ is a subgraph $H$ of $G$ such that for every pair of vertices $u,v\in S\times V$, their round-trip distance in $H$ is at most $k$ times of their round-trip distance in $G$. We show that for a graph $G(V,E)$ with $n$ vertices and $m$ edges, an $s$-sized source vertex set $S\subseteq V$ and an integer $k>1$, there exists an algorithm that in time $O(ms^{1/k}\log^5n)$ constructs an $S$-sourcewise round-trip spanner of stretch $O(k\log n)$ and $O(ns^{1/k}\log^2n)$ edges with high probability. Compared to the fast algorithms for constructing all-pairs round-trip spanners \cite{PRS+18,CLR+20}, our algorithm improve the running time and the number of edges in the spanner when $k$ is super-constant. Compared with the existing algorithm for constructing source-wise round-trip spanners \cite{ZL17}, 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 $O(ms^{1/k}\log^5n)$, at the expense of paying an extra $O(\log n)$ in the stretch. As an important building block of the algorithm, we develop a graph partitioning algorithm to partition $G$ into clusters of bounded radius and prove that for every $u,v\in S\times V$ at small round-trip distance, the probability of separating them in different clusters is small. The algorithm takes the size of $S$ as input and does not need the knowledge of $S$. With the algorithm and a reachability vertex size estimation algorithm, we show that the recursive algorithm for constructing standard round-trip spanners \cite{PRS+18} can be adapted to the source-wise setting.
翻译:在本文中, 我们研究的是快速构建源端圆程仪的问题。 在加权定向图表中, 源的顶点设定了$S\subseteq V$, $G( V, E), 美元源的圆程仪为$G$, 利差为$G$, 折价为$H$, 也就是说每一对脊椎( $, v\ in stime V) 美元, 它们的圆程距离以美元计算, 以美元计算, 以美元计算, 以美元计算其圆程距离, 以美元计算。 对于美元( V, E) 美元, 美元, 以美元, 美元, 以美元, 美元, 以美元, 以美元, 美元, 以美元, 以美元, 以美元, 以美元, 以美元, 以美元, 以美元, 以美元, 以美元, 以美元, 以美元, 以美元, 平价, 以美元, 以美元, 以美元, 以美元, 以美元, 以美元, 平价