所有配对2-在$\mathcal{O}(n^ω\log n)$时间内的可达性
All-Pairs 2-Reachability in $\mathcal{O}(n^ω\log n)$ Time
摘要：
在$2$ -可达性问题中，我们给出了一个有向图$G$，我们希望确定对于给定的一对顶点$u$和$v$，从$u$到$v$是否有两条（边或顶点）不相交的路径。在本文中，我们提出了一个算法，计算$2$ -可达性信息的所有对顶点在$\mathcal{O}(n^ω\log n)$时间，其中$n$是顶点的数量和$ω$是矩阵乘法指数。因此，我们表明，所有对$2$ -可达性的运行时间仅在传递闭包的$\log$因子内。此外，我们的算法产生一个见证人（即，分离边或分离顶点），其中$2$ -可达性不成立。通过处理这些见证，我们可以在$\mathcal{O}(n^2)$额外的时间内计算$G$的所有边和顶点支配树，这反过来又使我们能够在$\mathcal{O}(1)$时间内回答各种连接查询。例如，我们可以在恒定时间内测试，对于任何一对查询顶点$u$和$v$以及任何查询边$e$，是否存在从$u$到$v$的避开边$e$的路径，或者对于任何查询顶点$u$、$v$和$w$，是否存在从$u$到$v$的避开顶点$w$的路径。
英文摘要：
In the $2$-reachability problem we are given a directed graph $G$ and we wish to determine if there are two (edge or vertex) disjoint paths from $u$ to $v$, for a given pair of vertices $u$ and $v$. In this paper, we present an algorithm that computes $2$-reachability information for all pairs of vertices in $\mathcal{O}(n^ω\log n)$ time, where $n$ is the number of vertices and $ω$ is the matrix multiplication exponent. Hence, we show that the running time of all-pairs $2$-reachability is only within a $\log$ factor of transitive closure. Moreover, our algorithm produces a witness (i.e., a separating edge or a separating vertex) for all pair of vertices where $2$-reachability does not hold. By processing these witnesses, we can compute all the edge- and vertex-dominator trees of $G$ in $\mathcal{O}(n^2)$ additional time, which in turn enables us to answer various connectivity queries in $\mathcal{O}(1)$ time. For instance, we can test in constant time if there is a path from $u$ to $v$ avoiding an edge $e$, for any pair of query vertices $u$ and $v$, and any query edge $e$, or if there is a path from $u$ to $v$ avoiding a vertex $w$, for any query vertices $u$, $v$, and $w$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
