Given an $n$-vertex $m$-edge digraph $G = (V,E)$ and a set $S \subseteq V$, $|S| = n^{\sigma}$ (for some $0 < \sigma \le 1$) of designated sources, the $S \times V$-direcahability problem is to compute for every $s \in S$, the set of all the vertices reachable from $s$ in $G$. Known naive algorithms for this problem either run a BFS/DFS separately from every source, and as a result require $O(m \cdot n^{\sigma})$ time, or compute the transitive closure of $G$ in $\tilde O(n^{\omega})$ time, where $\omega < 2.371552\ldots$ is the matrix multiplication exponent. Hence, the current state-of-the-art bound for the problem on graphs with $m = \Theta(n^{\mu})$ edges in $\tilde O(n^{\min \{\mu + \sigma, \omega \}})$. Our first contribution is an algorithm with running time $\tilde O(n^{1 + \tiny{\frac{2}{3}} \omega(\sigma)})$ for this problem, where $\omega(\sigma)$ is the rectangular matrix multiplication exponent. Using current state-of-the-art estimates on $\omega(\sigma)$, our exponent is better than $\min \{2 + \sigma, \omega \}$ for $\tilde \sigma \le \sigma \le 0.53$, where $1/3 < \tilde \sigma < 0.3336$ is a universal constant. Our second contribution is a sequence of algorithms $\mathcal A_0, \mathcal A_1, \mathcal A_2, \ldots$ for the $S \times V$-direachability problem. We argue that under a certain assumption that we introduce, for every $\tilde \sigma \le \sigma < 1$, there exists a sufficiently large index $k = k(\sigma)$ so that $\mathcal A_k$ improves upon the current state-of-the-art bounds for $S \times V$-direachability with $|S| = n^{\sigma}$, in the densest regime $\mu =2$. We show that to prove this assumption, it is sufficient to devise an algorithm that computes a rectangular max-min matrix product roughly as efficiently as ordinary $(+, \cdot)$ matrix product. Our algorithms heavily exploit recent constructions of directed shortcuts by Kogan and Parter.

翻译：给定一个 $n$ 个顶点 $m$ 条边的有向图 $G = (V,E)$ 和指定源点集 $S \subseteq V$（其中 $|S| = n^{\sigma}$，$\sigma$ 满足 $0 < \sigma \le 1$），$S \times V$ 有向可达性问题是计算每个源点 $s \in S$ 在 $G$ 中可到达的所有顶点集合。已知的朴素算法要么对每个源点分别执行 BFS/DFS，从而需要 $O(m \cdot n^{\sigma})$ 时间，要么计算 $G$ 的传递闭包，花费 $\tilde O(n^{\omega})$ 时间（其中 $\omega < 2.371552\ldots$ 为矩阵乘法指数）。因此，对于边数 $m = \Theta(n^{\mu})$ 的图，当前最优复杂度界为 $\tilde O(n^{\min \{\mu + \sigma, \omega \}})$。本文的第一个贡献是提出一个运行时间为 $\tilde O(n^{1 + \frac{2}{3}\omega(\sigma)})$ 的算法，其中 $\omega(\sigma)$ 为矩形矩阵乘法指数。利用当前对 $\omega(\sigma)$ 的最优估计，当 $\tilde \sigma \le \sigma \le 0.53$ 时，我们的指数优于 $\min\{2 + \sigma, \omega\}$，这里 $1/3 < \tilde \sigma < 0.3336$ 是一个通用常数。本文的第二个贡献是为 $S \times V$ 有向可达性问题设计了一组算法序列 $\mathcal A_0, \mathcal A_1, \mathcal A_2, \ldots$。我们论证，在特定假设下，对于每个 $\tilde \sigma \le \sigma < 1$，存在足够大的索引 $k = k(\sigma)$，使得在边最稠密情形 $\mu =2$ 中，$\mathcal A_k$ 改进了当前 $|S| = n^{\sigma}$ 的 $S \times V$ 有向可达性问题的最优复杂度界。我们表明，证明该假设只需设计一个算法，使其计算矩形最大-最小矩阵乘积的效率大致等同于普通 $(+, \cdot)$ 矩阵乘积。本文算法充分利用了 Kogan 和 Parter 最近构造的有向捷径技术。