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 $S \times V$-direachability 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 $G = (V,E)$ 和一个指定源点集 $S \subseteq V$，$|S| = n^{\sigma}$（其中 $0 < \sigma \le 1$），$S \times V$ 有向可达性问题要求计算每个 $s \in S$ 在 $G$ 中可达的所有顶点集合。已知的朴素算法要么分别从每个源点运行 BFS/DFS，从而需要 $O(m \cdot n^{\sigma})$ 时间，要么在 $\tilde O(n^{\omega})$ 时间内计算 $G$ 的传递闭包，其中 $\omega < 2.371552\ldots$ 是矩阵乘法指数。因此，对于 $m = \Theta(n^{\mu})$ 条边的图，当前该问题的最优边界为 $\tilde O(n^{\min \{\mu + \sigma, \omega \}})$。我们的第一个贡献是针对该问题的算法，运行时间为 $\tilde O(n^{1 + \frac{2}{3} \omega(\sigma)})$，其中 $\omega(\sigma)$ 是矩形矩阵乘法指数。利用当前关于 $\omega(\sigma)$ 的最优估计，对于 $\tilde \sigma \le \sigma \le 0.53$，我们的指数优于 $\min \{2 + \sigma, \omega\}$，其中 $1/3 < \tilde \sigma < 0.3336$ 是一个普适常数。我们的第二个贡献是针对 $S \times V$ 有向可达性问题的算法序列 $\mathcal A_0, \mathcal A_1, \mathcal A_2, \ldots$。我们论证，在我们引入的特定假设下，对于每个 $\tilde \sigma \le \sigma < 1$，存在一个足够大的指标 $k = k(\sigma)$，使得在最稠密的情形 $\mu =2$ 下，$\mathcal A_k$ 改进了具有 $|S| = n^{\sigma}$ 的 $S \times V$ 有向可达性问题的当前最优边界。我们证明，要验证这一假设，只需设计一种能够以大致等同于普通 $(+, \cdot)$ 矩阵乘法的效率计算矩形最大-最小矩阵乘积的算法即可。我们的算法大量利用了 Kogan 和 Parter 近期提出的有向捷径构造。