Consider an undirected weighted graph $G = (V,E,w)$. We study the problem of computing $(1+\epsilon)$-approximate shortest paths for $S \times V$, for a subset $S \subseteq V$ of $|S| = n^r$ sources, for some $0 < r \le 1$. We devise a significantly improved algorithm for this problem in the entire range of parameter $r$, in both the classical centralized and the parallel (PRAM) models of computation, and in a wide range of $r$ in the distributed (Congested Clique) model. Specifically, our centralized algorithm for this problem requires time $\tilde{O}(|E| \cdot n^{o(1)} + n^{\omega(r)})$, where $n^{\omega(r)}$ is the time required to multiply an $n^r \times n$ matrix by an $n \times n$ one. Our PRAM algorithm has polylogarithmic time $(\log n)^{O(1/\rho)}$, and its work complexity is $\tilde{O}(|E| \cdot n^\rho + n^{\omega(r)})$, for any arbitrarily small constant $\rho >0$. In particular, for $r \le 0.313\ldots$, our centralized algorithm computes $S \times V$ $(1+\epsilon)$-approximate shortest paths in $n^{2 + o(1)}$ time. Our PRAM polylogarithmic-time algorithm has work complexity $O(|E| \cdot n^\rho + n^{2+o(1)})$, for any arbitrarily small constant $\rho >0$. Previously existing solutions either require centralized time/parallel work of $O(|E| \cdot |S|)$ or provide much weaker approximation guarantees. In the Congested Clique model, our algorithm solves the problem in polylogarithmic time for $|S| = n^r$ sources, for $r \le 0.655$, while previous state-of-the-art algorithms did so only for $r \le 1/2$. Moreover, it improves previous bounds for all $r > 1/2$. For unweighted graphs, the running time is improved further to $poly(\log\log n)$.

翻译：考虑一个非方向的加权G$ = (V,E,w) 。我们研究如何计算美元(1 ⁇ epsilon) 美元最短的路径 $S\time V$, 一个子集 $S\subsete V$ $ _S = n%r美元 美元 美元。 我们设计了一个显著改进的算法, 在整个参数范围内, 无论是在传统的中央和平行计算模式(PRAM) 美元, 还是在分布式的(Conged Crique) 模式中, 美元近于美元。 具体而言, 我们的中央算法需要时间 $\ subsetreet{O} + n% omga(r) 美元 。