Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication.

2020 
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 $0 0$. 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 $poly(\log\log n)$.
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