Parallel Reachability and Shortest Paths on Non-sparse Digraphs: Near-linear Work and Sub-square-root Depth

We present parallel algorithms for computing single-source reachability and shortest paths on directed $n$-vertex $m$-edge graphs using near-linear $\tilde{O}(m)$ work and $o(\sqrt{n})$ depth whenever $m\ge n^{1+o(1)}$. At the extreme of $m=\mathrm{\Omega }(n^2)$, our reachability and shortest path algorithms have depth only $n^{0.136}$ and $n^{0.25+o(1)}$, respectively. The state-of-the-art parallel algorithms with near-linear work for both problems require $\mathrm{\Omega }(\sqrt{n})$ depth in all density regimes.