Parallel Reachability and Shortest Paths on Non-sparse Digraphs: Near-linear Work and Sub-square-root Depth
For researchers in parallel graph algorithms, this work provides the first near-linear-work algorithms with sub-square-root depth for reachability and shortest paths on non-sparse digraphs, breaking a long-standing barrier.
They present parallel algorithms for single-source reachability and shortest paths on directed graphs that achieve near-linear work and sub-square-root depth for dense graphs, specifically depth as low as n^0.136 for reachability when m = Ω(n^2), improving over the previous Ω(√n) depth bound.
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 $m=Ω(n^{2})$, 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 $Ω(\sqrt{n})$ depth in all density regimes.