Faster single-source shortest paths with negative real weights via proper hop distance

The textbook algorithm for single-source shortest paths with real-valued edge weights runs in $O(mn)$ time on a graph with $m$ edges and $n$ vertices. A recent breakthrough algorithm by Fineman [Fin24] takes $\tilde{O}(m{n}^{8\mathrm{/}9})$ randomized time. We present an $\tilde{O}(m{n}^{4\mathrm{/}5})$ randomized time algorithm building on ideas from [Fin24].