Randomized $\tilde{O}(m\sqrt{n})$ Bellman-Ford from Fineman and the Boilermakers

A classical algorithm by Bellman and Ford from the 1950's computes shortest paths in weighted graphs on $n$ vertices and $m$ edges with possibly negative weights in $O(mn)$ time. Indeed, this algorithm is taught regularly in undergraduate Algorithms courses. In 2023, after nearly 70 years, Fineman \cite{fineman2024single} developed an $\tilde{O}(m n^{8/9})$ expected time algorithm for this problem. Huang, Jin and Quanrud improved on Fineman's startling breakthrough by providing an $\tilde{O}(m n^{4/5} )$ time algorithm. This paper builds on ideas from those results to produce an $\tilde{O}(m\sqrt{n})$ expected time algorithm. The simple observation that distances can be updated with respect to the reduced costs for a price function in linear time is key to the improvement. This almost immediately improves the previous work. To produce the final bound, this paper provides recursive versions of Fineman's structures.