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.

翻译：Bellman与Ford在20世纪50年代提出的经典算法可在$O(mn)$时间内计算具有$n$个顶点、$m$条边且边权可能为负的加权图中的最短路径。事实上，该算法已成为本科算法课程中的常规教学内容。2023年，在近70年之后，Fineman \cite{fineman2024single}针对该问题提出了一个期望时间为$\tilde{O}(m n^{8/9})$的算法。Huang、Jin与Quanrud在Fineman这一惊人突破的基础上，进一步提出了$\tilde{O}(m n^{4/5})$时间算法。本文基于这些成果的思想，提出了一个期望时间为$\tilde{O}(m\sqrt{n})$的算法。关键改进在于一个简单观察：利用价格函数的约化成本可在线性时间内更新距离。这几乎立即改进了先前的工作。为得到最终的时间界，本文给出了Fineman所提出结构的递归版本。