This paper presents a randomized algorithm for the problem of single-source shortest paths on directed graphs with real (both positive and negative) edge weights. Given an input graph with $n$ vertices and $m$ edges, the algorithm completes in $\tilde{O}(mn^{8/9})$ time with high probability. For real-weighted graphs, this result constitutes the first asymptotic improvement over the classic $O(mn)$-time algorithm variously attributed to Shimbel, Bellman, Ford, and Moore.

翻译：本文提出一种针对有向图上具有实数（既包含正数也包含负数）边权重的单源最短路径问题的随机化算法。对于包含$n$个顶点和$m$条边的输入图，该算法以高概率在$\tilde{O}(mn^{8/9})$时间内完成。对于实数权重图，这一结果构成了对经典$O(mn)$时间算法的首次渐近改进，该经典算法通常归功于Shimbel、Bellman、Ford和Moore。