In this work we revisit the fundamental Single-Source Shortest Paths (SSSP) problem with possibly negative edge weights. A recent breakthrough result by Bernstein, Nanongkai and Wulff-Nilsen established a near-linear $O(m \log^8(n) \log(W))$-time algorithm for negative-weight SSSP, where $W$ is an upper bound on the magnitude of the smallest negative-weight edge. In this work we improve the running time to $O(m \log^2(n) \log(nW) \log\log n)$, which is an improvement by nearly six log-factors. Some of these log-factors are easy to shave (e.g. replacing the priority queue used in Dijkstra's algorithm), while others are significantly more involved (e.g. to find negative cycles we design an algorithm reminiscent of noisy binary search and analyze it with drift analysis). As side results, we obtain an algorithm to compute the minimum cycle mean in the same running time as well as a new construction for computing Low-Diameter Decompositions in directed graphs.

翻译：本文重新探讨了基本的最短路径问题——单源最短路径（SSSP），并考虑了边权可能为负的情况。Bernstein、Nanongkai 和 Wulff-Nilsen 最近的一项突破性成果提出了一个近似线性的 $O(m \log^8(n) \log(W))$ 时间算法，用于处理负权 SSSP，其中 $W$ 是最小负权边绝对值的上界。本文将该运行时间改进为 $O(m \log^2(n) \log(nW) \log\log n)$，这几乎减少了六个对数因子。其中一些对数因子（如替换 Dijkstra 算法中使用的优先队列）较易消除，而另一些（如设计一种类似噪声二分搜索的算法并利用漂移分析来检测负环）则复杂得多。作为副产品，我们还获得了相同运行时间内计算最小环均值的算法，以及一种在有向图中构建低直径分解的新方法。