We present a randomized algorithm for the single-source shortest paths (SSSP) problem on directed graphs with arbitrary real-valued edge weights that runs in $n^{2+o(1)}$ time with high probability. This result yields the first almost linear-time algorithm for the problem on dense graphs ($m = Θ(n^2)$) and improves upon the best previously known bounds for moderately dense graphs ($m = ω(n^{1.306})$). Our approach builds on the hop-reduction via shortcutting framework introduced by Li, Li, Rao, and Zhang (2025), which iteratively augments the graph with shortcut edges to reduce the negative hop count of shortest paths. The central computational bottleneck in prior work is the cost of explicitly constructing these shortcuts in dense regions. We overcome this by introducing a new compression technique using auxiliary Steiner vertices. Specifically, we construct these vertices to represent large neighborhoods compactly in a structured manner, allowing us to efficiently generate and propagate shortcuts while strictly controlling the growth of vertex degrees and graph size.

翻译：本文提出一种针对具有任意实值边权的有向图的单源最短路径问题的随机算法，该算法以高概率在$n^{2+o(1)}$时间内运行。该结果首次为稠密图（$m = Θ(n^2)$）上的该问题提供了近乎线性时间的算法，并改进了先前对中等稠密图（$m = ω(n^{1.306})$）的最佳已知界。我们的方法基于Li、Li、Rao和Zhang（2025）提出的通过捷径进行跳数缩减的框架，该框架通过迭代地向图中添加捷径边来减少最短路径的负跳数。先前工作的核心计算瓶颈在于在稠密区域显式构建这些捷径的成本。我们通过引入一种使用辅助Steiner顶点的新压缩技术来克服这一瓶颈。具体而言，我们构建这些顶点以结构化方式紧凑地表示大型邻域，从而能够在严格控制顶点度数和图规模增长的同时，高效地生成和传播捷径。