We consider single-source shortest path algorithms that perform a sequence of relaxation steps whose ordering depends only on the input graph structure and not on its weights or the results of prior steps. Each step examines one edge of the graph, and replaces the tentative distance to the endpoint of the edge by its minimum with the tentative distance to the start of the edge, plus the edge length. As we prove, among such algorithms, the Bellman-Ford algorithm has optimal complexity for dense graphs and near-optimal complexity for sparse graphs, as a function of the number of edges and vertices in the given graph. Our analysis holds both for deterministic algorithms and for randomized algorithms that find shortest path distances with high probability.

翻译：我们考虑单源最短路径算法，这类算法执行一系列松弛步骤，其顺序仅依赖于输入图的结构，而不依赖于边的权重或先前步骤的结果。每个步骤检查图中的一条边，并将该边终点的暂定距离更新为其当前值与起点暂定距离加上边长度的较小值。正如我们所证明的，在这类算法中，Bellman-Ford算法在稠密图上具有最优复杂度，在稀疏图上具有近最优复杂度，这是基于给定图中边数和顶点数的函数。我们的分析适用于确定性算法和能以高概率找到最短路径距离的随机化算法。