The Bellman-Ford algorithm for single-source shortest paths repeatedly updates tentative distances in an operation called relaxing an edge. In several important applications a non-adaptive (oblivious) implementation is preferred, which means fixing the entire sequence of relaxations upfront, independent of the edge-weights. In a dense graph on $n$ vertices, the algorithm in its standard form performs $(1 + o(1))n^3$ relaxations. An improvement by Yen from 1970 reduces the number of relaxations by a factor of two. We show that no further constant-factor improvements are possible, and every non-adaptive deterministic algorithm based on relaxations must perform $(\frac{1}{2} - o(1))n^3$ steps. This improves an earlier lower bound of Eppstein of $(\frac{1}{6} - o(1))n^3$. Given that a non-adaptive randomized variant of Bellman-Ford with at most $(\frac{1}{3} + o(1))n^3$ relaxations (with high probability) is known, our result implies a strict separation between deterministic and randomized strategies, answering an open question of Eppstein.

翻译：用于单源最短路径的Bellman-Ford算法通过名为“松弛边”的操作反复更新暂定距离。在若干重要应用中，非自适应（ oblivious）实现更受青睐，这意味着预先固定整个松弛序列，且独立于边的权重。在包含$n$个顶点的稠密图中，该算法在标准形式下执行$(1 + o(1))n^3$次松弛操作。Yen在1970年提出的一项改进将松弛次数减半。我们证明不可能再获得任何常数因子的改进，且每个基于松弛的非自适应确定性算法必须执行$(\frac{1}{2} - o(1))n^3$步。这改进了Eppstein此前提出的$(\frac{1}{6} - o(1))n^3$下界。鉴于已知存在一种非自适应随机化Bellman-Ford变体，其松弛次数最多为$(\frac{1}{3} + o(1))n^3$（以高概率成立），我们的结果意味着确定性与随机化策略之间存在严格分离，从而回答了Eppstein的一个开放问题。