Non-adaptive Bellman-Ford: Yen's improvement is optimal

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))n³ 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 (1/2 - o(1))n³ steps. This improves an earlier lower bound of Eppstein of (1/6 - o(1))n³. Given that a non-adaptive randomized variant of Bellman-Ford with at most (1/3 + o(1))n³ relaxations (with high probability) is known, our result implies a strict separation between deterministic and randomized strategies, answering an open question of Eppstein.