Lower Bounds for Adaptive Relaxation-Based Algorithms for Single-Source Shortest Paths

We consider the classical single-source shortest path problem in directed weighted graphs. D.~Eppstein proved recently an $\mathrm{\Omega }(n^3)$ lower bound for oblivious algorithms that use relaxation operations to update the tentative distances from the source vertex. We generalize this result by extending this $\mathrm{\Omega }(n^3)$ lower bound to \emph{adaptive} algorithms that, in addition to relaxations, can perform queries involving some simple types of linear inequalities between edge weights and tentative distances. Our model captures as a special case the operations on tentative distances used by Dijkstra’s algorithm.