Generic Dijkstra: correctness and tractability

The recently-proposed generic Dijkstra algorithm finds shortest paths in networks with continuous and contiguous resources. The algorithm was proposed in the context of optical networks, but is applicable to networks with finite and discrete resources. The algorithm was published without a proof of correctness, and with a minor shortcoming. We provide that missing proof and offer a correction to the shortcoming. To prove the algorithm correct, we generalize the Bellman’s principle of optimality to algebraic structures with a partial ordering. By analyzing the size of the search space in the worst-case, we argue the stated problem is tractable. Thus we definitely answer a longstanding fundamental question of whether we can efficienlty find a shortest path in a network with discrete resources subject to the continuity and contiguity constraints: yes, we can.