Generic Dijkstra

TL;DR

This paper proves the correctness of the generic Dijkstra algorithm for shortest path problems with continuous and contiguous resources, extending its applicability and addressing previous shortcomings.

Contribution

It provides a formal proof of correctness, corrects a shortcoming, and generalizes the algorithm to algebraic structures with partial orderings.

Findings

The algorithm is correct with the provided proof.

The problem is computationally tractable in the worst case.

Efficient shortest path computation is possible under resource constraints.

Abstract

The recently-proposed generic Dijkstra algorithm finds shortest paths in networks with continuous and contiguous resources. While the algorithm was proposed in the context of optical networks (and is applicable to other networks with finite and discrete resources), we present the stated problem in a broader algorithmic setting of the greedy approach. 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 greedy approach and the Bellman 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 long-standing fundamental question of whether we can efficiently find a shortest path in a network with discrete resources subject to the continuity and contiguity constraints: yes, we can.