Dual Algorithms for the Shortest Path Tree Problem

We consider dual approaches for the Shortest Path Tree Problem. After a brief introduction to the problem, we review the most important dual algorithms which have been described in the literature for its solution, and propose a new family of dual ascent algorithms. In these algorithms, "local" and "global" dual updating operations are performed at the nodes in order to enlarge a partial shortest path tree, which at the beginning contains only the root node, until a shortest path tree is found. Several kinds of dual updating operations are proposed, which allow one to derive different dual algorithms from a general schema. One of them, in particular, which is based only on global operations, can be viewed as a dual interpretation of Dijkstra's classical algorithm. Due to their structure, all the proposed approaches are suitable for parallel implementations. They are also suitable for reoptimization approaches, when the computation of shortest paths from different root nodes is required.