Path-Value Functions for Which Dijkstra’s Algorithm Returns Optimal Mapping

Abstract

Dijkstra’s algorithm (DA) is one of the most useful and efficient graph-search algorithms, which can be modified to solve many different problems. It is usually presented as a tool for finding a mapping which, for every vertex v, returns a shortest-length path to v from a fixed single source vertex. However, it is well known that DA returns also a correct optimal mapping when multiple sources are considered and for path-value functions more general than the standard path-length. The use of DA in such general setting can reduce many image processing operations to the computation of an optimum-path forest with path-cost function defined in terms of local image attributes. In this paper, we describe the general properties of a path-value function defined on an arbitrary finite graph which, provably, ensure that Dijkstra’s algorithm indeed returns an optimal mapping. We also provide the examples showing that the properties presented in a 2004 TPAMI paper on the image foresting transform, which were supposed to imply proper behavior of DA, are actually insufficient. Finally, we describe the properties of the path-value function of a graph that are provably necessary for the algorithm to return an optimal mapping.