A Special Case of the Dynamization Problem for Least Cost Paths

Given a digraph G = (V; E) and a cost function C : E ! IR, which does not imply negative cost cycles, let us denote by G() the graph obtained from G by adding to the cost of each edge the positive constant ; then we want to compute the cost of the least cost path from a given origin r to each node v in the graph G() for diierent choices of 0, without having to run a least cost path algorithm everytime with a new cost function. Through a preprocessing of the given digraph based on the Bellmann-Ford algorithm, we will be able to obtain in time O((jEj + jV j)) the necessary information to generate a structure that will allow us to answer each query in time O(log), where is deened to be the length of the longest least cost path in the digraph. In particular we will see that the only possible candidates to become least cost paths in a graph G() are the least cost paths of bounded length in the original graph G; further we will show that nding which of these candidates will become least cost path in a G() and for which is equivalent to nding the lower envelope of at most lines, which, in turn, can be reduced to the lower convex hull problem for a set of ordered points.