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A {\em parametric weighted graph} is a graph whose edges are labeled withcontinuous real functions of a single common variable. For any instantiation ofthe variable, one obtains a standard edge-weighted graph. Parametric weightedgraph problems are generalizations of weighted graph problems, and arise invarious natural scenarios. Parametric weighted graph algorithms consist of twophases. A {\em preprocessing phase} whose input is a parametric weighted graph,and whose output is a data structure, the advice, that is later used by the{\em instantiation phase}, where a specific value for the variable is given.The instantiation phase outputs the solution to the (standard) weighted graphproblem that arises from the instantiation. The goal is to have the runningtime of the instantiation phase supersede the running time of any algorithmthat solves the weighted graph problem from scratch, by taking advantage of theadvice.  In this paper we construct several parametric algorithms for the shortestpath problem. For the case of linear function weights we present an algorithmfor the single source shortest path problem.

Two-phase algorithms for the parametric shortest path problem