Parametric Shortest Paths in a Linearly Interpolated Graph

We consider the parametric shortest paths problem in a linearly interpolated graph. Given two positively-weighted directed graphs $G_0=(V,E,{\omega }_0)$ and $G_1=(V,E,{\omega }_1),$ the linearly interpolated graph is the family of graphs $(1-\lambda )G_0+\lambda G_1$, parameterized by $\lambda \in [0,1]$. The problem is to compute all distinct parametric shortest paths. We compute a data structure in $\mathrm{\Theta }(k\mathrm{\mid }E\mathrm{\mid }\log \mathrm{\mid }V\mathrm{\mid })$ time, where~$k$ is the number of distinct parametric shortest paths over all~$\lambda \in [0,1]$ that exist for a nontrivial interval of parameters, each corresponding to a linear function in a maximal sub-interval of $[0,1]$. Using this data structure, a shortest path query takes~$\mathrm{\Theta }(\log k)$ time.