Parametric Shortest Paths in a Linearly Interpolated Graph

Abstract

We consider the parametric shortest paths problem in a linearly interpolated graph. Given two positively-weighted directed graphs $G_0=(V,E,ω_0)$ and $G_1=(V,E,ω_1),$ the linearly interpolated graph is the family of graphs $(1-λ)G_0+λG_1$, parameterized by $λ\in [0,1]$. The problem is to compute all distinct parametric shortest paths. We compute a data structure in $Θ(k|E|\log |V|)$ time, where~$k$ is the number of distinct parametric shortest paths over all~$λ\in [0,1]$ 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~$Θ(\log k)$ time.