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.

翻译：我们考虑线性插值图中的参数最短路径问题。给定两个带正权重的有向图 $G_0=(V,E,ω_0)$ 和 $G_1=(V,E,ω_1)$，线性插值图是由参数 $λ\in [0,1]$ 定义的图族 $(1-λ)G_0+λG_1$。该问题旨在计算所有不同的参数最短路径。我们提出了一种数据结构，其计算时间复杂度为 $Θ(k|E|\log |V|)$，其中~$k$ 是在所有~$λ\in [0,1]$ 中，对于非平凡参数区间内存在的不同参数最短路径的数量，每条路径对应于 $[0,1]$ 中某个最大子区间上的线性函数。利用该数据结构，最短路径查询的时间复杂度为~$Θ(\log k)$。