Parametric Shortest Paths in a Linearly Interpolated Graph

TL;DR

This paper introduces an efficient method to compute all parametric shortest paths in a linearly interpolated graph, enabling quick queries and handling multiple path variations across the parameter range.

Contribution

The authors develop a data structure that computes all distinct parametric shortest paths in a linearly interpolated graph efficiently, with query time logarithmic in the number of paths.

Findings

Data structure construction time is Θ(k|E| log |V|)

Query time for shortest paths is Θ(log k)

Handles multiple path variations over parameter range efficiently

Abstract

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 $\Theta(k|E|\log |V|)$ 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~$\Theta(\log k)$ time.