Algorithms for Shortest Path Tour Problem

Abstract

Carpooling route planning becomes an important problem with the growth of low-carbon traffic systems. When each passenger has multiple potential pick-up/drop-off locations, the problem will be more challenging. In the paper, we discussed a simplified carpooling route planning problem, namely the Shortest Path Tour Problem (SPTP), whose aim is to find a single-origin single-destination shortest path through an ordered sequence of disjoint node subsets. We propose Stage Dijkstra and Global Dijkstra algorithms to find the optimal shortest path, with the time complexity of $O\left(l\right(n+m)\log n)$ and $O\left(l\right(n+m)\log (ln\left)\right)$ respectively, where l represents the number of node subsets. To the best of our knowledge, $O\left(l\right(n+m)\log n)$ is the best time complexity of the exact algorithms for SPTP. Besides, the Stage Dijkstra and Global Dijkstra algorithms both have the linear space complexity, which is highly suitable for resource-constrained environments. Experiments conducted on large-scale road networks and synthetic datasets demonstrate the effectiveness and efficiency of our proposed algorithms in terms of running time and memory consumption.