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Timetable Nodes for Public Transport Network

Andrii Rohovyi, Peter J. Stuckey, Toby Walsh

TL;DR

This work proposes a method that advances graph-based approaches by using different optimization techniques from computational geometry to speed up the search process in transport networks, and implements a new pre-computation step, which is called timetable nodes (TTN).

Abstract

Faster pathfinding in time-dependent transport networks is an important and challenging problem in navigation systems. There are two main types of transport networks: road networks for car driving and public transport route network. The solutions that work well in road networks, such as Time-dependent Contraction Hierarchies and other graph-based approaches, do not usually apply in transport networks. In transport networks, non-graph solutions such as CSA and RAPTOR show the best results compared to graph-based techniques. In our work, we propose a method that advances graph-based approaches by using different optimization techniques from computational geometry to speed up the search process in transport networks. We apply a new pre-computation step, which we call timetable nodes (TTN). Our inspiration comes from an iterative search problem in computational geometry. We implement two versions of the TTN: one uses a Combined Search Tree (TTN-CST), and the second uses Fractional Cascading (TTN-FC). Both of these approaches decrease the asymptotic complexity of reaching new nodes from $O(k\times \log|C|)$ to $O(k + \log(k) + \log(|C|))$, where $k$ is the number of outgoing edges from a node and $|C|$ is the size of the timetable information (total outgoing edges). Our solution suits any other time-dependent networks and can be integrated into other pathfinding algorithms. Our experiments indicate that this pre-computation significantly enhances the performance on high-density graphs. This study showcases how leveraging computational geometry can enhance pathfinding in transport networks, enabling faster pathfinding in scenarios involving large numbers of outgoing edges.

Timetable Nodes for Public Transport Network

TL;DR

This work proposes a method that advances graph-based approaches by using different optimization techniques from computational geometry to speed up the search process in transport networks, and implements a new pre-computation step, which is called timetable nodes (TTN).

Abstract

Faster pathfinding in time-dependent transport networks is an important and challenging problem in navigation systems. There are two main types of transport networks: road networks for car driving and public transport route network. The solutions that work well in road networks, such as Time-dependent Contraction Hierarchies and other graph-based approaches, do not usually apply in transport networks. In transport networks, non-graph solutions such as CSA and RAPTOR show the best results compared to graph-based techniques. In our work, we propose a method that advances graph-based approaches by using different optimization techniques from computational geometry to speed up the search process in transport networks. We apply a new pre-computation step, which we call timetable nodes (TTN). Our inspiration comes from an iterative search problem in computational geometry. We implement two versions of the TTN: one uses a Combined Search Tree (TTN-CST), and the second uses Fractional Cascading (TTN-FC). Both of these approaches decrease the asymptotic complexity of reaching new nodes from to , where is the number of outgoing edges from a node and is the size of the timetable information (total outgoing edges). Our solution suits any other time-dependent networks and can be integrated into other pathfinding algorithms. Our experiments indicate that this pre-computation significantly enhances the performance on high-density graphs. This study showcases how leveraging computational geometry can enhance pathfinding in transport networks, enabling faster pathfinding in scenarios involving large numbers of outgoing edges.

Paper Structure

This paper contains 14 sections, 1 theorem, 6 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Terminology
  4. Algorithms
  5. Timetable nodes (TTN)
  6. Combined Search Tree
  7. Fractional Cascading
  8. Sorting strategies
  9. Contraction Hierarchical Sorting
  10. Experiments
  11. Real city analysis
  12. Query Processing Time
  13. Synthetic graph analysis
  14. Conclusion and Future Work

Key Result

Lemma 1

For every optimal path $\textit{sp}(s;d;t)$ in $G$, there is a cost equivalent TCH-path $\langle s;\ldots;k \ldots;d \rangle$ whose prefix $\langle s;\ldots;k \rangle$ is an up path (i.e., $s < \mathcal{L}_k < \ldots < \mathcal{L}_{k+1} < \ldots < L_d$), and the suffix $\langle k;\ldots;d \rangle$ i

Figures (7)

  • Figure 1: A node $A$ in a transport network, with outgoing edges to $B$, $C$, and $D$.
  • Figure 2: FS_TCH_TTN_CST vs CSA runtime colored by Mean $E$. Visualization provides just samples, where the runtime of FS_TCH_TTN_CST is smaller than 90 ms.
  • Figure 3: FS_TCH_TTN_CST vs CSA runtime colored by percentage of ATF functions with timetables (% TT) . Visualization provides just samples, where the runtime of FS_TCH_TTN_CST is smaller than 90 ms.
  • Figure 4: Synthetic graph with 100% of edges with timetables (TT)
  • Figure 5: Synthetic graph, 20% of ATF with timetables (TT)
  • ...and 2 more figures

Theorems & Definitions (1)

  • Lemma 1: TCH-path