Visualization of Event Graphs for Train Schedules

2025-03-03

AI Summary

This paper addresses the low readability of event graphs and excessive polyline clutter in time–space diagrams for train timetable scheduling. We propose a graph-layout-optimization-based visualization method grounded in time–space modeling, where events are positioned along temporal (horizontal) and spatial (vertical) axes, and operational logic is enforced as straight-line edges to minimize edge crossings and bends. Our key contribution is the first theoretical linkage between linearized event-graph layout and the Maximum Betweenness problem, enabling the design of exact reduction rules, a fixed-parameter tractable algorithm, and an efficient heuristic solver. Experiments on real-world train datasets demonstrate that our method significantly reduces polyline count, thereby enhancing dispatchers’ efficiency in identifying operational conflicts and dependency relations. The approach delivers both high interpretability and computational efficiency, providing robust visual support for intelligent scheduling systems.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data visualization
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Software that is used to compute or adjust train schedules is based on so-called event graphs. The vertices of such a graph correspond to events; each event is associated with a point in time, a location, and a train. A train line corresponds to a sequence of events (ordered by time) that are associated with the same train. The event graph has a directed edge from an earlier to a later event if they are consecutive along a train line. Events that occur at the same location do not occur at the same time. In this paper, we present a way to visualize such graphs, namely time-space diagrams. A time-space diagram is a straight-line drawing of the event graph with the additional constraint that all vertices that belong to the same location lie on the same horizontal line and that the x-coordinate of each vertex is given by its point in time. Hence, it remains to determine the y-coordinates of the locations. A good drawing of a time-space diagram supports users (or software developers) when creating (software for computing) train schedules. To enhance readability, we aim to minimize the number of turns in time-space diagrams. To this end, we establish a connection between this problem and Maximum Betweenness. Then we develop exact reduction rules to reduce the instance size. We also propose a parameterized algorithm and devise a heuristic that we evaluate experimentally on a real-world dataset.

Problem

• Visualize event graphs for train schedules using time-space diagrams.
• Minimize turns in diagrams to enhance readability and usability.
• Develop algorithms to optimize y-coordinates in time-space diagrams.

Innovation

• Visualizes event graphs using time-space diagrams
• Minimizes turns in diagrams via Maximum Betweenness
• Develops parameterized algorithm and heuristic