15. Maintenance of Multi-level Overlay Graphs for Timetable Queries

In railways systems the timetable is typically represented as a weighted digraph on which itinerary queries are answered by shortest path algorithms, usually running Dijkstra's algorithm. Due to the continuously growing size of real-world graphs, there is a constant need for faster algorithms and many techniques have been devised to heuristically speed up Dijkstra's algorithm. One of these techniques is the multi-level overlay graph, that has been recently introduced and shown to be experimentally efficient, especially when applied to timetable information. In many practical application major disruptions to the normal operation cannot be completely avoided because of the complexity of the underlying systems. Timetable information update after disruptions is considered one of the weakest points in current railway systems, and this determines the need for an effective online redesign and update of the shortest paths information as a consequence of disruptions. In this paper, we make a step forward toward this direction by showing some theoretical properties of multi-level overlay graphs that lead us to the definition of a new data structure for the dynamic maintenance of a multi-level overlay graph of a given graph G while weight decrease or weight increase operations are performed on G. Our solution is theoretically faster than the recomputation from scratch and allows fast queries.