We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.

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Route Planning in Transportation Networks

Algorithms for route planning in transportation networks have recently undergone a rapid development, leading to methods that are up to three million times faster than Dijkstra's algorithm. We give an overview of the techniques enabling this development and point out frontiers of ongoing research on more challenging variants of the problem that include dynamically changing networks, time-dependent routing, and flexible objective functions.

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Engineering Route Planning Algorithms

Minmax Robustness for Multi-objective Optimization Problems

We consider two approaches that model timetable information in public transportation systems as shortest-path problems in weighted graphs. In the time-expanded approach, every event at a station, e.g., the departure of a train, is modeled as a node in the graph, while in the time-dependent approach the graph contains only one node per station. Both approaches have been recently considered for (a simplified version of) the earliest arrival problem, but little is known about their relative performance. Thus far, there are only theoretical arguments in favor of the time-dependent approach. In this paper, we provide the first extensive experimental comparison of the two approaches. Using several real-world data sets, we evaluate the performance of the basic models and of several new extensions towards realistic modeling. Furthermore, new insights on solving bicriteria optimization problems in both models are presented. The time-expanded approach turns out to be more robust for modeling more complex scenarios, whereas the time-dependent approach shows a clearly better performance.

Efficient models for timetable information in public transportation systems

This paper surveys indoor spatial models developed for research fields ranging from mobile robot mapping, to indoor location-based services (LBS), and most recently to context-aware navigation services applied to indoor environments. Over the past few years, several studies have evaluated the potential of spatial models for robot navigation and ubiquitous computing. In this paper we take a slightly different perspective, consid- ering not only the underlying properties of those spatial models, but also to which degree the notion of context can be taken into account when delivering services in indoor environ- ments. Some preliminary recommendations for the development of indoor spatial models are introduced from a context-aware perspective. A taxonomy of models is then presented and assessed with the aim of providing a flexible spatial data model for navigation pur- poses, and by taking into account the context dimensions.

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Spatial models for context-aware indoor navigation systems: A survey

We study the problem of finding all Pareto-optimal solutions in a multi-criteria setting of the shortest path problem in time-dependent graphs. This has important applications in timetable information systems for train schedules. We present a new prototype to solve this problem in a fully realistic scenario based on a multi-criteria generalization of Dijkstra's algorithm. As optimization criteria we use travel time and number of train changes, as well as a new criterion "reliability of transfers".

The performance of the prototype and various speed-up techniques are analyzed experimentally on a large set of real test instances. In comparison with a base-line implementation, our prototype achieves significant speed-up factors of 20 with respect to the number of label creations and of 138 with respect to label insertions into the priority queue. We also compare our prototype with a time-expanded graph model.

Multi-criteria shortest paths in time-dependent train networks

We consider efficient algorithms for timetable information in public transportation systems undermultiple objectives like, for example, travel time, ticket costs, and number of interchanges between different means of transport. In this paper we focus on a fully realistic scenario in public railroad transport as it appears in practice while most previous work studied only simplified models.

Algorithmically this leads to multi-criteria shortest path problems in very large graphs. With several objectives the challenge is to find all connections which are potentially attractive for customers. To meet this informal goal we introduce the notion of relaxed Pareto dominance. Another difficulty arises from the fact that due to the complicated fare regulations even the single-criteria optimization problem of finding cheapest connections is intractable. Therefore, we have to work with fare estimations during the search for good connections.

In a cooperation with Deutsche Bahn Systems we realized this scenario in a prototypal implementation called PARETO based on a timeexpanded graph model. Computational experiments with our PARETO server demonstrate that the current central server of Deutsche Bahn AG often fails to give optimal recommendations for different user groups. In contrast, an important feature of the PARETO server is its ability to provide many attractive alternatives.

Finding all attractive train connections by multi-criteria Pareto search

The search for train connections in state-of-the-art commercial timetable information systems is based on a static schedule. Unfortunately, public transportation systems suffer from delays for various reasons. Thus, dynamic changes of the planned schedule have to be taken into account. A system that has access to delay information about trains (and uses this information within search queries) can provide valid alternatives in case a connection does not work. Additionally, it can be used to actively guide passengers as these alternatives may be presented before the passenger is already stranded at a station due to an invalid transfer.

In this work, we present an approach which takes a stream of delay information and schedule changes on short notice (partial train cancellations, extra trains) into account. Primary delays of trains may cause a cascade of so-called secondary delays of other trains which have to wait according to certain policies for delays between connecting trains. We introduce the concept of a dependency graph to efficiently calculate and update all primary and secondary delays. This delay information is then incorporated into a time-expanded search graph which has to be updated dynamically. These update operations are quite complex, but turn out to be not time-critical in a fully realistic scenario.

We finally present a case study with data provided by Deutsche Bahn AG, showing that this approach has been successfully integrated into the multi-criteria timetable information system MOTIS and can handle massive delay data streams instantly.

Efficient Timetable Information in the Presence of Delays

Ride-sharing facilitates cheap and eco-friendly mobility. In contrast to classical ride-sharing that basically works like a notice-board, dynamic ride-sharing allows a passenger to get a lift on a section of a driver's route and, if necessary, re-routes the driver. On a dynamic ride-sharing platform, drivers provide their routes and passengers specify queries consisting of departure and arrival location as well as a time for the journey. The platform computes suitable matches of driver routes and passenger queries, and proposes them to both parties. State-of-the-Art platforms for public transport routing and dynamic ride-sharing provide unimodal connections but do not combine both transport modes. The challenge is that the driver routes are not static but could be changed significantly if the driver accepts the detour to pick up the passenger and drop him / her off at his / her destination. Thus, a driver's route may result in a number of dynamic ride-sharing offers, namely all possible connections between pick up and drop off points with an acceptable detour for the driver. In this paper, we present a solution that integrates dynamic ride-sharing into our existing multi-criteria intermodal travel information system. We solve two challenges: First, we allow dynamic ride-sharing between two train rides by connecting public transport stations by dynamic ride-sharing offers of drivers. For this, we integrate driver offers into our graph model, which represents the public transport timetable. Second, we find suitable dynamic ride-sharing offers of drivers who can take the passenger from his / her start location to a public transport station or from a station to the queried destination location. In our computational study, compared to unimodal train connections, we obtain a significant improvement of the results by combining public transport and dynamic ride-sharing.

A Multi-modal Routing Approach Combining Dynamic Ride-sharing and Public Transport

Disposition management solves the decision problem whether a train should wait for incoming delayed trains or not. This problem has a highly dynamic nature due to a steady stream of update information about delayed trains. A dispatcher has to solve a global optimization problem since his decisions have an effect on the whole network, but he takes only local decisions for subnetworks (for few stations and only for departure events in the near future). In this paper, we introduce a new model for an optimization tool. Our implementation includes as building blocks (1) routines for the permanent update of our graph model subject to incoming delay messages, (2) routines for forecasting future arrival and departure times, (3) the update of passenger flows subject to several rerouting strategies (including dynamic shortest path queries), and (4) the simulation of passenger flows. The general objective is the satisfaction of passengers. We propose three different formalizations of objective functions to capture this goal. Experiments on test data with the train schedule of German Railways and real delay messages show that our disposition tool can compute waiting decisions within a few seconds. In a test with artificial passenger flows it is fast enough to handle the typical amount of decisions which have to be taken within a period of 15 minutes in real time.

Mathias Schnee

Papers

Multi-criteria shortest paths in time-dependent train networks

Finding all attractive train connections by multi-criteria Pareto search

Efficient Timetable Information in the Presence of Delays

A Multi-modal Routing Approach Combining Dynamic Ride-sharing and Public Transport

Passenger flow-oriented train disposition