Dominik Schultes

Engineering Route Planning Algorithms

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 Dijkstras 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.

Highway hierarchies hasten exact shortest path queries

We present a new speedup technique for route planning that exploits the hierarchy inherent in real world road networks. Our algorithm preprocesses the eight digit number of nodes needed for maps of the USA or Western Europe in a few hours using linear space. Shortest (i.e. fastest) path queries then take around eight milliseconds to produce exact shortest paths. This is about 2 000 times faster than using Dijkstra’s algorithm.

Combining hierarchical and goal-directed speed-up techniques for dijkstra's algorithm

In recent years, highly effective hierarchical and goal-directed speed-up techniques for routing in large road networks have been developed. This article makes a systematic study of combinations of such techniques. These combinations turn out to give the best results in many scenarios, including graphs for unit disk graphs, grid networks, and time-expanded timetables. Besides these quantitative results, we obtain general insights for successful combinations.

Exact Routing in Large Road Networks Using Contraction Hierarchies

Contraction hierarchies are a simple approach for fast routing in road networks. Our algorithm calculates exact shortest paths and handles road networks of whole continents. During a preprocessing step, we exploit the inherent hierarchical structure of road networks by adding shortcut edges. A subsequent modified bidirectional Dijkstra algorithm can then find a shortest path in a fraction of a millisecond, visiting only a few hundred nodes. This small search space makes it suitable to implement it on a mobile device. We present a mobile implementation that also handles changes in the road network, like traffic jams, and that allows instantaneous routing without noticeable delay for the user. Also, an algorithm to calculate large distance tables is currently the fastest if based on contraction hierarchies.

Dynamic highway-node routing

We introduce a dynamic technique for fast route planning in large road networks. For the first time, it is possible to handle the practically relevant scenarios that arise in present-day navigation systems: When an edge weight changes (e.g., due to a traffic jam), we can update the preprocessed information in 2-40ms allowing subsequent fast queries in about one millisecond on average. When we want to perform only a single query, we can skip the comparatively expensive update step and directly perform a prudent query that automatically takes the changed situation into account. If the overall cost function changes (e.g., due to a different vehicle type), recomputing the preprocessed information takes typically less than two minutes. The foundation of our dynamic method is a new static approach that generalises and combines several previous speedup techniques. It has outstandingly low memory requirements of only a few bytes per node.

Engineering fast route planning algorithms

Algorithms for route planning in transportation networks have recently undergone a rapid development, leading to methods that are up to one million times faster than Dijkstras algorithm. We outline ideas, algorithms, implementations, and experimental methods behind this development. We also explain why the story is not over yet because dynamically changing networks, flexible objective functions, and new applications pose a lot of interesting challenges.

Bidirectional A* search for time-dependent fast paths

The computation of point-to-point shortest paths on time-dependent road networks has many practical applications, but there have been very few works that propose efficient algorithms for large graphs. One of the difficulties of route planning on time-dependent graphs is that we do not know the exact arrival time at the destination, hence applying bidirectional search is not straightforward; we propose a novel approach based on A with landmarks (ALT) that starts a search from both the source and the destination node, where the backward search is used to bound the set of nodes that have to be explored by the forward search. Extensive computational results show that this approach is very effective in practice if we are willing to accept a small approximation factor, resulting in a speed-up of several times with respect to Dijkstras algorithm while finding only slightly suboptimal solutions.

Mobile Route Planning

We provide an implementation of an exact route planning algorithm on a mobile device that answers shortest-path queries in a road network of a whole continent instantaneously, i.e., with a delay of about 100 ms which is virtually not observable for a human user. Our main algorithmic contribution of is a highly compressed blocked representation of the underlying hierarchical graph and a new fast yet compact route reconstruction data structure. Our representation exploits the locality properties of the graph using a very simple algorithm that does not use any a priori information.

Combining hierarchical and goal-directed speed-up techniques for Dijkstra's algorithm

In [1], basic speed-up techniques for Dijkstras algorithm have been combined. The key observation in their work was that it is most promising to combine hierarchical and goal-directed speed-up techniques. However, since its publication, impressive progress has been made in the field of speed-up techniques for Dijkstras algorithm and huge data sets have been made available. Hence, we revisit the systematic combination of speed-up techniques in this work, which leads to the fastest known algorithms for various scenarios. Even for road networks, which have been worked on heavily during the last years, we are able to present an improvement in performance. Moreover, we gain interesting insights into the behavior of speed-up techniques when combining them.

Engineering an External Memory Minimum Spanning Tree Algorithm

We develop an external memory algorithm for computing minimum spanning trees. The algorithm is considerably simpler than previously known external memory algorithms for this problem and needs a factor of at least four less I/Os for realistic inputs.