An overlay graph of a given graph G =(V,E) on a subset S ⊆ V is a graph with vertex set S that preserves some property of G. In particular, we consider variations of the multi-level overlay graph used in [21] to speed up shortestpath computations. In this work, we follow up and present general vertex selection criteria and strategies of applying these criteria to determine a subset S inducing an overlay graph. The main contribution is a systematic experimental study where we investigate the impact of selection criteria and strategies on multi-level overlay graphs and the resulting speed-up achieved for shortest-path queries. Depending on selection strategy and graph type, a centrality index criterion, a criterion based on planar separators, and vertex degree turned out to be good selection criteria.

Computing a shortest path from one node to another in a directed graph is a very common task in practice. This problem is classically solved by Dijkstra's algorithm. Many techniques are known to speed up this algorithm heuristically, while optimality of the solution can still be guaranteed. In most studies, such techniques are considered individually.

Shortest-path computation is a frequent task in practice. Owing to ever-growing real-world graphs, there is a constant need for faster algorithms. In the course of time, a large number of techniques to heuristically speed up Dijkstra’s shortest-path algorithm have been devised. This work reviews the multi-level technique to answer shortest-path queries exactly [SWZ02, HSW06], which makes use of a hierarchical decomposition of the input graph and precomputation of supplementary information. We develop this preprocessing to the maximum and introduce several ideas to enhance this approach considerably, by reorganizing the precomputed data in partial graphs and optimizing them individually. To answer a given query, certain partial graphs are combined to a search graph, which can be explored by a simple and fast procedure. Experiments confirm query times of less than 200 µs for a road graph with over 15 million vertices.

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We study a graph-augmentation problem arising from a technique applied in recent approaches for route planning. Many such methods enhance the graph by inserting shortcuts, i.e., additional edges (u,v) such that the length of (u,v) is the distance from u to v. Given a weighted, directed graph G and a number c ∈ Z>0, the shortcut problem asks how to insert c shortcuts into G such that the expected number of edges that are contained in an edge-minimal shortest path from a random node s to a random node t is minimal. In this work, we study the algorithmic complexity of the problem and give approximation algorithms for a special graph class. Further, we state ILP-based exact approaches and show how to stochastically evaluate a given shortcut assignment on graphs that are too large to do so exactly. Submitted:

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.

During the last years, impressive speed-up techniques for DIJKSTRA’s algorithm have been developed. Unfortunately, recent research mainly focused on road networks. However, fast algorithms are also needed for other applications like timetable information systems. Even worse, the adaption of recently developed techniques to timetable information is more complicated than expected. In this work, we check whether results from road networks are transferable to timetable information. To this end, we present an extensive experimental study of the most prominent speed-up techniques on different types of inputs. It turns out that recently developed techniques are much slower on graphs derived from timetable information than on road networks. In addition, we gain amazing insights into the behavior of speed-up techniques in general.

We consider a generalization of the point-to-point (and single-source) shortest path problem to instances where the shortest path must satisfy a formal language constraint. Given an alphabet Σ, a (directed) network G whose edges are weighted and Σ-labeled, and a regular grammar L ⊆ Σ ∗ , the Regular Language Constrained Shortest Path problem consists of finding a shortest path p in G complying with the additional constraint that l(p) ∈ L. Here l(p) denotes the unique word given by concatenating the Σ-labels of the edges along the path p. In this chapter, we summarize our recent results and present new theoretical and experimental results for the Regular Language Constrained Shortest problem. We also present extensions of several speedup techniques developed earlier for the standard point-to-point shortest path problem. These speed-up techniques are integrated within the basic algorithmic framework to yield new algorithms for the problem. In order to evaluate the performance of the basic algorithm and its extensions, we have performed preliminary experimental analysis. Through our experiments, we study the scalability of the algorithm with respect to the network size as well as with respect to the constraining language complexity. Further, we study the effectiveness of speed-up techniques such as goal-directed and bidirectional search when applied to the Regular Language Constrained Shortest problem. 1

We give an overview of models and efficient algorithms for optimally solving timetable information problems like “given a departure and an arrival station as well as a departure time, which is the connection that arrives as early as possible at the arrival station?” Two main approaches that transform the problems into shortest path problems are reviewed, including issues like the modeling of realistic details (e.g., train transfers) and further optimization criteria (e.g., the number of transfers). An important topic is also multi-criteria optimization, where in general all attractive connections with respect to several criteria shall be determined. Finally, we discuss the performance of the described algorithms, which is crucial for their application in a real system.

A travel booking office has timetables giving arrival and departure times for all scheduled trains, including their origins and destinations. A customer presents a starting city and demands a route with perhaps several train connections taking him to his destination as early as possible. The booking office must find the best route for its customers. This problem was first considered in the theory of algorithms by George Minty [Min58], who reduced it to a problem on directed weighted graphs: find a path from a given source to a given target such that the consecutive weights on the path are nondecreasing and the last weight on the path is minimized. Minty gave the first algorithm for the single source version of the problem, in which one finds minimum last weight nondecreasing paths from the source to every other vertex. In this paper we give the first linear time algorithm for this problem. We also define an all pairs version for the problem and give a strongly polynomial truly subcubic algorithm for it. 1