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Contraction hierarchies: Faster and simpler . . .
, 2008
"... We present a route planning technique solely based on the concept of node contraction. We contract or remove one node at a time out of the graph and add shortcut edges to the remaining graph to preserve shortest paths distances. The resulting contraction hierarchy (CH), the original graph plus short ..."
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Cited by 120 (34 self)
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We present a route planning technique solely based on the concept of node contraction. We contract or remove one node at a time out of the graph and add shortcut edges to the remaining graph to preserve shortest paths distances. The resulting contraction hierarchy (CH), the original graph plus shortcuts, also defines an order of “importance ” among all nodes through the node selection. We apply a modified bidirectional Dĳkstra algorithm that takes advantage of this node order to obtain shortest paths. The search space is reduced by relaxing only edges leading to more important nodes in the forward search and edges coming from more important nodes in the backward search. Both search scopes eventually meet at the most important node on a shortest path. We use a simple but extensible heuristic to obtain the node order: a priority queue whose priority function for each node is a linear combination of several terms, e.g. one term weights nodes depending on the sparsity of the remaining graph after the contraction. Another term regards the already contracted nodes to allow a more uniform contraction. Depending on the application we can select the combination of the priority terms to obtain the required hierarchy.
Engineering Route Planning Algorithms
 ALGORITHMICS OF LARGE AND COMPLEX NETWORKS. LECTURE NOTES IN COMPUTER SCIENCE
, 2009
"... 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 ..."
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Cited by 82 (39 self)
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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, timedependent routing, and flexible objective functions.
Combining Hierarchical and GoalDirected SpeedUp Techniques for Dijkstra’s Algorithm
 PROCEEDINGS OF THE 7TH WORKSHOP ON EXPERIMENTAL ALGORITHMS (WEA’08), VOLUME 5038 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... In recent years, highly effective hierarchical and goaldirected speedup techniques for routing in large road networks have been developed. This paper makes a systematic study of combinations of such techniques. These combinations turn out to give the best results in many scenarios, including graphs ..."
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Cited by 60 (24 self)
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In recent years, highly effective hierarchical and goaldirected speedup techniques for routing in large road networks have been developed. This paper 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 timeexpanded timetables. Besides these quantitative results, we obtain general insights for successful combinations.
Highway dimension, shortest paths, and provably efficient algorithms
 IN ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA’10) JANUARY 1719, 2010
, 2010
"... Computing driving directions has motivated many shortest path heuristics that answer queries on continental scale networks, with tens of millions of intersections, literally instantly, and with very low storage overhead. In this paper we complement the experimental evidence with the first rigorous p ..."
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Cited by 40 (3 self)
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Computing driving directions has motivated many shortest path heuristics that answer queries on continental scale networks, with tens of millions of intersections, literally instantly, and with very low storage overhead. In this paper we complement the experimental evidence with the first rigorous proofs of efficiency for many of the heuristics suggested over the past decade. We introduce the notion of highway dimension and show how low highway dimension gives a unified explanation for several seemingly different algorithms.
Engineering multilevel overlay graphs for shortestpath queries
 IN: PROCEEDINGS OF THE EIGHT WORKSHOP ON ALGORITHM ENGINEERING AND EXPERIMENTS (ALENEX06), SIAM
, 2006
"... 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 multilevel overlay graph used in [21] to speed up shortestpath computations. In this work, we follow up and present general verte ..."
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Cited by 35 (7 self)
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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 multilevel 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 multilevel overlay graphs and the resulting speedup achieved for shortestpath 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.
Engineering Fast Route Planning Algorithms
, 2007
"... 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 Dijkstra’s algorithm. We outline ideas, algorithms, implementations, and experimental methods behind this development. We also explai ..."
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Cited by 33 (4 self)
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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 Dijkstra’s 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.
Probabilistic path queries in road networks: traffic uncertainty aware path selection
 In EDBT
, 2010
"... Path queries such as “finding the shortest path in travel time from my hotel to the airport ” are heavily used in many applications of road networks. Currently, simple statistic aggregates such as the average travel time between two vertices are often used to answer path queries. However, such simpl ..."
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Cited by 27 (0 self)
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Path queries such as “finding the shortest path in travel time from my hotel to the airport ” are heavily used in many applications of road networks. Currently, simple statistic aggregates such as the average travel time between two vertices are often used to answer path queries. However, such simple aggregates often cannot capture the uncertainty inherent in traffic. In this paper, we study how to take traffic uncertainty into account in answering path queries in road networks. To capture the uncertainty in traffic such as the travel time between two vertices, the weight of an edge is modeled as a random variable and is approximated by a set of samples. We propose three novel types of probabilistic path queries using basic probability principles: (1) a probabilistic path query like “what are the paths from my hotel to the airport whose travel time is at most 30 minutes with a probability of at least 90%?”; (2) a weightthreshold topk path query like “what are the top3 paths from my hotel to the airport with the highest probabilities to take at most 30 minutes?”; and (3) a probabilitythreshold topk path query like “what are the top3 shortest paths from my hotel to the airport whose travel time is guaranteed by a probability of at least 90%? ” To evaluate probabilistic path queries efficiently, we develop three efficient probability calculation methods: an exact algorithm, a constant factor approximation method and a sampling based approach. Moreover, we devise the P * algorithm, a bestfirst search method based on a novel hierarchical partition tree index and three effective heuristic evaluation functions. An extensive empirical study using real road networks and synthetic data sets shows the effectiveness of the proposed path queries and the efficiency of the query evaluation methods.
Computing ManytoMany Shortest Paths Using Highway Hierarchies
, 2007
"... We present a fast algorithm for computing all shortest paths between source nodes s ∈ S and target nodes t ∈ T. This problem is important as an initial step for many operations research problems (e.g., the vehicle routing problem), which require the distances between S and T as input. Our approach i ..."
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Cited by 26 (5 self)
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We present a fast algorithm for computing all shortest paths between source nodes s ∈ S and target nodes t ∈ T. This problem is important as an initial step for many operations research problems (e.g., the vehicle routing problem), which require the distances between S and T as input. Our approach is based on highway hierarchies, which are also used for the currently fastest speedup techniques for shortest path queries in road networks. We show how to use highway hierarchies so that for example, a 10 000 × 10 000 distance table in the European road network can be computed in about one minute. These results are based on a simple basic idea, several refinements, and careful engineering of the approach. We also explain how the approach can be parallelized and how the computation can be restricted to computing only the k closest connections.
Highway hierarchies star
 9TH DIMACS IMPLEMENTATION CHALLENGE
, 2006
"... We study two speedup techniques for route planning in road networks: highway hierarchies (HH) and goal directed search using landmarks (ALT). It turns out that there are several interesting synergies. Highway hierarchies yield a way to implement landmark selection more efficiently and to store landm ..."
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Cited by 25 (12 self)
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We study two speedup techniques for route planning in road networks: highway hierarchies (HH) and goal directed search using landmarks (ALT). It turns out that there are several interesting synergies. Highway hierarchies yield a way to implement landmark selection more efficiently and to store landmark information more space efficiently than before. ALT gives queries in highway hierarchies an excellent sense of direction and allows some pruning of the search space. For computing shortest distances and approximately shortest travel times, this combination yields a significant speedup over HH alone. We also explain how to compute actual shortest paths very efficiently.
TRANSIT— ultrafast shortestpath queries with lineartime preprocessing
 In 9th DIMACS Implementation Challenge [1
, 2006
"... {bast,funke,dmatijev} at mpiinf dot mpg dot de We introduce the concept of transit nodes, as a means for preprocessing a road network, with given coordinates for each node and a travel time for each edge, such that pointtopoint shortestpath queries can be answered extremely fast. The transit nod ..."
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Cited by 21 (1 self)
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{bast,funke,dmatijev} at mpiinf dot mpg dot de We introduce the concept of transit nodes, as a means for preprocessing a road network, with given coordinates for each node and a travel time for each edge, such that pointtopoint shortestpath queries can be answered extremely fast. The transit nodes are a set of nodes, as small as possible, with the property that every shortest path that is nonlocal in the sense that it covers a certain not too small euclidean distance passes through at least on of these nodes. With such a set and precomputed distances from each node in the graph to its few, closest transit nodes, every nonlocal shortest path query becomes a simple matter of combining information from a few table lookups. For the US road network, which has about 24 million nodes and 58 million edges, we achieve a worstcase query processing time of about 10 microseconds (not milliseconds) for 99 % of all queries. This improves over the best previously reported times by two orders of magnitude. 1