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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.
Engineering Highway Hierarchies
, 2006
"... Highway hierarchies exploit hierarchical properties inherent in realworld road networks to allow fast and exact pointtopoint shortestpath queries. A fast preprocessing routine iteratively performs two steps: first, it removes edges that only appear on shortest paths close to source or target; s ..."
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Cited by 69 (6 self)
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Highway hierarchies exploit hierarchical properties inherent in realworld road networks to allow fast and exact pointtopoint shortestpath queries. A fast preprocessing routine iteratively performs two steps: first, it removes edges that only appear on shortest paths close to source or target; second, it identifies lowdegree nodes and bypasses them by introducing shortcut edges. The resulting hierarchy of highway networks is then used in a Dijkstralike bidirectional query algorithm to considerably reduce the search space size without losing exactness. The crucial fact is that ‘far away ’ from source and target it is sufficient to consider only highlevel edges. Various experiments with realworld road networks confirm the performance of our approach. On a 2.0 GHz machine, preprocessing the network of Western Europe, which consists of about 18 million nodes, takes 13 minutes and yields 48 bytes of additional data per node. Then, random queries take 0.61 ms on average. If we are willing to accept slower query times (1.10 ms), the memory usage can be decreased to 17 bytes per node. We can guarantee that at most 0.014 % of all nodes are visited during any query. Results for US road networks are similar. Highway hierarchies can be combined with goaldirected search, they can be extended to answer manytomany queries, and they are a crucial ingredient for other speedup techniques, namely for transitnode routing and highwaynode routing.
Dynamic highwaynode routing
 In Proc. 6th Workshop on Experimental and Efficient Algorithms. LNCS
, 2007
"... Abstract. 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 presentday navigation systems: When an edge weight changes (e.g., due to a traffic jam), we can update the preproc ..."
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Cited by 48 (6 self)
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Abstract. 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 presentday navigation systems: When an edge weight changes (e.g., due to a traffic jam), we can update the preprocessed information in 2–40 ms 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. 1
Mobile Route Planning
"... We provide an implementation of an exact route planning algorithm on a mobile device that answers shortestpath queries in a road network of a whole continent instantaneously, i.e., with a delay of about 100ms, which is virtually not observable for a human user. We exploit spatial and hierarchical ..."
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Cited by 19 (2 self)
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We provide an implementation of an exact route planning algorithm on a mobile device that answers shortestpath queries in a road network of a whole continent instantaneously, i.e., with a delay of about 100ms, which is virtually not observable for a human user. We exploit spatial and hierarchical locality properties to design a significantly compressed externalmemory graph representation, which can be traversed efficiently and which occupies only a few hundred megabytes for road networks with up to 34 million nodes. Next to the accuracy, the computational speed, and the low space requirements, the simplicity of our approach is a fourth argument that suggests an application of our implementation in car navigation systems.
Experimental Study on SpeedUp Techniques for Timetable Information Systems
 PROCEEDINGS OF THE 7TH WORKSHOP ON ALGORITHMIC APPROACHES FOR TRANSPORTATION MODELING, OPTIMIZATION, AND SYSTEMS (ATMOS 2007
, 2007
"... During the last years, impressive speedup 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 recentl ..."
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Cited by 18 (10 self)
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During the last years, impressive speedup 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 speedup 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 speedup techniques in general.
Bidirectional A ∗ Search for TimeDependent Fast Paths
"... Abstract. The computation of pointtopoint shortest paths on timedependent 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 timedependent graphs is that we do not know ..."
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Cited by 16 (10 self)
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Abstract. The computation of pointtopoint shortest paths on timedependent 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 timedependent 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 speedup of several times with respect to Dijkstra’s algorithm while finding only slightly suboptimal solutions. 1
Route Planning with Flexible Objective Functions
 In Proceedings of the 12th Workshop on Algorithm Engineering and Experiments (ALENEX’10), 124–137. SIAM
"... Abstract We present the first fast route planning algorithm that answers shortest paths queries for a customizable linear combination of two different metrics, e. g. travel time and energy cost, on large scale road networks. The precomputation receives as input a directed graph, two edge weight fun ..."
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Cited by 10 (4 self)
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Abstract We present the first fast route planning algorithm that answers shortest paths queries for a customizable linear combination of two different metrics, e. g. travel time and energy cost, on large scale road networks. The precomputation receives as input a directed graph, two edge weight functions t(e) and c(e), and a discrete interval [L, U ]. The resulting flexible query algorithm finds for a parameter p ∈ [L, U ] an exact shortest path for the edge weight t(e)+p·c(e). This allows for different tradeoffs between the two edge weight functions at query time. We apply precomputation based on node contraction, which adds all necessary shortcuts for any parameter choice efficiently. To improve the node ordering, we developed the new concept of gradual parameter interval splitting. Additionally, we improve performance by combining node contraction and a goaldirected technique in our flexible scenario.
Online Computation of Fastest Path in TimeDependent Spatial Networks
, 2011
"... The problem of pointtopoint fastest path computation in static spatial networks is extensively studied with many precomputation techniques proposed to speedup the computation. Most of the existing approaches make the simplifying assumption that traveltimes of the network edges are constant. Howe ..."
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Cited by 8 (5 self)
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The problem of pointtopoint fastest path computation in static spatial networks is extensively studied with many precomputation techniques proposed to speedup the computation. Most of the existing approaches make the simplifying assumption that traveltimes of the network edges are constant. However, with realworld spatial networks the edge traveltimes are timedependent, where the arrivaltime to an edge determines the actual traveltime on the edge. In this paper, we study the online computation of fastest path in timedependent spatial networks and present a technique which speedsup the path computation. We show that our fastest path computation based on a bidirectional timedependent A * search significantly improves the computation time and storage complexity. With extensive experiments using real datasets (including a variety of large spatial networks with real traffic data) we demonstrate the efficacy of our proposed techniques for online fastest path computation.
K∗: A heuristic search algorithm for finding the k shortest paths
, 2011
"... We present a directed search algorithm, called K ∗ , for finding the k shortest paths between a designated pair of vertices in a given directed weighted graph. K ∗ has two advantages compared to current kshortestpaths algorithms. First, K ∗ operates onthefly, which means that it does not require ..."
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Cited by 7 (1 self)
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We present a directed search algorithm, called K ∗ , for finding the k shortest paths between a designated pair of vertices in a given directed weighted graph. K ∗ has two advantages compared to current kshortestpaths algorithms. First, K ∗ operates onthefly, which means that it does not require the graph to be explicitly available and stored in main memory. Portions of the graph will be generated as needed. Second, K ∗ can be guided using heuristic functions. We prove the correctness of K ∗ and determine its asymptotic worstcase complexity when using a consistent heuristic to be the same as the state of the art, O(m + n log n + k), with respect to both runtime and space, where n is the number of vertices and m is the number of edges of the graph. We present an experimental evaluation of K ∗ by applying it to route planning problems as well as counterexample generation for stochastic model checking. The experimental results illustrate that due to the use of heuristic, onthefly search K ∗ can use less time and memory compared to the most efficient kshortestpaths algorithms known so far.