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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 117 (31 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 80 (34 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 68 (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.
A HubBased Labeling Algorithm for Shortest Paths on Road Networks
, 2010
"... Abstract. Abraham et al. [SODA 2010] have recently presented a theoretical analysis of several practical pointtopoint shortest path algorithms based on modeling road networks as graphs with low highway dimension. They also analyze a labeling algorithm. While no practical implementation of this a ..."
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Cited by 46 (15 self)
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Abstract. Abraham et al. [SODA 2010] have recently presented a theoretical analysis of several practical pointtopoint shortest path algorithms based on modeling road networks as graphs with low highway dimension. They also analyze a labeling algorithm. While no practical implementation of this algorithm existed, it has the best time bounds. This paper describes an implementation of the labeling algorithm that is faster than any existing method on continental road networks. 1
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 32 (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.
In Transit to Constant Time ShortestPath Queries in Road Networks
"... When you drive to somewhere ‘far away’, you will leave your current location via one of only a few ‘important’ traffic junctions. Starting from this informal observation, we develop an algorithmic approach—transit node routing— that allows us to reduce quickestpath queries in road networks to a sma ..."
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Cited by 15 (4 self)
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When you drive to somewhere ‘far away’, you will leave your current location via one of only a few ‘important’ traffic junctions. Starting from this informal observation, we develop an algorithmic approach—transit node routing— that allows us to reduce quickestpath queries in road networks to a small number of table lookups. We present two implementations of this idea, one based on a simple grid data structure and one based on highway hierarchies. For the road map of the United States, our best query times improve over the best previously published figures by two orders of magnitude. Our results exhibit various tradeoffs between average query time (6 µs to 63 µs), preprocessing time (62 min to 1200 min), and storage overhead (27 bytes/node to 247 bytes/node).
Graph Indexing of Road Networks for Shortest Path Queries with Label Restrictions
"... The current widespread use of locationbased services and GPS technologies has revived interest in very fast and scalable shortest path queries. We introduce a new shortest path query type in which dynamic constraints may be placed on the allowable set of edges that can appear on a valid shortest pa ..."
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Cited by 13 (0 self)
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The current widespread use of locationbased services and GPS technologies has revived interest in very fast and scalable shortest path queries. We introduce a new shortest path query type in which dynamic constraints may be placed on the allowable set of edges that can appear on a valid shortest path (e.g., dynamically restricting the type of roads or modes of travel which may be considered in a multimodal transportation network). We formalize this problem as a specific variant of formal language constrained shortest path problems, which we call the Kleene Language Constrained Shortest Paths problem. To efficiently support this type of dynamically constrained shortest path query for largescale datasets, we extend the hierarchical graph indexing technique known as Contraction Hierarchies. Our experimental evaluation using the North American road network dataset (with over 50 million edges) shows an average query speed and search space improvement of over 3 orders of magnitude compared to the naïve adaptation of the standard Dijkstra’s algorithm to support this query type. We also show an improvement of over 2 orders of magnitude compared to the only previouslyexisting indexing technique which could solve this problem without additional preprocessing. 1.
Faster Batched Shortest Paths in Road Networks
 ATMOS
, 2011
"... We study the problem of computing batched shortest paths in road networks efficiently. Our focus is on computing paths from a single source to multiple targets (onetomany queries). We perform a comprehensive experimental comparison of several approaches, including new ones. We conclude that a new ..."
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Cited by 9 (5 self)
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We study the problem of computing batched shortest paths in road networks efficiently. Our focus is on computing paths from a single source to multiple targets (onetomany queries). We perform a comprehensive experimental comparison of several approaches, including new ones. We conclude that a new extension of PHAST (a recent onetoall algorithm), called RPHAST, has the best performance in most cases, often by orders of magnitude. When used to compute distance tables (manytomany queries), RPHAST often outperforms all previous approaches.
Transit Node Routing Reconsidered?
"... Abstract. Transit Node Routing (TNR) is a fast and exact distance oracle for road networks. We show several new results for TNR. First, we give a surprisingly simple implementation fully based on Contraction Hierarchies that speeds up preprocessing by an order of magnitude approaching the time for j ..."
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Cited by 9 (2 self)
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Abstract. Transit Node Routing (TNR) is a fast and exact distance oracle for road networks. We show several new results for TNR. First, we give a surprisingly simple implementation fully based on Contraction Hierarchies that speeds up preprocessing by an order of magnitude approaching the time for just finding a Contraction Hierarchies (which alone has two orders of magnitude larger query time). We also develop a very effective purely graph theoretical locality filter without any compromise in query times. Finally, we show that a specialization to the online manytoone (or onetomany) shortest path further speeds up query time by an order of magnitude. This variant even has better query time than the fastest known previous methods which need much more space. 1 Introduction and Related Work Route planning in road networks has seen a lot of results from the algorithm engineering community in recent years. With Dijkstra’s seminal algorithm being the baseline, a number of techniques preprocess the static input graph to achieve drastic speedups. Contraction Hierarchies (CH) [1,2] is a speeduptechnique that has a convenient tradeoff between preprocessing effort and query efficiency. Road network with millions of nodes
Computing Single Source Shortest Paths using SingleObjective Fitness Functions
, 2009
"... Runtime analysis of evolutionary algorithms has become an important part in the theoretical analysis of randomized search heuristics. The first combinatorial problem where rigorous runtime results have been achieved is the wellknown single source shortest path (SSSP) problem. Scharnow, Tinnefeld an ..."
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Cited by 8 (5 self)
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Runtime analysis of evolutionary algorithms has become an important part in the theoretical analysis of randomized search heuristics. The first combinatorial problem where rigorous runtime results have been achieved is the wellknown single source shortest path (SSSP) problem. Scharnow, Tinnefeld and Wegener [PPSN 2002, J. Math. Model. Alg. 2004] proposed a multiobjective approach which solves the problem in expected polynomial time. They also suggest a related singleobjective fitness function. However, it was left open whether this does solve the problem efficiently, and, in a broader context, whether multiobjective fitness functions for problems like the SSSP yield more efficient evolutionary algorithms. In this paper, we show that the single objective approach yields an efficient (1+1) EA with runtime bounds very close to those of the multiobjective approach.