## TRANSIT— ultrafast shortest-path queries with linear-time preprocessing (2006)

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Venue: | In 9th DIMACS Implementation Challenge [1 |

Citations: | 14 - 1 self |

### BibTeX

@INPROCEEDINGS{Bast06transit—ultrafast,

author = {Holger Bast and Stefan Funke and Domagoj Matijevic},

title = {TRANSIT— ultrafast shortest-path queries with linear-time preprocessing},

booktitle = {In 9th DIMACS Implementation Challenge [1},

year = {2006}

}

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### Abstract

{bast,funke,dmatijev} at mpi-inf 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 point-to-point shortest-path 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 non-local 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 non-local 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 worst-case 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

### Citations

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Citation Context ...e will take an understanding of the basic workings of Dijkstra’s algorithm as well as the associated standard terminology (relaxing an edge, settling a node) for granted. For details, see for example =-=[2]-=-. Our benchmark throughout this paper will be the US road network, which has about 24 million nodes and 58 million edges. Edge lengths are travel times, so that shortest paths are actually paths with ... |

1427 | A note on two problems in connexion with graphs
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Citation Context ... previously reported times by two orders of magnitude. 1 Introduction The classical way to compute the shortest path between two given nodes in a graph with given edge lengths is Dijkstra’s algorithm =-=[3]-=-. The asymptotic running time of Dijkstra’s algorithm is O(m + n log m), where n is the number of nodes, and m is the number of edges. For graphs with constant degree, like the road networks we consid... |

205 | Approximate distance oracles
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Citation Context ...n). There is strong theoretical evidence that, without preprocessing and without any assumptions on the graph except that the edge lengths be non-negative, Dijkstra’s algorithm is essentially optimal =-=[13]-=-. In the following we will take an understanding of the basic workings of Dijkstra’s algorithm as well as the associated standard terminology (relaxing an edge, settling a node) for granted. For detai... |

97 | Computing the shortest path: A ∗ search meets graph theory
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(Show Context)
Citation Context ...bout 5 milliseconds (which happens to be the average processing time for the local queries, too). This is still competitive with the processing times reported in [11] and its closest competitors [10] =-=[4]-=- [5]. All of these schemes, however, do not output edges along the shortest path, nor can they be easily modified to do so without a severe slowdown in query time. This is because in all of these work... |

67 |
Highway Hierarchies Hasten Exact Shortest Path Queries
- Sanders, Schultes
- 2005
(Show Context)
Citation Context ...vels with a compression scheme and they use lower bounds, based on precomputed distances to a few landmarks, to allow for a more goal-directed search. They report running times comparable to those of =-=[10]-=-. Their space consumption is somewhat higher though, because every node in the network has to store distances to all landmarks. Most recently, Sanders and Schultes [11] have presented the so far best ... |

58 | Reach for A*: Efficient point-to-point shortest path algorithms
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Citation Context ...h time and space is necessary to uncompress those edges. Their variant is also inherently bidirectional, so both goal-direction as well as one-to-many queries are not easily added. Goldberg et al. in =-=[5]-=- combine edge levels with a compression scheme and they use lower bounds, based on precomputed distances to a few landmarks, to allow for a more goal-directed search. They report running times compara... |

52 |
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Citation Context ...e could not yet come close to their extremely fast preprocessing time, our cost-only scheme beats their query time by two orders of magnitude. Möhring et al. [9, 7], and, in independent work, Lauther =-=[8]-=- explored edge signs as a means to achieve very fast query times. Intuitively, an edge sign says whether that edge is on a shortest path to a particular region of the graph. In an extreme case, an edg... |

51 |
Reach-based routing: A new approach to shortest path algorithms optimized for road networks
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Citation Context ...RANSIT in more detail in Section 4. 3 Related Work We give a quick survey of work directly relevant to the problem of preprocessing road networks for subsequent fast shortest-path querying. Gutman in =-=[6]-=- proposes a general concept of edge levels. Consider an edge e that appears ”in the middle” of a shortest paths, shortest with respect to travel time, between two nodes that are a certain distance d a... |

46 |
Engineering highway hierarchies
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- 2006
(Show Context)
Citation Context ... orders of magnitudes faster than arbitrary random queries. Our processing times for the non-local queries beat the best previously reported figure of about 1 millisecond, due to Sanders and Schultes =-=[11]-=-, by two orders of magnitude. When the full path, with all its edges, is to be output, we achieve an average query processing time of about 5 milliseconds on the US road network. This latter result st... |

43 |
Partitioning graphs to speedup Dijkstra’s algorithm
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(Show Context)
Citation Context ...l shortest path edges efficiently. While we could not yet come close to their extremely fast preprocessing time, our cost-only scheme beats their query time by two orders of magnitude. Möhring et al. =-=[9, 7]-=-, and, in independent work, Lauther [8] explored edge signs as a means to achieve very fast query times. Intuitively, an edge sign says whether that edge is on a shortest path to a particular region o... |

23 | Highway hierarchies star
- Delling, Sanders, et al.
- 2006
(Show Context)
Citation Context .... All of these schemes, do not output edges along the shortest path, because they use some kind of compression of subpaths. In their most recent works, Sanders and Schultes have resolved this problem =-=[4]-=- [15]. Many previous works provided a figure that showed the dependency of the processing time of a query on the Dijkstra rank of that query, which is the number of nodes Dijkstra’s algorithm would ha... |

20 |
Acceleration of Shortest Path and Constrained Shortest Path Computation
- Köhler, Möhring, et al.
(Show Context)
Citation Context ...l shortest path edges efficiently. While we could not yet come close to their extremely fast preprocessing time, our cost-only scheme beats their query time by two orders of magnitude. Möhring et al. =-=[9, 7]-=-, and, in independent work, Lauther [8] explored edge signs as a means to achieve very fast query times. Intuitively, an edge sign says whether that edge is on a shortest path to a particular region o... |

15 | D.: HighPerformance Multi-Level Graphs
- Delling, Holzer, et al.
- 2006
(Show Context)
Citation Context ...ed on highway hierarchies, are given in a joint follow-up paper [2]. 4sFigure 2: Transit neighborhood of a cell in a 64 × 64 subdivision of the US. In retrospect, the work of [12] (which later became =-=[3]-=-) can be taken as another alternative to computing transit nodes. In a nutshell, they use a hierarchy of separators to partition a given road network (making use of its almost-planarity). Their separa... |

14 |
transit to constant time shortest-path queries in road networks
- In
- 2007
(Show Context)
Citation Context ... complex algorithm and a higher space consumption. A joint paper presenting and comparing both approaches, our simple geometric one and the one based on highway hierarchies, is to appear at ALENEX’07 =-=[1]-=-. In our conclusions in Section 6, we give some details on why the tranist node idea goes together with highway hierarchies particulary well. 4 The TRANSIT algorithm 4.1 Intuition The basic intuition ... |

10 |
Design and Implementation of an Efficient Hierarchical Speed-up Technique for Computation of Exact Shortest
- Müller
- 2006
(Show Context)
Citation Context ...etric one and the one based on highway hierarchies, are given in a joint follow-up paper [2]. 4sFigure 2: Transit neighborhood of a cell in a 64 × 64 subdivision of the US. In retrospect, the work of =-=[12]-=- (which later became [3]) can be taken as another alternative to computing transit nodes. In a nutshell, they use a hierarchy of separators to partition a given road network (making use of its almost-... |

8 |
Almost Constant Time Shortest-Path Queries via Transit Nodes, volume 74
- Robust
- 2009
(Show Context)
Citation Context ...k for undirected graphs. A generalization to directed graphs is not trivial but feasible. The highway hierarchies from Sanders and Schultes, in particular their combination with the transit node idea =-=[15]-=-, also work for directed graphs. A more difficult open problem is how to make our data structure dynamic, that is, how to update our data structures in response to only small changes in the graph, lik... |