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## Engineering multi-level overlay graphs for shortest-path queries (2006)

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Venue: | IN: PROCEEDINGS OF THE EIGHT WORKSHOP ON ALGORITHM ENGINEERING AND EXPERIMENTS (ALENEX06), SIAM |

Citations: | 35 - 7 self |

### Citations

2386 | Random graphs
- Bollobás
- 1985
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Citation Context ...ollows (cf. [Schulz 2005] for further information). At top level, compute a classic Erdös-Réyni random graph with n ′ vertices and m ′ edges selected uniformly at random from all possible edges (cf. [=-=Bollobás 1985-=-]); if it is not connected, repeat this step. The remaining l ′ − 1 levels of the hierarchy are constructed in a recursive fashion: For the connected graph/component currently considered, introduce c ... |

2386 | A Note on Two Problems in Connexion with Graphs. Numerische Mathematik 1:269–271
- Dijkstra
- 1959
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Citation Context ...ts. 1.1 Related Work There are numerous approaches to speed up single-pair shortest-path computation (cf. [Willhalm and Wagner 2007] for a survey), where most of them improve on Dijkstra’s algorithm [=-=Dijkstra 1959-=-]. A few speed-up techniques can be applied immediately, e.g., goal-directed and bidirectional search [Ahuja et al. 1993]. However, much better speed-up factors are reached when some precomputed infor... |

2042 |
flows. Theory, algorithms, and applications
- Ahuja, Magnanti, et al.
- 1993
(Show Context)
Citation Context ...and Wagner 2007] for a survey), where most of them improve on Dijkstra’s algorithm [Dijkstra 1959]. A few speed-up techniques can be applied immediately, e.g., goal-directed and bidirectional search [=-=Ahuja et al. 1993-=-]. However, much better speed-up factors are reached when some precomputed information can be used. Since in most scenarios both time and storage requirements inhibit advance computation of all shorte... |

711 | Leda: A platform for combinatorial and geometric computing
- Mehlhorn, Naher
- 1999
(Show Context)
Citation Context ...ance when applied for shortest-path search. Further experiments contrast basic and extended multi-level graphs with different numbers of levels. Our code is written in C++, based on the LEDA library [=-=Näher and Mehlhorn 1999-=-], and compiled with the GNU compiler (version 3.3); as underlying shortestpath routine we use Dijkstra’s algorithm. The experiments were carried out on several 64-bit AMD Opteron machines, clocked at... |

167 | Computing the shortest path: A search meets graph theory.
- Goldberg, Harrelson
- 2005
(Show Context)
Citation Context ...ng time at least quadratic in the number of ver-tices) and no additional information at all. Also, combinations of speed-up techniques are possible [7, 11, 20]. One class of preprocessing techniques =-=[6, 8, 15, 16, 18, 20, 22]-=- prunes the search space of Dijkstra’s algorithm: some of the edges and vertices Dijkstra’s algorithm would consider can safely be ignored since by the additional information, the algorithm knows that... |

97 |
Highway Hierarchies Hasten Exact Shortest Path Queries
- Sanders, Schultes
- 2005
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Citation Context ...he ‘boundary vertices’ stores the shortest-path lengths between these vertices (to this end, also shortest-path overlay graphs would be appropriate).s4 · Martin Holzer et al. Highway Hierarchies. In [=-=Sanders and Schultes 2005-=-; 2006], the auxiliary graph is computed by first determining a neighborhood of each vertex. Then a level of so-called highway edges is introduced, i.e., edges (v, w) for some shortest path (u, . . .,... |

86 | Hierarchical encoded path views for path query processing: An optimal model and its performance evaluation.
- Jing, Huang, et al.
- 1998
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Citation Context ...is enriched with additional edges corresponding to shortest paths). We now discuss several approaches of the latter kind and point out their relationship to ours. Hierarchical Encoded Path Views. In [=-=Jing et al. 1998-=-], shortest-path computation in the context of navigation systems is studied, where there is also the need for efficiently maintaining shortest paths themselves, or path views—as opposed to the mere d... |

77 | Reach for A∗: Efficient point-to-point shortest path algorithms.
- Goldberg, Kaplan, et al.
- 2006
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Citation Context ...(c) SIAM, 2006; this is the author's version, the definite one is part of the ALENEX'06 proceedingsstices) and no additional information at all. Also, combinations of speed-up techniques are possible =-=[7, 11, 20]-=-. One class of preprocessing techniques [6, 8, 15, 16, 18, 20, 22] prunes the search space of Dijkstra’s algorithm: some of the edges and vertices Dijkstra’s algorithm would consider can safely be ign... |

75 |
Reach-based Routing: A New Approach to Shortest Path Algorithms Optimized for Road Networks
- Gutman
- 2004
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Citation Context ...ng time at least quadratic in the number of ver-tices) and no additional information at all. Also, combinations of speed-up techniques are possible [7, 11, 20]. One class of preprocessing techniques =-=[6, 8, 15, 16, 18, 20, 22]-=- prunes the search space of Dijkstra’s algorithm: some of the edges and vertices Dijkstra’s algorithm would consider can safely be ignored since by the additional information, the algorithm knows that... |

71 |
An Extremely Fast, Exact Algorithm for Finding Shortest Paths in Static Networks with Geographical Background.
- Lauther
- 2004
(Show Context)
Citation Context ...ng time at least quadratic in the number of ver-tices) and no additional information at all. Also, combinations of speed-up techniques are possible [7, 11, 20]. One class of preprocessing techniques =-=[6, 8, 15, 16, 18, 20, 22]-=- prunes the search space of Dijkstra’s algorithm: some of the edges and vertices Dijkstra’s algorithm would consider can safely be ignored since by the additional information, the algorithm knows that... |

69 | Engineering Highway Hierarchies - Sanders, Schultes - 2006 |

69 | Dijkstra’s algorithm on-line: An empirical case study from public railroad transport.
- Schulz, Wagner, et al.
- 2000
(Show Context)
Citation Context ...small and—informally speaking—decompose the graph in a balanced way. Moreover, we show how to construct random graphs that allow for a decomposition into balanced components through few vertices. In [=-=Schulz et al. 2000-=-], it is shown that the multi-level approach with one additional level can be successfully applied to public-transportation networks; [Schulz et al. 2002] then provides a generalization to multiple le... |

65 |
An Efficient Path Computation Model for Hierarchically Structured Topological Road Maps,
- Jung, Pramanik
- 2002
(Show Context)
Citation Context ...given source to the given target. Another common approach, which makes use of an auxiliary graph and is also followed in this paper, is to hierarchically decompose the graph into connected components =-=[9, 12, 13, 20, 21]-=-. Such techniques exploit the fact that—if the source s and the target t are not in the same component at some level—any s-t-path will pass through the border of the respective components. Additional ... |

64 | Partitioning Graphs to Speedup Dijkstra’s Algorithm.
- Mohring, Schilling, et al.
- 2006
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Citation Context ...shortest-path search toward the target [Goldberg and Harrelson 2005], or to prune the search space at vertices that are known not to form part of a shortest path requested [Gutman 2004; Lauther 2004; =-=Möhring et al. 2005-=-; Wagner and Willhalm 2003]. Another usage, also followed in this paper, is precomputing an auxiliary graph in which shortestpath calculations take place (mostly, the original graph is enriched with a... |

59 | Geometric Speed-Up Techniques for Finding Shortest Paths in Large Sparse Graphs - Wagner, Willhalm - 2003 |

48 | Dynamic Highway-Node Routing.
- Schultes, Sanders
- 2007
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Citation Context ...ically that only those parts of the multi-level graph have to be recomputed which belong to components (due to the decomposition through selected vertices) affected by an update. A very recent study [=-=Schultes and Sanders 2007-=-] considers basic multi-level graphs as defined in the work at hand but with selected vertices taken from a Highway Hierarchies precomputation. The most conspicuous difference is that large parts of t... |

32 | Using multi-level graphs for timetable information in railway systems
- Schulz, Wagner, et al.
- 2002
(Show Context)
Citation Context ...a given graph G = (V, E) on a subset S ⊆ V is a graph with vertex set S and edges corresponding to shortest paths in G. In particular, we consider variations of the multi-level overlay graph used in [=-=Schulz et al. 2002-=-] to speed up shortest-path computation. In this work, we follow up and present several vertex selection criteria, along with two general strategies of applying these criteria, to determine a subset S... |

29 | T.: Combining speed-up techniques for shortest-path computations
- Holzer, Schulz, et al.
- 2004
(Show Context)
Citation Context ...represent a trade-off between precomputational effort and resulting average speed-up. Also, many combinations of speed-up techniques have proven successful [Delling et al. 2007; Goldberg et al. 2006; =-=Holzer et al. 2004-=-; Schulz et al. 2000]. Preprocessed information can be employed to guide the shortest-path search toward the target [Goldberg and Harrelson 2005], or to prune the search space at vertices that are kno... |

25 | Highway hierarchies star
- Delling, Sanders, et al.
- 2006
(Show Context)
Citation Context ...ion of all shortest paths, such approaches represent a trade-off between precomputational effort and resulting average speed-up. Also, many combinations of speed-up techniques have proven successful [=-=Delling et al. 2007-=-; Goldberg et al. 2006; Holzer et al. 2004; Schulz et al. 2000]. Preprocessed information can be employed to guide the shortest-path search toward the target [Goldberg and Harrelson 2005], or to prune... |

13 | High-performance multi-level graphs
- Delling, Holzer, et al.
(Show Context)
Citation Context ...ion of all shortest paths, such approaches represent a trade-off between precomputational effort and resulting average speed-up. Also, many combinations of speed-up techniques have proven successful [=-=Delling et al. 2007-=-; Goldberg et al. 2006; Holzer et al. 2004; Schulz et al. 2000]. Preprocessed information can be employed to guide the shortest-path search toward the target [Goldberg and Harrelson 2005], or to prune... |

13 | Engineering shortest paths and layout algorithms for large graphs - WILLHALM |

9 | Timetable information and shortest paths
- SCHULZ
(Show Context)
Citation Context ...s reached. Another important parameter will be the maximum number of adjacent vertices of a component over all components, denoted by A. Due to lack of space, proofs are omitted; they can be found in =-=[19]-=-. 4.1 Size of the Multi-Level Graph and the Search Space Thefirstlemmashowsthatformultig h i l j klevel graphs that exhibit some regular hierarchical structure, the total number of additional edges c... |

8 | Engineering planar separator algorithms.
- Holzer, Schulz, et al.
- 2009
(Show Context)
Citation Context ...ntrality indexes, reach (RCH) [Gutman 2004], closeness (CLO), betweenness (BET), and betweenness approximation (BAP) [Brandes and Erlebach 2005]; and one involving a planar-separator algorithm (PLS) [=-=Holzer et al. 2005-=-].sEngineering Multi-Level Overlay Graphs for Shortest-Path Queries · 13 Random (RND). Vertices are selected uniformly at random. Degree (DEG). Vertices with the highest degrees are selected. Percenta... |

6 |
D.: In Transit to Constant Shortest-Path Queries in Road Networks
- Bast, Funke, et al.
- 2007
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Citation Context ...graph. This allows for individual optimization of each part, which was shown to reduce the overall amount of edges with road graphs considerably. Transit Node Routing. A similar idea is exploited in [=-=Bast et al. 2007-=-], which was developed independently of our approach: One basic observation is that from a fixed vertex, virtually all long-distance shortest paths leave a local area around it through a small number ... |

5 |
Hierarchical speed-up techniques for shortest-path algorithms
- Holzer
- 2003
(Show Context)
Citation Context ...given source to the given target. Another common approach, which makes use of an auxiliary graph and is also followed in this paper, is to hierarchically decompose the graph into connected components =-=[9, 12, 13, 20, 21]-=-. Such techniques exploit the fact that—if the source s and the target t are not in the same component at some level—any s-t-path will pass through the border of the respective components. Additional ... |

2 | Dynamic Speed-Up Techniques for Dijkstra’s Algorithm - Bauer - 2006 |

2 |
Shortest path speedup techniques, in Algorithmic Methods for Railway Optimization
- Willhalm, Wagner
- 2006
(Show Context)
Citation Context ...ion 5 provides an extensive experimental study. 1.1 Related Work There are numerous approaches to speed up single-pair shortest-path computations, most of them improving Dijkstra’s algorithm [5] (see =-=[24]-=- for a survey): On the one hand there are speedup techniques that can be applied without any preprocessed information, for example, goal-directed and bidirectional search [1]. On the other hand, much ... |

1 |
A graph generator
- Borgi, Graf, et al.
- 2005
(Show Context)
Citation Context ...tely twice as many in the largest srail graph. Sample instances of these graph classes can be found in Figures 6, 8, and 9, respectively. The generator for the ci and del graphs is available on-line [=-=Borgi et al. 2005-=-]. 2 We are grateful to the companies PTV AG, Karlsruhe, and HaCon, Hannover, for providing us with road and railway data, respectively, for scientific purposes. 3 Note that terms like railway, train,... |

1 | Reach for A*: Efficient point-to-point 24 · Martin Holzer et al. shortest path algorithms - Goldberg, Kaplan, et al. |