## Point-to-Point Shortest Path Algorithms with Preprocessing

Citations: | 4 - 0 self |

### BibTeX

@MISC{Goldberg_point-to-pointshortest,

author = {Andrew V. Goldberg},

title = {Point-to-Point Shortest Path Algorithms with Preprocessing},

year = {}

}

### OpenURL

### Abstract

Abstract. This is a survey of some recent results on point-to-point shortest path algorithms. This classical optimization problem received a lot of attention lately and significant progress has been made. After an overview of classical results, we study recent heuristics that solve the problem while examining only a small portion of the input graph; the graph can be very big. Note that the algorithms we discuss find exact shortest paths. These algorithms are heuristic because they perform well only on some graph classes. While their performance has been good in experimental studies, no theoretical bounds are known to support the experimental observations. Most of these algorithms have been motivated by finding paths in large road networks. We start by reviewing the classical Dijkstra’s algorithm and its bidirectional variant, developed in 1950’s and 1960’s. Then we review A* search, an AI technique developed in 1970’s. Next we turn our attention to modern results which are based on preprocessing the graph. To be practical, preprocessing needs to be reasonably fast and not use too much space. We discuss landmark- and reach-based algorithms as well as their combination. 1

### Citations

1444 | A note on two problems in connexion with graphs
- Dijkstra
- 1959
(Show Context)
Citation Context ...efine d(v) to be exact if it is equal to the distance from s to v. If one always selects a vertex v such that, at the selection time, d(v) is exact, then each vertex is scanned at most once. Dijkstra =-=[5]-=- and independently Dantzig [3] observed that if ℓ is non-negative and v is a labeled vertex with the smallest distance label, then d(v) is exact. The labeling method with the minimum label selection r... |

980 |
A formal basis for the heuristic determination of minimum cost paths
- Hart, Nilsson, et al.
- 1968
(Show Context)
Citation Context ... that we can think of π(v) as a lower bound on the distance from v to t. The second lemma allows us to combine feasible lower bound functions to obtain a better one. 3 A ∗ Search The A∗ search method =-=[6, 20]-=- was originally designed to speed up search in large, sometimes implicitly represented, graphs, such as game graphs. This algorithm is also known as heuristic search or goal-directed search. The idea ... |

816 |
Linear Programming and Extensions
- Dantzig
- 1963
(Show Context)
Citation Context ...s equal to the distance from s to v. If one always selects a vertex v such that, at the selection time, d(v) is exact, then each vertex is scanned at most once. Dijkstra [5] and independently Dantzig =-=[3]-=- observed that if ℓ is non-negative and v is a labeled vertex with the smallest distance label, then d(v) is exact. The labeling method with the minimum label selection rule is known as Dijkstra’s alg... |

770 |
Flows in networks
- Ford, Fulkerson
- 1962
(Show Context)
Citation Context ...tion t. The goal is to find a shortest path from s to t. Let dist(v,w) denote the shortest-path distance from vertex v to vertex w with respect to ℓ. The labeling method for the shortest path problem =-=[25, 26]-=- finds shortest paths from the source to all vertices in the graph. The method works as follows (see e.g. [37]). For every vertex v it maintains a distance label d(v), a parent p(v), and a status S(v)... |

603 |
Data Structures and Networks Algorithms
- Tarjan
- 1983
(Show Context)
Citation Context ...ex v to vertex w with respect to ℓ. The labeling method for the shortest path problem [25, 26] finds shortest paths from the source to all vertices in the graph. The method works as follows (see e.g. =-=[37]-=-). For every vertex v it maintains a distance label d(v), a parent p(v), and a status S(v) ∈ {unreached,labeled,scanned}. Initially d(v) = ∞, p(v) = nil, and S(v) = unreached for every vertex v. The m... |

577 | Fibonacci heaps and their uses in improved network optimization algorithms - Fredman, Tarjan - 1987 |

142 | Shortest path algorithms: Theory and experimental evaluation
- Cherkassky, Goldberg, et al.
- 1996
(Show Context)
Citation Context ...aphs of roughly comparable size but with different structure. One is a 400 × 400 planar grid with adjacent vertices connected by arcs with lengths chosen uniformly and independently from the interval =-=[1,16000]-=-. The grid is directed: length of arcs (v,w) and (w,v) are chosen independently. It has 160000 vertices and 638400 arcs. The other graph is a road network of the San Francisco Bay area. This graph has... |

109 |
An appraisal of some shortest-path algorithms
- Dreyfus
- 1969
(Show Context)
Citation Context ... the reverse graph, the graph with every arc reversed and obtain a shortest path as the reversal of the path found. One can combine the forward and the reverse algorithms. The bidirectional algorithm =-=[3, 7, 30]-=- alternates between running the two algorithms, each maintaining its own set of distance labels. Let ds(v) and dt(v) be the distance labels of v maintained by the forward and the reverse algorithms, r... |

98 | Computing the shortest path: A* search meets graph theory
- GOLDBERG, HARRELSON
(Show Context)
Citation Context ...h that πt(t) = π ′ t(t) = 0 and, for any vertex v, π ′ t(v) ≥ πt(v) (i.e., π ′ t dominates πt). If ties are broken consistently when selecting the next vertex to scan, the following holds. Theorem 1. =-=[13]-=- The set of vertices scanned by A ∗ search using π ′ t is contained in the set of vertices scanned by A ∗ search using πt. 3.1 Bidirectional A ∗ search We combine A∗ search and bidirectional search as... |

94 | Undirected Single-Source Shortest Paths with Positive Integer Weights in Linear Time - Thorup - 1999 |

87 |
Compact oracles for reachability and approximate distances in planar digraphs
- Thorup
(Show Context)
Citation Context ...anar graphs with slightly super-linear preprocessing space. The best bound in this context appears in [8]. Algorithms for approximate shortest paths that use preprocessing have been studied; see e.g. =-=[2, 23, 39]-=-. Some of the work on exact algorithms with preprocessing includes [14, 16, 15, 18, 19, 24, 27, 29, 32–35, 40]. Here we address only the approaches based on A ∗ search with landmark-based lower bounds... |

60 | Shortest Path Algorithms: An Evaluation using Real Road Networks - Zhan, Noon - 1998 |

59 | Reach for A ∗ : Efficient Point-to-Point Shortest Path Algorithms - Goldberg, Kaplan, et al. - 2006 |

57 | Shortest path algorithms - Gallo, Pallottino - 1988 |

53 | Planar graphs, negative weight edges, shortest paths, and near linear time
- Fakcharoenphol, Rao
- 2001
(Show Context)
Citation Context ...r the general P2P problem, but there are non-trivial results for the special case of undirected planar graphs with slightly super-linear preprocessing space. The best bound in this context appears in =-=[8]-=-. Algorithms for approximate shortest paths that use preprocessing have been studied; see e.g. [2, 23, 39]. Some of the work on exact algorithms with preprocessing includes [14, 16, 15, 18, 19, 24, 27... |

53 | An extremely fast exact algorithm for finding shortest paths in static networks with geographical background - Lauther - 2004 |

53 | Geometric speed-up techniques for finding shortest paths in large sparse graphs - Wagner, Willhalm - 2003 |

52 |
Reach-based routing: A new approach to shortest path algorithms optimized for road networks
- Gutman
(Show Context)
Citation Context ... details. Other, more minor, improvements and optimizations of alt have been discussed in [16, 18]. 4 Reach-Based Pruning In this section we discuss the notion of reach, originally proposed by Gutman =-=[19]-=- and further developed in [16, 18]. The definition of reach may seem odd at first,sbut becomes natural when one sees how it is used to prune Dijkstra’s search. To simplify the presentation, we assume ... |

51 |
Bi-directional Search
- POHL
- 1971
(Show Context)
Citation Context ...linear time bounds, and in practice, where running times are within a small constant factor of the breadth-first search time. The P2P problem with no preprocessing has been addressed, for example, in =-=[21, 31, 36, 42]-=-. With preprocessing, no nontrivial theoretical bound is known for the general P2P problem, but there are non-trivial results for the special case of undirected planar graphs with slightly super-linea... |

46 |
Engineering highway hierarchies
- Sanders, Schultes
- 2006
(Show Context)
Citation Context ...e we address only the approaches based on A ∗ search with landmark-based lower bounds [14, 18], the notion of reach [19], and their combination [16, 15]. For a related highway hierarchy approach, see =-=[32, 33]-=-. The paper is organized as follows. We give definitions and background in Section 2. Section 3 reviews A ∗ search and describes its landmark-based implementation, alt. Section 4 introduces the concep... |

43 | Partitioning graphs to speedup Dijkstra’s algorithm - Möhring, Schilling, et al. |

41 | A computational study of routing algorithms for realistic transportation networks - Jacob, Marathe, et al. |

37 |
Computing point-to-point shortest paths from external memory
- GOLDBERG, WERNECK
- 2005
(Show Context)
Citation Context ... Some of the work on exact algorithms with preprocessing includes [14, 16, 15, 18, 19, 24, 27, 29, 32–35, 40]. Here we address only the approaches based on A ∗ search with landmark-based lower bounds =-=[14, 18]-=-, the notion of reach [19], and their combination [16, 15]. For a related highway hierarchy approach, see [32, 33]. The paper is organized as follows. We give definitions and background in Section 2. ... |

34 | A simple shortest path algorithm with linear average time, in - Goldberg |

34 |
Network Flow Theory
- FORD
- 1956
(Show Context)
Citation Context ...tion t. The goal is to find a shortest path from s to t. Let dist(v,w) denote the shortest-path distance from vertex v to vertex w with respect to ℓ. The labeling method for the shortest path problem =-=[25, 26]-=- finds shortest paths from the source to all vertices in the graph. The method works as follows (see e.g. [37]). For every vertex v it maintains a distance label d(v), a parent p(v), and a status S(v)... |

33 |
Compact roundtrip routing in directed networks
- Cowen, Wagner
(Show Context)
Citation Context ...anar graphs with slightly super-linear preprocessing space. The best bound in this context appears in [8]. Algorithms for approximate shortest paths that use preprocessing have been studied; see e.g. =-=[2, 23, 39]-=-. Some of the work on exact algorithms with preprocessing includes [14, 16, 15, 18, 19, 24, 27, 29, 32–35, 40]. Here we address only the approaches based on A ∗ search with landmark-based lower bounds... |

28 | Single-source shortest-paths on arbitrary directed graphs in linear average-case time - Meyer - 2001 |

22 | Using multi-level graphs for timetable information - Schulz, Weihe - 2002 |

21 |
A fast algorithm for finding better routes by ai search techniques
- Ikeda, Tsu, et al.
- 2004
(Show Context)
Citation Context ...linear time bounds, and in practice, where running times are within a small constant factor of the breadth-first search time. The P2P problem with no preprocessing has been addressed, for example, in =-=[21, 31, 36, 42]-=-. With preprocessing, no nontrivial theoretical bound is known for the general P2P problem, but there are non-trivial results for the special case of undirected planar graphs with slightly super-linea... |

20 | Shortest path algorithms: Engineering aspects, in - Goldberg |

20 | Acceleration of Shortest Path and Constrained Shortest Path Computation - Köhler, Möhring, et al. |

18 |
Preprocessing an undirected planar network to enable fast approximate distance queries
- Klein
- 2002
(Show Context)
Citation Context ...anar graphs with slightly super-linear preprocessing space. The best bound in this context appears in [8]. Algorithms for approximate shortest paths that use preprocessing have been studied; see e.g. =-=[2, 23, 39]-=-. Some of the work on exact algorithms with preprocessing includes [14, 16, 15, 18, 19, 24, 27, 29, 32–35, 40]. Here we address only the approaches based on A ∗ search with landmark-based lower bounds... |

18 |
Finding the shortest route between two points in a network
- NICHOLSON
- 1966
(Show Context)
Citation Context ... the reverse graph, the graph with every arc reversed and obtain a shortest path as the reversal of the path found. One can combine the forward and the reverse algorithms. The bidirectional algorithm =-=[3, 7, 30]-=- alternates between running the two algorithms, each maintaining its own set of distance labels. Let ds(v) and dt(v) be the distance labels of v maintained by the forward and the reverse algorithms, r... |

14 | Shortest--Route Methods: 1 - Denardo, Fox - 1979 |

14 | Better landmarks within reach
- Goldberg, Kaplan, et al.
(Show Context)
Citation Context ...ncludes [14, 16, 15, 18, 19, 24, 27, 29, 32–35, 40]. Here we address only the approaches based on A ∗ search with landmark-based lower bounds [14, 18], the notion of reach [19], and their combination =-=[16, 15]-=-. For a related highway hierarchy approach, see [32, 33]. The paper is organized as follows. We give definitions and background in Section 2. Section 3 reviews A ∗ search and describes its landmark-ba... |

14 | Implementations of Dijkstra's Algorithm Based on Multi-Level Buckets - Goldberg, Silverstein - 1995 |

13 |
A comparison between label-setting and label-correcting algorithms for computing one-to-one shortest paths
- Zahn, Noon
(Show Context)
Citation Context ...linear time bounds, and in practice, where running times are within a small constant factor of the breadth-first search time. The P2P problem with no preprocessing has been addressed, for example, in =-=[21, 31, 36, 42]-=-. With preprocessing, no nontrivial theoretical bound is known for the general P2P problem, but there are non-trivial results for the special case of undirected planar graphs with slightly super-linea... |

10 |
An approach to automatic problem-solving
- Doran
- 1967
(Show Context)
Citation Context ... that we can think of π(v) as a lower bound on the distance from v to t. The second lemma allows us to combine feasible lower bound functions to obtain a better one. 3 A ∗ Search The A∗ search method =-=[6, 20]-=- was originally designed to speed up search in large, sometimes implicitly represented, graphs, such as game graphs. This algorithm is also known as heuristic search or goal-directed search. The idea ... |

8 |
Fast and exact shortest path queries using highway hierarchies
- Schultes
- 2005
(Show Context)
Citation Context ...e we address only the approaches based on A ∗ search with landmark-based lower bounds [14, 18], the notion of reach [19], and their combination [16, 15]. For a related highway hierarchy approach, see =-=[32, 33]-=-. The paper is organized as follows. We give definitions and background in Section 2. Section 3 reviews A ∗ search and describes its landmark-based implementation, alt. Section 4 introduces the concep... |