Shortest path problem

In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
This is analogous to the problem of finding the shortest path between two intersections on a road map: the graph's vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of its road segment.
Contents
Definition
The shortest path problem can be defined for graphs whether undirected, directed, or mixed. It is defined here for undirected graphs; for directed graphs the definition of path requires that consecutive vertices be connected by an appropriate directed edge.
Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices such that is adjacent to for . Such a path is called a path of length from to . (The are variables; their numbering here relates to their position in the sequence and needs not to relate to any canonical labeling of the vertices.)
Let be the edge incident to both and . Given a realvalued weight function , and an undirected (simple) graph , the shortest path from to is the path (where and ) that over all possible minimizes the sum When each edge in the graph has unit weight or , this is equivalent to finding the path with fewest edges.
The problem is also sometimes called the singlepair shortest path problem, to distinguish it from the following variations:
 The singlesource shortest path problem, in which we have to find shortest paths from a source vertex v to all other vertices in the graph.
 The singledestination shortest path problem, in which we have to find shortest paths from all vertices in the directed graph to a single destination vertex v. This can be reduced to the singlesource shortest path problem by reversing the arcs in the directed graph.
 The allpairs shortest path problem, in which we have to find shortest paths between every pair of vertices v, v' in the graph.
These generalizations have significantly more efficient algorithms than the simplistic approach of running a singlepair shortest path algorithm on all relevant pairs of vertices.
Algorithms
The most important algorithms for solving this problem are:
 Dijkstra's algorithm solves the singlesource shortest path problems.
 Bellman–Ford algorithm solves the singlesource problem if edge weights may be negative.
 A* search algorithm solves for single pair shortest path using heuristics to try to speed up the search.
 Floyd–Warshall algorithm solves all pairs shortest paths.
 Johnson's algorithm solves all pairs shortest paths, and may be faster than Floyd–Warshall on sparse graphs.
 Viterbi algorithm solves the shortest stochastic path problem with an additional probabilistic weight on each node.
Additional algorithms and associated evaluations may be found in Cherkassky et al.^{1}
Road networks
A road network can be considered as a graph with positive weights. The nodes represent road junctions and each edge of the graph is associated with a road segment between two junctions. The weight of an edge may correspond to the length of the associated road segment, the time needed to traverse the segment or the cost of traversing the segment. Using directed edges it is also possible to model oneway streets. Such graphs are special in the sense that some edges are more important than others for long distance travel (e.g. highways). This property has been formalized using the notion of highway dimension.^{2} There are a great number of algorithms that exploit this property and are therefore able to compute the shortest path a lot quicker than would be possible on general graphs.
All of these algorithms work in two phases. In the first phase, the graph is preprocessed without knowing the source or target node. This phase may take several days for realistic data and some techniques. The second phase is the query phase. In this phase, source and target node are known. The running time of the second phase is generally less than a second. The idea is that the road network is static, so the preprocessing phase can be done once and used for a large number of queries on the same road network.
The algorithm with the fastest known query time is called hub labeling and is able to compute shortest path on the road networks of Europe or the USA in a fraction of a microsecond.^{3} Other techniques that have been used are:
 ALT
 Arc Flags
 Contraction hierarchies
 Transit Node Routing
 Reach based Pruning
 Labeling
Singlesource shortest paths
Directed unweighted graphs
Algorithm  Time complexity  Author 

Breadthfirst search  O(E) 

This list is incomplete; you can help by expanding it.
Directed acyclic graphs
Directed graphs with nonnegative weights
Algorithm  Time complexity  Author 

O(V^{2}EL)  Ford 1956  
Bellman–Ford algorithm  O(VE)  Bellman 1958, Moore 1959 
O(V^{2} log V)  Dantzig 1958, Dantzig 1960, Minty (cf. Pollack & Wiebenson 1960), Whiting & Hillier 1960  
Dijkstra's algorithm with list  O(V^{2})  Leyzorek et al. 1957, Dijkstra 1959 
Dijkstra's algorithm with modified binary heap  O((E + V) log V)  
. . .  . . .  . . . 
Dijkstra's algorithm with Fibonacci heap  O(E + V log V)  Fredman & Tarjan 1984, Fredman & Tarjan 1987 
O(E log log L)  Johnson 1982, Karlsson & Poblete 1983  
Gabow's algorithm  O(E log_{E/V} L)  Gabow 1983b, Gabow 1985b 
O(E + V√log L)  Ahuja et al. 1990 

This list is incomplete; you can help by expanding it.
Planar directed graphs with nonnegative weights
Directed graphs with arbitrary weights
Algorithm  Time complexity  Author 

Bellman–Ford algorithm  O(VE)  Bellman 1958, Moore 1959 

This list is incomplete; you can help by expanding it.
Planar directed graphs with arbitrary weights
Allpairs shortest paths
This section requires expansion. (August 2013) 
The allpairs shortest paths problem for unweighted directed graphs was introduced by Shimbel (1953), who observed that it could be solved by a linear number of matrix multiplications, taking a total time of O(V^{4}).
Subsequent algorithms handle edge weights (which may possibly be negative), and are faster. The Floyd–Warshall algorithm takes O(V^{3}) time, and Johnson's algorithm (a combination of the Bellman–Ford and Dijkstra algorithms) takes O(VE + V^{2} log V).
Applications
Shortest path algorithms are applied to automatically find directions between physical locations, such as driving directions on web mapping websites like Mapquest or Google Maps. For this application fast specialized algorithms are available.^{4}
If one represents a nondeterministic abstract machine as a graph where vertices describe states and edges describe possible transitions, shortest path algorithms can be used to find an optimal sequence of choices to reach a certain goal state, or to establish lower bounds on the time needed to reach a given state. For example, if vertices represents the states of a puzzle like a Rubik's Cube and each directed edge corresponds to a single move or turn, shortest path algorithms can be used to find a solution that uses the minimum possible number of moves.
In a networking or telecommunications mindset, this shortest path problem is sometimes called the mindelay path problem and usually tied with a widest path problem. For example, the algorithm may seek the shortest (mindelay) widest path, or widest shortest (mindelay) path.
A more lighthearted application is the games of "six degrees of separation" that try to find the shortest path in graphs like movie stars appearing in the same film.
Other applications, often studied in operations research, include plant and facility layout, robotics, transportation, and VLSI design".^{5}
Related problems
For shortest path problems in computational geometry, see Euclidean shortest path.
The travelling salesman problem is the problem of finding the shortest path that goes through every vertex exactly once, and returns to the start. Unlike the shortest path problem, which can be solved in polynomial time in graphs without negative cycles, the travelling salesman problem is NPcomplete and, as such, is believed not to be efficiently solvable for large sets of data (see P = NP problem). The problem of finding the longest path in a graph is also NPcomplete.
The Canadian traveller problem and the stochastic shortest path problem are generalizations where either the graph isn't completely known to the mover, changes over time, or where actions (traversals) are probabilistic.
The shortest multiple disconnected path ^{6} is a representation of the primitive path network within the framework of Reptation theory.
The widest path problem seeks a path so that the minimum label of any edge is as large as possible.
Linear programming formulation
There is a natural linear programming formulation for the shortest path problem, given below. It is very simple compared to most other uses of linear programs in discrete optimization, however it illustrates connections to other concepts.
Given a directed graph (V, A) with source node s, target node t, and cost w_{ij} for each edge (i, j) in A, consider the program with variables x_{ij}
 minimize subject to and for all i,
The intuition behind this is that is an indicator variable for whether edge (i, j) is part of the shortest path  1 when it is, and 0 if it is not. We wish to select the set of edges with minimal weight, subject to the constraint that this set forms a path from s to t (represented by the equality constraint: for all vertices except s and t the number of incoming and outcoming edges that are part of the path must be the same (ie, that it should be a path from s to t).
This LP has the special property that it is integral; more specifically, every basic optimal solution (when one exists) has all variables equal to 0 or 1, and the set of edges whose variables equal 1 form an st dipath. See Ahuja et al.^{7} for one proof, although the origin of this approach dates back to mid20th century.
The dual for this linear program is
 maximize y_{t} − y_{s} subject to for all ij, y_{j} − y_{i} ≤ w_{ij}
and feasible duals correspond to the concept of a consistent heuristic for the A* algorithm for shortest paths. For any feasible dual y the reduced costs are nonnegative and A* essentially runs Dijkstra's algorithm on these reduced costs.
See also
 IEEE 802.1aq
 Flow network
 Shortest path tree
 Euclidean shortest path
 Minplus matrix multiplication
 Bidirectional search, an algorithm that finds the shortest path between two vertices on a directed graph
References
 ^ Cherkassky, Boris V.; Goldberg, Andrew V.; Radzik, Tomasz (1996). "Shortest paths algorithms: theory and experimental evaluation". Mathematical Programming. Ser. A 73 (2): 129–174. doi:10.1016/00255610(95)000216. MR 1392160.
 ^ Abraham, Ittai; Fiat, Amos; Goldberg, Andrew V.; Werneck, Renato F. "Highway Dimension, Shortest Paths, and Provably Efficient Algorithms". ACMSIAM Symposium on Discrete Algorithms, pages 782793, 2010.
 ^ Abraham, Ittai; Delling, Daniel; Goldberg, Andrew V.; Werneck, Renato F. research.microsoft.com/pubs/142356/HLTR.pdf "A HubBased Labeling Algorithm for Shortest Paths on Road Networks". Symposium on Experimental Algorithms, pages 230241, 2011.
 ^ Sanders, Peter (March 23, 2009). Fast route planning. Google Tech Talk.
 ^ Chen, Danny Z. (December 1996). "Developing algorithms and software for geometric path planning problems". ACM Computing Surveys 28 (4es): 18. doi:10.1145/242224.242246.
 ^ Kroger, Martin (2005). "Shortest multiple disconnected path for the analysis of entanglements in two and threedimensional polymeric systems". Computer Physics Communications 168 (168): 209–232. doi:10.1016/j.cpc.2005.01.020.
 ^ Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin (1993). Network Flows: Theory, Algorithms and Applications. Prentice Hall. ISBN 013617549X.
 Bellman, Richard (1958). "On a routing problem". Quarterly of Applied Mathematics 16: 87–90. MR 0102435.
 Cormen, Thomas H.; Leiserson, Charles E., Rivest, Ronald L., Stein, Clifford (2001) [1990]. "SingleSource Shortest Paths and AllPairs Shortest Paths". Introduction to Algorithms (2nd ed.). MIT Press and McGrawHill. pp. 580–642. ISBN 0262032937.
 Dijkstra, E. W. (1959). "A note on two problems in connexion with graphs". Numerische Mathematik 1: 269–271. doi:10.1007/BF01386390.
 Fredman, Michael Lawrence; Tarjan, Robert E. (1984). "Fibonacci heaps and their uses in improved network optimization algorithms". 25th Annual Symposium on Foundations of Computer Science. IEEE. pp. 338–346. doi:10.1109/SFCS.1984.715934. ISBN 081860591X.
 Fredman, Michael Lawrence; Tarjan, Robert E. (1987). "Fibonacci heaps and their uses in improved network optimization algorithms". Journal of the Association for Computing Machinery 34 (3): 596–615. doi:10.1145/28869.28874.
 Leyzorek, M.; Gray, R. S.; Johnson, A. A.; Ladew, W. C.; Meaker, S. R., Jr.; Petry, R. M.; Seitz, R. N. (1957). Investigation of Model Techniques — First Annual Report — 6 June 1956 — 1 July 1957 — A Study of Model Techniques for Communication Systems. Cleveland, Ohio: Case Institute of Technology.
 Moore, E. F. (1959). "The shortest path through a maze". Proceedings of an International Symposium on the Theory of Switching (Cambridge, Massachusetts, 2–5 April 1957). Cambridge: Harvard University Press. pp. 285–292.
 Shimbel, Alfonso (1953). "Structural parameters of communication networks". Bulletin of Mathematical Biophysics 15 (4): 501–507. doi:10.1007/BF02476438.
Further reading
 D. Frigioni; A. MarchettiSpaccamela and U. Nanni (1998). "Fully dynamic output bounded single source shortest path problem". Proc. 7th Annu. ACMSIAM Symp. Discrete Algorithms. Atlanta, GA. pp. 212–221.
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