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.
The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of the segment.
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Graph Data Structure 4. Dijkstra’s Shortest Path Algorithm

Dijkstra's algorithm in 3 minutes

3.6 Dijkstra Algorithm  Single Source Shortest Path  Greedy Method

Lecture 15: SingleSource Shortest Paths Problem

4.2 All Pairs Shortest Path (FloydWarshall)  Dynamic Programming
Transcription
Definition
The shortest path problem can be defined for graphs whether undirected, directed, or mixed. Here it is defined for undirected graphs; for directed graphs the definition of path requires that consecutive vertices be connected by an appropriate directed edge.^{[1]}
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 where is 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
Several well known algorithms exist for solving this problem and its variants.
 Dijkstra's algorithm solves the singlesource shortest path problem with nonnegative edge weight.
 Bellman–Ford algorithm solves the singlesource problem if edge weights may be negative.
 A* search algorithm solves for singlepair 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, Goldberg & Radzik (1996).
Singlesource shortest paths
Undirected graphs
Weights  Time complexity  Author 

_{+}  O(V^{2})  Dijkstra 1959 
_{+}  O((E + V) log V)  Johnson 1977 (binary heap) 
_{+}  O(E + V log V)  Fredman & Tarjan 1984 (Fibonacci heap) 
O(E)  Thorup 1999 (requires constanttime multiplication) 
Unweighted graphs
Algorithm  Time complexity  Author 

Breadthfirst search  O(E + V) 
Directed acyclic graphs (DAGs)
An algorithm using topological sorting can solve the singlesource shortest path problem in time Θ(E + V) in arbitrarilyweighted DAGs.^{[2]}
Directed graphs with nonnegative weights
The following table is taken from Schrijver (2004), with some corrections and additions. A green background indicates an asymptotically best bound in the table; L is the maximum length (or weight) among all edges, assuming integer edge weights.
Weights  Algorithm  Time complexity  Author 

Ford 1956  
Bellman–Ford algorithm  Shimbel 1955, Bellman 1958, Moore 1959  
Dantzig 1960  
Dijkstra's algorithm with list  Leyzorek et al. 1957, Dijkstra 1959, Minty (see Pollack & Wiebenson 1960), Whiting & Hillier 1960  
Dijkstra's algorithm with binary heap  Johnson 1977  
Dijkstra's algorithm with Fibonacci heap  Fredman & Tarjan 1984, Fredman & Tarjan 1987  
Quantum Dijkstra algorithm with adjacency list  Dürr et al. 2006^{[3]}  
Dial's algorithm^{[4]} (Dijkstra's algorithm using a bucket queue with L buckets)  Dial 1969  
Johnson 1981, Karlsson & Poblete 1983  
Gabow's algorithm  Gabow 1983, Gabow 1985  
Ahuja et al. 1990  
Thorup  Thorup 2004 
Directed graphs with arbitrary weights without negative cycles
Weights  Algorithm  Time complexity  Author 

Ford 1956  
Bellman–Ford algorithm  Shimbel 1955, Bellman 1958, Moore 1959  
JohnsonDijkstra with binary heap  Johnson 1977  
JohnsonDijkstra with Fibonacci heap  Fredman & Tarjan 1984, Fredman & Tarjan 1987, adapted after Johnson 1977  
Johnson's technique applied to Dial's algorithm^{[4]}  Dial 1969, adapted after Johnson 1977  
Interiorpoint method with Laplacian solver  Cohen et al. 2017  
Interiorpoint method with flow solver  Axiotis, Mądry & Vladu 2020  
Robust interiorpoint method with sketching  van den Brand et al. 2020  
interiorpoint method with dynamic minratio cycle data structure  Chen et al. 2022  
Based on lowdiameter decomposition  Bernstein, Nanongkai & WulffNilsen 2022  
Hoplimited shortest paths  Fineman 2023 
Directed graphs with arbitrary weights with negative cycles
Finds a negative cycle or calculates distances to all vertices.
Weights  Algorithm  Time complexity  Author 

Andrew V. Goldberg 
Planar graphs with nonnegative weights
Weights  Algorithm  Time complexity  Author 

Henzinger et al. 1997 
Allpairs shortest paths
The allpairs shortest path problem finds the shortest paths between every pair of vertices v, v' in the graph. 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 that takes a total time of O(V^{4}).
Undirected graph
Weights  Time complexity  Algorithm 

_{+}  O(V^{3})  Floyd–Warshall algorithm 
Seidel's algorithm (expected running time)  
Williams 2014  
_{+}  O(EV log α(E,V))  Pettie & Ramachandran 2002 
O(EV)  Thorup 1999 applied to every vertex (requires constanttime multiplication). 
Directed graph
Weights  Time complexity  Algorithm 

(no negative cycles)  Floyd–Warshall algorithm  
Williams 2014  
(no negative cycles)  Quantum search^{[5]}^{[6]}  
(no negative cycles)  O(EV + V^{2} log V)  Johnson–Dijkstra 
(no negative cycles)  O(EV + V^{2} log log V)  Pettie 2004 
O(EV + V^{2} log log V)  Hagerup 2000 
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.^{[7]}
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 represent 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.^{[8]}
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 longdistance travel (e.g. highways). This property has been formalized using the notion of highway dimension.^{[9]} 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. The second phase is the query phase. In this phase, source and target node are known. 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 US in a fraction of a microsecond.^{[10]} Other techniques that have been used are:
 ALT (A* search, landmarks, and triangle inequality)
 Arc flags
 Contraction hierarchies
 Transit node routing
 Reachbased pruning
 Labeling
 Hub labels
Related problems
For shortest path problems in computational geometry, see Euclidean shortest path.
The shortest multiple disconnected path ^{[11]} 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.
Other related problems may be classified into the following categories.
Paths with constraints
Unlike the shortest path problem, which can be solved in polynomial time in graphs without negative cycles, shortest path problems which include additional constraints on the desired solution path are called Constrained Shortest Path First, and are harder to solve. One example is the constrained shortest path problem,^{[12]} which attempts to minimize the total cost of the path while at the same time maintaining another metric below a given threshold. This makes the problem NPcomplete (such problems are not believed to be efficiently solvable for large sets of data, see P = NP problem). Another NPcomplete example requires a specific set of vertices to be included in the path,^{[13]} which makes the problem similar to the Traveling Salesman Problem (TSP). The TSP is the problem of finding the shortest path that goes through every vertex exactly once, and returns to the start. The problem of finding the longest path in a graph is also NPcomplete.
Partial observability
The Canadian traveller problem and the stochastic shortest path problem are generalizations where either the graph is not completely known to the mover, changes over time, or where actions (traversals) are probabilistic.^{[14]}^{[15]}
Strategic shortest paths
Sometimes, the edges in a graph have personalities: each edge has its own selfish interest. An example is a communication network, in which each edge is a computer that possibly belongs to a different person. Different computers have different transmission speeds, so every edge in the network has a numeric weight equal to the number of milliseconds it takes to transmit a message. Our goal is to send a message between two points in the network in the shortest time possible. If we know the transmissiontime of each computer (the weight of each edge), then we can use a standard shortestpaths algorithm. If we do not know the transmission times, then we have to ask each computer to tell us its transmissiontime. But, the computers may be selfish: a computer might tell us that its transmission time is very long, so that we will not bother it with our messages. A possible solution to this problem is to use a variant of the VCG mechanism, which gives the computers an incentive to reveal their true weights.
Negative cycle detection
In some cases, the main goal is not to find the shortest path, but only to detect if the graph contains a negative cycle. Some shortestpaths algorithms can be used for this purpose:
 The Bellman–Ford algorithm can be used to detect a negative cycle in time .
 Cherkassky and Goldberg^{[16]} survey several other algorithms for negative cycle detection.
General algebraic framework on semirings: the algebraic path problem
Many problems can be framed as a form of the shortest path for some suitably substituted notions of addition along a path and taking the minimum. The general approach to these is to consider the two operations to be those of a semiring. Semiring multiplication is done along the path, and the addition is between paths. This general framework is known as the algebraic path problem.^{[17]}^{[18]}^{[19]}
Most of the classic shortestpath algorithms (and new ones) can be formulated as solving linear systems over such algebraic structures.^{[20]}
More recently, an even more general framework for solving these (and much less obviously related problems) has been developed under the banner of valuation algebras.^{[21]}
Shortest path in stochastic timedependent networks
In reallife situations, the transportation network is usually stochastic and timedependent. In fact, a traveler traversing a link daily may experiences different travel times on that link due not only to the fluctuations in travel demand (origindestination matrix) but also due to such incidents as work zones, bad weather conditions, accidents and vehicle breakdowns. As a result, a stochastic timedependent (STD) network is a more realistic representation of an actual road network compared with the deterministic one.^{[22]}^{[23]}
Despite considerable progress during the course of the past decade, it remains a controversial question how an optimal path should be defined and identified in stochastic road networks. In other words, there is no unique definition of an optimal path under uncertainty. One possible and common answer to this question is to find a path with the minimum expected travel time. The main advantage of using this approach is that efficient shortest path algorithms introduced for the deterministic networks can be readily employed to identify the path with the minimum expected travel time in a stochastic network. However, the resulting optimal path identified by this approach may not be reliable, because this approach fails to address travel time variability. To tackle this issue some researchers use distribution of travel time instead of expected value of it so they find the probability distribution of total travelling time using different optimization methods such as dynamic programming and Dijkstra's algorithm .^{[24]} These methods use stochastic optimization, specifically stochastic dynamic programming to find the shortest path in networks with probabilistic arc length.^{[25]} The concept of travel time reliability is used interchangeably with travel time variability in the transportation research literature, so that, in general, one can say that the higher the variability in travel time, the lower the reliability would be, and vice versa.
In order to account for travel time reliability more accurately, two common alternative definitions for an optimal path under uncertainty have been suggested. Some have introduced the concept of the most reliable path, aiming to maximize the probability of arriving on time or earlier than a given travel time budget. Others, alternatively, have put forward the concept of an αreliable path based on which they intended to minimize the travel time budget required to ensure a prespecified ontime arrival probability.
See also
 Bidirectional search, an algorithm that finds the shortest path between two vertices on a directed graph
 Euclidean shortest path
 Flow network
 K shortest path routing
 Minplus matrix multiplication
 Pathfinding
 Shortest Path Bridging
 Shortest path tree
 TRILL (TRansparent Interconnection of Lots of Links)
References
Notes
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Further reading
 Altıntaş, Gökhan (2020). Exact Solutions of ShortestPath Problems Based on Mechanical Analogies: In Connection with Labyrinths. Amazon Digital Services LLC. ISBN 9798655831896.
 Frigioni, D.; MarchettiSpaccamela, A.; Nanni, U. (1998). "Fully dynamic output bounded single source shortest path problem". Proc. 7th Annu. ACMSIAM Symp. Discrete Algorithms. Atlanta, GA. pp. 212–221. CiteSeerX 10.1.1.32.9856.
 Dreyfus, S. E. (October 1967). An Appraisal of Some Shortest Path Algorithms (PDF) (Report). Project Rand. United States Air Force. RM5433PR. Archived (PDF) from the original on November 17, 2015. DTIC AD661265.