Shortest path (A, C, E, D, F) between vertices A and F in the weighted directed graph

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.

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 ${\displaystyle P=(v_{1},v_{2},\ldots ,v_{n})\in V\times V\times \cdots \times V}$ such that ${\displaystyle v_{i}}$ is adjacent to ${\displaystyle v_{i+1}}$ for ${\displaystyle 1\leq i<n}$. Such a path ${\displaystyle P}$ is called a path of length ${\displaystyle n-1}$ from ${\displaystyle v_{1}}$ to ${\displaystyle v_{n}}$. (The ${\displaystyle v_{i}}$ 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 ${\displaystyle E=\{e_{i,j}\}}$ where ${\displaystyle e_{i,j}}$ is the edge incident to both ${\displaystyle v_{i}}$ and ${\displaystyle v_{j}}$. Given a real-valued weight function ${\displaystyle f:E\rightarrow \mathbb {R} }$, and an undirected (simple) graph ${\displaystyle G}$, the shortest path from ${\displaystyle v}$ to ${\displaystyle v'}$ is the path ${\displaystyle P=(v_{1},v_{2},\ldots ,v_{n})}$ (where ${\displaystyle v_{1}=v}$ and ${\displaystyle v_{n}=v'}$) that over all possible ${\displaystyle n}$ minimizes the sum ${\displaystyle \sum _{i=1}^{n-1}f(e_{i,i+1}).}$ When each edge in the graph has unit weight or ${\displaystyle f:E\rightarrow \{1\}}$, this is equivalent to finding the path with fewest edges.

The problem is also sometimes called the single-pair shortest path problem, to distinguish it from the following variations:

• The single-source shortest path problem, in which we have to find shortest paths from a source vertex v to all other vertices in the graph.
• The single-destination 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 single-source shortest path problem by reversing the arcs in the directed graph.
• The all-pairs 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 single-pair shortest path algorithm on all relevant pairs of vertices.

Algorithms

The most important algorithms for solving this problem are:

• Dijkstra's algorithm solves the single-source shortest path problem with non-negative edge weight.
• Bellman–Ford algorithm solves the single-source 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, Goldberg & Radzik (1996).

Single-source shortest paths

Undirected graphs

Weights	Time complexity	Author
${\displaystyle \mathbb {R} }$+	O(V2)	Dijkstra 1959
${\displaystyle \mathbb {R} }$+	O((E + V) log V)	Johnson 1977 (binary heap)
${\displaystyle \mathbb {R} }$+	O(E + V log V)	Fredman & Tarjan 1984 (Fibonacci heap)
${\displaystyle \mathbb {N} }$	O(E)	Thorup 1999 (requires constant-time multiplication)

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
${\displaystyle \mathbb {R} }$		${\displaystyle O(V^{2}EL)}$	Ford 1956
${\displaystyle \mathbb {R} }$	Bellman–Ford algorithm	${\displaystyle O(VE)}$	Shimbel 1955, Bellman 1958, Moore 1959
${\displaystyle \mathbb {R} }$		${\displaystyle O(V^{2}\log {V})}$	Dantzig 1960
${\displaystyle \mathbb {R} }$	Dijkstra's algorithm with list	${\displaystyle O(V^{2})}$	Leyzorek et al. 1957, Dijkstra 1959, Minty (see Pollack & Wiebenson 1960), Whiting & Hillier 1960
${\displaystyle \mathbb {R} }$	Dijkstra's algorithm with binary heap	${\displaystyle O((E+V)\log {V})}$	Johnson 1977
${\displaystyle \mathbb {R} }$	Dijkstra's algorithm with Fibonacci heap	${\displaystyle O(E+V\log {V})}$	Fredman & Tarjan 1984, Fredman & Tarjan 1987
${\displaystyle \mathbb {R} }$	Quantum Dijkstra algorithm with adjacency list	${\displaystyle O({\sqrt {VE}}\log ^{2}{V})}$	Dürr et al. 2006[2]
${\displaystyle \mathbb {N} }$	Dial's algorithm[3] (Dijkstra's algorithm using a bucket queue with L buckets)	${\displaystyle O(E+LV)}$	Dial 1969
		${\displaystyle O(E\log {\log {L}})}$	Johnson 1981, Karlsson & Poblete 1983
	Gabow's algorithm	${\displaystyle O(E\log _{E/V}L)}$	Gabow 1983, Gabow 1985
		${\displaystyle O(E+V{\sqrt {\log {L}}})}$	Ahuja et al. 1990
${\displaystyle \mathbb {N} }$	Thorup	${\displaystyle O(E+V\log {\log {V}})}$	Thorup 2004

Directed graphs with arbitrary weights without negative cycles

Weights	Algorithm	Time complexity	Author
${\displaystyle \mathbb {R} }$		O(V 2EL)	Ford 1956
${\displaystyle \mathbb {R} }$	Bellman–Ford algorithm	O(VE)	Shimbel 1955, Bellman 1958, Moore 1959
${\displaystyle \mathbb {R} }$	Johnson-Dijkstra with binary heap	O(V (E + log V))	Johnson 1977
${\displaystyle \mathbb {R} }$	Johnson-Dijkstra with Fibonacci heap	O(V (E + log V))	Fredman & Tarjan 1984, Fredman & Tarjan 1987, adapted after Johnson 1977
${\displaystyle \mathbb {N} }$	Johnson's technique applied to Dial's algorithm[3]	O(V (E + L))	Dial 1969, adapted after Johnson 1977
${\displaystyle \mathbb {N} }$	Based on low-diameter decomposition	${\displaystyle O(E\log ^{8}V\log L)}$	Bernstein, Nanongkai & Wulff-Nilsen 2022

Directed graphs with arbitrary weights with negative cycles

Finds a negative cycle or calculates distances to all vertices.

Weights	Algorithm	Time complexity	Author
${\displaystyle \mathbb {Z} }$	Andrew V. Goldberg	${\displaystyle O(E{\sqrt {V}}\log {N})}$

Planar graphs with nonnegative weights

Weights	Algorithm	Time complexity	Author
${\displaystyle \mathbb {R} _{\geq 0}}$		${\displaystyle O(V)}$	Henzinger et al. 1997

All-pairs shortest paths

The all-pairs shortest path problem finds the shortest paths between every pair of vertices v, v' in the graph. The all-pairs 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(V⁴).

Undirected graph

Weights	Time complexity	Algorithm
${\displaystyle \mathbb {R} }$+	O(V3)	Floyd–Warshall algorithm
${\displaystyle \{1,\infty \}}$	${\displaystyle O(V^{\omega }\log V)}$	Seidel's algorithm (expected running time)
${\displaystyle \mathbb {N} }$	${\displaystyle O(V^{3}/2^{\Omega (\log V)^{1/2}})}$	Williams 2014
${\displaystyle \mathbb {R} }$+	O(EV log α(E,V))	Pettie & Ramachandran 2002
${\displaystyle \mathbb {N} }$	O(EV)	Thorup 1999 applied to every vertex (requires constant-time multiplication).

Directed graph

Weights	Time complexity	Algorithm
${\displaystyle \mathbb {R} }$ (no negative cycles)	O(V3)	Floyd–Warshall algorithm
${\displaystyle \mathbb {N} }$	${\displaystyle O(V^{3}/2^{\Omega (\log V)^{1/2}})}$	Williams 2014
${\displaystyle \mathbb {R} }$ (no negative cycles)	${\displaystyle O(V^{2.5}\log ^{2}{V})}$	Quantum search[4][5]
${\displaystyle \mathbb {R} }$ (no negative cycles)	O(EV + V2 log V)	Johnson–Dijkstra
${\displaystyle \mathbb {R} }$ (no negative cycles)	O(EV + V2 log log V)	Pettie 2004
${\displaystyle \mathbb {N} }$	O(EV + V2 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.^[6]

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 min-delay path problem and usually tied with a widest path problem. For example, the algorithm may seek the shortest (min-delay) widest path, or widest shortest (min-delay) 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.^[7]

Related problems

For shortest path problems in computational geometry, see Euclidean shortest path.

The shortest multiple disconnected path ^[10] 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.

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.^[16][17][18]

Most of the classic shortest-path algorithms (and new ones) can be formulated as solving linear systems over such algebraic structures.^[19]

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.^[20]

Shortest path in stochastic time-dependent networks

In real-life situations, the transportation network is usually stochastic and time-dependent. 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 (origin-destination matrix) but also due to such incidents as work zones, bad weather conditions, accidents and vehicle breakdowns. As a result, a stochastic time-dependent (STD) network is a more realistic representation of an actual road network compared with the deterministic one.^[21][22]

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 .^[23] These methods use stochastic optimization, specifically stochastic dynamic programming to find the shortest path in networks with probabilistic arc length.^[24] 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 α-reliable path based on which they intended to minimize the travel time budget required to ensure a pre-specified on-time 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
• Min-plus matrix multiplication
• Pathfinding
• Shortest Path Bridging
• Shortest path tree
• TRILL (TRansparent Interconnection of Lots of Links)

References

Further reading

• Altıntaş, Gökhan (2020). Exact Solutions of Shortest-Path Problems Based on Mechanical Analogies: In Connection with Labyrinths. Amazon Digital Services LLC. ISBN 9798655831896.
• Frigioni, D.; Marchetti-Spaccamela, A.; Nanni, U. (1998). "Fully dynamic output bounded single source shortest path problem". Proc. 7th Annu. ACM-SIAM 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. RM-5433-PR. Archived (PDF) from the original on November 17, 2015. DTIC AD-661265.