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In
graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
, the shortest path problem is the problem of finding a
path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire p ...
between two vertices (or nodes) in a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
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 Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
whether
undirected In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...
,
directed Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''D ...
, 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 A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire p ...
in an undirected graph is a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of vertices P = ( v_1, v_2, \ldots, v_n ) \in V \times V \times \cdots \times V such that v_i is adjacent to v_ for 1 \leq i < n. Such a path P is called a path of length n-1 from v_1 to v_n. (The 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 e_ be the edge incident to both v_i and v_j. Given a
real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an i ...
weight function f: E \rightarrow \mathbb, and an undirected (simple) graph G, the shortest path from v to v' is the path P = ( v_1, v_2, \ldots, v_n ) (where v_1 = v and v_n = v') that over all possible n minimizes the sum \sum_^ f(e_). When each edge in the graph has unit weight or f: E \rightarrow \, 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 Dijkstra's algorithm ( ) is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years ...
solves the single-source shortest path problem with non-negative edge weight. *
Bellman–Ford algorithm The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph. It is slower than Dijkstra's algorithm for the same problem, but more versatile, as it is ...
solves the single-source problem if edge weights may be negative. *
A* search algorithm A* (pronounced "A-star") is a graph traversal and path search algorithm, which is used in many fields of computer science due to its completeness, optimality, and optimal efficiency. One major practical drawback is its O(b^d) space complexity, ...
solves for single-pair shortest path using heuristics to try to speed up the search. *
Floyd–Warshall algorithm In computer science, the Floyd–Warshall algorithm (also known as Floyd's algorithm, the Roy–Warshall algorithm, the Roy–Floyd algorithm, or the WFI algorithm) is an algorithm for finding shortest paths in a directed weighted graph with p ...
solves all pairs shortest paths. *
Johnson's algorithm Johnson's algorithm is a way to find the shortest paths between all pairs of vertices in an edge-weighted directed graph. It allows some of the edge weights to be negative numbers, but no negative-weight cycles may exist. It works by using ...
solves all pairs shortest paths, and may be faster than Floyd–Warshall on
sparse graph In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction ...
s. *
Viterbi algorithm The Viterbi algorithm is a dynamic programming algorithm for obtaining the maximum a posteriori probability estimate of the most likely sequence of hidden states—called the Viterbi path—that results in a sequence of observed events, especiall ...
solves the shortest stochastic path problem with an additional probabilistic weight on each node. Additional algorithms and associated evaluations may be found in .


Single-source shortest paths


Undirected graphs


Unweighted graphs


Directed acyclic graphs (DAGs)

An algorithm using
topological sorting In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge ''uv'' from vertex ''u'' to vertex ''v'', ''u'' comes before ''v'' in the ordering. For ins ...
can solve the single-source shortest path problem in time in arbitrarily-weighted DAGs.


Directed graphs with nonnegative weights

The following table is taken from , 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.


Directed graphs with arbitrary weights without negative cycles


Directed graphs with arbitrary weights with negative cycles

Finds a negative cycle or calculates distances to all vertices.


Planar graphs with nonnegative weights


All-pairs shortest paths

The all-pairs shortest path problem finds the shortest paths between every pair of vertices , in the graph. The all-pairs shortest paths problem for unweighted directed graphs was introduced by , who observed that it could be solved by a linear number of matrix multiplications that takes a total time of .


Undirected graph


Directed graph


Applications

Shortest path algorithms are applied to automatically find directions between physical locations, such as driving directions on
web mapping Web mapping or an online mapping is the process of using maps, usually created through geographic information systems (GIS), on the Internet, more specifically in the World Wide Web (WWW). A web map or an online map is both served and consumed, t ...
websites like
MapQuest MapQuest (stylized as mapquest) is an American free online web mapping service. It was launched in 1996 as the first commercial web mapping service. MapQuest vies for market share with competitors such as Google Maps and Here. History MapQuest's ...
or
Google Maps Google Maps is a web mapping platform and consumer application offered by Google. It offers satellite imagery, aerial photography, street maps, 360° interactive panoramic views of streets ( Street View), real-time traffic conditions, and rou ...
. For this application fast specialized algorithms are available. If one represents a nondeterministic
abstract machine An abstract machine is a computer science theoretical model that allows for a detailed and precise analysis of how a computer system functions. It is analogous to a mathematical function in that it receives inputs and produces outputs based on pre ...
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 The Rubik's Cube is a Three-dimensional space, 3-D combination puzzle originally invented in 1974 by Hungarians, Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the Magic Cube, the puzzle was licensed by Rubik t ...
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 Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that fe ...
mindset, this shortest path problem is sometimes called the min-delay path problem and usually tied with a
widest path problem In graph algorithms, the widest path problem is the problem of finding a path between two designated vertices in a weighted graph, maximizing the weight of the minimum-weight edge in the path. The widest path problem is also known as the maximum ...
. 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 Six degrees of separation is the idea that all people are six or fewer social connections away from each other. As a result, a chain of "friend of a friend" statements can be made to connect any two people in a maximum of six steps. It is also k ...
" that try to find the shortest path in graphs like movie stars appearing in the same film. Other applications, often studied in
operations research Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve deci ...
, include plant and facility layout,
robotics Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrat ...
,
transportation Transport (in British English), or transportation (in American English), is the intentional movement of humans, animals, and goods from one location to another. Modes of transport include air, land (rail and road), water, cable, pipeline, ...
, and
VLSI Very large-scale integration (VLSI) is the process of creating an integrated circuit (IC) by combining millions or billions of MOS transistors onto a single chip. VLSI began in the 1970s when MOS integrated circuit (Metal Oxide Semiconductor) c ...
design.


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 one-way 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. 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. Other techniques that have been used are: * ALT (
A* search A* (pronounced "A-star") is a graph traversal and path search algorithm, which is used in many fields of computer science due to its completeness, optimality, and optimal efficiency. One major practical drawback is its O(b^d) space complexity, ...
, landmarks, and
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
) * Arc flags *
Contraction hierarchies In computer science, the method of contraction hierarchies is a speed-up technique for finding the shortest-path in a graph. The most intuitive applications are car-navigation systems: a user wants to drive from A to B using the quickest possible ...
* Transit node routing * Reach-based pruning * Labeling * Hub labels


Related problems

For shortest path problems in
computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...
, see
Euclidean shortest path The Euclidean shortest path problem is a problem in computational geometry: given a set of polyhedral obstacles in a Euclidean space, and two points, find the shortest path between the points that does not intersect any of the obstacles. Two d ...
. The shortest multiple disconnected path is a representation of the primitive path network within the framework of
Reptation theory A peculiarity of thermal motion of very long linear macromolecules in ''entangled'' polymer melts or concentrated polymer solutions is reptation. Derived from the word reptile, reptation suggests the movement of entangled polymer chains as bein ...
. The
widest path problem In graph algorithms, the widest path problem is the problem of finding a path between two designated vertices in a weighted graph, maximizing the weight of the minimum-weight edge in the path. The widest path problem is also known as the maximum ...
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, 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
NP-complete In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by tryi ...
(such problems are not believed to be efficiently solvable for large sets of data, see
P = NP problem The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved. The informal term ''quickly'', used abov ...
). Another
NP-complete In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by tryi ...
example requires a specific set of vertices to be included in the path, which makes the problem similar to the
Traveling Salesman Problem The travelling salesman problem (also called the travelling salesperson problem or TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each cit ...
(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 NP-complete.


Partial observability

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.


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 transmission-time of each computer (the weight of each edge), then we can use a standard shortest-paths algorithm. If we do not know the transmission times, then we have to ask each computer to tell us its transmission-time. 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 shortest-paths algorithms can be used for this purpose: * The
Bellman–Ford algorithm The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph. It is slower than Dijkstra's algorithm for the same problem, but more versatile, as it is ...
can be used to detect a negative cycle in time O(, V, , E, ). * Cherkassky and Goldberg 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 In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...
. Semiring multiplication is done along the path, and the addition is between paths. This general framework is known as the
algebraic 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. The problem of finding the shortest path between tw ...
. Most of the classic shortest-path algorithms (and new ones) can be formulated as solving linear systems over such algebraic structures. More recently, an even more general framework for solving these (and much less obviously related problems) has been developed under the banner of
valuation algebra The term "information algebra" refers to mathematical techniques of information processing. Classical information theory goes back to Claude Shannon. It is a theory of information transmission, looking at communication and storage. However, it has ...
s.


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. 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 Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. I ...
and
Dijkstra's algorithm Dijkstra's algorithm ( ) is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years ...
. These methods use
stochastic optimization Stochastic optimization (SO) methods are optimization methods that generate and use random variables. For stochastic problems, the random variables appear in the formulation of the optimization problem itself, which involves random objective funct ...
, specifically stochastic dynamic programming to find the shortest path in networks with probabilistic arc length. 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 Bidirectional search is a graph search algorithm that finds a shortest path from an initial vertex to a goal vertex in a directed graph. It runs two simultaneous searches: one forward from the initial state, and one backward from the goal, stopping ...
, an algorithm that finds the shortest path between two vertices on a directed graph *
Euclidean shortest path The Euclidean shortest path problem is a problem in computational geometry: given a set of polyhedral obstacles in a Euclidean space, and two points, find the shortest path between the points that does not intersect any of the obstacles. Two d ...
*
Flow network In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge. Often in operations res ...
*
K shortest path routing The ''k'' shortest path routing problem is a generalization of the shortest path routing problem in a given network. It asks not only about a shortest path but also about next ''k−1'' shortest paths (which may be longer than the shortest path) ...
*
Min-plus matrix multiplication Min-plus matrix multiplication, also known as distance product, is an operation on matrices. Given two n \times n matrices A = (a_) and B = (b_), their distance product C = (c_) = A \star B is defined as an n \times n matrix such that c_ = \min_^n ...
*
Pathfinding Pathfinding or pathing is the plotting, by a computer application, of the shortest route between two points. It is a more practical variant on solving mazes. This field of research is based heavily on Dijkstra's algorithm for finding the sh ...
* Shortest Path Bridging * Shortest path tree *
TRILL TRILL (Transparent Interconnection of Lots of Links) is an Internet Standard implemented by devices called TRILL switches. TRILL combines techniques from bridging and routing, and is the application of link-state routing to the VLAN-aware cus ...
(TRansparent Interconnection of Lots of Links)


References


Notes


Bibliography

* * * * * * * * * * * * * * * * * * * * * * Attributes Dijkstra's algorithm to Minty ("private communication") on p. 225. * Here: vol.A, sect.7.5b, p. 103 * * * * * *


Further reading

* * DTIC AD-661265. {{Authority control Network theory Graph distance Polynomial-time problems Computational problems in graph theory Edsger W. Dijkstra