View source for Shortest path problem

{{short description|Computational problem of graph theory}}
{{More footnotes needed|date=June 2009}}
[[File:Shortest path with direct weights.svg|thumb|upright=1.2|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 (graph theory)|path]] between two [[vertex (graph theory)|vertices]] (or nodes) in a [[Graph (discrete mathematics)|graph]] such that the sum of the [[Glossary of graph theory terms#weighted graph|weights]] of its constituent edges is minimized.<ref>{{Cite book |url=https://link.springer.com/book/10.1007/978-3-031-02574-7 |title=The Shortest-Path Problem |language=en |doi=10.1007/978-3-031-02574-7}}</ref>

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 or distance of each segment.

==Definition==
The shortest path problem can be defined for [[Graph (discrete mathematics)|graphs]] whether [[Graph (discrete mathematics)#Undirected graph|undirected]], [[Graph (discrete mathematics)#Directed graph|directed]], or [[Mixed graph|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.<ref>{{cite book |last1=Deo |first1=Narsingh |title=Graph Theory with Applications to Engineering and Computer Science |date=17 August 2016 |publisher=Courier Dover Publications |isbn=978-0-486-80793-5 |url=https://books.google.com/books?id=uk1KDAAAQBAJ |language=en}}</ref>

Two vertices are adjacent when they are both incident to a common edge.
A [[Path (graph theory)|path]] in an undirected graph is a [[sequence]] of vertices <math>P = ( v_1, v_2, \ldots, v_n ) \in V \times V \times \cdots \times V</math>
such that <math>v_i</math> is adjacent to <math>v_{i+1}</math> for <math>1 \leq i < n</math>.
Such a path <math>P</math> is called a path of length <math>n-1</math>
from <math>v_1</math> to <math>v_n</math>.
(The <math>v_i</math> 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 <math>E = \{e_{i, j}\}</math> where <math>e_{i, j}</math> is the edge incident to both <math>v_i</math> and <math>v_j</math>. Given a [[Function (mathematics)#Real function|real-valued]] weight function <math>f: E \rightarrow \mathbb{R}</math>, and an undirected (simple) graph <math>G</math>, the shortest path from <math>v</math> to <math>v'</math> is the path <math>P = ( v_1, v_2, \ldots, v_n )</math> (where <math>v_1 = v</math> and <math>v_n = v'</math>) that over all possible <math>n</math> minimizes the sum <math>\sum_{i =1}^{n-1} f(e_{i, i+1}).</math> When each edge in the graph has unit weight or <math>f: E \rightarrow \{1\}</math>, 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==
Several well known algorithms exist for solving this problem and its variants.
* [[Dijkstra's algorithm]] solves the single-source shortest path problem with only non-negative edge weights.
* [[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 graph]]s.
* [[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 {{harvtxt|Cherkassky|Goldberg|Radzik|1996}}.

==Single-source shortest paths==

===Undirected graphs===
{| class=wikitable
! Weights !! [[Time complexity]] !! Author
|-
| <math>\mathbb{R}</math><sub>+</sub> || ''O''(''V<sup>2</sup>'') || {{harvnb|Dijkstra|1959}}
|-
| <math>\mathbb{R}</math><sub>+</sub>  || ''O''((''E''&nbsp;+&nbsp;''V'')&nbsp;log&nbsp;''V'') || {{harvnb|Johnson|1977}} ([[binary heap]])
|-
| <math>\mathbb{R}</math><sub>+</sub> || ''O''(''E''&nbsp;+&nbsp;''V''&nbsp;log&nbsp;''V'') || {{harvnb|Fredman|Tarjan|1984}} ([[Fibonacci heap]])
|-
| <math>\mathbb{N}</math> || ''O''(''E'') || {{harvnb|Thorup|1999}} (requires constant-time multiplication)
|-
| <math>\mathbb{R}</math><sub>+</sub> || <math>O(E\sqrt{\log V \log \log V})</math> || {{harvnb|Duan|Mao|Shu|Yin|2023}}
|}

===Unweighted graphs===
{| class=wikitable
! Algorithm !! Time complexity !! Author
|-
| [[Breadth-first search]] || ''O''(''E''&nbsp;+&nbsp;''V'') ||
|}

===Directed acyclic graphs ===
An algorithm using [[Topological sorting#Application to shortest path finding|topological sorting]] can solve the single-source shortest path problem in time {{math|Θ(''E'' + ''V'')}} in arbitrarily-weighted directed acyclic graphs.<ref>{{harvnb|Cormen|Leiserson|Rivest|Stein|2001|p=655}}</ref>

===Directed graphs with nonnegative weights===
The following table is taken from {{harvtxt|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.
{| class=wikitable
! Weights !! Algorithm !! Time complexity !! Author
|-
| <math>\mathbb{R}</math> || || <math>O(V^2EL)</math> || {{harvnb|Ford|1956}}
|-
| <math>\mathbb{R}</math> || [[Bellman–Ford algorithm]] || <math>O(VE)</math> || {{harvnb|Shimbel|1955}}, {{harvnb|Bellman|1958}}, {{harvnb|Moore|1959}}
|-
| <math>\mathbb{R}</math> || || <math>O(V^2 \log{V})</math> || {{harvnb|Dantzig|1960}}
|-
| <math>\mathbb{R}</math> || [[Dijkstra's algorithm]] with list || <math>O(V^2)</math> || {{harvnb|Leyzorek|Gray|Johnson|Ladew|1957}}, {{harvnb|Dijkstra|1959}}, Minty (see {{harvnb|Pollack|Wiebenson|1960}}), {{harvnb|Whiting|Hillier|1960}}
|-
| <math>\mathbb{R}</math> || [[Dijkstra's algorithm]] with [[binary heap]] || <math> O((E+V)\log{V})</math> || {{harvnb|Johnson|1977}}
|- style="background: #d0ffd0"
| <math>\mathbb{R}</math> || [[Dijkstra's algorithm]] with [[Fibonacci heap]]||<math>O(E+V\log{V})</math> || {{harvnb|Fredman|Tarjan|1984}}, {{harvnb|Fredman|Tarjan|1987}}
|-
| <math>\mathbb{R}</math> || Quantum [[Dijkstra algorithm]] with adjacency list ||<math>O(\sqrt{VE}\log^2{V})</math> || Dürr et al. 2006<ref>{{Cite journal |last1=Dürr |first1=Christoph |last2=Heiligman |first2=Mark |last3=Høyer |first3=Peter |last4=Mhalla |first4=Mehdi |date=January 2006 |title=Quantum query complexity of some graph problems |journal=SIAM Journal on Computing |volume=35 |issue=6 |pages=1310–1328 |doi=10.1137/050644719 |arxiv=quant-ph/0401091 |s2cid=14253494 |issn=0097-5397}}</ref>
|-
| <math>\mathbb{N}</math> || Dial's algorithm<ref name="dial69">{{cite journal
   | last = Dial | first = Robert B.
   | title = Algorithm 360: Shortest-Path Forest with Topological Ordering [H]
   | journal = Communications of the ACM
   | volume = 12
   | issue = 11
   | pages = 632–633
   | year = 1969
   | doi = 10.1145/363269.363610
   | s2cid = 6754003
 | doi-access = free
   }}</ref> ([[Dijkstra's algorithm]] using a [[bucket queue]] with ''L'' buckets) || <math>O(E+LV)</math> || {{harvnb|Dial|1969}}
|-
|- style="background: #d0ffd0"
| || || <math>O(E\log{\log{L}})</math> || {{harvnb|Johnson|1981}}, {{harvnb|Karlsson|Poblete|1983}}
|-
| || [[Gabow's algorithm (single-source shortest paths)|Gabow's algorithm]] || <math>O(E\log_{E/V}L) </math>|| {{harvnb|Gabow|1983}}, {{harvnb|Gabow|1985}}
|- style="background: #d0ffd0"
| || || <math> O( E + V \sqrt{\log{L}})</math> || {{harvnb|Ahuja|Mehlhorn|Orlin|Tarjan|1990}}
|-
| <math>\mathbb{N}</math>|| Thorup || <math>O(E+V \log{\log{V}})</math>|| {{harvnb|Thorup|2004}}
|}
{{incomplete list|date=February 2011}}

===Directed graphs with arbitrary weights without negative cycles===
{| class=wikitable
! Weights !! Algorithm !! Time complexity !! Author
|-
| <math>\mathbb{R}</math> || || <math>O(V^2 E L)
</math>|| {{harvnb|Ford|1956}}
|-
| <math>\mathbb{R}</math> || [[Bellman–Ford algorithm]] || <math>O(VE)</math>|| {{harvnb|Shimbel|1955}}, {{harvnb|Bellman|1958}}, {{harvnb|Moore|1959}}
|-
| <math>\mathbb{R}</math> || [[Johnson's algorithm|Johnson-Dijkstra]] with [[binary heap]] || <math>O(V E + V \log V)</math>|| {{harvnb|Johnson|1977}}
|-
| <math>\mathbb{R}</math> || [[Johnson's algorithm|Johnson-Dijkstra]] with [[Fibonacci heap]] || <math>O(V E + V \log V)</math>|| {{harvnb|Fredman|Tarjan|1984}}, {{harvnb|Fredman|Tarjan|1987}}, adapted after {{harvnb|Johnson|1977}}
|-
| <math>\mathbb{Z}</math>|| [[Johnson's algorithm|Johnson's technique]] applied to Dial's algorithm<ref name="dial69" /> || <math>O(V(E+L))</math>|| {{harvnb|Dial|1969}}, adapted after {{harvnb|Johnson|1977}}
|-
|<math>\mathbb{Z}</math>
|[[Interior-point method]] with Laplacian solver
|<math>O(E^{10/7} \log^{O(1)} V \log^{O(1)} L)</math>
|{{harvnb|Cohen|Mądry|Sankowski|Vladu|2017|Ref=https://doi.org/10.1137/1.9781611974782.48}}
|-
|<math>\mathbb{Z}</math>
|[[Interior-point method]] with <math>\ell_p</math> flow solver
|<math>E^{4/3 + o(1)} \log^{O(1)} L</math>
|{{harvnb|Axiotis|Mądry|Vladu|2020|Ref=https://doi.org/10.1109/FOCS46700.2020.00018}}
|-
|<math>\mathbb{Z}</math>
|Robust [[interior-point method]] with sketching
|<math>O((E + V^{3/2}) \log^{O(1)} V \log^{O(1)} L)</math>
|{{harvnb|van den Brand|Lee|Nanongkai|Peng, Saranurak, Sidford, Song, Wang|2020|Ref=https://doi.org/10.1109/FOCS46700.2020.00090}}
|-
|<math>\mathbb{Z}</math>
| <math>\ell_1</math> [[interior-point method]] with dynamic min-ratio cycle data structure
|<math>O(E^{1+o(1)} \log L)</math>
|{{harvnb|Chen|Kyng|Liu|Peng|Probst|Sachdeva|2022|Ref=https://doi.org/10.1109/FOCS54457.2022.00064}}
|-
|<math>\mathbb{Z}</math>
|Based on low-diameter decomposition
|<math>O(E \log^8 V \log L)</math>
|{{harvnb|Bernstein|Nanongkai|Wulff-Nilsen|2022}}
|-
|<math>\mathbb{R}</math>
|Hop-limited shortest paths
|<math>O(E V^{8/9} \log^{O(1)} V)</math>
|{{harvnb|Fineman|2023|4=|Ref=https://arxiv.org/abs/2311.02520}}
|}
{{incomplete list|date=December 2012}}

===Directed graphs with arbitrary weights with negative cycles===
Finds a negative cycle or calculates distances to all vertices.
{| class=wikitable
! Weights !! Algorithm !! Time complexity !! Author
|-
| <math>\mathbb{Z}</math> || || <math>O(E\sqrt{V}\log{N})</math> ||Andrew V. Goldberg
|}

===Planar graphs with nonnegative weights===
{| class=wikitable
! Weights !! Algorithm !! Time complexity !! Author
|-
| <math>\mathbb{R}_{\geq 0}</math> || || <math> O( V )</math> || {{harvnb|Henzinger|Klein|Rao|Subramanian|1997}}
|}

== Applications ==
'''Network flows'''<ref>{{Cite book |last=Cormen |first=Thomas H. |title=Introduction to Algorithms |date=July 31, 2009 |publisher=MIT Press |year=2009 |isbn=9780262533058 |edition=3rd}}</ref> are a fundamental concept in graph theory and operations research, often used to model problems involving the transportation of goods, liquids, or information through a network. A network flow problem typically involves a directed graph where each edge represents a pipe, wire, or road, and each edge has a capacity, which is the maximum amount that can flow through it. The goal is to find a feasible flow that maximizes the flow from a source node to a sink node.

'''Shortest Path Problems''' can be used to solve certain network flow problems, particularly when dealing with single-source, single-sink networks. In these scenarios, we can transform the network flow problem into a series of shortest path problems.

=== Transformation Steps ===
<ref>{{Cite book |last=Kleinberg |first1=Jon |last2=Tardos |first2=Éva |title=Algorithm Design |publisher=Addison-Wesley |year=2005 |isbn=978-0321295354 |edition=1st |url=http://www.nytimes.com/2009/08/06/technology/06stats.html?_r=2&scp=1&sq=statistics&st=nyt}}</ref>
# '''Create a Residual Graph:'''
#* For each edge (u, v) in the original graph, create two edges in the residual graph:
#** (u, v) with capacity c(u, v)
#** (v, u) with capacity 0
#* The residual graph represents the remaining capacity available in the network.
# '''Find the Shortest Path:'''
#* Use a shortest path algorithm (e.g., Dijkstra's algorithm, Bellman-Ford algorithm) to find the shortest path from the source node to the sink node in the residual graph.
# '''Augment the Flow:'''
#* Find the minimum capacity along the shortest path.
#* Increase the flow on the edges of the shortest path by this minimum capacity.
#* Decrease the capacity of the edges in the forward direction and increase the capacity of the edges in the backward direction.
# '''Update the Residual Graph:'''
#* Update the residual graph based on the augmented flow.
# '''Repeat:'''
#* Repeat steps 2-4 until no more paths can be found from the source to the sink.

==All-pairs shortest paths==
The all-pairs shortest path problem finds the shortest paths between every pair of vertices {{mvar|v}}, {{mvar|v'}} in the graph.  The all-pairs shortest paths problem for unweighted directed graphs was introduced by {{harvtxt|Shimbel|1953}}, who observed that it could be solved by a linear number of matrix multiplications that takes a total time of {{math|''O''(''V''<sup>4</sup>)}}.

===Undirected graph===
{| class=wikitable
! Weights !! Time complexity !! Algorithm
|-
| <math>\mathbb{R}</math><sub>+</sub> || {{math|''O''(''V''<sup>3</sup>)}} || [[Floyd–Warshall algorithm]]
|-
| <math>\{1, \infty\}</math> || <math>O(V^\omega \log V)</math> || [[Seidel's algorithm]] (expected running time)
|-
| <math>\mathbb{N}</math> || <math>O(V^3/2^{\Omega(\log V)^{1/2}})</math> || {{harvnb|Williams|2014}}
|-
| <math>\mathbb{R}</math><sub>+</sub> || {{math|''O''(''EV''&nbsp;log&nbsp;α(''E'',''V''))}} || {{harvnb|Pettie|Ramachandran|2002}}
|-
| <math>\mathbb{N}</math> || {{math|''O''(''EV'')}} || {{harvnb|Thorup|1999}} applied to every vertex (requires constant-time multiplication).
|}

===Directed graph===
{| class=wikitable
! Weights !! Time complexity !! Algorithm
|-
| <math>\mathbb{R}</math> (no negative cycles) || <math>O(V^3)</math> || [[Floyd–Warshall algorithm]]
|-
| <math>\mathbb{N}</math> || <math>O(V^3/2^{\Omega(\log V)^{1/2}})</math> || {{harvnb|Williams|2014}}
|-
| <math>\mathbb{R}</math> (no negative cycles) || <math>O(V^{2.5}\log^2{V})</math> || [[Grover's algorithm|Quantum search]]<ref>{{Cite arXiv |last1=Dürr |first1=C. |last2=Høyer |first2=P. |date=1996-07-18 |title=A Quantum Algorithm for Finding the Minimum |eprint=quant-ph/9607014 }}</ref><ref>{{Cite arXiv |last1=Nayebi |first1=Aran |last2=Williams |first2=V. V. |date=2014-10-22 |title=Quantum algorithms for shortest paths problems in structured instances |class=quant-ph |eprint=1410.6220}}</ref>
|-
| <math>\mathbb{R}</math> (no negative cycles) || {{math|''O''(''EV''&nbsp;+&nbsp;''V''<sup>2</sup>&nbsp;log&nbsp;''V'')}} || [[Johnson's algorithm|Johnson–Dijkstra]]
|-
| <math>\mathbb{R}</math> (no negative cycles) || {{math|''O''(''EV''&nbsp;+&nbsp;''V''<sup>2</sup>&nbsp;log&nbsp;log&nbsp;''V'')}} || {{harvnb|Pettie|2004}}
|-
| <math>\mathbb{N}</math> || {{math|''O''(''EV''&nbsp;+&nbsp;''V''<sup>2</sup>&nbsp;log&nbsp;log&nbsp;''V'')}} || {{harvnb|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.<ref>{{Cite web |title=Fast route planning |first=Peter |last=Sanders |author-link=Peter Sanders (computer scientist) |date=March 23, 2009 |url=https://www.youtube.com/watch?v=-0ErpE8tQbw |work=Google Tech Talk |archive-url=https://ghostarchive.org/varchive/youtube/20211211/-0ErpE8tQbw| archive-date=2021-12-11 |url-status=live}}{{cbignore}}</ref>

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 [[Computer network|networking]] or [[Telecommunications network|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 [[Very-large-scale integration|VLSI]] design.<ref>{{cite journal |doi=10.1145/242224.242246 |title=Developing algorithms and software for geometric path planning problems |date=December 1996 |first=Danny Z. |last=Chen |journal=ACM Computing Surveys |volume=28 |issue=4es |s2cid=11761485 |at=Article 18 }}</ref>

===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.<ref>Abraham, Ittai; Fiat, Amos; [[Andrew V. Goldberg|Goldberg, Andrew V.]]; Werneck, Renato F. [http://research.microsoft.com/pubs/115272/soda10.pdf%20research.microsoft.com/pubs/115272/soda10.pdf "Highway Dimension, Shortest Paths, and Provably Efficient Algorithms"]. ACM-SIAM Symposium on Discrete Algorithms, pages 782–793, 2010.</ref> 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.<ref>Abraham, Ittai; Delling, Daniel; [[Andrew V. Goldberg|Goldberg, Andrew V.]]; Werneck, Renato F. [http://research.microsoft.com/pubs/142356/HL-TR.pdf research.microsoft.com/pubs/142356/HL-TR.pdf "A Hub-Based Labeling Algorithm for Shortest Paths on Road Networks"]. Symposium on Experimental Algorithms, pages 230–241, 2011.</ref> Other techniques that have been used are:

* ALT ([[A* search]], landmarks, and [[triangle inequality]])
* Arc flags
* [[Contraction hierarchies]]
* [[Transit node routing]]
* Reach-based pruning
* Labeling
* [[Hub labels]]

==Related problems==
For shortest path problems in [[computational geometry]], see [[Euclidean shortest path]].

The shortest multiple disconnected path <ref>{{cite journal |doi=10.1016/j.cpc.2005.01.020 |title=Shortest multiple disconnected path for the analysis of entanglements in two- and three-dimensional polymeric systems |year=2005 |first=Martin |last=Kroger |journal=Computer Physics Communications |volume=168 |issue=3 |pages=209–232 |bibcode=2005CoPhC.168..209K }}</ref> 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,<ref>{{cite journal |last1=Lozano |first1=Leonardo |last2=Medaglia |first2=Andrés L |title=On an exact method for the constrained shortest path problem |journal=Computers & Operations Research |date=2013 |volume=40 |issue=1 |pages=378–384 |doi=10.1016/j.cor.2012.07.008}}</ref> 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]] (such problems are not believed to be efficiently solvable for large sets of data, see [[P = NP problem]]). Another [[NP-complete]] example requires a specific set of vertices to be included in the path,<ref>{{cite book |last1=Osanlou |first1=Kevin |last2=Bursuc |first2=Andrei |last3=Guettier |first3=Christophe |last4=Cazenave |first4=Tristan |last5=Jacopin |first5=Eric |title=2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) |chapter=Optimal Solving of Constrained Path-Planning Problems with Graph Convolutional Networks and Optimized Tree Search |date=2019 |pages=3519–3525 |doi=10.1109/IROS40897.2019.8968113 |arxiv=2108.01036 |isbn=978-1-7281-4004-9 |s2cid=210706773 }}</ref> 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 [[Longest path problem|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 is not completely known to the mover, changes over time, or where actions (traversals) are probabilistic.<ref>{{cite journal |last1=Bar-Noy |first1=Amotz |last2=Schieber |first2=Baruch |title=The canadian traveller problem |journal=Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms |date=1991 |pages=261–270 |citeseerx=10.1.1.1088.3015 }}</ref><ref>{{cite conference |last1=Nikolova |first1=Evdokia |last2=Karger |first2=David R |title=Route planning under uncertainty: the Canadian traveller problem |book-title=Proceedings of the 23rd National Conference on Artificial Intelligence (AAAI) |pages=969–974 |url=https://www.aaai.org/Papers/AAAI/2008/AAAI08-154.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.aaai.org/Papers/AAAI/2008/AAAI08-154.pdf |archive-date=2022-10-09 |url-status=live}}</ref>

===Strategic shortest paths===
{{unreferenced section|date=December 2015}}
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 [[Vickrey–Clarke–Groves mechanism#quickest paths|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]] can be used to detect a negative cycle in time <math>O(|V||E|)</math>.
* Cherkassky and Goldberg<ref>{{Cite journal |last1=Cherkassky |first1=Boris V. |last2=Goldberg |first2=Andrew V. |date=1999-06-01 |title=Negative-cycle detection algorithms |url=https://doi.org/10.1007/s101070050058 |journal=Mathematical Programming |language=en |volume=85 |issue=2 |pages=277–311 |doi=10.1007/s101070050058 |s2cid=79739 |issn=1436-4646}}</ref> survey several other algorithms for negative cycle detection.

==General algebraic framework on semirings: the algebraic path problem==
{{expand section|date=August 2014}}
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]].<ref name="Pair1966">{{cite conference |first = Claude | last = Pair
| title= Sur des algorithmes pour des problèmes de cheminement dans les graphes finis |trans-title=On algorithms for path problems in finite graphs
| book-title = Théorie des graphes (journées internationales d'études) [Theory of Graphs (international symposium)]
| publisher = Dunod (Paris); Gordon and Breach (New York)
| date = 1967
| conference = Rome (Italy), July 1966
| page  = 271
| editor-last = Rosentiehl | editor-first = Pierre | editor-link = Pierre Rosenstiehl
| oclc = 901424694
}}</ref><ref>{{cite book
| first1 = Jean Claude | last1 = Derniame
| first2 = Claude | last2 = Pair
| title = Problèmes de cheminement dans les graphes |trans-title=Path Problems in Graphs
| publisher = Dunod (Paris)
| year = 1971
}}</ref><ref name="BarasTheodorakopoulos2010">{{cite book |first1=John |last1=Baras |first2=George |last2=Theodorakopoulos |title=Path Problems in Networks |url=https://books.google.com/books?id=fZJeAQAAQBAJ&pg=PA9|date=4 April 2010 |publisher=Morgan & Claypool Publishers |isbn=978-1-59829-924-3 |pages=9–}}</ref>

Most of the classic shortest-path algorithms (and new ones) can be formulated as solving linear systems over such algebraic structures.<ref name="GondranMinoux2008">{{cite book |first1=Michel |last1=Gondran |first2=Michel |last2=Minoux |title=Graphs, Dioids and Semirings: New Models and Algorithms |year=2008 |publisher=Springer Science & Business Media|isbn=978-0-387-75450-5|chapter=chapter 4}}</ref>

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]]s<!-- not just [[valuation (algebra)]]! -->.<ref name="PoulyKohlas2012">{{cite book |first1=Marc |last1=Pouly |first2=Jürg |last2=Kohlas |title=Generic Inference: A Unifying Theory for Automated Reasoning|year=2011|publisher=John Wiley & Sons |isbn=978-1-118-01086-0 |chapter=Chapter 6. Valuation Algebras for Path Problems}}</ref>

==Shortest path in stochastic time-dependent networks==
In real-life, a transportation network is usually stochastic and time-dependent. The travel duration on a road segment depends on many factors such as the amount of traffic (origin-destination matrix), road work, weather, accidents and vehicle breakdowns. A more realistic model of such a road network is a stochastic time-dependent (STD) network.<ref>Loui, R.P., 1983. Optimal paths in graphs with stochastic or multidimensional weights. Communications of the ACM, 26(9), pp.670-676.</ref><ref>{{cite journal |last1=Rajabi-Bahaabadi |first1=Mojtaba |first2=Afshin |last2=Shariat-Mohaymany |first3=Mohsen |last3=Babaei |first4=Chang Wook |last4=Ahn |title=Multi-objective path finding in stochastic time-dependent road networks using non-dominated sorting genetic algorithm |journal=Expert Systems with Applications |date=2015 |volume=42 |issue=12|pages=5056–5064 |doi=10.1016/j.eswa.2015.02.046 }}</ref>

There is no accepted definition of optimal path under uncertainty (that is, in stochastic road networks). It is a controversial subject, despite considerable progress during the past decade. One common definition is a path with the minimum expected travel time. The main advantage of this approach is that it can make use of efficient shortest path algorithms for deterministic networks. However, the resulting optimal path may not be reliable, because this approach fails to address travel time variability.

To tackle this issue, some researchers use travel duration distribution instead of its expected value. So, they find the probability distribution of total travel duration using different optimization methods such as [[dynamic programming]] and [[Dijkstra's algorithm]] .<ref>{{cite journal |last1=Olya |first1=Mohammad Hessam |title=Finding shortest path in a combined exponential – gamma probability distribution arc length |journal=International Journal of Operational Research |date=2014 |volume=21 |issue=1|pages=25–37 |doi=10.1504/IJOR.2014.064020 }}</ref> These methods use [[stochastic optimization]], specifically stochastic dynamic programming to find the shortest path in networks with probabilistic arc length.<ref>{{cite journal |last1=Olya |first1=Mohammad Hessam |title=Applying Dijkstra's algorithm for general shortest path problem with normal probability distribution arc length |journal=International Journal of Operational Research |date=2014 |volume=21 |issue=2|pages=143–154 |doi=10.1504/IJOR.2014.064541 }}</ref> The terms ''travel time reliability'' and ''travel time variability'' are used as opposites in the transportation research literature: the higher the variability, the lower the reliability of predictions.

To account for variability, researchers have suggested two alternative definitions for an optimal path under uncertainty. The ''most reliable path'' is one that maximizes the probability of arriving on time given a travel time budget. An ''α-reliable path'' is one that minimizes the travel time budget required to arrive on time with a given 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==

===Notes===
{{reflist|30em}}

===Bibliography===
{{sfn whitelist |CITEREFCormenLeisersonRivestStein2001}}
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{{refend}}

==Further reading==
*{{cite book
 |last=Altıntaş
 |first=Gökhan
 |date=2020
 |title= Exact Solutions of Shortest-Path Problems Based on Mechanical Analogies: In Connection with Labyrinths
 |url=https://books.google.com/books?id=oSvHzQEACAAJ
 |publisher= Amazon Digital Services LLC
 |isbn= 9798655831896
}}
* {{cite conference |title=Fully dynamic output bounded single source shortest path problem |first1=D. |last1=Frigioni |first2=A. |last2=Marchetti-Spaccamela |first3=U. |last3=Nanni |year=1998 |book-title=Proc. 7th Annu. ACM-SIAM Symp. Discrete Algorithms |pages=212–221 |location=Atlanta, GA |citeseerx = 10.1.1.32.9856}}
* {{cite report |last=Dreyfus |first=S. E. |title=An Appraisal of Some Shortest Path Algorithms |date=October 1967 |id=RM-5433-PR |publisher=United States Air Force |series=Project Rand |url=http://apps.dtic.mil/dtic/tr/fulltext/u2/661265.pdf|archive-url=https://web.archive.org/web/20151117014431/http://www.dtic.mil/dtic/tr/fulltext/u2/661265.pdf|url-status=live|archive-date=November 17, 2015}} DTIC AD-661265.

{{Graph traversal algorithms}}
{{Authority control}}

[[Category:Network theory]]
[[Category:Graph distance]]
[[Category:Polynomial-time problems]]
[[Category:Computational problems in graph theory]]
[[Category:Edsger W. Dijkstra]]