Shortest path problem#General algebraic framework on semirings: the algebraic path problem

The shortest path problem can be defined for graphs whether undirected, directed, or mixed. The definition for undirected graphs states that every edge can be traversed in either direction. Directed graphs require that consecutive vertices be connected by an appropriate directed edge.{{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}}

Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices $P\; =\; (\; v\_1,\; v\_2,\; \backslash ldots,\; v\_n\; )\; \backslash in\; V\; \backslash times\; V\; \backslash times\; \backslash cdots\; \backslash times\; V$ such that $v\_i$ is adjacent to $v\_\{i+1\}$ for . 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 relates to their position in the sequence and need not relate to a canonical labeling.)

Let $E\; =\; \backslash \{e\_\{i,\; j\}\backslash \}$ where $e\_\{i,\; j\}$ is the edge incident to both $v\_i$ and $v\_j$. Given a real-valued weight function $f:\; E\; \backslash rightarrow\; \backslash mathbb\{R\}$, and an undirected (simple) graph $G$, the shortest path from $v$ to $v\text{'}$ is the path $P\; =\; (\; v\_1,\; v\_2,\; \backslash ldots,\; v\_n\; )$ (where $v\_1\; =\; v$ and $v\_n\; =\; v\text{'}$) that over all possible $n$ minimizes the sum $\backslash sum\_\{i\; =1\}^\{n-1\}\; f(e\_\{i,\; i+1\}).$ When each edge in the graph has unit weight or $f:\; E\; \backslash rightarrow\; \backslash \{1\backslash \}$, 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.

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 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 {{harvtxt|Cherkassky|Goldberg|Radzik|1996}}.

An algorithm using topological sorting can solve the single-source shortest path problem in time {{math|Θ(E + V)}} in arbitrarily-weighted directed acyclic graphs.{{harvnb|Cormen|Leiserson|Rivest|Stein|2001|p=655}}

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.

{{incomplete list|date=February 2011}}

{{incomplete list|date=December 2012}}

Finds a negative cycle or calculates distances to all vertices.

Network flows{{Cite book |last=Cormen |first=Thomas H. |title=Introduction to Algorithms |date=July 31, 2009 |publisher=MIT Press |isbn=9780262533058 |edition=3rd}} 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.

{{Cite book |last1=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}}

1. Create a Residual Graph:
2. * For each edge (u, v) in the original graph, create two edges in the residual graph:
3. ** (u, v) with capacity c(u, v)
4. ** (v, u) with capacity 0
5. * The residual graph represents the remaining capacity available in the network.
6. Find the Shortest Path:
7. * 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.
8. Augment the Flow:
9. * Find the minimum capacity along the shortest path.
10. * Increase the flow on the edges of the shortest path by this minimum capacity.
11. * Decrease the capacity of the edges in the forward direction and increase the capacity of the edges in the backward direction.
12. Update the Residual Graph:
13. * Update the residual graph based on the augmented flow.
14. Repeat:
15. * Repeat steps 2-4 until no more paths can be found from the source to the sink.

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(V⁴)}}.

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.{{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}}

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.{{Cite book |chapter=K-Shortest Paths Q-Routing: A New QoS Routing Algorithm in Telecommunication Networks |first=S. |last=Hoceini |author2=A. Mellouk |author3=Y. Amirat |date=2005 |chapter-url=https://doi.org/10.1007/978-3-540-31957-3_21 |title=Networking - ICN 2005, Lecture Notes in Computer Science, Vol. 3421 |volume=3421 |pages=164–172 |publisher=Springer, Berlin, Heidelberg |doi=10.1007/978-3-540-31957-3_21 |isbn=978-3-540-25338-9 }}{{cbignore}}

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.{{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 }}

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.Abraham, Ittai; Fiat, Amos; 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. 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.Abraham, Ittai; Delling, Daniel; 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. 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

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

The shortest multiple disconnected path {{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 }} 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.

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,{{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}} 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,{{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 }} 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 NP-complete.

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.{{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 }}{{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}}

{{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 a variant of the VCG mechanism, which gives the computers an incentive to reveal their true weights.

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 $O(|V||E|)$.
• Cherkassky and Goldberg{{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}} survey several other algorithms for negative cycle detection.

{{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.{{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

}}{{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

}}{{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–}}

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

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.{{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}}

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.Loui, R.P., 1983. Optimal paths in graphs with stochastic or multidimensional weights. Communications of the ACM, 26(9), pp.670-676.{{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 }}

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 .{{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 }} These methods use stochastic optimization, specifically stochastic dynamic programming to find the shortest path in networks with probabilistic arc length.{{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 }} 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.

• 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)

{{reflist|30em}}

{{sfn whitelist |CITEREFCormenLeisersonRivestStein2001}}

{{refbegin}}

• {{cite journal |last2=Mehlhorn |first2=Kurt |last3=Orlin |first3=James |last4=Tarjan |first4=Robert E. |date=April 1990 |title=Faster algorithms for the shortest path problem |url=https://apps.dtic.mil/sti/pdfs/ADA194031.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://apps.dtic.mil/sti/pdfs/ADA194031.pdf |archive-date=2022-10-09 |url-status=live |journal=Journal of the ACM |publisher=ACM |volume=37 |issue=2 |pages=213–223 |last1=Ahuja |first1=Ravindra K. |author4-link=Robert Tarjan |doi=10.1145/77600.77615|hdl=1721.1/47994 |s2cid=5499589 |hdl-access=free }}
• {{cite conference

| last1 = Axiotis | first1 = Kyriakos

| last2 = Mądry | first2 = Aleksander

| last3 = Vladu | first3 = Adrian

| editor-last = Irani | editor-first = Sandy

| contribution = Circulation control for faster minimum cost flow in unit-capacity graphs

| doi = 10.1109/FOCS46700.2020.00018

| pages = 93–104

| publisher = IEEE

| title = 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16–19, 2020

| year = 2020| arxiv = 2003.04863

}}

• {{cite journal |last=Bellman |first=Richard |year=1958 |title=On a routing problem |journal=Quarterly of Applied Mathematics |volume=16 |pages=87–90 |mr=0102435 |author-link=Richard Bellman |doi=10.1090/qam/102435|doi-access=free }}
• {{Cite conference |last1=Bernstein |first1=Aaron |last2=Nanongkai |first2=Danupon |last3=Wulff-Nilsen |first3=Christian |chapter=Negative-Weight Single-Source Shortest Paths in Near-linear Time |date=2022 |title=2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) |pages=600–611 |publisher=IEEE |doi=10.1109/focs54457.2022.00063|arxiv=2203.03456 |isbn=978-1-6654-5519-0 |s2cid=247958461 }}
• {{cite conference

| last1 = van den Brand | first1 = Jan

| last2 = Lee | first2 = Yin Tat

| last3 = Nanongkai | first3 = Danupon

| last4 = Peng | first4 = Richard

| last5 = Saranurak | first5 = Thatchaphol

| last6 = Sidford | first6 = Aaron

| last7 = Song | first7 = Zhao

| last8 = Wang | first8 = Di

| editor-last = Irani | editor-first = Sandy

| contribution = Bipartite matching in nearly-linear time on moderately dense graphs

| doi = 10.1109/FOCS46700.2020.00090

| pages = 919–930

| publisher = IEEE

| title = 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16–19, 2020

| year = 2020| arxiv = 2009.01802

}}

• {{cite conference

| last1 = Chen | first1 = Li

| last2 = Kyng | first2 = Rasmus

| last3 = Liu | first3 = Yang P.

| last4 = Peng | first4 = Richard

| last5 = Gutenberg | first5 = Maximilian Probst

| last6 = Sachdeva | first6 = Sushant

| contribution = Maximum flow and minimum-cost flow in almost-linear time

| doi = 10.1109/FOCS54457.2022.00064

| pages = 612–623

| publisher = IEEE

| title = 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 – November 3, 2022

| year = 2022| arxiv = 2203.00671

}}

• {{cite conference

| last1 = Cohen | first1 = Michael B.

| last2 = Mądry | first2 = Aleksander

| last3 = Sankowski | first3 = Piotr

| last4 = Vladu | first4 = Adrian

| editor-last = Klein | editor-first = Philip N.

| contribution = Negative-weight shortest paths and unit capacity minimum cost flow in $\backslash tilde\; O(m^\{10/7\}\backslash log\; W)$ time

| doi = 10.1137/1.9781611974782.48

| pages = 752–771

| publisher = Society for Industrial and Applied Mathematics

| title = Proceedings of the Twenty-Eighth Annual ACM–SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16–19

| year = 2017}}

• {{Cite conference |last1=Duan|first1=Ran |last2=Mao |first2=Jiayi |last3=Shu |first3=Xinkai |last4=Yin |first4=Longhui |chapter=A Randomized Algorithm for Single-Source Shortest Path on Undirected Real-Weighted Graphs |date=2023 |title=2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) |pages=484–492 |publisher=IEEE |doi=10.1109/focs57990.2023.00035|arxiv=2307.04139 |isbn=979-8-3503-1894-4 |s2cid=259501045 }}
• {{Cite journal |last2=Goldberg |first2=Andrew V. |last3=Radzik |first3=Tomasz |year=1996 |title=Shortest paths algorithms: theory and experimental evaluation |url=http://ftp.cs.stanford.edu/cs/theory/pub/goldberg/sp-alg.ps.Z |journal=Mathematical Programming |series=Ser. A |volume=73 |issue=2 |pages=129–174 |doi=10.1016/0025-5610(95)00021-6 |mr=1392160 |last1=Cherkassky |first1=Boris V. |author2-link=Andrew V. Goldberg }}
• {{Introduction to Algorithms|edition=2|pages=580–642|chapter=Single-Source Shortest Paths and All-Pairs Shortest Paths}}
• {{cite journal |last=Dantzig |first=G. B. |date=January 1960 |title=On the Shortest Route through a Network |journal=Management Science |volume=6 |issue=2 |pages=187–190 |doi=10.1287/mnsc.6.2.187}}
• {{cite journal |year=1959 |title=A note on two problems in connexion with graphs |journal=Numerische Mathematik |volume=1 |pages=269–271 |doi=10.1007/BF01386390|last1=Dijkstra |first1=E. W. |s2cid=123284777 |author-link=Edsger W. Dijkstra }}
• {{cite conference

| last = Fineman | first = Jeremy T.

| editor1-last = Mohar | editor1-first = Bojan

| editor2-last = Shinkar | editor2-first = Igor

| editor3-last = O'Donnell | editor3-first = Ryan

| arxiv = 2311.02520

| contribution = Single-source shortest paths with negative real weights in $\backslash tilde\; O(mn^\{8/9\})$ time

| doi = 10.1145/3618260.3649614

| pages = 3–14

| publisher = Association for Computing Machinery

| title = Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, BC, Canada, June 24–28, 2024

| year = 2024}}

• {{cite report |last=Ford |first=L. R. |date=1956 |title=Network Flow Theory |place=Santa Monica, CA |publisher=RAND Corporation |id=P-923 |url=http://www.rand.org/pubs/papers/P923.html}}
• {{cite conference |last1=Fredman |first1=Michael Lawrence |author-link1=Michael Fredman |first2=Robert E. |last2=Tarjan |author-link2=Robert Tarjan |title=Fibonacci heaps and their uses in improved network optimization algorithms |conference=25th Annual Symposium on Foundations of Computer Science |year=1984 |publisher=IEEE |pages=338–346 |isbn=0-8186-0591-X |doi=10.1109/SFCS.1984.715934}}
• {{cite journal |last2=Tarjan |first2=Robert E. |year=1987 |title=Fibonacci heaps and their uses in improved network optimization algorithms |journal=Journal of the Association for Computing Machinery |volume=34 |issue=3 |pages=596–615 |doi=10.1145/28869.28874 |last1=Fredman |first1=Michael Lawrence |s2cid=7904683 |author-link1=Michael Fredman |author-link2=Robert Tarjan |doi-access=free }}
• {{cite conference

| last = Gabow | first = H. N. | author-link = Harold N. Gabow

| title = Scaling algorithms for network problems

| doi = 10.1109/SFCS.1983.68

| pages = 248–258

| book-title = Proceedings of the 24th Annual Symposium on Foundations of Computer Science (FOCS 1983)

| year = 1983

| url = http://www.eecs.umich.edu/%7Epettie/matching/Gabow-scaling-algorithms-for-network-problems.pdf

}}

• {{cite journal

| last = Gabow | first = Harold N. | author-link = Harold N. Gabow

| year = 1985

| title = Scaling algorithms for network problems

| journal = Journal of Computer and System Sciences

| volume = 31

| issue = 2

| pages = 148–168

| doi = 10.1016/0022-0000(85)90039-X

| mr = 828519

| doi-access = free

}}

• {{cite conference |title=Improved Shortest Paths on the Word RAM |last1=Hagerup |first1=Torben |year=2000 |pages=61–72 |book-title=Proceedings of the 27th International Colloquium on Automata, Languages and Programming |editor1-first=Ugo |editor1-last=Montanari |editor2-first=José D. P. |editor2-last=Rolim |editor3-first=Emo |editor3-last=Welzl |isbn=978-3-540-67715-4 |url=http://dl.acm.org/citation.cfm?id=686343&CFID=563073233&CFTOKEN=28801665}}
• {{cite journal

| last1 = Henzinger | first1 = Monika R.

| last2 = Klein | first2 = Philip

| last3 = Rao | first3 = Satish

| last4 = Subramanian | first4 = Sairam

| title = Faster Shortest-Path Algorithms for Planar Graphs

| journal = Journal of Computer and System Sciences

| volume = 55

| issue = 1

| pages = 3–23

| year = 1997

| doi = 10.1006/jcss.1997.1493

| doi-access = free

}}

• {{cite journal

| last = Johnson

| first = Donald B.

| author-link = Donald B. Johnson

| title = Efficient algorithms for shortest paths in sparse networks

| journal = Journal of the ACM

| volume = 24

| issue = 1

| pages = 1–13

| year = 1977

| doi=10.1145/321992.321993| s2cid = 207678246

| doi-access = free

}}

• {{cite journal

| last = Johnson | first = Donald B. | date = December 1981

| title = A priority queue in which initialization and queue operations take {{math|O(log log D)}} time

| journal = Mathematical Systems Theory

| volume = 15

| issue = 1

| pages = 295–309

| doi = 10.1007/BF01786986

| mr = 683047

| s2cid = 35703411 | author-link = Donald B. Johnson

}}

• {{cite journal

| last2 = Poblete | first2 = Patricio V.

| year = 1983

| title = An {{math|O(m log log D)}} algorithm for shortest paths

| journal = Discrete Applied Mathematics

| volume = 6

| issue = 1

| pages = 91–93

| doi = 10.1016/0166-218X(83)90104-X

| mr = 700028

| last1 = Karlsson | first1 = Rolf G.

| doi-access = free

}}

• {{cite book |title=Investigation of Model Techniques — First Annual Report — 6 June 1956 — 1 July 1957 — A Study of Model Techniques for Communication Systems |last2=Gray |first2=R. S. |last3=Johnson |first3=A. A. |last4=Ladew |first4=W. C. |last5=Meaker |first5=S. R. Jr. |last6=Petry |first6=R. M. |last7=Seitz |first7=R. N. |publisher=Case Institute of Technology |year=1957 |location=Cleveland, Ohio |last1=Leyzorek |first1=M.}}
• {{cite conference |last=Moore |first= E. F. |author-link=Edward F. Moore |year=1959 |title=The shortest path through a maze |pages=285–292 |book-title=Proceedings of an International Symposium on the Theory of Switching (Cambridge, Massachusetts, 2–5 April 1957) |publisher=Harvard University Press |location=Cambridge }}
• {{cite conference |url=https://archive.org/details/proceedingsofthi2002acms/page/267 |title=Computing shortest paths with comparisons and additions |last2=Ramachandran |first2=Vijaya |year=2002 |isbn=978-0-89871-513-2 |pages=[https://archive.org/details/proceedingsofthi2002acms/page/267 267–276] |last1=Pettie |first1=Seth |book-title=Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms }}
• {{cite journal |last=Pettie |first=Seth |date=26 January 2004 |title=A new approach to all-pairs shortest paths on real-weighted graphs |journal=Theoretical Computer Science |volume=312 |issue=1 |pages=47–74 |doi=10.1016/s0304-3975(03)00402-x|doi-access=free }}
• {{cite journal | last1=Pollack |first1= Maurice |last2=Wiebenson |first2=Walter |date=March–April 1960 |title=Solution of the Shortest-Route Problem—A Review |journal=Oper. Res. |volume=8 |issue=2 |pages=224–230 |doi=10.1287/opre.8.2.224}} Attributes Dijkstra's algorithm to Minty ("private communication") on p. 225.
• {{cite book| title=Combinatorial Optimization — Polyhedra and Efficiency| last=Schrijver| first=Alexander| publisher=Springer| year=2004| isbn=978-3-540-20456-5| series=Algorithms and Combinatorics| volume=24 |at=vol.A, sect.7.5b, p. 103}}
• {{cite journal |last=Shimbel |first=Alfonso |year=1953 |title=Structural parameters of communication networks |journal=Bulletin of Mathematical Biophysics |volume=15 |issue=4 |pages=501–507 |doi=10.1007/BF02476438}}
• {{cite conference |last=Shimbel |first=A. |title=Structure in communication nets |location=New York, NY |pages=199–203 |publisher=Polytechnic Press of the Polytechnic Institute of Brooklyn |book-title=Proceedings of the Symposium on Information Networks |year=1955}}
• {{cite journal |date=1999 |title=Undirected single-source shortest paths with positive integer weights in linear time |journal=Journal of the ACM |volume=46 |issue=3 |pages=362–394 |last1=Thorup |first1=Mikkel |doi=10.1145/316542.316548 |s2cid=207654795 |doi-access=free}}
• {{Cite journal |last=Thorup |first=Mikkel |date=2004 |title=Integer priority queues with decrease key in constant time and the single source shortest paths problem |journal=Journal of Computer and System Sciences |volume=69 |issue=3 |pages=330–353 |doi=10.1016/j.jcss.2004.04.003 |doi-access=free}}
• {{cite journal |last2=Hillier |first2=J. A. |date=March–June 1960 |title=A Method for Finding the Shortest Route through a Road Network |journal=Operational Research Quarterly |volume=11 |issue=1/2 |pages=37–40 |last1=Whiting |first1=P. D. |doi=10.1057/jors.1960.32}}
• {{cite conference |last=Williams |first=Ryan |author-link=Ryan Williams (computer scientist) |arxiv=1312.6680 |title=Faster all-pairs shortest paths via circuit complexity |doi=10.1145/2591796.2591811 |mr=3238994 |pages=664–673 |publisher=ACM |location=New York |book-title=Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC '14) |year=2014}}

{{refend}}

• {{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