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Shortest path problem

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Computational problem of graph theory
This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(June 2009) (Learn how and when to remove this message)
Shortest path (A, C, E, D, F), blue, between vertices A and F in the weighted directed graph

Ingraph theory, theshortest path problem is the problem of finding apath between twovertices (or nodes) in agraph such that the sum of theweights of its constituent edges is minimized.[1]

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.[2]

Definition

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The shortest path problem can be defined forgraphs whetherundirected,directed, ormixed. 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.[3]

Two vertices are adjacent when they are both incident to a common edge. Apath in an undirected graph is asequence of verticesP=(v1,v2,…,vn)∈V×V×⋯×V{\displaystyle P=(v_{1},v_{2},\ldots ,v_{n})\in V\times V\times \cdots \times V} such thatvi{\displaystyle v_{i}} is adjacent tovi+1{\displaystyle v_{i+1}} for1≤i<n{\displaystyle 1\leq i<n}. Such a pathP{\displaystyle P} is called a path of lengthn−1{\displaystyle n-1} fromv1{\displaystyle v_{1}} tovn{\displaystyle v_{n}}. (Thevi{\displaystyle v_{i}} are variables; their numbering relates to their position in the sequence and need not relate to a canonical labeling.)

LetE={ei,j}{\displaystyle E=\{e_{i,j}\}} whereei,j{\displaystyle e_{i,j}} is the edge incident to bothvi{\displaystyle v_{i}} andvj{\displaystyle v_{j}}. Given areal-valued weight functionf:E→R{\displaystyle f:E\rightarrow \mathbb {R} }, and an undirected (simple) graphG{\displaystyle G}, the shortest path fromv{\displaystyle v} tov′{\displaystyle v'} is the pathP=(v1,v2,…,vn){\displaystyle P=(v_{1},v_{2},\ldots ,v_{n})} (wherev1=v{\displaystyle v_{1}=v} andvn=v′{\displaystyle v_{n}=v'}) that over all possiblen{\displaystyle n} minimizes the sum∑i=1n−1f(ei,i+1).{\displaystyle \sum _{i=1}^{n-1}f(e_{i,i+1}).} When each edge in the graph has unit weight orf:E→{1}{\displaystyle f:E\rightarrow \{1\}}, this is equivalent to finding the path with fewest edges.

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

  • Thesingle-source shortest path problem, in which we have to find shortest paths from a source vertexv to all other vertices in the graph.
  • Thesingle-destination shortest path problem, in which we have to find shortest paths from all vertices in the directed graph to a single destination vertexv. This can be reduced to the single-source shortest path problem by reversing the arcs in the directed graph.
  • Theall-pairs shortest path problem, in which we have to find shortest paths between every pair of verticesv,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

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Several well-known algorithms exist for solving this problem and its variants.

Additional algorithms and associated evaluations may be found inCherkassky, Goldberg & Radzik (1996).

Single-source shortest paths

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Undirected graphs

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WeightsTime complexityAuthor
R{\displaystyle \mathbb {R} }+O(V2)Dijkstra 1959
R{\displaystyle \mathbb {R} }+O((E + V) log V)Johnson 1977 (binary heap)
R{\displaystyle \mathbb {R} }+O(E + V log V)Fredman & Tarjan 1984 (Fibonacci heap)
N{\displaystyle \mathbb {N} }O(E)Thorup 1999 (requires constant-time multiplication)
R{\displaystyle \mathbb {R} }+O(Elog⁡Vlog⁡log⁡V){\displaystyle O(E{\sqrt {\log V\log \log V}})}Duan et al. 2023

Unweighted graphs

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AlgorithmTime complexityAuthor
Breadth-first searchO(E + V)

Directed acyclic graphs

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An algorithm usingtopological sorting can solve the single-source shortest path problem in timeΘ(E +V) in arbitrarily-weighted directed acyclic graphs.[4]

Directed graphs with nonnegative weights

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The following table is taken fromSchrijver (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.

WeightsAlgorithmTime complexityAuthor
R{\displaystyle \mathbb {R} }O(V2EL){\displaystyle O(V^{2}EL)}Ford 1956
R{\displaystyle \mathbb {R} }Bellman–Ford algorithmO(VE){\displaystyle O(VE)}Shimbel 1955,Bellman 1958,Moore 1959
R{\displaystyle \mathbb {R} }O(V2log⁡V){\displaystyle O(V^{2}\log {V})}Dantzig 1960
R{\displaystyle \mathbb {R} }Dijkstra's algorithm with listO(V2){\displaystyle O(V^{2})}Leyzorek et al. 1957,Dijkstra 1959, Minty (seePollack & Wiebenson 1960),Whiting & Hillier 1960
R{\displaystyle \mathbb {R} }Dijkstra's algorithm withbinary heapO((E+V)log⁡V){\displaystyle O((E+V)\log {V})}Johnson 1977
R{\displaystyle \mathbb {R} }Dijkstra's algorithm withFibonacci heapO(E+Vlog⁡V){\displaystyle O(E+V\log {V})}Fredman & Tarjan 1984,Fredman & Tarjan 1987
R{\displaystyle \mathbb {R} }QuantumDijkstra algorithm with adjacency listO(VElog2⁡V){\displaystyle O({\sqrt {VE}}\log ^{2}{V})}Dürr et al. 2006[5]
R{\displaystyle \mathbb {R} }Dijkstra's-Bellman–Ford hybrid with adivide-and-conquer frontier reductionO(Elog2/3⁡V){\displaystyle O(E\log ^{2/3}{V})}Duan et al. 2025[6]
N{\displaystyle \mathbb {N} }Dial's algorithm[7] (Dijkstra's algorithm using abucket queue withL buckets)O(E+LV){\displaystyle O(E+LV)}Dial 1969
O(Elog⁡log⁡L){\displaystyle O(E\log {\log {L}})}Johnson 1981,Karlsson & Poblete 1983
Gabow's algorithmO(ElogE/V⁡L){\displaystyle O(E\log _{E/V}L)}Gabow 1983,Gabow 1985
O(E+Vlog⁡L){\displaystyle O(E+V{\sqrt {\log {L}}})}Ahuja et al. 1990
N{\displaystyle \mathbb {N} }ThorupO(E+Vlog⁡log⁡V){\displaystyle O(E+V\log {\log {V}})}Thorup 2004
This list isincomplete; you can help byadding missing items.(February 2011)

Directed graphs with arbitrary weights without negative cycles

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WeightsAlgorithmTime complexityAuthor
R{\displaystyle \mathbb {R} }O(V2EL){\displaystyle O(V^{2}EL)}Ford 1956
R{\displaystyle \mathbb {R} }Bellman–Ford algorithmO(VE){\displaystyle O(VE)}Shimbel 1955,Bellman 1958,Moore 1959
R{\displaystyle \mathbb {R} }Johnson-Dijkstra withbinary heapO(VE+Vlog⁡V){\displaystyle O(VE+V\log V)}Johnson 1977
R{\displaystyle \mathbb {R} }Johnson-Dijkstra withFibonacci heapO(VE+Vlog⁡V){\displaystyle O(VE+V\log V)}Fredman & Tarjan 1984,Fredman & Tarjan 1987, adapted afterJohnson 1977
Z{\displaystyle \mathbb {Z} }Johnson's technique applied to Dial's algorithm[7]O(V(E+L)){\displaystyle O(V(E+L))}Dial 1969, adapted afterJohnson 1977
Z{\displaystyle \mathbb {Z} }Interior-point method with Laplacian solverO(E10/7logO(1)⁡VlogO(1)⁡L){\displaystyle O(E^{10/7}\log ^{O(1)}V\log ^{O(1)}L)}Cohen et al. 2017
Z{\displaystyle \mathbb {Z} }Interior-point method withℓp{\displaystyle \ell _{p}} flow solverE4/3+o(1)logO(1)⁡L{\displaystyle E^{4/3+o(1)}\log ^{O(1)}L}Axiotis, Mądry & Vladu 2020
Z{\displaystyle \mathbb {Z} }Robustinterior-point method with sketchingO((E+V3/2)logO(1)⁡VlogO(1)⁡L){\displaystyle O((E+V^{3/2})\log ^{O(1)}V\log ^{O(1)}L)}van den Brand et al. 2020
Z{\displaystyle \mathbb {Z} }ℓ1{\displaystyle \ell _{1}}interior-point method with dynamic min-ratio cycle data structureO(E1+o(1)log⁡L){\displaystyle O(E^{1+o(1)}\log L)}Chen et al. 2022
Z{\displaystyle \mathbb {Z} }Based on low-diameter decompositionO(Elog8⁡Vlog⁡L){\displaystyle O(E\log ^{8}V\log L)}Bernstein, Nanongkai & Wulff-Nilsen 2022
R{\displaystyle \mathbb {R} }Hop-limited shortest pathsO(EV8/9logO(1)⁡V){\displaystyle O(EV^{8/9}\log ^{O(1)}V)}Fineman 2024
This list isincomplete; you can help byadding missing items.(December 2012)

Directed graphs with arbitrary weights with negative cycles

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Finds a negative cycle or calculates distances to all vertices.

WeightsAlgorithmTime complexityAuthor
Z{\displaystyle \mathbb {Z} }O(EVlog⁡N){\displaystyle O(E{\sqrt {V}}\log {N})}Andrew V. Goldberg

Planar graphs with nonnegative weights

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WeightsAlgorithmTime complexityAuthor
R≥0{\displaystyle \mathbb {R} _{\geq 0}}O(V){\displaystyle O(V)}Henzinger et al. 1997

Applications

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Network flows[8] 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

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[9]

  1. 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.
  2. 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.
  3. 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.
  4. Update the Residual Graph:
    • Update the residual graph based on the augmented flow.
  5. Repeat:
    • Repeat steps 2-4 until no more paths can be found from the source to the sink.

All-pairs shortest paths

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The all-pairs shortest path problem finds the shortest paths between every pair of verticesv,v' in the graph. The all-pairs shortest paths problem for unweighted directed graphs was introduced byShimbel (1953), who observed that it could be solved by a linear number of matrix multiplications that takes a total time ofO(V4).

Undirected graph

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WeightsTime complexityAlgorithm
R{\displaystyle \mathbb {R} }+O(V3)Floyd–Warshall algorithm
{1,∞}{\displaystyle \{1,\infty \}}O(Vωlog⁡V){\displaystyle O(V^{\omega }\log V)}Seidel's algorithm (expected running time usingfast matrix multiplication algorithms)
N{\displaystyle \mathbb {N} }O(V3/2Ω(log⁡V)1/2){\displaystyle O(V^{3}/2^{\Omega (\log V)^{1/2}})}Williams 2014
R{\displaystyle \mathbb {R} }+O(EV log α(E,V))Pettie & Ramachandran 2002
N{\displaystyle \mathbb {N} }O(EV)Thorup 1999 applied to every vertex (requires constant-time multiplication).

Directed graph

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WeightsTime complexityAlgorithm
R{\displaystyle \mathbb {R} } (no negative cycles)O(V3){\displaystyle O(V^{3})}Floyd–Warshall algorithm
N{\displaystyle \mathbb {N} }O(V3/2Ω(log⁡V)1/2){\displaystyle O(V^{3}/2^{\Omega (\log V)^{1/2}})}Williams 2014
R{\displaystyle \mathbb {R} } (no negative cycles)O(V2.5log2⁡V){\displaystyle O(V^{2.5}\log ^{2}{V})}Quantum search[10][11]
R{\displaystyle \mathbb {R} } (no negative cycles)O(EV + V2 log V)Johnson–Dijkstra
R{\displaystyle \mathbb {R} } (no negative cycles)O(EV + V2 log log V)Pettie 2004
N{\displaystyle \mathbb {N} }O(EV + V2 log log V)Hagerup 2000

Applications

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Shortest path algorithms are applied to automatically find directions between physical locations, such as driving directions onweb mapping websites likeMapQuest orGoogle Maps. For this application fast specialized algorithms are available.[12]

If one represents a nondeterministicabstract 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 aRubik'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 anetworking ortelecommunications mindset, this shortest path problem is sometimes called the min-delay path problem and usually tied with awidest path problem. For example, the algorithm may seek the shortest (min-delay) widest path, or widest shortest (min-delay) path.[13]

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 inoperations research, include plant and facility layout,robotics,transportation, andVLSI design.[14]

Road networks

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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.[15] 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.[16] Other techniques that have been used are:

Related problems

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For shortest path problems incomputational geometry, seeEuclidean shortest path.

The shortest multiple disconnected path[17] is a representation of the primitive path network within the framework ofReptation theory. Thewidest 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

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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 calledConstrained Shortest Path First, and are harder to solve. One example is the constrained shortest path problem,[18] 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 problemNP-complete (such problems are not believed to be efficiently solvable for large sets of data, seeP = NP problem). AnotherNP-complete example requires a specific set of vertices to be included in the path,[19] which makes the problem similar to theTraveling 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 offinding the longest path in a graph is also NP-complete.

Partial observability

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TheCanadian 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.[20][21]

Strategic shortest paths

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

Negative cycle detection

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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:

General algebraic framework on semirings: the algebraic path problem

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[icon]
This sectionneeds expansion. You can help byadding missing information.(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 asemiring. Semiring multiplication is done along the path, and the addition is between paths. This general framework is known as thealgebraic path problem.[23][24][25]

Most of the classic shortest-path algorithms (and new ones) can be formulated as solving linear systems over such algebraic structures.[26]

More recently, an even more general framework for solving these (and much less obviously related problems) has been developed under the banner ofvaluation algebras.[27]

Shortest path in stochastic time-dependent networks

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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.[28][29]

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 asdynamic programming andDijkstra's algorithm .[30] These methods usestochastic optimization, specifically stochastic dynamic programming to find the shortest path in networks with probabilistic arc length.[31] The termstravel time reliability andtravel 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. Themost reliable path is one that maximizes the probability of arriving on time given a travel time budget. Anα-reliable path is one that minimizes the travel time budget required to arrive on time with a given probability.

See also

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References

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Notes

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  1. ^Ortega-Arranz, Hector; Llanos, Diego R.; Gonzalez-Escribano, Arturo (2015).The Shortest-Path Problem. Synthesis Lectures on Theoretical Computer Science. Cham: Springer.doi:10.1007/978-3-031-02574-7.ISBN 978-3-031-01446-8.
  2. ^Guenin, Bertrand (2014).Gentle Introduction to Optimization. Jochen Koenemann, Levent Tunçel (1st ed.). West Nyack: Cambridge University Press. p. 27.ISBN 978-1-107-05344-1.
  3. ^Deo, Narsingh (17 August 2016).Graph Theory with Applications to Engineering and Computer Science. Courier Dover Publications.ISBN 978-0-486-80793-5.
  4. ^Cormen et al. 2001, p. 655
  5. ^Dürr, Christoph; Heiligman, Mark; Høyer, Peter; Mhalla, Mehdi (January 2006). "Quantum query complexity of some graph problems".SIAM Journal on Computing.35 (6):1310–1328.arXiv:quant-ph/0401091.doi:10.1137/050644719.ISSN 0097-5397.S2CID 14253494.
  6. ^Brubaker, Ben (2025-08-06)."New Method Is the Fastest Way To Find the Best Routes".Quanta Magazine. Retrieved2025-08-11.
  7. ^abDial, Robert B. (1969)."Algorithm 360: Shortest-Path Forest with Topological Ordering [H]".Communications of the ACM.12 (11):632–633.doi:10.1145/363269.363610.S2CID 6754003.
  8. ^Cormen, Thomas H. (July 31, 2009).Introduction to Algorithms (3rd ed.). MIT Press.ISBN 9780262533058.
  9. ^Kleinberg, Jon; Tardos, Éva (2005).Algorithm Design (1st ed.). Addison-Wesley.ISBN 978-0321295354.
  10. ^Dürr, C.; Høyer, P. (1996-07-18). "A Quantum Algorithm for Finding the Minimum".arXiv:quant-ph/9607014.
  11. ^Nayebi, Aran; Williams, V. V. (2014-10-22). "Quantum algorithms for shortest paths problems in structured instances".arXiv:1410.6220 [quant-ph].
  12. ^Sanders, Peter (March 23, 2009)."Fast route planning".Google Tech Talk.Archived from the original on 2021-12-11.
  13. ^Hoceini, S.; A. Mellouk; Y. Amirat (2005)."K-Shortest Paths Q-Routing: A New QoS Routing Algorithm in Telecommunication Networks".Networking - ICN 2005, Lecture Notes in Computer Science, Vol. 3421. Vol. 3421. Springer, Berlin, Heidelberg. pp. 164–172.doi:10.1007/978-3-540-31957-3_21.ISBN 978-3-540-25338-9.
  14. ^Chen, Danny Z. (December 1996). "Developing algorithms and software for geometric path planning problems".ACM Computing Surveys.28 (4es). Article 18.doi:10.1145/242224.242246.S2CID 11761485.
  15. ^Abraham, Ittai; Fiat, Amos;Goldberg, Andrew V.; Werneck, Renato F."Highway Dimension, Shortest Paths, and Provably Efficient Algorithms". ACM-SIAM Symposium on Discrete Algorithms, pages 782–793, 2010.
  16. ^Abraham, Ittai; Delling, Daniel;Goldberg, Andrew V.; Werneck, Renato F.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.
  17. ^Kroger, Martin (2005). "Shortest multiple disconnected path for the analysis of entanglements in two- and three-dimensional polymeric systems".Computer Physics Communications.168 (3):209–232.Bibcode:2005CoPhC.168..209K.doi:10.1016/j.cpc.2005.01.020.
  18. ^Lozano, Leonardo; Medaglia, Andrés L (2013). "On an exact method for the constrained shortest path problem".Computers & Operations Research.40 (1):378–384.doi:10.1016/j.cor.2012.07.008.
  19. ^Osanlou, Kevin; Bursuc, Andrei; Guettier, Christophe; Cazenave, Tristan; Jacopin, Eric (2019). "Optimal Solving of Constrained Path-Planning Problems with Graph Convolutional Networks and Optimized Tree Search".2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). pp. 3519–3525.arXiv:2108.01036.doi:10.1109/IROS40897.2019.8968113.ISBN 978-1-7281-4004-9.S2CID 210706773.
  20. ^Bar-Noy, Amotz; Schieber, Baruch (1991). "The canadian traveller problem".Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms:261–270.CiteSeerX 10.1.1.1088.3015.
  21. ^Nikolova, Evdokia; Karger, David R."Route planning under uncertainty: the Canadian traveller problem"(PDF).Proceedings of the 23rd National Conference on Artificial Intelligence (AAAI). pp. 969–974.Archived(PDF) from the original on 2022-10-09.
  22. ^Cherkassky, Boris V.; Goldberg, Andrew V. (1999-06-01)."Negative-cycle detection algorithms".Mathematical Programming.85 (2):277–311.doi:10.1007/s101070050058.ISSN 1436-4646.S2CID 79739.
  23. ^Pair, Claude (1967). "Sur des algorithmes pour des problèmes de cheminement dans les graphes finis" [On algorithms for path problems in finite graphs]. InRosentiehl, Pierre (ed.).Théorie des graphes (journées internationales d'études) [Theory of Graphs (international symposium)]. Rome (Italy), July 1966. Dunod (Paris); Gordon and Breach (New York). p. 271.OCLC 901424694.
  24. ^Derniame, Jean Claude; Pair, Claude (1971).Problèmes de cheminement dans les graphes [Path Problems in Graphs]. Dunod (Paris).
  25. ^Baras, John; Theodorakopoulos, George (4 April 2010).Path Problems in Networks. Morgan & Claypool Publishers. pp. 9–.ISBN 978-1-59829-924-3.
  26. ^Gondran, Michel; Minoux, Michel (2008). "chapter 4".Graphs, Dioids and Semirings: New Models and Algorithms. Springer Science & Business Media.ISBN 978-0-387-75450-5.
  27. ^Pouly, Marc; Kohlas, Jürg (2011). "Chapter 6. Valuation Algebras for Path Problems".Generic Inference: A Unifying Theory for Automated Reasoning. John Wiley & Sons.ISBN 978-1-118-01086-0.
  28. ^Loui, R.P., 1983. Optimal paths in graphs with stochastic or multidimensional weights. Communications of the ACM, 26(9), pp.670-676.
  29. ^Rajabi-Bahaabadi, Mojtaba; Shariat-Mohaymany, Afshin; Babaei, Mohsen; Ahn, Chang Wook (2015). "Multi-objective path finding in stochastic time-dependent road networks using non-dominated sorting genetic algorithm".Expert Systems with Applications.42 (12):5056–5064.doi:10.1016/j.eswa.2015.02.046.
  30. ^Olya, Mohammad Hessam (2014). "Finding shortest path in a combined exponential – gamma probability distribution arc length".International Journal of Operational Research.21 (1) 64020:25–37.doi:10.1504/IJOR.2014.064020.
  31. ^Olya, Mohammad Hessam (2014). "Applying Dijkstra's algorithm for general shortest path problem with normal probability distribution arc length".International Journal of Operational Research.21 (2) 64541:143–154.doi:10.1504/IJOR.2014.064541.

Bibliography

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Further reading

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Graph andtree traversal algorithms
Search
Shortest path
Minimum spanning tree
International
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