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[[File:Bellman-Ford worst-case example.svg|thumb|In this example graph, assuming that A is the source and edges are processed in the worst order, from right to left, it requires the full {{math||''V''|−1}} or 4 iterations for the distance estimates to converge. Conversely, if the edges are processed in the best order, from left to right, the algorithm converges in a single iteration.]]
Like [[Dijkstra's algorithm]], Bellman–Ford proceeds by [[Relaxation (iterative method)|relaxation]], in which approximations to the correct distance are replaced by better ones until they eventually reach the solution.
However, Dijkstra's algorithm uses a [[priority queue]] to [[Greedy algorithm|greedily]] select the closest vertex that has not yet been processed, and performs this relaxation process on all of its outgoing edges; by contrast, the Bellman–Ford algorithm simply relaxes ''all'' the edges, and does this <math>|V|-1</math> times, where <math>|V|</math> is the number of vertices in the graph.{{sfnp|Cormen|Leiserson|Rivest|Stein|2022|loc=Section 22.1}}
In each of these repetitions, the number of vertices with correctly calculated distances grows, from which it follows that eventually all vertices will have their correct distances. This method allows the Bellman–Ford algorithm to be applied to a wider class of inputs than Dijkstra's algorithm. The intermediate answers and the choices among equally short paths depend on the order of edges relaxed, but the final
Bellman–Ford runs in <math>O(|V|\cdot |E|)</math> [[Big O notation|time]], where <math>|V|</math> and <math>|E|</math> are the number of vertices and edges respectively.
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''// This implementation takes in a graph, represented as''
''// lists of vertices (represented as integers [0..n-1]) and
''// edges, and fills two arrays (distance and predecessor)
''// holding the shortest path from the source to each vertex''
distance := ''list'' of size ''n''
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'''if''' distance[u] + w < distance[v] '''then'''
predecessor[v] := u
''// A negative cycle exists;''
''// find a vertex on the cycle visited := ''list'' of size ''n'' initialized with '''false'''
visited[v] := '''true'''
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visited[u] := '''true'''
u := predecessor[u]
''// u is a vertex in a negative cycle,''
''// find the cycle itself'' ncycle := [u]
v := predecessor[u]
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== Proof of correctness ==
The correctness of the algorithm can be shown by [[mathematical induction|induction]]:<ref name="web.stanford.edu"/><ref>{{Cite journal |last=Dinitz |first=Yefim |last2=Itzhak |first2=Rotem |date=2017-01-01 |title=Hybrid Bellman–Ford–Dijkstra algorithm |url=https://www.sciencedirect.com/science/article/pii/S1570866717300011 |journal=Journal of Discrete Algorithms |volume=42 |pages=35–44 |doi=10.1016/j.jda.2017.01.001 |issn=1570-8667|url-access=subscription }}</ref>
'''Lemma'''. After ''i'' repetitions of ''for'' loop,
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Another improvement, by {{harvtxt|Bannister|Eppstein|2012}}, replaces the arbitrary linear order of the vertices used in Yen's second improvement by a [[random permutation]]. This change makes the worst case for Yen's improvement (in which the edges of a shortest path strictly alternate between the two subsets ''E<sub>f</sub>'' and ''E<sub>b</sub>'') very unlikely to happen. With a randomly permuted vertex ordering, the [[expected value|expected]] number of iterations needed in the main loop is at most <math>|V|/3</math>.<ref name=Sedweb>See Sedgewick's [http://algs4.cs.princeton.edu/44sp/ web exercises] for ''Algorithms'', 4th ed., exercises 5 and 12 (retrieved 2013-01-30).</ref>
{{harvtxt|Fineman|
== Notes ==
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| doi-access = free
}}
*{{cite conference|
*{{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 <math>\tilde O(mn^{8/9})</math> 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}}
=== Secondary sources ===
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*{{Cite book|first1=Jørgen |last1=Bang-Jensen|first2=Gregory|last2=Gutin|year=2000|title=Digraphs: Theory, Algorithms and Applications|edition=First |isbn=978-1-84800-997-4|chapter=Section 2.3.4: The Bellman-Ford-Moore algorithm|publisher=Springer |url=http://www.cs.rhul.ac.uk/books/dbook/}}
*{{cite journal|first=Alexander|last=Schrijver|title=On the history of combinatorial optimization (till 1960)|pages=1–68|publisher=Elsevier|journal=Handbook of Discrete Optimization|year=2005|url=http://homepages.cwi.nl/~lex/files/histco.pdf}}
*{{sfn whitelist|CITEREFCormenLeisersonRivestStein2022}}{{Introduction to Algorithms
*{{cite book | first1 = George T. | last1 = Heineman | first2 = Gary | last2 = Pollice | first3 = Stanley | last3 = Selkow | title= Algorithms in a Nutshell | publisher=[[O'Reilly Media]] | year=2008 | chapter=Chapter 6: Graph Algorithms | pages = 160–164 | isbn=978-0-596-51624-6 }}
*{{cite book|last1=Kleinberg|first1=Jon|author1-link=Jon Kleinberg|last2=Tardos|first2=Éva|author2-link=Éva Tardos|year=2006|title=Algorithm Design|___location=New York|publisher=Pearson Education, Inc.}}
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