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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.{{
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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== 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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*{{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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