Breadthfirst search
Breadthfirst search (BFS) is an algorithm for traversing or searching tree or graph data structures. It starts at the tree root (or some arbitrary node of a graph, sometimes referred to as a 'search key'[1]), and explores all of the neighbor nodes at the present depth prior to moving on to the nodes at the next depth level.
Order in which the nodes are expanded  
Class  Search algorithm 

Data structure  Graph 
Worstcase performance  
Worstcase space complexity 
Graph and tree search algorithms 

Listings 

Related topics 
It uses the opposite strategy as depthfirst search, which instead explores the node branch as far as possible before being forced to backtrack and expand other nodes.[2]
BFS and its application in finding connected components of graphs were invented in 1945 by Konrad Zuse, in his (rejected) Ph.D. thesis on the Plankalkül programming language, but this was not published until 1972.[3] It was reinvented in 1959 by Edward F. Moore, who used it to find the shortest path out of a maze,[4][5] and later developed by C. Y. Lee into a wire routing algorithm (published 1961).[6]
Pseudocode
Input: A graph Graph and a starting vertex root of Graph
Output: Goal state. The parent links trace the shortest path back to root
1 procedure BFS(G,start_v): 2 let Q be a queue 3 label start_v as discovered 4 Q.enqueue(start_v) 5 while Q is not empty 6 v = Q.dequeue() 7 if v is the goal: 8 return v 9 for all edges from v to w in G.adjacentEdges(v) do 10 if w is not labeled as discovered: 11 label w as discovered 12 w.parent = v 13 Q.enqueue(w)
More details
This nonrecursive implementation is similar to the nonrecursive implementation of depthfirst search, but differs from it in two ways:
 it uses a queue (First In First Out) instead of a stack and
 it checks whether a vertex has been discovered before enqueueing the vertex rather than delaying this check until the vertex is dequeued from the queue.
The Q queue contains the frontier along which the algorithm is currently searching.
Nodes can be labelled as discovered by storing them in a set, or by an attribute on each node, depending on the implementation.
Note that the word node is usually interchangeable with the word vertex.
The parent attribute of each node is useful for accessing the nodes in a shortest path, for example by backtracking from the destination node up to the starting node, once the BFS has been run, and the predecessors nodes have been set.
Breadthfirst search produces a socalled breadth first tree. You can see how a breadth first tree looks in the following example.
Example
The following is an example of the breadthfirst tree obtained by running a BFS on German cities starting from Frankfurt:
Analysis
Time and space complexity
The time complexity can be expressed as , since every vertex and every edge will be explored in the worst case. is the number of vertices and is the number of edges in the graph. Note that may vary between and , depending on how sparse the input graph is.[7]
When the number of vertices in the graph is known ahead of time, and additional data structures are used to determine which vertices have already been added to the queue, the space complexity can be expressed as , where is the cardinality of the set of vertices. This is in addition to the space required for the graph itself, which may vary depending on the graph representation used by an implementation of the algorithm.
When working with graphs that are too large to store explicitly (or infinite), it is more practical to describe the complexity of breadthfirst search in different terms: to find the nodes that are at distance d from the start node (measured in number of edge traversals), BFS takes O(b^{d + 1}) time and memory, where b is the "branching factor" of the graph (the average outdegree).[8]^{:81}
Completeness
In the analysis of algorithms, the input to breadthfirst search is assumed to be a finite graph, represented explicitly as an adjacency list or similar representation. However, in the application of graph traversal methods in artificial intelligence the input may be an implicit representation of an infinite graph. In this context, a search method is described as being complete if it is guaranteed to find a goal state if one exists. Breadthfirst search is complete, but depthfirst search is not. When applied to infinite graphs represented implicitly, breadthfirst search will eventually find the goal state, but depthfirst search may get lost in parts of the graph that have no goal state and never return.[9]
BFS ordering
An enumeration of the vertices of a graph is said to be a BFS ordering if it is the possible output of the application of BFS to this graph.
Let be a graph with vertices. Recall that is the set of neighbors of . For be a list of distinct elements of , for , let be the least such that is a neighbor of , if such a exists, and be otherwise.
Let be an enumeration of the vertices of . The enumeration is said to be a BFS ordering (with source ) if, for all , is the vertex such that is minimal. Equivalently, is a BFS ordering if, for all with , there exists a neighbor of such that .
Applications
Breadthfirst search can be used to solve many problems in graph theory, for example:
 Copying garbage collection, Cheney's algorithm
 Finding the shortest path between two nodes u and v, with path length measured by number of edges (an advantage over depthfirst search)[10]
 (Reverse) Cuthill–McKee mesh numbering
 Ford–Fulkerson method for computing the maximum flow in a flow network
 Serialization/Deserialization of a binary tree vs serialization in sorted order, allows the tree to be reconstructed in an efficient manner.
 Construction of the failure function of the AhoCorasick pattern matcher.
 Testing bipartiteness of a graph.
See also
References
 "Graph500 benchmark specification (supercomputer performance evaluation)". Graph500.org, 2010. Archived from the original on 20150326. Retrieved 20150315.
 Cormen Thomas H.; et al. (2009). "22.3". Introduction to Algorithms. MIT Press. Explicit use of et al. in:
author=
(help)  Zuse, Konrad (1972), Der Plankalkül (in German), Konrad Zuse Internet Archive. See pp. 96–105 of the linked pdf file (internal numbering 2.47–2.56).
 Moore, Edward F. (1959). "The shortest path through a maze". Proceedings of the International Symposium on the Theory of Switching. Harvard University Press. pp. 285–292. As cited by Cormen, Leiserson, Rivest, and Stein.
 Skiena, Steven (2008). "Sorting and Searching". The Algorithm Design Manual. Springer. p. 480. Bibcode:2008adm..book.....S. doi:10.1007/9781848000704_4. ISBN 9781848000698.
 Lee, C. Y. (1961). "An Algorithm for Path Connections and Its Applications". IRE Transactions on Electronic Computers.
 Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001) [1990]. "22.2 Breadthfirst search". Introduction to Algorithms (2nd ed.). MIT Press and McGrawHill. pp. 531–539. ISBN 0262032937.
 Russell, Stuart; Norvig, Peter (2003) [1995]. Artificial Intelligence: A Modern Approach (2nd ed.). Prentice Hall. ISBN 9780137903955.
 Coppin, B. (2004). Artificial intelligence illuminated. Jones & Bartlett Learning. pp. 79–80.
 Aziz, Adnan; Prakash, Amit (2010). "4. Algorithms on Graphs". Algorithms for Interviews. p. 144. ISBN 9781453792995.
 Knuth, Donald E. (1997), The Art of Computer Programming Vol 1. 3rd ed., Boston: AddisonWesley, ISBN 9780201896831
External links
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