In graph theory, breadth-first search is a strategy for searching in a graph when search
is limited to essentially two operations: visit and inspect a node of a graph; gain
access to visit the nodes that neighbor the currently visited node.
The BFS begins at a root node and inspects all the neighboring nodes.
Then for each of those neighbor nodes in turn, it inspects their neighbor nodes which were
unvisited, and so on.
Compare BFS with the equivalent, but more memory-efficient Iterative deepening depth-first
search and contrast with depth-first search.
Algorithm
The algorithm uses a queue data structure to store intermediate results as it traverses
the graph, as follows: Enqueue the root node
Dequeue a node and examine it If the element sought is found in this node,
quit the search and return a result.
Otherwise enqueue any successors that have not yet been discovered.
If the queue is empty, every node on the graph has been examined – quit the search and
return "not found".
If the queue is not empty, repeat from Step 2.
Pseudocode Input: A graph G and a root v of G
1 procedure BFS(G,v) is 2 create a queue Q
3 create a set V 4 add v to V
5 enqueue v onto Q 6 while Q is not empty loop
7 t ← Q.dequeue() 8 if t is what we are looking for then
9 return t 10 end if
11 for all edges e in G.adjacentEdges(t) loop 12 u ← G.adjacentVertex(t,e)
13 if u is not in V then 14 add u to V
15 enqueue u onto Q 16 end if
17 end loop 18 end loop
19 return none 20 end BFS
Features Space complexity
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.
If the graph is represented by an Adjacency list it occupies space in memory, while an
Adjacency matrix representation occupies . Time complexity
The time complexity can be expressed as since every vertex and every edge will be explored
in the worst case.
Note: may vary between and , depending on how sparse the input graph is.
Applications Breadth-first search can be used to solve
many problems in graph theory, for example: Finding all nodes within one connected component
Copying Collection, Cheney's algorithm Finding the shortest path between two nodes
u and v Testing a graph for bipartiteness
(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 re-constructed in an efficient
manner.
Construction of the failure function of the Aho-Corasick pattern matcher.
Finding connected components The set of nodes reached by a BFS form the
connected component containing the starting node.
Testing bipartiteness BFS can be used to test bipartiteness, by
starting the search at any vertex and giving alternating labels to the vertices visited
during the search.
That is, give label 0 to the starting vertex, 1 to all its neighbours, 0 to those neighbours'
neighbours, and so on.
If at any step a vertex has neighbours with the same label as itself, then the graph is
not bipartite.
If the search ends without such a situation occurring, then the graph is bipartite.
See also Depth-first search
Iterative deepening depth-first search Level structure
Lexicographic breadth-first search References
External links Breadth-First Explanation and Example
Open Data Structures - Section 12.3.1 - Bread-First Search