Breadth-first search

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Breadth-first search
Order in which the nodes get expanded
Order in which the nodes are expanded
Class Search algorithm
Data structure Graph
Worst case performance O(|E|) = O(b^d)
Worst case space complexity O(|V|) = O(b^d)

In graph theory, breadth-first search (BFS) is a strategy for searching in a graph when search is limited to essentially two operations: (a) visit and inspect a node of a graph; (b) 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.

Animated example of a breadth-first search

Algorithm[edit]

An example map of Germany with some connections between cities
The breadth-first tree obtained when running BFS on the given map and starting in Frankfurt

The algorithm uses a queue data structure to store intermediate results as it traverses the graph, as follows:

  1. Enqueue the root node
  2. 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 (the direct child nodes) that have not yet been discovered.
  3. If the queue is empty, every node on the graph has been examined – quit the search and return "not found".
  4. If the queue is not empty, repeat from Step 2.

Pseudocode[edit]

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

Analysis[edit]

Time and space complexity[edit]

The time complexity can be expressed as O(|V|+|E|) [1] since every vertex and every edge will be explored in the worst case. Note: O(|E|) may vary between O(1) and  O(|V|^2), depending on how sparse the input graph is.

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 O(|V|) where |V| is the cardinality of the set of vertices. If the graph is represented by an Adjacency list it occupies \Theta(|V|+|E|)[2] space in memory, while an Adjacency matrix representation occupies \Theta(|V|^2).[3]

Completeness and optimality[edit]

In the analysis of algorithms, the input to breadth-first 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. Breadth-first search is complete, but depth-first search is not: when applied to infinite graphs represented implicitly, it may get lost in parts of the graph that have no goal state and never return.[4]

A search method is optimal if it is guaranteed to find the best solution that exists. In other words, it will find the path to the goal state that involves the least number of steps. Breadth-first search is an optimal search method, but depth-first search is not[4]

Applications[edit]

Breadth-first search can be used to solve many problems in graph theory, for example:

Finding connected components[edit]

The set of nodes reached by a BFS (breadth-first search) form the connected component containing the starting node.

Testing bipartiteness[edit]

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 (visited) 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[edit]

References[edit]

  1. ^ Cormen, Thomas H., Charles E. Leiserson, and Ronald L. Rivest. p.597
  2. ^ Cormen, Thomas H., Charles E. Leiserson, and Ronald L. Rivest. p.590
  3. ^ Cormen, Thomas H., Charles E. Leiserson, and Ronald L. Rivest. p.591
  4. ^ a b Coppin, B. (2004). Artificial intelligence illuminated. Jones & Bartlett Learning. pp. 79–80.

External links[edit]