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)

Breadth-first 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 the neighbor nodes first, before moving to the next level neighbors. Compare BFS with the equivalent, but more memory-efficient Iterative deepening depth-first search and contrast with depth-first search.

BFS was invented in the late 1950s by E. F. Moore, who used it to find the shortest path out of a maze,[2] and discovered independently by C. Y. Lee as a wire routing algorithm (published 1961).[3][4]

Animated example of a breadth-first search

Pseudocode[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

Input: A graph G and a vertex v of G

Output: All vertices reachable from v labeled as discovered

A non-recursive implementation of BFS:

1  procedure BFS(G,v) is
2      let Q be a queue
3      Q.enqueue(v)
4      label v as discovered
5      while Q is not empty
6         vQ.dequeue()
7         process(v)
8         for all edges from v to w in G.adjacentEdges(v) do
9             if w is not labeled as discovered
10                 Q.enqueue(w)
11                label w as discovered

The non-recursive implementation is similar to depth-first search but differs from it in two ways: it uses a queue 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 function process is called on all nodes, and will have seen all nodes reachable by a length-n path from v before any of the nodes that are only reachable by a length-(n + 1) path.

Analysis[edit]

Time and space complexity[edit]

The time complexity can be expressed as O(|V|+|E|) [5] 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|)[6] space in memory, while an adjacency matrix representation occupies \Theta(|V|^2).[7]

When working with graphs that are too large to store explicitly (or infinite), it is more practical to describe the complexity of breadth-first 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(bd + 1) time and memory, where b is the "branching factor" of the graph (the average out-degree).[8]:81

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.[9]

Applications[edit]

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

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 neighbors, 0 to those neighbors' neighbors, and so on. If at any step a vertex has (visited) neighbors 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. ^ "Graph500 benchmark specification (for petascale-era supercomputer performance evaluation)". Graph500.org, 2010. 
  2. ^ Skiena, Steven (2008). The Algorithm Design Manual. Springer. p. 480. doi:10.1007/978-1-84800-070-4_4. 
  3. ^ Leiserson, Charles E.; Schardl, Tao B. (2010). A Work-Efficient Parallel Breadth-First Search Algorithm (or How to Cope with the Nondeterminism of Reducers) (PDF). ACM Symp. on Parallelism in Algorithms and Architectures. 
  4. ^ Lee, C. Y. (1961). "An Algorithm for Path Connections and Its Applications". IRE Transactions on Electronic Computers. 
  5. ^ Cormen, Thomas H., Charles E. Leiserson, and Ronald L. Rivest. p.597
  6. ^ Cormen, Thomas H., Charles E. Leiserson, and Ronald L. Rivest. p.590
  7. ^ Cormen, Thomas H., Charles E. Leiserson, and Ronald L. Rivest. p.591
  8. ^ Russell, Stuart; Norvig, Peter (2003) [1995]. Artificial Intelligence: A Modern Approach (2nd ed.). Prentice Hall. ISBN 978-0137903955. 
  9. ^ Coppin, B. (2004). Artificial intelligence illuminated. Jones & Bartlett Learning. pp. 79–80.

External links[edit]