Breadth-first search

Breadth-first search
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 it with the depth-first search.

Animated example of a breadth-first search

1 Algorithm
- 1.1 Pseudocode
2 Features
- 2.1 Space complexity
- 2.2 Time complexity
3 Applications
- 3.1 Finding connected components
- 3.2 Testing bipartiteness
4 See also
5 References
6 External links

Algorithm

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

Graph and tree search algorithms
α–β A* B* Backtracking Beam Bellman–Ford Best-first Bidirectional Borůvka Branch & bound BFS British Museum D* DFS Depth-limited Dijkstra Edmonds Floyd–Warshall Fringe Search Hill climbing IDA* Iterative deepening Kruskal Johnson Lexicographic BFS Prim SMA* Uniform-cost
Listings
Graph algorithms Search algorithms List of graph algorithms
Related topics
Dynamic programming Graph traversal Tree traversal Search games

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 (the direct child nodes) 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.

Note: Using a stack instead of a queue would turn this algorithm into a depth-first search.

Pseudocode

Input: A graph G and a root v of G

1  procedure BFS(G,v):2  create a queue Q3  enqueue v onto Q4  mark v5  while Q is not empty:6  t ← Q.dequeue()7  if t is what we are looking for:8  return t9  for all edges e in G.adjacentEdges(t) do12 u ← G.adjacentVertex(t,e)13 if u is not marked:14  mark u15  enqueue u onto Q16 return none

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 $O(|V|)$ where $|V|$ is the cardinality of the set of vertices. If the graph is represented by an Adjacency list it occupies $\Theta(|V|+|E|)$ ^[1] space in memory, while an Adjacency matrix representation occupies $\Theta(|V|^2)$ .^[2]

Time complexity

The time complexity can be expressed as $O(|V|+|E|)$ ^[3] since every vertex and every edge will be explored in the worst case. Note: $O(|E|)$ may vary between $O(|V|)$ and $O(|V|^2)$ , depending on how sparse the input graph is (assuming that the graph is connected).

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 (with path length measured by number of edges)
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.

Finding connected components

The set of nodes reached by a BFS (breadth-first search) 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 (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.

References

^ Cormen, Thomas H., Charles E. Leiserson, and Ronald L. Rivest. p.590
^ Cormen, Thomas H., Charles E. Leiserson, and Ronald L. Rivest. p.591
^ Cormen, Thomas H., Charles E. Leiserson, and Ronald L. Rivest. p.597

Knuth, Donald E. (1997), The Art Of Computer Programming Vol 1. 3rd ed., Boston: Addison-Wesley, ISBN 0-201-89683-4

External links

Breadth-First Explanation and Example

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