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Breadth-First Search Algorithm (BFS)

Definition:

Breadth-First Search Algorithm is a graph search algorithm. It begins from root node and continues to expand all neighbour node 1- level below.

In artificial intelligence, it is categorized as Uninformed (Blind) Search.

In worst case, the time complexity of this algorithm is O ( b ^d ). ( b: maximum branching factor of the search tree ;  d: depth of the least-cost solution )

In worst case, the space complexity of this algorithm is O ( b ^d ).

Properties of BFS:

  • Completed,
  • Finds the shallowest path,
  • Generates b + b^2 + b^3 + … + b^d = O( b^d ) nodes,
  • Space complexity (frontier size) is O (b ^ d ) .

How Does It Work?

  1. The root node is expanded.
  2. Then,  all successors of the root node are expanded,
  3. Then all their successors.

In detail: (from Wikipedia)

  1. Enqueue the root node,
  2. Dequeue a node
    • 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.

In the figure below, you can see the animated figure of BFS.

Animated_BFS

Animated Figure of BFS

In the figure below, you can see the order of nodes expanded (visited)

300px-Breadth-first-tree.svg

Order of Nodes Expandend in BFS

Animation:

For better understanding, go to this animation page.

Pseudocode:

1  procedure BFS(G,v) is
2      create a queue Q
3      create a vector set V
4      enqueue v onto Q
5      add v to V
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

 

References:

Artificial Intelligence  A Modern Approach, S.Russell, P. Norvig, Third Edition,

Figures from:  http://en.wikipedia.org