Breadth-First Search Traversal Algorithm

Breadth-first search is a way to find all the vertices reachable from the a given source vertex, s. Like depth first search, BFS traverse a connected component of a given graph and defines a spanning tree. Intuitively, the basic idea of the breath-first search is this: send a wave out from source s. The wave hits all vertices 1 edge from s. From there, the wave hits all vertices 2 edges from s. Etc. We use FIFO queue Q to maintain the wavefront: v is in Q if and only if wave has hit v but has not come out of v yet.

Overall Strategy of BFS Algorithm

Breadth-first search starts at a given vertex s, which is at level 0. In the first stage, we visit all the vertices that are at the distance of one edge away.When we visit there, we paint as "visited," the vertices adjacent to the start vertex s - these vertices are placed into level 1. In the second stage, we visit all the new vertices we can reach at the distance of two edges away from the source vertex s.These new vertices, which are adjacent to level 1 vertices and not previously assigned to a level, are placed into level 2, and so on. The BFS traversal terminates when every vertex has been visited.

To keep track of progress, breadth-first-search colors each vertex. Each vertex of the graph is in one of three states:

                       1. Undiscovered;

                       2. Discovered but not fully explored; and
                       3. Fully explored.

The state of a vertex, u, is stored in a color variable as follows:

      1. color[u] = White - for the "undiscovered" state,

      2. color [u] = Gray - for the "discovered but not fully explored" state, and
      3. color [u] = Black - for the "fully explored" state.

The BFS(G, s) algorithm develops a breadth-first search tree with the source vertex, s, as its root

The parent or predecessor of any other vertex in the tree is the vertex from which it was first discovered. For each vertex, v, the parent of v is placed in the variable π[v]. Another variable, d[v], computed by BFS contains the number of tree edges on the path from s to v. The breadth-first search uses a FIFO queue, Q, to store gray vertices.

Algorithm: Breadth-First Search Traversal

BFS(V, E, s)

1.         for each u in V − {s}             ▷ for each vertex u in V[G] except s.

2.             do color[u] ← WHITE

3.                 d[u] ← infinity

4.                 π[u] ← NIL

5.         color[s] ← GRAY                 ▷ Source vertex discovered

6.         d[s] ← 0                               ▷ initialize

7.         π[s] ← NIL                           ▷ initialize

8.         Q ← {}                                ▷ Clear queue Q

9.         ENQUEUE(Q, s)

10        while Q is non-empty

11.             do u ← DEQUEUE(Q)                      ▷ That is, u = head[Q]

12.                 for each v adjacent to u                  ▷ for loop for every node along with edge.

13.                         do if color[v] ← WHITE        ▷ if color is white you've never seen it before

14.                                 then  color[v] ← GRAY

15.                                          d[v] ← d[u] + 1
16.                                          π[v] ← u
17.                                          ENQUEUE(Q, v)
18.                 DEQUEUE(Q)
19.         color[u] ← BLACK