## Breadth First Search-

• Like Depth First Search (DFS), Breadth First Search (BFS) is a graph traversal algorithm.
• It is used for traversing or searching a graph in a systematic fashion.
• BFS uses a strategy that searches in the graph in breadth first manner whenever possible.

## Steps for Breadth First Search-

### Step-01:

Create and maintain 3 variables for each vertex of the graph.

For any vertex v of the graph, these 3 variables are-

#### 1. color[v]-

• This variable represents the color of the vertex v at the given point of time.
• The possible values of this variable are- WHITE, GREY and BLACK.
• WHITE color of the vertex signifies that it has not been discovered yet.
• GREY color of the vertex signifies that it has been discovered and it is being processed.
• BLACK color of the vertex signifies that it has been completely processed.

#### 2. π[v]-

• This variable represents the predecessor of vertex v.

#### 3. d[v]-

• This variable represents the distance of vertex v from the source vertex.

### Step-02:

For each vertex v of the graph except source vertex, initialize the variables as-

• color[v] = WHITE
• π[v] = NIL
• d[v] = ∞

For source vertex S, initialize the variables as-

• color[S] = GREY
• π[S] = NIL
• d[S] = 0

### Step-03:

Now, enqueue source vertex S in queue Q and repeat the following procedure until the queue Q becomes empty-

1. Dequeue vertex v from queue Q.

2. For all the adjacent white vertices u of vertex v,

do-

color[u] = GREY

d[u] = d[v] + 1

π[u] = v

Enqueue (Q,u)

3. Color vertex v to black.

The above steps are translated into the following algorithm.

## Algorithm-

#### BFS (V,E,s)

for each vertex v in V – {s}

do

color[v] ← WHITE

d[v] ← ∞

π[v] ← NIL

color[s] = GREY

d[s] ← 0

π[s] ← NIL

Q ← { }

ENQUEUE (Q,s)

While Q is non-empty

do v ← DEQUEUE (Q)

for each u adjacent to v

do if color[u] ← WHITE

then color[u] ← GREY

d[u] ← d[v] + 1

π[u] ← v

ENQUEUE (Q,u)

color[v] ← BLACK

## Time Complexity-

 The total running time for Breadth First Search is O (V+E).

## Problem-

Traverse the following graph using Breadth First Search Technique- Consider vertex S as the starting vertex.

## Solution-

### Step-01:

For all the vertices v except source vertex S of the graph, we will initialize the variables as-

• color[v] = WHITE
• π[v] = NIL
• d[v] = ∞

For source vertex S, we will initialize the variables as-

• color[S] = GREY
• π[S] = NIL
• d[S] = 0

We will enqueue source vertex S in the queue Q. ### Step-02:

• Dequeue vertex S from the queue Q
• For all adjacent white vertices ‘v’ (vertices R and W) of vertex S, we do-

1. color[v] = GREY
2. d[v] = d[S] + 1 = 0 + 1 = 1
3. π[v] = S
4. Enqueue all adjacent white vertices of S in queue Q

• color[S] = BLACK ### Step-03:

• Dequeue vertex W from the queue Q
• For all adjacent white vertices ‘v’ (vertices T and X) of vertex W, we do-

1. color[v] = GREY
2. d[v] = d[W] + 1 = 1 + 1 = 2
3. π[v] = W
4. Enqueue all adjacent white vertices of W in queue Q

• color[W] = BLACK ### Step-04:

• Dequeue vertex R from the queue Q
• For all adjacent white vertices ‘v’ (vertex V) of vertex R, we do-

1. color[v] = GREY
2. d[v] = d[R] + 1 = 1 + 1 = 2
3. π[v] = R
4. Enqueue all adjacent white vertices of R in queue Q

• color[R] = BLACK ### Step-05:

• Dequeue vertex T from the queue Q
• For all adjacent white vertices ‘v’ (vertex U) of vertex T, we do-

1. color[v] = GREY
2. d[v] = d[T] + 1 = 2 + 1 = 3
3. π[v] = T
4. Enqueue all adjacent white vertices of T in queue Q

• color[T] = BLACK ### Step-06:

• Dequeue vertex X from the queue Q
• For all adjacent white vertices ‘v’ (vertex Y) of vertex X, we do-

1. color[v] = GREY
2. d[v] = d[X] + 1 = 2 + 1 = 3
3. π[v] = X
4. Enqueue all adjacent white vertices of X in queue Q

• color[X] = BLACK ### Step-07:

• Dequeue vertex V from the queue Q
• There are no adjacent white vertices to vertex V.
• color[V] = BLACK ### Step-08:

• Dequeue vertex U from the queue Q
• There are no adjacent white vertices to vertex U.
• color[U] = BLACK ### Step-09:

• Dequeue vertex Y from the queue Q
• There are no adjacent white vertices to vertex Y.
• color[Y] = BLACK Since, all the vertices have turned black and the queue has got empty, so we stop.

This is how we traverse any given graph using Breadth First Search (BFS) technique.

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