Explain the breadth first search algorithm using the graph given below with v1 as the source vertex:

For each stage of the algorithm give:

i) ),v(d π )v( for each vertex, where )v(d is the distance from the source to the vertex v and is the predecessor of v.

ii) White and gray vertices

iii) Vertices in the queue

Also, give the breadth search tree.

To explain the Breadth-First Search (BFS) algorithm in terms of the graph with vertex v1v_1 as the source, here’s how the algorithm proceeds. BFS explores vertices level by level, starting from the source vertex. It uses a queue to track which vertex to visit next.

Initialization:

• Start from the source vertex v1v_1.
• Set d(v1)=0d(v_1) = 0 and π(v1)=null\pi(v_1) = \text{null} because v1v_1 is the starting point.
• All other vertices are initialized with d(v)=∞d(v) = \infty and π(v)=null\pi(v) = \text{null}.
• Mark all vertices as white, which means they are unvisited.

Algorithm Steps:

Step 1: Start with v1v_1

• Queue: [ v1v_1 ]
• d(v1)=0d(v_1) = 0, π(v1)=null\pi(v_1) = \text{null} (since it is the source)
• Gray: [ v1v_1 ] (currently being processed)
• White: All other vertices
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