Explain the breadth-first search algorithm for a graph.

The breadth-first search (BFS) algorithm is used to traverse or search a graph in a systematic manner. It starts at a given vertex (or node) and explores all the neighboring vertices at the current level before moving on to the next level.

Here is a step-by-step explanation of the BFS algorithm:

1. Start by selecting a starting vertex and mark it as visited.
2. Enqueue the starting vertex into a queue data structure.
3. While the queue is not empty, do the following steps:
a. Dequeue a vertex from the queue.
b. Process the dequeued vertex (e.g., print it or perform any desired operation).
c. Enqueue all the unvisited neighboring vertices of the dequeued vertex into the queue and mark them as visited.
4. Repeat steps 3 until the queue becomes empty.

By following this algorithm, BFS explores the graph level by level, visiting all the vertices at the current level before moving on to the next level. This ensures that all vertices reachable from the starting vertex are visited, and it also finds the shortest path between the starting vertex and any other vertex in an unweighted graph.

BFS can be implemented using a queue data structure to keep track of the vertices to be visited. Additionally, a visited array or set is used to mark the visited vertices and avoid revisiting them.

Overall, the breadth-first search algorithm is an efficient way to traverse or search a graph, especially when finding the shortest path or exploring all reachable vertices is required.