The runtime of Breadth-First Search (BFS) for finding the shortest path in a graph is typically represented as \(O(V + E)\), where \(V\) is the number of vertices (or nodes) in the graph, and \(E\) is the number of edges. This complexity holds true for both directed and undirected graphs.
### Why is it \(O(V + E)\)?
1. **Vertex Exploration**: BFS explores each vertex exactly once. During the process, it marks each vertex as visited to avoid revisiting. The exploration of each vertex has a constant time cost, leading to a total time cost of \(O(V)\).
2. **Edge Exploration**: BFS explores each edge in the graph exactly once in an undirected graph and twice in a directed graph (once for each direction). The exploration of each edge is necessary to find all vertices connected to a given vertex. This step ensures that BFS can traverse through all accessible parts of the graph from the starting vertex. The total time cost for exploring all edges is \(O(EAnswer:
Explanation:
