What's the time complexity of the breadth-first search algorithm?

The time complexity of the breadth-first search (BFS) algorithm is O(V + E).

Breadth-first search (BFS) is a graph traversal algorithm that explores all the vertices of a graph in breadth-first order, i.e., it explores all the vertices at the present depth before moving on to vertices at the next depth level. The time complexity of BFS is O(V + E), where V is the number of vertices and E is the number of edges in the graph.

The BFS algorithm starts at a root node and visits all the adjacent nodes. Then for each of those nearest nodes, it explores their unexplored neighbour nodes and so on, until it finds the target node. BFS uses a queue data structure to keep track of the nodes to be explored.

In terms of time complexity, BFS visits each vertex once and traverses each edge once. Therefore, the time complexity is linear in the size of the graph, i.e., the number of vertices plus the number of edges. This is represented as O(V + E).

The 'O' in the time complexity denotes the upper bound of the time required, which means the algorithm will not take more time than this. 'V' represents the number of vertices in the graph. As BFS traverses from one node to all its adjacent nodes, it visits each vertex once. Hence, 'V' is included in the time complexity.

'E' represents the number of edges in the graph. BFS explores each edge once when it checks for a connection from one node to its adjacent node. Therefore, 'E' is also included in the time complexity.

For further insight into how BFS fits within standard algorithms and their applications across various domains of computer science, you might find it helpful to explore Understanding and Applying Standard Algorithms. Additionally, an understanding of the Fundamental Computer Operations can enhance comprehension of the BFS algorithm's efficiency and role in solving problems. The use of BFS in more complex scenarios, such as in artificial intelligence for graph-based searches, is also discussed in depth in Graphs in AI.

A-Level Computer Science Tutor Summary: The time complexity of the Breadth-first search (BFS) algorithm is O(V + E), meaning it's dependent on the number of vertices (V) and edges (E) in the graph. BFS works by visiting each vertex and traversing each edge once, ensuring a comprehensive exploration of the graph. This makes its performance linear to the graph's size, providing a predictable upper time limit.