# Time complexity of Breadth First Search, why and how?

• MHB
• shivajikobardan
In summary, the time complexity for BFS on a tree with a goal node n is $O(b^d)$, where b is the branching factor and d is the depth of the tree. In the given example, b is 2 and d is 4, resulting in a time complexity of $O(2^4) = O(16)$.

#### shivajikobardan

Firstly, how is time complexity of BFS $O(b^d)$.
Say I have this tree with goal n, how do I calculate time complexity for it? Assume left to right traversal. I know the answer is a,b,c,x1,d,e,f,i,j,k,g,h,z,l,m,n. But I am not sure how to calculate time complexity here using the above formula

. The time complexity for BFS on this tree is $O(b^d)$ where b is the branching factor and d is the depth of the tree. In this case, b is 2 (for each node there are two child nodes) and d is 4 (the depth of the tree is 4 levels). Therefore the time complexity for BFS on this tree is $O(2^4) = O(16)$.

## 1. What is the time complexity of Breadth First Search (BFS)?

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. This means that the time taken by BFS to traverse a graph is directly proportional to the total number of vertices and edges in the graph.

## 2. Why is the time complexity of BFS O(V+E)?

The time complexity of BFS is O(V+E) because in the worst case scenario, BFS will visit every vertex and every edge in the graph. Therefore, the time taken by BFS is dependent on the total number of vertices and edges in the graph.

## 3. How is the time complexity of BFS calculated?

The time complexity of BFS is calculated by counting the number of operations performed by the algorithm. In BFS, each vertex is visited at most once and each edge is traversed exactly twice (once for each of its endpoints). Therefore, the time complexity can be expressed as O(V+E).

## 4. What is the significance of time complexity in BFS?

The time complexity of BFS is significant because it helps us understand the efficiency of the algorithm. It tells us how the running time of the algorithm will increase as the size of the graph increases. This allows us to compare the performance of different algorithms and choose the most efficient one for a given problem.

## 5. Can the time complexity of BFS be improved?

Yes, the time complexity of BFS can be improved in certain cases. For example, if we know that the graph is sparse (contains a relatively small number of edges compared to vertices), we can use an adjacency list representation of the graph instead of an adjacency matrix to reduce the time complexity to O(V+E). Additionally, using techniques like pruning or heuristic search can also improve the time complexity of BFS in certain scenarios.