The time complexity of Depth First Search (DFS) is O(V + E), where V is the number of vertices and E is the number of edges. The space complexity is O(V) for a recursive implementation.

Depth First Search (DFS)

Below are the detailed explanation of Time and Space Complexity of Depth First Search (DFS):

Time Complexity of Depth First Search (DFS):

Best Case of Depth First Search: O(V + E)

• The best-case time complexity of DFS occurs when the target node is found quickly after exploring only a few vertices and edges.
• In this scenario, the algorithm may terminate early without having to visit all vertices and edges.
• Thus, the time complexity is O(V + E), where V is the number of vertices and E is the number of edges.

Average Case of Depth First Search: O(V + E)

• The average-case time complexity of DFS is also O(V + E).
• This complexity holds across various graph structures and densities.
• DFS explores vertices and edges in a depth-first manner, which typically results in linear time complexity proportional to the sum of vertices and edges.

Worst Case of Depth First Search: O(V + E)

• The worst-case time complexity of DFS also remains O(V + E).
• In the worst case, DFS explores all vertices and edges reachable from the source node.
• The algorithm systematically traverses each edge and explores all vertices in a depth-first manner.
• Therefore, the time complexity is O(V + E) as it traverses all vertices and edges in the graph.

Auxiliary Space Complexity of Depth First Search (DFS):

The auxiliary space complexity of Depth-First Search (DFS) algorithm is O(V), where V is the number of vertices in the graph, this is due to the recursion stack or visited array.

Here’s why the auxiliary space complexity is O(V):

Recursion Stack:

• DFS uses a function call stack to track vertices.
• Space complexity depends on the maximum recursion depth.
• Maximum depth in a graph with V vertices is O(V).

Visited Array (Optional):

• May use a visited array to mark visited vertices.
• Size equals the number of vertices (V).