The time complexity of finding bridges using Tarjan's algorithm is O(V + E), where V is the number of vertices and E is the number of edges in the graph.
Here's why Tarjan's algorithm has this time complexity:
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Depth-First Search (DFS): The main component of Tarjan's algorithm is a depth-first search traversal of the graph. During the DFS traversal, each vertex and each edge is visited at most once. Thus, the time complexity of the DFS traversal is O(V + E).
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Bridge Identification: The process of identifying bridges occurs during the DFS traversal. For each edge visited during the traversal, the algorithm performs constant-time operations to compare the discovery time and low time of vertices. Therefore, the time complexity for identifying bridges does not exceed O(V + E).
Overall, the dominant factor in the time complexity is the DFS traversal, making the overall time complexity of Tarjan's algorithm O(V + E). This makes the algorithm efficient for practical use, even for large graphs.
