The time complexity of Iterative Depth-First Search (DFS) in terms of the number of vertices and edges depends on the representation of the graph and the specific implementation details. Here's a general analysis:
Let V be the number of vertices in the graph and E be the number of edges.
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Adjacency List Representation:
- In an adjacency list representation, each vertex maintains a list of its adjacent vertices.
- The time complexity of visiting each vertex and its adjacent vertices is O(V+E).
- In Iterative DFS, each vertex is visited once, and each edge is traversed once (either for discovering a new vertex or marking an already visited one).
- Therefore, the overall time complexity of Iterative DFS in terms of the number of vertices and edges is O(V+E).
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Adjacency Matrix Representation:
- In an adjacency matrix representation, the presence or absence of an edge between two vertices is stored in a matrix.
- Visiting all adjacent vertices of a given vertex takes O(V) time in the worst case.
- In Iterative DFS, each vertex is visited once, and each entry in the adjacency matrix is checked once.
- Therefore, the overall time complexity of Iterative DFS in terms of the number of vertices and edges is O(V2).
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Sparse Graphs:
- For sparse graphs (where E is much less than V2), the adjacency list representation is typically more efficient, resulting in a time complexity of O(V+E).
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Dense Graphs:
- For dense graphs (where E is close to V2), the adjacency matrix representation might be preferred despite its O(V2) time complexity.
In summary, the time complexity of Iterative DFS is typically O(V+E) for graphs represented using adjacency lists and O(V2) for graphs represented using adjacency matrices. The choice of representation affects the overall efficiency of Iterative DFS, with adjacency lists being more efficient for sparse graphs and adjacency matrices being more suitable for dense graphs.
