Comprehensive Report (14)

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Computer Science

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Nov 24, 2024

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Depth-First Search (DFS) is a fundamental algorithm used in graph traversal and searching. It explores as far as possible along each branch before backtracking. This algorithm is employed in various applications such as maze solving, topological sorting, connected components in a graph, and more. Let's delve into a detailed explanation of how DFS works. ### Basic Idea: DFS starts at a source node and explores as far as possible along each branch before backtracking. The algorithm maintains a stack or uses recursion to keep track of the vertices to visit. The key idea is to visit a vertex, mark it as visited, explore all its unvisited neighbors, and repeat the process until there are no more unvisited vertices. ### Steps of Depth-First Search: 1. **Initialization:** - Choose a starting vertex. If the graph is not connected, you might need to apply DFS to each connected component separately. - Create a data structure (usually a stack or recursion) to keep track of the visited vertices. 2. **Visit the Node:** - Mark the current node as visited. - Process the node – perform any action or collect information as needed. 3. **Explore Neighbors:** - For each unvisited neighbor of the current node: - Recursively apply DFS to the neighbor (if using recursion) or push it onto the stack (if using an iterative approach). 4. **Backtrack:** - If all neighbors are visited or there are no more unvisited nodes reachable from the current node, backtrack. - In an iterative approach, pop the stack to go back to the previous node. - In a recursive approach, the function returns, moving back up the call stack. 5. **Repeat:** - Repeat steps 2-4 until all nodes are visited. ### Pseudocode (Recursive): ```python function DFS(node): if node is not visited: mark node as visited process(node) for each neighbor in neighbors(node):

DFS(neighbor) ``` ### Pseudocode (Iterative with Stack): ```python function DFS(startNode): stack = new Stack() stack.push(startNode) while stack is not empty: current = stack.pop() if current is not visited: mark current as visited process(current) for each neighbor in neighbors(current): stack.push(neighbor) ``` ### Visualization: Consider a graph with vertices A, B, C, D, E, and edges (undirected) AB, AC, AD, BC, and CE. Starting from vertex A, the DFS would explore as follows: 1. A (visit A) 2. B (visit B) 3. C (visit C) 4. D (visit D) 5. E (visit E) ### Time Complexity: The time complexity of DFS is O(V + E), where V is the number of vertices and E is the number of edges in the graph. ### Space Complexity: The space complexity is O(V) for the recursive version (due to the call stack) and O(V) for the iterative version (using a stack). Depth-First Search is a versatile algorithm with applications in various domains. Understanding its principles is crucial for solving graph-related problems and algorithms.

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