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Depth-First Search (DFS) and Breadth-First Search (BFS) are two fundamental graph traversal algorithms used to explore nodes and edges in a graph. Both algorithms are essential in the design and analysis of algorithms and are used in various applications such as pathfinding, cycle detection, topological sorting, and many others. They differ mainly in the order in which they visit the vertices of a graph. Depth-First Search (DFS): Depth-First Search is a recursive algorithm that explores as far as possible along a branch before backtracking. It uses a stack data structure either implicitly through recursion or explicitly using a stack. DFS starts at a source node, explores one of its adjacent nodes, and continues this process until it reaches a node with no unvisited neighbors. It then backtracks and explores other unvisited paths. DFS can be implemented using recursion or an explicit stack. Algorithm for DFS (Recursive): ``` DFS(Graph G, vertex v) mark v as visited for each neighbor u of v in G if u is not visited DFS(G, u) ``` Algorithm for DFS (Using Stack): ``` DFS(G, start) create a stack S push start onto S while S is not empty vertex = pop S if vertex is not visited mark vertex as visited for each neighbor of vertex in reverse order if neighbor is not visited push neighbor onto S ``` Example: Consider the following undirected graph: ``` A / \ B C / \ D E ``` Adjacency list representation: * A → B, C * B → A, D * C → A, E * D → B * E → C Start DFS from vertex A. Step-by-step DFS traversal (recursive): 1. Start at A, mark A as visited. Stack: \[A] 2. Visit neighbor B, mark B as visited. Stack: \[A, _____ _____ _________ _____ __________ ______ ____.
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