Write an algorithm to implement depth-first search. How is depth-first search different from Breadth-first search? Also, write any two application of complete graph.

Algorithm for Depth-First Search:

1. Initialization:

   • Mark all vertices as unvisited.
   • Choose a starting vertex and mark it as visited.
   • Push it onto a stack.

2. Recursion:

   • While the stack is not empty:
     • Pop a vertex from the stack.
     • Visit it and mark it as visited.
     • For each of its unvisited neighbors:
     • Push the neighbor onto the stack.

Differences between Depth-First Search and Breadth-First Search:

• Traversal Order: DFS explores the graph in depth, going as far as possible along each branch before backtracking. BFS, on the other hand, explores the graph in breadth, visiting all neighbors of a vertex before moving on to the next level of the graph.
• Data Structure: DFS uses a stack to keep track of vertices that need to be explored. BFS uses a queue.
• Applications: DFS is often used for finding paths or cycles in a graph, while BFS is often used for finding the shortest path between two vertices.

Two Applications of Complete Graphs:

1. Scheduling: In scheduling problems, a complete graph can be used to represent the precedence constraints among tasks. The vertices of the graph represent the tasks, and the edges represent the precedence constraints. A complete graph ensures that all precedence constraints are represented, even if there are no direct dependencies between some tasks.

2. Network Routing: In network routing problems, a complete graph can be used to represent the network topology. The vertices of the graph represent the nodes in the network, and the edges represent the links between the nodes. A complete graph ensures that all possible paths between any two nodes are represented, even if some links are not directly connected.