How does the depth-first search algorithm function?

The depth-first search (DFS) algorithm explores a graph by visiting nodes as deeply as possible before backtracking.

The depth-first search algorithm is a technique used in computer science to traverse or search tree or graph data structures. The algorithm starts at the root (in case of a graph, an arbitrary node) and explores as far as possible along each branch before backtracking. This means it ventures deep into a graph by visiting a node's child nodes before visiting the node's sibling nodes.

DFS uses a stack data structure to remember to get back to the nodes to explore its branches. The process is simple. You start by pushing the root node into the stack. Then, you enter a loop and pop a node from the stack and push all its neighbours into the stack. You continue this process until the stack is empty.

However, there's a catch. You need to keep track of the nodes that you have visited. This is because graphs can have cycles, which means if you start at a node, you might end up at the same node. To avoid getting stuck in an infinite loop, you mark the nodes that you have visited. When you visit a node, you mark it as visited and then push all its unvisited neighbours into the stack.

DFS is a powerful algorithm that can be used to solve many problems such as finding a path between two nodes, detecting cycles in a graph, and finding a strongly connected component in a graph. It's worth noting that DFS is not the best algorithm for finding the shortest path between two nodes as it can traverse through more nodes than necessary. For such cases, algorithms like breadth-first search or Dijkstra's algorithm are more suitable.

In summary, the depth-first search algorithm is a systematic way of visiting all the nodes of a graph, prioritising depth over breadth. It uses a stack to keep track of nodes to visit next and backtracks when it reaches a node with no unvisited neighbours. It's a fundamental algorithm in computer science, with applications in various problems involving graphs and trees.