What is Depth-first search in graph theory?

Depth-first search is a graph traversal algorithm that explores as far as possible along each branch before backtracking.

Depth-first search (DFS) is a graph traversal algorithm that explores as far as possible along each branch before backtracking. It starts at a given vertex and visits all the vertices in the graph, marking each vertex as visited. The algorithm then backtracks to the previous vertex and continues the process until all vertices have been visited.

DFS can be implemented using either recursion or a stack. The recursive implementation is simpler and more elegant, but it may run out of stack space for large graphs. The stack-based implementation is more complex, but it can handle larger graphs without running out of stack space.

DFS can be used to solve a variety of graph problems, such as finding connected components, detecting cycles, and computing topological order. It can also be used to solve maze problems and to generate random mazes.

DFS has a time complexity of O(V+E), where V is the number of vertices and E is the number of edges in the graph. This is because each vertex and each edge is visited exactly once. However, the actual running time may depend on the data structure used to represent the graph and the specific implementation of the algorithm.

In conclusion, DFS is a powerful graph traversal algorithm that can be used to solve a variety of graph problems. It is relatively easy to implement and has a reasonable time complexity. A-Level Maths students should be familiar with DFS and its applications in graph theory.