How does the cycle detection algorithm function in graphs?

The cycle detection algorithm in graphs works by traversing nodes and checking for revisited nodes.

In more detail, the cycle detection algorithm is a method used to identify whether a given graph contains any cycles. A cycle in a graph is a non-empty trail in which the only repeated vertices are the first and last vertices. The algorithm works by traversing the graph, marking each visited node, and then checking if a node has been visited before. If a node is revisited, it means a cycle exists.

There are several ways to implement a cycle detection algorithm, but two of the most common methods are Depth-First Search (DFS) and Breadth-First Search (BFS).

In DFS, the algorithm starts at a random node (or a specified 'root' node), explores as far as possible along each branch before backtracking. It uses a stack data structure to remember to get back to branching nodes when a dead end is reached. If it comes across a node that's already in the stack, it means there's a cycle.

In BFS, the algorithm visits all the vertices of the graph or tree in breadth-first order, i.e., it visits all the nodes at the present depth before going to nodes at the next depth level. It uses a queue data structure. If a visited node is encountered again, a cycle is detected.

In both methods, a boolean visited array is used. For every visited node, it is marked true in the visited array. If the boolean array already contains true for the node, it means the node is revisited and hence a cycle is detected.

It's important to note that these algorithms only work for undirected graphs. For directed graphs, a variation of DFS called the "coloured DFS" can be used. In this method, three colours are used to mark the nodes: white (unvisited), grey (visited but not fully explored), and black (fully explored). If a grey node is encountered again, it indicates a cycle.

Understanding the cycle detection algorithm is crucial in computer science as it helps in solving many problems, such as detecting deadlocks in operating systems, finding infinite loops in programs, and in network routing.