cyclefinding

Cyclefinding refers to algorithms that determine whether a cycle exists in a data structure that can be traversed, such as a linked list, a sequence produced by iteration, or a graph. The problem often includes locating the cycle’s start, determining its length, and, in graphs, identifying cycle components. Cyclefinding techniques vary by domain, but share the goal of identifying repetition in a finite structure.

In linked lists, the classic solution is Floyd’s cycle-finding algorithm, or the tortoise and hare. Two pointers move at different speeds; if they meet, a cycle exists. To find the cycle’s entry point, one pointer is reset to the head and both pointers advance at the same speed until they meet again. The cycle’s length can be determined by continuing from the meeting point until the pointer returns to its position. Brent’s algorithm offers an alternative with similar goals and sometimes different performance characteristics.

In graphs, cycle detection depends on graph type. For directed graphs, depth-first search with color marking (white for unvisited, gray for in-progress, black for completed) detects back edges to gray vertices, signaling a directed cycle; the cycle can be reconstructed with a stack or by tracking predecessors. Kahn’s algorithm for topological sorting detects cycles when not all vertices can be removed. In undirected graphs, a DFS that avoids treating the immediate predecessor as a back edge detects cycles when encountering an already visited vertex; parent pointers can be used to reconstruct the cycle.

Complexity and variants vary by method. Floyd’s algorithm runs in O(n) time with O(1) extra space for linked lists. DFS-based graph methods typically run in O(n+m) time and O(n) space. Cyclefinding has applications in garbage collection, deadlock detection, and the analysis of iterative processes.