Cycle detection

In computer science, cycle detection is the algorithmic problem of finding a cycle in a sequence of iterated function values.

For any function ƒ that maps a finite set S to itself, and any initial value x₀ in S, the sequence of iterated function values

must eventually use the same value twice: there must be some i ≠ j such that x_i = x_j. Once this happens, the sequence must continue periodically, by repeating the same sequence of values from x_i to x_j−1. Cycle detection is the problem of finding i and j, given ƒ and x₀.

Example

The figure shows a function ƒ that maps the set S = {0,1,2,3,4,5,6,7,8} to itself. If one starts from x₀ = 2 and repeatedly applies ƒ, one sees the sequence of values

The cycle in this value sequence is 6, 3, 1.

Definitions

Let S be any finite set, ƒ be any function from S to itself, and x₀ be any element of S. For any i > 0, let x_i = ƒ(x_i−1). Let μ be the smallest index such that the value x_μ reappears infinitely often within the sequence of values x_i, and let λ (the loop length) be the smallest positive integer such that x_μ = x_λ+μ. The cycle detection problem is the task of finding λ and μ.

Cycle (graph theory)

In graph theory, there are several different types of object called cycles, principally a closed walk and a simple cycle; also, e.g., an element of the cycle space of the graph.

A closed walk consists of a sequence of vertices starting and ending at the same vertex, with each two consecutive vertices in the sequence adjacent to each other in the graph. In a directed graph, each edge must be traversed by the walk consistently with its direction: the edge must be oriented from the earlier of two consecutive vertices to the later of the two vertices in the sequence. The choice of starting vertex is not important: traversing the same cyclic sequence of edges from different starting vertices produces the same closed walk.

A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex, or as the set of edges in such a walk. The two definitions are equivalent in directed graphs, where simple cycles are also called directed cycles: the cyclic sequence of vertices and edges in a walk is completely determined by the set of edges that it uses. In undirected graphs the set of edges of a cycle can be traversed by a walk in either of two directions, giving two possible directed cycles for every undirected cycle. (For closed walks more generally, in directed or undirected graphs, the multiset of edges does not unambiguously determine the vertex ordering.) A circuit can be a closed walk allowing repetitions of vertices but not edges; however, it can also be a simple cycle, so explicit definition is recommended when it is used.