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In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values. For any function f that maps a finite set S to itself, and any initial value x 0 in S , the sequence of iterated function values
Therefore, the special case of the zero-weight cycle problem, on graphs with no negative cycle, has a polynomial-time algorithm. [1] In contrast, for graphs that contain negative cycles, detecting a simple cycle of weight exactly 0 is an NP-complete problem. [1] This is true even when the weights are integers of polynomial magnitude.
The figure below presents an example of the execution of the algorithm. In iteration =, all the three vertices detect the cycle (,,). The algorithm ensures that the cycle is reported only once by emitting the detected cycle only from the vertex with the least identifier value in the ordered sequence, which is the vertex 2 in the example.
If the graph is connected, it finds a minimum spanning tree. It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle. [2] The key steps of the algorithm are sorting and the use of a disjoint-set data structure to detect cycles. Its running time is dominated by the time to sort all of the ...
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an acyclic graph. A directed graph without directed cycles is called a directed ...
The problem of finding a Hamiltonian cycle or path is in FNP; the analogous decision problem is to test whether a Hamiltonian cycle or path exists. The directed and undirected Hamiltonian cycle problems were two of Karp's 21 NP-complete problems. They remain NP-complete even for special kinds of graphs, such as: bipartite graphs, [12]
Any vertex that is not on a directed cycle forms a strongly connected component all by itself: for example, a vertex whose in-degree or out-degree is 0, or any vertex of an acyclic graph. The basic idea of the algorithm is this: a depth-first search (DFS) begins from an arbitrary start node (and subsequent depth-first searches are conducted on ...
The first three stages of Johnson's algorithm are depicted in the illustration below. The graph on the left of the illustration has two negative edges, but no negative cycles. The center graph shows the new vertex q, a shortest path tree as computed by the Bellman–Ford algorithm with q as starting vertex, and the values h(v) computed at each other node as the length of the shortest path from ...