Best algorithm for detecting cycles in a directed graph [closed]

Answer 1

The simplest way to do it is to do a depth first traversal (DFT) of the graph.

If the graph has n vertices, this is a O(n) time complexity algorithm. Since you will possibly have to do a DFT starting from each vertex, the total complexity becomes O(n^2).

You have to maintain a stack containing all vertices in the current depth first traversal, with its first element being the root node. If you come across an element which is already in the stack during the DFT, then you have a cycle.

Answer 2

The way I do it is to do a Topological Sort, counting the number of vertices visited. If that number is less than the total number of vertices in the DAG, you have a cycle.

Answer 3

As you said, you have set of jobs, it need to be executed in certain order. Topological sort given you required order for scheduling of jobs(or for dependency problems if it is a direct acyclic graph). Run dfs and maintain a list, and start adding node in the beginning of the list, and if you encountered a node which is already visited. Then you found a cycle in given graph.

Answer 4

In my opinion, the most understandable algorithm for detecting cycle in a directed graph is the graph-coloring-algorithm.

Basically, the graph coloring algorithm walks the graph in a DFS manner (Depth First Search, which means that it explores a path completely before exploring another path). When it finds a back edge, it marks the graph as containing a loop.

For an in depth explanation of the graph coloring algorithm, please read this article: http://www.geeksforgeeks.org/detect-cycle-direct-graph-using-colors/

Also, I provide an implementation of graph coloring in JavaScript https://github.com/dexcodeinc/graph_algorithm.js/blob/master/graph_algorithm.js

Answer 5

Given that this is a schedule of jobs, I suspect that at some point you are going to sort them into a proposed order of execution.

If that's the case, then a topological sort implementation may in any case detect cycles. UNIX tsort certainly does. I think it is likely that it is therefore more efficient to detect cycles at the same time as tsorting, rather than in a separate step.

So the question might become, "how do I most efficiently tsort", rather than "how do I most efficiently detect loops". To which the answer is probably "use a library", but failing that the following Wikipedia article:

http://en.wikipedia.org/wiki/Topological_sorting

has the pseudo-code for one algorithm, and a brief description of another from Tarjan. Both have O(|V| + |E|) time complexity.

Answer 6

If DFS finds an edge that points to an already-visited vertex, you have a cycle there.