Question

I want an algorithm that gives one instance of a cycle in a directed graph if there is any. Can anyone show me a direction? In pseudo-code, or preferably, in Ruby?

I previously asked a similar question, and following the suggestions there, I implemented Kahn’s algorithm in Ruby that detects if a graph has a cycle, but I want not only whether it has a cycle, but also one possible instance of such cycle.

example_graph = [[1, 2], [2, 3], [3, 4], [3, 5], [3, 6], [6, 2]]

Kahn’s algorithm

def cyclic? graph
  ## The set of edges that have not been examined
  graph = graph.dup
  n, m = graph.transpose
  ## The set of nodes that are the supremum in the graph
  sup = (n - m).uniq
  while sup_old = sup.pop do
    sup_old = graph.select{|n, _| n == sup_old}
    graph -= sup_old
    sup_old.each {|_, ssup| sup.push(ssup) unless graph.any?{|_, n| n == ssup}}
  end
  !graph.empty?
end

The above algorithm tells whether a graph has a cycle:

cyclic?(example_graph) #=> true

but I want not only that but an example of a cycle like this:

#=> [[2, 3], [3, 6], [6, 2]]

If I were to output the variable graph in the above code at the end of examination, it will give:

#=> [[2, 3], [3, 4], [3, 5], [3, 6], [6, 2]]

which includes the cycle I want, but it also includes extra edges that are irrelevant to the cycle.

Answer 1

I asked the same question in the math stackexchange site, and got an answer. It turned out that Tarjan’s algorithm is good for solving this problem. I implemented it in Ruby as follows:

module DirectedGraph; module_function
    ## Tarjan's algorithm
    def strongly_connected_components graph
        @index, @stack, @indice, @lowlink, @scc = 0, [], {}, {}, []
        @graph = graph
        @graph.flatten(1).uniq.each{|v| strong_connect(v) unless @indice[v]}
        @scc
    end
    def strong_connect v
        @indice[v] = @index
        @lowlink[v] = @index
        @index += 1
        @stack.push(v)
        @graph.each do |vv, w|
            next unless vv == v
            if !@indice[w]
                strong_connect(w)
                @lowlink[v] = [@lowlink[v], @lowlink[w]].min
            elsif @stack.include?(w)
                @lowlink[v] = [@lowlink[v], @indice[w]].min
            end
        end
        if @lowlink[v] == @indice[v]
            i = @stack.index(v)
            @scc.push(@stack[i..-1])
            @stack = @stack[0...i]
        end
    end
end

So if I apply it to the example above, I get a list of strongly connected components of the graph:

example_graph = [[1, 2], [2, 3], [3, 4], [3, 5], [3, 6], [6, 2]]
DirectedGraph.strongly_connected_components(example_graph)
#=> [[4], [5], [2, 3, 6], [1]]

By selecting those components that are longer than one, I get the cycles:

DirectedGraph.strongly_connected_components(example_graph)
.select{|a| a.length > 1}
#=> [[2, 3, 6]]

And further if I select from the graph the edges whose both vertices are included in the components, I get the crucial edges that constitute the cycles:

DirectedGraph.strongly_connected_components(example_graph)
.select{|a| a.length > 1}
.map{|a| example_graph.select{|v, w| a.include?(v) and a.include?(w)}}
#=> [[[2, 3], [3, 6], [6, 2]]]