Strongly Connected Components

Strongly Connected Components

In this tutorial, you will learn how strongly connected components are formed. Also, you will find working examples of kosararju's algorithm in C, C++, Java and Python.

A strongly connected component is the portion of a directed graph in which there is a path from each vertex to another vertex. It is applicable only on a directed graph.

For example:

Let us take the graph below.

strongly connected components
Initial graph

The strongly connected components of the above graph are:

Strongly connected components
Strongly connected components

You can observe that in the first strongly connected component, every vertex can reach the other vertex through the directed path.

These components can be found using Kosaraju's Algorithm.


Kosaraju's Algorithm

Kosaraju's Algorithm is based on the depth-first search algorithm implemented twice.

Three steps are involved.

  1. Perform a depth first search on the whole graph.

    Let us start from vertex-0, visit all of its child vertices, and mark the visited vertices as done. If a vertex leads to an already visited vertex, then push this vertex to the stack.

    For example: Starting from vertex-0, go to vertex-1, vertex-2, and then to vertex-3. Vertex-3 leads to already visited vertex-0, so push the source vertex (ie. vertex-3) into the stack.
    strongly connected components
    DFS on the graph

    Go to the previous vertex (vertex-2) and visit its child vertices i.e. vertex-4, vertex-5, vertex-6 and vertex-7 sequentially. Since there is nowhere to go from vertex-7, push it into the stack.
    strongly connected components
    DFS on the graph

    Go to the previous vertex (vertex-6) and visit its child vertices. But, all of its child vertices are visited, so push it into the stack.
    strongly connected components
    Stacking

    Similarly, a final stack is created.
    strongly connected components
    Final Stack
  2. Reverse the original graph.
    reversed graph
    DFS on reversed graph
  3. Perform depth-first search on the reversed graph.

    Start from the top vertex of the stack. Traverse through all of its child vertices. Once the already visited vertex is reached, one strongly connected component is formed.

    For example: Pop vertex-0 from the stack. Starting from vertex-0, traverse through its child vertices (vertex-0, vertex-1, vertex-2, vertex-3 in sequence) and mark them as visited. The child of vertex-3 is already visited, so these visited vertices form one strongly connected component.
    reversed graph - strongly connected components
    Start from the top and traverse through all the vertices

    Go to the stack and pop the top vertex if already visited. Otherwise, choose the top vertex from the stack and traverse through its child vertices as presented above.
    strongly connected components
    Pop the top vertex if already visited
     
    reversed graph - strongly connected components
    Strongly connected component
  4. Thus, the strongly connected components are:
    strongly connected components
    All strongly connected components

Python, Java, C++ examples

# Kosaraju's algorithm to find strongly connected components in Python


from collections import defaultdict

class Graph:

    def __init__(self, vertex):
        self.V = vertex
        self.graph = defaultdict(list)

    # Add edge into the graph
    def add_edge(self, s, d):
        self.graph[s].append(d)

    # dfs
    def dfs(self, d, visited_vertex):
        visited_vertex[d] = True
        print(d, end='')
        for i in self.graph[d]:
            if not visited_vertex[i]:
                self.dfs(i, visited_vertex)

    def fill_order(self, d, visited_vertex, stack):
        visited_vertex[d] = True
        for i in self.graph[d]:
            if not visited_vertex[i]:
                self.fill_order(i, visited_vertex, stack)
        stack = stack.append(d)

    # transpose the matrix
    def transpose(self):
        g = Graph(self.V)

        for i in self.graph:
            for j in self.graph[i]:
                g.add_edge(j, i)
        return g

    # Print stongly connected components
    def print_scc(self):
        stack = []
        visited_vertex = [False] * (self.V)

        for i in range(self.V):
            if not visited_vertex[i]:
                self.fill_order(i, visited_vertex, stack)

        gr = self.transpose()

        visited_vertex = [False] * (self.V)

        while stack:
            i = stack.pop()
            if not visited_vertex[i]:
                gr.dfs(i, visited_vertex)
                print("")


g = Graph(8)
g.add_edge(0, 1)
g.add_edge(1, 2)
g.add_edge(2, 3)
g.add_edge(2, 4)
g.add_edge(3, 0)
g.add_edge(4, 5)
g.add_edge(5, 6)
g.add_edge(6, 4)
g.add_edge(6, 7)

print("Strongly Connected Components:")
g.print_scc()
// Kosaraju's algorithm to find strongly connected components in Java

import java.util.*;
import java.util.LinkedList;

class Graph {
	private int V;
	private LinkedList<Integer> adj[];

	// Create a graph
	Graph(int s) {
		V = s;
		adj = new LinkedList[s];
		for (int i = 0; i < s; ++i)
			adj[i] = new LinkedList();
	}

  // Add edge
	void addEdge(int s, int d) {
		adj[s].add(d);
	}

	// DFS
	void DFSUtil(int s, boolean visitedVertices[]) {
		visitedVertices[s] = true;
		System.out.print(s + " ");
		int n;

		Iterator<Integer> i = adj[s].iterator();
		while (i.hasNext()) {
			n = i.next();
			if (!visitedVertices[n])
				DFSUtil(n, visitedVertices);
		}
	}

	// Transpose the graph
	Graph Transpose() {
		Graph g = new Graph(V);
		for (int s = 0; s < V; s++) {
			Iterator<Integer> i = adj[s].listIterator();
			while (i.hasNext())
				g.adj[i.next()].add(s);
		}
		return g;
	}

	void fillOrder(int s, boolean visitedVertices[], Stack stack) {
		visitedVertices[s] = true;

		Iterator<Integer> i = adj[s].iterator();
		while (i.hasNext()) {
			int n = i.next();
			if (!visitedVertices[n])
				fillOrder(n, visitedVertices, stack);
		}
		stack.push(new Integer(s));
	}

	// Print strongly connected component
	void printSCC() {
		Stack stack = new Stack();

		boolean visitedVertices[] = new boolean[V];
		for (int i = 0; i < V; i++)
			visitedVertices[i] = false;

		for (int i = 0; i < V; i++)
			if (visitedVertices[i] == false)
				fillOrder(i, visitedVertices, stack);

		Graph gr = Transpose();

		for (int i = 0; i < V; i++)
			visitedVertices[i] = false;

		while (stack.empty() == false) {
			int s = (int) stack.pop();

			if (visitedVertices[s] == false) {
				gr.DFSUtil(s, visitedVertices);
				System.out.println();
			}
		}
	}

	public static void main(String args[]) {
		Graph g = new Graph(8);
		g.addEdge(0, 1);
		g.addEdge(1, 2);
		g.addEdge(2, 3);
		g.addEdge(2, 4);
		g.addEdge(3, 0);
		g.addEdge(4, 5);
		g.addEdge(5, 6);
		g.addEdge(6, 4);
		g.addEdge(6, 7);

		System.out.println("Strongly Connected Components:");
		g.printSCC();
	}
}
// Kosaraju's algorithm to find strongly connected components in C++

#include <iostream>
#include <list>
#include <stack>

using namespace std;

class Graph {
  int V;
  list<int> *adj;
  void fillOrder(int s, bool visitedV[], stack<int> &Stack);
  void DFS(int s, bool visitedV[]);

   public:
  Graph(int V);
  void addEdge(int s, int d);
  void printSCC();
  Graph transpose();
};

Graph::Graph(int V) {
  this->V = V;
  adj = new list<int>[V];
}

// DFS
void Graph::DFS(int s, bool visitedV[]) {
  visitedV[s] = true;
  cout << s << " ";

  list<int>::iterator i;
  for (i = adj[s].begin(); i != adj[s].end(); ++i)
    if (!visitedV[*i])
      DFS(*i, visitedV);
}

// Transpose
Graph Graph::transpose() {
  Graph g(V);
  for (int s = 0; s < V; s++) {
    list<int>::iterator i;
    for (i = adj[s].begin(); i != adj[s].end(); ++i) {
      g.adj[*i].push_back(s);
    }
  }
  return g;
}

// Add edge into the graph
void Graph::addEdge(int s, int d) {
  adj[s].push_back(d);
}

void Graph::fillOrder(int s, bool visitedV[], stack<int> &Stack) {
  visitedV[s] = true;

  list<int>::iterator i;
  for (i = adj[s].begin(); i != adj[s].end(); ++i)
    if (!visitedV[*i])
      fillOrder(*i, visitedV, Stack);

  Stack.push(s);
}

// Print strongly connected component
void Graph::printSCC() {
  stack<int> Stack;

  bool *visitedV = new bool[V];
  for (int i = 0; i < V; i++)
    visitedV[i] = false;

  for (int i = 0; i < V; i++)
    if (visitedV[i] == false)
      fillOrder(i, visitedV, Stack);

  Graph gr = transpose();

  for (int i = 0; i < V; i++)
    visitedV[i] = false;

  while (Stack.empty() == false) {
    int s = Stack.top();
    Stack.pop();

    if (visitedV[s] == false) {
      gr.DFS(s, visitedV);
      cout << endl;
    }
  }
}

int main() {
  Graph g(8);
  g.addEdge(0, 1);
  g.addEdge(1, 2);
  g.addEdge(2, 3);
  g.addEdge(2, 4);
  g.addEdge(3, 0);
  g.addEdge(4, 5);
  g.addEdge(5, 6);
  g.addEdge(6, 4);
  g.addEdge(6, 7);

  cout << "Strongly Connected Components:\n";
  g.printSCC();
}

Kosaraju's Algorithm Complexity

Kosaraju's algorithm runs in linear time i.e. O(V+E).


Strongly Connected Components Applications

  • Vehicle routing applications
  • Maps
  • Model-checking in formal verification