Comparison between Tarjan’s and Kosaraju’s Algorithm

Tarjan’s Algorithm: The Tarjan’s Algorithm is an efficient graph algorithm that is used to find the Strongly Connected Component(SCC) in a directed graph by using only one DFS traversal in linear time complexity.

Working:

• Perform a DFS traversal over the nodes so that the sub-trees of the Strongly Connected Components are removed when they are encountered.
• Then two values are assigned:
  • The first value is the counter value when the node is explored for the first time.
  • Second value stores the lowest node value reachable from the initial node which is not part of another SCC.
• When the nodes are explored, they are pushed into a stack.
• If there are any unexplored children of a node are left, they are explored and the assigned value is respectively updated.

Below is the program to find the SCC of the given graph using Tarjan’s Algorithm:

C++

// C++ program to find the SCC using
// Tarjan's algorithm (single DFS)
#include <iostream>
#include <list>
#include <stack>
#define NIL -1
using namespace std;

// A class that represents
// an directed graph
class Graph {
    // No. of vertices
    int V;

    // A dynamic array of adjacency lists
    list<int>* adj;

    // A Recursive DFS based function
    // used by SCC()
    void SCCUtil(int u, int disc[],
                 int low[], stack<int>* st,
                 bool stackMember[]);

public:
    // Member functions
    Graph(int V);
    void addEdge(int v, int w);
    void SCC();
};

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

// Function to add an edge to the graph
void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w);
}

// Recursive function to finds the SCC
// using DFS traversal
void Graph::SCCUtil(int u, int disc[],
                    int low[], stack<int>* st,
                    bool stackMember[])
{
    static int time = 0;

    // Initialize discovery time
    // and low value
    disc[u] = low[u] = ++time;
    st->push(u);
    stackMember[u] = true;

    // Go through all vertices
    // adjacent to this
    list<int>::iterator i;

    for (i = adj[u].begin();
         i != adj[u].end(); ++i) {
        // v is current adjacent of 'u'
        int v = *i;

        // If v is not visited yet,
        // then recur for it
        if (disc[v] == -1) {
            SCCUtil(v, disc, low,
                    st, stackMember);

            // Check if the subtree rooted
            // with 'v' has connection to
            // one of the ancestors of 'u'
            low[u] = min(low[u], low[v]);
        }

        // Update low value of 'u' only of
        // 'v' is still in stack
        else if (stackMember[v] == true)
            low[u] = min(low[u], disc[v]);
    }

    // head node found, pop the stack
    // and print an SCC

    // Store stack extracted vertices
    int w = 0;

    // If low[u] and disc[u]
    if (low[u] == disc[u]) {
        // Until stack st is empty
        while (st->top() != u) {
            w = (int)st->top();

            // Print the node
            cout << w << " ";
            stackMember[w] = false;
            st->pop();
        }
        w = (int)st->top();
        cout << w << "\n";
        stackMember[w] = false;
        st->pop();
    }
}

// Function to find the SCC in the graph
void Graph::SCC()
{
    // Stores the discovery times of
    // the nodes
    int* disc = new int[V];

    // Stores the nodes with least
    // discovery time
    int* low = new int[V];

    // Checks whether a node is in
    // the stack or not
    bool* stackMember = new bool[V];

    // Stores all the connected ancestors
    stack<int>* st = new stack<int>();

    // Initialize disc and low,
    // and stackMember arrays
    for (int i = 0; i < V; i++) {
        disc[i] = NIL;
        low[i] = NIL;
        stackMember[i] = false;
    }

    // Recursive helper function to
    // find the SCC in DFS tree with
    // vertex 'i'
    for (int i = 0; i < V; i++) {

        // If current node is not
        // yet visited
        if (disc[i] == NIL) {
            SCCUtil(i, disc, low,
                    st, stackMember);
        }
    }
}

// Driver Code
int main()
{
    // Given a graph
    Graph g1(5);
    g1.addEdge(1, 0);
    g1.addEdge(0, 2);
    g1.addEdge(2, 1);
    g1.addEdge(0, 3);
    g1.addEdge(3, 4);

    // Function Call to find SCC using
    // Tarjan's Algorithm
    g1.SCC();

    return 0;
}

Output:

4
3
1 2 0

Kosaraju’s Algorithm: The Kosaraju’s Algorithm is also a Depth First Search based algorithm which is used to find the SCC in a directed graph in linear time complexity. The basic concept of this algorithm is that if we are able to arrive at vertex v initially starting from vertex u, then we should be able to arrive at vertex u starting from vertex v, and if this is the situation, we can say and conclude that vertices u and v are strongly connected, and they are in the strongly connected sub-graph.

Working:

• Perform a DFS traversal on the given graph, keeping track of the finish times of each node. This process can be performed by using a stack.
• When the procedure of running the DFS traversal over the graph finishes, put the source vertex on the stack. In this way, the node with the highest finishing time will be at the top of the stack.
• Reverse the original graph by using an Adjacency List.
• Then perform another DFS traversal on the reversed graph with the source vertex as the vertex on the top of the stack. When the DFS running on the reversed graph finishes, all the nodes that are visited will form one strongly connected component.
• If any more nodes are left or remain unvisited, this signifies the presence of more than one strongly connected component on the graph.
• So pop the vertices from the top of the stack until a valid unvisited node is found. This will have the highest finishing time of all currently unvisited nodes.

Below is the program to find the SCC of the given graph using Kosaraju’s Algorithm:

C++

// C++ program to print the SCC of the
// graph using Kosaraju's Algorithm
#include <iostream>
#include <list>
#include <stack>
using namespace std;

class Graph {
    // No. of vertices
    int V;

    // An array of adjacency lists
    list<int>* adj;

    // Member Functions
    void fillOrder(int v, bool visited[],
                   stack<int>& Stack);
    void DFSUtil(int v, bool visited[]);

public:
    Graph(int V);
    void addEdge(int v, int w);
    void printSCCs();
    Graph getTranspose();
};

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

// Recursive function to print DFS
// starting from v
void Graph::DFSUtil(int v, bool visited[])
{
    // Mark the current node as
    // visited and print it
    visited[v] = true;
    cout << v << " ";

    // Recur for all the vertices
    // adjacent to this vertex
    list<int>::iterator i;

    // Traverse Adjacency List of node v
    for (i = adj[v].begin();
         i != adj[v].end(); ++i) {

        // If child node *i is unvisited
        if (!visited[*i])
            DFSUtil(*i, visited);
    }
}

// Function to get the transpose of
// the given graph
Graph Graph::getTranspose()
{
    Graph g(V);
    for (int v = 0; v < V; v++) {
        // Recur for all the vertices
        // adjacent to this vertex
        list<int>::iterator i;
        for (i = adj[v].begin();
             i != adj[v].end(); ++i) {
            // Add to adjacency list
            g.adj[*i].push_back(v);
        }
    }

    // Return the reversed graph
    return g;
}

// Function to add an Edge to the given
// graph
void Graph::addEdge(int v, int w)
{
    // Add w to v’s list
    adj[v].push_back(w);
}

// Function that fills stack with vertices
// in increasing order of finishing times
void Graph::fillOrder(int v, bool visited[],
                      stack<int>& Stack)
{
    // Mark the current node as
    // visited and print it
    visited[v] = true;

    // Recur for all the vertices
    // adjacent to this vertex
    list<int>::iterator i;

    for (i = adj[v].begin();
         i != adj[v].end(); ++i) {

        // If child node *i is unvisited
        if (!visited[*i]) {
            fillOrder(*i, visited, Stack);
        }
    }

    // All vertices reachable from v
    // are processed by now, push v
    Stack.push(v);
}

// Function that finds and prints all
// strongly connected components
void Graph::printSCCs()
{
    stack<int> Stack;

    // Mark all the vertices as
    // not visited (For first DFS)
    bool* visited = new bool[V];
    for (int i = 0; i < V; i++)
        visited[i] = false;

    // Fill vertices in stack according
    // to their finishing times
    for (int i = 0; i < V; i++)
        if (visited[i] == false)
            fillOrder(i, visited, Stack);

    // Create a reversed graph
    Graph gr = getTranspose();

    // Mark all the vertices as not
    // visited (For second DFS)
    for (int i = 0; i < V; i++)
        visited[i] = false;

    // Now process all vertices in
    // order defined by Stack
    while (Stack.empty() == false) {

        // Pop a vertex from stack
        int v = Stack.top();
        Stack.pop();

        // Print SCC of the popped vertex
        if (visited[v] == false) {
            gr.DFSUtil(v, visited);
            cout << endl;
        }
    }
}

// Driver Code
int main()
{
    // Given Graph
    Graph g(5);
    g.addEdge(1, 0);
    g.addEdge(0, 2);
    g.addEdge(2, 1);
    g.addEdge(0, 3);
    g.addEdge(3, 4);

    // Function Call to find the SCC
    // using Kosaraju's Algorithm
    g.printSCCs();

    return 0;
}

Output:

0 1 2 
3 
4

Time Complexity:
The time complexity of Tarjan’s Algorithm and Kosaraju’s Algorithm will be O(V + E), where V represents the set of vertices and E represents the set of edges of the graph. Tarjan’s algorithm has much lower constant factors w.r.t Kosaraju’s algorithm. In Kosaraju’s algorithm, the traversal of the graph is done at least 2 times, so the constant factor can be of double time. We can print the SCC in progress with Kosaraju’s algorithm as we perform the second DFS. While performing Tarjan’s Algorithm, it requires extra time to print the SCC after finding the head of the SCCs sub-tree.

Summary:
Both the methods have the same linear time complexity, but the techniques or the procedure for the SCC computations are fairly different. Tarjan’s method solely depends on the record of nodes in a DFS to partition the graph whereas Kosaraju’s method performs the two DFS (or 3 DFS if we want to leave the original graph unchanged) on the graph and is quite similar to the method for finding the topological sorting of a graph.