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Depth First Search or DFS for a Graph

  • Difficulty Level : Easy
  • Last Updated : 30 Nov, 2021
 

Depth First Traversal (or Search) for a graph is similar to Depth First Traversal of a tree. The only catch here is, unlike trees, graphs may contain cycles (a node may be visited twice). To avoid processing a node more than once, use a boolean visited array. 

Example: 

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Input: n = 4, e = 6 
0 -> 1, 0 -> 2, 1 -> 2, 2 -> 0, 2 -> 3, 3 -> 3 
Output: DFS from vertex 1 : 1 2 0 3 
Explanation: 
DFS Diagram: 
 



Input: n = 4, e = 6 
2 -> 0, 0 -> 2, 1 -> 2, 0 -> 1, 3 -> 3, 1 -> 3 
Output: DFS from vertex 2 : 2 0 1 3 
Explanation: 
DFS Diagram: 
 

Prerequisites: 
See this post for all applications of Depth First Traversal.

Approach: 
Depth-first search is an algorithm for traversing or searching tree or graph data structures. The algorithm starts at the root node (selecting some arbitrary node as the root node in the case of a graph) and explores as far as possible along each branch before backtracking. So the basic idea is to start from the root or any arbitrary node and mark the node and move to the adjacent unmarked node and continue this loop until there is no unmarked adjacent node. Then backtrack and check for other unmarked nodes and traverse them. Finally, print the nodes in the path.

Algorithm: 
Create a recursive function that takes the index of the node and a visited array.

  1. Mark the current node as visited and print the node.
  2. Traverse all the adjacent and unmarked nodes and call the recursive function with the index of the adjacent node.

Implementation: 
Below are implementations of simple Depth First Traversal. The C++ implementation uses an adjacency list representation of graphs. STL’s list container is used to store lists of adjacent nodes.

C++




// C++ program to print DFS traversal from
// a given vertex in a  given graph
#include <bits/stdc++.h>
using namespace std;
 
// Graph class represents a directed graph
// using adjacency list representation
class Graph {
public:
    map<int, bool> visited;
    map<int, list<int> > adj;
 
    // function to add an edge to graph
    void addEdge(int v, int w);
 
    // DFS traversal of the vertices
    // reachable from v
    void DFS(int v);
};
 
void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w); // Add w to v’s list.
}
 
void Graph::DFS(int v)
{
    // 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;
    for (i = adj[v].begin(); i != adj[v].end(); ++i)
        if (!visited[*i])
            DFS(*i);
}
 
// Driver code
int main()
{
    // Create a graph given in the above diagram
    Graph g;
    g.addEdge(0, 1);
    g.addEdge(0, 2);
    g.addEdge(1, 2);
    g.addEdge(2, 0);
    g.addEdge(2, 3);
    g.addEdge(3, 3);
 
    cout << "Following is Depth First Traversal"
            " (starting from vertex 2) \n";
    g.DFS(2);
 
    return 0;
}
 
// improved by Vishnudev C


Java




// Java program to print DFS
// mtraversal from a given given
// graph
import java.io.*;
import java.util.*;
 
// This class represents a
// directed graph using adjacency
// list representation
class Graph {
    private int V; // No. of vertices
 
    // Array  of lists for
    // Adjacency List Representation
    private LinkedList<Integer> adj[];
 
    // Constructor
    @SuppressWarnings("unchecked") Graph(int v)
    {
        V = v;
        adj = new LinkedList[v];
        for (int i = 0; i < v; ++i)
            adj[i] = new LinkedList();
    }
 
    // Function to add an edge into the graph
    void addEdge(int v, int w)
    {
        adj[v].add(w); // Add w to v's list.
    }
 
    // A function used by DFS
    void DFSUtil(int v, boolean visited[])
    {
        // Mark the current node as visited and print it
        visited[v] = true;
        System.out.print(v + " ");
 
        // Recur for all the vertices adjacent to this
        // vertex
        Iterator<Integer> i = adj[v].listIterator();
        while (i.hasNext()) {
            int n = i.next();
            if (!visited[n])
                DFSUtil(n, visited);
        }
    }
 
    // The function to do DFS traversal.
    // It uses recursive
    // DFSUtil()
    void DFS(int v)
    {
        // Mark all the vertices as
        // not visited(set as
        // false by default in java)
        boolean visited[] = new boolean[V];
 
        // Call the recursive helper
        // function to print DFS
        // traversal
        DFSUtil(v, visited);
    }
 
    // Driver Code
    public static void main(String args[])
    {
        Graph g = new Graph(4);
 
        g.addEdge(0, 1);
        g.addEdge(0, 2);
        g.addEdge(1, 2);
        g.addEdge(2, 0);
        g.addEdge(2, 3);
        g.addEdge(3, 3);
 
        System.out.println(
            "Following is Depth First Traversal "
            + "(starting from vertex 2)");
 
        g.DFS(2);
    }
}
// This code is contributed by Aakash Hasija


Python3




# Python3 program to print DFS traversal
# from a given given graph
from collections import defaultdict
 
# This class represents a directed graph using
# adjacency list representation
 
 
class Graph:
 
    # Constructor
    def __init__(self):
 
        # default dictionary to store graph
        self.graph = defaultdict(list)
 
    # function to add an edge to graph
    def addEdge(self, u, v):
        self.graph[u].append(v)
 
    # A function used by DFS
    def DFSUtil(self, v, visited):
 
        # Mark the current node as visited
        # and print it
        visited.add(v)
        print(v, end=' ')
 
        # Recur for all the vertices
        # adjacent to this vertex
        for neighbour in self.graph[v]:
            if neighbour not in visited:
                self.DFSUtil(neighbour, visited)
 
    # The function to do DFS traversal. It uses
    # recursive DFSUtil()
    def DFS(self, v):
 
        # Create a set to store visited vertices
        visited = set()
 
        # Call the recursive helper function
        # to print DFS traversal
        self.DFSUtil(v, visited)
 
# Driver code
 
 
# Create a graph given
# in the above diagram
g = Graph()
g.addEdge(0, 1)
g.addEdge(0, 2)
g.addEdge(1, 2)
g.addEdge(2, 0)
g.addEdge(2, 3)
g.addEdge(3, 3)
 
print("Following is DFS from (starting from vertex 2)")
g.DFS(2)
 
# This code is contributed by Neelam Yadav


C#




// C# program to print DFS traversal
// from a given graph
using System;
using System.Collections.Generic;
 
// This class represents a directed graph
// using adjacency list representation
class Graph {
    private int V; // No. of vertices
 
    // Array of lists for
    // Adjacency List Representation
    private List<int>[] adj;
 
    // Constructor
    Graph(int v)
    {
        V = v;
        adj = new List<int>[ v ];
        for (int i = 0; i < v; ++i)
            adj[i] = new List<int>();
    }
 
    // Function to Add an edge into the graph
    void AddEdge(int v, int w)
    {
        adj[v].Add(w); // Add w to v's list.
    }
 
    // A function used by DFS
    void DFSUtil(int v, bool[] visited)
    {
        // Mark the current node as visited
        // and print it
        visited[v] = true;
        Console.Write(v + " ");
 
        // Recur for all the vertices
        // adjacent to this vertex
        List<int> vList = adj[v];
        foreach(var n in vList)
        {
            if (!visited[n])
                DFSUtil(n, visited);
        }
    }
 
    // The function to do DFS traversal.
    // It uses recursive DFSUtil()
    void DFS(int v)
    {
        // Mark all the vertices as not visited
        // (set as false by default in c#)
        bool[] visited = new bool[V];
 
        // Call the recursive helper function
        // to print DFS traversal
        DFSUtil(v, visited);
    }
 
    // Driver Code
    public static void Main(String[] args)
    {
        Graph g = new Graph(4);
 
        g.AddEdge(0, 1);
        g.AddEdge(0, 2);
        g.AddEdge(1, 2);
        g.AddEdge(2, 0);
        g.AddEdge(2, 3);
        g.AddEdge(3, 3);
 
        Console.WriteLine(
            "Following is Depth First Traversal "
            + "(starting from vertex 2)");
 
        g.DFS(2);
        Console.ReadKey();
    }
}
 
// This code is contributed by techno2mahi


Javascript




<script>
 
// Javascript program to print DFS
// traversal from a given given
// graph
 
// This class represents a
// directed graph using adjacency
// list representation
class Graph
{
     
    // Constructor
    constructor(v)
    {
        this.V = v;
        this.adj = new Array(v);
        for(let i = 0; i < v; i++)
            this.adj[i] = [];
    }
     
    // Function to add an edge into the graph
    addEdge(v, w)
    {
         
        // Add w to v's list.
        this.adj[v].push(w);
    }
     
    // A function used by DFS
    DFSUtil(v, visited)
    {
         
        // Mark the current node as visited and print it
        visited[v] = true;
        document.write(v + " ");
  
        // Recur for all the vertices adjacent to this
        // vertex
        for(let i of this.adj[v].values())
        {
            let n = i
            if (!visited[n])
                this.DFSUtil(n, visited);
        }
    }
     
    // The function to do DFS traversal.
    // It uses recursive
    // DFSUtil()
    DFS(v)
    {
         
        // Mark all the vertices as
        // not visited(set as
        // false by default in java)
        let visited = new Array(this.V);
        for(let i = 0; i < this.V; i++)
            visited[i] = false;
  
        // Call the recursive helper
        // function to print DFS
        // traversal
        this.DFSUtil(v, visited);
    }
}
 
// Driver Code
g = new Graph(4);
  
g.addEdge(0, 1);
g.addEdge(0, 2);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(2, 3);
g.addEdge(3, 3);
 
document.write("Following is Depth First Traversal " +
               "(starting from vertex 2)<br>");
 
g.DFS(2);
 
// This code is contributed by avanitrachhadiya2155
 
</script>


Output: 



Following is Depth First Traversal (starting from vertex 2)
2 0 1 3

Complexity Analysis: 

  • Time complexity: O(V + E), where V is the number of vertices and E is the number of edges in the graph.
  • Space Complexity: O(V), since an extra visited array of size V is required.

 Handling A Disconnected Graph:

  • Solution: 
    This will happen by handling a corner case. 
    The above code traverses only the vertices reachable from a given source vertex. All the vertices may not be reachable from a given vertex, as in a Disconnected graph. To do a complete DFS traversal of such graphs, run DFS from all unvisited nodes after a DFS. 
    The recursive function remains the same.
  • Algorithm: 
    1. Create a recursive function that takes the index of the node and a visited array.
    2. Mark the current node as visited and print the node.
    3. Traverse all the adjacent and unmarked nodes and call the recursive function with the index of the adjacent node.
    4. Run a loop from 0 to the number of vertices and check if the node is unvisited in the previous DFS, call the recursive function with the current node.

Implementation: 

C++




// C++ program to print DFS
// traversal for a given given
// graph
#include <bits/stdc++.h>
using namespace std;
 
class Graph {
 
    // A function used by DFS
    void DFSUtil(int v);
 
public:
    map<int, bool> visited;
    map<int, list<int> > adj;
    // function to add an edge to graph
    void addEdge(int v, int w);
 
    // prints DFS traversal of the complete graph
    void DFS();
};
 
void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w); // Add w to v’s list.
}
 
void Graph::DFSUtil(int v)
{
    // 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;
    for (i = adj[v].begin(); i != adj[v].end(); ++i)
        if (!visited[*i])
            DFSUtil(*i);
}
 
// The function to do DFS traversal. It uses recursive
// DFSUtil()
void Graph::DFS()
{
    // Call the recursive helper function to print DFS
    // traversal starting from all vertices one by one
    for (auto i : adj)
        if (visited[i.first] == false)
            DFSUtil(i.first);
}
 
// Driver  Code
int main()
{
    // Create a graph given in the above diagram
    Graph g;
    g.addEdge(0, 1);
    g.addEdge(0, 9);
    g.addEdge(1, 2);
    g.addEdge(2, 0);
    g.addEdge(2, 3);
    g.addEdge(9, 3);
 
    cout << "Following is Depth First Traversal \n";
    g.DFS();
 
    return 0;
}
// improved by Vishnudev C


Java




// Java program to print DFS
// traversal from a given given
// graph
import java.io.*;
import java.util.*;
 
// This class represents a
// directed graph using adjacency
// list representation
class Graph {
    private int V; // No. of vertices
 
    // Array  of lists for
    // Adjacency List Representation
    private LinkedList<Integer> adj[];
 
    // Constructor
    @SuppressWarnings("unchecked") Graph(int v)
    {
        V = v;
        adj = new LinkedList[v];
        for (int i = 0; i < v; ++i)
            adj[i] = new LinkedList();
    }
 
    // Function to add an edge into the graph
    void addEdge(int v, int w)
    {
        adj[v].add(w); // Add w to v's list.
    }
 
    // A function used by DFS
    void DFSUtil(int v, boolean visited[])
    {
        // Mark the current node as visited and print it
        visited[v] = true;
        System.out.print(v + " ");
 
        // Recur for all the vertices adjacent to this
        // vertex
        Iterator<Integer> i = adj[v].listIterator();
        while (i.hasNext()) {
            int n = i.next();
            if (!visited[n])
                DFSUtil(n, visited);
        }
    }
 
    // The function to do DFS traversal. It uses recursive
    // DFSUtil()
    void DFS()
    {
        // Mark all the vertices as not visited(set as
        // false by default in java)
        boolean visited[] = new boolean[V];
 
        // Call the recursive helper function to print DFS
        // traversal starting from all vertices one by one
        for (int i = 0; i < V; ++i)
            if (visited[i] == false)
                DFSUtil(i, visited);
    }
 
    // Driver Code
    public static void main(String args[])
    {
        Graph g = new Graph(4);
 
        g.addEdge(0, 1);
        g.addEdge(0, 2);
        g.addEdge(1, 2);
        g.addEdge(2, 0);
        g.addEdge(2, 3);
        g.addEdge(3, 3);
 
        System.out.println(
            "Following is Depth First Traversal");
 
        g.DFS();
    }
}
// This code is contributed by Aakash Hasija


Python




'''Python program to print DFS traversal for complete graph'''
from collections import defaultdict
 
# this class represents a directed graph using adjacency list representation
 
 
class Graph:
    # Constructor
    def __init__(self):
        # default dictionary to store graph
        self.graph = defaultdict(list)
 
    # Function to add an edge to graph
    def addEdge(self, u, v):
        self.graph[u].append(v)
    # A function used by DFS
 
    def DFSUtil(self, v, visited):
        # Mark the current node as visited and print it
        visited.add(v)
        print(v)
 
        # recur for all the vertices adjacent to this vertex
        for neighbour in self.graph[v]:
            if neighbour not in visited:
                self.DFSUtil(neighbour, visited)
        # The function to do DFS traversal. It uses recursive DFSUtil
 
    def DFS(self):
        # create a set to store all visited vertices
        visited = set()
        # call the recursive helper function to print DFS traversal starting from all
        # vertices one by one
        for vertex in self.graph:
            if vertex not in visited:
                self.DFSUtil(vertex, visited)
# Driver code
# create a graph given in the above diagram
 
 
g = Graph()
g.addEdge(0, 1)
g.addEdge(0, 2)
g.addEdge(1, 2)
g.addEdge(2, 0)
g.addEdge(2, 3)
g.addEdge(3, 3)
g.DFS()
 
# Improved by Dheeraj Kumar


C#




// C# program to print DFS
// traversal from a given given
// graph
using System;
using System.Collections.Generic;
 
// This class represents a
// directed graph using adjacency
// list representation
public class Graph {
    private int V; // No. of vertices
 
    // Array of lists for
    // Adjacency List Representation
    private List<int>[] adj;
 
    // Constructor
    Graph(int v)
    {
        V = v;
        adj = new List<int>[ v ];
        for (int i = 0; i < v; ++i)
            adj[i] = new List<int>();
    }
 
    // Function to add an edge into the graph
    void addEdge(int v, int w)
    {
        adj[v].Add(w); // Add w to v's list.
    }
 
    // A function used by DFS
    void DFSUtil(int v, bool[] visited)
    {
        // Mark the current
        // node as visited and print it
        visited[v] = true;
        Console.Write(v + " ");
 
        // Recur for all the
        // vertices adjacent to this
        // vertex
        foreach(int i in adj[v])
        {
            int n = i;
            if (!visited[n])
                DFSUtil(n, visited);
        }
    }
 
    // The function to do
    // DFS traversal. It uses recursive
    // DFSUtil()
    void DFS()
    {
        // Mark all the vertices as not visited(set as
        // false by default in java)
        bool[] visited = new bool[V];
 
        // Call the recursive helper
        // function to print DFS
        // traversal starting from
        // all vertices one by one
        for (int i = 0; i < V; ++i)
            if (visited[i] == false)
                DFSUtil(i, visited);
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        Graph g = new Graph(4);
 
        g.addEdge(0, 1);
        g.addEdge(0, 2);
        g.addEdge(1, 2);
        g.addEdge(2, 0);
        g.addEdge(2, 3);
        g.addEdge(3, 3);
 
        Console.WriteLine(
            "Following is Depth First Traversal");
 
        g.DFS();
    }
}
 
// This code is contributed by PrinciRaj1992


Javascript




<script>
      // JavaScript program to print DFS
      // traversal from a given given
      // graph
 
      // This class represents a
      // directed graph using adjacency
      // list representation
      class Graph
      {
       
        // Constructor
        constructor(v) {
          this.V = v;
          this.adj = new Array(v).fill([]);
        }
 
        // Function to Add an edge into the graph
        AddEdge(v, w) {
          this.adj[v].push(w); // Add w to v's list.
        }
 
        // A function used by DFS
        DFSUtil(v, visited)
        {
         
          // Mark the current
          // node as visited and print it
          visited[v] = true;
          document.write(v + " ");
 
          // Recur for all the
          // vertices adjacent to this
          // vertex
          for (const n of this.adj[v]) {
            if (!visited[n]) this.DFSUtil(n, visited);
          }
        }
 
        // The function to do
        // DFS traversal. It uses recursive
        // DFSUtil()
        DFS()
        {
         
          // Mark all the vertices as not visited(set as
          var visited = new Array(this.V).fill(false);
 
          // Call the recursive helper
          // function to print DFS
          // traversal starting from
          // all vertices one by one
          for (var i = 0; i < this.V; ++i)
            if (visited[i] == false) this.DFSUtil(i, visited);
        }
      }
       
      // Driver Code
      var g = new Graph(4);
 
      g.AddEdge(0, 1);
      g.AddEdge(0, 2);
      g.AddEdge(1, 2);
      g.AddEdge(2, 0);
      g.AddEdge(2, 3);
      g.AddEdge(3, 3);
 
      document.write("Following is Depth First Traversal<br>");
 
      g.DFS();
       
      // This code is contributed by rdtank.
    </script>


Output: 

Following is Depth First Traversal
0 1 2 3 9

Complexity Analysis: 

  • Time complexity: O(V + E), where V is the number of vertices and E is the number of edges in the graph.
  • Space Complexity: O(V), since an extra visited array of size V is required.

https://youtu.be/Y40bRyPQQr0 

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