Depth First Search or DFS for a Graph

14 Jul, 2022
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: 
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:  
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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.
Mark the current node as visited and print the node.Traverse all the adjacent and unmarked nodes and call the recursive function with the index of the adjacent node.
Mark the current node as visited and print the node.
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++
Java
Python3
C#
Javascript
// 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 representationclass 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 codeint 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 program to print DFS// traversal from a given// graphimport java.io.*;import java.util.*; // This class represents a// directed graph using adjacency// list representationclass 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 program to print DFS traversal# from a given  graphfrom 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 diagramg = 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# program to print DFS traversal// from a given graphusing System;using System.Collections.Generic; // This class represents a directed graph// using adjacency list representationclass 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
<script> // Javascript program to print DFS// traversal from a given// graph // This class represents a// directed graph using adjacency// list representationclass 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 Codeg = 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: Create a recursive function that takes the index of the node and a visited array.Mark the current node as visited and print the node.Traverse all the adjacent and unmarked nodes and call the recursive function with the index of the adjacent node.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.
Create a recursive function that takes the index of the node and a visited array.Mark the current node as visited and print the node.Traverse all the adjacent and unmarked nodes and call the recursive function with the index of the adjacent node.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.
Create a recursive function that takes the index of the node and a visited array.
Mark the current node as visited and print the node.
Traverse all the adjacent and unmarked nodes and call the recursive function with the index of the adjacent node.
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++
Java
Python3
C#
Javascript
// C++ program to print DFS// traversal for a 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  Codeint 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 program to print DFS// traversal from a given// graphimport java.io.*;import java.util.*; // This class represents a// directed graph using adjacency// list representationclass 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 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,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):        # 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 print("Following is Depth First Traversal \n")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# program to print DFS// traversal from a given// graphusing System;using System.Collections.Generic; // This class represents a// directed graph using adjacency// list representationpublic 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
<script>      // JavaScript program to print DFS      // traversal from a 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 
Applications of DFS.
Breadth-First Traversal for a Graph
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