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:
Prerequisites: See this post for all applications of Depth First Traversal.
Following are implementations of simple Depth First Traversal. The C++ implementation uses adjacency list representation of graphs. STL‘s list container is used to store lists of adjacent nodes.
Solution:
- 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 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 index of adjacent node.
Implementation:
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, 9); g.addEdge(1, 2); g.addEdge(2, 0); g.addEdge(2, 3); g.addEdge(9, 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 |
Output:
Following is Depth First Traversal (starting from vertex 2) 2 0 1 9 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 is needed of size V.
Handling 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 the case of a Disconnected graph. To do 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 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 index of adjacent node.
- Run a loop from 0 to number of vertices and check if the node is unvisited in previous DFS then call the recursive function with 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 list ( 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 ) print "Following is Depth First Traversal" g.DFS() # This code is contributed by Neelam Yadav |
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 |
Output:
Following is Depth First Traversal 0 1 2 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 is needed of size V.
https://youtu.be/Y40bRyPQQr0
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