Transpose of a directed graph G is another directed graph on the same set of vertices with all of the edges reversed compared to the orientation of the corresponding edges in G. That is, if G contains an edge (u, v) then the converse/transpose/reverse of G contains an edge (v, u) and vice versa. Given a graph (represented as adjacency list), we need to find another graph which is the transpose of the given graph.
Example:
Input : figure (i) is the input graph.
Output : figure (ii) is the transpose graph of the given graph.
We traverse the adjacency list and as we find a vertex v in the adjacency list of vertex u which indicates an edge from u to v in main graph, we just add an edge from v to u in the transpose graph i.e. add u in the adjacency list of vertex v of the new graph. Thus traversing lists of all vertices of main graph we can get the transpose graph. Thus the total time complexity of the algorithm is O(V+E) where V is number of vertices of graph and E is the number of edges of the graph. Note : It is simple to get the transpose of a graph which is stored in adjacency matrix format, you just need to get the transpose of that matrix.
Implementation:
C++
// CPP program to find transpose of a graph.
#include <bits/stdc++.h>
using namespace std;
// function to add an edge from vertex source to vertex dest
void addEdge(vector<int> adj[], int src, int dest)
{
adj[src].push_back(dest);
}
// function to print adjacency list of a graph
void displayGraph(vector<int> adj[], int v)
{
for (int i = 0; i < v; i++) {
cout << i << "--> ";
for (int j = 0; j < adj[i].size(); j++)
cout << adj[i][j] << " ";
cout << "\n";
}
}
// function to get Transpose of a graph taking adjacency
// list of given graph and that of Transpose graph
void transposeGraph(vector<int> adj[],
vector<int> transpose[], int v)
{
// traverse the adjacency list of given graph and
// for each edge (u, v) add an edge (v, u) in the
// transpose graph's adjacency list
for (int i = 0; i < v; i++)
for (int j = 0; j < adj[i].size(); j++)
addEdge(transpose, adj[i][j], i);
}
int main()
{
int v = 5;
vector<int> adj[v];
addEdge(adj, 0, 1);
addEdge(adj, 0, 4);
addEdge(adj, 0, 3);
addEdge(adj, 2, 0);
addEdge(adj, 3, 2);
addEdge(adj, 4, 1);
addEdge(adj, 4, 3);
// Finding transpose of graph represented
// by adjacency list adj[]
vector<int> transpose[v];
transposeGraph(adj, transpose, v);
// displaying adjacency list of transpose
// graph i.e. b
displayGraph(transpose, v);
return 0;
}
// Java program to find the transpose of a graph
import java.util.*;
import java.lang.*;
import java.io.*;
class Graph
{
// Total number of vertices
private static int vertices = 5;
// Find transpose of graph represented by adj
private static ArrayList<Integer>[] adj = new ArrayList[vertices];
// Store the transpose of graph represented by tr
private static ArrayList<Integer>[] tr = new ArrayList[vertices];
// Function to add an edge from source vertex u to
// destination vertex v, if choice is false the edge is added
// to adj otherwise the edge is added to tr
public static void addedge(int u, int v, boolean choice)
{
if(!choice)
adj[u].add(v);
else
tr[u].add(v);
}
// Function to print the graph representation
public static void printGraph()
{
for(int i = 0; i < vertices; i++)
{
System.out.print(i + "--> ");
for(int j = 0; j < tr[i].size(); j++)
System.out.print(tr[i].get(j) + " ");
System.out.println();
}
}
// Function to print the transpose of
// the graph represented as adj and store it in tr
public static void getTranspose()
{
// Traverse the graph and for each edge u, v
// in graph add the edge v, u in transpose
for(int i = 0; i < vertices; i++)
for(int j = 0; j < adj[i].size(); j++)
addedge(adj[i].get(j), i, true);
}
public static void main (String[] args) throws java.lang.Exception
{
for(int i = 0; i < vertices; i++)
{
adj[i] = new ArrayList<Integer>();
tr[i] = new ArrayList<Integer>();
}
addedge(0, 1, false);
addedge(0, 4, false);
addedge(0, 3, false);
addedge(2, 0, false);
addedge(3, 2, false);
addedge(4, 1, false);
addedge(4, 3, false);
// Finding transpose of the graph
getTranspose();
// Printing the graph representation
printGraph();
}
}
// This code is contributed by code_freak
# Python3 program to find transpose of a graph.
# function to add an edge from vertex
# source to vertex dest
def addEdge(adj, src, dest):
adj[src].append(dest)
# function to print adjacency list
# of a graph
def displayGraph(adj, v):
for i in range(v):
print(i, "--> ", end = "")
for j in range(len(adj[i])):
print(adj[i][j], end = " ")
print()
# function to get Transpose of a graph
# taking adjacency list of given graph
# and that of Transpose graph
def transposeGraph(adj, transpose, v):
# traverse the adjacency list of given
# graph and for each edge (u, v) add
# an edge (v, u) in the transpose graph's
# adjacency list
for i in range(v):
for j in range(len(adj[i])):
addEdge(transpose, adj[i][j], i)
# Driver Code
if __name__ == '__main__':
v = 5
adj = [[] for i in range(v)]
addEdge(adj, 0, 1)
addEdge(adj, 0, 4)
addEdge(adj, 0, 3)
addEdge(adj, 2, 0)
addEdge(adj, 3, 2)
addEdge(adj, 4, 1)
addEdge(adj, 4, 3)
# Finding transpose of graph represented
# by adjacency list adj[]
transpose = [[]for i in range(v)]
transposeGraph(adj, transpose, v)
# displaying adjacency list of
# transpose graph i.e. b
displayGraph(transpose, v)
# This code is contributed by PranchalK
// C# program to find the transpose of a graph
using System;
using System.Collections.Generic;
class Graph
{
// Total number of vertices
private static int vertices = 5;
// Find transpose of graph represented by adj
private static List<int>[] adj = new List<int>[vertices];
// Store the transpose of graph represented by tr
private static List<int>[] tr = new List<int>[vertices];
// Function to add an edge from source vertex u to
// destination vertex v, if choice is false the edge is added
// to adj otherwise the edge is added to tr
public static void addedge(int u, int v, bool choice)
{
if(!choice)
adj[u].Add(v);
else
tr[u].Add(v);
}
// Function to print the graph representation
public static void printGraph()
{
for(int i = 0; i < vertices; i++)
{
Console.Write(i + "--> ");
for(int j = 0; j < tr[i].Count; j++)
Console.Write(tr[i][j] + " ");
Console.WriteLine();
}
}
// Function to print the transpose of
// the graph represented as adj and store it in tr
public static void getTranspose()
{
// Traverse the graph and for each edge u, v
// in graph add the edge v, u in transpose
for(int i = 0; i < vertices; i++)
for(int j = 0; j < adj[i].Count; j++)
addedge(adj[i][j], i, true);
}
// Driver code
public static void Main(String[] args)
{
for(int i = 0; i < vertices; i++)
{
adj[i] = new List<int>();
tr[i] = new List<int>();
}
addedge(0, 1, false);
addedge(0, 4, false);
addedge(0, 3, false);
addedge(2, 0, false);
addedge(3, 2, false);
addedge(4, 1, false);
addedge(4, 3, false);
// Finding transpose of the graph
getTranspose();
// Printing the graph representation
printGraph();
}
}
// This code is contributed by Rajput-Ji
<script>
// Javascript program to find transpose of a graph.
// function to add an edge from vertex
// source to vertex dest
function addEdge(adj, src, dest) {
adj[src].push(dest)
}
// function to print adjacency list
// of a graph
function displayGraph(adj, v) {
for (let i = 0; i < v; i++) {
document.write(i + "--> ")
for (let j = 0; j < adj[i].length; j++) {
document.write(adj[i][j] + " ")
}
document.write("<br>")
}
}
// function to get Transpose of a graph
// taking adjacency list of given graph
// and that of Transpose graph
function transposeGraph(adj, transpose, v) {
// traverse the adjacency list of given
// graph and for each edge (u, v) add
// an edge (v, u) in the transpose graph's
// adjacency list
for (let i = 0; i < v; i++)
for (let j = 0; j < adj[i].length; j++)
addEdge(transpose, adj[i][j], i)
}
// Driver Code
let v = 5
let adj = new Array(v).fill(0).map(() => new Array())
addEdge(adj, 0, 1)
addEdge(adj, 0, 4)
addEdge(adj, 0, 3)
addEdge(adj, 2, 0)
addEdge(adj, 3, 2)
addEdge(adj, 4, 1)
addEdge(adj, 4, 3)
// Finding transpose of graph represented
// by adjacency list adj[]
let transpose = new Array(v).fill(0).map(() => new Array())
transposeGraph(adj, transpose, v)
// displaying adjacency list of
// transpose graph i.e. b
displayGraph(transpose, v)
// This code is contributed by Saurabh Jaiswal
</script>
Output0--> 2
1--> 0 4
2--> 3
3--> 0 4
4--> 0
Time Complexity:
The time complexity of the addEdge function is O(1), as it simply appends an element to the vector.
The time complexity of the displayGraph function is O(V + E), where V is the number of vertices and E is the number of edges, as it needs to traverse the adjacency list of each vertex and print out the adjacent vertices.
The time complexity of the transposeGraph function is also O(V + E), where V is the number of vertices and E is the number of edges, as it needs to traverse the adjacency list of each vertex and add the corresponding edges to the transpose graph's adjacency list.
Therefore, the overall time complexity of the program is O(V + E).
Space complexity:
In terms of space complexity, the program uses two arrays of vectors to represent the original graph and its transpose, each of which has a size of V (the number of vertices). Additionally, the program uses a constant amount of space to store integer variables and temporary data structures. Therefore, the space complexity of the program is O(V).
Note that the space complexity of the program could be larger if the input graph has a large number of edges, as this would require more memory to store the adjacency lists.
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