Random Tree Generator Using Prüfer Sequence with Examples
Last Updated :
10 Jan, 2023
Given an integer N, the task is to generate a random labelled tree of N node with (N – 1) edges without forming cycle.
Note: The output generated below is random which may not match with the output generated by the code.
Examples:
Input: N = 3
Output:
1 3
1 2
Input: N = 5
Output:
3 2
4 3
1 4
1 5
This approach uses the Prüfer Sequence to generate random trees.
What is a Prüfer Sequence?
In combinatorial mathematics, the Prüfer sequence (also Prüfer code or Prüfer numbers) of a labelled tree is a unique sequence associated with the tree. The sequence for a tree on n vertices has length n – 2 and can be generated by a simple iterative algorithm.
If the number of nodes is N then the Prüfer Sequence is of length (N – 2) and each position can have N possible values. So the number of the possible labeled trees with N Nodes is N(N – 2).
How Random Trees are generated using Prüfer Sequence?
Generally, Random Tree Generation with N nodes is done in the following steps:
- Generate a Random Sequence
S = {s1, s2, s3.....sn-2}
where each element of the sequence si ? {1, 2, 3, … N} where repetition of elements is allowed
- Generate Tree from the generated Prüfer Sequence S:
- Create N nodes with values {1, 2, 3, … N}
- Find smallest element X such that X ? {1, 2, 3, … N} and X ? S
- Join Node with value X to the node with value s1
- Delete s1 from S
- Repeat the same process from step 2 with until the Prüfer Sequence is empty.
For Example:
- Number of Nodes = 3
- Then the Prüfer Sequence will be of length (N – 2), as in this case it will be of 1 and the possible values it can have {1, 2, 3}.
- Possible Random Sequences will be {{1}, {2}, {3}}.
Below is the implementation of the above approach.
C++
#include<bits/stdc++.h>
using namespace std;
void printTreeEdges(vector< int > prufer, int m)
{
int vertices = m + 2;
vector< int > vertex_set(vertices);
for ( int i = 0; i < vertices; i++)
vertex_set[i] = 0;
for ( int i = 0; i < vertices - 2; i++)
vertex_set[prufer[i] - 1] += 1;
cout<<( "\nThe edge set E(G) is:\n" );
int j = 0;
for ( int i = 0; i < vertices - 2; i++)
{
for (j = 0; j < vertices; j++)
{
if (vertex_set[j] == 0)
{
vertex_set[j] = -1;
cout<< "(" << (j + 1) << ", "
<< prufer[i] << ") " ;
vertex_set[prufer[i] - 1]--;
break ;
}
}
}
j = 0;
for ( int i = 0; i < vertices; i++)
{
if (vertex_set[i] == 0 && j == 0)
{
cout << "(" << (i + 1) << ", " ;
j++;
}
else if (vertex_set[i] == 0 && j == 1)
cout << (i + 1) << ")\n" ;
}
}
int ran( int l, int r)
{
return l + ( rand () % (r - l + 1));
}
void generateRandomTree( int n)
{
int length = n - 2;
vector< int > arr(length);
for ( int i = 0; i < length; i++)
{
arr[i] = ran(0, length + 1) + 1;
}
printTreeEdges(arr, length);
}
int main()
{
srand ( time (0));
int n = 5;
generateRandomTree(n);
return 0;
}
|
Java
import java.util.Arrays;
import java.util.Random;
class GFG {
static void printTreeEdges( int prufer[], int m)
{
int vertices = m + 2 ;
int vertex_set[] = new int [vertices];
for ( int i = 0 ; i < vertices; i++)
vertex_set[i] = 0 ;
for ( int i = 0 ; i < vertices - 2 ; i++)
vertex_set[prufer[i] - 1 ] += 1 ;
System.out.print( "\nThe edge set E(G) is:\n" );
int j = 0 ;
for ( int i = 0 ; i < vertices - 2 ; i++) {
for (j = 0 ; j < vertices; j++) {
if (vertex_set[j] == 0 ) {
vertex_set[j] = - 1 ;
System.out.print( "(" + (j + 1 ) + ", "
+ prufer[i] + ") " );
vertex_set[prufer[i] - 1 ]--;
break ;
}
}
}
j = 0 ;
for ( int i = 0 ; i < vertices; i++) {
if (vertex_set[i] == 0 && j == 0 ) {
System.out.print( "(" + (i + 1 ) + ", " );
j++;
}
else if (vertex_set[i] == 0 && j == 1 )
System.out.print((i + 1 ) + ")\n" );
}
}
static void generateRandomTree( int n)
{
Random rand = new Random();
int length = n - 2 ;
int [] arr = new int [length];
for ( int i = 0 ; i < length; i++) {
arr[i] = rand.nextInt(length + 1 ) + 1 ;
}
printTreeEdges(arr, length);
}
public static void main(String[] args)
{
int n = 5 ;
generateRandomTree(n);
}
}
|
Python3
import random
def print_tree_edges(prufer, m):
vertices = m + 2
vertex_set = [ 0 ] * vertices
for i in range (vertices):
vertex_set[i] = 0
for i in range (vertices - 2 ):
vertex_set[prufer[i] - 1 ] + = 1
print ( "\nThe edge set E(G) is:" )
j = 0
for i in range (vertices - 2 ):
for j in range (vertices):
if vertex_set[j] = = 0 :
vertex_set[j] = - 1
print ( "({}, {})" . format (j + 1 , prufer[i]), end = " " )
vertex_set[prufer[i] - 1 ] - = 1
break
j = 0
for i in range (vertices):
if vertex_set[i] = = 0 and j = = 0 :
print ( "({}, " . format (i + 1 ), end = "")
j + = 1
elif vertex_set[i] = = 0 and j = = 1 :
print ( "{})" . format (i + 1 ))
def generate_random_tree(n):
length = n - 2
arr = [ 0 ] * length
for i in range (length):
arr[i] = random.randint( 1 , length + 1 )
print_tree_edges(arr, length)
n = 5
generate_random_tree(n)
|
C#
using System;
class GFG
{
static void printTreeEdges( int []prufer, int m)
{
int vertices = m + 2;
int []vertex_set = new int [vertices];
for ( int i = 0; i < vertices; i++)
vertex_set[i] = 0;
for ( int i = 0; i < vertices - 2; i++)
vertex_set[prufer[i] - 1] += 1;
Console.Write( "\nThe edge set E(G) is:\n" );
int j = 0;
for ( int i = 0; i < vertices - 2; i++)
{
for (j = 0; j < vertices; j++)
{
if (vertex_set[j] == 0)
{
vertex_set[j] = -1;
Console.Write( "(" + (j + 1) + ", "
+ prufer[i] + ") " );
vertex_set[prufer[i] - 1]--;
break ;
}
}
}
j = 0;
for ( int i = 0; i < vertices; i++)
{
if (vertex_set[i] == 0 && j == 0)
{
Console.Write( "(" + (i + 1) + ", " );
j++;
}
else if (vertex_set[i] == 0 && j == 1)
Console.Write((i + 1) + ")\n" );
}
}
static void generateRandomTree( int n)
{
Random rand = new Random();
int length = n - 2;
int [] arr = new int [length];
for ( int i = 0; i < length; i++)
{
arr[i] = rand.Next(length + 1) + 1;
}
printTreeEdges(arr, length);
}
public static void Main(String[] args)
{
int n = 5;
generateRandomTree(n);
}
}
|
Javascript
<script>
function printTreeEdges(prufer,m)
{
let vertices = m + 2;
let vertex_set = new Array(vertices);
for (let i = 0; i < vertices; i++)
vertex_set[i] = 0;
for (let i = 0; i < vertices - 2; i++)
vertex_set[prufer[i] - 1] += 1;
document.write( "<br>The edge set E(G) is:<br>" );
let j = 0;
for (let i = 0; i < vertices - 2; i++) {
for (j = 0; j < vertices; j++) {
if (vertex_set[j] == 0) {
vertex_set[j] = -1;
document.write( "(" + (j + 1) + ", "
+ prufer[i] + ") " );
vertex_set[prufer[i] - 1]--;
break ;
}
}
}
j = 0;
for (let i = 0; i < vertices; i++) {
if (vertex_set[i] == 0 && j == 0) {
document.write( "(" + (i + 1) + ", " );
j++;
}
else if (vertex_set[i] == 0 && j == 1)
document.write((i + 1) + ")<br>" );
}
}
function generateRandomTree(n)
{
let length = n - 2;
let arr = new Array(length);
for (let i = 0; i < length; i++) {
arr[i] = Math.floor(Math.random()*(length + 1)) + 1;
}
printTreeEdges(arr, length);
}
let n = 5;
generateRandomTree(n);
</script>
|
Output:
The edge set E(G) is:
(2, 4) (4, 3) (3, 1) (1, 5)
Time Complexity: O(N*N)
Auxiliary Space: O(N)
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