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++
// C++ Implementation for random
// tree generator using Prufer Sequence
#include<bits/stdc++.h>
using namespace std;
// Prints edges of tree
// represented by give Prufer code
void printTreeEdges(vector<int> prufer, int m)
{
int vertices = m + 2;
vector<int> vertex_set(vertices);
// Initialize the array of vertices
for (int i = 0; i < vertices; i++)
vertex_set[i] = 0;
// Number of occurrences of vertex in code
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;
// Find the smallest label not present in
// prufer[].
for (int i = 0; i < vertices - 2; i++)
{
for (j = 0; j < vertices; j++)
{
// If j+1 is not present in prufer set
if (vertex_set[j] == 0)
{
// Remove from Prufer set and print
// pair.
vertex_set[j] = -1;
cout<<"(" << (j + 1) << ", "
<< prufer[i] << ") ";
vertex_set[prufer[i] - 1]--;
break;
}
}
}
j = 0;
// For the last element
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";
}
}
// generate random numbers in between l an r
int ran(int l, int r)
{
return l + (rand() % (r - l + 1));
}
// Function to Generate Random Tree
void generateRandomTree(int n)
{
int length = n - 2;
vector<int> arr(length);
// Loop to Generate Random Array
for (int i = 0; i < length; i++)
{
arr[i] = ran(0, length + 1) + 1;
}
printTreeEdges(arr, length);
}
// Driver Code
int main()
{
srand(time(0));
int n = 5;
generateRandomTree(n);
return 0;
}
// Java Implementation for random
// tree generator using Prufer Sequence
import java.util.Arrays;
import java.util.Random;
class GFG {
// Prints edges of tree
// represented by give Prufer code
static void printTreeEdges(int prufer[], int m)
{
int vertices = m + 2;
int vertex_set[] = new int[vertices];
// Initialize the array of vertices
for (int i = 0; i < vertices; i++)
vertex_set[i] = 0;
// Number of occurrences of vertex in code
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;
// Find the smallest label not present in
// prufer[].
for (int i = 0; i < vertices - 2; i++) {
for (j = 0; j < vertices; j++) {
// If j+1 is not present in prufer set
if (vertex_set[j] == 0) {
// Remove from Prufer set and print
// pair.
vertex_set[j] = -1;
System.out.print("(" + (j + 1) + ", "
+ prufer[i] + ") ");
vertex_set[prufer[i] - 1]--;
break;
}
}
}
j = 0;
// For the last element
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");
}
}
// Function to Generate Random Tree
static void generateRandomTree(int n)
{
Random rand = new Random();
int length = n - 2;
int[] arr = new int[length];
// Loop to Generate Random Array
for (int i = 0; i < length; i++) {
arr[i] = rand.nextInt(length + 1) + 1;
}
printTreeEdges(arr, length);
}
// Driver Code
public static void main(String[] args)
{
int n = 5;
generateRandomTree(n);
}
}
import random
def print_tree_edges(prufer, m):
vertices = m + 2
vertex_set = [0] * vertices
# Initialize the array of vertices
for i in range(vertices):
vertex_set[i] = 0
# Number of occurrences of vertex in code
for i in range(vertices - 2):
vertex_set[prufer[i] - 1] += 1
print("\nThe edge set E(G) is:")
j = 0
# Find the smallest label not present in prufer[].
for i in range(vertices - 2):
for j in range(vertices):
# If j+1 is not present in prufer set
if vertex_set[j] == 0:
# Remove from Prufer set and print pair.
vertex_set[j] = -1
print("({}, {})".format(j + 1, prufer[i]), end=" ")
vertex_set[prufer[i] - 1] -= 1
break
j = 0
# For the last element
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
# Generate random array
for i in range(length):
arr[i] = random.randint(1, length + 1)
print_tree_edges(arr, length)
# Driver code
n = 5
generate_random_tree(n)
// C# Implementation for random
// tree generator using Prufer Sequence
using System;
class GFG
{
// Prints edges of tree
// represented by give Prufer code
static void printTreeEdges(int []prufer, int m)
{
int vertices = m + 2;
int []vertex_set = new int[vertices];
// Initialize the array of vertices
for (int i = 0; i < vertices; i++)
vertex_set[i] = 0;
// Number of occurrences of vertex in code
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;
// Find the smallest label not present in
// prufer[].
for (int i = 0; i < vertices - 2; i++)
{
for (j = 0; j < vertices; j++)
{
// If j + 1 is not present in prufer set
if (vertex_set[j] == 0)
{
// Remove from Prufer set and print
// pair.
vertex_set[j] = -1;
Console.Write("(" + (j + 1) + ", "
+ prufer[i] + ") ");
vertex_set[prufer[i] - 1]--;
break;
}
}
}
j = 0;
// For the last element
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");
}
}
// Function to Generate Random Tree
static void generateRandomTree(int n)
{
Random rand = new Random();
int length = n - 2;
int[] arr = new int[length];
// Loop to Generate Random Array
for (int i = 0; i < length; i++)
{
arr[i] = rand.Next(length + 1) + 1;
}
printTreeEdges(arr, length);
}
// Driver Code
public static void Main(String[] args)
{
int n = 5;
generateRandomTree(n);
}
}
// This code is contributed by 29AjayKumar
<script>
// Javascript Implementation for random
// tree generator using Prufer Sequence
// Prints edges of tree
// represented by give Prufer code
function printTreeEdges(prufer,m)
{
let vertices = m + 2;
let vertex_set = new Array(vertices);
// Initialize the array of vertices
for (let i = 0; i < vertices; i++)
vertex_set[i] = 0;
// Number of occurrences of vertex in code
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;
// Find the smallest label not present in
// prufer[].
for (let i = 0; i < vertices - 2; i++) {
for (j = 0; j < vertices; j++) {
// If j+1 is not present in prufer set
if (vertex_set[j] == 0) {
// Remove from Prufer set and print
// pair.
vertex_set[j] = -1;
document.write("(" + (j + 1) + ", "
+ prufer[i] + ") ");
vertex_set[prufer[i] - 1]--;
break;
}
}
}
j = 0;
// For the last element
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 to Generate Random Tree
function generateRandomTree(n)
{
let length = n - 2;
let arr = new Array(length);
// Loop to Generate Random Array
for (let i = 0; i < length; i++) {
arr[i] = Math.floor(Math.random()*(length + 1)) + 1;
}
printTreeEdges(arr, length);
}
// Driver Code
let n = 5;
generateRandomTree(n);
// This code is contributed by unknown2108
</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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