Random Tree Generator Using Prüfer Sequence with Examples
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 untill 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(int prufer[], int m) { int vertices = m + 2; 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; int arr[length]; // Loop to Generate Random Array for (int i = 0; i < length; i++) { arr[i] = ran(0, pow(2, length + 1)) + 1; } printTreeEdges(arr, length); } // Driver Code int main() { srand(time(0)); int n = 5; generateRandomTree(n); return 0; } // This code is contributed by Arnab Kundu |
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Java
// 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); } } |
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C#
// 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 |
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Output:
The edge set E(G) is: (2, 4) (4, 3) (3, 1) (1, 5)
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