Kth ancestor of a node in an N-ary tree using Binary Lifting Technique

Given a vertex V of an N-ary tree and an integer K, the task is to print the Kth ancestor of the given vertex in the tree. If there does not exist any such ancestor then print -1.

Examples:

Input: K = 2, V = 4

Output: 1
2nd parent of vertex 4 is 1.

Input: K = 3, V = 4

Output: -1

Approach: The idea is to use Binary Lifting Technique. This technique is based on the fact that every integer can be represented in binary form. Through pre-processing, a sparse table table[v][i] can be calculated which stores the 2ith parent of the vertex v where 0 ≤ i ≤ log2N. This pre-processing takes O(NlogN) time.
To find the Kth parent of the vertex V, let K = b0b1b2…bn be an n bit number in the binary representation, let p1, p2, p3, …, pj be the indices where bit value is 1 then K can be represented as K = 2p1 + 2p2 + 2p3 + … + 2pj. Thus to reach Kth parent of V, we have to make jumps to 2pth1, 2pth2, 2pth3 upto 2pthj parent in any order. This can be done efficiently through the sparse table calculated earlier in O(logN).



Below is the implementation of the above approach:

C++

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// CPP implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Table for storing 2^ith parent
int **table;
  
// To store the height of the tree
int height;
  
// initializing the table and
// the height of the tree
void initialize(int n)
{
    height = (int)ceil(log2(n));
    table = new int *[n + 1];
}
  
// Filling with -1 as initial
void preprocessing(int n)
{
    for (int i = 0; i < n + 1; i++)
    {
        table[i] = new int[height + 1];
        memset(table[i], -1, sizeof table[i]);
    }
}
  
// Calculating sparse table[][] dynamically
void calculateSparse(int u, int v)
{
    // Using the recurrence relation to
    // calculate the values of table[][]
    table[v][0] = u;
    for (int i = 1; i <= height; i++)
    {
        table[v][i] = table[table[v][i - 1]][i - 1];
  
        // If we go out of bounds of the tree
        if (table[v][i] == -1)
            break;
    }
}
  
// Function to return the Kth ancestor of V
int kthancestor(int V, int k)
{
    // Doing bitwise operation to
    // check the set bit
    for (int i = 0; i <= height; i++)
    {
        if (k & (1 << i))
        {
            V = table[V][i];
            if (V == -1)
                break;
        }
    }
    return V;
}
  
// Driver Code
int main()
{
    // Number of vertices
    int n = 6;
  
    // initializing
    initialize(n);
  
    // Pre-processing
    preprocessing(n);
  
    // Calculating ancestors of v
    calculateSparse(1, 2);
    calculateSparse(1, 3);
    calculateSparse(2, 4);
    calculateSparse(2, 5);
    calculateSparse(3, 6);
  
    int K = 2, V = 5;
    cout << kthancestor(V, K) << endl;
  
    return 0;
}
  
// This code is contributed by
// sanjeev2552

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Java

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// Java implementation of the approach
import java.util.Arrays;
  
class GfG {
  
    // Table for storing 2^ith parent
    private static int table[][];
  
    // To store the height of the tree
    private static int height;
  
    // Private constructor for initializing
    // the table and the height of the tree
    private GfG(int n)
    {
  
        // log(n) with base 2
        height = (int)Math.ceil(Math.log10(n) / Math.log10(2));
        table = new int[n + 1][height + 1];
    }
  
    // Filling with -1 as initial
    private static void preprocessing()
    {
        for (int i = 0; i < table.length; i++) {
            Arrays.fill(table[i], -1);
        }
    }
  
    // Calculating sparse table[][] dynamically
    private static void calculateSparse(int u, int v)
    {
  
        // Using the recurrence relation to
        // calculate the values of table[][]
        table[v][0] = u;
        for (int i = 1; i <= height; i++) {
            table[v][i] = table[table[v][i - 1]][i - 1];
  
            // If we go out of bounds of the tree
            if (table[v][i] == -1)
                break;
        }
    }
  
    // Function to return the Kth ancestor of V
    private static int kthancestor(int V, int k)
    {
  
        // Doing bitwise operation to
        // check the set bit
        for (int i = 0; i <= height; i++) {
            if ((k & (1 << i)) != 0) {
                V = table[V][i];
                if (V == -1)
                    break;
            }
        }
        return V;
    }
  
    // Driver code
    public static void main(String args[])
    {
        // Number of vertices
        int n = 6;
  
        // Calling the constructor
        GfG obj = new GfG(n);
  
        // Pre-processing
        preprocessing();
  
        // Calculating ancestors of v
        calculateSparse(1, 2);
        calculateSparse(1, 3);
        calculateSparse(2, 4);
        calculateSparse(2, 5);
        calculateSparse(3, 6);
  
        int K = 2, V = 5;
        System.out.print(kthancestor(V, K));
    }
}

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Python3

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# Python3 implementation of the approach
import math
  
class GfG :
  
    # Private constructor for initializing
    # the table and the height of the tree
    def __init__(self, n):
      
        # log(n) with base 2
        # To store the height of the tree
        self.height = int(math.ceil(math.log10(n) / math.log10(2)))
          
        # Table for storing 2^ith parent
        self.table = [0] * (n + 1)
      
    # Filling with -1 as initial
    def preprocessing(self):
        i = 0
        while ( i < len(self.table)) :
            self.table[i] = [-1]*(self.height + 1)
            i = i + 1
          
    # Calculating sparse table[][] dynamically
    def calculateSparse(self, u, v):
      
        # Using the recurrence relation to
        # calculate the values of table[][]
        self.table[v][0] = u
        i = 1
        while ( i <= self.height) :
            self.table[v][i] = self.table[self.table[v][i - 1]][i - 1]
  
            # If we go out of bounds of the tree
            if (self.table[v][i] == -1):
                break
            i = i + 1
          
    # Function to return the Kth ancestor of V
    def kthancestor(self, V, k):
        i = 0
  
        # Doing bitwise operation to
        # check the set bit
        while ( i <= self.height) :
            if ((k & (1 << i)) != 0) :
                V = self.table[V][i]
                if (V == -1):
                    break
            i = i + 1
          
        return V
      
# Driver code
  
# Number of vertices
n = 6
  
# Calling the constructor
obj = GfG(n)
  
# Pre-processing
obj.preprocessing()
  
# Calculating ancestors of v
obj.calculateSparse(1, 2)
obj.calculateSparse(1, 3)
obj.calculateSparse(2, 4)
obj.calculateSparse(2, 5)
obj.calculateSparse(3, 6)
  
K = 2
V = 5
print(obj.kthancestor(V, K))
      
# This code is contributed by Arnab Kundu

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C#

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// C# implementation of the approach 
using System;
  
class GFG
{
      
    class GfG 
    
      
        // Table for storing 2^ith parent 
        private static int [,]table ; 
      
        // To store the height of the tree 
        private static int height; 
      
        // Private constructor for initializing 
        // the table and the height of the tree 
        private GfG(int n) 
        
      
            // log(n) with base 2 
            height = (int)Math.Ceiling(Math.Log10(n) / Math.Log10(2)); 
            table = new int[n + 1, height + 1]; 
        
      
        // Filling with -1 as initial 
        private static void preprocessing() 
        
            for (int i = 0; i < table.GetLength(0); i++)
            
                for (int j = 0; j < table.GetLength(1); j++)
                {
                    table[i, j] = -1; 
                }
            
        
      
        // Calculating sparse table[,] dynamically 
        private static void calculateSparse(int u, int v) 
        
      
            // Using the recurrence relation to 
            // calculate the values of table[,] 
            table[v, 0] = u; 
            for (int i = 1; i <= height; i++)
            
                table[v, i] = table[table[v, i - 1], i - 1]; 
      
                // If we go out of bounds of the tree 
                if (table[v, i] == -1) 
                    break
            
        
      
        // Function to return the Kth ancestor of V 
        private static int kthancestor(int V, int k) 
        
      
            // Doing bitwise operation to 
            // check the set bit 
            for (int i = 0; i <= height; i++) 
            
                if ((k & (1 << i)) != 0)
                
                    V = table[V, i]; 
                    if (V == -1) 
                        break
                
            
            return V; 
        
      
        // Driver code 
        public static void Main() 
        
            // Number of vertices 
            int n = 6; 
      
            // Calling the constructor 
            GfG obj = new GfG(n); 
      
            // Pre-processing 
            preprocessing(); 
      
            // Calculating ancestors of v 
            calculateSparse(1, 2); 
            calculateSparse(1, 3); 
            calculateSparse(2, 4); 
            calculateSparse(2, 5); 
            calculateSparse(3, 6); 
      
            int K = 2, V = 5; 
            Console.Write(kthancestor(V, K)); 
        
    
}
  
// This code is contributed by AnkitRai01

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Output:

1

Time Complexity: O(NlogN) for pre-processing and logN for finding the ancestor.

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