使用二元提升技术的 N 叉树中节点的第 K 个祖先

给定 N 元树的顶点V和整数K ，任务是打印树中给定顶点的^第K^个祖先。如果不存在任何这样的祖先，则打印-1 。
例子：

Input: K = 2, V = 4 
 

Output: 1 
2^nd parent of vertex 4 is 1.
Input: K = 3, V = 4 
 

Output: -1 
 

方法：这个想法是使用二元提升技术。这种技术基于这样一个事实，即每个整数都可以用二进制形式表示。通过预处理，可以计算出一个稀疏表table[v][i] ，其中存储了顶点v 的^第2^个父节点，其中0 ≤ i ≤ log ₂ N 。这个预处理需要O(NlogN)时间。
为了找到顶点V 的^第K^个父节点，让K = b ₀ b ₁ b ₂ …b _n是二进制表示中的n位数字，让p ₁ , p ₂ , p ₃ , …, p _j是索引其中位值为1，则K可以表示为K = 2 ^{p ₁} + 2 ^{p ₂} + 2 ^{p ₃} + … + 2 ^{p _j} 。因此，为了到达V 的^第K^个父节点，我们必须以任意顺序跳转到2 ^{pth ₁} 、 2 ^{pth ₂} 、 2 ^{pth ₃}到2 ^{pth _j}父节点。这可以通过之前在O(logN)中计算的稀疏表有效地完成。
下面是上述方法的实现：

C++

// CPP implementation of the approach
#include 
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

Java

// 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));
    }
}

Python3

# 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

C#

// 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

Javascript

    // Javascript implementation of the approach
     
    // Table for storing 2^ith parent
    let table;
   
    // To store the height of the tree
    let height;
       
    // initializing the table and
    // the height of the tree
    function initialize(n)
    {
        height = Math.ceil(Math.log2(n));
    }
   
    // Filling with -1 as initial
    function preprocessing()
    {
        table = new Array(n + 1);
        for (let i = 0; i < n + 1; i++) {
            table[i] = new Array(height + 1);
            for (let j = 0; j < height + 1; j++) {
                table[i][j] = -1;
            }
        }
    }
   
    // Calculating sparse table[][] dynamically
    function calculateSparse(u, v)
    {
   
        // Using the recurrence relation to
        // calculate the values of table[][]
        table[v][0] = u;
        for (let 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
    function kthancestor(V, k)
    {
   
        // Doing bitwise operation to
        // check the set bit
        for (let i = 0; i <= height; i++) {
            if ((k & (1 << i)) != 0) {
                V = table[V][i];
                if (V == -1)
                    break;
            }
        }
        return V;
    }
     
    // Number of vertices
    let n = 6;
 
    // Calling the constructor
    initialize(n);
 
    // Pre-processing
    preprocessing();
 
    // Calculating ancestors of v
    calculateSparse(1, 2);
    calculateSparse(1, 3);
    calculateSparse(2, 4);
    calculateSparse(2, 5);
    calculateSparse(3, 6);
 
    let K = 2, V = 5;
    document.write(kthancestor(V, K));
 
// This code is contributed by divyeshrabadiya07.

输出：

1

时间复杂度： O(NlogN) 用于预处理和 logN 用于查找祖先。

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