由 K 的幂组成的序列的第 N 个子集，按其总和的递增顺序排列

给定两个整数N和K ，任务是从由 K 的幂生成的子集序列中找到第N^个子集，即 {1, K ¹ , K ² , K ³ , …..} 使得子集被排列按照它们总和的递增顺序，任务是从序列中找到第N^个子集。

例子：

Input: N = 5, K = 3 
Output: 1 9 
Explanation: 
The sequence of subsets along with their sum are:

• Subset = {1}, Sum = 1
• Subset = {3}, Sum = 3
• Subset = {1, 3}, Sum = 4
• Subset = {9}, Sum = 9
• Subset = {1, 9}, Sum = 10

Therefore, the subset at position 5 is {1, 9}.

Input: N = 4, K = 4 
Output: 16

方法：
让我们参考下面给出的 K = 3 所需的序列：

从上面的序列可以看出，子集{3}的位置为2，子集{9}的位置为4，子集{27}的位置为8，以此类推。子集{1, 3}、{1, 9}、{1, 27}分别占据位置3、5和9。因此，可以通过找到小于或等于N的最接近的 2 的幂来获得所需的^第N^个子集的所有元素。

Illustration: 
N = 6, K = 3
1st iteration:

1. p = log₂(6) = 2
2. 3² = 9, Subset = {9}
3. N = 6 % 4 = 2

2nd iteration:

1. p = log₂(2) = 1
2. 3¹ = 3, Subset = {3, 9}
3. N = 2 % 2 = 0

Therefore, the required subset is {3, 9}

请按照以下步骤解决问题：

• 计算小于或等于N的最接近的2幂，例如p 。因此， p = log ₂ N 。
• 现在，子集的元素将是 K ^p 。将其插入到子集的前面。
• 更新N到N％2^页。
• 重复上述步骤直到N变为0，然后打印得到的子集。

下面是上述方法的实现：

C++

// C++ program for the above approach
#include 
#include 
using namespace std;
#define lli long long int
 
// Function to print the
// required N-th subset
void printSubset(lli n, int k)
{
    vector answer;
    while(n > 0)
    {
 
        // Nearest power of 2<=N
        lli p = log2(n);
         
        // Now insert k^p in the answer
        answer.push_back(pow(k, p));
         
        // update n
        n %= (int)pow(2, p);
    }
 
    // Print the ans in sorted order
    reverse(answer.begin(), answer.end());
    for(auto x: answer)
    {
        cout << x << " ";
    }
}
 
// Driver Code
int main()
{
    lli n = 5;
    int k = 4;
    printSubset(n, k);
}
 
// This code is contributed by winter_soldier

Java

// Java program for above approach
import java.util.*;
import java.lang.*;
import java.io.*;
class GFG
{
 
  // Function to print the 
  // required N-th subset 
  static void printSubset(long n, int k)
  {
    ArrayList answer = new ArrayList<>();
    while(n > 0)
    {
 
      // Nearest power of 2<=N
      long p = (long)(Math.log(n) / Math.log(2));;
 
      // Now insert k^p in the answer
      answer.add((long)(Math.pow(k, p)));
 
      // update n
      n %= (int)Math.pow(2, p);
    }
 
    // Print the ans in sorted order
    Collections.sort(answer);
    for(Long x: answer)
    {
      System.out.print(x + " ");
    }
  }
 
  // Driver function
  public static void main (String[] args)
  {
    long n = 5;
    int k = 4;
    printSubset(n, k);
  }
}
 
// This code is contributed by offbeat

Python3

# Python3 program for
# the above approach
import math
 
# Function to print the
# required N-th subset
def printSubset(N, K):
    # Stores the subset
    answer = ""
    while(N > 0):
        # Nearest power of 2 <= N
        p = int(math.log(N, 2))
        # Insert K ^ p in the subset
        answer = str(K**p)+" "+answer
        # Update N
        N = N % (2**p)
         
    # Print the subset
    print(answer)
     
# Driver Code
N = 5
K = 4
printSubset(N, K)

C#

// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG {
 
  // Function to print the
  // required N-th subset
  static void printSubset(int n, int k)
  {
    List answer = new List();
    while(n > 0)
    {
 
      // Nearest power of 2<=N
      int p = (int)Math.Log(n,2);
 
      // Now insert k^p in the answer
      answer.Add((int)Math.Pow(k, p));
 
      // update n
      n %= (int)Math.Pow(2, p);
    }
 
    // Print the ans in sorted order
    answer.Reverse();
    foreach(int x in answer)
    {
      Console.Write(x + " ");
    }
  }
 
  // Driver code
  static void Main() {
    int n = 5;
    int k = 4;
    printSubset(n, k);
  }
}
 
// This code is contributed by divyeshrabadiya07.

Javascript

// Javascript program for the above approach
 
// Function to print the
// required N-th subset
function printSubset(n, k)
{
    var answer = [];
    while(n > 0)
    {
 
        // Nearest power of 2<=N
        var p = parseInt(Math.log2(n));
         
        // Now insert k^p in the answer
        answer.push(Math.pow(k, p));
         
        // update n
        n %= parseInt(Math.pow(2, p));
    }
 
    // Print the ans in sorted order
    answer.sort();
    //reverse(answer.begin(), answer.end());
    for(var i=0;i<answer.length;i++)
    {
        document.write(answer[i] + " ");
    }
}
 
var n = 5;
var k = 4;
printSubset(n, k);
 
//This code is contributed by SoumikMondal

输出

1 16 

时间复杂度： O(logN)
辅助空间： O(1)

方法：

• 将计数和 x 初始化为 0。此外，还有一个向量来存储子集的元素。
• 当 n 大于 0 时执行以下操作。
  • 设置 x = n & 1，用于查找数字的最后一位是否已设置。
  • 现在，如果 n 不为 0，则将元素 3^计数推入子集。
  • 借助右移 1 个单位将数字 n 减少 2。
  • 将计数值增加 1。
• 最后，数组中的元素是第 N 个子集的元素。

下面是上述方法的实现：

C++

// C++ program to print subset
// at the nth position ordered
// by the sum of the elements
#include 
using namespace std;
 
// Function to print the elements of
// the subset at pos n
void printsubset(int n,int k)
{
    //  Initialize count=0 and x=0
    int count = 0, x = 0;
   
    // create a vector for
    // storing the elements
    // of subsets
    vector vec;
   
    // doing until all the
    // set bits of n are used
    while (n) {
        x = n & 1;
       
        // this part is executed only
        // when the last bit is
        // set
        if (x) {
            vec.push_back(pow(k, count));
        }
       
        // right shift the bit by one position
        n = n >> 1;
       
        // increasing the count each time by one
        count++;
    }
   
    // printing the values os elements
    for (int i = 0; i < vec.size(); i++)
        cout << vec[i] << " ";
}
 
// Driver Code
int main()
{
    int n = 7,k=4;
    printsubset(n,k);
    return 0;
}
 
// This code is contributed by shivkant

Java

// Java program to print subset
// at the nth position ordered
// by the sum of the elements
import java.util.*;
import java.lang.*;
class GFG{
   
// Function to print the
// elements of the subset
// at pos n
static void printsubset(int n,
                        int k)
{
  // Initialize count=0 and x=0
  int count = 0, x = 0;
 
  // Create a vector for
  // storing the elements
  // of subsets
  ArrayList vec =
            new ArrayList<>();
 
  // Doing until all the
  // set bits of n are used
  while (n != 0)
  {
    x = n & 1;
 
    // This part is executed only
    // when the last bit is
    // set
    if (x != 0)
    {
      vec.add((int)Math.pow(k,
                            count));
    }
 
    // Right shift the bit
    // by one position
    n = n >> 1;
 
    // Increasing the count
    // each time by one
    count++;
  }
 
  // Printing the values os elements
  for (int i = 0; i < vec.size(); i++)
    System.out.print(vec.get(i) + " ");
}
 
// Driver function
public static void main (String[] args)
{
  int n = 7, k = 4;
  printsubset(n, k);
}
}
 
// This code is contributed by offbeat

蟒蛇3

# Python3 program to print subset
# at the nth position ordered
# by the sum of the elements
import math
 
# Function to print the elements of
# the subset at pos n
def printsubset(n, k):
     
    # Initialize count=0 and x=0
    count = 0
    x = 0
 
    # Create a vector for
    # storing the elements
    # of subsets
    vec = []
 
    # Doing until all the
    # set bits of n are used
    while (n > 0):
        x = n & 1
         
        # This part is executed only
        # when the last bit is
        # set
        if (x):
            vec.append(pow(k, count))
     
        # Right shift the bit by one position
        n = n >> 1
     
        # Increasing the count each time by one
        count += 1
 
    # Printing the values os elements
    for item in vec:
        print(item, end = " ")
 
# Driver Code
n = 7
k = 4
 
printsubset(n, k)
 
# This code is contributed by Stream_Cipher

C＃

// C# program to print subset
// at the nth position ordered
// by the sum of the elements
using System.Collections.Generic;
using System;
 
class GFG{
 
// Function to print the
// elements of the subset
// at pos n
static void printsubset(int n, int k)
{
     
    // Initialize count=0 and x=0
    int count = 0, x = 0;
     
    // Create a vector for
    // storing the elements
    // of subsets
    List vec = new List();
     
    // Doing until all the
    // set bits of n are used
    while (n != 0)
    {
        x = n & 1;
     
        // This part is executed only
        // when the last bit is
        // set
        if (x != 0)
        {
            vec.Add((int)Math.Pow(k, count));
        }
     
        // Right shift the bit
        // by one position
        n = n >> 1;
     
        // Increasing the count
        // each time by one
        count++;
    }
     
    // Printing the values os elements
    for(int i = 0; i < vec.Count; i++)
        Console.Write(vec[i] + " ");
}
 
// Driver code
public static void Main ()
{
    int n = 7, k = 4;
     
    printsubset(n, k);
}
}
 
// This code is contributed by Stream_Cipher

Javascript

    // Javascript program to print subset
    // at the nth position ordered
    // by the sum of the elements
     
    // Function to print the
    // elements of the subset
    // at pos n
    function printsubset(n, k)
    {
 
        // Initialize count=0 and x=0
        let count = 0, x = 0;
 
        // Create a vector for
        // storing the elements
        // of subsets
        let vec = [];
 
        // Doing until all the
        // set bits of n are used
        while (n != 0)
        {
            x = n & 1;
 
            // This part is executed only
            // when the last bit is
            // set
            if (x != 0)
            {
                vec.push(Math.pow(k, count));
            }
 
            // Right shift the bit
            // by one position
            n = n >> 1;
 
            // Increasing the count
            // each time by one
            count++;
        }
 
        // Printing the values os elements
        for(let i = 0; i < vec.length; i++)
            document.write(vec[i] + " ");
    }
     
    let n = 7, k = 4;
      
    printsubset(n, k);
 

输出

1 4 16