dsal 元素数量，它与整数相乘使数组中的每对成为完全平方

元素数量，它与整数相乘使数组中的每对成为完全平方

原文：https://www.geeksforgeeks.org/count-of-elements-to-be-multiplied-with-integers-to-make-each-pair-of-array-a-perfect-square/

给定包含正整数的数组arr[]，任务是找到要在数组上执行的最小操作数，以使数组的每个数成为超级幂。

在每个操作中，我们可以将数组的任何元素乘以整数。

超级幂定义为数组中的一个数字，当乘以数组中除自身以外的任何其他数字时，就生成一个完美的平方。

示例：

输入：arr[] = {2, 2, 2}

输出：0

说明：

我们不需要在数组上执行任何操作，因为选择任何元素（2）并将其与其他索引（2）中的任何元素相乘，我们得到一个完美的平方（4）。

输入：arr[] = {2, 4, 6}

输出：2

说明：

我们需要执行以下两个操作：

首先，我们将索引 1 处的元素与整数 2 相乘。数组变为{2, 8, 6}。

其次，我们将索引 2 的元素与整数 3 相乘。数组变为{2, 8, 18}。

现在，所有要素都已成为超级大国。数组中任何两个数字相乘得到一个完美的平方。

即2 * 8 = 16、2 * 16 = 36和8 * 18 = 144。

方法：由于我们需要检查所有数字的质因数，所以我们的想法是首先预先计算所有数字的唯一质因数并将其存储在哈希映射中。然后，我们创建一个变量来存储质因数在数组的每个元素中出现的次数。我们还需要观察到，我们需要找到转换数组的最小步骤数。因此，我们计算向量中的奇数个数和偶数质数的个数，取最小者为准。可以计算以下步骤来找到答案：

我们首先需要计算spf[]数组。这是存储所有元素的最小质因数的数组。
然后，我们需要找到唯一的主要因素并将其存储在哈希映射中。
现在，遍历哈希映射以找到一个数的唯一质数，然后遍历给定数组中的元素以计算质数的频率。
现在，初始化两个变量，它们存储出现次数为偶数的质数的频率，另一个变量存储为奇数的质数的频率。
现在，我们需要在最终变量中添加两个变量中的最小值，这是为该质数获得超能力的最小运算。
对数组中的所有数字重复上述步骤。

下面是上述方法的实现：

C++

// C++ program to find the minimum 
// number of steps to modify the 
// array such that the product 
// of any two numbers in the 
// array is a perfect square 

#include <iostream> 
#include <unordered_map> 
#include <vector> 
using namespace std; 

// Function to find the smallest 
// prime factor of the elements 
void spf_array(int spf[]) 
{ 

    // Initializing the first element 
    // of the array 
    spf[1] = 1; 

    // Loop to add the remaining 
    // elements to the array 
    for (int i = 2; i < 1000; i++) 

        // Marking the smallest prime 
        // factor for every 
        // number to be itself 
        spf[i] = i; 

    // Separately marking spf for 
    // every even number as 2 
    for (int i = 4; i < 1000; i += 2) 
        spf[i] = 2; 

    for (int i = 3; i * i < 1000; i++) { 

        // Checking if i is prime 
        if (spf[i] == i) { 

            // Marking SPF for all the 
            // numbers divisible by i 
            for (int j = i * i; j < 1000; j += i) 

                // Marking spf[j] if it is not 
                // previously marked 
                if (spf[j] == j) 
                    spf[j] = i; 
        } 
    } 
} 

// Function to find the minimum 
// number of steps to modify the 
// array such that the product 
// of any two numbers in the 
// array is a perfect square 
int minimum_operation(int b[], int d, 
                    int spf[]) 
{ 

    // Map created to store 
    // the unique prime numbers 
    unordered_map<int, int> m; 
    int i = 0; 

    // Variable to store the 
    // minimum number of operations 
    int c = 0; 

    // Loop to store every 
    // unique prime number 
    for (i = 0; i < d; i++) { 
        int x = b[i]; 
        while (x != 1) { 
            x = x / spf[x]; 
            if (m[spf[x]] == 0) { 
                m[spf[x]] = 1; 
            } 
        } 
    } 

    // Erasing 1 as a key because 
    // it is not a prime number 
    m.erase(1); 

    // Iterating through the hash 
    for (auto x : m) { 

        // Two variables used for 
        // counting the frequency 
        // of prime is even or odd 
        int e = 0, o = 0; 

        // First prime number 
        int j = x.first; 

        // Iterating the number D 
        for (i = 0; i < d; i++) { 

            // check if prime is a 
            // factor of the element 
            // in the array 
            if (b[i] % j == 0) { 
                int h = 0; 
                int g = b[i]; 

                // Loop for calculating the 
                // frequency of the element 
                while (g != 0) { 
                    if (g % j != 0) { 
                        break; 
                    } 
                    g = g / j; 
                    h = h + 1; 
                } 

                // Check for frequency 
                // odd or even 
                if (h % 2 == 0) { 
                    e = e + 1; 
                } 
                else { 
                    o = o + 1; 
                } 
            } 
            else { 

                // If it is not a factor of the 
                // element, then it is automatically 
                // even 
                e = e + 1; 
            } 
        } 

        // Storing the minimum of two variable 
        // even or odd 
        c = c + min(o, e); 
    } 
    return c; 
} 

// Driver code 
int main() 
{ 
    int spf[1001]; 

    // Input array 
    int b[] = { 1, 4, 6 }; 

    // Creating shortest prime 
    // factorisation array 
    int d = sizeof(b) / sizeof(b[0]); 
    spf_array(spf); 

    cout << minimum_operation(b, d, spf) 
        << endl; 
}

Java

// Java program for the above approach
import java.io.*;
import java.util.*;

class GFG{

// Function to find the smallest
// prime factor of the elements
static void spf_array(int spf[])
{

    // Initializing the first element
    // of the array
    spf[1] = 1;

    // Loop to add the remaining
    // elements to the array
    for(int i = 2; i < 1000; i++)

        // Marking the smallest prime
        // factor for every
        // number to be itself
        spf[i] = i;

    // Separately marking spf for
    // every even number as 2
    for(int i = 4; i < 1000; i += 2)
        spf[i] = 2;

    for(int i = 3; i * i < 1000; i++) 
    {

        // Checking if i is prime
        if (spf[i] == i)
        {

            // Marking SPF for all the
            // numbers divisible by i
            for(int j = i * i; j < 1000; j += i)

                // Marking spf[j] if it is not
                // previously marked
                if (spf[j] == j)
                    spf[j] = i;
        }
    }
}

// Function to find the minimum
// number of steps to modify the
// array such that the product
// of any two numbers in the
// array is a perfect square
static int minimum_operation(int b[], int d,
                             int spf[])
{

    // Map created to store
    // the unique prime numbers
    Map<Integer, Integer> m=new HashMap<>();
    int i = 0;

    // Variable to store the
    // minimum number of operations
    int c = 0;

    // Loop to store every
    // unique prime number
    for(i = 0; i < d; i++)
    {
        int x = b[i];
        while (x != 1) 
        {
            x = x / spf[x];
            if (m.get(spf[x]) == null)
            {
                m.put(spf[x],1); 
            }
        }
    }

    // Erasing 1 as a key because
    // it is not a prime number
    m.remove(1);

    // Iterating through the hash
    for(Map.Entry<Integer,
                  Integer> x : m.entrySet()) 
    {

        // Two variables used for
        // counting the frequency
        // of prime is even or odd
        int e = 0, o = 0;

        // First prime number
        int j = x.getKey();

        // Iterating the number D
        for(i = 0; i < d; i++)
        {

            // Check if prime is a
            // factor of the element
            // in the array
            if (b[i] % j == 0)
            {
                int h = 0;
                int g = b[i];

                // Loop for calculating the
                // frequency of the element
                while (g != 0) 
                {
                    if (g % j != 0)
                    {
                        break;
                    }
                    g = g / j;
                    h = h + 1;
                }

                // Check for frequency
                // odd or even
                if (h % 2 == 0) 
                {
                    e = e + 1;
                }
                else
                {
                    o = o + 1;
                }
            }
            else
            {

                // If it is not a factor of the
                // element, then it is automatically
                // even
                e = e + 1;
            }
        }

        // Storing the minimum of two variable
        // even or odd
        c = c + Math.min(o, e);
    }
    return c;
}

// Driver Code
public static void main (String[] args)
{
    int[] spf = new int[1001];

    // Input array
    int b[] = { 1, 4, 6 };

    // Creating shortest prime
    // factorisation array
    int d = b.length;
    spf_array(spf);

    System.out.print(minimum_operation(b, d, spf));
}
}

// This code is contributed by offbeat

C

// C# program for 
// the above approach
using System;
using System.Collections.Generic;
class GFG{

// Function to find the smallest
// prime factor of the elements
static void spf_array(int []spf)
{
  // Initializing the first element
  // of the array
  spf[1] = 1;

  // Loop to add the remaining
  // elements to the array
  for(int i = 2; i < 1000; i++)

    // Marking the smallest prime
    // factor for every
    // number to be itself
    spf[i] = i;

  // Separately marking spf for
  // every even number as 2
  for(int i = 4; i < 1000; i += 2)
    spf[i] = 2;

  for(int i = 3; i * i < 1000; i++) 
  {
    // Checking if i is prime
    if (spf[i] == i)
    {
      // Marking SPF for all the
      // numbers divisible by i
      for(int j = i * i; j < 1000; j += i)

        // Marking spf[j] if it is not
        // previously marked
        if (spf[j] == j)
          spf[j] = i;
    }
  }
}

// Function to find the minimum
// number of steps to modify the
// array such that the product
// of any two numbers in the
// array is a perfect square
static int minimum_operation(int []b, 
                             int d, int []spf)
{
  // Map created to store
  // the unique prime numbers
  Dictionary<int, 
             int> m = new Dictionary<int, 
                                   int>();
  int i = 0;

  // Variable to store the
  // minimum number of operations
  int c = 0;

  // Loop to store every
  // unique prime number
  for(i = 0; i < d; i++)
  {
    int x = b[i];
    while (x != 1) 
    {
      x = x / spf[x];
      if (!m.ContainsKey(spf[x]))
      {
        m.Add(spf[x], 1); 
      }
    }
  }

  // Erasing 1 as a key because
  // it is not a prime number
  m.Remove(1);

  // Iterating through the hash
  foreach(KeyValuePair<int,
                       int> x in m) 
  {
    // Two variables used for
    // counting the frequency
    // of prime is even or odd
    int e = 0, o = 0;

    // First prime number
    int j = x.Key;

    // Iterating the number D
    for(i = 0; i < d; i++)
    {
      // Check if prime is a
      // factor of the element
      // in the array
      if (b[i] % j == 0)
      {
        int h = 0;
        int g = b[i];

        // Loop for calculating the
        // frequency of the element
        while (g != 0) 
        {
          if (g % j != 0)
          {
            break;
          }
          g = g / j;
          h = h + 1;
        }

        // Check for frequency
        // odd or even
        if (h % 2 == 0) 
        {
          e = e + 1;
        }
        else
        {
          o = o + 1;
        }
      }
      else
      {
        // If it is not a factor of the
        // element, then it is automatically
        // even
        e = e + 1;
      }
    }

    // Storing the minimum of two variable
    // even or odd
    c = c + Math.Min(o, e);
  }
  return c;
}

// Driver Code
public static void Main(String[] args)
{
  int[] spf = new int[1001];

  // Input array
  int []b = {1, 4, 6};

  // Creating shortest prime
  // factorisation array
  int d = b.Length;
  spf_array(spf);

  Console.Write(minimum_operation(b, d, spf));
}
}

// This code is contributed by shikhasingrajput

输出：

时间复杂度：O(N * log(N))，其中N是输入数组的大小。

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dsal 元素数量，它与整数相乘使数组中的每对成为完全平方

元素数量，它与整数相乘使数组中的每对成为完全平方

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Java

C

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