最小化使所有素数索引元素成为素数所需的交换

给定一个大小为N的数组arr[] 。任务是找到重新排列数组所需的最小交换次数，以使所有以素数为索引的元素都是素数，如果无法完成任务，则打印“ -1 “

例子：

Input:  N = 5, arr[] = {1, 2, 3, 4, 5}
Output: 0
Explanation: All the prime indices {2, 3, 5} (one-based indexing) have prime elements present on them. Therefore, minimum swaps required is 0.

Input:  N = 5, arr[] = {2, 7, 8, 5, 13}
Output: 1
Explanation: swap 8 and 5 once, to get all the prime numbers at prime indices.

方法：该任务可以使用埃拉托色尼筛法解决。

• 遍历数组并检查当前索引是否为素数，如果是素数，则检查此索引上存在的元素。
• 观察可以看出，只有当数组中素数的总数大于或等于数组中素数索引的总数时，才有可能实现所需的配置。
• 如果这个条件成立，那么交换的最小数量等于没有素数的素数索引的数量。

下面是上述方法的实现：

C++

#include 
using namespace std;
 
const int mxn = 1e4 + 1;
bool prime[mxn + 1];
 
// Function to pre-calculate the prime[]
// prime[i] denotes whether
// i is prime or not
void SieveOfEratosthenes()
{
    memset(prime, true, sizeof(prime));
 
    for (int p = 2; p * p <= mxn; p++) {
        // If prime[p] is not changed,
        // then it is a prime
        if (prime[p] == true) {
            // Update all multiples
            // of p greater than or
            // equal to the square of it
            // numbers which are multiple
            // of p and are less than p^2
            // are already been marked.
            for (int i = p * p; i <= mxn; i += p)
                prime[i] = false;
        }
    }
}
 
// Function to count minimum number
// of swaps required
int countMin(int arr[], int n)
{
    // To count the minimum number of swaps
    // required to convert the array into
    // perfectly prime
    int cMinSwaps = 0;
 
    // To count total number of prime
    // indexes in the array
    int cPrimeIndices = 0;
 
    // To count the total number of
    // prime numbers in the array
 
    int cPrimeNos = 0;
 
    for (int i = 0; i < n; i++) {
        // Check whether index
        // is prime or not
 
        if (prime[i + 1]) {
            cPrimeIndices++;
 
            // Element is not prime
            if (!prime[arr[i]])
                cMinSwaps++;
            else
                cPrimeNos++;
        }
        else if (prime[arr[i]]) {
            cPrimeNos++;
        }
    }
 
    // If the total number of prime numbers
    // is greater than or equal to the total
    // number of prime indices, then it is
    // possible to convert the array into
    // perfectly prime
    if (cPrimeNos >= cPrimeIndices)
        return cMinSwaps;
    else
        return -1;
}
// Driver Code
int main()
{
    // Pre-calculate prime[]
    SieveOfEratosthenes();
 
    int n = 5;
    int arr[5] = { 2, 7, 8, 5, 13 };
 
    cout << countMin(arr, n);
    return 0;
}

Java

// Java program for the above approach
import java.io.*;
 
class GFG {
    static boolean prime[] = new boolean[(int)1e4 + 2];
    static void SieveOfEratosthenes()
    {
        // Create a boolean array "prime[0..n]" and
        // initialize all entries it as true. A value in
        // prime[i] will finally be false if i is Not a
        // prime, else true.
 
        for (int i = 0; i < (int)1e4 + 2; i++)
            prime[i] = true;
 
        for (int p = 2; p * p < (int)1e4 + 2; p++)
        {
           
            // If prime[p] is not changed, then it is a
            // prime
            if (prime[p] == true)
            {
               
                // Update all multiples of p
                for (int i = p * p; i < (int)1e4 + 2;
                     i += p)
                    prime[i] = false;
            }
        }
    }
   
    // Function to count minimum number
    // of swaps required
    static int countMin(int[] arr, int n)
    {
       
        // To count the minimum number of swaps
        // required to convert the array into
        // perfectly prime
        int cMinSwaps = 0;
 
        // To count total number of prime
        // indexes in the array
        int cPrimeIndices = 0;
 
        // To count the total number of
        // prime numbers in the array
 
        int cPrimeNos = 0;
 
        for (int i = 0; i < n; i++) {
            // Check whether index
            // is prime or not
 
            if (prime[i + 1]) {
                cPrimeIndices++;
 
                // Element is not prime
                if (prime[arr[i]] == false)
                    cMinSwaps++;
                else
                    cPrimeNos++;
            }
            else if (prime[arr[i]]) {
                cPrimeNos++;
            }
        }
 
        // If the total number of prime numbers
        // is greater than or equal to the total
        // number of prime indices, then it is
        // possible to convert the array into
        // perfectly prime
        if (cPrimeNos >= cPrimeIndices)
            return cMinSwaps;
        else
            return -1;
    }
   
    // Driver Code
    public static void main(String[] args)
    {
       
        // Pre-calculate prime[]
        SieveOfEratosthenes();
 
        int n = 5;
        int arr[] = { 2, 7, 8, 5, 13 };
 
        System.out.println(countMin(arr, n));
    }
}
 
// This code is contributed by Potta Lokesh

Python3

# Python program for the above approach
import math
 
mxn = 10000 + 1
prime = [True for _ in range(mxn + 1)]
 
# Function to pre-calculate the prime[]
# prime[i] denotes whether
# i is prime or not
def SieveOfEratosthenes():
    global prime
    for p in range(2, int(math.sqrt(mxn)) + 1):
       
                # If prime[p] is not changed,
                # then it is a prime
        if (prime[p] == True):
                        # Update all multiples
                        # of p greater than or
                        # equal to the square of it
                        # numbers which are multiple
                        # of p and are less than p^2
                        # are already been marked.
 
            for i in range(p*p, mxn+1, p):
                prime[i] = False
 
 
# Function to count minimum number
# of swaps required
def countMin(arr, n):
 
        # To count the minimum number of swaps
        # required to convert the array into
        # perfectly prime
    cMinSwaps = 0
 
    # To count total number of prime
    # indexes in the array
    cPrimeIndices = 0
 
    # To count the total number of
    # prime numbers in the array
 
    cPrimeNos = 0
 
    for i in range(0, n):
                # Check whether index
                # is prime or not
 
        if (prime[i + 1]):
            cPrimeIndices += 1
 
            # Element is not prime
            if (not prime[arr[i]]):
                cMinSwaps += 1
            else:
                cPrimeNos += 1
 
        elif (prime[arr[i]]):
            cPrimeNos += 1
 
    # If the total number of prime numbers
    # is greater than or equal to the total
    # number of prime indices, then it is
    # possible to convert the array into
    # perfectly prime
    if (cPrimeNos >= cPrimeIndices):
        return cMinSwaps
    else:
        return -1
 
# Driver Code
if __name__ == "__main__":
 
    # Pre-calculate prime[]
    SieveOfEratosthenes()
 
    n = 5
    arr = [2, 7, 8, 5, 13]
 
    print(countMin(arr, n))
 
    # This code is contributed by rakeshsahni

C#

// C# program for the above approach
using System;
 
class GFG
{
    static bool[] prime = new bool[(int)1e4 + 2];
    static void SieveOfEratosthenes()
    {
       
        // Create a boolean array "prime[0..n]" and
        // initialize all entries it as true. A value in
        // prime[i] will finally be false if i is Not a
        // prime, else true.
        for (int i = 0; i < (int)1e4 + 2; i++)
            prime[i] = true;
 
        for (int p = 2; p * p < (int)1e4 + 2; p++) {
 
            // If prime[p] is not changed, then it is a
            // prime
            if (prime[p] == true) {
 
                // Update all multiples of p
                for (int i = p * p; i < (int)1e4 + 2;
                     i += p)
                    prime[i] = false;
            }
        }
    }
 
    // Function to count minimum number
    // of swaps required
    static int countMin(int[] arr, int n)
    {
 
        // To count the minimum number of swaps
        // required to convert the array into
        // perfectly prime
        int cMinSwaps = 0;
 
        // To count total number of prime
        // indexes in the array
        int cPrimeIndices = 0;
 
        // To count the total number of
        // prime numbers in the array
 
        int cPrimeNos = 0;
 
        for (int i = 0; i < n; i++) {
            // Check whether index
            // is prime or not
 
            if (prime[i + 1]) {
                cPrimeIndices++;
 
                // Element is not prime
                if (prime[arr[i]] == false)
                    cMinSwaps++;
                else
                    cPrimeNos++;
            }
            else if (prime[arr[i]]) {
                cPrimeNos++;
            }
        }
 
        // If the total number of prime numbers
        // is greater than or equal to the total
        // number of prime indices, then it is
        // possible to convert the array into
        // perfectly prime
        if (cPrimeNos >= cPrimeIndices)
            return cMinSwaps;
        else
            return -1;
    }
 
    // Driver Code
    public static void Main()
    {
 
        // Pre-calculate prime[]
        SieveOfEratosthenes();
 
        int n = 5;
        int[] arr = { 2, 7, 8, 5, 13 };
 
        Console.Write(countMin(arr, n));
    }
}
 
// This code is contributed by subhammahato348.

Javascript

const mxn = 1e4 + 1;
let prime = new Array(mxn + 1);
 
// Function to pre-calculate the prime[]
// prime[i] denotes whether
// i is prime or not
function SieveOfEratosthenes() {
  prime.fill(true);
 
  for (let p = 2; p * p <= mxn; p++)
  {
   
    // If prime[p] is not changed,
    // then it is a prime
    if (prime[p] == true)
    {
     
      // Update all multiples
      // of p greater than or
      // equal to the square of it
      // numbers which are multiple
      // of p and are less than p^2
      // are already been marked.
      for (let i = p * p; i <= mxn; i += p) prime[i] = false;
    }
  }
}
 
// Function to count minimum number
// of swaps required
function countMin(arr, n)
{
 
  // To count the minimum number of swaps
  // required to convert the array into
  // perfectly prime
  let cMinSwaps = 0;
 
  // To count total number of prime
  // indexes in the array
  let cPrimeIndices = 0;
 
  // To count the total number of
  // prime numbers in the array
 
  let cPrimeNos = 0;
 
  for (let i = 0; i < n; i++) {
    // Check whether index
    // is prime or not
 
    if (prime[i + 1]) {
      cPrimeIndices++;
 
      // Element is not prime
      if (!prime[arr[i]]) cMinSwaps++;
      else cPrimeNos++;
    } else if (prime[arr[i]]) {
      cPrimeNos++;
    }
  }
 
  // If the total number of prime numbers
  // is greater than or equal to the total
  // number of prime indices, then it is
  // possible to convert the array into
  // perfectly prime
  if (cPrimeNos >= cPrimeIndices) return cMinSwaps;
  else return -1;
}
 
// Driver Code
 
// Pre-calculate prime[]
SieveOfEratosthenes();
 
let n = 5;
let arr = [2, 7, 8, 5, 13];
 
document.write(countMin(arr, n));
 
// This code is contributed by gfgking.

输出

1

时间复杂度：O(mxn(log(log(mxn)))
辅助空间：O(mxn)