Maximize difference between sum of prime and non-prime array elements by left shifting of digits minimum number of times

Given an array arr[] of size N, the task is to find the maximum difference between the sum of the prime numbers and the sum of the non-prime numbers present in the array, by left shifting the digits of array elements by 1 minimum number of times.

Examples:

Input: arr[] = {541, 763, 321, 716, 143}
Output: Difference = 631, Count of operations = 6
Explanation: 
Operation 1: Left shift the digits of arr[1] (= 763). Therefore, arr[1] becomes 637.
Operation 2: Left shift the digits of arr[1] (= 637). Therefore, arr[1] becomes 376.
Operation 3: Left shift the digits of arr[2] (= 321). Therefore, arr[2] becomes 213.
Operation 4: Left shift the digits of arr[2] (= 213). Therefore, arr[2] becomes 132.
Operation 5: Left shift the digits of arr[3] (= 716). Therefore, arr[3] becomes 167.
Operation 6: Left shift the digits of arr[4] (= 143). Therefore, arr[4] becomes 431.
Therefore, Sum of prime array elements = 541 + 167 + 431 = 1139.
Therefore, sum of non-prime array elements = 376 + 132 = 508.
Therefore, difference = 1139 – 508 = 631.

Input: {396, 361, 359, 496, 780}
Output: Difference = 236, Count of operations = 4
Explanation:
Operation 1: Left shift the digits of arr[1] (= 361). Therefore, arr[1] becomes 613.
Operation 2: Left shift the digits of arr[2] (= 359). Therefore, arr[2] becomes 593.
Operation 3: Left shift the digits of arr[4] (= 780). Therefore, arr[4] becomes 807.
Operation 4: Left shift the digits of arr[4] (= 807). Therefore, arr[4] becomes 078.
Therefore, required difference = 613 + 593 – 496 – 78 – 396 = 236.

Approach: The given problem can be solved greedily. If it is possible to convert an element into one or more than one prime number, then take the maximum of them. Otherwise, try to minimize the element by using all the possible rotations.
Follow the steps below to solve the problem:

• Initialize two variables, say ans and cost, to store the maximum difference and the minimum number of operations required respectively.
• Traverse the array arr[] using a variable i and perform the following steps:
  • Initialize variables maxPrime and minRotation as -1 to store the maximum prime number and minimum number that can be obtained from arr[i] through left rotations.
  • Generate all left rotations of the number arr[i].
    • If arr[i] is a prime number exceeding maxPrime, then update maxPrime to arr[i] and the cost accordingly.
  • If the value of maxPrime remains unchanged, find the value of minRotation by similarly generating all the left rotations.
  • Add the value of arr[i] to ans.
• After completing the above steps, print the value of ans and cost as the result.

Below is the implementation of the above approach:

C++14

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to check if a
// number is prime or not
bool isPrime(int n)
{

    // Base cases
    if (n <= 1)
        return false;
    if (n <= 3)
        return true;

    // Check if the number is
    // a multiple of 2 or 3
    if (n % 2 == 0 || n % 3 == 0)
        return false;

    int i = 5;

    // Iterate until square root of n
    while (i * i <= n) {

        // If n is divisible by both i and i + 2
        if (n % i == 0 || n % (i + 2) == 0)
            return false;
        i = i + 6;
    }
    return true;
}

// Function to left shift a number
// to maximize its contribution
pair<int, int> rotateElement(int n)
{

    // Convert the number to string
    string strN = to_string(n);

    // Stores the maximum prime number
    // that can be obtained from n
    int maxPrime = -1;

    // Store the required
    // number of operations
    int cost = 0;

    string temp = strN;

    // Check for all the left
    // rotations of the number
    for (int i = 0; i < strN.size(); i++) {

        // If the number is prime, then
        // take the maximum among them
        if (isPrime(stoi(temp)) && stoi(temp) > maxPrime) {
            maxPrime = stoi(temp);
            cost = i;
        }

        // Left rotation
        temp = temp.substr(1) + temp[0];
    }

    int optEle = maxPrime;

    // If no prime number can be obtained
    if (optEle == -1) {
        optEle = INT_MAX;
        temp = strN;

        // Check all the left
        // rotations of the number
        for (int i = 0; i < strN.size(); i++) {

            // Take the minimum element
            if (stoi(temp) < optEle) {
                optEle = stoi(temp);
                cost = i;
            }

            // Left rotation
            temp = temp.substr(1) + temp[0];
        }
        optEle *= (-1);
    }

    return { optEle, cost };
}

// Function to find the maximum sum
// obtained using the given operations
void getMaxSum(int arr[], int N)
{

    // Store the maximum sum and
    // the number of operations
    int maxSum = 0, cost = 0;

    // Traverse array elements
    for (int i = 0; i < N; i++) {
        int x = arr[i];

        // Get the optimal element and the
        // number of operations to obtain it
        pair<int, int> ret = rotateElement(x);

        int optEle = ret.first, optCost = ret.second;

        // Increment the maximum difference
        // and number of operations required
        maxSum += optEle;
        cost += optCost;
    }

    // Print the result
    cout << "Difference = " << maxSum << " , "
         << "Count of operations = " << cost;
}

// Driven Program
int main()
{
    // Given array arr[]
    int arr[] = { 541, 763, 321, 716, 143 };

    // Store the size of the array
    int N = sizeof(arr) / sizeof(arr[0]);

    // Function call
    getMaxSum(arr, N);

    return 0;
}

Java

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

class GFG {

    // Function to check if a
    // number is prime or not
    static boolean isPrime(int n)
    {

        // Base cases
        if (n <= 1)
            return false;
        if (n <= 3)
            return true;

        // Check if the number is
        // a multiple of 2 or 3
        if (n % 2 == 0 || n % 3 == 0)
            return false;

        int i = 5;

        // Iterate until square root of n
        while (i * i <= n) {

            // If n is divisible by both i and i + 2
            if (n % i == 0 || n % (i + 2) == 0)
                return false;
            i = i + 6;
        }
        return true;
    }

    // Function to left shift a number
    // to maximize its contribution
    static int[] rotateElement(int n)
    {

        // Convert the number to string
        String strN = Integer.toString(n);

        // Stores the maximum prime number
        // that can be obtained from n
        int maxPrime = -1;

        // Store the required
        // number of operations
        int cost = 0;

        String temp = strN;

        // Check for all the left
        // rotations of the number
        for (int i = 0; i < strN.length(); i++) {

            // If the number is prime, then
            // take the maximum among them
            if (isPrime(Integer.parseInt(temp))
                && Integer.parseInt(temp) > maxPrime) {
                maxPrime = Integer.parseInt(temp);
                cost = i;
            }

            // Left rotation
            temp = temp.substring(1) + temp.charAt(0);
        }

        int optEle = maxPrime;

        // If no prime number can be obtained
        if (optEle == -1) {
            optEle = Integer.MAX_VALUE;
            temp = strN;

            // Check all the left
            // rotations of the number
            for (int i = 0; i < strN.length(); i++) {

                // Take the minimum element
                if (Integer.parseInt(temp) < optEle) {
                    optEle = Integer.parseInt(temp);
                    cost = i;
                }

                // Left rotation
                temp = temp.substring(1) + temp.charAt(0);
            }
            optEle *= (-1);
        }
        return new int[] { optEle, cost };
    }

    // Function to find the maximum sum
    // obtained using the given operations
    static void getMaxSum(int arr[])
    {

        // Store the maximum sum and
        // the number of operations
        int maxSum = 0, cost = 0;

        // Traverse array elements
        for (int x : arr) {

            // Get the optimal element and the
            // number of operations to obtain it
            int ret[] = rotateElement(x);

            int optEle = ret[0], optCost = ret[1];

            // Increment the maximum difference
            // and number of operations required
            maxSum += optEle;
            cost += optCost;
        }

        // Print the result
        System.out.println("Difference = " + maxSum + " , "
                           + "Count of operations = "
                           + cost);
    }

    // Driver Code
    public static void main(String[] args)
    {

        // Given array arr[]
        int arr[] = { 541, 763, 321, 716, 143 };

        // Function call
        getMaxSum(arr);
    }
}

Python3

# Python program for the above approach

# Function to check if a
# number is prime or not
def isPrime(n):

    # Base cases
    if (n <= 1):
        return False
    if (n <= 3):
        return True

    # Check if the number is
    # a multiple of 2 or 3
    if (n % 2 == 0 or n % 3 == 0):
        return False

    i = 5

    # Iterate until square root of n
    while(i * i <= n):

        # If n is divisible by both i and i + 2
        if (n % i == 0 or n % (i + 2) == 0):
            return False
        i = i + 6

    return True

# Function to left shift a number
# to maximize its contribution
def rotateElement(n):

        # Convert the number to string
    strN = str(n)

    # Stores the maximum prime number
    # that can be obtained from n
    maxPrime = -1

    # Store the required
    # number of operations
    cost = 0

    temp = str(strN)

    # Check for all the left
    # rotations of the number
    for i in range(len(strN)):

        # If the number is prime, then
        # take the maximum among them
        if isPrime(int(temp)) and int(temp) > maxPrime:
            maxPrime = int(temp)
            cost = i

        # Left rotation
        temp = temp[1:]+temp[:1]

    optEle = maxPrime

    # If no prime number can be obtained
    if optEle == -1:
        optEle = float('inf')
        temp = str(strN)

        # Check all the left
        # rotations of the number
        for i in range(len(strN)):

            # Take the minimum element
            if int(temp) < optEle:
                optEle = int(temp)
                cost = i

            # Left rotation
            temp = temp[1:]+temp[:1]
        optEle *= (-1)

    return (optEle, cost)

# Function to find the maximum sum
# obtained using the given operations
def getMaxSum(arr):

    # Store the maximum sum and
    # the number of operations
    maxSum, cost = 0, 0

    # Traverse array elements
    for x in arr:

        # Get the optimal element and the
        # number of operations to obtain it
        optEle, optCost = rotateElement(x)

        # Increment the maximum difference
        # and number of operations required
        maxSum += optEle
        cost += optCost

    # Print the result
    print('Difference =', maxSum, ',',
          'Count of operations =', cost)


# Driver Code

arr = [541, 763, 321, 716, 143]
getMaxSum(arr)

C#

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

class GFG {

  // Function to check if a
  // number is prime or not
  static bool isPrime(int n)
  {

    // Base cases
    if (n <= 1)
      return false;
    if (n <= 3)
      return true;

    // Check if the number is
    // a multiple of 2 or 3
    if (n % 2 == 0 || n % 3 == 0)
      return false;

    int i = 5;

    // Iterate until square root of n
    while (i * i <= n) {

      // If n is divisible by both i and i + 2
      if (n % i == 0 || n % (i + 2) == 0)
        return false;
      i = i + 6;
    }
    return true;
  }

  // Function to left shift a number
  // to maximize its contribution
  static int[] rotateElement(int n)
  {

    // Convert the number to string
    string strN = n.ToString();

    // Stores the maximum prime number
    // that can be obtained from n
    int maxPrime = -1;

    // Store the required
    // number of operations
    int cost = 0;

    string temp = strN;

    // Check for all the left
    // rotations of the number
    for (int i = 0; i < strN.Length; i++) {

      // If the number is prime, then
      // take the maximum among them
      if (isPrime(Int32.Parse(temp))
          &&  Int32.Parse(temp) > maxPrime) {
        maxPrime =  Int32.Parse(temp);
        cost = i;
      }

      // Left rotation
      temp = temp.Substring(1) + temp[0];
    }

    int optEle = maxPrime;

    // If no prime number can be obtained
    if (optEle == -1) {
      optEle = 2147483647;
      temp = strN;

      // Check all the left
      // rotations of the number
      for (int i = 0; i < strN.Length; i++) {

        // Take the minimum element
        if (Int32.Parse(temp) < optEle) {
          optEle = Int32.Parse(temp);
          cost = i;
        }

        // Left rotation
        temp = temp.Substring(1) + temp[0];
      }
      optEle *= (-1);
    }
    return new int[] { optEle, cost };
  }

  // Function to find the maximum sum
  // obtained using the given operations
  static void getMaxSum(int []arr)
  {

    // Store the maximum sum and
    // the number of operations
    int maxSum = 0, cost = 0;

    // Traverse array elements
    foreach (int x in arr) {

      // Get the optimal element and the
      // number of operations to obtain it
      int[] ret = rotateElement(x);

      int optEle = ret[0], optCost = ret[1];

      // Increment the maximum difference
      // and number of operations required
      maxSum += optEle;
      cost += optCost;
    }

    // Print the result
    Console.Write("Difference = " + maxSum + " , "+ "Count of operations = "
                  + cost);
  }

  // Driver Code
  public static void Main()
  {

    // Given array arr[]
    int []arr = { 541, 763, 321, 716, 143 };

    // Function call
    getMaxSum(arr);
  }
}

// This code is contributed by ipg2016107.

Javascript

<script>

    // JavaScript program for the above approach

    // Function to check if a
    // number is prime or not
    function isPrime(n)
    {

        // Base cases
        if (n <= 1)
            return false;
        if (n <= 3)
            return true;

        // Check if the number is
        // a multiple of 2 or 3
        if (n % 2 == 0 || n % 3 == 0)
            return false;

        let i = 5;

        // Iterate until square root of n
        while (i * i <= n) {

            // If n is divisible by both i and i + 2
            if (n % i == 0 || n % (i + 2) == 0)
                return false;
            i = i + 6;
        }
        return true;
    }

    // Function to left shift a number
    // to maximize its contribution
    function rotateElement(n)
    {

        // Convert the number to string
        let strN = n.toString();

        // Stores the maximum prime number
        // that can be obtained from n
        let maxPrime = -1;

        // Store the required
        // number of operations
        let cost = 0;

        let temp = strN;

        // Check for all the left
        // rotations of the number
        for (let i = 0; i < strN.length; i++) {

            // If the number is prime, then
            // take the maximum among them
            if (isPrime(parseInt(temp)) && parseInt(temp) > maxPrime)
            {
                maxPrime = parseInt(temp);
                cost = i;
            }

            // Left rotation
            temp = temp.substr(1) + temp[0];
        }

        let optEle = maxPrime;

        // If no prime number can be obtained
        if (optEle == -1) {
            optEle = Number.MAX_VALUE;
            temp = strN;

            // Check all the left
            // rotations of the number
            for (let i = 0; i < strN.length; i++) {

                // Take the minimum element
                if (parseInt(temp) < optEle) {
                    optEle = parseInt(temp);
                    cost = i;
                }

                // Left rotation
                temp = temp.substr(1) + temp[0];
            }
            optEle *= (-1);
        }

        return [ optEle, cost ];
    }

    // Function to find the maximum sum
    // obtained using the given operations
    function getMaxSum(arr,N)
    {

        // Store the maximum sum and
        // the number of operations
        let maxSum = 0, cost = 0;

        // Traverse array elements
        for (let i = 0; i < N; i++) {
            let x = arr[i];

            // Get the optimal element and the
            // number of operations to obtain it
            let ret = rotateElement(x);

            let optEle = ret[0], optCost = ret[1];

            // Increment the maximum difference
            // and number of operations required
            maxSum += optEle;
            cost += optCost;
        }

        // Print the result
         document.write("Difference = " + maxSum + " , "
            + "Count of operations = " + cost);
    }

    // Driven Program

    // Given array arr[]
    let arr = [ 541, 763, 321, 716, 143 ];

    // Store the size of the array
    let N = arr.length

    // Function call
    getMaxSum(arr, N);

</script>

Output:

Difference = 631 , Count of operations = 6

Time Complexity: O(X^2 * sqrt(X)), where n is the number of digits in the input number. 
Auxiliary Space: O(1)