Minimum operations to make all elements equal using the second array

Given two arrays A[] and B[] both having N elements. Find the minimum number of operations to make all elements of A equal to the second array B. An operation comprises of:

A[i] = A[i] - B[i], 0 <= i <= n-1 

Note: If it’s not possible to make array elements equal print -1.

Examples:

Input: A[] = {5, 7, 10, 5, 15}, B[] = {2, 2, 1, 3, 5}
Output: 8
Explanation:
Following are the operations:
1) Choosing index 1 -> 5 5 10 5 15
2) Choosing index 2 -> 5 5 9 5 15
3) Choosing index 2 -> 5 5 8 5 15
4) Choosing index 2 -> 5 5 7 5 15
5) Choosing index 2 -> 5 5 6 5 15
6) Choosing index 2 -> 5 5 5 5 15
7) Choosing index 4 -> 5 5 5 5 10
8) Choosing index 4 -> 5 5 5 5 5

Input: A[] = {3, 5, 8, 2}, B[] = {1, 2, 1, 1}
Output: 12



Approach:

  • It can be observed that the maximum possible value if all elements of A can be made equal is the minimum element of A.
  • So we can iterate over the final value x, when all elements of A become equal. Using above observation – 0 <= x <= min(A[i]). For each x, traverse A and check whether A[i] can be made equal to x using B[i] :
    A[i] - k*B[i] = x
    Take mod with B[i] on both sides
    A[i] %B[i] = x %B[i]  ->  
    must be satisfied for A[i] to be converted to x
    

    If condition is satisfied, then number of operations required for this element = (A[i] – x)/B[i]

Below is the implementation of the above approach:

C++

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// C++ implementation to find the
// minimum operations make all elements 
// equal using the second array
  
#include <bits/stdc++.h>
  
using namespace std;
  
// Function to find the minimum 
// operations required to make
// all elements of the array equal
int minOperations(int a[], int b[], int n)
{
    // Minimum element of A[]
    int minA = *min_element(a, a + n);
  
    // Traverse through all final values
    for (int x = minA; x >= 0; x--) {
          
        // Variable indicating 
        // whether all elements
        // can be converted to x or not
        bool check = 1;
  
        // Total operations
        int operations = 0;
          
        // Traverse through 
        // all array elements
        for (int i = 0; i < n; i++) {
            if (x % b[i] == a[i] % b[i]) {
                operations += 
                    (a[i] - x) / b[i];
            }
  
            // All elements can't 
            // be converted to x
            else {
                check = 0;
                break;
            }
        }
        if (check)
            return operations;
    }
    return -1;
}
  
// Driver Code
int main()
{
    int N = 5;
    int A[N] = { 5, 7, 10, 5, 15 };
    int B[N] = { 2, 2, 1, 3, 5 };
  
    cout << minOperations(A, B, N);
    return 0;
}

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Java

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// Java implementation to find the
// minimum operations make all elements 
// equal using the second array
import java.util.*;
class GFG{
  
// Function to find the minimum 
// operations required to make
// all elements of the array equal
static int minOperations(int a[], int b[], int n)
{
    // Minimum element of A[]
    int minA = Arrays.stream(a).min().getAsInt();
  
    // Traverse through all final values
    for (int x = minA; x >= 0; x--) 
    {
          
        // Variable indicating 
        // whether all elements
        // can be converted to x or not
        boolean check = true;
  
        // Total operations
        int operations = 0;
          
        // Traverse through 
        // all array elements
        for (int i = 0; i < n; i++) 
        {
            if (x % b[i] == a[i] % b[i]) 
            {
                operations += (a[i] - x) / b[i];
            }
  
            // All elements can't 
            // be converted to x
            else 
            {
                check = false;
                break;
            }
        }
        if (check)
            return operations;
    }
    return -1;
}
  
// Driver Code
public static void main(String[] args)
{
    int N = 5;
    int A[] = { 5, 7, 10, 5, 15 };
    int B[] = { 2, 2, 1, 3, 5 };
  
    System.out.print(minOperations(A, B, N));
}
}
  
// This code is contributed by AbhiThakur

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C#

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// C# implementation to find the
// minimum operations make all elements 
// equal using the second array
using System;
using System.Linq;
  
class GFG{
  
// Function to find the minimum 
// operations required to make
// all elements of the array equal
static int minOperations(int []a, int []b, int n)
{
      
    // Minimum element of A[]
    int minA = a.Max();
  
    // Traverse through all final values
    for(int x = minA; x >= 0; x--) 
    {
         
       // Variable indicating 
       // whether all elements
       // can be converted to x or not
       bool check = true;
         
       // Total operations
       int operations = 0;
         
       // Traverse through 
       // all array elements
       for(int i = 0; i < n; i++) 
       {
          if (x % b[i] == a[i] % b[i]) 
          {
              operations += (a[i] - x) / b[i];
          }
            
          // All elements can't 
          // be converted to x
          else
          {
              check = false;
              break;
          }
       }
       if (check)
           return operations;
    }
    return -1;
}
  
// Driver Code
public static void Main(string[] args)
{
    int N = 5;
    int []A = { 5, 7, 10, 5, 15 };
    int []B = { 2, 2, 1, 3, 5 };
  
    Console.WriteLine(minOperations(A, B, N));
}
}
  
// This code is contributed by SoumikMondal

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Output:

8

Time Complexity: O(N * min(Ai))

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