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Minimize sum of changes in Array such that ratio of new element to array sum is at most p:q

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  • Last Updated : 29 Mar, 2022
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Given an array A of n integers and two integers p and q, the task is to increase some (or all) of the values in array A in such a way that ratio of each element (after first element) with respect to the total sum of all elements before the current element remains less than or equal to p/q. Return the minimum sum of changes to be done.

Examples:

Input: n = 4, A = {20100, 1, 202, 202}, p = 1, q = 100
Output: 99
Explanation: 50 is added to the first element and 49 to the second element, so that the resulting array becomes {20150, 50, 202, 202}.
50/20150 <= 1/100
202/(20150+50) <=1/100
202/(20150+50+202)<=1/100
Therefore, the condition is satisfied for all the elements and hence 50+49=99 would be the answer. 
There are other answers possible as well, but we need to find out the minimum possible sum.

Input: n = 3, A = {1, 1, 1}, p = 100, q = 100
Output: 0

 

Approach: The problem can be easily solved using the concepts of prefix sum and binary search

  • It can be clearly observed that
    • if the condition stated in the problem is achieved for a certain sum of changes (say S),
    • then it is always possible to achieve the condition for all the numbers greater than S and
    • cannot be achieved for all numbers less than S.
    • So, we can apply binary search to find the answer.
  • Also, it must be carefully observed that instead of distributing S(sum of changes) over different elements, if we just add it to first element, that would not affect the answer.

For example, in the first example above, if 99 is added to the first element, the resultant array will still meet the required condition.

Follow the steps below to solve the problem – 

  • Make a prefix sum array of the given array.
  • Using binary search on range 0 to INT_MAX find the minimum possible answer.

NOTE: Division may lead to overlapping values and errors. 

So, instead of comparison like (a/b)<=(c/d), we will do (a*d)<=(b*c).

Below is the implementation of the above approach:

C++




// C++ program for find minimum
// sum of changes in an array
#include <bits/stdc++.h>
using namespace std;
 
// function to check if the candidate sum
// satisfies the condition of the problem
bool isValid(int candidate, int pre[], int n, int A[],
             int p, int q)
{
    // flag variable to check wrong answer
    bool flag = true;
    for (int i = 1; i < n; i++) {
 
        // Now for each element, we are checking
        // if its ratio with sum of all previous
        // elements + candidate is greater than p/q.
        // If so, we will return false.
        int curr_sum = pre[i - 1] + candidate;
        if (A[i] * q > p * curr_sum) {
            flag = false;
            break;
        }
        // comparing like A[i]/(curr_sum)>p/q
        // will be error prone.
    }
    return flag;
}
 
int solve(int n, int A[], int p, int q)
{
 
    // declaring and constructing
    // prefix sum array
    int pre[n];
    pre[0] = A[0];
    for (int i = 1; i < n; i++) {
        pre[i] = A[i] + pre[i - 1];
    }
 
    // setting lower and upper bound for
    // binary search
    int lo = 0, hi = INT_MAX, ans = INT_MAX;
 
    // since minimum answer is needed,
    // so it is initialized with INT_MAX
    while (lo <= hi) {
 
        // calculating mid by using (lo+hi)/2
        // may overflow in certain cases
        int mid = lo + (hi - lo) / 2;
 
        // checking if required ratio would be
        // achieved by all elements if "mid" is
        // considered as answer
        if (isValid(mid, pre, n, A, p, q)) {
            ans = mid;
            hi = mid - 1;
        }
        else {
            lo = mid + 1;
        }
    }
    return ans;
}
 
// Driver Function
int main()
{
    int n, p, q;
    n = 4, p = 1, q = 100;
    int A[] = { 20100, 1, 202, 202 };
 
    // printing the required answer
    cout << solve(n, A, p, q) << endl;
}


Java




// Java program for find minimum
// sum of changes in an arraykage whatever //do not write package name here */
import java.io.*;
 
class GFG {
 
  // function to check if the candidate sum
  // satisfies the condition of the problem
  static Boolean isValid(int candidate, int pre[], int n, int A[],
                         int p, int q)
  {
     
    // flag variable to check wrong answer
    Boolean flag = true;
    for (int i = 1; i < n; i++) {
 
      // Now for each element, we are checking
      // if its ratio with sum of all previous
      // elements + candidate is greater than p/q.
      // If so, we will return false.
      int curr_sum = pre[i - 1] + candidate;
      if (A[i] * q > p * curr_sum) {
        flag = false;
        break;
      }
      // comparing like A[i]/(curr_sum)>p/q
      // will be error prone.
    }
    return flag;
  }
 
  static int solve(int n, int A[], int p, int q)
  {
 
    // declaring and constructing
    // prefix sum array
    int pre[] = new int[n];
    pre[0] = A[0];
    for (int i = 1; i < n; i++) {
      pre[i] = A[i] + pre[i - 1];
    }
 
    // setting lower and upper bound for
    // binary search
    int lo = 0, hi = Integer.MAX_VALUE, ans = Integer.MAX_VALUE;
 
    // since minimum answer is needed,
    // so it is initialized with INT_MAX
    while (lo <= hi) {
 
      // calculating mid by using (lo+hi)/2
      // may overflow in certain cases
      int mid = lo + (hi - lo) / 2;
 
      // checking if required ratio would be
      // achieved by all elements if "mid" is
      // considered as answer
      if (isValid(mid, pre, n, A, p, q)) {
        ans = mid;
        hi = mid - 1;
      }
      else {
        lo = mid + 1;
      }
    }
    return ans;
  }
 
  // Driver Function
  public static void main (String[] args)
  {
    int n = 4, p = 1, q = 100;
    int A[] = { 20100, 1, 202, 202 };
 
    // printing the required answer
    System.out.println(solve(n, A, p, q));
  }
}
 
// This code is contributed by hrithikgarg03188.


Python3




# Python code for the above approach
import sys
 
# function to check if the candidate sum
# satisfies the condition of the problem
def isValid(candidate, pre, n, A, p, q) :
     
    # flag variable to check wrong answer
    flag = True
    for i in range(1, n) :
  
        # Now for each element, we are checking
        # if its ratio with sum of all previous
        # elements + candidate is greater than p/q.
        # If so, we will return false.
        curr_sum = pre[i - 1] + candidate
        if (A[i] * q > p * curr_sum) :
            flag = False
            break
         
        # comparing like A[i]/(curr_sum)>p/q
        # will be error prone.
     
    return flag
 
  
def solve(n, A, p, q) :
  
    # declaring and constructing
    # prefix sum array
    pre = [0] * 100
    pre[0] = A[0]
    for i in range(1, n) :
        pre[i] = A[i] + pre[i - 1]
     
  
    # setting lower and upper bound for
    # binary search
    lo = 0
    hi = sys.maxsize
    ans = sys.maxsize
  
    # since minimum answer is needed,
    # so it is initialized with INT_MAX
    while (lo <= hi) :
  
        # calculating mid by using (lo+hi)/2
        # may overflow in certain cases
        mid = lo + (hi - lo) // 2
  
        # checking if required ratio would be
        # achieved by all elements if "mid" is
        # considered as answer
        if (isValid(mid, pre, n, A, p, q)) :
            ans = mid
            hi = mid - 1
         
        else :
            lo = mid + 1
         
    return ans
  
# Driver Function
n = 4
p = 1
q = 100
A = [ 20100, 1, 202, 202 ]
  
# printing the required answer
print(solve(n, A, p, q))
 
# This code is contributed by code_hunt.


C#




// C# program for find minimum
// sum of changes in an array
using System;
class GFG
{
 
  // function to check if the candidate sum
  // satisfies the condition of the problem
  static bool isValid(int candidate, int []pre, int n, int []A,
                      int p, int q)
  {
 
    // flag variable to check wrong answer
    bool flag = true;
    for (int i = 1; i < n; i++) {
 
      // Now for each element, we are checking
      // if its ratio with sum of all previous
      // elements + candidate is greater than p/q.
      // If so, we will return false.
      int curr_sum = pre[i - 1] + candidate;
      if (A[i] * q > p * curr_sum) {
        flag = false;
        break;
      }
      // comparing like A[i]/(curr_sum)>p/q
      // will be error prone.
    }
    return flag;
  }
 
  static int solve(int n, int []A, int p, int q)
  {
 
    // declaring and constructing
    // prefix sum array
    int []pre = new int[n];
    pre[0] = A[0];
    for (int i = 1; i < n; i++) {
      pre[i] = A[i] + pre[i - 1];
    }
 
    // setting lower and upper bound for
    // binary search
    int lo = 0, hi = Int32.MaxValue, ans = Int32.MaxValue;
 
    // since minimum answer is needed,
    // so it is initialized with INT_MAX
    while (lo <= hi) {
 
      // calculating mid by using (lo+hi)/2
      // may overflow in certain cases
      int mid = lo + (hi - lo) / 2;
 
      // checking if required ratio would be
      // achieved by all elements if "mid" is
      // considered as answer
      if (isValid(mid, pre, n, A, p, q)) {
        ans = mid;
        hi = mid - 1;
      }
      else {
        lo = mid + 1;
      }
    }
    return ans;
  }
 
  // Driver Function
  public static void Main()
  {
    int n = 4, p = 1, q = 100;
    int []A = { 20100, 1, 202, 202 };
 
    // printing the required answer
    Console.WriteLine(solve(n, A, p, q));
  }
}
 
// This code is contributed by Samim Hossain Mondal.


Javascript




<script>
    // JavaScript  program for find minimum
    // sum of changes in an array
    const INT_MAX = 2147483647;
 
    // function to check if the candidate sum
    // satisfies the condition of the problem
    const isValid = (candidate, pre, n, A, p, q) => {
     
        // flag variable to check wrong answer
        let flag = true;
        for (let i = 1; i < n; i++) {
 
            // Now for each element, we are checking
            // if its ratio with sum of all previous
            // elements + candidate is greater than p/q.
            // If so, we will return false.
            let curr_sum = pre[i - 1] + candidate;
            if (A[i] * q > p * curr_sum) {
                flag = false;
                break;
            }
             
            // comparing like A[i]/(curr_sum)>p/q
            // will be error prone.
        }
        return flag;
    }
 
    let solve = (n, A, p, q) => {
 
        // declaring and constructing
        // prefix sum array
        let pre = new Array(n).fill(0);
        pre[0] = A[0];
        for (let i = 1; i < n; i++) {
            pre[i] = A[i] + pre[i - 1];
        }
 
        // setting lower and upper bound for
        // binary search
        let lo = 0, hi = INT_MAX, ans = INT_MAX;
 
        // since minimum answer is needed,
        // so it is initialized with INT_MAX
        while (lo <= hi) {
 
            // calculating mid by using (lo+hi)/2
            // may overflow in certain cases
            let mid = lo + parseInt((hi - lo) / 2);
 
            // checking if required ratio would be
            // achieved by all elements if "mid" is
            // considered as answer
            if (isValid(mid, pre, n, A, p, q)) {
                ans = mid;
                hi = mid - 1;
            }
            else {
                lo = mid + 1;
            }
        }
        return ans;
    }
 
    // Driver Function
    let n, p, q;
    n = 4, p = 1, q = 100;
    let A = [20100, 1, 202, 202];
 
    // printing the required answer
    document.write(solve(n, A, p, q));
 
// This code is contributed by rakeshsahni
 
</script>


 
 

Output

99

 

Time Complexity: O(n log(INT_MAX))
Auxiliary space: O(n)

 


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