Minimize subtraction of Array elements to make X at most 0

Given a number X, and an array arr[] of length N containing the N numbers. The task is to find the minimum number of operations required to make X non-positive. In one operation:

• Select any one number Y from the array and reduce X by Y. 
• Then make Y = Y/2 (take floor value if Y is odd).
• If it is not possible to make X non-positive, return -1.

Examples:

Input: N = 3, arr[] = {3, 4, 12}, X = 25
Output:  4
Explanation: Operation 1: Y=12,   X reduces to 13,  Y becomes 6, arr[]: {3, 4, 6}
Operation 2: Y = 6, X reduces to 7, Y becomes 3, arr[]: {3, 4, 3}
Operation 3: Y = 4, X reduces to 3, Y becomes 2, arr[]: {3, 2, 3}
Operation 4: Y = 3, X reduces to 0, Y becomes 1, arr[]: {1, 2, 3}
Total operations will be 4.

Input:  N = 3, arr[] = {11, 11, 110}, X = 11011
Output: -1
Explanation: It is impossible to make X non-positive

Approach: This problem can be solved using max-heap (priority queue) based on the following idea:

To minimize subtraction, it is optimal to subtract the maximum value each time. For this reason use a max-heap so that the maximum value numbers remain on top and perform the operation using the topmost element and keep checking if the number becomes non-positive or not.

Follow the below illustration for a better understanding.

Illustration:

Consider arr[] = {3, 4, 12} and X = 25

So max heap (say P) = [3, 4, 12]

1st step:
        => Maximum element (Y) = 12.
        => Subtract 12 from 25. X = 25 – 12 = 13. Y = 12/2 = 6.
        => P = {3, 4, 6}.

2nd step:
        => Maximum element (Y) = 6.
        => Subtract 6 from 13. X = 13 – 6 = 7. Y = 6/2 = 3.
        => P = {3, 3, 4}.

3rd step:
        => Maximum element (Y) = 4.
        => Subtract 4 from 7. X = 7 – 4 = 3. Y = 4/2 = 2.
        => P = {2, 3, 3}.

4th step:
        => Maximum element (Y) = 3.
        => Subtract 3 from 3. X = 3 – 3 = 0. Y = 3/2 = 1.
        => P = {1, 2, 3}.

X is non-positive. So operations required = 4

Follow the steps to solve the problem:

• Create a max-heap (implemented by priority queue)and store all the numbers in it.
• Perform the following until the priority queue is not empty and the X is positive.
  • Use the number having the maximum value. This will be the number on top of the priority queue.
  • Remove the top number from the priority queue and perform the operation as stated in the problem.
  • After performing the operation, if Y is positive add it back to the priority queue.
  • Increment the answer by 1 every time.
• After the completion of the above process, if X is positive then it is impossible to make it non-positive and thus return -1.
• Otherwise, return the answer.

Below is the implementation of the above approach.

C++

// C++ code to implement the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to find minimum operations
int minimumOperations(int N, int X,
                      vector<int> nums)
{

    // Initialize answer as zero
    int ans = 0;

    // Create Max-Heap using
    // Priority-queue
    priority_queue<int> pq;

    // Put all nums in the priority queue
    for (int i = 0; i < N; i++)
        pq.push(nums[i]);

    // Execute the operation on num with
    // max value until nums are left
    // and X is positive
    while (!pq.empty() and X > 0) {
        if (pq.top() == 0)
            break;

        // Increment the answer everytime
        ans++;

        // num with maximum value
        int num = pq.top();

        // Removing the num
        pq.pop();

        // Reduce X's value and num's
        // value as per the operation
        X -= num;
        num /= 2;

        // If the num's value is positive
        // insert back in the
        // priority queue
        if (num > 0)
            pq.push(num);
    }

    // If X's value is positive then it
    // is impossible to make X
    // non-positive
    if (X > 0)
        return -1;

    // Otherwise return the number of
    // operations performed
    return ans;
}

// Drivers code
int main()
{
    int N = 3;
    vector<int> nums = { 3, 4, 12 };
    int X = 25;

    // Function call
    cout << minimumOperations(N, X, nums);
    return 0;
}

Java

// Java code to implement the above approach
import java.io.*;
import java.util.*;

class GFG
{

  // Function to find minimum operations
  public static int minimumOperations(int N, int X,
                                      int nums[])
  {

    // Initialize answer as zero
    int ans = 0;

    // Create Max-Heap using
    // Priority-queue
    PriorityQueue<Integer> pq = new PriorityQueue<>(
      Collections.reverseOrder());

    // Put all nums in the priority queue
    for (int i = 0; i < N; i++)
      pq.add(nums[i]);

    // Execute the operation on num with
    // max value until nums are left
    // and X is positive
    while (!pq.isEmpty() && X > 0) {
      if (pq.peek() == 0)
        break;

      // Increment the answer everytime
      ans++;

      // num with maximum value
      int num = pq.peek();

      // Removing the num
      pq.poll();

      // Reduce X's value and num's
      // value as per the operation
      X -= num;
      num /= 2;

      // If the num's value is positive
      // insert back in the
      // priority queue
      if (num > 0)
        pq.add(num);
    }

    // If X's value is positive then it
    // is impossible to make X
    // non-positive
    if (X > 0)
      return -1;

    // Otherwise return the number of
    // operations performed
    return ans;
  }

  // Driver Code
  public static void main(String[] args)
  {
    int N = 3;
    int nums[] = { 3, 4, 12 };
    int X = 25;

    // Function call
    System.out.print(minimumOperations(N, X, nums));
  }
}

// This code is contributed by Rohit Pradhan

Python3

# Python code to implement the above approach
# importing heapq module for implementing max heap
import heapq as hq
# Function to find minimum operations


def minimumOperations(N, X, nums):
    # Initialize answer as zero
    ans = 0
    # assigning nums to pq
    pq = nums
    # converting pq to max heap
    # using heapq module
    hq._heapify_max(pq)
    # Execute the operation on num with
    # max value until nums are left
    # and X is positive
    while (len(pq) > 0 and X > 0):
        if pq[len(pq)-1] == 0:
            break
        # Increment the answer everytime
        ans += 1
        # num with maximum value
        # first element of max heap
        # contains maximum value
        num = pq[0]
        # deleting one maximum element
        # from max heap
        pq[0] = pq[-1]
        pq.pop()
        # again converting current pq
        # to max heap in O(n) time
        hq._heapify_max(pq)
        # Reduce X's value and num's
        # value as per the operation
        X -= num
        num //= 2
        # If the num's value is positive
        # insert back in the
        # priority queue
        if num > 0:
            pq.append(num)
        hq._heapify_max(pq)
    # If X's value is positive then it
    # is impossible to make X
    # non-positive
    if X > 0:
        return -1
    # Otherwise return the number of
    # operations performed
    return ans


# Drivers code
N = 3
nums = [3, 4, 12]
X = 25
# Function call
print(minimumOperations(N, X, nums))

C#

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

class GFG {

  // Function to find minimum operations
  public static int minimumOperations(int N, int X,
                                      int[] nums)
  {

    // Initialize answer as zero
    int ans = 0;

    // Create Max-Heap using
    // Priority-queue
    List<int> pq = new List<int>();

    // Put all nums in the priority queue
    for (int i = 0; i < N; i++){
      pq.Add(nums[i]);
      pq.Sort();
      pq.Reverse();
    }

    // Execute the operation on num with
    // max value until nums are left
    // and X is positive
    while (pq.Count != 0 && X > 0) {
      if (pq[0] == 0)
        break;

      // Increment the answer everytime
      ans++;

      // num with maximum value
      int num = pq[0];

      // Removing the num
      pq.RemoveAt(0);

      // Reduce X's value and num's
      // value as per the operation
      X -= num;
      num /= 2;

      // If the num's value is positive
      // insert back in the
      // priority queue
      if (num > 0){
        pq.Add(num);
        pq.Sort();
        pq.Reverse();
      }
    }

    // If X's value is positive then it
    // is impossible to make X
    // non-positive
    if (X > 0)
      return -1;

    // Otherwise return the number of
    // operations performed
    return ans;
  }

  // Driver Code
  public static void Main()
  {
    int N = 3;
    int[] nums = { 3, 4, 12 };
    int X = 25;

    // Function call
    Console.WriteLine(minimumOperations(N, X, nums));
  }
}

// This code is contributed by sanjoy_62.

Javascript

<script>

//  JavaScript code to implement the above approach

//  Function to find minimum operations
function minimumOperations(N, X,nums){

    //  Initialize answer as zero
    let ans = 0

    //  Create Max-Heap using
    //  Priority-queue
    let pq = []

    //  Put all nums in the priority queue
    for(let i=0;i<N;i++)
        pq.push(nums[i])

    pq.sort((a,b)=>a-b)

    //  Execute the operation on num with
    //  max value until nums are left
    //  && X is positive
    while (pq.length>0 && X > 0){
        if (pq[pq.length-1]== 0)
            break

        //  Increment the answer everytime
        ans += 1

        //  num with maximum value
        let num = pq[pq.length-1]

        //  Removing the num
        pq.pop()

        //  Reduce X's value && num's
        //  value as per the operation
        X -= num
        num = Math.floor(num/2)

        //  If the num's value is positive
        //  insert back in the
        //  priority queue
        if (num > 0)
            pq.push(num)

        pq.sort()
    }

    //  If X's value is positive then it
    //  is impossible to make X
    //  non-positive
    if (X > 0)
        return -1

    //  Otherwise return the number of
    //  operations performed
    return ans
}

//  Drivers code
let N = 3
let nums = [ 3, 4, 12 ]
let X = 25

//  Function call
document.write(minimumOperations(N, X, nums))

//  This code is contributed by shinjanpatra

<script>

Output

4

Time Complexity: O(N * log N)
Auxiliary Space: O(N)