Reduce given three Numbers by decrementing in Pairs

Given three integers A, B and C. In one operation, choose any two of the three integers, subject to the condition that both of them should be greater than 0, and reduce them by 1. The task is to find the maximum number of operations that can be performed until at least two of them becomes 0.

Examples:

Input: A = 1, B = 3, C = 1
Output: 2
Explanation:
Operation 1: Choose A and B, reduce both by 1. Current values: A = 0, B = 2, C = 1
Operation 2: Choose B and C, reduce both by 1. Current Values: A = 0, B = 1, C = 0
No more opeartions are possible as any pair chosen will have at least one 0 in it.

Input: A = 8, B = 1, C = 4
Output: 5

Approach: The idea is to arrange the given numbers in decreasing order find the maximum number of operations on the basis of the following condition:
Case 1: When A ≥ (B + C)



  • Choose pair A and B for B times. As a result, after B operations the current status will be A = (A – B) and B = 0.
  • As A ≥ (B + C) which implies (A – B) ≥ C. So the pair A and C can be chosen for C operations, and the current status will be A = (A – B – C), B = 0 and C = 0.
  • Total operations performed = (B + C)

Case 2: When A < (B + C)

  • Try to make A, B, C equal after performing some operations.
  • First make A and B equal. For this choose A and C for performing (A – B) operations. Let the updated values be named A1, B1 and C1. The values A1, B1 and C1 will be:

    A1 = A – (A – B)
    B1 = B
    C1 = C – (A – B)

  • The number of operations performed = (A – B).
  • A1 and B1 are equal. So, choose the pair A1 and B1 for (A1 – C1) operations.
  • Let A2, B2 and C2 be the updated values of A, B and C after the above operation. The values of A2, B2 and C2 will be same and that will be:

    A2 = A1 – (A1 – C1) = C1 = (C – A + B)
    B2 = C – A + B
    C2 = C – A + B

  • Let the total number of operations performed as of now be Z. So the value of Z will be:

    Z = (A – B) + (A1 – C1) = (A – B) + (B – C + A – B)= 2A – B – C

  • As A2 = B2 = C2, then there arises two cases:
    1. A2, B2, C2 are even: For every set of 3 operations on the pairs (A2, B2), (B2, C2), and (C2, A2) the count of the A2, B2 and C2 decreases by 2.
      Let A2 = B2 = C2 = 4. Let the operations that can be performed be X. So, X = (4 + 4 + 4) / 2 = 6. Thus the value of X can be generalised as:

      X = \frac{(A_2 + B_2 + C_2)}{2} = \frac{(3 * (C - A + B))}{2}

    2. A2, B2, C2 are odd: For every set of 3 operations on the pairs (A2, B2), (B2, C2), and (C2, A2) the count of the A2, B2 and C2 decreases by 2, finally the values A2, B2 and C2 reach 1, 1 and 1 respectively. Here one additional opeartion can be performed.
      Let A2 = B2 = C2 = 5. After performing 6 operations, A2 = B2 = C2 = 1. Here one more operation can be performed. Therefore total opeartions that can be performed are 7 (6+1). Let the operations that can be performed be Y. So, Y = floor((5 + 5 + 5) / 2) = 7. Thus the value of Y can be generalised as:

      Y = \frac{(A_2 + B_2 + C_2)}{2} = \frac{(3 * (C - A + B))}{2}

  • Since from the above steps X = Y, therefore total number of possible cases can be given by:

    Total number of possible cases = (Z + X) = (Z + Y) = (A + B + C) / 2.

Below is the implementation of the above approach:

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// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the minimum number
// operations
int solution(int A, int B, int C)
{
    int arr[3];
  
    // Insert the three numbers in array
    arr[0] = A, arr[1] = B, arr[2] = C;
  
    // Sort the array
    sort(arr, arr + 3);
  
    // Case 2
    if (arr[2] < arr[0] + arr[1])
        return ((arr[0] + arr[1]
                 + arr[2])
                / 2);
    // Case 1
    else
        return (arr[0] + arr[1]);
}
  
// Driver Code
int main()
{
    // Given A, B, C
    int A = 8, B = 1, C = 5;
  
    // Function Call
    cout << solution(A, B, C);
    return 0;
}

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Output:

6


Time Complexity: O(1)
Space Complexity: O(1)

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