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Minimum operations required to reduce N to 0 by either replacing N with N/M or incrementing M by 1

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  • Last Updated : 11 Nov, 2021
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Given two integers N and M, the task is to calculate the minimum number of operations required to reduce N to 0 using the following operations:

  • Replace N with (N/M).
  • Increment the value of M by 1.

Example:

Input: N = 9, M = 2
Output: 4
Explanation: The given example can be solved by following the below sequence of operations:

  • In 1st operation, replace N with (N/M), i.e, N = 9/2 = 4.
  • In 2nd operation, again replace N with N/M, i.e, N = 4/2 = 2.
  • In 3rd operation, increment M by 1, i.e, M = M+1 = 2+1 = 3.
  • In 4th operation, replace N with N/M, i.e, N = 2/3 = 0.

Hence, the number of required operations is 4 which is the minimum possible.

Input: N = 15, M = 1
Output: 5

 

Approach: The given problem can be solved by observing the fact that the most optimal choice of operations is to increment the value of M let’s say x times and then reduce the value of N to N / (M+x) until it becomes 0. To find the best case, iterate over all values of x in the range [0, √N] using a variable i and calculate the number of steps required to reduce N to 0 by dividing it by (M+i). Keep track of the minimum number of operations over all possible values of (M+i) in a variable ans, which is the required value. 

Below is the implementation of the above approach:

C++




// C++ Program of the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the minimum count of
// operations to reduce N to 0 using M
int findMinimum(int N, int M)
{
    // If N is already 0
    if (N == 0) {
        return 0;
    }
 
    // Stores the minimum count of operations
    int ans = INT_MAX;
 
    // Loop to iterate in the range [0, √N]
    for (int i = 0; i * i <= N; i++) {
 
        // Edge case to prevent infinite looping
        if (M == 1 && i == 0) {
            continue;
        }
 
        // Stores the current count of moves
        int count = i;
        int tempN = N;
 
        // Number of operations required to
        // reduce N to 0 by dividing by M + i
        while (tempN != 0) {
            tempN /= (M + i);
            count++;
        }
 
        // Update the final count
        ans = min(count, ans);
    }
 
    // Return answer
    return ans;
}
 
// Driver code
int main()
{
    int N = 9;
    int M = 2;
 
    cout << findMinimum(N, M);
 
    return 0;
}


Java




/*package whatever //do not write package name here */
 
import java.io.*;
 
class GFG {
 
    public static int findMinimum(int N, int M)
    {
        // If N is already 0
        if (N == 0) {
            return 0;
        }
 
        // Stores the minimum count of operations
        int ans = 1000000007;
 
        // Loop to iterate in the range [0, √N]
        for (int i = 0; i * i <= N; i++) {
 
            // Edge case to prevent infinite looping
            if (M == 1 && i == 0) {
                continue;
            }
 
            // Stores the current count of moves
            int count = i;
            int tempN = N;
 
            // Number of operations required to
            // reduce N to 0 by dividing by M + i
            while (tempN != 0) {
                tempN /= (M + i);
                count++;
            }
 
            // Update the final count
            if(count < ans){
              ans = count;
            }
        }
 
        // Return answer
        return ans;
    }
 
    // Driver code
 
    public static void main(String[] args)
    {
        int N = 9;
        int M = 2;
 
        System.out.println(findMinimum(N, M));
    }
}
 
// This code is contributed by maddler.


Python3




# Python Program to implement
# the above approach
 
# Function to find the minimum count of
# operations to reduce N to 0 using M
def findMinimum(N, M):
 
    # If N is already 0
    if (N == 0):
        return 0
 
    # Stores the minimum count of operations
    ans = 10**9
 
    # Loop to iterate in the range[0, √N]
    i = 0
    while(i * i <= N):
        i += 1
 
        # Edge case to prevent infinite looping
        if (M == 1 and i == 0):
            continue
 
        # Stores the current count of moves
        count = i
        tempN = N
 
        # Number of operations required to
        # reduce N to 0 by dividing by M + i
        while (tempN != 0):
            tempN = tempN // (M + i)
            count += 1
 
        # Update the final count
        ans = min(count, ans)
 
    # Return answer
    return ans
 
# Driver code
N = 9
M = 2
 
print(findMinimum(N, M))
 
# This code is contributed by Saurabh Jaiswal


C#




/*package whatever //do not write package name here */
 
using System;
 
class GFG
{
 
    public static int findMinimum(int N, int M)
    {
        // If N is already 0
        if (N == 0)
        {
            return 0;
        }
 
        // Stores the minimum count of operations
        int ans = 1000000007;
 
        // Loop to iterate in the range [0, √N]
        for (int i = 0; i * i <= N; i++)
        {
 
            // Edge case to prevent infinite looping
            if (M == 1 && i == 0)
            {
                continue;
            }
 
            // Stores the current count of moves
            int count = i;
            int tempN = N;
 
            // Number of operations required to
            // reduce N to 0 by dividing by M + i
            while (tempN != 0)
            {
                tempN /= (M + i);
                count++;
            }
 
            // Update the final count
            if (count < ans)
            {
                ans = count;
            }
        }
 
        // Return answer
        return ans;
    }
 
    // Driver code
 
    public static void Main()
    {
        int N = 9;
        int M = 2;
 
        Console.WriteLine(findMinimum(N, M));
    }
}
 
// This code is contributed by Saurabh Jaiswal


Javascript




<script>
       // JavaScript Program to implement
       // the above approach
 
       // Function to find the minimum count of
       // operations to reduce N to 0 using M
       function findMinimum(N, M)
       {
        
           // If N is already 0
           if (N == 0) {
               return 0;
           }
 
           // Stores the minimum count of operations
           let ans = Number.MAX_VALUE;
 
           // Loop to iterate in the range [0, √N]
           for (let i = 0; i * i <= N; i++) {
 
               // Edge case to prevent infinite looping
               if (M == 1 && i == 0) {
                   continue;
               }
 
               // Stores the current count of moves
               let count = i;
               let tempN = N;
 
               // Number of operations required to
               // reduce N to 0 by dividing by M + i
               while (tempN != 0) {
                   tempN = Math.floor(tempN / (M + i));
                   count++;
               }
 
               // Update the final count
               ans = Math.min(count, ans);
           }
 
           // Return answer
           return ans;
       }
 
       // Driver code
       let N = 9;
       let M = 2;
 
       document.write(findMinimum(N, M));
 
   // This code is contributed by Potta Lokesh
   </script>


Output

4

Time complexity: O(N*log N ) 
Auxiliary Space: O(1)


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