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Minimize steps to reach K from 0 by adding 1 or doubling at each step

Given a positive integer K, the task is to find the minimum number of operations of the following two types, required to change 0 to K.

Examples: 

Input: K = 1 
Output:
Explanation: 
Step 1: 0 + 1 = 1 = K

Input: K = 4 
Output:
Explanation: 
Step 1: 0 + 1 = 1, 
Step 2: 1 * 2 = 2, 
Step 3: 2 * 2 = 4 = K 

Approach: 

Uses dynamic programming to solve this problem. In this case, the subproblems are the minimum number of operations required to reach each number from 1 to k.

Initializes an array dp[] of size k+1 to store the minimum number of operations required to reach each number from 1 to k. Iterates through the numbers from 1 to k, and for each number i, it considers two options:

  • Adding 1 to the previous number i-1. This is represented by dp[i] = dp[i-1] + 1.
  • Doubling the previous even number i/2. This is represented by dp[i] = min(dp[i], dp[i/2] + 1).

The first option is straightforward. To reach number i, you can simply add 1 to the previous number i-1. The second option, if i is even, you can reach it by doubling the previous even number i/2. This is because doubling an even number always results in an even number.

By considering both options for each number, we’ll ensures that it finds the minimum number of operations required to reach k. The final result is stored in dp[k], which represents the minimum number of operations needed to reach k from 1.

Step-by-step approach:

Below is the implementation of the above approach: 




// C++ program to implement the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to find minimum operations
int minOperation(int k)
{
    // vector dp is initialised
    // to store the steps
    vector<int> dp(k + 1, 0);
 
    for (int i = 1; i <= k; i++) {
 
        dp[i] = dp[i - 1] + 1;
 
        // For all even numbers
        if (i % 2 == 0) {
            dp[i]
                = min(dp[i],
                      dp[i / 2] + 1);
        }
    }
    return dp[k];
}
 
// Driver Code
int main()
{
    int k = 12;
    cout << minOperation(k);
}




// Java program to implement the above approach
class GFG{
     
// Function to find minimum operations
static int minOperation(int k)
{
     
    // dp is initialised
    // to store the steps
    int dp[] = new int[k + 1];
 
    for(int i = 1; i <= k; i++)
    {
       dp[i] = dp[i - 1] + 1;
        
       // For all even numbers
       if (i % 2 == 0)
       {
           dp[i] = Math.min(dp[i], dp[i / 2] + 1);
       }
    }
    return dp[k];
}
 
// Driver Code
public static void main (String []args)
{
    int K = 12;
    System.out.print( minOperation(K));
}
}
 
// This code is contributed by chitranayal




// C# program to implement the above approach
using System;
class GFG{
     
// Function to find minimum operations
static int minOperation(int k)
{
     
    // dp is initialised
    // to store the steps
    int []dp = new int[k + 1];
 
    for(int i = 1; i <= k; i++)
    {
        dp[i] = dp[i - 1] + 1;
             
        // For all even numbers
        if (i % 2 == 0)
        {
            dp[i] = Math.Min(dp[i], dp[i / 2] + 1);
        }
    }
    return dp[k];
}
 
// Driver Code
public static void Main()
{
    int K = 12;
    Console.Write(minOperation(K));
}
}
 
// This code is contributed by Nidhi_Biet




// Javascript implementation of the above approach
 
// Function to find minimum operations
function minOperation(k)
{
       
    // dp is initialised
    // to store the steps
    let dp = Array.from({length: k+1}, (_, i) => 0);
   
    for(let i = 1; i <= k; i++)
    {
       dp[i] = dp[i - 1] + 1;
          
       // For all even numbers
       if (i % 2 == 0)
       {
           dp[i] = Math.min(dp[i], dp[i / 2] + 1);
       }
    }
    return dp[k];
}
 
  // Driver Code   
    let K = 12;
    document.write( minOperation(K));
  
 // This code is contributed by target_2.




# Python3 program to implement the above approach
 
# Function to find minimum operations
def minOperation(k):
     
    # dp is initialised
    # to store the steps
    dp = [0] * (k + 1)
 
    for i in range(1, k + 1):
        dp[i] = dp[i - 1] + 1
 
        # For all even numbers
        if (i % 2 == 0):
            dp[i]= min(dp[i], dp[i // 2] + 1)
 
    return dp[k]
 
# Driver Code
if __name__ == '__main__':
     
    k = 12
     
    print(minOperation(k))
 
# This code is contributed by mohit kumar 29

Output
5

Time Complexity: O(k)
Auxiliary Space: O(k)


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