Minimum Steps to Reduce a Number to 1 Using Dynamic Programming

Problem Statement: Given a positive integer ‘N’, our objective is to compute and return the minimum number of steps needed to reduce ‘N’ to 1. We have three permissible operations:

1. Subtract 1 from it.
2. If ‘N’ is divisible by 2, divide it by 2.
3. If ‘N’ is divisible by 3, divide it by 3.

Understanding the Approach: To efficiently solve this problem, we’ll employ dynamic programming to compute the minimum steps needed for each integer from 1 to ‘N’. We’ll iteratively calculate the minimum steps required to reduce each number to 1, leveraging the results of previously computed numbers.

Algorithm Overview:

1. Initialize a list, dp, to store the minimum steps needed for each number from 1 to ‘N’.
2. Set base case: dp[1] = 0, as it takes 0 steps to reduce 1 to 1.
3. Iterate over each number from 2 to ‘N’:
   • Compute the minimum steps needed to reduce the current number:
     • Subtract 1 from the previous number: dp[i] = dp[i – 1] + 1.
     • If the current number is divisible by 2, compare the minimum steps between subtracting 1 and dividing by 2: dp[i] = min(dp[i], dp[i // 2] + 1).
     • If the current number is divisible by 3, compare the minimum steps between the previous steps and dividing by 3: dp[i] = min(dp[i], dp[i // 3] + 1).
4. The minimum steps to reduce ‘N’ to 1 is stored in dp[N].

Implementation in Python:

def min_steps_to_one(N):
    dp = [0] * (N + 1)
    dp[1] = 0
    
    for i in range(2, N + 1):
        dp[i] = dp[i - 1] + 1
        if i % 2 == 0:
            dp[i] = min(dp[i], dp[i // 2] + 1)
        if i % 3 == 0:
            dp[i] = min(dp[i], dp[i // 3] + 1)
    
    return dp[N]

# Main function
if __name__ == "__main__":
    T = int(input().strip())  # Read the number of test cases
    for _ in range(T):
        N = int(input().strip())  # Read the value of N for the current test case
        print(min_steps_to_one(N))  # Find and print the minimum steps

Conclusion: With the aid of dynamic programming, we’ve successfully cracked the code to determine the minimum steps required to reduce a positive integer to 1. By mastering this technique, you’re equipped to tackle similar challenges with confidence and efficiency. Keep exploring the realms of algorithmic problem-solving and unleash your potential. Happy coding!