Minimum number of given operations required to reduce a number to 2
Given a positive integer N, the task is to reduce N to 2 by performing the following operations minimum number of times:
- Operation 1: Divide N by 5, if N is exactly divisible by 5.
- Operation 2: Subtract 3 from N.
If it is not possible, print -1.
Input: N = 28
Explanation: Operation 1: Subtract 3 from 28. Therefore, N becomes 28 – 3 = 25.
Operation 2: Divide 25 by 5. Therefore, N becomes 25 / 5 = 5.
Operation 3: Subtract 3 from 5. Therefore, N becomes 5 – 3 = 2.
Hence, the minimum number of operations required is 3.
Explanation: Operation 1: Divide 10 by 5, so n becomes 10/5=2.
Hence, the minimum operations required is 1.
Naive Approach: The idea is to recursively compute the minimum number of steps required.
- If the number is not divisible by 5, then subtract 3 from n and recur for the modified value of n, adding 1 to the result.
- Else make two recursive calls, one by subtracting 3 from n and the other by diving n by 5 and return the one with the minimum number of operations, adding 1 to the result.
Time Complexity: O(2n)
Auxiliary Space: O(1)
Efficient Approach: To optimize the above approach, the idea is to use dynamic programming. Follow these steps to solve this problem.
- Create an array, say dp[n+1] to store minimum operations and initialize all the entries with INT_MAX, where dp[i] stores the minimum number of steps required to reach 2 from i.
- Handle the base case by initializing dp as 0.
- Iterate in the range [2, n] using the variable i
- If the value of i*5 ≤ n, then update dp[i*5] to minimum of dp[i*5] and dp[i]+1.
- If the value of i+3 ≤ n, then update dp[i+3] to minimum of dp[i+3] and dp[i]+1.
- Print the value of dp[n] as the result.
Below is the implementation of the above approach:
Time Complexity: O(n)
Auxiliary Space: O(n)