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650. 2 Keys Keyboard
Difficulty:Medium
Topics:Math, Dynamic Programming
There is only one character 'A' on the screen of a notepad. You can perform one of two operations on this notepad for each step:
- Copy All: You can copy all the characters present on the screen (a partial copy is not allowed).
- Paste: You can paste the characters which are copied last time.
Given an integer n, returnthe minimum number of operations to get the character 'A' exactly n times on the screen.
Example 1:
- Input:n = 3
- Output:3
- Explanation:Initially, we have one character 'A'.
- In step 1, we use Copy All operation.
- In step 2, we use Paste operation to get 'AA'.
- In step 3, we use Paste operation to get 'AAA'.
Example 2:
Example 3:
Example 2:
Constraints:
Hint:
- How many characters may be there in the clipboard at the last step if n = 3? n = 7? n = 10? n = 24?
Solution:
We need to find the minimum number of operations to get exactly n characters 'A' on the screen. We'll use a dynamic programming approach to achieve this.
Understanding the Problem:
- We start with one 'A' on the screen.
- We can either "Copy All" (which copies the current screen content) or "Paste" (which pastes the last copied content).
- We need to determine the minimum operations required to have exactly n characters 'A' on the screen.
Dynamic Programming Approach:
- Use a dynamic programming (DP) array dp where dp[i] represents the minimum number of operations required to get exactly i characters on the screen.
- Initialize dp[1] = 0 since it takes 0 operations to have one 'A' on the screen.
- For each number of characters i from 2 to n, calculate the minimum operations by checking every divisor of i. If i is divisible by d, then:
- The number of operations needed to reach i is the sum of the operations to reach d plus the operations required to multiply d to get i.
Steps to Solve:
- Initialize a DP array with INF (or a large number) for all values except dp[1].
- For each i from 2 to n, iterate through possible divisors of i and update dp[i] based on the operations needed to reach i by copying and pasting.
Let's implement this solution in PHP:650. 2 Keys Keyboard
Explanation:
- Initialization:dp is initialized with a large number (PHP_INT_MAX) to represent an initially unreachable state.
- Divisor Check:For each number i, check all divisors d. Update dp[i] by considering the operations required to reach d and then multiplying to get i.
- Output:The result is the value of dp[n], which gives the minimum operations required to get exactly n characters on the screen.
This approach ensures we compute the minimum operations efficiently for the given constraints.
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