Problem Statement
2 Keys Keyboard
Initially, there is only one character 'A' on the screen. You can perform 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, return the minimum number of operations to get exactly n 'A's on the screen.
Examples
Example 1:
Input: n = 3
Output: 3
Explanation: Initially, we have one character 'A'.
Step 1: Copy All => 'A'
Step 2: Paste => 'AA'
Step 3: Paste => 'AAA'
Example 2:
Input: n = 1
Output: 0
Explanation: We already have one 'A' on the screen, so no operations are needed.
Constraints
- 1 <= n <= 1000
Problem Breakdown
To solve this problem, we need to:
- This problem can be solved using dynamic programming or by finding the prime factorization of n.
- For the dynamic programming approach, we can define dp[i] as the minimum number of operations to get i 'A's on the screen.
- For each i, we can try to paste j copies (where j is a divisor of i) and see which one gives the minimum operations.
- Alternatively, we can observe that the minimum number of operations is the sum of the prime factors of n (counting duplicates).
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
2 Keys Keyboard
Initially, there is only one character 'A' on the screen. You can perform 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, return the minimum number of operations to get exactly n 'A's on the screen.
Examples
Example 1:
Initially, we have one character 'A'. Step 1: Copy All => 'A' Step 2: Paste => 'AA' Step 3: Paste => 'AAA'
Example 2:
We already have one 'A' on the screen, so no operations are needed.
Problem Breakdown
Constraints:
- 1 <= n <= 1000
Key Insights:
This problem can be solved using dynamic programming or by finding the prime factorization of n.
For the dynamic programming approach, we can define dp[i] as the minimum number of operations to get i 'A's on the screen.
For each i, we can try to paste j copies (where j is a divisor of i) and see which one gives the minimum operations.
Alternatively, we can observe that the minimum number of operations is the sum of the prime factors of n (counting duplicates).
Real-World Application
This problem has several practical applications:
Text Editing
Optimizing copy-paste operations in text editors to minimize keystrokes.
Resource Allocation
Finding the most efficient way to allocate resources when only certain operations are allowed.
Process Optimization
Minimizing the number of steps in a process where each step has specific constraints.