This content originally appeared on DEV Community and was authored by MD ARIFUL HAQUE
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, return the 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:
- Input: n = 1
- Output: 0
Example 3:
- Input: n = 10
- Output: 7
Example 2:
- Input: n = 24
- Output: 9
Constraints:
1 <= n <= 1000
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
ncharacters'A'on the screen.
- We start with one
-
Dynamic Programming Approach:
- Use a dynamic programming (DP) array
dpwheredp[i]represents the minimum number of operations required to get exactlyicharacters on the screen. - Initialize
dp[1] = 0since it takes 0 operations to have one'A'on the screen. - For each number of characters
ifrom 2 ton, calculate the minimum operations by checking every divisor ofi. Ifiis divisible byd, then:- The number of operations needed to reach
iis the sum of the operations to reachdplus the operations required to multiplydto geti.
- The number of operations needed to reach
- Use a dynamic programming (DP) array
-
Steps to Solve:
- Initialize a DP array with
INF(or a large number) for all values exceptdp[1]. - For each
ifrom 2 ton, iterate through possible divisors ofiand updatedp[i]based on the operations needed to reachiby copying and pasting.
- Initialize a DP array with
Let's implement this solution in PHP: 650. 2 Keys Keyboard
<?php
// Example usage
echo minOperations(3); // Output: 3
echo minOperations(1); // Output: 0
echo minOperations(10) . "\n"; // Output: 7
echo minOperations(24) . "\n"; // Output: 9
?>
Explanation:
-
Initialization:
dpis initialized with a large number (PHP_INT_MAX) to represent an initially unreachable state. -
Divisor Check: For each number
i, check all divisorsd. Updatedp[i]by considering the operations required to reachdand then multiplying to geti. -
Output: The result is the value of
dp[n], which gives the minimum operations required to get exactlyncharacters on the screen.
This approach ensures we compute the minimum operations efficiently for the given constraints.
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This content originally appeared on DEV Community and was authored by MD ARIFUL HAQUE
MD ARIFUL HAQUE | Sciencx (2024-08-19T18:51:28+00:00) 650. 2 Keys Keyboard. Retrieved from https://www.scien.cx/2024/08/19/650-2-keys-keyboard/
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