651. 4 Keys Keyboard

Description

Imagine you have a special keyboard with the following keys:

Key 1: (A): Print one 'A' on screen.

Key 2: (Ctrl-A): Select the whole screen.

Key 3: (Ctrl-C): Copy selection to buffer.

Key 4: (Ctrl-V): Print buffer on screen appending it after what has already been printed.

Now, you can only press the keyboard for N times (with the above four keys), find out the maximum numbers of 'A' you can print on screen.

Example 1:

Input: N = 3
Output: 3
Explanation:
We can at most get 3 A's on screen by pressing following key sequence:
A, A, A

Example 2:

Input: N = 7
Output: 9
Explanation:
We can at most get 9 A's on screen by pressing following key sequence:
A, A, A, Ctrl A, Ctrl C, Ctrl V, Ctrl V

Note:

1. 1 <= N <= 50
2. Answers will be in the range of 32-bit signed integer.

Solution

DP, time O(n), space O(n)

We use i steps to reach maxA(i) then use the remaining n - i steps to reach n - i - 1 copies of maxA(i)

For example:
A, A, A, Ctrl A, Ctrl C, Ctrl V, Ctrl V
Here we have n = 7 and we used i = 3 steps to reach AAA
Then we use the remaining n - i = 4 steps: Ctrl A, Ctrl C, Ctrl V, Ctrl V, to reach n - i - 1 = 3 copies of AAA

We either don’t make copies at all, in which case the answer is just n, or if we want to make copies, we need to have 3 steps reserved for Ctrl A, Ctrl C, Ctrl V so i can be at most n - 3

class Solution {
    public int maxA(int n) {
        int[] dp = new int[n + 1];
        
        for (int i = 1; i <= n; ++i) {
            dp[i] = i;
            for (int j = i - 3; j > 0; --j) {
                dp[i] = Math.max(dp[i], (i - j - 1) * dp[j]);
            }
        }
        
        return dp[n];
    }
}