375. Guess Number Higher or Lower II

We are playing the Guess Game. The game is as follows:

I pick a number from 1 to n. You have to guess which number I picked.

Every time you guess wrong, I’ll tell you whether the number I picked is higher or lower.

However, when you guess a particular number x, and you guess wrong, you pay $x. You win the game when you guess the number I picked.

Example:

n = 10, I pick 8.

First round:  You guess 5, I tell you that it's higher. You pay $5.
Second round: You guess 7, I tell you that it's higher. You pay $7.
Third round:  You guess 9, I tell you that it's lower. You pay $9.

Game over. 8 is the number I picked.

You end up paying $5 + $7 + $9 = $21.

Given a particular n ≥ 1, find out how much money you need to have to guarantee a win.

此题可以看做枚举到第 i 个数时，且第 i 个数不是期望的数值，则分别要在 1~i-1 和 i +1 到 n之间找满足所需的最大的惩罚；

int dfs_help(vector<vector<int>>& dp, int left, int right){
if (left >= right)return 0;
if (dp[left][right])return dp[left][right];

int res = INT_MAX;
for (int i = left; i <= right; i++){
int temp = i + max(dfs_help(dp, left, i - 1), dfs_help(dp, i + 1, right));
res = min(res, temp);
}
dp[left][right] = res;
return res;
}

int getMoneyAmount(int n) {
vector<vector<int>> dp(n + 1, vector<int>(n + 1, 0));
dfs_help(dp, 1, n);
return dp[1]
;
}