Guess Number Higher or Lower II问题及解法

问题描述：

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.

示例:

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.

问题分析：

本题的题意要理解清楚：

For each number x in range[i~j]
we do: result_when_pick_x = x + max{DP([i~x-1]),
DP([
a42e
x+1, j])}
--> // the max means whenever you choose
a number, the feedback is always bad and therefore leads you to a worse branch.
then we get DP([i~j]) = min{xi,
... ,xj}
--> // this min makes sure that you
are minimizing your cost.

过程详见代码：

vector<vector<int>> dp(n + 1, vector<int>(n + 1, 0));
for (int j = 2; j <= n; j++){
for (int i = j - 1; i>0; i--){
int globalMin = INT_MAX;
for (int k = i + 1; k<j; k++){
int localMax = k + max(dp[i][k - 1], dp[k + 1][j]);
globalMin = min(globalMin, localMax);
}
dp[i][j] = i + 1 == j ? i : globalMin;
}
}
return dp[1]
;