375. Guess Number Higher or Lower II**-优快云博客

本文探讨了一种猜数字游戏的最佳策略，目标是最小化可能的最大损失。通过动态规划的方法，从递归和自底向上两种角度实现了算法，并提供了具体实现代码。

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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.

Hint:

The best strategy to play the game is to minimize the maximum loss you could possibly face. Another strategy is to minimize the expected loss. Here, we are interested in thefirst scenario.
Take a small example (n = 3). What do you end up paying in the worst case?
Check out this article if you're still stuck.
The purely recursive implementation of minimax would be worthless for even a small n. You MUST use dynamic programming.
As a follow-up, how would you modify your code to solve the problem of minimizing the expected loss, instead of the worst-case loss?

dynamic programming:

up-bottom:

public class Solution {
    public int getMoneyAmount(int n) {
        int[][] table = new int[n+1][n+1];
        return dp(table,1,n);
    }
    private int dp(int[][] table, int s, int e){
        if(s>=e) return 0;
        if(table[s][e]!=0) return table[s][e];
        int res =  Integer.MAX_VALUE;
        for(int x=s;x<=e;x++){
            int temp = x+Math.max(dp(table,s,x-1), dp(table,x+1,e));
            res= Math.min(temp,res);
        }
        table[s][e]=res;
        return res;
    }
}

bottom-up:

public class Solution {
    public int getMoneyAmount(int n) {
        int[][] table = new int[n+1][n+1];
        for(int j=2;j<=n;j++){
            for(int i=j-1;i>=0;i--){
                int globalmin = Integer.MAX_VALUE;
                for(int k=i+1;k<j;k++){
                    int temp = k+Math.max(table[i][k-1],table[k+1][j]);
                    globalmin=Math.min(globalmin, temp);
                }
                table[i][j]=i+1==j?i:globalmin;
            }
            
        }
        return table[1][n];
    }
}

总结：关键是拆分问题。