Leetcode 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.

Analysis:
The problem can be done by DFS + memorization.
Note that the problem asks for the worst case scenario since you need to guarantee a win. In the worst case scenario, you need to guess wrong for each time expect for there is only 1 number. 

Code (Java):

class Solution {
    public int getMoneyAmount(int n) {
        int[][] dp = new int[n + 1][n + 1];
        
        getMoneyAmountHelper(1, n, dp);
        
        return dp[1][n];
    }
    
    private int getMoneyAmountHelper(int start, int end, int[][] dp) {
        if (start >= end) {
            return 0;
        }
        
        if (dp[start][end] != 0) {
            return dp[start][end];
        }
        
        int minCost = Integer.MAX_VALUE;
        for (int i = start; i <= end; i++) {
            int cost = i + Math.max(getMoneyAmountHelper(start, i - 1, dp), 
                                    getMoneyAmountHelper(i + 1, end, dp));
            minCost = Math.min(minCost, cost);
        }
        
        dp[start][end] = minCost;
        
        return minCost;
    }
}