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):

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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;     } }