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.

Solution: DFS (MiniMax)

class Solution(object):
    def getMoneyAmount(self, n):
        """
        :type n: int
        :rtype: int
        """
        memo = [[0] * (n + 1) for _ in xrange(n + 1)]
        return self.min_money(1, n, memo)

    def min_money(self, begin, end, memo):
        if begin >= end:
            return 0
        if memo[begin][end]:
            return memo[begin][end]

        min_money = float('inf')
        for num in xrange(begin, end + 1):
            left = self.min_money(begin, num - 1, memo)
            right = self.min_money(num + 1, end, memo)
            money = num + max(left, right)
            min_money = min(min_money, money)
        memo[begin][end] = min_money
        return min_money

Lessons:

• When we pick a number x, we recursively calculate the money needed for both left side (1 to x-1) and right side (x+1 to n). Pick the worse side (more money), and plus x, that is the money needed if x is our first pick.
• Calculate the money needed for every number as the first pick. Return the minimum one.