Guess Number Higher or Lower II

Question

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.

Tags

• Dynamic Programming
• Minmax

Thought

This problem is harder than its easier one: Guess Number Higher or Lower, because this question requires the dynamic programming.

We can consider this question as a gaming problem that each player wants to defeat his or her opponent. For the gaming problem, we usually use the Minmax algorithm for Dynamic Programming.

Code

class Solution(object):
    def getMoneyAmount(self, n):
        """
        :type n: int
        :rtype: int
        """
        # dynamic programming
        dp_table = [[0] * (n + 1) for _ in range(n + 1)]
        for lo in xrange(n, 0, -1):
            for hi in xrange(lo + 1, n + 1):
                dp_table[lo][hi] = min(i + max(dp_table[lo][i - 1], dp_table[i + 1][hi]) for i in xrange(lo, hi))
        return dp_table[1][n]