花花酱 LeetCode 799. Champagne Tower

题目大意：往一个香槟塔（i层有i个杯子）倒入n个杯子容量的香槟之后，求指定位置杯子中酒的容量。

Problem:

https://leetcode.com/problems/champagne-tower/description/

We stack glasses in a pyramid, where the first row has 1 glass, the second row has 2 glasses, and so on until the 100th row.  Each glass holds one cup (250ml) of champagne.

Then, some champagne is poured in the first glass at the top.  When the top most glass is full, any excess liquid poured will fall equally to the glass immediately to the left and right of it.  When those glasses become full, any excess champagne will fall equally to the left and right of those glasses, and so on.  (A glass at the bottom row has it’s excess champagne fall on the floor.)

For example, after one cup of champagne is poured, the top most glass is full.  After two cups of champagne are poured, the two glasses on the second row are half full.  After three cups of champagne are poured, those two cups become full – there are 3 full glasses total now.  After four cups of champagne are poured, the third row has the middle glass half full, and the two outside glasses are a quarter full, as pictured below.

Now after pouring some non-negative integer cups of champagne, return how full the j-th glass in the i-th row is (both i and j are 0 indexed.)

 

 

Note:

• poured will be in the range of [0, 10 ^ 9].
• query_glass and query_row will be in the range of [0, 99].

Idea: DP + simulation

define dp[i][j] as the volume of champagne will be poured into the j -th glass in the i-th row, dp[i][j] can be greater than 1.

Init dp[0][0] = poured.

Transition: if dp[i][j] > 1, it will overflow, the overflow part will be evenly distributed to dp[i+1][j], dp[i+1][j+1]

if dp[i][j] > 1:
dp[i+1][j] += (dp[i][j] – 1) / 2.0
dp[i+1][j + 1] += (dp[i][j] – 1) / 2.0

Answer: min(1.0, dp[query_row][query_index])

Solution 1:

C++

Time complexity: O(100*100)

Space complexity: O(100*100)

Pull

 

C++

Time complexity: O(rows * 100)

Space complexity: O(100)

Push

Pull