[LeetCode] 312. Burst Balloons_hard tag: 区间Dynamic Programming
Given n
balloons, indexed from 0
to n-1
. Each balloon is painted with a number on it represented by array nums
. You are asked to burst all the balloons. If the you burst balloon i
you will get nums[left] * nums[i] * nums[right]
coins. Here left
and right
are adjacent indices of i
. After the burst, the left
and right
then becomes adjacent.
Find the maximum coins you can collect by bursting the balloons wisely.
Note:
- You may imagine
nums[-1] = nums[n] = 1
. They are not real therefore you can not burst them. - 0 ≤
n
≤ 500, 0 ≤nums[i]
≤ 100
Example:
Input:[3,1,5,8]
Output:167
nums = [3,1,5,8] --> [3,5,8] --> [3,8] --> [8] --> []
Explanation:
coins = 3*1*5 + 3*5*8 + 1*3*8 + 1*8*1 = 167
这个题目的思路也是利用 区间Dynamic Programming, 思想跟[LeetCode] 877. Stone Game == [LintCode] 396. Coins in a Line 3_hard tag: 区间Dynamic Programming, 博弈类似,
动态表达式为:
dp[i][j] 是得到的 [i,j] 里面气球的最大值
for k in [i,j]:
value_m = ballons[i-1] * ballons[k] * ballons[j+1]
dp[i][j] = max(dp[i][j], dp[i][k-1] + value_m + dp[k+1][j])
init:
dp[i][i] = ballons[i-1]* ballons[i]* ballons[i+1]
dp[i][j] = 0 if j < i
1. Constraints
1) size [0, 500]
2) element [0, 100]
2. Ideas
Dynamic Programming T: O(n^2) S; O(n^2)
3. Code
class Solution:
def burstBallons(self, nums):
n = len(nums)
ballons = [1] + nums + [1] # plus the corner case
dp = [[0]*(n+2) for _ in range(n+2)] # for corner case
flag = [[0]*(n+2) for _ in range(n+2)]
def helper(l, r):
if flag[l][r]:
return dp[l][r]
if l == r:
dp[l][r] = ballons[l-1] * ballons[l] * ballons[l+1]
elif l < r:
for k in range(l, r+1): # k belongs [l, r], so it is r + 1
value_m = ballons[l-1] * ballons[k] * ballons[r+1]
value_l = helper(l, k-1)
value_r = helper(k+1, r)
dp[l][r] = max(dp[l][r], value_l + value_m + value_r)
flag[l][r] = 1
return dp[l][r]
return helper(1, n)
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