Burst Balloons

07/17/2016 Divide and Conquer Dynamic Programming

Question

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:

1. You may imagine nums[-1] = nums[n] = 1. They are not real therefore you can not burst them.
2. 0 ≤ n ≤ 500, 0 ≤ nums[i] ≤ 100

Example:

Given [3, 1, 5, 8]

Return 167

    nums = [3,1,5,8] --> [3,5,8] -->   [3,8]   -->  [8]  --> []
coins =  3*1*5      +  3*5*8    +  1*3*8      + 1*8*1   = 167


Solution

Result: Accepted Time: 12 ms

Here should be some explanations.

#define MAXN 256
int dp[MAXN][MAXN]={0},num[MAXN];
int max(int a,int b){return a > b? a:b;}
int maxCoins(int* nums, int nsz) {
for(int i = 0 ; i < nsz; i++ )
num[i+1] = nums[i];
num[0] = num[nsz+1] = 1;
for(int i = 0; i <= nsz; i++)
for(int j = 1,k=i+j; k<= nsz;j++,k++)
{
dp[j][k] = 0;
for(int t = j; t <= k; t++)
dp[j][k] = max(dp[j][k],dp[j][t-1]+dp[t+1][k]+num[j-1]*num[t]*num[k+1]);
}
return dp[1][nsz];
}


Complexity Analytics

• Time Complexity: $O(n^2)$
• Space Complexity: $O(n^2)$