Burst Balloons

Hard Subject: 2-D Dynamic Programming

Time Complexity

O(N^3)

Space Complexity

O(N^2)

Problem Description

Problem Statement

You are given n balloons, indexed from 0 to n - 1. Each balloon is painted with a number on it represented by an array nums. You are asked to burst all the balloons. If you burst the ith balloon, you will get nums[i - 1] * nums[i] * nums[i + 1] coins. Return the maximum coins you can collect by bursting the balloons cleverly.

Example 1:

Input: nums = [3,1,5,8]
Output: 167

Optimal Solution

Python

Approach: Interval DP

Add a 1 to the beginning and end of the array. dp[i][j] represents the max coins we can get from bursting balloons between i and j (exclusive). We iterate over interval lengths, and for each interval, we guess the last balloon to burst.

class Solution:
    def maxCoins(self, nums: List[int]) -> int:
        nums = [1] + nums + [1]
        n = len(nums)
        dp = [[0] * n for _ in range(n)]

        for length in range(2, n):
            for i in range(n - length):
                j = i + length
                for k in range(i + 1, j):
                    dp[i][j] = max(dp[i][j], 
                                   nums[i] * nums[k] * nums[j] + dp[i][k] + dp[k][j])
        return dp[0][n-1]