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Burst Balloons

Certification: Advanced Level Accuracy: 100% Submissions: 1 Points: 15

Write a Java program to find the maximum coins you can collect by bursting balloons. 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. If i-1 or i+1 goes out of bounds of the array, then treat it as if there is a balloon with a 1 painted on it.

Return the maximum coins you can collect by bursting the balloons wisely.

Example 1
    
  
  • Input: nums = [3,1,5,8]
    • 
  
  • Output: 167
    • 
  
  • Explanation: 
    Burst balloon with value 1, getting 3*1*5 = 15 coins. 
    Remaining balloons: [3,5,8]
Burst balloon with value 5, getting 3*5*8 = 120 coins. 
    Remaining balloons: [3,8]
Burst balloon with value 3, getting 1*3*8 = 24 coins. 
    Remaining balloons: [8]
Burst balloon with value 8, getting 1*8*1 = 8 coins. 
    No balloons remain.
Total coins = 15 + 120 + 24 + 8 = 167
    •
Example 2
    
  
  • Input: nums = [1,5]
    • 
  
  • Output: 10
    • 
  
  • Explanation:
    Burst balloon with value 1, getting 1*1*5 = 5 coins. 
    Remaining balloons: [5]
Burst balloon with value 5, getting 1*5*1 = 5 coins. 
    No balloons remain.
Total coins = 5 + 5 = 10
    •
Constraints
    
  
  • n == nums.length
    • 
  
  • 1 <= n <= 300
    • 
  
  • 0 <= nums[i] <= 100
    • 
  
  • Time Complexity: O(n³)
    • 
  
  • Space Complexity: O(n²)
    •
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Hint
Solution

Solution Hints

    
  
  • Use dynamic programming to solve this problem
    • 
  
  • Instead of considering which balloon to burst first, consider which balloon to burst last
    • 
  
  • Create a new array with 1s at the beginning and end to handle boundary cases
    • 
  
  • Define dp[i][j] as the maximum coins that can be obtained by bursting all balloons between i and j (inclusive)
    • 
  
  • Use a bottom-up approach to fill the dp table
    • 
  
  • The final answer will be dp[1][n], where n is the number of balloons
    •

Steps to solve by this approach:

 Step 1: Prepare the input by adding 1s at the beginning and end of the array to handle boundary cases.

 Step 2: Create a 2D DP array where dp[i][j] represents the maximum coins obtainable by bursting all balloons from index i to  j.

 Step 3: Use a bottom-up approach, starting with subarrays of length 1 and gradually increasing to the entire array.

 Step 4: For each subarray (i,j), consider each balloon k between i and j as the last one to burst.

 Step 5: When k is the last balloon to burst, the coins obtained are newNums[i-1] * newNums[k] * newNums[j+1].

 Step 6: Add to this the maximum coins from the left subarray (i,k-1) and right subarray (k+1,j).
 Step 7: Return dp[1][n] as the final answer, which represents the maximum coins from the entire array.

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