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Burst Balloons

Certification: Advanced Level Accuracy: 0% Submissions: 0 Points: 15

Write a C# program to implement the MaxCoins(int[] nums) function. 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.

Example 1
    

  • Input: nums = [3, 1, 5, 8]
    • 

  • Output: 167
    • 

  • Explanation:

      

    • Step 1: Consider all possible orders of bursting the balloons.
      • 

    • Step 2: For the optimal order:

        

      • Burst balloon with value 1: 3 * 1 * 5 = 15 coins
        • 

      • Remaining balloons: [3, 5, 8]
        • 

      • Burst balloon with value 5: 3 * 5 * 8 = 120 coins
        • 

      • Remaining balloons: [3, 8]
        • 

      • Burst balloon with value 3: 1 * 3 * 8 = 24 coins
        • 

      • Remaining balloons: [8]
        • 

      • Burst balloon with value 8: 1 * 8 * 1 = 8 coins
        • 

      

      • 

    • Step 3: Calculate the total coins: 15 + 120 + 24 + 8 = 167
      • 

    • Step 4: Return the maximum possible coins, which is 167.
      • 

    •
Example 2
    

  • Input: nums = [1, 5]
    • 

  • Output: 10
    • 

  • Explanation:

      

    • Step 1: Consider all possible orders of bursting the balloons.
      • 

    • Step 2: For the optimal order:

        

      • Burst balloon with value 1: 1 * 1 * 5 = 5 coins
        • 

      • Remaining balloons: [5]
        • 

      • Burst balloon with value 5: 1 * 5 * 1 = 5 coins
        • 

      

      • 

    • Step 3: Calculate the total coins: 5 + 5 = 10
      • 

    • Step 4: Return the maximum possible coins, which is 10.
      • 

    •
Constraints
    

  • n == nums.length
    • 

  • 1 ≤ n ≤ 500
    • 

  • 0 ≤ nums[i] ≤ 100
    • 

  • Time Complexity: O(n³) where n is the number of balloons
    • 

  • Space Complexity: O(n²) for the dynamic programming table
    •
ArraysDynamic Programming IBMeBay
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Hint
Solution

Solution Hints

    
  
  • Use dynamic programming with a bottom-up approach
    • 
  
  • Consider the problem from a different angle: instead of thinking about which balloon to burst first, think about which one to burst last
    • 
  
  • Create a 2D array to store the maximum coins you can get by bursting all balloons between indices i and j
    • 
  
  • For each possible range, try each balloon as the last one to burst
    • 
  
  • Compute the value recursively and fill the DP table
    •

Steps to solve by this approach:

 Step 1: Create a new array with 1s at both ends to handle boundary cases.
 Step 2: Initialize a 2D dp array where dp[i][j] represents the maximum coins obtained from bursting all balloons from index i to j.
 Step 3: Iterate through all possible subarray lengths.
 Step 4: For each subarray, try each balloon as the last one to burst.
 Step 5: When choosing the last balloon k to burst, add the coins from bursting k (left-1 * k * right+1) plus the maximum coins from the left and right subarrays.
 Step 6: Fill the dp table bottom-up, starting with smaller subarrays.
 Step 7: Return dp[1][n], which represents the maximum coins from bursting all original balloons.

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