Burst Balloon to maximize coins

We have been given N balloons, each with a number of coins associated with it. On bursting a balloon i, the number of coins gained is equal to A[i-1]*A[i]*A[i+1]. Also, balloons i-1 and i+1 now become adjacent. Find the maximum possible profit earned after bursting all the balloons. Assume an extra 1 at each boundary.

Examples:

Input : 5, 10
Output : 60
Explanation - First Burst 5, Coins = 1*5*10
              Then burst 10, Coins+= 1*10*1
              Total = 60

Input : 1, 2, 3, 4, 5
Output : 110

A recursive solution is discussed here. We can solve this problem using dynamic programming.
First, consider a sub-array from indices Left to Right(inclusive).
If we assume the balloon at index Last to be the last balloon to be burst in this sub-array, we would say the coined gained to be-A[left-1]*A[last]*A[right+1].
Also, the total Coin Gained would be this value, plus dp[left][last – 1] + dp[last + 1][right], where dp[i][j] means maximum coin gained for sub-array with indices i, j.
Therefore, for each value of Left and Right, we need find and choose a value of Last with maximum coin gained, and update the dp array.
Our Answer is the value at dp[1][N].

Python3

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# Python program burst balloon problem. 
  
def getMax(A):
    N = len(A)
    A = [1] + A + [1]# Add Bordering Balloons
    dp = [[0 for x in range(N + 2)] for y in range(N + 2)]# Declare DP Array
      
    for length in range(1, N + 1):
        for left in range(1, N-length + 2):
            right = left + length -1
  
            # For a sub-array from indices left, right
            # This innermost loop finds the last balloon burst
            for last in range(left, right + 1):
                dp[left][right] = max(dp[left][right], \
                                      dp[left][last-1] + \
                                      A[left-1]*A[last]*A[right + 1] + \
                                      dp[last + 1][right])
    return(dp[1][N])
  
# Driver code
A = [1, 2, 3, 4, 5]
print(getMax(A))

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Output:

110


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