Burst Balloons Leetcode Problem 312 [Python Solution]

Welcome to another Python problem-solving session.

In this post, we'll tackle a challenging problem from LeetCode, the Burst Balloons problem.

It's categorized as a "Hard" problem and falls under the domain of 2-D Dynamic Programming.

We'll explore this problem step by step, providing a detailed Python solution.

Problem Name: Burst Balloons
Difficulty: Hard
Category: 2-D Dynamic Programming
Companies: Google

Keyword: Burst Balloons LeetCode

Question Link

Problem Overview

In the Burst Balloons problem, you are given n balloons indexed from 0 to n - 1.

Each balloon is painted with a number represented by an array called nums.

You are asked to burst all the balloons.

When you burst the ith balloon, you will gain nums[i - 1] * nums[i] * nums[i + 1] coins.

If i - 1 or i + 1 goes out of bounds of the array, you should treat it as if there is a balloon with a 1 painted on it.

Your task is to determine the maximum number of coins you can collect by bursting the balloons wisely.

Let's illustrate this with an example.

Example:

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

Explanation:

1. Bursting balloons in this sequence:

   • [3, 1, 5, 8] → [3, 5, 8] → [3, 8] → [8] → []

2. Calculate the total coins collected:

   • 3*1*5 + 3*5*8 + 1*3*8 + 1*8*1 = 167

Constraints:

• n == nums.length
• 1 <= n <= 300
• 0 <= nums[i] <= 100

Understanding the Problem Constraints

Before we dive into the solution, let's first understand the constraints provided in the problem.

• n is the number of balloons, and it is constrained to be between 1 and 300.
• The values in the nums array are integers ranging from 0 to 100.

Burst Balloons LeetCode Problem Solution

Now, let's explore the solution to the Burst Balloons problem.

1. Bruteforce Approach

At first glance, you might think of using a brute force approach, where you consider different orders in which to pop the balloons.

However, this approach results in a time complexity of O(n!), which is not efficient.

We can do much better by using dynamic programming.

The key insight is to consider which balloon to pop last, which can lead to an efficient approach.

2. An Efficient Approach: Dynamic Programming

We will use a dynamic programming approach to solve this problem efficiently.

We'll create a 2D array, dp, where dp[left][right] represents the maximum number of coins that can be obtained by bursting balloons within the range [left, right] of the input array nums.

This will help us avoid redundant calculations.

Here's the Python code for our efficient approach:

def maxCoins(nums):
    # Add implicit 1s at the beginning and end of the input array
    nums = [1] + nums + [1]
    n = len(nums)

    # Initialize a cache to store computed results
    cache = {}

    # Helper function to calculate the maximum coins in a subarray
    def burst(left, right):
        # Check if the result is already cached
        if (left, right) in cache:
            return cache[(left, right)]

        # Base case: If the subarray contains only one balloon, return its value
        if left + 1 == right:
            return 0

        max_coins = 0

        # Try bursting each balloon in the subarray
        for pivot in range(left + 1, right):
            coins = nums[left] * nums[pivot] * nums[right]
            coins += burst(left, pivot) + burst(pivot, right)
            max_coins = max(max_coins, coins)

        # Cache the result for this subarray
        cache[(left, right)] = max_coins
        return max_coins

    # Start the dynamic programming process
    return burst(0, n - 1)

In this code, we first add implicit 1s at the beginning and end of the nums array to account for the boundary balloons.

The cache dictionary is used to store and retrieve previously computed results to avoid redundant calculations.

We define a helper function burst(left, right) that recursively calculates the maximum coins for the subarray defined by the left and right boundaries.

The function explores various balloon popping sequences and returns the maximum coins that can be obtained for the given subarray.

The dynamic programming process starts with a call to burst(0, n - 1), where n is the length of the modified nums array.

This covers the entire problem, and the result is returned.

Time and Space Complexity

Now, let's analyze the time and space complexity of our solution.

Time Complexity: Our dynamic programming approach has a time complexity of O(n^3) due to the nested loops in the burst function.

For each subarray of size n, we explore all possible balloon popping sequences, resulting in a cubic time complexity.

Space Complexity: The space complexity is O(n^2) due to the 2D cache dictionary, which stores results for various subarrays.

The rest of the space usage is dominated by the implicit 1s added to the nums array.

Reasoning Behind Our Approach

The key insight behind our approach is to consider the order in which we pop the balloons to maximize the total number of coins.

We use dynamic programming to break down the problem into subproblems, where we determine the optimal order for popping balloons within specific subarrays.

By adding implicit 1s to the nums array and caching the results for subproblems, we avoid redundant calculations and efficiently compute the maximum coins achievable.

The dynamic programming process allows us to consider all possible balloon popping sequences while maximizing the total reward.

Our approach ensures that we make the best choices when deciding which balloon to pop last, ultimately leading to the optimal solution.

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Conclusion

In this post, we tackled the challenging Burst Balloons problem from LeetCode using a dynamic programming approach.

We developed an efficient Python solution to maximize the total number of coins obtained by popping balloons in the optimal order.

We explored the problem constraints, presented our approach, and provided a detailed Python code solution.

Our dynamic programming technique allowed us to break down the problem into manageable subproblems, and we used caching to store intermediate results, resulting in an efficient solution.

This problem is an excellent example of how dynamic programming can be applied to complex scenarios to find optimal solutions.

If you found this

post helpful, feel free to explore more coding challenges and enhance your problem-solving skills.

Happy coding!