Burst Balloons

Burst Balloons

What is this problem about?

The Burst Balloons interview question is a complex optimization problem. You are given n balloons, each with a number on it. If you burst balloon i, you gain nums[i-1] * nums[i] * nums[i+1] coins. After bursting, the neighbors of the burst balloon become adjacent. You need to find the maximum coins you can collect by bursting all balloons. This Burst Balloons coding problem is a classic example of Interval Dynamic Programming.

Why is this asked in interviews?

Tech giants like Google, Meta, and Uber use this to test a candidate's mastery of Dynamic Programming. It requires a non-obvious transformation: instead of thinking about which balloon to burst first, you think about which balloon to burst last in a given interval. It tests your ability to break a problem down into independent sub-problems.

Algorithmic pattern used

This problem follows the Array, Dynamic Programming interview pattern, specifically Matrix Chain Multiplication style DP. You define dp[i][j] as the maximum coins you can get from bursting all balloons in the interval (i, j). The base cases are empty intervals, and you build up by trying every possible "last balloon to burst" in the range.

Example explanation

Suppose nums = [3, 1, 5]. Add 1s to the ends: [1, 3, 1, 5, 1]. To find max coins for (1, 3, 1, 5, 1):

• If 1 is the last to burst in the range [3, 1, 5]: Coins = dp[left_of_1] + dp[right_of_1] + (1 * 1 * 1). (Note: the boundaries are the 1s we added).
• We iterate through all choices (3, 1, 5) being the last one to burst and take the maximum. The "last burst" logic is key because it ensures the sub-problems are independent.

Common mistakes candidates make

• Greedy approach: Trying to burst the smallest or largest balloon first. This doesn't work because the order matters in complex ways.
• Top-down without memoization: Attempting to solve recursively without storing results, leading to O(N!) complexity.
• Incorrect boundary handling: Forgetting to pad the array with 1 at both ends to handle the edge cases for the first and last balloons.

Interview preparation tip

Master "Interval DP." It is frequently used for problems where the result of an action depends on the state of the surrounding elements. Other similar problems include "Minimum Cost to Cut a Stick" or "Optimal Binary Search Tree."