Skip to content

🎈 Dynamic Programming: Burst Balloons

📝 Problem Description

You are given n balloons, indexed from 0 to n - 1. Each balloon has a number painted on it, represented by an array nums. You are asked to burst all the balloons. If you burst the ith balloon, you get nums[i - 1] * nums[i] * nums[i + 1] coins. If i - 1 or i + 1 are out of bounds, treat them as 1. Find the maximum coins you can collect.

Real-World Application

This problem is a classic application of Interval Dynamic Programming, used in resource scheduling and optimization where the order of operations significantly impacts the outcome, often modeling scenarios like demolition ordering or sequence dependency management.

🛠️ Constraints & Edge Cases

  • \(1 \le n \le 500\)
  • \(0 \le nums[i] \le 100\)
  • Edge Cases to Watch:
    • Empty or single-balloon arrays.
    • Balloons with 0 value (they can be useful to burst last).

🧠 Approach & Intuition

The Aha! Moment

Instead of deciding which balloon to burst first (which has complex dependencies), decide which balloon to burst last. If a balloon k is the last one to burst in a range (i, j), the balloons (i, k-1) and (k+1, j) are already burst, and the neighbors of k are simply i-1 and j+1.

🐢 Brute Force (Naive)

Trying every sequence of bursting balloons results in \(O(N!)\) time complexity, which is computationally infeasible for even moderate \(N\).

🐇 Optimal Approach

We use Interval DP. Let dp[i][j] be the maximum coins obtained by bursting all balloons between indices i and j. For each k in range [i, j] as the last balloon: - coins = nums[i-1] * nums[k] * nums[j+1] - dp[i][j] = max(dp[i][j], dp[i][k-1] + coins + dp[k+1][j])

🧩 Visual Tracing

graph LR
    subgraph Range [i...j]
    I[i] --> K{k - last}
    K --> J[j]
    end
    L[i-1] -->|multiplier| K
    R[j+1] -->|multiplier| K

💻 Solution Implementation

(Implementation details need to be added...)

⏱️ Complexity Analysis

  • Time Complexity: \(\mathcal{O}(N^3)\) — We have two nested loops for the interval size and start index, plus a loop for the split point k.
  • Space Complexity: \(\mathcal{O}(N^2)\) — To store the DP table.

🎤 Interview Toolkit

  • Alternative: Could be modeled as a recursive approach with memoization (top-down), which is often more intuitive to implement.
  • Related Patterns: This is similar to Matrix Chain Multiplication or optimal Binary Search Tree construction.
  • [Regular Expression Matching](../regular_expression_matching/PROBLEM.md)
  • [Edit Distance](../edit_distance/PROBLEM.md)