Problem Statement
Balloon Popping Strategy
You're playing a carnival game where you need to burst balloons to earn coins. There are n balloons arranged in a row, each with a number painted on it.
When you burst a balloon, you earn coins equal to: the number on the balloon you burst multiplied by the numbers on the adjacent balloons (left and right). After bursting a balloon, its adjacent balloons become adjacent to each other.
For example, if you have balloons with numbers [3,1,5,8] and you burst the balloon with number 1, you would get 3×1×5 = 15 coins, and the remaining balloons would be [3,5,8].
To handle edge cases, you can imagine there are two additional balloons with the number 1 at the beginning and end of the row, but you cannot burst these imaginary balloons.
Your goal is to find the maximum number of coins you can collect by bursting all the balloons in the optimal order.
Examples
Example 1:
Input: nums = [3,1,5,8]
Output: 167
Explanation: Burst balloons in this order to maximize coins:
- Burst balloon with number 1: 3×1×5 = 15 coins, remaining balloons: [3,5,8]
- Burst balloon with number 5: 3×5×8 = 120 coins, remaining balloons: [3,8]
- Burst balloon with number 3: 1×3×8 = 24 coins, remaining balloons: [8]
- Burst balloon with number 8: 1×8×1 = 8 coins, no remaining balloons
Total coins: 15 + 120 + 24 + 8 = 167
Example 2:
Input: nums = [1,5]
Output: 10
Explanation: Burst balloons in this order to maximize coins:
- Burst balloon with number 1: 1×1×5 = 5 coins, remaining balloons: [5]
- Burst balloon with number 5: 1×5×1 = 5 coins, no remaining balloons
Total coins: 5 + 5 = 10
Constraints
- n == nums.length
- 1 <= n <= 300
- 0 <= nums[i] <= 100
Problem Breakdown
To solve this problem, we need to:
- The order in which you burst the balloons matters significantly
- Greedy approaches (like always bursting the balloon with the smallest or largest number) don't work for this problem
- This problem has optimal substructure, making it suitable for dynamic programming
- Instead of thinking about which balloon to burst first, consider which balloon to burst last
- The problem can be reformulated as: for each subarray, which balloon should be the last one to burst to maximize coins?
- Using a bottom-up approach helps build solutions for larger subarrays from smaller ones
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Balloon Popping Strategy
You're playing a carnival game where you need to burst balloons to earn coins. There are n balloons arranged in a row, each with a number painted on it.
When you burst a balloon, you earn coins equal to: the number on the balloon you burst multiplied by the numbers on the adjacent balloons (left and right). After bursting a balloon, its adjacent balloons become adjacent to each other.
For example, if you have balloons with numbers [3,1,5,8] and you burst the balloon with number 1, you would get 3×1×5 = 15 coins, and the remaining balloons would be [3,5,8].
To handle edge cases, you can imagine there are two additional balloons with the number 1 at the beginning and end of the row, but you cannot burst these imaginary balloons.
Your goal is to find the maximum number of coins you can collect by bursting all the balloons in the optimal order.
Examples
Example 1:
Burst balloons in this order to maximize coins: - Burst balloon with number 1: 3×1×5 = 15 coins, remaining balloons: [3,5,8] - Burst balloon with number 5: 3×5×8 = 120 coins, remaining balloons: [3,8] - Burst balloon with number 3: 1×3×8 = 24 coins, remaining balloons: [8] - Burst balloon with number 8: 1×8×1 = 8 coins, no remaining balloons Total coins: 15 + 120 + 24 + 8 = 167
Example 2:
Burst balloons in this order to maximize coins: - Burst balloon with number 1: 1×1×5 = 5 coins, remaining balloons: [5] - Burst balloon with number 5: 1×5×1 = 5 coins, no remaining balloons Total coins: 5 + 5 = 10
Problem Breakdown
Constraints:
- n == nums.length
- 1 <= n <= 300
- 0 <= nums[i] <= 100
Key Insights:
The order in which you burst the balloons matters significantly
Greedy approaches (like always bursting the balloon with the smallest or largest number) don't work for this problem
This problem has optimal substructure, making it suitable for dynamic programming
Instead of thinking about which balloon to burst first, consider which balloon to burst last
The problem can be reformulated as: for each subarray, which balloon should be the last one to burst to maximize coins?
Using a bottom-up approach helps build solutions for larger subarrays from smaller ones
Real-World Application
This problem has several practical applications:
Game Strategy Optimization
Developing optimal strategies for games where the order of actions affects the final score.
Resource Allocation
Planning the sequence of resource utilization to maximize overall efficiency or profit.
Decision Tree Analysis
Evaluating different decision paths to find the optimal sequence of choices.
Operations Research
Optimizing complex operational processes where the order of operations impacts the outcome.