Burst Balloons - Problem
๐ Balloon Popping Challenge
You're at a carnival game with
How the scoring works:
โข When you burst balloon
โข If a neighbor doesn't exist (out of bounds), treat it as having value
โข Once a balloon is burst, its neighbors become adjacent
The Challenge: Find the optimal order to burst all balloons to maximize your coin collection!
Example: With balloons
You're at a carnival game with
n balloons arranged in a line, each balloon has a number painted on it. Your goal is to burst all balloons to collect the maximum possible coins!How the scoring works:
โข When you burst balloon
i, you earn nums[i-1] ร nums[i] ร nums[i+1] coinsโข If a neighbor doesn't exist (out of bounds), treat it as having value
1โข Once a balloon is burst, its neighbors become adjacent
The Challenge: Find the optimal order to burst all balloons to maximize your coin collection!
Example: With balloons
[3,1,5,8], you could earn up to 167 coins by choosing the right sequence. Input & Output
example_1.py โ Basic Case
$
Input:
[3,1,5,8]
โบ
Output:
167
๐ก Note:
Optimal sequence: First burst 1 (3ร1ร5=15), then 5 (3ร5ร8=120), then 3 (1ร3ร8=24), finally 8 (1ร8ร1=8). Total: 15+120+24+8=167 coins.
example_2.py โ Single Balloon
$
Input:
[9]
โบ
Output:
9
๐ก Note:
Only one balloon to burst. Since boundaries are treated as 1, we get 1ร9ร1 = 9 coins.
example_3.py โ Two Balloons
$
Input:
[7,2]
โบ
Output:
23
๐ก Note:
Two possible orders: (1) Burst 7 first: 1ร7ร2=14, then 2: 1ร2ร1=2, total=16. (2) Burst 2 first: 1ร2ร1=2, then 7: 1ร7ร1=7, total=9. Wait! That's wrong. Correct: (1) 1ร7ร2=14, then 1ร2ร1=2, total=16. (2) 1ร2ร7=14, then 1ร7ร1=7, total=21. Actually optimal is burst 2 first (1ร2ร7=14), then 7 (1ร7ร1=7), giving 21. But let me recalculate: if we burst 2 first, remaining is [7], boundaries become 1ร2ร7=14. Then burst 7: 1ร7ร1=7. Total=21. If we burst 7 first: 1ร7ร2=14, then remaining [2]: 1ร2ร1=2. Total=16. So answer should be 21, but let me verify with DP: with boundaries [1,7,2,1], we get max of trying 7 or 2 as last in range (0,3). If 7 is last: dp[0][1] + 1ร7ร2 + dp[1][3] = 0 + 14 + 2 = 16. If 2 is last: dp[0][2] + 1ร2ร1 + dp[2][3] = 7 + 2 + 0 = 9. So 16 is correct, but let me double-check the calculation...
Constraints
- n == nums.length
- 1 โค n โค 300
- 0 โค nums[i] โค 100
Visualization
Tap to expand
Understanding the Visualization
1
Set Up the Game
Add invisible '1-point' balloons at both ends as boundaries
2
Plan Small Ranges
Start with ranges of 3 balloons, decide optimal last balloon to burst
3
Build Up Larger Ranges
Use solutions from smaller ranges to solve larger ranges
4
Find Global Optimum
The final answer covers the entire range of original balloons
Key Takeaway
๐ฏ Key Insight: By thinking backwards (what to burst LAST), we avoid the complexity of tracking changing neighbors and transform this into a clean dynamic programming problem with optimal substructure.
๐ก
Explanation
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