> For the complete documentation index, see [llms.txt](https://wentao-shao.gitbook.io/leetcode/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://wentao-shao.gitbook.io/leetcode/graph-search/312.burst-balloons.md).

# 312.Burst-Balloons

## 312. Burst Balloons

## 题目地址

<https://leetcode.com/problems/burst-balloons/>

## 题目描述

```
Given n balloons, indexed from 0 to n-1. Each balloon is painted with a number on it represented by array nums. You are asked to burst all the balloons. If the you burst balloon i you will get nums[left] * nums[i] * nums[right] coins. Here left and right are adjacent indices of i. After the burst, the left and right then becomes adjacent.

Find the maximum coins you can collect by bursting the balloons wisely.

Note:
You may imagine nums[-1] = nums[n] = 1. They are not real therefore you can not burst them.
0 ≤ n ≤ 500, 0 ≤ nums[i] ≤ 100

Example:
Input: [3,1,5,8]
Output: 167 
Explanation: nums = [3,1,5,8] --> [3,5,8] -->   [3,8]   -->  [8]  --> []
             coins =  3*1*5      +  3*5*8    +  1*3*8      + 1*8*1   = 167
```

## 代码

### Approach #1 DFS

The best score after adding the `i`th balloon is given by:

`nums[left] * nums[i] * nums[right] + dp(left, i) + dp(i, right)`

```java
class Solution {
  public int maxCoins(int[] nums) {
    // reframe the problem
        int n = nums.length + 2;
    int[] new_nums = new int[n];

    for (int i = 0; i < nums.length; i++) {
      new_nums[i + 1] = nums[i];
    }

    new_nums[0] = new_nums[n - 1] = 1;

    int[][] memo = new int[n][n];

    return dfs(memo, new_nums, 0, n - 1);
  }

  private int dfs(int[][] memo, int[] nums, int left, int right) {
    if (left + 1 == right)    return 0;

    if (memo[left][right] > 0)    return memo[left][right];

    int ans = 0;
    for (int i = left + 1; i < right; i++) {
      ans = Math.max(ans, nums[left] * nums[i] * nums[right]
                    + dfs(memo, nums, left, i)
                    + dfs(memo, nums, i, right));
    }
    memo[left][right] = ans;
    return ans;
  }

}
```

### Approach #2 Dynamic Programming Bottom-Up

`dp[left][right] = Math.max(dp[left][right], new_nums[left] * new_nums[i] * new_nums[right] dp[left][i] + dp[i][right]);`

```java
class Solution {
  public int maxCoins(int[] nums) {
    int n = nums.length + 2;
    int[] new_nums = new int[n];

    for (int i = 0; i < nums.length; i++) {
      new_nums[i + 1] = nums[i];
    }

    new_nums[0] = new_nums[n - 1] = 1;


    int[][] dp = new int[n][n];

    for (int left = n - 2; left >= 0; left--) {
      for (int right = left + 2; right < n; right++) {
        for (int i = left + 1; i < right; i++) {
          dp[left][right] = Math.max(dp[left][right], 
                                    new_nums[left] * new_nums[i] * new_nums[right] + dp[left][i] + dp[i][right]);
        }
      }
    }
    return dp[0][n - 1];
  }
}
```


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter:

```
GET https://wentao-shao.gitbook.io/leetcode/graph-search/312.burst-balloons.md?ask=<question>&goal=<endgoal>
```

`ask` is the immediate question: it should be specific, self-contained, and written in natural language.
`goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal.

The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.