# Best Time to Buy and Sell Stock with Transaction Fee

<https://leetcode.com/problems/best-time-to-buy-and-sell-stock-with-transaction-fee/description/>

> Your are given an array of integers prices, for which the i-th element is the price of a given stock on day i; and a non-negative integer fee representing a transaction fee.
>
> You may complete as many transactions as you like, but you need to pay the transaction fee for each transaction. You may not buy more than 1 share of a stock at a time (ie. you must sell the stock share before you buy again.)
>
> Return the maximum profit you can make.
>
> Example 1:
>
> Input: prices = \[1, 3, 2, 8, 4, 9], fee = 2
>
> Output: 8
>
> Explanation: The maximum profit can be achieved by:
>
> Buying at prices\[0] = 1
>
> Selling at prices\[3] = 8
>
> Buying at prices\[4] = 4
>
> Selling at prices\[5] = 9
>
> The total profit is ((8 - 1) - 2) + ((9 - 4) - 2) = 8.

## Thoughts

max问题，且每天的状态是从前面一天跳转过来的，用DP。action分别为buy和sell。难点在于能想到要针对action分别维持两个不同类型的状态。buy\[i] = max(sell\[i - 2] - price, buy(i - 1)), 表示前i天的任意序列中以buy为结尾的最大利润，可以分别是前面i-1天中买的，也可以是在第i天买的。sell\[i] = max(buy(i - 1) + price - fee, sell(i - 1)), 表示前i天的任意序列中以sell为结尾的最大利润。同理可以是当天卖出，也可以是前面卖出。全局最优肯定不能以持有结束，只会是sell\[N-1]。

## Code

```cpp
class Solution {
public:
    int maxProfit(vector<int>& prices, int fee) {
        const int N = prices.size(); 
        vector<int> buy(N), sell(N);
        buy[0] = -prices[0];
        for (int i = 1; i < N; ++i) {
            buy[i] = max(buy[i - 1], sell[i - 1] - prices[i]);
            sell[i] = max(sell[i - 1], buy[i - 1] + prices[i] - fee);
        }
        return sell[N - 1];
    }
};
```

## Analysis

Errors:

1. 不是buy和sell都有fee, 两者选一个加入fee即可.

时间复杂度O(N), 空间可以优化成O(1).


---

# Agent Instructions: 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:

```
GET https://hao-fu-1.gitbook.io/oj/dynamic_programming_i/duo-zhuang-tai-dp/best-time-to-buy-and-sell-stock-with-transaction-fee.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
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.