# 714. Best Time to Buy and Sell Stock with Transaction Fee

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] = 1Selling at prices[3] = 8Buying at prices[4] = 4Selling at prices[5] = 9The total profit is ((8 - 1) - 2) + ((9 - 4) - 2) = 8.
```

**Note:**

`0 < prices.length <= 50000`.

`0 < prices[i] < 50000`.

`0 <= fee < 50000`.

```cpp
// Dynamic programming
// buy[i] = max(buy[i - 1], sell[i - 1] - prices[i])
// sell[i] = max(sell[i - 1], buy[i - 1] + prices[i] - fee)
int maxProfit(vector<int>& prices, int fee) { // time: O(n); space: O(1)
    long preBuy = INT_MIN, buy = INT_MIN, sell = 0;
    for (int price : prices) {
        preBuy = buy;
        buy = max(buy, sell - price);
        sell = max(sell, preBuy + price - fee);
    }
    return (int)sell;
}
```

```cpp
// DP with 1D vector
int maxProfit(vector<int>& prices, int fee) { // time: O(n); space: O(n)
    int n = prices.size();
    if (n <= 1) return 0;
    vector<int> buy(n, 0), sell(n, 0);
    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.back();
}
```

```cpp
// Space-Optimized DP
int maxProfit(vector<int>& prices, int fee) { // time: O(n); space: O(1)
    int n = prices.size();
    if (n <= 1) return 0;
    int buy = -prices[0], sell = 0, preBuy;
    for (int i = 1; i < n; ++i) {
        preBuy = buy;
        buy = max(buy, sell - prices[i]);
        sell = max(sell, preBuy + prices[i] - fee);
    }
    return sell;
}
```

```cpp
// Space-Optimized DP
int maxProfit(vector<int>& prices, int fee) { // time: O(n); space: O(1)
    int n = prices.size();
    if (n <= 1) return 0;
    int buy = -prices[0], sell = 0, preSell;
    for (int i = 1; i < n; ++i) {
        preSell = sell;
        sell = max(sell, buy + prices[i] - fee);
        buy = max(buy, preSell - prices[i]);
    }
    return sell;
}
```

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[121.-best-time-to-buy-and-sell-stock](121.-best-time-to-buy-and-sell-stock)
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[122.-best-time-to-buy-and-sell-stock-ii](122.-best-time-to-buy-and-sell-stock-ii)
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