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

You are given an array prices where prices[i] is the price of a given stock on the ith day, and an integer fee representing a transaction fee.

Find the maximum profit you can achieve. You may complete as many transactions as you like, but you need to pay the transaction fee for each transaction.

Note: You may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again).

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.

Example 2:

Input: prices = [1,3,7,5,10,3], fee = 3
Output: 6

Constraints:

• 1 < prices.length <= 5 * 104
• 0 < prices[i], fee < 5 * 104

class Solution {
    public int maxProfit(int[] prices, int fee) {
        int cash = 0, hold = -prices[0];
        for (int i = 1; i < prices.length; i++) {
            cash = Math.max(cash, hold + prices[i] - fee);
            hold = Math.max(hold, cash - prices[i]);
        }
        return cash;
    }
}

Intuition and Algorithm

At the end of the i-th day, we maintain cash, the maximum profit we could have if we did not have a share of stock, and hold, the maximum profit we could have if we owned a share of stock.

To transition from the i-th day to the i+1-th day, we either sell our stock cash = max(cash, hold + prices[i] - fee) or buy a stock hold = max(hold, cash - prices[i]). At the end, we want to return cash. We can transform cash first without using temporary variables because selling and buying on the same day can't be better than just continuing to hold the stock.

很有道理，分为第i天结束时手里有stock和没stock，第i+1天要么买要么卖，更新对应的max值即可。