Best Time to Buy and Sell Stock with Transaction Fee

题目
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.

答案
This problem adds in another layer of complexity to common dynamic programming question, as your best buying point depends on the best selling point. These two information needs to be stored in different arrays, or variable.

Explanation:
keep track of the best buying point and best selling point
1 does buying at ith day make it a worse buying decision than the best buying point? if yes, don't buy it.
2 does selling at ith day makes it a worse selling decision than best selling point? if yes, don't sell it

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