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.

Note:

• 0 < prices.length <= 50000.
• 0 < prices[i] < 50000.
• 0 <= fee < 50000.

还是买卖股票的最佳时间问题，这题每一次买卖时会有交易费。

解法：DP。第i天的利润分成两个，用两个dp数组分别进行计算，buy[i], sell[i]。

初始值：buy[0]=-prices[0]， sell[0]=0

公式：

buy[i] = Math.max(buy[i - 1], sell[i - 1] - prices[i])

第i天买，如果第i-1天是买，就不能买了，利润是buy[i-1]。如果i-1天是卖，就可以买，利润是sell[i-1] - prices[i]。

sell[i] = Math.max(sell[i - 1], buy[i - 1] + prices[i])

第i天卖，如果第i-1天是卖，就不能卖了，利润是sell[i-1]。如果i-1天是买，就可以卖，利润是buy[i - 1] + prices[i]。

Most consistent ways of dealing with the series of stock problems

Java: pay the fee when buying the stock

public int maxProfit(int[] prices, int fee) {

        if (prices.length <= 1) return 0;

        int days = prices.length, buy[] = new int[days], sell[] = new int[days];

        buy[0]=-prices[0]-fee;

        for (int i = 1; i<days; i++) {

            buy[i] = Math.max(buy[i - 1], sell[i - 1] - prices[i] - fee); // keep the same as day i-1, or buy from sell status at day i-1

            sell[i] = Math.max(sell[i - 1], buy[i - 1] + prices[i]); // keep the same as day i-1, or sell from buy status at day i-1

        }

        return sell[days - 1];

    }

Java: pay the fee when selling the stock

public int maxProfit(int[] prices, int fee) {

        if (prices.length <= 1) return 0;

        int days = prices.length, buy[] = new int[days], sell[] = new int[days];

        buy[0]=-prices[0];

        for (int i = 1; i<days; i++) {

            buy[i] = Math.max(buy[i - 1], sell[i - 1] - prices[i]); // keep the same as day i-1, or buy from sell status at day i-1

            sell[i] = Math.max(sell[i - 1], buy[i - 1] + prices[i] - fee); // keep the same as day i-1, or sell from buy status at day i-1

        }

        return sell[days - 1];

    }　

class Solution(object):

    def maxProfit(self, prices, fee):

        """

        :type prices: List[int]

        :type fee: int

        :rtype: int

        """

        cash, hold = 0, -prices[0]

        for i in xrange(1, len(prices)):

            cash = max(cash, hold+prices[i]-fee)

            hold = max(hold, cash-prices[i])

        return cash

class Solution {

public:

    int maxProfit(vector<int>& prices, int fee) {

        int s0 = 0, s1 = INT_MIN;

        for(int p:prices) {

            int tmp = s0;

            s0 = max(s0, s1+p);

            s1 = max(s1, tmp-p-fee);

        }

        return s0;

    }

};