Problem

Say you have an array for which the ith element is the price of a given stock on day i.

Design an algorithm to find the maximum profit. You may complete at most two transactions.

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

Example 1:

Input: [3,3,5,0,0,3,1,4]
Output: 6
Explanation: Buy on day 4 (price = 0) and sell on day 6 (price = 3), profit = 3-0 = 3.
             Then buy on day 7 (price = 1) and sell on day 8 (price = 4), profit = 4-1 = 3.

Example 2:

Input: [1,2,3,4,5]
Output: 4
Explanation: Buy on day 1 (price = 1) and sell on day 5 (price = 5), profit = 5-1 = 4.
             Note that you cannot buy on day 1, buy on day 2 and sell them later, as you are
             engaging multiple transactions at the same time. You must sell before buying again.

Example 3:

Input: [7,6,4,3,1]
Output: 0
Explanation: In this case, no transaction is done, i.e. max profit = 0.

问题描述

给定一个股票每天的售价，如果你可以完成最多两次交易，求最大的收益。注意，你必须要卖出股票才能买入新的股票。

样例

输入: [3, 3, 5, 0, 0, 3, 1, 4]
输出: 6
解释: 在第四天购入股票并在第六天卖出股票，收益为 3。然后在第七天购入股票，在第八天卖出股票，
收益为 3。总收益为 6，为最大收益。

解法 1

这道题与Best Time to Buy and Sell Stock 及 Best Time to Buy and Sell Stock 2 不同，这两道题分别是完成最多一次交易和完成任意多次交易。而这道题，要求是最多完成两次交易。我们的难点即在于如何处理完成两次交易的情形。

我们利用与前两题类似的方法来做，但这道题由于可以完成最多两次交易，故我们需要四个数组 dp1、dp2、dp3、dp4: