123. Best Time to Buy and Sell Stock III

1. Description

You are given an array prices where prices[i] is the price of a given stock on the ith day.
Find the maximum profit you can achieve. You may complete at most two transactions.
Note: You may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again).

2. Example

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

3. Constraints

• 1 <= prices.length <= $10^5$
• 0 <= prices[i] <= $10^5$

4. Solutions

Dynamic Programming

n = prices.size()
Time complexity: O(n)
Space complexity: O(1)

class Solution {
public:
    int maxProfit(const vector<int>& prices) {
        vector<vector<int>> profit(2, vector<int>(4));
        // 0 first trans with stock
        // 1 first trans sell stock
        // 2 second trans with stock
        // 3 second trans sell stock
        profit[0] = {-prices[0], 0, -prices[0], 0};
        for (int i = 1; i < prices.size(); ++i) {
            profit[1][0] = max(profit[0][0], -prices[i]);
            profit[1][1] = max(profit[0][1], profit[0][0] + prices[i]);
            profit[1][2] = max(profit[0][2], profit[0][1] - prices[i]);
            profit[1][3] = max(profit[0][3], profit[0][2] + prices[i]);

            swap(profit[0], profit[1]);
        }

        return *max_element(profit[0].begin(), profit[0].end());
    }
};