188. Best Time to Buy and Sell Stock IV

1. Description

You are given an integer array prices where prices[i] is the price of a given stock on the ith day, and an integer k.
Find the maximum profit you can achieve. You may complete at most k transactions: i.e. you may buy at most k times and sell at most k times.
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: k = 2, prices = [2,4,1]
Output: 2
Explanation: Buy on day 1 (price = 2) and sell on day 2 (price = 4), profit = 4-2 = 2.

Example 2

Input: k = 2, prices = [3,2,6,5,0,3]
Output: 7
Explanation: Buy on day 2 (price = 2) and sell on day 3 (price = 6), profit = 6-2 = 4. Then buy on day 5 (price = 0) and sell on day 6 (price = 3), profit = 3-0 = 3.

3. Constraints

• 1 <= k <= 100
• 1 <= prices.length <= 1000
• 0 <= prices[i] <= 1000

4. Solutions

Dynamic Programming

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

class Solution {
public:
    int maxProfit(int k, vector<int> &prices) {
        vector<pair<int, int>> profits(k, {numeric_limits<int>::min(), 0});
        for (int price : prices) {
            profits[0].first = max(profits[0].first, -price);
            profits[0].second = max(profits[0].second, profits[0].first + price);
            for (int i = 1; i < k; ++i) {
                profits[i].first = max(profits[i].first, profits[i - 1].second - price);
                profits[i].second = max(profits[i].second, profits[i].first + price);
            }
        }

        return profits.back().second;
    }
};