Dynamic Programming Questions Medium
Dynamic Programming can be used to solve the Best Time to Buy and Sell Stock problem by breaking it down into smaller subproblems and using the optimal solutions of these subproblems to find the optimal solution for the overall problem.
To solve this problem using Dynamic Programming, we can define an array dp of size n, where n is the number of days in the stock market. Each element dp[i] represents the maximum profit that can be obtained by buying and selling the stock up to day i.
We can initialize dp[0] as 0, as there is no profit on the first day. Then, for each subsequent day i, we can calculate dp[i] by considering two possibilities:
1. If we sell the stock on day i, we need to find the minimum price to buy the stock before day i. So, we iterate through all the previous days j (0 <= j < i) and update dp[i] as the maximum of dp[i] and prices[i] - prices[j] + dp[j-1]. Here, prices[i] - prices[j] represents the profit obtained by selling the stock on day i after buying it on day j, and dp[j-1] represents the maximum profit obtained before day j.
2. If we do not sell the stock on day i, then dp[i] will be the same as dp[i-1], as we are not making any transaction on day i.
Finally, the maximum profit that can be obtained by buying and selling the stock can be found by returning dp[n-1], where n is the number of days in the stock market.
By using this approach, we can solve the Best Time to Buy and Sell Stock problem in O(n^2) time complexity, where n is the number of days in the stock market. However, this can be optimized to O(n) time complexity by keeping track of the minimum price seen so far and updating the maximum profit accordingly.