Dynamic Programming Questions Medium
In the context of the Best Time to Buy and Sell Stock problem, the concept of optimal substructure refers to the property that an optimal solution to the problem can be constructed from optimal solutions to its subproblems.
In the Best Time to Buy and Sell Stock problem, we are given an array of stock prices representing the prices of a stock on different days. The goal is to find the maximum profit that can be obtained by buying and selling the stock at most once.
To understand the optimal substructure, let's consider a subproblem where we have to find the maximum profit that can be obtained by buying and selling the stock within a specific range of days, say from day i to day j.
Now, if we divide this subproblem into smaller subproblems, we can observe that the optimal solution to the original problem can be obtained by finding the maximum profit among all possible subproblems.
For example, let's say we divide the subproblem from day i to day j into two subproblems: from day i to day k and from day k+1 to day j, where i <= k < j. The optimal solution to the original problem can be obtained by finding the maximum profit among the following three possibilities:
1. The maximum profit obtained by buying and selling within the subproblem from day i to day j.
2. The maximum profit obtained by buying and selling within the subproblem from day i to day k.
3. The maximum profit obtained by buying and selling within the subproblem from day k+1 to day j.
By considering all possible divisions of the subproblem, we can find the maximum profit that can be obtained by buying and selling the stock within the given range of days.
This property of optimal substructure allows us to solve the Best Time to Buy and Sell Stock problem efficiently using dynamic programming. We can break down the problem into smaller subproblems, solve them independently, and then combine their solutions to obtain the optimal solution to the original problem.