Dynamic Programming Questions Medium
In the context of the House Robber 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.
The House Robber problem involves a row of houses, each containing a certain amount of money. The goal is to maximize the amount of money that can be robbed without robbing adjacent houses.
To understand the optimal substructure, let's consider a simple example with three houses: House 1, House 2, and House 3. We need to determine the maximum amount of money that can be robbed.
If we choose to rob House 1, we cannot rob House 2 or House 3. In this case, the maximum amount of money that can be robbed is the amount in House 1.
If we choose not to rob House 1, we have two options: rob House 2 or skip to House 3. We need to consider which option yields the maximum amount of money. If we rob House 2, we cannot rob House 3. If we skip House 2 and rob House 3, we can add the amount in House 3 to the maximum amount of money that can be robbed from the remaining houses. We then compare the two options and choose the one that gives us the maximum amount.
From this example, we can observe that the optimal solution to the House Robber problem can be obtained by considering the optimal solutions to its subproblems. In this case, the subproblems are the maximum amount of money that can be robbed from the first house, the first two houses, and so on.
By using dynamic programming, we can store the solutions to the subproblems in an array and build up the solution to the main problem by considering the optimal solutions to the subproblems. This approach allows us to avoid redundant calculations and efficiently solve the House Robber problem.
In summary, the concept of optimal substructure in the context of the House Robber problem means that the optimal solution to the problem can be constructed by considering the optimal solutions to its subproblems.