Dynamic Programming Questions Medium
In the context of the Coin Change 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 Coin Change problem involves finding the minimum number of coins needed to make a certain amount of change. For example, given a set of coins with different denominations and a target amount, the goal is to determine the minimum number of coins required to make that amount.
Optimal substructure comes into play when breaking down the problem into smaller subproblems. Let's say we have a set of coins {1, 2, 5} and we want to make a change for the amount 10. We can consider the following scenarios:
1. If we use a coin of denomination 1, the remaining amount to be made is 10 - 1 = 9.
2. If we use a coin of denomination 2, the remaining amount to be made is 10 - 2 = 8.
3. If we use a coin of denomination 5, the remaining amount to be made is 10 - 5 = 5.
Now, we can observe that the minimum number of coins required to make the remaining amounts (9, 8, and 5) can be found using the same approach. This is the optimal substructure property, where the optimal solution to the original problem can be obtained by combining optimal solutions to its subproblems.
By solving the subproblems recursively and storing their solutions, we can avoid redundant calculations and improve the efficiency of the algorithm. This is the essence of dynamic programming, where we break down a complex problem into simpler subproblems and solve them in a bottom-up manner, utilizing the optimal substructure property.
In summary, the concept of optimal substructure in the Coin Change problem means that the optimal solution to the problem can be constructed by combining optimal solutions to its subproblems, allowing us to solve the problem efficiently using dynamic programming techniques.