Imagine you are given a set of coins with denominations [1, 2, 5, 10] and you need to make change for 15. Discuss how dynamic programming can be applied to find the minimum number of coins required.
- Apply a brute-force approach by trying all possible combinations of coins.
- Implement a recursive solution without memoization.
- Use dynamic programming to build a table to store the minimum number of coins for each amount.
- Utilize a greedy algorithm to select the largest denomination at each step.
Dynamic programming can be applied to find the minimum number of coins required by creating a table that represents the minimum number of coins needed for each amount from 0 to the target amount. The table is filled iteratively, considering the optimal substructure of the problem. This ensures that the solution for smaller subproblems is used to build the solution for the larger problem, resulting in an efficient and optimal solution.
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